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Advances in High Energy Physics

Guest Editors: Shaaban Khalil, Gordon Kane, Ignatios Antoniadis, and Stefano Moretti Supersymmetry, Supergravity, and Superstring Phenomenology Advances in High Energy Physics

Supersymmetry, Supergravity, and Superstring Phenomenology

This is a special issue published in “Advances in High Energy Physics.” All articles are open access articles distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Editorial Board

Luis A. Anchordoqui, USA Frank Filthaut, Netherlands Anastasios Petkou, Greece T. Asselmeyer-Maluga, Germany Chao-Qiang Geng, Taiwan Alexey A. Petrov, USA Marco Battaglia, Switzerland Philippe Gras, France Thomas Rössler, Sweden Botio Betev, Switzerland Xiaochun He, USA Juan José Sanz-Cillero, Spain Lorenzo Bianchini, Switzerland Filipe R. Joaquim, Portugal Edward Sarkisyan-Grinbaum, USA Burak Bilki, USA KyungK.Joo,RepublicofKorea Sally Seidel, USA Adrian Buzatu, UK Aurelio Juste, Spain George Siopsis, USA Rong-Gen Cai, China Michal Kreps, UK Luca Stanco, Italy Duncan L. Carlsmith, USA Ming Liu, USA Satyendra Thoudam, Netherlands Ashot Chilingarian, Armenia Enrico Lunghi, USA Smarajit Triambak, South Africa Anna Cimmino, Belgium Piero Nicolini, Germany Elias C. Vagenas, Kuwait Andrea Coccaro, Switzerland Seog H. Oh, USA Nikos Varelas, USA Shi-Hai Dong, Mexico Sergio Palomares-Ruiz, Spain YauW.Wah,USA Edmond C. Dukes, USA Giovanni Pauletta, Italy Amir H. Fatollahi, Iran Yvonne Peters, UK Contents

Supersymmetry, Supergravity, and Superstring Phenomenology Shaaban Khalil, Gordon Kane, Ignatios Antoniadis, and Stefano Moretti Volume 2016, Article ID 3595120, 1 page

Geometric Algebra Techniques in Flux Compactifications Calin Iuliu Lazaroiu, Elena Mirela Babalic, and Ioana Alexandra Coman Volume 2016, Article ID 7292534, 42 pages

Structural Theory and Classification of 2D Adinkras Kevin Iga and Yan X. Zhang Volume 2016, Article ID 3980613, 12 pages

Aspects of Moduli Stabilization in Type IIB String Theory Shaaban Khalil, Ahmad Moursy, and Ali Nassar Volume 2016, Article ID 4303752, 17 pages

A 𝑄-Continuum of Off-Shell Supermultiplets Tristan Hübsch and Gregory A. Katona Volume 2016, Article ID 7350892, 11 pages

MSSM Dark Matter in Light of Higgs and LUX Results W. Abdallah and S. Khalil Volume 2016, Article ID 5687463, 10 pages

Helical Phase Inflation and Monodromy in Supergravity Theory Tianjun Li, Zhijin Li, and Dimitri V. Nanopoulos Volume 2015, Article ID 397410, 12 pages

Reciprocity and Self-Tuning Relations without Wrapping Davide Fioravanti, Gabriele Infusino, and Marco Rossi Volume 2015, Article ID 762481, 21 pages

Exploring New Models in All Detail with ËÊÀ Florian Staub Volume2015,ArticleID840780,126pages

Vacuum Condensates as a Mechanism of Spontaneous Supersymmetry Breaking Antonio Capolupo and Marco Di Mauro Volume 2015, Article ID 929362, 6 pages

A Chargeless Complex Vector Matter Field in Supersymmetric Scenario L. P. Colatto and A. L. A. Penna Volume 2015, Article ID 986570, 8 pages Hindawi Publishing Corporation Advances in High Energy Physics Volume 2016, Article ID 3595120, 1 page http://dx.doi.org/10.1155/2016/3595120

Editorial Supersymmetry, Supergravity, and Superstring Phenomenology

Shaaban Khalil,1 Gordon Kane,2 Ignatios Antoniadis,3,4 and Stefano Moretti5

1 Center for Fundamental Physics, Zewail City of Science and Technology, Giza, Egypt 2Department of Physics, University of Michigan, Ann Arbor, MI, USA 3Department of Physics, LPTHE, Sorbonne Universite,UPMC,Paris,France´ 4University of Bern, Bern, Switzerland 5Department of Physics, University of Southampton, Southampton, UK

Correspondence should be addressed to Shaaban Khalil; [email protected]

Received 14 December 2016; Accepted 14 December 2016

Copyright © 2016 Shaaban Khalil et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.

Supersymmetry, supergravity, and superstring are amongst stabilization in type IIB string theory compactification with the most popular research topics in particle physics. Super- fluxes. Another paper in this special issue describes supersymmetry is a generalization of the space-time symmetries of multiplets wherein a continuously variable “tuning parame- quantum field theory that links the matter particles with the ter” modifies the supersymmetry transformations. Another force-carrying particles and implies that there are additional paper studies the constraints imposed on the Minimal Super- superparticles necessary to complete the symmetry. Super- symmetric Standard Model (MSSM) parameter space by the gravity is the theory that combines the principles of super- Large Hadron Collider (LHC) Higgs mass measurements symmetry and general relativity. It naturally includes gravity and gluino mass lower bound. Another paper studies helical along with the other fundamental forces (the electromagnetic phase inflation which realizes “monodromy inflation” in force, the weak nuclear force, in turn already unified in the supergravity theory. Another paper considers scalar Wilson electroweak interactions, and the strong nuclear force). String operators of 𝑁=4Supersymmetric Yang-Mills (SYM) theo- theory is the leading candidate for a theory that unifies all ries at high spin and generic twist operators in the multicolor fundamentalforcesinnatureinaconsistentscheme.Italso limit. Another paper author gives an overview about the provides a consistent framework for the theory of quantum features that the Mathematica package SARAH provides to gravity. Compactified string/M-theories make testable pre- study new supersymmetric models. Another paper reviews a dictions about our four-dimensional world. possible mechanism for the spontaneous breaking of super- The phenomenology of supersymmetry, supergravity, symmetry, based on the presence of vacuum condensates. and superstring is thus very rich and covers many topics: Another paper constructs and studies a formulation of a flavour physics and CP violation, Higgs and collider physics, chargeless complex vector matter field in a supersymmetric modelbuildingbeyondtheStandardModel,andastroparticle framework. physics and cosmology. Some recent developments in these theories, each with important applications to particle physics Shaaban Khalil and/or cosmology, are the main theme of this special issue. Gordon Kane One of the papers of this special issue discusses the Ignatios Antoniadis constrained generalized Killing spinors, which characterize Stefano Moretti supersymmetric flux compactifications of supergravity theories, using geometric algebra techniques. Another paper presents a study on what are called Adinkras, which are combinatorial objects developed to study (1-dimensional) supersymmetry representations. Another paper reviews moduli Hindawi Publishing Corporation Advances in High Energy Physics Volume 2016, Article ID 7292534, 42 pages http://dx.doi.org/10.1155/2016/7292534

Research Article Geometric Algebra Techniques in Flux Compactifications

Calin Iuliu Lazaroiu,1 Elena Mirela Babalic,2 and Ioana Alexandra Coman3

1 Institute for Basic Science, Center for Geometry and Physics, Pohang 790-784, Republic of Korea 2Horia Hulubei National Institute for Physics and Nuclear Engineering, Department of Theoretical Physics, Strada Reactorului No. 30, P.O. BOX MG-6, 077125 Magurele, Romania 3DESY,TheoryGroup,Notkestrasse85,Building2a,22607Hamburg,Germany

Correspondence should be addressed to Elena Mirela Babalic; [email protected]

Received 12 May 2015; Accepted 10 September 2015

Academic Editor: Shaaban Khalil

Copyright © 2016 Calin Iuliu Lazaroiu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.

We study “constrained generalized Killing (s)pinors,” which characterize supersymmetric flux compactifications of supergravity theories. Using geometric algebra techniques, we give conceptually clear and computationally effective methods for translating supersymmetry conditions into differential and algebraic constraints on collections of differential forms. In particular, we give a synthetic description of Fierz identities, which are an important ingredient of such problems. As an application, we show how our approach can be used to efficiently treat N =1compactification of M-theory on eight manifolds and prove that we recover results previously obtained in the literature.

1. Introduction The purpose of this paper is to draw attention to the fact that many of the issues mentioned above can be resolved A fundamental problem in the study of flux compactifications using ideas inspired by a certain incarnation of the theory of of 𝑀-theory and string theory is to give efficient geometric Clifford bundles known as “geometric algebra,” which goes descriptions of supersymmetric backgrounds in the pres- backto[8,9](seealso[10–14]foranintroduction)—an ence of fluxes. This leads, in particular cases, to beautiful approach which provides a powerful language and efficient connections [1, 2] with the theory of 𝐺-structures, while in techniques, thus affording a more unified and systematic more general situations it translates to difficult mathematical description of flux compactifications and of supergravity and problems involving novel geometric realizations of superstring compactifications in general. In particular, we show symmetry algebras (see [3–6] for some examples). that the geometric analysis of supersymmetry conditions for When approaching this subject, one may be struck by the flux backgrounds (including the algebra of those Fierz iden- somewhat ad hoc nature of the methods usually employed, tities relevant for the analysis) can be formulated efficiently which signals a lack of unity in the current understanding in this language, thereby uncovering structure whose impli- of the subject. This is largely due to the intrinsic difficulty cations have remained largely unexplored. We mention here in finding unifying principles while keeping computational that our methods have a (nontrivial) connection with the 𝐺- complexity under control. In particular, one confronts the structure and exceptional generalized geometry approaches, lack of general and structurally clear descriptions of Fierz which were previously shown to be useful when studying flux identities, the fact that phenomena and methods which are compactifications. This connection will be discussed at length sometimes assumed to be “generic” turn out, upon closer in a different publication. inspection, to be relevant only under simplifying assump- Though the scope and applications of our approach are tions, and the insufficient mathematical development of the much wider, we will focus here on the study of what we call subject of “spin geometry [7] in the presence of fluxes.” “constrained generalized Killing (CGK) (s)pinor equations,” 2 Advances in High Energy Physics which distill the mathematical description of supersymmetry pinors through certain quadratic relations holding in (a conditions for flux backgrounds. A constrained generalized certain subalgebra of) the Kahler-Atiyah¨ algebra of the Killing (s)pinor is simply a (s)pinor satisfying conditions underlying manifold. We also reformulate the constrained of the type 𝐷𝑚𝜉=𝑄1𝜉 = ⋅⋅⋅𝜒 =𝑄 𝜉=0,where generalized Killing pinor equations in this language and dis- 𝑆 𝐷𝑚 =∇𝑚 +𝐴𝑚 is some connection on a bundle 𝑆 of (s)pinors cuss some aspects of the differential and algebraic structure 𝑆 (which generally differs from the connection ∇ induced resulted from this analysis, thereby extending the well-known 𝑚 theory of Killing forms. In Section 6, we apply this formalism on 𝑆 by the Levi-Civita connection ∇𝑚 of the underlying to the study of N =1compactification of 𝑀-theory on pseudo-Riemannian manifold) while 𝑄𝑗 are some globally defined endomorphisms of 𝑆. Such equations are abundant eightmanifolds.WeconcludeinSection7withafewremarks in flux compactifications of supergravity (see, e.g., [15, 16]), on further directions. Appendix summarize various technical where 𝜉 is the internal part of a supersymmetry generator detailsandmakecontactwithpreviouswork.Thephysics- while the equations themselves are the conditions that the oriented reader can start with Section 6, before delving into compactification preserves the supersymmetry generated by the technical and theoretical details of the other sections. 𝜉 𝐴 𝑄 .Thequantities 𝑚 and 𝑗 arethencertainalgebraic K R C combinations of gamma matrices with coefficients dependent Notations. We let denoteoneofthefields or of real on the metric and fluxes. An example with a single algebraic orcomplexnumbers.Weworkinthesmoothdifferential constraint 𝑄𝜉 =0 (arising in a compactification of eleven- category, so all manifolds, vector bundles, maps, morphisms dimensional supergravity) is discussed in Section 6, which of bundles, differential forms, and so forth are taken tobe smooth. We further assume that our connected and smooth the reader can consult first as an illustration motivating the 𝑀 formal developments taken up in the rest of the paper. manifolds are paracompact and of finite Lebesgue dimension, so that we have partitions of unity of finite covering Using geometric algebra techniques, we show how such 𝑉 K supersymmetry conditions can be translated efficiently and dimension subordinate to any open cover. If is a -vector bundle over 𝑀,weletΓ(𝑀, 𝑉) denote the space of smooth briefly into a system of differential and algebraic con- ∞ (C )sectionsof𝑉.WealsoletEnd(𝑉) = Hom(𝑉,𝑉) = straints for a collection of inhomogeneous differential forms ∗ 𝑉⊗𝑉 denote the K-vector bundle of endomorphisms of expressed as (s)pinor bilinears, thus displaying the underly- ∗ 𝑉,where𝑉 = Hom(𝑉, OK) is the dual vector bundle to 𝑉 ing structure in a form which is conceptually clear as well as O K 𝑀 highly amenable to computation. The conditions which we while K denotes the trivial -line bundle on .Theunital ring of smooth K-valued functions defined on 𝑀 is denoted obtain on differential forms provide a generalization of the C∞(𝑀, R)=Γ(𝑀,O ) K well-known theory of Killing forms, which could be studied by K .Thetensorproductof -vector spaces and K-vector bundles is denoted by ⊗, while the tensor in more depth through methods of Kahler-Cartan¨ theory ∞ C (𝑀, K) ⊗ ∞ [17]—even though we will not pursue that avenue in the product of modules over is denoted by C (𝑀,R); Γ(𝑀, 𝑉 ⊗𝑉)=Γ(𝑀,𝑉)⊗ ∞ Γ(𝑀, 𝑉 ) present work. We also touch on our implementation of this hence 1 2 1 C (𝑀,R) 2 .Setting def ∗ def ∗ approach using various symbolic computation systems. 𝑇K𝑀 =𝑇𝑀⊗OK and 𝑇K𝑀 =𝑇𝑀⊗OK, the space of K- As an example, Section 6 applies such techniques to the valued smooth inhomogeneous globally defined differential study of flux compactifications of 𝑀-theory on eight mani- def ∗ forms on 𝑀 is denoted by ΩK(𝑀) = Γ(𝑀, ∧𝑇K𝑀) and is a folds preserving N =1supersymmetry in 3 dimensions— ∞ Z-graded module over the commutative ring C (𝑀, R).The a class of solutions which was analyzed through direct fixed rank components of this graded module are denoted methodsin[3,4].Inthatsetting,wehaveasinglealge- 𝑘 𝑘 ∗ 𝑚 by Ω (𝑀) = Γ(𝑀, ∧ 𝑇 𝑀) (𝑘 = 0⋅⋅⋅𝑑,where𝑑 is the braic condition 𝑄𝜉,with =0 𝑄 = (1/2)𝛾 𝜕𝑚Δ− K K 𝑚𝑝𝑞𝑟 𝑝 (9) (9) dimension of 𝑀). (1/288)𝐹𝑚𝑝𝑞𝑟𝛾 − (1/6)𝑓𝑝𝛾 𝛾 −𝜅𝛾 and 𝐴𝑚 = 𝑝 (9) 𝑝𝑞𝑟 (9) The kernel and image of any K-linear map 𝑇:Γ(𝑀, (1/4)𝑓𝑝𝛾𝑚 𝛾 +(1/24)𝐹𝑚𝑝𝑞𝑟𝛾 +𝜅𝛾𝑚𝛾 .Weshowhowour 𝑉1)→Γ(𝑀,𝑉2) will be denoted by K(𝑇) and I(𝑇);these methods can be used to recover the results of [3] in a synthetic are K-linear subspaces of Γ(𝑀,1 𝑉 ) and Γ(𝑀,2 𝑉 ),respectively. and computationally efficient manner, while giving a more ∞ In the particular case when 𝑇 is a C (𝑀, R)-linear map (i.e., complete and general analysis. We express all equations in ∞ when it is a morphism of C (𝑀, R)-modules), the subspaces terms of certain combinations of iterated contractions and ∞ K(𝑇) and I(𝑇) are C (𝑀, R)-submodules of Γ(𝑀,1 𝑉 ) and wedge products which are known as “generalized prod- Γ(𝑀,2 𝑉 ), respectively—even in those cases when 𝑇 is not ucts” and whose conceptual role and origin is explained in induced by any bundle morphism from 𝑉1 to 𝑉2.Wealways Section 3. The reader can, at this point, pause to take a look denote a morphism 𝑓:𝑉1 →𝑉2 of K-vector bundles and the at Section 6.2, which should provide an illustration of the ∞ C (𝑀, R)-linear map Γ(𝑀,1 𝑉 )→Γ(𝑀,𝑉2) induced by it techniques developed in this paper. between the modules of sections by the same symbol. Because The paper is organized as follows. In Section 2, we define of this convention, we clarify that the notations K(𝑓) ⊂ and discuss constrained generalized Killing (s)pinors. In Γ(𝑀,1 𝑉 ) and I(𝑓) ⊂ Γ(𝑀,2 𝑉 ) denote the kernel and the Section 3, we recall the geometric algebra description of 𝑓 Clifford bundles asahler-Atiyah K¨ bundles while in Section 4 image of the corresponding map on sections Γ(𝑀,1 𝑉 ) 󳨀→ ∞ we explain how pinor bundles are described in this approach. Γ(𝑀,2 𝑉 ),whichinthiscaseareC (𝑀, R)-submodules Using our realization of spin geometry, Section 5 presents of Γ(𝑀,1 𝑉 ) and Γ(𝑀,2 𝑉 ), respectively. In general, there a synthetic formulation of Fierz rearrangement identities does not exist either any subbundle ker𝑓 of 𝑉1 such that for pinor bilinears, which encodes identities involving four K(𝑓) = Γ(𝑀, ker𝑓) or any subbundle im𝑓 of 𝑉2 such that Advances in High Energy Physics 3

∗ I(𝑓) = Γ(𝑀, im𝑓)—though there exist sheaves ker𝑓 and given above. The fiber of Cl(𝑇K𝑀) at a point 𝑥∈𝑀is the ∗ ∗ im𝑓 with the corresponding properties. Clifford algebra Cl(𝑇K,𝑥𝑀) = Cl(𝑇𝑥 𝑀) ⊗R K of the quadratic Given a pseudo-Riemannian metric 𝑔 on 𝑀 of signature ∗ ∗ def ∗ vector space (𝑇K,𝑥, 𝑔̂K,𝑥),where𝑇K,𝑥 =𝑇𝑥 𝑀⊗R K and 𝑔̂K,𝑥 (𝑝, 𝑞),welet(𝑒𝑎)𝑎=1⋅⋅⋅𝑑 (where 𝑑=dim 𝑀) denote a local denotes the K-valued bilinear pairing induced by 𝑔̂𝑥.The frame of 𝑇𝑀,definedonsomeopensubset𝑈 of 𝑀.Welet ev ∗ 𝑎 ∗ even Clifford bundle Cl (𝑇K𝑀) over K is the subbundle of 𝑒 betheduallocalcoframe(= local frame of 𝑇 𝑀), which ∗ 𝑎 𝑎 𝑎 𝑏 𝑎𝑏 𝑎𝑏 algebras of Cl(𝑇K𝑀) whosefibersaretheevensubalgebras satisfies 𝑒 (𝑒𝑏)=𝛿𝑏 and 𝑔(𝑒̂ ,𝑒 )=𝑔 ,where(𝑔 ) is the ev ∗ ∗ 𝑎 ♯ Cl (𝑇K,𝑥𝑀) ⊂ Cl(𝑇K,𝑥𝑀). Our point of view on (s)pinor inverse of the matrix (𝑔𝑎𝑏). The contragradient frame (𝑒 ) (𝑒 ) bundles is that taken in [18]. Namely, we define a bundle of and contragradient coframe 𝑎 ♯ are given by K-pinors over 𝑀 to be a K-vector bundle 𝑆 over 𝑀 which (𝑇∗𝑀) 𝑎 ♯ 𝑎𝑏 is a bundle of modules over the Clifford bundle Cl K . (𝑒 ) =𝑔 𝑒𝑏, Similarly, a bundle of K-spinors is a bundle of modules over ev ∗ (1) the even Clifford bundle Cl (𝑇 𝑀).Ofcourse,abundle (𝑒 ) =𝑔 𝑒𝑏, K 𝑎 ♯ 𝑎𝑏 of K-pinors is automatically a bundle of K-spinors. Hence any pinor is naturally a spinor but the converse need not ♯ where the subscript and superscript denote the (mutually hold.Inthispaper,wefocusonthecaseofpinors.Apinor 𝑇 𝑀 𝑇∗𝑀 inverse) musical isomorphisms between K and K bundle 𝑆 will be called a pin bundle if the underlying fiberwise ∗ given, respectively, by lowering and raising indices with the representation of Cl(𝑇K𝑀) is irreducible, that is, if each of 𝑎 ⋅⋅⋅𝑎 def 𝑎 𝑎 def 𝑔 𝑒 1 𝑘 =𝑒1 ∧⋅⋅⋅∧𝑒 𝑘 𝑒 =𝑒 ∧⋅⋅⋅∧ the fibers of 𝑆 is a simple module over the corresponding metric .Weset and 𝑎1⋅⋅⋅𝑎𝑘 𝑎1 𝑒 𝑘=0⋅⋅⋅𝑑 K fiber of the Clifford bundle. Similarly, a spin bundle is a spinor 𝑎𝑘 for any .Ageneral -valued inhomogeneous 𝜔∈Ω (𝑀) bundle for which the underlying fiberwise representation of form K expands as follows: ev ∗ Cl (𝑇K𝑀) is irreducible. Later on, we will sometimes denote 𝑑 𝑑 1 𝑔K by 𝑔, and so forth, in order to simplify notation. (𝑘) (𝑘) 𝑎1⋅⋅⋅𝑎𝑘 𝜔=∑𝜔 =𝑈 ∑ 𝜔𝑎 ⋅⋅⋅𝑎 𝑒 , (2) 𝑘! 1 𝑘 𝑘=0 𝑘=0 Remark 1. Physics terminology is often imprecise with the distinction between spinors and pinors which we are making where the symbol =𝑈 means that the equality holds only after restriction of 𝜔 to 𝑈 and where we used the expansion: here and throughout this paper. Physically, one typically assumes that (𝑀, 𝑔) is both oriented and time-oriented 1 (𝑘) (𝑘) 𝑎1⋅⋅⋅𝑎𝑘 and one is concerned with objects transforming in rep- 𝜔 =𝑈 𝜔𝑎 ⋅⋅⋅𝑎 𝑒 . (3) ↑ 𝑘! 1 𝑘 resentations of the orthochronous part Spin (𝑝, 𝑞) of the spin group Spin(𝑝, 𝑞) and thus in vector bundles associated 𝜔(𝑘) ∈ C∞(𝑈,K) ↑ The locally defined smooth functions 𝑎1⋅⋅⋅𝑎𝑘 with a principal bundle with fiber Spin (𝑝, 𝑞) which is a 𝜔 ↑ (the “strict coefficient functions” of )arecompletelyanti- doublecoveroftheprincipalSO(𝑝, 𝑞)-bundle consisting 𝑎 ⋅⋅⋅𝑎 𝑀 symmetric in 1 𝑘. Given a pinor bundle on with of those pseudo-orthonormal frames of (𝑀, 𝑔) which are 𝛾 underlying fiberwise representation of the Clifford bundle both oriented and time-oriented. Due to the issue of time- 𝑇∗𝑀 of K , the corresponding gamma “matrices” in the coframe orientability, what matters in many physics applications is not 𝑎 𝑎 def 𝑎 𝑒 are denoted by 𝛾 =𝛾(𝑒), while the gamma matrices a spin structure in the standard mathematical sense (see [19] def for a recent discussion with applications to string theory) but in the contragradient coframe (𝑒𝑎)♯ are denoted by 𝛾𝑎 = 𝑏 rather a “time-oriented” spin structure. 𝛾((𝑒𝑎)♯)=𝑔𝑎𝑏𝛾 . We will occasionally assume that the frame (𝑒𝑎) is pseudo-orthonormal in the sense that 𝑒𝑎 satisfy Constrained Generalized Killing (S)Pinors.LetusfixaK-pinor or K-spinor bundle 𝑆 over 𝑀,alinearconnection𝐷 on 𝑆,and 𝑔 (𝑒 ,𝑒 )(=𝑔 ) =𝜂 , 𝑎 𝑏 𝑎𝑏 𝑎𝑏 (4) a finite collection of bundle endomorphisms 𝑄1,...,𝑄𝜒 ∈ Γ(𝑀, End(𝑆)). where (𝜂𝑎𝑏) is a diagonal matrix with 𝑝 diagonal entries equal +1 𝑞 −1 to and diagonal entries equal to . Definition 2. A constrained generalized Killing (CGK) (s)pinor over 𝑀 is a section 𝜉∈Γ(𝑀,𝑆)which satisfies the constrained 𝐷𝜉 =𝑄 𝜉=⋅⋅⋅= 2. Constrained Generalized Killing (S)Pinors generalized Killing (s)pinor equations 1 𝑄𝜒𝜉=0.Wesaythat𝐷𝜉 =0 is the 𝐷-flatness or generalized (𝑀, 𝑔) The Basic Setup. Let be a connected pseudo- Killing (GK) (s)pinor equation satisfied by 𝜉 while 𝑄1𝜉= Riemannianmanifold(assumedtobesmoothandparacom- ⋅⋅⋅ = 𝑄𝜒𝜉=0are the algebraic constraints (or 𝑄-constraints) 𝑑=𝑝+𝑞 𝑝 𝑞 pact) of dimension ,where and are, respectively, satisfied by 𝜉. the numbers of positive and negative eigenvalues of 𝑔.We ∗ endow the cotangent bundle 𝑇 𝑀 with the metric 𝑔̂ induced When the algebraic constraints are trivial (𝜒=0 𝑔 K = R C by .Setting or , we similarly endow the bundle or, equivalently, when all 𝑄𝑗 vanish), one deals with the ∗ def ∗ 𝐷𝜉 =0 𝐷 𝑇K𝑀 =𝑇𝑀⊗OK with the metric 𝑔̂K induced by extension generalized Killing (GK) spinor equation .Since can ∗ ∗ 𝑆 𝑆 of scalars. Of course, we have 𝑇R𝑀=𝑇𝑀 and 𝑔̂R = be written as the sum ∇ +𝐴of the spinorial connection ∇ ∗ ∗ 𝑔̂.LetCl(𝑇K𝑀) = Cl(𝑇 𝑀) ⊗ OK be the Clifford bundle induced on 𝑆 by the Levi-Civita connection of (𝑀, 𝑔) and an ∗ defined by 𝑇K𝑀—when the latter is endowed with the metric End(𝑆)-valued one-form 𝐴 on 𝑀, the GK (s)pinor equations 4 Advances in High Energy Physics canbeviewedasadeformationoftheparallel (s)pinor (s)pinors, 𝜉 is replaced by its internal part (a section of 𝑆 equation ∇ 𝜉=0, the deformation being parameterized by some (s)pinor bundle—which now plays the role of 𝑆— 𝐴. Our terminology is inspired by the fact that the choice defined on the compactification space) while 𝐷 induces a 𝑚 𝐴𝑚 =−𝜆𝛾𝑚 (with 𝜆 a real parameter and 𝛾 ∈ Γ(𝑀, End(𝑆)) connection defined on this internal (s)pinor bundle as well the gamma “matrices” in some local coframe of (𝑀, 𝑔))leads as a further algebraic constraint—thereby leading once again to the ordinary Killing (s)pinor equations ∇𝑚𝜉=𝜆𝛾𝑚𝜉. to a system of equations of constrained generalized Killing type, which is now defined on the internal space. Finally, the Remark 3. In flux compactifications of supergravity, back- supersymmetry equations for type IIB supergravity in ten grounds admitting constrained generalized Killing spinors dimensions (with Minkowski signature) can be formulated1 canbeusedtoconstructsupersymmetriccompactifications, (see, e.g., [16]) in terms of sections of the real vector bundle + + provided that the equations of motion for all fields present in 𝑆=𝑆⊕𝑆 of Majorana-Weyl spinor doublets, with a the background are also satisfied. supercovariant connection 𝐷 definedonthisbundleaswell as two endomorphisms 𝑄1, 𝑄2 of 𝑆. The condition 𝐷𝜉 =0 Connection to Supergravity and String Theories. Constrained for sections 𝜉 of 𝑆 is the requirement that the supersymmetry generalized Killing (s)pinors arise naturally in supergravity variation of the gravitino vanishes in the background, while and string theory. In particular, they arise in supersymmet- the conditions 𝑄1𝜉=𝑄2𝜉=0are, respectively, the require- ric flux compactifications of string theory, 𝑀-theory, and ments that the supersymmetry variations of the axionino and various supergravity theories. In such setups, 𝜉 is a (s)pinor dilatino vanish. When considering a compactification down of spin 1/2 defined on the background pseudo-Riemannian from ten dimensions, the constraints 𝑄1𝜉=𝑄2𝜉=0descend manifold and corresponds to the generator of supersymme- to similar constraints for the internal part of 𝜉, while the try transformations of the underlying supergravity action constraint 𝐷𝜉 =0 induces both a differential and an algebraic (or string theory effective action) while the constrained constraint for the internal part; hence the compactification generalized Killing (s)pinor equations are the conditions procedure produces a differential constraint while increasing that the supersymmetry generated by 𝜉 is preserved by the the number of algebraic constraints, the resulting equations background. The connection 𝐷 on 𝑆 and the endomorphisms being again of constrained generalized Killing type, but 𝑄𝑗 arefixedbytheprecisedataofthebackground,thatis,by formulated for sections of some bundle of (s)pinors defined the metric and fluxes defining that background. For example, over the internal space of the compactification. the supersymmetry equations of eleven-dimensional supergravity involve the supercovariant connection 𝐷,whichacts Some Mathematical Observations. Let us for simplicity con- on sections of the bundle 𝑆 of Majorana spinors (a.k.a. real sider the case of a single algebraic constraint (𝑄𝜉). =0 Let ∞ pinors) defined in eleven dimensions; this corresponds to K(𝑄) denote the C (𝑀, R)-submodule of smooth solutions the differential constraint 𝐷𝜉, =0 without any algebraic to the equation 𝑄𝜉 =0 and let K(𝐷) denote the K-vector constraint. When considering a compactification of eleven- subspace of smooth solutions to the equation 𝐷𝜉.Then =0 dimensional supergravity down to a lower-dimensional space the K-vector subspace K(𝐷, 𝑄) of smooth solutions to the admitting Killing (s)pinors, the internal part (which now CGK spinor equations equals the intersection K(𝐷)∩K(𝑄). plays the role of 𝜉)ofthegeneratorofthesupersymmetry In general, the dimension of the subspace ker(𝑄𝑥)⊂𝑆𝑥 of the variation is a section of some bundle of (s)pinors (which now fiber of 𝑆 at a point 𝑥 may jump as 𝑥 varies inside 𝑀,so𝑄 does plays the role of 𝑆) defined over the internal space, while not admit a subbundle of 𝑆 as its kernel (in fact, this is one the condition of preserving the supersymmetry generated reason why smooth vector bundles do not form an Abelian by the tensor product of this internal generator and some category)—even though it does admit a kernel in the category Killing (s)pinor of the noncompact part of the background of sheaves over the ringed space associated with 𝑀.On induces a differential (generalized Killing) constraint as well the other hand, a simple argument2 using parallel transport as an algebraic constraint for the internal part of the super- shows that any linearly independent (over K) collection of symmetry generator. A specific example arising from eleven- smooth solutions 𝜉1,...,𝜉𝑠 of the generalized Killing (s)pinor dimensional supergravity is discussed in Section 6 below. equation must be linearly independent everywhere; that is, Similarly, the supersymmetry equations for IIA supergravity the vectors 𝜉1(𝑥),...,𝜉𝑠(𝑥) must be linearly independent in in ten dimensions (with Minkowski signature) can be written the fiber 𝑆𝑥 for any point 𝑥 of 𝑀. In particular, there exists a in terms of a supercovariant connection 𝐷 defined on the real K-vector subbundle 𝑆𝐷 of 𝑆 such that rkK𝑆𝐷 = dimKK(𝐷) ∞ vector bundle 𝑆 of Majorana spinors (a.k.a. real pinors) in and such that Γ(𝑀,𝐷 𝑆 )=K(𝐷)K ⊗ C (𝑀, R);infact, ten dimensions and an endomorphism 𝑄 of 𝑆;wehave𝑆= any basis of the space of solutions K(𝐷) of the generalized + − ± 𝑆 ⊕𝑆 where 𝑆 are the bundles of Majorana-Weyl spinors Killing (s)pinor equations provides a global frame for 𝑆𝐷 of positive and negative chirality. The condition 𝐷𝜉 =0 for (which, therefore, must be a trivial vector bundle). Since the the supersymmetry generator (a section 𝜉 of 𝑆, with positive restriction of 𝐷 to 𝑆𝐷 is flat, the bundle 𝑆𝐷 is sometimes and negative chirality components 𝜉±—which are sections referred to as “the 𝐷-flat vector subbundle of 𝑆.” The con- ± of 𝑆 —such that 𝜉=𝜉+ +𝜉−) is the requirement that the dition that the generalized Killing (s)pinor equations admit supersymmetry variation of the gravitino vanishes, while the exactly 𝑠 linearly independent solutions over K (i.e., the condition 𝑄𝜉 =0 encodes vanishing of the supersymmetry condition dimKK(𝐷) =) 𝑠 amounts to the requirement variation of the dilatino. When considering compactifications that 𝑆𝐷 has rank 𝑠; in particular, this imposes well-known on some internal space down to some space admitting Killing topological constraints on 𝑆. A similar argument shows Advances in High Energy Physics 5 that there exists a (topologically trivial) K-vector subbundle noncommutative composition of quantum observables; the 𝑆𝐷,𝑄 ⊂𝑆𝐷 ⊂𝑆such that rkK𝑆𝐷,𝑄 = dimKK(𝐷, 𝑄) and such classical limit corresponds to taking the scale of the metric ∞ that Γ(𝑀,𝐷,𝑄 𝑆 )=K(𝐷, 𝑄)K ⊗ C (𝑀, R). to infinity while the expansion of the geometric product As mentioned in the introduction, a basic problem in the into generalized products can be viewed as a semiclassical analysis of flux compactifications (which is also of mathe- expansion. In the classical limit, the geometric product matical interest in its own right) is to find efficient meth- reduces to the wedge product and the Kahler-Atiyah¨ algebra ods for translating constrained generalized Killing (s)pinor reduces to the exterior algebra of 𝑀,whichplaystherole equations for some collection 𝜉1,...,𝜉𝑠 of sections of 𝑆 into a of the classical algebra of observables. Section 3.3 discusses system of algebraic and differential conditions for differential certain (anti-)automorphisms of the Kahler-Atiyah¨ algebra forms which are constructed as bilinears in 𝜉1,...,𝜉𝑠.In which will be used intensively later on while Section 3.4 gives Section 5, we show how the geometric algebra formalism some properties of the left and right multiplication operators can be used to provide an efficient and conceptually clear in this algebra. In Section 3.5, we give a brief discussion of the solution to this problem. Before doing so, however, we have decomposition of an inhomogeneous form into parts parallel to recall the basics of the geometric algebra approach to spin and perpendicular to a normalized one-form and of the geometry, which we proceed to do next. interplay of this decomposition with the geometric product. Section 3.6 explains the role played by the volume form and introduces the “twisted Hodge operator,”acertainvariantof 3. The Kähler-Atiyah Bundle of a Pseudo- the ordinary Hodge operator which is natural from the point Riemannian Manifold of view of the Kahler-Atiyah¨ algebra. Section 3.7 discusses the eigenvectors of the twisted Hodge operator, which we This section lays out the basics of the geometric algebra call “twisted (anti-)self-dual forms”; these will play a crucial formalism and develops some specialized aspects which will role in later considerations. In Section 3.8, we recall the be needed later on. In Sections 3.1 and 3.2, we start with algebraic classification of the fiber type of the Clifford/Kahler-¨ the Clifford bundle of the cotangent bundle of a pseudo- Atiyah bundle, which is an obvious application of the well- Riemannian manifold (𝑀, 𝑔),viewedasabundleofunital known classification of Clifford algebras. We pay particular and associative—but noncommutative—algebras which is attention to the “nonsimple case”—the case when the fibers naturally associated with (𝑀, 𝑔). The basic idea of “geometric of the Kahler-Atiyah¨ bundle fail to be simple as associative algebra” is to use a certain isomorphic realization of the algebras over the base field. In Section 3.9, we discuss the Cliffordbundleinwhichtheunderlyingvectorbundleis spaces of twisted (anti-)self-dual forms in the nonsimple case, identified with the exterior bundle of 𝑀. In this realiza- showing that—in this case—they form two-sided ideals of tion, the multiplication of the Clifford bundle transports the Kahler-Atiyah¨ algebra. We also give a description of such to a fiberwise multiplication of the exterior bundle; when forms in terms of rank truncations, which is convenient in endowed with this associative but noncommutative multipli- certain computations even though it is not well behaved with cation, the exterior bundle becomes a bundle of associative respect to the geometric product. In Section 3.10, we show algebras known as the Kahler-Atiyah¨ bundle.Inturn,the that, in the presence of a globally defined one-form 𝜃 of noncommutative multiplication of the Kahler-Atiyah¨ bundle unit norm, the spaces of twisted self-dual and twisted anti- induces an associative but noncommutative multiplication self-dual forms are isomorphic (as unital associative algebras ⬦ (which we denote by and call the geometric product)on !) with the space of those inhomogeneous forms which are inhomogeneous differential forms. The resulting associative orthogonal to 𝜃—a space which always forms a subalgebra algebra is known as the Kahler-Atiyah¨ algebra of (𝑀, 𝑔) of the Kahler-Atiyah¨ algebra. We also show that the compo- and can be viewed as a certain deformation of the exterior nents of an inhomogeneous form which are orthogonal and algebra which is parameterized by the metric 𝑔 of 𝑀.The parallel to 𝜃 determine each other when the form is twisted Kahler-Atiyah¨ algebra is an associative and unital algebra over ∞ (anti-)self-dual and give explicit formulas for the relation the commutative and unital ring C (𝑀, K) of smooth K- between these components in terms of what we call the valued functions defined on 𝑀—so in particular it is a K- ∞ “reduced twisted Hodge operator.” Some of the material of algebra upon considering the embedding K ⊂ C (𝑀, K) this section is “well known” at least in certain circles, though which is defined by associating with each element of K the the literature tends to be limited in its treatment of general corresponding constant function. The geometric product has dimensions and signatures and of certain other aspects. The an expansion in terms of so-called “generalized products,” reader who is familiar with geometric algebra may wish which form a collection of binary operations acting on to concentrate on Sections 3.5, 3.6, 3.7, 3.9, and 3.10 and inhomogeneous forms. In turn, the generalized products can especially on our treatment of parallelism and orthogonality be described as certain combinations of contractions and for twisted (anti-)self-dual forms, which is important for wedge products. The expansion of the geometric product into applications. generalizedproductscanbeinterpreted(undercertainglobal conditions on (𝑀, 𝑔)) as a form of “partial quantization” 3.1. Preparations: Wedge and Generalized of a spin system—the role of the Planck constant being Contraction Operators played by the inverse of the overall scale of the metric. In this interpretation, the Kahler-Atiyah¨ algebra is the quantum The Grading Automorphism. Let 𝜋 be that involutive ∞ algebra of observables while the geometric product is the C (𝑀, K)-linear automorphism of the exterior algebra 6 Advances in High Energy Physics

𝑑 𝑘 (ΩK(𝑀), ∧) which is uniquely determined by the property so the rank decomposition ΩK(𝑀) = ⊕𝑘=0ΩK(𝑀) is an that it acts as minus the identity on all one-forms. Thus orthogonal direct sum decomposition with respect to this 1 pairing. Notice that the restriction of ⟨, ⟩to ΩK(𝑀) coincides 𝑑 𝜋 𝜋 (𝜔) def= ∑ (−1)𝑘 𝜔(𝑘), with (11). Also notice that is self-adjoint with respect to the pairing ⟨, ⟩: 𝑘=0 (5) ⟨𝜋 𝜔 ,𝜂⟩ = ⟨𝜔, 𝜋 (𝜂)⟩ ,∀𝜔,𝜂∈Ω𝑀 . 𝑑 ( ) K ( ) (13) (𝑘) (𝑘) 𝑘 ∀𝜔 = ∑𝜔 ∈ΩK (𝑀) , where 𝜔 ∈ΩK (𝑀) . 𝑘=0 Interior Products. The ⟨, ⟩-adjoints of the left and right wedge 𝜄𝑅 𝜄𝐿 Taking wedge products from the left and from the right with product operators (6) are denoted by 𝜔 and 𝜔 and are called ∞ some inhomogeneous form 𝜔∈ΩK(𝑀) defines C (𝑀, R)- the right and left generalized contraction (or interior product) 𝐿 𝑅 operators, respectively: linear operators ∧𝜔 and ∧𝜔: 𝐿 𝑅 𝐿 ⟨∧𝜔 (𝜂) , 𝜌⟩ =𝜔 ⟨𝜂,𝜄 (𝜌)⟩ , ∧𝜔 (𝜂) = 𝜔 ∧ 𝜂, 𝑅 ⟨∧𝑅 (𝜂) , 𝜌⟩ =𝐿 ⟨𝜂,𝜄 (𝜌)⟩ , (14) ∧𝜔 (𝜂)=𝜂∧𝜔, (6) 𝜔 𝜔

∀𝜔, 𝜂 ∈ΩK (𝑀) ∀𝜔,𝜂,𝜌∈ΩK (𝑀) . which satisfy the following identities by virtue of the fact that Properties (7) translate into the wedge product is associative 𝜄𝐿 ∘𝜄𝐿 =𝜄𝐿 , 𝜔1 𝜔2 𝜔1∧𝜔2 ∧𝐿 ∘∧𝐿 =∧𝐿 , 𝜔1 𝜔2 𝜔1∧𝜔2 𝑅 𝑅 𝑅 𝜄𝜔 ∘𝜄𝜔 =𝜄𝜔 ∧𝜔 , 𝑅 𝑅 𝑅 1 2 2 1 ∧𝜔 ∘∧𝜔 =∧𝜔 ∧𝜔 , (15) 1 2 2 1 𝐿 𝑅 𝑅 𝐿 (7) 𝜄𝜔 ∘𝜄𝜔 =𝜄𝜔 ∘𝜄𝜔 , ∧𝐿 ∘∧𝑅 =∧𝑅 ∘∧𝐿 , 1 2 2 1 𝜔1 𝜔2 𝜔2 𝜔1 ∀𝜔1,𝜔2 ∈ΩK (𝑀) , ∀𝜔1,𝜔2 ∈ΩK (𝑀) while relation (8) is equivalent to as well as the following relation, which encodes graded- 𝐿 𝑘 𝑅 𝑅 𝑘 𝐿 𝑘 commutativity of the wedge product: 𝜄𝜔 =𝜋 ∘𝜄𝜔 ⇐⇒ 𝜄 𝜔 =𝜋 ∘𝜄𝜔,∀𝜔∈ΩK (𝑀) . (16) ∧𝐿 =∧𝑅 ∘𝜋𝑘 ⇐⇒ ∧ 𝑅 =∧𝐿 ∘𝜋𝑘,∀𝜔∈Ω𝑘 (𝑀) . ∧𝐿,𝑅 = 𝜄𝐿,𝑅 = 𝜔 𝜔 𝜔 𝜔 K (8) We also have 1𝑀 idΩK(𝑀) and 1𝑀 idΩK(𝑀).Together 𝐿 𝑅 with (7) and (15), this shows that ∧ and ∧ define a structure 𝐿 𝑅 of (ΩK(𝑀), -bimodule∧) on ΩK(𝑀) while 𝜄 and 𝜄 define The Inner Product.Let⟨, ⟩denote the symmetric nondegen- ∞ another (ΩK(𝑀), -bimodule∧) structure on the same space. erate C (𝑀, R)-bilinear pairing (known as the inner product These two bimodule structures are adjoint to each other with of inhomogeneous forms) induced by the metric 𝑔 on the respect to the pairing ⟨, ⟩. exterior bundle. To be precise, this pairing is defined through 𝐿 𝑅 Identities (8) and (16) show that ∧𝜔 and ∧𝜔 determine 𝜄𝐿 𝜄𝑅 ⟨𝛼1 ∧⋅⋅⋅∧𝛼𝑘,𝛽1 ∧⋅⋅⋅∧𝛽𝑙⟩ each other while 𝜔 and 𝜔 also determine each other. From (9) nowonwechoosetoworkwithleft wedge-multiplication =𝛿 (𝑔(𝛼̂ ,𝛽 ) ), ∀𝛼,𝛽 ∈Ω1 (𝑀) , 𝑘𝑙 det 𝑖 𝑗 𝑖,𝑗=1⋅⋅⋅𝑘 𝑖 𝑗 K def 𝐿 ∧𝜔 =∧𝜔 (17) a relation which fixes the convention used later in our and with the following generalized contraction operator: computations (cf. Section 6) via the normalization property: 𝑅 𝜄𝜔 =𝜄𝜏(𝜔) ⇐⇒ ⟨𝜔 ∧ 𝜂, 𝜌⟩ =𝜏(𝜔) ⟨𝜂,𝜄 (𝜌)⟩ , (18) ⟨1𝑀,1𝑀⟩=1. (10)

∗ which satisfy Here, 𝑔̂ is the metric induced by 𝑔 on 𝑇K𝑀,whichgivesthe 𝑎 𝑏 1 ∧ ∘∧ =∧ , following pairing on one-forms 𝛼=𝛼𝑎𝑒 , 𝛽=𝛽𝑏𝑒 ∈ΩK(𝑀): 𝜔1 𝜔2 𝜔1∧𝜔2

𝑎𝑏 𝑎 𝑏 𝜄𝜔 ∘𝜄𝜔 =𝜄𝜔 ∧𝜔 , (19) 𝑔(𝛼,𝛽)=𝑔̂ 𝛼𝑎𝛽𝑏 for 𝛼=𝛼𝑎𝑒 ,𝛽=𝛽𝑏𝑒 . (11) 1 2 1 2

∀𝜔1,𝜔2 ∈ΩK (𝑀) The fixed rank components of ΩK(𝑀) are mutually orthogo- ⟨, ⟩ nal with respect to the pairing : as well as ⟨𝜔, 𝜂⟩ = 0, ∀𝜔∈Ω𝑘 (𝑀) ,∀𝜂∈Ω𝑙 (𝑀) ,∀𝑘=𝑙,̸ ∧ =𝜄 = K K (12) 1𝑀 1𝑀 idΩK(𝑀) (20) Advances in High Energy Physics 7 and thus define two different structures of left module onthe 3.2. Definition and First Properties of the Ω (𝑀) (Ω (𝑀), ∧) space K over the ring K . Kahler-Atiyah¨ Algebra Wedge and Interior Product with a One-Form.Forlater The Geometric Product. Following an idea originally due to 1 ∗ reference, let us consider the case when 𝜃∈ΩK(𝑀) is a Chevalley and Riesz [8, 9], we identify Cl(𝑇K𝑀) with the ∗ one-form. Recall that the metric 𝑔 induces mutually inverse exterior bundle ∧𝑇K𝑀, thus realizing the Clifford product 1 ♯ “musical isomorphisms” ♯: Γ(𝑀,K 𝑇 𝑀) → ΩK(𝑀) and : as the geometric product, which is the unique fiberwise Ω1 (𝑀) → Γ(𝑀, 𝑇 𝑀) associative, unital, and bilinear binary composition4 ⬦: K K defined by raising and lowering of ∧𝑇∗𝑀× ∧𝑇∗𝑀→∧𝑇∗𝑀 indices, respectively: K 𝑀 K K whose induced action on sections (which we again denote by ⬦) satisfies the following 1 def 𝑎 𝑎 𝑏 relations for all 𝜃∈ΩK(𝑀) and all 𝜔∈ΩK(𝑀): 𝑋=𝑋 𝑒𝑎 󳨐⇒ 𝑋 ♯ =𝑋𝑎𝑒 , where 𝑋𝑎 =𝑔𝑎𝑏𝑋 , (21) 𝜃⬦𝜔=𝜃∧𝜔+𝜄𝜃𝜔, 𝑎 ♯ 𝑎 𝑎 def 𝑎𝑏 𝜃=𝜃𝑎𝑒 󳨐⇒ 𝜃 =𝜃 𝑒𝑎, where 𝜃 =𝑔 𝜃𝑏. (29) 𝜋 (𝜔) ⬦𝜃=𝜃∧𝜔−𝜄𝜃𝜔. These isomorphisms satisfy Equations (29) determine the geometric composition of any two inhomogeneous forms via the requirement that the 𝑔 (𝑋, 𝑌) = 𝑔(𝑋̂ ,𝑌)=⟨𝑋,𝑌⟩, ∞ ♯ ♯ ♯ ♯ geometric product is associative and C (𝑀, K)-bilinear. ∗ The unit of the fiber Cl(𝑇K,𝑥𝑀) at a point 𝑥∈𝑀 ∀𝑋, 𝑌 ∈ Γ (𝑀,K 𝑇 𝑀) , 0 ∗ corresponds to the element 1∈K =∧𝑇K,𝑥𝑀,whichisthe ♯ ♯ (22) ∗ ∗ 𝑔(𝜃 ,𝜃 )=𝑔(𝜃̂ ,𝜃 )=⟨𝜃,𝜃 ⟩, unit of the associative algebra (∧𝑇K,𝑥𝑀,𝑥 ⬦ )≈Cl(𝑇K,𝑥𝑀). 1 2 1 2 1 2 ∗ Hence the unit section of the Clifford bundle Cl(𝑇K𝑀) is 1 1 :𝑀→K ∀𝜃1,𝜃2 ∈ΩK (𝑀) . identified with the constant function 𝑀 given by 1𝑀(𝑥) = 1 for all 𝑥∈𝑀.Throughthisconstruction, We have theCliffordbundleisidentifiedwiththebundleofalgebras (∧𝑇∗𝑀, ⬦) 𝑋 =𝑋⌟𝑔, K ,whichisknown[10]astheKahler-Atiyah¨ bundle ♯ of (𝑀, 𝑔). When endowed with the geometric product, the (23) Ω (𝑀) K 𝜃=𝜃♯⌟𝑔, space K of all inhomogeneous -valued smooth forms on 𝑀 becomes a unital and associative (but noncommutative) ∞ 𝑋⌟ algebra (ΩK(𝑀), ⬦) over the ring C (𝑀, R),knownasthe where denotes the ordinary left contraction of a tensor (𝑀, 𝑔) with a vector field. It is not hard to see that the left Kahler-Atiyah¨ algebra of .TheunitoftheKahler-¨ Atiyah algebra is the constant function 1𝑀.Wehaveaunital contraction 𝜄𝜃 with a one-form coincides with the ordinary ∞ ♯ ♯ isomorphism of associative algebras over C (𝑀, R) between left contraction 𝜃 ⌟ with the vector field 𝜃 : ∞ ∗ (ΩK(𝑀), ⬦) and the C (𝑀, R)-algebra Γ(𝑀, Cl(𝑇K𝑀)) of ♯ 1 all smooth sections of the Clifford bundle. The Kahler-Atiyah¨ 𝜄𝜃𝜔=𝜃⌟𝜔, ∀𝜃K ∈Ω (𝑀) ,∀𝜔∈ΩK (𝑀) . (24) algebra can be viewed as a Z2-graded associative algebra with Since 𝜃∧𝜃=0, properties (19) imply3 even and odd parts given by Ωev (𝑀) def=⊕ Ω𝑘 (𝑀) , ∧𝜃 ∘∧𝜃 =𝜄𝜃 ∘𝜄𝜃 =0. (25) K 𝑘=even K (30) ♯ Ωodd (𝑀) def=⊕ Ω𝑘 (𝑀) , Furthermore, the similar property of 𝜃 ⌟ implies that 𝜄𝜃 is an K 𝑘=odd K odd derivation of the exterior algebra: since it is easy to check the inclusions: 𝜄 (𝜔 ∧ 𝜂) = (𝜄 𝜔)∧𝜂+𝜋(𝜔) ∧𝜄 𝜂, ev ev ev 𝜃 𝜃 𝜃 ΩK (𝑀) ⬦ΩK (𝑀) ⊂ΩK (𝑀) , (26) ∀𝜔, 𝜂 ∈ΩK (𝑀) . odd odd ev ΩK (𝑀) ⬦ΩK (𝑀) ⊂ΩK (𝑀) ,

ev odd odd (31) ΩK (𝑀) ⬦ΩK (𝑀) ⊂ΩK (𝑀) , Local Expressions.If𝑒𝑎 is an arbitrary local frame of 𝑀 with 𝑎 𝑎 𝑎 dual coframe 𝑒 (thus 𝑒 (𝑒𝑏)=𝛿𝑏 ), we let 𝑔𝑎𝑏 =𝑔(𝑒𝑎,𝑒𝑏) and odd ev odd 𝑎𝑏 𝑎 𝑏 𝑎𝑏 𝑎 𝑎 ♯ ΩK (𝑀) ⬦ΩK (𝑀) ⊂ΩK (𝑀) . 𝑔 = 𝑔(𝑒̂ ,𝑒 ),sowehave𝑔 𝑔𝑏𝑐 =𝛿𝑐 .Thevectorfields(𝑒 ) 𝑎 ♯ 𝑎 Z satisfy (𝑒 ) ⌟𝑔=𝑒 andaregivenexplicitlyby However, it is not a -graded algebra since the geometric productoftwoformsofdefiniterankneednotbeaform 𝑎 ♯ 𝑎𝑏 𝑃 = (1/2)(1 + 𝜋) 𝑃 = (𝑒 ) =𝑔 𝑒𝑏; (27) of definite rank. We let ev and odd (1/2)(1 − 𝜋) be the complementary idempotents associated they form the contragradient local frame defined by (𝑒𝑎).We with the decomposition into even and odd parts: 𝑏 ♯ 𝑎 ♯ 𝑏 ♯ 𝑎𝑏 have 𝑒𝑎 =𝑔𝑎𝑏(𝑒 ) and 𝑔((𝑒 ) ,(𝑒 ) )=𝑔 .Thus 𝑃ev (𝜔) =𝜔ev, 𝑎 ♯ 𝑎𝑏 (32) 𝜄𝑒𝑎 =(𝑒 ) ⌟=𝑔 𝑒𝑏⌟⇐⇒𝑒𝑎⌟=𝑔𝑎𝑏𝜄𝑒𝑏 . (28) 𝑃odd (𝜔) =𝜔odd, 8 Advances in High Energy Physics

ev where 𝜔=𝜔ev +𝜔odd ∈ΩK(𝑀),with𝜔ev ∈ΩK (𝑀) and For forms of definite ranks, the graded ⬦-commutator and odd ⬦ 𝜔odd ∈ΩK (𝑀). graded -anticommutator are of course defined through

def 𝑝𝑞 Generalized Products: Connection with Quantization of Spin [[𝜔, 𝜂]]−,⬦ =𝜔⬦𝜂−(−1) 𝜂⬦𝜔, Systems.Thegeometricproduct⬦ canbeviewedasa deformation of the wedge product (parameterized by the def 𝑝𝑞 (38) [[𝜔, 𝜂]]+,⬦ =𝜔⬦𝜂+(−1) 𝜂⬦𝜔, metric 𝑔) and reduces to the latter in the limit 𝑔→ ∞;inthislimit,theKahler-Atiyah¨ algebra reduces to the 𝑝 𝑞 ∀𝜔 ∈ ΩK (𝑀) ,∀𝜂∈ΩK (𝑀) , exterior algebra (ΩK(𝑀), .∧) Under some mild assumptions, the geometric product can be described quite elegantly in being extended by linearity to the entire space ΩK(𝑀). the language of supermanifolds, as the star product induced Using(35)onecaneasilydeducethefollowingrelationfor by fiberwise Weyl quantization of a pure spin system [20– homogeneous forms: 23]. For this, consider the parity-changed tangent bundle 𝑝 𝑘(𝑝−𝑘+1)+[𝑘/2] Π𝑇𝑀 of 𝑀 (a supermanifold with body 𝑀)andintroduce 𝑝𝑞 (−1) 𝑎 𝜂⬦𝜔=(−1) ∑ 𝜔∧ 𝜂, odd coordinates 𝜁 on the fibers of Π𝑇𝑈, corresponding to 𝑘! 𝑘 𝑎 𝑘=0 (39) acoframe𝑒 of 𝑀 defined on a small enough open subset 𝑈⊂𝑀 𝑝 𝑞 . Inhomogeneous differential forms (2) correspond ∀𝜔 ∈ ΩK (𝑀) ,∀𝜂∈ΩK (𝑀) ,𝑝≤𝑞. to functions defined on Π𝑇𝑀 having the following local expansion: We will mostly use, instead of ∧𝑘, the so-called generalized products △𝑘, which are defined by rescaling the contracted 𝑑 1 wedge products: (𝑘) 𝑎1 𝑎𝑘 𝑓𝜔 (𝑥,) 𝜁 =𝑈 ∑ 𝜔𝑎 ⋅⋅⋅𝑎 (𝑥) 𝜁 ⋅⋅⋅𝜁 . (33) 𝑘! 1 𝑘 1 𝑘=0 △ = ∧ . 𝑘 𝑘! 𝑘 (40) Thisallowsustorepresentthegeometricproductthroughthe fermionic analogue ⋆ of the Moyal product, using a certain These have the advantage that the various factorial prefactors “vertical” [23] quantization procedure5: in the expansions above disappear when those expansions are reexpressed in terms of generalized products. ⃖ ⃗ Expansions (35) and (37) can also be obtained directly 𝑎𝑏 𝜕 𝜕 𝑓 ⋆𝑓 =𝑓 =𝑓 (𝑔 )𝑓. (34) from the definition of the geometric product using (29), 𝜔 𝜂 𝜔⬦𝜂 𝜔 exp 𝜕𝜁𝑎 𝜕𝜁𝑏 𝜂 which shows that the purely mathematical identities given abovealsoholdirrespectiveofanyinterpretationthroughthe Expanding the exponential in (34) gives the following expres- theory of quantization of spin systems. sions for two general inhomogeneous forms 𝜔, 𝜂 ∈ΩK(𝑀) (cf. [24–26]): 3.3. (Anti-)Automorphisms of the Kahler-Atiyah¨ Algebra. [𝑑/2] Direct computation shows that 𝜋 is an involutive automor- (−1)𝑘 𝜔⬦𝜂= ∑ 𝜔∧ 𝜂 phism (known as the main or grading automorphism) of the (2𝑘)! 2𝑘 𝑘=0 Kahler-Atiyah¨ bundle (a property which, in the limit 𝑔→∞, (35) recovers the well-known fact that 𝜋 is also an automorphism [(𝑑−1)/2] (−1)𝑘+1 + ∑ 𝜋 (𝜔) ∧ 𝜂, of the exterior bundle). Theahler-Atiyah K¨ bundle also admits (2𝑘 + 1)! 2𝑘+1 an involutive antiautomorphism 𝜏 (known as the main 𝑘=0 antiautomorphism or as reversion), which is given by ∞ where the binary C (𝑀, R)-bilinear operations ∧𝑘 are def 𝜏 (𝜔) = (−1)𝑘(𝑘−1)/2 𝜔, ∀𝜔 ∈Ω𝑘 (𝑀) . (41) the contracted wedge products [24–27], defined iteratively K through ∗ Itistheuniqueantiautomorphismof(∧𝑇K𝑀, ⬦) which acts 𝜏(𝜃) =𝜃 𝜔∧ 𝜂=𝜔∧𝜂, trivially on all one-forms (i.e., which satisfies for 0 any form 𝜃 of rank one). Direct computation (or the fact (36) 𝑎𝑏 that the exterior product is recovered from the diamond 𝜔∧𝑘+1𝜂=𝑔 (𝑒𝑎⌟𝜔) ∧𝑘 (𝑒𝑏⌟𝜂) =𝑎𝑏 𝑔 (𝜄𝑒𝑎 𝜔) ∧𝑘 (𝜄𝑒𝑏 𝜂) . productinthelimitofinfinitemetric)showsthat𝜏 is also an ∗ antiautomorphism of the exterior bundle (𝑇 𝑀, ∧).Wealso We also have the following expansions for the graded ⬦- K notice that 𝜋 and 𝜏 commute. All in all, we have the relations: commutator and graded ⬦-anticommutator of 𝜔 with 𝜂: 𝜋∘𝜏=𝜏∘𝜋, [(𝑑−1)/2] (−1)𝑘+1 (42) [[𝜔, 𝜂]] =2 ∑ 𝜋 (𝜔) ∧2𝑘+1𝜂, 𝜋∘𝜋=𝜏∘𝜏=idΩ (𝑀). −,⬦ (2𝑘 + 1)! K 𝑘=0 (37) ev(𝑇∗𝑀) [𝑑/2] Note that Cl K identifies with the subbundle of unital (−1)𝑘 ev ∗ 𝑘 ∗ [[𝜔, 𝜂]] =2∑ 𝜔∧ 𝜂. subalgebras ∧ 𝑇K𝑀=⊕𝑘=even∧ 𝑇K𝑀 of the Kahler-¨ +,⬦ (2𝑘)! 2𝑘 ev 𝑘=0 Atiyah bundle, whose space of smooth sections ΩK (𝑀) can Advances in High Energy Physics 9

be described as the eigenspace of 𝜋 corresponding to the The first identity in (49) shows that the operator 𝜄𝜃 is an odd ∞ eigenvalue +1: C (𝑀, R)-linear derivation (in fact, an odd differential— ev since 𝜄𝜃 ∘𝜄𝜃 =0)oftheKahler-Atiyah¨ algebra: ΩK (𝑀) = K (1−𝜋) = {𝜔∈ΩK (𝑀) |𝜋(𝜔) =𝜔} . (43)

𝜄𝜃 (𝜔 ⬦ 𝜂)𝜃 =𝜄 (𝜔) ⬦𝜂+𝜋(𝜔) ⬦𝜄𝜃 (𝜂) , 3.4. The Left and Right Geometric Multiplication Operators. (50) 𝐿 𝑅 C∞(𝑀, R) 1 Let 𝜔, 𝜔 be the -linear operators of left and right ∀𝜔, 𝜂 ∈ΩK (𝑀) ,∀𝜃∈Ω(𝑀) . multiplication with 𝜔∈ΩK(𝑀) in the Kahler-Atiyah¨ algebra: In the limit 𝑔→∞,thispropertyrecovers(26).Noticethat 𝐿 (𝜂) def=𝜔⬦𝜂, 𝜔 ∧𝜃 is not a derivation of the Kahler-Atiyah¨ algebra; however, it satisfies ∧𝜃 ∘∧𝜃 =0. def (44) 𝑅𝜔 (𝜂) =𝜂⬦𝜔, 1 3.5. Orthogonality and Parallelism. Let 𝜃∈Ω (𝑀) be a fixed ∀𝜔, 𝜂 ∈Ω (𝑀) . K K one-form which satisfies the normalization condition: These satisfy 𝑔̂ (𝜃,) 𝜃 =1 that is 𝜄𝜃𝜃=1. (51) 𝐿 ∘𝑅 =𝑅 ∘𝐿 , 𝜔1 𝜔2 𝜔2 𝜔1 This condition is equivalent to 𝐿 ∘𝐿 =𝐿 , 𝜔1 𝜔2 𝜔1⬦𝜔2 (45) 𝜃⬦𝜃=1, 𝑅 ∘𝑅 =𝑅 (52) 𝜔1 𝜔2 𝜔2⬦𝜔1 𝜃∧𝜃 =0 as a consequence of associativity of the geometric product. a fact which follows from (29) and from the identity 𝜃⬦𝜃=𝜄 𝜃 We also have (which, together, imply 𝜃 ). 𝜔∈Ω(𝑀) 𝐿 ∘𝜋=𝜋∘𝐿 , We say that an inhomogeneous form K is 𝜔 𝜋(𝜔) parallel to 𝜃 (we write 𝜃‖𝜔)if𝜃∧𝜔 =0and orthogonal to 𝜃 (we write 𝜃⊥𝜔)if𝜄𝜃𝜔=0.Thus 𝑅𝜔 ∘𝜋=𝜋∘𝑅𝜋(𝜔), 𝐿 ∘𝜏=𝜏∘𝑅 , def 𝜔 𝜏(𝜔) (46) 𝜃‖𝜔⇐⇒ 𝜔 ∈ K (∧𝜃), (53) 𝜏∘𝐿𝜔 =𝑅𝜏(𝜔) ∘𝜏, def 𝜃⊥𝜔⇐⇒ 𝜔 ∈ K (𝜄𝜃), ∀𝜔 ∈ ΩK (𝑀) , where we remind the reader that K(𝐴) denotes the kernel 𝜋 𝜏 since is an involutive algebra automorphism while is an of any K-linear operator 𝐴:ΩK(𝑀) → ΩK(𝑀).Properties involutive antiautomorphism. Identity (29) can be written as (25) imply I(𝜄𝜃)⊂K(𝜄𝜃) and I(∧𝜃)⊂K(∧𝜃),whereI(𝐴) K 𝐴:Ω(𝑀) → 𝐿𝜃 =∧𝜃 +𝜄𝜃, denotes the image of any -linear operator K ΩK(𝑀). These inclusions are in fact equalities, as we will see 𝑅𝜃 ∘𝜋=∧𝜃 −𝜄𝜃, (47) in a moment. 1 ∀𝜃 ∈ ΩK (𝑀) , Proposition 5. Any inhomogeneous differential form 𝜔∈ Ω (𝑀) being equivalent to K decomposes uniquely as 1 𝜔=𝜔 +𝜔 , ∧ = (𝐿 +𝑅 ∘𝜋), ‖ ⊥ (54) 𝜃 2 𝜃 𝜃 𝜃‖𝜔 𝜃⊥𝜔 1 where ‖ and ⊥.Moreover,theparalleland 𝜄 = (𝐿 −𝑅 ∘𝜋), (48) orthogonal parts of 𝜔 are given by 𝜃 2 𝜃 𝜃 1 𝜔 =𝜃∧(𝜄𝜔) , ∀𝜃 ∈ Ω (𝑀) . ‖ 𝜃 K (55) This shows that the operators ∧𝜃 and 𝜄𝜃 (and thus—given 𝜔⊥ =𝜄𝜃 (𝜃∧𝜔) . properties (19)—also the operators ∧𝜔 and 𝜄𝜔 for any 𝜔∈ Ω (𝑀) ∞ def def K ) are determined by the geometric product. In fact, the C (𝑀, R)-linear operators 𝑃‖ =∧𝜃 ∘𝜄𝜃 and 𝑃⊥ = 𝜄𝜃 ∘∧𝜃 are complementary idempotents: Remark 4. Equation (29) implies

1 𝑃‖ +𝑃⊥ = idΩ (𝑀), 𝜄 𝜔= [[𝜃,]] 𝜔 , K 𝜃 2 −,⬦ 𝑃 ∘𝑃 =𝑃, 1 ‖ ‖ ‖ (49) (56) 𝜃∧𝜔= [[𝜃,]] 𝜔 +,⬦ , 2 𝑃⊥ ∘𝑃⊥ =𝑃⊥, 1 ∀𝜃 ∈ ΩK (𝑀) ,∀𝜔∈ΩK (𝑀) . 𝑃‖ ∘𝑃⊥ =𝑃⊥ ∘𝑃‖ =0. 10 Advances in High Energy Physics

Proof. The statements of the proposition follow immediately As an immediate corollary of the proposition, we find the from the fact that ∧𝜃 and 𝜄𝜃 are nilpotent and because 𝜄𝜃 is an well-known equalities odd derivation of the wedge product, which implies K (𝜄𝜃)=I (𝜄𝜃), (58) 𝜄𝜃 (𝜃∧𝜔) =𝜔−𝜃∧(𝜄𝜃𝜔) , (57) K (∧𝜃)=I (∧𝜃) whereweusedthenormalizationcondition(51). as well as the characterizations:

𝜃‖𝜔⇐⇒𝜔=𝜃∧𝛼 with 𝛼∈ΩK (𝑀) ⇐⇒ 𝜃 ⬦ 𝜔 = −𝜋 (𝜔) ⬦𝜃⇐⇒𝜔∈K (𝐿𝜃 +𝑅𝜃 ∘𝜋), (59) 𝜃⊥𝜔⇐⇒𝜔=𝜄𝜃𝛽 with 𝛽∈ΩK (𝑀) ⇐⇒ 𝜃 ⬦ 𝜔 =𝜋 (𝜔) ⬦𝜃⇐⇒𝜔∈K (𝐿𝜃 −𝑅𝜃 ∘𝜋),

whereweusedrelations(48).Thus𝜃‖𝜔iff 𝜔 graded The Top Component of an Inhomogeneous Form. The parallel anticommutes with 𝜃 and 𝜃⊥𝜔iff 𝜔 graded commutes with part of 𝜔∈ΩK(𝑀) can be written as 𝜃 in the Kahler-Atiyah¨ algebra. 𝜔‖ =𝜃∧𝜔⊤, (65) Behavior with respect to the Geometric Product. Consider the ∞ where following C (𝑀, R)-submodules of ΩK(𝑀): def ⊥ 𝜔⊤ =𝜄𝜃𝜔∈ΩK (𝑀) . (66) Ω‖ (𝑀) def={𝜔∈Ω (𝑀) |𝜃‖𝜔}, K K 𝜔 (60) This shows that determines and is determined by the two ⊥ def inhomogeneous forms 𝜔⊥ and 𝜔⊤,bothofwhichbelongto ΩK (𝑀) ={𝜔∈ΩK (𝑀) |𝜃⊥𝜔}. ⊥ ΩK(𝑀). In fact, any 𝜔∈ΩK(𝑀) can be written uniquely in Using the characterizations in (59), we find the form ⊥ 𝜃‖𝜔,𝜃⊥𝜂󳨐⇒𝜃‖(𝜔⬦𝜂), 𝜃‖(𝜂⬦𝜔), 𝜔=𝜃∧𝛼+𝛽 with 𝛼, 𝛽K ∈Ω (𝑀) ; (67) ∞ namely, we have 𝛼=𝜔⊤ and 𝛽=𝜔⊥.ThisgivesaC (𝑀, R)- 𝜃‖𝜔,𝜂󳨐⇒𝜃⊥(𝜔⬦𝜂), (61) linear isomorphism: 𝜃⊥𝜔,𝜂󳨐⇒𝜃⊥(𝜔⬦𝜂), 𝜄𝜃+𝑃⊥ ⊥ ⊥ ΩK (𝑀) 󳨀󳨀󳨀󳨀→Ω (𝑀) ⊕Ω (𝑀) . (68) which translate into which sends 𝜔∈ΩK(𝑀) into the pair (𝜔⊤,𝜔⊥) and whose (𝜔 ⬦ 𝜂) =𝜔 ⬦𝜂 +𝜔 ⬦𝜂, ⊥ ‖ ‖ ⊥ ⊥ ‖ inverse sends a pair (𝛼, 𝛽) with 𝛼, 𝛽K ∈Ω (𝑀) into the form (62) (67). Since 𝜔⊤ is orthogonal to 𝜃,wehave𝜃∧𝜔⊤ =𝜃⬦ (𝜔 ⬦ 𝜂) =𝜔 ⬦𝜂 +𝜔 ⬦𝜂 . ⊥ ‖ ‖ ⊥ ⊥ 𝜔⊤ =𝜋(𝜔⊤)⬦𝜃and thus 𝜔‖ =𝜃⬦𝜔⊤. It follows that the decomposition of 𝜔 can be written entirely in terms of the We thus have the inclusions: geometric product: ‖ ‖ ⊥ ΩK (𝑀) ⬦ΩK (𝑀) ⊂ΩK (𝑀) , 𝜔=𝜃⬦𝜔⊤ +𝜔⊥. (69) Ω⊥ (𝑀) ⬦Ω⊥ (𝑀) ⊂Ω⊥ (𝑀) , K K K An easy computation using this formula gives (63) ‖ ⊥ ‖ (𝜔 ⬦ 𝜂) =𝜔 ⬦𝜂 +𝜋(𝜔 )⬦𝜂 , ΩK (𝑀) ⬦ΩK (𝑀) ⊂ΩK (𝑀) , ⊥ ⊥ ⊥ ⊤ ⊤ (70) ⊥ ‖ ‖ (𝜔 ⬦ 𝜂) =𝜔 ⬦𝜂 +𝜋(𝜔 )⬦𝜂 . ΩK (𝑀) ⬦ΩK (𝑀) ⊂ΩK (𝑀) . ⊤ ⊤ ⊥ ⊥ ⊤ 𝜄 (1 )=0 𝜃⊥ Together with the identity 𝜃 𝑀 (which shows that 3.6. The Volume Form and the Twisted Hodge 1 Ω⊥(𝑀) 𝑀), the last property in (61) shows that K is a unital Duality Operator subalgebra of the Kahler-Atiyah¨ algebra. Notice that characterizations (59) imply that the involu- The Ordinary Volume Form. From now on, we will assume tions 𝜋 and 𝜏 preserve parallelism and orthogonality to 𝜃: 𝑑 ∗ that 𝑀 is oriented (in particular, the K-line bundles Λ 𝑇K𝑀 𝜃‖𝜔󳨐⇒𝜃‖𝜋(𝜔) , are trivial for K = R, C). Consider the volume form determined on 𝑀 by the metric and by this orientation, which 𝜃⊥𝜔󳨐⇒𝜃⊥𝜋(𝜔) , has the following expression in a local frame defined on 𝑈⊂ (64) 𝑀: 𝜃‖𝜔󳨐⇒𝜃‖𝜏(𝜔) , 1 󵄨 󵄨 𝑎 ⋅⋅⋅𝑎 √󵄨 󵄨 1 𝑑 vol𝑀 =𝑈 󵄨det 𝑔󵄨𝜖𝑎 ⋅⋅⋅𝑎 𝑒 . (71) 𝜃⊥𝜔󳨐⇒𝜃⊥𝜏(𝜔) . 𝑑! 1 𝑑 Advances in High Energy Physics 11

Here, det 𝑔 is the determinant of the matrix (𝑔(𝑒𝑎,𝑒𝑏))𝑎,𝑏=1⋅⋅⋅𝑑 The Modified Volume Form. Consider the following K-valued 𝜖 𝑀 while 𝑎1⋅⋅⋅𝑎𝑑 are the local coefficients of the Ricci density— top form on : 1 2 ⋅⋅⋅ 𝑑 defined as the signature of the permutation ( 𝑎 𝑎 ⋅⋅⋅ 𝑎 ).The 1 2 𝑑 ] def=𝑐 (K) , volume form satisfies 𝑝,𝑞 vol𝑀 𝑞+𝑑(𝑑−1)/2 𝑞+[𝑑/2] 1, K = R (78) vol𝑀 ⬦ vol𝑀 = (−1) = (−1) def { if 𝑐 (K) = where 𝑝,𝑞 { 𝑞+[𝑑/2] 𝑖 , if K = C, {+1, if 𝑝−𝑞≡4 0, 1 ⇐⇒ 𝑝 − 𝑞 ≡8 0, 1, 4, 5 (72) { = { −1, 𝑝−𝑞≡ 2, 3 ⇐⇒ 𝑝 − 𝑞 ≡ 2, 3, 6, 7, which satisfies { if 4 8 𝑞+[𝑑/2] {(−1) 1𝑀, if K = R where we used the congruences: ] ⬦ ] = { (79) +1 , K = C. { 𝑀 if 𝑑 (𝑑−1) 𝑑 ≡ [ ], 2 2 2 We have the normalization property

𝑝−𝑞 𝑞 (73) {(−1) 1 , K = R 𝑑 (𝑑−1) { , if 𝑑=even 𝑀 if 𝑞+ ≡ 2 ⟨], ]⟩ = { (80) 2 {𝑝−𝑞−1 (−1)[𝑑/2] 1 , K = C 2 { , 𝑑= . { 𝑀 if { 2 if odd and the identity We remind the reader that 𝑝 and 𝑞 denote the number 𝑑−1 𝑑−1 of positive and negative eigenvalues of the metric tensor, 𝐿] =𝑅] ∘𝜋 ⇐⇒ ] ⬦𝜔=𝜋 (𝜔) ⬦ ], respectively. (81) ∀𝜔 ∈ ΩK (𝑀) , The Ordinary Hodge Operator.Recallthattheordinary ∞ 𝜋𝑑−1 𝑑−1 C (𝑀, R)-linear Hodge operator ∗ is defined through where represents the composition of copies of the main automorphism 𝜋:

𝜔 ∧ (∗𝜂) = ⟨𝜔, 𝜂⟩ vol𝑀, {idΩ (𝑀), if 𝑑=odd (74) 𝜋𝑑−1 = K ∀𝜔, 𝜂 ∈Ω𝑘 (𝑀) , ∀𝑘=0⋅⋅⋅𝑑 { (82) K 𝜋, 𝑑= . { if even and satisfies the following properties, which we list for In particular, ] is a central element of the Kahler-Atiyah¨ convenience of the reader: algebra iff 𝑑 is odd. 𝑞 𝜔∧𝜂=(−1) ⟨𝜂, ∗ 𝜔⟩ , ∞ vol𝑀 The Twisted Hodge Operator. Let us define the (C (𝑀, R)- ∗:Ω̃ (𝑀) → Ω (𝑀) ∀𝜔 ∈ Ω𝑘 (𝑀) ,∀𝜂∈Ω𝑑−𝑘 (𝑀) , ∀𝑘=0⋅⋅⋅𝑑, linear) twisted Hodge operator K K through the formula: 𝑞 ⟨∗𝜔, ∗𝜂⟩ = (−1) ⟨𝜔, 𝜂⟩ , ∀𝜔, 𝜂∈Ω (𝑀) , def ∗𝜔̃ =𝜔⬦],∀𝜔∈ΩK (𝑀) . (83) 𝑞 vol𝑀 =∗1𝑀 ⇐⇒ ∗ vol𝑀 = (−1) 1𝑀, (75) Identity (79) shows that (unlike what happens for the ∗𝜔 =𝜔 𝜄 vol𝑀,∀𝜔∈Ω(𝑀) , ordinary Hodge operator) the square of the twisted Hodge operator is always a scalar multiple of the identity: ∗∘∗=(−1)𝑞 𝜋𝑑−1, {(−1)𝑞+[𝑑/2] , K = R 𝑑 idΩK(𝑀) if ∗∘𝜋=(−1) 𝜋∘∗. ∗∘̃ ∗=̃ { (84) , K = C. {idΩK(𝑀) if We also note the identity A simple computation shows that the twisted and ordinary [𝑑/2] 𝑑−1 𝜏∘∗=(−1) ∗∘𝜏∘𝜋 , (76) Hodge operators are related through ∗=𝑐̃ 𝑝,𝑞 (K) ∗∘𝜏. (85) which follows by direct computation upon using the congru- ence: In particular, the ordinary Hodge operator admits the repre- 𝑘 (𝑘−1) (𝑑−𝑘)(𝑑−𝑘−1) sentation: + 2 2 1 ∗𝜔=𝜏(𝜔) ⬦ vol𝑀 = 𝜏 (𝜔) ⬦ ], (77) 𝑐 (K) 𝑑 (𝑑−1) 𝑝,𝑞 (86) ≡2 +𝑘(𝑑−1) . 2 ∀𝜔 ∈ ΩK (𝑀) . 12 Advances in High Energy Physics

3.7. Twisted (Anti-)Self-Dual Forms. Let us assume that K = expansion (2) satisfies ∗𝜔̃ = ±𝜔 iff its nonstrict coefficients C or that K = R and 𝑝−𝑞≡4 0, 1,sothatthetwisted satisfy the conditions: Hodge operator ∗̃ squares to the identity. In this case, the 𝜔(𝑘) twisted Hodge operator has real eigenvalues equal to ±1 and 𝑎1⋅⋅⋅𝑎𝑘 we can consider inhomogeneous real forms belonging to the 𝑘(𝑑−𝑘) (−1) 󵄨 󵄨 𝑎 ⋅⋅⋅𝑎 (𝑑−𝑘) corresponding eigenspaces. A form 𝜔∈ΩK(𝑀) will be called √󵄨 󵄨 𝑑 𝑘+1 =± 𝑐𝑝,𝑞 (K) 󵄨det 𝑔󵄨𝜖𝑎 ⋅⋅⋅𝑎 𝜔𝑎 ⋅⋅⋅𝑎 , (92) twisted self-dual if ∗𝜔̃ = +𝜔 and twisted anti-self-dual if (𝑑−𝑘)! 1 𝑘 𝑘+1 𝑑 ± def ∗𝜔̃ = −𝜔.WeletΩK(𝑀) ={𝜔∈ΩK(𝑀) | 𝜔 ⬦ ] = ∀𝑘=0,...,𝑑. ∞ ±𝜔} ⊂ ΩK(𝑀) be the C (𝑀, R)-submodules of twisted self- dual and twisted anti-self-dual forms on 𝑀. We note here for future reference the expansions for the Hodge dual and the twisted Hodge dual of any 𝑘-form 𝜔: ± def The Ideals ΩK(𝑀). The elements 𝑝± = (1/2)(1 ± ]) are 𝑘(𝑑−𝑘) (−1) 󵄨 󵄨 𝑎 ⋅⋅⋅𝑎 √󵄨 󵄨 1 𝑘 complementary idempotents of the Kahler-Atiyah¨ algebra: (∗𝜔)𝑎 ⋅⋅⋅𝑎 = 󵄨det 𝑔󵄨𝜖𝑎 ⋅⋅⋅𝑎 𝜔𝑎 ⋅⋅⋅𝑎 , 𝑘+1 𝑑 (𝑑−𝑘)! 󵄨 󵄨 𝑘+1 𝑑 1 𝑘 𝑝 ∘𝑝 =𝑝 , ± ± ± (∗𝜔̃ ) 𝑎𝑘+1⋅⋅⋅𝑎𝑑 (93) 𝑝+ +𝑝− =1𝑀, (87) 𝑘(𝑑−𝑘) (−1) 󵄨 󵄨 𝑎 ⋅⋅⋅𝑎 √󵄨 󵄨 1 𝑘 𝑝 ∘𝑝 =0. = 𝑐𝑝,𝑞 (K) 󵄨det 𝑔󵄨𝜖𝑎 ⋅⋅⋅𝑎 𝜔𝑎 ⋅⋅⋅𝑎 . ± ∓ (𝑑−𝑘)! 󵄨 󵄨 𝑑 𝑘+1 1 𝑘 Notice that these idempotents are central only when ] is def 3.8. Algebraic Classification of Fiber Types. The fibers of central, that is, only when 𝑑 is odd. The operators 𝑃± =𝑅𝑝 ± the Kahler-Atiyah¨ bundle are isomorphic with the Clifford defined through right ⬦-multiplication with these elements algebra ClK(𝑝, 𝑞) = Cl(𝑝, 𝑞) ⊗R K, whose classification is well def 1 known. For K = C,wehaveanisomorphismofalgebras 𝑃± (𝜔) =𝜔⬦𝑝± = (𝜔±𝜔⬦]) def 2 ClC(𝑝, 𝑞) ≈ ClC(𝑑, 0) = ClC(𝑑) and the classification (88) 1 depends only on the mod 2 reduction of 𝑑;forK = R, (𝜔 ∈ Ω (𝑀))⇐⇒𝑃 = (1±∗̃) 8 𝑝−𝑞 K ± 2 it depends on the mod reduction of .TheSchur algebra SK(𝑝, 𝑞) is the largest division algebra contained in are complementary idempotents in the algebra of endomor- K(𝑝, 𝑞) ∞ the center of Cl ; it is determined up to isomorphism phisms of the C (𝑀, R)-module ΩK(𝑀): of algebras, being isomorphic with R, C,orH.TheClifford algebra is either simple (in which case it is isomorphic 𝑃2 =𝑃, ± ± with a matrix algebra Mat(Δ K(𝑑), SK(𝑝, 𝑞)))oradirect sum of two central simple algebras (namely, the direct sum 𝑃+ ∘𝑃− =𝑃− ∘𝑃+ =0, (89) Mat(Δ K(𝑑), SK(𝑝, 𝑞)) ⊕ Mat(Δ K(𝑑), SK(𝑝, 𝑞))), where the 𝑃 +𝑃 = . positive integers Δ K(𝑑) are given by well-known formulas + − idΩK(𝑀) recalled below. We say that the Clifford algebra is normal if its ± Therefore, the images ΩK(𝑀) =± 𝑃 (ΩK(𝑀)) = ΩK(𝑀)𝑝± Schur algebra is isomorphic to the base field. It is convenient are complementary left ideals of the Kahler-Atiyah¨ algebra, forourpurposetoorganizethevariouscasesaccordingtothe giving the direct sum decomposition: isomorphism type of the Schur algebra and to whether the + − Clifford algebra is simple or not. ΩK (𝑀) =ΩK (𝑀) ⊕ΩK (𝑀) . (90) When K = C.Inthiscase,theSchuralgebraisalways (Ω± (𝑀), ⬦) In particular, K are associative subalgebras of the isomorphic with C (so ClC(𝑝, 𝑞) ≈ ClC(𝑑) is always normal) Kahler-Atiyah¨ algebra. These subalgebras have units (given ] ⬦ ] =+1 Δ (𝑑) = 2[𝑑/2] 𝑝 𝑑 and we always have and C .Moreover by ±)iff is odd, in which case they are two-sided ideals we have the following: of (ΩK(𝑀), ⬦). (i) The algebra is simple iff 𝑑=even, in which case 𝑎 Local Characterization. Withrespecttoalocalcoframe𝑒 ClC(𝑑) ≈ Mat(Δ C(𝑑), C) and ] is noncentral. 𝑈⊂𝑀 aboveanopensubset , we have the expansions: (ii) The algebra is nonsimple iff 𝑑=odd, in which case (𝑑) ≈ (Δ (𝑑), C)⊕ (Δ (𝑑), C) ] 𝑎 ⋅⋅⋅𝑎 1 󵄨 󵄨 𝑎 ⋅⋅⋅𝑎 𝑎 ⋅⋅⋅𝑎 ClC Mat C Mat C and is 1 𝑘 √󵄨 󵄨 1 𝑘 𝑘+1 𝑑 ∗(𝑒 )= 󵄨det 𝑔󵄨𝜖 𝑎 ⋅⋅⋅𝑎 𝑒 , (𝑑−𝑘)! 󵄨 󵄨 𝑘+1 𝑑 central. 𝑎 ⋅⋅⋅𝑎 ∗(𝑒̃ 1 𝑘 ) (91) When K = R,wehavethefollowing:

1 󵄨 󵄨 𝑎 ⋅⋅⋅𝑎 𝑎 ⋅⋅⋅𝑎 Δ (𝑑) √󵄨 󵄨 𝑘 1 𝑘+1 𝑑 (1) The Schur algebras and the numbers R are as = 𝑐𝑝,𝑞 (K) 󵄨det 𝑔󵄨𝜖 𝑎 ⋅⋅⋅𝑎 𝑒 , (𝑑−𝑘)! 𝑘+1 𝑑 follows (see, e.g., [28]):

𝑎𝑏 where indices are raised with 𝑔 . Using (91), one easily (i) S ≈ R (normal case), which occurs iff 𝑝− [𝑑/2] checks that an inhomogeneous form 𝜔∈ΩK(𝑀) with 𝑞≡8 0, 1, 2 and we have Δ R(𝑑) = 2 . Advances in High Energy Physics 13

Table 1: Properties of ] according to the mod 8 reduction of 𝑝−𝑞 where for the case K = R. At the intersection of each row and column, we def indicate the values of 𝑝−𝑞(mod 8) for which the modified volume Ω< (𝑀) =⊕[𝑑/2]Ω𝑘 (𝑀) , ] K 𝑘=0 K form has the corresponding properties. In parentheses, we also (97) indicate the isomorphism type of the Schur algebra for that value Ω> (𝑀) def=⊕𝑑 Ω𝑘 (𝑀) . of 𝑝−𝑞(mod 8). The real Clifford algebra cl(𝑝, 𝑞) is nonsimple iff K 𝑘=[𝑑/2]+1 K 𝑝−𝑞≡8 1, 5, which corresponds to the upper left corner of the table. C∞(𝑀, R) Notice that ] is central iff 𝑑 is odd. The corresponding complementary -linear idempotent operators 𝑃<,𝑃> :ΩK(𝑀)→ΩK(𝑀) are given by K = R ] ⬦ ] =+1 ] ⬦ ] =−1 ] is central 1(R), 5(H)3(C), 7(C) 𝑃 (𝜔) def=𝜔<, ] 0(R) 4(H)2(R) 6(H) < is not central , , (98) def > 𝑃> (𝜔) =𝜔,

(ii) S ≈ C (almost complex case), which occurs iff 𝑑 (𝑘) (𝑘) 𝑘 [𝑑/2] where, for any 𝜔=∑𝑘=0 𝜔 ∈ΩK(𝑀) (with 𝜔 ∈ΩK(𝑀)), 𝑝−𝑞≡8 3, 7 and we have Δ R(𝑑) = 2 . < > we define 𝜔 (the lower truncation of 𝜔)and𝜔 (the upper (iii) S ≈ H (quaternionic case), which occurs iff 𝑝− 𝜔 [𝑑/2]−1 truncation of )through 𝑞≡8 4, 5, 6 and we have Δ R(𝑑) = 2 . [𝑑/2] < def (𝑘) (2)Thesimpleandnonsimplecasesoccurasfollows: 𝜔 = ∑ 𝜔 , 𝑘=0 (𝑝, 𝑞) 𝑝−𝑞≡ 0, 2, 3, 4, 6, 7 (99) (i) Cl is simple iff 8 . 𝑑 > def (𝑘) (ii) Cl(𝑝, 𝑞) is nonsimple iff 𝑝−𝑞≡8 1, 5.Inthiscase, 𝜔 = ∑ 𝜔 . we always have ] ⬦ ] =+1and ] is central. 𝑘=[𝑑/2]+1

The situation when K = R is summarized in Table 1. For both We have K = R and K = C, the Clifford algebra is nonsimple iff ] is 𝑃> +𝑃< = idΩ (𝑀), central and satisfies ] ⬦ ] =1.Inthiscase—forbothK = K R and K = C—the Clifford algebra admits two inequivalent 𝑃 ∘𝑃 =𝑃 ∘𝑃 =0, K > < < > irreducible representations by -linear operators, which are (100) related by the main automorphism of the Clifford algebra and 𝑃> ∘𝑃> =𝑃>, both of which are nonfaithful; their Schur algebra equals C 𝑃 ∘𝑃 =𝑃. when K = C but may equal either R or H when K = R. < < < 𝜖 When 𝜔 is twisted (anti-)self-dual (i.e., 𝜔∈Ω (𝑀) with 𝜖= 3.9. Twisted (Anti-)Self-Dual Forms in the Nonsimple Case. In K ±1), we have ∗𝜔̃ = 𝜖𝜔, which implies this subsection, let us assume that we are in the nonsimple case. Then ] ⬦ ] =+1and (since 𝑑 is odd in the nonsimple > < 𝜖 𝜔 =𝜖∗(𝜔̃ ), ∀𝜔∈Ω (𝑀) . (101) case) ] is a central element of the Kahler-Atiyah¨ algebra: K Hence in this case 𝜔 can be reconstructed from its lower ] ⬦𝜔=𝜔⬦],∀𝜔∈ΩK (𝑀) . (94) truncation as Using the fact that ] is central, an easy computation shows 𝜔=𝜔< +𝜖∗(𝜔̃ <)=2𝑃 (𝜔<)=𝑃 (2𝑃 (𝜔)), 𝑃 𝑃 𝜖 𝜖 < that + and − are (nonunital) algebra endomorphisms of the (102) 𝜖 Kahler-Atiyah¨ algebra; in fact ∀𝜔 ∈ ΩK (𝑀) . 𝑃 (𝜔 ⬦ 𝜂) =𝑃 (𝜔) ⬦𝑃 (𝜂) = 𝑃 (𝜔) ⬦𝜂 𝜖 ± ± ± ± It follows that the restriction of 2𝑃< to the subspace ΩK(𝑀) ∞ gives a C (𝑀, R)-linear bijection between this subspace and =𝜔⬦𝑃± (𝜂) , ∀𝜔,K 𝜂∈Ω (𝑀) , (95) < the subspace ΩK(𝑀),withinversegivenbytherestrictionof < 𝑃𝜖 to ΩK(𝑀). We define the twisted (anti-)self-dual prolonga- 𝑃± (1𝑀)=𝑝±. < tion of a form 𝜔∈ΩK(𝑀) through Ω± (𝑀) In this case, K are complementary two-sided ideals def < of the Kahler-Atiyah¨ algebra (indeed, 𝑝± are central); in 𝜔± =𝑃± (𝜔) ,∀𝜔∈ΩK (𝑀) . (103) ± particular, (ΩK(𝑀), ⬦) are unital algebras, their units being 𝜔∈Ω< (𝑀) given by 𝑝±. Of course, the form K canberecoveredfromits two prolongations as 𝜔=𝜔+ +𝜔−. 𝑑 Truncation and Prolongation. Since isoddinthenonsimple < case, we have the decomposition: The Truncated Algebra (ΩK(𝑀), X±). We stress that 𝑃< does not preserve the geometric product on its entire domain < > < ΩK (𝑀) =ΩK (𝑀) ⊕ΩK (𝑀) , (96) of definition ΩK(𝑀);infact,itsimageΩK(𝑀) is not 14 Advances in High Energy Physics

𝜔(𝑘) ∈ C∞(𝑈,K) asubalgebraoftheKahler-Atiyah¨ algebra since it is not stable where 𝑎1⋅⋅⋅𝑎𝑘 areasin(3).Thecoefficientswith with respect to ⬦-multiplication. To cure this problem, we 𝑘 ≥ [𝑑/2]+1 are determined by those with 𝑘 ≤ [𝑑/2] through use the linear isomorphisms mentioned above to transfer 𝜔 𝜖 relations (92). The lower truncation of such has the local the multiplication ⬦ of the unital subalgebra ΩK(𝑀) to an < expansion: associative and unital multiplication X𝜖 defined on ΩK(𝑀) through [𝑑/2] 1 < (𝑘) 𝑎1⋅⋅⋅𝑎𝑘 𝜔 =𝑈 ∑ 𝜔𝑎 ⋅⋅⋅𝑎 𝑒 . (111) < 𝑘! 1 𝑘 𝜔 X𝜖𝜂=2𝑃< (𝑃𝜖 (𝜔) ⬦𝑃𝜖 (𝜂)) ∈ ΩK (𝑀) ⇐⇒ 𝑘=0

𝑃𝜖 (𝜔 X𝜖𝜂) =𝜖 𝑃 (𝜔) ⬦𝑃𝜖 (𝜂) , (104) We also note the explicit expressions: < 1 ∀𝜔, 𝜂 ∈Ω (𝑀) . 𝑎1⋅⋅⋅𝑎𝑘 𝑎1⋅⋅⋅𝑎𝑘 𝑎𝑘⋅⋅⋅𝑎1 𝑎𝑘+1⋅⋅⋅𝑎𝑑 K 𝑒 = (𝑒 ±𝑐𝑝,𝑞 (K) 𝜖 𝑎 ⋅⋅⋅𝑎 𝑒 ) , (112) ± 2 𝑘+1 𝑑 Since 𝑃𝜖 is a morphism of algebras on its entire domain 𝑐 (K) of definition ΩK(𝑀),wehave𝑃𝜖(𝜔) ⬦𝜖 𝑃 (𝜂) =𝜖 𝑃 (𝜔 ⬦ 𝜂),so where 𝑝,q was defined in (78). (104) gives 3.10. Orthogonality in the Nonsimple Case. Assuming that we 𝜔 X 𝜂=2𝑃 (𝑃 (𝜔 ⬦ 𝜂)) , ∀𝜔,< 𝜂∈Ω (𝑀) . 𝜖 < 𝜖 K (105) are in the nonsimple case, let us consider the situation when 1 we have a distinguished one-form 𝜃∈ΩK(𝑀) which satisfies Since 2𝑃𝜖(𝜔 ⬦ 𝜂) = (1∗)(𝑃 +𝜖̃ <(𝜔 ⬦ 𝜂)> +𝑃 (𝜔 ⬦ 𝜂)) and the normalization condition 𝜄𝜃𝜃=1. ∗∘𝑃̃ < =𝑃> ∘ ∗̃, this implies Ω⊥(𝑀) Ω𝜖 (𝑀) 𝜔 X𝜖𝜂=𝑃< (𝜔 ⬦ 𝜂)∗𝑃 +𝜖̃ > (𝜔 ⬦ 𝜂) The Isomorphism of Algebras between K and K . For any 𝜔∈ΩK(𝑀), consider the decomposition 𝜔=𝜔‖ +𝜔⊥ < > (106) =(𝜔⬦𝜂) +𝜖∗[(𝜔⬦𝜂)̃ ], into parts 𝜔‖ =𝜃∧(𝜄𝜃𝜔) and 𝜔⊥ =𝜄𝜃(𝜃 ∧ 𝜔) = 𝜔‖ −𝜔 parallel and perpendicular to 𝜃.Since𝜃‖] (indeed, we have 𝜃∧] =0) a formula which can be used to implement the product X𝜖 in a and ∗𝜔̃ = 𝜔 ⬦ ], properties (61) imply symbolic computation system. Combining everything shows that we have mutually inverse isomorphisms of algebras: 𝜃‖∗(𝜔̃ ⊥), (113) 𝑃 | 𝜖 Ω< (𝑀) 𝜃⊥∗(𝜔̃ ‖), < 󳨀󳨀󳨀󳨀󳨀󳨀󳨀K → 𝜖 (ΩK (𝑀) , X𝜖) ←󳨀󳨀󳨀󳨀󳨀󳨀󳨀 (ΩK (𝑀) ,⬦). (107) 2𝑃<|Ω𝜖 (𝑀) K which gives < Thus (ΩK(𝑀), X𝜖) provides a model for the unital associative (∗𝜔̃ )‖ = ∗(𝜔̃ ⊥), 𝜖 algebra (ΩK(𝑀), ⬦). (114) (∗𝜔̃ )⊥ = ∗(𝜔̃ ‖). Local Expansions. Let us further assume that 𝑑≥3. 𝑎 1 Ω𝜖 (𝑀) Ω⊥(𝑀) Then the covector fields 𝑒 ∈ΩK(𝑀) defined by a local The subalgebras K and K of the Kahler-Atiyah¨ pseudo-orthonormal frame above 𝑈⊂𝑀belong to the algebra can be identified with each other using the operator < 2𝑃 𝜔 2𝜔 subspace ΩK(𝑀) and we consider their twisted (anti-)self- ⊥, which takes a twisted (anti-)self-dual form into ⊥. dual prolongations: Indeed, if ∗𝜔̃ = 𝜖𝜔 (with 𝜖=±1), then ∗(𝜔̃ ‖)=𝜖𝜔⊥ and ∗(𝜔̃ ⊥)=𝜖𝜔‖. Hence the last of (62) implies 𝑎 def 𝑎 ± 𝑒 =𝑃± (𝑒 )∈Ω (𝑀) . (108) ± K ̃ ̃ (𝜔⬦𝜂)⊥ = ∗ (𝜔⊥) ⬦ ∗ (𝜂⊥) +𝜔⊥ ⬦𝜂⊥ =2𝜔⊥ ⬦𝜂⊥, (115) 𝑎1⋅⋅⋅𝑎𝑘 𝑎1 𝑎𝑘 𝑎1 𝑎𝑘 Since 𝑒 =𝑒 ∧⋅⋅⋅∧𝑒 =𝑒 ⬦⋅⋅⋅⬦𝑒 and since 𝑃± are endomorphisms of the Kahler-Atiyah¨ algebra, we find that since ∗(𝜔̃ ⊥)⬦∗(𝜂̃ ⊥)=𝜔⊥ ⬦ ] ⬦𝜂⊥ ⬦ ] =𝜔⊥ ⬦𝜂⊥ ⬦ ] ⬦ ] = 𝑎 ⋅⋅⋅𝑎 the prolongations of 𝑒 1 𝑘 are given by ⬦-monomials in the 𝜔⊥ ⬦𝜂⊥, where we used the fact that ] is central in the Kahler-¨ 𝑎 prolongations of 𝑒 : Atiyah algebra and that it squares to 1𝑀.Thus

𝑎 ⋅⋅⋅𝑎 def 𝑎 ⋅⋅⋅𝑎 𝑎 𝑎 1 𝑘 1 𝑘 1 𝑘 (𝜔 ⬦ 𝜂) =2𝜔⊥ ⬦𝜂⊥ ⇐⇒ 𝑒± =𝑃± (𝑒 )=𝑒± ⬦⋅⋅⋅⬦𝑒± . (109) ⊥

𝑎1⋅⋅⋅𝑎𝑘 2𝑃 (𝜔 ⬦ 𝜂) = 2 (2𝑃 𝜔) ⬦ (2𝑃 𝜂) , In particular, the twisted (anti-)self-dual forms {𝑒± |1≤ ⊥ ⊥ ⊥ (116) 𝑎 < ⋅⋅⋅ < 𝑎 ≤𝑑,𝑘=0,...,[𝑑/2]} 1 𝑘 constitute a basis of ∀𝜔, 𝜂 ∈Ω𝜖 (𝑀) . ∞ ± 𝑎1⋅⋅⋅𝑎𝑘 K the free C (𝑈,K)-module ΩK(𝑈) (since {𝑒 |1≤𝑎1 < ⋅⋅⋅ < 𝑎𝑘 ≤ 𝑑, 𝑘 = 0,...,[𝑑/2]}form a basis of the module < We also have ΩK(𝑈) and since the operation of taking the prolongation is ∞ an isomorphism of C (𝑈,K)-modules). In fact, any twisted 2𝑃 (𝑝 )=𝑃 (1±]) =𝑃 (1 )=1 , ± ⊥ ± ⊥ ⊥ 𝑀 𝑀 (117) (anti-)self-dual form 𝜔∈ΩK(𝑀) (cf. (2), (3)) expands as 𝜃‖] [𝑑/2] whereweusedthefactthat .Thesepropertiesshow 1 2𝑃 | 𝜖 (𝑘) 𝑎1⋅⋅⋅𝑎𝑘 ± that the restriction ⊥ Ω (𝑀) is a unital morphism of algebras 𝜔=𝑈 2 ∑ 𝜔𝑎 ⋅⋅⋅𝑎 𝑒± ,∀𝜔∈ΩK (𝑀) , (110) K 𝑘! 1 𝑘 𝜖 ⊥ 𝑘=0 from (ΩK(𝑀), ⬦) to (ΩK(𝑀), ⬦). An easy computation shows Advances in High Energy Physics 15 that it is an isomorphism whose inverse equals the restriction To find explicit expressions for the parallel and perpendicular ⊥ of 𝑃𝜖 to ΩK(𝑀). It follows that we have mutually inverse unital parts of ∗𝜔̃ ,noticethat∗𝜔̃ = 𝜔 ⬦ ] =𝜔⬦]⊤ ⬦ 𝜃 = 𝜃 ∧ 𝜋(𝜔 ⬦ isomorphisms of algebras: ]⊤)−𝜄𝜃𝜋(𝜔 ⬦ ]⊤) = 𝜃 ∧ (𝜋(𝜔) ⬦ ]⊤)−𝜄𝜃(𝜋(𝜔) ⬦ ]⊤),where we used (29) and the fact that 𝜋(]⊤)=+]⊤.Thus 2𝑃⊥|Ω𝜖 (𝑀) 𝜖 K ⊥ (Ω (𝑀) ,⬦)󳨀←󳨀󳨀󳨀󳨀󳨀󳨀󳨀󳨀󳨀󳨀󳨀󳨀→󳨀 (Ω (𝑀) ,⬦). ̃ K K (118) (∗𝜔)‖ =𝜃∧[𝜋(𝜔) ⬦ ]⊤]=𝜃∧[(𝜋(𝜔) ⬦ ]⊤)⊥], 𝑃𝜖|Ω⊥(𝑀) K (129) ̃ (∗𝜔)⊥ =−𝜄𝜃 [𝜋 (𝜔) ⬦ ]⊤]=−𝜄𝜃 [(𝜋 (𝜔) ⬦ ]⊤)‖]. Combining with the results of Section 3.9, we have thus found 𝜖 twoisomorphicmodelsfortheunitalsubalgebra(Ω (𝑀), ⬦): K The decomposition 𝜔=𝜔‖ +𝜔⊥ and the fact that 𝜄𝜃]⊤ =0 𝜃⊥] 𝑃 | (thus ⊤) imply (using (61)) 𝜖 Ω< (𝑀) 2𝑃⊥|Ω𝜖 (𝑀) < 󳨀󳨀󳨀󳨀󳨀󳨀󳨀K → 𝜖 󳨀󳨀󳨀󳨀󳨀󳨀󳨀K → ⊥ (ΩK (𝑀) , X𝜖) ←󳨀󳨀󳨀󳨀󳨀󳨀󳨀 (ΩK (𝑀) ,⬦) ←󳨀󳨀󳨀󳨀󳨀󳨀󳨀 (ΩK (𝑀) ,⬦) . (119) 2𝑃<|Ω𝜖 (𝑀) 𝑃𝜖|Ω⊥(𝑀) 𝜃‖(𝜔 ⬦ ] ), K K ‖ ⊤ (130) The Reduced Twisted Hodge Operator. Since 𝜃‖],wecan 𝜃⊥(𝜔⊥ ⬦ ]⊤), write and(using(64)andthefactthat𝜋(]⊤)=+1) ] =𝜃∧]⊤, (120) 𝜃 ‖ (𝜋‖ (𝜔 )⬦]⊤), where the reduced volume form ]⊤ is defined through (131) 𝜃⊥(𝜋(𝜔⊥)⬦]⊤). def ]⊤ =𝜄𝜃] =𝜃⬦] = ] ⬦𝜃. (121) These relations show that Thelasttwoequalitiesin(121)followfrom(29)andfromthe (𝜔 ⬦ ] ) =𝜔 ⬦ ] , fact that (in the nonsimple case) ] is central in the Kahler-¨ ⊤ ‖ ‖ ⊤ (132) Atiyah algebra (since 𝑑 is odd in this case). Multiplying the (𝜔 ⬦ ]⊤) =𝜔⊥ ⬦ ]⊤ last equation with 𝜃 in the Kahler-Atiyah¨ algebra and using ⊥ 𝜃⬦𝜃=𝑔(𝜃,̂ 𝜃) =1 the fact that give as well as ] = ] ⬦𝜃=𝜃⬦] . ⊤ ⊤ (122) (𝜋 (𝜔) ⬦ ]⊤)‖ =𝜋(𝜔‖)⬦]⊤, (133) Notice the identity: (𝜋 (𝜔) ⬦ ]⊤)⊥ =𝜋(𝜔⊥)⬦]⊤. ]2 =+1 , ⊤ 𝑀 (123) Combining the last relation with (129) gives which follows from (121) using the fact that ] is central, the (∗𝜔̃ ) =𝜃∧[𝜋(𝜔 )⬦] ], 𝜃 ] ⬦ ] = ‖ ⊥ ⊤ normalization condition for and the property (134) +1 𝑀, which always holds in the nonsimple case. Defining the (∗𝜔̃ )⊥ =−𝜄𝜃 [𝜋 ‖(𝜔 )⬦]⊤]. reduced twisted Hodge operator ∗̃0 through Equations (134) and (114) read def ∗̃0𝜔 =𝜋(𝜔) ⬦ ]⊤, (124) (∗𝜔̃ )‖ = ∗(𝜔̃ ⊥)=𝜃∧∗̃0 (𝜔⊥), ∀𝜔 ∈ Ω (𝑀) ⇐⇒ ∗̃ =𝑅 ∘𝜋, K 0 ]⊤ (135) (∗𝜔̃ )⊥ = ∗(𝜔̃ ‖)=−𝜄𝜃∗̃0 (𝜔‖), we have 𝜋(]⊤)=]⊤, so (123) implies while (133) gives ∗̃0 ∘ ∗̃0 =+idΩ (𝑀). (125) K ̃ ̃ (∗0𝜔)‖ = ∗0 (𝜔‖), For later reference, we note the identities (where we use (122) (136) ̃ ̃ and the fact that 𝜋(]⊤)=]⊤) (∗0𝜔)⊥ = ∗0 (𝜔⊥). [𝜋, 𝑅 ] =[𝜋,∗̃ ] =0 ]⊤ −,∘ 0 −,∘ (126) In particular, we have [∗̃ ,𝑃] =[∗̃ ,𝑃 ] =0 as well as 0 ‖ −,∘ 0 ⊥ −,∘ (137) ̃ ̃ ∗=∧̃ ∘ ∗̃ ∘𝑃 −𝜄 ∘ ∗̃ ∘𝑃 =∧ ∘𝑃 ∘ ∗̃ −𝜄 ∘𝑃 ∘ ∗̃ [𝐿𝜃, ∗0]+,∘ =[𝑅𝜃, ∗0]+,∘ =0, (127) and 𝜃 0 ⊥ 𝜃 0 ‖ 𝜃 ⊥ 0 𝜃 ‖ 0, which gives whichfollowbyeasycomputation.Using(48),thelast identities imply the following anticommutation relations, ∗=∧̃ 𝜃 ∘ ∗̃0 −𝜄𝜃 ∘ ∗̃0 (138) which will be important below: upon using ∧𝜃 ∘𝑃‖ =𝜄𝜃 ∘𝑃⊥ =0⇔∧𝜃 ∘𝑃⊥ =∧𝜃 and 𝜄𝜃 ∘𝑃‖ =𝜄𝜃. ̃ ̃ [∧𝜃, ∗0]+,∘ =[𝜄𝜃, ∗0]+,∘ =0. (128) The relations above imply the following. 16 Advances in High Energy Physics

def Lemma 6. Consider the operators 𝛼𝜃 =∧𝜃 ∘ ∗̃0 and 𝛽𝜃 = Corollary 8. The following statements are equivalent for any 𝜔∈Ω (𝑀) −𝜄𝜃 ∘ ∗̃0.Then K . (a) 𝜔 is twisted (anti-)self-dual; that is, ∗𝜔̃ = 𝜖𝜔 for 𝜖=±1. 𝜔 =𝜄𝜔 𝜔 =𝜄(𝜃∧𝜔) 𝛼𝜃 ∘𝛼𝜃 =𝛽𝜃 ∘𝛽𝜃 =0, (b) The inhomogeneous forms ⊤ 𝜃 and ⊥ 𝜃 satisfy the equation 𝛼𝜃 ∘𝛽𝜃 =𝑃‖, (139) 𝜔⊥ =𝜖∗̃0𝜔⊤ ⇐⇒ 𝜔 ⊤ =𝜖∗̃0𝜔⊥. (145) 𝛽𝜃 ∘𝛼𝜃 =𝑃⊥. 𝜔 𝜔 Proof. The statement follows by direct computation using In this case, ⊥ and ⊤ determine each other and thus any 𝜔 𝜔 𝜔 properties (128) and (125). of them determines .Explicitly, ⊤ determines through the formula Notice that (135) takes the form 𝜔=(∧𝜃 +𝜖∗̃0)(𝜔⊤), (146) (∗𝜔̃ )‖ = ∗(𝜔̃ ⊥)=𝛼𝜃 (𝜔⊥), (140) while 𝜔⊥ determines 𝜔 through (∗𝜔̃ )⊥ = ∗(𝜔̃ ‖)=𝛽𝜃 (𝜔‖). 𝜔=( +𝜖∧ ∘ ∗̃ )(𝜔 ). For reader’s convenience, we also list a few other properties idΩK(𝑀) 𝜃 0 ⊥ (147) which follow immediately from the above: ⊥ ∼ The corollary shows that the maps ∧𝜃 +𝜖∗̃0 :ΩK(𝑀) 󳨀→ 𝛼𝜃 ∘𝑃‖ =0󳨐⇒𝛼𝜃 ∘𝑃⊥ =𝛼𝜃, 𝜖 ∞ ΩK(𝑀) are isomorphisms of C (𝑀, K)-modules, whose 𝜔→𝜔 𝑃 ∘𝛼 =0󳨐⇒𝑃 ∘𝛼 =𝛼, inverses are given by ⊤. We stress that these maps are ⊥ 𝜃 ‖ 𝜃 𝜃 not isomorphisms of algebras. 𝛽 ∘𝑃 =0󳨐⇒𝛽 ∘𝑃 =𝛽, 𝜃 ⊥ 𝜃 ‖ 𝜃 𝜑 (141) The Morphism 𝜖. For later reference, consider the C∞(𝑀, R) 𝑃‖ ∘𝛽𝜃 =0󳨐⇒𝑃⊥ ∘𝛽𝜃 =𝛽𝜃, -linear operator:

𝑃‖ ∘ ∗=𝛼̃ 𝜃, def ⊥ 𝜑𝜖 =2𝑃⊥ ∘𝑃𝜖 :ΩK (𝑀) 󳨀→ Ω K (𝑀) (148)

𝑃⊥ ∘ ∗=𝛽̃ 𝜃. which acts as follows on 𝜔=𝜃⬦𝜔⊤ +𝜔⊥ =𝜃∧𝜔⊤ +𝜔⊥ ∈ Proposition 7. Let 𝜔∈ΩK(𝑀). Then the following state- ΩK(𝑀): ments are equivalent. (a) 𝜔 is twisted (anti-)self-dual; that is, ∗𝜔̃ = 𝜖𝜔 for 𝜖=±1. 𝜑𝜖 (𝜔) =𝜖∗̃0 (𝜔⊤)+𝜔⊥ =𝜖]⊤ ⬦𝜔⊤ +𝜔⊥, (149) (b) The components 𝜔‖ and 𝜔⊥ satisfy the following equivalent relations: whereweused(144)andwenoticedthat]⊤ ⬦𝜔⊤ =𝜋(𝜔⊤)⬦ ]⊤ = ∗̃0(𝜔⊤) (since 𝜔⊤ and ]⊤ are orthogonal to 𝜃 and since 𝜔 =𝜖𝜃∧∗̃ (𝜔 ), ‖ 0 ⊥ rk]⊤ =𝑑−1is even). We have 𝜑𝜖(𝜃) =] 𝜖 ⊤ (since 𝜃⊤ =1 (142) ⊥ and 𝜃⊥ =0)and𝜑𝜖(𝜔) = 𝜔 for all 𝜔∈ΩK(𝑀); in fact, these 𝜔⊥ =−𝜖𝜄𝜃∗̃0 (𝜔‖). properties determine 𝜑𝜖.Onehasthefollowing.

In this case, 𝜔‖ and 𝜔⊥ determine each other and thus any of Proposition 9. The map 𝜑𝜖 is idempotent; that is, 𝜑𝜖 ∘𝜑𝜖 = them determines 𝜔. 𝜑𝜖. Moreover, it is a (unital) morphism of algebras from ⊥ (ΩK(𝑀), ⬦) to (Ω (𝑀), ⬦). Proof. The fact that the two relations listed in (142) are equivalent to each other is an immediate consequence of the Proof. Idempotency follows by noticing that 𝜑𝜖|Ω⊥(𝑀) = lemma. The rest of the proposition follows from (138). K ⊥ 𝜑 idΩK(𝑀). The fact that 𝜖 is a morphism of algebras follows 𝑃 𝑃 𝜑 Recalling definition (66), we have 𝜃⊥𝜔⊤ and 𝜔‖ =∧𝜃𝜔⊤, since both ⊥ and 𝜖 are such. Finally, unitality of 𝜖 follows so the decomposition of 𝜔 reads 𝜔=𝜃∧𝜔⊤ +𝜔⊥.Using(128), by computing: we find 𝜑𝜖 (1𝑀) =𝑃⊥ (1𝑀 +𝜖]) =1𝑀, (150) 𝛽𝜃 ∘∧𝜃 =𝑃⊥ ∘ ∗̃0 = ∗̃0 ∘𝑃⊥, (143) where we used 𝑃⊥(1𝑀)=1𝑀 and 𝑃⊥(])=0. which implies 𝜑 𝛽𝜃 (𝜔‖) = ∗̃0 (𝜔⊤) 󳨐⇒ (∗𝜔̃ )⊥ = ∗̃ (𝜔‖) = ∗̃0 (𝜔⊤) , (144) It is clear that 𝜖 is surjective. Moreover, the last proposition of the previous paragraph implies where in the last equality we used (140). Hence the previous −𝜖 proposition has the following. K (𝜑𝜖)=ΩK (𝑀) . (151) Advances in High Energy Physics 17

𝜖 It follows that 𝜑𝜖 restricts to an isomorphism from ΩK(𝑀) bundle of pseudo-orthonormal frames of 𝑇K𝑀 to a certain ⊥ to ΩK(𝑀)—which, of course, equals the isomorphism extension of the Clifford (a.k.a. Lipschitz) group over K.In 2𝑃⊥|Ω𝜖 (𝑀) of diagram (118). Notice the relations: this case, the choices globally available for 𝑆 depend—up to K 𝑐 isomorphism—on the choices of a Clifford structure and can 𝑃 ∘𝜑 =𝑃, 𝜖 𝜖 𝜖 be obtained from such a structure by applying the associated bundle construction. The map induced on sections satisfies 𝜑𝜖 ∘𝑃𝜖 =𝜑𝜖, (152) 𝛾 (𝜔⬦𝜂) =𝛾(𝜔) ∘𝛾(𝜂) ,∀𝜔,𝜂∈Ω(𝑀) 𝑃⊥ ∘𝜑𝜖 =𝜑𝜖, K (154)

𝜑𝜖 ∘𝑃⊥ =𝑃⊥, as well as where the first equality follows from the fact that 2𝑃𝜖 ∘𝑃⊥ 𝜖 𝛾(1𝑀)=id𝑆, (155) restricts to the identity on ΩK(𝑀) (see diagram (118)). Also notice the property: where id𝑆 ∈ Γ(𝑀, End(𝑆)) denotes the identity section of the (𝑆) 𝜑𝜖 (]) =𝜖1𝑀, (153) bundle End . In the language of “vertical quantization” of spin systems, which follows by direct computation upon using 𝑃⊥(1𝑀)= 2 the (𝐿 -completion of the) space Γ(𝑀, 𝑆) of smooth sections 1𝑀, 𝑃⊥(])=0and the fact that ] ⬦ ] =1𝑀. of 𝑆 plays the role of the Hilbert space (when K = R,one of course has to consider the complexification of 𝑆 instead). 4. Describing Bundles of Pinors In this interpretation, 𝛾 plays the role of quantization map, givingamorphismfromthealgebraofquantumobservables In this section, we discuss the realization of pin bundles 2 of the system (which is the Kahler-Atiyah¨ algebra) to the (𝐿 - within the geometric algebra formalism—focusing especially completion of the) algebra (Γ(𝑀, End(𝑆)), ,whichplaysthe∘) on the nonsimple case, when the irreducible pin represen- role of algebra of “vertical” operators acting in the Hilbert tations are nonfaithful. Section 4.1 discusses an approach space.Thesestatementscanbemadequitepreciseprovided to pinor bundles which is particularly well adapted to the that certain global conditions are imposed on (𝑀, 𝑔),butthe geometric algebra formalism. In this approach (which, in rigorous treatment of this issue falls outside of the scope of some ways, goes back to Dirac; see [18, 29] for a beautiful this paper. treatment), one defines pinors as sections of a bundle 𝑆 of modules over the Kahler-Atiyah¨ algebra, the fiberwise Notation 1. If (𝑒𝑎)𝑎=1⋅⋅⋅𝑑 is a local frame of 𝑇𝑀 above an module structure being described by a morphism 𝛾: 𝑈⊂𝑀 (𝑒𝑎) (Ω (𝑀), ⬦) →( (𝑆), ∘) open subset (with dual coframe 𝑎=1⋅⋅⋅𝑑), then any K End of bundles of algebras. For the inhomogeneous differential form 𝜔 on 𝑀 expands locally as case when 𝛾 is irreducible on the fibers, the well-known def 𝑎 𝑎 𝑎1⋅⋅⋅𝑎𝑘 representation theory relevant for this construction and its in (2). We define 𝛾 =𝛾(𝑒)∈Γ(𝑈,End(𝑆)),so𝛾(𝑒 )= def 𝑎1⋅⋅⋅𝑎𝑘 𝑎1 𝑎𝑘 relation with the fiber type classification of the Kahler-¨ 𝛾 =(1/𝑘!)𝜖𝑎 ⋅⋅⋅𝑎 𝛾 ∘⋅⋅⋅∘𝛾 ∈ Γ(𝑈,End(𝑆)) (the 1 𝑘 𝑎 𝑎 Atiyah bundle is recalled in Section 4.2, paying attention to complete antisymmetrization of the composition 𝛾 1 ∘⋅⋅⋅∘𝛾 𝑘 ). characterizing the kernel and image of 𝛾.InSection4.3,we −1 We have introduce a certain “partial inverse” 𝛾 of 𝛾, which provides 𝑑 a sort of “vertical dequantization” map. Section 4.4 discusses def 1 𝛾 (𝑘) 𝑎1⋅⋅⋅𝑎𝑘 Ω (𝑀) 𝛾 𝛾 (𝜔) = 𝑈 ∑ 𝜔𝑎 ⋅⋅⋅𝑎 𝛾 . (156) atraceonthesubalgebra K ,whichisrelatedby to the 𝑘! 1 𝑘 natural fiberwise trace on the pin bundle. 𝑘=0 𝑎 The locally defined sections 𝛾 ∈ Γ(𝑈,End(𝑆)) correspond to 4.1. Basic Considerations. We define a bundle of K-pinors to ∗ physicists’ “gamma matrices.” be a bundle 𝑆 of modules over the Clifford bundle Cl(𝑇K𝑀), that is, a K-vector bundle all of whose fibers 𝑆𝑥 (𝑥∈𝑀)are ∗ 𝑆 modules over the corresponding Clifford algebras Cl(𝑇 𝑀). 4.2. Representation Theory. Let be a pin bundle with K,𝑥 𝛾:(Ω (𝑀), ⬦) →( (𝑆), ∘) In our language, such a bundle is simply a bundle of modules underlying morphism K End . ∗ over the Kahler-Atiyah¨ bundle (∧𝑇K𝑀, ⬦).Intheparticular ∗ 𝛾 case when the morphism 𝛾 : (∧𝑇 𝑀, ⬦) → End(𝑆) Injectivity and Surjectivity of . It is important to note that K 𝛾 induces an irreducible representation of the Clifford algebra need not be fiberwise injective or surjective; that is, the ∗ 𝛾 :(Λ𝑇∗ 𝑀, ⬦ )≈ (𝑇∗ 𝑀) → Cl(𝑇 𝑀) on each fiber End(𝑆𝑥), a bundle of pinors will morphisms of algebras 𝑥 K,𝑥 𝑥 Cl K,𝑥 K,𝑥 (𝑆 ) be called a pin bundle.BundlesofK-spinors and K-spin End 𝑥 need not be injective or surjective. The following ∗ bundles are defined similarly, but replacing Cl(𝑇K𝑀) with characterization is convenient in this regard: ev ∗ Cl (𝑇 𝑀), that is, replacing the Kahler-Atiyah¨ bundle with K (i) 𝛾 is fiberwise injective iff the fiber ofahler- theK¨ its even subbundle. Physically, smooth sections of a (s)pin Atiyah bundle is simple as an associative algebra. bundle describe K-valued (s)pinors of spin 1/2 defined over the manifold 𝑀. As explained in [18, 29], a pin bundle 𝑆 in (ii) 𝛾 is fiberwise surjective iff the Schur algebra of the 𝑐 our sense exists on 𝑀 iff 𝑀 admits a so-called Clifford - fiber of the Kahler-Atiyah¨ bundle is isomorphic with structure, that is, a “reduction” of the structure group of the thebasefieldK. 18 Advances in High Energy Physics

Table 2: Fiberwise character of 𝛾 for the case K = R.Atthe (i) A simple Clifford algebra admits (up to K-linear intersection of each row and column, we indicate the values of 𝑝−𝑞 equivalence) a single nontrivial irreducible represen- (mod 8) for which the map induced by 𝛾 on each fiber of the Kahler-¨ tation by K-linear operators, whose dimension equals Atiyah algebra has the corresponding properties. In parentheses, we Δ K(𝑑)dimK SK(𝑝, 𝑞). also indicate the isomorphism type of the Schur algebra for that value of 𝑝−𝑞(mod 8). Note that 𝛾 is fiberwise surjective exactly (ii) A nonsimple Clifford algebra admits (up to K-linear for the normal case, that is, when the Schur algebra is isomorphic equivalence) two nontrivial irreducible representa- with R. tions by K-linear operators, whose real dimensions Δ (𝑑) S (𝑝, 𝑞) K = R Injective Noninjective are both equal to K dimR K .Thetworep- resentations map the Clifford volume element deter- Surjective 0(R), 2(R)1(R) 3(C) 7(C) 4(H) 6(H)5(H) mined by some given orientation into a sign factor Nonsurjective , , , times the identity operator of the representation space and are distinguished from one another by the value This gives the following classification. of that signed factor.

The Case K = C.Then𝛾 is always fiberwise surjective, being 4.3. A Partial Inverse of 𝛾 in the Nonsimple Case. In the fiberwise injective iff 𝑑 is even. nonsimplecase,thebundlemorphism𝛾 has the property: K = R 𝛾 𝑝− The Case .Then is fiberwise surjective iff 𝛾 (]) =𝜖훾id푆, (161) 𝑞≡8 0, 1, 2.Itisfiberwiseinjectiveiff𝑝−𝑞≡8 0, 2, 3, 4, 6, 7. This is summarized in Table 2. where 𝜖훾 ∈{−1,1}is a sign factor which we will call the signature of 𝛾. Direct computation using (161) gives The Schur Bundle and the Image of 𝛾.TheSchuralgebra SK(𝑝, 𝑞) of Section 3.8 is realized naturally in the represen- 𝛾∘𝑃휖 =𝛾, tation space. Picking a point 𝑥 on 𝑀,letΣ푥 be the subalgebra 𝛾 (162) of (End(𝑆푥), ∘) consisting of those endomorphisms 𝑇푥 ∈ 𝛾∘𝑃−휖 =0, End(𝑆푥) which commute with any operator lying in the image 𝛾 of 𝛾푥: which implies that 𝛾 vanishes when restricted to the sub- def −휖𝛾 ∗ 휖𝛾 ∗ bundle ∧ 𝑇K𝑀 and that its restriction to ∧ 𝑇K𝑀 gives Σ푥 ={𝑇푥 ∈ End (𝑆푥)|[𝛾푥 (𝜔푥),𝑇푥]−,∘ =0,∀𝜔푥 (157) an isomorphism between this latter subbundle of alge- ∗ ∈∧𝑇 𝑀} . brasandthesubbundleofalgebras(EndΣ(𝑆), ∘) of End(𝑆). 푥 −휖𝛾 ∗ We have ker(𝛾) = ∧ 𝑇K𝑀. Hence the corresponding Then Σ푥 is isomorphic with SK(𝑝, 𝑞) for all 𝑥∈𝑀.The maponsections(whichwedenoteagainby𝛾)hasker- −휖 휖 Σ 𝑥 𝑀 𝛾 𝛾 bundle determined by 푥 when varies on will be nel K(𝛾) =K Ω (𝑀) while its restriction to ΩK (𝑀) called the Schur bundle of 𝛾; it is a bundle of subalgebras of gives an isomorphism between this latter subalgebra of the ( (𝑆), ∘) 𝑆 End .Ofcourse,thespace 푥 can be viewed as a left Kahler-Atiyah¨ algebra and the subalgebra Γ(𝑀, EndΣ(𝑆)) of Σ푥-module via the obvious action of the elements of Σ푥— (Γ(𝑀, End(𝑆)), .∘) whereby 𝑆 canbeviewedasabundleofmodulesoverthe ∗ 훾 ∗ 훾 Schur bundle. The image 𝛾(Λ𝑇K𝑀) of 𝛾 coincides with the The Subbundle ∧ 𝑇K𝑀 and the Subalgebra ΩK(𝑀).Letus subbundle EndΣ(𝑆) ⊂ End(𝑆) whose fiber at 𝑥∈𝑀is given introduce notation which will allow us to treat all cases by uniformly: (𝑆) EndΣ 푥 (i) In the nonsimple case (when the signature 𝜖훾 is (158) 훾 ∗ def 휖 ∗ 훾 def ∧ 𝑇 𝑀 =∧𝛾 𝑇 𝑀 Ω (𝑀) = ={𝑇푥 ∈ End (𝑆) |[𝑇푥,𝑈푥]−,∘ =0,∀𝑈∈Σ푥}, defined), we let K K and K 휖𝛾 ΩK (𝑀). whileitsspaceofgloballydefinedsmoothsectionsis ∧훾𝑇∗𝑀 def=∧𝑇∗𝑀 Γ(𝑀, (𝑆))={𝑇∈Γ(𝑀, (𝑆)) | [𝑇, 𝑉] (ii) In the simple case, we let K K and EndΣ End −,∘ 훾 def (159) Ω (𝑀) =ΩK(𝑀). =0,∀𝑉∈Γ(𝑀, Σ)}. K 훾 훾 ∗ In both cases, we have ΩK(𝑀) = Γ(𝑀, ∧ 𝑇K𝑀).Noticethat We also note that the image of the map induced by 𝛾 on 훾 ∗ ∧ 𝑇 𝑀 is always a subbundle of unital algebras of the Kahler-¨ sections is given by K 훾 Atiyah bundle while ΩK(𝑀) is always a unital subalgebra of 𝛾(ΩK (𝑀))=Γ(𝑀,EndΣ (𝑆)). (160) the Kahler-Atiyah¨ algebra. The Partial Inverse of 𝛾. Consider the bundle isomorphism: Irreducible Algebra Representations of the Fiber of the Kahler-¨ ∼ Atiyah Bundle. We end by recalling some well-known facts 󵄨EndΣ(푆) 훾 ∗ 𝛾󵄨∧𝛾푇∗푀 :∧ 𝑇K𝑀 󳨀→ EndΣ (𝑆) (163) from the representation theory of Clifford algebras: K Advances in High Energy Physics 19

훾 훾 obtained by restricting the domain of definition of 𝛾 to the 4.4. Trace on ΩK(𝑀). The subalgebra ΩK(𝑀) admits a 훾 ∗ ∞ 휖 ∞ subbundle ∧ 𝑇K𝑀 of the exterior bundle and the codomain C (𝑀, R)-linear map S :ΩK(𝑀) → C (𝑀, K) given by of definition to the subbundle EndΣ(𝑆) of End(𝑆).Welet (0) S (𝜔) =𝜔 𝑁푝,푞rkK (𝑆) , (171) def −1 ∼ −1 󵄨EndΣ(푆) 훾 ∗ 𝛾 =(𝛾󵄨 𝛾 ∗ ) : EndΣ (𝑆) 󳨀→ ∧ 𝑇K𝑀 (164) ∧ 푇K푀 (0) where 𝜔 is the rank 0 component of 𝜔 (see expansion 𝑆 K 𝑁 be the inverse of (163). The corresponding maps on sections (2)), is any of the -pinor bundles, and 푝,푞 equals 1 2 give the mutually inverse isomorphisms of algebras displayed or according to whether the corresponding fiberwise 𝑁 𝑆 in the following diagram: representation is faithful or not (notice that 푝,푞rkK is the dimension of the smallest faithful representation of the Γ(𝑀,EndΣ(𝑆)) 훾| 𝛾 fiberwise Clifford algebra). One has Ω (𝑀) 훾 K (Ω 𝑀 ,⬦)󳨀󳨀󳨀󳨀󳨀󳨀󳨀󳨀󳨀→ (Γ (𝑀, 𝑆 ),∘). (165) 훾 K ( ) ←󳨀󳨀󳨀󳨀󳨀󳨀󳨀󳨀󳨀 EndΣ ( ) S (𝜔) = (𝛾 (𝜔)), ∀𝜔∈Ω (𝑀) 훾−1 tr K (172) as well as Notation 2. We define S (1푀)=𝑁푝,푞dimK (𝑆) , ̌ def −1 훾 𝑇 =𝛾 (𝑇) ∈Ω (𝑀) ,∀𝑇∈Γ(𝑀,EndΣ (𝑆)). (166) K 훾 (173) S (𝜔 ⬦ 𝜂) = S (𝜂 ⬦ 𝜔) , ∀𝜔, 𝜂∈ΩK (𝑀) . −1 In the context of “vertical quantization” of spin systems, 𝛾 plays the role of a (partial) “vertical dequantization map,” so ̌ 5. The Fierz Isomorphism and Generalized 𝑇 in (166) is the dequantization of a “vertical” operator 𝑇 Killing Forms acting in the Hilbert space. The partial inverse of 𝛾 allows us to transfer statements about operators acting on pinors In this section, we take up the issue of translating constrained to statements about differential forms—an observation which generalized Killing pinor equations into conditions on dif- will be used intensively in what follows. Notice the relations: ferential forms. To simplify presentation, we will assume from the outset that the Schur algebra is isomorphic with 𝛾−1 ∘𝛾=𝑃 , K K = C 휖𝛾 the base field ,sothateither or we are in the normal case with K = R. We start in Section 5.1 with a 𝛾∘𝛾−1 = , idEndΣ(푆) discussion of the bundle of bipinors. Section 5.2 considers a (167) certain isomorphism of bundles of algebras (which we will 𝛾∘𝑃 =𝛾, 휖𝛾 call the Fierz isomorphism) that provides an identification 훾 ∗ ofthebundleofbipinorswiththebundle(∧ 𝑇 𝑀, ⬦) and 𝛾∘𝑃 =0. K −휖𝛾 allows for a concise description of those Fierz identities which involve four pinors. This construction makes essential use of a choice of bilinear and nondegenerate “admissible” −1 Local Expression for 𝛾 . Considering a local pseudo- form B on the pin bundle (such inner products were 푎 푎 푎 orthonormal coframe 𝑒 and recalling that 𝛾(𝑒 )=𝛾 ,wefind classifiedin[30],seealso[31]).Section5.3extractssomebasic properties of this isomorphism which will be useful later on. −1 푎 푎 𝛾 (𝛾 )=𝑒훾, (168) In Section 5.4, we give a brief discussion of completeness relations for the endomorphism algebra of the pin bundle. wherewehaveset Section 5.5 gives an explicit local expansion of the Fierz isomorphism which depends on the choice of a local pseudo- {𝑒푎 , orthonormal coframe. In Section 5.6, we show how algebraic 푎 휖𝛾 ifweareinthenon-simplecase 𝑒훾 = { (169) constraints on pinors translate very directly into constraints 𝑒푎, , { ifweareinthesimplecase on differential forms if one uses the basic properties of the Fierz isomorphism. Section 5.7 takes up the problem of trans- 𝑒푎 with 휖𝛾 defined as in (108). Relation (168) implies lating generalized Killing pinor equations into conditions on differential forms. Using the Fierz isomorphism, we show −1 푎 ⋅⋅⋅푎 𝛾 (𝛾 1 𝑘 ) thatanyconnectiononthepinbundlewhichiscompatible with B defines a certain algebra connection onahler- theK¨ 푎 ⋅⋅⋅푎 𝑒 1 𝑘 , 푎 ⋅⋅⋅푎 { 휖 ifweareinthenon-simplecase (170) Atiyah bundle (i.e., a linear connection which is a fiberwise =𝑒1 𝑘 𝛾 훾 { 푎 ⋅⋅⋅푎 derivation of the geometric product) such that the Fierz 𝑒 1 𝑘 , , { ifweareinthesimplecase isomorphism is flat with respect to the connections induced on its domain and codomain. Using this property, we show 푎1⋅⋅⋅푎𝑘 where 𝑒± aredefinedin(109)andweusedthefactthat how one can easily translate generalized Killing conditions 푎1⋅⋅⋅푎𝑘 푎1 푎𝑘 𝛾 =𝛾 ∘⋅⋅⋅∘𝛾 for all mutually distinct 𝑎1 ⋅⋅⋅𝑎푘,thefact on pinors into differential constraints on forms defined −1 that 𝛾 is an isomorphism of algebras when corestricted to on 𝑀. Section 5.8 gives another form of such differential its image and (for the nonsimple case) identity (109). constraints, which is used in Appendix B for comparison with 20 Advances in High Energy Physics

푎 푎 푎 푡 푎 the component approach outlined in [3]. In Section 5.9, we which means that 𝛾 =𝛾(𝑒) satisfy (𝛾 ) =𝜖B𝛾 ,arelation discuss some basic aspects of the algebrodifferential system of which implies constraints on inhomogeneous forms which results from our 푎1⋅⋅⋅푎𝑘 푡 푘 푎𝑘⋅⋅⋅푎1 analysis. As expected, our formulation allows one to extract (𝛾 ) =𝜖B𝛾 , (180) basic structural properties of this system, thereby providing 푘 푘 푎 푏 푎푏 a starting point for a natural generalization of the classical where 𝜖B =(𝜖B) .Since[𝛾 ,𝛾 ]+,∘ =2𝜂 ,wealsohave theory of Killing forms. Finally, Section 5.10 considers the particular cases of one and two independent constrained 푎 −1 def 푏 (𝛾 ) =𝛾푎, where 𝛾푎 =𝜂푎푏𝛾 , (181) generalized Killing pinors with a definite and symmetric (𝑑) Spin -invariant metric, the first of which is relevant to the which in turn gives application discussed in Section 6. 푎 ⋅⋅⋅푎 −1 (𝛾 1 𝑘 ) =𝛾 . 푎𝑘⋅⋅⋅푎1 (182) 5.1. Bipinor Algebras. Let 𝑆 be a pin bundle over (𝑀, 𝑔) with ∗ underlying morphism 𝛾 : (∧𝑇K𝑀, ⬦) →(End(𝑆), .∘) Combining the above, we find

푎 ⋅⋅⋅푎 −1 푡 푡 푘 Admissible Bilinear Pairings on the Pin Bundle. It is well 1 𝑘 ((𝛾 ) ) =(𝛾푎 ⋅⋅⋅푎 ) =𝜖B𝛾푎 ⋅⋅⋅푎 , (183) known that 𝑆 carries so-called admissible nondegenerate 𝑘 1 1 𝑘 B 𝑆 bilinear pairings whose action on sections of satisfies which implies 耠耠 B (𝛾 (𝜔) 𝜉, 𝜉 )=B (𝜉, 𝛾 (𝜏 (𝜔))𝜉 ), 푎 ⋅⋅⋅푎 −1 耠 푘 耠 B B ((𝛾 1 𝑘 ) 𝜉, 𝜉 )=𝜖 B (𝜉, 𝛾 𝜉 ). B 푎1⋅⋅⋅푎𝑘 (184) 耠耠 B (𝜉 ,𝜉)=𝜎 B (𝜉, 𝜉 ), (174) B These relations will be useful later. ∀𝜔 ∈ Ω (𝑀) ,∀𝜉,𝜉耠 ∈Γ(𝑀,) 𝑆 K The Isomorphism 𝐸. The nondegenerate pairing B induces a ∼ ∗ as well as another property which can be found in [31] but will bundle isomorphism 𝜌:𝑆󳨀→𝑆 , whose action on sections is notberelevantforwhatfollows.Intheformulasabove,we given by used the following antiautomorphism of the Kahler-Atiyah¨ def algebra: 𝜌 (𝜉) (𝜉耠) = B (𝜉耠,𝜉) ,∀𝜉,𝜉耠 ∈Γ(𝑀,) 𝑆 . (185)

{𝜏, if 𝜖B =+1 (1−휖B)/2 Ontheotherhand,wehaveanaturalbundleisomorphism 𝜏B =𝜋 ∘𝜏={ (175) ∗ ∼ 𝜋∘𝜏, 𝜖 =−1, 𝑞:𝑆⊗𝑆 󳨀→ End(𝑆), given on sections as follows: { if B def where 𝜏 is the reversion antiautomorphism defined in (41). 𝑞(𝜉⊗𝜂)(𝜉耠) =𝜂(𝜉耠)𝜉, The numbers 𝜖B (the type of B)and𝜎B (the symmetry of (186) B)equal+1 or −1, depending on 𝑝, 𝑞 andtheprecisechoice ∀𝜉, 𝜉耠 ∈Γ(𝑀,) 𝑆 ,∀𝜂∈Γ(𝑀,𝑆∗). of B; such bilinear pairings were classified in [30, 31]. Notice that the first equation in (174) implies The two maps above combine to give a bundle isomorphism def ∼ def 耠 푡 𝐸 =𝑞∘( ⊗𝜌): 𝑆⊗𝑆 󳨀→ (𝑆) 𝐸 󸀠 =𝐸(𝜉⊗𝜉)∈ 𝛾 (𝜔) =𝛾(𝜏B (𝜔)), ∀𝜔∈ΩK (𝑀) ; (176) id푆 End .Setting 휉,휉 耠 Γ(𝑀, End(𝑆)) for all 𝜉, 𝜉 ∈ Γ(𝑀, 𝑆),wehave that is, 푡 𝐸휉 ,휉 ∘𝐸휉 ,휉 = B (𝜉3,𝜉2)𝐸휉 ,휉 , () ∘𝛾=𝛾∘𝜏 , 1 2 3 4 1 4 B (177) (187) 푡 ∀𝜉1,𝜉2,𝜉3,𝜉4 ∈Γ(𝑀,) 𝑆 , where 𝑇 denotes the transpose of 𝑇∈Γ(𝑀,End(𝑆)) with B respect to , which is defined through an identity which follows from the explicit form: 耠 푡 耠耠 B (𝑇𝜉, 𝜉 )=B (𝜉, 𝑇 𝜉 ), ∀𝜉,𝜉 ∈Γ(𝑀,) 𝑆 . (178) 耠耠耠耠耠 𝐸휉,휉󸀠 (𝜉 )=B (𝜉 ,𝜉 )𝜉 (188) 푡 푡 푡 This operation satisfies (𝑇 ) =𝑇and (id푆) = id푆,thefirst without making use of the signed symmetry property of B. identity being a consequence of the signed symmetry prop- Note that 𝐸 depends on the choice of B (since 𝜌 does). erty of B (the second identity listed in (174)). The operation 푡 ∞ 𝑇→𝑇of taking the B-transpose defines a C (𝑀, R)- 𝑆 (Γ(𝑀, (𝑆)), ∘) The Bundle of Bipinors and the Bipinor Algebra of . The linear antiautomorphism of the algebra End . bundle isomorphism 𝐸 allows us to transfer the fiberwise composition of operators from End(𝑆) to an associative and Local Expressions. Given a local pseudo-orthonormal 푎 bilinear fiberwise composition ∙ definedonthebundle of coframe 𝑒 above 𝑈⊂𝑀, the first of properties (174) bipinors 𝑆⊗𝑆, whose action on sections takes the form amounts to 푎 耠 푎 耠耠 def −1 B (𝛾 𝜉, 𝜉 )=𝜖BB (𝜉, 𝛾 𝜉 ), ∀𝜉,𝜉 ∈Γ(𝑈,𝑆) , (179) 𝑢∙V =𝐸 (𝐸 (𝑢) ∘𝐸(V)) ,∀𝑢,V ∈Γ(𝑀, 𝑆) ⊗𝑆 . (189) Advances in High Energy Physics 21

This operation satisfies which identifies the bipinor algebra with the subalgebra Ω훾 (𝑀) 𝐸̌ (𝜉 ⊗𝜉 )∙(𝜉 ⊗𝜉 )=B (𝜉 ,𝜉 )𝜉 ⊗𝜉 , K of the Kahler-Atiyah¨ algebra. Note that depends on 1 2 3 4 3 2 1 4 B (190) thechoiceofadmissibleform . ∀𝜉1,𝜉2,𝜉3,𝜉4 ∈Γ(𝑀,) 𝑆 . ̌ def ̌ Fierz Identities Involving Four Pinors. Setting 𝐸휉,휉󸀠 = 𝐸(𝜉 ⊗ The composition ∙ makes the bundle of bipinors into a bundle 耠 −1 훾 耠 𝜉 )=𝛾 (𝐸 󸀠 )∈Ω (𝑀) 𝜉, 𝜉 ∈ Γ(𝑀, 𝑆) of unital associative algebras which is isomorphic with the 휉,휉 K (for ), (187) implies Ω훾 (𝑀) bundle of algebras (End(𝑆), ;ofcourse,theunitsectionid∘) 푆 the following identity in the subalgebra K of the Kahler-¨ of End(𝑆) maps to the unit section of 𝑆⊗𝑆, which we denote Atiyah algebra: def −1 by I =𝐸(id푆).TheunitalassociativealgebraΓ(𝑀, 𝑆 ⊗ 𝐸̌ ⬦ 𝐸̌ = B (𝜉 ,𝜉 ) 𝐸̌ , 𝑆) = Γ(𝑀, 𝑆) ⊗C∞(푀,R) Γ(𝑀, 𝑆) consisting of smooth sections 휉1,휉2 휉3,휉4 3 2 휉1,휉4 of the bipinor bundle will be called the bipinor algebra of 𝑆;it (195) ∞ ∀𝜉 ,𝜉 ,𝜉 ,𝜉 ∈Γ(𝑀,) 𝑆 . is an algebra over the ring C (𝑀, R). 1 2 3 4

∞ The Bipinor C (𝑀, R)-Algebra of a Submodule of Sections. If Equation (195) is the condensed expression of Fierz identities K is any K-linear subspace of Γ(𝑀, 𝑆), then the set involving four pinors. These identities simply express the fact 𝛾 𝐸̌ def 耠耠 that (and thus ) is an isomorphism of bundles of algebras, K ⊗C∞(푀,R) K ={𝜉⊗𝜉|𝜉,𝜉 ∈ K} rather than simply an isomorphism of vector bundles—and (191) are,infact,equivalenttothispropertyoncefiberwiselinearity ⊂Γ(𝑀, 𝑆) ⊗𝑆 ≈Γ(𝑀,) 𝑆 ⊗ ∞ Γ (𝑀,) 𝑆 C (푀,R) of 𝛾 is assumed. The construction of 𝐸̌ is summarized in is a K-linear subspace of Γ(𝑀, 𝑆 ⊗ 𝑆). the commutative diagram (196), which applies provided that ∞ When K ⊂ Γ(𝑀, 𝑆) is a submodule of the C (𝑀, R)- SK(𝑝, 𝑞) ≈ K. In the diagram, we show the action of the module Γ(𝑀, 𝑆), then the subspace K ⊗C∞(푀,R) K ⊂ various morphisms on sections. Consider Γ(𝑀, 𝑆⊗𝑆) is a (generally nonunital) subalgebra of the bipinor algebra of 𝑆, which we will call the bipinor algebra of K.This idS ⊗𝜌 ∗ ∞ Γ(M, S ⊗ S) Γ(M, S ⊗K S ) associative algebra defined over C (𝑀, R) depends on the ∼ B E q choice of . In particular, the space of smooth global sections Ě ∼ ∼ ∼ Γ(𝑀, 𝐾) 𝐾 𝑆 C∞(𝑀, R) (196) of any vector subbundle of is a - 𝛾 ∼ Γ(𝑀, 𝑆) Ω (M) Γ(M, End (S)) submodule of and the corresponding bipinor algebra K 𝛾 K Γ(𝑀, 𝐾) ⊗C∞(푀,R) Γ(𝑀,𝐾)=Γ(𝑀,𝐾⊗𝐾)will be called the bipinor algebra of 𝐾. Notation 3. Let K ⊂ Γ(𝑀, 𝑆) be any K-linear subspace of The Bipinor K-Algebra of a B-Flat Subspace of Sections. Γ(𝑀, 𝑆).TheimageoftheK-linear subspace K ⊗C∞(푀,R) K ⊂ Another interesting case arises when the K-linear subspace Γ(𝑀, 𝑆 ⊗ 𝑆) through the Fierz isomorphism will be denoted K ⊂ Γ(𝑀, 𝑆) is B-flat,bywhichwemeanthatK satisfies by the condition: B (𝜉,耠 𝜉 ) 𝑀, ̌ def ̌ 훾 is a constant function on K = 𝐸 (K ⊗C∞(푀,R) K) ⊂ΩK (𝑀) (197) (192) 耠 ∀𝜉, 𝜉 ∈ K. 훾 and is a K-linear subspace of ΩK(𝑀). In this case, K⊗C∞(푀,R)K is a K-subalgebra of the bipinor C∞(𝑀, R) algebra (Γ(𝑀, 𝑆 ⊗ 𝑆),, ∙) which we will call the (flat) bipinor The Fierz -Algebra of a Submodule of Sections. K K When the subspace K ⊂ Γ(𝑀, 𝑆) is a submodule -algebra determined by . ∞ of the C (𝑀, R)-module Γ(𝑀, 𝑆), then the subspace K ⊗C∞(푀,R) K ⊂ Γ(𝑀, 𝑆 ⊗ 𝑆) is a (generally nonunital) ∞ 5.2. The Fierz Isomorphism and Fierz Identities: Fierz Algebras subalgebra (over C (𝑀, R))ofthebipinoralgebraof𝑆.Its ̌ image (197) through the Fierz isomorphism is a (generally The Fierz Isomorphism 𝐸. Let us now assume that we are in ∞ 훾 nonunital) subalgebra of the C (𝑀, R)-algebra (Ω (𝑀), ⬦), one of the cases when the Schur algebra is isomorphic with K which we will call the Fierz subalgebra determined by K. thebasefield.Thenonecanalsotransporttothebundleof ∞ ∼ This algebra over C (𝑀, R) encodes the Fierz identities 𝛾−1 : (𝑆) 󳨀→(∧𝑇∗𝑀)훾 bipinors the isomorphism End K to get between bilinears constructed from pinors which belong to an isomorphism of bundles of algebras: K. A particular case arises when K = Γ(𝑀, 𝐾) where 𝐾⊂𝑆 𝑆 ̌ def −1 ∼ 훾 ∗ is some vector subbundle of —in which situation 𝐸 =𝛾 ∘𝐸:(𝑆⊗𝑆,∙) 󳨀→ ( ∧ 𝑇K𝑀, ⬦) , (193) K ⊗C∞(푀,R) K is the bipinor algebra of 𝐾.Thecorresponding ̌ which we will call the Fierz isomorphism.Onsections, Fierz subalgebra K will then be called the Fierz subalgebra ∞ this induces a C (𝑀, R)-linear isomorphism of algebras determined by the subbundle 𝐾. With the further assumption (denoted,asusual,bythesamesymbol): that B is a scalar product (as happens in the application of Section 6), the morphism 𝐸 canthenbeusedtoidentifythe ̌ def −1 ∼ 훾 ∞ 𝐸 =𝛾 ∘𝐸:(Γ (𝑀, 𝑆) ⊗𝑆 ,∙) 󳨀→ ( Ω K (𝑀) ,⬦), (194) Fierz algebra of 𝐾 with the C (𝑀, K)-algebra Γ(𝑀, End(𝐾)) 22 Advances in High Energy Physics of globally defined endomorphisms of the bundle 𝐾;in We also notice the relation: particular,theFierzsubalgebraisunitalinsuchcases. 푡 ̌ ̌ (𝐸휉,휉󸀠 ) =𝜎B𝐸휉󸀠,휉 ⇐⇒ 𝜏 B (𝐸휉,휉󸀠 )=𝜎B𝐸휉󸀠,휉, K B (205) The Fierz -Algebra of a -Flat Subspace of Sections. When ∀𝜉, 𝜉耠 ∈Γ(𝑀,) 𝑆 , K is a B-flat K-linear subspace of Γ(𝑀, K),thevector space K ⊗C∞(푀,R) K is a K-subalgebra of the bipinor algebra which follows from the signed symmetry of B together (Γ(𝑀, 𝑆⊗𝑆),. ∙) It follows that its image (197) through the Fierz 훾 with definition (188). The last identity can also be written as isomorphism is a K-subalgebra of the algebra (ΩK(𝑀), ⬦), follows: K which will be called the (flat)Fierz -algebra determined by ()푡 ∘𝐸=𝐸∘ ⇐⇒ 𝜏 ∘ 𝐸=̌ 𝐸∘̌ , K. transpB B transpB (206) def where transpB =𝜎Btransp and transp :Γ(𝑀,𝑆⊗𝑆)→ ∞ 5.3. Some Properties of the Fierz Isomorphism. Asimple Γ(𝑀, 𝑆⊗𝑆) is the C (𝑀, R)-linear operator which is defined computation using (188) shows that the following identities as follows on decomposable elements: hold for any 𝑇∈Γ(𝑀,End(𝑆)): 耠耠耠 transp (𝜉 ⊗ 𝜉 )=𝜉 ⊗𝜉, ∀𝜉,𝜉 ∈Γ(𝑀,) 𝑆 . (207) 𝑇∘𝐸휉,휉󸀠 =𝐸푇휉,휉󸀠 , One easily checks that transpB is an antiautomorphism of the 𝐸휉,휉󸀠 ∘𝑇=𝐸휉,푇𝑡휉󸀠 , (198) bipinor algebra. For later reference, note the identity:

tr ∘𝐿푇 ∘𝐸=B ∘(𝑇⊗id푆)⇐⇒ ∀𝜉, 𝜉耠 ∈Γ(𝑀,) 𝑆 ; 耠耠 (208) (𝑇 ∘ 𝐸 󸀠 )=B (𝑇𝜉, 𝜉 ), ∀𝜉,𝜉 ∈Γ(𝑀,) 𝑆 , that is, tr 휉,휉

𝐿푇 ∘𝐸=𝐸∘(𝑇⊗id푆), which will be useful below. (199) 푡 𝑅푇 ∘𝐸=𝐸∘(id푆 ⊗𝑇), 5.4. Local Completeness Relations for the Endomorphism AlgebraofthePinBundle.Let tr : End(𝑆) → OK be 6 where 𝐿푇 and 𝑅푇 are the operators of left and right multipli- the natural trace on End(𝑆) (which is a morphism of K- −1 cation with 𝑇 in the algebra (Γ(𝑀, End(𝑆)), .∘) Applying 𝛾 vector bundles). Recall that we assume the Schur algebra ̌ def −1 훾 K to identities (198) and setting 𝑇 =𝛾 (𝑇) ∈ Ω (𝑀) give to be isomorphic with the base field . Then a convenient K generating set of local sections for the vector bundle End(𝑆) ̌ ̌ ̌ 𝑈⊂𝑀 𝑇⬦𝐸휉,휉󸀠 = 𝐸푇휉,휉󸀠 , above a sufficiently small open subset is given (see, 푎1⋅⋅⋅푎𝑘 e.g., [28]) by the operators {𝛾 | 𝑘 = 0,...,𝑑, 11 ≤𝑎 < (200) 푎 푎 ̌ ̌ ̌ ⋅⋅⋅ < 𝑎푘 ≤𝑑},where𝑑=𝑝+𝑞and 𝛾 =𝛾(𝑒) with 𝐸 󸀠 ⬦ 𝑇=𝐸 𝑡 󸀠 , 푎 휉,휉 휉,푇 휉 (𝑒 ) a pseudo-orthonormal local coframe of 𝑀 above 𝑈.The 푡 that is, (substituting 𝑇→𝑇 into the second equation) following identity (the “completeness relation”) holds: ̌ Δ (𝑑) 푑 1 𝑇⬦𝐸̌ 󸀠 = 𝐸̌ 󸀠 , R 푎𝑘⋅⋅⋅푎1 휉,휉 푇휉,휉 𝑇=푈 ∑ tr (𝛾 ∘𝑇)𝛾푎 ⋅⋅⋅푎 , 2푑 𝑘! 1 𝑘 (201) 푘=0 (209) 푡 ̌ 𝐸̌ 󸀠 ⬦(𝑇) = 𝐸̌ 󸀠 . 휉,휉 휉,푇휉 ∀𝑇 ∈ Γ (𝑈, End (𝑆)) , This also reads [푑/2] where Δ R(𝑑) = 2 for S ≈ R or S ≈ C (see Section 3.8). ̌ ̌ 𝐿 ̌ ∘ 𝐸=𝐸∘(𝑇⊗id푆), 푇 K (𝑆) ⊗ (𝑆) (202) Remark 10. The -vector bundles End End and ̌ ̌ 푡 End(End(𝑆)) canbeidentifiedthroughthebundleisomor- 𝑅 ̌ ∘ 𝐸=𝐸∘(id푆 ⊗𝑇), 푇 phism W : End(𝑆) ⊗ End(𝑆) → End(End(𝑆)),whichactsas where 𝐿푇̌ and 𝑅푇̌ are the operators of left and right multi- follows on sections: ̌ plication with 𝑇 in the Kahler-Atiyah¨ algebra. Equation (177) W (𝐴⊗𝐵)(𝑇) =𝐴tr (𝐵∘𝑇) , −1 푡 −1 implies 𝛾 ∘() =𝜏B ∘𝛾 ;thatis, (210) ∀𝐴, 𝐵, 𝑇 ∈Γ (𝑀, End (𝑆)) . −1 푡 −1 푡 ̌ ̌ 𝛾 (𝑇 )=𝜏B (𝛾 (𝑇))⇐⇒(𝑇) =𝜏B (𝑇) , Using this isomorphism, we transport the composition ∘ (203) of End(End(𝑆)) to an associative composition ∙ defined on ∀𝑇 ∈ Γ (𝑀, End (𝑆)) . End(𝑆) ⊗ End(𝑆), whose action on sections is given by 耠耠 We can thus write the second relation of (200) in the form (𝐴⊗𝐵) ∙(𝐴 ⊗𝐵)

𝐸̌ 󸀠 ⬦𝜏 (𝑇)̌ = 𝐸̌ 󸀠 , −1 耠耠 휉,휉 B 휉,푇휉 = W (W (𝐴⊗𝐵) ∘ W (𝐴 ⊗𝐵)) , (211) (204) 耠耠耠 ∀𝑇 ∈ Γ (𝑀, End (𝑆)) ,∀𝜉,𝜉∈Γ(𝑀,) 𝑆 . ∀𝐴, 𝐴 ,𝐵,𝐵 ∈Γ(𝑀, End (𝑆)) . Advances in High Energy Physics 23

An easy computation shows that ∙ has the explicit form: 5.6. Expressing the Algebraic Constraints through Differential Forms. Considerthecasewhenwehaveasinglealgebraic 耠耠耠耠 (𝐴⊗𝐵) ∙(𝐴 ⊗𝐵)=tr (𝐴 ∘𝐵)𝐴⊗𝐵, constraint 𝑄𝜉,where =0 𝑄∈Γ(𝑀,End(𝑆)).AsinSection2, (212) we let K(𝑄) denote the space of solutions of the algebraic 耠耠 ∞ ∀𝐴, 𝐴 ,𝐵,𝐵 ∈Γ(𝑀, End (𝑆)) . constraint, which is a submodule of the C (𝑀, K)-module Γ(𝑀, 𝑆).Recallthattheregenerallyexistsnosubbundleof𝑆 K(𝑄) The unit section idEnd(푆) ofthebundleEnd(End(𝑆)) cor- whose space of smooth sections equals . def W I = W−1( ) ∞ responds via to the unit section idEnd(푆) The Dequantized Constraint and the C (𝑀, R)-Subalgebra of (End(𝑆) ⊗ End(𝑆), .Expression(212)showsthatthe∙) of Constrained Inhomogeneous Forms. The inhomogeneous completeness relation is equivalent to the following decom- form position of the unit I|푈 of the algebra (Γ(𝑈,End(𝑆) ⊗ (𝑆)), ∙) ̌ −1 훾 End : 𝑄=𝛾 (𝑄) ∈ΩK (𝑀) (219) 푑 1 𝑄 푎𝑘⋅⋅⋅푎1 will be called the dequantization of .Relation(199)gives I =푈 ∑ 𝛾푎 ⋅⋅⋅푎 ⊗𝛾 . (213) 𝑘! 1 𝑘 푘=0 𝐿푄 ∘𝐸=𝐸∘(id푆 ⊗𝑄), (220) 5.5. Explicit Expansion of the Fierz Isomorphism When the 𝑅푄𝑡 ∘𝐸=𝐸∘(𝑄⊗id푆), Schur Algebra Equals the Base Field. Given a local pseudo- orthonormal coframe of 𝑀, explicit expansions for the 𝐿 ,𝑅 𝑡 Γ(𝑀, (𝑆)) ̌ where 푄 푄 are—as above—the operators on End isomorphism 𝐸 canbederivedusingtheresultsof[28](see 푡 given by left and right composition with 𝑄 and 𝑄 .Using also [32]) for any choices of the base field K and of the the fact that 𝐸 is an isomorphism as well as the identities signature type (𝑝, 𝑞). A complete discussion is quite involved K(𝑄 ⊗ id푆)=K(𝑄)C ⊗ ∞(푀,R) Γ(𝑀, 𝑆) and K(id푆 ⊗𝑄)= given the different behavior in various cases and will be taken Γ(𝑀, 𝑆) ⊗C∞(푀,R) K(𝑄),wefindK(𝑄) ⊗C∞(푀,R) K(𝑄) = up in detail in a different publication. Below, we will consider K(𝑄 ⊗ id푆)∩K(id푆 ⊗𝑄)and only the case when the Schur algebra is isomorphic with the base field K,thatis,thecasewhenK = C and the case when 𝐸(K (𝑄) ⊗C∞(푀,R) K (𝑄))=K (𝐿푄)∩K (𝑅Q𝑡 ). (221) K = R with 𝑝−𝑞≡8 0, 1, 2. In this case, the local completeness relation (209) holds. −1 Applying 𝛾 to both sides of this relation gives the following Applying it to the endomorphism 𝑇=𝐸휉,휉󸀠 and using relation description of the Fierz algebra of the submodule of sections (208), we find the local expansion: ∞ K(𝑄), which we will call the C (𝑀, R)-algebra of 𝑄- constrained inhomogeneous forms: 1 푑 1 耠 푎1⋅⋅⋅푎𝑘 𝐸휉,휉󸀠 =푈 ∑ B (𝛾푎 ⋅⋅⋅푎 𝜉, 𝜉 )𝛾 , 2[(푑+1)/2] 𝑘! 𝑘 1 ̌ def ̌ 푘=0 (214) K푄 = 𝐸(K (𝑄) ⊗C∞(푀,R) K (𝑄))

∀𝜉, 𝜉耠 ∈Γ(𝑈,𝑆) , 훾 = K (𝐿 ̌ )∩K (𝑅 ̌ )∩Ω (𝑀) 푄 휏B(푄) K (222) which implies the following local expansion of the Fierz −1 = K (𝐿푄̌ +𝑅휏 (푄)̌ )∩K (𝐿푄̌ −𝑅휏 (푄)̌ ) isomorphism upon applying 𝛾 to both sides: B B 훾 1 ∩ΩK (𝑀) . 𝐸̌ 󸀠 = ̌ 󸀠 , 휉,휉 2[(푑+1)/2] E휉,휉 (215) Here, 𝐿푄̌ , 𝑅푄̌ are the left and right ⬦-multiplication operators 훾 of Section 3.4 and the second equality above is obvious. With an identity holding in ΩK(𝑀),where these definitions, we have the equivalence: 푑 ̌ ̌(푘) ̌ ̌ 耠 E휉,휉󸀠 = ∑E휉,휉󸀠 (216) 𝐸휉,휉󸀠 ∈ K푄 ⇐⇒ 𝜉 ⊗ 𝜉 ∈ K (𝑄) ⊗C∞(푀,R) K (𝑄) ; (223) 푘=0 that is, as in (2), with ̌ ̌ ̌ ̌ 𝑄⬦𝐸휉,휉󸀠 = 𝐸휉,휉󸀠 ⬦𝜏B (𝑄) = 0 ⇐⇒ (푘) 1 (푘) ̌ ̌ 耠 푎1⋅⋅⋅푎𝑘 E휉,휉󸀠 =푈 E푎 ⋅⋅⋅푎 (𝜉, 𝜉 )𝑒훾 , (217) 𝑘! 1 𝑘 𝜉=0 (224) where, using (184), we have 耠 or 𝜉 =0 ̌(푘) 耠耠 푘 耠 E푎 ⋅⋅⋅푎 (𝜉, 𝜉 ) = B (𝛾푎 ⋅⋅⋅푎 𝜉, 𝜉 ) =𝜖BB (𝜉,푎 𝛾 ⋅⋅⋅푎 𝜉 ) . (218) 耠 1 𝑘 𝑘 1 1 𝑘 or 𝑄𝜉 = 𝑄𝜉 =0. 24 Advances in High Energy Physics

Behavior under 𝜏B. The equivalence (224) can be written Remark 11. Using (204) and (215) and separating ranks shows in the following form, which follows by applying 𝜏B to the that (224) can be written as follows: second equation in the left hand side 耠 𝜉⊗𝜉 ∈ K (𝑄) ⊗C∞(푀,R) K (𝑄) ⇐⇒

𝑄⬦̌ 𝐸̌ 耠 = 𝑄⬦𝜏̌ (𝐸̌ 耠 )=0⇐⇒ (228) 휉,휉 B 휉,휉 (푘) (푘) ̌ = ̌ =0, ∀𝑘=0⋅⋅⋅𝑑, (225) E푄휉,휉耠 E휉,푄휉耠耠 𝜉⊗𝜉 ∈ K (𝑄) ⊗C∞(푀,R) K (𝑄) which amounts to the following description upon using the explicit form (218) of Ě: or (using (205)) in the form 耠 𝜉⊗𝜉 ∈ K (𝑄) ⊗ ∞ K (𝑄) ⇐⇒ ̌ ̌ ̌ ̌ C (푀,R) 𝑄⬦𝐸휉,휉耠 = 𝑄⬦𝐸휉耠,휉 =0⇐⇒ (226) B (𝜉,푡 𝑄 ∘𝛾 𝜉耠)=B (𝜉, 𝛾 ∘𝑄𝜉耠)=0, (229) 耠 푎1⋅⋅⋅푎푘 푎1⋅⋅⋅푎푘 𝜉⊗𝜉 ∈ K (𝑄) ⊗C∞(푀,R) K (𝑄) . ∀𝜉, 𝜉耠 ∈ K (𝑄) . In fact, relation (206) and the obvious fact that K(𝑄) ⊗C∞(푀,R) K(𝑄) is invariant under the antiauto- In Appendix B.1, we show that (229) are equivalent to certain morphism transpB imply that the algebra of 𝑄-constrained relations which were first discussed in [3] for the special case 𝑑=𝑝=8,𝑞=0 differential forms is invariant under 𝜏B: .

̌ ̌ 5.7. Conditions on Differential Forms Implied by the Gen- 𝜏B (K푄) = K푄. (227) eralized Killing Equation. In our formulation, a Clifford 푆 푚 푆 connection ∇ = d𝑥 ⊗∇푚 on 𝑆 is a connection which satisfies

ad ad [∇푆 ,𝛾(𝜔)] =𝛾(∇ 𝜔) , ∀𝜔 ∈Ω (𝑀) ⇐⇒ ( ∇ 푆 ) ∘𝛾=𝛾∘∇ ⇐⇒ 𝑃 ∘∇ =𝛾−1 ∘(∇푆 ) ∘𝛾, 푚 −,∘ 푚 K 푚 푚 휖훾 푚 푚 (230)

∗ 푑 where ∇푚 is the connection induced on ∧𝑇K𝑀 by the Levi- Notation 4. We let K(𝐷) =푚=1 ∩ K(𝐷푚)⊂Γ(𝑀,𝑆)be 푆 ad K Civita connection of (𝑀, 𝑔) and (∇푚) : Γ(𝑀, End(𝑆)) → the finite-dimensional -linear subspace of all generalized 푆 Killing pinors with respect to 𝐷. Γ(𝑀, End(𝑆)) is the connection induced by ∇푚 on End(𝑆):

ad (∇푆 ) (𝑇) def=[∇푆 ,𝑇] =∇푆 ∘𝑇−𝑇∘∇푆 , The Dequantized Connection. In the following, we consider 푚 푚 −,∘ 푚 푚 only the case when the Schur algebra is isomorphic with the (231) 𝐷 𝑆 ∀𝑇 ∈ Γ (𝑀, (𝑆)) . base field. Then any connection on canbewrittenas End follows:

Notice that 𝑃휖 ∘∇is the connection induced by ∇ on 푆 푆 ̌ 훾 𝐷푚 =∇푚 +𝐴푚 =∇푚 +𝛾(𝐴푚), (233) ∗ 휖훾 the subbundle (∧𝑇K𝑀) and that this induced connection 푆 ad is determined by (∇ ) through property (230). In the where 푆 following, we take ∇ to be the connection on 𝑆 induced by 𝐴̌ =𝛾−1 (𝐴 )∈Ω훾 (𝑀) the Levi-Civita connection of (𝑀, 𝑔);itiswellknownthat 푚 푚 K (234) 푆 ∇ is a Clifford connection in the sense discussed above7.A 푆 are inhomogeneous differential forms on 𝑀.TheClifford discussion of this and other properties of ∇ in index language 푆 connection property (230) of ∇ implies (which also serves to fix our conventions and leads to another derivation of certain identities extracted in this paper) can ad 𝐷 ∘𝛾=𝛾∘D푚 ⇐⇒ be found in Appendix A. Equation (230) is compatible with 푚 the fact that ∇푚 is a derivation of the Kahler-Atiyah¨ algebra −1 ad 𝑃휖 ∘ D푚 =𝛾 ∘𝐷푚 ∘𝛾⇐⇒ (235) (ΩK(𝑀), ⬦)—apropertywhichcanbecheckedbydirect 훾 computation using the fact that ∇푚 is an even derivation [𝐷 ,𝛾(𝜔)] =𝛾( 𝜔) , ∀𝜔 ∈Ω (𝑀) , of the exterior algebra which is compatible with the metric. 푚 −,∘ D푚 K ad Similarly, we consider the connection 𝐷 induced by 𝐷 on End(𝑆): where the derivation D푚 (which we will call the adjoint dequantized connection)oftheKahler-Atiyah¨ algebra def ad (ΩK(𝑀), ⬦) is defined through 𝐷푚 (𝑇) =𝐷푚 ∘𝑇−𝑇∘𝐷푚, (232) 𝜔 def=∇𝜔+[𝐴̌ ,𝜔] ,∀𝜔∈Ω(𝑀) . ∀𝑇 ∈ Γ (𝑀, End (𝑆)) . D푚 푚 푚 −,⬦ K (236) Advances in High Energy Physics 25

̌ 훾 ̌ 훾 Since ∇푚] =0and 𝐴푚 ∈ΩK(𝑀),wehave[𝐴푚,ΩK(𝑀)]−,⬦ ⊂ whichinturnimplies(237)uponusingtheequation Ω훾 (𝑀) (Ω훾 (𝑀)) ⊂ Ω훾 (𝑀) 𝑃 ∘ ∘𝛾−1 =𝛾−1 ∘𝐷ad K and D푚 K K . Composing (235) with 휖훾 D푚 푚 , which follows upon composing −1 −1 𝛾 from both sides gives the following relation which will be the last equality in (235) with 𝛾 from the right and using the used below: 𝛾∘𝛾−1 = 𝐴 property idEndK(푆).Notethat 푚 can be combined −1 −1 ad into the object: D푚 ∘𝛾 =𝛾 ∘𝐷푚 . (237) 푚 1 훾 𝐴=𝑒 ⊗𝐴 ∈Ω (𝑀) ⊗ ∞ Ω (𝑀) , −1 푚 K C (푀,R) K (238) To arrive at (237), we noticed that D푚 ∘𝛾 (EndK(𝑆)) = 𝐷ad(Ω훾 (𝑀)) ⊂ Ω훾 (𝑀) 𝑃 ∘ ∘𝛾−1 = ∘𝛾−1 푚 K K implies 휖훾 D푚 D푚 , while D푚 can be combined into the map:

푚 푚 ̌ ad 1 D =𝑒 ⊗ D푚 =∇+𝑒 ⊗(𝐴푚) :ΩK (𝑀) 󳨀→ Ω K (𝑀) ⊗C∞(푀,R) ΩK (𝑀) , (239)

where The K-Algebra of Generalized Killing Forms. We define the ad K-algebra of generalized Killing forms to be the following K- (𝐴̌ ) (𝜔) ﬂ [𝐴̌ ,𝜔] (Ω훾 (𝑀), ⬦) 푚 푚 −,⬦ (240) subalgebra of K :

푚 def and we used 𝑒 ⊗∇푚 =∇. ̌ 훾 −1 ad K퐷 = K (D) ∩ΩK (𝑀) =𝛾 (K (𝐷 )) . (247) Flatness of the Fierz Isomorphism. For the remainder of this ̌ K ad paper, we will assume that 𝐷 is compatible with B in the usual The elements of 퐷 will be called generalized Killing forms. sense that B is 𝐷-flat: Relation (243) implies that the flat Fierz K-algebra 耠耠耠 dB (𝜉, 𝜉 )=B (𝐷𝜉, 𝜉 )+B (𝜉, 𝐷𝜉 )⇐⇒ ̌ def ̌ K (𝐷) = 𝐸(K (𝐷) ⊗C∞(푀,R) K (𝐷)) (248) 𝜕 B (𝜉, 𝜉耠)=B (𝐷 𝜉,耠 𝜉 )+B (𝜉, 𝐷 𝜉耠), 푚 푚 푚 (241) defined by the B-flat subspace K(𝐷) ⊂ Γ(𝑀, 𝑆) is a K ∀𝜉, 𝜉耠 ∈Γ(𝑀,) 𝑆 . subalgebra of the -algebra of generalized Killing forms: Ǩ (𝐷) ⊂ Ǩ . In this case, the K-linear subspace K(𝐷) ⊂ Γ(𝑀, 𝑆) is B-flat, 퐷 (249) so it defines a flat bipinor K-algebra K(𝐷)C ⊗ ∞(푀,R) K(𝐷) ⊂ ∼ ∗ In particular, we have Γ(𝑀, 𝑆 ⊗ 𝑆). Furthermore, the isomorphism 𝜌:𝑆󳨀→𝑆 𝐷∗ ∘𝜌=𝜌∘𝐷 𝐷∗ ̌ 耠 satisfies 푚 푚 (where 푚 is the dual connection) D푚𝐸휉,휉耠 =0, ∀𝜉,𝜉 ∈ K (𝐷) . (250) ∗ ∼ while the natural isomorphism 𝑞:𝑆⊗𝑆 󳨀→ End(𝑆) satisfies ad ∗ 𝐷푚 ∘𝑞 = 𝑞∘(𝐷푚 ⊗ id푆 + id푆 ⊗𝐷푚). It follows that the 8 Behavior under 𝜏B. Spin(𝑑)-invariance of B implies that B isomorphism 𝐸=𝑞∘(id푆 ⊗𝜌)satisfies 푆 is flat with respect to the connection ∇ : ad 𝐷푚 ∘𝐸=𝐸∘(𝐷푚 ⊗ id푆 + id푆 ⊗𝐷푚). (242) 耠 푆 耠 푆 耠 dB (𝜉, 𝜉 )=B (∇ 𝜉, 𝜉 )+B (𝜉, ∇ 𝜉 )⇐⇒ In particular, we have 耠 푆 耠 푆 耠 𝜕 B (𝜉, 𝜉 )=B (∇ 𝜉, 𝜉 )+B (𝜉, ∇ 𝜉 ), (251) ad 푚 푚 푚 𝐸 (K (𝐷) ⊗C∞(푀,R) K (𝐷)) ⊂ K (𝐷 ) , (243) ∀𝜉, 𝜉耠 ∈Γ(𝑀,) 𝑆 , wherewehaveintroducedthefollowingK-subalgebra of (Γ(𝑀, End(𝑆)), :∘) whichiseasilyseentoimplytheproperty:

푑 푡 ad def ad 푆 ad 푆 ad 푡 K (𝐷 ) = ⋂ K (𝐷푚 ) . (244) [(∇푚) (𝑇)] =(∇푚) (𝑇 ), 푚=1 (252) −1 ∀𝑇 ∈ Γ (𝑀, End (𝑆)) . On the other hand, 𝛾 satisfies (237). Together with (242), 𝐸̌ this implies that the Fierz isomorphism satisfies Together with assumption (241), identity (251) implies that ̌ ̌ 𝐴푚 are B-antisymmetric endomorphisms of 𝑆: D푚 ∘ 𝐸=𝐸∘(𝐷푚 ⊗ id푆 + id푆 ⊗𝐷푚). (245) 𝐴푡 =−𝐴 ⇐⇒ B (𝐴 𝜉,耠 𝜉 )=−B (𝜉, 𝐴 𝜉耠), Therefore, we find 푚 푚 푚 푚 (253) ̌ ̌ ̌ 耠耠 𝐸 耠 = 𝐸 耠 + 𝐸 耠 ,∀𝜉,𝜉∈Γ(𝑀,) 𝑆 . ∀𝜉, 𝜉 ∈Γ(𝑀,) 𝑆 . D푚 휉,휉 퐷푚휉,휉 휉,퐷푚휉 (246) 26 Advances in High Energy Physics

푡 In turn, these properties imply the relation [𝐴푚,𝑇]−,∘ = Using this identity in the definition (236) of D푚 gives [𝐴 ,𝑇푡] 𝐷ad 푚 −,∘,so 푚 satisfies ̌ ̌ D푚E휉,휉耠 =∇푚E휉,휉耠 푡 𝐷ad (𝑇푡)=(𝐷ad (𝑇)) , 푚 푚 푑 1 (262) 耠 푎1⋅⋅⋅푎푘 (254) − ∑ B (𝜉, [𝐴푚,𝛾푎 ⋅⋅⋅푎 ] 𝜉 )𝑒훾 . ad 푡 푡 ad 𝑘! 1 푘 −,∘ 푘=0 ∀𝑇 ∈ Γ (𝑀, End (𝑆)) ⇐⇒ 𝐷 푚 ∘ () = () ∘𝐷푚 . −1 ̌ Setting 𝑇=𝛾(𝜔)and applying 𝛾 to both sides give Consider now relation (246), written in terms of E휉,휉耠 :

耠 D푚 (𝜏B (𝜔))=𝜏B (D푚 (𝜔)), ̌ 耠 = ̌ 耠 + ̌ 耠 ,∀𝜉,𝜉∈Γ(𝑀,) 𝑆 . D푚E휉,휉 E퐷푚휉,휉 E휉,퐷푚휉 (263) (255) 훾 ∀𝜔 ∈ ΩK (𝑀) ⇐⇒ D푚 ∘𝜏B =𝜏B ∘ D푚. Substituting (262) in the left hand side, this becomes

K ∇ ̌ 耠 = ̌ 耠 + ̌ 耠 In particular, the -algebra of generalized Killing forms is 푚E휉,휉 E퐷푚휉,휉 E휉,퐷푚휉 invariant under 𝜏B: 푑 1 (264) ̌ ̌ 耠 푎1⋅⋅⋅푎푘 𝜏 (K ) = K , + ∑ B (𝜉, [𝐴푚,𝛾푎 ⋅⋅⋅푎 ] 𝜉 )𝑒 . B 퐷 퐷 (256) 𝑘! 1 푘 −,∘ 훾 푘=0 a property (by virtue of (206)) it shares with the flat Fierz K- ̌ Separating ranks, we conclude that (246) is equivalent to the algebra K(𝐷): following system of identities: 𝜏 (Ǩ (𝐷)) = Ǩ (𝐷) . B (257) ∇ ̌ (𝜉,耠 𝜉 )= ̌ (𝐷 𝜉,耠 𝜉 ) 푚E푎1⋅⋅⋅푎푘 E푎1⋅⋅⋅푎푘 푚 D Together with (205), identity (255) implies that -flatness of ̌ 耠 ̌ ̌ + E푎 ⋅⋅⋅푎 (𝜉,푚 𝐷 𝜉 ) 𝐸휉,휉耠 and D-flatness of 𝐸휉耠,휉 are equivalent statements, so that 1 푘 it suffices to require only one of the two. (265) + B (𝜉, [𝐴 ,𝛾 ] 𝜉耠), 푚 푎1⋅⋅⋅푎푘 −,∘ 5.8. Alternate Form of the Differential Constraints. Consider ∀𝑘=0⋅⋅⋅𝑑, the following local expansion, which results by applying (209) to [𝑇,휉,휉 𝐸 耠 ]−,∘,where𝑇∈Γ(𝑀,End(𝑆)): 耠 where 𝜉, 𝜉 ∈ Γ(𝑀, 𝑆) are arbitrary. In particular, we have

[𝑇, 𝐸 耠 ] 휉,휉 −,∘ ∇ ̌ (𝜉,耠 𝜉 )=B (𝜉, [𝐴 ,𝛾 ] 𝜉耠), 푚E푎1⋅⋅⋅푎푘 푚 푎1⋅⋅⋅푎푘 −,∘ 1 푑 1 (266) 푎푘⋅⋅⋅푎1 耠 = ∑ tr (𝛾 ∘[𝑇,𝐸휉,휉耠 ] )𝛾푎 ⋅⋅⋅푎 , (258) ∀𝑘=0⋅⋅⋅𝑑, ∀𝜉,𝜉 ∈ K (𝐷) , [푑/2] 𝑘! −,∘ 1 푘 2 푘=0

耠 whichagreeswith(B.12)(seeAppendixB.2). ∀𝑇 ∈ Γ (𝑀, End (𝑆)) ,∀𝜉,𝜉∈Γ(𝑀,) 𝑆 . 5.9. The K-Algebra of Constrained Generalized Killing Forms. An easy computation using cyclicity of tr and identity (208) 𝜒=1 gives As before, we consider the case of the CGK equations: 푆 푚 푎푘⋅⋅⋅푎1 푎푘⋅⋅⋅푎1 耠 𝐷𝜉 = 𝑄𝜉 =0, 𝐷=∇ +𝐴,𝐴= 𝑥 ⊗𝐴 . (𝛾 ∘[𝑇,𝐸 耠 ] )=−B ([𝑇, 𝛾 ] 𝜉, 𝜉 ), with d 푚 (267) tr 휉,휉 −,∘ −,∘ (259) Let K(𝐷, 𝑄) = K(𝐷) ∩ K(𝑄) denote the (finite- so that (258) becomes dimensional) K-linear subspace of Γ(𝑀, 𝑆) consisting of all K [𝑇,휉,휉 𝐸 耠 ] solutions to (267). We define the -algebra of constrained −,∘ generalized Killing (CGK) forms determined by 𝐷 and 𝑄 to K (Ω훾 (𝑀), ⬦) 1 푑 1 be the following -subalgebra of K : 푎푘⋅⋅⋅푎1 耠 =− ∑ B ([𝑇, 𝛾 ] 𝜉, 𝜉 )𝛾푎 ⋅⋅⋅푎 , (260) [푑/2] 𝑘! −,∘ 1 푘 def 2 푘=0 ̌ ̌ ̌ K퐷,푄 = K퐷 ∩ K푄. (268) 耠 ∀𝑇 ∈ Γ (𝑀, End (𝑆)) ,∀𝜉,𝜉∈Γ(𝑀,) 𝑆 . ̌ In general, K퐷,푄 is a nonunital K-algebra.Thediscussionof ̌ the previous subsections shows that the flat Fierz K-algebra Setting 𝑇=𝐴푚 =𝛾(𝐴푚) in (260) and applying the morphism −1 determined by the B-flat subspace K(𝐷, 𝑄) ⊂ Γ(𝑀, 𝑆) 𝛾 to both sides give ̌ ̌ def ̌ [𝐴̌ , 耠 ] K (𝐷,) 𝑄 = 𝐸(K (𝐷,) 𝑄 ⊗C∞(푀,R) K (𝐷,) 𝑄 ) (269) 푚 E휉,휉 −,⬦

푑 (261) isasubalgebraoftheK-algebra of CGK forms: 1 푎 ⋅⋅⋅푎 耠 푎 ⋅⋅⋅푎 =−∑ B ([𝐴 ,𝛾 푘 1 ] 𝜉, 𝜉 )𝑒 1 푘 . 𝑘! 푚 −,∘ 훾 ̌ ̌ 푘=0 K (𝐷,) 𝑄 ⊂ K퐷,푄. (270) Advances in High Energy Physics 27

This property of the Fierz isomorphism depends essentially the essential conditions imposed by having a fixed number on the assumption that B is 𝐷-flat (an assumption which is of unbroken supersymmetries in a flux compactification, satisfied in the application discussed in Section 6). formulated in the language of geometric algebra. When expanding the geometric product into generalized products Expression for a Basis of Solutions of the CGK Pinor Equations. using (35), the innocently looking equations (276) and (277) Let 𝑠=dimKK(𝐷, 𝑄) denote the K-dimension of the space take on a form which may seem rather formidable when 𝑠 is of solutions to the CGK equations. Choosing a basis (𝜉푖)푖=1⋅⋅⋅푠 sufficiently large (see Section 6 for an example). Of course, of such solutions, we set the language of geometric algebra allows one to study such systems starting directly from the synthetic expressions (276) ̌ def ̌ 1 ̌ 훾 𝐸푖푗 = 𝐸휉 ,휉 = E푖푗 ∈Ω (𝑀) , (271) and (277), which shows that the problem of characterizing 푖 푗 2[(푑+1)/2] K the solutions of such equations belongs most properly to where (cf. equation (218)) the intersection between Kahler-Cartan¨ theory [17] and the theory of noncommutative associative algebras—a point 푑 of view on flux compactifications, which, in our opinion, ̌ ̌(푘) E푖푗 = ∑E푖푗 , (272) could lead to a deeper understanding of various problems 푘=0 pertaining to that subject. with Truncated Model of the K-Algebra of CGK Forms in the (푘) def (푘) 1 (푘) Nonsimple Case. Recall that, in the nonsimple case, we can ̌ ̌ ̌ 푎1⋅⋅⋅푎푘 휖 < E푖푗 = E휉 ,휉 =푈 E푎 ⋅⋅⋅푎 (𝜉푖,𝜉푗)𝑒훾 , (Ω (𝑀), ⬦) (Ω (𝑀), X ) 푖 푗 𝑘! 1 푘 realize as the truncated algebra 휖 , where 𝜖∈{−1,+1}is the signature of 𝛾.Aninhomogeneous ̌(푘) (273) 𝜔∈Ω휖(𝑀) ∗𝜔̃ = 𝜖𝜔 E푎 ⋅⋅⋅푎 (𝜉푖,𝜉푗)=B (𝛾푎 ⋅⋅⋅푎 𝜉푖,𝜉푗) form satisfies andiscalledtwisted 1 푘 푘 1 self-dual if 𝜖=+1and twisted anti-self-dual if 𝜖=−1. 푘 =𝜖 B (𝜉 ,𝛾 𝜉 ) Such forms can be uniquely decomposed as 𝜔=2𝑃휖(𝜔<)= B 푖 푎1⋅⋅⋅푎푘 푗 < 𝜔< +𝜖∗𝜔̃ <,where𝜔< ∈Ω (𝑀) has rank smaller than [𝑑/2].In ̌ giving the following expression for the homogeneous form- particular, the forms 𝐸푖푗 discussed above can be decomposed valued bilinears: ̌ ̌ < ̌ < uniquely as 𝐸푖푗 = 𝐸푖푗 +𝜖∗̃𝐸푖푗 . Similarly, we have the unique < < < < (푘) 1 푘 푎 ⋅⋅⋅푎 ̌ ̌ ̌ ̌ ̌ ̌ ̌ 1 푘 decompositions 𝑄=𝑄 +𝜖∗̃𝑄 and 𝐴푚 = 𝐴푚 +𝜖∗̃𝐴푚,where E푖푗 =푈 𝜖BB (𝜉푖,𝛾푎 ⋅⋅⋅푎 𝜉푗) 𝑒훾 . (274) 𝑘! 1 푘 ̌ ad,< 𝜖=𝜖훾. Let us define a derivation 𝐷푚 of (Ω(𝑀), X휖) through Since 𝜉푖 ⊗𝜉푗 form a basis of the K-vector space ad,< def < K(𝐷, 𝑄)C ⊗ ∞(푀,R) K(𝐷, 𝑄), the inhomogeneous differential ̌ ̌ 𝐷푚 (𝜔) =∇푚𝜔+2[𝐴푚,𝜔] ̌ ̌ −,X휖 forms 𝐸푖푗 form a basis of the K-vector space K(𝐷, 𝑄). Inclusion (270) amounts to the following system of equations ̌ < < (279) ̌ =∇푚𝜔+([𝐴푚,𝜔] ) =(D푚𝜔) , for the inhomogeneous differential forms 𝐸푖푗 : −,⬦ < ̌ ̌ ̌ ̌ ̌ ∀𝜔 ∈ Ω (𝑀) , D푚𝐸푖푗 = 𝑄⬦𝐸푖푗 = 𝐸푖푗 ⬦𝜏B (𝑄) = 0, (275) ∀𝑖,𝑗=1⋅⋅⋅𝑠. where, as usual, the (anti-)commutator with respect to X휖 is defined as follows:

These can also be written as follows: < [𝜔, 𝜂]±,X =𝜔X휖𝜂±𝜂X휖𝜔, ∀𝜔, 𝜂∈Ω (𝑀) . (280) ̌ ̌ ̌ 휖 D푚𝐸푖푗 = 𝑄⬦𝐸푖푗 =0, ∀𝑖,𝑗=1⋅⋅⋅𝑠 (276) We have upon applying 𝜏B to the last equation in (275) and using the < 𝜏 (𝐸̌ )=𝜖𝐸̌ relation B 푖푗 B 푗푖 (cf. (205)). The inhomogeneous ([𝜔, 𝜂]±,⬦) =𝑃< ([𝜔, 𝜂]±,⬦)=2[𝜔<,𝜂<]−,X , ̌ 휖 (281) differential forms 𝐸푖푗 also satisfy the Fierz identities: ∀𝜔, 𝜂 ∈Ω (𝑀) ̌ ̌ ̌ 𝐸푖푗 ⬦ 𝐸푘푙 = B푘푗 𝐸푖푙, ∀𝑖,𝑗,𝑘,𝑙=1⋅⋅⋅𝑠, (277) since 2𝑃< is a morphism of algebras from (Ω(𝑀), ⬦) to ̌ ad,< where we defined the following constants: (Ω(𝑀), X휖). Noticing that [∇푚,𝑃<]−,∘ =0,wefindthat𝐷푚 satisfies def B = B (𝜉 ,𝜉 ). (278) 푖푗 푖 푗 ̌ ad,< 𝑃< ∘ D푚 = 𝐷푚 ∘𝑃<, (282) The algebrodifferential system consisting of (276) and (277) can be taken as the basis for extending the classical theory of which generalizes (279). Killing forms, a subject which is of mathematical interest in Using the commutation relation [∇푚,𝑃휖]−,∘ =0and the its own right. It provides a synthetic geometric description of mutually inverse isomorphisms of diagram (107), it is easy to 28 Advances in High Energy Physics see that the CGK pinor equations (276) are equivalent to the considered in Section 6, though in that case we prefer to truncated CGK pinor equations: take the squared norm B(𝜉, 𝜉) of 𝜉 to equal 2, for ease of comparison with the results of [3]. This produces an extra ̌ ad,< ̌ < ̌ < ̌ < 𝐷푚 𝐸푖푗 = 𝑄 X휖𝐸푖푗 =0, ∀𝑖,𝑗=1⋅⋅⋅𝑠. (283) factoroftwointherighthandsideof(287). 𝑠=2 Indeed, applying 2𝑃< to (276) gives (283) while applying The Case of Two CGK Pinors. When , the flat Fierz ̌ ̌ ̌ 𝑃휖 to (283) gives (276). The first of the truncated CGK K-algebra K(𝐷, 𝑄) is generated by 𝐸12 and 𝐸21 with the ̌ < relations pinor equations can also be written in the form ∇푚𝐸푖푗 = < −2[𝐴̌ , 𝐸̌ ] 2𝑃 ̌ ⬦2 ̌ ⬦2 푚 푖푗 −,X휖 . On the other hand, applying < to (277) 𝐸12 = 𝐸21 =0, (288) gives the truncated geometric Fierz identities: ̌ ̌ ̌ ̌ ̌ ̌ ̌ ̌ 𝐸12 ⬦ 𝐸21 ⬦ 𝐸12 − 𝐸12 = 𝐸21 ⬦ 𝐸12 ⬦ 𝐸21 − 𝐸21 =0. (289) < < 1 < 𝐸̌ X 𝐸̌ = B 𝐸̌ , ∀𝑖,𝑗,𝑘,𝑙=1⋅⋅⋅𝑠. 푖푗 휖 푘푙 2 푘푗 푖푙 (284) ̌ ̌ ̌ ̌ ̌ ̌ ̌ ̌ We have 𝐸11 = 𝐸12 ⬦ 𝐸21, 𝐸22 = 𝐸21 ⬦ 𝐸12 and 𝐶=𝐸11 + ̌ ̌ ̌ ̌ ̌ ̌ ̌ 𝐸22 = 𝐸12 ⬦ 𝐸21 + 𝐸21 ⬦ 𝐸12.Since𝐸21 =𝜏(𝐸12),wecan 5.10. A Particular Case: When K = R and B Is a Scalar ̌ in fact generate the entire algebra from 𝐸12 up to applying Product. Consider the particular case when K = R and 𝜏.Usingthefactthatthefirstandsecondequationin(288) B is symmetric (thus 𝜎B =+1) and positive-definite with as well as the first and second equation in (289) are related 𝜖B =+1(this happens, e.g., in the application considered through reversion, we find that the entire system is equivalent in Section 6). In this case, we can choose 𝜉1,...,𝜉푠 such that to the following Fierz relations: B푖푗 =𝛿푖푗 . Then relations (277) show that the K-algebra ̌ ⬦2 K(𝐷, 𝑄) is isomorphic with the algebra Mat(𝑠, R) of square 𝐸̌ =0, def 12 real matrices of dimension 𝑠,theunitbeinggivenby𝐶̌ = (290) 푠 ̌ ̌ ̌ ̌ ̌ ̌ ∑푖=1 𝐸푖푖.AnisomorphismtoMat(𝑠, R) is given by 𝐸푖푗 →𝑒푖푗 , 𝐸12 ⬦𝜏(𝐸12)⬦𝐸12 = 𝐸12, where 𝑒푖푗 ∈ Mat(𝑠, R) is the matrix whose only nonvanishing entry equals 1 and is found on the 𝑖th row and 𝑗th column: together with the following 𝑄-and𝐷-constraints: ̌ ̌ ̌ ̌ (𝑒 ) =𝛿 𝛿 . 𝑄⬦𝐸12 = 𝐸12 ⬦𝜏(𝑄) = 0, 푖푗 푘푙 푖푘 푗푙 (285) (291) ̌ ⬦2 ̌ ̌ ̌ ̌ ̌ ̌ ̌ ⬦2 ̌ ̌ ⬦2 𝐸̌ =0. We have 𝐶 = 𝐶, 𝐶⬦𝐸푖푗 = 𝐸푖푗 ⬦ 𝐶=𝐸푖푗 , 𝐸푖푖 = 𝐸푖푖, 𝐸푖푗 =0 D푚 12 𝑖 =𝑗̸ 𝐸̌ ⬦ 𝐸̌ = 𝐸̌ K Ǩ (𝐷, 𝑄) for .Since 푖푗 푗푖 푖푖,the -algebra is Remark 12. The algebra Mat(2, R)≈Cl(2, 0) is also generated ̌ generated by the elements (𝐸푖푗 )푖=푗̸ ,withtherelations: by two real Pauli matrices, for example, by

𝐸̌ ⬦ 𝐸̌ =0, ∀𝑖=𝑗,𝑗̸ =𝑘,𝑘̸ =𝑙,̸ 01 푖푗 푘푙 𝜎 =[ ]=𝑒 +𝑒 , 1 10 12 21 𝐸̌ ⬦ 𝐸̌ = 𝐸̌ , ∀𝑖, 𝑗, 𝑘 , 푖푗 푗푘 푖푘 distinct (292) (286) ̌ ̌ ̌ ̌ 10 𝐸푖푗 ⬦ 𝐸푗푖 ⬦ 𝐸푖푘 = 𝐸푖푘,∀𝑖=𝑗,𝑖̸ =𝑘,̸ 𝜎 =[ ]=𝑒 −𝑒 , 3 0−1 11 22 𝐸̌ ⬦ 𝐸̌ ⬦ 𝐸̌ = 𝐸̌ ,∀𝑖=𝑗,𝑗̸ =𝑘.̸ 푖푗 푗푘 푘푗 푖푗 2 2 subject to the relations 𝜎1 =𝜎3 =1and 𝜎1𝜎3 +𝜎3𝜎1 =0. ̌ This system of generators and relations can be reduced Therefore, the flat Fierz K-algebra K(𝐷, 𝑄) is also generated ̌ further (see the example below) upon using the identity 𝐸푗푖 = by the inhomogeneous differential forms: 𝜏(𝐸̌ ) 𝜎 =+1 푖푗 , where we used the fact that B and we noticed ̌ ̌ ̌ Σ1 = 𝐸12 + 𝐸21, that 𝜏B =𝜏(since 𝜖B =+1). (293) Σ̌ = 𝐸̌ − 𝐸̌ , The Case of One CGK Pinor.When𝑠=1(a single unit norm 3 11 22 solution 𝜉 of the CGK pinor equations, which is unique up ̌ subject to the relations to sign), the flat Fierz K-algebra K(𝐷, 𝑄) has dimension one, R R ⬦2 ⬦2 being isomorphic with as a unital -algebra. It is generated Σ̌ − Σ̌ =0, ̌ ̌ 1 3 by the single basis element 𝐸=𝐸휉,휉 (which is the unit of Ǩ (𝐷, 𝑄) ̌ ̌ ̌ ̌ ), with the relation Σ1 ⬦ Σ3 + Σ3 ⬦ Σ1 =0, 𝐸⬦̌ 𝐸=̌ 𝐸.̌ (287) (294) ̌⬦3 ̌ Σ1 = Σ1, The condition 𝜏(𝐸)̌ = 𝐸̌ implies that certain rank components ̌ ̌⬦3 ̌ of 𝐸 vanish identically. This situation occurs in the example Σ3 = Σ3. Advances in High Energy Physics 29

̌ ̌ ̌ ̌ ̌ ̌ ̌ Noticing that Σ1 ⬦ Σ3 = 𝐸21 − 𝐸12,wehave𝐶=𝐸11 + 𝐸22 = supersymmetry generator 𝜂̃ (which is a Majorana spinor field ̌⬦2 ̌⬦2 1/2 𝑆̃ Σ1 = Σ3 and of spin )isasmoothsectionofthepinbundle ,whichis arank32 real vector bundle defined on 𝑀̃. 1 𝐸̌ = (𝐶+̌ Σ̌ ), As in [3], we consider compactification down to an AdS3 11 3 2 2 space of cosmological constant Λ=−8𝜅,where𝜅 is a positive real parameter; this includes the Minkowski case as ̌ 1 ̌ ̌ ̃ 𝐸22 = (𝐶−Σ3), the limit 𝜅→0.Thus𝑀=𝑁×𝑀,where𝑁 is an oriented 2 R3 (295) 3-manifold diffeomorphic with and carrying the AdS3 1 metric while 𝑀 is an oriented Riemannian eight-manifold 𝐸̌ = (Σ̌ − Σ̌ ⬦ Σ̌ ), ̃ 12 2 1 1 3 whose metric we denote by 𝑔.Themetricon𝑀 is a warped product: ̌ 1 ̌ ̌ ̌ 𝐸21 = (Σ1 + Σ1 ⬦ Σ3). 2 2Δ 2 2 2 푚 푛 2 d̃𝑠11 =𝑒 d𝑠11, where d𝑠11 = d𝑠3 +𝑔푚푛d𝑥 d𝑥 . (299)

Here, the warp factor Δ is a smooth function defined on 𝑀 6. Example: Flux Compactifications of 2 while d𝑠3 is the squared length element on 𝑁.Forthefield Eleven-Dimensional Supergravity on Eight 𝐺̃ Manifolds strength ,weusetheansatz: 𝐺=𝑒̃ 3Δ𝐺 𝐺= ∧𝑓+𝐹, In this section, we illustrate the methods developed in the with vol3 (300) present paper by applying them to the case of N =1warped 𝑓=𝑓 𝑒푚 ∈Ω1(𝑀) 𝐹 = (1/4!)𝐹 𝑒푚푛푝푞 ∈Ω4(𝑀) compactifications of eleven-dimensional supergravity on where 푚 , 푚푛푝푞 , 𝑁 eight manifolds down to an AdS3 space, a situation which was and vol3 is the volume form of .SmallLatinindicesfrom studiedthroughdirectmethodsin[3,4].Aftersomebasic themiddleofthealphabetrunfrom1 to 8 and correspond to preparations in Section 6.1, Section 6.2 gives our translation a choice of frame on 𝑀.For𝜂̃,weusetheansatz: of the generalized Killing pinor equations into a system Δ/2 of algebraic and differential constraints on inhomogeneous 𝜂=𝑒̃ 𝜂 with 𝜂=𝜓⊗𝜉, (301) forms defined on the compactification space and shows how our approach allows one to recover the results of [3]. where 𝜉 isaMajoranaspinorofspin1/2 (a.k.a. a real pinor) on the internal space 𝑀 and 𝜓 isaMajoranaspinorontheAdS3 𝑁 𝜉 6.1. Preparations. Consider supergravity on an 11-manifold space .Mathematically, is a section of the pinor bundle of 𝑀, which is a real vector bundle of rank 16 defined on 𝑀̃ with Lorentzian metric 𝑔̃ (of “mostly plus” signature). 𝑀, carrying a fiberwise representation of the Clifford algebra Besides the metric, the action of the theory contains the (8, 0) 𝑝−𝑞≡ 0 𝑝=8 𝑞=0 𝐺∈̃ Cl .Since 8 for and , this corresponds three-form potential with four-form field strength tothesimplenormalcaseofSection4.Inparticular,the 4 ̃ ̃ ∗ Ω (𝑀) and the gravitino Ψ푀, which is a real pinor of spin 𝛾 : (∧𝑇 𝑀, ⬦) →( (𝑆), ∘) ̃ corresponding morphism End of 3/2. We assume that (𝑀, 𝑔)̃ is spinnable. For supersymmetric bundlesofalgebrasisanisomorphism;thatis,itisbijective 푚 푚 bosonic backgrounds, both the gravitino VEV and its super- on the fibers. We set 𝛾 =𝛾(𝑒 ) forsomelocalframeof𝑀. symmetry variation must vanish, which requires that there In dimension eight with Euclidean signature, there exists an exists at least one solution 𝜂̃ to the equation: admissible [30] (and thus Spin(8)-invariant) bilinear pairing B on the pin bundle 𝑆 with 𝜎B =+1and 𝜖B =+1which is ̃ def ̃ 𝛿휂̃Ψ푀 = D푀𝜂=0,̃ (296) a scalar product (i.e., it is fiberwise symmetric and positive- definite). 0 10 D̃ where uppercase indices run from to and 푀 is the Assuming that 𝜓 is a Killing pinor on the AdS3 space, the supercovariant connection: supersymmetry condition (296) decomposes into a system consisting of the following constraints for 𝜉: ̃ def ̃푆 D푀 =∇푀 𝐷 𝜉=0, (297) 푚 1 푁푃푄푅 푁푃푄 (302) − (𝐺̃ 𝛾̃ −8𝐺̃ 𝛾̃ ). 𝑄𝜉 = 0, 288 푁푃푄푅 푀 푀푁푃푄 푀 Here, 𝛾̃ are the gamma matrices of Cl(10, 1) in that 32- where 𝐷푚 is a linear connection on 𝑆 and 𝑄∈Γ(𝑀,End(𝑆)) dimensional real (Majorana) irreducible representation for is a globally defined endomorphism of the vector bundle 𝑆. (12) def 1 11 As in [3, 4], we do not require that 𝜉 has definite chirality (i.e., which 𝛾̃ = 𝛾̃ ⋅⋅⋅𝛾̃ =+1and 𝜉 need not satisfy the Weyl condition). The space of solutions ̃ 1 of (302) is a finite-dimensional R-linear subspace K(𝐷, 𝑄) ∇푆 =𝜕 + Ω̃ 𝛾̃푁푃 푀 푀 4 푀푁푃 (298) of the space Γ(𝑀, 𝑆) of smooth sections of 𝑆.Ofcourse,this subspace is trivial for generic metrics 𝑔 and fluxes 𝐹 and 𝑓 on ̃ istheconnectiononthepinbundle𝑆 induced by the 𝑀, since the generic compactification of the type we consider Levi-Civita connection of (𝑀,̃ 𝑔̃). The eleven-dimensional breaks all supersymmetry. The interesting problem is to find 30 Advances in High Energy Physics

(9) (9) thosemetricsandfluxeson𝑀 for which some fixed amount We have 𝜉=𝜉+ +𝜉− and 𝛾 𝜉=𝜉+ −𝜉− with 𝛾 𝜉± =±𝜉±. of supersymmetry is preserved in three dimensions, that is, The operators P± are complementary B-orthoprojectors: for which the space K(𝐷, 𝑄) has some given nonvanishing 2 dimension, which we denote by 𝑠.Thecase𝑠=1(which P± = P±, corresponds to N =1supersymmetry in three dimensions) was studied in [3, 4] and will be reconsidered below. Direct P±P∓ =0, computation using the compactification ansatz gives the (307) following expressions for 𝐷 and 𝑄 (which are equivalent to P+ + P− =1, thosederivedinreference[3];seetheremarkbelowforease 푡 of comparison): (P±) = P±.

푆 In particular, one has 𝐷푚 =∇푚 +𝐴푚, 1 1 1 (303) 𝜀+ = 𝜉, 𝐴 = 𝑓 𝛾 푝 ∘𝛾(9) + 𝐹 𝛾푝푞푟 +𝜅𝛾 ∘𝛾(9), √ 푚 4 푝 푚 24 푚푝푞푟 푚 2 (308) 1 1 1 − 1 (9) 𝑄= 𝛾푚𝜕 Δ− 𝐹 𝛾푚푝푞푟 − 𝑓 𝛾푝 ∘𝛾(9) 𝜀 = 𝛾 𝜉, 2 푚 288 푚푝푞푟 6 푝 √2 (304) + − (9) + −𝜅𝛾(9), so 𝜀 and 𝜀 =𝛾 𝜀 are not independent. For reader’s convenience, we note the identities:

(9) 1 8 푎 ⋅⋅⋅푎 푎 ⋅⋅⋅푎 where 𝛾 =𝛾 ∘⋅⋅⋅∘𝛾 .Noticethatthelasttermsin(303) P 𝛾 1 푘 P =0󳨐⇒B (𝜉 ,𝛾 1 푘 𝜉 )=0, 휖1 휖2 휖1 휖2 and (304) depend on the cosmological constant of the AdS3 space and that they vanish in the Minkowski limit 𝜅→0. 푘+1 when 𝜖1𝜖2 = (−1) , Given some desired amount of supersymmetry which we want to be preserved in three dimensions (i.e., given some 푎1⋅⋅⋅푎푘 푎1⋅⋅⋅푎푘 P휖 𝛾 P휖 = P휖 𝛾 󳨐⇒ desired dimensionality 𝑠 of the space of solutions to (302)), 1 2 1 (309) the general aim is to reformulate (302) as equations on differ- 푎 ⋅⋅⋅푎 푎 ⋅⋅⋅푎 (푘) B (𝜉 ,𝛾 1 푘 𝜉 )=B (𝜉, P 𝛾 1 푘 𝜉) , ̌ 耠 푎1⋅⋅⋅푎푚 휖 휖 휖 耠 =(1/𝑘!)B(𝜉, 𝛾 𝜉 )𝑒 1 2 1 ential forms E휉,휉 푚1⋅⋅⋅푚푘 constructed 耠 as bilinear combinations of pinors 𝜉, 𝜉 which satisfy (302). 푘 when 𝜖1𝜖2 = (−1) , The pinor bilinears will be constrained by Fierz identities. The ̌(푘) translation to equations on the differential forms E휉,휉耠 can be def def where 𝜖1,𝜖2 ∈{−1,+1}and P±1 = P±, 𝜉±1 =𝜉±.Since achieved directly by starting from the following equivalent 耠 𝜖B =+1, (184) implies reformulation of the algebraic constraints 𝑄𝜉 = 𝑄𝜉 =0 푎 ⋅⋅⋅푎 푡 푘(푘−1)/2 푎 ⋅⋅⋅푎 (𝛾 1 푘 ) = (−1) 𝛾 1 푘 , (310) B (𝜉, (𝑄푡 ∘𝛾 ±𝛾 ∘𝑄)𝜉耠)=0 푚1⋅⋅⋅푚푘 푚1⋅⋅⋅푚푘 (305) 푎 ⋅⋅⋅푎 which shows that 𝛾 1 푘 is B-symmetric for 𝑘 = 0,1,4,5,8 𝐷 𝜉=𝐷 𝜉耠 =0 and B-antisymmetric for 𝑘 = 2, 3, 6, 7.Inparticular,wehave and treating the constraints 푚 푚 through the (9) method outlined in [3]. The theoretical basis of that approach that 𝛾 is B-symmetric. is explained in detail in Appendix B, where we also show how that method is equivalent to the more general approach 6.2. The Case of N =1Supersymmetry in 3 Dimensions. which we have developed in the present paper. In the next Let us consider the CGK pinor equations (302) on the subsection,wewillillustrateourapproachinthesimplestcase Riemannian 8-manifold (𝑀, 𝑔), assuming that the space of 𝑠=1 N =1 ( supersymmetry in three dimensions, see [3]), solutions has dimension 𝑠=1over R.Since𝑝−𝑞≡8 0,weare so we will require that (302) admits one nontrivial solution 𝜉. in the normal simple case. This case is characterized by two admissible pairings B± on 𝑆,whichhavethepropertiesgiven Remark 13. For easy comparison with [3], we note that loc. 𝜖 =±1 𝜎 =+1 in [31]; that is, B± and B± . Since any choice cit. uses a redundant parameterization of the degrees of of admissible pairing leads to the same result, we choose to 𝜉 freedom described by , which is given by the following work with B = B+ for convenience; this satisfies 𝜖B =+1 𝑆 sections of : and 𝜎B =+1. We can assume that B is a scalar product on 𝑆 and we denote the corresponding norm by ‖‖. ± def 1 𝜀 = (𝜉 ±𝜉 ) , As in Section 5.9 (see equation (271)), consider the inho- √2 + − (306) mogeneous differential form:

(9) ± ∓ 8 which satisfy 𝛾 𝜀 =𝜀 .Here,𝜉± = P±𝜉 are the positive and def 1 (푘) 𝐸̌ = ∑ ̌ ∈Ω(𝑀) def (9) 16 E (311) negative chirality components of 𝜉,withP± = (1/2)(1±𝛾 ). 푘=0 Advances in High Energy Physics 31 defined by a nontrivial solution 𝜉 of (302). Equation (274) would lead to the homogeneous form-valued bilinears: gives ̌(0) ̌(8) E− = B− (𝜉,) 𝜉 =𝜆=B (𝜉, 𝛾 (]) 𝜉) = E ⬦ ], (푘) 1 푚 ⋅⋅⋅푚 푘 ̌ = B (𝜉, 𝛾 𝜉) 𝑒 1 푘 ∈Ω (𝑀) , (3) 1 E 푚 ⋅⋅⋅푚 ̌ 푎1⋅⋅⋅푎3 𝑘! 1 푘 E =− B− (𝜉,푎 𝛾 ⋅⋅⋅푎 𝜉) 𝑒 (312) − 3! 1 3 ∀𝑎 ,...,𝑎 =1⋅⋅⋅8, ∀𝑘=0⋅⋅⋅8. 1 (5) 1 푘 푎1⋅⋅⋅푎3 ̌ =− B (𝜉,푎 𝛾 ⋅⋅⋅푎 ∘𝛾(]) 𝜉) 𝑒 =−E ⬦ ], 3! 1 3 Using the properties of B and relations (178) and (184), one (4) 1 ̌ 푎1⋅⋅⋅푎4 (2) (3) (6) (7) E− = B− (𝜉,푎 𝛾 ⋅⋅⋅푎 𝜉) 𝑒 easily checks that Ě = Ě = Ě = Ě =0while the 4! 1 4 nonvanishing bilinears are 1 (4) 푎1⋅⋅⋅푎4 ̌ = B (𝜉,푎 𝛾 ⋅⋅⋅푎 ∘𝛾(]) 𝜉) 𝑒 = E ⬦ ], 4! 1 4 (317) (0) ̌ = B (𝜉,) 𝜉 , (7) 1 E ̌ 푎1⋅⋅⋅푎7 E =− B− (𝜉,푎 𝛾 ⋅⋅⋅푎 𝜉) 𝑒 − 7! 1 7 ̌(1) 푎 E = B (𝜉,푎 𝛾 𝜉) 𝑒 , 1 (1) 푎1⋅⋅⋅푎7 ̌ =− B (𝜉,푎 𝛾 ⋅⋅⋅푎 ∘𝛾(]) 𝜉) 𝑒 =−E ⬦ ], 7! 1 7 (4) 1 푎 ⋅⋅⋅푎 ̌ = B (𝜉, 𝛾 𝜉) 𝑒 1 4 , E 푎1⋅⋅⋅푎4 (8) 1 4! (313) ̌ 푎1⋅⋅⋅푎8 E = B− (𝜉,푎 𝛾 ⋅⋅⋅푎 𝜉) 𝑒 =𝛽] − 8! 1 8 (5) 1 푎 ⋅⋅⋅푎 ̌ = B (𝜉, 𝛾 𝜉) 𝑒 1 5 , 1 (0) E 푎1⋅⋅⋅푎5 푎1⋅⋅⋅푎8 ̌ 5! = B (𝜉,푎 𝛾 ⋅⋅⋅푎 ∘𝛾(]) 𝜉) 𝑒 = E ⬦ ]. 8! 1 8 (8) 1 ̌ 푎1⋅⋅⋅푎8 E = B (𝜉,푎 𝛾 ⋅⋅⋅푎 𝜉) 𝑒 = B (𝜉,(9) 𝛾 𝜉) ], Thus, one could also write the generator: 8! 1 8 𝑁 (푘) 𝐸̌ = ∑ ̌ ] − 푑 E− (318) where isthevolumeform.Thustheinhomogeneousform 2 푘 which generates the Fierz algebra expands as which would lead to the same result up to twisted Hodge 1 (0) (1) (4) (5) (8) duality—since the form-valued bilinears constructed using 𝐸=̌ [ ̌ + ̌ + ̌ + ̌ + ̌ ] . B 16 E E E E E (314) − are connected through Hodge duality with those constructed using B+.

Note that this case also admits nonvanishing 3- and 7-form Following the notations and conventions of [3], we define bilinears, which are proportional to the Hodge duals of the def 1 (0) 5- and 1-forms, respectively (but they do not appear in the 𝑎 = Ě , generator). These other form-valued bilinears can also be 2 expressed as in (274), but upon using the other admissible def 1 (1) B 𝐾 = ̌ , pairing −, which is in fact not necessary for this analysis. 2E Remark 14. If one wanted to write the form-valued pinor def 1 (4) 𝑌 = Ě , (319) bilinears in terms of the other admissible pairing B− = B+ ∘ 2 ( ⊗𝛾(])) id푆 ,thenonewouldbeinterestedinconstructing 1 (5) 𝑍 def= ̌ , 2E (푘) def 1 푘 ̌ 푎1⋅⋅⋅푎푘 E = (𝜖B ) B− (𝜉,푎 𝛾 ⋅⋅⋅푎 𝜉) 𝑒 def 1 (8) − 𝑘! − 1 푘 𝑊 = ̌ =𝑏], 2E 1 푘 푎 ⋅⋅⋅푎 1 푘 (315) = (−1) B (𝜉,푎 𝛾 ⋅⋅⋅푎 ∘𝛾(]) 𝜉) 𝑒 2𝑎 𝜉 𝑏 𝑘! 1 푘 where is the squared norm of and asmoothfunction on 𝑀. With these notations, (314) becomes 푘 ∈Ω (𝑀) ,∀𝑎,...,𝑎 =1⋅⋅⋅8, ∀𝑘=0⋅⋅⋅8, 2 1 푘 𝐸=̌ [𝑎 + 𝐾+𝑌+𝑍+𝑏]]. 16 (320) which, upon using 𝜖B =−1and 𝜎B =+1and the well- − − Note that we use the five-form 𝑍 instead of its sign-reversed known formula, def Hodge dual 𝜙 =−∗𝑍,whichwasusedin[3].Weprefer def (−1)푘(푘−1)/2 𝜉=𝜉 ⊕𝜉 𝜉 𝜖± = 푎푘+1⋅⋅⋅푎푑 (푑+1) to work with + − rather than with ± and 𝛾푎 ⋅⋅⋅푎 = 𝜖푎 ⋅⋅⋅푎 𝛾푎 ⋅⋅⋅푎 ∘𝛾 (316) √ 1 푘 (𝑑−𝑘)! 1 푘 푘+1 푑 (1/ 2)(𝜉+ ±𝜉−), which were used in loc. cit. All in all, we 32 Advances in High Energy Physics have the following correspondence with the notations and where 𝜏 is the reversion defined in (41). During our calcula- conventions of [3]: tions we will use the following identities which hold in any 𝜔∈ 1 (9) dimension and signature for any homogeneous forms 𝜉 = P 𝜉, P = (1 ± 𝛾 ), Ω푟(𝑀) 𝜂∈Ω푠(𝑀) ± ± where ± 2 and 1 𝜀+ = 𝜉, 𝜔∧∗𝜂=(−1)푟(푠−1) ∗𝜄 𝜂, 𝑟≤𝑠, √ 휏(휔) when 2 (328) 𝜄 (∗𝜂) = (−1)푟푠 ∗(𝜏(𝜔) ∧𝜂), 𝑟+𝑠≤𝑑 1 휔 when 𝜀− = 𝛾(9)𝜉, √2 and of the following: 푡 푡 1 푡 𝐾푚 =𝜉 𝛾푚𝜉− =𝜉P+𝛾푚𝜉= 𝜉 𝛾푚𝜉, + 2 [(푚+1)/2] 푚 (−1) [𝜋 (𝜔)]△푚 [∗𝜏 (𝜔)] 푡 푡 −(∗𝑍) = 𝜙 =𝜉 𝛾 𝜉 =𝜉P 𝛾 𝜉 (321) [(푚耠+1)/2] 푑−1 푚푛푝 푚푛푝 + 푚푛푝 − + 푚푛푝 = (−1) [∗𝜏 (𝜔)] △푚耠 [𝜋 (𝜂)] (329) 1 푡 (9) [(푚耠耠+1)/2] 푚耠耠 = 𝜉 𝛾 𝛾 𝜉, = (−1) ∗𝜏[𝜋 (𝜔) △ 耠耠 𝜂] , 2 푚푛푝 푚 耠耠耠耠耠耠 ±푡 ± for 𝜔−𝑚=̃ 𝜂−𝑚̃ =𝑚 , where 𝑚, 𝑚 ,𝑚 >0. 𝑌푚푛푝푟 =𝜀 𝛾푚푛푝푟𝜀

1 푡 푡 We also remind the reader that one can uniquely decompose = (𝜉 𝛾 𝜉 +𝜉 𝛾 𝜉 ) 푘 2 + 푚푛푝푟 + − 푚푛푝푟 − any 𝑘-form 𝜔∈Ω(𝑀) into parallel and orthogonal parts 1 with respect to any fixed 1-form 𝜃∈Ω(𝑀) such that 𝜔= 1 푡 푘−1 푘 = 𝜉 𝛾 𝜉. 𝜔⊥ +𝜔‖,where𝜔‖ =𝜃∧𝜔⊤ with 𝜔⊤ ∈Ω (𝑀), 𝜔⊥ ∈Ω(𝑀), 2 푚푛푝푟 and 𝜃△1𝜔⊤ =𝜃△1𝜔⊥ =0(see Section 3.5). Choosing 𝜃=𝐾, The dequantizations of 𝐴푚 and 𝑄 are given by one has 𝑌=𝑌⊥ +𝑌‖ and 𝑍=𝑍⊥ + 𝑍‖.Notethat] = ]‖ and def −1 1 1 ̌ 𝐾=𝐾‖. 𝐴푚 =𝛾 (𝐴푚)= 𝜄(푒 ) 𝐹+ ((𝑒푚)♯ ∧𝑓)⬦] 4 푚 ♯ 4 Given the decomposition of 𝑌,itsHodgedualmusttake the form ∗𝑌 = ∗(𝑌⊥)+∗(𝑌‖)=𝛼1𝑌‖ +𝛼2𝑌⊥ for some +𝜅(𝑒푚) ⬦ ], ∞ ♯ (322) 𝛼1,𝛼2 ∈𝐶 (𝑀, R), since the Hodge dual of any component parallel to 𝜃 is orthogonal to 𝜃 while the Hodge dual of any def −1 1 1 1 𝜃 𝜃 𝑄̌ =𝛾 (𝑄) = dΔ− 𝑓⬦] − 𝐹−𝜅]. component orthogonal to is parallel to andsincethereare 2 6 12 no other four-form-valued pinor bilinears that could appear As shown in Section 5, the CGK pinor equations imply the in the right hand side. In the current example (see Table 1), following conditions for 𝐸̌: the volume form ] is twisted central (i.e., we have ] ⬦𝜔 = 𝜋(𝜔) ⬦ ] for any inhomogeneous differential form 𝜔)and 𝑄⬦̌ 𝐸=0,̌ (323) satisfies ]⬦] =+1. The latter property amounts to ∗∗𝑌 =, 𝑌 which leads to 𝛼2 =1/𝛼1 and implies ∇ 𝐸=−[̌ 𝐴̌ , 𝐸]̌ . 푚 푚 −,⬦ (324) 1 In turn, relations (324) imply ∗𝑌 = 𝛼 𝑌 + 𝑌 , 𝛼 ∈𝐶∞ (𝑀, R) . 1 ‖ 𝛼 ⊥ with 1 (330) 𝐸=−(𝑒̌ ) ∧[𝐴̌ , 𝐸]̌ . 1 d 푚 ♯ 푚 −,⬦ (325) In this case, (277) amounts to only one quadratic relation for Expanding (323) into rank components gives the follow- the inhomogeneous form 𝐸̌: ing conditions, which are equivalent to the “useful relations” discussed in Appendix C of [3]: 𝐸⬦̌ 𝐸=2𝑎̌ 𝐸,̌ (326) which encodes the relevant Fierz identities between the form 1 − 𝜄 𝑌+𝜄 𝐾 − 2𝜅𝑏 =0, bilinears constructed from 𝜉. 6 퐹 dΔ We remind the reader of the following relations (see the 1 𝑏 previous sections): − 𝜄 𝑍− 𝑓+𝑎 Δ=0, 6 퐹 3 d ∗𝜔 = 𝜏 (𝜔) ⬦ ] =𝜄휔], 1 1 dΔ∧𝐾− 𝐹△3𝑌+ 𝜄푓 ∗ 𝑍=0, ∗∗𝜔=𝜋(𝜔) , (327) 6 3 1 1 1 ∀𝜔 ∈ Ω (𝑀) , − 𝜄 ∗𝑌+2𝜅∗𝑍+ 𝜄 𝐹+𝜄 𝑌− 𝐹△ 𝑍=0, 3 푓 6 퐾 dΔ 6 3 Advances in High Energy Physics 33

1 1 𝑏 𝑎 1 𝐹△ 𝑌−2𝜅∗𝑌+ 𝑓∧∗𝑍− ∗𝐹+𝜄 𝑍− 𝐹 ∇ 𝑌=2𝜅(𝑒 ) ∧∗𝑍+ ((𝑒 ) ∧𝑓)△ ∗𝑌 6 2 3 6 dΔ 6 푚 푚 ♯ 2 푚 ♯ 1 =0, 1 1 + 𝐾∧(𝜄(푒 ) 𝐹) + (𝜄(푒 ) 𝐹)2 △ 𝑍, 2 푚 ♯ 2 푚 ♯ 1 1 1 − 𝐹∧𝐾+dΔ∧𝑌+ 𝐹△2𝑍− 𝑓∧∗𝑌=0, 1 6 6 3 ∇ 𝑍=−2𝜅(𝑒 ) ∧∗𝑌+ ∗((𝑒 ) ∧ 𝐾∧𝑓) 푚 푚 ♯ 2 푚 ♯ 1 1 dΔ∧𝑍+ 𝜄푓 ∗ 𝐾+ 𝐹△1𝑌=0, 1 1 3 6 + (𝑒푚) ∧𝑓∧∗𝑍− (𝜄(푒 ) 𝐹) △1𝑌 2 ♯ 2 푚 ♯ 𝑎 1 − ∗𝑓+2𝜅∗𝐾+ 𝐹△1𝑍+𝑏∗(dΔ) =0, 𝑏 3 6 − (𝑒 ) ∧∗𝐹, 2 푚 ♯ 1 1 2𝑎𝜅] − 𝑓∧∗𝐾+ 𝑌∧𝐹=0. (333) 3 6 (331) which in turn implies the following constraints representing the rank components of (325): 1 Remark 15. It will be useful to Hodge dualize relations (331), d𝑏=2𝜅𝐾+ 𝜄∗푍𝐹, which gives 2 1 1 d𝐾=− 𝐹△3𝑌+𝜄푓 ∗ 𝑍, − 𝐹∧∗𝑌+ Δ∧∗𝐾−2𝜅𝑏] =0, 2 6 d (334) 𝑌=𝐹△𝑍 − 2𝑓 ∧ ∗𝑌 −2𝐹∧ 𝐾, 1 𝑏 d 2 − 𝐹∧∗𝑍− ∗𝑓+𝑎∗ Δ=0, 6 3 d 3 d𝑍= 𝐹△1𝑌+3𝜄푓 ∗ 𝐾. 1 1 2 −𝜄 ∗ 𝐾− ∗(𝐹△ 𝑌) − 𝑓∧𝑍=0, dΔ 6 3 3 The Hodge duals of (333) 1 1 1 𝜕 𝑏] =2𝜅(𝑒 ) ∧∗𝐾− (𝜄 𝐹) ∧ 𝑍 𝑓∧𝑌−2𝜅𝑍− 𝐾∧∗𝐹− Δ∧∗𝑌 푚 푚 ♯ (푒푚)♯ 3 6 d 2 1 1 ∇ ∗ 𝐾=−2𝜅𝑏∗(𝑒 ) + (𝑒 ) ∧𝑓∧𝑍 − ∗(𝐹△ 𝑍) = 0, 푚 푚 ♯ 푚 ♯ 6 3 2 1 1 1 𝑏 − (𝜄(푒 ) 𝐹) ∧ ∗𝑌 ∗(𝐹△ 𝑌) − 2𝜅𝑌 + 𝜄 𝑍− 𝐹+ Δ∧∗𝑍 (332) 2 푚 ♯ 6 2 3 푓 6 d 1 𝑎 ∇ ∗𝑌=2𝜅𝜄 𝑍+ ((𝑒 ) ∧𝑓)△ 𝑌 − ∗𝐹=0 푚 (푒푚)♯ 푚 ♯ 1 6 2 1 1 1 1 + 𝜄 ((𝑒 ) ∧∗𝐹) − 𝜄 ∗𝐹+𝜄 ∗𝑌− ∗(𝐹△ 𝑍) − 𝜄 𝑌=0 퐾 푚 ♯ (335) 6 퐾 dΔ 6 2 3 푓 2 1 1 1 − (𝜄 𝐹) △ ∗ 𝑍 − ∗(𝐹△ 𝑌) − 𝜄 ∗ 𝑍− 𝑓∧𝐾=0 (푒푚)♯ 1 6 1 dΔ 3 2 1 𝑎 1 ∇ ∗ 𝑍=−2𝜅𝜄 𝑌− (𝑒 ) ∧ 𝐾∧𝑓 − 𝑓+2𝜅𝐾+ 𝜄 𝐹+𝑏 Δ=0 푚 (푒푚)♯ 푚 ♯ 3 6 ∗푍 d 2 1 1 1 1 + 𝜄 𝑍+ (𝜄 𝐹) △ ∗𝑌 − 𝜄 𝐾+ 𝜄 ∗𝐹+2𝜅𝑎=0, ((푒푚)♯∧푓) (푒푚)♯ 2 3 푓 6 푌 2 2 1 − 𝑏𝜄(푒 ) 𝐹 whereweused(328)and(329). 2 푚 ♯ lead to another set of differential constraints: Similarly, the rank expansion of (324) gives d (∗𝐾) = −16𝜅𝑏] −2𝐹∧∗𝑌, 1 𝜕푚𝑏=2𝜅𝜄(푒 ) 𝐾− ∗[(𝜄(푒 ) 𝐹) ∧ 𝑍] , 1 푚 ♯ 2 푚 ♯ (∗𝑌) =10𝜅𝑍−2𝑓∧𝑌+ 𝐾∧∗𝐹 d 2 1 (336) ∇ 𝐾 = −2𝜅𝑏 (𝑒 ) − ∗((𝑒 ) ∧𝑓∧𝑍) 3 푚 푚 ♯ 2 푚 ♯ + ∗(𝐹△ 𝑍) , 2 3 1 + (𝜄(푒 ) 𝐹)3 △ 𝑌, (∗𝑍) = −2𝑏𝐹 − 8𝜅𝑌 +∗(𝐹△ 𝑌) + 2𝜄 𝑍. 2 푚 ♯ d 2 푓 34 Advances in High Energy Physics

Finally, expanding (326) and using the properties of ],wefind Given expansions (338) and (339), identity (337) gives the following relations when separated into rank components: 𝐾⬦𝐾+𝐾⬦𝑌+𝑌⬦𝐾+𝐾⬦𝑍+𝑍⬦𝐾+𝑌⬦𝑌 󵄩 󵄩2 󵄩 󵄩2 󵄩 󵄩2 󵄩 󵄩2 󵄩𝐾󵄩 + 󵄩𝑌 󵄩 + 󵄩𝑌 󵄩 + 󵄩𝑍󵄩 +𝑏2 = 15𝑎2, 2 󵄩 󵄩 󵄩 ⊥󵄩 󵄩 ‖󵄩 󵄩 󵄩 + 𝑍⬦𝑍+𝑌⬦𝑍+𝑍⬦𝑌+2𝑏∗𝑌+𝑏 (337) 𝑌 △ 𝑍=7𝑎𝐾, =15𝑎2 + 14𝑎𝐾+14𝑎𝑌+14𝑎𝑍 + 14𝑎𝑏]. ⊥ 4

2𝐾△1𝑍−𝑌‖△2𝑌‖ −2𝑌‖△2𝑌⊥ −𝑌⊥△2𝑌⊥ We write below the expansions for all those geometric (340) products (see (35)) that appear in (337). Let us first list the − 𝑍△3𝑍+2𝑏∗𝑌=14𝑎𝑌, geometricproductsinvolving𝐾: 𝐾∧𝑌⊥ −𝑌⊥△2𝑍=7𝑎𝑍, 󵄩 󵄩2 󵄩 󵄩 𝐾⬦𝐾=󵄩𝐾󵄩 , 2𝑌‖ ∧𝑌⊥ + 𝑍△1𝑍 = 14𝑎𝑏].

𝐾⬦𝑌=𝐾∧𝑌⊥ + 𝐾△1𝑌‖, In order to solve system (340), we introduce the notations: 𝑌⬦𝐾=𝐾∧𝑌⊥ − 𝐾△1𝑌‖, (338) 𝐾∧𝑌⊥ =𝑒𝑍, 𝐾⬦𝑍=𝐾∧𝑍⊥ + 𝐾△1𝑍‖ = 𝐾△1𝑍, 𝐾△1𝑌‖ =𝑓∗𝑍, 𝑍⬦𝐾=−𝐾∧𝑍⊥ + 𝐾△1𝑍‖ = 𝐾△1𝑍. 𝐾△1𝑍=𝑔𝑌⊥,

Notice that 𝐾∧𝑍⊥ =0, since there is no nontrivial six-form 𝑌‖ ∧𝑌⊥ =ℎ], pinor bilinear that can be constructed in this case. Thus, we 𝑍 =0 𝑍=𝑍 must have ⊥ ,whichmeans ‖,arelationwhichwe 𝑌‖△2𝑌‖ =𝑦1𝑌⊥, use when performing the following expansions: 𝑌‖△2𝑌⊥ =𝑦2𝑌‖, 𝑌⬦𝑌=2𝑌 ∧𝑌 −𝑌△ 𝑌 −2𝑌△ 𝑌 ‖ ⊥ ‖ 2 ‖ ‖ 2 ⊥ 𝑌 △ 𝑌 =𝑦𝑌 , ⊥ 2 ⊥ 3 ⊥ (341) 󵄩 󵄩2 󵄩 󵄩2 −𝑌⊥△2𝑌⊥ + 󵄩𝑌‖󵄩 + 󵄩𝑌⊥󵄩 , 𝑌‖△1𝑍=𝑛∗𝐾,

𝑌⬦𝑍=−𝑌‖△1𝑍‖ −𝑌⊥△2𝑍‖ +𝑌‖△3𝑍‖ 𝑌⊥△2𝑍=𝑟𝑍, +𝑌⊥△4𝑍‖, (339) 𝑌‖△3𝑍=𝑠∗𝑍,

𝑍⬦𝑌=𝑌‖△1𝑍‖ −𝑌⊥△2𝑍‖ −𝑌‖△3𝑍‖ 𝑌⊥△4𝑍=𝑡𝐾,

+𝑌⊥△4𝑍‖, 𝑍△1𝑍=𝑢], 󵄩 󵄩2 𝑍⬦𝑍=𝑍‖△1𝑍‖ − 𝑍‖△3𝑍‖ + 󵄩𝑍󵄩 . 󵄩 󵄩 𝑍△3𝑍=𝑥𝑌⊥,

We mention here that expansions (331)–(339) were obtained where 𝑒, 𝑓, 𝑔, ℎ, 𝑦1, 𝑦2, 𝑦3, 𝑛, 𝑟, 𝑠, 𝑡, 𝑢, 𝑥 are smooth functions Ricci using a package of procedures which we created using on 𝑀 which we want to determine in terms of 𝑎 and 𝑏. [33]. We have also verified these relations (in tensorial form) Using (341), system (340) leads to four independent Cadabra using [34]. relations for ranks 0, 1, 5, 8 and two independent identities for We deduce from the Fierz identities in normal cases that the parallel and perpendicular components of rank 4: any generalized product of form-valued pinor bilinears can be expressed as a form-valued pinor bilinear. In (339), we 󵄩 󵄩2 󵄩 󵄩2 󵄩 󵄩2 󵄩 󵄩2 𝑏2 + 󵄩𝐾󵄩 + 󵄩𝑌 󵄩 + 󵄩𝑌 󵄩 + 󵄩𝑍󵄩 =15𝑎2, omitted to write down those terms which are tautologically 󵄩 󵄩 󵄩 ‖󵄩 󵄩 ⊥󵄩 󵄩 󵄩 zeroduetothegradedantisymmetryofthewedgeproduct. 𝑡=7𝑎, We also omitted writing some other vanishing terms such as 𝑌⊥ ∧𝑌⊥, which has rank 8 and thus must be proportional to 𝑒−𝑟=7𝑎, the volume form but vanishes since the volume form does (342) nothaveanorthogonalcomponent.Similarly,𝑌‖△1𝑍‖ =0 2ℎ + 𝑢 = 14𝑎𝑏, since it should be a parallel 7-form and thus proportional to 𝑏𝛼1 −𝑦2 =7𝑎, (∗𝐾)‖ =∗(𝐾⊥),but𝐾⊥ =0.Furthermore,𝑌⊥△1𝑍‖ =0since 𝑍 =0 2𝑏 it is an orthogonal 5-form, thus proportional to ⊥ .For +2𝑔−(𝑦 +𝑦)−𝑥=14𝑎. 𝑌 △ 𝑍 =0 𝑌 △ 𝑍 =0 1 3 similar reasons, ⊥ 3 ‖ and ‖ 4 ‖ . 𝛼1 Advances in High Energy Physics 35

Using associativity of the geometric product which need not be independent. We thus find that the Fierz identities are equivalent to the algebraic system of (342)– (𝐾⬦𝐾)⬦𝑌=𝐾⬦(𝐾⬦𝑌), (344), which can be solved using Mathematica5 and give 𝑒=𝑎, (𝐾⬦𝑌)⬦𝑌=𝐾⬦(𝑌⬦𝑌) , 𝑓=𝑏, (𝐾⬦𝑍) ⬦ 𝑍=𝐾⬦(𝑍⬦𝑍) , 𝑎2 −𝑏2 𝑔= , (343) 𝑎 (𝐾⬦𝑌)⬦𝑍=𝐾⬦(𝑌⬦𝑍) , ℎ = 7𝑎𝑏, (𝑌⬦𝑌) ⬦ 𝑍=𝑌⬦(𝑌⬦𝑍) , 6𝑏2 𝑦 =− , 1 𝑎 (𝐾⬦𝑍)⬦𝑌=𝐾⬦(𝑍⬦𝑌) 𝑦2 =−6𝑎, as well as (338), (339), and (341), we find the relations 𝑦3 =−6𝑎, 󵄩 󵄩2 󵄩 󵄩 󵄩𝐾󵄩 =𝑒𝑔, 𝑛=7𝑏, 𝑟=−6𝑎, 𝑒=𝑓𝛼1, 𝑠=−6𝑏, (345) 𝑒𝑛 + 𝑓𝑡 =2ℎ, 𝑡=7𝑎, 𝑒𝑟 + 𝑓𝑠 =𝑒(𝑦 +𝑦), 1 3 𝑢=0,

𝑒𝑠 + 𝑟𝑓2 =2𝑦 𝑓, 6(𝑎2 −𝑏2) 󵄩 󵄩2 󵄩 󵄩2 𝑥=− , 󵄩𝑌‖󵄩 + 󵄩𝑌⊥󵄩 =𝑒𝑡+𝑓𝑛, 𝑎 𝑎 󵄩 󵄩2 𝛼 = , 󵄩 󵄩 1 󵄩𝑍󵄩 =𝑡𝑔, 𝑏 󵄩 󵄩2 󵄩 󵄩 2 2 𝑢=0, 󵄩𝐾󵄩 =𝑎 −𝑏 , 󵄩 󵄩2 2 𝑔𝑟 = 𝑥𝑒, 󵄩𝑌‖󵄩 =7𝑏, 󵄩 󵄩2 2 𝑓𝑥=𝑠𝑔, 󵄩𝑌⊥󵄩 =7𝑎, 󵄩 󵄩2 󵄩 󵄩2 󵄩 󵄩2 (344) 󵄩 󵄩 2 2 󵄩 󵄩 󵄩 󵄩 󵄩𝑍󵄩 =7(𝑎 −𝑏 ). 𝑓 󵄩𝑍󵄩 =𝑛󵄩𝐾󵄩 , 󵄩 󵄩 Substituting this solution into (341) gives 󵄩 󵄩2 󵄩 󵄩2 𝑒 󵄩𝑍󵄩 =𝑡󵄩𝐾󵄩 , 1 󵄩 󵄩 󵄩 󵄩 𝑍= 𝐾∧𝑌 , 𝑎 ⊥ 2ℎ + 2𝑠𝑦 =𝑛𝑒+2𝑟𝑠+𝑓𝑡, 2 1 ∗𝑍= 𝐾△1𝑌‖, 󵄩 󵄩2 󵄩 󵄩2 2 2 𝑏 󵄩𝑌‖󵄩 + 󵄩𝑌⊥󵄩 +𝑟(𝑦1 +𝑦3)=𝑛𝑓+𝑟 +𝑠 +𝑡𝑒, 1 ∗𝐾= 𝑌‖△1𝑍, 𝑡(𝑦1 +𝑦3) = 𝑟𝑡 + 𝑛𝑠, 7𝑏

󵄩 󵄩2 𝑌 ∧𝑌 =7𝑎𝑏], 󵄩 󵄩 ‖ ⊥ 𝑛 󵄩𝐾󵄩 =𝑔ℎ, 6𝑏2 󵄩 󵄩2 𝑌 △ 𝑌 =− 𝑌 , 󵄩 󵄩2 󵄩 󵄩 ‖ 2 ‖ 𝑎 ⊥ 𝑔 󵄩𝑌⊥󵄩 =𝑡󵄩𝐾󵄩 , 𝑌‖△2𝑌⊥ =−6𝑎𝑌‖, 2𝑦2𝑛=𝑛𝑟+𝑡𝑠,

𝑌⊥△2𝑌⊥ =−6𝑎𝑌⊥, 𝑦3 =𝑟, 𝑎 𝑏 ∗𝑌 = 𝑌‖ + 𝑌⊥, 𝑦2 =𝑠𝛼1, 𝑏 𝑎 36 Advances in High Energy Physics

󵄩 󵄩2 󵄩 󵄩 2 2 𝑎=1 𝑏= 𝜁 𝐾=( 𝜁)−1𝐾 𝑍=( 𝜁)−1𝑍 󵄩𝐾󵄩 =𝑎 −𝑏 , For , sin , cos ,and cos ,this result becomes 󵄩 󵄩2 󵄩𝑌 󵄩 =7𝑏2, 󵄩 ‖󵄩 ‖𝐾‖2 =1, 󵄩 󵄩2 󵄩𝑌 󵄩 =7𝑎2, (358) 󵄩 ⊥󵄩 ‖∗𝑍‖2 = ‖𝑍‖2 =7. 󵄩 󵄩2 󵄩 󵄩 2 2 󵄩𝑍󵄩 =7(𝑎 −𝑏 ), 𝑏 𝑎 𝑌=𝑌 +𝑌 = 𝐾∧∗𝑍+ 𝜄 𝑍, ProofofRelation(349).The second relation in (328) gives ‖ ⊥ 󵄩 󵄩2 󵄩 󵄩2 퐾 󵄩 󵄩 󵄩 󵄩 𝜄 (∗𝑍) = −∗(𝐾∧𝑍) 𝐾∧𝑍 = (1/ 2𝜁)𝐾∧𝑍 󵄩𝐾󵄩 󵄩𝐾󵄩 퐾 ,where cos must be a 6-form bilinear in 𝜉 since 𝐾 and 𝑍 are bilinears in 𝜉 and since 𝑍△3𝑍=−6𝜄퐾𝑍. the Fierz identities allow us to reduce 𝐾∧𝑍 to a pinor bilinear. (346) But there is no nontrivial 6-form pinor bilinear which can be constructedinourcase,andthuswemusthave𝐾∧𝑍=0. 2 Taking (as in [3]) 𝑎=1and 𝑏=sin 𝜁 (since 1−𝑏 >0for This proves relation (349). the norms to be positive) we find agreement with the results Proof of Relation (350). Starting from the expression 𝑌= in loc. cit. (up to a sign issue which is discussed in [35]), as 2 2 (𝑏/‖𝐾‖ )𝐾∧∗𝑍 + (𝑎/‖𝐾‖ )𝜄 𝑍 found in (346), taking the for example, 퐾 normalization used in [3, 4] and the notations in (347), (348), ‖𝐾‖2 =1, one finds (347) 1 def 𝑌= (sin 𝜁𝐾∧∗𝑍+𝜄 𝑍) 𝐾 = ( 𝜁)−1 𝐾, 2𝜁 퐾 where cos cos (359) 2 ‖∗𝑍‖ =7, =𝜄퐾𝑍−(∗𝑍) ∧𝐾sin 𝜁, (348) def −1 thus proving relation (350). where 𝑍 = (cos 𝜁) 𝑍, 𝜄 (∗𝑍) =0, Proof of Relation (351). The third algebraic constraint listed in 퐾 (349) (331)

𝑌=𝜄퐾𝑍−(∗𝑍) ∧𝐾sin 𝜁, (350) 1 1 dΔ∧𝐾− 𝐹△3𝑌+ 𝜄푓 ∗ 𝑍=0⇐⇒ 3Δ 6 3 d (𝑒 𝐾) = 0, (351) (360) 1 6Δ 3dΔ∧𝐾= 𝐹△3𝑌−𝜄푓 ∗ 𝑍 𝐾∧d (𝑒 𝜄퐾𝑍) = 0, (352) 2

−6Δ 6Δ 𝑒 d (𝑒 ∗ 𝑍) = ∗𝐹 −𝐹 sin 𝜁+4𝜅𝑌, (353) and the second differential constraint listed in (334) 1 𝑒−3Δ (𝑒3Δ 𝜁) = 𝑓 − 4𝜅𝐾, 𝐾=− 𝐹△ 𝑌+𝜄 ∗ 𝑍 d sin (354) d 2 3 푓 (361) −12Δ 12Δ 𝑒 d (𝑒 vol7 cos 𝜁) = −8𝜅vol7 ∧𝐾sin 𝜁, (355) imply the relation 3 Δ∧𝐾+ 𝐾=0⇐⇒ 𝐾=3𝐾∧ Δ, with vol7 defined as in [3]: d d d d (362) 1 1 whichiseasilyseentobeequivalentto(351). = 𝜙∧𝜄 ∗𝜙=− (∗𝑍) ∧𝜄 𝑍=−𝜄 ] vol7 7 퐾 7 퐾 퐾 (356) ProofofRelation(352). Using(350),onecanwrite(352)inthe =−∗𝐾. equivalent form:

It is easy to see that the first four of these conditions follow 𝐾∧[6dΔ∧𝑌+d𝑌−sin 𝜁 (∗𝑍) ∧ d𝐾]=0, (363) directly from the Fierz identities (346), while the last four can be obtained using the algebraic constraints (331) and the where we used 𝐾∧𝐾=0.Relations(362)and𝐾∧𝐾=0 differential constraints (334). imply 𝐾∧(sin 𝜁(∗𝑍) ∧ d𝐾) =,so(352)reducesto 0 𝐾∧(6 Δ∧𝑌+ 𝑌) =0. Proof of Relations (347) and (348). From (346), we have d d (364)

󵄩 󵄩2 Toprove(364),noticethataddingthesixthalgebraiccon- 󵄩𝐾󵄩 =𝑎2 −𝑏2, 󵄩 󵄩 straint listed in (331) (357) 󵄩 󵄩2 󵄩𝑍󵄩 =7(𝑎2 −𝑏2). 󵄩 󵄩 6dΔ∧𝑌=𝐹∧𝐾−𝐹△2𝑍+2𝑓∧∗𝑌 (365) Advances in High Energy Physics 37 to the third differential constraint listed in (334) 7. Conclusions and Further Directions

d𝑌=−2𝐹∧𝐾 +𝐹△2𝑍 −2𝑓∧∗𝑌, (366) We showed that geometric algebra techniques can be used to give a highly synthetic, conceptually transparent, and gives computationally efficient reformulation of the constrained Killing pinor equations, which constitute the condition that a 6dΔ∧𝑌+d𝑌=−𝐹∧𝐾, (367) flux background preserves a given amount of supersymmetry. Thisformulationclearlydisplaysthealgebraicanddifferential whichinturnimplies(364)uponusingtheidentity𝐾∧𝐾= structure governing the supersymmetry conditions, leading 0. to a description which opens the way for unified studies of flux backgrounds aimed at uncovering their deeper structure. Proof of Relation (353). Relation (353) is equivalent to We showed that our general formalism recovers results and methods which were used in [3] and therefore that it provides a powerful way to extend them. Our formulation is highly 6dΔ∧(∗𝑍) + d (∗𝑍) = ∗𝐹 −𝐹 sin 𝜁+4𝜅𝑌. (368) amenable to implementation in various symbolic computational packages specialized in tensor algebra, and we touched To prove (368), notice that the fifth relation in (332), which on two particular implementations which we have carried out is the Hodge dual of the fifth algebraic relation listed in (331), Ricci Mathematica5 Cadabra 6 using [33] in ,aswellas [34]. when multiplied by gives Here we illustrated our approach with the case of the most general compactifications of 𝑀-theory which preserve 6 Δ∧∗𝑍=∗𝐹+𝐹 𝜁+12𝜅𝑌−∗(𝐹△ 𝑌) d sin 2 N =1supersymmetry in three dimensions [3, 4], showing (369) −2𝜄 𝑍. how the results derived through different methods in loc. cit. 푓 can be recovered through our techniques. We stress that the methods introduced in this paper have much wider applica- On the other hand, the fourth differential relation listed in bility than the example considered in Section 6, leading to (336) gives promising new directions in the study of supergravity backgrounds and supergravity actions. In particular, we believe (∗𝑍) = −2𝐹 𝜁−8𝜅𝑌+∗(𝐹△ 𝑌) + 2𝜄 𝑍. d sin 2 푓 (370) that many computationally difficult issues in the subject could be understood much better by using such techniques. The Adding (369) and (370) gives (368). This finishes the proof of connection with a certain form of geometric quantization (353). (which we have only touched upon) also leads to interesting ideas, problems, and directions for further research, which Proof of Relation (354). Relation (354) is obviously equivalent are currently under investigation. Further applications of our to approach can be found in [35–38].

3 (sin 𝜁) dΔ+dsin𝜁=𝑓−4𝜅𝐾. (371) Appendix To prove (371), notice that the Hodge dual of the eighth A. Identities Satisfied by the Covariant algebraic constraint listed in (331) (i.e., the eight relation in Derivative of Pinors (332)), after multiplication by 3,gives Let (𝑀, 𝑔) be a pseudo-Riemannian manifold endowed 1 푚 3 ( 𝜁) Δ=𝑓−6𝜅𝐾− 𝜄 𝐹. with a local coordinate system (𝑥 ) and a local pseudo- sin d 2 ∗푍 (372) orthonormal frame (𝑒푎) (vielbein) of (𝑇𝑀, 𝑔), both defined above an open set 𝑈⊂𝑀. In this Appendix, both 𝑚 and 𝑎 On the other hand, the first differential constraint listed in run from 1 to dim 𝑀. As usual, pseudo-orthonormality of 𝑒푎 (334) gives means 𝑔(𝑒푎,𝑒푏)=𝜂푎푏,where𝜂푎푏 is a diagonal matrix all of 푎 whose diagonal entries equal +1 or −1.Welet𝑒 denote the 1 푎 푎 dsin𝜁=2𝜅𝐾 + 𝜄∗푍𝐹. (373) dual coframe of 𝑀, defined through 𝑒 (𝑒푏)=𝛿푏 ; it is a local 2 ∗ frame of 𝑇 𝑀 which is pseudo-orthonormal with respect ∗ 푎 푏 푎푏 Adding (372) and (373) gives (371), which finishes the proof to the metric 𝑔̂ induced on 𝑇 𝑀;thatis,𝑔(𝑒̂ ,𝑒 )=𝜂 푎푏 푎 def 푚 푎 of relation (354). where 𝜂 𝜂푏푐 =𝛿푐 .Wehave𝜕푚 = 𝜕/𝜕𝑥 =𝑒푚(𝑥)𝑒푎 and 푚 푎 푚 𝑒푎 =𝑒푎 (𝑥)𝜕푚 for some locally defined functions 𝑒푚, 𝑒푎 which 푎 푚 푎 푎 푛 푛 Proof of Relation (355). Using the first relation in (336) and satisfy 𝑒푚𝑒푏 =𝛿푏 and 𝑒푚𝑒푎 =𝛿푚. This implies 𝑔(𝑒푎,𝑒푏)= first relation in (332) one finds 푚 푛 푎 푏 𝑒푎 𝑒푏𝑔푚푛 =𝜂푎푏 and 𝑔(𝜕푚,𝜕푛)=𝑒푚𝑒푛𝜂푎푏 =𝑔푚푛.Anytensor ⊗푝 ∗ ⊗푞 field 𝑡∈Γ(𝑇𝑀 ⊗(𝑇 𝑀) ) of type (𝑝, 𝑞) expands as follows: 12dΔ∧∗𝐾 + d ∗ 𝐾 =8𝜅sin 𝜁], (374) 푚 ⋅⋅⋅푚 푛 푛 𝑡= 𝑡 1 푝 𝜕 ⊗⋅⋅⋅⊗𝜕 ⊗ 𝑥 1 ⊗⋅⋅⋅⊗ 𝑥 푞 푈 푛1⋅⋅⋅푛푞 푚1 푚푝 d d which, when expressing the volume form ] in terms of vol7 푎 ⋅⋅⋅푎 (A.1) 1 푝 푏1 푏푞 =𝑡 𝑒푎 ⊗⋅⋅⋅⊗𝑒푎 ⊗𝑒 ⊗⋅⋅⋅⊗𝑒 , defined in (356), leads to (355). 푏1⋅⋅⋅푏푞 1 푝 38 Advances in High Energy Physics

푚1⋅⋅⋅푚푝 푏 ⋅⋅⋅푏 𝑡 ∇ 𝜔=(∇ 𝜔) 𝑒 1 푘 where the locally defined coefficient functions 푛1⋅⋅⋅푛푞 and For a differential form (2), this gives 푚 푚 푏1⋅⋅⋅푏푘 , 푎 ⋅⋅⋅푎 𝑡 1 푝 with 푏1⋅⋅⋅푏푞 are related through (∇ 𝜔) =𝜔 푎 ⋅⋅⋅푎 푎 푛 푚 ⋅⋅⋅푚 푚 푏 ⋅⋅⋅푏 푏1⋅⋅⋅푏푘;푚 1 푝 푎1 푝 푛1 푞 1 푝 1 푘 𝑡 =𝑒 ⋅⋅⋅𝑒푚 𝑒 ⋅⋅⋅𝑒 𝑡푛 ⋅⋅⋅푛 ⇐⇒ 푏1⋅⋅⋅푏푞 푚1 푝 푏1 푏푞 1 푞 푘 (A.9) (A.2) 푎 푚 ⋅⋅⋅푚 푚 푏 푎 ⋅⋅⋅푎 =𝜕 𝜔 − ∑Ω 𝜔 , 1 푝 푚1 푝 푏1 푞 1 푝 푚 푏 ⋅⋅⋅푏 푚푏 푏 ⋅⋅⋅푏 ,푎,푏 ⋅⋅⋅푏 𝑡푛 ⋅⋅⋅푛 =𝑒 ⋅⋅⋅𝑒푎 𝑒 ⋅⋅⋅𝑒푛 𝑡 . 1 푘 푠 1 푡−1 푡+1 푘 1 푞 푎1 푝 푛1 푞 푏1⋅⋅⋅푏푞 푠=1

Here and below, indices denoted by letters chosen from the while for a polyvector field (A.3), we find ∇푚𝛼= 푎 ⋅⋅⋅푎 (∇ 𝛼) 1 푘 𝑒 ∧⋅⋅⋅∧𝑒 middle of the Latin alphabet refer to the coordinate frame 푚 푎1 푎푘 ,with defined by (𝜕푚) while indices denoted by letters chosen from the beginning of the Latin alphabet refer to the local pseudo- 푎1⋅⋅⋅푎푘 def 푎1⋅⋅⋅푎푝 (∇푚𝛼) =𝛼;푚 orthonormal frame (vielbein) defined by (𝑒푎). 𝑘 𝜔∈Ω푘(𝑀) 푘 (A.10) A differential -form expands locally as in 푎 ⋅⋅⋅푎 푎 푎 ⋅⋅⋅푎 ,푎,푎 ⋅⋅⋅푎 푘 =𝜕 𝛼 1 푘 + ∑Ω 푠 𝛼 1 푠−1 푠+1 푘 . (2). Similarly, a polyvector field 𝛼∈Γ(𝑀,∧𝑇𝑀) expands 푚 푚푎 locally as follows: 푠=1 1 𝑆 𝑀 ∇푆 푎1⋅⋅⋅푎푘 Let be a pin bundle over . Recall that the connection 𝛼=푈 𝛼 (𝑥) 𝑒푎 ∧⋅⋅⋅∧𝑒푎 , (A.3) 𝑘! 1 푘 induced by the Levi-Civita connection ∇ takes the form

푆 ̂ with coefficients functions which are totally antisymmetric in ∇푚 =𝜕푚 + Ω푚, the indices. (A.11) ̂ 1 푎푏 1 푎푏 Let ∇ be the Levi-Civita connection of (𝑀, 𝑔).Its where Ω푚 = Ω푚푎푏𝛾 = 𝑔(𝑒푎,∇푚𝑒푏)𝛾 ∈Γ(𝑀, End (𝑆)) ; 휌 4 4 Christoffel symbols Γ푚푛 in the given local coordinates are 휌 푏 푎 defined through ∇푚(𝜕푛)=Γ푚푛𝜕휌, while its coefficients Ω푚푎 this acts on sections of 𝑆 in the obvious manner. Here, 𝛾 ∈ with respect to the given coordinate system and vielbein Γ(𝑀, End(𝑆)) are the gamma operators associated with the 푏 (𝑒푎) are determined by the expansion ∇푚(𝑒푎)=Ω푚푎𝑒푏.Here coframe ,whichsatisfy def and below, we set ∇푚 =∇휕 . The two sets of connection 푚 [𝛾푎,𝛾푏] =2𝜂푎푏. coefficients are related through +,∘ (A.12)

푎 푎 휆 푛 휆 푎 Ω =𝑒 𝑒 Γ −𝑒 𝜕 𝑒 ⇐⇒ def 푏 푚푏 푛 푏 푚휆 푏 푚 휆 We will also use the operators 𝛾푎 =𝜂푎푏𝛾 ,whichsatisfy (A.4) [𝛾푎,𝛾푏]+,∘ =2𝜂푎푏. In what follows, we recall some basic Γ푛 =𝑒푛𝑒푏 Ω푎 +𝑒푛𝜕 𝑒푎. 푆 푚휆 푎 휆 푚푏 푎 푚 휆 properties of ∇ .

The fact that ∇ is torsion-free amounts to the conditions Algebraic Identities.Let𝑉 be a finite-dimensional K-vector def Γ휌 =Γ휌 ⇐⇒ Ω =−Ω , space. For 𝑋, 𝑌 ∈ End(𝑉),weset[𝑋, 𝑌]휖 = 𝑋𝑌+𝜖𝑌𝑋, 푚푛 푛푚 푚푎푏 푚푏푎 (A.5) def where 𝜖∈{−1,+1},sothat[𝑋, 𝑌]+1 =[𝑋,𝑌]+,∘ is the anti- where Ω푚푎푏 is defined through def commutator of 𝑋 with 𝑌, while [𝑋, 𝑌]−1 =[𝑋,𝑌]−,∘ is the def 푐 commutator. We start with the following trivial observation. Ω푚푎푏 =𝜂푎푐Ω푚푏 =𝑔(𝑒푎,∇푚𝑒푏)=−𝑔(∇푚𝑒푎,𝑒푏). (A.6) Lemma A.1. For any 𝐴, 𝐵, 𝐶∈ Mat(𝑛, K) and any 𝜖∈ With respect to the vielbein, the covariant derivative of a {−1, +1},wehave (𝑝, 𝑞)-tensor (A.1) takes the form [𝐴𝐵,] 𝐶 =𝐴[𝐵,] 𝐶 −𝜖[𝐴,] 𝐶 𝐵. 푎 ⋅⋅⋅푎 휖 휖 휖 (A.13) 1 푝 푏1 푏푞 ∇푚𝑡=(∇푚𝑡) 𝑒푎 ⊗⋅⋅⋅⊗𝑒푎 ⊗𝑒 ⊗⋅⋅⋅⊗𝑒 , 푏1⋅⋅⋅푏푞 1 푝 (A.7) Proposition A.2. The following identities hold for all 𝑝 =𝑞̸ : where 푘 푎 ⋅⋅⋅푎 def 푎 ⋅⋅⋅푎 푝 푞 푎1⋅⋅⋅푎푘 푞푎푠 푎1⋅⋅⋅푎푠−1 푝푎푠+1⋅⋅⋅푎푘 1 푝 1 푝 [𝛾 ∘𝛾 ,𝛾 ]−,∘ =2∑𝜂 𝛾 (∇푚𝑡)푏 ⋅⋅⋅푏 =𝑡푏 ⋅⋅⋅푏 ;푚 1 푞 1 푞 푠=1 (A.14) 푝 −(𝑝←→𝑞), 푎 ⋅⋅⋅푎 푎 ⋅⋅⋅푎 ,푎,푎 ⋅⋅⋅푎 1 푝 푎푠 1 푠−1 푠+1 푝 =𝜕푚𝑡 + ∑Ω 𝑡 푏1⋅⋅⋅푏푞 푚푎 푏1⋅⋅⋅푏푞 푠=1 (A.8) 푘 푝 푞 푞 푝푝耠 [𝛾 ∘𝛾 ,𝛾 ] =2∑𝛿 𝜂 𝛾 耠 푎1⋅⋅⋅푎푘 −,∘ 푎푠 푎1⋅⋅⋅푎푠−1 푝 푎푠+1⋅⋅⋅푎푘 푞 푠=1 푎 ⋅⋅⋅푎 (A.15) − ∑Ω푎 𝑡 1 푝 . 푚푏푡 푏1⋅⋅⋅푏푡−1,푎,푏푡+1⋅⋅⋅푏푞 푡=1 −(𝑝←→𝑞). Advances in High Energy Physics 39

Proof. We prove only the first identity, since it immediately Proposition A.2 has the following immediate conse- implies the second upon lowering indices with 𝜂. Applying quence, which follows by using the antisymmetry property 푘−1 the lemma with 𝜖=(−1) ,wefind (A.5) of Ω푚푝푞.

푝 푞 푎1⋅⋅⋅푎푘 푝 푞 푎1⋅⋅⋅푎푘 [𝛾 ∘𝛾 ,𝛾 ]−,∘ =𝛾 ∘[𝛾 ,𝛾 ](−1)푘−1,∘ Proposition A.3. The following identities hold:

푘 푝 푎1⋅⋅⋅푎푘 푆 푎 ⋅⋅⋅푎 1 푏푐 푎 ⋅⋅⋅푎 + (−1) [𝛾 ,𝛾 ](−1)푘−1,∘ (A.16) [∇ ,𝛾 1 푘 ] = Ω [𝛾 ,𝛾 1 푘 ] 푚 −,∘ 4 푚푏푐 −,∘ 푞 ∘𝛾 . 푘 푎푠 푎1⋅⋅⋅푎푠−1 푝푎푠+1⋅⋅⋅푎푘 =−∑Ω푚 푝𝛾 , A simple computation (or a mathematical induction argu- 푠=1 ment) gives (A.22) 푆 1 푏푐 [∇ ,𝛾푎 ⋅⋅⋅푎 ] = Ω푚푏푐 [𝛾 ,𝛾푎 ⋅⋅⋅푎 ] 푘 푚 1 푘 −,∘ 4 1 푘 −,∘ 푝 푎1⋅⋅⋅푎푘 푠−1 푝푎푠 푎1⋅⋅⋅푎̂푠⋅⋅⋅푎푘 [𝛾 ,𝛾 ](−1)푘−1,∘ =2∑ (−1) 𝜂 𝛾 , (A.17) 푠=1 푘 =+∑Ω 푝 𝛾 . 푚 푎푠 푎1⋅⋅⋅푎푠−1 푝푎푠+1⋅⋅⋅푎푘 where the hat indicates that the corresponding index is 푠=1 missing. Using this equation and its counterpart with 𝑝 𝑘 𝜔 replaced by 𝑞 in (A.16) gives Consider an arbitrary -form as in (2) and an arbitrary polyvector field 𝛼 as in (A.3). We define endomorphisms of 푝 푞 푎1⋅⋅⋅푎푘 𝑆 [𝛾 ∘𝛾 ,𝛾 ]−,∘ the pin bundle via 1 푘 푎1⋅⋅⋅푎푘 푠−1 푞푎 푝 푎 ⋅⋅⋅푎̂ ⋅⋅⋅푎 𝛾 (𝜔) = 𝜔푎 ⋅⋅⋅푎 𝛾 , =2∑ (−1) 𝜂 푠 𝛾 ∘𝛾 1 푠 푘 𝑘! 1 푘 (A.18) 푠=1 1 푎1⋅⋅⋅푎푘 𝛾(∇푚𝜔) = (∇푚𝜔) 𝛾 , 푘 푎1⋅⋅⋅푎푘 푠−1 푝푎 푎 ⋅⋅⋅푎̂ ⋅⋅⋅푎 푞 𝑘! +2∑ (−1) 𝜂 푠 𝛾 1 푠 푘 ∘𝛾 . (A.23) 1 푠=1 푎1⋅⋅⋅푎푘 𝛾 (𝛼) = 𝛼 𝛾푎 ⋅⋅⋅푎 , 𝑘! 1 푘 We next use the following identities, which can be checked by 1 9 푎1⋅⋅⋅푎푘 mathematical induction : 𝛾(∇푚𝛼) = (∇푚𝛼) 𝛾푎 ⋅⋅⋅푎 . 𝑘! 1 푘 푝 푎 ⋅⋅⋅푎̂ ⋅⋅⋅푎 𝛾 ∘𝛾 1 푠 푘 Proposition A.3 implies the following. 푠−1 푎 ⋅⋅⋅푎 푝푎 ⋅⋅⋅푎 = (−1) 𝛾 1 푠−1 푠+1 푘 푆 Proposition A.4. ∇ satisfies the following identities for any 푠−1 𝜔 𝛼 푡−1 푝푎 푝 푎 ⋅⋅⋅푎̂ ⋅⋅⋅푎̂ ⋅⋅⋅푎 differential form and any polyvector field : +2∑ (−1) 𝜂 푡 𝛾 ∘𝛾 1 푡 푠 푘 , 푡=1 [∇푆 ,𝛾(𝜔)] =𝛾(∇ 𝜔) , (A.19) 푚 −,∘ 푚 푎 ⋅⋅⋅푎̂ ⋅⋅⋅푎 푞 𝛾 1 푠 푘 ∘𝛾 (A.24) [∇푆 ,𝛾(𝛼)] =𝛾(∇ 𝛼) . 푚 −,∘ 푚 푘−푠 푎 ⋅⋅⋅푎 푞푎 ⋅⋅⋅푎 = (−1) 𝛾 1 푠−1 푠+1 푘 Proof. Relations (A.24) follow by using Proposition A.4, 푘 (A.9), and (A.10) as well as an obvious relabeling of dummy 푘−푡 푞푎푡 푞 푎1⋅⋅⋅푎̂푠⋅⋅⋅푎̂푡⋅⋅⋅푎푘 +2∑ (−1) 𝜂 𝛾 ∘𝛾 . indices. 푡=푠+1 Observation. The first identity in Proposition A.4 is the well- Inserting these in (A.18) gives 푆 known statement that ∇ is a Clifford connection on 𝑆 in the 푘 sense typically used in spin geometry (see, e.g., [39]). 푝 푞 푎1⋅⋅⋅푎푘 푞푎푠 푎1⋅⋅⋅푎푠−1 푝푎푠+1⋅⋅⋅푎푘 [𝛾 ∘𝛾 ,𝛾 ]−,∘ =2∑𝜂 𝛾 푠=1 (A.20) B. Component Approach to Pinor Bilinears −(𝑝←→𝑞)+𝑇, In this appendix, we show that the abstract equations (224) where the term 𝑇 is given by and (246) are equivalent to a set of equations which were found in [3] via component calculations pertaining to the 𝑇=4(𝑇1 −𝑇2) , (A.21) particular case considered in loc. cit.

푠+푡 푝푎푡 푞푎푠 푎1⋅⋅⋅푎̂푡⋅⋅⋅푎̂푠⋅⋅⋅푎푘 with 𝑇1 =∑1≤푡<푠≤푘(−1) 𝜂 𝜂 𝛾 and 𝑇2 = B.1. Alternate Form of the Algebraic Constraints. Taking linear 푠+푡 푞푎푡 푝푎푠 푎1⋅⋅⋅푎̂푠⋅⋅⋅푎̂푡⋅⋅⋅푎푘 ∑1≤푠<푡≤푘(−1) 𝜂 𝜂 𝛾 .Since𝑇2 =𝑇1|푝↔푞,we combinations of (223) (equivalently, using the fact that 𝑇=0 K(𝐿 ̌ )∩K(𝑅 ̌ )=K(𝐿 ̌ +𝑅 ̌ )∩K(𝐿 ̌ −𝑅 ̌ ) have and the first relation (A.14) is proved. 푄 휏B(푄) 푄 휏B(푄) 푄 휏B(푄) ) 40 Advances in High Energy Physics shows that the algebraic constraints derived in Section 5 can are not independent, being equivalent to the first two. We also be written in the following form: conclude that the useful relations of [3] are equivalent to the particular incarnation of our algebraic constraints for the case B (𝜉, (𝑄푡 ∘𝛾 ±𝛾 ∘𝑄)𝜉耠) 푎1⋅⋅⋅푎푘 푎1⋅⋅⋅푎푘 considered in loc. cit. (B.1) 푡 푡 耠 =𝜉 (𝑄 ∘𝛾푎 ⋅⋅⋅푎 ±𝛾푎 ⋅⋅⋅푎 ∘𝑄)𝜉 =0. 1 푘 1 푘 B.2. Alternate Derivation of the Differential Constraints. For simplicity, let us consider only the case K = R.Recallthat These equations are equivalent—for the particular case con- the pin bundle 𝑆 of a pseudo-Riemannian manifold (𝑀, 𝑔) sidered there—with a set of conditions used in [3]. Starting is endowed with admissible bilinear pairings B,which,in from equation (2.26) of that reference, let us show that the particular, satisfy (174). 푡 “useful relations” of [3, Appendix C] are equivalent to our Ω̂ =−Ω̂ Ω̂ algebraic constraints. Recall that [3] deals with the case of Since 푚 푚,wefindthat 푚 is anti-self-adjoint with 𝜉 respect to the pairing B, which means that the pin covariant Majorana spinors in eight Euclidean dimensions, a case 푆 derivative ∇ induced by the Levi-Civita connection of (𝑀, 𝑔) which was also the subject of our application in Section 6. 푆 As explained in that section, we prefer to work directly with is compatible with B in the sense that this pairing is ∇ -flat: 𝜉 𝜀 =(1/√2)𝜉 𝜀 = rather than with the quantities + and − 耠 푆 耠 푆 耠 (9) 𝜕 B (𝜉, 𝜉 )=B (∇ 𝜉, 𝜉 )+B (𝜉, ∇ 𝜉 ), (1/√2)𝛾 𝜉 used in loc. cit. (which provide a redundant 푚 푚 푚 parameterization of 𝜉). (B.7) ∀𝜉, 𝜉耠 ∈Γ(𝑀,) 𝑆 . When expressed in terms of 𝜉, (2.26) of [3] is equivalent to 𝑄𝜉,where =0 𝑄 is given in (304). Choosing a local frame 𝐷 =∇푆 +𝐴 = of 𝑆,wecanthinkof𝑄 as a locally defined matrix-valued The deformed pin connection takes the form 푚 푚 푚 ̂ 푡 function and of 𝜉 as a column matrix with entries given by 𝜕푚 + Ω푚 +𝐴푚.Since𝐴푚 =−𝐴푚,itfollowsthat𝐴푚 (and locally defined smooth functions. When 𝑄𝜉,wealsohave =0 Ω̂ +𝐴 푡 푡 thus also 푚 푚) is again anti-self-adjoint with respect to 𝜉 𝑄 =0,where the scalar product on 𝑆; as a consequence, the deformed pin connection is also compatible with this pairing: 푡 1 푛 1 푛 (9) 1 푛푝푞푟 𝑄 = (𝜕푛Δ) 𝛾 + 𝑓푛𝛾 ∘𝛾 − 𝐹푛푝푞푟𝛾 2 6 288 耠耠耠 (B.2) 𝜕푚B (𝜉, 𝜉 )=B (𝐷푚𝜉, 𝜉 )+B (𝜉,푚 𝐷 𝜉 ), −𝜅𝛾(9). (B.8) ∀𝜉, 𝜉耠 ∈Γ(𝑀,) 𝑆 . 𝑄𝜉 =0 𝜉푡𝑄푡 =0 The two equations and imply the relations 耠耠 Replacing 𝜉 with 𝑇𝜉 in the last equation (where 𝑇∈ 푡 𝜉 𝑇∘𝑄𝜉=0, (B.3) Γ(𝑀, EndR(𝑆)) is arbitrary) gives

𝜉푡𝑄푡 ∘𝑇𝜉=0, 耠耠 (B.4) 𝜕푚B (𝜉, 𝑇𝜉 )=B (𝐷푚𝜉, 𝑇𝜉 ) 𝑇∈ (16, R)≈ (8, 0) where Mat Cl is a general Clifford matrix. + B (𝜉, [𝐷 ,𝑇] 𝜉耠) 𝛾(9) 푚 −,∘ Using relations (308) and the fact that anticommutes with (B.9) 𝛾1,...,𝛾8 耠 ,itiseasytocheckthatthe“usefulrelations”(C.1)– + B (𝜉, 𝑇𝐷푚𝜉 ), (C.3) given in Appendix C of [3] take the following form when expressed in terms of 𝜉: ∀𝜉, 𝜉耠 ∈Γ(𝑀,) 𝑆 , 1 1 𝐹 𝜉푡 [𝛾푝푞푟푠,𝑇] 𝜉− (𝜕 Δ) 𝜉푡 [𝛾푚,𝑇] 𝜉 288 푝푞푟푠 ∓ 2 푚 ∓ which immediately implies the following statement. (B.5) 耠 1 Lemma B.1. When 𝐷푚𝜉=𝐷푚𝜉 =0,wehave +𝜅𝜉푡 [𝛾(9),𝑇] 𝜉− 𝑓 𝜉푡 [𝛾 𝛾(9),𝑇] 𝜉=0, ∓ 6 푚 푚 ± 𝜕 B (𝜉, 𝑇𝜉耠)=B (𝜉, [𝐷 ,𝑇] 𝜉耠), 1 푚 푚 −,∘ 𝐹 𝜉푡 [𝛾푝푞푟푠,𝑇𝛾(9)] 𝜉 (B.10) 288 푝푞푟푠 ∓ ∀𝑇 ∈ Γ (𝑀, EndR (𝑆)). 1 푡 푚 (9) 푡 (9) (9) − (𝜕푚Δ) 𝜉 [𝛾 ,𝑇𝛾 ] 𝜉+𝜅𝜉 [𝛾 ,𝑇𝛾 ] 𝜉 (B.6) (푘) 푘 2 ∓ ∓ Let now Ě ∈Ω(𝑀) be a 𝑘-form defined through

1 푡 (9) (9) − 𝑓 𝜉 [𝛾 𝛾 ,𝑇𝛾 ] 𝜉=0. (푘) def 耠 푚 푚 ± ̌ = B (𝜉, 𝛾 𝜉 )󳨐⇒ 6 E푎1⋅⋅⋅푎푘 푎1⋅⋅⋅푎푘 (B.11) (푘) It is now clear that the first identity in (B.5) is equivalent ̌ 耠 푎1⋅⋅⋅푎푘 E = B (𝜉,푎 𝛾 ⋅⋅⋅푎 𝜉 )𝑒 , to the difference (B.3) − (B.4) and the second is equivalent 1 푘 to the sum (B.3) + (B.4), while the two identities in (B.6) (𝑒 ) (𝑀, 𝑔) ∓ 𝑇 𝑇𝛾(9) where 푎 is a local pseudo-orthonormal frame of and are equivalent to (B.3) (B.4) where is replaced by . 𝛾 = 𝛾((𝑒 ) )=𝜂 𝛾푏 One can easily check that the third and fourth identities 푎 푎 ♯ 푎푏 . Advances in High Energy Physics 41

耠 Proposition B.2. When 𝐷푚𝜉=𝐷푚𝜉 =0,wehave Endnotes ̌(푘) ̌(푘) (∇푚E ) = E 1. They can, of course, also be formulated using complex 푎 ⋅⋅⋅푎 푎1⋅⋅⋅푎푘;푚 1 푘 Weyl spinors. (B.12) 耠 = B (𝜉, [𝐴 ,𝛾 ] 𝜉 ). 2. Indeed, the values of 𝜉1,...,𝜉푠 at two points 𝑥, 𝑦 of 𝑀 푚 푎1⋅⋅⋅푎푘 −,∘ are related through the parallel transport of 𝐷 along 𝑇=𝛾 some curve connecting 𝑥 and 𝑦 in 𝑀.Sincetheparallel Proof. Applying Lemma B.1 to 푎1⋅⋅⋅푎푘 gives transport gives a linear isomorphism between the fibers (푘) 𝜕 ̌ = B (𝜉, [𝐷 ,𝛾 ] 𝜉耠) 𝑆푥 and 𝑆푦 of 𝑆, it follows that any linear dependence rela- 푚E푎1⋅⋅⋅푎푘 푚 푎1⋅⋅⋅푎푘 −,∘ tion within 𝜉1(𝑥),...,𝜉푠(𝑥) also holds—with the same 푆 耠 coefficients—within 𝜉1(𝑦),...,𝜉푠(𝑦).Since𝑥 and 𝑦 are = B (𝜉, [∇ ,𝛾푎 ⋅⋅⋅푎 ] 𝜉 ) 푚 1 푘 −,∘ arbitrary, this would give a linear dependence relation K 耠 over (i.e., with constant coefficients) between the + B (𝜉, [𝐴푚,𝛾푎 ⋅⋅⋅푎 ] 𝜉 ) 𝜉 ,...,𝜉 1 푘 −,∘ (B.13) globally defined sections 1 푠,whichcontradicts our assumptions. 푘 = B (𝜉, ∑Ω 푝 𝛾 𝜉) 𝜄 ∘𝜄 =∧ ∘∧ =0 𝜔∈ 푚 푎푠 푎1⋅⋅⋅푎푠−1 푝푎푠+1⋅⋅⋅푎푘 3. In general, we have 휔 휔 휔 휔 for any 푠=1 odd ΩK (𝑀). + B (𝜉, [𝐴 ,𝛾 ] 𝜉耠), 4. Ofcourse,wecanview⬦ as a section of the vector bundle 푚 푎1⋅⋅⋅푎푘 −,∘ ∗ ∗ ∗ Hom(∧𝑇K𝑀⊗∧𝑇K𝑀, K∧𝑇 𝑀). where we used the second identity stated in Proposition A.3 5. This is most easily explained in the case when 𝑑=dim 𝑀 of Appendix A. The conclusion follows upon moving the first is even; namely, 𝑑=2𝑟. Then “vertical” fermionic Weyl term of the last expression to the left hand side and applying quantization (in which one quantizes only along the (A.9). odd directions of Π𝑇𝑀, while treating the body 𝑀 as classical) can be performed by choosing an almost com- Competing Interests plex structure on 𝑀, which induces a decomposition ∗ 𝑇C𝑀=𝑊⊕𝑊 of the complexified tangent bundle The authors declare that they have no competing interests. of 𝑀,with𝑊 a complex vector bundle. This allows us to define fermionic Fock representations at each point Acknowledgments 𝑥∈𝑀given by annihilation and creation operators def √ † def √ a푘 =(1/ 2)(𝛾푘 +𝑖𝛾푟+푘), a푘 =(1/ 2)(𝛾푘 −𝑖𝛾푟+푘) (where This work was supported by the CNCS projects PN-II-RU- 𝑖 = 1⋅⋅⋅𝑟 𝑥 TE (Contract no. 77/2010), PN-II-ID-PCE (Contract nos. ) defined at , with coherent states given by def −∑푑/2 푧푘a† 50/2011 and 121/2011), and PN 09 37 01 02/2009. The work |𝑧⟩푥 =𝑒 푘=1 푘 |0⟩푥,where|0⟩푥 is the vacuum at 𝑥 and 푘 푘 푟+푘 of Calin-Iuliu Lazaroiu was also supported by the Research 𝑧 = (1/2)(𝜁 +𝑖𝜁 ) are odd complex coordinates along Center Program of IBS (Institute for Basic Science) in Korea the fibers of Π𝑇C𝑀. Identifying ΩC(𝑀) with the algebra 푘 (Grant CA1205-01). Calin-Iuliu Lazaroiu and Elena-Mirela of complex functions on Π𝑇𝑀,wehavea푘 =𝜕/𝜕𝑧 Babalic thank the Center for Geometry and Physics, Institute † 푘 and a푘 = 𝑧 , so the bundle of spin Fock spaces can be for Basic Science and Pohang University of Science and ∗ ∗ identified with the subbundle ∧𝑊 of ∧𝑇C𝑀.Thestar Technology (POSTECH), Korea, and especially Jae-Suk Park product ⋆ takes the form for providing excellent conditions at various stages during the preparation of this work, through the research visitor 𝑓 =𝑓 ⋆𝑓 program affiliated with Grant no. CA1205-1. The Center for 휔⬦휂 휔 휂 Geometry and Physics is supported by the Government of ⃖ ⃗ ⃖ ⃗ (∗) 푘푙 𝜕 𝜕 푘푙 𝜕 𝜕 Korea through the Research Center Program of IBS (Institute =𝑓 (𝑔 +𝑔 )𝑓, 휔 exp 𝜕𝑧푘 푙 푘 𝜕𝑧푙 휂 for Basic Science). Calin-Iuliu Lazaroiu also thanks Perimeter 𝜕𝑧 𝜕𝑧 Institute for hospitality and for providing an excellent and stimulating research environment during the last stages of which agrees with (34). the preparation of this paper. Research at Perimeter Insti- 6. Thismapcanbedefinednaturallyonthebundleofendo- tute is supported by the Government of Canada through morphisms of any vector bundle, making no reference Industry Canada and by the Province of Ontario through the whatsoever to any bilinear pairing on the bundle. Ministry of Economic Development and Innovation. Calin- Iuliu Lazaroiu thanks Lilia Anguelova for interest and for 7. The same is true for any metric but torsion-full connec- stimulating discussions as well as for critical input during tion. the final stages of this project. Ioana-Alexandra Coman 8. Note that 𝐷푚 ⊗ id푆 + id푆 ⊗𝐷푚 is the connection induced acknowledges the student scholarship from the Dinu Patriciu by 𝐷푚 on 𝑆⊗𝑆. Foundation “Open Horizons,” which supported part of her studies. 9. NoticethatthereisnoEinsteinsummationover𝑝 or 𝑞. 42 Advances in High Energy Physics

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Research Article Structural Theory and Classification of 2D Adinkras

Kevin Iga1 and Yan X. Zhang2

1 Natural Science Division, Pepperdine University, 24255 Pacific Coast Highway, Malibu, CA 90263, USA 2Department of Mathematics, University of California, 970 Evans Hall, Berkeley, CA 94720-3840, USA

Correspondence should be addressed to Kevin Iga; [email protected]

Received 6 August 2015; Revised 3 January 2016; Accepted 11 January 2016

Academic Editor: Torsten Asselmeyer-Maluga

Copyright © 2016 K. Iga and Y. X. Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.

Adinkras are combinatorial objects developed to study (1-dimensional) supersymmetry representations. Recently, 2D Adinkras have been developed to study 2-dimensional supersymmetry. In this paper, we classify all 2D Adinkras, confirming a conjecture of T. Hubsch.¨ Along the way, we obtain other structural results, including a simple characterization of Hubsch’s¨ even-split doubly even codes.

1. Introduction the super-Poincare´ algebra in two dimensions. Note that many of the motivations for studying 1D SUSY apply here Despite supersymmetry being of theoretical interest since the as well. In the progression from simple to difficult, this is 1970s, there has not been a careful mathematical classification a logical next step. It also has a very easy Lorentz group of off-shell supersymmetric field theories. Many supermul- symmetry and a lack of gauge fields, while incorporating a tipletshavebeendiscoveredinanadhocfashion.Manyof few elements that are of interest in higher dimensions. Two- these theories are only known on-shell, and it was not clear dimensional SUSY also is of interest to superstring theory. which of these had off-shell counterparts [1–3]. The approach One approach to off-shell 2D SUSY is to study the process in [3] was to consider 1D theories (i.e., representations of of dimensional reduction from 2 to 1 dimension and to use the super-Poincare´ algebra in one dimension, i.e., supersym- the results from the 1D classification, to determine graphical metric quantum mechanics). This is reasonable for several objectsthatcapturetherelevantrepresentationtheoreticdata reasons: first, it makes sense to solve a problem starting with in this new setting. The graphical “calculus” idea is very useful simpler cases, and 1D has a trivial Lorentz group structure, because once the fundamental physics ideas are instilled not to mention lack of gauge fields. Second, supermultiplets into the definitions, we only need to perform combinatorial in higher dimensions (perhaps dimensions of interest like manipulations and very little algebra. This has led to the 4or10)canbedimensionallyreducedto1D,andsothe development of 2D Adinkras [11, 12]. 1D reduction can serve as a starting point for classifying In this paper, we completely characterize 2D Adinkras, higher dimensional theories. Third, this 1D classification is a guided by the approach and conjectures set forth in [12]. compelling mathematical question, in its own right. The main result settles Hubsch’s¨ Conjecture (the formulation In 2004, Faux and Gates developed Adinkras (what we in [12] is slightly different: see the Appendix for details) call 1D Adinkras in this paper) to study off-shell super- in Theorem 21. Essentially, this says that these 2D Adinkras multiplets in one dimension. There have been a number come from two 1D Adinkras: one describing the left-moving of developments that have led to the classification of 1D supersymmetries, and the other describing the right-moving Adinkras [4–9]. From this, the classification of off-shell 1D supersymmetries. Every 2D Adinkra is a product of these supersymmetric theories was outlined in [10]. 1DAdinkras,followedbyvertexswitchesandaquotienting Based on the success of this program, there have been a operation.Thisallowsustouseourknowledgeof1D few recent approaches to using Adinkra-like ideas to study Adinkras to completely understand 2DAdinkras. 2 Advances in High Energy Physics

We begin in Section 2 by recalling the definition of (1D) Adinkras and some of their features, reviewing the code 3 associated with an Adinkra [6] and the concept of vertex switching [8, 9]. As this paper is a mostly self-contained work of combinatorial classification, we do not discuss (or require from the reader) the physics and representation theory h 2 background relating to 1D Adinkras; the interested reader may see the Appendix and the aforementioned references for more information along these lines. Instead, Section 2’s goal is to provide the minimum background to understand and 1 manipulate Adinkras as purely combinatorial objects. Then, Sections 3–5 discuss 2D Adinkras: the definition, some basic constructions, and characterizing their codes. In Section6,weprovethemaintheorem,Hubsch’s¨ Conjecture Figure 1: Example of a 1D Adinkra with 3 colors. The coloring is mentioned above. represented by colors on the edges. The dashing is represented by having a dashed edge if 𝜇(𝑒) =1 andasolidedgeif𝜇(𝑒) =0.The Finally, Section 7, guided by the main theorem, sum- grading is represented by the vertical height as indicated on the axis marizes the basic structure of 2D Adinkras, including a on the left. (computable but impractical due to combinatorial explosion) scheme to generate all 2D Adinkras. We end with some remarks in Section 8. we require the parity of 𝜇 on every two-colored simple cycle to be odd. Such a dashing 𝜇 is called admissible, 2. Preliminaries (4) ℎ:𝑉→Z is a map called the grading.Werequirethat 2.1. 1D Adinkras. Adinkras in[4,9,13]willbereferredtoas if (V,𝑤)∈,then 𝐸 |ℎ(V)−ℎ(𝑤)|=1.Equivalently,ℎ 1D Adinkras in this paper, since they relate to supersymmetry provides a height function that makes (𝑉,𝐸) into the in 1 dimension. In this section, we review a definition of 1D Hasse diagram of a ranked poset. Adinkras and give some tools from previous work on their structural theory. The material in this section is mainly found Figure 1 gives an example of a 1D Adinkra. in [6, 9], with minor paraphrasing. 2.2. Structural Aspects of 1D Adinkras. Let 𝐴 be a 1D Adinkra Definition 1 (1D Adinkras). Let 𝑛 be a nonnegative integer. A with 𝑛 colors, with vertex set 𝑉.Forall𝑖∈[𝑛], define 𝑞𝑖 : 1D Adinkra with 𝑛 colors is (𝑉,𝐸,𝑐,𝜇,ℎ)where 𝑉→𝑉such that, for all V ∈𝑉, 𝑞𝑖(V) is the unique vertex joined to V by an edge of color 𝑖.In[6],itwasshownthatthe (1) (𝑉,𝐸) is a finite undirected graph (called the under- map 𝑞𝑖 is a graph isomorphism (in fact, an involution) from lying graph of the Adinkra) with vertex set (in [4, 13], the underlying graph of 𝐴 to itself which preserves colors. 𝑞𝑖’s there is also a bipartition of the vertices, where some commutewitheachother.Thesefactscanbeusedtocombine vertices are represented by open circles and called 𝑛 the 𝑞1,...,𝑞𝑛 maps into an action of Z on the graph (𝑉,𝐸) bosons, and other vertices are represented by filled 2 underlying the Adinkra in the following way. circles and called fermions. This is not necessary to include in our definition, because the bipartition can 𝑛 (𝑉,𝐸) ℎ Definition 2. The action of Z2 on the graph underlying be obtained directly by taking the grading modulo theAdinkraisgivenonverticesby 2, which is a bipartition by property (4) below) 𝑉 and 𝐸 edge set , 𝑥1 𝑥𝑛 (𝑥1,...,𝑥𝑛) V =𝑞1 ∘⋅⋅⋅∘𝑞𝑛 (V) . (2) (2) 𝑐 : 𝐸 → [𝑛]:={1,...,𝑛}is a map called the coloring. We require that, for every V ∈𝑉and 𝑖∈[𝑛],there Intuitively, the action of a sequence of bits, for instance, exists exactly one 𝑤∈𝑉so that (V,𝑤) ∈ 𝐸 and 11001, on a vertex is obtained by following edges with colors 𝑐(V,𝑤) =. 𝑖 We also require that every two-colored that correspond to 1’s in the sequence (in this case, colors 1, 2, 4 simplecyclebeoflength (A simple cycle is one which and 5). The fact that the 𝑞𝑖’s commute implies that the order does not repeat vertices other than the starting vertex; of the colors does not matter. 𝑛 A two-colored cycleisonewherethesetofcolorsofthe The Adinkra 𝐴 is connected if and only if Z2 action is edges has cardinality 2), transitive on the vertex set of 𝐴.Inthiscasethestabilizersof all vertices are equal (in general the stabilizers of two points (3) 𝜇:𝐸→Z2 ={0,1}is a map called the dashing.The inthesameorbitareconjugate;hereweknowmoresincethe parity of 𝜇 on a cycle given by vertices (V0,...,V𝑘) is group is abelian). Define 𝐶(𝐴),thecode of the Adinkra 𝐴,to defined as the sum be this stabilizer. This is a binary linear code of length 𝑛 (i.e., a 𝑛 𝑘−1 linear subspace of Z2). As these are the only types of codes we use,fromnowonwesimplysaycode to mean “binary linear ∑𝜇(V𝑖, V𝑖+1) (mod 2) , (1) 𝑖=0 code.” Advances in High Energy Physics 3

(a) (b) (c)

4 4 Figure 2: (a) The colored graph 𝐼 .(b)Thequotient𝐼 /{0000, 1111}.(c)Asthecode{0000, 1111} is doubly even, there exists an Adinkra with the quotient as its underlying graph by Theorem 3.

v We call the elements of a code codewords.Theweight of a v codeword 𝑤 isthenumberof1’sintheword.Acodeiscalled even if all its codewords have even weight. A code is called doubly even if all its codewords have weight divisible by 4. An exampleofadoublyevencodeisthespan⟨111100, 001111⟩, 2 which has 2 =4elements. An example of a code that is even but not doubly even is the 1-dimensional code ⟨11⟩. Codes are surprisingly relevant to the structural theory of A A Adinkras; in fact, one should basically think of the underlying (a) (b) graph of an Adinkra as a doubly even code, as we now see. Figure 3: A vertex switching at V turns the Adinkra 𝐴 on the left into the Adinkra 𝐴 on the right. The two Adinkras have the same 2.3. Quotients. We now know that the stabilizer of our action dashing except precisely the edges that are incident to V.Notethat on the graph is a code. We can also go in the opposite in both cases each face of the cube has an odd number of dashed 𝑛 𝑛 direction: let 𝐼 ,theHamming cube,bethegraphwith2 edges. vertices labeled by strings of length 𝑛 using the alphabet {0, 1}, with an edge between two vertices V and 𝑤 if and only if they 𝑛 differ in exactly one place. There is a natural coloring on 𝐼 : there exists an admissible dashing. A constructive proof of just color each edge by the coordinate where the two vertices existencecanbefoundin[7].See[9]foranenumerationof 𝑛 𝑛 differ. Now, codes in Z2 act on 𝐼 by bitwise addition modulo all admissible dashings for any doubly even code. 2, and these are isomorphisms that preserve colors. A natural operation to consider on a colored graph Γ = (𝑉,𝐸,𝑐)and a 2.4. Vertex Switching. Vertex switching was first introduced group 𝐶 acting on 𝑉 via graph isomorphisms that preserve in the context of Adinkras in [13] and is more thoroughly set colors is the quotient Γ/𝐶, where the vertices are defined to in its context in [8, 9]. be orbits in 𝑉/𝐶,andwehavean𝑐-colored edge (V,𝑤)if and 󸀠 󸀠 󸀠 only if there is at least one 𝑐-colored edge (V ,𝑤 )∈𝐸with V Definition 4 (vertex switching). Given an Adinkra 𝐴,and 󸀠 in the orbit V and 𝑤 in the orbit 𝑤. avertexV of 𝐴, we define vertex switching at V to be the operation on 𝐴 that returns a new Adinkra 𝐴 with the same 𝐼𝑛/𝐶 Theorem 3. is the colored graph of some 1D Adinkra if vertices, edges, coloring, and grading but a new dashing 𝜇 so 𝐶 and only if is a doubly even code. that

See [6] for the original proof of this result. See [9] for {1−𝜇(𝑒) , if 𝑒 is incident to V, a more general treatment of quotienting by a code and a 𝜇 (𝑒) = { (3) 𝜇 𝑒 , . slightly extended correspondence (note that the quotient Γ/𝐶 { ( ) otherwise does not necessarily retain nice properties of Γ;itdoesnot even have to be a simple graph. It may also have edges with We leave to the reader to check that 𝜇 is still an admissible different colors between two vertices. Part of the work here dashing; since the vertices, edges, coloring, and grading(s) is to show that these pathologies do not happen when 𝐶 is a are the same, 𝐴 remains an Adinkra (Figure 3). We also use doubly even code) between graph properties of the quotient 𝐴 𝑛 a vertex switching of to refer to a composition of vertex 𝐼 /𝐶 and properties of the code 𝐶. 𝐴 4 switchings at various vertices of . Figure 2 provides an example of a quotient of 𝐼 by a code that obeys this theorem. Encoded within the proof of In [14], vertex switching was first applied to dashings Theorem 3 is the fact that if 𝐶 is a doubly even code; then in Adinkras from a point of view inspired by Seidel’s two 4 Advances in High Energy Physics

(1, 1)

(1, 0) (0, 1)

(0, 0)

Figure 4: A 2D Adinkra with (2, 2) colors. The grading coordinates aregivennexttothenodesas(ℎ𝐿,ℎ𝑅). graphs [15]. (In Seidel’s setting, vertex switching switched the existence of edges, not the sign of edges; this can be seen as equivalent to our definition applied to the complete (a) (b) graph.Thetypeofvertexswitchingwedointhispaper 2 1 is sometimes called vertex switching on signed graphs in Figure 5: Constructing the D Adinkra (b) from two smaller D literature for disambiguation.) An enumeration of vertex Adinkras (a) as a product. Note that the dashings are all “consistent” with the smaller Adinkras, except for the right-moving edges on the switching classes leading to counting the number of dashings upper-left “boundary” of the rectangle; these correspond to right- of any 1D Adinkra can be found in [9]. moving edges where the grading corresponding to the first Adinkra has height 1. 3. 2D Adinkras

Just as 1D Adinkras were used to study 1D supersymmetry, 2 2 2 completely characterize D Adinkras. As a first step, we define Gates and Hubsch¨ developed DAdinkrastostudy D the natural notion of products in the following section. supersymmetry [11, 12]. We use a definition here that is equivalent to the one found there. (The main notational difference is a kind of change of coordinates: there, nodes are 4. Products labeled by mass dimension,whichisℎ𝐿 +ℎ𝑅,andspin,which One important way to produce a 2DAdinkrawith(𝑝, 𝑞) is ℎ𝑅 −ℎ𝐿. Mass dimension is the units of mass associated with the field, where 𝑐=ℏ=1and spin is the eigenvalue of colors is to take the product of two 1D Adinkras (one with 𝑝 colors, and the other with 𝑞 colors), using the following 𝑥𝜕𝑡 +𝑡𝜕𝑥.) A 2D Adinkra is similar to a 1D Adinkra except that some construction. colors are called “left-moving” and the other colors called Construction 6. Let 𝑝 and 𝑞 be nonnegative integers. Let “right-moving.” Edges are called “left-moving” if they are 𝐴1 =(𝑉1,𝐸1,𝑐1,𝜇1,ℎ1) be a 1D Adinkra with 𝑝 colors and let colored by left-moving colors and are called right-moving 𝐴2 =(𝑉2,𝐸2,𝑐2,𝜇2,ℎ2) be a 1D Adinkra 𝑞 colors. We define otherwise. Furthermore, there are two gradings, one that is the product of these Adinkras 𝐴1 ×𝐴2 as the following 2D affected by the left-moving edges and the other for the right- Adinkra with (𝑝, 𝑞) colors: movingedges.Moreformally,wehavethefollowing.

𝐴1 ×𝐴2 = (𝑉, 𝐸 , 𝑐, 𝜇,ℎ 1,ℎ2) , (4) Definition 5 (2D Adinkras). Let 𝑝 and 𝑞 be nonnegative integers. A 2D Adinkra with (𝑝, 𝑞) colors is a 1D Adinkra where 𝑉=𝑉1 ×𝑉2 and there are two kinds of edges in 𝐸: (𝑉,𝐸,𝑐,𝜇,ℎ) with 𝑝+𝑞colors, and two grading functions ℎ :𝑉→ ℎ :𝑉→ 𝐿 Z and 𝑅 Z so that we have the following: (i) For every edge 𝑒 in 𝐸1 connecting vertices V and 𝑤∈ 𝑉1, and for every vertex 𝑥∈𝑉2,wehaveanedgein𝐸 (i) ℎ(V)=ℎ𝐿(V)+ℎ𝑅(V). between vertices (V,𝑥)and (𝑤, 𝑥) in 𝑉=𝑉1 ×𝑉2 of 𝑒 𝑐(𝑒) ≤𝑝 𝑒 (ii) Let be an edge. If then is called a left- color 𝑐1(𝑒) and dashing 𝜇1(𝑒). 𝑐(𝑒) >𝑝 moving edge;if then it is called a right-moving 𝑒 𝐸 V 𝑤∈ 𝑝 (ii) For every edge in 2 connecting vertices and edge.Similarly,thefirst colors are called left-moving 𝑉 𝑥∈𝑉 𝐸 𝑞 2 and for every vertex 1,wehaveanedgein colors and the last colors are called right-moving (𝑥, V) (𝑥, 𝑤) 𝑉=𝑉×𝑉 colors. between vertices and in 1 2 of color 𝑝+𝑐2(𝑒) and dashing 𝜇2(𝑒) +1 ℎ (𝑥)mod ( 2). (iii) if (V,𝑤)is a left-moving edge, then |ℎ𝐿(V)−ℎ𝐿(𝑤)| = 1 and ℎ𝑅(V)=ℎ𝑅(𝑤).If(V,𝑤)is a right-moving edge, SeeFigure5foranexample. then |ℎ𝑅(V)−ℎ𝑅(𝑤)| = 1 and ℎ𝐿(V)=ℎ𝐿(𝑤). This definition is intended to be a graph-theoretic version See Figure 4 for an example of a 2DAdinkra.Themain of the tensor product construction in Z2-graded represen- goal of this paper is to follow the program set out in [12] and tations (see the Appendix for more details). The edges that Advances in High Energy Physics 5

come from 𝐸1 give rise to left-moving edges, and the edges Theorem 11. If 𝐴 isaconnected2DAdinkrawith(𝑝, 𝑞) colors, that come from 𝐸2 give rise to right-moving edges. It follows then 𝐶(𝐴) is an ESDE. easily that for every vertex V in 𝐴1 ×𝐴2 and for every color in [𝑛] there is a unique edge in 𝐴1 ×𝐴2 incident to V.The We now prove the converse of this theorem. That is, given fact that two-colored simple cycles have length 4 follows an ESDE code, there exists connected 2D Adinkra with that from following cases, depending on whether the colors are code.ThisprocedureisanalogoustotheValiseAdinkrasin1D both left-moving, both right-moving, or one of each. The [4, 13], in that the possible values of each component (ℎ𝐿,ℎ𝑅) parity condition for an Adinkra also follows from considering of the bigrading are as small as possible, that is, two values. these cases. The properties related to the bigrading are 𝐶 straightforward. We then have the following. Construction 12. Let be an ESDE code. We will describe a construction that provides a 2D Adinkra with code 𝐶, called 𝐶 Proposition 7. Given Adinkras 𝐴1 and 𝐴2, 𝐴1 ×𝐴2 is a 2D the Valise 2D Adinkra. First, since is doubly even, there Adinkra. exists a connected 1D Adinkra 𝐴 with code 𝐶(𝐴) =𝐶 by Theorem 3. Fix a vertex 0 of 𝐴. Now for every vertex V there 𝑝 𝑞 𝑛 Definition 8 (extending codes). Let and be nonnegative x ∈ Z x0=V ℎ𝐿(V)= 𝑝 𝑛 is a vector 2 so that . Then define integers and let 𝑛=𝑝+𝑞.Define𝑍𝐿 : Z2 → Z2 to be the wt𝐿(x)(mod 2) and ℎ𝑅(V)=wt𝑅(x)(mod 2).Notethatthese function that appends 𝑞 zeros, so that, for instance, if 𝑝=4 functions are well-defined since 𝐶 is ESDE. Then (ℎ𝐿,ℎ𝑅) is and 𝑞=3,then𝑍𝐿(1011) = 1011000. Likewise, define 𝑍𝑅 : 𝐴 2 𝑞 𝑛 a bigrading for ,makingita D Adinkra. An example of Z2 → Z2 to be the function that prepends 𝑝 zeros. the kind of 2D Adinkra that arises from this construction is Our most common use of this notation is as follows: if 𝐶 is Figure 4. a binary block code of length 𝑝,wewrite𝑍𝐿(𝐶) for the image under 𝑍𝐿.Likewise,if𝐶 is a binary block code of length 𝑞,we We therefore have the following. 𝑍 (𝐶) 𝑍 write 𝑅 for the image under 𝑅. 𝑛 Theorem 13. For a code 𝐶⊂Z2, there exists a 2D Adinkra 𝐴 Proposition 9. Let 𝐴1 and 𝐴2 be as above. Then with 𝐶(𝐴) =𝐶 if and only if 𝐶 is a ESDE code.

𝐶(𝐴1 ×𝐴2)=𝑍𝐿 (𝐶 (𝐴1)) ⊕ 𝑍𝑅 (𝐶 (𝐴2)) . (5) The structure of the ESDE relates to interesting features of the colored graph of 𝐴.Let𝐴𝐿 be the 1D Adinkra with 𝑝 𝑁 Proof. Let (V1, V2)∈𝐴1 ×𝐴2.Letx ∈ Z2 .Wecanwritex = colors that consists of only the left-moving edges of 𝐴.Let x𝐿 + x𝑅,wherex𝐿 is zero in the last 𝑞 bits and x𝑅 is zero in the 𝐴𝑅 be the 1D Adinkra with 𝑞 colors that consists of only the first 𝑝 bits. Now right-moving edges of 𝐴 (where we shift the colors so that they range from 1 to 𝑞 instead of 𝑝+1to 𝑝+𝑞).Pickavertex x (V1, V2) = (x𝐿 + x𝑅)(V1, V2) = (x𝐿V1, x𝑅V2) . (6) 0 0 in 𝐴.Let𝐴𝐿 be the connected component of 𝐴𝐿 containing 0 0 and let 𝐴 be the connected component of 𝐴𝑅 containing This means that x(V1, V2)=(V1, V2) if and only if x𝐿V1 = V1 and 𝑅 0 x𝑅V2 = V2.Sox ∈𝐶(𝐴1 ×𝐴2) if and only if x𝐿 ∈𝑍𝐿(𝐶(𝐴1)) . 𝐴0 𝐴0 and x𝑅 ∈𝑍𝑅(𝐶(𝐴2)). We now see that the codes for 𝐿 and 𝑅 (with an appropriate number of 0s added to the left or right as necessary) provide important linear subspaces of 𝐶(𝐴). 5. Codes for 2D Adinkras Proposition 14. The following hold: Let 𝐴 be a connected 2DAdinkrawith(𝑝, 𝑞) colors. Then there is a doubly even code 𝐶(𝐴) associated with 𝐴.Butasa 𝑍 (𝐶 (𝐴0 )) = 𝐶 (𝐴) ∩𝑍 ( 𝑝), 2D Adinkra, we make a distinction between the first 𝑝 colors 𝐿 𝐿 𝐿 Z2 and the last 𝑞 colors, which for a code translates to the first (7) 𝑍 (𝐶 (𝐴0 )) = 𝐶 (𝐴) ∩𝑍 ( 𝑞). 𝑝 bits and the last 𝑞 bits. A natural question is “knowing that 𝑅 𝑅 𝑅 Z2 a 1D Adinkra can be enriched into a 2D Adinkra, what else In other words, the codewords that are zero in the last can we say about its code?” In this section, we address this 0 question. 𝑞 bits are precisely the codewords from 𝐶(𝐴𝐿) with 𝑞 zeros appended to the right; and the codewords that are zero in the Definition 10 (weights for left-moving and right-moving 𝑝 𝐶(𝐴0 ) 𝑝 𝑛 first bits are precisely the codewords from 𝑅 with colors; ESDE codes). Recall that, for any vector x ∈ Z2, zeros prepended to the left. the weight of x, denoted by wt(x),isthenumberof1’s in x. ( ) 1 𝑝 Likewise, wt𝐿 x is the the number of ’s in the first bits and ∈𝑍(𝐶(𝐴0 )) =𝑍( ) ∈ ( ) 1 𝑞 𝐶 Proof. If x 𝐿 𝐿 ,sothatx 𝐿 y for some y wt𝑅 x is the number of ’s in the last bits of x.Letacode , 𝐶(𝐴0 ) ∈𝑍( 𝑝) 0=0 (𝑝, 𝑞) 𝐿 ,Thentriviallyx 𝐿 Z2 .Furthermore,y in alongwiththeparameters ,becalledaneven-split doubly 𝐴0 𝐴 𝑍 ( )= 0=0 even (ESDE) code if 𝐶 is doubly even and all codewords x in 𝐿.In , 𝐿 y x does exactly the same thing, so x in 𝐴. Therefore x ∈ 𝐶(𝐴). 𝐶 have both wt𝐿(x) and wt𝑅(x) even. 𝑝 Conversely if x ∈ 𝐶(𝐴)𝐿 ∩𝑍 (Z2 ), then by the definition This definition of ESDE codes is due to [12], which also of the group action there is a path in 𝐴 from 0 to 0 following 𝑝 proves the following. the colors corresponding to the 1s in x.Sincex ∈𝑍𝐿(Z2 ),we 6 Advances in High Energy Physics have that this path only consists of left-moving colors and so 0 lies in 𝐴𝐿. 0 The proof for 𝑍𝑅(𝐶(𝐴𝑅)) is similar.

Corollary 15. The following hold:

0 0 𝑍𝐿 (𝐶 (𝐴𝐿)) ⊕ 𝑍𝑅 (𝐶 (𝐴𝑅)) ⊂ 𝐶 (𝐴) . (8)

Comparing with Proposition 9, this corollary says that the 0 0 code for 𝐴𝐿 ×𝐴𝑅 is a linear subspace of the code for 𝐴. (a) (b) Simply for brevity (and thus, readability) in later descrip- 󸀠 2 tions, we define using the product construction above 𝐴 = Figure 6: (a) A D Adinkra with dashes omitted. (b) Picking any 0 0 󸀠 0 0 vertex and looking at connected components give us a pair of 1D 𝐴 ×𝐴 𝐶 =𝑍(𝐶(𝐴 )) ⊕ 𝑍 (𝐶(𝐴 )) 0 0 𝐿 𝑅 and the code 𝐿 𝐿 𝑅 𝑅 .Also, Adinkras 𝐴 and 𝐴 , the product of which contains as a quotient 𝐶 𝐶(𝐴) 𝐶(𝐴󸀠)=𝐶󸀠 𝐶󸀠 ⊂𝐶 𝐿 𝑅 we write for .Then and . the original Adinkra; in this case we quotient via a 1-dimensional 1 code 𝐾 of size 2 =2, obtaining the desired 8 × 4/2 = 16 vertices. Lemma 16. There exists a binary linear block code 𝐾 so that

󸀠 𝐶=𝐶 ⊕𝐾. (9) Example 18. Let 𝑝=4and 𝑞=2and let the generating matrix for 𝐶 be Proof. From Corollary 15 and basic linear algebra, there exists 𝑛 111100 a vector subspace 𝐾 of Z2 that is a vector space complement [ ]. (10) 󸀠 001111 of 𝐶 in 𝐶. 0 Then 𝑍𝐿(𝐶(𝐴𝐿)) has generating vector m =[1111|00] 𝑍 (𝐶(𝐴0 )) = { } Note that 𝐾 is not necessarily uniquely defined. It is, and 𝑅 𝑅 0 , the trivial code. We therefore see that 󸀠 0 however, uniquely defined up to adding vectors in 𝐶 .Soa 𝐴𝐿 is a 1D Adinkra with four colors with code with generating 𝐶/𝐶󸀠 0 more invariant approach would be to use instead of vector [1111] and 𝐴𝑅 isa1DAdinkrawithtwocolors 𝐾,but𝐾 has the advantage of being a code, therefore more with trivial code. The code 𝐾 canbechosentobegenerated concrete for computational purposes. by k =[0011|11].Anotherchoicefor𝐾 would have The interpretation of 𝐾 canbeobtainedbyexaminingthe been the code generated by 0 0 0 0 set 𝑉 =𝐴𝐿 ∩𝐴𝑅.Since𝐴𝐿 has only left-moving edges 0 0 k + m =[110011]. (11) and 𝐴𝑅 has only right-moving edges, 𝑉 has no edges at 0 all: only vertices. Furthermore, for every V ∈𝑉,wehave See Figure 6 for this example. To use Theorem 17, we start 0 0 3 ℎ𝐿(V)=ℎ𝐿(0) and ℎ𝑅(V)=ℎ𝑅(0) so all of the vertices in 𝑉 at andfollowanedgeofcolor andthenanedgeofcolor havethesamebigrading. 4, which uses the left-moving edges in 001111.Thisbringsus 0 0 Wenowshowthatthereisabijectionbetween𝐾 and 𝑉 . to a new vertex, which is in 𝑉 .Thisvertexand0 itself are the 𝑛 𝑉0 𝐾 As before, for every x ∈ Z2,wewritex = x𝐿 + x𝑅,wherex𝐿 is two elements of , corresponding to the two elements of . zero in the last 𝑞 bits and x𝑅 is zero in the first 𝑝 bits. Using this notation, we have the following theorem. Example 19. Let 𝑝=𝑞=4and consider the following gener- 0 ating matrix for 𝐶: Theorem 17. The map Ψ:𝐾→𝑉 given by Ψ(x)=x𝐿0 is a bijection. 11001100 [ ] [ 11111111] Proof. If x ∈ 𝐾 ⊂ 𝐶(𝐴),then(x𝐿 + x𝑅)0=0,sox𝐿0=x𝑅0.So [ ] 0 0 0 [ ] . (12) Ψ(x)∈𝐴𝐿 ∩𝐴𝑅 =𝑉. [ 00111100] To prove Ψ is one-to-one, suppose Ψ(x)=Ψ(y).Then 0 [ 10101010] x𝐿0=y𝐿0. Therefore x𝐿 + y𝐿 ∈𝐶(𝐴𝐿).Likewisex𝑅 + y𝑅 ∈ 0 󸀠 󸀠 ,..., 𝐶(𝐴𝑅).Sox + y ∈𝐶.Since𝐶 ∩𝐾={0},wehavethatΨ is If we let x1 x4 betherowsofthismatrix,wesee one-to-one. that x1 + x3 has zeros on the right side of the vertical 0 0 To prove Ψ is onto, let V ∈𝑉.SinceV ∈𝐴𝐿, there exists line. Other than the zero word, no other combination has 0 𝑍 (𝐶(𝐴0 )) x𝐿 with last 𝑞 bits zero, so that x𝐿0=V in 𝐴𝐿.Likewisethere all zeros on the right side, so 𝐿 𝐿 has generating 0 [1111|0000] 𝐶(𝐴0 ) exists x𝑅 with first 𝑝 bits zero, so that x𝑅0=V in 𝐴𝑅.Then matrix/vector and 𝐿 has gener- 󸀠 ating matrix/vector [1111]. x = x𝐿 + x𝑅 ∈ 𝐶(𝐴).Since𝐶(𝐴) =𝐶 ⊕𝐾,wecanwrite 󸀠 󸀠 + + x = c + k, where c ∈𝐶 and k ∈𝐾.Bythefactthat𝐶 = Likewise we can find x1 x2 x3 which is the unique 0 0 0 nonzero codeword with all zeros on the left side, and so 𝑍𝐿(𝐶(𝐴𝐿)) ⊕ 𝑍𝑅(𝐶(𝐴𝑅)),wehavethatc𝐿 ∈𝑍𝐿(𝐶(𝐴𝐿)) and 0 𝑍𝑅(𝐶(𝐴𝑅)) has generating matrix/vector so c𝐿0=0.ThenΨ(k)=k𝐿0=(x𝐿 − c𝐿)0=x𝐿(c𝐿(0)) = x𝐿(0) = V. [ 00001111] (13) Advances in High Energy Physics 7

0 0 󸀠 󸀠 0 0 and 𝐶(𝐴𝑅) has generating matrix [1111].Then 𝐴𝐿 and and 𝐶 =𝐶(𝐴)=𝑍𝐿(𝐶(𝐴𝐿)) ⊕ 𝑍𝑅(𝐶(𝐴𝑅)).Let𝐾 be a code 0 󸀠 𝐴𝑅 are both 1D Adinkras with 4 colors with code generated such that 𝐶=𝐶 ⊕𝐾(the existence of which is guaranteed by by 1111,and𝐾 canbetakentobe,forinstance, Lemma 16).

00111100 Theorem 21. Let 𝐴 be a connected 2D Adinkra. Then there is [ ]. (14) 𝐹 𝐾 𝐹(𝐴󸀠) 10101010 a vertex switching andanactionofthecode on that preserves colors, dashing, and bigrading, such that Standard arguments in linear algebra allow us to choose the generating basis for 𝐶 toconsistofageneratingbasisfor 󸀠 0 0 𝐹(𝐴 ) 𝑍𝐿(𝐶(𝐴 )),thenageneratingbasisfor𝑍𝑅(𝐶(𝐴 )),andthen 𝐴≅ (16) 𝐿 𝑅 𝐾 a generating basis for 𝐾. In this case, we would write 11110000 [ ] as an isomorphism of 2D Adinkras (i.e., as an isomorphism of [ 00001111] [ ] , graphs that preserves colors, dashing, and bigrading). [ 00111100] (15) We prove Theorem 21 in two steps, by first constructing [ 10101010] a (color and bigrading preserving) graph isomorphism Φ:̃ 󸀠 where the horizontal lines separate the three subspaces. Each 𝐴 /𝐾 → 𝐴 andthenfindingasuitablevertexswitching𝐹. 4 line has weight a multiple of ,wherethefirsttwolineshave 󸀠 Theorem 22. The code 𝐾 acts on 𝐴 via color preserving the 1s all on one side or the other, while the last two lines (the 󸀠 𝐾 isomorphisms to produce a quotient 𝐴 /𝐾.Thereisacolor ones responsible for ) have the 1s split on both sides in a way 󸀠 preserving graph epimorphism Φ:𝐴 →𝐴that sends (0, 0) that both sides have even weight. 󸀠 𝑉0 0 (00110000)0 (10100000)0 to 0.Thisdescendsto𝐴 /𝐾 to produce a color preserving graph Then has four elements: , , , ̃ 󸀠 and (10010000)0. isomorphism Φ:𝐴/𝐾 →. 𝐴 𝑛 Proof. Let 𝐼 be the colored Hamming cube: that is, a graph While not necessary for proving the main theorem of this 𝑛 paper, [12] also asked how to classify ESDE codes. It turns with vertex set {0, 1} and two vertices are connected with out that the answer is fairly concise. To do this, it is useful to an edge of color 𝑖 if they differ only in bit 𝑖.Recallfrom 𝑛 𝑝+𝑞 Theorem 3 that every connected Adinkra is, as a colored extend the notion of the splitting of into .Inparticular, 𝑛 graph, the quotient of 𝐼 by the code for the Adinkra. So we instead of insisting that the left-moving colors be written as 𝑛 𝑛 󸀠 󸀠 the first 𝑝 bits, we partition [𝑛] into [𝑛] = 𝐿 ∪𝑅 such that have 𝐼 /𝐶 ≅ 𝐴 and 𝐼 /𝐶 ≅𝐴. These are isomorphisms as |𝐿| = 𝑝 |𝑅| = 𝑞 𝑛 𝐶 colored graphs. They can be chosen so that 0 = (0,...,0)is and .Wefix and a doubly even code and 󸀠 𝐿 𝑅 𝐶 sent to 0 in 𝐴 and (0, 0) in 𝐴 ,respectively. then characterize which partitions into and make an 󸀠 Now 𝐾 is a doubly even code, and 𝐾∩𝐶 =0,so𝐾 ESDE. 𝑛 󸀠 󸀠 acts on 𝐼 /𝐶 and on 𝐴 in a way that nontrivial elements of Theorem 20. Given a doubly even code 𝐶 of length 𝑛,thereis 𝐾 move vertices a distance greater than 2.Bythecontentof ⊥ a bijection between codewords in 𝐶 and (ordered) partitions the proof of the extension of Theorem 2.3 in [9], this means 𝑛−𝑘 𝑛 󸀠 󸀠 𝐿∪𝑅that make 𝐶 into an ESDE. There are 2 such partitions. we can quotient the colored graph 𝐼 /𝐶 and thus 𝐴 ,by𝐾. We then have the following commutative diagram of colored Proof. Consider a partition 𝐿∪𝑅that makes 𝐶 into an ESDE, graphs: andconsiderthecodeword𝑤 that is defined to have 1 at all the positions in 𝐿 and 0 at all the positions in 𝑅.By definition of ESDE codes, all codewords in 𝐶 have an even number of 1’s in the support of 𝑤,whichisequivalentto n 󳰀 i1 󳰀 saying that 𝑤 is orthogonal to all the codewords in 𝐶.Thus, I /C ≅ A ⊥ ⊥ 𝑤∈𝐶.Conversely,forany𝑤∈𝐶, 𝑤 is orthogonal to all codewords in 𝐶 and thus gives an ESDE. Thus, there 𝜋1 𝜋2 is a bijection between the two sets. Note these are ordered 𝑅 i2 partitions; the codeword which has 1 at all the positions in (In/C󳰀)/K A󳰀/K and 0 otherwisewouldgivethesamepartition,butinreversed ≅ order.

6. Proof of Main Theorem A standard argument gives In this section we prove the main theorem of the paper. This 2 𝐴 (𝑝, 𝑞) refers to a connected DAdinkra with colors. As in 𝑛 󸀠 0 𝐴 𝐴0 𝐴0 (𝐼 /𝐶 ) 𝐼𝑛 𝐼𝑛 Section 5 we pick a vertex in and define 𝐿 and 𝑅.We ≅ = , 󸀠 0 0 󸀠 (17) use Construction 6 to define 𝐴 =𝐴𝐿 ×𝐴𝑅 and let 𝐶=𝐶(𝐴) 𝐾 (𝐶 ⊕𝐾) 𝐶 8 Advances in High Energy Physics and adding this to the above commutative diagram, we then Φ (resp., Φ̃) is a color and bigrading preserving epimorphism have (resp., isomorphism).

n 󳰀 i1 󳰀 I /C ≅ A Proof. This theorem builds on Theorem 22. Lemma 27 means 𝐾 𝐴󸀠 𝜋1 𝜋2 that the action of on preserves the bigrading. Lemma 26 provides the rest of this theorem. i (In/C󳰀)/K 2 󳰀 Φ ≅ A /K Unfortunately, it is too much to expect Φ to preserve the ≅≅i Φ̃ 󸀠 0 0 4 dashing or even that the dashing on 𝐴 =𝐴 ×𝐴 is invariant i 𝐿 𝑅 In/(C󳰀 ⊕K) 3 A 𝐾 𝐴󸀠/𝐾 ≅ under the action of (so that could have an obviously well-defined dashing). However, if we allow the operation of ̃ −1 ̃ ̃ where Φ=𝑖3 ∘𝑖4 ∘𝑖2 and Φ=Φ∘𝜋2.ThenΦ is an vertex switching,thenwecanbasicallyaccomplishthesegoals, isomorphism of colored graphs, and Φ is an epimorphism giving 𝐹 of Theorem 21. 0 0 of colored graphs. Standard diagram chasing shows that Consider the dashing 𝜇 on 𝐴.Thisrestrictsto𝐴𝐿 and 𝐴𝑅, Φ(0, 0) = 0 󸀠 0 . and Construction 6 produces a dashing 𝜇1 on 𝐴 =𝐴𝐿 × 0 󸀠 𝐴𝑅. The graph homomorphism Φ:𝐴 →𝐴pulls back the It will be useful to have the following result. 󸀠 dashing 𝜇 to 𝜇2 on 𝐴 .While𝜇1 and 𝜇2 can be different, they 𝑛 Lemma 23. If x ∈ Z2,thenΦ(x(V1, V2)) = xΦ(V1, V2). agreeonthefollowingpartsoftheAdinkra. 𝜇 𝜇 𝐴0 ×{0} Proof. For x = e𝑖,thevectorthatis1 in component 𝑖 and Lemma 29. The dashings 1 and 2 agree on 𝐿 and on 0 0 otherwise, this lemma is the statement that Φ is color- {0} × 𝐴𝑅. preserving. By composing many maps of this type, we get the 𝑛 0 statement for all vectors x ∈ Z2. Proof. The construction of 𝜇1 gives each edge in 𝐴𝐿 ×{0} the 0 same dashing as in 𝐴𝐿 under the association of every edge V ∈𝐴0 Φ(V, 0) = V 𝑤∈𝐴0 Lemma 24. For all 𝐿, ,andforall 𝑅, (V,𝑤)with ((V, 0), (𝑤, 0)).Lemma24showsthatthesameis Φ(0, 𝑤) = 𝑤 Φ 𝐴0 ×{0} 0 .Inparticular, restricted to 𝐿 is an true for 𝜇2. Therefore 𝜇1 and 𝜇2 agree on 𝐴𝐿 ×{0},likewise isomorphism onto its image and likewise for Φ restricted to {0} × 𝐴0 0 for 𝑅. {0} × 𝐴𝑅. Lemma 30. Two dashings 𝜇 and 𝜇 on an Adinkra 𝐴 have 𝑝 Proof. Let x ∈ Z2 be such that x0=V.ByLemma23, the same parity on all cycles (This type of result has a natural Φ(V,0)= Φ(𝑍𝐿(x)(0, 0)) = 𝑍𝐿(x)Φ(0, 0) = 𝑍𝐿(x)0=V.The reformulation with homological algebra, done in independent proof for 𝑤 is similar. ways by the first author’s work using cubical cohomology [8] and the second author’s work using CW-complexes [9]. In either 󸀠 Lemma 25. Let (V,𝑤) ∈ 𝐴 ,with𝑤=x0.ThenΦ(V,𝑤) = formulation, having parities of 𝜇 and 𝜇 agree on cycles is 𝑍𝑅(x)V. equivalent to 𝜇−𝜇2 =0in cohomology.) if and only if there is a vertex switching on 𝐴 that turns 𝜇 into 𝜇. Proof. We have Φ(V, 0) = V from Lemma 24. Acting on both sides with 𝑍𝑅(x),andusingLemma23,theresultfollows. Proof. Since vertex switching preserves parity on any cycle, the“if”directionistrivialanditsufficestoprovetheother Lemma 26. The graph epimorphism Φ preserves the bigrad- direction. ing. Assume that 𝜇 and 𝜇 havethesameparityonallcycles.It 𝐴 󸀠 suffices to prove the statement for connected, since we can Proof. Let (V,𝑤)∈𝐴 with 𝑤=x0. By Lemma 25, we get repeat our argument on each connected component of 𝐴. Next,wewillprovethat,foranytree𝑇 that is a subgraph ℎ𝐿 (Φ (V,𝑤)) =ℎ𝐿 (𝑍𝑅 (x) V). (18) of 𝐴, there exists a vertex switching 𝐹 on 𝐴 so that 𝐹(𝜇) and 𝜇 𝑇 Since 𝑍𝑅(x) follows right-moving colors, this does not affect agree on . This can be proved by induction of the number 󸀠 0 0 𝑇 𝑇 ℎ𝐿,andsotheaboveisequaltoℎ𝐿(V).In𝐴 =𝐴 ×𝐴 ,thisis of vertices in . The base case of one vertex is trivial. If has 𝐿 𝑅 V 𝑇 ℎ𝐿(V,𝑤). Therefore Φ preserves ℎ𝐿.Thefactthatitpreserves more than one vertex, then there is a leaf in incident to 𝑒 𝑇 𝑇 V ℎ𝑅 is proved similarly. only one edge in .Let 0 be the tree with the vertex and edge 𝑒 omitted. Then 𝑇0 has one fewer vertex than 𝑇 so by the 0 0 𝐹 𝐴 Lemma 27. If x ∈𝐾,and(V1, V2)∈𝐴 ×𝐴 ,thenx(V1, V2) inductive hypothesis, there is a vertex switching 0 on so 𝐿 𝑅 𝐹 (𝜇) 𝜇 𝑇 𝐹 (𝜇)(𝑒) ≠ 𝜇(𝑒) 𝐹 has the same bigrading as (V1, V2). that 0 and agree on 0.If 0 ,thenlet be the vertex switching 𝐹0 followed by a vertex switching at Proof. This follows from Lemma 26 and the fact that 𝐾⊂ V; otherwise let 𝐹=𝐹0.Then𝐹(𝜇) and 𝐹0(𝜇) agree on 𝑇0,so 𝐹(𝜇) 𝜇 𝑇 𝐶(𝐴),soifx ∈𝐾,thenxΦ(V1, V2)=Φ(V1, V2). agrees with on all of . Now in the case where 𝑇 is a spanning tree (so that it 󸀠 Theorem 28. The code 𝐾 acts on 𝐴 via color and bigrading is maximal), we claim that 𝐹(𝜇) and 𝜇 agree on all of 𝐴. 󸀠 preserving isomorphisms to produce a quotient 𝐴 /𝐾.Themap Consider any edge 𝑒 not in 𝑇. This edge completes at least one Advances in High Energy Physics 9

cycle with edges in 𝑇 (otherwise 𝑇 was not a spanning tree). j Sincethetwocycleshavethesameparityin𝐹(𝜇) and 𝜇 by 𝐹(𝜇) 𝜇 assumption, and the and agree on all edges in the cycle cj cj+1 except for 𝑒,theymustagreeon𝑒 as well. Thus, 𝐹(𝜇) = 𝜇 on all of 𝐴. cj−1 cj+2 󸀠 j−2 j−1 j+1 j+2 Nowwereturntothetwodashings𝜇1 and 𝜇2 on 𝐴 .Based on what we just proved, the following lemma will assure the cj+1 cj existence of a vertex switching that sends 𝜇1 to 𝜇2.Itusesthe 𝐴0 ×{0} {0} × 𝐴0 󳰀 fact that these two dashings agree on 𝐿 and 𝑅 j (Lemma 29). In terms of the cubical cohomology, this result Figure 7: An adjacent swap of colors. If we swap colors in position is a kind of Kunneth¨ theorem. 󸀠 𝑗 and 𝑗+1,vertexV𝑗 will be replaced by vertex V𝑗,andthepathgets 󸀠 Lemma 31. The parities of 𝜇1 and 𝜇2 agree on all cycles of 𝐴 . modified from the upper path to the lower path in the diagram. The parity of the path changes by exactly 1 modulo 2 because the square Proof. Let our cycle be (V0, V1,...,V𝑘) with V0 = V𝑘.Wefirst above has odd parity. consider the case where V0 = 0. For this proof, we define a color sequence of a path to (𝑐(V , V ), 𝑐(V , V ),...,𝑐(V , V )) bethesequenceofcolors 0 1 1 2 𝑘−1 𝑘 𝑝 V ≠ 0 𝐴0 ×𝐴0 V Now we consider loops where 0 .Since 𝐿 𝑅 is of edges along the path. Note that, given a starting vertex 0 𝑝 0 V 𝑝 V connected, there is a path 0 from to 0.Takethepath 0, and a color sequence, there is a unique path that starts at 0 𝑝 𝑝−1 𝑝 with that color sequence (Recall in Section 2.2 we treated this followed by , then followed by 0 (meaning 0 traversed in 𝑛 0 sequence of colors as a Z action on the underlying graph, theoppositesense).Thisisaloopstartingandendingin ,but 2 𝑝 where the order did not matter. In this proof we are not just the parity of a dashing is the same as that of , since every new 𝑝 𝑝−1 traversingthegraphbutalsokeepingtrackofthesignofthe edge in 0 is counterbalanced by a new edge in 0 . Therefore dashings,sowehavetokeepinmindtheorder.)Thisfollows the parities of 𝜇1 and 𝜇2 agree on all loops. by applying induction to property (2) of the definition of an Adinkra. We begin with the color sequence for the cycle We are now ready to put everything together and prove (V0,...,V𝑘). We will now describe a series of modifications our main theorem. to this cycle, described by modifying the color sequence. The idea is to perform a “bubble sort,” by iteratively swapping adjacent colors until the left-moving colors are all at the ProofofTheorem21.Let 𝐴 be a connected 2DAdinkra,and 0 0 󸀠 󸀠 ̃ beginningandtheright-movingcolorsareallattheend. define 𝐴𝐿, 𝐴𝑅, 𝐴 , 𝐶 , 𝐾, Φ,andΦ as above. 󸀠 First, given a color sequence Use Construction 6 to construct the 2DAdinkra𝐴 = 0 0 𝐴𝐿 ×𝐴𝑅 with dashing 𝜇1.ByTheorems22and28,thereis (𝑐 ,...,𝑐 ,𝑐,𝑐 ,𝑐 ,...,𝑐 ), 󸀠 1 𝑗−1 𝑗 𝑗+1 𝑗+2 𝑘 (19) a graph homomorphism Φ:𝐴 →𝐴that preserves colors and bigrading. If we take the dashing 𝜇 from 𝐴 and pull it 󸀠 an adjacent swap results in a color sequence back using Φ to a dashing 𝜇2 on 𝐴 ,thenLemmas31and30 together give the existence of a vertex switching 𝐹 sending 𝜇1 (𝑐1,...,𝑐𝑗−1,𝑐𝑗+1,𝑐𝑗,𝑐𝑗+2,...,𝑐𝑘). (20) to 𝜇2. 󸀠 󸀠 By Theorem 28, 𝐾 acts on 𝐴 to produce 𝐴 /𝐾,awell- Modifying a color sequence in this way leads to a new path defined 2D Adinkra without dashing and an isomorphism ̃ 󸀠 from 0.ThepathisunchangeduptoV𝑗−1, but by the definition Φ:𝐴/𝐾 → 𝐴 that preserves colors and the bigrading. Since 󸀠 of Adinkras, property (2),thepathreturnstoV𝑗+1 so it is only 𝜇2 is invariant under 𝐾,weobtain𝐹(𝐴 )/𝐾, a well-defined 󸀠 󸀠 V𝑗 that has changed (see Figure 7). Thus, the new path is still 2D Adinkra with dashing. Since 𝐹(𝐴 ) and 𝐴 only differ in ̃ acyclestartingat0.Theeffectontheparityofanydashingis, dashing, Φ is still an isomorphism that preserves colors and 𝜇 𝜇 𝐴 by property (3),toadd1modulo2.Inparticular,𝜇1 and 𝜇2 bigrading. Since 2 is obtained by pulling back from ,this are both affected in the same way. isomorphism preserves dashing as well. It is straightforward to find a series of adjacent swaps so that the left-moving colors are moved to the beginning of the 7. The Structure of 2D Adinkras color sequence. Then the resulting path starts from 0,stays 0 in 𝐴𝐿 ×{0}, and then follows right-moving edges, ending in Theorem 21 is very powerful; we immediately know alot 0. Since the right-moving edges end in 0,itmustbethatthe about what a 2D Adinkra must look like. Let the support 0 2 (ℎ ,ℎ ) right-moving edges are in {0} × 𝐴𝑅.ByLemma29,𝜇1 and 𝜇2 of a D Adinkra (and/or its bigrading function 𝐿 𝑅 )be are equal here, and so their parities on this modified path are defined as the range of (ℎ𝐿,ℎ𝑅), its bigrading function. Then the same. Therefore, their parities on the original loop were we have the following. the same. 10 Advances in High Energy Physics

Corollary 32. Let 𝐴 be a connected 2D Adinkra. The support 𝐴=(𝐴1×𝐴2)/𝐾 using Construction 6. This produces a graph of 𝐴 is a rectangle. That is, there exist integers 𝑥0, 𝑥1, 𝑦0,and with colors and a bigrading. The invariance of the gradings 𝑦1 such that the support is ℎ1 and ℎ2 makes this bigrading well defined under the

2 quotient. {(𝑖, 𝑗) ∈ Z |𝑥0 ≤𝑖≤𝑥1,𝑦0 ≤𝑗≤𝑦1}. (21) Finally,weputanadmissibledashingon𝐴.Thereisagain a finite number of possible dashings, so this is doable via 0 Proof. Since 𝐴𝐿 is a connected 1D Adinkra and edges change an exhaustive process. We can obtain dashings on the 1D 𝑛 ℎ𝐿 by 1, there are integers 𝑥0 and 𝑥1 so that the range of the Adinkra 𝐼 /𝐶 and use Φ to pull them back to 𝐴.Recallthe 0 grading is {𝑖 |0 𝑥 ≤𝑖≤𝑥1}.Likewise𝐴𝑅 has a range of discussion after Theorem 3 for relevant results. grading {𝑗 |0 𝑦 ≤𝑗≤𝑦1} for some integers 𝑦0 and 𝑦1. 󸀠 0 0 By Construction 6, the support of 𝐴 =𝐴𝐿 ×𝐴𝑅 is Theorem 35. Every2DAdinkracanbeobtainedbythis construction. {(𝑖,) 𝑗 |𝑥0 ≤𝑖≤𝑥1,𝑦0 ≤𝑗≤𝑦1} . (22) Proof. Given any 2DAdinkra𝐴,thereisanESDCcode𝐶. Φ 𝐴 0 0 Since preserves the bigrading, this is the support of as Pick a vertex 0 and define 𝐴𝐿 and 𝐴𝑅 as in Section 5. well. 0 0 Restrict the gradings ℎ𝐿 and ℎ𝑅 onto 𝐴𝐿 and 𝐴𝑅.Note that if 𝑔∈𝐶,then𝜋𝐿(𝑔)V =𝜋𝑅(𝑔)V,andsoℎ𝐿(𝜋𝐿(𝑔)V)= Proposition 33. Let 𝐴 be a connected 2D Adinkra. All ℎ𝐿(V) and ℎ𝑅(𝜋𝐿(𝑔)V)=ℎ𝑅(V). Therefore ℎ𝐿 restricted connected components of 𝐴𝐿 (and, resp., 𝐴𝑅) are isomorphic 0 to 𝐴𝐿 is invariant under 𝜋𝐿(𝐾).Likewiseℎ𝑅 restricted to as graded posets. 0 𝐴𝑅 is invariant under 𝜋𝑅(𝐾). The dashings, as described Proof. Consider a connected component of 𝐴𝐿.SupposeV is in [7], can be obtained by choosing the specific quotient 𝑛 𝐼𝑛/𝐶 𝐴 avertexinthisconnectedcomponent.Thenthereisax ∈ Z2 . Theorem 21 gives a description of in terms of this construction. so that x0=V.Thenthemap𝑓:𝐴𝐿 →𝐴𝐿 with 𝑓(𝑤) = x(𝑤) is a color-preserving graph isomorphism. Thus, it sends Example 36. Consider the code given by the generating connected components onto connected components and, 0 matrix/vector [1 1 | 1 1].Then𝐶𝐿 and 𝐶𝑅 are trivial, with in particular, 𝐴 to the connected component containing 𝐿 𝑝=𝑞=2,and𝐾 is generated by [1111]. V. As graphs, Adinkras 𝐴1 and 𝐴2 are both isomorphic to 2 From these results and the results of the previous section, 𝐼 .Therearenormally2 ways (up to relabeling of vertices) to 2 we now fully know what a 2DAdinkralookslike.Allofthe put a rank function on 𝐼 : graphical data of a 2D Adinkra is basically dictated by the connected components (one from left-moving colors and one from right-moving colors) of any single vertex and how they 3 aregluedtogether(thisiswhat𝐾 encodes); then these “slices” are put together into a rectangle.

7.1. Constructing 2D Adinkras. An alternate way to view Theorem 21 is as a way to construct all 2D Adinkras. h 2 Construction 34. First choose any doubly even code and any codeword in its orthogonal complement. Theorem 20 shows that this is exactly the amount of data we need to create an ESDC code 𝐶.Write

𝐶=𝐶𝐿 ⊕𝐶𝑅 ⊕𝐾, (23) 1

𝑝 𝑞 where 𝐶𝐿 =𝐶∩𝑍𝐿(Z2 ) and 𝐶𝑅 =𝐶∩𝑍𝑅(Z2).Usethe quotient construction involved in Theorem 3 to create the 1D 𝑝 𝑞 However, the height function must be invariant under Adinkras 𝐴1 =𝐼/𝜋𝐿(𝐶𝐿) and 𝐴2 =𝐼/𝜋𝑅(𝐶𝑅),where𝜋𝐿 : 𝜋 (𝐶) = 𝜋 (𝐶) = ⟨11⟩ 𝑛 → 𝑝 𝑝 𝜋 : 𝑛 → 𝑞 𝐿 𝑅 . In other words, moving once with an Z2 Z2 is projection onto the first bits and 𝑅 Z2 Z2 edge of both colors should not change the grading. Thus, both is projection onto the last 𝑞 bits. 𝐴 𝐴 ℎ 𝐴 1 and 2 can only be graded via the rank function depicted Second, we need a grading 1 on 1 that is invariant on the right. So they must look like this, after assigning colors: under 𝜋𝐿(𝐾) and a grading ℎ2 on 𝐴2 that is invariant under 𝜋𝑅(𝐾). There are a finite number of rank functions to consider for each graph, so it is definitely possible in principle to generate all gradings (For specific algorithms, one can use either the “hanging gardens” construction in [4] or consider the vertices of the “rank family poset” from [9].) though this is expectedtobealargesetforhigher𝑛.Constructthequotient Advances in High Energy Physics 11

We now take the product 𝐴1 ×𝐴2,whichhas4×4=16 fields, which have not yet played a role in this discussion. In vertices: onedimensiongaugefieldscanbegaugedawaytozero.In twodimensionsthisisnotthecase,butthecorresponding field strengths automatically satisfy Bianchi identities, and so the Adinkra formalism works well in this case. But in higher dimensions, gauge fields will be more difficult to avoid.) Beyond this, it is hoped that this work will generally help develop our knowledge of supersymmetry in two dimensions. Many mathematicians may appreciate Adinkras simply as nice combinatorial objects with lots of structure, with surprising links to coding theory, switching graphs, and graph coloring. This view presents additional questions, less relevant to the physics but still mathematically interesting, in the spirit of Theorem 20: (i) For example, it would be good to know how many To construct 𝐴1 ×𝐴2/𝐾,recallthat𝐾 is generated by [1111] ESDE’s are “compatible” with a 1DAdinkra𝐴 (i.e., , which means that each vertex is identified with 2 𝐴󸀠 the vertex that is obtained by following all four colors once. there is some DAdinkra with the same underly- 16/2 = 8 ing graph and rankings as 𝐴,withtherequiredESDE Weshouldnowgetagraphwith vertices.Ifweputan 1 admissible dashing on it, we obtain a complete 2DAdinkra. as its code) or with the family of DAdinkraswiththe One such choice recovers our Adinkra from Figure 4: same underlying graph. (ii) Enumerating all 2DAdinkrasisalsoanaturalgoal, though fairly ambitious (We still do not know how 𝑛 to count 1DAdinkraswith𝐼 as underlying graph 𝑛 beyond small 𝑛.For𝐼 /𝐶 with nontrivial 𝐶,wehave almost no data! The work in [9] basically settles dashings completely and gives some structural results on rankings, but counting rankings completely remains a very difficult problem, related to the chromatic polynomial for some families of 𝐶.), but counting all 2D Adinkras under some natural constraints may be fruitful. (iii) How often does the main theorem require no addi- 𝐹 8. Conclusion and Future Work tional vertex switching (i.e., is the identity)? In general, when given a dashing, we can ask related In this work, we have continued in the vein of [12] to questions about the minimum number of vertex study 2D Adinkras and provided stuctural results to study switches needed to produce a dashing with a well- them combinatorially. Describing already-known worldsheet defined quotient. supermultiplets in these terms could lead to new insights about these supermultiplets and lead to the discovery of new worldsheet supermultiplets. Appendix 4 One of the motivations for [13] was the idea of studying - Relation to Hübsch’s Original Language dimensional or higher-dimensional supermultiplets dimen- 1 sionally reduced to dimension. Likewise, the present work The statement of Theorem 21 is a bit different from the 2 allows us to study the dimensional reduction to dimensions, statement of the conjecture in [12]. There, the language was which might carry important information about the original partly in terms of representations of the 2D SUSY algebra supermultiplet. instead of graphs. As in [16], we could also consider nonadinkraic worldsheet supermultiplets. It would be interesting to see if, as in Construction A.1 ((off-shell) [12, Construction 2.1]). Let 𝑅+ the case of one dimension, there is a continuum of worldsheet and 𝑅− denote off-shell representations of two copies of the supermultiplets. Another direction to extend these results 1D SUSY algebra with 𝑝 and 𝑞 colors, respectively, and let is local supersymmetry, as in the case of supergravity or 𝑍 be a symmetry of 𝑅+ ⊗𝑅−, as a representation. The 𝑍- superconformal theories, which could be of importance to quotient of the tensor product (𝑅+ ⊗𝑅−)/𝑍 is then an off-shell superstrings. representation of 2D SUSY algebra with (𝑝, 𝑞) colors. Ofcourse,oneobviousstepistogotothreeorfour dimensions. It is expected that 𝑆𝑂(1, 𝑑 −1) representations The conjecture in [12] then says that every Adinkraic will play a role, and this work may begin to intersect [17, 18]. representation of 2DSUSYwith(𝑝, 𝑞) colors is obtained by (Higher dimensions is also a natural place to consider gauge this construction. 12 Advances in High Energy Physics

The relation between this conjecture and Theorem 21 will [5]C.F.Doran,M.G.Faux,S.J.GatesJr.,T.Hubsch,¨ and K. M. be apparent once we establish the following relationships. Iga, “Relating doubly-even error-correcting codes, graphs, and None of these facts are new to this paper, but this information irreducible representations of n-supersymmetry,” in Discrete is collected here for the convenience of the reader: and Computational Mathematics,F.Liu,G.M.N’Guerekata,D. Pokrajac, X. Shi, J. Sun, and X. Xia, Eds., chapter 5, pp. 53–71, (i) The relationship between off-shell representations of Nova Science, Hauppauge, NY, USA, 2008. SUSY and Adinkras is as follows: this is the central [6]C.F.Doran,M.G.Faux,J.Gatesetal.,“Codesandsuper- idea behind the original paper on Adinkras [13], and symmetry in one dimension,” Advances in Theoretical and so we do not go into detail here. The idea is that each Mathematical Physics,vol.15,no.6,pp.1909–1970,2011. vertex V of the Adinkra corresponds to a field 𝑓V in the [7]C.F.Doran,M.G.Faux,S.J.GatesJr.etal.,“Topologytypesof SUSY representation, and 𝑄𝑖 acts on fields by the edge Adinkras and the corresponding representations of N-extended of color 𝑖, with possible derivatives depending on the supersymmetry,” http://arxiv.org/abs/0806.0050. grading or bigrading (as the case may be), and with an [8]C.F.Doran,K.Iga,andG.D.Landweber,“Anapplicationof extra minus sign if the corresponding edge is dashed. cubical cohomology to adinkras and supersymmetry representations,” http://arxiv.org/abs/1207.6806. (ii) The relationship between 𝐴1 ×𝐴2 and 𝑅+ ⊗𝑅− is as [9]Y.X.Zhang,“Adinkrasformathematicians,”Transactions of the follows: the definition of 𝑅+ ⊗𝑅− as a representation of 2DSUSYthat American Mathematical Society,vol.366,no.6,pp.3325–3355, 2014. 𝑄𝑖 (𝑓V ⊗𝑓𝑤) =𝑄𝑖 (𝑓V) ⊗𝑓𝑤 (A.1) [10] C. F. Doran, T. Hubsch,¨ K. M. Iga, and G. D. Landweber, “On general off-shell representations of world line (1D) supersym- if 𝑖≤𝑝and metry,” Symmetry,vol.6,no.1,pp.67–88,2014. [11] S. J. Gates Jr. and T. Hubsch,¨ “On dimensional extension of |ℎ(V)| 𝑄𝑖 (𝑓V ⊗𝑓𝑤)=(−1) 𝑓V ⊗𝑄𝑖 (𝑓𝑤) (A.2) supersymmetry: from worldlines to worldsheets,” Advances in Theoretical and Mathematical Physics,vol.16,no.6,pp.1619– if 𝑖>𝑝. This is the standard way in which tensor 1667, 2012. products are defined in Z2-graded algebras [19, 20]. [12] T. Hubsch,¨ “Weaving worldsheet supermultiplets from the Then Construction 6 mimics this definition on the worldlines within,” Advances in Theoretical and Mathematical level of Adinkras. Physics,vol.17,no.5,pp.903–974,2013. V 𝑓 [13] M. G. Faux and S. J. Gates Jr., “Adinkras: a graphical technology (iii) A vertex switching at corresponds to replacing V for supersymmetric representation theory,” Physical Review D, −𝑓 𝑄 (𝑓 ) 𝑓 with V. Then all equations involving 𝑖 V or V vol. 71, no. 6, pp. 1–21, 2005. will get an extra minus sign. This reverses all dashings V [14] B. L. Douglas, S. J. Gates, B. L. Segler, and J. B. Wang, “Auto- on edges connected to . morphism properties and classification of adinkras,” Advances (iv) A quotient defined in Theorem 21 is a symmetry 𝑍 of in Mathematical Physics, vol. 2015, Article ID 584542, 17 pages, the representation. 2015. [15] J. J. Seidel, “Asurvey of two-graphs,”in Colloquio Internazionale sulleTeorieCombinatorie(Rome,1973),TomoI,AttideiCon- Conflict of Interests vegni Lincei no 17, pp. 481–511, Accademia Nazionale dei Lincei, Rome,Italy,1976. The authors declare that there is no conflict of interests regarding the publication of this paper. [16] T. Hubsch¨ and G. A. Katona, “A Q-continuum of off-shell supermultiplets,” Advances in High Energy Physics,vol.2016, Article ID 7350892, 11 pages, 2016. Acknowledgments [17] M. G. Faux, K. Iga, and G. D. Landweber, “Dimensional enhancement via supersymmetry,” Advances in Mathematical The authors wish to thank Charles Doran, Sylvester Gates, Physics, vol. 2011, Article ID 259089, 45 pages, 2011. Tristan Hubsch,¨ and Richard Eager for helpful conversation. [18] M. G. Faux and G. D. Landweber, “Spin holography via dimensional enhancement,” Physics Letters B,vol.681,no.2,pp. References 161–165, 2009. [19]R.BottandL.W.Tu,Differential Forms in Algebraic Topology, [1] J. Wess and J. Bagger, Supersymmetry and Supergravity, Prince- vol. 82 of Graduate Texts in Mathematics, Springer, New York, ton University Press, Princeton, NJ, USA, 1992. NY, USA, 1982. [2] S. J. Gates Jr., J. Phillips, W. Linch, and L. Rana, “The fun- [20] D. S. Freed, Five Lectures on Supersymmetry, American Mathe- damental supersymmetry challenge remains,” Gravitation and matical Society, Providence, RI, USA, 1999. Cosmology,vol.9,no.1-2,pp.96–100,2002. [3] S. J. Gates Jr., W.Linch, and J. Phillips, “When superspace is not enough,” http://arxiv.org/abs/hep-th/0211034. [4]C.F.Doran,M.G.Faux,S.J.GatesJr.,T.Hubsch,¨ K. M. Iga, and G. D. Landweber, “On graph-theoretic identifications of Adinkras, supersymmetry representations and superfields,” International Journal of Modern Physics A,vol.22,no.5,pp.869– 930, 2007. Hindawi Publishing Corporation Advances in High Energy Physics Volume 2016, Article ID 4303752, 17 pages http://dx.doi.org/10.1155/2016/4303752

Review Article Aspects of Moduli Stabilization in Type IIB String Theory

Shaaban Khalil, Ahmad Moursy, and Ali Nassar

Center for Fundamental Physics, Zewail City of Science and Technology, Giza 12588, Egypt

Correspondence should be addressed to Ahmad Moursy; [email protected]

Received 1 October 2015; Revised 28 December 2015; Accepted 30 December 2015

Academic Editor: Elias C. Vagenas

We review moduli stabilization in type IIB string theory compactification with fluxes. We focus on KKLT and Large Volume Scenario (LVS). We show that the predicted soft SUSY breaking terms in KKLT model are not phenomenological viable. In LVS, the following result for scalar mass, gaugino mass, and trilinear term is obtained: 𝑚0 =𝑚1/2 =−𝐴0 =𝑚3/2, which may account for Higgs mass limit if 𝑚3/2 ∼ O(1.5) TeV. However, in this case, the relic abundance of the lightest neutralino cannot be consistent with the measured limits. We also study the cosmological consequences of moduli stabilization in both models. In particular, the associated inflation models such as racetrack inflation and Kahler¨ inflation are analyzed. Finally, the problem of moduli destabilization and the effect of string moduli backreaction on the inflation models are discussed.

1. Introduction models, the 10D gauge group (𝐸8 ×𝐸8 or 𝑆𝑂(32))isbrokento Standard-Model-like gauge groups by turning on background Ever since the invention of the Kaluza-Klein mechanism, gaugefieldsontheinternalspace.Thechiralfermionsare it was realized that extradimensional models are plagued obtained from the dimensional reduction of the 10D gaugino with massless scalar fields when compactified to 4D. In the and the number of the generations is half the Euler charac- 5 original Kaluza-Klein construction, the radius of the 𝑥 - teristic of the internal space [2]. These compactifications are 𝜇 circle, 𝑅(𝑥 ) is not fixed by the dynamics and appears as purely geometric and give rise to an N =1supersymmetric ascalarfieldwithnopotentialintheeffective4Dtheory. (SUSY) 4D theories with large number of moduli. Other Thisisagenericfeatureofmostcompactificationsofhigher- compactifications which leads to N =1SUSY in 4D are dimensional gravitational/Yang-Mills theories. The parame- type II theories on Calabi-Yau orientifolds with D-branes ters which describe the shape and size of the compactification and fluxes, M-theory on manifolds with 𝐺2 holonomy, and manifold give rise to massless scalar moduli with no potential F-theory on Calabi-Yau fourfold (see [3] for a review and attreelevelin4D(flatdirections),thatis,nonstabilized references therein). moduli fields. These moduli are gravitationally coupled and One essential fact about our universe is that it has a small as such their existence would be in conflict with experiment positive cosmological constant, that is, a de Sitter space-time. (see[1]forareview). The moduli scalar potential 𝑉(𝜙) should be such that its String theory being a candidate for a unified theory of minimum produces the observed value of the cosmological the forces of nature must be able to reproduce the physics of constant. This turns out to be a very difficult problem since our real world which is 4-dimensional with a small positive de Sitter vacua are known to break supersymmetry. The first cosmological constant and chiral gauge interactions. This attempt at finding realistic vacua of string theory was mainly amounts to finding de Sitter vacua of string theory with concerned with Minkowski and Anti-de Sitter vacua [2, 4– allmodulistabilized.Theseminalworkin[2]constructed 6].Itwasshownthatturningonmagneticfluxesinthe the first models of Grand Unified Theories (GUTs) from internal manifold leads to a nontrivial warp factor and a non- Calabi-Yau compactification of the heterotic string. In these Kahler¨ geometry. The situation improved drastically since 2 Advances in High Energy Physics

𝑚 the introduction of D-branes as nonperturbative objects in where 𝑦 are the coordinates on M with the metric 𝑔𝑚𝑛(𝑦) string theory. This resulted in the celebrated KKLT scenario and the requirement of Poincaresymmetryof´ 𝑀4 still allows in which a moduli fixing mechanism was introduced [7]. The for a warp factor which depends on M only. LargeVolumeScenariofollowedafter[8].Therein,onecan We also require an N =1supersymmetry in 4D turn on a vacuum expectation value for the fluxes in the since this is phenomenologically appealing and gives more internal space without breaking the 4D Lorentz invariance. analytic control. For 𝑀4, one considers homogenous and In this paper, we review the flux compactifications and isotropic maximally symmetric solutions which implies that moduli stabilization in type IIB string theory. The Kachru- the Riemann tensor takes the form Kallosh-Linde-Trivedi (KKLT) [7] scenario was a major step 𝑅 =𝑐(𝑔 𝑔 −𝑔 𝑔 ), in constructing dS vacua with all the moduli stabilized 𝜇]𝜌𝜆 𝜇𝜌 ]𝜆 𝜇𝜆 ]𝜌 (2) including the volume modulus. The Large Volume Scenario 𝑐 𝑔𝜇𝜌𝑔]𝜆 (LVS) by Quevedo et al. [9, 10] was proposed as an alter- where is fixed by contracting both sides with and 𝑅/12 native to stabilize all the moduli with the volume moduli turns out to be equal to .Theconstantscalarcurvature 𝑅 𝑅=0 𝑅<0 𝑅>0 stabilizedatextremelylargevolume.Realisticmodelsof could be (Minkowski), (AdS), or (dS). M moduli stabilization must come as close as possible to the When the radius of curvature of is large compared to observed phenomenology at low energy and also to account the Planck scale, one can use the supergravity approximation for cosmological inflation at high energy scales. From this of string theory. In order to have a supersymmetric back- point of view, we analyze the low energy phenomenology of ground, the supergravity transformations of the fermions both KKLT and LVS. Weemphasize that these models provide must vanish [2] specific set of soft SUSY breaking terms, which are not 𝛿𝜀 (Fermions) =0. (3) phenomenologically viable. In addition, we study the impact of these scenarios on inflation. It turns out that inflation could For heterotic string theory, the supergravity variations of the destabilize the moduli again. This problem has been analyzed fermions are given by [2] in details in [11, 12]. On the other hand, moduli stabilization may have backreaction effects on the inflationary potential, 1 𝛿𝜓𝑀 =∇𝑀𝜀− H𝑀𝜀, which could change the inflationary parameters. 4 This paper is organized as follows. In Section 2, we review 1 1 𝛿𝜆 =− ¡𝜕Φ𝜀 + 𝜀, the flux compactification and moduli stabilization in type IIB 2 4H (4) string theory. Section 3 is devoted to KKLT model and its variants, where we discussed several examples for vacuum 1 𝛿𝜉= − F𝜀, uplifting. In Section 4, we present the LVS as an alternative 2 scenario that overcomes some of the KKLT drawbacks. where 𝜓𝑀,𝜆,and𝜉 are the gravitino, dilatino, and the The phenomenological implications and SUSY breaking soft 𝐻 =𝛾 𝐻 termsofthesetwomodelsarestudiedinSection5.In gaugino, respectively. The -flux is H𝑀 𝑁𝑃 𝑀𝑁𝑃 and H =𝛾𝑀𝑁𝑃 𝐻𝑀𝑁𝑃 ;andF =𝛾𝑀𝑁𝐹𝑀𝑁 is Yang-Mills field Section 6, we highlight the cosmological implications of these 𝐸 ×𝐸 𝑆𝑂(32) two models. Finally, we state our conclusions in Section 7. strength of the 8 8 or gauge fields in 10D. The Bianchi identity of 𝐻 is given by 𝛼󸀠 𝑑𝐻 = [ (𝑅∧𝑅) − (𝐹∧𝐹)] . (5) 2. Flux Compactifications and Moduli 4 Tr Tr Stabilization in String Theory Since 𝑑𝐻 is exact, that is, zero in cohomology, then the We start by reviewing the heterotic string compactification cohomology classes of Tr(𝑅 ∧ 𝑅) and Tr(𝐹 ∧ 𝐹) are the same. which has been considered to connect string theory to The vanishing of the above variations for a given spinor four-dimensional physics [2]. Then, we discuss type IIB 𝜀 will put some restrictions on the background fields and fluxcompactifications,wherethecomplexstructuremoduli in particular on the geometry and topology of M.The (CSM)andthedilatonarestabilizedbytheRRandNS-NS compactification of heterotic string theory with vanishing 𝐻- 3-form fluxes. flux (or vanishing torsion) was first studied in [2, 5] and it leads to the following conditions on the string background: 2.1. Heterotic String Compactification. To compactify string 𝛿𝜓 =0󳨀→𝐺 =𝜂 , theory down to 4D, one looks for vacuum solutions of 𝜇 𝜇] 𝜇] (6) the form 𝑀10 =𝑀4 ×𝑀6,where𝑀4 is assumed to 𝐴(𝑦) 𝑒 = constant. have 4D Poincare´ invariance and 𝑀6 (or simply M)isa compact internal 6D Euclidean space. The most general That is, the external space 𝑀4 is Minkowski with a constant metric compatible with these requirements can be written as warp factor. [2] The dilatino variation gives 𝐴(𝑦) 𝑒 𝜂𝜇] 0 𝜕𝑚Φ=0. (7) 𝐺𝑀𝑁 =( ), (1) 0𝑔(𝑦) 𝑚𝑛 That is, the dilaton is constant over M. Advances in High Energy Physics 3

The gravitino variation 𝛿𝜓𝑚 gives In the string frame, 𝑆𝑏, 𝑆cs,and𝑆𝑙 are given by

∇𝑚𝜀=0. (8) 1 10 −2𝜙 2 This equation says that M admits a covariantly constant 𝑆 = ∫ 𝑑 𝑥√−𝑔 {𝑒 [R +4(∇𝜙) ] 𝑏 (2𝜋)7 𝛼󸀠4 spinor. The integrability condition resulting from the above equation implies that M is Ricci flat 2 𝐹2 1 𝐹̃ 𝑅 =0. − 1 − 𝐺 ⋅ 𝐺 − 5 }, 𝑚𝑛 (9) 2 2⋅3! 3 3 4⋅5! Hence, the first Chern class of M vanishes 1 1 𝑆 = ∫ 𝑒𝜙𝐶 ∧𝐺 ∧ 𝐺 , (12) 𝑐 = [R] . cs 7 󸀠4 4 3 3 1 2𝜋 (10) 4𝑖 (2𝜋) 𝛼 It was conjectured by Calabi and proved by Yau that Ricci- 𝑝+1 −𝜙 𝑆𝑙 = ∑ (− ∫ 𝑑 𝜉𝑇𝑝𝑒 √−𝑔 flat compactahler K¨ manifolds with 𝑐1 =0admit a metric sources R4×Σ with 𝑆𝑈(3) holonomy. These metrics come in families and are 𝑇 parameterized by continuous parameters 𝑖 which defines the +𝜇𝑝 ∫ 𝐶𝑝+1), 4 shape and sizes of M.Theparameters𝑇𝑖 appear as scalar fields R ×Σ (moduli) in 4D with no potential and a major goal in string theory is to generate a potential which stabilizes these moduli where 𝑇𝑝 and 𝜇𝑝 are, respectively, the tension and charge of in a way that is consistent with observations. the D𝑝-brane. The string tension is expressed in terms of Onecandescribethe4DN =1models resulting string length as from the heterotic string compactification in terms of an effective SUSY theory. This theory is characterized bya 1 𝑙 2 𝛼󸀠 = = ( 𝑠 ) . Kahler¨ potential 𝐾, a gauge kinetic function 𝑓,anda 2 (13) 𝑀st 2𝜋 superpotential 𝑊. The tree-level superpotential 𝑊 does not 𝑊 fixthemoduli.Duetononrenormalizationtheorems, is 𝐹̃ not renormalized at any order in perturbation theory [14, 15]. The 5-form 5 is defined as This means that if supersymmetry is unbroken at tree level, ̃ 1 1 it will remain unbroken at all orders of perturbation theory. 𝐹5 =𝑑𝐶4 − 𝐶2 ∧𝐻3 + 𝐹3 ∧𝐵2, (14) Nonperturbative effects such as gaugino condensation [16] 2 2 can correct the superpotential and fix some of the moduli. It is worth mentioning that there is an alternative way whichisselfdualandsatisfiesBianchiidentity to stabilize the moduli if the string model is nonsupersym- ̃ metric. In this case, perturbative corrections can generate a 𝑑𝐹5 =𝐻3 ∧𝐹3. (15) one-loop potential for the moduli, for example, a vacuum energy. For example, in the nonsupersymmetric tachyon- One would like to consider warped compactifications of free heterotic string models studied in [17–19], the one- type IIB on a compact manifold M.Themetricansatzfora loop vacuum energy was shown to be finite and extremized 4D warped compactification is given by [22] at the symmetry-enhancement points in the moduli space. 9 The one-loop potential can stabilize some of the moduli 2 𝑀 𝑁 fields. Another way to generate masses for the moduli is by 𝑑𝑠10 = ∑ 𝐺𝑀𝑁𝑑𝑥 𝑑𝑥 the breaking of supersymmetry through a Scherk-Schwartz 𝑀,𝑁=0 (16) mechanism [20]. 2𝐴(𝑦) 𝜇 ] −2𝐴(𝑦) 𝑚 𝑛 =𝑒 𝜂𝜇]𝑑𝑥 𝑑𝑥 +𝑒 𝑔𝑚𝑛𝑑𝑦 𝑑𝑦 . 2.2. Type IIB Compactification. We now turn our discussion to type IIB string theory. The massless bosonic spectrum The 10D Einstein equation of motion is of type IIB consists of the metric 𝑔𝑀𝑁,RR0-form𝐶0,and 𝜙 𝑆= 2 1 scalar dilaton which are combined into the axiodilaton 𝑅 =𝜅 (𝑇 − 𝐺 𝑇) , (17) −𝜙 −𝜙 𝑀𝑁 𝑀𝑁 8 𝑀𝑁 𝐶0 +𝑖𝑒 , where the string coupling is given by 1/𝑔𝑠 =𝑒 . In addition, the spectrum contains RR 2-form 𝐶2 and 4-form where 𝐶4,aswellastheNS2-form𝐵2. It is convenient to combine the RR and NS 3-forms 𝐹3 =𝑑𝐶2 and 𝐻3 =𝑑𝐵2 into 𝑇 =𝑇sugra +𝑇𝑙 𝐺3 =𝐹3 −𝑆𝐻3. The classical action of type IIB supergravity 𝑀𝑁 𝑀𝑁 𝑀𝑁 (18) 0 𝑆IIB is divided into a bulk action 𝑆𝑏,theChern-Simonsaction 𝑆cs, and contributions from the D-brane sources 𝑆𝑙 [21] (more is the total stress tensor of supergravity plus the localized precisely, 𝑆𝑙 represents the action of the localized sources for objects; that is the case of a D𝑝-brane wrapping a (𝑝−3)-cycle Σ) 𝑙 2 𝛿𝑆𝑙 𝑆0 =𝑆 +𝑆 +𝑆. 𝑇 =− . (19) IIB 𝑏 cs 𝑙 (11) 𝑀𝑁 √−𝐺 𝛿𝐺𝑀𝑁 4 Advances in High Energy Physics

In this regard, the space-time components of the latter which is minimized at 𝑅=𝑁,andif𝑁 islargethecurvatureis action reduce to small and the supergravity approximation is reliable [25, 27]. 2𝐴 In string theory, additional ingredients beside the fluxes are 𝑒 󵄨 󵄨2 2 4𝐴 󵄨 󵄨2 −6𝐴 2 󵄨 4𝐴󵄨 needed to construct stable vacua. These ingredients are the ∇ 𝑒 = 󵄨𝐺3󵄨 +𝑒 (|𝜕𝛼| + 󵄨𝜕𝑒 󵄨 ) 12 Im 𝜏 D-branes and orientifold planes [28]. (20) 2 𝜅 𝑙 The main idea of flux compactification is that there are + 10 𝑒2𝐴 (𝑇𝑚 −𝑇𝜇) , solutions of the string tree-level equations in which some 2 𝑚 𝜇 of the 𝑝-form fields are nonzero in the vacuum. In these where 𝛼 is a function on the compact space. Integrating both constructions, one needs to make sure that the backreaction sides of this equation over the compact manifold M,theleft- of the flux on the geometry does not take us outside the hand side gives zero since it is a total derivative. If there are no supergravity approximation. This turns out to be possible localized sources, then the right-hand side is a sum of positive [5] with the introduction of a warp factor varying over the terms and vanishes only if 𝛼 and 𝐴 are constants and 𝐺3 =0. internal manifold and hence the new geometry is conformal This is the familiar no-go theorem of flux compactifications to the nonflux case. The fluxes, which can be turned on, are 𝐻 [23, 24]. However, the existence of localized sources in string the RR fluxes of type II and the 3 flux. In this case, the theory like orientifold planes can balance the contribution quantization condition on the fluxes is coming from fluxes to give a nontrivial warp factor. This was 1 𝐹 ∈ Z, realized in string theory in [22]. The setup in [22] allows for 𝑝 ∫ 𝑝+1 (25) 𝑙 Σ a stabilization of complex structure moduli by turning on RR 𝑠 𝑝+1 andNSfluxesintheinternalspace[25,26]. where the integrality of the cohomology classes of 𝐹𝑝+1 is due to Dirac’s charge quantization. The nonvanishing 2.3. Type IIB Fluxes and Moduli Stabilization. One way to see of the cohomology classes of these 𝑝-form fields leads to the problem in a simple setting is nicely reviewed in [25, 27] obstructions which lifts some of the flat directions of the in a toy model and we review it here. One can generate a compactification; that is, it leads to potential which freezes potential for the moduli by turning on fluxes in the internal some of the moduli. space. The potential in 4D results from the Maxwell term of In the presence of sources, the modified Bianchi identity the fluxes now reads [28] 𝜇 ] 𝜇 ] 𝑉=∫ √−𝑔𝑔 1 1 ⋅⋅⋅𝑔 𝑝 𝑝 𝐹 𝐹 2 𝑎 2 𝑂3 𝜇1⋅⋅⋅𝜇𝑝 ]1⋅⋅⋅]𝑝 𝑑𝐹̃ =𝐻 ∧𝐹 +2𝜅 𝜇 ∑𝜋 +2𝜅 𝑄 𝜇 𝜋 . M 5 3 3 10 3 6 10 3 3 6 (26) (21) 𝑎 = ∫ 𝐹 ∧∗𝐹 , 𝑝 𝑝 Integrating this equation over the compact internal manifold, one gets the tadpole cancelation condition [28] where 𝑔 is the metric of the internal space. The metric 𝑔 will depend on the moduli of the internal space and, after 𝑁flux +𝑁D3 +𝑄3𝑁𝑂3 =0, (27) doing the integral on M, one gets a potential for the moduli 𝑉(𝜙). For example, consider a 6D Maxwell-Einstein theory where 𝑆2 𝐹 compactified on a two-sphere with a nonzero flux of 2 1 𝑆2 𝑁 = ∫ 𝐻 ∧𝐹. piercing flux 4 3 3 (28) 𝑙𝑠 M ∫ 𝐹=𝑁. (22) 𝑆2 The type IIB string theory will be compactified on Calabi-Yau orientifolds in order to obtain a 4D N =1 This flux contributes a positive energy to the effective 4D model. It turned out that one needs to make an orientifold potentialwhichcanthenbalancethenegativecontribution projection in order to have supersymmetric compactification 2 coming from the curvature of 𝑆 . More specifically, the [28]. The orientifold action projects out one of the two contribution to the effective potential coming from the flux gravitinos and breaks the N =2SUSY down to N = originates from the Maxwell term in 6D 1. The orientifold projection also introduces O-planes with a background charge and a negative energy density which 2 𝑁 balances the contribution of D-branes and leads to stable 𝑉 (𝑅) = ∫ 𝐹 ∧∗𝐹 ∼ , (23) 2 2 𝑅6 compactification. The RR and NS-NS 3-form fluxes are restricted via the integral cohomology which determines the 2 where the determinant of the metric contributes a factor of 𝑅 quantized background fluxes as follows: 2 and two metric contractions contribute a factor of 1/𝑅 while 4 1 the transformation to the Einstein frame gives a factor 1/𝑅 . ∫ 𝐹 =𝑛 ∈ Z, 2 󸀠 3 𝑎 Therefore, total 4D potential takes the form (2𝜋) 𝛼 Σ𝑎 (29) 𝑁2 1 1 𝑉 (𝑅) = − , ∫ 𝐻3 =𝑚𝑏 ∈ Z, 6 4 (24) 2 󸀠 𝑅 𝑅 (2𝜋) 𝛼 Σ𝑏 Advances in High Energy Physics 5

where Σ푎,푏 are3cyclesoftheCalabi-Yaumanifold.Inthis The above discussion shows that the no-scale structure case, tadpole condition (27) reads does not fix the value of the volume modulus 𝑇;thatis, the modulus 𝑇 is a flat direction. The stabilization of 𝑇 is 𝜒 (M) 1 =𝑁 + ∫ 𝐻 ∧𝐹, of uttermost importance in order for string theory to make D3 4 耠2 3 3 (30) 24 (2𝜋 )𝛼 M contact with realistic models. This issue will be addressed in the upcoming sections. where 𝑁D3 is the net number of (D3−D3) branes filling the 𝜒(M) noncompact dimensions and is the Euler characteristic 3. KKLT and Its Variants of the elliptically fibred Calabi-Yau fourfold M. The compactification to 4D on a Calabi-Yau manifold In order to stabilize the volume modulus 𝑇,anonperturbative N =1 with orientifold planes will give rise to an supergravity superpotential was considered by Kachru, Kallosh, Linde, 𝐾 theory which is characterized by a Kahler¨ potential ,a and Trivedi (KKLT) [7]. The source of these nonperturba- 𝑊 𝑓 superpotential , and a gauge kinetic function [29– tive terms could be either D3-brane instantons or gaugino 31]. The tree-level Kahler¨ potential is given by the Weil- condensation from the nonabelian gauge sector on the D7- Petersson metric using the Kaluza-Klein reduction of type IIB branes. As advocated in the previous section, the dilaton and supergravity the complex structure moduli are stabilized at a high scale by the flux induced superpotential and hence their contribution 𝐾=−3 [(𝑇 + 𝑇)] − [(𝑆 + 𝑆)] ln ln to the superpotential is a constant 𝑊0.Thus,thetotaleffective (31) superpotential is given by − ln [−𝑖 ∫ Ω∧Ω] . M −푎푇 𝑊=𝑊0 +𝐴𝑒 . (36) 𝑇 Here, represents the volume modulus and is one of the 𝑎=2𝜋 2𝜋/𝑁 Kahler¨ moduli. The conditions on the fluxes in type IIBare The coefficient or is the correction arising from D3 instantons or 𝑆𝑈(𝑁) gaugino condensation, and 𝐴 O(1) ∗𝐺3 =𝑖𝐺3, is constant of order . In addition, the Kahler¨ potential is given by 0,3 (32) 𝐺3 =0. 𝐾=−3ln [𝑇 + 𝑇] . (37) Since the hodge ∗ depends on the metric, then one expects the above conditions can fix the geometric moduli except for Here, 𝑇=𝜏+𝑖𝜓,where𝜏 is the volume modulus of the inter- theoverallscaleofthemetricsincethehodge∗ is conformally nal manifold and 𝜓 is the axionic part. A supersymmetric invariant in six dimensions. This leaves the overall volume minimum is obtained by solving the equation of the compactification manifold undetermined. These conditions can be derived from a superpotential given by the 𝐷푇𝑊=0󳨐⇒ Gukov-Vafa-Witten (GVW) form [32] −푎휏 2 (38) 𝑊 =−𝐴𝑒 0 (1 + 𝑎𝜏 ), 0 3 0 𝑊=∫ Ω∧𝐺3. (33) M where 𝜏0 is the value of 𝜏 that minimizes the scalar potential. This superpotential depends on the complex structure mod- Substituting this solution in the potential uli through Ω and is independent of the Kahler¨ moduli. 𝑁=1 The supergravity scalar potential is given by 3 󵄨 󵄨2 𝑉=𝑒퐾 ( 󵄨𝐷 𝑊󵄨 −3 𝑊 2), 2 󵄨 푇 󵄨 | | (39) (𝑇 + 𝑇) 퐾 푎푏 2 𝑉=𝑒 (∑𝑔 𝐷푎𝑊𝐷푏𝑊 −3|𝑊| ) , (34) 푎,푏 one finds the following negative minimum:

𝑎, 𝑏 2 2 −2푎휏 where runoverallthemoduli.Duetotheno-scale 󵄨 𝑎 𝐴 𝑒 0 AdS 퐾 2 󵄨 structure of the Kahler¨ potential (31), the sum over Kahler¨ 𝑉0 =(−3𝑒 |𝑊| )󵄨 =− . (40) 2 󵄨휏0 6𝜏 moduli cancels the term 3|𝑊| and the potential (34) reduces 0 to the no-scale structure [33–35] The scalar potential as a function of 𝜏=Re(𝑇) is given by (the imaginary part Im(𝑇) = 𝜓 is frozen at zero) 𝑉 =𝑒퐾 (∑𝑔푖푗𝐷 𝑊𝐷 𝑊) , no-scale 푖 푗 (35) −2푎휏 푎휏 푖,푗 𝑎𝐴𝑒 (𝐴 (𝑎𝜏) +3 +3𝑊𝑒 ) 𝑉 (𝜏) = 0 . (41) 6𝜏2 where 𝑖, 𝑗 runoverthedilatonandcomplexstructuremoduli. Accordingly, the dilaton and complex structure moduli can It is important to uplift this AdS minimum to a be stabilized in a supersymmetric minimum by solving the Minkowski or a dS minimum, as shown in Figure 1, in equation 𝐷푎𝑊=0which may have a solution for generic order to have realistic models. The uplift of the above AdS choice of the flux. In this case, 𝑊=𝑊0 =0at the minimum. vacuum to a dS one will break SUSY where one needs another 6 Advances in High Energy Physics

2 where Σ is the 4 cycles on which D7-branes wrap, 𝑇7 is the tension of D7-branes, and 𝐸 measures the strength of the flux. 1 Accordingly, the D-term scalar potential is given by 𝑔2 2 푌푀 2 2𝜋 𝐸 󵄨 󵄨2 𝑉퐷 = 𝐷 = ( + ∑ 𝑞푖 󵄨𝐶푖󵄨 ) , (46) V 2 𝜏 𝜏 30 35 40 45 V0 𝜏 where 𝐶푖 are charged matter fields with charges 𝑞푖.Thesefields −1 can be minimized at 𝐶푖 =0and accordingly the full potential will be

−2 𝑉=𝑉퐹 +𝑉퐷 (47)

3 2 3 Vup =E2/𝜏 𝑉 =2𝜋𝐸/𝜏 2 with 퐷 and we end up with similar behavior to Vup =E2/𝜏 V the KKLT uplifted potential. AdS Another approach for uplifting uses the F-term uplifting 29 Figure 1: The scalar potential 𝑉(𝜏) (multiplied by 10 )with𝑊0 = [13, 44]. In this case, SUSY will be broken spontaneously in −12 −10 , 𝐴=1,and𝑎=1. The blue curve shows the AdS minimum, the F-term moduli sector which in turn generates an uplift while the green and red curves exhibit the uplifting to dS minimum term for the AdS KKLT stabilized volume. For example, in 𝛿𝑉 =𝐸 /𝜏3 𝐸 /𝜏2 𝐸 = 3.5 × 10−25 𝐸 = via 3 , 2 ,respectively,with 3 and 2 [13], the Kahler¨ potential contains a modulus 𝑇 and a matter 1.13 × 10−26 . field 𝐶 andhastheform 2 𝐾=−3ln (𝑇 + 𝑇) + |𝐶| . (48) contribution to the potential which usually has dependence −푛 like 𝜏 [36] The effective superpotential [13] is given by −훼 푇 푛 𝑊=∑𝜔 (𝐶) 𝑒 𝑖 +𝜙(𝐶) , 󵄨 AdS󵄨 𝜏0 푖 (49) 𝛿𝑉 (𝜏) ≈ 󵄨𝑉 󵄨 . (42) 푖 󵄨 0 󵄨 𝜏푛 where 𝜔푖(𝐶) and 𝜙(𝐶) arefunctionsofthematterfields In this case, a new minimum is obtained due to shifting 𝜏0 to resulting due to integrating out heavy fields with index 𝑖 that 𝜏耠 =𝜏 +𝜀 𝜀 0 0 ,where is given by runs over the gaugino condensates. The scalar potential will be minimized at [13] 1 𝜀≃ . 𝑎2𝜏 (43) 𝐶=0, 0 (50) 𝑇=𝑇0. Since the consistency of the KKLT requires that 𝜏0,𝑎𝜏0 ≫1 [7], the shift in the minima is much small and we can calculate Figure 2 shows the shape of the potential near the minimum 𝜏 physical quantities such as masses in terms of 0. 𝑇=𝑇0, 𝐶=0. This model is different from the approach There are many proposals for such uplifting, for example, studied in [48], where the effects of charged chiral fields that adding anti-D3-branes [7], D-term uplift [37–43], F-term reside on D3-branes and D7-branes [49] were considered. In uplift [13, 44], and Kahler¨ uplift [41, 45–47]. In the original that case, new 𝑇-dependence will be generated in the Kahler¨ KKLT scenario [7], some anti-D3-branes were added which potential contributes along with additional part to the scalar potential 󵄨 󵄨2 󵄨⟨𝐶3⟩󵄨 𝐸 𝐾=−3ln (𝑇 + 𝑇) − 3 ln (𝑆 + 𝑆) + 𝛿𝑉 = 3 . (44) (𝑇 + 𝑇) 𝜏3 (51) 󵄨 󵄨2 󵄨⟨𝐶7⟩󵄨 One of the drawbacks of this mechanism is that SUSY is + , broken explicitly due to the addition of the anti-D3-branes. (𝑆 + 𝑆) In this case, the effective 4D theory cannot be recast into 𝐶 𝐶 the standard form of 4D supergravity and this in turn makes where 3, 7 are charged chiral matter fields. In addition, the itverydifficulttohavealowenergyeffectivetheory[37]. superpotential does not contain any nonperturbative effects An uplifting mechanism via a D-term scalar potential was 𝑊=𝐴+𝐵𝑆. (52) proposed in [37] where the possible fluxes of gauge fields living on the D7-branes were used. In this case, the fluxes Therefore, an AdS minimum is obtained which is uplifted by induce a term in the 4D effective action of the form aD-termassociatedwiththegaugesymmetryofthematter fields. 2𝜋𝐸2 𝑇 ∫ 𝑑4𝑦 𝑔 𝐹푚푛𝐹 = , Another way for uplifting to dS vacua of the volume stabi- 7 √ 8 푚푛 3 (45) Σ 𝜏 lized moduli is by Kahler¨ uplift models where the perturbative Advances in High Energy Physics 7

2.02 𝜏 2.00 1.98

0.04 0.10 0.03 4 0.05 0.02

) ,C)

C 0 0.00 V(𝜏, 𝜓) 0.01 2

Re( V(T 0.00 0 −0.05 0.05 0.10 0.05 0.00 0.00 −0.05 −0.10 𝜓 −0.05 Im(C) −0.10

(a) (b)

2 Figure 2: The minimum of the potential of the scenario [13] at 𝑇=2, 𝐶=0. (a) corresponds to 𝑉(𝑇, 0) rescaled by 1/𝜖 and (b) corresponds 4 2 to 𝑉(𝑇0,𝐶)rescaled by 10 /𝜖 .

corrections to the Kahler¨ potential will paly an important the scalar potential will be simplified to the form role in constructing dS vacua [41, 45–47]. Now, we consider 1,1 2,1 −𝑊 𝑎3𝐴 2𝐶 𝑒−푥 a model proposed in [46] where ℎ =1and ℎ >1so that 𝑉 (𝑥) ≃ 0 ( − ). 1,1 2,1 2 2 (57) the Euler characteristic is 𝜒=2(ℎ −ℎ )<0.Inthiscase, 2𝛾 9𝑥9/2 𝑥 the Kahler¨ potential and superpotential are In order to have stable dS vacuum, 𝐶 must satisfy the ̂ constraint [46] ̂ 耠3 𝜉 𝐾=−2ln (V +𝛼 ), 2 3.65 ≲ 𝐶 ≲ 3.89. (58) (53) −푎 푇 𝑖 𝑖 𝐶 < 3.65 𝑊=𝑊0 + ∑𝐴푖𝑒 , ThisisclarifiedinFigure3,where,forvaluesof ,we 푖 have AdS minimum and, for values of 𝐶 > 3.89,thevolume is destabilized. ̂ 3/2 where V = 𝛾(𝑇 + 𝑇) is the normalized volume, with 𝛾= √ 3/(2√𝜅) and 4. Moduli Stabilization in Large 𝜒 (M) 𝜁 (3) 3/2 Volume Scenario ̂𝜉=− (𝑆+𝑆) . √ 3 (54) 4 2 (2𝜋) Another alternative scenario for moduli stabilization based 𝜏 𝑁=1 on a Large Volume Scenario has been proposed by Quevedo Accordingly, the dependence of supergravity scalar et al. [9, 10] in order to overcome some of the drawbacks of potential is given by [46, 50, 51] theKKLTmodel.Basically,increasingthenumberofKahler¨ 𝛼耠 𝑉 (𝑡) moduli will worsen the situation when corrections are neglected. The LVS was built on the proposal that the number of complex structure moduli is bigger than the number of 퐾 푇푇 2 2 −2푎푡 −푎푡 2,1 1,1 =𝑒 (𝐾 [𝑎 𝐴 𝑒 +(−𝑎𝐴𝑒 𝑊𝐾푇 + c.c)] Kahler¨ moduli; that is, ℎ >ℎ >1,aswellastheinclusion 耠耠3 (55) of 𝛼 corrections. O(𝛼 ) contribution to the Kahler¨ potential 2 2 (after integrating out the dilaton and the complex structure ̂𝜉 +7̂𝜉V̂ + V̂ +3̂𝜉 |𝑊|2), moduli) is given by [50–52] ̂ ̂ 2 (V̂ − 𝜉)(𝜉+2V̂) 3/2 (𝑆 + 𝑆) [ −3휙/2 𝜉 ] 𝐾훼󸀠 =−2log 𝑒 V + ( ) +𝐾cs, (59) where 𝜓 stabilizes at 𝜓=𝑛𝜋/𝑎for 𝑛 = 0, 1, ..Usingthe . 2 2 ̂ ̂ −푎푡 [ ] approximations V ≫ 𝜉 and |𝑊0|≫𝐴𝑒 and defining the 3 quantities with 𝜉=−𝜁(3)𝜒(M)/2(2𝜋) and 𝜙 is the type IIB dilaton. Here, V is defined as the classical volume of the manifold M 𝑥=𝑎⋅𝑡, which is given by −27𝑊 ̂𝜉𝑎3/2 (56) 𝐶= 0 , 3 1 푖 푗 푘 √ V =∫ 𝐽 = 𝜅푖푗푘 𝑡 𝑡 𝑡 , (60) 64 2𝛾𝐴 M 6 8 Advances in High Energy Physics

10 where the volume moduli are 𝑇4 =𝜏4 +𝑖𝜓4 and 𝑇4 =𝜏5 +𝑖𝜓5 𝑡 𝜏 =𝑡2/2 𝜏 =(𝑡 +6𝑡 )2/2 8 and the link to 푖 is given by 4 1 and 5 1 5 .In this respect, the Kahler¨ potential and the superpotential, after 6 fixing the dilaton and complex structure moduli, are given in the string frame by V 4 𝜉 𝐾=𝐾 −2 (V + ), 2 cs log 2 (65)

−(푎4/푔𝑠)푇4 −(푎5/푔𝑠)푇5 2 4 6 8 10 𝑊=𝑊0 +𝐴4𝑒 +𝐴5𝑒 . −2 𝜏 In the large volume limit, V ∼𝜏5 ≫𝜏4 >1,thebehavior C=3.97 C=3.65 of the scalar potential is given by [9, 10] C=3.89 C=3.50 2 C=3.75 −2푎4휏4/푔𝑠 √𝜏4 (𝑎4𝐴4) 𝑒 𝑉(V,𝜏4)= Figure 3: Potential of one modulus model for the Kahler¨ uplift. The V 𝐶 (66) red curves correspond to the limits on the parameter for a dS −푎 휏 /푔 2 𝑊 𝜏 𝑎 𝐴 𝑒 4 4 𝑠 𝜉𝑊 vacuum, while the dotted one corresponds to 𝐶 in the destabilization − 0 4 4 4 + 0 . region. V2 V3

Minimizing the potential (66), one finds the following: where 𝐽 is the Kahler¨ class and 𝑡푖 are the moduli that measure 𝜕𝑉 𝜕𝑉 𝑖 = 1,2,...,ℎ = =0. (67) theareasof2cycleswith 1,1. The complexified 𝜕V 𝜕𝜏4 Kahler¨ moduli are defined as 𝑇푗 ≡𝜏푗 +𝑖𝜓푗,where𝜏푗 are the four-cycle moduli defined by the relation andsolvingthetwoequationsinthetwovariables,𝜏4, V,one cangetoneequationin𝜏4 1 푘 푙 𝜏푗 =𝜕푡 V = 𝜅푗푘푙𝑡 𝑡 . (61) 𝑗 2 3𝐵 𝐵 1 (1±√1− 3 1 )( −2𝑎𝜏 ) = (1−𝑎𝜏 ) , Since the superpotential is not renormalized at any order 3/2 2 4 4 4 4 (68) 耠 𝐵2𝜏4 in perturbation theory, it will not receive 𝛼 corrections. But there is a possibility of nonperturbative corrections, which where may depend on the Kahler¨ moduli (as in the KKLT model) via 2 󵄨 󵄨2 D3-brane instantons or gaugino condensation from wrapped 𝐵1 ∼𝑎4 󵄨𝐴4󵄨 , D7-branes. Accordingly, the superpotential is given by 󵄨 󵄨 𝐵 ∼𝑎 󵄨𝐴 𝑊 󵄨 , −푎 푇 2 4 󵄨 4 0󵄨 (69) 𝑊=𝑊 + ∑𝐴 𝑒 𝑖 𝑖 , 0 푖 (62) 푖 󵄨 󵄨2 𝐵3 ∼𝜉󵄨𝑊0󵄨 . 𝐴 𝑊 where 푖 are model dependent constants and again 0 is the 𝑎 𝜏 ≫1 value of the superpotential due to the geometric flux after Using the assumption 4 4 ,whichisnecessarytoneglect stabilizing the dilaton and the complex structure moduli. In higher order instanton corrections [9], the solution is given this respect, the scalar potential will take the form [50, 51] by 2/3 4𝐵3𝐵1 퐾 푇𝑗푇𝑘 (푎𝑗푇𝑗+푎𝑘푇𝑘) 𝜏 =( ) , 𝑉=𝑒 [𝐾 (𝑎푗𝐴푗𝑎푘𝐴푘𝑒 4 2 𝐵2 (70) 1/3 푖푎𝑗푇𝑗 −푖푎𝑘푇𝑘 +𝑎푗𝐴푗𝑒 𝑊𝜕 𝐾−𝑎푘𝐴푘𝑒 𝑊𝜕푇 𝐾) + 3𝜉 (63) 𝐵2 4𝐵3𝐵1 푎 (4퐵 퐵 /퐵2)2/3 푇𝑘 𝑗 V = ( ) 𝑒 4 3 1 2 . 2𝐵 𝐵2 (𝜉2 +7𝜉V + V2) 1 2 2 ⋅ 𝑊 ]≡𝑉 +𝑉 +𝑉󸀠 , 2 | | 푛푝1 푛푝2 훼 (V −𝜉)(2V +𝜉) Substituting for 𝐵푖 by their expressions, we have

耠 2/3 where 𝛼 correction is encoded in 𝑉훼󸀠 . The simplest example 𝜏4 ∼ (4𝜉) , that can realize the notion of LVS [9, 10] is the orientifold of 4 1,1 2,1 1/3 󵄨 󵄨 (71) 𝑃 ℎ =2 ℎ = 272 𝜉 󵄨𝑊0󵄨 푎 휏 /푔 [1,1,1,6,9] for which and and therefore the V ∼ 𝑒 4 4 𝑠 . volume is given by 𝑎4𝐴4

1 3/2 3/2 V = (𝜏5 −𝜏4 ) , (64) Therefore, the potential possesses an AdS minimum at 9√2 exponentially large volume V since it approaches zero from Advances in High Energy Physics 9

mechanisms. The gravitino mass at the dS minimum is given by

Small volumes 󵄨 𝑚 =𝑒퐾/2 |𝑊|󵄨 󳨐⇒ 3/2 󵄨dS

𝑎𝐴 −푎휏 𝑊 (74) Large volume 𝑚 ≃ 𝑒 0 ≃ 0 . 3/2 1/2 3/2 3(2𝜏0) (2𝜏0)

Therefore, we have gravitino mass of order ∼1TeV if (𝑎𝜏0)∼ −12 32 and hence 𝑊0 ∼10 .Intermsoftheshifts𝜀 (given in (43)) of 𝜏0, an approximate expression for 𝐷푇𝑊 near the dS minimum is given by Figure 4: Swiss-cheese structure in Large Volume Scenario. 3√2 𝐷푇𝑊(𝜏0 +𝜀)=(𝐷푇𝑊)휏 𝜀≃𝑊푇,푇𝜀= 𝑚3/2, (75) 𝑎√𝜏0 below in the limit 𝜏5 →∞and 𝜏4 ∝ log(V).Namely,inthe latter limit, the potential has the form 𝑊2 which is of the same order as the gravitino mass. Accordingly, 𝑉= 0 (𝐶 √ (V) −𝐶 (V) +𝜉V). (72) the soft SUSY breaking terms are given by V3 1 ln 2 ln

Therefore, the negativity of the potential requires a very 2 󵄨 2 |𝑊| 󵄨 2 large ln(V). Still one has to uplift this minimum by one of 𝑚 = 󵄨 =𝑚 , 0 (2𝜏)3 󵄨 3/2 the mechanisms mentioned in Section 3. This result can be 󵄨dS generalizedtomorethantwomoduliwhereoneofthemtakes 󵄨 𝑚 √2𝜏 𝜕 ∗ 󵄨 3/2 a large limit while other moduli stay small. This structure will 𝑚 = 𝐷 𝑊 (𝑇) ( 𝑓 )󵄨 ≃ , (76) 1/2 6 푇 𝜕𝑇 ln Re 󵄨 𝑎𝜏 form what is called the Swiss-cheese form of the CY manifold, 󵄨dS 󵄨 as depicted in Figure 4. In this case, the volume will take the 1 󵄨 3𝑚3/2 𝐴 =− 𝐷 𝑊 󵄨 =− , form 0 푇 ℎ󵄨 √2𝜏 󵄨dS 𝑎𝜏 V =𝜏3/2 − ∑𝜏3/2. 푏 s,푖 (73) 푖 wherethegaugekineticfunction𝑓푎푏 canbechosensuch that 𝑓푎푏(𝑇) = 𝑓(𝑇)𝛿푎푏, which will lead to universal gaugino 5. SUSY Breaking and masses. The gauge kinetic function is considered to have Phenomenological Consequences alineardependanceonthemodulusfieldwhichcanbe derived from the reduction of Dirac-Born-Infeld action for In this section, we will study SUSY breaking in the mod- an unmagnetized brane [62, 63]. ulisectorandthepropertiesofthecorrespondingsoft The soft terms indicate that SUSY breaking in KKLT isa terms (more general expressions of soft term for generic special example of the constrained MSSM (CMSSM), where superpotentials and Kahler¨ potentials in supergravity and all the soft terms are given in terms of one free parameter stringmodelsaregivenin[53,54].)thatareinducedinthe (𝑚3/2) which is of order TeV.Note that 𝑎𝜏0 is fixed as 𝑎𝜏0 ≃32. observable sector, and then we summarize the phenomeno- It is well known that this type of soft terms cannot account for logical consequences. SUSY breaking in models of KKLT the experimental constraints imposed by the Large Hadron compactification type with phenomenological consequences Collider (LHC) and relic abundance of the lightest SUSY has been extensively studied in [36, 40, 55–59], while, for LVS particle. Even if we relax the Dark matter constraints, the type, SUSY breaking was studied in [10, 60–65]. We would Higgsmasslimit(∼125 GeV) and gluino mass limit (≳1.4 TeV) like to remark that in studying MSSM-like theories arising will imply 𝑚3/2 ≃ O(30) TeV. Thus, all SUSY spectrum will be from large volume compactifications one faces the issue of the quiteheavywhichisbeyondtheLHCsensitivity.Afeature validity of perturbation theory. In large volume models, one of this model is the fact that gauginos are lighter than the has to worry about the Kaluza-Klein towers which become sfermions by at least one order of magnitude. However, if light when the compactification volume is large. If some of one checks the parameter space for such set of soft terms, he these KK modes are charged under the gauge symmetry, they finds that tadpole equations at the TeV scale are not satisfied. can push the gauge and Yukawa couplings to large values and Namely, condition [66] the perturbative control is lost. 𝑀2 (i) Soft Terms in KKLT Scenario. As shown in models of KKLT 2 푍 2 2 𝜇 + ≃ −0.1𝑚0 +2𝑚1/2, (77) type in Section 3, SUSY is broken by one of the uplifting 2 10 Advances in High Energy Physics

2 cannot account for positive 𝜇 . Therefore, the above set of soft 130 terms failed to describe TeV scale phenomenology. 125

(ii) Soft Terms in LVS Scenario. Let us now consider the 120 same scenario for SUSY breaking in LVS model, namely, with 4 𝑃[1,1,1,6,9] geometry. In this case, we have 115 (GeV) h

𝜉 m 𝐾=−2 (V + ), 110 log 2 (78) 105 −(푎4/푔𝑠)푇4 −(푎5/푔𝑠)푇5 𝑊=𝑊0 +𝐴4𝑒 +𝐴5𝑒 . 100 500 1000 1500 2000 2500 Therefore, the gravitino mass will be given by m3/2 (GeV) 2 󵄨 󵄨 퐾/2 󵄨 𝑔푠 󵄨𝑊0󵄨 𝑚 =𝑒 |𝑊|󵄨 ∼ 𝑀 . Figure 5: Higgs mass 𝑚ℎ as a function of the gravitino mass 𝑚3/2. 3/2 󵄨dS 푝 (79) V√4𝜋 The region in left to the green line is disallowed by the gluino mass V constraint, while the area between the red lines shows the region for It is remarkable that depending on the large volume ,the whichtheHiggsmassliesintherangeof124–126 GeV. gravitino mass could be TeV or much larger. Considering the ̃ Kahlermetricoftheobservablesectortohavetheform¨ 𝐾훼훽 = ̃ ̃ 13 15 𝐾훼𝛿훼훽,with𝐾훼 being constant, we have scalar soft masses of Accordingly, at large volumes V ∼10 –10 , the soft masses the following form: arerelatedtothegravitinomassasfollows: 2 2 𝑚 ≃𝑚 =𝑚 , 𝑚훼 =𝑚3/2, (80) 0 1/2 3/2 (86) 𝐴 ≃−𝑚 . wherewehaveneglectedtheverytinycosmologicalconstant 0 3/2 value 𝑉0.Inthelargevolumelimit,V ∼𝜏5 ≡𝜏푏 ≫𝜏4 ≡𝜏푠 ≫ These soft terms are generated at string scale; therefore, 1,theKahler¨ metric and its inverse are given by we have to run them to the electroweak (EW) scale and 1 1 impose the EW symmetry breaking conditions to analyze the − [ V V5/3 ] corresponding spectrum. In this case, condition (77) reads 𝐾푚푛 ≃ [ 1 1 ] , − 2 5/3 4/3 𝑀 [ V V ] (81) 𝜇2 + 푍 ≃ −0.1𝑚2 +2𝑚2 , (87) 2 3/2 3/2 VV2/3 𝐾푚푛 ≃[ ]. 2/3 4/3 which can be satisfied easily at the TeV scale. The above V V set of soft terms is special case of CMSSM. Therefore, the Higgsmasslimitrequires𝑚1/2 to be of order 1.5 TeV, as The F-terms can be calculated from showninFigure5,wheretheHiggsmass𝑚ℎ is plotted versus 푚 퐾/2 푚푛 the gravitino mass 𝑚3/2.Withsuchheavyvaluesof𝑚0 and 𝐹 =𝑒 𝐾 𝐷푛𝑊. (82) 𝑚1/2, the SUSY spectrum will be quite heavy and the relic Hence, the approximate dependence of the F-terms on the abundance of the lightest neutralino is not consistent with volume is given by Planck’s results [67]. 1 𝐹4 ∼ , V 6. Cosmological Consequences (83) 1 Oneoftheimportantconsequencesofthemodulistabiliza- 𝐹5 ∼ . V1/3 tion is its effect on inflation. It turned out that the scalar potential of the modulus field in KKLT is not suitable to Consider a linear dependence of the gauge kinetic function account for single-field inflation since the modulus potential on the Kahler¨ moduli [62, 63]; therefore, the gaugino masses is not flat enough to allow for slow roll [68, 69]. Namely, are given by the real part is not protected by a shift symmetry which 𝜂 2 will result in an -problem. Although the axionic part is 𝑔푠 𝑊0𝑀푝 1 1 𝑚 ∼ ( + ). (84) subjected to shift symmetry, the field dependent mass matrix 1/2 √4𝜋 V V5/3 is not diagonal with a significant mixing. Therefore, the shift symmetry is violated and the 𝜂-problem persists (for a review Also, the universal A-term is given by forstringinflationinthelightofrecentobservations,see,e.g., −𝑔2𝑊 𝑀 [70]). 푚 푠 0 푝 𝐴0 ≃𝐹 𝐾푚 ∼ . (85) In this section, we discuss how this problem can be √4𝜋V evaded in what is called the racetrack superpotential.Also, Advances in High Energy Physics 11

1.675

1.670 2.0 1.5 )

) 1.665

𝜓 , 1.0 saddle 𝜓

0.5 ,

V(𝜏 160 0.0 1.660 V(𝜏 20 140 1.655 0 120 𝜏 𝜓 −20 115 120 125 130 135 100 𝜏 (a) (b)

Figure 6: (a) corresponds to the racetrack scalar potential and (b) corresponds to the variation of the potential in the neighborhood of the 16 −12 saddle point (all are multiplied by 10 ). The parameter values are 𝐴 = 0.02, 𝐵 = 0.035, 𝑎 = 2𝜋/100, 𝑏=2𝜋/90, 𝐸2 = 4.14668 × 10 ,and −5 𝑊0 =−4×10 .

inflation viaahler K¨ moduli is discussed. In addition, we The racetrack inflation model predicts spectral index 𝑛푠 = 14 explain the destabilization problem that arises in KKLT and 0.96,inflationscale𝑀inf ∼10 GeV,and tiny tensor to scalar 4 −8 LVS.Finally,westudytheeffectofmodulibackreactionon ratio 𝑟≃(𝑀inf /𝑀GUT) ∼10 . inflationary scenarios. 6.2. Kahler¨ Moduli Inflation in the LVS. The idea of Kahler¨ 6.1. Racetrack Inflationary Model. In the racetrack model, the moduli inflation [71] is to produce an inflationary potential superpotential is given by [68, 69] similar to the form −푎푇 −푏푇 𝑉 (𝜑) =𝑉 (1−𝑘𝛽𝑒−푘휑) , 𝑊=𝑊0 +𝐴𝑒 +𝐵𝑒 , (88) inf 0 (91) where the additional nonperturbative term may be obtained where 𝛽 and 𝑘 are positive constants. from gaugino condensation in a theory with a product of In the case of multimoduli Calabi-Yau geometries, the gauge groups such as 𝑆𝑈(𝑁) × 𝑆𝑈(𝑀) gauge groups; hence, Calabi-Yau volume can take the form 𝑎=2𝜋/𝑀and 𝑏=2𝜋/𝑁. This is known as the racetrack 푛 𝛼 [ 3/2 3/2] model. In this case, the global SUSY minimum is located at V = (𝑇 + 𝑇) − ∑𝜆푗 (𝑇푗 + 𝑇푗) , (92) 2√2 𝑀𝑁 𝑀𝐵 [ 푗=1 ] 𝑇0 = log (− ). (89) 𝑀−𝑁 𝑁𝐴 where 𝑇=𝜏+𝑖𝜓is responsible for the large volume and 𝑇 =𝜏 +𝑖𝜓 𝜏 𝛼 𝜆 The inflationary potential is given by the F-term scalar 푗 푗 푗,where 푗 are the blow-ups. Here, and 푗 are potential added to the uplifting term as follows: model dependent positive constants. In this case, the Kahler¨ potential is given by −4휏(푎+푏) 𝐸2 𝑒 2 2휏(푎+2푏) 𝜉 𝑉inf = + {𝑎𝐴 (𝑎𝜏) +3 𝑒 𝐾=𝐾 −2 [V + ], 𝜏2 6𝜏2 cs ln 2 (93) 3휏(푎+푏) +𝑒 [𝐴𝐵 (2𝑎𝑏𝜏 +3 (𝑎+𝑏)) (cos (𝜏 (𝑎−𝑏))) whilethesuperpotentialisgivenby (90) 푏휏 푎휏 푛 +3𝑎𝐴𝑊𝑒 (𝑎𝜓) + 3𝑏𝐵𝑊 𝑒 (𝑏𝜓)] −푎𝑖푇𝑖 0 cos 0 cos 𝑊=𝑊0 + ∑𝐴푖𝑒 , (94) 푖=2 +𝑏𝐵2 (𝑏𝜏) +3 𝑒2휏(2푎+푏)}. where 𝑎푖 =2𝜋/𝑔푠𝑁. The minimum of the scalar potential V ∼𝜏≫𝜏 Figure 6 shows the scalar potential with two degenerate exists at the large volume limit 푗 and the scalar minima and one saddle point located at 𝜓saddle =0.At potential has the form [71] this saddle point, the potential has a minimum in the 𝜏 2 8(𝑎푖𝐴푖) √𝜏푖 −2푎 휏 𝑎 𝐴 −푎 휏 direction and a maximum in the 𝜓 direction. The inflaton is 𝑉=∑ 𝑒 𝑖 𝑖 − ∑4 푖 푖 𝑊 𝜏 𝑒 𝑖 𝑖 3V𝜆 𝛼 V2 0 푖 considered to be rolling slowly from initial conditions near 푖 푖 푖 (95) the saddle point on the inflationary trajectory. Namely, the 3𝜉𝑊2 initial motion is in 𝜓 direction and the inflationary path is + 0 . determined numerically [68, 69]. 4V3 12 Advances in High Energy Physics

The inflation can occur away from the minimum and the 10 inflaton is considered to be one of the small moduli 𝜏푛.The volume V and the small moduli, other than the inflaton, 8 are guaranteed to be stabilized to their minima during the 6 inflation [71]. Therefore, the inflationary potential is given by V [71] 4 4𝑎 𝐴 𝑊 𝜏 𝑒−푘휑 𝑉 =𝑉 − 푛 푛 0 푛 , (96) 2 inf 0 V2 30 35 40 45 3 V with 𝑉0 =𝛽𝑊0/V which is constant during the inflation. B −10 −2 𝜏 This model predicts 0, 967 > 𝑛푠 > 0.960 and 𝑟∼10 for the number of e-folding 50 < 𝑁푒 <60with inflationary scale Vdestab 13 V 𝑀inf ∼10 GeV. dS VdS 6.3. Destabilization Problem. In the KKLT scenario, the Figure 7: The purple curve shows the dS vacuum, while the modulusmassisgivenby[36] green one displays the destabilization of the modulus due to large 󵄨 contribution of vacuum energy density. 𝑉耠耠 (𝜏)󵄨 2 󵄨 𝑚2 = 󵄨 = (𝐷 𝑊) (𝜏𝑊 −2𝑊 )󵄨 . 휏 󵄨 푇 휏 푇푇 푇 󵄨 (97) 2𝐾 󵄨 9 󵄨휏0 푇푇 󵄨휏0 In this case, there will be a competition between the runaway Therefore, it is linked to 𝑚3/2 by the relation −푛 dependence 𝜏 and the barrier 𝑉퐵. Therefore, protecting 󵄨 the volume modulus from destabilization will impose the √2𝜏 󵄨 𝑚 = 𝑊 󵄨 =2𝑎𝜏𝑚 , condition 휏 9 푇푇󵄨 0 3/2 (98) 󵄨휏0 2 𝑉inf ≲𝑉퐵 ∼3𝑚3/2 󳨐⇒ which is approximately two orders of magnitude greater than (103) 𝑚 ≳𝐻. theTeVscalegravitinomass.Incasethereisaninflaton 3/2 ∼ 13 field different from modulus field with mass 10 GeV, the On the other hand, in the LVS case, there is an extra modulus may perturb GUT scale inflation. It turns out that suppression by large volume V [72] a problem arises due to the conflict between the requirement of a high energy scale (GUT scale) inflationary phenomenon 𝑚2 𝑀2 𝑉 ∼ 3/2 푝 ∼𝑚3 𝑀 , (104) and low energy (TeV scale) SUSY phenomenology in the 퐵 V 3/2 푝 context of KKLT. The conflict is stemming from the constraint [11, 12] and correspondingly we have the constraint 𝐻≲𝑚3/2, 𝑚3/2 ≳𝐻, (99) 3/2 (105) where 𝐻 is the Hubble parameter. This constraint originates which will not improve the situation much. In this regard, from the fact that the barrier 𝑉퐵 from the runaway direction, TeV scale gravitino mass which is favored by low energy as exhibited in Figure 7, is a bit less than the magnitude of the phenomenology implies disfavored nontraditional low scale AdS inflation. The property of 𝐻≲𝑚3/2 seems to be a common AdS minimum |𝑉0 | since the uplift effect is negligible for large 𝜏.Hence,wehave property between many inflationary models in stabilization scenarios in string theories [72–74]. This problem is some- AdS 2 times called the Kallosh-Linde problem. The latter originates 𝑉퐵 ≲𝑉 ≃3𝑚 . (100) 0 3/2 fromthefactthatcompatiblelargescalemodelsofinflation Therefore, recovering the Planck mass, we have require a very large gravitino mass, where

2 2 𝑚휑 ≪𝐻≲𝑚3/2. (106) 𝑉퐵 ∼𝑚3/2𝑀푝. (101)

If we included an inflaton, there will be a contribution to the The KL Model and Strong Moduli Stabilization.Theproblem 푛 overall inflationary potential by a term of the form 𝑉(𝜑)/𝜏 of destabilization can be evaded in models of strong moduli due to the fact that the modulus couples to all sources of stabilization like the KL model [11, 12]. In this scenario, the energy. Accordingly, the total inflationary potential at the volume is still determined by one modulus 𝑇,butthereisan KKLT dS minimum is given by additional nonperturbative term contributing to the racetrack superpotential 𝑉 (𝜑) 𝑉 (𝜑) 𝑉inf =𝑉dS (𝜏) + ≃ . (102) −푎푇 −푏푇 𝜏푛 𝜏푛 𝑊=𝑊0 +𝐴𝑒 −𝐵𝑒 . (107) Advances in High Energy Physics 13

In this respect, the imaginary part stabilizes at the origin again, whereas the potential of the real part 𝜏 is given by

𝑒−2휏(푎+푏) (𝑎𝐴𝑒푏휏 −𝑏𝐵𝑒푎휏)(𝑒푏휏 (𝐴 (𝑎𝜏) +3 +3𝑤0𝑒푎휏)−𝐵𝑒푎휏 (𝑏𝜏) +3 ) 𝑉 (𝜏) = . (108) 6𝜏2

This potential possesses two minima; one is a metastable It is worth mentioning that the uplifting effect here is so supersymmetric Minkowski vacuum at small and cannot exceed the barrier between the dS and the 1 𝑎𝐴 AdS vacua which is one of the strengths of this model. On 𝜏 = ( ), the other hand, a weakness of the approach is the lack of 0 𝑎−𝑏ln 𝑏𝐵 (109) interpretation of the origin of the scale of the shift Δ𝑊 which −13 and the other is a deeper AdS one as shown in Figure 8. The should be around 10 , if we are seeking TeV scale gravitino KL modulus squared mass is given by [36] mass. 2𝑎𝐴𝑏𝐵 (𝑎−𝑏) 𝑎𝐴 −(푎+푏)/(푎−푏) 𝑎𝐴 𝑚2 = ( ) ( ). (110) 휏 9 𝑏𝐵 ln 𝑏𝐵 6.4. Impact of String Moduli Backreaction. Here, we give a brief overview of the effect of string moduli backreaction Therefore, for 𝐴=𝐵=1, 𝑎 = 0.1,and𝑏 = 0.05, 𝑚 ∼4×1015 ontheinflationarymodelsanditslinktoSUSYbreaking. we find 휏 GeV which means that the volume To study such effects on the inflation, models of strong modulusmassismuchlargerthantheinflatonmass(𝑚휑 ∼ 13 moduli stabilization are favored such as the KL model. In 10 GeV) and accordingly will be frozen quickly during the thelattermodel,themoduliareheavy;inparticular,the inflation without perturbing the inflaton dynamics. Clearly, modulus mass is larger than the Hubble parameter. The this is much featured than the KKLT case since the gravitino moduli backreaction effect on the inflation and its link to mass vanishes at the SUSY minimum. Therefore, the hight SUSY breaking were studied in many research papers [75– of the barrier from the runaway direction is independent 79]. For a single modulus field case, the impact of the of the gravitino mass; hence, the Hubble parameter is also stabilized volume modulus field on large and small field independent of the gravitino mass. In this case, KL model can inflation was studied in [75]. In this respect, the totalahler K¨ account for high scale inflation. potential and superpotential are given by In order to obtain interesting phenomenology at the TeV scale for SUSY breaking without conflict with the high scale 𝐾=𝐾 (𝑇 + 𝑇) + 𝐾 (𝜙 , 𝜙 ) inflation requirements, a constant shift Δ𝑊 is added to the mod inf 훼 훼 superpotential [36] and it is supposed to be of the order of the weak scale. In this case, the value of 𝜏 minimizing the =−𝜅log (𝑇 + 𝑇) +inf 𝐾 , (115) potential will be shifted to 𝜏0 +𝛿𝜏. The SUSY vacuum is obtained by solving the equation of motion 𝑊=𝑊mod (𝑇) +𝑊inf (𝜙훼),

𝐷푇𝑊(𝜏0 +𝛿𝜏)=0, (111) where 𝜙훼 are chiral superfields related to the inflationary which implies scenario and the constant 𝜅=1for heterotic dilaton and 3Δ𝑊 𝜅=3for type IIB Kahlermodulus.Accordingly,theF-term¨ 𝛿𝜏 = . (112) scalar potential can be written as 2𝜏0𝑊푇,푇 (𝜏0)

퐾inf −휅 퐾̃ Correspondingly, the AdS minimum is independent of the 𝑉=𝑒 𝑉mod (𝑇) + (2𝜏) 𝑉inf (𝜙훼)+𝑒 𝑉(𝑇,𝜙훼), (116) sign of Δ𝑊 andisapproximatedto 3 (Δ𝑊)2 𝑉AdS ≃− , where 0 3 (113) 8𝜏0 퐾 푇푇 󵄨 󵄨2 󵄨 󵄨2 𝑉 =𝑒 mod [𝐾 󵄨𝐷 𝑊 󵄨 −3󵄨𝑊 󵄨 ], where the above expressions are obtained using 𝑊푇(𝜏0)=0, mod 󵄨 푇 mod󵄨 󵄨 mod󵄨 𝑊(𝜏0)=Δ𝑊,and𝑊푇(𝜏0 + 𝛿𝜏) = 푇,푇𝛿𝜏𝑊 (𝜏0).TheAdS (117) 2 2 퐾inf 훼훼 󵄨 󵄨 󵄨 󵄨 vacuum can be uplifted to a dS one by mechanisms similar to 𝑉inf =𝑒 [𝐾 󵄨𝐷훼𝑊inf 󵄨 −3󵄨𝑊inf 󵄨 ]. thoseintheKKLTmodelandhenceSUSYwillbebrokenwith the gravitino becoming massive. In this respect, the gravitino 𝜏 +𝛿𝑇 mass is given in terms of Δ𝑊 as follows: In this case, the potential is minimized at 0 ,dueto the effect of the inflationary large positive energy density, 󵄨 󵄨 𝜏 󵄨𝑉AdS󵄨 3/2 where 0 corresponds to the SUSY Minkowski minimum, √ 󵄨 0 󵄨 1 𝑎−𝑏 𝑚3/2 ≃ = ( ) |Δ𝑊| . (114) if the scenario contains moduli only. Since the modulus 3 2√2 ln (𝑎𝐴/𝑏𝐵) is very heavy, it stabilizes quickly to its minimum and 14 Advances in High Energy Physics

Another example is the large field inflation which was 1.0 successfully embedded in supergravity. In this case, a shift symmetry is used to avoid the 𝜂-problem as well as consid- 0.5 ering stabilizer field 𝑆 [82–87]. In this regard, consider a form of the Kahler¨ potential and superpotential given by V 20 40 60 80 100 1 󵄨 󵄨2 2 4 𝐾inf =− 󵄨𝜑−𝜑󵄨 + |𝑆| −𝜁|𝑆| , (122) 𝜏 2 −0.5 𝑊inf = 𝑆𝑓(𝜑) , (123) −1.0 where the quartic term in the Kahler¨ potential causes stabi- 7 Figure 8: The F-term scalar potential (multiplied by 10 )forKL lizer 𝑆 to get mass much larger than the Hubble parameter model possesses both metastable Minkowski vacuum which is and so stabilizes quickly to the origin. Theahler K¨ potential 𝐴=𝐵=1𝑎 = 0.1 supersymmetric and another AdS minimum, for , , is independent of the real part Re(𝜑) =푟 𝜑 ,duetotheshift 𝑏 = 0.05 and . symmetry and, hence, it will correspond to the inflaton. The 𝑆 field and the imaginary part of 𝜑 will be stabilized at zero and hence the inflationary potential (which is given in pure the inflationary potential gets corrections after setting 𝑇 to its 2 inflation scenario case by 𝑉(𝜑푟)=|𝑓(𝜑푟)| )getcorrectionsas minimum as follows [75]: follows [75]: −3 𝜅 𝑉=(2𝜏) 𝑉 − {𝑊 [𝑉 2 0 inf 3휅/3 inf inf 𝑉̃ 𝑉̃ 2(2𝜏 ) 𝑚 𝑉 (𝜑 ) = 𝑉̃ [1−𝜅 −𝜅2 ] , 0 푇 푟 2 2 2 (124) 𝑚푇 𝑚푇𝑚푆 퐾inf 훼훼 +𝑒 𝐾 𝜕훼𝑊inf 𝐷훼𝑊inf ]+h.c.} (118) ̃ 2 휅 where 𝑉(𝜑푟)=|𝑓(𝜑푟)| /(2𝜏0) .For𝑓(𝜑) =,wegetthe 𝑚𝜑 퐾 𝜅𝑒 inf 󵄨 󵄨 − 󵄨𝐾훼훼𝐷 𝑊 𝜕 𝑊 󵄨 . case of chaotic inflation. It is worth noting that the leading 2휅 󵄨 훼 inf 훼 inf 󵄨 2 2 correction term here is proportional to 1/𝑚푇,whereasinthe (2𝜏0) 𝑚푇 case of hybrid inflation it is proportional to 1/𝑚푇. Now, consider a case of small field inflation such as the F-term Hybrid inflation [80, 81], where the Kahler¨ potential 7. Conclusions andsuperpotentialaregivenby In this minireview, we have analyzed the problem of mod- 󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨2 𝐾inf = 󵄨𝜑󵄨 + 󵄨𝜙1󵄨 + 󵄨𝜙2󵄨 , uli stabilization in type IIB string theory with positive (119) vacuum energy. We focused on KKLT and Large Volume 2 𝑊=𝜆𝜑(𝑀 −𝜙1𝜙2), Scenarios, where geometrical fluxes and nonperturbative superpotentials are required to stabilize complex structure where 𝜑 is the inflaton and 𝜙1, 𝜙2 correspond to the waterfall moduli, dilation, and Kahler¨ moduli. We also discussed fields. The hybrid inflation scenario supposes that, for values some possible mechanisms for uplifting the AdS minimum of 𝜑>𝜑푐 =𝑀, the potential is minimized at 𝜙1 =𝜙2 =0and, and making it a metastable de Sitter ground state. We have therefore, the inflationary superpotential is effectively given derived the soft SUSY breaking terms in these models. We by showedthatinKKLTthesetermsarenotconsistentwith

2 electroweak breaking conditions and hence they are not 𝑊inf =𝜆𝑀𝜑. (120) phenomenologically viable, while in LVS we found that the scalar masses and gaugino masses and trilinear terms are Accordingly, the corrected inflationary potential due to the universal and are given in terms of gravitino mass 𝑚3/2. moduli backreaction will contain a linear term in the inflaton We emphasized that a very heavy spectrum with 𝑚3/2 ∼ [75] 1.5 TeV is required to account for the lightest Higgs mass limit. However, in this case, the relic abundance of the lightest 2𝜅√𝑉0 𝑉 (𝜑) =𝑉 [1− (𝜑)] +Δ𝑉 +𝑉 , neutralino is not consistent with the measured limits. We inf 0 𝑚 Re 1 sug (121) 푇 also studied inflation scenarios associated with the moduli stabilization. We considered two examples of racetrack and where Δ𝑉1 is the one-loop correction to the inflation potential Kahler¨ inflation. Finally, we commented on the problem of and 𝑉sug represents the contribution due to supergravity, moduli destabilization and moduli backreaction effects on which has small effect and can be neglected. Here, 𝑉0 is the 2 inflation. 𝑉 = 𝜆̃ 𝑀4 rescaled vacuum energy during inflation given by 0 , The recent developments in the KKLT and the LVS are ̃2 2 휅 with 𝜆 =𝜆/(2𝜏0) . In this case, the value of the spectral worth mentioning. One of the updates is the anti-D3-brane index can be improved; 𝑛푠 ≃ 0.96,comparedtoitsvaluein upliftviaanilpotentsuperfield[88].InthecaseoftheLVS,the thepurehybridscenarios;𝑛푠 = 0.98. So, we do not need to go universal gaugino masses are of the same order as the scalar 3 to nonminimal Kahler¨ scenario [80, 81]. massesandhavelowerlimits∼10TeV implied from the Advances in High Energy Physics 15

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Research Article A 𝑄-Continuum of Off-Shell Supermultiplets

Tristan Hübsch1,2 and Gregory A. Katona1,3

1 Department of Physics & Astronomy, Howard University, Washington, DC 20059, USA 2Department of Physics, University of Central Florida, Orlando, FL 32816, USA 3Affine Connections, LLC, College Park, MD 20740, USA

Correspondence should be addressed to Tristan Hubsch;¨ [email protected]

Received 4 August 2015; Revised 15 October 2015; Accepted 22 October 2015

Academic Editor: Stefano Moretti

Copyright © 2016 T. Hubsch¨ and G. A. Katona. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.

Within each supermultiplet in the standard literature, supersymmetry relates its bosonic and fermionic component fields in a fixed way, particularly to the selected supermultiplet. Herein, we describe supermultiplets wherein a continuously variable “tuning parameter” modifies the supersymmetry transformations, effectively parametrizing a novel “𝑄-continuum” of distinct finite- dimensional off-shell supermultiplets, which may be probed already with bilinear Lagrangians that couple to each other andto external magnetic fields, two or more of these continuously many supermultiplets, each “tuned” differently. The dependence onthe tuning parameters cannot be removed by any field redefinition, rendering this “𝑄-moduli space”observable.

“Discreteness is the refuge of the clumsy.” Jorge Hazzan

1. Introduction, Results, and Synopsis turned out to relate the classification problem to encryption and coding theory [23–25]. Supersymmetry has been studied for over forty years [1, 2], Although this research program uncovered trillions of has had successful application in nuclear physics [3, 4] and off-shell supermultiplets of world-line 𝑁-extended super- critical phenomena [5, 6], and has recently found applications symmetry, we show herein—corroborated by concurrent also in condensed matter physics: see the recent reviews [7, 8], research [26]—that they provide merely a discrete subset for example. In quantum applications, the supermultiplets of a vast 𝑄-continuum,givingrisetoa𝑄-moduli space. must be off-shell, that is, free of any (space) time-differential Furthermore, we prove herein that this novel 𝑄-continuum constraint that could play the role of the Euler-Lagrange is physically observable. (classical) equation of motion. The long-standing challenge of This may well come as a surprise, since both the con- a systematic classification of off-shell supermultiplets [9, 10] tinuous Lie algebras and the various discrete symmetry has been addressed with significant success in the last decade groups familiar from physics applications all have discrete or so; see [11–19] and references therein. One of the pivotal sequences of inequivalent unitary and linear and finite- ideas enabling this recent development was the use of graph- dimensional representations. Reference [27] showed that the theoretical methods [20–22] in assessing the structure of infinite sequence of quotient supermultiplets specified in [21] the supersymmetry transformations within the world-line defines a similarly infinite sequence of ever larger unitary, dimensional reduction of off-shell supermultiplets, which linear, and finite-dimensional off-shell representations of 2 Advances in High Energy Physics

𝑁⩾3-extended world-line supersymmetry, and [28] finds 𝑄1 𝑄2 𝑄3 𝑄4 highly nontrivial and continuously variable dynamics for the 𝜙1 𝜓1 𝜓2 𝜓3 𝜓4 simplest of these supermultiplets, even with only bilinear 𝜙2 𝜓4 −𝛼𝜓6 𝜓3 +𝛼𝜓5 −𝜓2 −𝛼𝜓8 −𝜓1 +𝛼𝜓7 Lagrangians. 𝐹 𝜓̇ −𝜓̇ −𝜓̇ 𝜓̇ Our present results, however, radically extend this line 3 3 4 1 2 ̇ ̇ ̇ ̇ of research. By proving that each of these supermultiplets is 𝐹4 𝜓2 −𝜓1 𝜓4 −𝜓3 merely a special member in a continuum of distinct super- 𝜙5 𝜓5 𝜓6 𝜓7 𝜓8 multiplets, we prove that already bilinear Lagrangians of [28] 𝜙 𝜓 𝜓 −𝜓 −𝜓 can couple continuously many distinct supermultiplets. The 6 8 7 6 5 ̇ ̇ ̇ ̇ coupling constants are therefore functions over this novel 𝐹7 𝜓7 −𝜓8 −𝜓5 𝜓6 𝑄 -moduli space, providing access to physically probe and 𝐹 𝜓̇ −𝜓̇ 𝜓̇ −𝜓̇ observe this 𝑄-continuum. 8 6 5 8 7 𝑁=4 For simplicity and concreteness, we focus on the - 𝑄 𝑄 𝑄 𝑄 extended world-line supersymmetry algebra without central 1 2 3 4 ̇ ̇ (2) charges: 𝜓1 𝑖𝜙1 −𝑖𝐹4 −𝑖𝐹3 −𝑖𝜙2 −𝑖𝛼𝐹8

{𝑄𝐼,𝑄𝐽}=2𝑖𝛿𝐼𝐽𝜕𝜏, ̇ ̇ 𝜓2 𝑖𝐹4 𝑖𝜙1 −𝑖𝜙2 −𝑖𝛼𝐹8 𝑖𝐹3 (1) 𝜓 𝑖𝐹 𝑖𝜙̇ +𝑖𝛼𝐹 𝑖𝜙̇ −𝑖𝐹 [𝜕𝜏,𝑄𝐼]=0, 3 3 2 8 1 4 ̇ ̇ where 𝑖𝜕𝜏 is the Hamiltonian (in the familiar ℏ=1=𝑐units) 𝜓4 𝑖𝜙2 +𝑖𝛼𝐹8 −𝑖𝐹3 𝑖𝐹4 𝑖𝜙1 𝑄 ,...,𝑄 and 1 4 are the supercharges, four real generators of 𝜓 𝑖𝜙̇ −𝑖𝐹 −𝑖𝐹 −𝑖𝜙̇ supersymmetry. For concreteness, we focus on a particular 5 5 8 7 6 set of supermultiplets (see (2) below) which were adapted ̇ ̇ 𝜓6 𝑖𝐹8 𝑖𝜙 −𝑖𝜙 𝑖𝐹7 from [28] by replacing one of the component bosons with 5 6 its 𝜏-derivative and renaming the component fields. Our ̇ ̇ 𝜓7 𝑖𝐹7 𝑖𝜙6 𝑖𝜙5 −𝑖𝐹8 present results then apply equally well not only to the 𝑁=3 supermultiplet of [27, 28] but also to the infinite sequence of ̇ ̇ 𝜓8 𝑖𝜙6 −𝑖𝐹7 𝑖𝐹8 𝑖𝜙5 ever larger supermultiplets constructed therein. Our present focus on world-line supersymmetry should nevertheless have Omitting the fourth supersymmetry, 𝑄4, would result in a implications for all supersymmetry, since, (a) by dimensional minimal example of this 𝑄-continuum; see Main Theorem of reduction, (1) is an integral part and common denominator [26]. The inclusion of 𝑄4, however, proves that our results are of every supersymmetric theory, (b) it is directly relevant in not an artifact of “too few supersymmetries” and also affords diverse fields in physics, from candidates for the fundamental possible extensions to higher-dimensional spacetimes to be description of 𝑀-theory [29] to the phenomenology of explored separately. topological insulators and graphene [30], and (c) it shows The supermultiplet (2) may be depicted (graphical depic- up in the Hilbert space of every supersymmetric quantum tions of supersymmetry transformation rules are a time- theory. We defer the exploration of these implications to a tested intuitive tool [31] but have been rigorously formalized subsequent effort. only recently [21], and we adopt those conventions) in the The paper is organized as follows. Section 2 defines manner of Figure 1. Component fields are depicted as nodes an illustrative 1-parameter family of indecomposable off- and the 𝑄-transformations between them are depicted as shell, unitary, and finite-dimensional supermultiplets and connecting edges, variously colored to correspond to the four identifies the novel 𝑄-continuum and the corresponding 𝑄- supercharges 𝑄𝐼; these are drawn solid (dashed) to depict the moduli space, {𝛼 ∈ R}. Section 3 then explicitly constructs positive (negative) signs in (2). This graphical rendition of the Lagrangians that, even though being just bilinear in fields, (1) supermultiplet (2) at once reveals that the supermultiplet (2) inextricably depend on the tuning parameter 𝛼,(2)pairwise consists of two identical submultiplets, (𝜙1,𝜙2 |𝜓1,...,𝜓4 | couple continuously many inequivalent supermultiplets, and 𝐹3,𝐹4) and (𝜙5,𝜙6 |𝜓5,...,𝜓8 |𝐹7,𝐹8),whichsupersym- (3) provide for physical probing of this 𝑄-continuum by metryconnectsbytheone-waytransformations.Suchone- coupling to external magnetic fields. Section 4 provides a way transformations are exemplified by the fact that 𝑄1(𝜙2) token example of such nontrivial dynamics which essentially contains 𝜓6,but𝑄1(𝜓6) does not contain 𝜙2;thisisdepicted depend on the tuning parameter 𝛼,andourconclusionsare by the tapering edges crossing the dashed vertical divider in summarized in Section 5. Figure 1. Surprisingly—and radically extending our previous work 2. The 𝑄-Continuum of onthetopic[27,28]—wefindthattheseone-way𝑄- Off-Shell Supermultiplets transformations admit a continuous “tuning parameter.” Denoting 𝛼∈R in the tabulation (2), its value is in no way We proceed by way of a concrete example, introducing the restricted by the supersymmetry algebra relations (1)! That following 1-parameter family of variations of the off-shell is, the supersymmetry transformations (2) close the algebraic supermultiplet from [28]: relations (1) on every given component field with no need Advances in High Energy Physics 3

F4 F3 F8 F7

𝜓1 𝜓2 𝜓3 𝜓4 𝜓5 𝜓6 𝜓7 𝜓8

𝜙1 𝜙2 𝜙5 𝜙6

Q1 Q3 Q Q2 4

Figure 1: A graphical depiction of the 𝑁=4world-line supermultiplet (2).

of any 𝜏-differential condition and for each possible value of parametrizing this 𝑄-continuum and providing a coarse 𝛼∈R separately. The tabulation (2) is thus a continuous 1- parametrization for the corresponding 𝑄-moduli space. parameterfamilyofproperoff-shellrepresentationsof𝑁= 4-extended supersymmetry on the world-line. By contrast, 3. Lagrangians the trillions of supermultiplets reported in [22, 25] as well as those of [27, 28] and all known supermultiplets [1, 32] form We now turn to show that the supersymmetry tuning param- at most discrete sequences. eter 𝛼 does show up in the dynamics, is observable, and The supermultiplet (2) then is one of the simplest exam- makes any two such supermultiplets, each with a different 𝛼- ples of the 𝑄-continuum; see also [26]. In turn the real line value, usefully inequivalent in the sense of [34]: using super- {𝛼 ∈ R} is the corresponding coarse 𝑄-moduli space. Explicit multiplets with different tuning parameter values permits choices of the Lagrangian will determine corresponding constructing Lagrangians which could not be constructed actions of a mapping class group, Γ,whereby{𝛼 ∈ R}/Γ without this variation. becomes the true (and model-dependent) 𝑄-moduli space; To prove this, we construct sufficiently general Lagran- see below. gians for direct use in classical applications and in quantum The special value 𝛼=0decomposes the supermultiplet models using the corresponding Hamiltonian, 𝐻 fl 𝑝⋅𝑞−𝐿̇ ,or (2) into two separate off-shell (2|4|2)-dimensional super- via the partition functional 𝑍[𝜙∗] fl ∫D[𝜙]exp{𝑖 ∫ d𝜏𝐿[𝜙∗ + ̇ ̇ multiplets, both of which being the world-line dimensional 𝜙, 𝜙∗ + 𝜙,...]}. reduction of the familiar chiral supermultiplet [1, 21]. When 𝛼 =0̸ , the off-shell supermultiplet (2) cannot be decomposed 3.1. Simple Kinetic Terms. Following the procedure employed as a direct sum of two separate supermultiplets. The off-shell in[28],weusethefactthatanyLagrangianoftheform supermultiplets of 𝑁=3-extended supersymmetry considered in [27, 28] may be similarly extended to depend on a pre- 𝐿 fl −𝑄4𝑄3𝑄2𝑄1𝑘 (𝜙, 𝜓, 𝐹) (3) cisely analogous tuning parameter, dialing the “magnitude” of 𝑄 𝐼 the one-way 𝐼-transformations connecting the two halves is automatically supersymmetric, since its 𝛿𝑄 fl 𝑖𝜖 𝑄𝐼- of the supermultiplet; see Figure 1. Those supermultiplets are transformation necessarily produces a total 𝜏-derivative. This closely related to the 𝛼=1version of (2): except for some is the direct adaptation of the construction of the so-called renaming of component fields, one merely needs to drop 𝐷-terms in standard treatments of supersymmetry [1, 2]. the fourth supersymmetry and replace 𝐹4 󳨃→𝜙4 =∫d𝐹4, Dimensional analysis dictates that for kinetic-type effectively lowering the corresponding node (top, left) to the Lagrangians we need 𝑘(𝜙, 𝜓, 𝐹) to be bilinear in the bottomlevelinthegraphinFigure1.This,however,obstructs component fields 𝜙1,𝜙2,𝜙5,𝜙6;thiswillproducetermsof ̇ ̇ ̇ ̇ dimensional extension even to just world-sheet supersym- the form 𝜙𝑎𝜙𝑏, 𝑖𝜓𝛼𝜓𝛽, 𝐹𝐴𝐹𝐵,and𝜙𝑎𝐹𝐵 as appropriate for metry [33], providing our main motivation to consider (2) kinetic terms. Table 1 lists the individually supersymmetric instead of the slightly simpler supermultiplet of [27, 28]. Lagrangian summands obtained this way after dropping Explicit attempts verify that no local component field total 𝜏-derivatives. As shown, the ten bilinear functions redefinition can remove the parameter 𝛼 from the super- 𝑎,𝑏 𝑘(𝜙) =𝑘 𝜙𝑎𝜙𝑏 result in six linearly independent terms, so symmetry transformations (2). As discussed subsequently, we define all efforts to eliminate 𝛼 from the 𝑄-transformations must 1 involve nonlocal transformations; see also (6) below. 𝐿KE fl − 𝑄 𝑄 𝑄 𝑄 (𝐴 𝜙 2 +𝐴 𝜙 2 +𝐴 𝜙 2 𝐴⃗ 4 4 3 2 1 1 1 2 2 3 5 The table (2) thus defines a 1-parameter continuum of (4) indecomposable off-shell, unitary, and linear representa- +2𝐴 𝜙 𝜙 +2𝐴 𝜙 𝜙 +2𝐴 𝜙 𝜙 ), tions of world-line 𝑁=4-extended supersymmetry, 𝛼, 4 1 2 5 1 5 6 1 6 4 Advances in High Energy Physics

Table 1: Manifestly supersymmetric kinetic Lagrangian terms for the 𝛼-supermultiplet.

4 𝜙𝑖𝜙𝑗 −𝑄 (𝜙𝑖𝜙𝑗) ﬂ −𝑄4𝑄3𝑄2𝑄1(𝜙𝑖𝜙𝑗) 2 ̇ 2 ̇ 2 2 2 ̇ ̇ ̇ ̇ (1/2)𝜙1 +(𝜙1) +(𝜙2 +𝛼𝐹8) +𝐹3 +𝐹4 +𝑖𝜓1𝜓1 +𝑖𝜓2𝜓2 +𝑖𝜓3𝜓3 +𝑖𝜓4𝜓4 +(𝜙̇ −𝛼𝐹)2 +(𝜙̇ )2 +(𝐹 +𝛼𝜙̇ )2 +(𝐹 +𝛼𝜙̇ )2 +𝑖(𝜓 −𝛼𝜓)(𝜓̇ −𝛼𝜓̇ ) + 𝑖(𝜓 +𝛼𝜓)(𝜓̇ +𝛼𝜓̇ ) + 𝑖(𝜓 +𝛼𝜓)(𝜓̇ +𝛼𝜓̇ )+ (1/2)𝜙 2 1 7 2 3 5 4 6 1 7 1 7 2 8 2 8 3 5 3 5 2 ̇ ̇ 𝑖(𝜓4 −𝛼𝜓6)(𝜓4 −𝛼𝜓6) 2 ̇ 2 ̇ 2 2 2 ̇ ̇ ̇ ̇ (1/2)𝜙5 +(𝜙5) +(𝜙6) +𝐹7 +𝐹8 +𝑖𝜓6𝜓6 +𝑖𝜓5𝜓5 +𝑖𝜓8𝜓8 +𝑖𝜓7𝜓7 ‡ ̇ ̇ ̇ ̇ ̇ ̇ ̇ ̇ ̇ ̇ ̇ ̇ ̇ ̇ 𝜙1𝜙2 2𝜙1𝜙2 −2(𝜙1 −𝛼𝐹7)(𝜙2 +𝛼𝐹8)+2(𝐹3 −𝛼𝜙5)𝐹4 −2(𝐹4 +𝛼𝜙6)𝐹3 +2𝑖𝜓1(𝜓4 −𝛼𝜓6)+2𝑖𝜓2(𝜓3 +𝛼𝜓5)−2𝑖𝜓3(𝜓2 +𝛼𝜓8)−2𝑖𝜓4(𝜓1 −𝛼𝜓7) ̇ ̇ ̇ ̇ ̇ ̇ ̇ ̇ 𝜙1𝜙5 +2𝜙1𝜙5 +2(𝜙2 +𝛼𝐹8)𝜙6 +2𝐹3𝐹7 +2𝐹4𝐹8 +2𝑖𝜓1𝜓5 +2𝑖𝜓2𝜓6 +2𝑖𝜓3𝜓7 +2𝑖𝜓4𝜓8 ̇ ̇ ̇ ̇ ̇ ̇ ̇ ̇ 𝜙1𝜙6 +2𝜙1𝜙6 −2(𝜙2 +𝛼𝐹8)𝜙5 −2𝐹3𝐹8 +2𝐹4𝐹7 +2𝑖𝜓1𝜓8 +2𝑖𝜓2𝜓7 −2𝑖𝜓3𝜓6 −2𝑖𝜓4𝜓5 4 2 4 2 4 2 4 2 4 4 4 Also, 𝑄 (𝜙6 )≃𝑄(𝜙5 ), 𝑄 (𝜙6 )≃𝑄(𝜙5 ), 𝑄 (𝜙2𝜙6)≃𝑄(𝜙1𝜙5), 𝑄 (𝜙5𝜙6)≃0. ‡ 4 The −𝑄 (𝜙1𝜙2) entry is not simplified further to facilitate comparison with Table 2.

𝐹 −𝛼𝜙̇ andwereadofftheactualsummandsfromTable1tosave 𝜙̃ ﬂ √1+𝛼2𝜙 , 𝐹̃ ﬂ 8 2 , space. For example, 2 2 8 √1+𝛼2

1 2 1 2 1 1 (6) 𝐿KE = (𝜙̇ ) + (𝜙̇ +𝛼𝐹) + 𝐹 2 + 𝐹 2 (1,0,1,0,0,0) 2 1 2 2 8 2 3 2 4 would completely eliminate the appearance of the continuous 1 2 1 2 1 1 𝛼 + (𝜙̇ ) + (𝜙̇ ) + 𝐹 2 + 𝐹 2 tuning parameter from the “standard-looking” Lagrangian 2 5 2 6 2 7 2 8 (5) and would thus seem to render the supermultiplets (2) (5) 𝛼 𝑖 𝑖 𝑖 𝑖 withvariousvaluesofthetuningparameter physically + 𝜓 𝜓̇ + 𝜓 𝜓̇ + 𝜓 𝜓̇ + 𝜓 𝜓̇ equivalent to each other. We note in passing that field 2 1 1 2 2 2 2 3 3 2 4 4 redefinition (6) complicates the transformation table (2), the 𝑖 𝑖 𝑖 𝑖 + 𝜓 𝜓̇ + 𝜓 𝜓̇ + 𝜓 𝜓̇ + 𝜓 𝜓̇ effect of which is that the partition-crossing edges in the 2 6 6 2 5 5 2 8 8 2 7 7 graph in Figure 1 become regular, “two-way” edges, hiding the reducibility of the supermultiplet (2). defines the “standard-looking” kinetic terms for this super- In fact, the 𝛼-dependence can be eliminated from all multiplet. Herein, the local component field redefinition Lagrangians of the particular form (4) by “diagonalizing” to normal modes. To see this, note that all such Lagrangians can (𝜙 ,𝐹 ) 󳨃󳨀→ ( 𝜙̃ , 𝐹̃ ), 2 8 2 8 be written in the matrix form:

∘ 𝑇 ∘ 𝐿KE = Φ ⋅ K ⋅ Φ+𝑖Ψ𝑇 ⋅ M ⋅ Ψ,̇ 𝐴⃗ (7) ∘ Φ=(𝜙1,𝜙2,𝜙5,𝜙6 |𝐹3,𝐹4,𝐹7,𝐹8), Ψ=(𝜓1,...,𝜓8), Φ ﬂ DΦ, D ﬂ diag [𝜕𝜏,...,𝜕𝜏 |1,...,1].

All such Lagrangians can be “diagonalized” by local field This “diagonalizes” the kinetic terms (7): ∘ (Φ,̃ Ψ)̃ = (BΦ, FΨ) B𝑇B = K redefinitions defined so that 𝑇 𝑇 ∘ ∘ 𝑇 and F F = M.Here,K is manifestly symmetric and defines 𝐿KE = Φ̃ ⋅ 1 ⋅ Φ+𝑖̃ Ψ̃ ⋅ 1 ⋅ Ψ,̃̇ ( 8 )=28 𝐴⃗ 2 linearly independent bosonic bilinear terms. Taking (9) 8 ∘ 𝜕 M ( )=28 ∘ modulo total 𝜏-derivatives, defines 2 linearly Φ=̃ BΦ, Ψ=̃ FΨ. independent fermionic bilinear terms. It then follows that F where is an orthogonal basis transformation of the fermions 𝜓𝛼 → ̃ 𝛽 ∘ −1 Straightforward computation then shows that 𝜓𝛼 =Λ𝛼 𝜓𝛽,andB is of the general form (with B ﬂ D BD): 󸀠 ∘ ∘ OL𝜕 Φ A O [Φ ] 𝜏 B =[ ], if =[ ][ ] 󸀠 RO Ψ C E [⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟Ψ ] is a supersymmetry of (7) ∘ A O B =[ ], C𝜕 E ∘ 󸀠 ∘ (10) 𝜏 (8) ̃ −1 ̃ [Φ ] BO OL𝜕𝜏 B O [Φ] 𝑗 =[ ][ ][ ] . 𝜙 󳨀→ 𝜙̃ =𝐴 𝜙 , then [ 󸀠] [ ] 𝑖 𝑖 𝑖 𝑗 Ψ̃ OF RO OF−1 Ψ̃ [⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟] [ ] 𝐹 󳨀→ 𝐹̃ =𝐶 𝑗𝜙̇ +𝐸 𝐵𝐹 . 𝐴 𝐴 𝐴 𝑗 𝐴 𝐵 is a supersymmetry of (9) Advances in High Energy Physics 5

Since det[B]=√det[K] and det[F]=√det[M],thetrans- tuning parameters 𝛼 and 𝛽. To see this and motivated by the formation analysis of (4), consider even the very simple analogue of (5):

KE 1 ∘ OL𝜕𝜏 𝐿 =− 𝑄 𝑄 𝑄 𝑄 (𝜙 𝜑 +𝜙 𝜑 ), Q =[ ]󳨀→ mix. 2 4 3 2 1 1 1 5 5 RO 𝐿KE = 𝜙̇ 𝜑̇ +(𝜙̇ +𝛼𝐹)(𝜑̇ +𝛽𝐺 )+𝐹𝐺 +𝐹𝐺 (14) −1 (11) mix. 1 1 2 8 2 8 3 3 4 4 ∘ BO OL𝜕 B O Q̃ =[ ][ 𝜏][ ] −1 ̇ ̇ ̇ ̇ OF RO OF + 𝜙5𝜑5 + 𝜙6𝜑6 +𝐹7𝐺7 +𝐹8𝐺8 +⋅⋅⋅ , where the ellipses indicate that the fermionic bilinear terms [K] =0̸ ≠ needed in (10) is well defined if and only if det were omitted. As in (6), the local field redefinition det[M]. Finally, since the nonzero coefficients in the final, “diag- ̃ √ 𝜙2 fl 1+𝛼𝛽𝜙2, onalized” form of the kinetic Lagrangian (9) are all unity, it follows that all explicit dependence on the continuous ̇ 𝐹8 −𝛽𝜙 tuning parameter 𝛼 can be hidden from the simple kinetic 𝐹̃ fl 2 , 8 √1+𝛼𝛽 Lagrangian (9) precisely if det[K] =0̸ ≠ det[M]. (15) However, the explicit dependence on the continuous 𝜑̃ fl √1+𝛼𝛽𝜑 , tuning parameter 𝛼 does not vanish from the supersym- 2 2 metry transformations (2). This implies that simple kinetic 𝐺 −𝛼𝜑̇ ̃ 8 2 Lagrangians (4) may well have a continuum of supersym- 𝐺8 fl metries, not just a discrete number of supersymmetries as √1+𝛼𝛽 proven recently for “flat” kinetic Lagrangians [35, Appendix KE C.2]. Since the “kinetic-diagonalizing” field redefinition (9) turns 𝐿mix. into complicates the supersymmetry transformation rules (2) KE ̇ ̃̇ ̇ ̇ ̇ and virtually all of the subsequent computations, we do 𝐿mix. = 𝜙1𝜑̇1 + 𝜙2𝜑̃2 +𝐹3𝐺3 +𝐹4𝐺4 + 𝜙5𝜑̇5 + 𝜙6𝜑̇6 notpresumeitattheoutsetbutaccountforthisfreedom (16) ̃ ̃ subsequently; see below. +𝐹7𝐺7 + 𝐹8𝐺8 +⋅⋅⋅ 𝛼 𝛽 3.2. Unhiding the Tuning Parameter. We now turn to specify hiding both and . The transformation (15) preserves the 𝛼𝛽 > −1 some generalizations of the simple Lagrangian (4), where the reality of the component fields only if and becomes 𝛼𝛽 → −1 explicit occurrences of 𝛼 can no longer be hidden by local undefined (diverges) when .Also,thetransforma- tion (15) does not hide the 𝛼-dependence in (4) or the 𝛽- field redefinitions. 4 2 2 dependence in −𝑄 (𝜑1 +𝜑5 ). Thus, by explicit counter- Mixing. Consider two separate supermultiplets of the type (2), example, we have the following theorem. and label their separately variable tuning parameters 𝛼 and 𝛽, respectively: Theorem 1. There do exist Lagrangians of the generic type (13) in which no well-defined, real, local field redefinition can hide the explicit 𝛼-and𝛽-dependence. (𝜙1,𝜙2,𝜙5,𝜙6 |𝜓1,...𝜓7 |𝐹3,𝐹4,𝐹7,𝐹8)𝛼 , (12) The expression (13) then provides a 6+6+14= 26- (𝜑1,𝜑2,𝜑5,𝜑6 |𝜒1,...𝜒7 |𝐺3,𝐺4,𝐺7,𝐺8)𝛽 . parameter continuous family of bilinear Lagrangians for the two distinct 1-parameter families of supermultiplets: one 2 Now consider even just the bilinear coupling Lagrangians of such family for every choice of the pair (𝛼, 𝛽) ∈ R ! the form Generic choices in this 26-dimensional parameter space (𝛼) (𝛽) (𝛼,𝛽) (𝛼) (𝛽) {𝐴⃗ , 𝐴⃗ , 𝐴⃗ } 𝐿KE fl −𝑄 𝑄 𝑄 𝑄 (𝐴⃗ ⋅ 𝑓(𝜙)+⃗ 𝐴⃗ ⋅ 𝑔⃗ (𝜑) define Lagrangians that depend irremov- 𝐴;𝛼,𝛽⃗ 4 3 2 1 ably on both tuning parameters 𝛼 and 𝛽 and so provide for (13) dynamical responses that can be used to observe the values of ⃗ (𝛼,𝛽) ⃗ + 𝐴 ⋅ ℎ(𝜙,𝜑)), 𝛼 and 𝛽 and indeed any difference between them. This then is the practical distinction between (𝜙 | 𝜓 |𝛼 𝐹) and (𝜑 | 2 (𝛼) (𝛽) 𝜒|𝐺)𝛽. For each value of the tuning parameters (𝛼, 𝛽) ∈ R , 𝐴⃗ ⋅ 𝑓(𝜙)⃗ 𝐴⃗ ⋅ 𝑔(𝜑)⃗ where are the bilinear terms (4) and they provide a distinct and usefully inequivalent pair of off- terms are constructed in a precisely analogous way but for shell representations of 𝑁=4-extended supersymmetry. (𝜑|𝜒|𝐺) the supermultiplet 𝛽, involving a corresponding Recall that the familiar chiral and twisted-chiral super- independent set of six coefficients. fields afford constructing 𝜎-models with target spaces that (𝛼,𝛽) ⃗ Finally, 𝐴⃗ ⋅ ℎ(𝜙, 𝜑) represents the mixing terms, con- cannot be described using only chiral superfields [36, 37]. In structed as a general linear combination of the fourteen precise analogy, (𝜙 | 𝜓 |𝛼 𝐹) and (𝜑|𝜒|𝐺)𝛽 jointly afford analogously constructed terms, listed in Table 2, and it is the Lagrangians that cannot be constructed using only copies inclusion of these terms that can obstruct the hiding of the of one of the two, except that (2) presents a continuum of 6 Advances in High Energy Physics

Table 2: Fourteen bilinear “𝐷-term”-type manifestly supersymmetric Lagrangian terms that couple the 𝛼-supermultiplet with the 𝛽- supermultiplet and showcase the appearance of these tuning parameters.

4 𝜙𝑖𝜑𝑗 −𝑄 (𝜙𝑖𝜑𝑗) ﬂ −𝑄4𝑄3𝑄2𝑄1(𝜙𝑖𝜑𝑗) ̇ ̇ 𝜙1𝜑1 +2𝜙1𝜑̇1 +2(𝜙2 +𝛼𝐹8)(𝜑̇2 +𝛽𝐺8)+2𝐹3𝐺3 +2𝐹4𝐺4 +2𝑖𝜓1𝜒̇1 +2𝑖𝜓2𝜒̇2 +2𝑖𝜓3𝜒̇3 +2𝑖𝜓4𝜒̇4 ̇ ̇ 𝜙1𝜑2 +2𝜙1𝜑̇2 −2(𝜙2 +𝛼𝐹8)(𝜑̇1 −𝛽𝐺7)−2𝐹3(𝐺4 +𝛽𝜑̇6)+2𝐹4(𝐺3 −𝛽𝜑̇5)+2𝑖𝜓1(𝜒̇4 −𝛽𝜒̇6)+2𝑖𝜓2(𝜒̇3 +𝛽𝜒̇5)−2𝑖𝜓3(𝜒̇2 +𝛽𝜒̇8)−2𝑖𝜓4(𝜒̇1 −𝛽𝜒̇7) ̇ ̇ 𝜙1𝜑5 +2𝜙1𝜑̇5 +2(𝜙2 +𝛼𝐹8)𝜑̇6 +2𝐹3𝐺7 +2𝐹4𝐺8 +2𝑖𝜓1𝜒̇5 +2𝑖𝜓2𝜒̇6 +2𝑖𝜓3𝜒̇7 +2𝑖𝜓4𝜒̇8 ̇ ̇ 𝜙1𝜑6 +2𝜙1𝜑̇6 −2(𝜙2 +𝛼𝐹8)𝜑̇5 −2𝐹3𝐺8 +2𝐹4𝐺7 +2𝑖𝜓1𝜒̇8 +2𝑖𝜓2𝜒̇7 −2𝑖𝜓3𝜒̇6 −2𝑖𝜓4𝜒̇5 ̇ ̇ ̇ ̇ 𝜙2𝜑1 +2𝜙2𝜑̇1 −2(𝜙1 −𝛼𝐹7)(𝜑̇2 +𝛽𝐺8)+2(𝐹3 −𝛼𝜙5)𝐺4 −2(𝐹4 +𝛼𝜙6)𝐺3 +2𝑖(𝜓4 −𝛼𝜓6)𝜒̇1 +2𝑖(𝜓3 +𝛼𝜓5)𝜒̇2 −2𝑖(𝜓2 +𝛼𝜓8)𝜒̇3 −2𝑖(𝜓1 −𝛼𝜓7)𝜒̇4 ̇ ̇ ̇ ̇ +2(𝜙1 −𝛼𝐹7)(𝜑̇1 −𝛽𝐺7)+2𝜙2𝜑̇2 +2(𝐹3 +𝛼𝜙5)(𝐺3 +𝛽𝜑̇5)+2(𝐹4 +𝛼𝜙6)(𝐺4 +𝛽𝜑̇6)+2𝑖(𝜓1 −𝛼𝜓7)(𝜒̇1 −𝛽𝜒̇7)+2𝑖(𝜓2 + 𝜙2𝜑2 𝛼𝜓8)(𝜒̇2 +𝛽𝜒̇8)+2𝑖(𝜓3 +𝛼𝜓5)(𝜒̇3 +𝛽𝜒̇5)+2𝑖(𝜓4 −𝛼𝜓6)(𝜒̇4 −𝛽𝜒̇6) ̇ ̇ ̇ ̇ 𝜙2𝜑5 +2𝜙2𝜑̇5 −2(𝜙1 −𝛼𝐹7)𝜑̇6 +2(𝐹3 +𝛼𝜙5)𝐺8 −2(𝐹4 +𝛼𝜙6)𝐺7 + 2𝑖(𝜓1 −𝛼𝜓7)𝜒̇8 + 2𝑖(𝜓2 +𝛼𝜓8)𝜒̇7 − 2𝑖(𝜓3 +𝛼𝜓5)𝜒̇6 − 2𝑖(𝜓4 −𝛼𝜓6)𝜒̇5 ̇ ̇ ̇ ̇ 𝜙2𝜑6 +2𝜙2𝜑̇6 +2(𝜙1 −𝛼𝐹7)𝜑̇5 +2(𝐹3 +𝛼𝜙5)𝐺7 +2(𝐹4 +𝛼𝜙6)𝐺8 + 2𝑖(𝜓1 −𝛼𝜓7)𝜒̇5 + 2𝑖(𝜓2 +𝛼𝜓8)𝜒̇6 + 2𝑖(𝜓3 +𝛼𝜓5)𝜒̇7 + 2𝑖(𝜓4 −𝛼𝜓6)𝜒̇8 ̇ ̇ 𝜙5𝜑1 +2𝜙5𝜑̇1 +2𝜙6(𝜑̇2 +𝛽𝐺8)+2𝐹7𝐺3 +2𝐹8𝐺4 +2𝑖𝜓6𝜒̇2 +2𝑖𝜓5𝜒̇1 +2𝑖𝜓8𝜒̇4 +2𝑖𝜓7𝜒̇3 ̇ ̇ 𝜙5𝜑2 +2𝜙5𝜑̇2 −2𝜙6(𝜑̇1 −𝛽𝐺7)−2𝐹7(𝐺4 +𝛽𝜑̇6)+2𝐹8(𝐺3 +𝛽𝜑̇5)−2𝑖𝜓6(𝜒̇3 −𝛽𝜒̇5)+2𝑖𝜓5(𝜒̇4 −𝛽𝜒̇6)−2𝑖𝜓8(𝜒̇1 −𝛽𝜒̇7)+2𝑖𝜓7(𝜒̇2 −𝛽𝜒̇8) ̇ ̇ 𝜙5𝜑5 +2𝜙5𝜑̇5 +2𝜙6𝜑̇6 +2𝐹7𝐺7 +2𝐹8𝐺8 −2𝑖𝜓6𝜒̇6 −2𝑖𝜓5𝜒̇5 −2𝑖𝜓8𝜒̇8 −2𝑖𝜓7𝜒̇7 ̇ ̇ 𝜙5𝜑6 +2𝜙5𝜑̇6 −2𝜙6𝜑̇5 −2𝐹7𝐺8 +2𝐹8𝐺7 +2𝑖𝜓6𝜒̇7 +2𝑖𝜓5𝜒̇8 −2𝑖𝜓8𝜒̇5 −2𝑖𝜓7𝜒̇6 ̇ ̇ 𝜙6𝜑1 +2𝜙6𝜑̇1 −2𝜙5(𝜑̇2 +𝛽𝐺8)+2𝐹7𝐺4 −2𝐹8𝐺3 −2𝑖𝜓6𝜒̇3 −2𝑖𝜓5𝜒̇4 +2𝑖𝜓8𝜒̇1 +2𝑖𝜓7𝜒̇2 ̇ ̇ 𝜙6𝜑2 +2𝜙6𝜑̇2 +2𝜙5(𝜑̇1 −𝛽𝐺7)+2𝐹7(𝐺3 +𝛽𝜑̇5)+2𝐹8(𝐺4 +𝛽𝜑̇6)+2𝑖𝜓6(𝜒̇2 +𝛽𝜒̇8)+2𝑖𝜓5(𝜒̇1 −𝛽𝜒̇7)+2𝑖𝜓8(𝜒̇4 −𝛽𝜒̇6)+2𝑖𝜓7(𝜒̇3 +𝛽𝜒̇5) 4 4 4 4 Also, 𝑄 (𝜙6𝜑5)≃𝑄(𝜙5𝜑6), 𝑄 (𝜙6𝜑6)≃𝑄(𝜙5𝜑5). such usefully inequivalent supermultiplets, not just a discrete 𝛼, 𝛽 do parametrize the supersymmetry action within the set of two! The most general bilinear “kinetic” Lagrangian supermultiplet (2), in a way not unlike within the formalism providing for pairwise coupling of the continuously many of “projective superspace” [42, 43]; see [44, 45] for the relation inequivalent, off-shell supermultiplets of this type is then a to the more general “harmonic superspace.” However, in all double 𝑄-moduli space integral: those efforts, all supersymmetric Lagrangians are localized to special values of those parameters, whereas Lagrangians such 𝐿KE ﬂ ∫ 𝛼 𝛽𝑤 (𝛼, 𝛽)KE 𝐿 , 𝛼, 𝛽 bilinear d d 𝐴⃗ 𝐴;𝛼,𝛽⃗ (17) as (13) are supersymmetric for every choice of ,andthe integral (17) is then also supersymmetric. where 𝑤𝐴⃗ (𝛼, 𝛽) is some appropriate weight function ensuring Since an analogous continuous parameter 𝛼 =1̸ may be theconvergenceofthedoubleintegraloverthecoarsemoduli introduced in the 𝑁=3supermultiplets studied in [27, 28], 2 space, R𝛼,𝛽. Depending on the particular choice of the those specific supermultiplets are also just special members of 𝑄 Lagrangian, that is, a choice of the various 𝐴⃗ -parameters in separate -continua of off-shell supermultiplets, all usefully (13), certain different values in the (𝛼, 𝛽)-plane will produce inequivalent in the same sense. physically equivalent dynamics, generating a corresponding 3.3. Super-Zeeman Terms. WenowturntoLagrangianterms “mapping class group,” Γ𝐴⃗ , of symmetries. In particular, Section 3.1 shows that there do exist proper local field that are still bilinear but where dimensional analysis requires (𝛼) (𝛽) an overall dimension-full parameter of the kind that may be redefinitions that can hide the 6+6parameters 𝐴⃗ , 𝐴⃗ in (𝛼,𝛽) identified as a Larmor-like frequency, coupling the supermul- (13), but it is not clear how many of the 14 parameters 𝐴⃗ — tiplet (2) to external magnetic fields [28, 46]. if any—can be hidden this way; a precise determination In general, we seek functions 𝑓(𝜙, 𝜓, 𝐹) such that each of of the “mapping class group” and corresponding physical 𝑄 𝑄 𝑄 𝑓 (𝜙, 𝜓, 𝐹), equivalences (dualities) will have to remain open for now. 3 2 1 The weight function will have to be invariant with respect to 𝑄 𝑄 𝑄 𝑓 (𝜙, 𝜓, 𝐹), Γ 4 2 1 this 𝐴⃗ , seems likely to be model-dependent, and so is also (18) beyond our present scope. Suffice it here to mention that, 𝑄4𝑄3𝑄1𝑓 (𝜙, 𝜓, 𝐹), in familiar cases (such as the Deligne-Mumford-Teichmuller¨ theory for Riemann surfaces, the moduli spaces of Calabi- 𝑄4𝑄3𝑄2𝑓 (𝜙, 𝜓, 𝐹) Yau manifolds, and the (super)string landscape [38–41]), the analogue of this Γ𝐴⃗ is discrete and the analogue of the quotient vanishes modulo total derivatives. Then, the six quadratic 2 R𝛼,𝛽/Γ𝐴⃗ is a compact, albeit singular space. derivatives The Lagrangians (17) depend on the continuous tun- 𝑄2𝑄1𝑓 (𝜙, 𝜓, 𝐹), ing parameters 𝛼, 𝛽 differently from all previously studied 𝛼, 𝛽 supersymmetric Lagrangians. The tuning parameters are 𝑄3𝑄1𝑓 (𝜙, 𝜓, 𝐹), not the familiar coefficients parametrizing the choice of the Lagrangian as 𝐴𝑖’s above are. Instead, the tuning parameters 𝑄3𝑄2𝑓 (𝜙, 𝜓, 𝐹), Advances in High Energy Physics 7

Table 3: The 𝑄3𝑄2𝑄1-transforms of bosonic bilinear terms, modulo total 𝜏-derivatives.

𝜙𝑖𝜙𝑗 −𝑖𝑄3𝑄2𝑄1(𝜙𝑖𝜙𝑗) ̇ ̇ (1/2)𝜙1𝜙1 +𝜙1𝜓4 −(𝜙2 +𝛼𝐹8)𝜓1 +𝐹3𝜓2 −𝐹4𝜓3 ̇ ̇ ̇ ̇ (1/2)𝜙2𝜙2 +(𝜙1 −𝛼𝐹7)(𝜓4 −𝛼𝜓6)−𝜙2(𝜓1 −𝛼𝜓7)+(𝐹3 +𝛼𝜙5)(𝜓2 +𝛼𝜓8)−(𝐹4 +𝛼𝜙6)(𝜓3 +𝛼𝜓5) (1/2)𝜙 𝜙 ̇ ̇ 5 5 +𝜙5𝜓8 − 𝜙6𝜓5 +𝐹7𝜓6 −𝐹8𝜓7 } } subtract (1/2)𝜙 𝜙 +𝜙̇ 𝜓 − 𝜙̇ 𝜓 +𝐹𝜓 −𝐹𝜓 6 6 5 8 6 5 7 6 8 7 } ̇ ̇ ̇ ̇ 𝜙1𝜙2 +𝛼[𝜙1𝜓7 +(𝜙2 +𝛼𝐹8)𝜓6 +𝐹3𝜓5 +𝐹4𝜓8 − 𝜙5𝜓3 − 𝜙6𝜓2 −𝐹7𝜓1 −𝐹8𝜓4] 𝜙 𝜙 ̇ ̇ ̇ ̇ 1 5 +𝜙1𝜓8 −(𝜙2 +𝛼𝐹8)𝜓5 +𝐹3𝜓6 −𝐹4𝜓7 + 𝜙5𝜓4 − 𝜙6𝜓1 +𝐹7𝜓2 −𝐹8𝜓3 } } subtract 𝜙 𝜙 +𝜙̇ 𝜓 −(𝜙̇ +𝛼𝐹)𝜓 +𝐹𝜓 −𝐹𝜓 + 𝜙̇ 𝜓 − 𝜙̇ 𝜓 +𝐹𝜓 −𝐹𝜓 2 6 1 8 2 8 5 3 6 4 7 5 4 6 1 7 2 8 3 } 𝜙 𝜙 ̇ ̇ ̇ ̇ 1 6 −𝜙1𝜓5 −(𝜙2 +𝛼𝐹8)𝜓8 +𝐹3𝜓7 +𝐹4𝜓6 + 𝜙5𝜓1 + 𝜙6𝜓4 −𝐹7𝜓3 −𝐹8𝜓2 } } add 𝜙 𝜙 +𝜙̇ 𝜓 +(𝜙̇ +𝛼𝐹)𝜓 −𝐹𝜓 −𝐹𝜓 − 𝜙̇ 𝜓 − 𝜙̇ 𝜓 +𝐹𝜓 +𝐹𝜓 2 5 1 5 2 8 8 3 7 4 6 5 1 6 4 7 3 8 2 }

𝜙5𝜙6 𝜕𝜏(𝜙5𝜓5 −𝜙6𝜓8)≃0

𝑄4𝑄1𝑓 (𝜙, 𝜓, 𝐹), with the terms 𝑍𝑖 listed in Table 4. Of these, only the last term contains the expression 𝑄4𝑄2𝑓 (𝜙, 𝜓, 𝐹), ̇ 𝐵4𝑍4 =⋅⋅⋅+𝛼𝐵4𝜙5𝜙6 +⋅⋅⋅ 𝑄4𝑄3𝑓 (𝜙, 𝜓, 𝐹) 1 (22) (19) ≃⋅⋅⋅+ 𝛼𝐵 (𝜙 𝜙̇ − 𝜙̇ 𝜙 )+⋅⋅⋅ 2 4 5 6 5 6 are all manifestly supersymmetric. When applying 𝛿𝑄 =𝑖𝜖⋅ 𝑄, 𝑄𝐼 from 𝛿𝑄 either equals one of the two 𝑄𝐼’s used in the which in Lagrangian physics may be interpreted as the definition(19)andsoproduces𝑖𝜕𝜏 by (1) or does not and so coupling of the magnetic field 𝐵4 to the angular momentum reproduces one of the expressions (18) and again a total 𝜏- of rotation in the (𝜙5,𝜙6)-plane—if the bosons 𝜙5,𝜙6 are derivative by assumption (18). Such terms remind us of the interpreted as Cartesian coordinates in the target space. so-called 𝐹-terms in standard treatments of supersymmetry The elimination of the auxiliary fields 𝐹3,𝐹4,𝐹7,𝐹8 (and [1, 2]. 𝐺3,𝐺4,𝐺7,𝐺8) by means of their equations of motion is We again restrict our attention to bilinear terms for expected to induce additional terms of the type (22) owing simplicity, and Table 3 presents the linearly independent such to the mixing of the auxiliary fields with the 𝜏-derivatives of terms, obtained by applying only the first batch of three the propagating fields 𝜙𝑖. This justifies the identification of the supercharges. The other three expressions (18) each produce terms (21) and the supersymmetric version of the 𝐵⋅⃗ 𝐿⃗ terms analogous results with a pattern virtually identical to the one exhibiting the Zeeman effect. shown in Table 3. The last-row entry, 𝜙5𝜙6, results in a total 𝜏- derivative all by itself, and simple row operations (indicated Summary. The four terms in Table 4 together with their (𝜙 | bybraces)showthatwecanformthreemore.Thismeansthat 𝜓|𝐹)𝛼 →(𝜑|𝜒|𝐺)𝛽 counterparts and the 26-parameter each of the twenty-four terms Lagrangian (13) then form the most general, 34-parameter family of bilinear Lagrangians 1 2 2 𝑄𝐼𝑄𝐽 (𝜙5 −𝜙6 ), 2 𝐿KE +𝐿SZ +𝐿SZ 𝐴;𝛼,𝛽⃗ 𝐵;𝛼⃗ 𝐵;𝛽⃗ (23) 𝑄 𝑄 (𝜙 𝜙 −𝜙 𝜙 ), 𝐼 𝐽 1 5 2 6 (20) for two different supermultiplets from the family (2). 𝑄 𝑄 (𝜙 𝜙 +𝜙 𝜙 ), 𝐼 𝐽 1 6 2 5 ManyofthesummandsinTables1,2,and4havenegative 𝑄 𝑄 (𝜙 𝜙 ) signs and so would—if used on their own—contribute nega- 𝐼 𝐽 5 6 tively to the kinetic energy, that is, induce nonpositivity of the kinetic energy and nonunitarity in general. However, when is a supersymmetric Lagrangian contribution. This list turns they are used jointly with the first three supersymmetric sets out to be repetitive and contains only four linearly indepen- of kinetic terms in Table 1 (which are positive-definite), it is dent expressions, listed in Table 4. The most general super- clear that unitarity constrains the coefficients 𝐴𝑖 in (4) so that Zeeman type Lagrangian bilinear in the component fields of 𝐴4,𝐴5,𝐴6 as well as the coefficients of the Lagrangian sum- the (𝜙|𝜓|𝐹)𝛼 supermultiplet is therefore mands from Tables 2 and 4 should be sufficiently smaller than 𝐴 ,𝐴 ,𝐴 𝐿SZ ﬂ 𝐵 𝑍 +𝐵 𝑍 +𝐵 𝑍 +𝐵 𝑍 , 1 2 3. This is similar to the analogous case examined in 𝐵;𝛼⃗ 1 1 2 2 3 3 4 4 (21) [28]. 8 Advances in High Energy Physics

Table 4: Super-Zeeman bilinear contributions, modulo total 𝜏-derivatives.

𝑍1 ﬂ 𝜙5𝐹7 +𝜙6𝐹8 +𝑖𝜓6𝜓8 −𝑖𝜓6𝜓8

𝑍2 ﬂ 𝜙5𝐹8 −𝜙6𝐹7 +𝑖𝜓6𝜓5 +𝑖𝜓8𝜓7

𝑍3 ﬂ 𝜙1𝐹7 +𝜙2𝐹8 +𝜙5𝐹3 +𝜙6𝐹4 −𝑖𝜓1𝜓7 +𝑖𝜓2𝜓8 +𝑖𝜓3𝜓5 −𝑖𝜓4𝜓6 ̇ 𝑍4 ﬂ 𝜙1𝐹8 −𝜙2𝐹7 −𝜙6𝐹3 +𝜙5𝐹4 −𝑖𝜓1𝜓6 +𝑖𝜓2𝜓5 −𝑖𝜓3𝜓8 +𝑖𝜓4𝜓7 +𝛼(𝜙5𝜙6 −𝑖𝜓6𝜓7 −𝑖𝜓5𝜓8)

2 Requiring positivity of the kinetic energy, and unitarity The special case when 𝑎3 =−𝛼𝑎1 must be treated separately. more generally, restricts these parameters to an open neigh- Substituting these back into the Lagrangian yields borhood in this 34-dimensional parameter space. Most of 󵄨 KE 󵄨 𝑎1 ̇ 2 𝑎1𝑎3 ̇ 2 𝑎3 ̇ 2 the corresponding models depend explicitly on the tuning 𝐿 󸀠 󵄨 = 𝜙1 + 𝜙2 + 𝜙5 𝐴⃗ ;𝛼󵄨𝐹 2 2(𝑎 +𝑎𝛼2) 2 parameter 𝛼∈R (and 𝛽 for two copies of the supermultiplet 𝐴 3 1 (2), etc.) besides the dependence on the coefficients of these 2 2 𝛼 𝛽 𝑎 2 𝐵 𝐵 summands. This parameter (and for two copies, etc.) + 3 𝜙̇ − 4 𝜙 2 − 4 𝜙 2 6 2 1 2 then provides a genuine, observable characteristic of the 2 2(𝑎3 +𝑎1𝛼 ) 2𝑎3 supermultiplet (2). We conclude that the supermultiplets (2) which differ only in a different choice of the parameter 𝛼 2 2 𝐵 2 𝐵 2 cannot be regarded as physically equivalent in general. This − 4 𝜙 − 4 𝜙 +⋅⋅⋅ (26) 2𝑎 5 2𝑎 6 dependence on the tuning parameter 𝛼 becomes only more 1 1 𝐷 complex in the general (not just bilinear) “ -term” (3) and 𝑎 𝛼 𝐹 + 1 𝐵 (𝜙̇ 𝜙 −𝜙 𝜙̇ ) “ -term” (18)-(19) Lagrangian summands. 2 4 1 2 1 2 𝑎3 +𝑎1𝛼 2 𝛼 4. Sample Dynamics − 𝐵 (𝜙̇ 𝜙 −𝜙 𝜙̇ )+⋅⋅⋅ 2 4 5 6 5 6 To illustrate the intricate dependence on 𝛼, that is,onthe choice from among the continuously many inequivalent which induces a coupling of the external angular momentum supermultiplets, consider the sum of the Lagrangian sum- also to the angular momentum in the (𝜙1,𝜙2)-plane but with ⃗ 󸀠 𝑎 /(𝑎 +𝑎𝛼2) mands (4) with 𝐴 =(𝑎1,0,𝑎3,0,0,0)and those in (21) with 1 3 1 times the magnitude of the interaction in the 󸀠 ⃗ (𝜙5,𝜙6)-plane. 𝐵 = (0, 0, 0, 𝐵4),andfocusonlyonthebosonicfields: The Euler-Lagrange equations are 𝑎 2 KE 1 ̇ 2 ̇ 2 2 2 𝐿 󸀠 = [𝜙 +(𝜙 +𝛼𝐹) +𝐹 +𝐹 ] 𝛼𝑎 𝐵 𝐵 𝐴⃗ ;𝛼 2 1 2 8 3 4 0=𝑎𝜙̈ + 1 4 𝜙̇ + 4 𝜙 , 1 1 2 2 2 1 𝛼 𝑎1 +𝑎3 𝛼 𝑎1 +𝑎3 𝑎3 ̇ 2 ̇ 2 2 2 + [𝜙5 + 𝜙6 +𝐹7 +𝐹8 ]+⋅⋅⋅+𝐵4 [𝜙1𝐹8 (24) 2 𝑎 𝑎 𝛼𝑎 𝐵 𝐵 2 0= 1 3 𝜙̈ − 1 4 𝜙̇ + 4 𝜙 ; 𝛼 2 2 2 1 2 ̇ ̇ 𝛼 𝑎1 +𝑎3 𝛼 𝑎1 +𝑎3 𝑎3 −𝜙2𝐹7 −𝜙6𝐹3 +𝜙5𝐹4 + (𝜙5𝜙6 − 𝜙5𝜙6)]+⋅⋅⋅ , 2 (27) 𝐵 2 where the ellipses denote the omitted fermionic terms. The 0=𝑎𝜙̈ −𝛼𝐵 𝜙̇ + 4 𝜙 , 3 5 4 6 𝑎 5 external magnetic field, 𝐵4, is here coupled only to the angular 1 momentum in the (𝜙5,𝜙6)-plane. 2 The Euler-Lagrange equations of motion for 𝐹3,𝐹4,𝐹7,𝐹8 ̈ ̇ 𝐵4 0=𝑎3𝜙6 +𝛼𝐵4𝜙5 + 𝜙6. are of course algebraic: 𝑎1 𝐵 𝜙 𝐹 = 4 6 , The two coupled pairs of differential equations both describe 3 𝐵 𝑎1 a similar response to the external magnetic field 4.Alittle surprisingly perhaps, the frequencies in the solutions to both 𝐵 𝜙 4 5 pairs are (system (27) produces four decoupled 4th-order, 𝐹4 =− , 𝑎1 linear equations with constant coefficients, which are easily 𝑒𝑖𝜔𝑡 𝐵 𝜙 solved using trial -like functions) 𝐹 = 4 2 , (25) 7 𝑎 3 √ 2 2 2𝑎3 +𝑎1𝛼 ±𝛼√𝑎1 (4𝑎3 +𝑎1𝛼 ) 𝛼𝑎 𝜙̇ +𝐵 𝜙 𝜔 =𝜔 ﬂ 𝐵 (28) 𝐹 =− 1 2 4 1 , 12± 56± √2𝑎 𝑎 4 8 2 1 3 𝛼 𝑎1 +𝑎3 and clearly depend on 𝛼;notethat𝜔𝑖𝑗− (𝛼) =𝑖𝑗+ 𝜔 (−𝛼).This 𝑎 =−𝛼̸ 2𝑎 . 3 1 proves that the value of the tuning parameter 𝛼,effectively Advances in High Energy Physics 9 limited to 𝛼⩾0,isphysicallyobservablethroughprobing terms from Table 2, where the explicit dependence with external magnetic fields and that distinctly “tuned” on the tuning parameter(s) 𝛼(𝛽,...)cannot always be supermultiplets of type (2) are observably (and so usefully) eliminated. inequivalent [34]. These frequencies acquire complex or (3) It couples such supermultiplets to external magnetic 𝑎 ,𝑎 purely imaginary values for certain choices of 1 3,and fields inducing a variant of the super-Zeeman effect, 𝛼 , describing, respectively, attenuated/boosted oscillatory or given as 4-parameter linear combinations of the hyperbolic motion. The frequencies are real for terms from Table 4, where the explicit dependence 4𝑎 on the tuning parameter(s) 𝛼(𝛽,...)cannot always be ((𝛼 =0̸ ) ,(𝑎 ⩾− 3 ),(𝑎 >0)) 1 𝛼2 3 eliminated. (29) (4) The multidimensional parameter space of the Lagran- ∨((𝛼=0) ,(𝑎1,𝑎3 >0)). gians (23) has at least one open neighborhood, where the kinetic energy is guaranteed to be positive, indi- Furthermore, the frequencies (28) are incommensurate 𝑎 ,𝑎 𝛼 cating unitarity of the corresponding quantum the- for most choices of 1 3,and .Thatis,theratio ory.

2 2 Using the constructions described in Section 3, these Lagran- 𝜔 𝜔 2𝑎3 +𝑎1𝛼 +𝛼√𝑎1 (4𝑎3 +𝑎1𝛼 ) 12+ = 56+ = √ (30) gians can be generalized to include (a) higher-order interac- 𝜔12− 𝜔56− 2𝑎 +𝑎𝛼2 −𝛼√𝑎 (4𝑎 +𝑎𝛼2) tion terms and (b) couplings to additional and all differently 3 1 1 3 1 tuned supermultiplets from the family (2). Section 4 then demonstrates that, except for very special is not an integer for most choices of the coefficients in the choices within this parameter space, the Lagrangians explic- Lagrangian 𝑎1,𝑎3 and the tuning parameter 𝛼;theorbits itly depend on the tuning parameter 𝛼,andalso𝛽 in (13), are space-filling Bowditch/Lissajous-like figures. Therefore, of the supermultiplet (2), in ways that have direct dynamical the dynamics of the supermultiplet (2) governed by the Lag- consequences,andobservablyaffecttheresponseofthese rangian (24) exhibits nonrepetitive (pseudorandom or chao- supermultiplets to probing by external magnetic fields. tic) oscillatory motion for most of the parameter values Furthermore, the wealth and diversity of even just the (29). This effect reminds us of the nonrepetitive (chaotic) bilinear coupling/mixing terms listed in Table 2 indicate dynamics also found for a similar supermultiplet of 𝑁=3 that supermultiplets with a different choice of the tuning supersymmetry on the world-line [28]. parameter are indeed observably different and so usefully 5. Conclusions inequivalentinthesenseof[34].Thesameanalysisapplies justaswellfortheinfinitesequenceofsupermultiplets We have presented a 1-parameter continuous family of off- discussed in [27]. shell supermultiplets (2) of 𝑁=4world-line supersym- We thus have clear proof by explicit example that inequiv- metry, which radically generalizes the study of the discrete alent off-shell supermultiplets (unitary, finite-dimensional 𝑁 sequence of off-shell supermultiplets [27, 28]. In fact, all of linear representations) of world-line -extended supersym- the qualitative conclusions from the present study of (2) apply metry without central extensions form a physically observ- 𝑄 just as well to a similar 𝑄-continuum of 𝑁=3off-shell able continuum. We have explicitly parametrized this - 𝛼 supermultiplets within which the supermultiplets [27, 28] are continuum in terms of the “tuning parameter” appearing special cases. explicitly in (2) and have shown that even simple, bilinear The supermultiplet (2) exhibits an explicit, continuously Lagrangians such as (23) provide pairwise coupling between variable tuning parameter, labeled 𝛼,thevalueofwhich continuously many inequivalent such off-shell supermulti- controls the relative “magnitude” in the binomial results of plets. This in turn provides a way of physically probing the 𝑄 applying the supercharges to the component fields. By virtue variation in the dynamics over this novel -moduli space. of the existence of these binomial terms, the supermultiplet Quite clearly, just as all bilinear Lagrangians (23) depend 𝜎 (2)maybethoughtofasanetworkofAdinkras[21]connected quadratically on the tuning parameters, general -model by one-way edges, as depicted in Figure 1. Lagrangians including but not limited to (3) will exhibit more 𝑄 For two distinct members from this continuous family of general variation over the -moduli space. However, already off-shell supermultiplets, we have constructed a multiparam- the dynamics governed by Lagrangians (23) restricted to eter family of general bilinear Lagrangians (23) which has the purely bilinear terms exhibit physically observable and highly 𝛼 following characteristics: nonlinear dependence on —as exhibited in the frequencies (28)—albeit being derived from Lagrangian (24) that is itself (1) It generalizes the “standard” kinetic terms (as in (5), only quadratic in 𝛼. 𝛼→0 with ) into a 6-parameter family of Lagran- Such variations in dynamics are at the core of stud- gians (4) but which all by itself may be simplified ies such as the Deligne-Mumford-Teichmuller¨ theory for back to the “standard” form by means of local field (super)string world-sheets, moduli spaces of Calabi-Yau redefinitions. compactifications, and the (super)string landscape [38–41]. (2) It mixes two off-shell supermultiplets of the same type We have herein shown that conceptually similar moduli (2), each with a different value of the tuning parame- spaces also emerge, in a qualitatively similar manner, in the ter, given as 14-parameter linear combinations of the representation theory of supersymmetry. 10 Advances in High Energy Physics

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[34] T. Hubsch,¨ “Linear and chiral superfields are usefully inequivalent,” Classical and Quantum Gravity,vol.16,no.9,pp.L51–L54, 1999. [35] S. J. Gates Jr., T. Hubsch,¨ and K. Stiffler, “Adinkras and SUSY holography,” International Journal of Modern Physics A,vol.29, no. 7, Article ID 1450041, 14 pages, 2014. [36] J. Gates Jr., “Superspace formulation of new nonlinear sigma models,” Nuclear Physics B,vol.238,no.2,pp.349–366,1984. [37] S.J.GatesJr.,C.M.Hull,andM.Rocek,ˇ “Twisted multiplets and new supersymmetric non-linear 𝜎-models,” Nuclear Physics B, vol.248,no.1,pp.157–186,1984. [38] J. Polchinski, String Theory, Cambridge Monographs on Math- ematical Physics, Cambridge University Press, Cambridge, UK, 1998. [39] T. Hubsch,¨ Calabi-Yau Manifolds, World Scientific Publishing, River Edge, NJ, USA, 2nd edition, 1994. [40] C. Vafa, “The string landscape and the swampland,” http://arxiv .org/abs/hep-th/0509212. [41] H. Ooguri and C. Vafa, “On the geometry of the string landscape and the swampland,” Nuclear Physics. B,vol.766,no.1–3,pp.21– 33, 2007. [42] U. Lindstrom¨ and M. Rocek,ˇ “New hyperkahler¨ metrics and new supermultiplets,” Communications in Mathematical Phy- sics, vol. 115, no. 1, pp. 21–29, 1988. [43] U. Lindstrom¨ and M. Rocek,ˇ “𝑁=2super Yang-Mills theory in projective superspace,” Communications in Mathematical Physics,vol.128,no.1,pp.191–196,1990. [44] S. M. Kuzenko, “Projective superspace as a double-punctured harmonic superspace,” International Journal of Modern Physics A,vol.14,no.11,pp.1737–1757,1999. [45] D. Jain and W. Siegel, “Deriving projective hyperspace from harmonic,” Physical Review D,vol.80,no.4,ArticleID045024, 2009. [46] C. F. Doran, M. G. Faux, S. J. Gates Jr., T. Hubsch,¨ K. M. Iga, and G. D. Landweber, “Super-Zeeman embedding models on N-supersymmetric world-lines,” Journal of Physics A,vol.42, no. 6, Article ID 065402, 2009. Hindawi Publishing Corporation Advances in High Energy Physics Volume 2016, Article ID 5687463, 10 pages http://dx.doi.org/10.1155/2016/5687463

Research Article MSSM Dark Matter in Light of Higgs and LUX Results

W. Abdallah1,2 and S. Khalil1 1 Center for Fundamental Physics, Zewail City of Science and Technology, 6th of October City, Giza 12588, Egypt 2Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt

Correspondence should be addressed to W. Abdallah; [email protected]

Received 26 September 2015; Accepted 6 December 2015

Academic Editor: Enrico Lunghi

Copyright © 2016 W. Abdallah and S. Khalil. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.

The constraints imposed on the Minimal Supersymmetric Standard Model (MSSM) parameter space by the Large Hadron Collider (LHC) Higgs mass limit and gluino mass lower bound are revisited. We also analyze the thermal relic abundance of lightest neutralino, which is the Lightest Supersymmetric Particle (LSP). We show that the combined LHC and relic abundance constraints rule out most of the MSSM parameter space except a very narrow region with very large tan 𝛽 (∼50). Within this region, we emphasize that the spin-independent scattering cross section of the LSP with a proton is less than the latest Large Underground Xenon (LUX) limit by at least two orders of magnitude. Finally, we argue that nonthermal Dark Matter (DM) scenario may relax the constraints imposed on the MSSM parameter space. Namely, the following regions are obtained: 𝑚0 ≃ O(4) TeV and 𝑚1/2 ≃ 600 GeV for low tan 𝛽 (∼10); 𝑚0 ∼𝑚1/2 ≃ O(1) TeV or 𝑚0 ≃ O(4) TeV and 𝑚1/2 ≃ 700 GeV for large tan 𝛽 (∼50).

1. Introduction mass 𝑚1/2, universal trilinear coupling 𝐴0,andtheratioof the vacuum expectation values of Higgs bosons tan 𝛽.In The most recent observations by the Planck satellite con- addition, due to 𝑅-parity conservation, SUSY particles are firmed that 26.8% of the universe content is in the form of produced or destroyed only in pairs and therefore the LSP is DM and the usual visible matter only accounts for 5% [1]. absolutely stable, implying that it might constitute a possible The LSP remains one of the best candidates for the DM [2,3]. candidate for DM, as first suggested by Goldberg in 1983 [12]. It is a Weakly Interacting Massive Particle (WIMP) that can So although the original motivation of SUSY has nothing to naturally account for the observed relic density of DM. do with the DM problem, it turns out that it provides a stable Despite the absence of direct experimental verification, neutral particle and, hence, a candidate for solving the DM Supersymmetry (SUSY) is still the most promising candidate problem. for a unified theory beyond the Standard Model (SM). SUSY The landmark discovery of the SM-like Higgs boson at is a generalization of the space-time symmetries of the quan- the LHC, with mass ∼125 GeV [13, 14], might be an indication tumfieldtheorythatlinksthematterparticles(quarksand for the presence of SUSY. Indeed, the MSSM predicts that leptons) with the force-carrying particles and implies that there is an upper bound of 130 GeV on the Higgs mass. there are additional “superparticles” necessary to complete However, this mass of lightest Higgs boson implies that the the symmetry. In this regard, SUSY solves the problem of the SUSY particles are quite heavy. This may justify the negative quadraticdivergenceintheHiggssectoroftheSMinavery searches for SUSY at the LHC run-I [15–18]. However, it is elegant natural way. The most simple supersymmetric exten- clearly generating a new “little hierarchy problem.” sion of the SM, which is the most widely studied, is known as Moreover, the relic density data [1] and upper limits on the MSSM [4–11]. In this model, certain universality of soft the DM scattering cross sections on nuclei (LUX [19] and SUSY breaking terms is assumed at grand unification scale. other direct detection experiments [20, 21]) impose stringent Therefore, the SUSY spectrum is determined by the following constraints on the parameter space of the MSSM [22–25]. four parameters: universal scalar mass 𝑚0, universal gaugino In fact, combining the collider, astrophysics, and rare decay 2 Advances in High Energy Physics

constraints [26–36] almost rules out the MSSM. It is tempting 𝑆𝑈(3)𝐶 ×𝑆𝑈(2)𝐿 × 𝑈(1)𝑌 quantum numbers, we have therefore to explore well motivated extensions of the MSSM, additional terms that can be written as such as NMSSM [37, 38] and BLSSM [39, 40], which may 𝑊󸀠 =𝜆 𝐿 𝐿 𝐸𝑐 +𝜆󸀠 𝐿 𝑄 𝐷𝑐 +𝜆󸀠󸀠 𝐷𝑐𝐷𝑐𝑈𝑐 alleviate the little hierarchy problem of the MSSM through 𝑖𝑗𝑘 𝑖 𝑗 𝑘 𝑖𝑗𝑘 𝑖 𝑗 𝑘 𝑖𝑗𝑘 𝑖 𝑗 𝑘 (2) additional contributions to Higgs mass [37, 38, 41] and also +𝜇𝐿 𝐻 . provide new DM candidates [42–45] that may account for the 𝑖 𝑖 2 relic density with no conflict with other phenomenological These terms violate baryon and lepton number explicitly and constraints. lead to proton decay at unacceptable rates. To forbid these In this paper, we analyze the constraints imposed by the terms, a new symmetry, called 𝑅-parity, is introduced, which Higgs mass limit and the gluino lower bound, which are the 3𝐵+𝐿+2𝑆 is defined as 𝑅𝑃 =(−1) ,where𝐵 and 𝐿 are baryon and most stringent collider constraints, on the constrained MSSM lepton number and 𝑆 isthespin.Therearetworemarkable (minimal SUGRA model, hereafter referred to as MSSM) phenomenological implications of the presence of 𝑅-parity: parameter space. In particular, these constraints imply that (i) SUSY particles are produced or destroyed only in pair; (ii) the gaugino mass, 𝑚1/2, resides within the mass range: the LSP is absolutely stable and, hence, it might constitute a 620 GeV ≲𝑚1/2 ≲ 2000 GeV,while the other parameters are possible candidate for DM. much less constrained. We study the effect of the measured In the MSSM, a certain universality of soft SUSY breaking DM relic density on the MSSM allowed parameter space. 𝑀 =3×1016 We emphasized that in this case all parameter space is ruled terms at grand unification scale 𝑋 GeV is assumed. These terms are defined as 𝑚0,theuniversalscalar out except for few points around tan 𝛽∼50, 𝑚0 ∼1TeV, 𝑚 ∼ 1.5 soft mass, 𝑚1/2,theuniversalgauginomass,𝐴0,theuniversal and 1/2 TeV. We also investigate the direct detection 𝐵 rate of the LSP at these allowed points in light of the latest trilinear coupling, , and the bilinear coupling (the soft LUX result. Finally, we show that if one assumes nonstandard mixing between the Higgs scalars). In order to discuss the physical implication of soft SUSY breaking at low energy, scenario of cosmology with low reheating temperature, where 𝑀 theLSPmayreachequilibriumbeforethereheatingtime, we need to renormalize these parameters from 𝑋 down to electroweak scale, which has been performed using SARAH then the relic abundance constraints on (𝑚0,𝑚1/2) can be significantly relaxed. [46],andthespectrumhasbeencalculatedusingSPheno[47, The paper is organized as follows. In Section 2, we briefly 48]. In addition, the MSSM contains another two free SUSY (𝑚 ,𝑚 ) parameters: 𝜇 and tan 𝛽=⟨𝐻2⟩/⟨𝐻1⟩.Twoofthesefree introduce the MSSM and study the constraints on 0 1/2 𝜇 𝐵 plane from Higgs and gluino mass experimental limits. In parameters, and , can be determined by the electroweak Section3,westudythethermalrelicabundanceoftheLSP breaking conditions: in the allowed region of parameter space. We show that the 2 2 2 2 𝑚𝐻 −𝑚𝐻 tan 𝛽 𝑀 combined LHC and relic abundance constraints rule out most 𝜇2 = 1 2 − 𝑍 , 2 (3) of the parameter space except the case of very large tan 𝛽. tan 𝛽−1 2 We also provide the expected rate of direct LSP detection at −2𝑚2 these points with large tan 𝛽 and TeV masses. Section 4 is 2𝛽 = 3 . sin 2 2 (4) devotedtononthermalscenarioofDMandhowitcanrelax 𝑚1 +𝑚2 the constraints imposed on MSSM parameter space. Finally, we give our conclusions in Section 5. Thus, the MSSM has only four independent free parameters, 𝑚0,𝑚1/2,𝐴0, tan 𝛽, besides the sign of 𝜇, which determine the whole spectrum. 2. MSSM after the LHC Run-I In the MSSM, the mass of the lightest Higgs state can be The particle content of the MSSM is three generations of approximated, at the one-loop level, as [49–52] (chiral) quark and lepton superfields; the (vector) superfields 2 4 𝑚2 𝑚2 2 2 3𝑔 𝑚𝑡 ̃𝑡 ̃𝑡 are necessary to gauge 𝑆𝑈(3)𝐶 ×𝑆𝑈(2)𝐿 × 𝑈(1)𝑌 gauge of the 𝑚 ≤𝑀 + ( 1 2 ). ℎ 𝑍 2 2 2 log 4 (5) SM, and two (chiral) 𝑆𝑈(2) doublet Higgs superfields. The 16𝜋 𝑀𝑊 sin 𝛽 𝑚𝑡 introduction of a second Higgs doublet is necessary in order to cancel the anomalies produced by the fermionic members Therefore, if one assumes that the stop masses are of order of the first Higgs superfield and also to give masses to both up TeV, then the one-loop effect leads to a correction of order O(100) and down type quarks. The interactions between Higgs and GeV, which implies that matter superfields are described by the superpotential MSSM √ 2 2 𝑐 𝑐 𝑐 𝑚ℎ ≲ (90 GeV) + (100 GeV) ≃ 135 GeV. (6) 𝑊=ℎ𝑈𝑄𝐿𝑈𝐿𝐻2 +ℎ𝐷𝑄𝐿𝐷𝐿𝐻1 +ℎ𝐿𝐿𝐿𝐸𝐿𝐻1 (1) The two-loop corrections reduce this upper bound by few +𝜇𝐻1𝐻2. GeVs [53–55]. Hence, the MSSM predicts the following upper 𝑐 𝑐 Here, 𝑄𝐿 contains 𝑆𝑈(2) (s)quark doublets and 𝑈𝐿, 𝐷𝐿 are the bound for the Higgs mass: 𝑚ℎ ≲130GeV,which was consist- corresponding singlets, (s)lepton doublets and singlets reside ent with the measured value of Higgs mass (of order 125 GeV) 𝑐 in 𝐿𝐿 and 𝐸𝐿,respectively.𝐻1 and 𝐻2 denote Higgs super- at the LHC [13, 14]. fields with hypercharge 𝑌=∓1/2. Further, due to the fact In Figure 1, we display the contour plot of the SM- that Higgs and lepton doublet superfields have the same like Higgs boson: 𝑚ℎ ∈[124, 126] GeV in (𝑚0,𝑚1/2) Advances in High Energy Physics 3

tan𝛽 = 10, 𝜇 > 0,A =0GeV tan𝛽 = 10, 𝜇 > 0,A = 2000 GeV 6000 0 6000 0

5000 5000

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3000 3000 (GeV) (GeV) 0 0 m m 2000 2000

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5000 5000

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Figure 1: MSSM parameter space for tan 𝛽=10(a) and 50 (b) with 𝐴0 =0and 2 TeV. The green region indicates 124 ≲ 𝑚ℎ ≲ 126 GeV. The blue region is excluded because the lightest neutralino is not the LSP. The pink region is excluded due to the absence of radiative electroweak 2 symmetry breaking (𝜇 becomes negative). The gray shadow lines denote the excluded area because of 𝑚𝑔̃ < 1.4 TeV.

plane for different values of 𝐴0 and tan 𝛽. It is remarkable Furthermore, this region is shown with dashed lines in that the smaller the value of 𝐴0 is, the smaller the value Figure 1. of 𝑚1/2 is needed to satisfy this value of Higgs mass. It 𝑚 is also clear that the scalar mass 0 remains essentially 3. Dark Matter Constraints on unconstrained by Higgs mass limit. It can vary from few MSSM Parameter Space hundred GeVs to few TeVs. Such large values of 𝑚1/2 seem to imply a quite heavy SUSY spectrum, much heavier than 3.1. The LSP as Dark Matter Candidate. The neutralinos 𝜒𝑖 the lower bound imposed by direct searches at the LHC 𝑖 = 1, 2, 3, 4 experiments in centre of mass energies √𝑠=7,8TeV and ( )arethephysical(mass)superpositionsoftwo −1 fermionic partners of the two neutral gauge bosons, called total integrated luminosity of order 20 fb .Furthermore, 0 0 𝐵̃ 𝑊̃ the LHC lower limit on the gluino mass, 𝑚𝑔̃ ≳ 1.4 TeV gaugino (bino) and 3 (wino), and of the two neutral 𝑚 < 620 ̃ 0 ̃0 [56, 57], excluded the values of 1/2 GeV which Higgs bosons, called Higgsinos 𝐻1 and 𝐻2.Theneutralino wasallowedbyHiggsmassconstraintsfor𝑚0 >4TeV. mass matrix is given by [58–61]

𝑀1 0−𝑀𝑍 cos 𝛽 sin 𝜃𝑊 𝑀𝑍 sin 𝛽 sin 𝜃𝑊

0𝑀2 𝑀𝑍 cos 𝛽 cos 𝜃𝑊 −𝑀𝑍 sin 𝛽 cos 𝜃𝑊 ( ) 𝑀𝑁 = , (7) −𝑀𝑍 cos 𝛽 sin 𝜃𝑊 𝑀𝑍 cos 𝛽 cos 𝜃𝑊 0−𝜇

( 𝑀𝑍 sin 𝛽 sin 𝜃𝑊 −𝑀𝑍 sin 𝛽 cos 𝜃𝑊 −𝜇 0 ) 4 Advances in High Energy Physics

1.00 where 𝑀1 and 𝑀2 are related due to the universality of the gaugino masses at the grand unification scale, 𝑀1 = 2 2 (3𝑔1/5𝑔2)𝑀2,where𝑔1, 𝑔2 are the gauge couplings of 𝑈(1)𝑌 and 𝑆𝑈(2)𝐿, respectively. This Hermitian matrix is diagonalized by a unitary transformation of the neutralino fields, 0.99 diag † 𝑀𝑁 =𝑁𝑀𝑁𝑁. The lightest eigenvalue of this matrix and the corresponding eigenstate, say 𝜒, has good chance of being the LSP. The lightest neutralino will be a linear combination

g 0.98 of the original fields: f ̃0 ̃0 ̃0 ̃0 𝜒=𝑁11𝐵 +𝑁12𝑊 +𝑁13𝐻1 +𝑁14𝐻2. (8)

The phenomenology and cosmology of the neutralino are 0.97 governed primarily by its mass and composition. A useful parameter for describing the neutralino composition is the 2 2 gaugino “purity” function 𝑓𝑔 =|𝑁11| +|𝑁12| [58–61]. If 𝑓𝑔 > 0.5 𝑓 < 0.5 , then the neutralino is primarily gaugino and if 𝑔 , 0.96 then the neutralino is primarily Higgsino. Actually, if |𝜇| > |𝑀 |≥𝑀 300 400 500 600 700 800 900 2 𝑍, the two lightest neutralino states will be deter- m (GeV) mined by the gaugino components; similarly, the light char- 𝜒 |𝜇| < |𝑀 | gino will be mostly a charged wino, while if 2 ,the Figure 2: The mass of lightest neutralino versus the purity function two lighter neutralinos and the lighter chargino are all mostly in the region of parameter space allowed by gluino and Higgs mass Higgsinos, with mass close to |𝜇|.Finally,if|𝜇| ≃ 2|𝑀 |,the limits. states will be strongly mixed. Here, two remarks are in order. (i) The abovementioned 3/2 −𝑚𝜒/𝑇 constraints in 𝑚1/2 fromHiggsmasslimitandgluinomass 𝑔𝜒(𝑚𝜒𝑇/2𝜋) 𝑒 .Here,𝑚𝜒 and 𝑔𝜒 are the mass and the lower bound imply that 𝑚𝜒 ≳ 240 GeV, which is larger number of degrees of freedom of the LSP,respectively. Finally, than the limits obtained from direct searches at the LHC. 𝑠 is the entropy density. In the standard cosmology, the Hub- 2 Moreover, an upper bound of order one TeV is also obtained ble parameter 𝐻 is given by 𝐻(𝑇) =√ 2𝜋 𝜋𝑔∗/45(𝑇 /𝑀𝑃𝑙), 19 (from Higgs mass constraint). (ii) In this region of allowed where 𝑀𝑃𝑙 =1.22×10 GeV and 𝑔∗ isthenumberofrela- parameter space, the LSP is essentially pure bino, as shown in tivistic degrees of freedom, for MSSM 𝑔∗ ≃ 228.75.Letus 𝜇 Figure 2. This can be easily understood from the fact that - introduce the variable 𝑥=𝑚𝜒/𝑇 and define 𝑌=𝑛𝜒/𝑠 with parameter, determined by the radiative electroweak breaking eq 𝑌eq =𝑛𝜒 /𝑠.Inthiscase,theBoltzmannequationisgivenby condition, (3), is typically of order 𝑚0 and hence it is much heavier than the gaugino mass 𝑀1. 𝑑𝑌 1 𝑑𝑠 ann 2 2 = ⟨𝜎 V⟩(𝑌 −𝑌 ). (10) 𝑑𝑥 3𝐻 𝑑𝑥 𝜒 eq 3.2. Relic Density. As advocated in the previous section, the 𝜒 In radiation domination era, the entropy, as a function of the LSP in MSSM, the lightest neutralino ,isaperfectcandidate temperature, is given by for DM. Here, we assume that 𝜒 was in thermal equilibrium 2 with the SM particles in the early universe and decoupled 2𝜋 3 −3 𝑠 (𝑥) = 𝑔∗ (𝑥) 𝑚𝜒𝑥 , (11) when it was nonrelativistic. Once 𝜒 annihilation rate Γ𝜒 = 45 𝑠 ann ⟨𝜎𝜒 V⟩𝑛𝜒 dropped below the expansion rate of the universe, which is deduced from the fact that 𝑠 = (𝜌 + 𝑝)/𝑇 and 𝑔∗ Γ ≤𝐻 𝑠 𝜒 , the LSP particles stop to annihilate and fall out of is the effective degrees of freedom for the entropy density. equilibrium and their relic density remains intact till now. The ann Therefore, one finds above ⟨𝜎 V⟩ refers to thermally averaged total cross section 𝜒 𝑑𝑠 3𝑠 for annihilation of 𝜒𝜒 into lighter particles times the relative =− . (12) velocity, V. 𝑑𝑥 𝑥 The relic density is then determined by the Boltzmann 𝑔 ≃𝑔 Thus, with assuming ∗ ∗𝑠 , the following expression for equationfortheLSPnumberdensity(𝑛𝜒) and the law of the Boltzmann equation for the LSP number density is entropy conservation: obtained: ann 𝑑𝑛𝜒 2 2 𝑑𝑌 𝜋𝑔 ⟨𝜎 V⟩ =−3𝐻𝑛 −⟨𝜎annV⟩[(𝑛 ) −(𝑛eq) ], =−√ ∗ 𝑀 𝑚 𝜒 (𝑌2 −𝑌2 ). (13) 𝑑𝑡 𝜒 𝜒 𝜒 𝜒 𝑑𝑥 45 𝑃𝑙 𝜒 𝑥2 eq (9) 𝑑𝑠 If one considers the s-wave and p-wave annihilation =−3𝐻𝑠, ann 𝑑𝑡 processes only, the thermal average ⟨𝜎𝜒 V⟩ then shows as eq where 𝑛 is the LSP equilibrium number density which, 6𝑏 𝜒 ann 𝜒 𝑇 𝑛eq = ⟨𝜎 V⟩≃𝑎 + , (14) as a function of temperature ,isgivenby 𝜒 𝜒 𝜒 𝑥 Advances in High Energy Physics 5

𝜒 f

𝜒 f 𝜒 f

Z f̃ A

𝜒 f 𝜒 f 𝜒 f

Figure 3: Feynman diagrams contributing to early-universe neutralino 𝜒 annihilation into fermions through sfermions, 𝑍-gauge boson, and Higgs.

where 𝑎𝜒 and 𝑏𝜒 are the s-wave and p-wave contributions of neutralino, taking into account the possibility of having coan- annihilation processes, respectively. The relic density of the nihilation with the next-to-lightest supersymmetric particle, DM candidate is given by which is typically the lightest stau. Note that this type of coan- nihilation is not included in the approximated expressions in 𝑚 𝑠 𝑌 (∞) Ωℎ2 = 𝜒 0 𝜒 , (14)–(17). In this figure, the red regions correspond to a relic 2 (15) 𝜌𝑐/ℎ abundance within the measured limits [1]: 2 −41 3 2 −47 4 0.09 < Ωℎ <0.14. (18) where 𝑠0 = 2282.15 × 10 GeV , 𝜌𝑐 = 8.0992ℎ ×10 GeV , and by solving the Boltzmann equation, one can find 𝑌𝜒(∞) It is noticeable that, with low tan 𝛽 (∼10), this region corre- as follows [62]: sponds to light 𝑚1/2 (<500 GeV), where significant coanni-

−1 hilation between the LSP and stau took place. However, this 1 𝑎 3𝑏 possibilityisnowexcludedbytheHiggsandgluinomass 𝑌 (∞) = ( 𝜒 + 𝜒 ) , 𝜒 2 (16) constraints [64]. At large tan 𝛽, another region is allowed 𝜆𝜒 𝑥(𝑇 ) 𝑥 (𝑇 ) 𝑓 𝑓 duetopossibleresonancedueto𝑠-channel annihilation of 𝑇 𝜆 =𝑠(𝑚)/𝐻(𝑚 ) the DM pair into fermion-antifermion via the pseudoscalar where 𝑓 is the freeze-out temperature, 𝜒 𝜒 𝜒 , 𝐴 𝑀 ≃2𝑚 𝐴 =0 𝑥(𝑇 ) Higgs boson at 𝐴 𝜒 [65]. For 0 ,averysmall and 𝑓 is given by part of this region is allowed by the Higgs mass constraint, while for large 𝐴0 (∼2TeV)slightenhancementofthispart can be achieved. In Figure 5, we zoom in on this region to [𝛼𝜒𝜆𝜒𝑐 (𝑐+2) 6𝑏𝜒 ] 𝑥(𝑇𝑓)=ln [ (𝑎𝜒 + )] , (17) show the explicit dependence of the relic abundance on the √𝑥(𝑇 ) 𝑥(𝑇𝑓) 𝛽 [ 𝑓 ] LSPmassandlargevaluesoftan .Ascanbeseenfromthis figure, there is no point that can satisfy the relic abundance 𝛼 =(45/2𝜋4)√𝜋/8(𝑔 /𝑔 (𝑇 )) 𝑐=1/2 stringent constraints with tan 𝛽<30. where 𝜒 𝜒 ∗𝑠 𝑓 ;thevalue results in a typical accuracy of about 5–10% more than sufficient for our purposes here. 3.3. Direct Detection. Perhapsthemostnaturalwayofsearch- The lightest neutralino may annihilate into fermion- ing for the neutralino DM is provided by direct experi- + − + − antifermion (𝑓𝑓), 𝑊 𝑊 , 𝑍𝑍, 𝑊 𝐻 , 𝑍𝐴, 𝑍𝐻, 𝑍ℎ,and ments, where the effects induced in appropriate detectors by + − 𝐻 𝐻 and all other contributions of neutral Higgs. For a neutralino-nucleus elastic scattering may be measured. The elastic-scattering cross section of the LSP with a given nucleus bino-like LSP, that is, 𝑁11 ≃1and 𝑁1𝑖 ≃0, 𝑖 = 2, 3, 4,one has two contributions: spin-dependent contribution arising finds that the relevant annihilation channels are the fermion- 𝑍 𝑞̃ antifermion ones, as shown in Figure 3, and all other chan- from and exchange diagrams and spin-independent nels are instead suppressed. Also, the annihilation process (scalar) contribution due to the Higgs and squark exchange mediated by 𝑍-gauge boson is suppressed due to the small diagrams, which is typically suppressed. The effective scalar 2 2 interaction of neutralino with a quark is given by 𝑍𝜒𝜒 coupling ∝𝑁13 −𝑁14, except at the resonance when 𝑚 ∼𝑀/2 𝜒 𝑍 , which is no longer possible due to the above- Lscalar =𝑓𝑞𝜒𝜒𝑞𝑞, (19) mentioned constraints. Furthermore, one finds that the t- channel annihilation (first Feynman diagram in Figure 3) is where 𝑓𝑞 is the neutralino-quark effective coupling. The predominantly into leptons through the exchanges of the scalar cross section of the neutralino scattering with target ̃ ̃ three slepton families (𝑙𝐿, 𝑙𝑅),with𝑙=𝑒,𝜇,𝜏.Thesquarks nucleus at zero momentum transfer is given by [2] exchanges are suppressed due to their large masses. 2 SI 4𝑚𝑟 2 In Figure 4, we display the constraint from the observed 𝜎0 = (𝑍𝑓𝑝 + (𝐴−𝑍) 𝑓𝑛) , (20) 2 𝜋 limits of Ωℎ on the plane (𝑚0-𝑚1/2) for 𝐴0 = 0, 2000 GeV, tan 𝛽 = 10, 50,and𝜇>0. Here, we used micrOMEGAs [63] where 𝑍 and 𝐴−𝑍are the number of protons and neu- to compute the complete relic abundance of the lightest trons, respectively, 𝑚𝑟 =𝑚𝑁𝑚𝜒/(𝑚𝑁 +𝑚𝜒),where𝑚𝑁 is 6 Advances in High Energy Physics

tan𝛽 = 10, 𝜇 > 0,A =0GeV tan𝛽 = 10, 𝜇 > 0,A0 = 2000 GeV 6000 0 6000

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Figure 4: LSP relic abundance constraints (red regions) on (푚0-푚1/2) plane for tan 훽 and 퐴0 as in Figure 1. The LUX result is satisfied by the yellow region. The other color codes are as in Figure 1.

0.14

0.13 LUX 10−45

0.12 ) 2 2 −47 Ωh 10 0.11 (cm p SI 𝜎 0.10 10−49 0.09 500 550 600 650 700 750 800 m (GeV) 10−51 𝜒 10 20 50 100 200 500 1000 2000 Figure 5: The relic abundance versus the mass of the LSP for m𝜒 (GeV) different values of tan 훽. Red points indicate 40 ≤ tan 훽≤50and blue points 30 ≤ tan 훽<40. All points satisfy the abovementioned Figure 6: Spin-independent scattering cross section of the LSP with constraints. aprotonversusthemassoftheLSPwithintheregionallowedbyall constraints (from the LHC and relic abundance). the nucleus mass, and 푓𝑝, 푓𝑛 are the neutralino coupling to protons and neutrons, respectively. The differential scalar 푞 2 cross section for nonzero momentum transfer can now be where V is the neutrino velocity and 퐹(푞 ) is the form factor written: [2]. In Figure 6, we display the MSSM prediction for spin- 휎SI independent scattering cross section of the LSP with a proton 푑휎SI 0 2 2 2 2 2 2 2 = 퐹 (푞 ) ,0<푞<4푚V , 𝑝 4𝑚 V 2 2 2 𝑟 (21) 휎 =∫ 𝑟 (푑휎 /푑푞2)| 푑푞2 푑푞 4푚𝑟 V ( SI 0 SI 𝑓𝑛=𝑓𝑝 ) after imposing the LHC Advances in High Energy Physics 7

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Figure 7: LSP nonthermal relic abundance constraints (red regions) on (푚0-푚1/2) plane for tan 훽 and 퐴0 as in Figure 1. The color codes are as in Figure 1. and relic abundance constraints. It is clear that our results for Here, the reheating temperature is defined as [62] 휎푝 SI are less than the recent LUX bound (blue curve) by at least 1/4 90 1/2 two orders of magnitude. This would explain the negative 푇 =( ) (Γ 푀 ) , RH 2 휙 푃푙 (23) resultsofdirectsearchessofar. 휋 푔∗ (푇RH) Γ 4. Nonthermal Dark Matter and where the decay width 휙 is given by 3 MSSM Parameter Space 1 푚 Γ = 휙 . (24) 휙 2 In the previous section, we assumed standard cosmology 2휋 Λ 푇 scenario where the reheating temperature RH is very large; The scale Λ is the effective suppression scale, which is of order namely, 푇RH ≫푇푓 ≃10GeV.However, the only constraint on the grand unification scale 푀푋. Therefore, for scalar field with the reheating temperature, which could be associated with 7 −11 mass 푚휙 ≃10GeV, one finds Γ휙 ≃10 GeV,andinour decay of any scalar field, 휙, not only the inflaton field, is calculations, we have used 푔∗ = 10.75 due to the consid- 푇RH ≳1MeV in order not to spoil the successful predictions eration of a low reheating temperature scenario. ofbigbangnucleosynthesis. In Figure 7, we show the constraints imposed on the A detailed analysis of the relic density with a low reheat- MSSM (푚0-푚1/2) planeincaseofnonthermalrelicabun- ing temperature has been carried out in [66]. It was empha- ann dance of the LSP for tan 훽 = 10, 50 and 퐴0 =0,2TeV. In this sized that, for a large annihilation cross section, ⟨휎 V⟩≳ 10−14 −2 plot, we also imposed the LHC constraints, namely, the Higgs GeV so that the neutralino reaches equilibrium mass limit and the gluino mass lower bound, similar to the before reheating, and if there are a large number of neutrali- 휙 case of thermal scenario. It is clear from this figure that the nos produced by the scalar field decay, then the relic density stringent constraints imposed on the MSSM parameter space is estimated as [67] by thermal relic abundance are now relaxed and now low 훽 ∼ 푚 (∼O(4) ) 3푚 Γ 2 tan ( 10)isallowedbutwithveryheavy 0 TeV 2 휒 휙 ℎ Ωℎ = . and 푚1/2 ≃ 600 GeV. In addition, the following two regions 2 3 ann 휌 /푠 (22) 2(2휋 /45) 푔∗ (푇RH)푇RH ⟨휎휒 V⟩ 푐 0 are now allowed with large tan 훽 (∼50): (i) 푚0 ∼푚1/2 ≃ 8 Advances in High Energy Physics

O(1) TeV; (ii) 푚0 ≃ O(4) TeV and 푚1/2 ≃ 700 GeV. The SUSY [7]H.E.HaberandG.Kane,“Thesearchforsupersymmetry: spectrum associated with these regions of parameters space probing physics beyond the standard model,” Physics Reports, could be striking signature for nonthermal scenario at the vol. 117, no. 2–4, pp. 75–263, 1985. LHC. [8] S. P. Martin, “A Supersymmetry Primer,” Advanced Series on Directions in High Energy Physics,vol.21,pp.1–153,2010, Advanced Series on Directions in High Energy Physics, vol. 18, 5. Conclusion pp. 1–98, 1998. [9]D.J.H.Chung,L.L.Everett,G.L.Kane,S.F.King,J.D.Lykken, We have studied the constraints imposed on the MSSM and L. T. Wang, “The soft supersymmetry-breaking Lagrangian: parameter space by the Higgs mass limit and the gluino lower theory and applications,” Physics Reports,vol.407,no.1–3,pp.1– bound, which are the most stringent collider constraints 203, 2005. obtained from the LHC run-I at energy 8 TeV. We showed [10] M. Drees, P. Roy, and R. M. 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Also vation of a new boson at a mass of 125 GeV with the CMS two allowed regions are now associated with large tan 훽 (∼ experiment at the LHC,” Physics Letters B,vol.716,no.1,pp. 30–61, 2012. 50); namely, 푚0 ∼푚1/2 ≃ O(1) TeV or 푚0 ≃ O(4) TeV and [15]G.Aad,B.Abbott,J.Abdallahetal.,“SummaryoftheATLAS 푚1/2 ≃ 700 GeV. experiment’s sensitivity to supersymmetry after LHC Run 1— interpreted in the phenomenological MSSM,” JournalofHigh Conflict of Interests Energy Physics,vol.2015,article134,2015. [16] A. Gaz, “SUSY searches at CMS,” In press, http://arxiv.org/abs/ The authors declare that there is no conflict of interests 1411.1886. regarding the publication of this paper. [17] I. Melzer-Pellmann and P. Pralavorio, “Lessons for SUSY from the LHC after the first run,” The European Physical Journal C, vol.74,article2801,2014. Acknowledgments [18] N. 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Research Article Helical Phase Inflation and Monodromy in Supergravity Theory

Tianjun Li,1,2 Zhijin Li,3 and Dimitri V. Nanopoulos3,4,5

1 State Key Laboratory of Theoretical Physics and Kavli Institute for Theoretical Physics China (KITPC), Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China 2School of Physical Electronics, University of Electronic Science and Technology of China, Chengdu 610054, China 3George P. and Cynthia W. Mitchell Institute for Fundamental Physics and Astronomy, Texas A&M University, College Station, TX 77843, USA 4Astroparticle Physics Group, Houston Advanced Research Center (HARC), Mitchell Campus, The Woodlands, TX 77381, USA 5Division of Natural Sciences, Academy of Athens, 28 Panepistimiou Avenue, 10679 Athens, Greece

Correspondence should be addressed to Zhijin Li; [email protected]

Received 3 July 2015; Accepted 17 November 2015

Academic Editor: Ignatios Antoniadis

Copyright © 2015 Tianjun Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.

We study helical phase inflation which realizes “monodromy inflation” in supergravity theory. In the model, inflation is driven by the phase component of a complex field whose potential possesses helicoid structure. We construct phase monodromy based on explicitly breaking global 𝑈(1) symmetry in the superpotential. By integrating out heavy fields, the phase monodromy from single complex scalar field is realized and the model fulfills natural inflation. The phase-axion alignment is achieved from explicitly symmetry breaking and gives super-Planckian phase decay constant. The 𝐹-term scalar potential provides strong field stabilization for all the scalars except inflaton, which is protected by the approximate global 𝑈(1) symmetry. Besides, we show that helical phase inflation can be naturally realized in no-scale supergravity with 𝑆𝑈(2, 1)/𝑆𝑈(2) ×𝑈(1) symmetry since the supergravity setup needed for phase monodromy is automatically provided in the no-scale Kahler¨ potential. We also demonstrate that helical phase inflation can be reduced to another well-known supergravity inflation model with shift symmetry. Helical phase inflation is free from the UV-sensitivity problem although there is super-Planckian field excursion, and it suggests that inflation can be effectively studied based on supersymmetric field theory while a UV-completed framework is not prerequisite.

1. Introduction where 𝑀𝑃 is the reduced Planck mass; otherwise, inflation cannotbetriggeredorlastforasufficientlongperiod. Inflation plays a crucial role in the early stage of our universe However, as a scalar field, the inflaton is expected to obtain [1–3], and supersymmetry was found to be necessary for large quantum loop-corrections on the potential which can inflation soon after its discovery. A simple argument is that break the slow-roll conditions unless there is extremely fine the inflation process is triggered close to the unification scale tuning. Supersymmetry is a natural way to eliminate such in Grand Unified Theory (GUT) [4, 5]. At this scale physics quantum corrections. By introducing supersymmetry, the theory is widely believed to be supersymmetric. To realize flatness problem can be partially relaxed but not completely the slow-roll inflation, it requires strict flat conditions on the solved since supersymmetry is broken during inflation. 𝑉(𝜙) 𝜙 𝑚 potential of inflaton .Themassofinflaton 𝜙 should Moreover, gravity plays an important role in inflation, so be significantly smaller than the inflation energy scale dueto it is natural to study inflation within supergravity the- the slow-roll parameter ory. 󸀠󸀠 𝑚2 Once combining the supersymmetry and gravity theory 2 𝑉 𝜙 together, the flatness problem reappears known as 𝜂 prob- 𝜂≡𝑀𝑃 ≃ ≪1, (1) 𝑉 3𝐻2 lem. 𝑁=1supergravity in four-dimensional space-time 2 Advances in High Energy Physics is determined by three functions: Kahler¨ potential 𝐾,super- effect is likely to introduce extra terms which are suppressed potential 𝑊, and gauge kinetic function. The 𝐹-term scalar by the Planck mass and then irrelevant in the low energy 𝐾 potential contains an exponential factor 𝑒 .Intheminimal scale, while for a super-Planckian field, the irrelevant terms 𝐾 supergravity with 𝐾=ΦΦ, the exponential factor 𝑒 become important and may introduce significant corrections introduces a term on the inflaton mass at Hubble scale, or even destroy the inflation process. In this sense, the which breaks the slow-roll condition (1). To realize inflation predictions just based on the effective field theory are not 𝐾 in supergravity, the large contribution from 𝑒 to scalar trustable.Amoredetaileddiscussionontheultraviolet mass should be suppressed, which needs a symmetry in (UV) sensitivity of the inflation process is provided in 𝐾.Intheminimalsupergravity,𝜂 problem can be solved [22]. by introducing shift symmetry in the Kahler¨ potential as A lot of works have been proposed to realize inflation proposedbyKawasaki,Yamaguchi,andYanagida(KYY)[6]: based on the UV-completed theory, for example, in [23– 𝐾 is invariant under the shift Φ→Φ+𝑖𝐶.Consequently, 30]. However, to realize inflation in string theory, it needs 𝐾 𝐾 is independent of Im(Φ),soisthefactor𝑒 in the 𝐹- to address several difficult problems such as moduli stabi- 𝛼󸀠 term potential. By employing Im(Φ) as inflaton, its mass is lization, Minkowski or de Sitter vacuum, and -andhigher 𝐾 not affected by 𝑒 andthenthereisno𝜂 problem anymore. string loop-corrections on the Kahler¨ potential. However, one The shift symmetry can be slightly broken; in this case, there may doubt whether such difficult UV-completed framework is still no 𝜂 problemandthemodelgivesabroadrangeof is necessary for inflation. In certain scenario, the super- tensor-to-scalar ratios r [7, 8]. 𝜂 problem is automatically Planckian field excursion does not necessarily lead to the solved in no-scale supergravity because of 𝑆𝑈(𝑁, 1)/𝑆𝑈(𝑁)× physical field above the Planck scale. A simple example is 𝑈(1) symmetry in the Kahlerpotential.Historically,theno-¨ thephaseofacomplexfield.Thephasefactor,likeapseudo- scale supergravity was proposed to get vanishing cosmol- Nambu-Goldstone boson (PNGB), can be shifted to any value ogy constant. At classical level, the potential is strictly flat without any effect on the energy scale. By employing the guaranteed by 𝑆𝑈(𝑁, 1)/𝑆𝑈(𝑁) ×𝑈(1) symmetry of the phase as an inflaton, the super-Planckian field excursion Kahler¨ potential, which meanwhile protects the no-scale type is not problematic at all as there is no polynomial higher inflation away from 𝜂 problem. Moreover, 𝑆𝑈(𝑁, 1)/𝑆𝑈(𝑁)× order quantum gravity correction for the phase component. 𝑈(1) symmetry has rich structure that allows different types Besides helical phase inflation, inflationary models using of inflation. Thus, inflation based on no-scale supergravity PNGB as an inflaton have been studied [31–39]. For natural hasbeenextensivelystudied[9–19].Inthiswork,wewillshow inflation, it requires super-Planckian axion decay constant, that,inno-scalesupergravitywith𝑆𝑈(2, 1)/𝑆𝑈(2) ×𝑈(1) which can be obtained by aligned axions [34] (the axion 𝑈(1) alignment relates to 𝑆𝑛 symmetry among Kahler¨ moduli [40]) symmetry, one can pick up subsector, together with 𝑈(1) the superpotential phase monodromy to realize helical phase or anomalous gauge symmetry with large condensation inflation. gaugegroup[41].Inhelicalphaseinflation,aswillbe 𝜂 shown later, the phase monodromy in superpotential can Recently, it was shown that problem can be naturally be easily modified to generate natural inflation and also solved in helical phase inflation [20]. This solution employs a 𝑈(1) realize the super-Planckian phase decay constant, which global symmetry, which is a trivial fact in the minimal is from the phase-axion alignment hidden in the process 𝐾=ΦΦ supergravity with . Using the phase of a complex of integrating out heavy fields. Furthermore, all the extra Φ 𝜂 field as an inflaton, problem is solved due to the global fields are consistently stabilized based on the helicoid poten- 𝑈(1) Φ symmetry. The norm of needs to be stabilized; tial. otherwise, it will generate notable isocurvature perturbations Like helical phase inflation, “monodromy inflation” was that contradict the observations. However, it is a nontrivial proposed to solve the UV sensitivity of large field inflation Φ task to stabilize the norm of while keeping the phase light [42, 43]. In such model, the inflaton is identified as an axion as the norm and phase couple with each other. In that work, obtained from 𝑝-form field after string compactifications. the field stabilization and quadratic inflation are realized The inflaton potential arises from the DBI action of branes via a helicoid type potential. The inflationary trajectory isa or coupling between axion and fluxes. During inflation, the helix line, and this is the reason for the name “helical phase axion rotates along internal cycles and reduces the axion inflation.” In addition, the superpotential of helical phase potential slowly, while all the other physical parameters are inflation realizes monodromy in supersymmetric field theory. unaffected by the axion rotation. Interesting realization of Furthermore, helical phase inflation gives a method to avoid monodromy inflation is the axion alignment [34], which was the dangerous quantum gravity effect on inflation. proposed to get super-Planckian axion decay constant for The single field slow-roll inflation agrees with recent natural inflation, and it was noticed that this mechanism observations [4, 5]. Such kind of inflation admits a relation- actually provides an axion monodromy in [44–46]. Actually, ship between the inflaton field excursion and the tensor-to- a similar name “helical inflation” was firstly introduced in scalar ratio, which is known as the Lyth bound [21]. It suggests [45] for an inflation model with axion monodromy. However, that, to get large tensor-to-scalar ratio, the field excursion amajordifferenceshouldbenoted;the“helical”structure during inflation should be much larger than the Planck mass. in [45] is to describe the alignment structure of two axions, The super-Planckian field excursion challenges the validity, while the “helical” structure in our model is from a single in the Wilsonian sense, of inflationary models described by complex field with stabilized field norm. The physical picture effective field theory. At Planck scale, the quantum gravity of axion monodromy is analogical to the superpotential 𝑊 in Advances in High Energy Physics 3 helical phase inflation. For 𝑊, there is monodromy around a global 𝑈(1) symmetry in the Kahler¨ potential 𝐾=ΦΦ.This the singularity Φ=0: global 𝑈(1) symmetry is employed in helical phase inflation. 𝜃 2𝜋𝑖 As the Kahler¨ potential is independent of the phase ,the Φ󳨀→Φ𝑒 , potential of phase 𝜃 is not affected by the exponential factor 𝑒𝐾 𝜂 𝑊 (2) . Consequently, there is no problem for phase inflation. 𝑊󳨀→𝑊+2𝜋𝑖 . However, the field stabilization becomes more subtle. All the Φ log extra fields except inflaton have to be stabilized for single field inflation,butnormallythephaseandnormofacomplexfield The phase monodromy, together with 𝑈(1) symmetry in couple with each other and then it is very difficult to stabilize the Kahler¨ potential, provides flat direction for inflation. normwhilekeepingphaselight. In the following, we will show that this monodromy is The physical picture of helical phase inflation is that corresponding to the global 𝑈(1) symmetry explicitly broken the phase evolves along a flat circular path with constant, by the inflation term. or almost constant, radius—the field magnitude, and the Inthiswork,wewillstudyhelicalphaseinflationfrom potential decreases slowly. So even before writing down the several aspects in detail. Firstly, we will show that the phase explicit supergravity formula, one can deduce that phase monodromy in the superpotential, which leads to the helicoid inflation, if realizable, should be particular realization of structure of inflaton potential, can be effectively generated complex phase monodromy, and there exists a singularity in by integrating out heavy fields in supersymmetric field the superpotential that generates the phase monodromy. Such theory. Besides quadratic inflation, the phase monodromy singularity further indicates that the model is described by an for helical phase inflation can be easily modified to realize effective theory. natural inflation, in which the process of integrating out Helical phase inflation is realized in the minimal superheavy fields fulfills the phase-axion alignment indirectly gravity with the Kahler¨ potential and leads to super-Planckian phase decay constant with consistent field stabilization as well. We also show that helical 2 phaseinflationcanbereducedtotheKYYinflationbyfield 𝐾=ΦΦ+𝑋𝑋−𝑔(𝑋𝑋) (4) redefinition; however, there is no such field transformation that can map the KYY model back to helical phase inflation. and superpotential Furthermore, we show that the no-scale supergravity with 𝑆𝑈(2, 1)/𝑆𝑈(2)×𝑈(1) symmetry provides a natural frame for 𝑋 𝑊=𝑎 ln (Φ) . (5) helical phase inflation, as 𝑆𝑈(2, 1)/𝑆𝑈(2) ×𝑈(1) symmetry Φ of no-scale Kahler¨ potential already combines the symmetry factors needed for phase monodromy. Moreover, we argue The global 𝑈(1) symmetry in 𝐾 is broken by the superpoten- that helical phase inflation is free from the UV-sensitivity tial with a small factor 𝑎;when𝑎→0, 𝑈(1) symmetry is problem. restored. Therefore, the superpotential with small coefficient This paper is organized as follows. In Section 2, we review is technically natural [47], which makes the model technically the minimal supergravity construction of helical phase infla- stable against radiative corrections. As discussed before, the tion. In Section 3, we present the realization of phase mon- superpotential 𝑊 is singular at Φ=0and exhibits a phase odromy based on supersymmetric field theory. In Section 4, monodromy natural inflation as a special type of helical phase inflation is 2𝜋𝑖 studied. In Section 5, the relationship between helical phase Φ󳨀→Φ𝑒 , inflation and the KYY model is discussed. In Section 6, we 𝑋 (6) study helical phase inflation in no-scale supergravity with 𝑊󳨀→𝑊+2𝜋𝑎𝑖 . 𝑆𝑈(2, 1)/𝑆𝑈(2)×𝑈(1) symmetry. In Section 7, we discuss how Φ helical phase inflation dodges the UV-sensitivity problem of Φ largefieldinflation.ConclusionisgiveninSection8. The theory is well defined only for away from the singularity. During inflation, the field 𝑋 is stabilized at 𝑋=0,and 2. Helical Phase Inflation the scalar potential is simplified as 𝑁=1 In four dimensions, supergravity is determined by 2 1 𝐾 𝑊 𝑉=𝑒ΦΦ𝑊 𝑊 =𝑎2𝑒𝑟 (( 𝑟)2 +𝜃2), the Kahler¨ potential ,superpotential ,andgaugekinetic 𝑋 𝑋 𝑟2 ln (7) function. The 𝐹-term scalar potential is given by 𝑖𝜃 𝜇 𝐾 𝑖𝑗 where Φ=𝑟𝑒, and the kinetic term is 𝐿𝐾 =𝜕𝜇𝑟𝜕 𝑟+ 𝑉=𝑒 (𝐾 𝐷𝑖𝑊𝐷𝑗𝑊−3𝑊𝑊) . (3) 2 𝜇 𝑟 𝜕𝜇𝜃𝜕 𝜃. Interestingly, in the potential (7), both the norm- 2 𝐾 𝑒𝑟 (1/𝑟2) ( 𝑟)2 Torealizeinflationinsupergravity,thefactor𝑒 in the above dependent factor and ln reach the minimum at 𝑟=1 𝑟 formula is an obstacle as it makes the potential too steep for .Thephysicalmassofnorm is a sufficient long slow-roll process. This is the well-known 𝜂 󵄨 𝜂 1 𝜕2𝑉󵄨 1 problem. To solve problem, usually one needs a symmetry 𝑚2 = 󵄨 =(2+ )𝑉; 𝑟 2 󵄨 2 𝐼 (8) in the Kahler¨ potential. In the minimal supergravity, there is 2 𝜕𝑟 󵄨𝑟=1 𝜃 4 Advances in High Energy Physics

𝑟=1 𝜙 therefore, the norm is strongly stabilized at during infla- 1 x tion and the Lagrangian for the inflaton is 0 −1 𝜇 2 2 𝐿=𝜕𝜇𝜃𝜕 𝜃−𝑒𝑎 𝜃 , (9) 15 which gives quadratic inflation driven by the phase of complex field Φ. In the above simple example given by (4) and (5), the field stabilization is obtained from the combination of 10 𝐾 supergravity correction 𝑒 and the pole 1/Φ in 𝑊, besides an ) 2 2 2 𝑟 a accidental agreement that both the factor (1/𝑟 )𝑒 and the 2 2 ( 𝑟) 𝑟=1 5 term ln obtain their minima at , while for more V(10 general helical phase inflation, such accidental agreement is not guaranteed. For example, one may get the following inflaton potential: 0 2 𝐾(𝑟) 1 𝑟 2 −1 𝑉=𝑒 ((ln ) +𝜃 ), (10) 𝑟2 Λ 0 1 𝜙 𝐾(𝑟) 2 y in which the coefficient 𝑒 (1/𝑟 ) admits a minimum at 𝐾(𝑟) 2 2 2 𝑟0 ∼Λbut 𝑟0 =Λ̸ .Inthiscase,thecoefficient𝑒 (1/𝑟 ) Figure 1: The helicoid structure of potential (13) scaled by 10 𝑎 . still gives a mass above the Hubble scale for 𝑟, while ⟨𝑟⟩ is In the graph, parameters 𝑏 and Λ are set to be 𝑏 = 0.1 and Λ=1, slightly shifted away from 𝑟0 in the early stage of inflation, and respectively. 2 after inflation 𝑟 evolves to Λ rapidly. Also, the term (ln(𝑟/Λ)) gives a small correction to the potential and inflationary 2 observables, so this correction is ignorable comparing with the potential, at the lowest order, is proportional to 𝑏 .After the contributions from the super-Planckian valued phase canonical field normalization, the inflaton potential takes the unless it is unexpectedly large. form

1 2 2 2 Potential Deformations.IntheKahler¨ potential, there are cor- 𝑉 (𝜃) = 𝑚 (2𝑏 +𝜃 ), 2 𝜃 (14) rections from the quantum loop effect, while the superpoten- 𝑊 3+𝑖 tial is nonrenormalized. Besides, when coupled with heavy in which the higher order terms proportional to 𝑏 are Φ fields, the Kahler¨ potential of receives corrections through ignored. With regard to the inflationary observations, take integrating out the heavy fields. Nevertheless, because of the tensor-to-scalar ratio r, for example, as the global 𝑈(1) symmetry in the Kahler¨ potential, these 2 2 −2 corrections can only affect the field stabilization, while phase 32𝜃𝑖 8 𝑏 inflation is not sensitive to these corrections. r = ≈ (1 + ) , (15) (𝜃2 +2𝑏2)2 𝑁 2𝑁 Given a higher order correction on the Kahler¨ potential 𝑖

2 2 where 𝜃𝑖 is the phase when inflation starts and 𝑁 ∈ (50, 60) 𝐾=ΦΦ +𝑏(ΦΦ) +𝑋𝑋 −𝑔(𝑋𝑋) , (11) is the 𝑒-folding number. So the correction from higher order term is insignificant for 𝑏<1. one may introduce an extra parameter Λ in the superpotential 𝑋 Φ 3. Monodromy in Supersymmetric 𝑊=𝑎 . (12) Φ ln Λ Field Theory Based on the same argument, it is easy to see the scalar As discussed before, phase inflation naturally leads to the potential reduces to phase monodromy (in mathematical sense) in the superpotential. The phase monodromy requires singularity, which 2 4 1 𝑉=𝑎2𝑒𝑟 +𝑏𝑟 (( 𝑟− Λ)2 +𝜃2) . means the superpotential proposed for phase inflation should 2 ln ln (13) 𝑟 be an effective theory. It is preferred to show how such phase 2 4 monodromy appears from a more “fundamental” theory at 𝑒𝑟 +𝑏𝑟 (1/𝑟2) 𝑟2 =2/(1+ The factor reaches its minimum at 0 higher scale. In [20], the monodromy needed for phase √ 1+8𝑏),below𝑀𝑃 for 𝑏>0. To get the “accidental inflation is realized based on the supersymmetric field theory, agreement” it needs the parameter Λ=𝑟0,andtheninflation in which the monodromy relates to the soft breaking of a is still driven by the phase with exact quadratic potential. global 𝑈(1) symmetry. Without Λ, the superpotential comes back to (5) and the Historically, the monodromy inflation as an attractive scalar potential is shown in Figure 1 with 𝑏 = 0.1.During method to realize super-Planckian field excursion was first 2 inflation ⟨𝑟⟩ ≃0 𝑟 for small 𝑏,theterm(ln 𝑟) contribution to proposed, in a more physical sense, for axions arising from Advances in High Energy Physics 5 string compactifications [42]. In the inflaton potential, the wasanobstaclefornaturalinflationasitleadstotheaxion only factor that changes during axion circular rotation is decaystoosmallforinflation,whileinthisscenariolarge𝛼 from the DBI action of branes. In [44–46], the generalized is helpful for phase inflation to stabilize the axion. Therefore, axion alignment mechanisms are considered as particular for the physical process at scale below the mass scale of three realization of axion monodromy with the potential from heavy fields, the only unfixed degrees of freedom are 𝑋 and nonperturbative effects. We will show that such kind of Φ, which can be described by an effective field theory with axionmonodromycanalsobefulfilledbythesuperpotential threeheavyfieldsintegratedout.Thecoupling𝜎𝑋Ψ(𝑇 −𝛿) phasemonodromy[20],eventhoughitisnotshowninthe is designed for inflation and hierarchically smaller than the effective superpotential after integrating out the heavy fields. extra terms in 𝑊0. Therefore, to describe inflation process, the Furthermore, all the extra fields can be consistently stabilized. heavy fields need to be integrated out. The more “fundamental” field theory for the superpoten- To integrate out heavy fields, we need to consider the 𝐹- tial in (5) is terms again. The 𝐹-term flatness of fields 𝑌 and 𝑍 gives −𝛼𝑇 𝐹 =𝑒−𝛼𝑇 −𝛽Ψ+𝐾 𝑊 =0, 𝑊0 =𝜎𝑋Ψ(𝑇−𝛿) +𝑌(𝑒 −𝛽Ψ)+𝑍(ΨΦ − 𝜆) , (16) 𝑌 𝑌 0 (20) 𝐹 =ΨΦ−𝜆+𝐾 𝑊 =0. where the coupling constants for the second and third terms 𝑍 𝑍 0 are taken to be 1 for simplicity, and a small hierarchy is Near the vacuum 𝑌=𝑍≈0≪𝑀𝑃,theabove assumed between the first term and the last two terms; that 𝐾 𝑊 −𝛼𝑇 supergravity corrections 𝑌(𝑍) 0 areignorable,andthen is, 𝜎≪1.Thecoupling𝑌𝑒 isassumedtobeaneffective the 𝐹-term flatness conditions reduce to these for global description of certain nonperturbative effects. Similar forms supersymmetry. This is gained from the fact that although the can be obtained from 𝐷-brane instanton effect in type string inflation dynamics are subtle, the inflation energy density is theory (for a review, see [48]), besides the coefficient 𝛼∝ close to the GUT scale, far below the Planck scale. Solving 1/𝑓 ≫ 1 in Planck unit, since 𝑓≪1is the decay constant the 𝐹-term flatness equations in (20), we obtain the effective and should be significantly lower than the Planck scale. For superpotential 𝑊 in (5). 𝑊 𝑈(1) the last two terms of 0,thereisaglobal symmetry: Based on the above construction, it is clear that the phase 𝑊 𝑈(1) 𝑊 −𝑖𝑞𝜃 monodromy in is from the transformation of 0, Ψ󳨀→Ψ𝑒 , and the pole of superpotential (5) at Φ=0arises from the Ψ 𝑖𝑞𝜃 integration process. The heavy field is integrated out based Φ󳨀→Φ𝑒 , on the 𝐹-term flatness conditions when ⟨Φ⟩ ≫ ⟨Ψ⟩, while 2 (17) if Φ→0, Ψ becomes massless from |𝐹𝑍| and it is illegal 𝑌󳨀→𝑌𝑒𝑖𝑞𝜃, to integrate out a “massless” field. For inflation, the condition ⟨Φ⟩ ≫ ⟨Ψ⟩ 𝑖𝑞𝜃 is satisfied so the theory with superpotential (5) 𝑇󳨀→𝑇+ , is reliable. 𝛼 As to the inflation term, a question appears: as global 𝑈(1) which is anomalous and explicitly broken by the first term. is explicitly broken by the first term in (16) at inflation Phase rotation of the stabilizer field 𝑋 has no effect on scale, why is the phase light while the norm is much heavier? The supergravity correction to the scalar potential plays a inflation and so is ignored here. The phase monodromy of 𝐾 superpotential 𝑊 in (6) originates from the 𝑈(1) rotation of crucialroleatthisstage.Thecoefficient𝑒 appears in the 𝑊 scalar potential, and because of 𝑈(1) symmetry in the Kahler¨ 0: 𝐾 potential, the factor 𝑒 is invariant under 𝑈(1) symmetry but Ψ󳨀→Ψ𝑒−𝑖2𝜋, increases exponentially for a large norm. Here, the Kahler¨ potential of 𝑇 should be shift invariant; that is, 𝐾=𝐾(𝑇+ 1 (18) 𝑊 󳨀→ 𝑊 +𝑖2𝜋𝜎 𝑋Ψ. 𝑇) instead of the minimal type. Otherwise, the exponential 0 0 𝛼 𝐾 factor 𝑒 depends on phase as well and the phase rotation will (𝑇) As shown in [20], the supersymmetric field theory with be strongly fixed, like the norm component or Re . When integrating out the heavy fields, they should be superpotential 𝑊0 admits the Minkowski vacuum at replaced both in superpotential and in Kahler¨ potential by ⟨𝑋⟩ = ⟨𝑌⟩ = ⟨𝑍⟩ =0, the solutions from vanishing 𝐹-term equations. So, different from the superpotential, the Kahler¨ potential obtained in this ⟨𝑇⟩ =𝛿, way is slightly different from the minimal case given in (4). There are extra terms like 1 −𝛼𝛿 (19) ⟨Ψ⟩ = 𝑒 , 𝜆2 𝛽 ΨΨ= , 𝑟2 𝛼𝛿 (21) ⟨Φ⟩ =𝜆𝛽𝑒 , 1 𝐾(𝑇+𝑇) = 𝐾 ( ( 𝑟)2), 𝛼2 ln with ⟨Φ⟩ ≫ ⟨Ψ⟩ so that near the vacuum the masses of 𝑌, 𝑍,andΨ are much larger than Φ; besides, the effective mass where |Φ| = 𝑟. Nevertheless, since 𝜆≪1and 𝛼≫1,these of 𝑇 isalsolargenearthevacuumduetolarge𝛼.Large𝛼 terms are rather small and have little effect on phase inflation, 6 Advances in High Energy Physics as shown in the last section. Furthermore, the quantum Near vacuum, fields 𝑌,𝑍, Ψ,and𝑇 obtain large effective loop effects during integrating out heavy fields can introduce masses above inflation scale while 𝑋, Φ are much lighter. At corrections to the Kahler¨ potential as well. However, because the inflation scale, the heavy fields should be integrated out. of 𝑈(1) symmetry built in the Kahler¨ potential, these terms, The 𝐹-term flatness conditions for fields 𝑌 and 𝑍 are together with (21), can only mildly affect the field stabiliza- −𝛽𝑇 tion, and phase inflation is not sensitive to the corrections 𝐹𝑌 =𝑒 −𝜇Ψ=0, in Kahler¨ potential. As to the superpotential, it is protected (26) 𝐹 =ΨΦ−𝜆=0, by the nonrenormalized theorem and free from radiative 𝑍 corrections. in which the supergravity corrections 𝐾𝑌/𝑍𝑊1 are neglected as both 𝑌 and 𝑍 get close to zero during inflation. Integrating 4. Natural Inflation in Helical Phase Inflation out Ψ and 𝑇 from (26), we obtain the effective superpotential 𝑊󸀠 𝑊 In the superpotential 𝑊0, the inflation term is perturbative from 1: coupling of complex field 𝑇 which shifts under global 𝑈(1);an 𝛼/𝛽 interesting modification is to consider inflation given by non- 󸀠 𝑋 𝜇𝜆 𝑊 =𝜎𝜆 (( ) −𝛿). (27) perturbative coupling of 𝑇. Such term gives a modified 𝑈(1) Φ Φ phasemonodromyinthesuperpotential.Anditleadstonat- ural inflation as a special type of helical phase inflation with Given 𝛼≪𝛽, the effective superpotential contains a term phase-axion alignment, which is similar to the axion-axion with fractional power. Inflation driven by complex potential alignment mechanism proposed in [34] for natural inflation with fractional power was considered in [49] to get sufficient with super-Planckian axion decay constant. Specifically, it large axion decay constant. Here, the supersymmetric field canbeshownthatthephasemonodromyrealizedinsuper- monodromy naturally leads to the superpotential with frac- symmetric field theory has similar physical picture to the tional power, which arises from the small hierarchy of axion modified axion alignment mechanism provided in [44, 45]. decay constants in two nonperturbative terms. 𝑊 To realize natural inflation, the superpotential 0 in (16) Helical phase inflation is described by the effective super- 󸀠 just needs to be slightly modified: potential 𝑊 ; the role of phase-axion alignment is not clear 󸀠 −𝛼𝑇 −𝛽𝑇 from 𝑊 since it is hidden in the procedure of integrating out 𝑊1 =𝜎𝑋Ψ(𝑒 −𝛿)+𝑌(𝑒 −𝜇Ψ) (22) heavy fields. +𝑍(ΨΦ − 𝜆) , The 𝐹-term flatness conditions (26) fix four degrees of freedom; for the extra degree of freedom, they correspond to in which 1≪𝛼≪𝛽.Againthereisaglobal𝑈(1) the transformations free from constraints (26): symmetry in the last two terms of 𝑊1,andthefieldstransfer under 𝑈(1) like in (17). The first term, which is hierarchically Ψ󳨀→Ψ𝑒−𝑢−𝑖V, smaller, breaks the 𝑈(1) symmetry explicitly; besides, a shift symmetry of 𝑇 is needed in the Kahler¨ potential. The Φ󳨀→Φ𝑒𝑢+𝑖V, (28) monodromy of 𝑊1 under a circular 𝑈(1) rotation is 𝑢 𝑖V −𝑖2𝜋 𝑇󳨀→𝑇+ + . Ψ󳨀→Ψ𝑒 , 𝛽 𝛽 −𝛼𝑇 −2𝜋𝑖(𝛼/𝛽) (23) 𝑊1 󳨀→ 𝑊1 +𝜎𝑋Ψ𝑒 (𝑒 −1). Parameter 𝑢 corresponds to the norm variation of complex The supersymmetric field theory given by (22) admits the field Φ,whichisfixedbythesupergravitycorrectiononthe 𝐾(ΦΦ) following supersymmetric Minkowski vacuum: scalar potential 𝑒 . Parameter V relates to 𝑈(1) transfor- ⟨𝑋⟩ = ⟨𝑌⟩ = ⟨𝑍⟩ =0, mation, which leads to the phase monodromy from the first term of 𝑊1. The scalar potential, including the inflation term, 1 Ψ Φ ⟨𝑇⟩ =− ln 𝛿, depends on the superpositions among phases of , and the 𝛼 axion Im(𝑇), which are constrained as in (28). Among these 1 (24) Φ ⟨Ψ⟩ = 𝛿𝛽/𝛼, fields, the phase of has the lightest mass after canonical field 𝜇 normalization. Similar physical picture appears in the axion- ⟨Φ⟩ =𝜆𝜇𝛿−𝛽/𝛼. axion alignment where inflation is triggered by the axion superpositionalongtheflatdirection. The parameters are set to satisfy conditions After integrating out the heavy fields,ahler theK¨ potential is 𝛿𝛽/𝛼 ∼𝜆𝜇, 𝜆2 𝐾=ΦΦ+ +⋅⋅⋅ . (29) 𝜆≪1, (25) ΦΦ 𝛿<1, As 𝜆≪1,theKahler¨ potential is dominated by ΦΦ;as 2 so that ⟨Ψ⟩ ≪ ⟨Φ⟩ and ⟨𝑇⟩ > 0. discussed before, the extra terms like 𝜆 /ΦΦ only give small Advances in High Energy Physics 7 corrections to the field stabilization. Helical phase inflation can be simply described by the supergravity 𝐾=ΦΦ+𝑋𝑋+..., 10 𝑋 (30) 𝑊=𝑎 (Φ−𝑏 −𝑐), Φ 8

) 𝑏=𝛼/𝛽≪1 𝑐≈1 2 where and .Thescalarpotentialgivenby 6 the above Kahler¨ potential and superpotential is V(a 2 2 𝑎 4 𝑉=𝑒𝑟 (𝑟−2𝑏 +𝑐2 −2𝑐𝑟−𝑏 (𝑏𝜃)), (31) 𝑟2 cos

𝑖𝜃 in which we have used Φ≡𝑟𝑒 .Asusual,thenorm𝑟 couples 1 −1 with the phase in the scalar potential. To generate single 0 0 field inflation, the norm component needs to be stabilized 𝜙 𝜙y with heavy mass above the Hubble scale while the phase x −1 1 2 component remains light. Due to the phase-norm coupling, Figure 2: The helicoid structure of potential (31) scaled by 𝑎 .The thephaseandnormarelikelytoobtaincomparableeffective parameters with 𝑐 = 0.96 and 𝑏 = 0.1 areadoptedinthegraph.It masses during inflation so it is difficult to realize single is shown that the local valley locates around 𝑟≈1.Notethatthe field inflation. However, it is a bit different in helical phase potential gets flatter at the top of the graph. inflation. The above scalar potential can be rewritten as follows: 2 2 2 model. The physics in these two solutions are obviously dif- 𝑟2 𝑎 −𝑏 2 𝑟2 4𝑎 𝑐 𝑏 𝑉=𝑒 (𝑟 −𝑐) +𝑒 ( 𝜃) . (32) ferent. For helical phase inflation, the solution employs 𝑈(1) 𝑟2 𝑟2+𝑏 sin 2 symmetry in Kahler¨ potential of the minimal supergravity, ⟨𝑟⟩ = 𝑟 =𝑐−1/𝑏 𝜃=0 and the superpotential admits a phase monodromy arising So its vacuum locates at 0 and . Besides, 𝑈(1) 𝑟2 2+𝑏 from the global transformation: the coefficient of the phase term 𝑒 (4𝑐/𝑟 ) reaches its minimal value at 𝑟1 = √1 + 𝑏/2.For𝑐≈1,wehavethe 𝐾=ΦΦ+𝑋𝑋+⋅⋅⋅ , 𝑟 ≈𝑟 ≈1 𝐾 approximation 0 1 . The extra terms in give 𝑋 (33) small corrections to 𝑟0 and 𝑟1,buttheapproximationisstill 𝑊=𝑎 ln Φ. valid. So with the parameters in (25), the norm 𝑟 in the Φ two terms of scalar potential 𝑉 can be stabilized at the close The inflation is driven by the phase of complex field Φ region 𝑟≈1separately. If the parameters are tuned so that and several special virtues appear in the model. For the 𝑟0 =𝑟1, then the norm of complex field is strictly stabilized KYY model with shift symmetry [6], theahler K¨ potential is at the vacuum value during inflation. Without such tuning, adjusted so that it admits a shift symmetry along the direction a small difference between 𝑟0 and 𝑟1 is expected but the of Im(𝑇): shift of 𝑟 during inflation is rather small and inflation is still 1 2 approximate to the single field inflation driven by the phase 𝐾= (𝑇+𝑇) +𝑋𝑋+⋅⋅⋅, 2 2 term ∝(sin(𝑏/2)𝜃) .Thehelicoidstructureofthepotential (34) (31) is shown in Figure 2. 𝑊=𝑎𝑋𝑇, It is known that, to realize aligned axion mechanism in supergravity [50], it is very difficult to stabilize the moduli as and inflation is driven by Im(𝑇).Theshiftsymmetryis they couple with the axions. In [40], the moduli stabilization endowed with axions so this mechanism is attractive for axion is fulfilled with gauged anomalous 𝑈(1) symmetries, since inflation. 𝑈(1) 𝐷-terms only depend on the norm |Φ| or Re(𝑇) and Here, we will show that although the physical pictures are then directly separate the norms and phases of matter fields. much different in helical phase inflation and KYY type model, In helical phase inflation, the modulus and matter fields just considering the lower order terms in the Kahler¨ potential except the phase are stabilized at higher scale or by the of redefined complex field, helical phase inflation can reduce supergravity scalar potential. Only the phase can be an to the KYY model. inflaton candidate because of the protection from the global Because the phase of Φ rotates under the global 𝑈(1) 𝑈(1) symmetry in Kahler¨ potential and approximate 𝑈(1) transformation, to connect helical phase inflation with the KYY model, a natural guess is to take the following field symmetry in superpotential. 𝑇 redefinition Φ=𝑒 ;thenhelicalphaseinflation(33)becomes 1 2 5. Helical Phase Inflation and the KYY Model 𝐾=𝑒𝑇+𝑇 +⋅⋅⋅=1+𝑇+𝑇+ (𝑇+𝑇) +⋅⋅⋅ , 2 𝜂 problem in supergravity inflation can be solved both in (35) −𝑇 helical phase inflation and in the shift symmetry in KYY 𝑊=𝑎𝑋𝑇𝑒 . 8 Advances in High Energy Physics

The Kahler¨ manifold of complex field 𝑇 is invariant under the The Kahler¨ potential with 𝑆𝑈(2, 1)/𝑆𝑈(2) ×𝑈(1) symme- holomorphic Kahler¨ transformation try is

𝐾(𝑇,𝑇) 󳨀→ 𝐾 (𝑇, 𝑇) + 𝐹 (𝑇) + 𝐹(𝑇) , ΦΦ (36) 𝐾=−3 (𝑇 + 𝑇− ). ln 3 (39) in which 𝐹(𝑇) is a holomorphic function of 𝑇.Tokeep the whole Lagrangian also invariant under the Kahler¨ trans- In the symmetry of the Kahler¨ manifold, there is 𝑈(1) formation, the superpotential transforms under the Kahler¨ subsector, the phase rotation of complex field Φ,whichcanbe transformation employed for helical phase inflation. Besides, the modulus 𝑇 should be stabilized during inflation, which can be fulfilled by −𝐹(𝑇) 𝑊󳨀→𝑒 𝑊. (37) introducing extra terms on 𝑇. As a simple example of no-scale helical phase inflation, here we follow the simplification in [9] For the supergravity model in (35), taking the Kahler¨ trans- that the modulus 𝑇 has already been stabilized at ⟨𝑇⟩ =. 𝑐 formation with 𝐹(𝑇) = −(1/2) −𝑇,theKahler¨ potential and Different from the minimal supergravity, the kinetic term superpotential become given by the no-scale Kahler¨ potential is noncanonical:

1 2 𝜇 𝐾= (𝑇+𝑇) +⋅⋅⋅, 𝐿𝐾 =𝐾ΦΦ𝜕𝜇Φ𝜕 Φ 2 (38) 2𝑐 𝜇 2 𝜇 (40) 𝑊=𝑎√𝑒𝑋𝑇, = (𝜕𝜇𝑟𝜕 𝑟+𝑟 𝜕𝜇𝜃𝜕 𝜃) , (2𝑐 − 𝑟2/3)2 which is just the KYY model (34) with higher order cor- 𝑖𝜃 rections in the Kahler¨ potential. These higher order terms in which Φ≡𝑟𝑒 is used. The 𝐹-term scalar potential is vanish after field stabilization and have no effect on inflation 𝑇 󵄨 󵄨2 Φ=𝑒 󵄨 󵄨2 󵄨𝑊 󵄨 process. The field relation also gives a map between 𝑉=𝑒−(2/3)𝐾 󵄨𝑊 󵄨 = 󵄨 Φ󵄨 , 󵄨 Φ󵄨 2 (41) the simplest helical phase inflation and the KYY model, such (𝑇+𝑇−ΦΦ/3) as the inflaton, arg(Φ) → Im(𝑇), and field stabilization |Φ| = 1 → Re(𝑇) =. 0 where the superpotential 𝑊 is a holomorphic function of Nevertheless, helical phase inflation is not equivalent to superfield Φ and 𝑇=⟨𝑇⟩=𝑐. It requires a phase monodromy the KYY model. There are higher order corrections to the in superpotential 𝑊 for phase inflation; the simple choice is Kahler¨ potential in the map from helical phase inflation to KYY model, which have no effect on inflation after field 𝑎 Φ 𝑊= . stabilization but indicate different physics in two models. By Φ ln Λ (42) dropping these terms, certain information is lost so the map is irreversible. Specifically, the inverse function 𝑇=ln Φ of The scalar potential given by this superpotential is 𝑇 the field redefinition Φ=𝑒 cannot reproduce helical phase 2 2 inflation from the KYY model, unless one introduces higher 9𝑎 𝑟 2 𝑉= ((ln ) +𝜃 ) . (43) dimensional operators in the Kahler¨ potential exactly the (6𝑐 − 𝑟2)2 𝑟4 𝑒Λ same as given in the expansion (35). Such strict constraint on each higher dimensional operator is not guaranteed by any As in the minimal supergravity, the norm and phase of known rule and so is unphysical. As there is no pole or singu- complex field Φ are separated in the scalar potential. For the 2 2 4 larity in the KYY model, it is unlikely to introduce pole and 𝑟-dependent coefficient factor 1/(6𝑐 − 𝑟 ) 𝑟 , its minimum singularity at origin with phase monodromy through well- √ 2 locates at 𝑟0 = 3𝑐 and another term (ln(𝑟/𝑒Λ)) reaches its defined field redefinition. Actually, the singularity with phase minimum at 𝑟1 =𝑒Λ. Given that the parameter Λ is tuned so monodromy in the superpotential indicates rich physics in that 𝑟0 =𝑟1, in the radial direction, the potential has a global thescaleaboveinflation. minimum at 𝑟=𝑟0. Similar to helical phase inflation in the minimal supergravity, the potential (43) also shows helicoid 6. Helical Phase Inflation in structure, as presented in Figure 3. No-Scale Supergravity The physical mass of 𝑟 in the region near the vacuum is 2 󵄨 The no-scale supergravity is an attractive frame for GUT (2𝑐 − 𝑟2/3) 2 󵄨 2 𝜕 𝑉󵄨 1 2 scale phenomenology; it is interesting to realize helical 𝑚𝑟 = 󵄨 =(4+ )𝐻 , (44) 4𝑐 𝜕𝑟2 󵄨 2𝜃2 phase inflation in no-scale supergravity. Generally, the 󵄨𝑟=√3𝑐 Kahler¨ manifold of the no-scale supergravity is equipped with 𝑆𝑈(𝑁, 1)/𝑆𝑈(𝑁) ×𝑈(1) symmetry. For the no-scale where 𝐻 is the Hubble constant during inflation. So the norm supergravity with exact 𝑆𝑈(1, 1)/𝑈(1) symmetry, without 𝑟 is strongly stabilized at 𝑟0.If𝑟0 and 𝑟1 are not equal but extra fields no inflation can be realized, so the case with close to each other, 𝑟 will slightly shift during inflation but 𝑆𝑈(2, 1)/𝑆𝑈(2) ×𝑈(1) symmetry is the simplest one that itsmassremainsabovetheHubblescaleandinflationisstill admits inflation. approximately the single field inflation. Advances in High Energy Physics 9

with 𝛼≫1and 𝜆≪1.Thefieldnorm𝑟 is stabilized by 2 2 3 minimizing the coefficient 1/𝑟 (𝑐 − 𝑟 /3) and the parameter 𝑐 from ⟨𝑇⟩ determines the scale of ⟨𝑟⟩ = √3𝑐/2. This is different from helical phase inflation in the minimal super- 30 gravity, which minimizes the norm based on the coefficient 𝑟2 2 𝑒 (1/𝑟 ), and then the scale of ⟨𝑟⟩ is close to 𝑀𝑃, the unique energy scale of supergravity corrections. Similarly, combining 20 ) theno-scaleKahler¨ manifold with phase monodromy in (22), 4

2 we can obtain natural inflation.

10 V(a 7. UV-Sensitivity and Helical Phase Inflation −1.0 0 For the slow-roll inflation, the Lyth bound [21] provides a −0.5 1.0 relationship between the tensor-to-scalar ratio r and the field 0.5 0.0 excursion Δ𝜙.Roughlyitrequires 𝜙 0.0 x (√ 3c) 0.5 −0.5 3c) (√ Δ𝜙 1/2 𝜙 y r 1.0 −1.0 ⩾( ) . (47) 𝑀𝑃 0.01 2 4 Figure 3: The helicoid structure of potential (43) scaled by 𝑎 /𝑐 .In the graph, the parameter Λ is tuned so that 𝑟0 =𝑟1.Thecomplex To get large tensor-to-scalar ratio r ⩾0.01,suchasinchaotic field Φ has been rescaled by √3𝑐, and the scale of field norm at local inflation or natural inflation, the field excursion should be valley is determined by the parameter 𝑐 instead of the Planck mass. much larger than the Planck mass. The super-Planckian field excursion makes the description based on the effective field theory questionable. In the Wilsonian sense, the low Instead of stabilizing 𝑇 independently with the inflation energy field theory is an effective theory with higher order process, we can realize helical phase inflation from the no- corrections introduced by the physics at the cut-off scale, scale supergravity with dynamical 𝑇.Thereisanaturalreason like quantum gravity, and these terms are irrelevant in the for such consideration. The phase monodromy requires effective field theory since they are suppressed by the cut-off matter fields transforming as rotations under 𝑈(1) and also energy scale. However, for the inflation process, the inflaton amodulus𝑇 as a shift under 𝑈(1).TheKahler¨ potential of 𝑇 has super-Planckian field excursion, which is much larger is shift invariant. Interestingly, for the no-scale supergravity than the cut-off scale. Thus, the higher order terms cannot be with 𝑆𝑈(2, 1)/𝑆𝑈(2) ×𝑈(1) symmetry, as shown in (39), the suppressed by the Planck mass and may affect the inflationary Kahler¨ potential 𝐾 is automatically endowed with the shift observations significantly. And then they may significantly symmetry and global 𝑈(1) symmetry: affect inflation or even destroy the inflation process. For 𝑇󳨀→𝑇+𝑖𝐶, example, considering the corrections (45) Φ󳨀→Φ𝑒𝑖𝜃. 𝜙 𝑖 Δ𝑉 = 𝑐 𝑉( ) +⋅⋅⋅, (48) 𝑖 𝑀 Therefore, the no-scale Kahler¨ potential fits with the phase 𝑃 monodromy in (16) and (22) initiatively. 𝑉 𝑐 𝑧∈𝑋,𝑌,𝑍 to the original inflaton potential ,aslongas 𝑖 are of the The Kahlerpotentialsofsuperfields¨ are 10−𝑖 𝜙∼𝑂(10)𝑀 of the minimal type 𝑧𝑧, while for Ψ,itsKahler¨ potential order , in the initial stage of inflation 𝑃,the higher order terms can be as large as the original potential can be the minimal type ΨΨ, or the no-scale type 𝐾= 𝑉.Soforlargefieldinflation,itissensitivetothephysicsat −3 ln(𝑇 + 𝑇−(ΦΦ+ΨΨ)/3), which extends the symmetry of 𝑆𝑈(3, 1)/𝑆𝑈(3) ×𝑈(1) thecut-offscaleandthepredictionsofinflationjustbasedon Kahler¨ manifold to .Inthisscenario, effective field theory are questionable. theprocesstointegrateoutheavyfieldsisthesameas In consideration of the UV sensitivity of large field |𝑊 |2 before. Besides, the potential of phase proportional to 𝑋 inflation, a possible choice is to realize inflation in UV- is insensitive to the formula of Kahler¨ potential due to completed theory, like string theory (for a review, see [22]). 𝑈(1) symmetry. The major difference appears in the field To realize inflation in string theory, there are a lot of problems stabilization. The scalar potential from phase monodromy in to solve besides inflation, such as the moduli stabilization, 󸀠 (16) is Minkowski/de Sitter vacua, and effects of 𝛼 -andstringloop- 󵄨 󵄨2 1 𝑉=𝑒𝐾 󵄨𝑊 󵄨 = corrections. Alternatively, in the bottom-up approach, one 󵄨 𝑋󵄨 3 (ln 𝑟2/𝛼+𝑐−𝑟2/3 − 𝜆2𝑟2/3) may avoid the higher order corrections by introducing an 1 1 extra shift symmetry in the theory. The shift symmetry is ⋅ (( 𝑟− Λ)2 +𝜃2)≈ 2 ln ln 3 (46) technically natural and safe under quantum loop-corrections. 𝑟 (𝑐 − 𝑟2/3) However, the global symmetry can be broken by the quantum 1 gravity effect. So it is still questionable whether the shift ⋅ (( 𝑟− Λ)2 +𝜃2), 𝑟2 ln ln symmetry can safely evade the higher corrections like in (48). 10 Advances in High Energy Physics

The UV-completion problem is dodged in helical phase (ii) The phase excursion requires phase monodromy in inflation. Since the super-Planckian field excursion is the the superpotential. So helical phase inflation provides, phaseofacomplexfieldandthephasecomponentisnot in the mathematical sense, a new type of monodromy directly involved in the gravity interaction, there are no in supersymmetric field theory. dangerous high-order corrections like in (48) for the phase (iii) The singularity in the superpotential, together with potential. The inflaton evolves along the helical trajectory and the supergravity scalar potential, provides strong field doesnotrelatetothephysicsintheregionabovethePlanck stabilization which is consistent with phase inflation. scale. For helical phase inflation in the minimal supergravity, the norm of field is stabilized at the marginal point of the (iv) The super-Planckian field excursion is realized by Planck scale, where the supergravity correction on the scalar the phase of a complex field instead of any other potential gets important based on which the norm of complex “physical” fields that directly couple with gravity. So field can be strongly stabilized. The extra corrections are there are no polynomial higher order corrections for likely to appear in the Kahler¨ potential; however, they can thephaseandthusinflationisnotsensitivetothe only slightly affect the field stabilization while the phase quantum gravity corrections. inflation is protected by the global 𝑈(1) symmetry in the Kahler¨ potential; in consequence, helical phase inflation is To summarize, helical phase inflation introduces a new not sensitive to these corrections at all. For the helical phase type of inflation that can be effectively described by super- inflation in no-scale supergravity, the norm of complex field is symmetric field theory at the GUT scale. Generically, the stabilized at the scale of the modulus ⟨𝑇⟩ instead of the Planck super-Planckian field excursion makes the inflationary pre- scale;onecansimplyadjustthescaleof⟨𝑇⟩ to keep the model dictions based on effective field theory questionable, since away from super-Planckian region. the higher order corrections from quantum gravity are Helical phase inflation is free from the UV-sensitivity likely to affect the inflation process significantly. One of the problem, and it is just a typical physical process at the solutions is to realize inflation in a UV-completed theory, like GUT scale with special superpotential that admits phase string theory; nevertheless, there are many difficult issues in monodromy. So it provides an inflationary model that can be string theory to resolve before realizing inflation completely. reliablystudiedjustinsupersymmetricfieldtheory. Helical phase inflation is another simple solution to the UV- sensitivity problem. It is based on the supersymmetric field theory and the physics are clear and much easier to control. 8. Discussions and Conclusion Furthermore, helical phase inflation makes the unification of inflation theory with GUT more natural, since both of them In this work, we have studied the details of helical phase infla- 16 are triggered at the scale of 10 GeV and can be effectively tion from several aspects. Helical phase inflation is realized studied based on supersymmetric field theory. in supergravity with global 𝑈(1) symmetry. 𝑈(1) symmetry Besides,wehaveshownthathelicalphaseinflationalso is built in the Kahler¨ potential so that helical phase inflation relates to several interesting developments in inflation theory. can be realized by the ordinary Kahler¨ potentials, such as the It can be easily modified for natural inflation and realize minimal or no-scale types. Helical phase inflation directly the phase-axion alignment indirectly, which is similar to leadstothephasemonodromyinthesuperpotential,whichis the axion-axion alignment mechanism for super-Planckian singular and an effective field theory arising from integrating axion decay constant [34]. The phase-axion alignment is out heavy fields. The phase monodromy originates from 𝑈(1) not shown in the final supergravity model which exhibits rotation of the superpotential at higher scale. Generically, the explicit phase monodromy only. However, the phase-axion superpotential can be separated into two parts 𝑊=𝑊𝐼 +𝑊𝑆, alignment is hidden in the process when the heavy fields are where 𝑊𝑆 admits the global 𝑈(1) symmetry while 𝑊𝐼 breaks it integrating out. For 𝜂 problem in the supergravity inflation, explicitly at scale much lower than 𝑊𝑆. Under 𝑈(1) rotation, there is another well-known solution, the KYY model with the inflation term 𝑊𝐼 is slightly changed, which realizes the shift symmetry. We showed that through field redefinition phasemonodromyintheeffectivetheoryandintroduces helicalphaseinflationgivenby(4)and(5)reducestothe a flat potential along the direction of phase rotation. By KYY model, where the higher order corrections in the Kahler¨ breaking the global 𝑈(1) symmetry in different ways, we may potential have no effect on inflation process. However, there get different kinds of inflation such as quadratic inflation, is no inverse transformation from the KYY model to helical natural inflation, or the other types of inflation that are not phase inflation since no well-defined field redefinition can presented in this work. introduce the pole and phase singularity needed for phase An amazing fact of helical phase inflation is that it monodromy. Helical phase inflation can be realized in no- deeply relates to several interesting points of inflation and scale supergravity. The no-scaleahler K¨ potential automati- naturally combines them in a rather simple potential with cally provides the symmetry needed by phase monodromy. helicoid structure. The features of helical phase inflation can In the no-scale supergravity, the norm of complex field is be summarized as follows: stabilized at the scale of the modulus. Our inflation models are constructed within the super- (i) The global 𝑈(1) symmetry is built in the Kahler¨ gravitytheorywithglobal𝑈(1) symmetry broken explicitly potential, so helical phase inflation provides a natural by the subleading order superpotential term. So it is just solution to 𝜂 problem. typical GUT scale physics and indicates that a UV-completed Advances in High Energy Physics 11

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Research Article Reciprocity and Self-Tuning Relations without Wrapping

Davide Fioravanti,1 Gabriele Infusino,2,3 and Marco Rossi4

1 Sezione INFN di Bologna, Dipartimento di Fisica e Astronomia, Universita` di Bologna, Via Irnerio 46, 40126 Bologna, Italy 2Dipartimento di Fisica dell’Universita` della Calabria, Arcavacata, Rende, 87036 Cosenza, Italy 3Laboratoire Jean Alexandre Dieudonne,´ UniversiteNiceSophiaAntipolis,06100Nice,France´ 4Dipartimento di Fisica dell’Universita` della Calabria and INFN, Gruppo Collegato di Cosenza, Arcavacata, Rende, 87036 Cosenza, Italy

Correspondence should be addressed to Davide Fioravanti; [email protected]

Received 6 August 2015; Accepted 15 October 2015

Academic Editor: Stefano Moretti

Copyright © 2015 Davide Fioravanti et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.

WeconsiderscalarWilsonoperatorsofN = 4 SYM at high spin, 𝑠, and generic twist in the multicolor limit. We show that the corresponding (non)linear integral equations (originating from the asymptotic Bethe Ansatz equations) respect certain 𝑛 “reciprocity” and functional “self-tuning” relations up to all terms 1/𝑠(ln𝑠) (inclusive) at any fixed ’t Hooft coupling 𝜆.Ofcourse, this relation entails straightforwardly the well-known (homonymous) relations for the anomalous dimension at the same order in 𝑠. On this basis we give some evidence that wrapping corrections should enter the nonlinear integral equation and anomalous 2 2 dimension expansions at the next order (ln𝑠) /𝑠 , at fixed ’t Hooft coupling, in such a way to reestablish the aforementioned relation (which fails otherwise).

1. Introduction, Aims, and Results where D is a (light-cone) covariant derivative acting in all thepossiblewaysonthe𝐿 complex bosonic fields Z,the One of the major achievements of modern theoretical physics trace ensuring gauge invariance. The Lorentz spin of these is the so-called AdS/CFT correspondence [1–3] and its operators is 𝑠 and 𝐿 coincides with the twist, that is, the description in terms of integrability tools [4–19]. In fact, classical dimension minus the spin. The AdS/CFT correspon- being a strong/weak coupling duality, the nonperturbative, dence relates operators (1) to spinning folded closed strings 5 5 exact, though not necessarily explicit (as a simple example on AdS5 × S spacetime, with AdS5 and S angular momenta we can mention, just with reference to the present paper, that 𝑠 and 𝐿,respectively[20,21]. the following nonlinear integral equation (which governs the One of the several reasons for the large interest in these spectrum) is not explicitly solvable), nature of integrability is operators is their similarity with twist operators in QCD, of incomparable value and utility. In particular, the spectrum where, maybe, the scalars are substituted by fermions, that N =4 of anomalous dimensions of composite operators in is,thequarks,orgaugefields:becauseofintegrabilityin super Yang-Mills (SYM) theory ought to correspond to the N =4these cases would be dealt with in an analogous energy spectrum of states in type IIB superstring theory in manner [22, 23]. Similarities among the two theories give the × 5 AdS5 S , and both must be described by an integrable possibility to believe that QCD could take many advantages of system. a full all-loop solution of its supersymmetric counterpart. In N =4 Among the different sectors of multicolor SYM QCD, in the framework of Partonic Model, the Lorentz spin (perturbatively closed under renormalisation), one of the 𝑠 istheconjugatedvariable,intheMellintransform(ofthe (2) most studied ones is the so-called sl scalar twist sector. This splitting function, for instance, which gives the anomalous is spanned by local composite operators of single trace form: dimension), to the Bjorken variable 𝑥, namely, the fraction ofthehadronmomentumcarriedbythesingleparton(of (D𝑠Z𝐿)+⋅⋅⋅ , Tr (1) course, the coupling does run in QCD, unlike what happens 2 Advances in High Energy Physics in the maximally supersymmetric theory). In this context, Relations (3) and (4), both in QCD and in N =4SYM, two regimes emerge naturally: 𝑥→0,governedbythe aredeveloped,checkedinvariouscases,anddiscussedin[27– BFKL equations [24], and 𝑥→1, corresponding exactly 37].Recently,theyhavebeenproven,restrictively,totwisttwo to large values of the Lorentz spin, 𝑠→∞.Propertiesof operators, but in a generic conformal field theory, in [38], this second (called quasielastic) regime can be deduced by with some arguments for their validity in nonconformal the- large spin results in three-loop twist 2 QCD calculations. ories at the end. Clearly, they provide important information In particular, we can highlight two main features about on the high spin expansion of anomalous dimension of twist anomalous dimension of twist operators: operators. Unlike QCD, in N =4SYM, it is possible to obtain better and more suitable results as we can consider these (1) The leading term has a logarithmic scale: relations into the framework of integrability. The latter was firstly discovered in the planar limit for the purely bosonic 𝛾 (𝑠) ∼ ln 𝑠, 𝑠 󳨀→ ∞. (2) so(6) sector at one loop [4]; then it was extended to all the gauge theory sectors and to all loops [5–10]. In specific, it (2) Subleading terms obey hidden relations, the Moch- was found that every composite operator can be thought Vermaseren-Vogt constraints [25, 26]: in brief, terms of as a state of a “spin chain,” whose Hamiltonian is the 𝑠/𝑠 1/𝑠 proportional to ln and are completely deter- dilatation operator itself, although the latter does not have 𝑠 𝑠0 mined by terms proportional to ln and .These an explicit expression of the spin chain form, but for the constraints are related with spacetime reciprocity of first few loops. Nevertheless, the spectrum of infinitely long deep inelastic scattering and its crossed version of operators has turned out to be exactly described by a set of 𝑒+𝑒− annihilation into hadrons. Asymptotic Bethe Ansatz (ABA) equations [5–10]. On the other hand, anomalous dimensions of operators with finite N =4gauge theory shares at large 𝑠 these features, and, quantum numbers depend not only on ABA data but also besides, allows us an understanding of their origin and thus on finite size “wrapping” corrections [11, 12]. Subsequent possible extension to QCD. In specific, the asymptotic large progress has shown that a set of Thermodynamic Bethe 𝑠 series of the anomalous dimensions are believed to be 2 2 Ansatz (TBA) equations [13–17] or an equivalent 𝑌-system constrained by nonperturbative (in 𝜆=8𝜋𝑔 ,the’tHooft of functional equations [18], together with certain additional multicolor coupling) functional relations that work for any information [19], provides a solid ground for exact (any finitevalueofthetwist𝐿.Tobemoreprecise,conformal length, any coupling) predictions on anomalous dimensions symmetry implies that anomalous dimensions 𝛾(𝑔, 𝐿, 𝑠) of of planar N =4SYM. twist operators are functions of the conformal spin: this Despite this impressive progress, we believe that it is translates into the following “self-tuning” functional relation still important to define the largest domain of composite [27, 28]: operators for which the “simpler” ABA equations give the 1 𝛾 (𝑔, 𝐿, 𝑠) = 𝑃(𝑠+ 𝛾(𝑔,𝐿,𝑠)). correct anomalous dimensions, especially in connexion with 2 (3) other well-established relevant equations. In fact, we intended this to be the main aim of this paper and the most natural Additionally, this has to be meant asymptotically in the sense settingtoperformthisstudytobethereformulationofABA that the function (the function 𝑃 in (3) actually depends on equations as one (Non)linear Integral Equation (NLIE) [39]. the twist of the operator as well) Generically and sketchily, for operators composed of 𝐿 elementary fields, ABA gives the correct perturbative ∞ 𝑎(𝑛) ( 𝐶 (𝑠)) 𝑃 (𝑠) = ∑ ln , expansion of the anomalous dimension up to 𝐿−1loops. 2𝑛 (4) 𝑛=0 𝐶 (𝑠) Starting from 𝐿 loops, “wrapping” diagrams, which are not taken into account by ABA, start to contribute. In this 𝑠 is represented by a series in via the conformal Casimir general framework, the high spin limit of fixed twist operators 𝐿 𝐿 seems to offer a better scenario. Perturbative (up to six 𝐶 (𝑠)2 =(𝑠+ −1)(𝑠+ ) 2 2 (5) loops) computations [40–42] for short (twist two and three) operators show that wrapping diagrams (which enter from only. Relation (4) is equivalent to the so-called reciprocity four loops on) actually give contributions which in the high 2 2 symmetry 𝑥→−𝑥, but for the Mellin space variable 𝑠,and spin limit behave as 𝑂((ln 𝑠) /𝑠 ).Itisthennaturaltoaskif (𝑛) an important piece of information is that the function 𝑎 has such property extends to higher (and possibly to all) orders the form of an upper truncated Laurent series: of perturbation theory. In this paper we want to provide

𝑀 evidence in favour of this picture, by using the self-tuning (𝑛) (𝑛) 𝑚 and reciprocity properties. In order to do that, we first rewrite 𝑎 (ln 𝐶 (𝑠)) = ∑ 𝑏𝑚 (𝑔, 𝐿) (ln 𝐶 (𝑠)) ,𝑀<∞;(6) 𝑚=−∞ (Section 2) the ABA equations as NLIEs for the counting function. Then, in Section 3, we specialise ourselves to the (𝑛) that is, 𝑎 (ln 𝐶(𝑠)) depends on 𝑠 only through powers of minimal anomalous dimension state and go to the high spin ln 𝐶(𝑠). For twist two and three negative powers of ln 𝐶(𝑠) are limit,whilekeepingthetwistfinite:uponcomputingthe (𝑛) absent and 𝑎 is a polynomial; for generic twist, however, positions of the external holes and the effect of the nonlinear one has to cope with the infinite Laurent series (6). terms, we write a linear integral equation equivalent to ABA Advances in High Energy Physics 3

𝑛 up to the orders 1/𝑠(ln 𝑠) , 𝑛∈Z, 𝑛≥−1(inclusive). In being the expression of the 𝑟th charge in terms of the rapidity Section 4, we use this linear integral equation to compute 𝑢. Operators (1) of twist 𝐿 correspond to zero momentum at the same order of 𝑠, but at all values of the coupling, states of the sl(2) spin chain described by an even number the ABA prediction for the minimal anomalous dimension. 𝑠 of real Bethe roots 𝑢𝑘 which satisfy (7). For a state described Then, in Section 5, we show the latter to satisfy the self-tuning by the set of Bethe roots {𝑢𝑘}, 𝑘 = 1,...,𝑠,theeigenvalueof and reciprocity relations. Interestingly, we also find that the the 𝑟th charge is solution of the linear integral equation respects suitable self- 𝑠 tuning and reciprocity relations (up to this order in 𝑠). Finally, 𝑄𝑟 (𝑔, 𝐿,) 𝑠 = ∑𝑞𝑟 (𝑢𝑘) . (12) we provide some arguments supporting the idea that at high 𝑘=1 spin wrapping corrections affect twist operators starting from 2 2 orders (ln 𝑠) /𝑠 , so that self-tuning and reciprocity relations In particular (asymptotic) anomalous dimension of (1) is stillhold(andlikelyalsoamodified(non)linearintegral 2 equation). 𝛾(𝑔,𝐿,𝑠)=𝑔 𝑄2 (𝑔,𝐿,𝑠). (13)

2. From the ABA to the NLIE Let us focus (in this section; from Section 2 on, we will restrict to the minimal anomalous dimension state) on states As planned in the Introduction, we start from the ABA described by positions of roots which are symmetric with equations [5–10] for the sl(2) sector of N =4SYM: respect to the origin. These are in particular zero momentum states. An efficient way to treat states described by solutions 2 𝐿 𝑢 +𝑖/2 𝐿 1+𝑔2/2𝑥− (𝑢 ) to a (possibly large) number of (algebraic) Bethe Ansatz ( 𝑘 ) ( 𝑘 ) 2 𝑢𝑘 −𝑖/2 1+𝑔2/2𝑥+ (𝑢 ) equations consists in writing one nonlinear integral equation 𝑘 completely equivalent to them (cf. [45] and references therein 2 𝑠 𝑢 −𝑢 −𝑖 1−𝑔2/2𝑥+ (𝑢 )𝑥− (𝑢 ) for the idea without holes degree of freedom). The nonlinear = ∏ 𝑘 𝑗 ( 𝑘 𝑗 ) (7) integral equation is satisfied by the counting function 𝑍(𝑢), 𝑢 −𝑢 +𝑖 2 − + 𝑗=1 𝑘 𝑗 1−𝑔 /2𝑥 (𝑢𝑘)𝑥 (𝑢𝑗) which in the case (7) reads as 𝑗=𝑘̸ 𝑠 2𝑖𝜃(𝑢 ,𝑢 ) 𝑍 (𝑢) =Φ(𝑢) − ∑𝜙(𝑢,𝑢 ), ⋅𝑒 𝑘 𝑗 , 𝑘 (14) 𝑘=1 where where ± 𝑖 𝑥 (𝑢𝑘)=𝑥(𝑢𝑘 ± ), 2 Φ (𝑢) =Φ0 (𝑢) +Φ𝐻 (𝑢) , (15) 2 𝑢 2𝑔 𝜙 (𝑢, V) =𝜙0 (𝑢−V) +𝜙𝐻 (𝑢, V) , 𝑥 (𝑢) = [1+√1− ] , (8) 2 𝑢2 [ ] with 2 2 𝜆=8𝜋𝑔 , Φ0 (𝑢) =−2𝐿arctan 2𝑢, 𝜆 being the ’t Hooft coupling. The so-called dressing factor 1+𝑔2/2𝑥− (𝑢)2 𝜃(𝑢, V) Φ𝐻 (𝑢) =−𝑖𝐿ln ( ), [10, 43, 44] is given by 1+𝑔2/2𝑥+ (𝑢)2 ∞ ∞ 𝜃 (𝑢, V) = ∑ ∑𝛽𝑟,𝑟+1+2] (𝑔) 𝜙0 (𝑢−V) =2arctan (𝑢−V) , (16) 𝑟=2 ]=0 (9) 𝜙𝐻 (𝑢, V) ⋅[𝑞𝑟 (𝑢) 𝑞𝑟+1+2] (V) −𝑞𝑟 (V) 𝑞𝑟+1+2] (𝑢)], 1−𝑔2/2𝑥+ (𝑢) 𝑥− (V) 𝛽 (𝑔) = 𝑔2𝑟+2]−121/2−𝑟−]𝑐 (𝑔) =−2𝑖[ ( )+𝑖𝜃(𝑢, V)]. the functions 𝑟,𝑟+1+2] 𝑟,𝑟+1+2] being ln 1−𝑔2/2𝑥− (𝑢) 𝑥+ (V)

𝛽𝑟,𝑟+1+2] (𝑔) It follows from its definition that the counting function 𝑍(𝑢) ∞ 𝑔2𝑟+2]+2𝜇 (𝑟−1)(𝑟+2]) is a monotonously decreasing function. In addition, in the =2∑ (−1)𝑟+𝜇+1 𝑟+𝜇+] limit 𝑢→±∞,since 𝜇=] 2 2𝜇 + 1 (10) 4 2𝜇 + 1 2𝜇 + 1 𝜙 (𝑢, V) +𝜙(𝑢,V − ) 󳨀→ ± 2 𝜋 − ⋅( )( )𝜁(2𝜇+1) 𝑢 𝜇−𝑟−] +1 𝜇−] 2𝑖𝑔2 1 1 𝑞 (𝑢) + ( − ) (17) and 𝑟 , 𝑢 𝑥− (V) 𝑥+ (V) 𝑟−1 𝑟−1 𝑖 1 1 1 𝑞𝑟 (𝑢) = [( ) − ( ) ] , (11) +𝑂( ), 𝑟−1 𝑥+ (𝑢) 𝑥− (𝑢) 𝑢3 4 Advances in High Energy Physics

+∞ 𝑓̂(𝑘) = ∫ 𝑑𝑢𝑒−𝑖𝑘𝑢𝑓(𝑢) one has the asymptotic behaviour and pass to Fourier transforms −∞ , keeping in mind that 𝑢󳨀→±∞,

(18) −|𝑘|/2 𝐿+2𝑠+𝛾(𝑔,𝐿,𝑠) 1 ̂ 2𝜋𝐿𝑒 𝑍 (𝑢) 󳨀→ ∓ (𝐿+𝑠) 𝜋+ +𝑂( ). Φ0 (𝑘) =− , 𝑢 𝑢3 𝑖𝑘 2𝜋𝐿 Φ̂ (𝑘) = 𝑒−|𝑘|/2 [1 − 𝐽 (√2𝑔𝑘)] , 𝐻 𝑖𝑘 0 𝑖𝑍(𝜐𝑘) This means that there are 𝐿+𝑠real points 𝜐𝑘 such that 𝑒 = 2𝜋𝑒−|𝑘| 𝐿+1 𝜙̂ (𝑘) = , (−1) . It is a simple consequence of the definition of 𝑍(𝑢) 0 𝑖𝑘 that 𝑠 of them coincide with the Bethe roots 𝑢𝑘.ForBethe 𝑒−(|𝑡|+|𝑘|)/2 ∞ (22) equations (2) Bethe roots are all real and are all contained in 𝜙̂ (𝑘,) 𝑡 =−8𝑖𝜋2 [∑𝑟 (−1)𝑟+1 𝐽 (√2𝑔𝑘) 𝐽 (√2𝑔𝑡) 𝐻 𝑘 |𝑡| 𝑟 𝑟 an interval [−𝑏, 𝑏] of the real line. The remaining 𝐿 points 𝑟=1 are called “holes” [39, 46–52]; they also are real and they 1− (𝑘𝑡) ∞ ∞ ⋅ sgn + (𝑡) ∑ ∑𝑐 (𝑔) (−1)𝑟+] will be denoted as 𝑥ℎ. One should distinguish between 𝐿−2 sgn 𝑟,𝑟+1+2] 2 𝑟=2 ]=0 “internal” or “small” holes 𝑥ℎ, ℎ = 1,...,𝐿−2,whichreside [−𝑏, 𝑏] inside the interval , and two “external” or “large” holes √ √ √ √ ⋅(𝐽𝑟−1 ( 2𝑔𝑘)𝑟+2 𝐽 ] ( 2𝑔𝑡)𝑟−1 −𝐽 ( 2𝑔𝑡)𝑟+2 𝐽 ] ( 2𝑔𝑘))] . 𝑥𝐿−1 =−𝑥𝐿,with𝑥𝐿 >𝑏. Wefinally remark that anomalous dimension appears (18) in the limit 𝑢→∞of the counting function. We will come back to this fact in Appendix A. Asweareinpresenceofholes,wemayfollowthe We obtain the equation extension of the idea as developed in [53] and make use of the 󸀠 Cauchy theorem to obtain a simple integral formula (𝑍 (V)= (𝑑/𝑑V)𝑍(V); cf. also [54] for more details on the following 𝜎̂ (𝑘) formulae): 𝑖𝑘 𝑖𝑘𝑒−|𝑘| = Φ̂ (𝑘) −2 𝐿̂ (𝑘) −|𝑘| −|𝑘| 𝑠 𝐿 1−𝑒 1−𝑒 ∑𝑂(𝑢𝑘)+∑𝑂(𝑥ℎ) 𝑖𝑘 +∞ 𝑑𝑡 + ∫ 𝜙̂ (𝑘,) 𝑡 [𝜎̂ (𝑡) −2𝑖𝑡𝐿̂ (𝑡)] 𝑘=1 ℎ=1 1−𝑒−|𝑘| 4𝜋2 𝐻 −∞ (23) +∞ 𝑑V =−∫ 𝑂 (V) 𝑍󸀠 (V) (19) 𝐿 𝑖𝑘 𝑖𝑘𝑥 −∞ 2𝜋 + ∑𝑒 ℎ 𝜙̂ (𝑘) 1−𝑒−|𝑘| 0 +∞ ℎ=1 𝑑V 𝑑 + + ∫ 𝑂 (V) [1 + (−1)𝐿 𝑒𝑖𝑍(V−𝑖0 )]. Im ln 𝐿 +∞ −∞ 𝜋 𝑑V 𝑖𝑘 𝑑𝑡 𝑖𝑡𝑥 + ∑ ∫ 𝑒 ℎ 𝜙̂ (𝑘,) 𝑡 , −|𝑘| 2𝜋 𝐻 1−𝑒 ℎ=1 −∞ Application of (19) to the derivative of (14) gives

𝑍(𝑘)̂ +∞ 𝑑V 𝑑 and for the equation 𝑍󸀠 (𝑢) =Φ󸀠 (𝑢) + ∫ 𝜙 (𝑢, V) 𝑍󸀠 (V) −∞ 2𝜋 𝑑𝑢 𝐿 𝑑 ̂ + ∑ 𝜙(𝑢,𝑥 ) 𝑍 (𝑘) 𝑑𝑢 ℎ (20) ℎ=1 1 𝑒−|𝑘| +∞ = Φ̂ (𝑘) −2 𝐿̂ (𝑘) 𝑑V 𝑑 𝑑 𝐿 𝑖𝑍(V−𝑖0+) −|𝑘| −|𝑘| − ∫ 𝜙 (𝑢, V) Im ln [1 + (−1) 𝑒 ]. 1−𝑒 1−𝑒 −∞ 𝜋 𝑑𝑢 𝑑V 1 +∞ 𝑑𝑡 + ∫ 𝜙̂ (𝑘,) 𝑡 𝑖𝑡 [𝑍̂(𝑡) −2𝐿̂ (𝑡)] 1−𝑒−|𝑘| 4𝜋2 𝐻 −∞ (24) We introduce the notations 𝐿 1 𝑖𝑘𝑥 + ∑𝑒 ℎ 𝜙̂ (𝑘) 1−𝑒−|𝑘| 0 𝜎 (𝑢) =𝑍󸀠 (𝑢) , ℎ=1 𝐿 +∞ (21) 1 𝑑𝑡 𝑖𝑡𝑥 󸀠 𝑑 𝐿 𝑖𝑍(𝑢−𝑖0+) + ∑ ∫ 𝑒 ℎ 𝜙̂ (𝑘,) 𝑡 , 𝐿 (𝑢) = Im ln [1 + (−1) 𝑒 ], 1−𝑒−|𝑘| 2𝜋 𝐻 𝑑𝑢 ℎ=1 −∞ Advances in High Energy Physics 5 which is the nonlinear integral equation for the counting and the restriction to 𝑘>0allow to write the equation for function 𝑍(𝑢), describing states of the sl(2) sector. We will 𝑆(𝑘) in the alternative way: find it convenient to introduce the following function: 𝐿 𝑆 (𝑘) = (1 − 𝐽 (√2𝑔𝑘)) 𝑘 0 (|𝑘| /2) 2𝑖𝑘𝑒−|𝑘| +∞ 𝑆 (𝑘) = sinh {𝜎̂ (𝑘) + 𝐿̂ (𝑘) 2 𝑑𝑡 −𝑡/2 ̂ √ √ 𝜋 𝑘 −|𝑘| −𝑔 ∫ 𝑒 𝐾( 2𝑔𝑘, 2𝑔𝑡) | | 1−𝑒 0 𝜋

𝜋𝐿 −|𝑘|/2 𝜋𝑡 2𝑖𝑡 𝑖𝑡 (30) + (1 − 𝑒 ) (25) ⋅[ 𝑆 (𝑡) − 𝐿̂ (𝑡) + Φ̂ (𝑡) (|𝑘| /2) −𝑡 −𝑡 0 sinh sinh (𝑡/2) 1−𝑒 1−𝑒 −|𝑘| 𝐿 2𝜋𝑒 𝐿 − ∑ [ 𝑘𝑥 −1]}, 𝑖𝑡 𝑖𝑡𝑥 1−𝑒−|𝑘| cos ℎ +( 𝜙̂ (𝑡) +2𝜋)∑𝑒 ℎ ]. ℎ=1 1−𝑒−𝑡 0 ℎ=1

Equations (30) and (26) are our starting points for study- because, in Appendix A, we show that it satisfies the simple ing ABA contributions to anomalous dimension of twist relation operators. As planned in the Introduction, we will consider the minimal anomalous dimension state, go to the high spin limit, and determine the predictions of ABA for the 𝛾(𝑔,𝐿,𝑠) 𝑛 lim 𝑆 (𝑘) = . (26) anomalous dimension up to orders 1/𝑠(ln 𝑠) , 𝑛≥−1.We 𝑘→0 2 therefore discuss in next section all the simplifications that (30) undergoes in the high spin limit. The function (25) satisfies the nonlinear equation 3. Ground State and High Spin Limit 𝐿 𝑖𝑘 𝑆 (𝑘) = (1 − 𝐽 (√2𝑔𝑘)) + In this section we start our study of the minimal anomalous |𝑘| 0 1−𝑒−|𝑘| dimension state. For this state the positions of the internal +∞ holes are as close as possible to the origin; that is, they satisfy 𝑑𝑡 ̂ ⋅ ∫ 𝜙𝐻 (𝑘,) 𝑡 the relations −∞ 2𝜋 𝑍 (𝑥 ) =𝜋(2ℎ+1−𝐿) , ℎ=1,...,𝐿−2, 𝐿 −|𝑡|/2 ̂ (27) ℎ (31) ∑ (cos 𝑡𝑥ℎ −1)+𝐿(1−𝑒 )−(𝑖𝑡/𝜋) 𝐿 (𝑡) ⋅[ ℎ=1 ] 1−𝑒−|𝑡| while the positions of the two external holes are determined after solving the equations 𝑖𝑘 +∞ 𝑑𝑡 |𝑡| + ∫ 𝜙̂ (𝑘,) 𝑡 𝑆 (𝑡) . −|𝑘| 𝐻 𝑍 (𝑥 ) =−𝑍(𝑥 ) =𝜋(𝑠+𝐿−1) . 1−𝑒 −∞ 2𝜋 2 sinh (|𝑡| /2) 𝐿−1 𝐿 (32)

It follows that the positions of the Bethe roots 𝑢𝑙 are all greater Now, the introduction of the “magic kernel” [10] in modulus than the positions of the internal holes; that is, |𝑢𝑙|>𝑥ℎ, ℎ = 1,...,𝐿 −. 2 For our convenience we order 󸀠 Bethe roots 𝑢𝑙 in such a way that 𝑢𝑙 <𝑢𝑙󸀠 if 𝑙<𝑙. 2 ∞ In the following we will find useful to integrate over 𝐾(𝑡,𝑡̂ 󸀠)= [∑𝑛𝐽 (𝑡) 𝐽 (𝑡󸀠) 𝑡𝑡󸀠 𝑛 𝑛 theregioninwhichBetherootsarecontained.Itisthen 𝑛=1 very important to make the most convenient choice for the (28) ∞ ∞ “extrema” of integration, which naturally identify the points 𝑘+𝑙 󸀠 ±𝑏 𝑢 /𝑢 𝑍(𝑢 /𝑢 ) = ∓𝜋(𝑠+ +2∑ ∑ (−1) 𝑐2𝑘+1,2𝑙+2 (𝑔)2𝑘 𝐽 (𝑡) 𝐽2𝑙+1 (𝑡 )] , which separate the last/first root 𝑠 1 ( 𝑠 1 𝑘=1 𝑙=0 𝐿−3)) from the positive/negative external hole 𝑥𝐿/𝑥𝐿−1:we choose 𝑏 such that theuseoftheproperty,validfor𝑘>0, 𝑍 (±𝑏) =∓𝜋(𝑠+𝐿−2) . (33)

Then, we perform our analysis of the minimal anomalous +∞ ̂ dimension state in the high spin limit. We have to remark that ∫ 𝑑𝑡𝜙𝐻 (𝑘,) 𝑡 𝑓 (𝑡) −∞ in this limit the set of operators (1) has been the object of an extensive activity [10, 39, 46–52, 55–67]; also in perturbative +∞ =8𝑖𝜋2𝑔2 ∫ 𝑑𝑡𝑒−(𝑡+𝑘)/2𝐾(̂ √2𝑔𝑘, √2𝑔𝑡) 𝑓 (𝑡) , (29) QCD, see [68–73]. In the high spin limit, the position of the 0 internal holes is proportional to 1/ ln 𝑠, so it is very close to the origin: they will be determined by using (31) in Section 4. On 𝑓 (𝑡) =𝑓(−𝑡) , the other hand, in order to estimate the position of the two 6 Advances in High Energy Physics external or “large” holes, we have to evaluate the counting andwhereweusedtherelation[76–79] 𝑍(𝑢) ±𝑏 𝑏∼𝑠 function near the points , , delimiting the 𝑠 𝐿−2 interval in which Bethe roots reside. The result we will find, −2∑ (𝑢 −𝑢 )−2∑ (𝑢 −𝑥 ) 𝑂(𝑠) 𝑂(𝑠0) arctan 𝑙 𝑘 arctan 𝑙 ℎ at their leading orders and , 𝑘=1 ℎ=1

𝑠 𝐿−2 𝑥𝐿 =−𝑥𝐿−1 +𝜋∑ sgn (𝑢𝑙 − 𝑢𝑘)+𝜋∑ sgn (𝑢𝑙 − 𝑥ℎ) 𝑠 𝐿−1+𝛾(𝑔,𝐿,𝑠) 1 (34) 𝑘=1 ℎ=1 = [1 + +𝑂( )] , 𝑘=𝑙̸ √2 2𝑠 𝑠2 1 𝑠 𝑢 −𝑢 +𝑖 1 𝐿−2 𝑢 −𝑥 +𝑖 = ∑ 𝑙 𝑘 + ∑ 𝑙 ℎ (38) 𝑖 ln 𝑢 −𝑢 −𝑖 𝑖 ln 𝑢 −𝑥 −𝑖 is proved in next subsection. We have to mention that the 𝑘=1 𝑙 𝑘 ℎ=1 𝑙 ℎ same formula (34) was found for twist two in [65], by using 𝑘=𝑙̸ results of [74, 75]. However, as far as we have understood, 𝑏 1 𝜋 󸀠 resultsof[74,75]areprovedonlyatoneandtwoloops. =2∫ 𝑑V𝜌 (V) 𝑃 + 𝜌 (𝑢𝑙) coth 𝜋𝜌 (𝑢𝑙) Therefore, we would like to give a different and more general −𝑏 𝑢𝑙 − V 𝑠 proof of (34). 1 +𝑂( ). 𝑠2 3.1. Position of the External Holes. When the spin is large, Bethe roots near the two “extrema” ±𝑏 scale with 𝑠.Inthe We remark that, in order to obtain the last equality in (38), it proximity of ±𝑏, it is therefore convenient to rescale the is crucial to transform the sum into an integration from −𝑏 to variable 𝑢 of the counting function 𝑍(𝑢):wewillwrite𝑢=𝑢𝑠, 𝑏,where𝑏 = 𝑏/𝑠 satisfies (33). If, for instance, we transform where 𝑢 will stay finite. Analogously, we will define 𝑏=𝑏𝑠, the sum over Bethe roots and holes into an integration from 𝑏 the first 𝑢1 =−𝑢𝑠 to the last 𝑢𝑠 root,weobtainanextra𝑂(1/𝑠) with finite. From the definitions (14) and (16) of the counting 1/(𝑢 −𝑢 )+1/(𝑢 +𝑢 ) function, we have term 𝑙 𝑠 𝑙 𝑠 in the last line of (38): specifically, 1 𝑠 𝑢 −𝑢 +𝑖 1 𝐿−2 𝑢 −𝑥 +𝑖 𝑠 ∑ 𝑙 𝑘 + ∑ 𝑙 ℎ 𝑍 (𝑢𝑠) =−2𝐿 2𝑢𝑠 −2∑ (𝑢𝑠 − 𝑢 𝑠) 𝑖 ln 𝑢 −𝑢 −𝑖 𝑖 ln 𝑢 −𝑥 −𝑖 arctan arctan 𝑘 𝑘=1 𝑙 𝑘 ℎ=1 𝑙 ℎ 𝑘=1 𝑘=𝑙̸ (35) 𝑢 𝛾(𝑔,𝐿,𝑠) 1 𝑠 1 𝜋 󸀠 (39) + +𝑂( ). =2∫ 𝑑V𝜌 (V) 𝑃 + 𝜌 (𝑢𝑙) coth 𝜋𝜌 (𝑢𝑙) 2 𝑢 − V 𝑠 𝑢𝑠 𝑠 −𝑢𝑠 𝑙 1 1 1 We observe that the only “higher loops” effect is in the last + + 𝑂( ). 𝑠(𝑢 − 𝑢 ) 𝑠(𝑢 + 𝑢 ) 𝑠2 term, proportional to the anomalous dimension. For 𝑢= 𝑙 𝑠 𝑙 𝑠 𝑢𝑙,where𝑢𝑙𝑠=𝑢𝑙 is a Bethe root, we expand the various 𝑠 Sticking to formula (38), we remember that for the minimal functions for large andevaluatethesumovertheBetheroots anomalous dimension state and with our ordering of Bethe containedin(35)asanintegraltermplusan“anomaly”[76– roots the value of the counting function on a generic root 𝑢𝑙 79]. We obtain is given by the simple formula

𝑠 𝐿−2 𝑍(𝑢𝑙𝑠) 𝑍(𝑢 𝑠) = −𝜋 ∑ (𝑢 − 𝑢 )−𝜋∑ (𝑢 − 𝑥 ). 𝑙 sgn 𝑙 𝑘 sgn 𝑙 ℎ (40) 𝛾(𝑔,𝐿,𝑠)+𝐿 𝑘=1 ℎ=1 𝑘=𝑙̸ =−𝜋𝐿sgn (𝑢𝑙)+ 𝑢𝑙𝑠 Property (40) allows to simplify equation (36) as follows: 𝑏 1 𝜋 +2∫ 𝑑V𝜌 (V) 𝑃 + 𝜌󸀠 (𝑢 ) 𝜋𝜌 (𝑢 ) 4−𝐿+𝛾(𝑔,𝐿,𝑠) 𝑢 − V 𝑠 𝑙 coth 𝑙 0=−2𝜋 (𝑢 )+ −𝑏 𝑙 (36) sgn 𝑙 𝑢𝑙𝑠 𝑠 𝐿−2 −𝜋∑ (𝑢 − 𝑢 )−𝜋∑ (𝑢 − 𝑥 ) 𝑏 1 𝜋 sgn 𝑙 𝑘 sgn 𝑙 ℎ +2∫ 𝑑V𝜌 V 𝑃 + 𝜌󸀠 (𝑢 ) 𝜋𝜌 (𝑢 ) (41) 𝑘=1 ℎ=1 ( ) 𝑙 coth 𝑙 𝑘=𝑙̸ −𝑏 𝑢𝑙 − V 𝑠 𝜋 1 1 1 +2(𝐿−2) [ (𝑢 )− ]+𝑂( ), +𝑂( ). sgn 𝑙 2 𝑠2 2 𝑢𝑙𝑠 𝑠 0 At the leading order, 𝑂(𝑠 ),weknowthattheequationtobe 𝑥 =𝑥 /𝑠 where ℎ ℎ , satisfied, for all 𝑢,is 1 𝑑 𝑏 1 𝜌 (𝑢) =− 𝑍 (𝑢𝑠) , (37) 0=−2𝜋sgn (𝑢) +2∫ 𝑑V𝜌 (V) 𝑃 (42) 2𝜋𝑠 𝑑𝑢 −𝑏 𝑢−V Advances in High Energy Physics 7

whose solution is the well-known [55, 80] density We now pass to determine the position 𝑥𝐿 = 𝑥𝐿𝑠, 𝑥𝐿 > 𝑏, of the positive external hole. We first compute (35) for |𝑢| > 2 |𝑏| |𝑢| −𝑏| | = 𝑂(1) 2 (more precisely, ): 1 𝑏+√𝑏 − 𝑢2 𝜌 (𝑢) = ln ( ) . (43) 𝜋 𝑢 𝑠 𝑍 (𝑢𝑠) =−2𝐿arctan 2𝑢𝑠 −2∑ arctan (𝑢𝑠 − 𝑢𝑘𝑠) 𝑘=1 Using (43), we give an estimate of the last term in (41), 𝛾(𝑔,𝐿,𝑠) 1 + +𝑂( ) 𝑢𝑠 𝑠2 𝜋 𝜌󸀠 (𝑢 ) 𝜋𝜌 (𝑢 ) (50) 𝑠 𝑙 coth 𝑙 𝐿+𝛾 2 𝑠 1 =−(𝐿+𝑠) 𝜋 sgn (𝑢) + + ∑ (44) 𝑢𝑠 𝑠 𝑢−𝑢 1 1 1 2 1 𝑘=1 𝑘 = [ − − ]+𝑂( ), 𝑠 2𝑏+2𝑢 2𝑏−2𝑢 𝑢 𝑠2 1 𝑙 𝑙 𝑙 +𝑂( ). 𝑠2 whichallowstofindthefunction𝜌(𝑢) which satisfies (41): The sum over the Bethe roots is evaluated as 2 2 1 𝑏+√𝑏 − 𝑢2 𝜌 (𝑢) = ( ) 2 𝑠 1 𝑏 1 𝐿−2 1 𝜋 ln 𝑢 ∑ =2∫ 𝑑V𝜌 (V) −2∑ 𝑠 𝑢−𝑢 𝑢−V 𝑢𝑠 −𝑥 𝑘=1 𝑘 −𝑏 ℎ=1 ℎ (45) 1 (2 + 𝛾 (𝑔, 𝐿, 𝑠) −𝐿)𝛿 (𝑢) +𝛿(𝑢+𝑏) + 𝛿𝑢− ( 𝑏) +𝑂( ) − 𝑠2 2𝑠 (51) 1 𝑏 1 2𝐿 − 4 +𝑂( ). 2 =2∫ 𝑑V𝜌 (V) − 𝑠 −𝑏 𝑢−V 𝑢𝑠 1 +𝑂( ). Using the form (45) of the solution, we can determine the 𝑠2 position of the extremum 𝑏 through the relation We now insert (45) into (51) and use the result, valid for |𝑢| > 𝑏 𝑍 (𝑏) −𝑍(−𝑏) 𝐿−2 𝑏 ∫ 𝑑𝑢𝜌 (𝑢) =− =1+ , (46) : −𝑏 2𝜋𝑠 𝑠 2 2 𝑏 𝑑V 𝑏+√𝑏 − V2 whereweused(33),whichgives ∫ ln ( ) −𝑏 𝑢−V V 1 𝐿−1+𝛾(𝑔,𝐿,𝑠) 1 𝑏= (1 + )+𝑂( ). (52) 2 (47) 2 2 2𝑠 𝑠 𝑖𝑢√1−(𝑏 /𝑢2)+𝑏 =𝑖𝜋ln . 2 We remark that if we had transformed the sum over Bethe 𝑖𝑢√1−(𝑏 /𝑢2)−𝑏 roots and holes into an integration from 𝑢1 =−𝑢𝑠 to 𝑢𝑠 according to (39), by repeating all the steps until (47) we wouldhavefoundthepositionofthelargestrootatleading Inserting the resulting expression for (51) into (50), we and subleading order: this result is eventually arrive at the formula 1 𝑢 =𝑏+𝑂( ), 2 𝑠 𝑠 (48) 𝑍 (𝑢𝑠) =−(𝐿+𝑠) 𝜋 (𝑢) + sgn 𝑢𝑠

󸀠 1 1 1 which, in particular, allows to give an estimate for 𝑍 (𝑏): − ( + ) 2𝑠 𝑢+𝑏 𝑢−𝑏 (53) 𝑍 (𝑏) −𝑍(𝑢) 𝜋 2 𝑍󸀠 (𝑏) ∼ 𝑠 ∼ ∼𝑂(𝑠) . 𝑖𝑢√1−(𝑏 /𝑢2)+𝑏 𝑏−𝑢 𝑂 (1/𝑠) (49) 1 𝑠 +2𝑖ln +𝑂( ), 2 𝑠2 𝑖𝑢√1−(𝑏 /𝑢2)−𝑏 󸀠 We will use this result for 𝑍 (𝑏) in next subsection. 8 Advances in High Energy Physics which is certainly valid for |𝑢| −𝑏| | = 𝑂(1).Now,theposition In our equation (30) nonlinearity appears in the following of the (positive) external holes is fixed by the condition integral: 𝑍(𝑥𝐿𝑠) = −(𝐿 + 𝑠)𝜋. +𝜋 Imposing that on (53), we find 𝑁𝐿̂ (𝑘) +∞ 𝑑𝑡 2𝑖𝑡 (56) =𝑔2 ∫ 𝑒−𝑡/2𝐾(̂ √2𝑔𝑘, √2𝑔𝑡) 𝐿̂ (𝑡) . −𝑡 0 𝜋 1−𝑒 1 𝑥 = √2 𝑏+𝑂( )󳨐⇒ 𝐿 𝑠2 It is convenient to pass to the coordinate space and to define (54) −𝛼𝑡 1 𝐿−1+𝛾(𝑔,𝐿,𝑠) 1 +∞ 𝑑𝑡 2𝑖𝑡𝑒 𝐼𝛼 (𝑢) =−2∫ 𝑡𝑢 𝐿̂ (𝑡) 𝑥𝐿 = (1 + )+𝑂( ). cos −𝑡 √2 2𝑠 𝑠2 0 2𝜋 1−𝑒 (57) +∞ 𝑑V = ∫ [𝜓󸀠 (𝛼−𝑖𝑢+𝑖V) −𝜓󸀠 (𝛼+𝑖𝑢−𝑖V)]𝐿(V) . −∞ 𝑖𝜋

We can keep 𝛼 generic, having in mind that the case 𝛼=1/2 𝐿=2 Weobservethatsuchresultagreeswhen with the zeroes is relevant for our case (56): of the transfer matrix which one can obtain from expressions contained in [74, 75]. This is an important check for our +∞ 𝑑𝑘 𝑁𝐿 (𝑢) =2∫ cos 𝑘𝑢𝑁𝐿̂ (𝑘) findings. 0 2𝜋 (58) +∞ 𝑢 V =−∫ 𝑑V𝐾( , )𝐼1/2 (V) , 0 √2𝑔 √2𝑔 3.2. High Spin Limit of Nonlinear Terms. Another important simplification occurring for large spin concerns the nonlinear where 𝐿̂󸀠(𝑡) term (containing ) which appears in (30). In this subsec- +∞ +∞ tion we extend to all loops the result of [47–52]. Some of the 𝑢 V 2 𝑑𝑘 𝑑𝑡 𝐾( , )=8𝑔 ∫ ∫ cos 𝑘𝑢 resultsofthissectionhavebeenalreadyannounced,butnot √2𝑔 √2𝑔 0 2𝜋 0 2𝜋 (59) completelyproved,in[81].Wewillfillthatgaphere:atthe end, we are able to show that ⋅ cos 𝑡V𝐾(̂ √2𝑔𝑘, √2𝑔𝑡) . 𝛼 𝛼 𝛼 𝛼 In general we split 𝐼 (𝑢) as 𝐼 (𝑢) =in 𝐼 (𝑢) +out 𝐼 (𝑢),where 𝑏 𝑑V 𝐼𝛼 (𝑢) = ∫ [𝜓󸀠 (𝛼−𝑖𝑢+𝑖V) −𝜓󸀠 (𝛼+𝑖𝑢−𝑖V)] +∞ 𝑑𝑡 2𝑖𝑡 in −𝑏 𝑖𝜋 𝑔2 ∫ 𝑒−𝑡/2𝐾(̂ √2𝑔𝑘, √2𝑔𝑡) 𝐿̂ (𝑡) −𝑡 0 𝜋 1−𝑒 ⋅𝐿(V) , (55) 1 𝛼 =2𝑔2 2𝐾(̂ √2𝑔𝑘, 0) + 𝑂( ). 𝐼 (𝑢) (60) ln 𝑠2 out 𝑑V = ∫ [𝜓󸀠 (𝛼−𝑖𝑢+𝑖V) −𝜓󸀠 (𝛼+𝑖𝑢−𝑖V)] |V|>𝑏 𝑖𝜋 ⋅𝐿(V) . This means that, in our approximation, nonlinearity effects in 𝛼 (30) are under control. Then, 𝐼in(𝑢) is evaluated using formula (2.17) of [82]:

1 𝜓󸀠 (𝛼−𝑖𝑢+𝑖𝑏) −𝜓󸀠 (𝛼+𝑖𝑢−𝑖𝑏) −𝜓󸀠 (𝛼−𝑖𝑢−𝑖𝑏) +𝜓󸀠 (𝛼+𝑖𝑢+𝑖𝑏) 1 𝐼𝛼 (𝑢) =−𝑖𝐵 ( ) +𝑂( ) in 2 2 𝑍󸀠 (𝑏) 𝑍󸀠 (𝑏)3 (61) 2𝐵 (1/2) 𝑢−𝑏 𝑢+𝑏 1 = 2 [ − ]+𝑂( ). 𝑍󸀠 (𝑏) 𝛼2 + (𝑢−𝑏)2 𝛼2 + (𝑢+𝑏)2 𝑍󸀠 (𝑏)3

󸀠 𝛼 Now, we remember that 𝑍 (𝑏) = 𝑂(𝑏) (see (49)); in addi- 𝐼 (𝑢) tion, in the high spin limit we are allowed to consider 𝛼 2 𝑑V 󸀠 󸀠 𝑢≪𝑠. Therefore, we conclude that 𝐼in(𝑢) = 𝑂(1/𝑠 ) and, = ∫ [𝜓 (𝛼−𝑖𝑢+𝑖V) −𝜓 (𝛼+𝑖𝑢−𝑖V)]𝐿(V) consequently, |V|>𝑏 𝑖𝜋 Advances in High Energy Physics 9

1 +𝑂( ). Therefore, 𝑠2 (62) 𝛼 +∞ Since we can restrict 𝐼 (𝑢) to |𝑢| ≪,wedevelopthe 𝑠 𝜓 𝑢 V 1/2 V 𝑁𝐿 (𝑢) =−∫ 𝑑V𝐾( , )𝐼 (V) functions in the integrand for large .Weobtain 0 √2𝑔 √2𝑔 4 +∞ 𝑑V 1 𝐼𝛼 (𝑢) =− ∫ 𝐿 (V) +𝑂( ) , |𝑢| ≪𝑠. Λ 𝑢 V 2 (63) 𝜋 𝑏 V 𝑠 =2ln 2 ∫ 𝑑V𝐾( , ) 0 √2𝑔 √2𝑔 (71) Integrating by parts we can write down +∞ 𝑢 V 1/2 4 4 +∞ − ∫ 𝑑V𝐾( , )𝐼 (V) 𝛼 󸀠 Λ √2𝑔 √2𝑔 𝐼 (𝑢) = ln 𝑏𝐿 (𝑏) + ∫ 𝑑V ln V𝐿 (V) 𝜋 𝜋 𝑏 (64) 1 1 +𝑂( ), +𝑂( ), |𝑢| ≪𝑠. 𝑠2 𝑠2 We then use the fact that 𝐿(𝑏) =0 and the identity +∞ 𝑑V +∞ 𝑑V where Λ∼𝑠is a cutoff such that for 𝑢<Λapproximation (69) 𝑥 =−∫ V𝑍󸀠 (V) + ∫ V𝐿󸀠 (V) 𝑠→+∞ ln 𝐿 2𝜋 ln 𝜋 ln (65) canbeused.When , in the first integral, we replace 𝑏 𝑏 Λ with +∞; in the second integral we estimate 𝐾 using (70) 1/2 󸀠 to obtain and 𝐼 (V) by means of (57), using that 𝜓 (𝑧) ∼ 1/𝑧 for large +∞ 𝑧: 𝛼 2 󸀠 𝐼 (𝑢) =4ln 𝑥𝐿 + ∫ 𝑑V ln V𝑍 (V) . (66) 𝜋 𝑏 In order to perform the integration in (66), we need an 󸀠 V 󳨀→ + ∞ 󳨐⇒ estimate of 𝑍 (V) when V >𝑏. In Appendix B we prove that 𝑢 V 1 󸀠 4𝑏 1 1 𝐾( , )∼ , 𝑍 (V) =− +𝑂( ), V >𝑏. (67) √2𝑔 √2𝑔 V2 (72) V √V2 −𝑏2 𝑏3 1 Integration in (66) is then performed exactly: 𝐼1/2 (V) ∼ , V 8𝑏 +∞ ln V 1 − ∫ 𝑑V =−4ln 𝑏−4ln 2. (68) 𝜋 𝑏 V √V2 −𝑏2 √ which therefore imply that Plugging (68) into (66) and using the equality 𝑥𝐿 = 2𝑏 + 𝑂(1/𝑠),weobtain 1 𝐼𝛼 (𝑢) =−2 2+𝑂( ), |𝑢| ≪𝑠. ln 𝑠2 (69) Λ󳨀→∞󳨐⇒ 𝐾 +∞ 𝑢 V 1 1 (73) Passing now to the kernel , its evaluation in the coordinate ∫ 𝑑V𝐾( , )𝐼1/2 (V) ∼ ∼ . 2 2 space shows the following behaviour: Λ √2𝑔 √2𝑔 Λ 𝑠 𝑢 V 1 𝑔4 𝐾( , )=− ln [1 − ], √2𝑔 √2𝑔 𝜋2 4𝑥 (𝑢)2 𝑥 (V)2 Putting all together we find out that |𝑢| , |V| ≥ √2𝑔,

𝑢 V 1 𝐾( , )=− 1 √2𝑔 √2𝑔 2𝜋2 𝑁𝐿̂ (𝑘) =2𝑔2 2𝐾(̂ √2𝑔𝑘, 0) + 𝑂( ). ln 𝑠2 (74) (70) 𝑔2𝑒2𝑖 arcsin(𝑢/√2𝑔) ⋅ ln ([1 + ] 2𝑥 (V)2 4. High Spin Results from ABA: 𝑔2𝑒−2𝑖 arcsin(𝑢/√2𝑔) Up to Order 1/𝑠 ⋅[1+ ]) , 2 2𝑥 (V) Having analysed all the simplifications occurring in the high spin limit, let us come back to (30). We insert formula (34) |𝑢| ≤ √2𝑔, |V| ≥ √2𝑔. for the position of the external holes, use relation (55) for the 10 Advances in High Energy Physics nonlinear term, and work out all the “known” terms. We end the anomalous dimension is wrapping independent at its 0 up with the following integral equation: leading orders ln 𝑠 and (ln 𝑠) . After these first considerations, we come back to (75): the 2 ̂ √ (−1) (0) 𝑆 (𝑘) =4𝑔 ln 𝑠𝐾( 2𝑔𝑘, 0) “known” terms driving the equations for 𝑆 (𝑘) and 𝑆 (𝑘) +∞ 𝑑𝑡 are contained in the first two lines of (75). The structure of +4𝑔2 ∫ 𝐾̂∗ (√2𝑔𝑘, √2𝑔𝑡) such driving terms implies the following equalities between 𝑡 0 𝑒 −1 densities: 2𝑔2 + (𝐿+𝛾(𝑔,𝐿,𝑠)−1)𝐾(̂ √2𝑔𝑘, 0) 𝑓(𝑔) 𝑠 𝑆(−1)̌ (𝑘) = 𝑆(−1) (𝑘) , 2 𝐿 √ 2 ̂ √ (79) + [1 − 𝐽0 ( 2𝑔𝑘)] + 4𝑔 𝛾𝐸𝐾( 2𝑔𝑘, 0) 𝑓 (𝑔, 𝐿) + 𝐿 −1 𝑘 𝑆(0)̌ (𝑘) = sl 𝑆(−1) (𝑘) , 2 +𝑔2 (𝐿−2) (75) +∞ 𝑡/2 which translate in terms of anomalous dimensions to the −𝑡/2 ̂ 1−𝑒 ⋅ ∫ 𝑑𝑡𝑒 𝐾(√2𝑔𝑘, √2𝑔𝑡) equalities [65, 85] 0 sinh (𝑡/2) 𝐿−2 +∞ ∑ [ 𝑡𝑥 −1] 1 2 −𝑔2 ∫ 𝑑𝑡𝐾(̂ √2𝑔𝑘, √2𝑔𝑡) ℎ=1 cos ℎ 𝛾(−1)̌ (𝑔, 𝐿) = [𝑓 (𝑔)] , 0 sinh (𝑡/2) 2 (80) +∞ (0) 1 2 −𝑡/2 𝑡 𝛾̌ (𝑔, 𝐿) = 𝑓(𝑔)[𝐿−1+𝑓 (𝑔, 𝐿)] . −𝑔 ∫ 𝑑𝑡𝑒 𝐾(̂ √2𝑔𝑘, √2𝑔𝑡) 𝑆 (𝑡) 2 sl 0 sinh (𝑡/2)

−1 −∞ (𝑛) +𝑂(𝑠 (ln 𝑠) ), It is possible to obtain analogous relations for 𝛾̌ (𝑔, 𝐿) (𝑛) expressed in terms of the 𝛾 (𝑔, 𝐿),for𝑛≥1. The first step is 𝐾̂∗(𝑡,󸀠 𝑡 )=𝐾(𝑡,̂ 󸀠 𝑡 )−𝐾(𝑡,̂ 0) where .Theparticularformof a standard procedure for integral equations with a separable the known terms in (75), together with condition (31) for the kernel, called Neumann expansion [86], applied to (75) for internal holes, suggests that 𝑆(𝑘) expands in (inverse) powers 𝑆(𝑘) −1 −∞ : (with 𝑂(𝑠 (ln 𝑠) ) we denote terms going to zero faster than 1/𝑠 times any inverse power of ln 𝑠.) of ln 𝑠: ∞ 𝐽 (√2𝑔𝑘) 𝑆 𝑘 = ∑S (𝑔,𝐿,𝑠) 𝑝 , ∞ (𝑛) ∞ (𝑛)̌ ( ) 𝑝 𝑆 (𝑘) 𝑆 (𝑘) 𝑝=1 𝑘 𝑆 (𝑘) = ∑ 𝑛 + ∑ 𝑛 𝑛=−1 (ln 𝑠) 𝑛=−1 𝑠 (ln 𝑠) (76) S𝑝 (𝑔, 𝐿, 𝑠) +𝑂(𝑠−1 ( 𝑠)−∞). ln ∞ 1 1 (81) = ∑ [𝑆(𝑛) (𝑔, 𝐿) + 𝑆(𝑛)̌ (𝑔, 𝐿) + 𝑂( )] ( 𝑠)−𝑛 󳨐⇒ 𝑝 𝑠 𝑝 𝑠2 ln And, consistently with (76), the condition for the internal 𝑛=−1 holes (31) is solved by the following Ansatz on their positions: (𝑛) √ (𝑛) 𝛾 (𝑔, 𝐿) = 2𝑔𝑆1 (𝑔, 𝐿) , ∞ 𝛼̌ 𝑛,ℎ −𝑛 −1 −∞ 𝛾(𝑛)̌ (𝑔, 𝐿) = √2𝑔𝑆(𝑛)̌ (𝑔, 𝐿) . 𝑥ℎ = ∑ (𝛼𝑛,ℎ + ) (ln 𝑠) +𝑂(𝑠 (ln 𝑠) ). (77) 1 𝑛=1 𝑠

For the anomalous dimension 𝛾(𝑔, 𝐿, 𝑠) = 2𝑆(0), therefore, This procedure is fully explained in this application in we have the expansion Appendix C. For 𝑛 = 1, 2, 3, 4, 5 we obtain ∞ 𝛾(𝑛) (𝑔, 𝐿) 𝛾(𝑔,𝐿,𝑠)=𝑓(𝑔) 𝑠+𝑓 (𝑔, 𝐿) + ∑ 𝑆(1)̌ (𝑔, 𝐿) = 0, ln sl 𝑛 𝑝 𝑛=1 (ln 𝑠) (78) 𝐿−2 𝑆(−1) (𝑔) ∞ 𝛾(𝑛)̌ (𝑔, 𝐿) 𝑆(2)̌ (𝑔, )=2𝜋𝑆̃(1) (𝑔) ∑𝛼 𝛼̌ + 𝑝 + ∑ +𝑂(𝑠−1 ( 𝑠)−∞), 𝑝 L 𝑝 1,ℎ 1,ℎ 2 𝑛 ln ℎ=1 𝑛=−1 𝑠 (ln 𝑠) ⋅𝛾(2) (𝑔, 𝐿) , where the scaling functions 𝑓(𝑔), 𝑓sl(𝑔, 𝐿) appear also in othercontexts:forinstance,𝑓(𝑔) is twice the cusp anomalous 𝐿−2 dimensionofWilsonloops[83,84]. (3)̌ ̃(1) 𝑆𝑝 (𝑔, 𝐿) = 2𝜋𝑆𝑝 (𝑔) ∑ (𝛼2,ℎ𝛼̌1,ℎ +𝛼1,ℎ𝛼̌2,ℎ) For our purposes, it is important to remark that the strong ℎ=1 coupling limits of 𝑓(𝑔) [63, 64] and 𝑓sl(𝑔, 𝐿) [65, 66] agree (−1) with string theory computations. This shows that such func- 𝑆𝑝 (𝑔) + 𝛾(3) (𝑔, 𝐿) , tions are actually wrapping independent and, consequently, 2 Advances in High Energy Physics 11

(4)̌ ̃(1) (𝑛) 𝑆𝑝 (𝑔, 𝐿) = 2𝜋𝑆𝑝 (𝑔) 𝑆𝑝 (𝑔, 𝐿), found in [87] and reported in Appendix F, ending up with the following simple and compact expressions: 𝐿−2 𝜋 ⋅ ∑ (𝛼 𝛼̌ +𝛼 𝛼̌ +𝛼 𝛼̌ )− 𝑆̃(2) (𝑔) 1,ℎ 3,ℎ 2,ℎ 2,ℎ 3,ℎ 1,ℎ 3 𝑝 ℎ=1 (1)̌ 𝑆𝑝 (𝑔, 𝐿) = 0, 𝐿−2 𝑆(−1) (𝑔) 3 𝑝 (4) (−1) ⋅ ∑ (𝛼1,ℎ) 𝛼̌1,ℎ + 𝛾 (𝑔, 𝐿) , 𝑆 (𝑔) 2 𝑆(2)̌ (𝑔, 𝐿) = 𝑝 𝛾(2) (𝑔, 𝐿) − 𝑓 (𝑔)(2) 𝑆 (𝑔, 𝐿) , ℎ=1 𝑝 2 𝑝 (5)̌ ̃(1) 𝑆𝑝 (𝑔, 𝐿) = 2𝜋𝑆𝑝 (𝑔) 𝑆(−1) (𝑔) (3)̌ 𝑝 (3) 𝑆𝑝 (𝑔, 𝐿) = 𝛾 (𝑔, 𝐿) 𝐿−2 𝜋 2 ⋅ ∑ (𝛼 𝛼̌ +𝛼 𝛼̌ +𝛼 𝛼̌ +𝛼 𝛼̌ )− 1,ℎ 4,ℎ 2,ℎ 3,ℎ 3,ℎ 2,ℎ 4,ℎ 1,ℎ 3 −(𝑓 (𝑔, 𝐿) + 𝐿 −(2) 1)𝑆 (𝑔, 𝐿) ℎ=1 sl 𝑝 𝐿−2 3 ̃(2) 2 3 − 𝑆(3) (𝑔, 𝐿) 𝑓 (𝑔), ⋅ 𝑆𝑝 (𝑔) ∑ (3 (𝛼1,ℎ) 𝛼2,ℎ𝛼̌1,ℎ +(𝛼1,ℎ) 𝛼̌2,ℎ) 2 𝑝 ℎ=1 (85) 𝑆(−1) (𝑔) 𝛾(4) (𝑔, 𝐿) 𝑆(−1) (𝑔) 𝑆(4)̌ (𝑔, 𝐿) = 𝑝 −2𝑆(4) (𝑔, 𝐿) 𝑓 (𝑔) + 𝑝 𝛾(5) (𝑔, 𝐿) , 𝑝 2 𝑝 2 3 (82) − 𝑆(3) (𝑔, 𝐿) (𝑓 (𝑔, 𝐿) + 𝐿 − 1), 2 𝑝 sl 𝑆̃(1)(𝑔) 𝑆̃(2)(𝑔) where 𝑝 and 𝑝 belong to a set of “reduced 𝑆(−1) (𝑔) (5)̌ 𝑝 (5) 5 (5) coefficients,” satisfying the system (C.6), reported also in 𝑆𝑝 (𝑔, 𝐿) = 𝛾 (𝑔, 𝐿) − 𝑆𝑝 (𝑔, 𝐿) 𝑓 (𝑔) Appendix C. 2 2 Theseexpressionsarestillquiteinvolved,buttheycan −2𝑆(4) (𝑔, 𝐿) (𝑓 (𝑔, 𝐿) + 𝐿 −1) be significantly simplified. This can be done through the 𝑝 sl following steps. (2) (2) −𝑆𝑝 (𝑔, 𝐿) 𝛾 (𝑔, 𝐿) . (i) After introducing the notation 𝑑𝑟 𝜎 (𝑢=0) 𝑑𝑢𝑟 For anomalous dimensions, such relations read (83) ∞ 𝜎̌(𝑛) 1 = ∑ (𝜎(𝑛) + 𝑟 +𝑂( )) ( 𝑠)−𝑛 , 𝛾(1)̌ (𝑔, 𝐿) = 0, 𝑟 2 ln 𝑛=−1 𝑠 𝑠 (2) 𝑓(𝑔) (2) we “invert” relation (31), expressing 𝛼𝑚,ℎ and 𝛼̌𝑚,ℎ in 𝛾̌ (𝑔, 𝐿) =− 𝛾 (𝑔, 𝐿) , 2 terms of the densities and their derivatives in zero, (𝑛) (𝑛) 𝜎 𝜎̌ (3) (3) that is, in terms of the coefficients 𝑟 and 𝑟 .In 𝛾̌ (𝑔, 𝐿) = −𝑓 (𝑔)𝛾 (𝑔, 𝐿) performing this procedure we use techniques and results of [87]. Then, we plug the obtained expressions −(𝑓 (𝑔, 𝐿) + 𝐿 − 1)𝛾(2) (𝑔, 𝐿) , sl for 𝛼𝑚,ℎ, 𝛼̌𝑚,ℎ in (82). Detailed calculations are shown inAppendixD,wherewehavealsolistedthefull (4) 3 (4) (𝑛)̌ 𝛾̌ (𝑔, 𝐿) =− 𝑓(𝑔)𝛾 (𝑔, 𝐿) (86) expressions for the first 𝑆𝑝 (𝑔, 𝐿) (relations (D.4)). 2 3 (ii) Then, we use the following relations, proven in − (𝑓 (𝑔, 𝐿) + 𝐿 − 1)𝛾(3) (𝑔, 𝐿) , Appendix E: 2 sl 𝜎̌(−1) 𝜎̌(−1) 𝛾(5)̌ (𝑔, 𝐿) = −2𝑓 (𝑔)𝛾(5) (𝑔, 𝐿) 𝑓(𝑔)=2 0 =2 2 , 𝜎(−1) 𝜎(−1) 0 2 −2(𝑓 (𝑔,𝐿)+𝐿−1)𝛾(4) (𝑔, 𝐿) sl 𝜎̌(0) 𝑓 (𝑔, 𝐿) =2 0 − (𝐿−1) , (2) 2 sl (−1) (84) −(𝛾 (𝑔, 𝐿)) . 𝜎0 𝜎̌(2) 𝜎(2)𝜎̌(−1) 𝛾(2) (𝑔, 𝐿) =2 0 +4 0 0 . Relations (80) and (86) are the prediction of ABA for 𝜎(−1) (−1) 2 𝑛 0 (𝜎0 ) anomalous dimensions at various orders 1/𝑠(ln 𝑠) , 𝑛= −1,...,5. In the next section we will show that they agree With the help of these formulae it is possible to compare with predictions coming from self-tuning and reciprocity the complicated relations (D.4) with analogous results for relations. 12 Advances in High Energy Physics

5. 1/𝑠 Contributions from the basis of our results, we are naturally led to make the Functional Relations following proposal for a self-tuning relation involving the coefficients of the Neumann expansion of the function 𝑆(𝑘): Self-tuning and reciprocity relations were summarised in 1 S (𝑔, 𝑠,) 𝐿 = P (𝑠+ 𝛾 (𝑔, 𝑠,)) 𝐿 , formulae (3), (4), and (5). We remember notations (78) for 𝑝 𝑝 2 (91) the high spin expansion of the anomalous dimension: where P𝑝(𝑠) satisfies a high spin expansion analogous to (4): ∞ 𝛾(𝑛) (𝑔, 𝐿) ∞ (𝑛) 𝛾(𝑔,𝐿,𝑠)=𝑓(𝑔)ln 𝑠+𝑓 (𝑔, 𝐿) + ∑ 𝑛 𝑎 (ln 𝐶 (𝑠)) sl (ln 𝑠) P (𝑠) = ∑ 𝑝 , 𝑛=1 𝑝 2𝑛 (92) (87) 𝑛=0 𝐶 (𝑠) ∞ 𝛾(𝑛)̌ (𝑔, 𝐿) + ∑ +𝑂(𝑠−1 ( 𝑠)−∞). where 𝐶(𝑠) is given by (5). In particular, formula (91) has 𝑠 ( 𝑠)𝑛 ln 𝑛=−1 ln the advantage to furnish immediately the self-tuning (and reciprocity) relations for all the higher conserved charges Comparing (87) with (3), (4), and (5), we obtain that the [88]: 𝑃(𝑠) leading terms of should read 1 𝑄𝑟 (𝑔,𝐿,𝑠)=𝑃𝑟 (𝑠 + 𝛾 (𝑔, 𝑠, 𝐿)) . (93) ∞ 𝛾(𝑛) (𝑔, 𝐿) 2 𝑃 (𝑠) =𝑓(𝑔) 𝐶 (𝑠) +𝑓 (𝑔, 𝐿) + ∑ ln sl 𝑛 Furthermore, we may suppose that, because of integrability, 𝑛=1 (ln 𝐶 (𝑠)) (88) the statement above is equivalent to the self-tuning (and 1 +𝑂( ). reciprocity) of the counting function (91). 𝐶2 Remark 2. Concerning the leading terms in the high spin Developing 𝐶(𝑠) inthesameregime expansion (78) of the minimal anomalous dimension, we already commented that, since the strong coupling limit of 2 𝐿 𝐿 𝐶 (𝑠) =(𝑠+ −1)(𝑠+ )󳨐⇒ 𝑓(𝑔) and 𝑓sl(𝑔, 𝐿) agrees with string theory calculations (and 2 2 many gauge loop calculations), there are good reasons to (89) 𝐿−1 1 believe that anomalous dimension at the orders ln 𝑠 and 𝐶 (𝑠) =𝑠+ +𝑂( ), ( 𝑠)0 2 𝑠 ln is free from wrapping contributions. Then, if we suppose that self-tuning and reciprocity are exact symmetries, (−1) (0) and putting together (3), (88), and (89), we end up with the from (80) and (90), it follows that also 𝛾̌ and 𝛾̌ are following prediction for the anomalous dimension: wrapping-free. Then, one could expect that all the terms that (−1) (𝑛) areinbetween𝑓sl(𝑔, 𝐿) and 𝛾̌ (the various 𝛾 (𝑔, 𝐿))are ∞ 𝛾(𝑛) (𝑔, 𝐿) 𝛾(𝑔,𝐿,𝑠)=𝑓(𝑔) 𝑠+𝑓 (𝑔, 𝐿) + ∑ also not affected by wrapping. If we suppose this, use again ln sl 𝑛 𝑛=1 (ln 𝑠) self-tuning and reciprocity and compare (86) with (90), we (𝑛) areabletoconcludethatallthefunctions𝛾̌ (𝑔, 𝐿) do not 𝑠 1 + ln [𝑓 (𝑔)]2 + 𝑓(𝑔)(𝐿−1+𝑓 (𝑔, 𝐿)) depend on wrapping either. 2𝑠 2𝑠 sl Even if we are aware that our arguments do not provide ∞ (𝑛) ∞ aproofofthefactthatathighspinwrappingdiagramsstart 𝑓(𝑔) 𝛾 (𝑔, 𝐿) 2 2 + ∑ − ∑𝑛 contributing at orders (ln 𝑠) /𝑠 ,wehoweverthinkthatour 2𝑠 ( 𝑠)𝑛 (90) 𝑛=1 ln 𝑛=1 results provide some nonperturbative hints of this property. 𝛾(𝑛) (𝑔, 𝐿) ⋅ [𝑓 (𝑔) 𝑠+𝑓 (𝑔, 𝐿) + 𝐿 −1 Remark 3. Inthepreviousremarkwegavesomeevidencein 𝑛+1 ln sl 𝑠→+∞ 𝐿 2𝑠 (ln 𝑠) favourofthefactthatwhen and the twist is fixed for all values of the coupling constant wrapping diagrams (𝑚) 2 2 ∞ 𝛾 (𝑔, 𝐿) start contributing at the order (ln 𝑠) /𝑠 .Wewouldliketo + ∑ ]+𝑂(𝑠−1 ( 𝑠)−∞). 𝑚 ln stress that this conclusion depends on the particular order in 𝑚=1 (ln 𝑠) which limits are performed: indeed, we first sent 𝑠→+∞, 1/𝑠( 𝑠)𝑛 𝑛=−1,...,5 with 𝐿 and 𝑔 fixed;then,possibly,wecouldhavemadethe Working out this formula for orders ln , , 𝑔→+∞ we find formulae which coincide with (80), for 𝑛=−1,0and limit . Obviously, our situation is different from 𝑛=1,...,5 what happens in the peculiar regime of semiclassical strings: with (86), for . Therefore, our findings in Section 4 𝑠 𝑔 agreewithself-tuningandreciprocitypredictions.Wewould in this case, indeed, and go together to infinity, with their S = 𝑠/2√2𝜋𝑔 like to stress the fact that this conclusion holds for all values of ratio kept fixed, and wrapping contribution 1/ S the coupling 𝑔; that is, it is a nonperturbative statement on the enters already at the order ln [89–92]. high spin expansion of (asymptotic) anomalous dimension. 6. Conclusions Remark 1. Formulae (79) and (85) seem to indicate that more generally functional relations similar to (3) and (4) should We studied the high spin limit of twist operators in the sl(2) hold for (the high spin expansion of) the function 𝑆(𝑘).On sector of N =4SYM. Using ABA equations rewritten as one Advances in High Energy Physics 13

NLIE, we computed the minimal anomalous dimension up On the other hand, if we want to compute the anomalous 𝑛 to orders 1/𝑠(ln 𝑠) —in detail: Section 4 and formulae (86)— dimension 𝛾(𝑔, 𝐿, 𝑠) of a state described by a solution of provingeventuallythatourresultssatisfy(forallvaluesof the ABA equations, we have to compute the sum 𝛾= 𝑔 2 𝑠 the coupling ) the self-tuning and reciprocity properties (3), 𝑔 ∑𝑘=1 𝑒(𝑢𝑘).Usingformula(19)weobtain (4), and (5). As a consequence, in Remark 2 above, we could 𝐿 +∞ 𝑑V give some clues supporting the idea that in the high spin limit 𝛾(𝑔,𝐿,𝑠)=−𝑔2 ∑𝑒(𝑥 )−𝑔2 ∫ 𝑒 (V) 𝑍󸀠 (V) 2 2 ℎ 2𝜋 wrapping corrections start contributing at order (ln 𝑠) /𝑠 for ℎ=1 −∞ 𝐿 (A.5) any twist . +∞ 2 𝑑V 𝑑 𝐿 𝑖𝑍(V−𝑖0+) In addition, as a byproduct of our analysis we provided +𝑔 ∫ 𝑒 (V) Im ln [1 + (−1) 𝑒 ], also the following new results: −∞ 𝜋 𝑑V (1) Exact connection between a nonlinear function of the which can be written also, in terms of Fourier transforms, as +∞ 𝑑𝑡 counting function and the (asymptotic) anomalous 𝛾(𝑔,𝐿,𝑠)=−𝑔2 ∫ 𝑒̂(𝑡) [𝜎̂ (𝑡) −2𝑖𝑡𝐿̂ (𝑡)] 2 dimension (formula (26) and Appendix A). −∞ 4𝜋

(2) Evaluation of the external holes position at the (sub- 𝐿 +∞ (A.6) 0 2 𝑑𝑡 𝑖𝑡𝑥 leading) order 𝑠 (Section 3.1). −𝑔 ∑ ∫ 𝑒 ℎ 𝑒̂(𝑡) . −∞ 2𝜋 (3)Proposalforself-tuningandreciprocityrelationssat- ℎ=1 isfied by the function (directly related to the counting Comparing (A.6) with (A.2), we gain relation (26). function) 𝑆(𝑘) (Remark 1 of Section 5 and formulae For an alternative proof of (26), we first notice that (91) and (92). +∞ 󸀠 𝜎̂ (0) = ∫ 𝑑𝑢𝑍 (𝑢) =−2𝜋(𝐿+𝑠) . (A.7) Eventually, we are confident that further analysis of the −∞ last issue may shed light on how to construct, order by order Then, using (18) we find that, when 𝑢→±∞, in 𝑠, an “effective” (non)linear integral equation which takes 𝐿+2𝑠+𝛾(𝑔,𝐿,𝑠) 1 into account the wrapping corrections as well. 𝐿 (𝑢) 󳨀→ +𝑂( ). (A.8) 2𝑢 𝑢3 Appendices Using these property, one finds that 𝑖𝜋 lim 𝐿̂ (𝑘) =∓ [𝐿 + 2𝑠 + 𝛾 (𝑔, 𝐿, 𝑠)], (A.9) A. Connection between the Counting Function 𝑘→0± 2 and (Asymptotic) Anomalous Dimension Inserting (A.7) and (A.9) into (25), we find again (26). This It is convenient to rewrite (27) in the following form: alternative proof emphasizes the fact that the information on the anomalous dimension comes entirely from the term 𝐿(𝑘)̂ , 𝐿 𝑖𝑘 𝑍(𝑢) 𝑆 (𝑘) = [1 − 𝐽 (√2𝑔𝑘)] + which is a nonlinear function of the counting function : |𝑘| 0 2𝜋 |𝑘| lim 𝐿̂ (𝑘) 𝑘→0± 𝐿 +∞ |𝑘|/2 𝑑𝑡 𝑖𝑡𝑥 ⋅𝑒 ∑ ∫ 𝑒 ℎ 𝜙̂ (𝑘,) 𝑡 +∞ 2𝜋 𝐻 (A.1) −𝑖𝑘𝑢 𝐿 𝑖𝑍(𝑢−𝑖0+) ℎ=1 −∞ = lim ∫ 𝑑𝑢𝑒 Im ln [1 + (−1) 𝑒 ] (A.10) 𝑘→0± −∞ 𝑖𝑘 +∞ 𝑑𝑡 + 𝑒|𝑘|/2 ∫ 𝜙̂ (𝑘,) 𝑡 [𝜎̂ (𝑡) −2𝑖𝑡𝐿̂ (𝑡)]. 𝜋 2 𝐻 =± [𝐿 + 2𝑠 + 𝛾 (𝑔, 𝐿, 𝑠)]. 2𝜋 |𝑘| −∞ 4𝜋 2𝑖

Then, we compute (A.1) at 𝑘=0.Weobtain Curiously (and, perhaps, interestingly), formula (A.10) looks similartotheTBAexpressionforthefreeenergy.Wehave +∞ 𝑑𝑡 checked formula (A.10) in various particular cases (twist 2, 3, 𝑆 (0) =−𝑔2 ∫ 𝑒̂(𝑡) [𝜎̂ (𝑡) −2𝑖𝑡𝐿̂ (𝑡)] 2 −∞ 8𝜋 one, and two loops) for which explicit solutions [93] of ABA equations were found1. 𝐿 +∞ (A.2) 2 𝑑𝑡 𝑖𝑡𝑥 −𝑔 ∑ ∫ 𝑒 ℎ 𝑒̂(𝑡) , 4𝜋 𝑍󸀠(𝑢) ℎ=1 −∞ B. Evaluation of at High Spin and Large 𝑢 where Using definition (14), we write the counting function 𝑍(𝑢) in 2√2𝜋 𝑒̂(𝑡) = 𝑒−|𝑡|/2𝐽 (√2𝑔𝑡) the region of large 𝑢∼𝑏: 𝑔𝑡 1 (A.3) 𝛾+𝐿 𝑠 𝑍 (𝑢) =−𝐿𝜋+ −2∑ (𝑢 − 𝑢 ) 𝑢 arctan 𝑘 is the Fourier transform of 𝑘=1 (B.1) 1 1 1 𝑒 (𝑢) =𝑞2 (𝑢) =𝑖[ − ]. (A.4) +𝑂( ). 𝑥+ (𝑢) 𝑥− (𝑢) 𝑏2 14 Advances in High Energy Physics

We add and subtract the sum over the internal holes and thus Indeed,ifweinsert(B.6)intotheintegralof(B.5)andusethe obtain integration formula 𝛾+𝐿 𝑠 𝑍 (𝑢) =−𝐿𝜋+ −2∑ (𝑢 − 𝑢 ) 𝑢 arctan 𝑘 𝑘=1 𝑏 𝑑V 𝑏+√𝑏2 − V2 𝐿−2 2𝐿 − 4 ∫ ln −2∑ (𝑢 − 𝑥 )+𝜋(𝐿−2) − (B.2) −𝑏 𝑧−V 𝑏−√𝑏2 − V2 arctan ℎ 𝑢 ℎ=1 (B.7) √𝑏2 −𝑧2 −𝑏 1 =𝑖𝜋sgn (Im 𝑧) ln ,𝑧∉[−𝑏,] 𝑏 , +𝑂( ). √𝑏2 −𝑧2 +𝑏 𝑏2 Then, the use of (19) gives 𝛾−𝐿+4 𝑍 (𝑢) =−2𝜋+ in the right-hand side of this last equation (with 𝑧=𝑢±𝑖), 𝑢 when 0<𝑢<𝑏,weareleftwith 𝑏 𝑑V 󸀠 󸀠 + ∫ arctan (𝑢−V) [𝑍 (V) −2𝐿 (V)] (B.3) −𝑏 𝜋 1 √𝑏2 − (𝑢+𝑖)2 +𝑏 √𝑏2 − (𝑢−𝑖)2 +𝑏 +𝑂( ). 𝑏2 − ln − ln √𝑏2 − (𝑢+𝑖)2 −𝑏 √𝑏2 − (𝑢−𝑖)2 −𝑏 Evaluation of the nonlinear term is done using formula (2.17) (B.8) 1 𝑏+√𝑏2 −𝑢2 1 of [82]: +𝑂( )=−2ln +𝑂( ), 𝑏2 √ 2 2 𝑏2 𝛾−𝐿+4 𝑏 𝑑V 𝑏− 𝑏 −𝑢 𝑍 (𝑢) =−2𝜋+ + ∫ arctan (𝑢−V) 𝑢 −𝑏 𝜋 ∞ (2𝜋)2𝑘+1 1 ⋅𝑍󸀠 (V) +2∑ 𝐵 ( ) which matches (B.6). Plugging approximation (B.6) into (B.5) (2𝑘 + 2)! 2𝑘+2 2 󸀠 𝑘=0 and letting 𝑢>𝑏,wefind𝑍 (𝑢) in this domain. Application of (B.7) gives 𝑑 𝑥=𝑍(𝑏) ⋅[ (𝑢 −(−1) 𝑍 (𝑥))] 2𝑘+1 arctan 𝑑𝑥 𝑥=𝑍(−𝑏)

1 𝛾−𝐿+4 (B.4) √𝑏2 − (𝑢+𝑖)2 +𝑏 +𝑂( )=−2𝜋+ 󸀠 𝑏2 𝑢 𝑍 (𝑢) =−ln √ 2 2 𝑏 𝑏 − (𝑢+𝑖) −𝑏 𝑑V 󸀠 + ∫ arctan (𝑢−V) 𝑍 (V) −𝑏 𝜋 2 √𝑏2 − (𝑢−𝑖) +𝑏 1 (B.9) 𝜋 1 1 − +𝑂( ), + [ − ] ln 𝑏3 6𝑍󸀠 (𝑏) 1+(𝑢−𝑏)2 1+(𝑢+𝑏)2 √𝑏2 − (𝑢−𝑖)2 −𝑏 1 +𝑂( ), 𝑢>𝑏, 𝑏2 where we neglected higher order terms in the sum over 𝑘 󸀠 since 𝑍 (𝑏) ∼ 𝑂(𝑏) (49). It is convenient to pass to the derivative of 𝑍: which, since 𝑢>𝑏≫1, is expanded as follows: 𝛾−𝐿+4 𝑏 𝑑V 1 𝑍󸀠 (𝑢) =− + ∫ 𝑍󸀠 (V) 2 2 𝑢 −𝑏 𝜋 1+(𝑢−V) (B.5) 4𝑏 1 1 1 𝑍󸀠 (𝑢) =− +𝑂( ). +𝑂( ). 3 (B.10) 𝑏2 𝑢 √𝑢2 −𝑏2 𝑏 For large 𝑢,butstill0<𝑢<𝑏, the solution to (B.5) is √ 2 2 󸀠 𝑏+ 𝑏 −𝑢 C. Neumann Expansion for 𝑆(𝑘) 𝑍 (𝑢) =−2ln +𝜋(𝛾−𝐿+4)𝛿(𝑢) 𝑏−√𝑏2 −𝑢2 (B.6) Let one see how the Neumann expansion for 𝑆(𝑘) works. This 1 +𝑂( ). isastandardprocedure[86]inthecaseofanintegralequation 𝑏2 with separable kernel: Advances in High Energy Physics 15

∞ √ 𝐽𝑝 ( 2𝑔𝑘) Plugging (C.4) into (C.2) we obtain (𝑛≥1) 𝑆 (𝑘) = ∑S𝑝 (𝑔, 𝐿, 𝑠) , 𝑝=1 𝑘 𝑛 𝜋𝑟 𝑆(𝑛)̌ (𝑔, 𝐿) = −2𝜋∑𝑆̃(𝑟/2) (𝑔) S𝑝 (𝑔, 𝐿, 𝑠) 𝑝 𝑝 cos 𝑟=1 2

∞ 𝑗 1 1 𝐿−2 𝑛−𝑟+1 𝑚 𝑛−𝑟+1 (𝑛) (𝑛)̌ ∑ℎ=1 (∏𝑚=1 (𝛼𝑚,ℎ) ) ∑𝑚󸀠=1 𝑗𝑚󸀠 (𝛼̌𝑚󸀠,ℎ/𝛼𝑚󸀠,ℎ) = ∑ [𝑆𝑝 (𝑔, 𝐿) + 𝑆𝑝 (𝑔, 𝐿) + 𝑂( )] ⋅ ∑ 2 (C.1) 𝑛−𝑟+1 (C.5) 𝑛=−1 𝑠 𝑠 ∏ 𝑗 ! {𝑗1,...,𝑗𝑛−𝑟+1} 𝑚=1 𝑚 −𝑛 ⋅ (ln 𝑠) 󳨐⇒ 𝑆(−1) (𝑔) 𝑛−𝑟+1 𝑛−𝑟+1 + 𝑝 𝛾(𝑛) (𝑔, 𝐿) , ∑ 𝑗 =𝑟, ∑ 𝑚𝑗 =𝑛, 2 𝑚 𝑚 (𝑛) √ (𝑛) 𝑚=1 𝑚=1 𝛾 (𝑔, 𝐿) = 2𝑔𝑆1 (𝑔, 𝐿) ,

(𝑛) (𝑛) 𝛾̌ (𝑔, 𝐿) = √2𝑔𝑆̌ (𝑔, 𝐿) . ̃(𝑟) 1 where the “reduced coefficients” 𝑆𝑝 (𝑔) satisfy the systems (see [52]) The Neumann expansion transforms the linear integral equation for 𝑆(𝑘) into a set of linear infinite system. In particular, 𝑆(𝑛)̌ (𝑔, 𝐿) 𝑛≥1 2 ̃(𝑟) (𝑟) 𝑝 , ,satisfythesystem 𝑆2𝑝−1 (𝑔) = I2𝑝−1 (𝑔) (𝑛)̌ ∞ 𝑆2𝑝−1 (𝑔, 𝐿) ̃(𝑟) −2(2𝑝−1)∑ 𝑍2𝑝−1,𝑚 (𝑔) 𝑆𝑚 (𝑔) , (C.6) √ (𝑛) 𝑚=1 = 2𝑔𝛿𝑝,1𝛾 (𝑔, 𝐿) ∞ (𝑟) (𝑟) 𝑚 (𝑟) (𝑛) 𝑆̃ (𝑔) = I (𝑔) − 4𝑝 ∑ 𝑍 (𝑔) (−1) 𝑆̃ (𝑔) , +∞ 𝑑𝑡 𝑃̌ (𝑔, 𝑡) 𝐽 (√2𝑔𝑡) 2𝑝 2𝑝 2𝑝,𝑚 𝑚 −(2𝑝−1)∫ 2𝑝−1 𝑚=1 0 𝑡 sinh (𝑡/2) ∞ (𝑛)̌ with −2(2𝑝−1)∑ 𝑍2𝑝−1,𝑚 (𝑔) 𝑆𝑚 (𝑔, 𝐿) , 𝑚=1 (C.2) +∞ 𝑑ℎ 𝐽 (√2𝑔ℎ) (𝑛)̌ I(𝑟) (𝑔) = 𝑝 ∫ ℎ2𝑟−1 𝑝 . 𝑆2𝑝 (𝑔, 𝐿) 𝑝 (C.7) 0 2𝜋 sinh (ℎ/2) +∞ ̌(𝑛) √ 𝑑𝑡 𝑃 (𝑔, 𝑡)2𝑝 𝐽 ( 2𝑔𝑡) =−2𝑝∫ 𝛼̌ 0 𝑡 sinh (𝑡/2) D. Solution of Internal Holes Equation: 𝑛,ℎ ∞ Expressed in Terms of the Densities and 𝑚 (𝑛)̌ −4𝑝∑ 𝑍2𝑝,𝑚 (𝑔) (−1) 𝑆𝑚 (𝑔, 𝐿) , Their Derivatives 𝑚=1 We want to extract from (31), explicit expressions for 𝛼𝑚,ℎ ̌(𝑛) 𝛼̌ where 𝑃 (𝑔, 𝑡) appears in the expansion and 𝑚,ℎ in terms of the densities and their derivatives, using the notation (83). We can use techniques and results 𝐿−2 of [87]. In particular, relations (14) of [87] are still valid, after 𝑃(𝑠,𝑔,𝑡)= ∑ [cos 𝑡𝑥ℎ −1] substituting in them ℎ=1 ∞ ̌(𝑛) (𝑛) 𝑃 (𝑔, 𝑡) −𝑛 (C.3) 𝛼̌𝑚,ℎ = ∑ (𝑃 (𝑔, 𝑡) + ) (ln 𝑠) 𝛼𝑚,ℎ 󳨀→ 𝛼 𝑚,ℎ + , 𝑛=1 𝑠 𝑠 (D.1) −1 −∞ 𝜎̌(𝑛) +𝑂(𝑠 (ln 𝑠) ), 𝜎(𝑛) 󳨀→ 𝜎 (𝑛) + 𝑟 . 𝑟 𝑟 𝑠 which follows from (77). More explicitly, inserting (77) into (C.3), we obtain We can then solve for 𝛼̌𝑚,ℎ.Weobtain 𝑛 𝜋𝑟 𝑃̌(𝑛) (𝑔, 𝑡) = ∑𝑡𝑟 cos 2 𝑟=1 𝑝 (−1) 𝑝−𝑟+1 (−1) 𝜎 𝑗 󸀠 (𝛼̌ 󸀠 ) 𝜎̌ [ 𝑟 𝑚 𝑚 ,ℎ 𝑟 𝐿−2 𝑛−𝑟+1 𝑗𝑚 𝑛−𝑟+1 𝛼̌𝑝+1,ℎ =−∑ ∑ + ∑ (∏ (𝛼 ) )∑ 𝑗 󸀠 (𝛼̌ 󸀠 /𝛼 󸀠 ) (−1) (−1) ℎ=1 𝑚=1 𝑚,ℎ 𝑚󸀠=1 𝑚 𝑚 ,ℎ 𝑚 ,ℎ 𝜎 (𝛼 󸀠 ) 𝜎 ⋅ ∑ , 𝑟=1 [ 0 𝑚󸀠=1 𝑚 ,ℎ 0 ∏𝑛−𝑟+1𝑗 ! (C.4) {𝑗1,...,𝑗𝑛−𝑟+1} 𝑚=1 𝑚 (−1) (−1) 𝑝−𝑟+1 𝑗𝑚 𝑛−𝑟+1 𝑛−𝑟+1 𝜎𝑟 𝜎0̌ (𝛼𝑚,ℎ) ∑ 𝑗 =𝑟, ∑ 𝑚𝑗 =𝑛. − ] ∑ ∏ 𝑚 𝑚 (−1) 2 𝑗 ! 𝑚=1 𝑚=1 (𝜎 ) {𝑗 ,...,𝑗 } 𝑚=1 𝑚 0 ] 1 𝑝−𝑟+1 16 Advances in High Energy Physics

𝑝−1 𝑝−𝑙 (𝑙) 𝑝−𝑟−𝑙+1 (𝑙) 𝜎 𝑗𝑚󸀠 (𝛼̌𝑚󸀠,ℎ) 𝜎̌ Now, inserting (D.3) into (82), we can derive expressions for − ∑ ∑ [ 𝑟−1 ∑ + 𝑟−1 (𝑛) (𝑛) (𝑛) (−1) (−1) 𝑆̌ (𝑔, 𝐿) 𝜎 𝜎̌ 𝑛=2,...,5 𝜎 (𝛼 󸀠 ) 𝜎 𝑝 ,intermsof 𝑟 and 𝑟 :for ,weobtain 𝑙=0 𝑟=1 [ 0 𝑚󸀠=1 𝑚 ,ℎ 0 𝑆(−1) (𝑔) 3 𝑗 𝑝 2𝜋 𝜎(𝑙) 𝜎̌(−1) 𝑝−𝑟−𝑙+1 (𝛼 ) 𝑚 𝑆(2)̌ (𝑔, 𝐿) = 𝛾(2) (𝑔, 𝐿) − 𝑆̃(1) (𝑔) − 𝑟−1 0 ] ∑ ∏ 𝑚,ℎ , 𝑝 2 3 𝑝 (−1) 2 𝑗 ! (𝜎 ) {𝑗 ,...,𝑗 } 𝑚=1 𝑚 0 ] 1 𝑝−𝑟−𝑙+1 𝜎̌(−1) ⋅ 0 (𝐿−1)(𝐿−2)(𝐿−3) , 3 (−1) (−1) −𝜋𝜎̌ (2ℎ+1−𝐿) (𝜎0 ) 𝛼̌ = 0 , 1,ℎ 2 (𝜎(−1)) (−1) 0 𝑆 (𝑔) 2𝜋3 𝑆(3)̌ (𝑔, 𝐿) = 𝑝 𝛾(3) (𝑔, 𝐿) + (D.2) 𝑝 2 3 𝑆̃(1) (𝑔) 3𝜎(0)𝜎̌(−1) ⋅ 𝑝 [ 0 0 − 𝜎̌(0)] (𝐿−1)(𝐿−2)(𝐿 3 (−1) 0 where the coefficients 𝑗𝑚 ofthefirsttermintheright-hand (−1) 𝜎 𝑝−𝑟+1 𝑝−𝑟+1 (𝜎0 ) 0 side satisfy ∑𝑚=1 𝑗𝑚 =𝑟+1, ∑𝑚=1 𝑚𝑗𝑚 =𝑝+1, while the 𝑝−𝑟−𝑙+1 −3) , coefficients 𝑗𝑚 related to the second term satisfy ∑𝑚=1 𝑗𝑚 = 𝑝−𝑟−𝑙+1 𝑟, ∑ 𝑚𝑗𝑚 =𝑝−𝑙.Thefirst𝛼̌𝑚,ℎ are (−1) 𝑚=1 𝑆 (𝑔) 2𝜋3 𝑆(4)̌ (𝑔, 𝐿) = 𝑝 𝛾(4) (𝑔, 𝐿) + 𝑝 2 3

(−1) (−1) (−1) 𝜋 (2ℎ+1−𝐿) 𝜎0̌ 2 5𝜎 𝜎̌ 𝛼̌ =− , 1 𝜋 (1) 2 0 1,ℎ 2 ⋅ { [𝑆̃ (𝑔) ( (𝜎(−1)) (−1) 4 6𝜎(−1) 𝑝 𝜎(−1) 0 (𝜎0 ) 0 0 𝜋 (2ℎ+1−𝐿) 𝜎̌(0) 0 (−1) (−1) (2) (5+3𝐿(𝐿−4)) 𝛼̌2,ℎ =− ̃ (−1) 2 − 𝜎2̌ )+𝜎0̌ 𝑆𝑝 (𝑔)] (𝜎0 ) 5

2𝜋 (2ℎ+1−𝐿) 𝜎(0)𝜎̌(−1) 2𝜎(0)𝜎̌(−1) + 0 0 , +3𝜎(0)𝑆̃(1) (𝑔)𝜎 [ ̌(0) − 0 0 ]} (𝐿−1)(𝐿 3 0 𝑝 0 (−1) (𝜎(−1)) 𝜎 0 0 (D.4) 2 −2)(𝐿−3) , 2𝜎(0)𝜎̌(0) 3(𝜎(0)) 𝜎̌(−1) [ 0 0 0 0 ] 𝛼̌3,ℎ =𝜋(2ℎ+1−𝐿) − (−1) (−1) 3 (−1) 4 𝑆 (𝑔) 3 (𝜎 ) (𝜎 ) (5) 𝑝 (5) 2𝜋 [ 0 0 ] 𝑆̌ (𝑔, 𝐿) = 𝛾 (𝑔, 𝐿) + 𝑝 2 3 𝜋3 (2ℎ + 1 − 𝐿)3 𝜎̌(−1) 4𝜎(−1)𝜎̌(−1) − [ 2 − 0 0 ] , 1 { 𝜋2 𝜎(0)𝜎̌(−1) 6 (−1) 4 (−1) 5 [̃(1) 2 0 (𝜎 ) (𝜎 ) ⋅ { 𝑆𝑝 (𝑔) ( [ 0 0 ] (−1) 3 (−1) 2 𝜎(−1) (𝜎0 ) {2(𝜎0 ) [ 0 3 (D.3) 4(𝜎(0)) 𝜎̌(−1) 𝜎(0)𝜎̌(−1) 𝜎(−1)𝜎̌(0) 𝜎̌(−1) [ 0 0 0 2 2 0 2 𝛼̌4,ℎ =𝜋(2ℎ+1−𝐿) + + − (−1) 5 𝜎(−1) 𝜎(−1) 5 [ (𝜎0 ) 0 0

(0) 2 (0) (−1) (0) (−1) 3(𝜎 ) 𝜎̌ (2) (−1) (2) 6𝜎2 𝜎0 𝜎0̌ (2) (0) 0 0 2𝜎 𝜎̌ 𝜎̌ − )+𝑆̃ (𝑔) (5𝜎̌ − + 0 0 − 0 ] 2 𝑝 0 4 3 2 (−1) (−1) (−1) (−1) (𝜎0 ) (𝜎0 ) (𝜎0 ) (𝜎0 ) ] (0) (−1) (−1) (0) (−1) 3 3 𝜎0 𝜎0̌ (5+3𝐿(𝐿−4)) (1) (2) 2𝜋 (2ℎ+1−𝐿) 5𝜎2 𝜎0 𝜎0̌ − )] + 𝑆̃ (𝑔) [−𝜎̌ + [− (−1) 𝑝 0 6 𝜎 3 3 (−1) 0 ] [ [ (𝜎0 ) (0) 2 (0) 𝜎(0)𝜎̌(−1) 𝜎(−1)𝜎̌(0) 𝜎(0)𝜎̌(−1) 3𝜎(2)𝜎̌(−1) 6(𝜎 ) 𝜎̌ + 0 2 + 2 0 + 2 0 + 0 0 − 0 0 5 5 5 (−1) 2 (−1) (−1) (−1) 𝜎 (𝜎(−1)) (𝜎0 ) (𝜎0 ) (𝜎0 ) 0 0

3 𝜎̌(0) 10 (𝜎(0)) 𝜎̌(−1) } − 2 ] . + 0 0 ] (𝐿−1)(𝐿−2)(𝐿−3) . (−1) 4 (−1) 3 } 4(𝜎0 ) ] (𝜎0 ) ]} Advances in High Energy Physics 17

E. Useful Relations Using the position of the external holes (34) and computing 𝜎(𝑢) 𝑢=0 (𝑛) at ,weobtain It is possible to express certain ratios among coefficients 𝜎𝑟 𝜎̌(𝑛) 𝑓(𝑔) 𝑓 (𝑔, 𝐿) 𝛾(𝑛)(𝑔, 𝐿) and 𝑟 in terms of functions , sl ,and . 𝑠 We now find some of these relations, which are useful in order 𝜎 (0) =−4 𝑠−4𝛾 −4𝐿 2+𝐺(0) −2𝑓(𝑔)ln to prove (85). ln 𝐸 ln 𝑠 Let us start with (2.14) of [87] and (C.4). Comparing 𝑓 (𝑔,𝐿)+𝐿−1 ∞ 𝛾(𝑛) (𝑔, 𝐿) them, we find the following relation: −2 sl −2∑ ( 𝑠)−𝑛 𝑠 𝑠 ln 𝜎̌(−1) 𝑛=1 ̌(2) 0 (2) (E.7) 𝑃 (𝑔, 𝑡) = −2 𝑃 (𝑔, 𝑡) . (E.1) (𝑛) 𝜎(−1) ∞ +∞ 𝑒−|𝑘| 𝑃̌ (𝑔, 𝑘) 0 + ∑ ∫ 𝑑𝑘 (𝑃(𝑛) (𝑔, 𝑘) + ) −|𝑘| −∞ 1−𝑒 𝑠 Now, developing 𝑆(𝑘) according to relation (76), 𝑛=1 ⋅ ( 𝑠)−𝑛 +𝑂(𝑠−1 ( 𝑠)−∞). ∞ ∞ ( 𝑠)−𝑛 ln ln 𝑆 (𝑘) = ∑ 𝑆(𝑛) (𝑘)( 𝑠)−𝑛 + ∑ 𝑆(𝑛)̌ (𝑘) ln ln 𝑠 𝑛=−1 𝑛=−1 (E.2) −1 −∞ 𝐺(0) 𝑆(𝑘) +𝑂(𝑠 (ln 𝑠) ), It is obvious that, expanding inthesamewayof in (E.2), relations (E.3) are also valid for the corresponding 𝐺(0) and, using integral equation (75) together with (78), it is coefficients of . Using these relations and also (E.1) it possible to obtain the following relations: is possible to find, from (E.7) and remembering (83), the following relations: 𝑓(𝑔) 𝑆(−1)̌ (𝑘) = 𝑆(−1) (𝑘) , 2 (−1) 𝜎0̌ (0) 𝑓 (𝑔, 𝐿) + 𝐿 −1 (−1) 𝑓(𝑔)=2 , 𝑆̌ (𝑘) = sl 𝑆 (𝑘) , (E.3) (−1) 2 𝜎0 𝛾(2) (𝑔, 𝐿) 𝜎̌(0) 𝑆(2)̌ (𝑘) = 𝑆(−1) (𝑘) −𝑓(𝑔)𝑆(2) (𝑘) . 𝑓 (𝑔, 𝐿) =2 0 − (𝐿−1) , 2 sl (−1) (E.8) 𝜎0 𝜎(𝑘)̂ For what concerns we have the exact expression 𝜎̌(2) 𝜎(2)𝜎̌(−1) 𝛾(2) (𝑔, 𝐿) =2 0 +4 0 0 . 2𝜋𝐿𝑒−|𝑘|/2 2𝜋𝐿𝑒−|𝑘| 𝜎(−1) (−1) 2 𝜎̂ (𝑘) =− + 0 (𝜎0 ) 1−𝑒−|𝑘| 1−𝑒−|𝑘| 2𝜋𝑒−|𝑘| 𝐿 2𝑖𝑘𝑒−|𝑘| + ∑ ( 𝑘𝑥 −1)− 𝐿̂ (𝑘) (E.4) 𝜎(𝑢) 𝑢=0 1−𝑒−|𝑘| cos ℎ 1−𝑒−|𝑘| Computing from (E.6) the second derivative of at , ℎ=1 it is also possible to show that + 𝐺̂(𝑘) , ̌(−1) where 𝜎2 𝑓(𝑔)=2 . (E.9) 𝜎(−1) ̂ 𝜋 |𝑘| 2 𝐺 (𝑘) = 𝑆 (𝑘) . (E.5) sinh (|𝑘| /2) (𝑛) Then, applying inverse Fourier transform, we obtain F. Explicit Expressions for 𝑆𝑝 (𝑔,𝐿) with 1 1 𝑛 = 1, 2, 3, 4, 5 𝜎 (𝑢) =𝐿[𝜓( −𝑖𝑢)+𝜓( +𝑖𝑢)] 2 2 (𝑛) We report here the expressions of the functions 𝑆𝑝 (𝑔, 𝐿) − (𝐿−2) [𝜓 (1−𝑖𝑢) +𝜓(1+𝑖𝑢)] with 𝑛 = 1, 2, 3, 4, 5 in terms of the densities (and their derivativesinzero)andthesolutionsofthe“reducedsystems” −𝜓(1−𝑖𝑥𝐿 −𝑖𝑢)−𝜓(1+𝑖𝑥𝐿 −𝑖𝑢) ̃(𝑛) 𝑆𝑝 (𝑔). The general method to obtain them and results for 𝑛 = 1, 2, 3, 4 are shown in [87] −𝜓(1−𝑖𝑥𝐿 +𝑖𝑢)−𝜓(1+𝑖𝑥𝐿 +𝑖𝑢) (E.6) 𝑢2 −[2ln 2+𝑂( )] (1) 𝑠2 𝑆𝑝 (𝑔,) 𝐿 =0, (F.1)

+∞ −|𝑘| 𝜋3 𝑖𝑘𝑢 𝑒 (2) ̃(1) + ∫ 𝑑𝑘𝑒 𝑃(𝑠,𝑔,𝑘)+𝐺(𝑢) . 𝑆𝑝 (𝑔, 𝐿) = (𝐿−3)(𝐿−2)(𝐿−1) 𝑆𝑝 (𝑔) , 1−𝑒−|𝑘| (−1) 2 (F.2) −∞ 3(𝜎0 ) 18 Advances in High Energy Physics

𝜋3𝜎(0) Σ(𝑘)̂ 𝑆(3) (𝑔, 𝐿) = −2 0 (𝐿−3)(𝐿−2)(𝐿−1) that the function which satisfies the BES equation 𝑝 3 Σ(0̂ +)=𝜋𝛾(𝑔,𝐿,𝑠)| 3(𝜎(−1)) such that ln 𝑠 in our notations reads 0 (F.3) −|𝑘| ⋅ 𝑆̃(1) (𝑔) , 󵄨 2𝑒 󵄨 𝑝 Σ̂ (𝑘) = 𝜎̂ (𝑘)󵄨 + 𝑖𝑘𝐿̂ (𝑘)󵄨 , (∗) 𝐻 󵄨ln 𝑠 1−𝑒−|𝑘| 𝐻 󵄨ln 𝑠 2 { 𝜋2 (𝜎(0)) (4) [ 0 where the label 𝐻 means that only higher than one loop 𝑆𝑝 (𝑔, 𝐿) = 2𝜋 (𝐿−3)(𝐿−2)(𝐿−1) { (−1) 4 contributions have to be included. Then, in addition to {[ 2(𝜎0 ) what we would call “higher than one loop density of roots,” that is, 𝜎̂𝐻(𝑘), in BES density (∗),thereisalso 𝜋4𝜎(−1) − 2 5+3𝐿 𝐿−4 ] 𝑆̃(1) (𝑔) a term nonlinear in the counting function 𝑍.Thisterm 5 ( ( )) 𝑝 (F.4) 90 (𝜎(−1)) canbeestimatedatlarge𝑠 by using (69) to be (almost 0 ] 2 everywhere) 𝑂(1/𝑠 ): therefore, as far as the order ln 𝑠 is 4 concerned, it is almost everywhere negligible, with the 𝜋 (2) } − (5+3𝐿(𝐿−4)) 𝑆̃ (𝑔) , exception of the point 𝑘=0, where one experiences (−1) 4 𝑝 } 360 (𝜎0 ) } the noncommutativity between the limits 𝑠→+∞and 𝑘→0.However,thenonlineartermhastobekeptin (5) 𝑆𝑝 (𝑔, 𝐿) = (𝐿−3)(𝐿−2)(𝐿−1) definition (∗) of the BES density, since it gives the entire information on anomalous dimensions (see (A.10)), due (−1) (0) (0) { 5𝜋5 𝜎 𝜎 𝜋5 𝜎 to the fact that 𝜎̂𝐻(0) = 𝜎(0)̂ − 𝜎|̂ 1loop(0) = 0. ⋅ [( 2 0 − 2 ) { 6 5 𝐽 3 (−1) 3 (𝜎(−1)) 2. We use the notation ( 𝑛 is a Bessel function) {[ (𝜎0 ) 0 (5+3𝐿(𝐿−4)) +∞ 𝑑𝑡 𝐽 (√2𝑔𝑡) 𝐽 (√2𝑔𝑡) ⋅ 𝑍 (𝑔) = ∫ 𝑛 𝑚 . (∗∗) 15 𝑛,𝑚 𝑡 (F.5) 0 𝑡 𝑒 −1 3 4𝜋3 (𝜎(0)) 2𝜋3𝜎(2) +(− 0 − 0 )] 𝑆̃(1) (𝑔) References 5 3 𝑝 3(𝜎(−1)) 3(𝜎(−1)) 0 0 ] [1] J. M. Maldacena, “The large N Limit of superconformal field theories and supergravity,” Advances in Theoretical and Math- 𝜋5𝜎(0) } 0 ̃(2) ematical Physics,vol.2,no.2,pp.231–252,1998. + 𝑆𝑝 (𝑔) (5+3𝐿(𝐿−4))} . (−1) 5 [2] S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, “Gauge theory 45 (𝜎0 ) } correlators from non-critical string theory,” Physics Letters B: Nuclear, Elementary Particle and High-Energy Physics,vol.428, Conflict of Interests no. 1-2, pp. 105–114, 1998. [3] E. Witten, “Anti de sitter space and holography,” Advances in The authors declare that there is no conflict of interests Theoretical and Mathematical Physics,vol.2,pp.253–291,1998. regarding the publication of this paper. [4] J. Minahan and K. Zarembo, “The bethe-ansatz for N =4super yang-mills,” JournalofHighEnergyPhysics, vol. 2003, article 03, 2003. Acknowledgments [5] N.BeisertandM.Staudacher,“TheN =4SYM integrable super spin chain,” Nuclear Physics B,vol.670,no.3,pp.439–463,2003. The authors thank gratefully B. Basso for scientific discus- [6] V. Kazakov, A. Marshakov, J. Minahan, and K. Zarembo, sions. As for travel financial support, INFN IS Grant GAST, “Classical/quantum integrability in AdS/CFT,” Journal of High the UniTo-SanPaolo research Grant no. TO-Call3-2012-0088, Energy Physics,vol.2004,no.5,article024,2004. the ESF Network HoloGrav (09-RNP-092 (PESC)), and [7] M. Staudacher, “The factorized S-matrix of CFT/AdS,” Journal the MPNS-COST Action MP1210 are kindly acknowledged. of High Energy Physics, vol. 2005, no. 5, article 054, 2005. Gabriele Infusino acknowledges E.U., Italian Republic and [8] N. Beisert, V. A. Kazakov, K. Sakai, and K. Zarembo, “The 5 Calabria Region, for funding through Regional Operative Algebraic curve of classical superstrings on 𝐴𝑑𝑠5 ×𝑆 ,” Commu- Program (ROP) Calabria ESF 2007/2013—IV Axis Human nications in Mathematical Physics,vol.263,no.3,pp.659–710, Capital-Operative Objective M2. 2006. [9] N. Beisert and M. Staudacher, “Long-range psu(2,2—4) Bethe ansatze¨ for gauge theory and strings,” Nuclear Physics B,vol.727, Endnotes no.1-2,pp.1–62,2005. [10] N. Beisert, B. Eden, and M. Staudacher, “Transcendentality and 1. Results in this Appendix could seem in contrast with BES crossing,” Journal of Statistical Mechanics,vol.2007,no.1,Article finding stating that anomalous dimension at order ln 𝑠 is ID P01021, 2007. givenbythevalueinzerooftheFouriertransformofthe [11] J. Ambjørn, R. A. Janik, and C. Kristjansen, “Wrapping interac- “higher than one loop density of roots,” which satisfies tions and a new source of corrections to the spin-chain/string BES equation. The solution to this apparent contrast is duality,” Nuclear Physics B,vol.736,no.3,pp.288–301,2006. Advances in High Energy Physics 19

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Review Article Exploring New Models in All Detail with SARAH

Florian Staub

TheoryDivision,CERN,1211Geneva23,Switzerland

Correspondence should be addressed to Florian Staub; [email protected]

Received 13 March 2015; Accepted 7 July 2015

Academic Editor: Gordon Kane

Copyright © 2015 Florian Staub. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.

IgiveanoverviewaboutthefeaturestheMathematica package SARAH provides to study new models. In general, SARAH can handle a wide range of models beyond the MSSM coming with additional chiral superfields, extra gauge groups, or distinctive features like Dirac gaugino masses. All of these models can be implemented in a compact form in SARAH and are easy to use: SARAH extracts all analytical properties of the given model like two-loop renormalization group equations, tadpole equations, mass matrices, and vertices. Also one- and two-loop corrections to tadpoles and self-energies can be obtained. For numerical calculations SARAH can be interfaced with other tools to get the mass spectrum, to check flavour or dark matter constraints, and to test the vacuum stability or to perform collider studies. In particular, the interface to SPheno allows a precise prediction of the Higgs mass in a given model comparable to MSSM precision by incorporating the important two-loop corrections. I show in great detail with the example of the B-L-SSM how SARAH together with SPheno, HiggsBounds/HiggsSignals, FlavorKit, Vevacious, CalcHep, MicrOmegas, WHIZARD,andMadGraph can be used to study all phenomenological aspects of a model.

1. Introduction before LHC has started, have been ruled out. This has caused more interest in nonminimal SUSY models. Beyond-MSSM Supersymmetry (SUSY) has been the top candidate for model can provide many advantages compared to the MSSM: beyond standard model (BSM) physics for many years [1–3]. they address not only the two issues mentioned so far. A more This has many reasons. SUSY solves the hierarchy problem complete list of good reasons to take a look on extensions of of the standard model (SM) [4, 5], provides a dark matter theMSSMisasfollows. candidate [6–8], leads to gauge coupling unification9 [ –15], and gives an explanation for electroweak symmetry breaking (EWSB) [16, 17]. One might even consider the measured (i) Naturalness. The Higgs mass in SUSY is not a free HiggsmassasafirsthintforSUSYsinceitfallsinthecorrect parameterlikeintheSM.IntheMSSMthetree- (𝑇) ballpark, while it could be much higher in the SM and other level mass is bounded from above by 𝑚ℎ <𝑚𝑍. BSMscenarios.BeforetheLHChasbeenturnedon,themain Thus, about one-third of the Higgs mass has to focus has been on the minimal supersymmetric extensions be generated radiatively to explain the observation. of the SM, the MSSM. The 105 additional parameters of this Heavy SUSY masses are needed to gain sufficiently model,mainlylocatedintheSUSYbreakingsector,canbe large loop corrections: that is, a soft version of the constrained by assuming a fundamental, grand unified theory hierarchy problem appears again. The need for large (GUT) and a specific mechanism for SUSY breaking [18– loop corrections gets significantly softened if 𝐹-or 27]. In these cases often four or five free parameters are 𝐷-terms are present which already give a push to the left and the model becomes very predictive. However, the tree-level mass [30–34]. negative results from SUSY searches at the LHC1 as well as the measured Higgs mass of about 125 GeV [28, 29]put (ii) SUSY Searches. The negative results from all SUSY largepressureonthesimplestscenarios.Wideregionsof searchesattheLHChaveputimpressivelimitson the parameter space, which had been considered as natural the Sparticle masses. However, the different searches 2 Advances in High Energy Physics

are based on certain assumptions like a stable, neu- [79–119]. Dirac gauginos are also attractive because tral, and colourless lightest SUSY particle (LSP), a they can weaken LHC search bounds [120–122]and sufficient mass splitting between the SUSY states, and flavour constraints [123–125]. so on. As soon as these conditions are no longer givenlikeinmodelswithbroken𝑅-parity, the limits Despite the large variety and flexibility of SUSY, many become much weaker [35–37]. Also in scenarios with dedicated public computer tools like Isajet [126– compressed spectra where SUSY states are nearly 132], Suspect [133], SoftSUSY [134–136], SPheno degenerated, the strong limits do often not apply [38– [137, 138], or FeynHiggs [139, 140] are restricted to the 49]. simplest realization of SUSY, the MSSM, or small extensions (iii) Neutrino Data. There is an overwhelming experimen- of it. Therefore, more generic tools are needed to allow tal evidence that neutrinos have masses and do mix studying of nonminimal SUSY models with the same among each other; see [50] and references therein. precision as the MSSM. This precision is needed to confront However, neutrino masses are not incorporated in also these models with the strong limits from SUSY searches, the MSSM. To do that, either one of the different flavour observables, dark matter observations, and Higgs 𝑅 measurements. The most powerful tool in this direction is seesaw mechanisms can be utilised or -parity must Mathematica SARAH SARAH be broken to allow a neutrino-neutralino mixing [51– the package [141–145]. has 62]. been optimized for an easy, fast, and exhaustive study of nonminimal SUSY models. While the first version of SARAH (iv) Strong CP-Problem.ThestrongCP-problemremains has been focused on the derivation of tree-level properties an open question not only in the SM but also in of a SUSY model, that is, mass matrices and vertices, the MSSM. In principle, for both models the same and interfacing this information with Monte-Carlo (MC) solution exists to explain the smallness of the Θ term tools, with the second version of SARAH the calculation of in QCD: the presence of a broken Peccei-Quinn (PQ) one-loopself-energiesaswellastwo-looprenormalization symmetry [63]. In its supersymmetric version PQ group equations (RGEs) has been automatized. With models predict not only an axion but also an axino version 3, SARAH became the first “spectrum-generator- which could be another DM candidate [64–70]. In generator”: all analytical information derived by SARAH can general, the phenomenological aspects of axion-axino be exported to Fortran code which provides a fully-fledged models are often even richer, in particular if the spectrum generator based on SPheno.Thisfunctionality DSFZ version is considered [71, 72]: the minimal, self- has been later extended by the FlavorKit [146]interface consistent supersymmetric DSFZ-axion model needs which allows a modular implementation of new flavour in total three additional superfields compared to the observables based on the tools FeynArts/FormCalc– MSSM [73]. SARAH–SPheno. Also different methods to calculate the (v) 𝜇-Problem. The superpotential of the MSSM involves two-loopcorrectionstotheHiggsstatesinanonminimal one parameter with dimension mass: the 𝜇-term. model are available with SPheno modules generated This term is not protected by any symmetry: that by SARAH today: the radiative contributions to CP even is, the natural values would be either exactly 0 or scalar masses at the two-loop level can be obtained by 𝑂(𝑀 ) using either the effective potential approach [147]based GUT . However, both extreme values are ruled out by phenomenological considerations. The opti- ongenericresultsgivenin[16], or a fully diagrammatic mal size of this parameter would be comparable to calculation [148]. Both calculations provide Higgs masses the electroweak scale. This could be obtained if the with a precision which is otherwise just available for the 𝜇-term is actually not a fundamental parameter but is MSSM. Beginning with SARAH 4,thepackageisnolonger generated dynamically. For instance, in singlet exten- restricted to SUSY models but can handle also a general, sions an effective 𝜇-term appears as consequence of renormalizable quantum field theory and provides nearly 𝑂(𝑀 ) the same features as for SUSY models. Today, SARAH can be SUSY breaking and is therefore naturally SUSY [30, 74]. used for SUSY and non-SUSY models to write model files for CalcHep/CompHep [149, 150], FeynArts/FormCalc (vi) Top-Down Approach. Starting with a GUT or String [151, 152], and WHIZARD/O’Mega [153, 154]aswellasin theory it is not necessarily clear that only the gauge the UFO format [155] which can be handled, for instance, sector and particle content of the MSSM are present by MadGraph 5 [156], GoSam [157], Herwig++ [158–160], at the low scale. Realistic UV completions come and Sherpa [161–163]. The modules created by SARAH often with many additional matter close to the SUSY for SPheno calculate the full one-loop and partially scale. In many cases also additional neutral and even two-loop-corrected mass spectrum, branching ratios and charged gauge bosons are predicted [75–78]. decays widths of all states, and many flavour and precision (vii) 𝑅-Symmetry. If one considers 𝑅-symmetric models, observables. Also an easy link to HiggsBounds [164, 165] Majorana gaugino masses are forbidden. To give and HiggsSignals [166] exists. Another possibility to get masses to the gauginos in these models, a coupling atailor-madespectrumgeneratorforanonminimalSUSY to a chiral superfield in the adjoint representation model based on SARAH is the tool FlexibleSUSY [167]. is needed. This gives rise to Dirac masses for the Finally, SARAH can also produce model files for Vevacious gauginos which are in agreement with 𝑅-symmetry [168]. The combination SARAH–SPheno–Vevacious Advances in High Energy Physics 3 provides the possibility to find the global minimum of the (3) chiral superfields, one-loop effective potential of a given model and parameter (4) superpotential. point. The range of models covered by SARAH is very That means that SARAH automatizes many steps to derive the broad. SARAH and its different interfaces have been Lagrangian from that input as follows: successfully used to study many different SUSY scenarios: 𝐹 singlet extensions with and without CP violation [169– (1) All interactions of matter fermions and the -terms 179], triplet extensions [180, 181], models with 𝑅-parity are derived from the superpotential. violation [182–188], different kinds of seesaw mechanisms (2) All vector boson and gaugino interactions as well as [58, 60–62, 189–194], models with extended gauge sectors 𝐷-terms are derived from gauge invariance. at intermediate scales [195–198] or the SUSY scale [34, 199– (3) All gauge fixing terms are derived by demanding that 203], models with Dirac gauginos [109, 111, 204–206]or scalar-vector mixing vanishes in the kinetic terms. vector-like states [207], and even more exotic extensions [208–211]. In addition, SARAH can be also very useful to (4) All ghost interactions are derived from the gauge perform studies in the context of the MSSM which cannot fixing terms. be done with any other public tool out of the box. That is (5) All soft-breaking masses for scalars and gauginos as the case, for instance, if new SUSY breaking mechanisms well as the soft-breaking counterparts to the superpo- should be considered [212–219] or if the presence of charge tential couplings are added automatically. and colour breaking minima should be checked [220, 221]. For the NMSSM, despite the presence of specialized tools Of course, the Lagrangian of the gauge eigenstates is not the like NMSSMTools [222], SoftSUSY [223], or NMSSMCalc final aim. Usually one is interested in the mass eigenstates [224], the SPheno version created by SARAH is the only after gauge symmetry breaking. To perform the necessary code providing two-loop corrections beyond 𝑂(𝛼𝑆(𝛼𝑡 +𝛼𝑏)) rotations to the new eigenstates, the user has to give some not relying on MSSM approximations [225]. Also the full more information: one-loop corrections to all SUSY states in the NMSSM have (1) definition of the fields which get a vacuum expecta- first been derived with SARAH [226]. tion value (VEV) to break gauge symmetries, Thispaperisorganizedasfollows.inthenextsectionan overview about the models supported by SARAH is given. In (2) definition of vector bosons, scalars, and fermions Section 3, I will discuss the possible analytical calculations which mix among each other. which can be done with SARAH and list the possible output Using this information, all necessary redefinitions and fields of the derived information for further evaluation. The main rotations are done by SARAH.Alsothegaugefixingtermsare part of this paper is a detailed example of how SARAH can be derived for the new eigenstates and the ghost interactions used to study all phenomenological aspects of a model. That is are added. For all eigenstates plenty of information can be done in Sections 4–8:inSection 4 the implementation of the derived by SARAH as explained in Section 3. Before coming B-L-SSM in SARAH is described, in Section 5 how the model to that, I will give more details about what kind of models can be understood at the analytical level in Mathematica and what features are supported by SARAH. is discussed, the SPheno output with all its features is presented in Section 6,inSection 7 I will show how other tools can be used together with SARAH and SPheno to study, 2.2. Supported Models and Features. As we have seen in the for instance, the dark matter and collider phenomenology, introduction, there are many possibilities to go beyond the and in Section 8, different possibilities to perform parameter widely studied MSSM. Each approach modifies the on or scans are presented. I summarize in Section 9.Throughout the other sector of the model. In general, possible changes the paper and in the given examples I will focus mainly on compared to the MSSM are (i) using other global symmetries SUSY models, but many statements apply one-to-one also to to extent the set of allowed couplings, (ii) adding chiral non-SUSY models. superfields, (iii) extending the gauge sector, (iv) giving VEVs to other particles compared to only the Higgs doublets, (v) adding Dirac masses for gauginos, (vi) considering non- 2. Models canonical terms like nonholomorphic soft-SUSY breaking SARAH SARAH interactions or Fayet-Iliopoulos 𝐷-terms. All of these roads 2.1. Input Needed by to Define a Model. is SARAH optimized for the handling of a wide range of SUSY models. can in principle be gone by and I will briefly discuss SARAH what is possible in the different sectors and which steps are The basic idea of wastogivetheuserthepossibilityto SARAH implement models in an easy, compact, and straightforward done by to get the Lagrangian. Of course, extending way. Most tasks to get the Lagrangian are fully automatized: the gauge sector or adding Dirac masses to gauginos comes it is sufficient to define just the fundamental properties ofthe inevitable with an extended matter sector as well. Thus, often several new effects appear together and can be covered model. That means that the necessary inputs to completely SARAH define the gauge eigenstates with all their interactions are by .

(1) global symmetries, 2.2.1. Global Symmetries. SARAH canhandleanarbitrary (2) gauge symmetries, number of global symmetries which are either 𝑍𝑁 or 𝑈(1) 4 Advances in High Energy Physics

symmetries. Also a continuous 𝑅-symmetry 𝑈(1)𝑅 is possi- Iamusinghereandinthefollowingcapitalletters𝐴 and 𝐵 to ble. Global symmetries are used in SARAH mainly for three label the gauge groups and small letter 𝑎, 𝑏,and𝑐 to label the different purposes. First, they help to constrain the allowed generators, vector bosons, and gauginos of a particular gauge couplings in the superpotential. However, SARAH does not group. The field strength tensor is defined as strictly forbid terms in the superpotential which violate a 𝐴,𝑎 𝐴,𝑎 𝐴,𝑎 𝑎𝑏𝑐 𝐴,𝑏 𝐴,𝑐 global symmetry. SARAH only prints a warning to point out 𝐹𝜇] =𝜕𝜇𝑉] −𝜕]𝑉𝜇 +𝑔𝐴𝑓𝐴 𝑉𝜇 𝑉] , (3) the potential contradiction. The reason is that such a term might be included on purpose to explain its tininess. Global and the covariant derivative is symmetries can also affect the soft-breaking terms written 𝑎 𝑎 𝑎𝑏𝑐 𝑏 𝑐 𝐷𝜇𝜆 =𝜕𝜇𝜆 +𝑔𝐴𝑓 𝐴 𝜆 . (4) down by SARAH. SARAH always tries to generate the most 𝐴 𝐴 𝜇 general Lagrangian and includes also soft-masses of the form 𝑎𝑏𝑐 2 ∗ Here, 𝑓𝐴 is the structure constant of the gauge group 𝐴. 𝑚 𝜙𝑖𝜙 for two scalars 𝜙𝑖, 𝜙𝑗 with identical charges. However, 𝑗 Plugging (3) in the first term of (2) leads to self-interactions these terms are dropped if they are forbidden by a global of three and four gauge bosons. In general, the procedure to symmetry. By the same consideration, Dirac gaugino mass obtain the Lagrangian from the vector and chiral superfields terms are written down or not. Finally, global symmetries MicrOmegas is very similar to [233]. Interested readers might check this are crucial for the output of model files for reference for more details. to calculate the relic density. For this output at least one unbroken discrete global symmetry must be present. GaugeInteractionsofMatterFields.Vectorsuperfieldsusually By modifying the global symmetries one can already go donotcomealonebutalsomatterfieldsarepresent.Iam beyond the MSSM without changing the particle content: going to discuss the possibilities to define chiral superfields 𝑍 𝑅 choosing a 3 (Baryon triality) instead of -parity [227–231], in Section 2.2.4. Here, I assume that a number of chiral lepton number violating terms would be allowed while the superfieldsarepresentandIwanttodiscussthegauge SARAH 𝑅 proton is still stable. comes not only with -parity interactions which are taken into account for those. First, violating models based on Baryon triality, but also with a the 𝐷-terms stemming from the auxiliary component of variant for Baryon number violation but conserved Lepton the superfield are calculated. These terms cause four scalar number is included. interactions and read 1 2 󵄨 ∗ 𝑎 󵄨2 L𝐷 = 𝑔 ∑ 󵄨(𝜙 𝑇 𝜙𝑗)󵄨 . 2.2.2. Gauge Sector 𝐴 𝐴 󵄨 𝑖 𝐴𝑟 󵄨 (5) 2 𝑖,𝑗 SARAH Gauge Groups. The gauge sector of a SUSY model in 𝑖, 𝑗 𝑇𝑎 is fixed by defining a set of vector superfields. SARAH Here, the sum is over all scalars in the model, and 𝐴𝑟 are the generators of the gauge group 𝐴 for an irreducible is not restricted to three vector superfields like in the 𝑟 𝑇𝑎 MSSM, but many more gauge groups can be defined. To representation .ForAbeliangroups 𝐴𝑟 simplify to the 𝑄𝐴 improve the power in dealing with gauge groups, SARAH has charges 𝜙 of the different fields. In addition, Abelian gauge linked routines from the Mathematica package Susyno groups can come also with another feature: a Fayet-Iliopoulos [232]. SARAH together with Susyno take care of all group- 𝐷-term [234]: theoretical calculations: the Dynkin and Casimir invariants 𝑔 L =𝜉 𝐴 ∑ (𝜙∗𝑄𝐴𝜙 ). are calculated, and the needed representation matrices as FI,𝐴 𝐴 2 𝑖 𝜙 𝑖 (6) well as Clebsch-Gordan coefficients are derived. This is done 𝑖 𝑈(1) (𝑁) not only for and SU gaugegroups,butalsofor This term can optionally be included in SARAH for any 𝑈(1). (𝑁) (2𝑁) SO and Sp and expectational groups can be used. The other gauge-matter interactions are those stemming ForallAbeliangroupsalsoaGUTnormalizationcanbe from the kinetic terms: given. This factor comes usually from considerations about L =−𝐷𝜇𝜙∗𝑖𝐷 𝜙 −𝑖𝜓†𝑖𝜎𝜇𝐷 𝜓 the embedding of a model in a greater symmetry group like kin 𝜇 𝑖 𝜇 𝑖 (7) SU(5) or SO(10). If a GUT normalization is defined for a 𝐴,𝑎 𝑎 group, it will be used in the calculation of the RGEs. The soft- with covariant derivatives 𝐷𝜇 ≡𝜕𝜇 −𝑖𝑔𝐴𝑉𝜇 (𝑇𝐴𝑟).TheSUSY breaking terms for a gaugino 𝜆 of a gauge group 𝐴 are usually counterparts of these interactions are those between gauginos included as and matter fermions and scalars: 1 √ ∗ 𝑎 𝑎 𝑎 𝑎 L =− 2𝑔𝐴 (𝜙𝑖 𝑇𝐴𝑟𝜓𝑗)𝜆𝐴 + h.c. (8) L ,𝜆 = 𝜆 𝜆 𝑀𝐴 + h.c. (1) GFS SB 𝐴 2 𝐴 𝐴 Gauge-Kinetic Mixing. The terms mentioned so far cover all Gauge Interactions. With the definition of the vector supergauge interactions which are possible in the MSSM. These are fields already the self-interactions of vector bosons as well derived for any other SUSY model in exactly the same way. as the interactions between vector bosons and gauginos are However, there is another subtlety which arises if more than fixed.Thosearetakentobe one Abelian gauge group is present. In that case 1 1 L =− 𝐹𝐴,𝑎𝐹𝐴,𝜇]𝑎 −𝑖𝜆†𝑎𝜎𝜇𝐷 𝜆𝑎 . L =− 𝜅𝐹𝐴 𝐹𝐵,𝜇] 𝐴 =𝐵̸ 4 𝜇] 𝐴 𝜇 𝐴 (2) 4 𝜇] (9) Advances in High Energy Physics 5

𝜇] 𝑎 are allowed for field strength tensors 𝐹 of two different Here, F𝐴 is usually a function involving partial derivatives 𝐴 𝐵 𝜅 𝑛×𝑛 𝑛 𝐴,𝑎 Abelian groups and [235]. is in general a matrix if of gauge bosons 𝑉𝜇 . SARAH uses 𝑅𝜉 gauge. This means that, SARAH Abelian groups are present. fully includes the effect of foranunbrokengaugesymmetry,thegaugefixingtermsare kinetic mixing independent of the number of Abelian groups. For this purpose SARAH is not working with field strength 1 󵄨 𝜇 𝐴,𝑎󵄨2 L =− 󵄨𝜕 𝑉 󵄨 . interactions like (9) but performs a rotation to bring the field GF 2𝑅 󵄨 𝜇 󵄨 (17) 𝜉𝐴 strength in a diagonal form. That is done by a redefinition of 𝜇 the vector Υ carrying all gauge fields 𝑉𝑋: For broken symmetries, the gauge fixing terms are chosen in a way where the mixing terms between vector bosons and Υ󳨀→√𝜅Υ. (10) scalars disappear from the Lagrangian. This generates usually terms of the form This rotation has an impact on the interactions of the gauge 1 󵄨 𝜇 𝐴 𝐴󵄨2 bosons with matter fields. In general, the interaction of a L ,𝑅 =− 󵄨𝜕 𝑉 +𝑅𝜉 𝑀𝐴𝐺 󵄨 . 𝜙 GF 𝜉 2𝑅 󵄨 𝜇 𝐴 󵄨 (18) particle with all gauge fields can be expressed by 𝜉𝐴 𝑇 ̃ 𝐴 𝐴 Θ𝜙 𝐺Υ, (11) Here, 𝐺 is the Goldstone boson of the vector boson 𝑉𝜇 with mass 𝑀𝐴. From the gauge fixing part, the interactions Θ 𝑄𝑥 𝜙 𝑎 where 𝜙 is a vector containing the charges 𝜙 of under of ghost fields 𝜂𝐴 are derived by 𝑈(1) 𝑥 𝐺̃ 𝑛×𝑛 all groups and is a diagonal matrix carrying L =−𝜂𝑎 (𝛿F𝑎 ). the gauge couplings of the different groups. After the rotation Ghost 𝐴 𝐴 (19) according to (10) the interaction part can be expressed by Here, 𝛿 assigns the operator for a BRST transformation. All Θ𝑇𝐺Υ steps to get the gauge fixing parts and the ghost interactions 𝜙 (12) are completely done automatically by SARAH and adjusted to the gauge groups in the model. with a general 𝑛×𝑛matrix 𝐺 which is no longer diagonal. In that way, the effect of gauge-kinetic mixing has been absorbed in “off-diagonal” gauge couplings. This means that 2.2.4. Matter Sector. Therecanbeupto99chiralsuperfields SARAH the covariant derivative in SARAH reads in a single SUSY model in .Allsuperfieldscancome with an arbitrary number of generations and can transform 𝑥 𝜇 as any irreducible representation with respect to the defined 𝐷𝜇𝜙=(𝜕𝜇 −𝑖∑𝑄𝜙𝑔𝑥𝑦 𝑉𝑦 )𝜙, (13) 𝑥,𝑦 gauge groups. In the handling of nonfundamental fields under a symmetry, SARAH distinguishes if the corresponding where 𝑥 and 𝑦 are running over all 𝑈(1) groups and 𝑔𝑥𝑦 are symmetry gets broken or not: for unbroken symmetries the entries of the matrix 𝐺. Gauge-kinetic mixing is included it is convenient to work with fields which transform as not only in the interactions with vector bosons, but also in vector under the symmetry with the appropriate length. For (3) the derivation of the 𝐷-terms. Therefore, the 𝐷-terms for the instance, a 6 under SU 𝑐 is taken to be Abelian sector in SARAH read 𝜙𝛼 𝛼=1,2,...,6. (20) ∗ 𝑇 ∗ L𝐷,𝑈(1) = ∑ (𝜙 𝜙𝑖)(𝐺 Θ𝜙 )(𝐺Θ𝜙 )(𝜙 𝜙𝑗) 𝑖 𝑖 𝑗 𝑗 (14) 𝑖𝑗 That is, it carries one charge index. In contrast, nonfundamental fields under a broken gauge symmetry are represented while the non-Abelian 𝐷-terms keep the standard form by tensor products of the fundamental representation. For (2) equation (5). Finally, also “off-diagonal” gaugino masses are instance, a 3 under SU 𝐿 is taken to be introduced. The soft-breaking part of the Lagrangian then 𝜙 𝑎, 𝑏 = 1, 2. reads 𝑎𝑏 (21) 1 2×2 L ⊃ ∑ 𝜆 𝜆 𝑀 + Thus, the triplet can be given as usual as matrix. SB,𝜆,𝑈(1) 𝑥 𝑦 𝑥𝑦 h.c. (15) 𝑥𝑦 2 For Abelian gauge groups not only one can define charges for superfields which are real numbers, but also variables can SARAH takes the off-diagonal gaugino masses to be symmet- be used for that. All interactions are then expressed keeping ric: 𝑀𝑥𝑦 =𝑀𝑦𝑥. these charges as free parameter. For all chiral superfield SARAH adds the soft-breaking masses. For fields appearing in 𝑁 generations, these are 2.2.3. Gauge Fixing Sector. All terms written down so far lead treated as Hermitian 𝑁×𝑁matrices. As written above, also to a Lagrangian which is invariant under a general gauge soft-terms mixing two scalars are included if allowed by all transformation. To break this invariance one can add “gauge symmetries. Hence, the soft-breaking mass terms read, in fixing” terms to the Lagrangian. The general form of these general, terms is ̃ † 2 1 󵄨 󵄨2 L = ∑𝛿 𝜙 𝑚 𝜙 + L =− 󵄨F𝑎 󵄨 . SB,𝜙 𝑖𝑗 𝑖 𝑖𝑗 𝑗 h.c. (22) GF 2 󵄨 𝐴󵄨 (16) 𝑖𝑗 6 Advances in High Energy Physics

Note that 𝑖, 𝑗 label different scalar fields; generation indices and interactions of matter fermions 𝛿̃ 𝜙 𝜙 are not shown. 𝑖𝑗 is 1, if fields 𝑖 and 𝑗 have exactly the 1 𝜕2𝑊̃ same transformation properties under all local and global L =− 𝜓 𝜓 + 𝑌 2 𝜕𝜙 𝜕𝜙 𝑖 𝑗 h.c. (28) symmetries, and otherwise 0. 𝑖 𝑗 𝑊̃ 𝑊 2.2.5. Models with Dirac Gauginos. Another feature which are derived. Here is the superpotential with all super- 𝜙̂ 𝜙 𝜓 became popular in the last years is models with Dirac fields 𝑖 replacedbytheirscalarcomponent 𝑖. 𝑖 is the 𝜙̂ 𝐴 fermionic component of that superfield. 𝑚 𝑖 𝜆 𝜓 gauginos. In these models mass terms 𝐷 𝐴 𝑖 between 𝐹 𝐷 𝜆 𝜓 Usually, the -and -terms and the soft-breaking terms gauginos 𝐴 and a fermionic component 𝑖 of the chiral for chiral and vector superfields fix the full scalar potential of 𝜙̂ superfield 𝑖 in the adjoint representation of the gauge group themodel.However,insomecasesalsononcanonicalterms 𝐴 𝐷 arepresent.Inaddition,alsonew -terms are introduced should be studied. These are, for instance, nonholomorphic in these models [98]. Thus, the new terms in the Lagrangian soft-terms: are L = 𝑇̃𝑖𝑗𝑘 𝜙 𝜙 𝜙∗. 𝜙̂ 𝐴 𝜙̂ 𝐴 SB,NH 𝑖 𝑗 𝑘 (29) L =−𝑚 𝑖 𝜆𝑎 𝜓 + √2𝑚 𝑖 𝜙 𝐷 . DG 𝐷 𝐴 𝑖 𝐷 𝑖 𝐴 (23) Those can be added as well and they are taken into account in 𝐷 𝐴 is the auxiliary component of the vector superfield of the the calculation of the vertices and masses and as consequence 𝐴 group .ToallowforDiracmassterms,thesemodelscome also in all loop calculations. However, they are not included always with an extended matter sector: to generate Dirac mass in the calculation of the RGEs because of the lack of generic terms for all MSSM gauginos at least one singlet, one triplet results in the literature. under SU(2), and one octet under SU(3) must be added. Furthermore, models with Dirac gauginos generate also new 2.2.7. Symmetry Breaking and VEVs. All gauge symmetries structures in the RGEs [236]. All of this is fully supported can also be broken. This is in general done by decomposing a in SARAH. complex scalar into its real components and a VEV: If Dirac masses for gauginos are explicitly turned on in SARAH, it will check for all allowed combinations of vector 1 𝑆 󳨀→ (V +𝜙 +𝑖𝜎). and chiral superfields which can generate Dirac masses and 𝑖 √2 𝑖 𝑖 𝑖 (30) which are consistent with all symmetries. For instance, in models with several gauge singlets, the bino might even get Assigning a VEV to a scalar is not restricted to colourless several Dirac mass terms. and neutral particles. Also models with spontaneous colour or charge breaking (CCB) can be studied with SARAH.Also 2.2.6. Superpotential, Soft-Terms, and Noncanonical Interac- explicit CP violation in the Higgs sector is possible. There tions. The matter interactions in SUSY models are usually are two possibilities to define that. Either a complex phase is fixed by the superpotential and the soft-SUSY breaking added SARAH terms. fully supports all renormalizable terms in the 1 𝑖𝜂 superpotential 𝑆𝑖 󳨀→ 𝑒 (V𝑖 +𝜙𝑖 +𝑖𝜎𝑖) √2 (31) 𝑊=𝑐𝐿 𝜙̂ +𝑐 𝑀𝑖𝑗 𝜙̂ 𝜙̂ +𝑐 𝑌𝑖𝑗𝑘 𝜙̂ 𝜙̂ 𝜙̂ 𝐿 𝑖 𝑖 𝑀 𝑖 𝑗 𝑇 𝑖 𝑗 𝑘 (24) or a VEV for the CP odd component is defined: and generates the corresponding soft-breaking terms: 1 𝑆 󳨀→ (V𝑅 +𝜙 +𝑖(V𝐼 +𝜎)) . 𝑖 √2 𝑖 𝑖 𝑖 𝑖 (32) 𝐿 =𝑐𝑡 𝜙 +𝑐 𝐵𝑖𝑗 𝜙 𝜙 +𝑐 𝑇𝑖𝑗𝑘 𝜙 𝜙 𝜙 + SB,𝑊 𝐿 𝑖 𝑖 𝑀 𝑖 𝑗 𝑇 𝑖 𝑗 𝑘 h.c. (25) Both options are possible in SARAH,evenifthefirstone 𝑐 𝑐 𝑐 𝐿, 𝑀,and 𝑇 are real coefficients. All parameters are treated might often be preferred. by default in the most general way by taking them as complex In the case of an extended gauge sector also additional tensors of appropriate order and dimension. If identical fields gauge bosons are present. Depending on the quantum num- SARAH are involved in the same coupling, derives also the bers of the states which get a VEV these gauge bosons might symmetry properties for the parameter. mix with the SM ones. Also this mixing is fully supported SARAH As discussed below, can also handle to some by SARAH. There is no restriction if the additional gauge 󸀠 extent nonrenormalizable terms with four superfields in the bosons are ultralight (dark photons) or much heavier (𝑍 , 󸀠 superpotential: 𝑊 -bosons). 𝑖𝑗𝑘𝑙 𝑊 =𝑐 𝑊 𝜙̂ 𝜙̂ 𝜙̂ 𝜙̂ . (26) NR 𝑊 𝑖 𝑗 𝑘 𝑙 2.2.8. Mixing in Matter Sector. Mixing between gauge eigen- From the superpotential, all the 𝐹-terms states to new mass eigenstate appears not only in the gauge but also in the matter sector. In general the mixing is 󵄨 ̃ 󵄨2 induced via bilinear terms in the Lagrangian between gauge 2 󵄨𝜕𝑊󵄨 |𝐹| = ∑ 󵄨 󵄨 (27) 󵄨 𝜕𝜙 󵄨 eigenstates. These bilinear terms either can be a consequence 𝑖 󵄨 𝑖 󵄨 of gauge symmetry breaking or can correspond to bilinear Advances in High Energy Physics 7

𝐴 superpotential or soft-terms. In general, four kinds of bilinear 𝑆 (𝑟) is the Dynkin index of a superfield transforming terms can show up in the matter part of the Lagrangian: as representation 𝑟 with respect to the gauge group 𝐴. When evaluating the RGEs from the low scale to the high 𝑖𝑗 ∗ 1 𝑖𝑗 1 𝑖𝑗 0 0 𝑖𝑗 1 2 scale the contribution is positive; when running down, it L =−𝑚 𝜙𝑖 𝜙𝑗 − 𝑚𝑅𝜑𝑖𝜑𝑗 − 𝑚𝑀Ψ𝑖 Ψ𝑗 −𝑚𝐷Ψ𝑖 Ψ𝑗 . (33) 𝐶 2 2 is negative. Equations (36) assume that the mass splitting between the components of the chiral superfield integrated 𝜙 𝜑 Ψ𝑥 (𝑥=0,1,2) Here, , , are vectors whose components are outisnegligible.Thatisoftenagoodapproximationforvery 𝜙 𝜑 Ψ0 gauge eigenstates. are complex and are real scalars, , heavy states. Nevertheless, SARAH canalsotakeintoaccount Ψ Ψ 1,and 2 are Weyl spinors. The rotation of complex scalars the mass splitting among the components if necessary. 𝜙 to mass eigenstates 𝜙 happensviaaunitarymatrix𝑈 which Also higher-dimensional operators can be initialized diagonalizes the matrix 𝑚𝐶. For real scalars the rotation is which give rise to terms like (26).However,thoseareonly doneviaarealmatrix𝑍 which diagonalizes 𝑚𝑅: partially supported in SARAH.ThismeansthatonlytheRGEs arecalculatedforthesetermsandtheresultinginteractions diag † 𝜙=𝑈𝜙𝐶 𝑀 =𝑈𝑚𝐶𝑈 , between two fermions and two scalars are included in the (34) Lagrangian. The six scalar interactions are not taken into diag 𝑇 𝜑 = 𝑍𝜑𝑅 𝑀 =𝑍𝑚𝑅𝑍 . account. This approach is, for instance, sufficient to work with the Weinberg operator necessary for neutrino masses We have to distinguish for fermions if the bilinear terms are [238, 239]. symmetric or not. In the symmetric case the gauge eigenstates 𝑚 arerotatedtoMajoranafermions.Themassmatrix 𝑀 is 2.3. Checks of Implemented Models. After the initialization of then diagonalized by one unitary matrix. In the second case, amodel SARAH provides functions to check the (self-) con- Ψ Ψ two unitary matrices are needed to transform 1 and 2 sistency of this model. The function CheckModel performs differently. This results in Dirac fermions. Both matrices the following checks. together diagonalize the mass matrix 𝑚𝐷: What Causes the Particle Content Gauge Anomalies?Gauge 0 0 diag ∗ −1 Ψ =𝑁Ψ 𝑀𝑀 =𝑁 𝑚𝑀𝑁 , anomalies are caused by triangle diagrams with three external (35) gauge bosons and internal fermions [240]. The corresponding 1 1 2 2 diag ∗ −1 (𝑁) Ψ =𝑉Ψ, Ψ =𝑈Ψ 𝑀𝐷 =𝑈 𝑚𝐷𝑉 . conditions for all SU 𝐴 groups to be anomaly free are ∑ [𝑇𝑎 (𝜓 )𝑇𝑎 (𝜓 )𝑇𝑎 (𝜓 )] = 0. SARAH Tr 𝐴𝑟 𝑖 𝐴𝑟 𝑖 𝐴𝑟 𝑖 (37) There is no restriction in of how many states do 𝑖 mix. The most extreme case is the one with spontaneous 𝑎 charge, colour, and CP violation where all fermions, scalars, Again, 𝑇𝐴𝑟(𝜓𝑖) are the generators for a fermion 𝜓𝑖 transform- and vector bosons mix among each other. This results in ing as irreducible representation 𝑟 under the gauge group ahugemassmatrixwhichwouldbederivedbySARAH. SU(𝑁)𝐴. The sum is taken over all chiral superfields. In Phenomenological more relevant models can still have a neu- the Abelian sector several conditions have to be fulfilled tralino sector mixing seven to ten states. That is done without depending on the number of 𝑈(1) gauge groups: SARAH any problem with . Information about the calculation 3 𝐴 3 SARAH 𝑈 (1) : ∑ (𝑄 ) =0, of the mass matrices in is given in Section 3.3. 𝐴 𝜓𝑖 𝑖

2 2.2.9. Superheavy Particles. Extensions of the MSSM can not 𝑈 (1) ×𝑈(1)2 : ∑𝑄𝐴 (𝑄𝐵 ) =0, 𝐴 𝐵 𝜓𝑖 𝜓𝑖 (38) only be present at the SUSY scale but also appear at much 𝑖 higher scales. These superheavy states have then only indirect 𝑈 (1) ×𝑈(1) ×𝑈(1) : ∑𝑄𝐴 𝑄𝐵 𝑄𝐶 =0. effects on the SUSY phenomenology compared to the MSSM: 𝐴 𝐵 𝐶 𝜓𝑖 𝜓𝑖 𝜓𝑖 they alter the RGE evolution and give a different prediction 𝑖 for the SUSY parameters. In addition, they can also induce The mixed condition involving Abelian and non-Abelian SARAH higher-dimensional operators which are important. groups is provides features to explore models with superheavy states: it 2 𝐴 𝑎 𝑎 is possible to change stepwise the set of RGEs which is used 𝑈 (1)𝐴 × SU (𝑁) : ∑𝑄 Tr [𝑇 (𝜓𝑖)𝑇 (𝜓𝑖)] = 0. 𝐵 𝜓𝑖 𝐵𝑟 𝐵𝑟 (39) to run the parameters numerically with SPheno. In addition, 𝑖 𝑀 the most important thresholds are included at the scale 𝑇 at Finally, conditions involving gravity G are which the fields of mass 𝑀 are integrated out. These are the 2 corrections to the gauge couplings and gaugino masses [237]: G ×𝑈(1)2 : ∑ (𝑄𝐴 ) =0, 𝐴 𝜓𝑖 𝑖 1 𝑀2 2 𝐴 𝐴 𝐵 𝑔𝐴 󳨀→ 𝑔 𝐴 (1 ± 𝑔 𝑆 (𝑟) ln ( )) , 2 𝐴 2 G ×𝑈(1)𝐴 ×𝑈(1)𝐵 : ∑𝑄𝜓 𝑄𝜓 =0, 16𝜋 𝑀𝑇 𝑖 𝑖 (40) (36) 𝑖 1 𝑀2 𝑀 󳨀→ 𝑀 (1 ± 𝑔2 𝑆𝐴 (𝑟) ( )) . G2 ×𝑈(1) : ∑𝑄𝐴 =0. 𝐴 𝐴 2 𝐴 ln 2 𝐴 𝜓𝑖 16𝜋 𝑀𝑇 𝑖 8 Advances in High Energy Physics

If one if these conditions is not fulfilled a warning is printed 3.1. Renormalization Group Equations. SARAH calculates the by SARAH.Ifsome𝑈(1) charges were defined as variable, the SUSY RGEs at the one- and two-loop level. In general, the conditions on these variables for anomaly cancellation are 𝛽-function of a parameter 𝑐 is parametrized by printed. 𝑑 1 1 𝑐≡𝛽 = 𝛽(1) + 𝛽(2). 𝑑𝑡 푐 16𝜋2 푐 (16𝜋2) 푐 (41) What Leads the Particle Content to the Witten (1) (2) Anomaly? SARAH checks that there is an even number 𝛽푐 and 𝛽푐 are the coefficients at one- and two-loop level. of SU(2) doublets. This is necessary for a model in order to For the gauge couplings the generic one-loop expression is befreeoftheWittenanomaly[241]. rather simple and reads (1) 3 𝛽푔 =𝑔퐴 (𝑆 (𝑅) −3𝐶(𝐺)) . (42) Are All Terms in the (Super)Potential in Agreement with Global 𝐴 and Local Symmetries?Asmentionedabove, SARAH does not 𝑆(𝑅) is the Dynkin index for the gauge group summed over forbid including terms in the superpotential which violate all chiral superfields charged under that group and 𝐶(𝐺) is global or gauge symmetries. However, it prints a warning if the Casimir of the adjoint representation of the group. The this happens. two-loopexpressionsaremorecomplicatedandareskipped here. They are, for instance, given in[242]. AreThereOtherTermsAllowedinthe(Super)Potentialby The starting points for the calculation of the RGEs for the SARAH Global and Local Symmetries? SARAH will print a list of superpotential terms in are the anomalous dimen- 𝛾 renormalizable terms which are allowed by all symmetries sions for all superfields. These can be also parametrized by but which have not been included in the model file. 1 (1) 1 (2) 𝛾̂ ̂ = 𝛾 + 𝛾 . 휙𝑖휙𝑗 2 휙̂ 휙̂ 2 휙̂ 휙̂ (43) 16𝜋 𝑖 𝑖 (16𝜋2) 𝑖 𝑗 Are All Unbroken Gauge Groups Respected? SARAH checks 𝑖, 𝑗 what gauge symmetries remain unbroken and if the defini- Iwanttostressagainthat are not generation indices tions of all rotations in the matter sector and of the Dirac but label the different fields. Generic formulas for the one- 𝛾(1) 𝛾(2) spinors are consistent with that. and two-loop coefficients and are given in [242]as well. SARAH includes the case of an anomalous dimension ̂ ̂ matrix with off-diagonal entries: that is, 𝜙푖 ≠ 𝜙푗.Thatis,for AreThereTermsintheLagrangianoftheMassEigenstates instance, necessary in models with vector-like quarks where Which Can Cause Additional Mixing between Fields?Ifin the superpotential reads the final Lagrangian bilinear terms between different matter ̂ ̂ ̂ ̂̂ eigenstates are present this means that not the entire mixing 𝑊⊃𝑌푢𝑢̂𝑞̂𝐻푢 +𝑌푈𝑈𝑞̂𝐻푢 +𝑀푈𝑈𝑈. (44) of states has been taken into account. SARAH checks if those 𝛾 terms are present and returns a warning showing the involved 푢̂푈̂ is not vanishing but receives already at one-loop contri- ∝𝑌𝑌 fields and the nonvanishing coefficients. butions 푢 푈. From the anomalous dimensions it is straightforward to get the 𝛽-functions of the superpotential terms: for a Are All Mass Matrices Irreducible?Ifmassmatricesareblock generic superpotential of the form equations (24) and (26) diagonal, a mixing has been assumed which is actually not 𝛽(푥) there. In that case SARAH will point this out. the coefficients are given by 𝛽(푥) ∼𝐿푎𝛾(푥), 퐿𝑖 푎휙̂ Are the Properties of All Particles and Parameters Defined 𝑖 Correctly? These are formal checks about the implementation 𝛽(푥) ∼𝑀푖푎𝛾(푥) +(𝑗←→𝑖), 푀𝑖𝑗 ̂ of a model. It is checked, for instance, if the number of PDGs 푎휙𝑗 fits the number of generations for each particle class, if LaTeX 𝛽(푥) ∼𝑌푖푗푎 𝛾(푥) + 𝑘←→𝑖 +(𝑘←→𝑗), names are defined for all particles and parameters, if the 𝑖𝑗𝑘 ̂ ( ) (45) 푌 푎휙𝑘 positions in a Les Houches spectrum file are defined for all parameters, and so forth. Not all of these warnings have to be 𝛽(푥) ∼𝑊푖푗푘푎 𝛾(푥) + (𝑙←→𝑖) +(𝑙←→𝑗) 푊𝑖𝑗𝑘𝑙 푎휙̂ addressed by the user, especially if he/she is not interested in 𝑙 theoutputwhichwouldfailbecauseofmissingdefinitions. + (𝑙←→𝑘) up to constant coefficients. In the soft-breaking sector SARAH includes also all standard terms of the form 3. Calculations and Output 1 1 −L =𝑡푖𝜙 + 𝐵푖푗 𝜙 𝜙 + 𝑇푖푗푘 𝜙 𝜙 𝜙 SB 푖 푖 푗 푖 푗 푘 SARAH can perform in its natural Mathematica environ- 2 3! ment many calculations for a model on the analytical level. 1 푖푗푘푙 1 2 푗 ∗푖 + 𝑄 𝜙 𝜙 𝜙 𝜙 + (𝑚 ) 𝜙 𝜙 (46) For an exhaustive numerical analysis usually one of the 4! 푖 푗 푘 푙 2 푖 푗 dedicated interfaces to other tools is the best approach. I give 1 in this section an overview about what SARAH calculates itself − 𝑀𝜆𝜆. and how that information is linked to other codes. 2 Advances in High Energy Physics 9

2 The generic expressions for 𝐵’s, 𝑇’s, 𝑚 ’s, and 𝑀’s up to two- some more results from the literature which became available looparegivenagainin[242]whichareusedby SARAH.The in the last few years. In the case of several 𝑈(1)’s, gauge- 𝛽-function for the linear soft-term 𝑡 is calculated using [243]. kinetic mixing can arise if the groups are not orthogonal. For the quartic soft-term 𝑄 the approach of [244]isadopted. Substitution rules to translate the results of [242]tothose In this approach 𝛾 is defined by including gauge-kinetic mixing were presented in [248]and have been implemented in SARAH2.Forinstance,toinclude 𝜕 𝜕 𝛾(푥) = (𝑀 𝑔2 −𝑇푙푚푛 ) 𝛾(푥) . gauge-kinetic mixing in the running of the gauge couplings 휙̂ 휙̂ 퐴 퐴 2 푙푚푛 휙̂ 휙̂ (47) 푖 푗 𝜕𝑔퐴 𝜕𝑌 푖 푗 and gaugino masses (42) and (52) canbeusedtogetherwith the substitutions: The 𝛽-functions for 𝑄 can then be expressed by 𝛾 and 𝛾: 𝑔3 𝑆 (𝑅) 󳨀→ 𝐺 ∑𝑉 𝑉푇, 퐴 휙̂ 휙̂ 𝛽(푥) =[𝑄푖푗푘푎 𝛾(푥) +2𝑊푖푗푘푎 𝛾(푥)]+(𝑙←→𝑖) 휙̂ 푄푖푗푘푙 푎휙̂ 푎휙̂ 푙 푙 (48) (53) 𝑔2 𝑀 𝑆 (𝑅) 󳨀→ 𝑀 ∑𝑉 𝑉푇 + ∑𝑉 𝑉푇𝑀. +(𝑙←→𝑗)+(𝑙←→𝑘) . 퐴 퐴 휙̂ 휙̂ 휙̂ 휙̂ 휙̂ 휙̂ In principle, the same approach can also be used for 𝐵 and 𝑇 Here, 𝐺 and 𝑀 are matrices carrying the gauge couplings termsaslongasnogaugesingletexistsinthemodel.Because 𝑈(1) SARAH and gaugino masses of all groups; see also Section 2.2, of this restriction, uses the more general expressions 𝑉 =𝐺푇𝑄 of [242]. and I introduced 휙̂ 휙̂. The sums are running over all ̂ The running of the Fayet-Iliopoulos 𝐷-term 𝜉 receives chiral superfields 𝜙. Also for all other terms involving gauge two contributions: couplings and gaugino masses appearing in the 𝛽-functions similar rules are presented in [248]whichareusedby SARAH. 𝛽(푥) (푥) 푔퐴 (푥) Furthermore, also the changes in the RGEs in the pres- 𝛽 = 𝜉퐴 +𝛽 . (49) 휉퐴 휉̂ 𝑔퐴 퐴 enceofDiracgauginomasstermsareknowntodayatthetwo- loop level; see [236]. SARAH makes use of [236]toobtain The first part is already fixed by the running of the gauge the 𝛽-functionsforthenewmassparametersaswellasto 𝛽(푥) include new contribution to the RGEs of tadpole terms in coupling of the Abelian group, the second part, 휉̂ ,isknown SARAH presence of Dirac gauginos. The 𝛽-functions of a Dirac mass even to three loops [245, 246]. has implemented the 휙휆̂ one- and two-loop results which are rather simple: terms 𝑚퐷 Ψ𝜆푖 arerelatedtotheanomalousdimensionofthe ̂ involved chiral superfield 𝜙,whosefermioniccomponentis 𝛽(1) =2𝑔 ∑ (𝑄퐴 𝑚2 )≡𝜎 , Ψ 휉̂ 퐴 휙푖 휙푖휙푖 1,퐴 , and to the running of the corresponding gauge coupling: 퐴 푖 (50) 𝛽푔 휙퐴̂ (1) 퐴 2 (1) 푎퐴 퐴 𝛽 휙퐴̂ =𝛾̂ 𝑚 + 𝑚 . (54) 𝛽 =−4𝑔퐴∑ (𝑄 𝑚 𝛾 )≡𝜎3,퐴 푚 휙푎 퐷 퐷 휉̂ 휙푖 휙푖휙푗 휙̂ 휙̂ 퐷 𝑔퐴 퐴 푖푗 푗 푖

𝜎 𝜎 𝛽 The tadpole term receives two new contributions from Fayet- 1 and 3 aretraceswhicharealsousedtoexpressthe - Iliopoulos terms discussed above and terms mimicking 𝐵 functions of the soft-scalar masses at one- and two-loop; see, insertions: for instance, [242]. (푥) (푥) (푥) (푥) Finally, the 𝛽-functions for the gaugino mass parameters 𝛽 =𝛽 +𝛽 +𝛽 . 푡,퐷퐺 푡 휉̂ 퐷 (55) are (푥) 𝑑 1 1 Thus, the only missing piece is 𝛽 which is also calculated ≡𝛽 = 𝛽(1) + 𝛽(2), 퐷 𝑑𝑡 푀 16𝜋2 푀 (16𝜋2) 푀 (51) by SARAH up to two-loop based on [236]. Finally, the set of SUSY RGEs is completed by using the 𝛽(푥) results of [249, 250] to get the gauge dependence in the where the expressions for 푀 are also given in [242, 243, 247]. 𝛽 𝛽(1) running of the VEVs. As a consequence, the -functions 푀 has actually a rather simple form similar to the one of the for the VEVs consist of two parts which are calculated gauge couplings. One finds independently by SARAH: (1) 2 𝛽 =2𝑔 (𝑆 (𝑅) −3𝐶(𝐺)) 𝑀 . (푥) 푆,(푥) 푆,(푥) 푀퐴 퐴 퐴 (52) 𝛽 =(𝛾 + 𝛾̂ ) V . V휙 휙푎 휙푎 푎 (56)

Therefore, the running of the gaugino masses is strongly 𝛾푆 𝜙 correlated with the one of the gauge couplings. Thus, for a is the anomalous dimension of the scalar which receives the VEV V휙. The gauge dependent parts which vanish in GUT model the hierarchy of the running gaugino masses is 푆 thesameastheoneforthegaugecouplings. Landau gauge are absorbed in 𝛾̂ . All details about this 푆,(푥) The expressions presented in the early works of[242, calculation and the generic results for 𝛾̂ are given in 243, 247] did actually not cover all possibilities and are not [249, 250]. sufficient for any possible SUSY models which can be imple- Iwanttomentionthat SARAH providesthesameaccu- mented in SARAH. Therefore, SARAH has implemented also racy also for the RGEs for a nonSUSY model by making 10 Advances in High Energy Physics

use of the generic results of [251–254]. These results are states which include Goldstone bosons also the 𝑅휉 dependent completed by [255] to cover gauge-kinetic mixing and again terms are included. by [249, 250] to include the gauge dependence of the running The mass matrices for fermions are calculated as of VEVs also in the non-SUSY case.

Output.TheRGEscalculatedbySARAH are outputted in −𝜕2L 𝑀퐹 = differentformats:(i)theyarewrittenintheinternal SARAH 푖푗 𝜕𝜓푥𝜓푦 (61) format in the output directory, (ii) they are included in 푖 푗 the LaTeX output in a much more readable format, (iii) they are exported into a format which can be used together 𝑥=𝑦=0 𝑥=1 𝑦=2 with NDSolve of Mathematica to solve the RGEs numer- with for Majorana fermions, and and ically within Mathematica, and (iv) they are exported for Dirac fermions. SARAH into Fortran code which is used by SPheno. calculates for all states which are rotated to mass eigenstates the mass matrices during the evaluation of a model. In addition, it checks if there are also particles where 3.2. Tadpole Equations. During the evaluation of a gauge and mass eigenstates are identical. In that case, it model, SARAH calculates “on the fly” all minimum calculates also the expressions for the masses of these states. conditions of the tree-level potential, the so-called tadpole equations.InthecaseofnoCPviolation,inwhichcomplex Output. The tree-level masses and mass matrices are also scalars are decomposed as exported to LaTeX files as well as to Fortran code 1 for SPheno. In addition, they are used in the Vevacious 𝑆 󳨀→ (V +𝜙 +𝑖𝜎) , 푖 √2 푖 푖 푖 (57) output to enable the calculation of the one-loop effective potential. The mass matrices can also be exported to the expressions the CalcHep model files if the user wants to calculate the CalcHep 𝜕𝑉 masses internally with instead of using them as 0= ≡𝑇푖 (58) input. 𝜕𝜙푖 are calculated. These are equivalent to 𝜕𝑉/𝜕V푖.Formodels 3.4. Vertices. Vertices are not automatically calculated dur- with CP violation in the Higgs sector, that is, either where ing the initialization of a model like it is done for mass complex phases appear between the real scalars or where matrices and tadpole equations. However, the calculation theVEVshaveanimaginarypart,SARAH calculates the canbestartedveryeasily.Ingeneral, SARAH is optimized minimum conditions with respect to the CP even and CP odd for the extraction of three- and four-point interactions components: with renormalizable operators. This means that usually only 𝜕𝑉 the following generic interactions are taken into account 0= ≡𝑇 , 휙푖 𝜕𝜙푖 in the calculations: interactions of two fermions or two (59) ghosts with one scalar or vector bosons (FFS, FFV, GGS, 𝜕𝑉 GGV 0= ≡𝑇 . and ), interactions of three or four scalars or vector 휎푖 SSS SSSS VVV VVVV 𝜕𝜎푖 bosons ( , , ,and ), and interactions of two scalars with one or two vector bosons (SSV and SSVV) 𝑇 ={𝑇 ,𝑇 } Thesetofalltadpoleequationsisinthiscase 푖 휙푖 휎푖 . or two vector bosons with one scalar (SVV). In this context, vertices not involving fermions are calcu- Output. The tadpole equations are exported into LaTeX lated by format as well as in Fortran code used by SPheno.This ensures that all parameter points evaluated by SPheno are at least sitting at a local minimum of the scalar potential. 𝜕3L Moreover,thetadpoleequationsareincludedinthemodel 𝑉(𝜂푎,𝜂푏,𝜂푐)≡𝑖 =𝐶Γ, files for Vevacious whichisusedtofindallpossible 𝜕𝜂푎𝜕𝜂푏𝜕𝜂푐 (62) solutions of them with respect to the different VEVs. 𝜕4L 𝑉(𝜂 ,𝜂 ,𝜂 ,𝜂 )≡𝑖 =𝐶Γ. 푎 푏 푐 푑 𝜕𝜂 𝜕𝜂 𝜕𝜂 𝜕𝜂 3.3. Masses and Mass Matrices. SARAH uses the definition 푎 푏 푐 푑 oftherotationsdefinedinthemodelfiletocalculatethe mass matrices for particles which mix. The mass matrices for 𝜂 scalars are calculated by Here, 푖 areeitherscalars,vectorbosons,orghosts.Theresults are expressed by a coefficient 𝐶 which is a Lorentz invariant −𝜕2L Γ 𝛾 𝑝 𝑔휇] 𝑀푆 = , and a Lorentz factor which involves 휇, 휇,or .Vertices 푖푗 ∗ (60) for Dirac fermions are first expressed in terms of Weyl 𝜕𝜙푖𝜕𝜙푗 fermions. The two vertices are then calculated separately. 1 2∗ where 𝜙 canbeeitherrealorcomplex:thatis,theresulting Taking two Dirac fermions 𝐹푎 =(Ψ푎 ,Ψ푎 ) and 𝐹푏 = 푆 1 2∗ 𝑀 corresponds to 𝑚퐶 or 𝑚푅 of (33). In the mass matrices of (Ψ푏 ,Ψ푏 ) and distinguishing the two cases for fermion-vector Advances in High Energy Physics 11 andfermion-scalarcouplings,theverticesarecalculatedand Figure 1 are included in SARAH. Each generic amplitude is expressed by parametrized by

M = Symmetry × Colour × Couplings 𝑉(𝐹 ,𝐹 ,𝑉)={𝑉(Ψ1∗,Ψ1,𝑉),𝑉(Ψ2,Ψ2∗,𝑉)} 푎 푏 푐 푎 푏 푐 푎 푏 푐 (64) × Loop-Function. 퐿 푅 ≡{𝐶 𝛾휇𝑃퐿,𝐶 𝛾휇𝑃푅}, (63) Here “Symmetry” and “Colour” are real factors. The loop 2 1 1∗ 2∗ 𝑉(𝐹푎,𝐹푏,𝑆푐)={𝑉(Ψ푎 ,Ψ푏 ,𝑆푐),𝑉(Ψ푎 ,Ψ푏 ,𝑉푐)} functions are expressed by standard Passarino-Veltman inte- grals 𝐴0 and 𝐵0 andsomerelatedfunctions:𝐵1, 𝐵22, 𝐹0, 𝐺0, 퐿 푅 𝐻 𝐵 ≡{𝐶 𝑃퐿,𝐶 𝑃푅}. 0,and 22 asdefinedin[256]. As first step to get the loop corrections, SARAH generates 𝑃 all possible Feynman diagrams with all field combinations Here, the polarization operators 퐿,푅 are used. possible in the considered model. The second step is to match The user can either calculate specific vertices for a SARAH these diagrams to the generic expressions. All calculations are particular set of external states or call functions where done without any assumption and always the most general derives all existing interactions from the Lagrangian. The first case is taken. For instance, the generic expression for a purely option might be useful to check the exact structure of single scalar contribution to the scalar self-energy reads vertices, while the second one is needed to get all vertices to write model files for other tools. M =𝑐 ×𝑐 × |𝑐|2 𝐵 (𝑝2,𝑚2 ,𝑚2 ). 푆푆푆 푆 퐹 0 푆1 푆2 (65) Output. The vertices are exported into many different 𝜙+ = ((𝐻−)∗,𝐻+) SARAH In the case of an external charged Higgs 푑 푢 formats. They are saved in the internal format together with down- and up-squarks in the loop the correc- andtheycanbewrittentoLaTeXfiles.Themainpur- tiontothechargedHiggsmassmatrixbecomes pose is the export into formats which can be used SARAH FeynArts 6 with other tools. writes model files for , 󵄨 󵄨2 󵄨 + ̃∗ 󵄨 2 2 2 WHIZARD/OMEGA,and CalcHep/CompHep as well as in M + ̃∗ =3×∑ ∑ 󵄨𝑐(𝜙푎 𝑢̃푖𝑑푗 )󵄨 𝐵0 (𝑝 ,𝑚푢̃ ,𝑚̃ ). 휙푎 푢̃푑 󵄨 󵄨 푖 푑푗 (66) the UFO format. The UFO format is supported by MadGraph, 푖=1 푗=1 Herwigg+, and Sherpa.Thus,bytheoutputofthevertices + ̃ ̃∗ into these different formats, SARAH provides an implemen- 𝑐(𝜙푎 𝑢푖𝑑푗 ) is the charged Higgs-sdown-sup vertex where the tation of a given model in a wide range of HEP tools. rotation matrix of the charged Higgs is replaced by the iden- In addition, SARAH generates also Fortran code to imple- tity matrix to get the projection on the gauge eigenstates. One ment all vertices in SPheno. can see that all possible combinations of internal generations are included: that is, also effects like flavour mixing are 2 completely covered. Also the entire 𝑝 dependence is kept. 3.5. One- and Two-Loop Corrections to Tadpoles and Self-Energies Output.Theone-loopexpressionsaresavedinthe SARAH Mathematica SARAH internal format and can be included in the 3.5.1. One-Loop Corrections. calculates the analytical LaTeX output. In addition, all self-energies and one-loop expressions for the one-loop corrections to the tadpoles and tadpoles are exported into Fortran code for SPheno.This the one-loop self-energies for all particles. For states which enables SPheno to calculate the loop-corrected masses for all are a mixture of several gauge eigenstates, the self-energy particlesasdiscussedbelow. matrices are calculated. For doing that, SARAH is working with gauge eigenstates as external particles but uses mass eigenstatesintheloop.Thecalculationsareperformedin 3.5.2. Two-Loop Corrections. It is even possible to go beyond one-loop with SARAH and to calculate two-loop contribu- DR-schemeusing’tHooftgauge.Inthecaseofnon-SUSY SARAH tions to the self-energies of real scalars. There are two models switches to MS-scheme. This approach is a equivalent approaches implemented in the SPheno interface generalization of the procedure applied in [256]totheMSSM. of SARAH to perform these calculations: an effective potential In this context, the following results are obtained: approach and a diagrammatic approach with vanishing external momenta. Because of the very complicated form of the (i) the self-energies Π of scalars and scalar mass matrices, results there is no output of the corresponding expressions in Mathematica 퐿 푅 푆 the or LaTeX format but the results are just (ii) the self-energies Σ , Σ ,andΣ for fermions and included in the Fortran code for numerical evaluation. I fermion mass matrices, will discuss both calculations a bit more. Π푇 (iii) the transversal self-energy of massive vector Effective Potential Calculation. The first calculation of the two- bosons. loop self-energies is based on the effective potential approach. The starting points of the calculation are the generic results The approach to calculate the loop corrections is as follows: for the two-loop effective potential given in16 [ ]. These all possible generic diagrams at the one-loop level shown in have been translated to four component notations and were 12 Advances in High Energy Physics

Figure 1: Generic diagrams included by SARAH to calculate one-loop tadpoles and self-energies.

The Need for Both Calculations. Since both calculations are based on a completely independent implementation and use a different approach they are very useful to perform cross- checks. For the MSSM and NMSSM both calculations reproduce exactly the results obtained by widely used routines based on [259–264]. However, for nonminimal SUSY models there are no references available to compare with. Thus, the only possibility to cross check the results is within SPheno and comparing of the two different methods. Figure 2: Generic diagrams included to calculate the effective potential at the two-loop level. These are the diagrams which do not Output. The two-loop expressions for the effective poten- vanish in the gaugeless limit. tial, the tadpoles, and the self-energies are just exported to SPheno at the moment to calculate the loop-corrected mass spectrum. implemented in SARAH.When SARAH creates the SPheno output it writes down the amplitude for all two-loop diagrams 3.6. Loop-Corrected Mass Spectrum. The information about which do not vanish in the gaugeless limit. This limit means the one- and two-loop corrections to the one- and two-point that contributions from broken gauge groups are ignored. The functions introduced in Section 3.5 canbeusedtocalculate remaining generic diagrams which are included are shown the loop-corrected mass spectrum. Sticking to approach of in Figure 2.Usingthesediagramsincludes,forinstance,all [256], the renormalized mass matrices (or masses) are related two-loop contributions which are also taken into account in to the tree-level mass matrices (or masses) and the self- the MSSM. To get the values for the two-loop self-energies energies as follows. and two-loop tadpoles, the derivatives of the potential with respect to the VEVs are taken numerically as proposed in 3.6.1. Loop-Corrected Masses [257]. There are two possibilities for this derivation imple- SARAH SPheno mented in / : (i) a fully numerical procedure Real Scalars.Forarealscalar𝜙, the one-loop, and in some which takes the derivative of the full effective potential with cases also two-loop, self-energies are calculated by SPheno. respect to the VEVs and (ii) a semianalytical derivation 2,(퐿) The loop-corrected mass matrix squared 𝑚 is related which takes analytically the derivative of the loop functions 휙 𝑚2,(푇) with respect to involved masses but derives the masses and to the tree-level mass matrix squared 휙 and the self- coupling numerically with respect to the VEVs. More details energies via about both methods and the numerical differences are given 𝑚2,(퐿) (𝑝2)=𝑚2,(푇) − R (Π(1퐿) (𝑝2)) in [147]. 휙 휙 휙 (67) (2퐿) Diagrammatic Calculation. A fully diagrammatic calcu- − R (Π휙 (0)). lation for two-loop contributions to scalar self-energies SARAH SPheno 𝑀2 (𝑝2) with – became available with [148]. In this The one-shell condition for the eigenvalue 휙푖 of the setup a set of generic expressions first derived in148 [ ]isused. 2,(퐿) 2 loop-corrected mass matrix 𝑚휙 (𝑝 ) reads All two-loop diagrams shown in Figure 3 areincludedinthe 2 limit 𝑝 =0. These are again the diagrams which do not 2 2 2 Det [𝑝푖 1 −𝑀휙 (𝑝 )] = 0. (68) vanish in general in the gaugeless limit. The results of148 [ ] 푖 have the advantage that the expressions which are derived 𝑀2 (𝑝2 = Astablesolutionof(68) for each eigenvalue 휙푖 from the effective potential are much simpler than taking the 2 2 𝑀휙 ) is usually just found via an iterative procedure. In this limit 𝑝 →0in other two-loop functions available in the 푖 𝑚2,(푇) literature [258]. The diagrammatic method gives completely approach one has to be careful how 휙 is defined: this is equivalent results to the effective potential calculation but is the tree-level mass matrix where the parameters are taken usually numerically more robust. attheminimumoftheeffectivepotentialevaluatedatthe Advances in High Energy Physics 13

Figure 3: Generic diagrams included to calculate the two two-loop corrections to tadpoles and scalars. These are the diagrams which do not vanish in the gaugeless.

(1퐿) † (1퐿) same loop level at which the self-energies are known. The Here, the eigenvalues of (𝑚Ψ ) 𝑚Ψ are used in (68) to get 𝑀2 (𝑝2 = the pole masses. physical masses are associated with the eigenvalues 휙푖 𝑀2 ) 휙푖 . In general, for each eigenvalue the rotation matrix is 2 slightly different because of the 𝑝 dependence of the self- 3.6.2. Renormalization Procedure. I have explained so far SPheno energies. The convention by SARAH and SPheno is that the how does calculate the one- and two-loop self- rotation matrix of the lightest eigenvalue is used in all further energies and how these are related to the loop-corrected calculations and the output. masses. Now, it is time to put this in a more global picture by describing step-by-step the entire renormalization procedure Complex Scalars. For a complex scalar 𝜂 the one-loop- that SPheno uses. correctedmassmatrixsquaredisrelatedtothetree-levelmass and the one-loop self-energy via (1) Everything starts with calculating the running parameters at the renormalization scale 𝑄=𝑀SUSY from 2,(1퐿) 2 (푇) (1퐿) 2 𝑚휂 (𝑝 )=𝑚휂 −Π휂 (𝑝푖 ). (69) the given input parameters. Either the parameters canbegivendirectlyat𝑀SUSY as input or they are The same on-shell condition (68) as for real scalars is used. fixed by some GUT conditions and a RGE running is performed. 𝑀SUSY itself either can be a fixed value or Vector Bosons. For vector bosons we have similar simple canbedynamicallychosen.Itiscommontochoose expressions as for scalar. The one-loop masses of real or the geometric mean of the stop masses because this complex vector bosons 𝑉 are given by usually minimizes the scale dependence of the Higgs 2,(1퐿) 2,(푇) 푇,(1퐿) 2 𝑚푉 =𝑚푉 − R (Π푉 (𝑝 )) . (70) mass prediction. Majorana Fermions. The one-loop mass matrix of a Majorana (2) Not all parameters are fixed by the input but (푇) some parameters are kept free. These parameters are fermion 𝜒 is related to the tree-level mass matrix 𝑚 and the 휒 arranged in a way that all further calculations are done different parts of the self-energies by at the minimum of the potential. For this purpose 1 𝑇 𝑚(1퐿) (𝑝2)=𝑚(푇) − [Σ휒 (𝑝2)+Σ휒,푇 (𝑝2) the tadpole equations 푖 aresolvedattree-levelwith 휒 휒 2 푆 푆 respect to these free parameters. 휒,푇 2 휒 2 (푇) (3)Assoonasallrunningparametersareknownatthe +(Σ퐿 (𝑝 )+Σ푅 (𝑝 )) 𝑚휒 (71) SUSY scale, they are used to calculate the tree-level (푇) 휒,푇 2 휒 2 mass spectrum. +𝑚휒 (Σ푅 (𝑝 )+Σ퐿 (𝑝 ))] . (4) The tree-level masses are used to calculate the self- Note that (𝑇) is used to assign tree-level values while 𝑇 푇 (푝),2 (푝) energies of the 𝑍-boson, Π (𝑀 ),where𝑀 is denotes a transposition. Equation (68) canalsobeusedfor 푍 푍 푍 2,(1퐿) (1퐿)∗ (1퐿) thepolemass. fermions by taking the eigenvalues of 𝑚휒 =𝑚휒 𝑚휒 . (푝) 푇 (푝),2 (5) 𝑀푍 , Π푍(𝑀푍 ) areusedtogetthetree-levelvalue Dirac Fermions. For a Dirac fermion Ψ one obtains the one- of the electroweak VEV V. V and the running value loop-corrected mass matrix via of tan 𝛽 are used to get tree-level VEVs V푢, V푑.Note 𝑚(1퐿) (𝑝2)=𝑚(푇) −Σ+ (𝑝2)−Σ+ (𝑝2)𝑚(푇) that in this step it is assumed that always two Higgs Ψ Ψ 푆 푅 Ψ doublets are present in SUSY models which give mass (72) −𝑚(푇)Σ+ (𝑝2). to up- and down-quark as well as leptons and gauge Ψ 퐿 bosons. 14 Advances in High Energy Physics

𝛼 2 𝑚 (6) Now, all tree-level parameters are known and the tree- SM 푆 푡 Δ𝛼푆 (𝜇) = (− log ), level masses and rotation matrices are recalculated 2𝜋 3 𝜇 using the obtained values. 𝛼 1 𝑚 Δ𝛼NP (𝜇) = 푆 ( − ∑𝑐 푖 ). (7) Tree-level masses, rotation matrices, and parameter 푆 2𝜋 2 푖 log 𝜇 are used to get all vertices at tree-level. The vertices 푖 and masses are then plugged in the expressions for (75) the one- and two-loop corrections to the tadpoles (푥) 𝛿𝑡푖 (𝑥=1,2). The conditions to work at the minimum of the effective potential are The sum runs over all particles 𝑖 which are not present in the SMandwhichareeitherchargedorcoloured.Thecoefficient (1) (2) 𝑐 𝑇푖 +𝛿𝑡푖 +𝛿𝑡푖 ≡0. (73) 푖 depends on the charge, respectively, colour representation, the generic type of the particle (scalar, fermion, and vector), and the degrees of freedom of the particle (real/complex These equations are again solved for the same set of boson, respectively, Majorana/Dirac fermion). parameters as at tree-level. (2) The next step is the calculation of the running ΘDR V (8) The self-energies for all particles are calculated at the Weinberg angle sin and electroweak VEV . For that the 𝛿𝑀2 𝛿𝑀2 𝑍 𝑊 highest available loop level as explained above. Note, one-loop corrections 푍 and 푊 to the -mass and - these calculations involve purely tree-level parame- mass are needed. And an iterative procedure is applied with DR SM ters but not the ones obtained from (73). Θ푊 =Θ푊 in the first iteration together with (9) Equations (67)–(72) areusedtogettheloop- corrected mass matrices for all particle. Now, the parameters coming from loop-corrected tadpoles are (1 − 2ΘDR) 2ΘDR 2 2 2 sin 푊 sin 푊 used to express the tree-level mass matrices. All V =(𝑀푍 +𝛿𝑀푍) , calculations are iterated until the on-shell condition 𝜋𝛼DR is satisfied for all masses. (76) DR 2 1 1 𝜋𝛼 sin ΘDR = − √ − . 푊 2 4 √2𝑀2 𝐺 (1 − 𝛿 ) 3.6.3. Thresholds. So far, I have not mentioned another 푍 퐹 푟 subtlety: in general, the running SM parameters depend on the SUSY masses. The reason is the thresholds used to match the running parameters to the measured ones. These 𝐺 𝛿 thresholds change, when the mass spectrum changes. There- Here, 퐹 is the Fermi constant and 푟 is defined by fore, the above procedure is iterated until the entire loop- corrected mass spectrum has converged. The calculation of the thresholds is also dynamically adjusted by SARAH to 𝛿𝑀2 𝛿𝑀2 𝛿 =𝜌 푊 − 푍 +𝛿 , include all new physics contributions. The general procedure 푟 2 2 푉퐵 (77) 𝑀푊 𝑀푍 to obtain the running gauge and Yukawa at 𝑀푍 is as follows. DR DR (1) The first step is the calculation of 𝛼 (𝑀푍), 𝛼푆 (𝑀푍) via where 𝛿푉퐵 are the corrections to the muon decay 𝜇→𝑒]푖]푗 𝛼(5),MS (𝑀 ) which are calculated at one-loop as well. The 𝜌 parameter is DR 푍 𝛼 (𝑀푍)= , calculated also at full one-loop and the known two-loop SM 1−Δ𝛼SM (𝑀 )−Δ𝛼NP (𝑀 ) 푍 푍 corrections are added. (74) (5),MS (3) With the obtained information, the running gauge 𝛼 (𝑀푍) 𝛼DR (𝑀 )= 푆 . couplings at 𝑀푍 are given by 푆 푍 SM NP 1−Δ𝛼푆 (𝑀푍)−Δ𝛼푆 (𝑀푍)

𝛼(5),MS 𝛼(5),MS Here, 푆 and are taken as input and receive √4𝜋𝛼DR (𝑀 ) corrections from the top loops as well as form new physics DR 푍 𝑔1 (𝑀푍)= , DR (NP): cos 𝜃푊 (𝑀푍) 𝛼 1 16 𝑚 SM 푡 √4𝜋𝛼DR (𝑀 ) (78) Δ𝛼 (𝜇) = ( − log ), DR 푍 2𝜋 3 9 𝜇 𝑔2 (𝑀푍)= , DR sin 𝜃푊 (𝑀푍) 𝛼 𝑚 NP 푖 Δ𝛼 (𝜇) = (−∑𝑐푖 log ), DR √ DR 𝑔3 (𝑀푍)= 4𝜋𝛼푆 (𝑀푍). 2𝜋 푖 𝜇 Advances in High Energy Physics 15

(4) The running Yukawa couplings are also calculated in 3.7. Decays and Branching Ratios. The calculation of decays an iterative way. The starting points are the running fermion widthsandbranchingratioscanbedonebyusingtheinter- masses in DR obtained from the pole masses given as input: face between SARAH and SPheno. SPheno modules created by SARAH calculate all two-body decays for SUSY and 3 DR,SM DR,2 DR,2 Higgs states as well as for additional gauge bosons. In 𝑚푙,휇,휏 =𝑚푙,휇,휏 (1 − (𝑔1 +𝑔2 )) , 128𝜋2 addition, the three-body decays of a fermion into three other fermions and of a scalar into another scalar and two fermions 𝛼DR 23𝛼DR,2 3 DR,SM 푆 푆 DR,2 are included. 𝑚 =𝑚푑,푠,푏 (1 − − + 𝑔 푑,푠,푏 3𝜋 72𝜋2 128𝜋2 2 In the Higgs sector, possible decays into two SUSY particles, leptons, and massive gauge bosons are calculated 13 at tree-level. For two quarks in the final state the dominant − 𝑔DR,2), 1152𝜋2 1 QCDcorrectionsduetogluonsareincluded[267]. The loop induced decays into two photons and gluons are fully calculated at leading-order (LO) with the dominant next-to- 𝛼DR 23𝛼DR,2 3 𝑚DR,SM =𝑚 (1 − 푆 − 푆 + 𝑔DR,2 (79) leading-order corrections known from the MSSM. For the 푢,푐 푢,푐 3𝜋 72𝜋2 128𝜋2 2 LO contributions all charged and coloured states in the given model are included in the loop: that is, new contributions rising in a model beyond the MSSM are fully covered at one- 7 ,2 − 𝑔DR ), loop. In addition, in the Higgs decays also final states with 1152𝜋2 1 ∗ ∗ off-shell gauge bosons (𝑍𝑍 , 𝑊𝑊 )areincluded.Theonly missing piece is the 𝛾𝑍 channel. The corresponding loops are DR,SM SARAH 𝑚푡 =𝑚푡 [1 not yet derived by and the partial width is set to zero. In contrast to other spectrum generators, SPheno mod- 1 ules by SARAH perform a RGE running to the mass scale of + (Δ𝑚(1),푞푐푑 +Δ𝑚(2),푞푐푑 +Δ𝑚(1),ew)] 16𝜋2 푡 푡 푡 thedecayingparticle.Thisshouldgiveamoreaccuratepre- diction for the decay width and branching ratios. However, with the user can also turn off this running and use always the parameters as calculated at 𝑀 inalldecaysasthisisdone 16𝜋𝛼DR 𝑀2 SUSY Δ𝑚(1),푞푐푑 =− 푆 (5 + 3 푍 ), by other codes. 푡 log 2 3 𝑚푡 Output. All necessary routines to calculate the two- and three- 64𝜋2𝛼DR,2 1 2011 2 body decays are included by default in the Fortran output Δ𝑚(2),푞푐푑 =− 푆 ( + + ln 푡 3 24 384𝜋2 12 for SPheno. (80) 2 𝜁 (3) 123 𝑀2 33 𝑀2 − + 푍 + ( 푍 ) ), 3.8. Higgs Coupling Ratios. With the discovery of the Higgs 2 2 log 2 2 log 2 8𝜋 32𝜋 𝑚푡 32𝜋 𝑚푡 boson at the LHC and the precise measurements of its mass and couplings to other particles, a new era of high energy 4 𝑀2 Δ𝑚(1),ew =− 𝑔DR,2 2ΘDR (5 + 3 푍 ). physics has started. Today, many SUSY models not only 푡 2 sin 푊 log 2 9 𝑚푡 have been confronted with the exclusion limits from direct searches, but also have to reproduce the Higgs properties The two-loop parts are taken from [265, 266]. The masses correctly. The agreement with respect to the mass can be arematchedtotheeigenvaluesoftheloop-correctedfermion easily read off a spectrum file. For the rates this is usually not mass matrices calculated as so easy. One can parametrize how “SM-like” the couplings of (1퐿) 2 (푇) ̃+ 2 ̃+ 2 (푇) a particular scalar are by considering the ratio 𝑚푓 (𝑝푖 )=𝑚푓 − Σ푆 (𝑝푖 )−Σ푅 (𝑝푖 )𝑚푓 (81) (푇) ̃+ 2 2 −𝑚푓 Σ퐿 (𝑝푖 ). SUSY 𝑐휙푋푌 𝑟휙푋푌 =( ) . (82) 𝑐SM Here, the pure QCD and QED corrections are dropped in ℎ푋푌 the self-energies Σ̃. Inverting this relation to get the running (푇) DR DR DR tree-level mass matrix 𝑚 leads to 𝑌푑 , 𝑌푢 ,and𝑌푒 . 푓 𝑐SUSY 𝜙 Since the self-energies depend also on the Yukawa matrices, Here, 휙푋푌 isthecalculatedcouplingbetweenascalar and 𝑋 𝑌 this calculation has to be iterated until a stable point is two SM particles and for a particular parameter point in a particular model. This coupling is normalized to the SM reached. Optionally, also the constraint that the CKM matrix 휙푋푌 is reproduced can be included in the matching. expectation for the same interaction. Nowadays, all 𝑟 are constrained to be rather close to 1 if 𝜙 should be associated Output. The calculation of the loop-corrected mass spectrum with the SM Higgs. SARAH uses the information which is and the thresholds is included in the SPheno output. already available from the calculation of the decays to obtain 16 Advances in High Energy Physics

휙푋푌 also values for 𝑟 .Ofcourse,alsoherethe𝛾𝑍 channel is hard-coded in SARAH but is provided by external files. This 휙훾푍 missing and 𝑟 is therefore put always to 0. makes it possible for the user to extent the list of flavour observables when necessary. The FlavorKit is an autom- Output. All necessary routines to calculate the Higgs cou- atization of the procedure presented in [268]toimplement Fortran 0 pling ratios are included by default in the output 𝐵푠,푑 →𝑙𝑙 in SARAH and SPheno. Users interested in the for SPheno. internal calculation might take a look at these two references. Also some other observables are calculated by the 3.9. Flavour and Precision Observables. Constraints for new combination SARAH–SPheno which are measured with high physics scenarios come not only from direct searches and precision: electric dipole moments (EDMs) and anomalous the Higgs mass observation but also from the measurement magnetic moments of leptons ((𝑔 − 2)푙)and𝛿𝜌 of processes which happen only very rarely in the SM 푇 2 푇 2 and/or which are known to have a very high accuracy. These Π푍푍 (𝑀푍) Π푊푊 (𝑀푊) 𝛿𝜌 = − . (83) areinparticularflavourviolationobservables.Whenusing 𝑀2 𝑀2 the SPheno output of SARAH, routines for the calculation of 푍 푊 many quark and lepton flavour (QFV and LFV) observables Output. When generating SPheno code with SARAH,the are generated. above-listed flavour and precision observables are included in the Fortran code. In addition, SARAH writes also LaTeX Lepton Flavour Violation. The radiative decays of a lepton into files with all contribution to the Wilson coefficients fromany ( (𝑙 →𝑙𝛾)) another lepton and a photon Br 푖 푗 and the purely possible diagram. leptonic three-body decays of leptons are included (Br(𝑙 → 耠 3𝑙 )).Alsoflavourviolatingdecaysofthe𝑍-boson (Br(𝑍 → 耠 3.10. Fine-Tuning. A measure for the electroweak fine-tuning 𝑙𝑙 )) are tested by SARAH/SPheno.Moreover,therearealso was introduced in [269, 270] semihadronic observables in the output: 𝜇-𝑒 conversion in ( (𝜇 − 𝑒, 𝑁)) 𝑁 Δ ≡ 𝐴 [Δ ], nuclei CR , where the considered nuclei are ( FT max bs 훼 =Al,Ti,Sr,Sb,Au,andPb),aswellasdecaysof𝜏’s into (𝜏 → 𝑙 + 𝑃) (𝑃=𝜋,𝜂,𝜂耠) 𝜕 𝑀2 𝛼 𝜕𝑀2 (84) pseudoscalars, Br with . Δ ≡ ln 푍 = 푍 . 훼 2 𝜕 ln 𝛼 𝑀푍 𝜕𝛼 Quark Flavour Violation.Theradiative𝐵-decay Br(𝐵 → −1 𝑋푠𝛾) and a set of 𝐵-decays stemming from four-fermion 𝛼 Δ 0 is a set of independent parameters. 훼 gives an estimate operators are calculated: Br(𝐵푠,푑 →𝑙𝑙),Br(𝐵 → 𝑠𝑙𝑙), of the accuracy to which the parameter 𝛼 must be tuned to Br(𝐵 → 𝑞]]),andBr(𝐵 → 𝐾𝜇𝜇).AlsoKoandecaysare + + get the correct electroweak breaking scale [271]. Using this considered (Br(𝐾 →] 𝜇 ),Br(𝐾 →𝜋]]),andBr(𝐾퐿 → definition the fine-tuning of a given model depends on the 𝜋0]])) Δ𝑀 Δ𝑀 𝜖 as well as CP observables ( 퐵푠,퐵푑 , 퐾,and 퐾). choice of what parameters are considered as fundamental and Finally, some decays which take place already at tree-level are at which scale they are defined. The approach by SARAH is included: namely, Br(𝐵 → 𝑙]) and Br(𝐷푠 →𝑙]). that it takes by default the scale at which the SUSY breaking The approach in SARAH to generate the routines to parameters are set. This corresponds to models where SUSY calculate all these observables is similar to the approach is broken by gravity to the scale of grand unification (GUT used for loop calculations needed for radiative corrections scale), while for models with gauge mediated SUSY breaking to the masses: generic formulas for all possible Feynman (GMSB) the messenger scale would be used. For simplicity, 3 𝑀 𝛼 diagrams which contribute to the Wilson coefficients of Icallboth Boundary.Thechoiceofthesetofparameters widely used dimension 5 and 6 operators are implemented is made by user. Usually, one uses in scenarios motivated by in SARAH. I show in Figures 4 and 5 only the topologies which supergravity the universal scalar and gaugino masses (𝑚0, are considered because these are already many. For each 𝑀1/2) as well as the parameters relating to the superpotential topology the amplitudes with all possible generic insertions terms and the corresponding soft-breaking terms (𝐵, 𝐴)to areincluded.Today,theseimplementationsaremainlybased calculate the fine-tuning. However, since also these parame- on the FlavorKit functionality discussed below. SARAH ters are related in specific models for SUSY breaking, it might generates all possible one-loop diagrams and uses the generic be necessary to consider even more fundamental parameters expression to get their amplitudes. In this context, not only like the gravitino mass 𝑚3/2. In addition, also the fine-tuning allpossibleparticlesintheloopareincludedbutalsoall with respect to the superpotential parameters themselves as different propagators for penguin diagrams are considered. well as to the strong coupling 𝛼푆 might be included because Thus, not only photonic penguins which are often considered theycanevensupersedethefine-tuninginthesoft-SUSY to be dominant in many processes are taken into account, but breaking sector. 耠 also all Higgs 𝑍 and—if existing—𝑍 penguins are generated. To calculate the fine-tuning in practice, an iteration of 𝑀 𝑀 After the calculation of the Wilson coefficients, these arethen the RGEs between SUSY and Boundary happens using the combined to calculate the observables. This can easily be done full two-loop RGEs. In each iteration one of the fundamental by using expressions from the literature which are usually parameters is slightly varied and the running parameters at 𝑀 model independent. SUSY are calculated. These parameters are used to solve the With the development of the FlavorKit [146]interface tadpoleequationsnumericallywithrespecttoallVEVsandto all information to calculate flavour observables is no longer recalculate the 𝑍-bosonmass.Togiveanevenmoreaccurate Advances in High Energy Physics 17

F1 B

Figure 4: All topologies considered by FlavorKit to calculate the Wilson coe cients of 2-fermion-1-boson operators. All possible generic combinations of the internal fields are taken into account.

F F1 3

F2 F4

Figure 5: All tree topologies considered by FlavorKit to calculate the Wilson coefficients of 4-fermion operators. All possible generic combinations of the internal fields are taken into account. estimate, also one-loop corrections to the 𝑍 mass stemming gets a fully functional spectrum generator for a new model. 푇 from Π푍 canbeincluded. The features of a spectrum generator created in this way are as follows. Output. A fine-tuning calculation is optionally included in the Fortran output for SPheno. (1) RGE Running.Thefulltwo-loopRGEsareincluded. (2) Precise Mass Spectrum Calculation. SPheno modules created by SARAH include the one-loop corrections to 3.11. Summary. SARAH derives a lot of information about allSUSYparticles.ForHiggsstatesthefullone-loop a given model. This information can be used in different and in addition dominant two-loop corrections are interfaces to study a model in all detail. In general, one can get included. (i) LaTeX files, (ii) a spectrum generator based on SPheno, and (iii) model files for different HEP tools. (3) Calculation of Decays. SPheno calculates all two- body decays for SUSY and Higgs states. In addition, LaTeX. All analytical information derived about a model the three-body decays of a fermion into three other can be exported to LaTeX files. These files provide ina fermions and the three-body decays of scalar into human readable format the following information: (i) list of another scalar and a pair of fermions are included. allsuperfieldsaswellascomponentfieldsforalleigenstates; (4) FlavorKit Interface. SPheno modules calculate out (ii) the superpotential and important parts of the Lagrangian of the box many flavour observables for a given like soft-breaking and gauge fixing terms added by SARAH; model. (iii) all mass matrices and tadpole equations; (iv) the full two-loop RGEs; (v) analytical expressions for the one-loop (5) CalculationofPrecisionObservables. SPheno does self-energies and tadpoles; (vi) all interactions and the cor- also calculate 𝛿𝜌, electromagnetic dipole moments, responding Feynman diagrams; and (vii) details about the and anomalous magnetic moments of leptons. SARAH implementation in . Separated files are also generated (6) Output for HiggsBounds and HiggsSignals. for the flavour observables showing all contributing diagrams SPheno generates all necessary files with the Higgs with their amplitudes. properties (masses, widths, and couplings to SM states) which are needed to run HiggsBounds Spectrum Generator. SARAH 3 has been the first “spectrum- and HiggsSignals. generator-generator”: using the derived information about the mass matrices, tadpole equations, vertices, loop (7) Estimate of the Fine-Tuning. SPheno modules can corrections, and RGEs for the given model SARAH writes calculatetheelectroweakfine-tuningwithrespectto Fortran source code for SPheno. Based on this code the user a set of defined parameters. 18 Advances in High Energy Physics

Model Files. In particular the vertex lists can be exported rich phenomenology [277]. In the 𝑅-parity conserving case, into several formats to create model files for FeynArts/ these models come with a new Higgs which mixes with the FormCalc, CalcHep/CompHep,andWHIZARD/O’Mega as MSSM one [199], they provide new dark matter candidates well as for MadGraph, Herwig++,orSherpa based on [278], and they can have an impact on the Higgs decays [279]. the UFO format. Also model files for Vevacious can be In both cases of broken and unbroken 𝑅-parity these models generated which include the tree-level potential as well as the have interesting consequences for LHC searches [280–283]. mass matrices to generate the one-loop effective potential. 4.1.1. Particle Content and Superpotential. We study the B-L minimal supersymmetric model where the SM gauge sector 4. Example—Part I: The -SSM and 𝑈(1) Its Implementation in SARAH is extended by a 퐵-퐿 G =𝑈(1) ⊗ (2) ⊗ (3) ⊗𝑈(1) I will discuss in this section and the subsequent ones the 푌 SU 퐿 SU 푐 퐵-퐿 (85) implementation of the B-L-SSM in SARAH and how all phenomenological aspects of this model can be studied. The B-L- and where 𝑅-parity is not broken by sneutrino VEVs. This SSM is for sure not the simplest extension of the MSSM, but it model is called the B-L-SSM. In this model the matter sector provides many interesting features. There are some subtleties of the MSSM is extended by three generations of right- 푐 in the implementation which will not show up in singlet handed neutrino superfields ̂] and two fields which are extensions, for instance. I hope that the examples presented 𝐵 𝐿 𝜂̂ 𝜂̂ in the following sections show how even such a complicated responsible for - breaking, and .Thesefieldscarry model can be studied with a very high precision without too lepton number 2 and are called “bileptons.” The chiral much effort. Applying the same methodology to other SUSY superfields and their quantum numbers are summarized in or non-SUSY models should be straightforward. However, Table 1.ThesuperpotentialoftheB-L-SSM is given by Icannotdiscussherealltopicswhichmightbeinteresting 푖푗 ̂ ̂ ̂ 푖푗 ̂ ̂ ̂ 푖푗 ̂ ̂ ̂ and useful for some models. In particular, I will not show 𝑊=𝑌푢 𝑈푖𝑄푗𝐻푢 −𝑌푑 𝐷푖𝑄푗𝐻푑 −𝑌푒 𝐸푖𝐿푗𝐻푑 how models with threshold scales are implemented. SARAH (86) ̂ ̂ 푖푗 ̂ ̂ 耠 ̂ 푖푗 supports thresholds where heavy particles get integrated out +𝜇𝐻푢𝐻푑 +𝑌] 𝐿푖𝐻푢̂]푗 −𝜇𝜂̂𝜂+𝑌푥 ̂]푖𝜂̂̂]푗. and where the gauge symmetries do either change or not. Users interested in that topic might take a look at the manual Here, 𝑖 and 𝑗 are generation indices and we suppressed all as well as the implementations of seesaw models4 or left-right colour and isospin indices. The first line is identical tothe symmetric models5. A brief summary of the general approach MSSM, and all new terms are collected in the second line of to include thresholds is also given in Appendix A.5. (86).After𝐵-𝐿 breaking a Majorana mass term 𝑌푥⟨𝜂⟩ for the InthefirstpartofthissectionIwillgiveashort right-handed neutrinos is generated. This term causes a mass introduction to the B-L-SSM,beforeIcometothetools splittingbetweentheCPevenandoddpartsofthecomplex which are going to be used. The implementation of the sneutrinos. B-L-SSM in SARAH is discussed in Section 4.4 and how The additional soft-breaking terms compared to the it is evaluated is shown in Section 4.5. The next sec- MSSM are tion explains what can be done with the model using Mathematica just . It is shown how the LaTeX output is 1 generated, how mass matrices and tadpole equations can be L = L −𝜆̃𝜆 ̃耠 𝑀 耠 − 𝜆 ̃耠 𝜆 ̃耠 𝑀 耠 SB,퐵-퐿-SSM SB,MSSM 퐵 퐵 퐵퐵 2 퐵 퐵 퐵 handled with Mathematica,andhowRGEsarecalculated and solved. Section 6 is about the interface between SARAH 2 󵄨 󵄨2 2 󵄨 󵄨2 2 푐 ∗ 푐 (87) −𝑚휂 󵄨𝜂󵄨 −𝑚휂 󵄨𝜂󵄨 −𝑚],푖푗 (̃]푖 ) ̃]푗 and SPheno. The mass spectrum calculation is explained, and what else can be obtained with SPheno is discussed: 푖푗 푐 ̃ 푖푗 푐 푐 −𝜂𝜂𝐵휇耠 +𝑇] 𝐻푢̃]푖 𝐿푗 +𝑇푥 𝜂̃]푖 ̃]푗. decays and branching ratios, flavour and precision observables, and the fine-tuning. Afterwards, we include more tools in our study in Section 7: HiggsBounds/HiggsSignals to The electroweak and 𝐵-𝐿 symmetry are broken to 𝑈(1)푒푚 by test Higgs constraints, Vevacious to check the vacuum the following VEVs of Higgs states and bileptons: stability, MicrOmegas to calculate the dark matter relic den- WHIZARD O Mega 1 sity and direct detections rates, / ’ to produce 𝐻0 = (𝑖𝜎 + V +𝜙 ), monojet events, and MadGraph to make a simple dilepton 푑 √2 푑 푑 푑 analysis. Section 8 is about parameter scans and how the 1 different tools can be interfaced. 𝐻0 = (𝑖𝜎 + V +𝜙 ), 푢 √2 푢 푢 푢 (88) 4.1. The Model. Supersymmetric models with an additional 1 𝑈(1) 𝜂= (𝑖𝜎휂 + V휂 +𝜙휂), 퐵-퐿 gauge symmetry at the TeV scale have recently √2 received considerable attention: they can explain neutrino 𝑅 1 data, they might help to understand the origin of -parity and 𝜂= (𝑖𝜎 + V +𝜙 ). its possible spontaneous violation [272–276],andtheyhavea √2 휂 휂 휂 Advances in High Energy Physics 19

Table 1: Chiral superfields of the B-L-SSMandtheirquantumnumbersunder𝑈(1)푌 ⊗𝑆𝑈(2)퐿 ⊗𝑆𝑈(3)푐 ⊗𝑈(1)푌𝑈(1)퐵−퐿. 1/2 (𝑈(1) ⊗𝑆𝑈(2) ⊗𝑆𝑈(3) ⊗𝑈(1) ) Superfield Spin 0 Spin Generations 푌 퐿 퐶 퐵-퐿 𝑄̂ 𝑄𝑄̃ 3 1/6 ⊗ 2 ⊗ 3 ⊗1/6 ̃푐 푐 𝐷̂ 𝑑 𝑑 3 1/3 ⊗ 1 ⊗ 3 ⊗−1/6 푐 푐 𝑈̂ 𝑢̃ 𝑢 3 −2/3 ⊗ 1 ⊗ 3 ⊗−1/6 𝐿̂ 𝐿𝐿̃ 3 −1/2 ⊗ 2 ⊗ 1 ⊗−1/2 푐 푐 𝐸̂ 𝑒̃ 𝑒 3 1⊗1 ⊗ 1 ⊗1/2 푐 푐 ̂] ̃] ] 3 0⊗1 ⊗ 1 ⊗1/2 ̂ ̃ 𝐻푑 𝐻푑 𝐻푑 1 −1/2 ⊗ 2 ⊗ 1 ⊗0 ̂ ̃ 𝐻푢 𝐻푢 𝐻푢 1 1/2 ⊗ 2 ⊗ 1 ⊗0 𝜂𝜂̂ 𝜂̃ 1 0⊗1 ⊗ 1 ⊗−1 ̂ ̃ 𝜂 𝜂 𝜂 1 0⊗1 ⊗ 1 ⊗1

In analogy to the MSSM definition tan 𝛽=V푢/V푑,wecall The gauge couplings are related284 by[ ]: 耠 the ratio of the two bilepton VEVs tan 𝛽 = V휂/V휂.As 𝑔푌푌𝑔퐵퐵 −𝑔푌퐵 𝑔퐵푌 mentioned above, the Majorana mass term causes a mass 𝑔1 = , √ 2 2 splitting between the CP even and odd parts of the right 𝑔퐵퐵 +𝑔퐵푌 sneutrinos. This makes it necessary to define 𝑔 𝑔 +𝑔 𝑔 𝑔=̃ 푌퐵 퐵퐵 퐵푌 푌푌 , (93) 푖 1 푖 푖 √𝑔2 +𝑔2 ̃] = (𝑖𝜎 +𝜙 ), 퐵퐵 퐵푌 푅 √2 푅 푅 (89) 1 𝑔 = √𝑔2 +𝑔2 . ̃]푖 = (𝑖𝜎푖 +𝜙푖 ). 퐵퐿 퐵퐵 퐵푌 퐿 √2 퐿 퐿 4.1.3. Mass Eigenstates. After electroweak and 𝐵-𝐿 breaking Here, 𝑖=1,2,3is the generation index of the sneutrinos. and also because of gauge-kinetic mixing there is a mixing between the neutral SUSY particles from the MSSM and the new sector. 4.1.2. Gauge-Kinetic Mixing. The particle content of the B-L- SSM gives rise to gauge-kinetic mixing because the matrix Neutral Gauge Bosons. In the gauge sector, the neutral gauge 3 耠 bosons 𝐵, 𝑊 ,and𝐵 mix. This gives three mass eigenstates 1 耠 𝛾 = ∑𝑄퐴𝑄퐵 𝛾, 𝑍,and𝑍 which are related by a rotation matrix which can 퐴퐵 16𝜋2 휙 휙 (90) 耠 휙 be expressed by two angles Θ and Θ : 𝐵 is not diagonal. The indices 𝐴 and 𝐵 run over all 𝑈(1) groups (𝑊) and 𝜙 over all superfields. For the particle content of Table 1 耠 we find 𝐵 (94) Θ − Θ耠 Θ Θ Θ耠 𝛾 1 11 4 cos 푊 cos 푊 sin 푊 sin 푊 sin 푊 𝛾= 𝑁( )𝑁. (91) =( Θ Θ Θ耠 − Θ Θ耠 )(𝑍 ). 16𝜋2 46 sin 푊 cos 푊 cos 푊 cos 푊 sin 푊 耠耠 𝑍耠 0 sin Θ푊 cos Θ푊 𝑁 contains the GUT normalization of the two Abelian gauge The entire mixing between the 𝐵-𝐿 andtheSMgaugeboson √3/5 𝑈(1) √3/2 𝑈(1) Θ耠 groups. We will take for 푌 and for 퐵-퐿. comes from which can be approximated by [285] From (91) we see that even if gauge-kinetic mixing vanishes 2𝑔̃√𝑔2 +𝑔2 at one scale, it is induced again by RGE running and must be 耠 1 2 tan 2Θ ≃ (95) included therefore. 푊 𝑔̃2 +16(𝑥/V)2 𝑔2 −𝑔2 −𝑔2 However, as long as the two Abelian gauge groups are 퐵퐿 2 1 unbroken, we can make a change of basis. This freedom is with V = √V2 + V2 and 𝑥=√V2 + V2. used to go to a basis where electroweak precision data is 푑 푢 휂 휂 respected in a simple way: by choosing a triangle form of the gauge coupling matrix, the bilepton contributions to the 𝑍 Neutral Higgs Bosons.IntheHiggssector,theCPevenand mass vanish: odd parts of the doublets mix with the corresponding CP eigenstates of the bileptons. This leads to four physical scalar Higgs particles: 𝑔푌푌 𝑔푌퐵 𝑔1 𝑔̃ ( )󳨀→( ). (92) 𝑔퐵푌 𝑔퐵퐵 0𝑔퐵퐿 (𝜙푑,𝜙푢,𝜙휂,𝜙휂)󳨀→(ℎ1,ℎ2,ℎ3,ℎ4) (96) 20 Advances in High Energy Physics and four pseudoscalars. Two pseudoscalars are physical and For the trilinear soft-breaking terms we use in analogy to the 耠 the other two are Goldstone bosons of the 𝑍 and 𝑍 : CMSSM the relation 푍 푍耠 ℎ ℎ (𝜎푑,𝜎푢,𝜎휂,𝜎휂) 󳨀→ (𝐺 ,𝐺 ,𝐴1,𝐴2) . (97) 𝑇푖 =𝐴0𝑌푖, 𝑖=𝑒,𝑑,𝑢,𝑥,], (107)

Neutralinos. There are seven neutralinos in the model which with a free parameter 𝐴0.Theremainingsoft-terms𝐵휇 and are an admixture of the three gauginos, the two Higgsinos, 𝐵휇耠 arenottakentobeinput,butthetadpoleequationsare,if 耠 and the two bileptinos: not stated otherwise, solved with respect to 𝜇,휇 𝐵 ,𝜇 ,and𝐵휇耠 . The advantage of this choice is that these parameters do not (𝜆 , 𝑊̃0, 𝐻̃0, 𝐻̃0,𝜆 , 𝜂,̃ 𝜂)̃ 󳨀→𝜒 ( ̃ ,...,𝜒̃0). 퐵̃ 푑 푢 퐵̃耠 1 7 (98) enter the RGE evaluation of all other terms and they can be considered independently. Neutrinos and Sneutrinos. There are six Majorana neutrinos To fix the gauge part of the 𝐵-𝐿 sector, we need two more 耠 which are a superposition of the left and right neutrino gauge input parameters. These are the mass of the 𝑍 and the ratio eigenstates: ofthebileptonVEVs.Finally,therearetwomoreYukawa-like 푖 푖 matrices, 𝑌] and 𝑌푥.Thus,thefullsetofinputparameteris (]퐿, ]푅)󳨀→]푗 𝑖=1,2,3; 𝑗=1,...,6. (99) 耠耠 𝑚0,𝑀1/2,𝐴0, tan 𝛽, tan 𝛽 , sign (𝜇) , sign (𝜇 ),𝑀푍耠 , Similarly, the sneutrinos are admixtures of left and right (108) sneutrino gauge eigenstates. Because of the mass splitting due 𝑌푥 and 𝑌]. to the Majorana mass term the CP even and odd parts mix separately, and there are 12 real mass eigenstates: The masses of the light neutrinos in this model are proportional to 𝑌]. This gives strong constraints and we can often (𝜙푖 ,𝜙푖 )󳨀→̃]푅 𝑖=1,2,3; 𝑗=1,...,6, 퐿 푅 푗 neglect 𝑌] because the entries must be tiny. (100) (𝜎푖 ,𝜎푖 )󳨀→̃]퐼 𝑖=1,2,3; 𝑗=1,...,6. 퐿 푅 푗 4.3. Setup. We want to study all aspects of the B-L-SSM: the mass spectrum, decays, flavour observables, Higgs con- 4.2. Constrained Model. The B-L-SSM has 55 additional straints, dark matter, vacuum stability, and collider physics. parameters, including all phases, compared to the general For this purpose, we not only make use of SARAH but also MSSM. Therefore, an organizing principle to relate these interface it with several other public tools. To simplify the parameters to reduce the number of free parameters is often following presentation, I am going to assume that all the helpful. We choose a CMSSM-like setup inspired by minimal following tools are installed in the same directory $PATH: supergravity. We set the boundary conditions at the GUT SARAH 4.5.0 PATH/SARAH scalewhichistakentobe (1) or newer in $ (http:// www.hepforge.org/downloads/sarah), 𝑔GUT =𝑔GUT =𝑔GUT. 1 2 퐵퐿 (101) (2) SPheno 3.3.6 or newer in $PATH/SPHENO (http:// www.hepforge.org/downloads/spheno), We assume that at the GUT gauge-kinetic mixing is absent SSP 1.2.0 PATH/SSP butgetsinducedbelowthatscale.Thisgivesasadditional (3) or newer in $ (http://www conditions: .hepforge.org/downloads/sarah), (4) PreSARAH 1.0.3 or newer in $PATH/PRESARAH 𝑔GUT =𝑔GUT =0. 푌퐵 퐵푌 (102) (http://www.hepforge.org/downloads/sarah), GUT GUT GUT (5) HiggsBounds 4.1.0 or newer in Thus, at the GUT scale the relations 𝑔1 =𝑔푌푌 and 𝑔퐵퐿 = $PATH/HIGGSBOUNDS (http://higgsbounds.hepforge 𝑔GUT hold. 퐵퐵 .org/downloads.html), In minimal supergravity all scalar and gaugino soft- masses unify at the GUT scale: that is, there are only two free (6) HiggsSignals 1.2.0 or newer in PATH/HIGGSSIGNALS parameters to fix all soft-masses: 𝑚0 and 𝑀1/2.Theboundary $ (http://higgsbounds.hepforge conditions at the GUT scale are .org/downloads.html), 2 2 2 2 2 (7) CalcHep 3.4.6 or newer in $PATH/CALCHEP 𝑚0 ≡𝑚퐻 =𝑚퐻 =𝑚휂 =𝑚휂, (103) 푑 푢 (http://theory.sinp.msu.ru/∼pukhov/calchep.html), 2 2 2 2 2 2 2 MicrOmegas 4.1 PATH/MICROMEGAS 𝑚01 ≡𝑚퐷 =𝑚푈 =𝑚푄 =𝑚퐸 =𝑚퐿 =𝑚], (104) (8) or newer in $ (http://lapth.cnrs.fr/micromegas/), 𝑀 ≡𝑀 =𝑀 =𝑀 =𝑀 1/2 1 2 3 퐵̃耠 (105) (9) WHIZARD 2.2.2 or newer in $PATH/WHIZARD (http://www.hepforge.org/downloads/whizard), with the identity matrix 1. We assume also that no gauge- MadGraph 5.2.2.2 PATH/MADGRAPH kinetic mixing is present in the gaugino sector at this scale: (10) or newer in $ that is, (https://launchpad.net/mg5amcnlo), (11) Vevacious 1.1.0 or newer in $PATH/VEVACIOUS 𝑀 耠 =0. 퐵퐵 (106) (http://www.hepforge.org/downloads/vevacious), Advances in High Energy Physics 21

(12) MadAnalysis 1.1.11 or newer in 4.4. Implementation of the 𝐵-𝐿-SSM. I assume in $PATH/MADANALYSIS (https://launchpad.net/mada- the following that the B-L-SSM was not delivered nalysis5). with SARAH and we have to implement it ourselves. All informationaboutthemodelissavedinthreedifferent files: B-L-SSM.m, parameters.m,andparticles.m It is possible to use the BSM Toolbox scripts [286]from whichhavetobelocatedinthesubdirectoryB-L-SSM http://sarah.hepforge.org/Toolbox.html to install most tools in the Model directory of SARAH.Onlythefirst at once via the included configure script. The Toolbox file, B-L-SSM.m,isabsolutelynecessaryandcontains contains also a script called butler to implement a model all physical information about the model: the in the different tools automatically based on the SARAH symmetries, particle content, superpotential, and mixing. implementation. This might be the convenient choice for the In parameters.m properties of all parameters can be user. However, I am going to use the explicit route and discuss defined, for example, LaTeX name, Les Houches block and theimplementationineachtoolinsomedetail. number, relations among parameters, and real/complex. In particles.m additional information about particles Installation of SARAH.TheinstallationofSARAH is areset:mass,width,electriccharge,PDG,LaTeXname, very simple: the package can be downloaded from output name, and so on. The optional information http://sarah.hepforge.org. After copying the tar file to in parameters.m and particles.m might be needed for the directory $PATH, it can be extracted. the different outputs of SARAH as discussed later.

$ cp [ Download−Directory ]/SARAH−X.Y.Z. tar . gz → 4.4.1. Main File: B-L-SSM.m. Usually the easiest way to $PATH/ implement a new model in SARAH is to start with an existing cd $ $PATH model file. For non-SUSY models the convenient choice is $tar−xf SARAH−X.Y.Z. tar . gz − − themodelfilefortheSMandforSUSYmodelstheonefor $ln s SARAH SARAH X.Y.Z the MSSM. Therefore, I am going to discuss the differences in themodelfileoftheB-L-SSM compared to the MSSM one. X.Y.Z hastobereplacedbytheversionwhichwas As reference, the corresponding parts in the MSSM model downloaded. In the last line a symbolic link from the fileareshownaswell.Inaddition,therearesomespecific directory SARAH to SARAH, X.Y.Z,iscreated.Thereisno flags listed in Appendix A.1 which can be used to turn on/off compilation necessary, but SARAH can directly be used with particular structures in the Lagrangian. Those are not needed any Mathematica version between 7 and 10. forourpurposeherebutmightbeusefulforothermodels. At the very beginning of a model file, some additional Installation of All Other Tools. For the installation of all other information about the model implementation can be given: tools, please check the manual or readme files of these tools. what is the name of the model (in LaTeX syntax as well as string without any special character), who is the author of the model file, and when was the last change?

B-L-SSM

3 Model‘Name = "BLSSM"; 4 Model‘NameLaTeX ="B-L-SSM"; 5 Model‘Authors = "L.Basso, F.Staub"; 6 Model‘Date = "2012-09-01";

Global Symmetries. SARAH supports 𝑍푁 as well as 𝑈(1) Usually, one considers the MSSM with conserved 𝑅-parity. global symmetries which are defined in the array Global. This discrete symmetry is defined via the following.

MSSM, B-L-SSM

18 Global[[1]] = {Z[2],RParity};

First, the kind of the symmetry is defined (Z[N] with SUSY coordinates are considered as well. There are two integer N,or U[1]). Note that 𝑍푁 symmetries are always possibilities to define the charges of SUSY fields with respect understood as multiplicative symmetries. The second to a global symmetry: entry gives a name to the symmetry. For the 𝑈(1) there is one specific name which can be used to define 𝑅- (1) If in the definition of the vector or chiral superfields, symmetries: RSymmetry.Inthatcase,the𝑅 charges of the which will be explained below, only one quantum 22 Advances in High Energy Physics

number is given per superfield per global symmetry charge for the superfield, the second one for the this number is used for the superfield itself but also scalar component, and the third one for the fermionic for component fields. component. For vector superfield, the second entry refers to the gaugino and the third one to the gauge boson. (2) If a list with three entries is given as charge for a vector or chiral superfield the following convention With these conventions a suitable definition of the global applies: for chiral superfields, the first entry is the symmetries for states with 𝑅-parity ±1 would be as follows.

MSSM, B-L-SSM

19 RpM = {-1, -1, 1}; 20 RpP = { 1, 1, -1};

In principle, the B-L-SSM has no global symmetry but 𝑅- Gauge Symmetries. The next step to define a SUSY model is 𝑈(1) parity is just a remnant of the broken 퐵-퐿.However, to fix the gauge sector. That is done by adding for each gauge it turns out to be helpful to keep the standard definition group one entry to the array Gauge. For SUSY models for for 𝑅-parity: we can use this 𝑍2 to get the relic density each entry in Gauge also the corresponding vector super- with MicrOmegas. Sometimes, one uses also matter parities fields are set automatically. For instance, the SM gauge group 𝑍 𝐵 𝐿 G = 𝑈(1) × (2) × (3) to have an additional 2 in models with gauged - [287, SM 푌 SU 퐿 SU 퐶 is set via the following. 288].

MSSM

24 (* Vector Superfields *) 25 Gauge[[1]]={B, U[1], hypercharge , g1, False, RpM}; 26 Gauge[[2]]={WB, SU[2], left, g2, True, RpM}; 27 Gauge[[3]]={G, SU[3], color, g3, False, RpM};

First, the names of the gauge fields are given6,andthesecond explained above: here, the vector superfields as well as its entry defines the kind of the group. The third entry gives fermionic components get 𝑅-parity −1, while the spin-1 states anametothegaugegroup7 and the fourth entry fixes the have 𝑅-parity +1. name of the corresponding gauge coupling. In the fifth entry For the B-L-SSM only a fourth entry is needed. We call it is defined if the group will be broken later. The last entry the group Bp:thatis,thevectorbosonwillbenamed VBp sets the global charges of the gauginos and vector bosons of by SARAH and the gaugino fBp.Forthegaugecouplingwe the corresponding vector superfield using the conventions chose as name gBL8. Thus, the definition of the B-L-SSM gaugesectorisasfollows.

B-L-SSM

24 (* Vector Superfields *) 25 Gauge[[1]]={B, U[1], hypercharge , g1, False, RpM}; 26 Gauge[[2]]={WB, SU[2], left, g2, True, RpM}; 27 Gauge[[3]]={G, SU[3], color, g3, False, RpM}; 28 Gauge[[4]]={Bp, U[1], BminusL , gBL, False, RpM};

Since this is the second 𝑈(1) besides hyper- Chiral Superfields. Chiral superfields in SARAH are defined charge, SARAH will generate two off-diagonal gauge via the array SuperFields. The conventions are as follows. couplings g1BL and gBL1 stemming from kinetic mixing: First, a name for the superfield is given, in the second see also Section 2. The soft-masses for the gauginos entry the number of generations is set, and in the third which are added by SARAH are MassB, MassWB, MassG, entry the names for the isospin components are defined. and MassBp. And as consequence of gauge-kinetic Afterwards, the transformation under the different gauge mixing MassBBp appearsaswell. groups is given, and the last entries set the charges under Advances in High Energy Physics 23 the global symmetries. To define the transformation with treated as vector which is the appropriate dimension. respect to Abelian groups the charge is given. For non- To do this SARAH makes use of the generators as Abelian groups the dimension is given as integer, where well as the Clebsch-Gordan-coefficient provided by conjugated representations are negative. If the dimension Susyno. is not unique, also the Dynkin labels can be defined. The treatment of higher-dimensional representation is different We have to deal here only with doublets under a broken for groups which get broken and those which stay unbroken. SU(2): that is, we need a vector of length two for the isospin components. (1) Representation with respect to groups which get The particle contents of the MSSM are the squark broken: in that case it is convenient to define higher- superfields q, d,and u,theleptonsuperfields l and e,and dimensional representations as tensor products of the the two superfields for the Higgs doublets Hd and Hu.We fundamental representation. have 3 generations for all matter fields and the usual charges (2) Representation with respect to groups which do with respect to the SM gauge groups. These are defined by the not get broken: in that case the representation is following.

MSSM

31 (* Chiral Superfields *) 32 SuperFields[[1]] = {q, 3, {uL, dL}, 1/6, 2, 3, RpM}; 33 SuperFields[[2]] = {l, 3, {vL, eL}, -1/2, 2, 1, RpM}; 34 SuperFields[[3]] = {Hd,1, {Hd0, Hdm}, -1/2, 2, 1, RpP}; 35 SuperFields[[4]] = {Hu,1, {Hup, Hu0}, 1/2, 2, 1, RpP};

37 SuperFields[[5]] = {d, 3, conj[dR], 1/3, 1, -3, RpM}; 38 SuperFields[[6]] = {u, 3, conj[uR], -2/3, 1, -3, RpM}; 39 SuperFields[[7]] = {e, 3, conj[eR], 1, 1, 1, RpM};

The SARAH conventions for the component fields of a 𝑈(1) charges as this is done usually by conventions for the superfield are as follows: scalars start with S and fermions hypercharge. Therefore, quark superfields have 𝐵-𝐿 charge with F: for example, the left down-squark is called SdL ±1/6 andleptonsuperfieldscarry±1/2.Theotherchanges and the right lepton FeR. The soft-masses for the scalars are the definition of the right sneutrino superfield vR which are mq2, ml2, mHd2,andsoforth. comes in three generations and which is a gauge singlet under 𝑈(1) C1 C2 In the B-L-SSM, we have first to define the 퐵-퐿 charge the SM gauge groups. The bileptons and are also SM of all MSSM fields and SARAH includes a factor 1/2 for all singlets and come with 𝐵-𝐿 charge ±1 according to the just- mentioned conventions.

B-L-SSM

31 (* Chiral Superfields *) 32 SuperFields[[1]] = {q, 3, {uL, dL}, 1/6, 2, 3, 1/6, RpM}; 33 SuperFields[[2]] = {l, 3, {vL, eL}, -1/2, 2, 1, -1/2, RpM}; 34 SuperFields[[3]] = {Hd,1, {Hd0, Hdm}, -1/2, 2, 1, 0, RpP}; 35 SuperFields[[4]] = {Hu,1, {Hup, Hu0}, 1/2, 2, 1, 0, RpP};

37 SuperFields[[5]] = {d, 3, conj[dR], 1/3, 1, -3, -1/6, RpM}; 38 SuperFields[[6]] = {u, 3, conj[uR], -2/3, 1, -3, -1/6, RpM}; 39 SuperFields[[7]] = {e, 3, conj[eR], 1, 1, 1, 1/2, RpM}; 40 SuperFields[[8]] = {vR,3, conj[vR], 0, 1, 1, 1/2, RpM};

42 SuperFields[[9]] = {C1, 1, C10, 0, 1, 1, -1, RpP}; 43 SuperFields[[10]] = {C2, 1, C20, 0, 1, 1, 1, RpP}; 24 Advances in High Energy Physics

The soft-terms for the new superfields are mvR2, mC12, term called \[Mu]. SARAH will add the corresponding soft- and mC22. terms called T[Yu], T[Yd], T[Ye],andB[\[Mu]].The Yukawas and the trilinear soft-terms are treated as complex 3 Superpotential.ThesuperpotentialoftheMSSMconsistsof × 3 matrices, and 𝜇 and 𝐵휇 are complex by default. The MSSM the Yukawa interaction Yu, Yd,and Ye as well as the 𝜇- superpotential is set in SARAH by the following.

MSSM

45 (* Superpotential *) 46 SuperPotential = Yu u.q.Hu - Yd d.q.Hd - Ye e.l.Hd + \[Mu] Hu.Hd;

We see here that we do not have to define the charge indices In the B-L-SSM three more terms are present in the super- and the contraction of those. All of this is done automatically potential: the neutrino Yukawa coupling Yv,thecoupling 耠 by SARAH and the term Yu u.q.Hu gets interpreted inter- between bileptons and right neutrino Yn,andthe𝜇 -term nally as called MuP. The corresponding soft-terms added by SARAH are T[Yv], T[Yn],and B[MuP].Alsothosearetreatedby default in the most general way. The full superpotential of the 𝑌푖푗 𝜖 𝛿 𝑢̂훼𝑞̂훽푎𝐻̂푏. 푢 푎푏 훼훽 푖 푗 푢 (109) B-L-SSMreadsasfollows.

B-L-SSM

49 (* Superpotential *) 50 SuperPotential = Yu u.q.Hu - Yd d.q.Hd - Ye e.l.Hd + 51 \[Mu] Hu.Hd + Yv vR.l.Hu - MuP C1.C2 + Yn vR.C1.vR;

Eigenstates. In the MSSM and the B-L-SSM we have to deal defined in the array NameOfStates. Sometimes, it might with two sets of eigenstates. First, we have the gauge eigen- be useful to use also intermediate states to make stepwise states which will always be present. These states are called rotations from the gauge to the final matter eigenstates. by default GaugeES. Second, we have the mass eigenstates Therefore, the length of NameOfStates is not restricted. after symmetry breaking which we call EWSB.Bothsetsare

MSSM, B-L-SSM

56 NameOfStates={GaugeES , EWSB};

Electroweak and 𝐵-𝐿 Symmetry Breaking.IntheMSSM, their CP even and odd parts. These definitions are done EWSB happens when the neutral components of the Higgs in the array DEFINITION[EWSB][VEVs].IntheMSSM doublets receive VEVs. In addition, the states are split in implementation in SARAH the factors of 1/√2 are chosen in a way that the electroweak (ew) VEV is about 246 GeV.

MSSM

87 (*--- VEVs ---- *) 88 DEFINITION[EWSB][VEVs]= 89 { {SHd0, {vd, 1/Sqrt[2]}, {sigmad , I/Sqrt[2]},{phid ,1/Sqrt[2]}}, 90 {SHu0, {vu, 1/Sqrt[2]}, {sigmau , I/Sqrt[2]},{phiu ,1/Sqrt[2]}} 91 };

In the B-L-SSM we have to extend Here,weusethesamenormalizationasfortheew DEFINITION[EWSB][VEVs] to give also VEVs to bileptons. VEVs. As mentioned in Section 4.1.1,alsothesneutrino Advances in High Energy Physics 25 will split into real and imaginary parts because of the VEVs. We could study spontaneous 𝑅-parity the Majorana mass term. We can also define this violation by adding a nonzero VEV for these splitting in DEFINITION[EWSB][VEVs] and keep 0 for particles.

B-L-SSM

95 (*--- VEVs ---- *) 96 DEFINITION[EWSB][VEVs]= 97 { {SHd0, {vd, 1/Sqrt[2]}, {sigmad , I/Sqrt[2]},{phid ,1/Sqrt[2]}}, 98 {SHu0, {vu, 1/Sqrt[2]}, {sigmau , I/Sqrt[2]},{phiu ,1/Sqrt[2]}}, 99 {SC10, {x1, 1/Sqrt[2]}, {sigma1 , I/Sqrt[2]},{phi1 ,1/Sqrt[2]}}, 100 {SC20, {x2, 1/Sqrt[2]}, {sigma2 , I/Sqrt[2]},{phi2 ,1/Sqrt[2]}}, 101 {SvL, {0, 0}, {sigmaL , I/Sqrt[2]},{phiL ,1/Sqrt[2]}}, 102 {SvR, {0, 0}, {sigmaR , I/Sqrt[2]},{phiR ,1/Sqrt[2]}} 103 };

Rotations in Gauge Sector.Theelectroweakgaugebosonsmix by the Weinberg angle and ZW and ZfW just involve con- in the MSSM as they do in the SM after EWSB. This mixing is stant factors. This parametrization can be defined later defined in the array DEFINITION[EWSB][GaugeSector] in parameters.m;seeSection 4.4.2. If no parametrization and is encoded by three matrices ZZ, ZW,and ZfW.These is given, SARAH will treat the matrices as general rotation matrices have a simple form: ZZ is usually parametrized matrices with the appropriate dimensions9 as follows.

MSSM

77 (*--- Gauge Sector ---- *) 78 DEFINITION[EWSB][GaugeSector] = 79 { {{VB,VWB[3]},{VP,VZ},ZZ}, 80 {{VWB[1],VWB[2]},{VWm,conj[VWm]},ZW}, 81 {{fWB[1],fWB[2],fWB[3]},{fWm,fWp,fW0},ZfW} 82 };

𝑈(1) In the B-L-SSM the vector boson of the 퐵-퐿 mixes but use another parametrization based on (94) as shown in with the 𝐵-boson and third component of the 𝑊-boson to Section 4.4.2. The mixing in the charged gauge boson and three neutral states. We call the rotation matrix again ZZ gaugino sector does not change compared to the MSSM.

B-L-SSM

83 (*--- Gauge Sector ---- *) 84 DEFINITION[EWSB][GaugeSector] = 85 { {{VB,VWB[3],VBp},{VP,VZ,VZp},ZZ}, 86 {{VWB[1],VWB[2]},{VWm,conj[VWm]},ZW}, 87 {{fWB[1],fWB[2],fWB[3]},{fWm,fWp,fW0},ZfW} 88 };

Rotations in Matter Sector.Allrotationsinthe involved gauge eigenstates are given and then the names matter sector of the MSSM are defined via the for the mass eigenstates and the rotation matrices are array DEFINITION[EWSB][MatterSector].Firstthe given. 26 Advances in High Energy Physics

MSSM

95 (*--- Matter Sector ---- *) 96 DEFINITION[EWSB][MatterSector]= 97 { {{SdL, SdR}, {Sd, ZD}}, 98 {{SuL, SuR}, {Su, ZU}}, 99 {{SeL, SeR}, {Se, ZE}}, 100 {{SvL}, {Sv, ZV}}, 101 {{phid, phiu}, {hh, ZH}}, 102 {{sigmad , sigmau}, {Ah, ZA}}, 103 {{SHdm,conj[SHup]},{Hpm,ZP}}, 104 {{fB, fW0, FHd0, FHu0}, {L0, ZN}}, 105 {{{fWm, FHdm}, {fWp, FHup}}, {{Lm,UM}, {Lp,UP}}}, 106 {{{FeL},{conj[FeR]}},{{FEL,ZEL},{FER,ZER}}}, 107 {{{FdL},{conj[FdR]}},{{FDL,ZDL},{FDR,ZDR}}}, 108 {{{FuL},{conj[FuR]}},{{FUL,ZUL},{FUR,ZUR}}} 109 };

Lines 97 and 98 set the mixing of the down- and up- rotation matrices follow. Thus, line 105 leads to the following squarks (Sd, Su), and the next two lines set the mixing relations: for charged and neutral sleptons (Se, Sv). Afterwards, the 𝑊̃− = ∑𝑈−∗𝜆−, rotations of the CP even, CP odd, and charged Higgs bosons 푗1 푗 follow (hh, Ah,and Hpm). Lines 104–108 are the mixing for 푗 L0 Lm Lp the fermions: neutralinos ( ), charginos ( / ), charged 𝐻̃− = ∑𝑈−∗𝜆−, leptons (FEL/FEL), down-quarks (FDL/FDL), and up-quarks 푑 푗2 푗 FUL FUR 푗 ( / ). One sees the different conventions used for the (111) mixing of Dirac fermions compared to Majorana fermions ̃+ +∗ + 𝑊 = ∑𝑈1푗 𝜆푗 , and scalars. For instance, the first line is interpreted as 푗 ̃+ +∗ + 𝐻푢 = ∑𝑈2푗 𝜆푗 . ̃ 퐷,∗ ̃ 푗 𝑑퐿,푖훼 = ∑𝑍푗푖 𝑑푗훼, 푗 − + (110) The two rotation matrices 𝑈 (Um)and𝑈 (Up)diagonalize 𝑑̃ = ∑𝑍퐷,∗𝑑̃ . 푅,푖훼 푗푖 푗훼 the chargino mass matrix. 푗+3 A few modifications are necessary in the B-L-SSM to include the additional mixing discussed in Section 4.1.3:the mixing of the sneutrinos has to be defined for CP even and Here, 𝑖 = 1, 2, 3 is the generation index for the gauge SvIm SvRe ̃ ̃ odd states separately ( , )andthebasisfortheCP eigenstates 𝑑퐿, 𝑑푅, 𝑗=1,...,6 is the generation index for the even and CP odd Higgs scalars and the neutralinos has been ̃ 퐷 mass eigenstates 𝑑,and𝑍 is the rotation matrix. 𝛼 is a colour extended by the 𝐵-𝐿 fields. Finally, a mixing of left and right index. In the case of charginos, first the two basis vectors are neutrinos to Majorana states is added. The corresponding defined, before the names for the mass eigenstates and the definitions in the model file of the B-L-SSM read as follows. B-L-SSM

109 (*--- Matter Sector ---- *) 110 DEFINITION[EWSB][MatterSector]= 111 { {{SdL, SdR}, {Sd, ZD}}, 112 {{SuL, SuR}, {Su, ZU}}, 113 {{SeL, SeR}, {Se, ZE}}, 114 {{phiL,phiR}, {SvRe, ZVR}}, 115 {{sigmaL ,sigmaR}, {SvIm, ZVI}}, 116 {{phid, phiu,phi1, phi2}, {hh, ZH}}, 117 {{sigmad , sigmau ,sigma1 ,sigma2}, {Ah, ZA}}, 118 {{SHdm,conj[SHup]},{Hpm,ZP}}, 119 {{fB, fW0, FHd0, FHu0,fBp,FC10,FC20}, {L0, ZN}}, 120 {{{fWm, FHdm}, {fWp, FHup}}, {{Lm,UM}, {Lp,UP}}}, 121 {{FvL,conj[FvR]},{Fvm,UV}}, 122 {{{FeL},{conj[FeR]}},{{FEL,ZEL},{FER,ZER}}}, 123 {{{FdL},{conj[FdR]}},{{FDL,ZDL},{FDR,ZDR}}}, 124 {{{FuL},{conj[FuR]}},{{FUL,ZUL},{FUR,ZUR}}} 125 }; Advances in High Energy Physics 27

Phases for Unrotated Fields. For states which donot mix there is still a phase which is not fixed. This is just the case for the gluino in both models. MSSM, B-L-SSM

128 DEFINITION[EWSB][Phases]= 129 { {fG, PhaseGlu} 130 };

With this definition, the physical gluino mass 𝑀푔̃ is related to tools and also by SPheno. Therefore, we have to define the the gaugino mass parameter 𝑀3 by relation between the two- and four-component fermions. SincetherearenoadditionalmasseigenstatesintheB-L- 푖휙푔̃ 𝑀푔̃ =𝑒 𝑀3. (112) SSM compared to the MSSM but just some states come with more generations, the definition of Dirac spinors is the same in both models. Dirac Spinors. While SARAH worksinternallyoftenwithWeyl spinors, Dirac spinors are commonly used by Monte-Carlo

MSSM, B-L-SSM

132 DEFINITION[EWSB][DiracSpinors]={ 133 Fd ->{ FDL, conj[FDR]}, 134 Fe ->{ FEL, conj[FER]}, 135 Fu ->{ FUL, conj[FUR]}, 136 Fv ->{ Fvm, conj[Fvm]}, 137 Chi ->{ L0, conj[L0]}, 138 Cha ->{ Lm, conj[Lp]}, 139 Glu ->{ fG, conj[fG]} 140 };

The first line defines a Dirac spinor for the down-quarks: properties of a parameter in the file parameters.m are the following. 𝑑퐿 𝑑=( ) (113) Description 𝑑∗ (i) : it defines a string to describe the 푅 parameter. while the last line sets the Majorana gluino: (ii) OutputName: it defines a string which is used for the parameter in the different outputs. No special 𝜆푔̃ characters should be used to be compatible with C++ 𝑔=(̃ ). ∗ (114) and Fortran. 𝜆푔̃ (iii) Real: it defines if a parameter should be considered as In principle, one can also add the definition of Dirac spinors real (True)orcomplex(False). Default is False. for the gauge eigenstates in exactly the same way. However, Form we are not interested in performing calculations for gauge (iv) :itcanbeusedformatricestodefineifitis Diagonal Scalar eigenstates and I skip this part here. Interested readers can diagonal ( )orascalar( ). By default take a look at Appendix C.1 to see the entire model file for the no assumption is made. SARAH implementation of the B-L-SSM in . (v) LaTeX: it defines the name of the parameter used in theLaTeXoutput.StandardLaTeXlanguageshouldbe 4.4.2. Parameter Definitions. When the changes used10. in B-L-SSM.m are done, the model is already ready to (vi) GUTnormalization: it defines a GUT normalization be used with SARAH. While in principle all calculations for an Abelian gauge coupling. in Mathematica can be performed, the different outputs need some more information. These are mostly formal (vii) Dependence: it defines a dependence on other points. All possible options which can be used to define the parameters which should always be used. 28 Advances in High Energy Physics

(viii) DependenceOptional: it defines a dependence (xiii) Value: it assigns a numerical value to the parameter. which is optionally used in analytical calculations. Many of the above definitions are just optional and are often (ix) DependenceNum: it defines a dependence which is not needed. I will show some parts of parameters.m to used in numerical calculations. define the properties of new parameters in the B-L-SSM or (x) DependenceSPheno: it defines a dependence which to change properties of MSSM parameters. However, I will is just used by SPheno. not show here all changes compared to the MSSM but will pick just some important and interesting cases. The full list of (xi) MatrixProduct: it can be used to express a matrix changes is given in Appendix C.2. as product of two other matrices. This can be used, for instance, to relate the CKM matrix to the quark Gauge Sector.Inthegaugesectorwehavethreenewgauge 𝐵 𝐿 rotation matrices. couplings: the one corresponding to the new - gauge group and the two gauge couplings induced by gauge-kinetic (xii) LesHouches: it defines the position of the parameter mixing. We define for all three couplings an output name, a in a Les Houches spectrum file. For matrices just the LaTeX name, and the position in the Les Houches file which name of the block is defined, while for scalars the will become important later. block and an entry have to be given: {block, number}.

parameters.m

1 {g1BL, {Description -> "Mixed Gauge Coupling 1", 2 LesHouches -> {gauge, 10}, 3 LaTeX -> "g_{Y B}", 4 OutputName -> gYB }}, 5 {gBL1, {Description -> "Mixed Gauge Coupling 2", 6 LesHouches -> {gauge, 11}, 7 LaTeX -> "g_{B Y}", 8 OutputName -> gBY}}, 9 {gBL, {Description -> "B-L-Coupling", 10 LaTeX -> "g_{B}", 11 GUTnormalization -> Sqrt[3/2], 12 LesHouches -> {gauge ,4}, 13 OutputName -> gBL }},

耠 Gaugecouplingsarebydefaulttakentobereal:thatis, We have seen in (94) how the 𝛾−𝑍−𝑍 rotation it is not necessary to define this explicitly. We also used matrix can be parametrized by two angles. To do this, we here √3/2 for the GUT normalization of the 𝐵-𝐿 charge. use the Dependence statement for the parameter ZZ used The GUT normalization of the two off-diagonal charges is to label the rotation in the neutral gauge sector as follows. automatically set by SARAH. Similarly, the properties of two new gaugino mass parameters are set; see Appendix C.2.

parameters.m

1 CW=Cos[ThetaW]; SW=Sin[ThetaW]; CWp=Cos[ThetaWp]; SWp=Sin[ThetaWp]; 2 {ZZ, {Description -> "Photon -Z-Z’ Mixing Matrix", 3 Dependence -> {{CW,-SW CWp, SW SWp }, 4 {SW, CW CWp, -CW SW }, 5 {0 , SWp, CWp }}, 6 Real ->True, 7 LaTeX -> "Z^{\\gamma Z Z’}", 8 LesHouches -> None, 9 OutputName -> ZZ }}, Advances in High Energy Physics 29

Here, we put this rotation matrix explicitly to real. The reason with SPheno is to use the numerical result for the rotation why we also add an output name is that this matrix shows up matrix calculated by SPheno when diagonalizing the vector internally in SPheno when it diagonalizes the gauge boson bosons mass matrix. To translate this matrix into the rotation mass matrix. angle, we use Since the rotation matrix is completely defined by the two 󵄨 훾푍푍耠 󵄨 rotation angles, it is not necessary to include it in a spectrum Θ耠 = 󵄨𝑍 󵄨 . arccos 󵄨 33 󵄨 (115) file. Therefore, LesHouches -> None is used. In (95) an Θ耠 approximation for the new angle was given. It might SPheno calculates this value, uses it internally to get the be useful to use this approximation sometimes in numeri- vertices, and writes it into the block ANGLES as entry 10 of Mathematica cal calculations in . Therefore, it is included. the spectrum file. The needed lines to define all that are as However, the better method for high precision calculations follows.

parameters.m

1 {ThetaWp , { LaTeX -> "{\\Theta ’}_W", 2 DependenceNum ->ArcTan[(2 g1BL Sqrt[g1^2+g2^2]) 3 /(g1BL^2 + 16 (x1^2+x2^2)/(vd^2+vu^2) → -g1^2-g2^2)]/2, 4 Real ->True, 5 DependenceSPheno -> ArcCos[Abs[ZZ[3,3]]], 6 OutputName -> TWp, 7 LesHouches -> {ANGLES ,10} }},

Rotations in Matter Sector.Inthemattersectorwehave short and just include the descriptions, the LaTeX commands, to define new rotation matrices for the CP even and odd theLesHouches,andoutputnames.Fortheneutrinorotation sneutrinos and for the neutrinos. The definitions are very matrix the entries read as follows.

parameters.m

1 {UV, {Description ->"Neutrino -Mixing-Matrix", 2 LaTeX -> "U^V", 3 LesHouches -> UVMIX, 4 OutputName -> UV }},

ZVR ZVI − sin 𝛼 cos 𝛼 Similarly, and are set. In addition, there are some 𝑍퐻 =( ). rotation matrices which change compared to the MSSM. cos 𝛼 sin 𝛼 There is no need to modify the definition for the neutralino rotation matrix because no assumptions are made for these (116) in the MSSM. However, the rotation matrices for scalar and pseudoscalar Higgs are parametrized in the MSSM by two This parametrization is used to express dependence option- angles: ally used in the MSSM. Since the rotation matrices have grown to 4×4matrices in the B-L-SSM, we can no longer − 𝛽 𝛽 make use of that. Therefore, we can take the definition of the 퐴 cos sin 𝑍 =( ), MSSM and overwrite the dependence by None. sin 𝛽 cos 𝛽

parameters.m

1 {ZH, { Description ->"Scalar -Mixing -Matrix", 2 DependenceOptional ->None}}, 3 {ZA, { Description ->"Pseudo -Scalar -Mixing -Matrix", 4 DependenceOptional ->None}}, 30 Advances in High Energy Physics

Weseehereafeaturewhichmakeslifeoftenmuchsimpler. $PATH/Models/particles.m. It is actually not necessary to give the full definitions for particles and parameters in the model files for each model. One can make use of global definitions which are defined in the files: The entries for the two Higgs rotation matrices in these files $PATH/Models/parameters.m, read as follows.

parameters.m

1 {{ Description -> "Scalar -Mixing -Matrix", 2 LaTeX -> "Z^H", 3 Real -> True, 4 DependenceOptional -> {{-Sin[\[Alpha]],Cos[\[Alpha]]}, 5 {Cos[\[Alpha]],Sin[\[Alpha]]}}, 6 LesHouches -> SCALARMIX , 7 OutputName -> ZH }},

9 {{ Description ->"Pseudo -Scalar -Mixing -Matrix", 10 LaTeX -> "Z^A", 11 Real -> True, 12 DependenceOptional -> {{-Cos[\[Beta]],Sin[\[Beta]]}, 13 {Sin[\[Beta]],Cos[\[Beta]]}}, 14 Value -> None, 15 LesHouches -> PSEUDOSCALARMIX , 16 OutputName -> ZA }},

We refereed to these in parameters.m of the B-L-SSM by (a) a numerical value can be given; Description using the same string as ,butweoverwrote (b) the keyword Automatic assigns that SARAH locally the dependence which changed. derives the tree-level expression for the mass from the Lagrangian. The mass is then cal- 4.4.3. Particles Definitions. We turn now to the particles. culated by using the values of the other To define the properties of any particle present in the parameters; model, several options are available which can be put in (c) the keyword LesHouches assigns that this particles.m . mass is calculated numerically by a spectrum generator like SPheno andprovidedviaaLes (i) Description: it defines a string for defining the Houches spectrum file. This is usually the pre- particle. ferred method because also loop corrections are included. (ii) PDG: it defines the PDG numbers of all generations. PDG.IX (iii) : it defines a nine-digit number of a particle (vii) OutputName: it defines the name used in the different supportingtheproposalin[289, 290]. By default, the outputs for other codes. entries of PDG areusedintheoutput11. (viii) LaTeX:itdefinesthenameoftheparticleusedinthe (iv) ElectricCharge: it defines the electric charge of a LaTeX output. particle in units of 𝑒. This information is exported to the CalcHep/CompHep and UFO model files. (ix) FeynArtsNr:itisthenumberassignedtotheparticle used in the FeynArts output. (v) Width: it can be used to define a fixed numerical value for the width. (x) LHPC: it defines the column and colour used for the particle in the steering file for (vi) Mass: it defines how MC tools obtain a numerical the LHPC Spectrum Plotter http://lhpc.hepforge value for the mass of the particle: .org/.Allcoloursavailablein gnuplot canbeused. Advances in High Energy Physics 31

(xi) Goldstone: for each massive vector boson the name demonstrate the procedure. The full changes compared to the of corresponding Goldstone boson is given. MSSM are given in Appendix C.3.

The properties of all particles appearing as either gauge or Gauge Eigenstates and Intermediate States. Intermediate states mass eigenstates, or just at intermediate steps, can be defined are those which do not show up in any Lagrangian. These in particles.m. Usually, the user is only interested in the are, for instance, the superfields which get decomposed output for the mass eigenstates. Therefore, it is not really in components fields or the real parts of complex scalars necessary to define the entire properties of all intermediate after symmetry breaking which directly mix to new mass states and the gauge eigenstates. The only input which is eigenstates. Also Weyl spinors are considered as intermediate usually helpful to have a nice looking LaTeX output is to because the output is always in terms of Dirac spinors. For define the LaTeX syntax for all particles which appear at all of these particles it is usually sufficient to define a LaTeX pdf any stage in the model. Again, I pick just some entries to name to have a nice looking file at the end.

particles.m

1 WeylFermionAndIndermediate = { 2 ... 3 (* Superfields *) 4 {vR, { Description -> "Right Neutrino Superfield" }}, 5 {C1, { LaTeX -> "\\hat{\\eta}" }}, 6 {C2, { LaTeX -> "\\hat{\\bar{\\eta}}" }},

8 (* Intermediate Scalars *) 9 {phi1, { LaTeX -> "\\phi_{\\eta}" }}, 10 {phi2, { LaTeX -> "\\phi_{\\bar{\\eta}}" }},

12 {sigma1 , { LaTeX -> "\\sigma_{\\eta}"}}, 13 {sigma2 , { LaTeX -> "\\sigma_{\\bar{\\eta}}" }},

15 ... 16 };

The properties of gauge eigenstates can be defined inthe sufficient to give just the LaTeX names. Of course, also all array ParticleDefinitions[GaugeES].However,since other information can be set as for the mass eigenstates if rarelycalculationsaredoneforthesestates,itisalsooften demanded.

particles.m

1 ParticleDefinitions[GaugeES] = { 2 ... 3 {SC10, {LaTeX -> "\\eta" }}, 4 {SC20, {LaTeX -> "\\bar{\\eta}"}}, 5 ...

Mass Eigenstates. More interesting are the mass eigenstates. with the new states which are not present in the MSSM: 耠 The additional information given for those is used in the the 𝑍 and the corresponding ghost as well as the real different output for SPheno andtheMC-tools.Webegin sneutrinos. 32 Advances in High Energy Physics

particles.m

1 {SvRe, { Description -> "CP-even Sneutrino", 2 LaTeX -> "\\nu^R", 3 OutputName -> "nR", 4 FeynArtsNr -> 41, 5 LHPC -> {5, "blue"}, 6 PDG->{1000012,1000014,1000016,2000012,2000014,2000016}, 7 PDG.IX ->{200000001,200000002,200000003, 8 200000004,200000005,200000006} }}, 9 {SvIm, { Description -> "CP-odd Sneutrino", 10 LaTeX -> "\\nu^I", 11 OutputName -> "nI", 12 FeynArtsNr -> 40, 13 LHPC -> {5, "turquoise"}, 14 PDG->{4000012,4000014,4000016,5000012,5000014,5000016}, 15 PDG.IX ->{202000001,202000002,202000003, 16 202000004,202000005,202000006}}}, 17 18 {VZp, { Description -> "Z’-Boson", 19 PDG -> {31}, 20 PDG.IX -> {122000002}, 21 Width -> Automatic , 22 Mass -> LesHouches , 23 FeynArtsNr -> 10, 24 LaTeX -> "{Z’}", 25 Goldstone -> Ah[{2}], 26 ElectricCharge -> 0, 27 OutputName -> "Zp"}} 28 {gZp, { Description -> "Z’-Ghost", 29 PDG -> 0, 30 PDG.IX -> 0, 31 Width -> 0, 32 Mass -> Automatic , 33 FeynArtsNr -> 10, 34 LaTeX -> "\\eta^{Z’}", 35 ElectricCharge -> 0, 36 OutputName -> "gZp"}},

Here, we defined that the second pseudoscalar (Ah[{2}])is to the MSSM. Since all MSSM particles are defined 耠 the Goldstone of the 𝑍 . For the ghost we used the PDG 0 to globally in $PATH/SARAH/Models/particles.m by make clear that this is not a physical state. the Description statement, we can make use of that but In addition, there are also particles where the just overwrite the lists of PDGs which become longer. The number of generations has increased with respect new CP even and odd Higgs states carry now four PDGs.

particles.m

Here, we used for the first two entries of the pseudoscalars 0. bosons. In a similar way, the additional PDGs for This means that these two states are not physical neutralinos Chi and neutrinos Fv are defined; see but are the Goldstones of the massive, neutral gauge Appendix C.3. Advances in High Energy Physics 33

4.5. Running the Model. Whenwearedonewiththemodel should appear and no error messages or warning during the files, the model is initialized in Mathematica via the follow- evaluation should show up. More detailed checks if the model ing. implementation is self-consistent and if the model is working fine can be carried out by the following command. <<[$PATH]/SARAH.m; Start["B-L-SSM"]; CheckModel[];

After about 1 minute the following message This function makes all checks listed in Section 2.3.Withour implementation above there will be some messages like the All Done. B-L-SSM is ready! following.

Lagrangian::PossibleMixing: Possible mixing between SvIm and SvRe → induced by the term: 1/4 I (2 (sum[j2,1,3,conj[ZVR[gt2,j2]] → sum[j1,1,3,conj[<<1>>] ml2[<<2>>]]]-sum[j2,1,3,conj[ZVI[<<2>>]] → sum[j1,1,3,Times[<<2>>]]]+sum[j2,<<2>>,conj[ZVR[gt2,Plus[<<2>>]]] → sum[j1,<<2>>,<<1>>]]-sum[j2,1,3,conj[ZVI[<<2>>]] → sum[j1,1,3,Times[<<2>>]]])+<<6>>)

Lagrangian::PossibleMixing: Possible mixing between hh and Ah → induced by the term: 1/2 I ((B[\[Mu]]-conj[B[\[Mu]]]) → (ZA[gt2,2] ZH[gt1,1]+ZA[gt2,1] ZH[gt1,2])+(B[MuP]-conj[B[MuP]]) → (ZA[gt2,4] ZH[gt1,3]+ZA[gt2,3] ZH[gt1,4]))

Lagrangian::PossibleMixing: Possible mixing between VP and VZ → induced by the term: gBL1 (x1^2+x2^2) Cos[ThetaW] (gBL1 → Cos[ThetaWp] Sin[ThetaW]-gBL Sin[ThetaWp]) ...

The last message is caused because it is not obvious from among the phases are satisfied. This mixing would be missed theLagrangianthatthereisno𝛾-𝑍 mixing: the relations inourimplementationsofar.Wejusthavetokeepinmind 耠 between the rotation angels Θ and Θ and all five involved that the model is supposed to be used for the CP conserving gauge couplings (𝑔푌푌, 𝑔퐵푌, 𝑔푌퐵 , 𝑔퐵퐵,and𝑔2) are not known case or just for parameters which satisfy conditions where the at this stage. Hence, it is not obvious from the Lagrangian that mixing between CP even and odd states vanishes. Of course, these terms cancel. one can also use the entries in parameters.m to define the Thereasonforthefirsttwomessagesisthat SARAH found parameters explicitly as real. terms in the Lagrangian of the mass eigenstates which seem Even if a study of CP violation is clearly beyond the scope 푅 퐼 to cause a mixing between ̃] and ̃] as well as between ℎ of this example, I want to show briefly what has to be changed ℎ and 𝐴 . The origin of this is that we did not define terms like to incorporate it. First, the definition of the VEVs has to be 耠 𝐵휇, 𝐵휇,or𝑌] explicitly as real. Therefore, SARAH considers changed by introducing relative phases between the scalars them to be complex. In the complex case a mixing between as follows. CP even and odd scalars would occur unless specific relations

1 (*--- VEVs ---- *) 2 DEFINITION[EWSB][VEVs]= 3 {{SHd0, {vd,1/Sqrt[2]}, {sigmad ,I/Sqrt[2]}, {phid ,1/Sqrt[2]}}, 4 {SHu0, {vu,1/Sqrt[2]}, {sigmau ,I/Sqrt[2]}, {phiu ,1/Sqrt[2]},{etaU}}, 5 {SC10, {x1,1/Sqrt[2]}, {sigma1 ,I/Sqrt[2]}, {phi1 ,1/Sqrt[2]},{etaX1}}, 6 {SC20, {x2,1/Sqrt[2]}, {sigma2 ,I/Sqrt[2]}, {phi2 ,1/Sqrt[2]},{etaX2}}, 7 {SvL, {0, 0}, {sigmaL ,I/Sqrt[2]}, {phiL ,1/Sqrt[2]}}, 8 {SvR, {0, 0}, {sigmaR ,I/Sqrt[2]}, {phiR ,1/Sqrt[2]},{etaR}} 9 }; 34 Advances in High Energy Physics

One could also work with complex VEVs as follows.

1 DEFINITION[EWSB][VEVs]= 2 {{SHd0, {vdR ,1/Sqrt[2]},{vdI,I/Sqrt[2]}, 3 {sigmad , I/Sqrt[2]},{phid ,1/Sqrt[2]}}, 4 {SHu0, {vuR ,1/Sqrt[2]},{vuI,I/Sqrt[2]}, 5 {sigmau , I/Sqrt[2]},{phiu ,1/Sqrt[2]}}, 6 ... 7 }

However,thisislesscommonandmainlysupposedtobeused the phases also the definition of the rotations must be changed for generating Vevacious model files. After introducing to respect the mixing between CP even and odd states as follows.

1 (*--- Matter Sector ---- *) 2 DEFINITION[EWSB][MatterSector]= 3 { ..., 4 {{phiL,phiR,sigmaL ,sigmaR}, {Sv, ZV}}, 5 {{phid, phiu,phi1, phi2, sigmad , sigmau ,sigma1 ,sigma2}, {hh, ZH}}, 6 ... }

The rotation matrices and mass eigenstates in particles.m states Sv need twelve PDGs, while hh includes two Gold- and parameters.m have to be adjusted accordingly: the stones and six physical scalars. Also assignments of the matrices ZVR and ZH wouldnolongerbereal,andthe Goldstones have to be adjusted in particles.m as follows.

1 {VZp, { Description -> "Z’-Boson", 2 Goldstone -> hh[{2}]}} 3 {VZ, { Description -> "Z-Boson", 4 Goldstone -> hh[{1}]}}

However,asIsaidthisisbeyondthescopeofthediscussion eigenstates. To get this output one has first to run the here. We start now to study the CP conserving version of this command ModelOutput which tells SARAH what it has to model in detail. calculate and for which eigenstates. We are just interested in the eigenstates after EWSB as follows.

5. Example—Part II: Masses, Vertices, ModelOutput[EWSB]; Tadpoles, and RGEs with Mathematica This command calculates always all vertices for the 5.1. LaTeX Output considered eigenstates. Loop corrections and RGEs are not included in the calculations by default. To include 5.1.1. General Information. Afterwearedonewiththeimple- them, the user can choose IncludeRGEs -> True mentation of the model and all consistency checks are passed, and IncludeLoopCorrections -> True.This we can generate a pdf filetogetafirstoverviewabout information will then also be included in the the B-L-SSM. Thetex . files generated by SARAH include LaTeX files. In principle, one can also use options all information about the model, for example, particle con- for ModelOutput to directly generate the LaTeX tent and superpotential, and all information which SARAH output and all other outputs for the different codes by derives like RGEs as well as masses and vertices for specific setting WriteTeX -> True, WriteFeynArts -> True, Advances in High Energy Physics 35

WriteCHep -> True, WriteWHIZARD -> True,andoption WriteSARAH -> True is that this includes helpful WriteUFO -> True.However,wewillnotmakeuseofthis information about the implementation: the names for all here but discuss each output separately. particles and parameters in SARAH aregiven,anditisshown When SARAH is done with all calculations, one can run howthepiecesarecalledinLaTeXandintheoutputfor the following. other codes. This output is given for all eigenstates. However, to follow the subsequent examples only the mass eigenstates MakeTeX[WriteSARAH ->True]; after EWSB are necessary. So, I skip the output for the other eigenstates and show here only the corresponding tables for We used here the option that not only the information the mass eigenstates. about the model and its physics is included, but also details Particles. The whole lists of fermions, scalars, vector bosons, about the implementation in SARAH are attached to the pdf. and ghosts are listed in Tables 2–5. One sees that not only Additional options which are available are as follows. the names of each particle are given which are used at the different stage, but also the indices which the parti- (i) FeynmanDiagrams: it defines if the Feynman dia- cles carry are defined. In the case of fermions, the Dirac grams should or should not be included in the spinors together with their Weyl components are listed. An output. By default they are included. To draw the alternative to get an overview about all particles during Feynman diagrams, SARAH makes use of the LaTeX the Mathematica session is to use the following. package feynmf [291]. (ii) ShortForm:itdefinesifashorternotation Particles[EWSB] for the vertices should be used by not using a equation separate environment for each vertex and Parameters. All parameters which are present at some stage in skipping Feynman diagrams. the B-L-SSM are listed in Table 6. This includes not only the fundamental parameters like gauge couplings, superpotential When SARAH is done with the output, the .tex files are couplings, and soft-breaking terms, but also rotation matrices stored in the following. and angles, VEVs, and auxiliary parameters which just show up via dependence defined in parameters.m.Onecanget − − $PATH/SARAH/Output/B L SSM/EWSB/TeX the entire list of parameters also during the Mathematica session by using the following. The main file which can be compiled with pdflatex is B-L-SSM EWSB.tex.InthecasewheretheFeynmandia- parameters grams are included, the compilation is a bit more complicated because mpost hastobeusedforeachdiagramafterthefirst run of pdflatex.Afterwards,asecondrunof pdflatex is 5.2. Extracting Mass Matrices, Tadpole Equations, and needed. SARAH provides a shell script MakePDF.sh which Vertices. We can finally do some physics. At first, we does take care of that. Thus, the easiest way to get the pdf is want to study the analytical properties of the B-L-SSM as follows. within Mathematica. For this purpose I will give some example how to extract mass matrices, vertices, or tadpole $ cd $PATH/SARAH/Output/B−L−SSM/EWSB/TeX equations and how to deal with them. To improve the $ chmod 755 MakePDF. sh readability I will give the input in Mathematica format but Mathematica $./MakePDF.sh the output of will be translated into LaTeX. If the user just wants to see the expressions without modifying them it is also possible to generate the entire LaTeX output for The second step is just necessary to make the script executable themodelandjustreadthe pdf asjustshownintheprevious if it is not. section.

5.1.2. Particles and Parameters of the B-L-SSM 5.2.1. Tadpole Equations. Westartwiththetadpoleequations. in SARAH. Thereasonwhywehavechosenthe All four minimum conditions are returned via

TadEquations={TadpoleEquation[vd], TadpoleEquation[vu], TadpoleEquation[x1], TadpoleEquation[x2]}

1 and read 0= V (2 (𝑔 𝑔 +𝑔 𝑔 )(V2 − V2) 8 푢 1 퐵푌 푌퐵 퐵 휂 휂 1 0=−V R𝐵 + V (2 (𝑔 𝑔 +𝑔 𝑔 )(V2 − V2) +(𝑔2 +𝑔2 +𝑔2)(V2 − V2 )) − V R (𝐵 ) 푢 휇 8 푑 1 퐵푌 푌퐵 퐵 휂 휂 1 푌퐵 2 푢 푑 푑 휇 2 2 2 2 2 2 󵄨 󵄨2 2 󵄨 󵄨2 +(𝑔1 +𝑔푌퐵 +𝑔2)(V푑 − V푢)) + V푑 (𝑚퐻 + 󵄨𝜇󵄨 ), + V (𝑚 + 󵄨𝜇󵄨 ), 푑 󵄨 󵄨 푢 퐻푢 󵄨 󵄨 36 Advances in High Energy Physics

Table 2: Fermions in the B-L-SSMafterelectroweaksymmetrybreaking.

LaTeX SARAH Output 𝜆− Lm [{generation}] − 푖 𝜒̃푖 =( ) Cha [{generation}]=( ) C +,∗ conj [Lp [{generation}]] 𝜆푖 𝜆0 L0 [{generation}] 0 푖 𝜒̃푖 =( ) Chi [generation]=( ) N 0,∗ conj [L0 [{generation}]] 𝜆푖

𝐷퐿,푖훼 FDL [{generation, color}] 𝑑푖훼 =( ) Fd [{generation, color}]=( ) d ∗ conj [FDR [{generation, color}]] 𝐷푅,푖훼

𝐸퐿,푖 FEL [{generation}] 𝑒푖 =( ) Fe [{generation}]=( ) e ∗ conj [FER [{generation}]] 𝐸푅,푖

𝑈퐿,푖훼 FUL [{generation, color}] 𝑢푖훼 =( ) Fu [{generation, color}]=( ) u ∗ conj [FUR [{ generation, color}]] 𝑈푅,푖훼

𝜆],푖 Fvm [{generation}] ]푖 =( ) Fv [{generation}]=( ) nu ∗ conj [Fvm [{generation}]] 𝜆],푖

𝜆푔,훼̃ fG [{color}] 𝑔̃훼 =( ) Glu [{color}] =( ) go ∗ conj fG color 𝜆푔,훼̃ [ [{ }]]

Table 3: Scalars in the B-L-SSMafterelectroweaksymmetrybreaking.

LaTeX SARAH Output LaTeX SARAH Output ̃ 𝑑푖훼 Sd[generation,color] sd 𝑢̃푖훼 Su[generation,color] su 푖 𝑒̃푖 Se[generation] se ]푖 SvIm[generation] nI 푅 ]푖 SvRe[generation] nR ℎ푖 hh[generation] h 0 − 𝐴푖 Ah[generation] Ah 𝐻푖 Hpm[generation] Hm,Hp

1 2 2 2 2 0=−V휂R𝐵휂 + V휂 (2 (𝑔퐵 +𝑔퐵푌)(V휂 − V휂) in the following. Alternatively, one can also use the content 4 of TadpleEquations[EWSB] which contains all tadpole 󵄨 󵄨2 +(𝑔 𝑔 +𝑔 𝑔 )(V2 − V2 )) + V (𝑚2 + 󵄨𝜇 󵄨 ), equations after EWSB separated by commas. To get the 1 퐵푌 푌퐵 퐵 푑 푢 휂 휂 󵄨 휂󵄨 same TadEquations as above, we make an equation out of any entry in TadpoleEquations[EWSB] by the following. 1 2 2 2 2 0= V휂 (−2 (𝑔퐵 +𝑔퐵푌)(V휂 − V휂) 4 TadEquations = Map[# == 0 &, TadpoleEquations[EWSB]]; 2 2 2 󵄨 󵄨2 +(𝑔 𝑔 +𝑔 𝑔 )(V − V )) + V (𝑚 + 󵄨𝜇 󵄨 ) 1 퐵푌 푌퐵 퐵 푢 푑 휂 휂 󵄨 휂󵄨 Let us gain some understanding of these equations. A 𝜇 𝜇耠 𝐵 𝐵耠 − V R (𝐵 ). convenient choice is to solve them for , , 휇,and 휇.For 휂 휂 simplicity we do this by restricting ourself to the real case (117) (conj[x ]->x)andworkinthetrianglebasiswhere𝑔푌퐵 disappears as follows. Isavedtheminanewlist TadEquations which we will use

solutionTad=Solve[TadEquations /. \[Mu] conj[\[Mu]] -> AbsMu2 /. MuP conj[MuP] -> AbsMuP2 /. conj[x_] -> x /. gBL1 -> 0, {B[\[Mu]], B[MuP], AbsMu2 , AbsMuP2}]

2 耠 2 2 In addition, we introduced new parameters for |𝜇| and |𝜇 | . equations in that way for |𝜇| ; (ii) if X and B[X] appear in There are mainly two reasons for this: (i) we can solve the the equations, Mathematica interprets B[X] as function Advances in High Energy Physics 37

Table 4: Vector bosons in the B-L-SSM after electroweak symmetry breaking.

LaTeX SARAH Output LaTeX SARAH Output

𝑔훼휌 VG[color,lorentz] g 𝛾휌 VP[lorentz] A 耠 𝑍휌 VZ[lorentz] Z 𝑍휌 VZp[lorentz] Zp − 𝑊휌 VWm[lorentz] Wm,Wp

Table 5: Ghost particles in the B-L-SSMafterelectroweaksymmetrybreaking.

LaTeX SARAH Output LaTeX SARAH Output 퐺 훾 𝜂훼 gG[color] gG 𝜂 gP gA 耠 𝜂푍 gZ gZ 𝜂푍 gZp gZp 𝜂− gWm gWm 𝜂+ gWmC gWpC

󵄨 󵄨2 1 −1 of X instead as an independent parameter. The consequence 󵄨𝜇耠󵄨 =− (−V2 + V2) (+4𝑚2V2 +𝑔 𝑔 V2 V2 󵄨 󵄨 휂 휂 휂 휂 푌퐵 퐵 푑 휂 is that it cannot solve the equations. To circumvent this, a 4 B[X]->BX replacement like would be necessary. −𝑔 𝑔 V2 V2 +2𝑔2 V4 −4𝑚2V2 +𝑔 𝑔 V2 V2 We are just interested for the moment in the solutions for 푌퐵 퐵 푢 휂 퐵 휂 휂 휂 푌퐵 퐵 푑 휂 2 耠 2 |𝜇| and |𝜇 | .Theyread 2 2 2 4 −𝑔푌퐵 𝑔퐵V푢V휂 −2𝑔퐵V휂). (118)

For a better understanding, we can make the approximation 󵄨 󵄨2 1 2 2 −1 2 2 2 4 2 4 󵄨𝜇󵄨 = (−V푢 + V푑) (+ 8𝑚퐻 V푑 +𝑔1V푑 +𝑔푌퐵 V푑 𝑔 =0 8 푑 of vanishing kinetic mixing ( 푌퐵 ): in this limit, as we will see below, the vector boson masses are given by 𝑀푍 = 2 4 2 2 2 4 2 4 2 4 2 2 2 耠 2 2 +𝑔2V푑 −8𝑚퐻 V푢 −𝑔1V푢 −𝑔푌퐵 V푢 −𝑔2V푢 √ 2 2 푢 (1/4)(𝑔1 +𝑔2)V and 𝑀푍 =𝑔퐵𝑥 ,withV = V푑 + V푢 and 2 2 2 2 2 2 𝑥=√V2 + V2 V → +2𝑔푌퐵 𝑔퐵V푑V휂 +2𝑔푌퐵 𝑔퐵V푢V휂 −2𝑔푌퐵 𝑔퐵V푑V휂 휂 휂. In addition, we make the replacements 푑 耠耠 V sin 𝛽, V푢 → V cos 𝛽, 𝑥1 →𝑥sin 𝛽 ,and𝑥2 →𝑥cos 𝛽 −2𝑔 𝑔 V2 V2), 耠耠 푌퐵 퐵 푢 휂 and express 𝛽, 𝛽 by tan 𝛽 (TB)andtan𝛽 (TBp) as follows.

SimplySolution=Simplify[solutionTad /. {x1 -> x Sin[BetaP], x2 -> x Cos[BetaP], vd -> v Sin[Beta], vu -> v Cos[Beta]} /. { Beta -> ArcTan[TB], BetaP -> ArcTan[TBp]} /. {g1BL ->0, v -> 2/Sqrt[(g1^2 + g2^2)] MZ, -> MZp/gBL} ]

We find quite simple expressions: The first expression is just the one of the MSSM. Thus, any new contribution to 𝜇 comes only from gauge-kinetic mixing. 󵄨 󵄨2 1 耠 󵄨𝜇󵄨 = (− (2𝑚2 +𝑀2 ) 2𝛽2 The equation for |𝜇 | looks very similar. However, the large 󵄨 󵄨 2 퐻푑 푍 tan 2(−1+tan 𝛽) ratio between 𝑀푍 and 𝑀푍耠 gives a much larger constraint: (119) 𝛽耠 2 2 for radiative symmetry breaking tan is usually restricted +2𝑚 +𝑀 ), 1 퐻푢 푍 to be close to to minimize the negative contributions. We Δ𝑚2 =𝑚2 −𝑚2 󵄨 󵄨2 1 can check this by assuming 휂 휂 together with 󵄨𝜇耠󵄨 = (− (2𝑚2 +𝑀2 ) 2𝛽耠 2 2 耠 2 󵄨 󵄨 耠 tan 𝑀 耠 = 2.5 𝑚 =1 |𝜇 | 󵄨 󵄨 2(−1+ 2𝛽耠) 휂 푍 푍 TeV and 휂 TeV . will then just tan 耠 2 (120) be a function of tan 𝛽 and Δ𝑚 .Usingthe ContourPlot 2 2 +2𝑚휂 +𝑀푍耠 ). function of Mathematica 38 Advances in High Energy Physics

Table 6: All parameters in the B-L-SSM with their names used internally by SARAH as well as the names for the LaTeX and other outputs.

LaTeX SARAH Output LaTeX SARAH Output LaTeX SARAH Output

𝑔1 g1 g1 𝑔2 g2 g2 𝑔3 g3 g3

𝑔퐵 gBL gBL 𝑔푌퐵 g1BL gYB 𝑔퐵푌 gBL1 gBY

𝜇휂 MuP MuP 𝐵휂 B[MuP] BMuP 𝜇\[Mu] Mu

𝐵휇 B[\[Mu]] Bmu 𝑌푑 Yd Yd 𝑇푑 T[Yd] Td

𝑌푒 Ye Ye 𝑇푒 T[Ye] Te 𝑌푢 Yu Yu

𝑇푢 T[Yu] Tu 𝑌푥 Yn Yx 𝑇푥 T[Yn] Tx 2 𝑌] Yv Yv 𝑇] T[Yv] Tv 𝑚푞 mq2 mq2 𝑚2 ml2 ml2 𝑚2 mHd2 mHd2 𝑚2 mHu2 mHu2 푙 퐻푑 퐻푢 2 2 2 𝑚푑 md2 md2 𝑚푢 mu2 mu2 𝑚푒 me2 me2 2 2 2 𝑚] mvR2 mv2 𝑚휂 mC12 mC12 𝑚휂 mC22 mC22

𝑀1 MassB M1 𝑀2 MassWB M2 𝑀3 MassG M3

𝑀퐵퐿 MassBp MBp 𝑀퐵퐵耠 MassBBp MBBp V푑 vd vd

V푢 vu vu V휂 x1 x1 V휂 x2 x2 耠 𝑍훾푍푍 ZZ ZZ 𝑍푊 ZW ZW 𝑍푊̃ ZfW ZfW 퐷 푈 𝜙푔̃ PhaseGlu pG 𝑍 ZD ZD 𝑍 ZU ZU 𝑍퐸 ZE ZE 𝑍푖 ZVI ZVI 𝑍푅 ZVR ZVR 𝑍퐻 ZH ZH 𝑍퐴 ZA ZA 𝑍+ ZP ZP 𝑁 ZN ZN 𝑈 UM UM 𝑉 UP UP 푉 푒 푒 𝑈 UV UV 𝑈퐿 ZEL ZEL 𝑈푅 ZER ZER 푑 푑 푢 𝑈퐿 ZDL ZDL 𝑈푅 ZDR ZDR 𝑈퐿 ZUL ZUL 푢 𝑈푅 ZUR ZUR 𝑒 eel Θ푊 ThetaW TW 耠 𝛽\[Beta] betaH Θ푊 ThetaWp TWp 𝛼푆 AlphaS aS 𝛼−1 aEWinv aEWinv V vv 𝛽耠 BetaP Bp 耠 ∗ 𝑥 vX vX tan(𝛽 ) TBetaP TBp Mass[𝑉𝑊𝑚] Mass[VWm] Mass[VWm]

𝐺푓 Gf Gf

ContourPlot[ AbsMuP2 /. SimplySolution[[1]] /. {mC12 -> mC22 - deltaM, MZp -> 2500} /. mC22 -> 10^6 {deltaM , 0, 10^6}, {TBp, 1, 1.5}, ContourLabels -> True, FrameLabel ->{\[CapitalDelta] Superscript[m, 2],"tan(\[Beta]’)"}] we get the plot shown in Figure 6. We see that the numbers mixing to obtain (119)-(120) is correct. To do that we check 耠耠 for |𝜇 | quickly drop and the entire area with tan 𝛽 > 1.1 is the vector boson mass matrix. That is done via the following ruled out. FullSimplify[ MassMatrix[VectorBoson] /. {x1 -> x Sin[BetaP], x2 -> x Cos[BetaP], vd -> v Sin[Beta], vu -> v Cos[Beta]]}]

5.2.2. Mass Matrices. Weturn now to the mass matrices. First, where all three possibilities (VectorBoson = VP, Ihavetoprovidetheproofthatourapproximationforthe VectorBoson = VZ,andVectorBoson = VZP)return vector boson masses in the limit of vanishing gauge-kinetic thesamemassmatrix:

1 1 1 𝑔2V2 +𝑔2 𝑥2 − 𝑔 𝑔 V2 𝑔 𝑔 V2 +𝑔 𝑔 𝑥2 4 1 퐵푌 4 1 2 4 1 푌퐵 퐵 퐵푌 1 1 1 ( − 𝑔 𝑔 V2 𝑔2V2 − 𝑔 𝑔 V2 ). 4 1 2 4 2 4 푌퐵 2 (121) 1 1 1 𝑔 𝑔 V2 +𝑔 𝑔 𝑥2 − 𝑔 𝑔 V2 𝑔2 V2 +𝑔2 𝑥2 4 1 푌퐵 퐵 퐵푌 4 푌퐵 2 4 푌퐵 퐵 Advances in High Energy Physics 39

1.5 1 2 2 2 2 𝑚휎 휎 = (2 (𝑔 +𝑔 )(−V + V ) 휂 휂 4 퐵 퐵푌 휂 휂

2 2 2 󵄨 󵄨2 +(𝑔 𝑔 +𝑔 𝑔 )(−V + V )) + 𝑚 + 󵄨𝜇 󵄨 , 1.4 1 퐵푌 푌퐵 퐵 푢 푑 휂 󵄨 휂󵄨 1 𝑚 = (−2 (𝑔2 +𝑔2 )(−V2 + V2) 휎휂휎휂 퐵 퐵푌 휂 휂 −3000000 −2000000 4 1.3 2 2 2 󵄨 󵄨2 +(𝑔 𝑔 +𝑔 𝑔 )(−V + V )) + 𝑚 + 󵄨𝜇 󵄨 㰀 1 퐵푌 푌퐵 퐵 푑 푢 휂 󵄨 휂󵄨 훽

tan tan (123) 1.2

2 2 耠 𝑚 (𝑍)𝜉 𝑚 (𝑍 )𝜉 耠 −1000000 1000000 and gauge fixing contributions 푍 and 푍 .We 1.1 see that the matrix is block diagonal. That means that there 300000 0 2000000 is no mixing between the CP odd components of the Higgs doublets and bileptons. However, this is a statement which 1.0 −4000000 is only strictly true at tree-level. Therefore, we kept the 0 200000 400000 600000 800000 1×106 mixing of both states in the model definition. One self- Δm2 consistency check to see that the Goldstone degrees of

耠 2 2 耠 2 freedom appear correctly is quickly done: this matrix should Figure 6: |𝜇 | in the (Δ𝑚 , tan 𝛽 ) plane using (120) with Δ𝑚 = 𝑚2 −𝑚2 𝑀 𝑚2 2 have two zero eigenvalues in Landau gauge. For this purpose 휂 휂.Ihavechosen 푍耠 =2.5TeVand 휂 =1TeV . and for simplicity we solve the tadpoles with respect to the soft-breaking scalar masses and plug the solution into the pseudoscalar mass matrix. In addition, we are going to Landau gauge (RXi[ ]-> 0). Thismatrixisblockdiagonalfor𝑔퐵푌 =𝑔푌퐵 =0with an upper 2×2matrix which is identical to the SM. Thus, we can see Solve[TadEquations , {mHd2, mHu2, mC12,mC22}]; MassMatrix[Ah] /. % /. RXi[_] -> 0; that gauge-kinetic mixing in this model is an important effect 耠 Eigenvalues[%] //. a_ conj[x_] + a_ x_ -> 2 a Re[x] because it leads to 𝑍-𝑍 mixing already at tree-level. In the scalar sector the mass matrix for the CP odd scalars ∗ In the last line, we replaced 𝑎𝑥 +𝑎𝑥 →R 2𝑎 (𝑥).The is printed via outcome is this handy list of eigenvalues: MassMatrix[Ah]

1 2 2 and reads {0, 0, (2V휂R (𝐵휂)+2V휂R (𝐵휂)) , 2V휂V휂 (124) 1 𝑚 R (𝐵 )0 0 (2V2 R (𝐵 )+2V2 R (𝐵 ))} . 휎푑휎푑 휇 푑 휇 푢 휇 2V푑V푢 R (𝐵휇)𝑚휎 휎 00 2 ( 푢 푢 ) 𝑚 0 = 퐴 00𝑚R (𝐵 ) 휎휂휎휂 휂 (122) So, we find the expected two massless modes. The physical 00R (𝐵 )𝑚 ( 휂 휎휂휎휂 ) pseudoscalars have at tree-level the same expressions as in the 𝐵 2 2 耠 MSSM by replacing the corresponding VEVs and -terms for +𝜉푍𝑚 (𝑍) +𝜉푍耠 𝑚 (𝑍 ) the 𝐵-𝐿 sector. The scalar mass matrix is given by with MassMatrix[hh]

and is a bit more complicated. We parametrize it by 1 2 2 𝑚휎 휎 = (2 (𝑔1𝑔퐵푌 +𝑔푌퐵 𝑔퐵)(−V + V ) 푑 푑 8 휂 휂

2 2 2 2 2 2 󵄨 󵄨2 +(𝑔 +𝑔 +𝑔 )(−V + V )) + 𝑚 + 󵄨𝜇󵄨 , 𝑚휙 휙 𝑚휙 휙 𝑚휙 휙 𝑚휙 휙 1 푌퐵 2 푢 푑 퐻푑 󵄨 󵄨 푑 푑 푢 푑 휂 푑 휂 푑 𝑚 𝑚 𝑚 𝑚 1 휙푑휙푢 휙푢휙푢 휙휂휙푢 휙휂휙푢 𝑚 = (2 (𝑔 𝑔 +𝑔 𝑔 )(−V2 + V2) ( ) (125) 휎푢휎푢 1 퐵푌 푌퐵 퐵 휂 휂 𝑚 𝑚 𝑚 𝑚 8 휙푑휙휂 휙푢휙휂 휙휂휙휂 휙휂휙휂 2 2 2 2 2 2 󵄨 󵄨2 𝑚 𝑚 𝑚 𝑚 +(𝑔 +𝑔 +𝑔 )(−V + V )) + 𝑚 + 󵄨𝜇󵄨 , 휙푑휙휂 휙푢휙휂 휙휂휙휂 휙휂휙휂 1 푌퐵 2 푑 푢 퐻푢 󵄨 󵄨 40 Advances in High Energy Physics

1 with 𝑚휙 휙 =− (𝑔1𝑔퐵푌 +𝑔푌퐵 𝑔퐵) V푑V휂, 푑 휂 2 1 𝑚 = (𝑔 𝑔 +𝑔 𝑔 ) V V , 휙푢휙휂 1 퐵푌 푌퐵 퐵 푢 휂 1 2 2 2 𝑚휙 휙 = (2 (𝑔1𝑔퐵푌 +𝑔푌퐵 𝑔퐵)(−V + V ) 푑 푑 8 휂 휂 𝑚 =−(𝑔2 +𝑔2 ) V V − R (𝐵 ), 휙휂휙휂 퐵 퐵푌 휂 휂 휂 2 2 2 2 2 2 󵄨 󵄨2 +(𝑔1 +𝑔푌퐵 +𝑔2)(3V푑 − V푢)) + 𝑚퐻 + 󵄨𝜇󵄨 , 푑 󵄨 󵄨 1 𝑚 = (−2 (𝑔2 +𝑔2 )(−3V2 + V2) 휙휂휙휂 퐵 퐵푌 휂 휂 1 2 2 2 4 𝑚휙 휙 =− (𝑔1 +𝑔푌퐵 +𝑔2) V푑V푢 − R (𝐵휇), 푑 푢 4 󵄨 󵄨2 +(𝑔 𝑔 +𝑔 𝑔 )(−V2 + V2 )) + 𝑚2 + 󵄨𝜇 󵄨 . 1 퐵푌 푌퐵 퐵 푑 푢 휂 󵄨 휂󵄨 1 2 2 2 2 𝑚휙 휙 = (2𝑔1𝑔퐵푌 (V − V )+2𝑔푌퐵 𝑔퐵 (V − V ) 푢 푢 8 휂 휂 휂 휂 (126) 2 2 2 2 2 2 󵄨 󵄨2 +(−𝑔1 −𝑔2 −𝑔푌퐵 )(V푑 −3V푢)) + 𝑚퐻 + 󵄨𝜇󵄨 , Onecanseethatthismatrixisingeneralnotblockdiagonal: 푢 that is, there is already a mixing between the MSSM doublets 1 and the bileptons at tree-level. However, all terms 𝑚휙 휙 with 𝑚 = (𝑔 𝑔 +𝑔 𝑔 ) V V , 푖 푗 휙푑휙휂 1 퐵푌 푌퐵 퐵 푑 휂 2 𝑖=𝑑,𝑢and 𝑗=𝜂,𝜂 are proportional to (𝑔1𝑔퐵푌 +𝑔푌퐵 𝑔퐵); 1 that is, this mixing is only visible if gauge-kinetic mixing is 𝑚휙 휙 =− (𝑔1𝑔퐵푌 +𝑔푌퐵 𝑔퐵) V푢V휂, 푢 휂 2 taken into account. That is another reason why gauge-kinetic mixing is in general a very important effect in this model. We 1 2 2 2 2 canalsotrytogetanestimateofthesizeofthismixing.For 𝑚휙 휙 = (2 (𝑔 +𝑔 )(3V − V ) 휂 휂 4 퐵 퐵푌 휂 휂 this purpose, we plug the solution of the tadpole equations in 𝑔 = 2 2 2 󵄨 󵄨2 theHiggsmassmatrixandfixsomenumericalvalues: 1 +(𝑔 𝑔 +𝑔 𝑔 )(−V + V )) + 𝑚 + 󵄨𝜇 󵄨 , 耠耠 2 12 1 퐵푌 푌퐵 퐵 푢 푑 휂 󵄨 휂󵄨 0.36, 𝑔2 = 0.63, 𝑔퐵퐿 =0.5, 𝜇=𝜇 =1TeV,𝐵휇 =𝐵휇 =1TeV , .

numMhh = MassMatrix[hh] /. solutionTad2 /. gBL1 ->0 /. conj[x_]->x /.{x1 -> x Sin[BetaP], x2 -> x Cos[BetaP], vd -> v Sin[Beta], vu → -> v Cos[Beta]} /. {BetaP -> ArcTan[1.1], Beta -> ArcTan[10.], v -> 246} /. {g1 -> 0.36, g2 -> 0.63, gBL -> 0.5} /. { B[\[Mu]] -> 10^6, \[Mu] -> 10^3,B[MuP] -> 10^6, MuP -> 10^3} numMhh is now the mass matrix which just depends on the thisfunctionextractsthetwolightestmassesaswellasthe bilepton VEV 𝑥 and the off-diagonal gauge coupling. We bilepton admixture of the lightest eigenstates and returns can write a simple function which diagonalizes this mass these three values as follows. matrix for given values of these two parameters. Furthermore,

FunctionHiggEigenvalues[xInput_ , g1BLinput_] := Block[{eig}, eig = Eigensystem[simpHH /. {x -> xInput , g1BL -> g1BLinput}]; bileptonFrac = Drop[eig[[2, -1]], {3, 4}]; bileptonFrac=bileptonFrac.bileptonFrac; Return[{Sqrt[eig[[1, -1]]], Sqrt[eig[[1, -2]]], bileptonFrac}]; ];

We can use this new function with the ContourPlot command of Mathematica as follows.

ContourPlot[FunctionHiggEigenvalues[x, g1BL][[NUMBER]], {x, 1000, 3000}, {g1BL, -0.1,0.1}, ContourLabels -> True, ImageSize -> 250] Advances in High Energy Physics 41 where NUMBER = 1, 2, 3 should be used to get all three MassMatrix[Se][[3, 3]] plots shown in Figure 7. We see at these plots that the mixing isespeciallylargewhenbothstatesarecloseinmassandcan We can rewrite the terms under the assumption that only 𝑂(10 ) be easily % and more. third generation Yuka was a nonzero. For this purpose, we We can now turn the sfermion sector. The matrices there expand the sum and put all entries of 𝑌푒 to zero but the areusuallyquitelengthy.Ijustwanttopickoutoneimportant (3, 3) one. In addition, we make the assumption that gauge- effect which we see in the diagonal entries of the charged kinetic mixing vanishes for simplicity and we can use the 𝜏̃ sleptons, for instance. The entry corresponding to 퐿 in the usual replacements for the VEVs as follows. 6×6mass matrix of the sfermions (Se)isshownviathe following.

% /. sum[a_, b_, c_, d_] :> Sum[d, {a, b, c}] /. Ye[3, 3] -> Ytau /. Ye[a__] -> 0 /. {x1 -> x Sin[BetaP], x2 -> x Cos[BetaP], vd -> v Sin[Beta], → vu -> v Cos[Beta]} /. {v -> 2/Sqrt[(g1^2 + g2^2)] MZ, x -> MZp/gBL, gBL1 -> 0, → g1BL -> 0} /. conj[x_] -> x

耠 The entries then read The important point is the appearance of 𝑀푍 which gives largenegativecontributionsbecausethelowerlimitonthis 耠 mass is about 2.5 TeV. Thus, tan 𝛽 must be close to 1 to minimize this term and to prevent tachyons. 1 − (2 (𝑔2 −𝑔2)𝑀2 2𝛽 The last mass matrix we want to check is the one of the 2 2 1 2 푍 cos 4(𝑔1 +𝑔2) neutralinos. This mass matrix is returned by 2 2 2 (127) −8𝑀푍𝑌휏 (sin 𝛽) MassMatrix[Chi] +(𝑔2 +𝑔2)(−4𝑚2 +𝑀2 2𝛽耠)) . 1 2 푙,33 푍耠 cos and reads

1 1 𝑀 0−𝑔 V 𝑔 V 𝑀 耠 −𝑔 V 𝑔 V 1 2 1 푑 2 1 푢 퐵퐵 퐵푌 휂 퐵푌 휂 1 1 ( 0𝑀 𝑔 V − 𝑔 V 000) ( 2 2 2 푑 2 2 푢 ) ( ) ( 1 1 1 ) (− 𝑔1V푑 𝑔2V푑 0−𝜇−𝑔푌퐵 V푑 00) ( 2 2 2 ) ( 1 1 1 ) . ( 𝑔 V − 𝑔 V −𝜇 0 𝑔 V 00) (128) ( 1 푢 2 푢 푌퐵 푢 ) ( 2 2 2 ) ( 1 1 ) ( 𝑀 耠 0−𝑔 V 𝑔 V 𝑀 −𝑔 V 𝑔 V ) 퐵퐵 2 푌퐵 푑 2 푌퐵 푢 퐵퐿 퐵 휂 퐵 휂 −𝑔퐵푌V휂 000−𝑔퐵V휂 0−𝜇휂

( 𝑔퐵푌V휂 000𝑔퐵V휂 −𝜇휂 0 )

The upper 4×4block is the one known from the MSSM. The bemadeliketheCPevenandoddmasssplittingforthe lower 3×3 block is the counterpart in 𝐵-𝐿 sector of this model. sneutrinos. Here, we make a similar observation as for the scalar Higgs mass matrix: both blocks are only coupled if gauge-kinetic 5.2.3. Vertices mixingistakenintoaccount. In the same way all other mass matrices of the model Single Vertices. We continue with vertices and show how they can be checked with SARAH and interesting observations can can be handled in SARAH. Let uss assume one is interested in 42 Advances in High Energy Physics

0.10 0.10

50 60 70 88 80 92 0.05 0.05

0.00 45 0.00 98

−0.05 −0.05 94 96 75 65 90 55

−0.10 85 −0.10 1000 1500 2000 2500 3000 1000 1500 2000 2500 3000

(a) (b) 0.10 0.8 0.2

0.40.6 0.05

0.00 1

−0.05 0.3 0.7

0.9 −0.10 0.1 0.5 1000 1500 2000 2500 3000

(c)

Figure 7: The two lightest eigenvalues of the scalar mass matrix at tree-level ((a) and (b)) and the bilepton fraction of the lightest eigenstates (c) in the (𝑥;푌퐵 𝑔 )plane.

퐻 the coupling between two up-quarks and the neutral CP even 𝑈퐿, 𝑈푅,and𝑍 are rotation matrices as shown in Table 6, scalars: 𝑖, 𝑗,and𝑘 are the generation indices of the external states, and 𝛼 and 𝛽 are the colour indices of the quarks. Vertex[{bar[Fd], Fd, hh}] One sees that only the projection on the second gauge eigenstate (𝑍푘2) contributes which corresponds to the up- and it gives Higgs. Thus, this is the same vertex as in the MSSM and the new 𝐵-𝐿 sector does not contribute here. That is 1 3 3 1−𝛾 −𝑖 𝛿 ∑𝑈푢,∗ ∑𝑈푢,∗ 𝑌 𝑍퐻 ( 5 ), different if one considers, for instance, the neutrino-scalar √ 훼훽 퐿,푗푏 푅,푖푎 푢,푎푏 푘2 2 2 푏=1 푎=1 vertex: (129) 1 3 3 1+𝛾 −𝑖 𝛿 ∑ ∑𝑌∗ 𝑈푢 𝑈푢 𝑍퐻 ( 5 ). √ 훼훽 푢,푎푏 푅,푗푎 퐿,푖푏 푘2 2 Vertex[{Fv,Fv, hh}] 2 푏=1 푎=1 Advances in High Energy Physics 43

耠 and it returns constraints on the angle Θ푊.Thatisofcoursethecaseforany 𝑍-interaction in this model.

3 3 3 All Vertices. It has been already mentioned that it is also 1 푉, ∗ 푉, ∗ 퐻 푉, ∗ −𝑖 (∑𝑈 ∑𝑈 𝑌 𝑍 + ∑𝑈 possible to calculate all vertices at once. The command to do √2 푗푏 푖3+푎 ],푎푏 푘2 푖푏 푏=1 푎=1 푏=1 this is as follows. 3 3 3 푉, ∗ 퐻 푉, ∗ 푉, ∗ MakeVertexList[EWSB] ⋅ ∑𝑈푗3+푎𝑌],푎푏𝑍푘2 +(∑𝑈푗3+푏 ∑𝑈푖3+푎𝑌푥,푎푏 푎=1 푏=1 푎=1 This creates lists 3 3 1−𝛾 + ∑𝑈푉, ∗ ∑𝑈푉, ∗ 𝑌 )𝑍퐻 )( 5 ), 푖3+푏 푗3+푎 푥,푎푏 푘3 2 SA‘VertexList[$TYPE] 푏=1 푎=1

1 3 3 TYPE −𝑖 (∑ ∑𝑌∗ 𝑈푉 𝑈푉𝑍퐻 for the different generic types of vertices ($ √ ],푎푏 푗3+푎 푖푏 푘2 (130) = SSS, SSSS, SSV, SSVV, SVV, FFS, FFV, VVV, VVVV, 2 푏=1 푎=1 GGS,and GGV). One can also play a bit with these lists. For 耠 3 3 instance, to get all fermion interactions with the 𝑍 one can ∗ 푉 푉 퐻 + ∑ ∑𝑌],푎푏𝑈푖3+푎𝑈푗푏𝑍푘2 use the Select command of Mathematica as follows. 푏=1 푎=1 Select[SA‘VertexList[FFV], FreeQ[#, VZp] == False &] 3 3 ∗ 푉 푉 +(∑ ∑𝑌푥,푎푏𝑈푗3+푎𝑈푖3+푏 푏=1 푎=1 Similarly, all scalar interactions can be extracted where the off-diagonal gauge couplings show up. 3 3 1+𝛾 + ∑ ∑𝑌∗ 𝑈푉 𝑈푉 )𝑍퐻 )( 5 ). 푥,푎푏 푖3+푎 푗3+푏 푘3 2 Select[SA‘VertexList[SSSS], 푏=1 푎=1 (FreeQ[#, g1BL] == False) &];

If one compares the length of this list with the length of all Wefindherealsoprojectionsonthethirdgaugeeigenstate four scalar interactions in general which comes from the ̂]𝜂̂̂]-term in the superpotential. In general, many vertices get modified with respect to the MSSM and a discussion of all effects is far beyond the scope of this Length[%] - Length[SA‘VertexList[SSSS]] paper. I just want to pick out one more vertex: the electron-𝑍 interaction, as follows. it turns out that this is the case for any vertex. That is of course not surprising because of the 𝐷-term contributions, but it Vertex[{bar[Fe],Fe, VZ}] underlines the importance of this effect again.

We find that this vertex receives important modification due 耠 5.3. Understanding the RGEs to the 𝑍-𝑍 mixing: 5.3.1. Analytical Results. The full two-loop RGEs of the B-L- SSM are calculated just via 𝑖 𝛿 (− (𝑔 +𝑔 ) Θ耠 Θ 2 푖푗 1 퐵푌 cos 푊 sin 푊 CalcRGEs[];

+𝑔 Θ Θ耠 +(𝑔 +𝑔 ) Θ耠 )(𝛾 2 cos 푊 cos 푊 푌퐵 퐵 sin 푊 휇 The options for CalcRGEs are as follows. 1−𝛾 ⋅ 5 ), (i) TwoLoop: it defines if two-loop RGEs should be 2 (131) calculated. This is done by default. 𝑖 − 𝛿 ((2𝑔 +𝑔 ) Θ耠 Θ 2 푖푗 1 퐵푌 cos 푊 sin 푊 (ii) ReadLists: it defines if the results from previous calculations should be read instead of calculating the 1+𝛾 −(2𝑔 +𝑔 ) Θ耠 )(𝛾 ⋅ 5 ). RGEs again. 푌퐵 퐵 sin 푊 휇 2 (iii) VariableGenerations: it defines if the generations of some particles should be treated as free Working in the triangle basis, the contributions from 𝑔퐵푌 parameters. The RGEs are then expressed in terms vanish. However, the coupling compared to SM expectation of NumberGenertions[X],where X isthenameof 耠 gets still modified by the presence of sin Θ푊.Thisgivesstrong the superfield. 44 Advances in High Energy Physics

耠 (iv) NoMatrixMultiplication:itcanbesetiftheRGEs (v) BetaBij: bilinear soft-breaking parameters𝐵 ( 휇, 𝐵휇), should not be expressed in terms of matrix multipli- Betam2ij 𝑚2 𝑚2 𝑚2 𝑚2 cation but by using sums over indices. (vi) :scalarsquaredmasses( 푞̃, 푑̃, 푢̃, 푒̃, 2 2 2 2 2 𝑚̃ , 𝑚퐻 , 𝑚퐻 , 𝑚휂,and𝑚휂), (v) IgnoreAt2Loop: it can be used to define parameters 푙 푑 푢 which should be put to zero in the two-loop calcula- (vii) BetaMi:gauginomasses(𝑀1, 𝑀2, 𝑀3, 𝑀퐵,and tion. 𝑀퐵퐵耠 ), (vi) WriteFunctionsToRun: it defines if a file should (viii) BetaGauge:gaugecouplings(𝑔1, 𝑔2, 𝑔3, 𝑔퐵, 𝑔퐵푌,and bewrittentoevaluatetheRGEsnumerically 𝑔푌퐵 ), in Mathematica. This is done by default and we are BetaVEVs V V V V going to make use of it. (ix) :VEVs( 푑, 푢, 휂,and 휂). When the calculation is finished, the full two-loop RGEs are BetaQijkl, BetaWijkl, BetaLi, BetaSLi, BetaDGi, saved in different arrays: and BetaFIi are empty in this model. All lists are three- dimensional arrays where each entry gives (i) the parameter, Gij 2 (i) : anomalous dimensions of all chiral superfields, (ii) the one-loop 𝛽-function (up to a factor 1/16𝜋 ), and (iii) 2 2 (ii) BetaYijk: trilinear superpotential parameters (𝑌푑, the two-loop 𝛽-function (up to a factor 1/(16𝜋 ) ). To check 𝑌푒, 𝑌푢, 𝑌푥,and𝑌]), the order in which the RGEs for the trilinear superpotential 耠 (iii) BetaMuij: bilinear superpotential parameters (𝜇, 𝜇 ), parameters are given, one can use Transpose[BetaYijk][[1]] (iv) BetaTijk: trilinear soft-breaking parameters (𝑇푑, 𝑇푒, 𝑇푢, 𝑇푥,and𝑇]), what returns

{Yd[i1,i2],Ye[i1,i2],Yu[i1,i2],Yn[i1,i2],Yv[i1,i2]}

𝛽(1) =−3𝑔3, Thus, if we want to see the one-loop RGE for the electron 푔3 3 Yukawa coupling we have to check (1) 3 2 √ 2 3 𝛽푔 = (11𝑔푌퐵 𝑔퐵 +4 10𝑔푌퐵 𝑔퐵 + 15𝑔퐵 BetaYijk[[2,2]] 퐵 5 √ √ 2 + 11𝑔1𝑔푌퐵 𝑔퐵푌 +2 10𝑔1𝑔퐵𝑔퐵푌 +2 10𝑔푌퐵 𝑔퐵푌 and get 2 + 15𝑔퐵𝑔퐵푌), 1 +3𝑌𝑌†𝑌 +𝑌𝑌†𝑌 − 𝑌 (3 (3√10𝑔 𝑔 푒 푒 푒 푒 ] ] 10 푒 1 퐵푌 (1) 3 √ 𝛽푔 = (𝑔1 (15𝑔퐵 +2 10𝑔푌퐵 )𝑔퐵푌 푌퐵 5 +3√10𝑔 𝑔 +5(2𝑔2 +𝑔2 +𝑔2 )+6𝑔2 (132) 푌퐵 퐵 2 퐵 퐵푌 1 2 √ +𝑔1 (11𝑔푌퐵 +2 10𝑔퐵) 2 † † +6𝑔 )−30Tr (𝑌푑𝑌 )−10Tr (𝑌푒𝑌 )) . 푌퐵 푑 푒 2 2 √ +𝑔푌퐵 (11𝑔푌퐵 +15𝑔퐵 +4 10𝑔푌퐵 𝑔퐵)) ,

One sees that gauge-kinetic mixing is also taken here into (1) 3 2 𝑔 =𝑔 =𝑔 =0 𝛽푔 = (11𝑔1𝑔퐵푌 account. In the limit 퐵 퐵푌 푌퐵 this reproduces the 퐵푌 5 well-known MSSM result. Maybe, more interesting are new √ 2 2 features in the gauge sector. To get just all one-loop RGEs at +𝑔1 (11𝑔푌퐵 𝑔퐵 +2 10 (2𝑔퐵푌 +𝑔퐵)) once, we can execute 2 2 √ +𝑔퐵푌 (15 (𝑔퐵 +𝑔퐵푌)+2 10𝑔푌퐵 𝑔퐵)) . BetaGauge /. {a_, b_, c_} -> {a, b} (133) that returns The expressions for 𝑔2 and 𝑔3 are just the MSSM results 𝑔 (1) 3 3 √ 2 but 1 gets modified by gauge-kinetic mixing. Solving these 𝛽푔 = (11𝑔1 +4 10𝑔1𝑔퐵푌 1 5 equations analytically is no longer possible but we are going to study them numerically. 2 2 √ +𝑔1 (11𝑔푌퐵 + 15𝑔퐵푌 +2 10𝑔푌퐵 𝑔퐵) Finally, we can also check analytical expressions for soft- breaking terms. We will do this at the example of the bilepton √ +𝑔푌퐵 (15𝑔퐵 +2 10𝑔푌퐵 )𝑔퐵푌), masses which are given in the last two entries of Betam2ij. Even more interesting is the difference between both 𝛽- 𝛽(1) =𝑔3, 푔2 2 functions: we see from (120) that a large mass splitting Advances in High Energy Physics 45

between both masses is needed to get radiative symmetry possibility to get a large mass splitting is large values for 𝑌푥 2 2 breaking. Thus, starting with the same values at the GUT and 𝑇푥 and soft-masses 𝑚] and 𝑚휂.Wewillcomebacktothis scale, the differences in the 𝛽-functions are crucial in order when we study the model with SPheno. to break 𝐵-𝐿 or not. To see the difference, we can use the following: 5.3.2. Numerical Results. The RGEs for the gauge coupling demand a closer look. However, the analytical expres- Betam2ij[[-1, 2]] - Betam2ij[[-2, 2]] sions at one-loop are already a bit complicated and it is hard to learn something from them. Before turning and find to the full numerical analysis of the entire set of RGEs with SPheno there is the possibility to study the RGEs 2√6𝑔 𝜎 +2√6𝑔 𝜎 −4 (𝑇∗𝑇 ) 퐵 1,4 퐵푌 1,1 Tr 푥 푥 in Mathematica first: CalcRGEs creates a file which con- (134) tains all 𝛽-functions in a format which can be used −4𝑚2 (𝑌 𝑌∗)−8 (𝑚2𝑌 𝑌∗). 휂 Tr 푥 푥 Tr ] 푥 푥 with NDSolve in Mathematica to solve the RGEs. This file is loaded via 𝜎1,4 and 𝜎1,1 are abbreviations for often appearing traces over scalar masses. These can be found in TraceAbbr: << "$PATH/SARAH/Output/B-L-SSM/RGEs/RunRGEs.m"

1 √ 2 2 2 𝜎1,1 = (4 15𝑔1 (−2 Tr (𝑚푢)−Tr (𝑚푙 )−𝑚퐻 In addition, also a function is provided to perform the RGE 20 푑 running. 2 2 2 2 +𝑚퐻 + Tr (𝑚푑)+Tr (𝑚푒 )+Tr (𝑚푞)) 푢 RunRGEs[values , start, finish , Options] −5√6𝑔 (−2𝑚2 +2 (𝑚2)−2 (𝑚2)+2𝑚2 퐵푌 휂 Tr 푙 Tr 푞 휂 The input is as follows: 2 2 2 2 values − Tr (𝑚푒 )−Tr (𝑚])+Tr (𝑚푑)+Tr (𝑚푢))) , (i) : all nonzero values for parameters at the scale (135) where the running starts, 1 √ 2 2 2 start 𝜎1,4 = (4 15𝑔푌퐵 (−2 Tr (𝑚푢)−Tr (𝑚푙 )−𝑚퐻 (ii) :logarithmofthescalewheretherunning 20 푑 starts, +𝑚2 + (𝑚2 )+ (𝑚2)+ (𝑚2)) finish 퐻푢 Tr 푑 Tr 푒 Tr 푞 (iii) :logarithmofthescalewheretherunning should end, −5√6𝑔 (−2𝑚2 +2 (𝑚2)−2 (𝑚2)+2𝑚2 퐵 휂 Tr 푙 Tr 푞 휂 (iv) Options: optionally two-loop contributions can be TwoLoop->False 2 2 2 2 turned off ( ). − Tr (𝑚푒 )−Tr (𝑚])+Tr (𝑚푑)+Tr (𝑚푢))) . Since the running at one-loop for 𝑔1, 𝑔2,and𝑔3 is the same as Ifonestartswithamodelinwhichallscalarsunifyaswehave in the MSSM in the limit of vanishing gauge-kinetic mixing, 16 in mind according to Section 4.2,onegets𝜎1,1 =𝜎1,4 =0. we should find the same unification at about 2⋅10 GeV.This ThisisaRGEinvariantanddoesalwayshold.Thus,theonly canbetestedvia

solutionMSSM = RunRGEs[{g1->0.46, g2->0.64, g3->1.09}, 3, 17, TwoLoop ->False]; Plot[{g1[t], g2[t], g3[t]} /. solutionMSSM , {t, 3, 17}, Frame ->True, Axes->False , FrameLabel ->{"log[Q/GeV]",Subscript["g", "i"]}]

Here, I used for the SM gauge couplings the DR values as g2[ Log [10 , 2 10^16]] /. solutionMSSM input which are found when including thresholds discussed in Section 3.6.3 with SUSY states of about 1 TeV. The plot → {0.697832} whichiscreatedviathesetwocommandsisshowninFigure 8 on the left. We find a value of about 0.7 of the gauge couplings We can now use this value and demand a strict unification at the unification scale. (𝑔1 =𝑔2 =𝑔3 =𝑔퐵)andrundowntheRGEs.

solutionBLSSM = RunRGEs[{g1->0.7, g2->0.7, g3->0.7, gBL->0.7}, 16, 3, → TwoLoop ->False]; 46 Advances in High Energy Physics

We can check what we get for 𝑔1 at 1 TeV. And actually the have to associate with the physical hypercharge coupling is naive try as follows.

g1[3] /. solutionBLSSM (g1[3]*gBL[3] - g1BL[3]*gBL1[3])/ Sqrt[gBL[3]^2 + gBL1[3]^2] /. solutionBLSSM returns 0.476. That is quite a bit away from the input value of This indeed returns exactly the input value of 0.46. We can 0.46 we started with. The reason is that we missed performing now plot the proper couplings via the rotations to go to the correct basis; see (93).Whatwe

Plot[{(g1[t]*gBL[t] - g1BL[t]*gBL1[t])/Sqrt[gBL[t]^2 + gBL1[t]^2], g2[t], g3[t], Sqrt[gBL[t]^2 + gBL1[t]^2], (g1BL[t]*gBL[t] + gBL1[t]*g1[t])/Sqrt[gBL[t]^2 + gBL1[t]^2]} /. solutionBLSSM , {t, 3, 17}, Frame -> True, Axes -> False, FrameLabel -> {"log[Q/GeV]", Subscript["g", "i"]}]

The plot is shown on the right in Figure 8. One sees herethat the off-diagonal coupling gets negative. The values of the five physical running couplings at 3 TeV are as follows.

{(g1[t]*gBL[t] - g1BL[t]*gBL1[t])/Sqrt[gBL[t]^2 + gBL1[t]^2], g2[t], g3[t], Sqrt[gBL[t]^2 + gBL1[t]^2], (g1BL[t]*gBL[t] + gBL1[t]*g1[t])/Sqrt[gBL[t]^2 + gBL1[t]^2]} /. t->Log[10,3000] /. solutionBLSSM

→ {0.464058, 0.640818, 1.07975, 0.442469, -0.142595}

Here, the GUT normalization is still included for the 𝑈(1) 𝑔̃ = −0.110454. couplings. The not normalized values are (136)

Withthesameprocedurewecouldnowalsostarttoanalyse 𝑔1 = 0.359458, therunningofthesuperpotentialtermsandthesoft-masses. An estimate of the running gaugino masses based on univer- 𝑔퐵퐿 = 0.541912, sal GUT values is obtained by

solGauginos = RunRGEs[{g1 -> 0.7, g2 -> 0.7, g3 -> 0.7, gBL -> 0.7, MassB -> 1000, MassWB -> 1000, MassG -> 1000, MassBp -> 1000, MassBBp -> 0}, 17, 3, TwoLoop -> False];

and we find come back to that in the next section. We also see that the off-diagonal terms run negative similar to the off-diagonal {MassB[t], MassWB[t], MassG[t], MassBp[t],MassBBp[t]} coupling. In the limit of vanishing kinetic mixing it is easy to /. t->Log[10,3000] /. solGauginos explain the hierarchy of the gaugino masses: combining (42) 2 → {480.986, 838.056, 2379.32, 399.549, -128.763} and (52) we see that the value 𝑀푖/𝑔푖 is a constant at one-loop: that is, 耠 The hierarchy is similar to the CMSSM, but the 𝐵 soft mass 𝑔2 issmallerthanallothergauginomasses.So,itmightbethat 𝑀 = 푖 𝑀 . 푖 𝑔2 1/2 (137) this particle would be a new dark matter candidate. I will GUT Advances in High Energy Physics 47

1.1 1.0 1.0 0.8 0.9 0.6 i

0.8 i g g 0.4 0.7 0.2 0.6 0.0 0.5

4 6 8 10 12 14 16 4 6 8 10 12 14 16 log[Q/GeV] log[Q/GeV]

(a) (b)

Figure 8: On the left: running of the three gauge couplings in the MSSM limit; on the right: running inthe 𝐵-𝐿-SSM including gauge-kinetic mixing.

Plugging in the numbers from (136) we get for the gaugino (i) “GUT Version.” In a GUT version of SPheno aRGE mass terms running between the electroweak, SUSY, and GUT scaleissupported.Theusercandefineappropriate 𝑀1 ≃ 439.5 GeV, boundary conditions at each of these three scales. Furthermore, also threshold effects by including addi- 𝑀 ≃ 838.0 , 2 GeV tional scales where heavy particles are integrated (138) out can optionally be included. Finally, the user 𝑀3 ≃ 2379.3 GeV, can define a condition which has to be satisfied to 𝑀퐵耠 ≃ 399.5 GeV. identify the GUT scale. The most common choice is the unification scale of gauge couplings, but also other choices like Yukawa unification are possible. In The values for 𝑀1 and 𝑀퐵耠 are a bit different because addition, these versions include also the possibility to (137) does not include the effect of gauge-kinetic mixing. define the entire input at the SUSY scale and skip the Nevertheless, there is a nice agreement with the numerical RGE running to the GUT scale. results. (ii) “Low Scale” Version. In a low scale version no RGE running is included, but the SPheno version expects 6. Example—Part III: Mass Spectrum, Decays, all free parameters to be given at the SUSY scale. Flavour Observables, and Fine-Tuning with SPheno I concentrate on the first option because we are interested in a GUT model. Actually, the input for the second version is 6.1. Calculating the Mass Spectrum with SPheno. We start much shorter and can be easily derived from the information nowtomakeuseofthedifferentoutputs SARAH provides to given here. The file to define the properties should be use the derived information about a model with other tools. called SPheno.m and must be located in Maybe, the most important interface is the one with SPheno whichgivesaveryflexible,fullyfunctional,andhighlyprecise $PATH/SARAH/ M o d e l s /B−L−SSM/ spectrum generator for the model under consideration. A similar functionality to get a tailor-made spectrum generator based on SARAH became available with FlexibleSUSY 13. as well. The entire file is shown in Appendix C.4 and I will Before we can use SPheno we have to provide an addi- discuss here the main parts of it. tional input file for SARAH. I will start with a description of Input Parameters.Wehavecollectedin(108) alistofallinput whatthisfileissupposedtodo. parameters we want to use. These parameters should be given to SPheno in our numerical studies via a Les Houches input 6.1.1. Defining the Boundary Conditions. In general, there are file. We make the choice that all input parameters which are two different kinds of SPheno versions the user can create not a matrix are included in the block MINPAR of the Les which need a different amount of input. Houches input file. This can be done via 48 Advances in High Energy Physics

SPheno.m

1 MINPAR={{1, m0}, 2 {2, m12}, 3 {3, TanBeta}, 4 {4, SignumMu}, 5 {5, Azero}, 6 {6, SignumMuP}, 7 {7, TBetaP}, 8 {8, MZp}};

The exact meaning of this definition will become clear when Houches file. This might be useful to remove parameters wediscusstheLesHouchesinputinSection 6.1.3. In addition, not present in the MSSM from MINPAR.Inthatcasethe itwouldalsobepossibletousethearray EXTPAR to define corresponding lines in SPheno.m wouldreadasfollows. input which will be given via the block EXTPAR in the Les

1 MINPAR={{1, m0}, 2 {2, m12}, 3 {3, TanBeta}, 4 {4, SignumMu}, 5 {5, Azero}, 6 EXTPAR={{101, SignumMuP}, 7 {102, TBetaP}, 8 {103, MZp}};

SARAH is completely agnostic concerning official SLHA con- In any case, three of the input parameters are always real. ventions for the MINPAR and EXTPAR blocks [292, 293]. It This information is set via just refers to the definition as given by the user. However, it might be helpful to stick for models which are covered by SLHA to the corresponding conventions.

SPheno.m

10 RealParameters = {TanBeta , TBetaP ,m0};

耠 This definition is especially for tan 𝛽 and tan 𝛽 importantly Tadpole Equations.Wechoosetosolvethetadpoleequations 耠耠 because we will use trigonometric functions with these with respect to 𝜇, 𝐵휇, 𝜇 ,and𝐵휇. parameters as argument. If not defined as real, Fortran might return NaNs.

SPheno.m

11 ParametersToSolveTadpoles = {B[\[Mu]],B[MuP],\[Mu],MuP}; Advances in High Energy Physics 49

SARAH Solve 𝑀 = 𝑚̃ 𝑚̃ In that way uses the command ically. A convenient choice is often SUSY √ 푡1 푡2 .Inour of Mathematica to get the analytical solutions and case the stops are part of the six generations of up-squarks. exports them to Fortran code. For other choices of Sincewehavealwayslarge𝐴-terms the stops will always parameters, no analytical solution might exist. In these cases correspond to the lightest and heaviest of the mixed states. it is possible to solve the equations numerically during the So we can put as renormalization scale 𝑀 = √𝑚푢̃ 𝑚푢̃ .If SPheno SUSY 1 6 run with . I give some more information about the hierarchy of the up-squark masses is not clear, another this in Appendix A.4.1.Onecouldgivealsothesolutions possibility would be to give an analytical expression of based on simplified assumptions or define some assumptions the determinant of the stop mass matrix as renormaliza- which should be used for solving the equations. This is briefly tion scale. This is, for instance, done in the MSSM by discussed in Appendix A.4.2. setting Renormalization Scale. SPheno either can be used with a fixed renormalization scale, or it calculates the scale dynam-

1 RenormalizationScale = Sqrt[(mq2[3, 3] + (vu^2*conj[Yu[3, 3]]*Yu[3, → 3])/2)*(mu2[3, 3] + (vu^2*conj[Yu[3, 3]]*Yu[3, → 3])/2)-((vd*\[Mu]*conj[Yu[3, 3]] - vu*conj[T[Yu][3, → 3]])*(vd*conj[\[Mu]]*Yu[3, 3] - vu*T[Yu][3, 3]))/2];

Here, we skipped 𝐷-terms which are negligible in the MSSM. In the very first iteration when the stop masses are However, this does no longer hold in the B-L-SSM because unknown, SPheno needs a crude first guess of the renor- of the new contributions from the bilepton VEVs. Hence, the 2 2 malization scale. We choose √𝑚0 +4𝑀1/2.However,alsoa definition of the renormalization scale using that approach constant like 1 TeV could be used. The two lines to define both would even be a bit more lengthy. We can keep in this model 𝑀 = 𝑚 𝑚 scales for the B-L-SSM are the simple expression SUSY √ 푢̃1 푢̃6 .

SPheno.m

13 RenormalizationScale = MSu[1]*MSu[6]; 14 RenormalizationScaleFirstGuess = m0^2 + 4 m12^2;

GUT Condition. The condition for the GUT scale in our running the general 2×2gauge coupling matrix is used: that model is 𝑔1 =𝑔2. However, because of gauge-kinetic mixing is, for the check of the GUT scale it is necessary to rotate it to onehastobecarefulbychoosingthecorrect𝑔1:inthe the triangle form.

SPheno.m

16 ConditionGUTscale = (g1*gBL-g1BL*gBL1)/Sqrt[gBL^2+gBL1^2] == g2;

We demand that gauge-kinetic mixing vanishes at the GUT coupling matrix is in the triangle form. In order to further scale and this condition will simplify to g1 == g2.Never- stabilize numerics in the first iterations, where the unification theless, one should keep the full form to stabilize numerics in might not be too good, we average 𝑔1 and 𝑔2 before we set the first few iterations. the couplings in the 𝐵-𝐿 sector and the off-diagonal ones. Boundary Conditions. Now, we define the boundary condi- All other entries just parametrize in an obvious form the tionsattheGUTscale.First,wemakesureagainthatthe𝑈(1) boundary conditions from (103) to (107). 50 Advances in High Energy Physics

SPheno.m

19 BoundaryHighScale={ 20 {g1,(g1*gBL-g1BL*gBL1)/Sqrt[gBL^2+gBL1^2]}, 21 {g1,Sqrt[(g1^2+g2^2)/2]}, 22 {g2,g1}, 23 {gBL, g1}, 24 {g1BL ,0}, 25 {gBL1 ,0}, 26 {T[Ye], Azero*Ye}, 27 {T[Yd], Azero*Yd}, 28 {T[Yu], Azero*Yu}, 29 {T[Yv], Azero*Yv}, 30 {T[Yn], Azero*Yn}, 31 {mq2, DIAGONAL m0^2}, 32 {ml2, DIAGONAL m0^2}, 33 {md2, DIAGONAL m0^2}, 34 {mu2, DIAGONAL m0^2}, 35 {me2, DIAGONAL m0^2}, 36 {mvR2, DIAGONAL m0^2}, 37 {mHd2, m0^2}, 38 {mHu2, m0^2}, 39 {mC12, m0^2}, 40 {mC22, m0^2}, 41 {MassB , m12}, 42 {MassWB ,m12}, 43 {MassG ,m12}, 44 {MassBp ,m12}, 45 {MassBBp ,0} 46 };

耠耠 Note the keyword DIAGONAL and the usage of the parame- V휂 and V휂 from the input values of tan 𝛽 and 𝑀푍.Finally, ters defined via MINPAR. the input parameters for 𝑌푥 and 𝑌] areusedhere.Since At the SUSY scale we rotate again the gauge couplings 𝑌푥 and 𝑌] are matrices, it is not possible to define them tothetrianglebasisbecausethe(2, 1) entry will not be via MINPAR or EXTPAR. Therefore, LHInput[x] is used. zero any more due to RGE effects. In addition, we calculate With this command, SPheno expects the parameters to be given via blocks YXIN and YNUIN in the Les Houches file14.

SPheno.m

48 BoundarySUSYScale = { 49 {g1T,(g1*gBL-g1BL*gBL1)/Sqrt[gBL^2+gBL1^2]}, 50 {gBLT, Sqrt[gBL^2+gBL1^2]}, 51 {g1BLT ,(g1BL*gBL+gBL1*g1)/Sqrt[gBL^2+gBL1^2]}, 52 {g1, g1T}, 53 {gBL, gBLT}, 54 {g1BL, g1BLT}, 55 {gBL1 ,0}, 56 {vevP, MZp/gBL}, 57 {betaP,ArcTan[TBetaP]}, 58 {x2,vevP*Cos[betaP]}, 59 {x1,vevP*Sin[betaP]}, 60 {Yv, LHInput[Yv]}, 61 {Yn, LHInput[Yn]} 62 }; Advances in High Energy Physics 51

The boundary conditions at the EWSB scale are similar to stabilize numerics in the first run up to the GUT scale, it is the ones at the SUSY scale and are skipped here. To further useful to give approximate values for the initialization of the gauge couplings which are not present in the SM.

SPheno.m

78 InitializationValues = { 79 {gBL, 0.5}, 80 {g1BL, -0.06}, 81 {gBL1, -0.06} 82 }

Decays.Finally,wetell SARAH that it should make use of the default conventions to write code to calculate two- and three- body decays with SPheno.

SPheno.m

97 ListDecayParticles = Automatic; 98 ListDecayParticles3B =Automatic;

By convention, the following decays are included that way: When executing MakeSPheno, SARAH calculates first all (i) all two-body decays of SUSY particles, Higgs states, the information which it needs: that is, it is not necessary that top quark, and additional vector bosons and (ii) three-body the user has done the calculation of vertices or RGEs before. decays of SUSY fermions in three other fermions and decays When SARAH is done, the source code for SPheno is stored of SUSY scalars in another scalar and two fermions. in $SPATH/SARAH/Output/B-L-SSM/EWSB/SPheno/.The compilation of this codes is done as follows: one has to enter Precision. We are mostly going to neglect neutrino masses the directory of the SPheno installation, a new subdirectory inthefollowing.However,ifastudyoftheneutrinophe- hastobecreated,andthecodemustbecopiedintothis nomenology should be included, it might be necessary to directory. calculate the masses of the neutrino eigenstates with a higher precisionasthisisusuallydonein SPheno.Thereasonis $ cd $PATH/SPHENO the potential large hierarchy between left and right neutrinos. $ mkdir BLSSM Details about this are given in Appendix A.3. $ cp $PATH/SARAH/Output/B−L−SSM/EWSB/SPheno/∗ BLSSM/

6.1.2. Obtaining and Running the SPheno Code. To obtain Afterwards, the code is compiled via the SPheno output we run in Mathematica after SARAH is loaded and the B-L-SSM is initialized at the command as $ make Model=BLSSM follows.

MakeSPheno[]; and a new executable SPhenoBLSSM is created which we will use in the following. The different options are as follows. 6.1.3.SettingtheInputforSPhenoBLSSM. An input file, (i) ReadLists->True:itcanbeusedifallverticesand by default called LesHouches.in.BLSSM, is needed to RGEs have already been calculated for the model and run SPhenoBLSSM. SARAH writes a template for that the former results should be used. filewhichhasbeencopiedtotheBLSSM subdirectory of SPheno together with the Fortran code. We move it to (ii) InputFile: it defines the name of the SPheno input the root directory of SPheno as follows. file. By default SPheno.m is used.

(iii) StandardCompiler->Compile: it defines the $ cp BLSSM/LesHouches . in .BLSSM . compiler which should be set as standard in the Makefile.Defaultis gfortran. By doing this we can work now from the SPheno main (iv) IncludeFlavorKit:itcanbeusedtodisablethe directoryandwedonothavetogivethefileasargumentwhen output of flavour observables based on FlavorKit. running SPheno.Thus, SPheno canbecalledvia 52 Advances in High Energy Physics

耠 Table 7: Input parameters for the example points EP1 and EP2. In addition, we use sign(𝜇) = sign(𝜇 )=1and 𝑌] =0.

耠 𝑚0 [GeV] 𝑀1/2 [GeV] tan 𝛽𝐴0 [GeV] tan 𝛽 𝑀푍耠 [GeV] diag(𝑌푥) EP1 1700 1500 7.0 −1400 1.20 2500 (0.42, 0.42, 0.38) EP2 1700 1500 7.0 −1400 1.06 4500 (0.42, 0.42, 0.05)

SPheno $ . / bin /SPhenoBLSSM However, before we can run we first have to define theinputparameters:forthatpurposewehavetofillthe template written by SARAH with numbers. I am going to Alternatively,onecankeeptheLesHouchesfileinthe BLSSM discuss the different blocks appearing in the Les Houches and work with it via file briefly. More details especially about the block SMINPUTS can also be found in the official references for SLHA292 [ , $ . / bin /SPhenoBLSSM BLSSM/ LesHouches . in .BLSSM 293]. At the very beginning, the block MODSEL is given.

LesHouches.in.BLSSM

1 Block MODSEL # 2 11 # 1/0: High/low scale input 3 21 # Boundary Condition 4 61 # Generation Mixing

This block fixes the general setup the user wants touse. 6: if put to 1,flavourviolationisallowedandalloff- diagonal entries in soft or superpotential parameters 1: it defines if a GUT scale input is used1 ( )orall can receive nonzero values. If put to 0,theCKM parameters should be given at the SUSY scale (0). matrix is taken to be the identity matrix. 2: in principle, it is possible to define in SPheno.m 12: this flag can be given optionally to fix the SUSY different boundary conditions for the GUT input, scaletoaconstantvalue. for instance, if different SUSY breaking mechanism SMINPUTS shouldbestudied.Thisflagcanbeusedtochooseone. The block contains all important values for the SM parameters like Fermi constant 𝐺퐹,strongcoupling Sincewehavenotmadeuseofthat,thisflaghasno 𝛼 (𝑀 ) 𝑍 effect here. constant 푆 푍 ,polemassofthe -boson, and third- generation fermion masses 𝑚푡, 𝑚푏,and𝑚휏.Alsoother 5: if put to 1,CPviolationisallowedandthephase parameters can be set as explained in the SLHA write-ups but of the CKM matrix is included. this is usually not necessary.

LesHouches.in.BLSSM

5 Block SMINPUTS # Standard Model inputs 6 2 1.166370E-05 # G_F,Fermi constant 7 3 1.187000E-01 # alpha_s(MZ) SM MSbar 8 4 9.118870E+01 # Z-boson pole mass 9 5 4.180000E+00 # m_b(mb) SM MSbar 10 6 1.735000E+02 # m_top(pole) 11 7 1.776690E+00 # m_tau(pole)

Weturnnowtotheinputtofixtheparameterpointwewant is highly motivated by the observations we made at the 耠 to study. I have chosen two points as examples, EP1 and EP2, analytical level: we need small tan 𝛽 and large 𝑚0, 𝐴0, with slightly different input. All necessary input parameters and 𝑌푥 to break 𝐵-𝐿 radiatively. One can test that the are given in Table 7. The neutrino Yukawa coupling is highly values of 𝑌푥 in Table 7 are close to the Landau pole. When constrained by neutrino data and we can ignore it here for using even larger values SPheno will stop with an error our purposes. It will become important if the user wants message. The values for EP1 are set in the Les Houches file to study lepton flavour violation, for instance. The input via Advances in High Energy Physics 53

LesHouches.in.BLSSM

12 Block MINPAR # Input parameters 13 1 1.7000000E+03 #m0 14 2 1.5000000E+03 # m12 15 3 7.0000000E+00 # TanBeta 16 4 1.0000000E+00 # SignumMu 17 5 -1.4000000E+03 # Azero 18 6 1.0000000E+00 # SignumMuP 19 7 1.2000000E+00 # TBetaP 20 8 2.5000000E+03 # MZp 21 Block YXIN # 22 1 1 4.2000000E-01 # Yx(1,1) 23 2 2 4.2000000E-01 # Yx(2,2) 24 3 3 3.8000000E-01 # Yx(3,3) 25 Block YVIN # 26 1 1 0.0000000E+00 # Yv(1,1) 27 2 2 0.0000000E+00 # Yv(2,2) 28 3 3 0.0000000E+00 # Yv(3,3)

Finally, we have some more switches in the block SPhenoInput as follows.

LesHouches.in.BLSSM

31 Block SPhenoInput # SPheno specific input 32 1-1 # error level 33 20 # SPA conventions 34 70 # Skip 2-loop Higgs corrections 35 83 # Method used for two - loop calculation 36 91 # Gaugeless limit used at two-loop 37 10 0 # safe -mode used at two-loop 38 11 1 # calculate branching ratios 39 13 1 # 3-Body decays: none (0), fermion (1), scalar (2), both (3) 40 12 1.000E-04 # write only branching ratios larger than this 41 15 1.000E-30 # write only decay if width larger than this 42 31 -1 # fixed GUT scale (-1: dynamical GUT scale) 43 32 0 # Strict unification 44 34 1.000E-04 # Precision of mass calculation 45 35 40 # Maximal number of iterations 46 37 1 # Set Yukawa scheme 47 38 2 # 1- or 2-Loop RGEs 48 50 0 # Majorana phases: use only positive masses 49 51 0 # Write Output in CKM basis 50 52 0 # Write spectrum in case of tachyonic states 51 55 1 # Calculate one loop masses 52 57 1 # Calculate low energy constraints 53 60 1 # Include possible , kinetic mixing 54 65 1 # Solution tadpole equation 55 75 1 # Write WHIZARD files 56 76 1 # Write HiggsBounds file 57 86 0. # Maximal width to be counted as invisible 58 510 0. # Write tree level values for tadpole solutions 59 515 0 # Write parameter values at GUT scale 60 520 1. # Write effective Higgs couplings (HB blocks) 61 525 0. # Write contributions to diphoton decay of Higgs 62 530 1. # Write Blocks for Vevacious 54 Advances in High Energy Physics

Theseflagscanbeusedtoadjustthecalculationsdone (v) Output for Other Codes.Theoutputs by SPheno and the output. All possible flags are listed in of HiggsBounds and HiggsSignals input Appendix B. Im explaining here just the most important files are switched on/off via flag 76, while flag 75 ones. is responsible for the output of the parameter file for WHIZARD.TheVevacious specific blocks in (i) Loop Level. To turn off all loop corrections to thespectrumfileareincludedorexcludedwithflag 55 0 the masses, flag is put to . The two-loop 530. corrections in the Higgs sector are turned on/off by flag 7. flag 8 chooses the method to calcu- (vi) Fine-Tuning. The calculation of the fine-tuning is 550 0 late the two-loop contributions: 1, purely numer- skipped when setting flag to . ical effective potential calculation; 2,semianalytical effective potential calculation; 3,diagrammatic 6.1.4. Calculating the Spectrum and Higgs Couplings calculation; 8/9, results from the literature (if avail- with SPheno. Whentheinputfileisfilledwithnumbers able). we can run the point as explained above. The entire (ii) Decays and Branching Ratios.Alldecaysareturned output including all parameters, masses, branching on/off via flag 11, while three-body decays can be ratios, and low-energy observables is saved by default SPheno.spc.BLSSM adjusted via flag 13. in . This file is rather lengthy and contains a lot of information. I will pick some parts of it and (iii) Precision Observables. The calculation of precision discuss them. and flavour observables is turned on and off using 57 Sometimes, it is convenient to work with input and output flag . files with other names. In that case, the names can be used as (iv) Gauge-Kinetic Mixing. SPheno can be forced to argument for SPhenoBLSSM. For instance, we can make two ignore gauge-kinetic mixing by setting flag 60 to 0. inputfilesforthepointsEP1andEP2andrunthemvia

$ . / bin /SPhenoBLSSM LesHouches . in . BLSSM EP1 SPheno . spc .BLSSM EP1 $ . / bin /SPhenoBLSSM LesHouches . in . BLSSM EP2 SPheno . spc .BLSSM EP2

The entire output is written to the files given as unification scale is a bit higher than that in the MSSM and second argument, that is, SPheno.spc.BLSSM EP1 that we have no strict unification here because there is a small and SPheno.spc.BLSSM EP2. offset of 𝑔3. This behaviour is also known from the MSSM and oneassumesthathigherordercorrectionsaswellasthreshold The Running Parameters and Mass Spectrum for EP1. corrections from super heavy particles are responsible for an First, we check if gauge coupling unification at about exact unification once taken into account. 16 10 GeV remains. From the block gaugeGUT we see that the

SPheno.spc.BLSSM (EP1)

38 Block gaugeGUT Q= 5.55462899E+16 # (GUT scale) 39 1 7.14501610E-01 # g1(Q)^DRbar 40 2 7.14501610E-01 # g2(Q)^DRbar 41 3 6.87174622E-01 # g3(Q)^DRbar 42 4 7.14501610E-01 # gBL(Q)^DRbar

The running gauge couplings at the SUSY scale are given in breaking. Note that 𝑔1 in this block is the value without GUT the block Gauge.TheSUSYscaleshownas𝑄 in the head normalization. Thus, it is related to 𝑔1 used in Section 5.3.2 of the block is about 3 TeV: that is, the stop masses used by a factor of √5/3.Wefindalsoanoff-diagonalcoupling tofixthescalearequiteheavy.Thatisaconsequenceof of −0.11 and 𝑔퐵 close to 0.55 as expected from our estimates the large 𝑚0 and 𝑀1/2 where we used to get radiative 𝐵-𝐿 with Mathematica;see(136). Advances in High Energy Physics 55

SPheno.spc.BLSSM (EP1)

51 Block GAUGE Q= 3.04424240E+03 # (SUSY Scale) 52 1 3.63178061E-01 #g1 53 2 6.44147228E-01 #g2 54 3 1.00570244E+00 #g3 55 4 5.48624610E-01 # gBL 56 10 -1.13408283E-01 # gYB 57 11 0.00000000E+00 # gBY

Many more blocks appear after the one for the gauge cou- and MSOFT. The first one contains the soft-terms of VEVs 耠 plings and contain all other running parameters at the SUSY and 𝜇 in the 𝐵-𝐿 sector, and the second one contains the soft- scale. We just want to take a look at two other blocks: BL terms of Higgs and gaugino masses from the MSSM part.

SPheno.spc.BLSSM (EP1)

58 Block BL Q= 3.04424240E+03 # (SUSY Scale) 59 1 1.59624321E+03 # MuP 60 2 4.75415053E+06 # BMuP 61 11 8.49277085E+05 # mC12 62 12 3.74696940E+06 # mC22 63 32 -2.21262550E+02 # MBBp 64 31 6.59015204E+02 # MBp 65 41 3.50066906E+03 #x1 66 42 2.91722422E+03 #x2 67 43 4.55684990E+03 #vX

SPheno.spc.BLSSM (EP1)

75 Block MSOFT Q= 3.04424240E+03 # (SUSY Scale) 76 21 4.00188260E+06 # mHd2 77 22 -4.80480045E+06 # mHu2 78 1 7.87335844E+02 #M1 79 2 1.32773211E+03 #M2 80 3 3.57502541E+03 #M3

2 𝑚 𝐵 耠 Weseeherethat 퐻푢 is negatives as it is expected to break Becauseofthelargevalueof 휇 the condition is fulfilled the electroweak symmetry. However, both soft-terms for the for this point and 𝐵-𝐿 is broken. Another observation is bileptons are actually positive and one might wonder if 𝐵-𝐿 that the blino soft mass 𝑀퐵耠 is lighter than the bino one as is really broken radiatively. It is broken because the full we already expected from Section 5.3.2.Actually,𝑀퐵耠 is also condition is not that a soft-term has to be negative, but that smaller than the other gaugino masses and the 𝜇-term not the determinant of the mass matrix has a negative eigenvalue shown here. So, one would expect that the lightest neutralino in the limit of vanishing VEVs. For the lower 2×2block of is a blino. However, we will see below that this is not the thescalarmassmatrixthisconditionreads case. 󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨2 Afterallblockswiththerunningparameters,allmasses 󵄨 耠󵄨 󵄨 耠󵄨 󵄨 󵄨 MASS (𝑚휂 + 󵄨𝜇 󵄨 )(𝑚휂 + 󵄨𝜇 󵄨 )−󵄨𝐵휇耠 󵄨 <0. (139) are printed in the block . 56 Advances in High Energy Physics

SPheno.spc.BLSSM (EP1)

264 Block MASS # Mass spectrum 265 # PDG code mass particle 266 1000001 3.30871123E+03 # Sd_1 267 1000003 3.60959661E+03 # Sd_2 268 1000005 3.62130996E+03 # Sd_3 269 2000001 3.62131463E+03 # Sd_4 270 2000003 3.73450325E+03 # Sd_5 271 2000005 3.73452286E+03 # Sd_6 272 1000002 2.67104525E+03 # Su_1 273 1000004 3.32204106E+03 # Su_2 274 1000006 3.61199579E+03 # Su_3 275 2000002 3.61201405E+03 # Su_4 276 2000004 3.73368042E+03 # Su_5 277 2000006 3.73369930E+03 # Su_6 278 1000011 1.76023835E+03 # Se_1 279 1000013 1.77259986E+03 # Se_2 280 1000015 1.77264366E+03 # Se_3 281 2000011 2.11331234E+03 # Se_4 282 2000013 2.11823100E+03 # Se_5 283 2000015 2.11824845E+03 # Se_6 284 1000012 4.96686479E+02 # SvRe_1 285 1000014 8.25895521E+02 # SvRe_2 286 1000016 8.25895521E+02 # SvRe_3 287 2000012 2.11130838E+03 # SvRe_4 288 2000014 2.11640817E+03 # SvRe_5 289 2000016 2.11642628E+03 # SvRe_6 290 4000012 2.11130838E+03 # SvIm_1 291 4000014 2.11640817E+03 # SvIm_2 292 4000016 2.11642628E+03 # SvIm_3 293 5000012 2.69896818E+03 # SvIm_4 294 5000014 2.99406918E+03 # SvIm_5 295 5000016 2.99406918E+03 # SvIm_6 296 25 1.24182913E+02 # hh_1 297 35 3.55295631E+02 # hh_2 298 9900025 3.06408049E+03 # hh_3 299 9900035 3.96149850E+03 # hh_4 300 36 3.06214679E+03 # Ah_3 301 9900036 3.09582809E+03 # Ah_4 302 37 3.06360579E+03 # Hpm_2 303 23 9.11887000E+01 #VZ 304 24 8.06613172E+01 # VWm 305 1 5.00000000E-03 # Fd_1 306 3 9.50000000E-02 # Fd_2 307 5 4.18000000E+00 # Fd_3 308 2 2.50000000E-03 # Fu_1 309 4 1.27000000E+00 # Fu_2 310 6 1.73500000E+02 # Fu_3 311 11 5.10998930E-04 # Fe_1 312 13 1.05658372E-01 # Fe_2 313 15 1.77669000E+00 # Fe_3 314 12 1.31876738E-11 # Fv_1 315 14 1.36424206E-11 # Fv_2 316 16 1.83674189E-11 # Fv_3 317 112 1.48043893E+03 # Fv_4 318 114 2.06846561E+03 # Fv_5 319 116 2.06846561E+03 # Fv_6 320 1000021 3.75512792E+03 # Glu 321 1000024 1.37525092E+03 # Cha_1 322 1000037 2.27493791E+03 # Cha_2 323 31 2.50003732E+03 # VZp 324 1000022 7.85565922E+02 # Chi_1 325 1000023 1.29973769E+03 # Chi_2 326 1000025 1.37521122E+03 # Chi_3 327 1000035 1.70198093E+03 # Chi_4 328 1000032 2.27046891E+03 # Chi_5 329 1000036 2.27465029E+03 # Chi_6 330 1000039 3.64727768E+03 # Chi_7 Advances in High Energy Physics 57

The squarks (Su and Sd) are very heavy, and also the charged that the input is taken to be the tree-level mass in the limit sleptons (Se)havemasseswellabove1TeV.However,there of vanishing gauge-kinetic mixing, while here the full one- are some light CP even sneutrinos (SvRe) while the CP loop mass including all mixing effects is shown. However, the odd ones are much heavier (SvIm). Thus, the mass splitting differences are very small and one has usually not to worry between the CP eigenstates of the sneutrinos is another inter- about them. esting and import aspect of the B-L-SSM. Actually, scrolling In the Higgs sector, we have the lightest scalar (hh 1)in down, we see that all other SUSY states like neutralinos (Chi) the mass range preferred by measurements. Also the second 푅 and charginos (Cha) are actually heavier than ̃]1 .Thus,the scalar ℎ2 is not much heavier. In contrast, the two heavier LSP, and therefore the DM candidate, is the lightest CP even scalarsaswellasthetwophysicalpseudoscalars(Ah)andthe sneutrino for this point. The hierarchy of the heavy neutrino charged Higgs (Hpm)areallabout3TeV. (Fv) reflects the input values of 𝑌푥. We see also that the Thisismaybeagoodtimetobrieflycommentonthe 耠 𝑍 mass is not exactly identical to the input. The reason is importance of the loop corrections in this model. We can turn off the two-loop corrections via

Block SPhenoInput # SPheno specific input ... 71 # Skip 2-loop Higgs corrections

𝑚(1퐿) 𝑚(1퐿) andweget ℎ1 = 115.9 GeV and ℎ1 = 356.0 GeV. Turning even off one-loop corrections via

55 0 # Calculate one loop masses

𝑚(푇) 𝑚(푇) we find ℎ1 =87.2GeVand ℎ2 = 353.9 GeV.The corrections MSSM-likedoublet.Toconfirmthisguess,wecancheck 𝑚 the mixing matrix in the scalar sector which is given in the for ℎ1 are similar as in the MSSM and one can guess already from these numbers that this state is mostly a block SCALARMIX as follows.

SPheno.spc.BLSSM (EP1)

516 Block SCALARMIX Q= 3.04424240E+03 #() 517 1 1 1.48960470E-01 #ZH(1,1) 518 1 2 9.88343642E-01 #ZH(1,2) 519 1 3 -2.39894518E-02 #ZH(1,3) 520 1 4 -2.03009771E-02 #ZH(1,4) 521 2 1 4.79675110E-03 #ZH(2,1) 522 2 2 3.09937950E-02 #ZH(2,2) 523 2 3 7.20236031E-01 #ZH(2,3) 524 2 4 6.93019794E-01 #ZH(2,4) 525 ...

퐻 −2 These are the entries of the 𝑍 rotation matrix and we 𝑂(10 )% 15,whilethesecondoneisnearlyapurebilepton. concentrate here one only on the first two eigenstates. We So, the mixing between both sectors for this point is moder- see that bilepton admixture of the first eigenstate is just ately small because the masses are not too close. 58 Advances in High Energy Physics

The mixing matrix for the neutralinos is shown inthe block NMIX as follows.

SPheno.spc.BLSSM (EP1)

555 Block NMIX Q= 3.04424240E+03 #() 556 1 1 9.95429472E-01 # Real(ZN(1,1),dp) 557 1 2 -1.92612245E-03 # Real(ZN(1,2),dp) 558 1 3 2.32548331E-02 # Real(ZN(1,3),dp) 559 1 4 -1.10389021E-02 # Real(ZN(1,4),dp) 560 1 5 -2.68824741E-02 # Real(ZN(1,5),dp) 561 1 6 -6.00307896E-02 # Real(ZN(1,6),dp) 562 1 7 6.42452196E-02 # Real(ZN(1,7),dp) 563 ...

So, we see that the lightest state has a bino fraction of term is smaller than the bino one. Why is not then the LSP −4 about 99%16, a wino fraction of 𝑂(10 %) 17,aHiggsino ablino?Tounderstandthis,wecancheckthemixingof fraction of roughly 0.07%18 and a similar blino fraction19, the second lightest neutralino. Since the default convention and finally a bileptino fraction of a bit less than 1%20.This by SPheno is that all Majorana masses are negative, some issurprisingbecausewehaveseenabovethattheblinosoft- entries of the rotation matrix are imaginary. Hence, we find 0 the composition of 𝜒̃2 in IMNMIX as follows.

SPheno.spc.BLSSM (EP1)

605 Block IMNMIX Q= 3.04424240E+03 #() 606 ... 607 2 1 -6.79846118E-02 # Aimag(ZN(2,1)) 608 2 2 3.37464672E-05 # Aimag(ZN(2,2)) 609 2 3 3.48780244E-03 # Aimag(ZN(2,3)) 610 2 4 1.61010850E-03 # Aimag(ZN(2,4)) 611 2 5 -6.48176532E-01 # Aimag(ZN(2,5)) 612 2 6 -2.47517778E-02 # Aimag(ZN(2,6)) 613 2 7 7.58035531E-01 # Aimag(ZN(2,7))

We see that in the 𝐵-𝐿 sector there is a large mixing between for the given parameter point are shown in the two blino and bileptino. The reason is that the mixing entries blocks HiggsBoundsInputHiggsCouplingsFermions are proportional to 𝑀푍耠 and not to 𝑀푍 as for the MSSM and HiggsBoundsInputHiggsCouplingsBosons.These part. Therefore, one does not have a pure blino that would blocks give the coupling of all Higgs states in the be lighter than the bino. One can check that, in the limit model normalized to the SM expectation as defined in 耠 of very heavy 𝜇 , the bileptinos decouple and the blino (82). is indeed the LSP [199]. All other rotation matrices are Thus, the values in these blocks for the lightest Higgs, givenaswellinthespectrumfilebutnotfurtherdiscussed which we associated with the SM-like state, should be close here. to one, while those for the second lightest Higgs are expected Higgs Couplings. Other important pieces of information to be much smaller. This is exactly what we observe as about the properties of all scalars and pseudoscalars follows. Advances in High Energy Physics 59

SPheno.spc.BLSSM (EP1)

805 Block HiggsBoundsInputHiggsCouplingsFermions # 806 1.21544807E+00 ... 3 25 5 5 # h_1 b b coupling 807 1.21544807E+00 ... 3 25 3 3 # h_1 s s coupling 808 9.94987651E-01 ... 3 25 6 6 # h_1 t t coupling 809 9.94987651E-01 ... 3 25 4 4 # h_1 c c coupling 810 1.21544807E+00 ... 3 25 15 15 # h_1 tau tau coupling 811 1.21544807E+00 ... 3 25 13 13 # h_1 mu mu coupling 812 1.22177746E-03 ... 3 35 5 5 # h_2 b b coupling 813 1.22177746E-03 ... 3 35 3 3 # h_2 s s coupling 814 9.79053104E-04 ... 3 35 6 6 # h_2 t t coupling 815 9.79053104E-04 ... 3 35 4 4 # h_2 c c coupling 816 1.22177746E-03 ... 3 35 15 15 # h_2 tau tau coupling 817 1.22177746E-03 ... 3 35 13 13 # h_2 mu mu coupling 818 ... 819 Block HiggsBoundsInputHiggsCouplingsBosons # 820 9.98814843E-01 3 25 24 24 # h_1 W W coupling 821 1.00445622E+00 3 25 23 23 # h_1 Z Z coupling 822 0.00000000E+00 3 25 23 22 # h_1 Z gamma coupling 823 1.00085782E+00 3 25 22 22 # h_1 gamma gamma coupling 824 9.89551529E-01 3 25 21 21 # h_1 g g coupling 825 0.00000000E+00 4 25 21 21 23 # h_1 g g Z coupling 826 9.83376076E-04 3 35 24 24 # h_2 W W coupling 827 1.10657726E-03 3 35 23 23 # h_2 Z Z coupling 828 0.00000000E+00 3 35 23 22 # h_2 Z gamma coupling 829 9.83659658E-04 3 35 22 22 # h_2 gamma gamma coupling 830 9.78781315E-04 3 35 21 21 # h_2 g g coupling 831 0.00000000E+00 4 35 21 21 23 # h_2 g g Z coupling 832 833 ...

Since the couplings squared to down-type quarks and leptons cross sections for a given Higgs mass have been obtained by for ℎ1 are about 20% larger than in the SM we can expect fitting the data from that this point does not explain the measurements too well. To quantize that we will use HiggsSignals later, see https://twiki.cern.ch/twiki/bin/view/LHCPhysics/ Section 7.1.2.Thesecouplingratioscanalsobeusedtoget CERNYellowReportPageAt7TeV, the production cross section of the Higgs states at the LHC https://twiki.cern.ch/twiki/bin/view/LHCPhysics/ in the different channels by rescaling the SM results. This CERNYellowReportPageAt8TeV. is done in the blocks HiggsLHC7 and HiggsLHC8.TheSM Since this fit is only valid in a finite range for the Higgs21 mass also the results are only given for Higgs masses which lie in the fit range of each channel.

SPheno.spc.BLSSM (EP1)

789 Block HiggsLHC7 # Higgs production cross section at LHC7 [pb] 790 1 25 1.54190769E+01 # Gluon fusion 791 2 25 1.22391804E+00 # Vector boson fusion 792 3 25 5.54211509E-01 # W-H production 793 4 25 3.24580015E-01 # Z-H production 794 5 25 8.74181618E-02 # t-t-H production 795 1 35 2.29725104E-03 # Gluon fusion 796 2 35 2.30558730E-04 # Vector boson fusion 797 Block HiggsLHC8 # Higgs production cross section at LHC8 [pb] 798 1 25 1.95249057E+01 # Gluon fusion 799 2 25 1.57753091E+00 # Vector boson fusion 800 3 25 7.08633453E-01 # W-H production 801 4 25 4.05291805E-01 # Z-H production 802 5 25 1.32262917E-01 # t-t-H production 803 1 35 3.22819525E-03 # Gluon fusion 804 2 35 3.49847940E-04 # Vector boson fusion 60 Advances in High Energy Physics

APointwithaLightScalar.Wewanttobrieflydiscussthe point, we have only to change three lines in the Les Houches particularities of the point EP2 given in Table 7.Torunthis input file compared to EP1 as follows.

LesHouches.in.BLSSM

1 Block MINPAR # Input parameters 2 ... 3 7 1.060E+00 # TBetaP 4 8 4.500000E+03 # MZp 5 Block YXIN # 6 ... 7 3 3 5.00000000E-02 #Yx(3,3)

The interesting aspect of this point is the Higgs sector which now contains a light scalar as follows.

SPheno.spc.BLSSM (EP2)

1 Block MASS # Mass spectrum 2 ... 3 25 7.78960599E+01 # hh_1 4 35 1.25607177E+02 # hh_2

This point is mostly a bilepton, of course. However, it has a doublet fraction of about 1% doublet as we can see fromthe scalar mixing matrix as follows.

SPheno.spc.BLSSM (EP2)

1 Block SCALARMIX Q= 3.04382955E+03 #() 2 1 1 2.41678467E-02 #ZH(1,1) 3 1 2 1.60726196E-01 #ZH(1,2) 4 1 3 -6.98258293E-01 # ZH(1,3) 5 1 4 -6.97150171E-01 # ZH(1,4) 6 2 1 -1.47075505E-01 # ZH(2,1) 7 2 2 -9.75679078E-01 # ZH(2,2) 8 2 3 -1.13971461E-01 # ZH(2,3) 9 2 4 -1.15886318E-01 # ZH(2,4) 10 ...

𝑚(1퐿) = 68.3 , Thus, it will be interesting to see if it passes all tests from Higgs ℎ1 GeV searches. We are going to check that with HiggsBounds 𝑚(2퐿) = 117.5 , in Section 7.1. Another important aspect of this point is the ℎ2 GeV Higgsmassatthedifferentlooplevels: 𝑚(2퐿) = 77.9 , ℎ1 GeV (푇) 𝑚 = 87.1 , (2퐿) ℎ1 GeV 𝑚 = 125.6 . ℎ2 GeV 𝑚(푇) = 184.3 , (140) ℎ2 GeV Advances in High Energy Physics 61

At tree-level the lighter mass is the doublet, and the heavier 6.2. Decay Widths and Branching. SPheno modules one is the bilepton. Thus, there is a level crossing at one-loop by SARAH do not only calculate the mass spectrum and compared to tree-level. The bilepton mass nearly changes by effective couplings for the Higgs scalars, but also provide a factor of 3 when including radiative corrections. Therefore, functions for calculating decays. To adjust the output of the to have some trust in the mass prediction, also two-loop decays, the important flags in the Les Houches input are as corrections for the bileptons are crucial for this point. These follows. corrections are even a bit larger than for the MSSM-like particle and give a push of about 10 GeV.

LesHouches.in.BLSSM

1 Block SPhenoInput # SPheno specific input 2 ... 3 11 1 # calculate branching ratios (BR) 4 13 1 # 3-Body decays: none (0), fermion (1), scalar → (2), both (3) 5 12 1.000E-04 # write only BR larger than this value 6 15 1.000E-30 # write only decay if width larger than this value

With flag 11 the entire calculation of all decays can calculation of the three-body decays can be time consuming be turned on and off. Note that for the HiggsBounds and is not always needed. With flag 12 alowerlimitofthe and HiggsSignals outputs discussed in Section 7.1 the branching ratios which should be shown in the SLHA file is calculation of at least two-body decays is essential. flag 13 given, while flag 15 gives a lower limit on the width to be canbeusedtoturnon/offthethree-bodydecaysseparately: listed in the SLHA file. only three-body decays either of fermions or of scalars For our discussion, we pick out some decays of specific or of both can be calculated. Usually, the scalar three- states which give some impression of the main differences body decays are quite time consuming because of the many compared to the MSSM. For instance, we have seen that for decay channels. This can sometimes be helpful because the EP1 the neutralino is actually not the LSP. So, it should decay. This is exactly what we find.

SPheno.spc.BLSSM (EP1)

1495 DECAY 1000022 3.45203467E-17 # Chi_1 1496 # BR NDA ID1 ID2 1497 1.00E+00 2 16 1000012 # BR(Chi_1 -> Fv_3 SvRe_1 )

Thewidthisverysmall.Thereasonisthatitissuppressed The second lightest neutralino consists mostly of 𝐵-𝐿 twice: by the small blino admixture to the neutralino and the states; that is, the coupling to the second lightest Higgs small right fraction of the neutrino. (bilepton) is larger than the lightest one. This explains why 𝜒̃2 mostlydecaysintothesecondlightestscalarbutnotthe lightest one despite the larger phase space.

SPheno.spc.BLSSM (EP1)

1498 DECAY 1000023 1.51576185E-03 # Chi_2 1499 # BR NDA ID1 ID2 1500 1.74E-03 2 1000022 25 # BR(Chi_2 -> Chi_1 hh_1 ) 1501 9.81E-01 2 1000022 35 # BR(Chi_2 -> Chi_1 hh_2 ) 1502 1.01E-02 2 1000022 23 # BR(Chi_2 -> Chi_1 VZ ) 62 Advances in High Energy Physics

Also the heavy neutrinos are expected to decay. Since these neutrino. Thus, the width of these states is much larger than are mostly right-handed states the coupling to the lightest the width of the lightest neutralino even if the involved neutralino and sneutrino is much larger than for the light particles seem to be similar.

SPheno.spc.BLSSM (EP1)

1598 DECAY 112 3.53565312E-02 # Fv_4

1599 # BR NDA ID1 ID2

1600 9.99E-01 2 1000022 1000012 # BR(Fv_4 -> Chi_1 SvRe_1 )

Another interesting topic in these models are the decays of the 98% of the final states are SM fermions. The reason is that 耠 𝑍 : new decay channels can alter significantly the width of the the important channels in right neutrinos are kinematically 耠耠 𝑍 andhaveanimpactonthelimitsof𝑍 masses from collider forbidden. searches [200, 282]. However, for our point EP1 we see that

SPheno.spc.BLSSM (EP1)

1433 DECAY 31 2.05629381E+01 # VZp

1434 # BR NDA ID1 ID2

1435 1.50E-03 2 1000022 1000023 # BR(VZp -> Chi_1 Chi_2 )

1436 6.39E-04 2 1000022 1000035 # BR(VZp -> Chi_1 Chi_4 )

1437 1.05E-01 2 -1 1 # BR(VZp -> Fd_1^* Fd_1 )

1438 1.05E-01 2 -3 3 # BR(VZp -> Fd_2^* Fd_2 )

1439 1.05E-01 2 -5 5 # BR(VZp -> Fd_3^* Fd_3 )

1440 1.17E-01 2 -11 11 # BR(VZp -> Fe_1^* Fe_1 )

1441 1.17E-01 2 -13 13 # BR(VZp -> Fe_2^* Fe_2 )

1442 1.17E-01 2 -15 15 # BR(VZp -> Fe_3^* Fe_3 )

1443 2.64E-02 2 -2 2 # BR(VZp -> Fu_1^* Fu_1 )

1444 2.64E-02 2 -4 4 # BR(VZp -> Fu_2^* Fu_2 )

1445 2.62E-02 2 -6 6 # BR(VZp -> Fu_3^* Fu_3 )

1446 7.61E-02 2 12 12 # BR(VZp -> Fv_1 Fv_1 )

1447 7.61E-02 2 14 14 # BR(VZp -> Fv_2 Fv_2 )

1448 7.61E-02 2 16 16 # BR(VZp -> Fv_3 Fv_3 )

1449 1.19E-03 2 25 23 # BR(VZp -> hh_1 VZ )

1450 1.84E-02 2 -24 24 # BR(VZp -> VWm VWm^* )

耠 This changes in EP2 where the 𝑍 is much heavier than the and is much more important than the channels in sleptons right neutrinos. Here, the BR in right neutrinos is about 10% which are also kinematically allowed for 𝑀푍耠 = 4.5 TeV. Advances in High Energy Physics 63

SPheno.spc.BLSSM (EP2)

1432 DECAY 31 4.18911354E+01 # VZp 1433 # BR NDA ID1 ID2 1434 ... 1435 6.74E-02 2 12 12 # BR(VZp -> Fv_1 Fv_1 ) 1436 6.74E-02 2 14 14 # BR(VZp -> Fv_2 Fv_2 ) 1437 6.74E-02 2 16 16 # BR(VZp -> Fv_3 Fv_3 ) 1438 1.01E-01 2 112 112 # BR(VZp -> Fv_4 Fv_4 ) 1439 ... 1440 4.47E-03 2 -1000011 1000011 # BR(VZp -> Se_1^* Se_1 ) 1441 4.32E-03 2 -1000013 1000013 # BR(VZp -> Se_2^* Se_2 ) 1442 4.32E-03 2 -1000015 1000015 # BR(VZp -> Se_3^* Se_3 ) 1443 1.33E-03 2 -2000011 2000011 # BR(VZp -> Se_4^* Se_4 ) 1444 1.26E-03 2 -2000013 2000013 # BR(VZp -> Se_5^* Se_5 ) 1445 1.26E-03 2 -2000015 2000015 # BR(VZp -> Se_6^* Se_6 ) 1446 7.94E-03 2 1000012 5000012 # BR(VZp -> SvRe_1 SvIm_4 ) 1447 6.82E-04 2 2000012 4000012 # BR(VZp -> SvRe_4 SvIm_1 ) 1448 6.45E-04 2 2000014 4000014 # BR(VZp -> SvRe_5 SvIm_2 ) 1449 6.45E-04 2 2000016 4000016 # BR(VZp -> SvRe_6 SvIm_3 ) 1450 1.71E-03 2 -24 24 # BR(VZp -> VWm VWm^* )

We go back to EP1 and change topics a bit: we are no longer to know how good the BR of the light Higgs reproduce the interested in new effects compared to the MSSM but we want SM expectations. The width and BRs for this point calculated by SPheno are as follows.

SPheno.spc.BLSSM (EP1)

1343 DECAY 25 4.36597968E-03 # hh_1 1344 # BR NDA ID1 ID2 1345 2.12E-03 2 22 22 # BR(hh_1 -> VP VP ) 1346 8.89E-02 2 21 21 # BR(hh_1 -> VG VG ) 1347 1.84E-02 2 23 23 # BR(hh_1 -> VZ VZ ) 1348 1.65E-01 2 24 -24 # BR(hh_1 -> VWm^* VWm_virt ) 1349 2.34E-04 2 -3 3 # BR(hh_1 -> Fd_2^* Fd_2 ) 1350 6.27E-01 2 -5 5 # BR(hh_1 -> Fd_3^* Fd_3 ) 1351 2.52E-04 2 -13 13 # BR(hh_1 -> Fe_2^* Fe_2 ) 1352 7.27E-02 2 -15 15 # BR(hh_1 -> Fe_3^* Fe_3 ) 1353 2.42E-02 2 -4 4 # BR(hh_1 -> Fu_2^* Fu_2 )

−4 These values have to be compared with those of the SM for BR (ℎ 󳨀→𝜇𝜇) = 2.24 ⋅ 10 , a Higgs mass of 124.2 GeV. The numbers for the SM can be −2 found online at BR (ℎ󳨀→𝑐𝑐) = 2.97 ⋅ 10 , −1 https://twiki.cern.ch/twiki/bin/view/LHCPhysics/ BR (ℎ󳨀→𝑠𝑠) = 2.51 ⋅ 10 , CERNYellowReportPageBR3 (ℎ 󳨀→𝑔𝑔) = 8.62 ⋅10−2, and the different BRs and the total width are BR −1 −3 BR (ℎ 󳨀→𝑏𝑏) = 5.89 ⋅ 10 , BR (ℎ 󳨀→𝛾𝛾) = 2.28 ⋅10 ,

−1 −3 BR (ℎ󳨀→𝜏𝜏) = 6.45 ⋅ 10 , BR (ℎ 󳨀→𝑍𝛾) = 1.47 ⋅10 , 64 Advances in High Energy Physics

−1 BR (ℎ󳨀→𝑊𝑊) = 2.02 ⋅ 10 , 6.3. Flavour and Precision Observables. The SPheno mod- SARAH −2 ules written by contain already out of the box the BR (ℎ󳨀→𝑍𝑍) = 2.45 ⋅ 10 , routines to calculate many quark and lepton flavour violating observables. In addition, also other observables like (𝑔 − 2)푙 Γℎ = 3.96 MeV. and 𝛿𝜌 are calculated. We are going to start with a short (141) discussion of the results which can be obtained just by running SARAH and SPheno outofthebox.Inasecond TheseBRsaresimilartotheoneswegotwithSPheno step, I show how the FlavorKit functionality [146]canbe but they do not agree exactly. Also the expected width is used to implement Wilson coefficients for new operators and smaller by about 10% than the one calculated by SPheno. howtousethesecoefficientstocalculatenewobservables The reason for this is the enhanced coupling of the Higgs with SPheno. to bottom quarks for this point which was already visible HiggsBounds from the blocks as discussed in the last 6.3.1. Observables Out of the Box. To turn the calculation of subsection. low-energy observables on, the Les Houches input file must contain the following.

LesHouches.in.BLSSM

1 Block SPhenoInput # SPheno specific input 2 ... 3 57 1 # Calculate low energy constraints

In that case the QFV and LFV observables are given in the EDMs are shown in SPhenoLowEnergy.Thisblockreadsfor blocks FlavorKitQFV and FlavorKitLFV. 𝛿𝜌, (𝑔−2)푙 and EP1 as follows.

SPheno.spc.BLSSM (EP1)

964 Block SPhenoLowEnergy # low energy observables 965 20 7.44947152E-16 # (g-2)_e 966 21 3.18490399E-11 # (g-2)_mu 967 22 9.02332214E-09 # (g-2)_tau 968 23 0.00000000E+00 # EDM(e) 969 24 0.00000000E+00 # EDM(mu) 970 25 0.00000000E+00 # EDM(tau) 971 39 -1.62703955E-04 # delta(rho)

WhiletheEDMsvanishbecausewedidnotincludeCPviola- SM prediction of SPheno with exclusion limits from exper- tion, the other observables do not receive a large contribution iment. The reason is that by taking this ratio uncertainties, in this model because of the heavy SUSY spectrum. So, let us for example, in the hadronic parameters, drop out. Also a turn to the flavour observables. constant shift just caused by missing higher order corrections −35 All LFV rates are smaller than 10 and can be inter- in SPheno does not lead to the false impression of a deviation preted as numerical zeros. For QFV there are, of course, the fromtheSMaslongastheratioiscloseto1.Weseeforour nonvanishing SM contributions which have to be taken into pointthatSUSYand𝐵-𝐿 contributions change the prediction account as well. For QFV observables, SPheno does not only of at most 1.5% compared to SM expectation. Thus, this point givetheabsolutesizeoftheobservablelikethecorresponding is in total agreement with all limits. The reasons for this branching ratio or mass splitting, but also give the observable are, of course, again the heavy sfermions in general and the normalized to the SM expectation. For this purpose SPheno weak coupling of the few light states. Even if these results actually calculates each observable internally twice. In the are not very exciting I show the entire output of SPheno second calculation all non-SM contributions are dropped. It forQFVobservablestogiveanoverviewofwhatis is more convenient to confront the value normalized to the calculated. Advances in High Energy Physics 65

SPheno.spc.BLSSM (EP1)

972 Block FlavorKitQFV # quark flavor violating observables 973 200 3.17688828E-04 # BR(B->X_s gamma) 974 201 1.00853596E+00 # BR(B->X_s gamma)/BR(B->X_s gamma)_SM 975 300 6.58819362E-04 # BR(D->mu nu) 976 301 9.99994713E-01 # BR(D->mu nu)/BR(D->mu nu)_SM 977 400 6.23567236E-03 # BR(Ds->mu nu) 978 401 9.99994126E-01 # BR(Ds->mu nu)/BR(Ds->mu nu)_SM 979 402 6.08680525E-02 # BR(Ds->tau nu) 980 403 9.98785723E-01 # BR(Ds->tau nu)/BR(Ds->tau nu)_SM 981 500 5.76751780E-07 # BR(B->mu nu) 982 501 9.99956660E-01 # BR(B->mu nu)/BR(B->mu nu)_SM 983 502 1.27192820E-04 # BR(B->tau nu) 984 503 9.91058613E-01 # BR(B->tau nu)/BR(B->tau nu)_SM 985 600 7.06814633E-01 # BR(K->mu nu) 986 601 9.99999621E-01 # BR(K->mu nu)/BR(K->mu nu)_SM 987 602 2.43654009E-05 # R_K = BR(K->e nu)/(K->mu nu) 988 603 2.39724507E-05 # R_K^SM = BR(K->e nu)_SM/(K->mu nu)_SM 989 1900 1.79264760E+01 # Delta(M_Bs) 990 1901 1.00256419E+00 # Delta(M_Bs)/Delta(M_Bs)_SM 991 1902 4.00744431E-01 # Delta(M_Bd) 992 1903 1.00272001E+00 # Delta(M_Bd)/Delta(M_Bd)_SM 993 4000 2.52888015E-15 # BR(B^0_d->e e) 994 4001 1.01492301E+00 # BR(B^0_d->e e)/BR(B^0_d->e e)_SM 995 4002 7.84581655E-14 # BR(B^0_s->e e) 996 4003 1.01404869E+00 # BR(B^0_s->e e)/BR(B^0_s->e e)_SM 997 4004 1.08030916E-10 # BR(B^0_d->mu mu) 998 4005 1.01492301E+00 # BR(B^0_d->mu mu)/BR(B^0_d->mu mu)_SM 999 4006 3.35173125E-09 # BR(B^0_s->mu mu) 1000 4007 1.01404869E+00 # BR(B^0_s->mu mu)/BR(B^0_s->mu mu)_SM 1001 4008 2.26121864E-08 # BR(B^0_d->tau tau) 1002 4009 1.01492426E+00 # BR(B^0_d->tau tau)/BR(B^0_d->tau tau)_SM 1003 4010 7.10840026E-07 # BR(B^0_s->tau tau) 1004 4011 1.01404999E+00 # BR(B^0_s->tau tau)/BR(B^0_s->tau tau)_SM 1005 5000 1.64101883E-06 #BR(B->see) 1006 5001 9.91378447E-01 #BR(B->see)/BR(B->see)_SM 1007 5002 1.59074205E-06 # BR(B-> s mu mu) 1008 5003 9.91246240E-01 # BR(B-> s mu mu)/BR(B-> s mu mu)_SM 1009 6000 1.10869943E-07 # BR(B -> K mu mu) 1010 6001 9.98828312E-01 # BR(B -> K mu mu)/BR(B -> K mu mu)_SM 1011 7000 4.14273034E-05 # BR(B->s nu nu) 1012 7001 9.99749124E-01 # BR(B->s nu nu)/BR(B->s nu nu)_SM 1013 7002 1.91819884E-06 # BR(B->D nu nu) 1014 7003 9.99750026E-01 # BR(B->D nu nu)/BR(B->D nu nu)_SM 1015 8000 1.30902890E-10 #BR(K^+->pi^+nunu) 1016 8001 9.99859854E-01 # BR(K^+ -> pi^+ nu nu)/BR(K^+ -> pi^+ → nu nu)_SM 1017 8002 3.06205871E-11 # BR(K_L -> pi^0 nu nu) 1018 8003 9.99751687E-01 # BR(K_L -> pi^0 nu nu)/BR(K_L -> pi^0 → nu nu)_SM 1019 9100 2.08773176E-15 # Delta(M_K) 1020 9102 1.00002301E+00 # Delta(M_K)/Delta(M_K)_SM 1021 9103 3.30952503E-03 # epsilon_K 1022 9104 1.00227602E+00 # epsilon_K/epsilon_K^SM

IhaveclaimedabovethatwecanneglecttheneutrinoYukawa To show this, we can try what happens if we turn on this couplings for most studies because they have hardly an coupling by using arbitrary values of 𝑂(0.01) for some entries impact on the calculation since they are highly constrained. of 𝑌]. 66 Advances in High Energy Physics

LesHouches.in.BLSSM

21 Block YVIN # 22 1 1 0.010000E+00 #Yv(1,1) 23 1 2 0.001000E+00 #Yv(1,2) 24 1 3 0.001000E+00 #Yv(1,3) 25 2 1 0.010000E+00 #Yv(2,1) 26 2 2 0.010000E+00 #Yv(2,2) 27 2 3 0.001000E+00 #Yv(2,3) 28 3 1 0.000000E+00 #Yv(3,1) 29 3 2 0.001000E+00 #Yv(3,2) 30 3 3 0.010000E+00 #Yv(3,3)

We find only a small impact on most masses and the QFV observables. However, the light neutrino masses are much too large. SPheno.spc.BLSSM

315 Block MASS # Mass spectrum 316 ... 317 12 5.40862949E-04 # Fv_1 318 14 2.46024560E-03 # Fv_2 319 16 5.03020429E-03 # Fv_3 320 ...

Thus,itwouldbenecessarytomaketherightneutrinomuch some flavour observables, in particular 𝜇-𝑒 conversion, are heavier to get a kind of seesaw suppression. However, also already in conflict with the experimental values shown in Table 8. SPheno.spc.BLSSM

1027 Block FlavorKitLFV # lepton flavor violating observables 1028 701 2.12125239E-14 # BR(mu->e gamma) 1029 702 8.56147942E-18 # BR(tau->e gamma) 1030 703 1.01032494E-17 # BR(tau->mu gamma) 1031 800 1.20856362E-11 #CR(mu-e,Al) 1032 801 2.17524735E-11 #CR(mu-e,Ti) 1033 802 2.94406896E-11 #CR(mu-e,Sr) 1034 803 3.31181138E-11 #CR(mu-e,Sb) 1035 804 1.78405656E-11 #CR(mu-e,Au) 1036 805 1.67814615E-11 #CR(mu-e,Pb) 1037 901 1.28640195E-12 #BR(mu->3e) 1038 902 6.31256061E-16 # BR(tau ->3e) 1039 903 9.44251544E-15 # BR(tau->3mu) 1040 904 4.22162097E-16 # BR(tau- -> e- mu+ mu-) 1041 905 1.23482830E-14 # BR(tau- -> mu- e+ e-) 1042 906 3.37021529E-24 # BR(tau- -> e+ mumu-) 1043 907 2.29773463E-23 # BR(tau- -> mu+ e- e-) 1044 1001 1.94197824E-17 #BR(Z->emu) 1045 1002 6.42222866E-19 # BR(Z->e tau) 1046 1003 1.23585715E-18 # BR(Z->mu tau) 1047 1101 1.94824828E-16 #BR(h->emu) 1048 1102 1.83418137E-15 # BR(h->e tau) 1049 1103 1.40346574E-15 # BR(h->mu tau) 1050 2001 9.16839984E-19 # BR(tau->e pi) 1051 2002 1.35797684E-19 # BR(tau->e eta) 1052 2003 1.49779387E-19 # BR(tau->e eta ’) 1053 2004 1.78707828E-18 # BR(tau->mu pi) 1054 2005 2.64136021E-19 # BR(tau->mu eta) 1055 2006 2.89107562E-19 # BR(tau->mu eta ’) Advances in High Energy Physics 67

Table 8: Current experimental bounds for some low-energy LFV and the BRs we are interested in are observables. Γ (𝑡 → 𝑞𝛾) LFV Bound BR (𝑡 󳨀→𝑞𝛾) = (145) −13 Γ 𝜇 → 𝑒𝛾 5.7 ×10 [294] tot −8 𝜏 → 𝑒𝛾 3.3 ×10 [295] with the total width Γ of the top quark. −8 tot 𝜏 → 𝜇𝛾 4.4 ×10 [295] One can also work in the chiral basis using the effective −12 𝜇 → 𝑒𝑒𝑒 1.0 ×10 [296] Lagrangian: −8 𝜏 → 𝜇𝜇𝜇 2.1 ×10 [297] 훾 훾 − − + − −8 L =𝐴 O +𝐴 O 𝜏 →𝑒𝜇 𝜇 2.7 × 10 [297] 푅 푅 퐿 퐿 (146) − − + − −8 𝜏 →𝜇𝑒 𝑒 1.8 × 10 [297] −8 with the operators 𝜏 → 𝑒𝑒𝑒 2.7 ×10 [297] − − −12 𝜇 , Ti →𝑒, Ti 4.3 × 10 [298] 훾 휇] O퐿 =𝜖휇𝑢(𝑝푞)[𝑖𝜎 𝑝]𝑃푅]𝑢(𝑝푡), 𝜇−, →𝑒−, 7×10−13 Au Au [299] 훾 휇] (147) O푅 =𝜖휇𝑢(𝑝푞)[𝑖𝜎 𝑝]𝑃퐿]𝑢(𝑝푡). The relations between the coefficients are just Thisbringsusalreadytoourendoftheshortexcur- SARAH sion to flavour observables which are included in 𝐴훾 =𝐴퐿 +𝐴푅, and SPheno.Idiscussnowwhatcanbedoneifyourfavourite (148) 𝐵 =𝐴 −𝐴 . observable is not yet calculated by SPheno. 훾 퐿 푅

6.3.2. Adding New Observables. I show now how Wearegoingtomakeuseofthisrelationinthefollowingby 𝐴 𝐴 𝐴 the FlavorKit functionality can be used to implement new first calculating 퐿, 푅, and translating them later into 훾 𝐵 observables in SPheno. To make it even more interesting we and 훾 to calculate the partial width according to (144). choose a process for which SPheno does not even know the Our to-do list is the following: Wilson coefficients. Namely, we decide to study the flavour (1) Get the generic expressions for 𝐴퐿 and 𝐴푅, violating, radiative decays of the top quark: (2) Implement those in SARAH, BR (𝑡 󳨀→𝑞𝛾) with 𝑞=𝑢,𝑐. (142) (3) Implement the formula for the BRs in SARAH, (4)Makesurethatallinformationisfound The process is interesting, because it is highly suppressed in by SARAH and included in the SPheno output. theSMbyGIMbutcanreceivelargecontributionsinSUSY models [300]. The transition amplitude can be expressed by Even if this sounds like a lot of work involving several [301] loop calculations and hacking some code, this is not the case at all. Each step is fully automatized. The user just M = 𝑢(𝑝 )[𝑖𝜎휇]𝑞 (𝐴 +𝐵 𝛾 )] 𝑢 (𝑝 )𝜖∗ (𝑞) 푞 ] 훾 훾 5 푡 휇 (143) has to create three small input files. The first input file is needed for PreSARAH. PreSARAH is a Mathematica pack- with the quark momenta 𝑝푞 and 𝑝푡.Thepartialwidthcanbe age which calculates one-loop amplitudes in a generic way expressed using 𝐴훾 and 𝐵훾 as using FeynArts and FormCalc and extracts the Wilson coefficients for the operators the user needs. The results are 3 𝑚2 −𝑚2 then translated into input files which can be used by SARAH. 1 푡 푞 󵄨 󵄨2 󵄨 󵄨2 Γ (𝑡 󳨀→𝑞𝛾) = ( ) (󵄨𝐴 󵄨 + 󵄨𝐵 󵄨 ) (144) TopPhotonQ.m 𝜋 𝑚 󵄨 훾󵄨 󵄨 훾󵄨 So,wecreatethefile with the following 푡 content. TopPhotonQ.m 1 NameProcess="TopPhotonQ"; 2 3 ConsideredProcess = "2Fermion1Vector"; 4 FermionOrderExternal={1,2}; 5 NeglectMasses={3}; 6 7 8 ExternalFields= {bar[TopQuark], TopQuark , Photon}; 9 CombinationGenerations = {{3,2}, {3,1}}; 10 11 12 AllOperators={ 13 {OTgQSL ,Op[7] Pair[ec[3],k[1]]}, 14 {OTgQSR ,Op[6] Pair[ec[3],k[1]]} 15 }; 16 17 OutputFile = "TopPhotonQ.m"; 18 19 Filters = {}; 68 Advances in High Energy Physics

This defines a new process called TopPhotonQ theoutputshouldbewrittenintothefile TopPhotonQ.m and which involves two fermions and one vector boson we do not want to filter out any diagrams (Filters = {}). (ConsideredProcess). The Fierz ordering of the external We run now this file in Mathematica with PreSARAH states is defined via FermionOrderExternal. PreSARAH similartohowwerunmodelswith SARAH. is absolutely agnostic concerning particle physics and it tries always to calculate the most general case. For us, this would <<$PATH/PreSARAH/PreSARAH.m mean that the results are a function of three masses: those Start["TopPhotonQ.m"]; of the fermions and the one of the vector boson. To make sure that in the generic expression the photon mass does The output file is located in the PreSARAH output directory NeglectMasses = {3} not show up, we put .Thereason and has to be copied to the FlavorKit directory of SARAH. is that the photon is the third particle defined in the list of Because of obvious reasons we choose the QFV/Operators ExternalFields all external states. The information in subdirectory. is not used at all by PreSARAH. PreSARAH just includes the information in the output used for SARAH. SARAH $ cp $PATH/PreSARAH/Output/TopPhotonQ .m → knows then what a “top quark” and a “photon” are. Also CombinationGenerations is not used by PreSARAH $PATH/SARAH/FlavorKit/QFV/Operators/ but just passed to SARAH and SPheno.Thislistcontainsall combinations of external generation indices for which the If we would have put the file in LFV/Operators,the SPheno coefficients are later calculated by . We need here coefficientswouldhavebeencalculatedat𝑄=𝑀푍 instead (3, 2) for top-charm and (3, 1) for top-up operators. The two of 𝑄 = 160 GeV. We are already done with steps (1) and (2) of operators from above are called OTgQSL (𝐴퐿)and OTgQSR the to-do list. Now we have to teach SPheno how to calculate (𝐴푅) and their expressions are given in FeynArts syntax thebranchingratios.Forthispurposeweneedtwofileswhich using ec[3] for the helicity of the third particle and k[1] we have to put into the following. as momentum of the first particle. The meaning of these symbols and how to use, for instance, Dirac matrices in $PATH/SARAH/FlavorKit/QFV/Observables/ the definition of operators are explained in the FeynArts Mathematica FlavorKit The first file is a steering file in syntax. It manual and briefly summarized in the reference defines the name for the process used by SARAH internally, aswell.Notethat,inthecaseofWilsoncoefficientsforQFV which operators are needed to calculate the process, and observables, SARAH will automatically generate for each X XSM which observables should show up in the spectrum file Wilson coefficient another coefficient which just later. includes SM contributions. Finally, we tell PreSARAH that

TqGamma.m

1 NameProcess = "TqGamma"; 2 NameObservables = {{BrTuGamma , 210, "BR(t->u gamma)"}, 3 {ratioTuGamma , 211, "BR(t->u gamma)/BR(t->c gamma)_SM"}, 4 {BrTcGamma , 212, "BR(t->c gamma)"}, 5 {ratioTcGamma , 213, "BR(t->c gamma)/BR(t->c gamma)_SM"}, 6 };

8 NeededOperators = {OTgQSL , OTgQSR , 9 OTgQSLSM , OTgQSRSM};

11 Body = "TqGamma.f90";

NameObservables is an array containing all observables the observables, are given in NeededOperators.Asmen- which should show up in the spectrum file. The first part tioned above, SARAH creates not only routines to calculate of each entry gives the name of a variable, the second one the Wilson coefficients including all new physics, but also thenumberusedintheLesHouchesblock FlavorKitQFV, coefficients in the SM limit. For our purpose, we need both andthethirdonethecommentwhichisusedintheLes sets of coefficients, because we want not only to calculate Houches file to make clear to which variable the number the BR but also to normalize it to the SM expectation shown belongs. All operators, which we need to calculate calculatedunderthesameassumptions.Thenameofanother Advances in High Energy Physics 69 file is given at the end of the steering file. This file con- can start at the stage of initializing variables needed for the tains the “body” of the Fortran routine to calculate the calculation of the observable. The entire file TqGamma.f90 observable TqGamma.f90. With “body” I mean that the head looksasfollows. of the routine is automatically generated by SARAH.Theuser

TqGamma.f90

1 Real(dp) :: width, widthSM , norm 2 Complex(dp) :: Agamma , Bgamma , AgammaSM , BgammaSM 3 Integer :: i1, gt1, gt2

5 Do i1=1,2

7 If (i1.eq.1) Then ! t -> u gamma 8 gt1 = 3 9 gt2 = 1 10 Elseif (i1.eq.2) Then ! t -> c gamma 11 gt1 = 3 12 gt2 = 2 13 End if

15 Agamma=OTgQSL(gt1,gt2)+OTgQSR(gt1,gt2) 16 Bgamma=OTgQSL(gt1,gt2)-OTgQSR(gt1,gt2) 17 AgammaSM=OTgQSLSM(gt1,gt2)+OTgQSRSM(gt1,gt2) 18 BgammaSM=OTgQSLSM(gt1,gt2)-OTgQSRSM(gt1,gt2)

20 norm=1/Pi*((mf_u(gt1)**2-mf_u(gt2)**2)/(2._dp*mf_u(gt1)))**3

22 width=norm*(Abs(Agamma)**2+Abs(Bgamma)**2) 23 widthSM=norm*(Abs(AgammaSM)**2+Abs(BgammaSM)**2)

27 If (i1.eq.1) Then 28 BrTuGamma = width/gTFu(3) 29 ratioTuGamma=width/widthSM 30 Elseif (i1.eq.2) Then 31 BrTcGamma = width/gTFu(3) 32 ratioTcGamma=width/widthSM 33 End if

35 End do

In the first three lines we initialize a few local variables we the coefficients for the SM part only. In line 20, we calculate 2 2 3 need. In the entire routine we make a loop over two iterations. the overall normalization factor ((𝑚푡 −𝑚푞)/(2𝑚푡)) /𝜋 and (3, 1) In the first iteration we pick up the generation indices plug everything into (144).Thevariable width in line 22 is (3, 2) for the top-u decay and in the second one the indices . the partial width including all contributions and widthSM gt1 gt2 Theseindicesaresavedinthevariables and which inline23isthepartialwidthonlywithSMcontributions. arethenusedasargumentforthecoefficients.Wefirst Ourfinalobservablesarethencalculatedusingthetotalwidth 𝐴 𝐵 𝐴 𝐴 express 훾 and 훾 by 퐿 and 푅 and do the same with of the top (gTFu(3)) or taking the ratio of both widths. 70 Advances in High Energy Physics

We are now done with step (3) of the to-do list. Step (4) To save time, I used the option ReadLists->True:that happens automatically if all files have been put into the is, SARAH reads the list with all analytical results from the correct directories. We can now generate the SPheno code previous run. The output has to be copied again in the BLSSM again. subdirectory of SPheno and can be compiled as usual. After running SPheno with the input for EP1 we get the new <<$PATH/SARAH.m entries in the block FlavorKitQFV. Start["B-L-SSM"]; MakeSPheno[ReadLists ->True];

SPheno.spc.BLSSM (EP1)

972 Block FlavorKitQFV # quark flavor violating observables 973 200 3.17688828E-04 # BR(B->X_s gamma) 974 210 2.98458324E-16 # BR(t->u gamma) 975 211 7.63379583E-01 # BR(t->u gamma)/BR(t->c gamma)_SM 976 212 4.32829318E-14 # BR(t->c gamma) 977 213 7.63840319E-01 # BR(t->c gamma)/BR(t->c gamma)_SM 978 ...

Obviously, the numbers and comments show up as expected. and agrees with our calculation within errors. BR(𝑡 → 𝑢𝛾) 2 We see some deviations from the SM prediction. However, is supposed to be suppressed by a factor |𝑉푢푏/𝑉푐푏| ≃ 7.9 ⋅ −3 this is still far away from the experimental limits which still 10 ,where𝑉푞푏 are the entries of the CKM matrix. That is allow branching ratios in the percent range [302]. To get large also similar to what we find. effects of this size one might search for points with light stops, for instance. One can also compare these numbers with the SM prediction in the literature. The partial width is strongly suppressed 6.4. Getting the Fine-Tuning. One of the main motiva- by GIM mechanism and therefore is rather sensitive on the tions for SUSY was naturalness: it solves the hierarchy values of the CKM matrix as well as on the running quark problem of the SM by stabilizing the unprotected Higgs masses in the loop. Therefore, the predicted rates in the SM mass. However, with the more and more severe limits on come with a sizable, theoretical error. In [301]SMprediction the SUSY masses, the question about the fine-tuning rises (𝑡 → 𝑞𝛾) for BR and other flavour violating top decays were again. SARAH and SPheno provide functions to calculate the given. The number for the radiative decays into a charm quark fine-tuning according to (84). The user can choose the list reads of parameters which should be included in the fine-tuning SPheno.m (𝑡 󳨀→𝑐𝛾) = (4.6+1.2 ± 0.4+1.6)×10−14 calculation. For this purpose, of the model has to BR −1.0 −0.5 (149) be extended by the following.

SPheno.m

99 IncludeFineTuning = True; 100 FineTuningParameters={ 101 {m0,1/2}, {m12,1}, {Azero ,1}, 102 {\[Mu],1}, {B[\[Mu]],1}, {MuP,1}, {B[MuP],1} 103 };

2 The list FineTuningParameters contains the parameters in terms of 𝑚0. We see that the fine-tuning can not only which are varied at the GUT and a numerical coefficient be calculated with respect to the input parameters defined to “normalize” the fine-tuning. The factor 1/2 for 𝑚0 is in MINPAR. Also other parameters which, for instance, are there because at the GUT scale the boundary conditions are fixed by the tadpoles equations can be included. In principle, Advances in High Energy Physics 71 one can also calculate the fine-tuning with respect to SM version. Exactly the same steps in Section 6.1.2 areusedfor parameters like the top Yukawa coupling (Yu[3,3])orthe that. When SPheno is compiled, the fine-tuning is calcu- strong interaction (g3)byincludingthoseinthelistabove. lated and included in spectrum file if the corresponding flag After editing SPheno.m,itisnecessarytoreproduce is set in the Les Houches input file. the SPheno code with SARAH and to compile the new

LesHouches.in.BLSSM

1 Block SPhenoInput # SPheno specific input 2 ... 3 550 1. # Calculate Fine-Tuning

Running our point EP1 with the new version before the list of decays. This block contains the following of SPhenoBLSSM, we find the block for the FineTuning entries.

SPheno.spc.BLSSM EP1

1112 Block FineTuning # 1113 0 1.30058900E+03 # Overall FT 1114 1 1.28554566E+02 #m0 1115 2 1.30058900E+03 # m12 1116 3 1.30781970E+02 # Azero 1117 4 1.25153715E+03 #\[Mu] 1118 5 1.04705641E+01 # B[\[Mu]] 1119 6 5.17087156E+02 # MuP 1120 7 4.87134742E+02 # B[MuP]

The overall fine-tuning is given 𝑛 the first entry coming fine-tuning. Moreover, one sees that even the additional with number 0. All other entries list the fine-tuning with parameters from the 𝐵-𝐿 sector can have quite some impact respect to the different parameters. This makes it obvious on the fine-tuning. The reason is that the tadpole equations what parameters contribute mostly to the fine-tuning. In our are coupled because of gauge-kinetic mixing. We can check example these are mainly 𝑀1/2 and 𝜇 which have a similar this assumption by using the flag

LesHouches.in.BLSSM

1 Block SPhenoInput # SPheno specific input 2 .. 3 60 0 # Include possible , kinetic mixing

to turn off gauge-kinetic mixing. The impact on the overall fine-tuning with respect to 𝑀1/2 becomes smaller because fine-tuning is moderately small. However, the contributions the off-diagonal gauge couplings and gaugino do not further 耠耠 of 𝜇 and 𝐵휇 do vanish in this limit as expected. Also the contribute to the running of the gauginos.

1112 Block FineTuning # 1113 0 1.27402534E+03 # Overall FT 1114 1 1.60632359E+01 #m0 1115 2 1.07256471E+03 # m12 1116 3 2.11712917E+02 # Azero 1117 4 1.27402534E+03 #\[Mu] 1118 5 1.14380486E+01 # B[\[Mu]] 1119 6 1.05503076E-08 # MuP 1120 7 3.38297907E-09 # B[MuP] 72 Advances in High Energy Physics

7. Example—Part IV: Higgs Constraints, BR H NP.dat, BR Hplus.dat, BR t.dat, effC.dat). Vacuum Stability, Dark Matter, and Collider While SLHA files can be used with HiggsBounds for Studies models with up to five neutral scalars, the separated files canbeusedwithevenuptonineneutralandninecharged 7.1. Checking Higgs Constraints with HiggsBounds scalars. Since the second input works in more cases I am and HiggsSignals. HiggsBounds and HiggsSignals going to concentrate on it. First, I discuss how exclusion are dedicated tools to study the Higgs properties of a given limits are checked with HiggsBounds;afterwardsIshow parameter point in a particular model. While HiggsBounds the usage of HiggsSignals. checks the parameter point against exclusion limits from 2 Higgs searches, HiggsSignals gives a 𝜒 value to express how good the point reproduces the Higgs measurements. 7.1.1. HiggsBounds. In the same directory in which In general, HiggsSignals and HiggsBounds can handle the SPheno spectrum file is located, also all other input different inputs: either the cross sections for all necessary files for HiggsBounds and HiggsSignals are saved processes can be given at the parton or hadron level, by SPheno.The(relative)pathtothisdirectoryhastobe or the effective couplings of the Higgs states to all SM given as last argument to HiggsBounds when executing it. particles are taken as input. In addition, the masses and Thus, working from the directory $PATH, HiggsBounds is widthsofallCPevenandoddaswellaschargedHiggs started via the following. states are always needed. SPheno provides all input for the effective coupling approach. The information is given $ ./ HiggsBounds/HiggsBounds LandH effC 6 1 SPHENO/ in the SLHA spectrum file and in addition in separated files (called MH GammaTot.dat, MHplus GammaTot.dat, From other directories, one can use absolute paths.

$ $PATH/HiggsBounds/HiggsBounds LandH effC 6 1 $PATH/SPHENO/

The other arguments are the data which should be used. Here, by SPheno would be SLHA which uses also the effective we have chosen LandH which incorporated data from LEP coupling approach. The numbers of neutral scalar22 and andhadroncolliders.OtheroptionswouldbeonlyL for charged scalars are given as integer. HiggsBounds checks all only LEP data, or onlyH foronlyTevatronandLHCdata, files for consistency and if no problem appears, it writes the or onlyP for only data which has been published. Then, results to a file called HiggsBounds results.txt in the weturnontheeffectivecouplinginput effC via separated same directory where the input is located. For our standard files. The other possible options with the data provided point EP1 the results look as follows.

HiggsBounds results.txt

1 # generated with HiggsBounds version 4.1.3 on 22.01.2015 at 08:41 2 # settings: LandH , effC 3 # 4 # column abbreviations 5 # n : line id of input 6 # Mh(i) : Neutral Higgs boson masses in GeV 7 # Mhplus(i) : Charged Higgs boson masses in GeV 8 # HBresult : scenario allowed flag (1: allowed , 0: → excluded , -1: unphysical) 9 # chan : most sensitive channel (see below). chan=0 if no → channel applies 10 # obsratio : ratio [sig x BR]_model/[sig x BR]_limit (<1: allowed , → >1: excluded) 11 # ncomb : number of Higgs bosons combined in most → sensitive channel 12 # 13 # channel numbers used in this file 14 # 682:(pp)->h1->ZZ->llll(lowmass) where h1 is → SM-like (CMS-PAS-HIG -13-002) 15 # (for full list of processes , see Key.dat) 16 # 17 #cols: n Mh(1) Mh(2) Mh(3) 18 1 124.183 355.296 3064.08 19 → Mh(4) Mh(5) Mh(6) Mhplus(1) → 3961.50 3062.15 3095.83 3063.61

→ HBresult chan obsratio ncomb → 1 682 0.525609 1 Advances in High Energy Physics 73

All information to understand the output is already given in about this. In these cases it might be helpful to provide an the file: HiggsBounds finds that the strongest constraints input file MHall uncertainties.dat which includes the come from 𝑝𝑝 →ℎ1 →𝑍𝑍→4𝑙at CMS (chan = 682) uncertaintiesintheHiggsmasscalculation.Iwillgivemore but the rate normalized to the exclusion limit (obsratio)is details about this in the HiggsSignals part. If this file is smaller than 1: that is, the point is allowed (HBresult = 1). provided, HiggsBounds runs several times varying all Higgs A list with all processes which are implemented and which masses in the range of their uncertainty and checks for the were checked is also written to Key.dat in the same direc- strongest constraints. tory. Checking Light Singlet.WecanalsorunthepointEP2which We are far away from any exclusion limit: that is, we do has a light singlet of just 79 GeV and find that also this point nothavetoworryaboutsmalluncertaintiesintheHiggs is allowed by all Higgs searches because of the highly reduced masses because they will not change the overall result. For coupling of this scalar to SM particles. points which are closer to the border, one has to think more HiggsBounds results.txt

1 # channel numbers used in this file 2 # 1:(ee)->(h1)Z->(bb-bar)Z → (hep-ex/0602042, table 14b (LEP)) 3 # (for full list of processes , see Key.dat) 4 # 5 #cols: n ... HBresult chan obsratio ncomb 6 # 7 1 ... 1 1 0.505675 1

The most dominant search channel comes from LEP, but the theHiggsmassandratemeasurements.Thesyntaxisvery rate is just about half of the one needed to rule this point out. similar to HiggsBounds and to run it with the data for our standard point we have to call from the directory $PATH. 7.1.2. HiggsSignals. HiggsSignals is the complement to HiggsBounds and checks how good a point reproduces

$ ./ HiggsSignals/HiggsSignals latestresults peak 2 effC 6 1 SPHENO/

2 Itwouldbepossibletousealsohereabsolutepaths.Thefirst 𝜒 method(peak for peak centred, mass for mass centred, three arguments are different compared to HiggsSignals or both)23, and (iii) parametrization of the Higgs mass and have the following meaning: (i) which experimental uncertainty (1:box, 2:Gaussian; 3:boxandGaussian).The data should be used (refers to the corresponding subdi- results are written into the file HiggsBounds results.txt rectory in $PATH/HIGGSSIGNALS/Expt tables/), (ii) the which reads, for EP1, as follows. HiggsSignals results.txt

1 # generated with HiggsSignals version 1.2.0 on 22.01.2015 at 11:07 2 # settings: latestresults , effC , peak , gaussian 3 # 4 # column abbreviations 5 # n : line id of input 6 # Mh(i) : Neutral Higgs boson masses in GeV 7 # Mhplus(i) : Charged Higgs boson masses in GeV 8 # csq(mu) :Chi ^2 from the signal strengths observables 9 # csq(mh) : Chi^2 from the Higgs mass observables 10 # csq(tot) : total Chi^2 11 # nobs(mu) : number of signal strength observables 12 # nobs(mh) : number of Higgs mass observables 13 # nobs(tot) : total number of observables 14 # Pvalue : Probability , given csq(tot) and ndf=nobs(tot)- 0 15 # 16 #cols: n Mh(1) Mh(2) Mh(3) 17 1 124.183 355.296 3064.08 18 → Mh(4) Mh(5) Mh(6) Mhplus(1) → 3961.50 3062.15 3095.83 3063.61

→ csq(mu) csq(mh) csq(tot) nobs(mu) → 124.810 5.34802 130.158 80

→ nobs(mh) nobs(tot) Pvalue → 4 84 0.933103E-03 74 Advances in High Energy Physics

Also, the HiggsSignals output is rather self-explaining. into a 𝑝 value (Pvalue)24.However,onewarningappears 2 HiggsSignals The important numbers are the 𝜒휇 (csq(mu))fortheHiggs intheterminalwhenrunning in that 2 way. rates, the 𝜒푚 (csq(mh)) for the Higgs mass, and the com- 2 bined 𝜒tot (csq(tot)). The combined one is also translated

Optional datafile SPHENO/MHall uncertainties .dat not found. Using → default values .

That means that the file MHall uncertainties.dat was The theoretical uncertainty for the corrections included in not found. That is not surprising because it was not created the SARAH-SPheno interface is expected to be similar to the by SPheno. This file contains the theoretical uncertainty oneintheMSSMusingstandardtwo-loopcorrections.Thus, of the Higgs mass prediction. Thus, to produce this file we put 3.0 GeV for the two light scalars and 1.0 GeV for all some estimate of the size of missing higher order correc- others25.WecreateMHall uncertainties.dat and put tions to the Higgs masses is needed. This is something it in the SPheno directory where also all other input files what SPheno cannot do automatically at the moment. for HiggsBounds/HiggsSignals are stored. The content However, HiggsSignals assumes no theoretical uncer- of MHall uncertainties.dat readsasfollows. tainty if the file is missing. That is, of course, unrealistic.

1 1 3.0 3.0 1.0 1.0 1.0 1.0 1.0

2 Now, running HiggsSignals againwefindthatthe𝜒 values become slightly smaller.

HiggsSignals results.txt

16 csq(mu) csq(mh) csq(tot) nobs(mu) 17 80.8857 3.46415 84.3498 80

18 → nobs(mh) nobs(tot) Pvalue → 4 84 0.468755

Light Singlet. We want to run also the second example point which has a light singlet. We keep our estimate of the theoretical uncertainty and find for this point the following.

HiggsSignals results.txt

16 #cols: n Mh(1) Mh(2) ... 17 1 77.8961 125.607 ...

19 → csq(mu) csq(mh) csq(tot) → 81.0244 1.35131 82.3757

→ nobs(mu) nobs(mh) nobs(tot) Pvalue → 80 4 84 0.529729 Advances in High Energy Physics 75

The couplings of the SM-like Higgs to the SM-fermions do Vevacious takes the tadpole equations, the polynomial 2 not change significantly between EP1 and EP2: that is, also 𝜒휇 part of the scalar potential, and all mass matrices as input. All stayed the same. However, the mass of the SM-like state is a of this information has to be expressed including all VEVs 2 which should be tested. This means that to check for charge bit closer to the best-fit of the measurements; hence, 𝜒푚 has slightly decreased. and colour breaking minima the stop and stau have to show up in the potential and mass matrices and the entire mixing triggered by these VEVs should be included. To take care 7.2. Checking the Vacuum Stability with Vevacious. The of all that, the corresponding input files can be generated parameter points EP1 and EP2 have passed the first checks. by SARAH as explained below. The mass spectrum looks promising and they are consistent with all bounds from flavour and Higgs physics. As next step we want to check if the points have really 7.2.1. Finding the Global Minimum without Sneutrino VEVs. a stable vacuum: since SPheno found a solution for the If one is just interested in the global minimum for the tadpole equations, we are sure that the given parameters case that no other VEVs are allowed, it is straightforward Vevacious areatleastatalocalminimumwithrespecttothescalar to get the model files for :thestandardimple- potential where the set {V푑, V푢,𝑥1,𝑥2} of VEVs is nonzero. mentation of the model can be used together with the MakeVevacious However, this does not ensure that this is also the global command . minimum. First, there might be a deeper minimum for <<$PATH/SARAH/SARAH.m; other values of {V푑, V푢,𝑥1,𝑥2}. Those minima are in general ruled out because they would predict another mass for Start["B-L-SSM"]; the 𝑍-boson. Another possibility is that other particles MakeVevacious[]; could receive VEVs as well. These can either be points with spontaneous 𝑅-parity violation where the sneutrino MakeVevacious comes with some options which I list for {V , V ,𝑥 ,𝑥 , V푖 , V푖 } get a VEV ( 푑 푢 1 2 ̃]퐿 ̃]푅 ), points where charge is completeness. However, we stick to the default settings. 푖 푖 broken by slepton VEVs ({V푑, V푢,𝑥1,𝑥2, V푒̃ , V푒̃ }), or points 퐿 푅 (i) ComplexParameters: it defines if specific param- wherechargeandcolourarebrokenbysquarkVEVs 푖 푖 푖 푖 eters should be treated as complex. By default, all ({V푑, V푢,𝑥1,𝑥2, V ̃ , V ̃ , V푢̃ , V푢̃ }). The last two possibilities are Vevacious 푑푅 푑푅 퐿 푅 parametersareassumedtoberealinthe completely forbidden and points would always be ruled out output. by that. However, the dangerous regions for charge or colour (ii) IgnoreParameters: it defines if a given set of breakingarethosewherethetrilinearsoft-termsarelarge parametersshouldbesettozerowhenwriting compared to the soft-masses in the stop or stau sector [22, the Vevacious model files. 303–313].ThisisnotthecaseforthepointsEP1andEP2 OutputFile andwedonothavetoworryaboutthat.Spontaneous𝑅- (iii) : it defines the name for the model files. BLSSM.vin parity violation is not completely forbidden and could lead By default is used. to a different phenomenology. However, in our approach it (iv) Scheme: it defines the renormalization scheme. For 耠 is also very likely that the electroweak VEV changes at the SUSY models SARAH uses DR and for non-SUSY MS global minimum where the sneutrinos gain nonzero VEVs. by default. Hence,suchascenarioisruledoutaswellbythe𝑍 mass. We are going to check the stability of the vacuum with neglecting One sees from the first option that the parameters are handled and with including the possibility of sneutrino VEVs. For this less general in the Vevacious output as this is usually done purposeweusethepackage Vevacious [168]. by SARAH.Thereasonisthattheevaluationofaparameter Vevacious is a tool to check for the global mini- point with Vevacious can be very time consuming. Thus, mum of the one-loop effective potential for a given model doing reasonable approximations might be an option to speed allowingforaparticularsetofnonzeroVEVs.Forthis this up. purpose Vevacious finds first all tree-level minima by As soon as the model file is created, it is convenient to using HOM4PS2 [314]. Afterwards, it minimizes the one-loop copy them to the model directory of the local Vevacious effective potential starting from these minima using minuit installation. In addition, one can also generate a new subdi- [315].Iftheinputminimumturnsoutnottobethe rectory which contains the SPheno spectrum files for the B- global one, life-time of meta-stable vacua can be calculated L-SSM used as input for Vevacious,aswellastheoutput using Cosmotransitions [316]. written by Vevacious.

$ cd $PATH/VEVACIOUS $ mkdir BLSSM/ $ cp $PATH/SARAH/Output/B−L−SSM/ V e v a c i o u s /BLSSM . v i n m o d e l s / $ cp $PATH/SPHENO/SPheno . spc .BLSSM BLSSM/ 76 Advances in High Energy Physics

These steps are just optional: the user can give in the to write this initialization file for a new study. The easiest way initialization file used by Vevacious, which is discussed in is to start with the file included in the Vevacious package a second, also paths to other locations of the model and in the subdirectory bin and edit it. spectrum file. Independent of the location of the files, one has

$ cd $PATH/VEVACIOUS/ b i n $ cp VevaciousInitialization .xml VevaciousInitialization BLSSM .xml

The only change we apply here is to give the paths for by Vevacious. I am going to assume here that these are HOM4PS2 (http://www.math.nsysu.edu.tw/∼leetsung/works/ installedinthesamedirectory$PATH as all other tools are. HOM4PS soft files/)andCosmoTransitions (http:// The other pieces of information needed are the location of chasm.uchicago.edu/cosmotransition) which are used the model and spectrum files as mentioned above.

VevaciousInitialization BLSSM.xml ...

$PATH/HOM4PS2/

...

$PATH/CosmoTransitions_package_v1.0.2

...

$PATH/Vevacious -1.1.01/models/BLSSM.vin

...

$PATH/Vevacious -1.1.01/BLSSM/SPheno.spc.BLSSM

For all other settings like what homotopy method should be $ ./../bin/Vevacious.exe → used, what is the tolerance to consider extrema as identical, −− input =./../ bin/VevaciousInitialization BLSSM .xml what the necessary survival probability is to label a meta- Vevacious stable point “long-lived,” and how should try to After about 30 s Vevacious is done with checking for get away from saddle points we keep the default values. Inter- the global minimum of the one-loop effective potential. Vevacious ested reader might take a look at the manual for Since it has not started CosmoTransitions to calculate the more details about these options. tunnelling time, the point is stable. This can also be seen from When all adjustments of the initialization file are done, the file SPheno.spc.BLSSM whereanewblockhasbeen Vevacious we can run on EP1 by calling from the directory appended. $PATH/Vevacious/BLSSM. Advances in High Energy Physics 77

SPheno.spc.BLSSM (EP1)

1542 BLOCK VEVACIOUSRESULTS # results from Vevacious 1543 0 0 1.00000000E+000 stable # stability of input 1544 0 1 -1.00000000E+000 unnecessary # tunneling time → in Universe ages / calculation type 1545 0 2 0.00000000E+000 1.0 # estimated best tunneling → temperature / survival probability at this temperature 1546 1 0 -5.25790342E+011 relative_depth # DSB vacuum → potential energy 1547 1 1 3.58009590E+001 vd # DSB vacuum VEV 1548 1 2 2.37944201E+002 vu # DSB vacuum VEV 1549 1 3 3.50031616E+003 x1 # DSB vacuum VEV 1550 1 4 2.91749649E+003 x2 # DSB vacuum VEV 1551 2 0 -5.25790342E+011 relative_depth # panic vacuum → potential 1552 2 1 3.58009590E+001 vd # panic vacuum VEV 1553 2 2 2.37944201E+002 vu # panic vacuum VEV 1554 2 3 3.50031616E+003 x1 # panic vacuum VEV 1555 2 4 2.91749649E+003 x2 # panic vacuum VEV

This block contains a flag to assign the stability[0,0] ( = 1: minimum are shown (entries [2,0]–[2,4]). Obviously, stable, [0,0] = 0 long-lived, [0,0] = -1:short- the entries [1,X] and [2,X] are identical. lived, [0,0] = -2: thermally excluded), the tunnelling If the user is interested in some more information about time if calculated (entry [0,1]), and the temperature all possible minima found at tree-level and one-loop, he/she at which tunnelling is likely to happen (entry [0,2]). can check the file Vevacious tree-level extrema.txt. Afterwards the input VEVs are repeated (entries [1,1]– This file contains all VEV combinations which are actually [1,4]) and the depth of the potential at the input a minimum of the tree-level potential. Also the depth of the minimum is given (entry [1,0]). Finally, the depths potential at each minimum is given at tree-level and one-loop of the global minimum together with the VEVs at that level.

Vevacious tree-level extrema.txt

1 { 2 { { vd->(-143.325), vu->(-953.137), x1->(0),x2 ->(0) }, 3 TreeLevelPotentialValue -> -13789506454.7, 4 EffectivePotentialValue -> -2.12169006e+13 } 5 { { vd->(-35.8005), vu->(-238.079), x1->(-3500.67), x2->(-2917.22) }, 6 TreeLevelPotentialValue -> -5.29194675525e+11, 7 EffectivePotentialValue -> -2.17385124968e+13 } 8 { { vd->(-35.8005), vu->(-238.079), x1->(3500.67), x2->(2917.22) }, 9 TreeLevelPotentialValue -> -5.29194675525e+11, 10 EffectivePotentialValue -> -2.17385124968e+13 } 11 { { vd->(0), vu->(0), x1->(-3503.34), x2->(-2919.45) }, 12 TreeLevelPotentialValue -> -5.29142228921e+11, 13 EffectivePotentialValue -> -2.17384087462e+13 } 14 { { vd->(0), vu->(0), x1 ->(0), x2->(0) }, 15 TreeLevelPotentialValue -> 0.0, 16 EffectivePotentialValue -> -2.12127237174e+13 } 17 { { vd->(0), vu->(0), x1->(3503.34), x2->(2919.45) }, 18 TreeLevelPotentialValue -> -5.29142228921e+11, 19 EffectivePotentialValue -> -2.17384087462e+13 } 20 { { vd->(35.8005), vu->(238.079), x1->(-3500.67), x2->(-2917.22) }, 21 TreeLevelPotentialValue -> -5.29194675525e+11, 22 EffectivePotentialValue -> -2.17385124968e+13 } 23 { { vd->(35.8005), vu->(238.079), x1->(3500.67), x2->(2917.22) }, 24 TreeLevelPotentialValue -> -5.29194675525e+11, 25 EffectivePotentialValue -> -2.17385124968e+13 } 26 { { vd->(143.325), vu->(953.137), x1->(0), x2->(0) }, 27 TreeLevelPotentialValue -> -13789506454.7, 28 EffectivePotentialValue -> -2.12169006e+13 } 29 } 78 Advances in High Energy Physics

Wecanseefromthatfilethatthereareactuallyfourminima We can do the same check for EP2 and find that also this which are not related by a phase transformation of the point is stable. VEVs. The minimum without symmetry breaking (all VEVs are zero) has a depth of 0 at tree-level as expected but receives large loop corrections. Nevertheless, the depth is 7.2.2. Checking for Spontaneous 𝑅-Parity Violation. As men- still much less than that for all other combinations where tioned above, one cannot be completely sure that the point there is at least one symmetry (electroweak or 𝐵-𝐿 is bro- is stable if Vevacious does not find a deeper minimum if ken). There is just one minimum where both symmetries the first check is passed. There is still the possibility that are broken and this corresponds to our input minimum. additional particles might receive VEVs. We are checking The full list of minima found at one-loop is given inthe here the possibility of spontaneous 𝑅-parity violation. For file Vevacious loop-corrected minima.txt. All addi- this purpose, it is necessary to create a new SARAH model tional minima listed there are small variations of the tree-level file. We call it B-L-SSM RpV.m.Thesimplestwayistotake ones. our B-L-SSM.m fileasbasisandapplythefollowingchanges.

BLSSM RpV.m Model‘Name = "BLSSMRpV";

DEFINITION[EWSB][VEVs]= { ..., {SvL, {vL[3], 1/Sqrt[2]}, {sigmaL , → \[ImaginaryI]/Sqrt[2]},{phiL ,1/Sqrt[2]}}, {SvR, {vR[3], 1/Sqrt[2]}, {sigmaR , → \[ImaginaryI]/Sqrt[2]},{phiR ,1/Sqrt[2]}}, };

(*--- Matter Sector ---- *) DEFINITION[EWSB][MatterSector]= { ... {{phid, phiu,phi1, phi2,phiL,phiR}, {hh, ZH}}, {{sigmad , sigmau ,sigma1 ,sigma2 ,sigmaL ,sigmaR}, {Ah, ZA}}, {{SHdm,conj[SHup],SeL, SeR},{Hpm,ZP}}, {{fB, fW0, FHd0, FHu0,fBp,FC10,FC20,FvL,conj[FvR]}, {L0, ZN}}, {{{fWm, FHdm,FeL}, {fWp, FHup,conj[FeR]}}, {{Lm,UM}, {Lp,UP}}} ... };

First,wechangethenameofthemodeltomakesurethatno the neutralinos with the neutrinos. We have to include files of the other implementations are overwritten. The main this mixing because Vevacious checks not only the tree- modification is to give VEVs to the left and right sneutrinos level potential but also the one-loop effective potential. as done by the changes in DEFINITION[EWSB][VEVs]. This mixing will give additional contributions to the However, we did not consider the most general case where one-loop corrections. If we are really just interested in all three generations get VEVs but restrict VEVs to the the Vevacious output, we can skip the modifications third generation only. Even in this case we have to deal of particles.m and parameters.m andcomeback with a 6-dimensional parameter space. In the general case, directly to your study with Vevacious: the remaining we would even have 10 VEVs and running Vevacious steps are the same as for the 𝑅-parity conserving would take significant longer. Therefore, it s always good case: (i) running the B-L-SSM RpV with SARAH, (ii) to check what degrees of freedom can be rotated away. running MakeVevacious[], (iii) copying the file to The other lines are a consequence of 𝑅-parity violation: the Vevacious installation, (iv) creating a new initialization a mixing between the CP even and odd sneutrinos and file VevaciousInitialization BLSSM RpV.xml with the Higgs scalars happens and the charged Higgs scalars the location of the model file, and (v) running Vevacious. mix with the charged sleptons. In the fermionic sector We find that both parameter points pass also this check. the charginos mix similarly with the charged leptons and For example, for EP1 the Vevacious outputreadsasfollows. Advances in High Energy Physics 79

SPheno.spc.BLSSM (EP1)

1542 BLOCK VEVACIOUSRESULTS # results from Vevacious version 1.1.01, documented in → arXiv:1307.1477, arXiv:1405.7376 (hep-ph) 1543 0 0 1.00000000E+000 stable # stability of input 1544 0 1 -1.00000000E+000 unnecessary # tunneling time ... 1545 0 2 0.00000000E+000 1.0 # ... tuneling temperature 1546 1 0 -5.24262779E+011 relative_depth # DSB vacuum ... 1547 1 1 0.00000000E+000 vL3 # DSB vacuum VEV 1548 1 2 0.00000000E+000 vR3 # DSB vacuum VEV 1549 1 3 3.58548559E+001 vd # DSB vacuum VEV 1550 1 4 2.26723606E+002 vu # DSB vacuum VEV 1551 1 5 3.49979972E+003 x1 # DSB vacuum VEV 1552 1 6 2.91786084E+003 x2 # DSB vacuum VEV 1553 2 0 -5.24262779E+011 relative_depth # panic vacuum ... 1554 2 1 0.00000000E+000 vL3 # panic vacuum VEV 1555 2 2 0.00000000E+000 vR3 # panic vacuum VEV 1556 2 3 3.58548559E+001 vd # panic vacuum VEV 1557 2 4 2.26723606E+002 vu # panic vacuum VEV 1558 2 5 3.49979972E+003 x1 # panic vacuum VEV 1559 2 6 2.91786084E+003 x2 # panic vacuum VEV

Iwanttoshowthatthingsarenotalwayssoboringandunex- Even with these modifications, all right sneutrino soft-terms pected things can happen in such complicated potentials. For are still positive. this purpose, I modify the input parameters for EP1 a bit:

𝐴0 = −1600 GeV, (150) 𝑌푥 = diag (0.39, 0.40, 0.41) .

SPheno.spc.BLSSM Block mv2 Q= 3.04177370E+03 # (SUSY Scale) 1 1 1.03252784E+06 # Real(mv2(1,1),dp) 2 2 9.42973968E+05 # Real(mv2(2,2),dp) 3 3 8.52847167E+05 # Real(mv2(3,3),dp)

Nevertheless, we find that at the global minimum 𝑅-parity is broken by sneutrino VEVs.

SPheno.spc.BLSSM

1542 BLOCK VEVACIOUSRESULTS # results from Vevacious version 1.1.02, documented in → arXiv:1307.1477, arXiv:1405.7376 (hep-ph) 1543 0 0 -2.00000000E+000 long-lived_but_thermally_excluded # 1544 0 1 4.89995748E+031 direct_path # tunneling time in → Universe ages / calculation type 1545 0 2 1.11583595E+003 0.0 # estimated best tunneling → temperature / survival probability → at this temperature 1546 1 0 -6.30122774E+011 relative_depth # DSB vacuum ... 1547 1 1 0.00000000E+000 vL3 # DSB vacuum VEV 1548 1 2 0.00000000E+000 vR3 # DSB vacuum VEV 1549 1 3 3.57990259E+001 vd # DSB vacuum VEV 1550 1 4 2.39981235E+002 vu # DSB vacuum VEV 1551 1 5 3.66612055E+003 x1 # DSB vacuum VEV 1552 1 6 3.05552200E+003 x2 # DSB vacuum VEV 1553 2 0 -6.99210151E+011 relative_depth # panic vacuum ... 1554 2 1 0.00000000E+000 vL3 # panic vacuum VEV 1555 2 2 1.57052814E+003 vR3 # panic vacuum VEV 1556 2 3 4.56069965E-001 vd # panic vacuum VEV 1557 2 4 3.14986273E+000 vu # panic vacuum VEV 1558 2 5 2.41594641E+003 x1 # panic vacuum VEV 1559 2 6 2.05701579E+003 x2 # panic vacuum VEV 80 Advances in High Energy Physics

We see that the stability is labelled as “long-lived but (i) FeynmanGauge: it defines if Feynman gauge should thermally excluded” ([0,0] = -2). This means that the be supported beside Landau gauge. This is done by point is long-lived at zero temperature but quickly decays if default. temperature effects are taken into account. In that case the CPViolation [0,2] (ii) : it defines if parameters should be han- entry shows at which temperature the tunnelling is dled as complex. By default, all parameters are treated likely to happen. Thus, this point is actually ruled out. as real because CalcHep is not really optimized for One sees at this example that the condition the usage of complex parameters and this option 2 should be used carefully. 𝑚̃]퐶 <0 (151) (iii) ModelNr:itnumbersthemodelfiles. SARAH starts by sometimes used in the literature for distinguishing 𝑅-parity default with 1. violation and conservation is not necessary. On the other (iv) CompHep:itcanbeusedtowritemodelfiles hand, it is also possible to find points where this condition is in CompHep instead of CalcHep format. fulfilled, but 𝑅-parity is still unbroken at the global minimum NoSplittingWith SARAH [275]; that is, it is also not sufficient. Therefore, one should (v) : does not decompose not rely on such simple minded conditions but perform four-scalar interactions in pairs of two scalar always a numerical check to test the vacuum stability. The interactions with auxiliary fields if particular fields same statement holds for charge and colour breaking minima: are involved. Such a decomposition is usually analytical thumb rules like [22, 303–306] done because of the implicit colour structure in CalcHep which does not allow four-point 2 2 󵄨 󵄨2 2 2 interactions of coloured states. To keep the model 𝐴휏 <3(𝑚퐻 + 󵄨𝜇󵄨 +𝑚휏 +𝑚휏 ), 푑 퐿 푅 SARAH (152) files shorter, makes the same decomposition 󵄨 󵄨2 𝐴2 <3(𝑚2 + 󵄨𝜇󵄨 +𝑚2 +𝑚2 ) also for noncoloured states. 푡 퐻푢 󵄨 󵄨 푡퐿 푡푅 (vi) NoSplittingOnly: one can define particles, for whichare,unfortunately,stillwidelyusedintheliteraturedon which SARAH does not decompose four-scalar inter- not bear up against numerical checks and turn out to be pretty actions in pairs of two scalar interactions with aux- useless [220, 221]. These conditions miss the large majority of iliary fields if only the given fields are involved the points which actually suffer from an unstable ew vacuum. interaction.

(vii) UseRunningCoupling: it defines if 𝛼푆 should run in 7.3. Calculating the Dark Matter Properties the model files. with MicrOmegas. As next step we want to study (viii) SLHAinput: it defines if parameter values should be thedarkmatter(DM)propertiesofthemodelby read from a spectrum file. using MicrOmegas. MicrOmegas is a tool which not only calculates the relic density for one or more dark matter (ix) CalculateMasses: it defines if tree-level masses candidates, but also gives cross sections for direct and indirect should be calculated internally by CalcHep. MicrOmegas DM searches. To enable these calculations, (x) RunSPhenoViaCalcHep: it writes C code to needs in general three inputs: run SPheno from the graphical interface CalcHep (1) the model files to implement a new model, of to calculate the spectrum on the fly. (2) a steering file to coordinate the different calculations, (xi) IncludeEffectiveHiggsVertices: it defines if (3) numerical values for all parameters. effective Higgs vertices ℎ𝛾𝛾 and ℎ𝑔𝑔 should be I will show step by step how these three points are addressed. included. (xii) DMcandidate1:itsetsthefirstDMcandidate. 7.3.1. Implementing New Models in MicrOmegas. The calcu- (xiii) DMcandidate2:itsetsoptionallyasecondDMcan- lations of the cross section and all necessary decay widths are didate. done by CalcHep which comes together with MicrOmegas. Thus, a new model in MicrOmegas is implemented by pro- For our example we can stick to the default options. I will just viding the corresponding CalcHep model files. This means comment on two important switches which demand a further that one can use the SARAH output for CalcHep to work explanation. with MicrOmegas. Mass Spectrum.ByusingSLHAinput -> True CalcHep <<$PATH/SARAH/SARAH.m; themodelfilesarewritteninawaythat and, respectively, MicrOmegas expect all input Start["B-L-SSM"]; parameters to be provided in a spectrum file which is MakeCHep[]; called SPheno.spc.BLSSM. CalcHep and MicrOmegas are going to read this file and extract all important By just running MakeCHep[], the default options are used. information using the SLHA+ functionality [317]fromit. However, there are several options to adjust the output. With the other options MicrOmegas/CalcHep expect either Advances in High Energy Physics 81

all masses and rotation matrices given in the file vars.mdl odd under the first 𝑍2 defined as global symmetry; (ii) for (SLHAinput -> False, CalculateMasses -> False), any other choices, one can give first the name of the global or all fundamental parameters (soft-terms, couplings, and symmetry and then the quantum number with respect to that VEVs) as input and diagonalizes the mass matrices internally symmetry GlobalSymmetry == Charge. (SLHAinput -> False, CalculateMasses -> True). When SARAH is finished with MakeCHep,the CalcHep model files are located in the directory Dark Matter Candidates. One can work either with one or two dark matter candidates in MicrOmegas.ThefirstDM $PATH/SARAH/Output/B-L-SSM/EWSB/CHep/. candidate is the lightest particle of all states having a particular charge under a discrete symmetry to define the symmetry and the charge, and the option DMcandidate1->Value is To implement the model in MicrOmegas,anewprojecthas used. There are two possibilities for Value:(i)whenset tobecreatedandthefileshavetobecopiedintheworking to Default, the DM candidate is the lightest odd particle directory of this project.

$ cd $PATH/MICROMEGAS $ ./ newProject BLSSM $ cd BLSSM $ cp $PATH/SARAH/Output/B−L−SSM/EWSB/CHep/∗ work/models

7.3.2. Setting Up the DM Calculations. To use the model rates. Those can be added as well to the main file provided with MicrOmegas a steering or “main” file has to be by SARAH or the user can write their own file. For this provided either in Fortran or C language and must purpose, it might be helpful to take a look at main.F be compiled. Examples for these files are delivered or main.c which show the different options to turn on with MicrOmegas and called main.F and main.c. SARAH specific calculations and outputs. writes also two examples which can be used for the following The steering files written by SARAH were copied together calculations. with all model files into the working directory of the current project. We can move it to the main project directory and (i) CalcOmega.cpp: this file calculates only the DM 2 compile it. relic density Ωℎ andprintstheresultatthescreen omg.out andintoafilecalled . $ mv work/models/CalcOmega with DDetection . cpp . (ii) CalcOmega with DDetection.cpp: this file $ make main=CalcOmega with DDetection . cpp 2 calculates the DM relic density Ωℎ andinaddition some direct detection rates: (i) spin independent Anewbinary CalcOmega with DDetection is now avail- cross section with proton and neutron in pb, (ii) able. The only missing pieces are the input parameters. spin dependent cross section with proton and neutron in pb, and (iii) recoil events in the 10–50 keV 73 131 23 127 region at Ge, Xe, Na, and nuclei. The I MicrOmegas SPheno output is also written into a file called omg.out. 7.3.3. Running with Spectrum Note that the syntax for the direct detection Files. Providing the numerical parameters is pretty easy MicrOmegas CalcHep SPheno calculations has been changed in MicrOmegas because / can read the compared to earlier versions. SARAH includes also spectrumfile.However,theusermustmakesurethatno afileCalcOmega with DDetection old.cpp complex rotation matrices show up in the spectrum file: in which is compatible with versions 2.X the case of Majorana matrices and no CP violation, there are of MicrOmegas. two equivalent outputs: (i) all Majorana masses are positive, but some entries of the corresponding rotation matrices are Wearegoingtochoosethesecondfilewhichincludesthe complex; (ii) all mixing matrices are real, but some masses calculation of direct detection rates. There are even more are negative. CalcHep can just handle the second case with calculations MicrOmegas can do like indirect detection real matrices. Hence, one has to use the flag as follows

LesHouches.in.BLSSM Block SPhenoInput # SPheno specific input ... 50 0 # Majorana phases: use only positive masses 82 Advances in High Energy Physics to get the spectrum according to that convention. Afterwards, compile all necessary annihilation channels of the DM thespectrumfilejusthastobemovedtothesamedirectory candidate for that particular parameter point. All further as CalcOmega with DDetection.Wecopyitthereand evaluations of similar points are done in a second or less. start the calculation as follows. However, as soon as new channels are needed, MicrOmegas hastocompilenewamplitudesandthecomputationslows $ cp $PATH/SPHENO/SPheno . spc .BLSSM . down extremely again. This can happen, for instance, if the $ ./ CalcOmega with DDetection DM candidate changes or if the second lightest state becomes closeinmassandcoannihilationhastobeincluded.Assoon The first run can take some time, even up to several hours as the run is done, we see the following on the screen. depending on the computer power: MicrOmegas has to

... Xf=2.43e+01 Omega hˆ2=1.22e−01

# Channels which contribute to 1/(omega) more than 1%. # Relative contributions in % are displayed 98% ˜nR1 ˜nR1 −>h2 h2

==== C a l c u l a t i o n o f CDM−n u c l e o n s a m p l i t u d e s ===== TREE LEVEL CDM−nucleon micrOMEGAs amplitudes : proton : SI 1.217E−10 SD 0.000E+00 neutron : SI 1.225E−10 SD 0.000E+00 BOX DIAGRAMS CDM−nucleon micrOMEGAs amplitudes : proton : SI 1.217E−10 SD 0.000E+00 neutron : SI 1.225E−10 SD 0.000E+00 CDM−nucleon cross sections [pb]: proton SI 6.453E−12 SD 0.000E+00 neutron SI 6.539E−12 SD 0.000E+00

======D i r e c t D etection ======73Ge: Total number of events=6.71E−07 /day/kg Number of events in 10 − 50 KeV region =3.56E−07 /day/kg 131Xe: Total number of events=1.07E−06 /day/kg Number of events in 10 − 50 KeV region =5.49E−07 /day/kg 23Na: Total number of events=6.66E−08 /day/kg Number of events in 10 − 50 KeV region =3.57E−08 /day/kg I127 : Total number of events=1.05E−06 /day/kg Number of events in 10 − 50 KeV region =5.48E−07 /day/kg

In the first line, the freeze-out temperature and the relic combining Planck, WMAP polarization, high-resolution density are given. We find that this point falls into the CMB data, and baryon acoustic oscillation results [318]. preferred 2𝜎 region: The important channels contributing to the annihilation follow in the next lines. This point is a bit boring, because the annihilation in two bileptons makes 98% of the entire annihilation. All other individual channels are not printed because they are below 1%. This threshold can be changed in CalcOmega with DDetection.cpp by changing the cut 0.1153 < Ω ℎ2 < 0.1221 CDM (153) to lower values. Advances in High Energy Physics 83

CalcOmega with DDetection.cpp

18 double cut = 0.01; // cut-off for channel output

The same information is also written in the file omg.out. The style of this file is inspired by the SLHA format.

omg.out

1 1 0.122031 # relic density 2 100 0.980976 # ~nR1 ~nR1 -> h2 h2 3 201 0.000000000006453 # 4 202 0.000000000000000 # 5 203 0.000000000006539 # 6 204 0.000000000000000 # 7 301 0.000001 # 8 302 0.000001 # 9 303 0.000000 # 10 304 0.000001 #

Because of this format, one can append this file to the Despite the many different channels which contribute to the spectrum file to save the dark matter results together with the annihilation, the relic density is much too high. This is not other pieces of information and read it later with a standard surprising because it is well known that for a neutralino SLHA parser. LSP often particular conditions are needed to fulfil the relic density bounds. Either a charged particle close in mass, or resonances, or a large Higgsino fraction are needed. This $ cat SPheno . spc .BLSSM > SPheno . spc .BLSSM with MO holds not only for a bino LSP in the CMSSM but also for a $ echo ’ B l o c k DARKMATTER #’ >> SPheno . spc . BLSSM with MO $ cat omg . out >> SPheno . spc .BLSSM with MO blino LSP in the constrained B-L-SSM as we have it here [278].

This is, for instance, done automatically when running scans with SSP and including MicrOmegas. WHIZARD The values shown for the direct detection rates can be 7.4. Monojet Events with . We change topics again compared with limits from experiments. For this purpose it and enter the wide field of collider studies with Monte −36 is helpful to multiply these values by a factor of 10 to get Carlo(MC)tools.Adetaileddiscussionofthisisbeyond 2 CalcHep WHIZARD the rates in cm which is usually used to present the direct the scope of this paper. Tools like , , (𝑚 ,𝜎) MadGraph, Herwig++,orSherpa are very powerful and detection limits in the DM plane. We can do the same for the EP2. This point has a offer a rather unlimited number of possibilities what can be neutralino LSP; that is, MicrOmegas has to compile again done. Therefore, I am just going to show at two examples how the output of SARAH canbeusedtogetherwith WHIZARD many channels and we have to wait again sometime for the MadGraph results. The output on the screen looks less promising. and to perform simple studies. As soon as a model is implemented in these tools and is working fine for one study, it can be used in the same way as all models Xf=2.15e+01 Omega hˆ2=2.66e+01 are delivered with the different tools. Thus, to become more familiar with these tools, one can check for the many # Channels which contribute to 1/(omega) more than 1%. examples and tutorials which can be found online. # Relative contributions in % are displayed Actually, there is one big advantage when working with −> 28% Ñ1 Ñ1 e3 E3 model files produced by SARAH:notonlythechosenMC 27% Ñ1 Ñ1 −>e2 E2 27% Ñ1 Ñ1 −>e1 E1 tool needs the model files containing all vertices but also 5% Ñ1 Ñ1 −>u3 U3 numerical values for all parameters have to be provided. This 3% Ñ1 Ñ1 −>Wm Wp can be a delicate task especially in supersymmetric models 2% Ñ1 Ñ1 −>u2 U2 coming with a lot of parameters and rotation matrices. When 2% Ñ1 Ñ1 −>u1 U1 using numerical values for all these parameters obtained 1% Ñ1 Ñ1 −>ZZ 1% Ñ1 Ñ1 −>h2 h2 with another code, one has to make always sure that the ... conventionswhichareusedinthemodelfileandtheseof the spectrum generator are identical. This problem is absent 84 Advances in High Energy Physics when working with model files produced by SARAH and events for the process spectrum files written by SARAH generated SPheno.Inthat case, the implementation of models in the MC tool and 𝑝𝑝 󳨀→ 𝑗𝜒̃1𝜒̃1 in SPheno isbasedonsinglemodelfilein SARAH.Thus,the (154) sameconventionsareusedforsureinbothparts. √𝑠=14TeV 𝑝 7.4.1. Introduction. WHIZARD [153] is a fast tree-level MC are generated. The jet 푇 and rapidity distributions are plotted WHIZARD generator for events at parton level. WHIZARD makes use using intrinsic functions. of O’Mega [154] to generate the matrix elements; that is, strictly speaking a model implementation in WHIZARD 7.4.2. Generating the Model Files for WHIZARD/O’Mega. For means that model files for WHIZARD and O’Mega have to the process we are interested in, we just need vertices be generated and included in both codes. SARAH is going which involve fermions. Thus, we can neglect all vertices to take care of both. I will first show how the WHIZARD which only come with scalars and vectors. This is some- and O’Mega model files are generated with SARAH and how times helpful because the compilation of the model files they are compiled. In the second step, I will show how the with WHIZARD/O’Mega can be quite time and memory con- parameters are passed from SPheno to WHIZARD.Finally, suming for complicated models. So, we run the following

MakeWHIZARD[Exclude ->{SSSS,SSVV,SSV,SVV,GGV,GGS,VVV,VVVV}] to include only FFV and FFS vertices. This shortcut is very (iii) WriteWHIZARD: it defines if the model files helpful for our purposes here to get quickly some results. for WHIZARD should be written. However, it has to be used carefully in order to make sure WOModelName that no relevant vertices are dropped. In the case that all (iv) : it defines the name for the model in vertices should be kept, there is another possibility to speed the output. up compilation a bit: usually, SARAH splits the entire list of (v) Version: it defines for which version of WHIZARD vertices in pieces containing 150 vertices and writes for each the files are generated. By default 2.2.0 is used. part a separate file. Especially for SSSS and SSS interactions even 150 vertices can cause a large file which needs sometime (vi) ReadLists: it defines if the information from a to be compiled. Thus, for complicated models where the former evaluation should be used. expressions for the vertices are lengthy, it might be helpful to go even for less couplings per file. That is done by the option MaximalCouplingsPerFile -> X with some 7.4.3. Compiling the Model Files. After the interface has integer X. A good choice for the full model files for the B- completed, the generated files are stored in the directory L-SSM is 50 or less. There are some more flags which can be used to adjust the WHIZARD output. The full list of options is $ PATH/SARAH/Output/B-L-SSM/EWSB/WHIZARD as follows. Omega/. (i) MaximalCouplingsPerFile: it defines the maximal number of vertices per file. In order to use the model with WHIZARD and O’Mega,the (ii) WriteOmega: it defines if the model files for O’Mega generated code must be compiled and installed. In most cases should be written. this is done by the following.

$ cd $PATH/SARAH/Output/B−L−SSM/EWSB/WHIZARD Omega $ ./ configure −− prefix=$PATH/WHIZARD/ WOCONFIG=$PATH/WHIZARD/ b i n / $make $makeinstall

If WHIZARD has not been installed globally in the home model just for one installation. For these cases the instal- directory of the current user, WHIZARD will not be able lation path has been defined via the --prefix option of tofindthebinaries.Thus,theWO CONFIG environment the configure script. More information on the available variable was used to point explicitly to the binaries. By options is shown with the following command. default, the configure script would install the compiled model into .whizard inthehomedirectoryoftheuser. If the user wants to have several WHIZARD installations or ./ configure −− help install WHIZARD locally, it might be better to provide a Advances in High Energy Physics 85

The configure script prints also another import information, namely, the name of the model which is used to load it in WHIZARD.

... ########################################################### configure : collecting models ###########################################################

found : blssm sarah

configure : writing whizard/Makefile . src configure : writing omega/Makefile . src

...

Thus, the model is called blssm sarah.Thisname public spectrum generators using SLHA conventions. could be changed by using the option WOModelName However, WHIZARD does not provide a possibility to of MakeWHIZARD. read spectrum files which go beyond that. Therefore, The model files produced by SARAH aresupposedtobe to link WHIZARD and SPheno,allSPheno modules used with WHIZARD2.x. The possibility to patch these files created by SARAH write the information about the for a use with WHIZARD1.x does exit in principle. However, parameters and masses into an additional file. This file I will not go into detail here and highly recommend to use iswrittenintheWHIZARD specific format and can be version 2. directly read by WHIZARD. In our example the file is called WHIZARD.par.BLSSM anditiswrittentothesame directory where SPheno writes the standard spectrum file. One just has to make sure that the corresponding flag is 7.4.4. Parameter Values from SPheno. WHIZARD is able turned on the Les Houches input for SPheno to get this to read Les Houches files for the MSSM generated by output.

LesHouches.in.BLSSM Block SPhenoInput # SPheno specific input ... 75 1 # Write WHIZARD files

The parameter file can then be included in the Sindarin input file for WHIZARD via the following.

include("$PATH/SPHENO/WHIZARD.par.BLSSM")

7.4.5. Sindarin Input and Running WHIZARD. WHIZARD modeltoapplycutsandeventomakeplotscanbeputinone comes with its own steering language called Sindarin. single input file. The input file BLSSM monojet.sin for our With Sindarin all settings to define a process in a specific example of monojets at the LHC might look as follows. 86 Advances in High Energy Physics

BLSSM monojet.sin

1 model = blssm_sarah

3 include("$PATH/SPHENO/WHIZARD.par.BLSSM")

5 Mu1 = 0. 6 Md1 = 0. 7 Mu2 = 0. 8 Md2 = 0.

10 alias parton = u1:u1bar:u2:u2bar:d2:d2bar:d1:d1bar:G 11 alias jet = parton

13 process monojet = parton , parton => jet, N1, N1

15 compile

17 sqrts = 14 TeV

19 beams = p, p => pdf_builtin 20 cuts = all Pt >= 50 GeV [jet] 21 integrate (monojet) { iterations = 5:20000 }

23 $description = "Monojets" 24 $y_label = "$N_{\textrm{events}}$" 25 $title = "Jet-$p_T$ in $pp\to j\tilde\chi^0\tilde\chi^0$" 26 $x_label = "$p_T(j)$/GeV" 27 histogram pt_jet (0 GeV, 1000 GeV, 10 GeV) 28 $title = "Jet rapidity in $pp\to j\tilde\chi^0\tilde\chi^0$" 29 $x_label = "$\eta(j)$" 30 histogram eta_jet (-5, 5, 0.1) 31 analysis = record pt_jet (eval Pt [extract index 1 [jet]]); 32 record eta_jet (eval Eta [extract index 1 [jet]])

34 simulate (monojet) { n_events = 100000 } 35 compile_analysis { $out_file = "monojet.dat" }

First, we set the model and tell WHIZARD where to find the several processes in one file, treat them separately, and run spectrum file written by SPheno. In general, the SPheno file one after the other. contains nonzero and different masses for all SM fermions. The next steps are to compile the process (line 15), set the However, to group fermions together into one object, those beam energy (line 17), and define the pdf set which should have to have the same masses. Therefore, we put all first- be used (line 19). We apply a 𝑝푡 cut on the jet of 50 GeV. The and second-generation quark masses explicitly to zero in process is now fully set and can be integrated (line 20). To lines 5–8. Afterwards, we can combine these quarks, their improve numerics, we use 5 iterations26. antiparticles, and the gluon into one object called parton. Lines22–32areusedtogeneratefiguresdirectlywhile For the final state we define another object jet which running WHIZARD.Thefigureswillshowahistogramofthe consists of the same particles. When we now define a jet 𝑝푇 from 0 to 1000 GeV in bins of 10 GeV and the rapidity process involving parton and jet, WHIZARD will generate of the jet from −5to−5inbinsof0.1.Notethat pt jet allnonvanishingsubprocessesonpartonlevel.Anamefor and eta jet areundefinedatthisstagebutarejustvariables. the process (monojet) is given. This name is used in the The analysis command is used to tell WHIZARD what is following to refer to this process. Thus, one can also define meant by both pt jet and eta jet.Inthelasttwolines, Advances in High Energy Physics 87 thenumberofeventsandthenamefortheoutputfileare The last line runs the executable whizard in the given. binary directory on our Sindarin input file. Note We save this file in the root directory of WHIZARD that we did not move BLSSM monojet.sin to the ($PATH/WHIZARD). However, running it in the same direc- subdirectory run BLSSM monojet.Thereasonisthat torywouldgivesomemessbecause WHIZARD produces sev- wemightwanttocleanthisdirectoryby rm * in order to eral output files. Therefore, we generate a new subdirectory make a new run with other settings. which contains at the end the entire WHIZARD output. After some time, WHIZARD is done and has created a pdf including both plots shown in Figure 9.Theoutput $ cd $PATH/WHIZARD directory includes also all events in the WHIZARD native $mkdirrunBLSSM monojet format called evx.Toturnontheoutputofotherfor- Sindarin $ cd run BLSSM monojet mats,itispossibletoaddtheflagstothe input file $ ./../bin/whizard BLSSM monojet . sin

1 sample_format =

where can be, for instance, lhef to get files in All files included in this directory have to be copied to a new the Les Houches accord event format. For a complete list of subdirectory in MadGraph’s model directory. all supported formats, I refer to the WHIZARD manual. cd $PATH/MADGRAPH/ mkdir m o d e l s /BLSSM cp $PATH/SARAH/Output/B−L−SSM/EWSB/UFO/∗ m o d e l s /BLSSM 7.5. Dilepton Analysis with MadGraph. As second example for doing a collider study with SARAH modelfiles,Iwillshow the usage of UFO model files with MadGraph [155]. The UFO Now, we can import this model in MadGraph and work with format is also supported by other tools like Herwig++ it. For this purpose one can either start the interactive mode or Sherpa and the user can pick his/her favourite MC by running program. The command to generate the UFO files is as follows. ./bin/mg5 aMC MakeUFO[] or one can make a short input file including all necessary As option one can give a list of generic vertices commands and give it as argument. which should not be included in the output similar to MakeWHIZARD: Exclude -> $LIST. By default, four ./bin/mg5 aMC Input pp−MuMu . t x t scalar vertices are excluded and this is sufficient for us to Input pp-MuMu.txt have a speedy output. The UFO model files for the B-L-SSM Here, I used a file which contains the are written to following lines. $PATH/SARAH/Output/B-L-SSM/EWSB/UFO.

Input pp-MuMu.txt

1 import model BLSSM -modelname 2 define p d1 d1bar d2 d2bar u1 u1bar u2 u2bar 3 generate p p > e2 e2bar 4 output ppMuMu 5 exit

InthefirstlineweimportthemodelinMadGraph.The to use the default naming conventions. However, this would option modelname is used to keep the names of the particles fail for this model, because there are more than two CP as given in the model files. By default, MadGraph will try even scalars and h3 canbeusedasnamefortheCPodd 88 Advances in High Energy Physics

0 0 Jet rapidity in pp → j휒̃0휒̃0 Jet-pT in pp → j휒̃ 휒̃ Monojets Monojets 5000 4000

4000 3000

3000 2000 events events N N 2000

1000 1000

0 0

0 200 400 600 800 1000 −4 −2 0 2 4

pT(j) (GeV) 휂(j) Data within bounds: Data within bounds:

⟨Observable⟩ = 288.6 ± 1.26 [nentries = 49655] ⟨Observable⟩ = −0.0017 ± 0.0037 [nentries = 100000] All data: All data:

⟨Observable⟩ = 944.5 ± 2.29 [nentries = 100000] ⟨Observable⟩ = −0.0017 ± 0.0037 [nentries = 100000] (a) (b)

0 0 Figure 9: Plots produced by WHIZARD for the monojet event 𝑝𝑝 →𝑗𝜒̃1 𝜒̃1 . On the left: transversal momentum distribution 𝑝푇(𝑗),onthe right: rapidity distributions 𝜂(𝑗). one as MadGraph wants to do27. We define a multiparticle can, for instance, set the beam type and energy, define the called p which consists of all light quarks. We can skip the renormalization scale, apply cuts, and fix the number of gluon because it will not contribute to our process. The muon events. is the second lepton which is called e2 and the antimuon is We want to use, of course, the spectrum file accordingly e2bar.Thus,inthethirdlinewegeneratethe as written by SPheno.However,thereisone process 𝑝𝑝 →𝜇 .Theoutputfor MadEvent is written to a caveat: MadEvent has problems with reading new subdirectory ppMuMu andweclose MadGraph when it the HiggsBounds specificblocksinthe SPheno spectrum is done via exit. fileHiggsBoundsInputHiggsCouplingsFermions ( and After MadGraph has created the output for MadEvent HiggsBoundsInputHiggsCouplingsBosons). If these and finished, we can enter the new subdirectory ppMuMu. blocks are included, MadEvent will not accept the file. The important settings to generate events are done viathe Therefore, we either modify the output by hand and delete files in the Cards-directory: the file param card.dat is these blocks or we regenerate the file by changing the options used to give the input for all parameters and run card.dat in the Les Houches input file. The HiggsBounds blocks are controls the event generations. In the last file, the user disabled by the flag as follows.

LesHouches.in.BLSSM Block SPhenoInput # SPheno specific input ... 11 1 # calculate branching ratios ... 520 0. # Write effective Higgs couplings

In addition, we turned on the decays just in case that this When we have the spectrum file in the correct was not done before: MadEvent is going to read the decay form, we can copy this file to the Cards directory blocks from SPheno to know the widths of all particles. If as param card.dat. those widths are not provided via the SLHA file it is necessary to calculate them first with MadEvent before generating $ cd $PATH/MADGRAPH/ppMuMu/ events. $ cp $PATH/SPHENO/SPheno . spc .BLSSM Cards/param card . dat Advances in High Energy Physics 89

Theothersettingswehavetodoaretodemandsmall million events and we want to apply a cut on the invariant modifications on the run-card: we want to generate one mass of leptons to get rid of the 𝑍-peak. The number of events is set here.

run card.dat

27 #******************************************************************* 28 # Number of events and rnd seed * 29 # Warning: Do not generate more than 1M events in a single run * 30 # If you want to run Pythia, avoid more than 50k events in a run. * 31 #******************************************************************* 32 1000000 = nevents ! Number of unweighted events requested

And the cuts are applied here. run card.dat

281 #******************************************************************* 282 # Minimum and maximum invariant mass for pairs * 283 # WARNING: for four lepton final state mmll cut require to have * 284 # different lepton masses for each flavor! * 285 #******************************************************************* 286 ... 287 ... 288 ... 289 200 = mmll ! min invariant mass of l+l- (same flavour) lepton pair

We are now ready to generate the events. This can either be ./bin/generate events 0 0 done again in the interactive mode by starting the following The two 0’s areusedasargumentbecausewedonotwant ./ bin/madevent to make any further modifications on the param-or run- card, and we also do not want to run pythia or any detector simulation. When starting MadEvent in that way a long list or we can directly start the event generation with the of warnings appears on the screen. following.

... WARNING: information about ”imuvmix [6 , 2]” is missing ( full block → missing) using default value : 0.0. ...

The reason is that the UFO model files by SARAH in general MadEvent will give a status update in a new browser can handle complex parameters. However, SPheno does window.Whenitisdone,theeventsaresavedintheLes only print the real parts if we do not turn on CP violation. Houches event format and can be processed further. 耠 The zeros for all imaginary parts are not given explicitly in Wearejustgoingtomakeaplottocheckifthe𝑍 peak the spectrum file. Thus, MadEvent does not find an input showsup.Thiscan,forinstance,bedonewith MadAnalysis for the imaginary parts and takes them as zero as it should. [319] which I assume here to be installed as well in $PATH. In addition, MadEvent prints a warning for each parameter We make another short input file called plotMuMu.txt and wherethishappens.Thus,wedonothavetoworryabout save it in $PATH/MADANALYSIS. The content of the file is the these many warnings. following. 90 Advances in High Energy Physics

plotMuMu.txt

1 import $PATH/MADGRAPH/ppMuMu/Events/run_01/unweighted_events.lhe.gz 2 plot M(mu+ mu-) 100 500 3000 [logX] 3 submit ppMuMu )

−1 35 ./bin/ma5 plotMuMu. txt fb 30 =10 MadAnalysis

int The output of is stored in L 25 $PATH/MADANALYSIS/ppMuMU and contains also the plot shown in Figure 10 with the expected peak at 2.5 TeV.

pairs ( pairs 20 −

mu 15

+ 8. Example—Part V: Making Scans 10 We have learned in the last sections how SARAH can be used 5 together with other tools to study all aspects of a model. Of 0 course, it is often not sufficient to consider just one single Number ofNumber mu 103 parameter point. SUSY models like the B-L-SSM have even M [mu+ mu−] (GeV/c2) in their constrained version a large parameter space which wants to be explored. Thus, at some point one has to start Figure 10: Invariant mass of the 𝜇-pair. making scans to check many different points. I will discuss two possibilities of how to perform scans: the first one is only using functions the Linux bash provides together with simple scripts28.Thatmightbesufficienttocheckquickly the dependence of a few observables on a single parameter. Afterwards, I will introduce the Mathematica package SSP which is a dedicated tool for more sophisticated scans. In the first line we import the unweighted events which are 8.1. Using Shell Scripts. Let us assume that one is just generated by MadGraph and which are saved by default in interested in the dependence of the two lightest Higgs 耠 the LHE format. In the second line, we make a histogram masses on tan 𝛽 in a small range starting from our param- oftheinvariantmassofthemuonpairinthemassrange eter point EP1. In principle, one does not need any addi- of 500 to 3000 GeV using 100 bins. For the 𝑥-axis we use tionalsoftwaretodoasmallscanbutLinux provides alogscale(logX). Finally, everything is submitted to be everything which is needed. For this purpose we create a evaluated by MadAnalysis and the output directory should file called LesHouches.in.BLSSM Template which is the be called ppMuMu.WerunMadAnalysis on that file as input file for EP1 with just one change: we replace the input 耠 follows. value for tan 𝛽 by a unique string as follows.

LesHouches.in.BLSSM Template Block MINPAR # Input parameters ... 6 1.0000000E+00 # SignumMuP 7 TBPINPUT # TBetaP 8 2.5000000E+03 # MZp

Now, we can write a short bash script which makes file LesHouches.in.BLSSM in that way. We run SPheno a loop over all numbers from 1.2 to 1.3 in steps of with that file and use grep and sed to extract the masses 0.01 using the seq command. For each value we use of the two lightest Higgs states. These numbers are “piped” the sed command to replace the string TBPINPUT into two files called results hX.dat with X = 1, 2 in LesHouches.in.BLSSM Template by the value using >>. The full script called RunSPheno.sh reads as oftheloopvariableandtogenerateacompleteinput follows. Advances in High Energy Physics 91

RunSPheno.sh

1 #!/bin/bash 2 rm results_h1.dat 3 rm results_h2.dat 4 for i in $(seq 1.20 0.01 1.30) 5 do 6 sed -e "s#TBPINPUT#$i#" LesHouches.in.BLSSM_Template > → LesHouches.in.BLSSM 7 rm SPheno.spc.BLSSM 8 ./bin/SPhenoBLSSM 9 mh1=‘cat SPheno.spc.BLSSM | grep "# hh_1" | grep -v DECAY | sed → ’s/$.*$ 25 $.*$\# hh_1/\2/g’‘ 10 mh2=‘cat SPheno.spc.BLSSM | grep "# hh_2" | grep -v DECAY | sed → ’s/$.*$ 35 $.*$\# hh_2/\2/g’‘ 11 echo "$i $mh1" >> results_h1.dat 12 echo "$i $mh2" >> results_h2.dat 13 done

We see here that a very handy method to extract single Otherwise, the paths must be adjusted accordingly. We run lines from the SPheno spectrum file is to use grep with the script via the following. the comments appearing in the spc file# ( ...). The sed commands after grep areusedtocutthePDGandthe $ ./ RunSPheno . sh commentappearinginthesamelineinthespectrumfile,that is, the variable mh1, just contains a real number at the end. When the script is finished, the file results h1.dat just 耠 The script has to be saved in the SPheno root directory contains in each line a pair of the tan 𝛽 value and of the where also LesHouches.in.BLSSM Template is located. corresponding Higgs mass.

results h1.dat

1 1.20 1.24183203E+02 2 1.21 1.24194534E+02 3 1.22 1.24203648E+02 4 ...

耠 We can plot both masses as function of tan 𝛽 ,forinstance, tributions. For this purpose we write a short input file with gnuplot whichisalsoincludedinmanyLinuxdis- (gnuplot mh.txt) as follows.

gnuplot mh.txt

1 set terminal postscript eps 25 color solid linewidth 3 enhanced; 2 set output ’TBpMh.eps’; 3 set xlabel ’tan(beta ‘)’; 4 set ylabel ’lightest Higgs masses ’; 5 set key off; 6 plot "results_h1.dat", "results_h2.dat"; 7 exit 92 Advances in High Energy Physics

450 be improved by using the full power gnuplot provides to polishthelayout,andsoon.However,Ithinkthereisnoneed 400 to invent the wheel again and again. There are public tools 350 which can be used for scanning and plotting. I will discuss briefly SSP now which is one of these tools. 300 8.2. Making Scans with SSP. Atoolwhichisoptimized 250 for parameter scans using SPheno and the other tools Mathematica SSP 200 discussed so far is the package Lightest masses Higgs (SARAH Scan and Plot). SSP provides functions for 150 simple random or grid scans but can also make use of intrinsic Mathematica functions to sample the parameter 100 space or to include constraints directly during the scan. I 1.18 1.2 1.22 1.24 1.26 1.28 1.3 want to discuss here two simple examples. First, a linear scan tan 훽㰀 in 𝑀푍耠 is Figure 11: Simple plot created with a shell script and gnuplot 耠 𝑀푍耠 ∈ [2.5, 4.0] TeV. (155) showing the two lightest scalar masses as function of tan 𝛽 . Second, I discuss a grid scan in the range

耠 Iusedherebasic gnuplot commands to adjust the output tan 𝛽 ∈ [1.20, 1.25] format (line 1), the name of the output file (TBpMH.eps), the (156) 𝑀 耠 ∈ [2.5, 3.0] . labels for the axes (lines 3 and 4), disabling the legend (line 푍 TeV 5) and plotting the content of the two files with our data (line All other parameters are set to the values of EP1. For more 6). We run gnuplot on that file as follows complicated scans one can also study the examples which are delivered with SSP. $ gnuplot gnuplot mh . txt 8.2.1. General Setup. First, we need a file which con- andgettheplotshowninFigure 11. tains information about the location and usage of all There are now many possibilities to improve this ansatz. the different tools. For this purpose, we rename the One can include easily in the script to run SPheno also other file DefaultSettings.in included in the SSP package codes; the scans can be varied by playing with seq, and more to DefaultSettings.in.BLSSM.Thecontentshouldlook observables can be stored; the appearance of the plot can like the following.

DefaultSettings.in.BLSSM

1 DEFAULT[SPheno] = "$PATH/SPHENO/bin/SPhenoBLSSM"; 2 DEFAULT[SPhenoInputFile] = "LesHouches.in.BLSSM"; 3 DEFAULT[SPhenoSpectrumFile] = "SPheno.spc.BLSSM";

5 DEFAULT[MicroOmegas] = → "$PATH/MICROMEGAS/BLSSM/CalcOmega_with_DDetection_MO4"; 6 DEFAULT[MicroOmegasInputFile] = "SPheno.spc.BLSSM"; 7 DEFAULT[MicroOmegasOutputFile] = "omg.out"; 8 DEFAULT[DarkMatterCandidate] = ALL;

10 DEFAULT[HiggsBounds] = "$PATH/HIGGSBOUNDS/HiggsBounds LandH effC 6 1"; 11 DEFAULT[HiggsSignals] = "$PATH/HIGGSBOUNDS/HiggsSignals → latestresults peak 2 effC 6 1";

13 DEFAULT[VevaciousBin] = "$PATH/VEVACIOUS/bin/Vevacious.exe"; 14 DEFAULT[VevaciousInit] = → "$PATH/VEVACIOUS1/bin/VevaciousInitialization_BLSSM.xml"; Advances in High Energy Physics 93

Of course, $PATH has to be replaced everywhere by the to include MicrOmegas inthescan.Inaddition,onecan installation directory of the different tools. The absolute define if MicrOmegas should only calculate the relic density path to the executable has to be defined for SPheno and if a specific particle is the LSP. In that case the PDG has thenamefortheinputandoutputhastobegiven.Also to be given, that is, either 1000022 for a neutralino LSP the path for the executable for MicrOmegas is set. The or 1000012 for a CP even sneutrino LSP. We use here ALL namesofthespectrumfileusedasinputandtheoutputfile to calculate the relic for any particle. One could also use the written by MicrOmegas are the other information necessary following

1 DEFAULT[DarkMatterCandidate] = 1000022 | 1000012;

to just consider a subset of particles. The lines below give in the scans is loaded. This is the file we have set up the commands to run HiggsBounds and HiggsSignals in the first step. Then, identifiers for all scans which we as explained in Section 7.1.Finally,torun Vevacious the want to make are defined using the list RunScans.We path to the executable as well as the desired initialization file just perform two scans here as said above which are havetobegivenasdoneinthelasttwolines. called MZpLinear and MZpTBpGrid,butthereisinprinci- plenolimitofhowmanyscansaredonewithinasinglefile.By default, SSP always runs SPheno.Wealsowanttoinclude here HiggsBounds and HiggsSignals and put therefore 8.2.2. Defining a Scan. Asecondinputfiledefinesthescanwe the flags to True.Toinclude MicrOmegas as well, it would want to make. SARAH also writes templates for this file dur- just be necessary to put also that flag to True.However, ing the SPheno output which could be used as starting point. this would slow down the scan significantly because different The file names of these templates start with SSP Template. LSPs show up in the range we have chosen and MicrOmegas We call the file for our examples here BLSSM TBpMZp.m.The would need a long time to compile all amplitudes. Hence, I different parts are as follows. skip it for the example here but for practical applications it At the very beginning, the file which contains the can easily be included. For the same reason, I have also not information about the installation of the codes involved included Vevacious in the scan.

BLSSM TBpMZp.m

1 (* SETUP *) 2 LoadSettings="DefaultSettings.m.BLSSM";

4 RunScans = {MZpLinear ,MZpTBpGrid};

6 DEFINITION[a_][IncludeMicrOmegas]=False; 7 DEFINITION[a_][IncludeVevacious]=False; 8 DEFINITION[a_][IncludeHiggsBounds]=True; 9 DEFINITION[a_][IncludeHiggsSignals]=True;

Note that we applied all definitions to any scan These are the blocks which we discussed in Section 6.1.3. definedinthisfilebecauseweusedDEFINITION[a ]. The main part of the input is setting the numbers To use different options for the different for these blocks and their different entries. In this scans, DEFINITION[$NAMEofSCAN] canbeused. context, the blocks are defined as arrays: first the Now, the main part which defines all input parameters blocknumberappears,andthenwecangivethe and ranges follows. SSP is very agnostic concerning the numerical value. Fixed values are assigned by the underlying model. Therefore, it is first necessary to tell SSP flag Value.Thus,theblocksMODSEL, SMINPUTS, what blocks are actually needed for a scan before the values and SPhenoInput which just come with fixed values canbedefined(DEFINITION[a ][Blocks] = {. . .}). read as follows. 94 Advances in High Energy Physics

BLSSM TBpMZp.m

10 DEFINITION[a_][Blocks]={MODSEL ,SMINPUTS ,SPhenoInput ,MINPAR ,YVIN,YXIN};

12 DEFINITION[a_][MODSEL]={

13 {{1},{Value ->1}},

14 {{2},{Value ->1}},

15 {{6},{Value ->0}}

16 };

17 DEFINITION[a_][SMINPUTS]={

18 {{2},{Value ->1.166390*10^-5}},

19 {{3},{Value ->0.1172}},

20 {{4},{Value ->91.18760}},

21 {{5},{Value ->4.2}},

22 {{6},{Value ->172.9}},

23 {{7},{Value ->1.777}}

24 };

25 DEFINITION[a_][SPhenoInput]={

26 {{1},{Value ->-1}}, (* error level *)

27 {{2},{Value ->0}}, (* SPA conventions *)

28 {{7},{Value ->0}},

29 {{8},{Value ->3}},

30 {{11},{Value ->1}}, (* Calculate widhts and BRs *)

31 {{12},{Value ->0.0001}}, (* minimal BR to write out *)

32 {{13},{Value ->0}}, (* Enable 3-body decays *)

33 {{34},{Value ->0.0001}}, (* precision of masses *)

34 {{38},{Value ->2}}, (* 1/2 - Loop RGEs *)

35 {{50},{Value ->0}},

36 {{51},{Value ->0}}, (* Switch to CKM matrix *)

37 {{55},{Value ->1}}, (* 1 - Loop masses *)

38 {{57},{Value ->0}}, (* low energy constraints *)

39 {{60},{Value ->1}}, (* Include possible , kinetic mixing *)

40 {{65},{Value ->1}}, (* Solution tadpole equation *)

41 {{75},{Value ->1}}, (* Write WHIZARD files *)

42 {{76},{Value ->1}}, (* Write HiggsBounds files *)

43 {{86},{Value ->0.}}, (* Maximal width to be counted as invisible in → Higgs decays; -1: only LSP *)

44 {{550},{Value ->1}}, (* Calculate Fine-Tuning *)

45 {{530},{Value ->1.}} (* Write Blocks for Vevacious *)

46 };

The other blocks showing up in the Les Houches file arethose to set the parameters for the scans. Those are definedina similar way as follows. Advances in High Energy Physics 95

BLSSM TBpMZp.m

48 DEFINITION[MZpLinear][MINPAR]={ 49 {{1},{Value ->1700}} (*m0*), 50 {{2},{Value ->1700}} (*m12*), 51 {{3},{Value ->7}} (*TanBeta*), 52 {{4},{Value ->1}} (*SignumMu*), 53 {{5},{Value ->-1400}} (*Azero*), 54 {{6},{Value ->1}} (*SignumMuP*), 55 {{7},{Value ->1.20}} (*TBetaP*), 56 {{8},{Min->2500, Max->4000, Steps ->40, Distribution ->LINEAR}} → (*MZp*)};

58 DEFINITION[MZpTBpGrid][MINPAR]={ 59 {{1},{Value ->1700}} (*m0*), 60 {{2},{Value ->1700}} (*m12*), 61 {{3},{Value ->7}} (*TanBeta*), 62 {{4},{Value ->1}} (*SignumMu*), 63 {{5},{Value ->-1400}} (*Azero*), 64 {{6},{Value ->1}} (*SignumMuP*), 65 {{7},{Min->1.2, Max->1.25, Steps ->15, Distribution ->LINEAR}} → (*TBetaP*), 66 {{8},{Min->2500, Max->3000, Steps ->15, Distribution ->LINEAR}} → (*MZp*)};

68 DEFINITION[a_][YVIN]={ 69 {{1,1},{Value ->0}} , 70 {{3,3},{Value ->0}} };

72 DEFINITION[a_][YXIN]={ 73 {{1,1},{Value ->0.42}} , 74 {{2,2},{Value ->0.42}} , 75 {{3,3},{Value ->0.30}} };

耠 We see that we can use for both scans exactly the same blocks a linear distribution. The grid scan is set by varying tan 𝛽 but for MINPAR. That means that the majority of blocks had and 𝑀푍耠 (MINPAR[7], MINPAR[8])withinthegivenlimits just to be defined once using again DEFINITION[a ].For and assuming linear distributions in both directions. There the two versions of MINPAR we gave the name of the scans are also other options possible, for example, a logarithmic as arguments. Also the scan ranges are defined easily. distribution (Distribution->LOG)orrandomdistribution The linear scan is set up by varying 𝑀푍耠 (MINPAR[8]) (Distribution->RANDOM). Also relations to other parame- in the range between 2500 and 4000 using 40 steps with ters can be given. For instance, to scale in a scan 𝑚0 the same way as 𝑀1/2,onecouldusethefollowing.

DEFINITION[MZpTBpGridZoom][MINPAR]={ {{1},{Min->1500, Max->3000, Steps ->25, Distribution ->LINEAR}} (*m0*), {{2},{Value->MINPAR[1]}} (*m12*) ... 96 Advances in High Energy Physics

For the possibility to perform basic Marcov-Chain Monte- style (BasicStyle)whichisappliedtoallfigures.Sinceour Carlo runs, to apply fits during the scan, or to make a figures are a composition of two, respectively, four plots, we sampling of the parameter space, I refer to the SSP manual have to define also styles with twoStyleDefault2 ( )and and the examples which come with SSP. four (StyleDefault4)colours.Thatisdonebymapping Finally, we want to get some figures automatically when our colours on the basic style. These definitions are so far the scan is finished. For the linear scan we want to plot (i) purely Mathematica commands.Whenthisisdone,the the two lightest CP even sneutrinos, (ii) the lightest CP even plotsthemselvesaredefinedquickly:weusetheoption P2D and odd sneutrino, (iii) the two lightest neutralino masses, of SSP for 2-dimensional plots, and set for each figure what and (iv) the composition of the lightest neutralino. To make parameters and observables should be shown, what style the plots a bit more appealing, we first generate a basic should be used, what the label of the 𝑦-axis that we need, and what the name for the output file should be.

BLSSM TBpMZp.m

78 BasicStyle={Frame ->True, Axes->False , 79 FrameTicksStyle -> Directive[Black, 10], 80 ImageSize -> 200, ImageMargins ->10, 81 Joined ->True,FrameLabel ->{UseLaTeX["$M_{Z’}$~[GeV]"],yAxis}}; 82 StyleDefault2 = Map[Join[BasicStyle ,{PlotStyle ->#}]&,{Red,Green}]; 83 StyleDefault4 = → Map[Join[BasicStyle ,{PlotStyle ->#}]&,{Red,Green,Blue,Black}];

85 DEFINITION[MZpLinear][Plots]={ 86 {P2D, {MINPAR[8],{MASS[1000012],MASS[1000014]}}, 87 StyleDefault2 /. yAxis->UseLaTeX["$m_{\\tilde → \\nu^R_{1,2}}$~[GeV]"], 88 "MZp_MSvRe.pdf"}, 89 {P2D, {MINPAR[8],{MASS[1000012],MASS[4000012]}}, 90 StyleDefault2 /. yAxis->UseLaTeX["$m_{\\tilde → \\nu^R_1},m_{\\tilde \\nu^I_1}$~[GeV]"], 91 "MZp_MSvRe1_SvIm1.pdf"}, 92 {P2D, {MINPAR[8],{Abs[MASS[1000022]],Abs[MASS[1000023]]}}, 93 StyleDefault2/. yAxis->UseLaTeX["$m_{\\tilde → \\chi^0_{1,2}}$~[GeV]"], 94 "MZp_MChi.pdf"}, 95 {P2D, {MINPAR[8],{Log[10,NMIX[1,1]^2+NMIX[1,2]^2], → Log[10,NMIX[1,3]^2+NMIX[1,4]^2], 96 Log[10,NMIX[1,5]^2 ], Log[10,NMIX[1,6]^2 +NMIX[1,7]^2]}}, 97 StyleDefault4/. yAxis->UseLaTeX["$\\log(Z^N_{i1})$~[GeV]"], 98 "MZp_ZN.pdf"} 99 };

NotethatIusedherethekeywordUseLaTeX together plots (contour plots) based on the Mathematica with LaTeX syntax for the labels. By doing this, SSP calls function ListContourPlot.Thus,onecansetthe the script fragmaster (http://www.ctan.org/tex-archive/ options for these plots by using the SetOptions support/fragmaster)whichmakesuseof psfrag to get nice command of Mathematica.Wearegoingtomake looking labels. four plots again: (i, ii) the two lightest Higgs 耠 Theplotsforthegridscanareactuallyeven masses in the (tan 𝛽 ,𝑀푍耠 ) plane and the results of simpler to define because we can use the same style (iii) HiggsBounds and (iv) HiggsSignals in the same for each plot. The flag P3D performs 3-dimensional plane. Advances in High Energy Physics 97

BLSSM TBpMZp.m

105 StyleDefault={Frame ->True, Axes->False , 106 FrameLabel ->{tan\[Beta],Subscript["M","Z’"]}, 107 FrameTicksStyle -> Directive[Black, 10], 108 ContourLabels ->True, ImageSize -> 200};

109

110 DEFINITION[MZpTBpGrid][Plots]={ 111 {P3D, {MINPAR[7],MINPAR[8], → MASS[25]},StyleDefault ,"tbp_MZp_Mass25.pdf"}, 112 {P3D, {MINPAR[7],MINPAR[8], → MASS[35]},StyleDefault ,"tbp_MZp_Mass35.pdf"}, 113 {P3D, {MINPAR[7],MINPAR[8], → HIGGSBOUNDS[10]},StyleDefault ,"tbp_MZp_HB10.pdf"}, 114 {P3D, {MINPAR[7],MINPAR[8], → HIGGSSIGNALS[10]},StyleDefault ,"tbp_MZp_HS10.pdf"} 115 };

When we are done with setting up the entire scan, it is started that we wanted to see. I will show these in Figure 12 for the by running Mathematica with the following. linear scan and in Figure 13 for the grid scan.

<<$PATH/SSP/SSP.m; 8.2.3. Using Data of a Scan in Mathematica. Of course, it is Start["BLSSM_TBpMZp.m"]; also possible to use the results of scans performed by SSP later in Mathematica.ForthispurposeSSP provides a MakeSubNum The output is stored in the subdirectories: function totranslatethedatasavedinthe SLHA or Mathematica format into a list of Mathematica substitutions. These substitutions can then be used to either $PATH/SSP/Output/MZpLinear, apply cuts or to extract points or to make more plots. To load and format the data of the grid scan, we can either $PATH/SSP/Output/MZpTBpGrid. use <<$PATH/SSP/SSP.m data=Get["$PATH/SSP/Output/MZpTBpGrid/Data.m"]; These directories contain not only the scan data saved SubData = MakeSubNum/@ data; in the Les Houches format (SpectrumFiles.spc) and Mathematica format (Data.m)butalsotheplots or

<<$PATH/SSP/SSP.m ReadSpectrumFile["$PATH/SSP/Output/MZpTBpGrid/SpectrumFiles.spc", → "ENDOFPARAMETERFILE"]; SubData = MakeSubNum/@ AllLesHouchesInput;

With both options we get an array of substitutions parameters or observables. Therefore, it often saves a lot of called SubData whichwecanbeused.However,there time and memory to extract the information from the big is one caveat: the data files for large scans are huge. These files which is actually needed and store that information in file include any information as calculated by SPheno and smaller files. This can be done, for instance, under Linux the other tools. Often, not all information is really needed, with a small shell script using again grep and the comments but one is only interested in the behaviour of a subset of appearing in each line of the SPheno output. 98 Advances in High Energy Physics

2000 2000

1500 1500 (GeV) 퐼 1 (GeV)

̃ m 푅 1,2

̃ , 푅 1

m 1000 1000

̃ m

500 500 2600 2800 3000 3200 2600 2800 3000 3200

Mz㰀 (GeV) Mz㰀 (GeV)

(a) (b) 0 1300

1200 −2

1100

) (GeV) −4 (GeV) N i1 0 1,2

1000 Z ̃ 휒 m 900 log( −6

800 −8

2600 2800 3000 3200 2600 2800 3000 3200

Mz㰀 (GeV) Mz㰀 (GeV)

Figure 12: The two lightest CP even sneutrino masses (top left), the lightest CP even and odd sneutrino mass (top right), the lightest CPeven sneutrino and the lightest neutralino mass (bottom left), and the decomposition of the lightest neutralino (bottom right). Plots are produced 耠 with SSP using EP1 and a variation of tan 𝛽 . extract.sh

1 #!/bin/bash

3 cat *$ | grep --regexp="Block MINPAR" \ 4 --regexp="# TBp" \ 5 --regexp="# MZp" \ 6 --regexp="Block MASS" \ 7 --regexp="# hh_1" \ 8 --regexp="# hh_2" \ 9 --regexp="# SvRe_1" \ 10 --regexp="# Chi_1" \ 11 --regexp="Block SCALARMIX" \ 12 --regexp="ZH(1,1)" \ 13 --regexp="ZH(1,2)" \ 14 --regexp="ZH(2,1)" \ 15 --regexp="ZH(2,2)" \ 16 --regexp="# FlavorKitQFV" \ 17 --regexp="# BR(B->X_s gamma)/BR(B->X_s gamma)_SM" \ 18 --regexp="ENDOFPARAMETERFILE" | grep -v DECAY > SmallSpectrum.out

This script takes as argument the name of the file containing in another file called SmallSpectrum.out.Wecallthis all spectra, extracts the data, and writes the necessary lines Advances in High Energy Physics 99

3000 3000 380 390

123.92 400 2900 123.93 2900

2800 2800 㰀㰀 z z M M 123.91 2700 2700 370 410

2600 2600

123.88 123.9 360 2500 123.89 2500 1.20 1.21 1.22 1.23 1.24 1.25 1.20 1.21 1.22 1.23 1.24 1.25 tan 훽 tan 훽

(a) (b) 3000 3000 0.5165 132 132.1 2900 0.5155 2900

0.516 2800 2800 㰀㰀 z z 132.3 M M 2700 0.515 2700 132.2 132.4 0.5145 2600 2600 0.514 132.5

2500 2500 1.20 1.21 1.22 1.23 1.24 1.25 1.20 1.21 1.22 1.23 1.24 1.25 tan 훽 tan 훽

2 Figure 13: The two lightest scalar Higgs masses (top row), theobsratio “ ”ascalculatedby HiggsBounds (bottom left), and the total 𝜒 as 耠 calculated by HiggsSignals.Plotsareproducedwith SSP using EP1 and making a grid scan in the (tan 𝛽 ;𝑀푍耠 ) plane.

script extract.sh andsaveitin$PATH.Itcanbeusedthen ./ extract SSP/Output/TBpMZpGrid/SpectrumFiles . spc via the following. The small spectrum file is now loaded much faster as follows

<<$PATH/SSP/SSP.m ReadSpectrumFile["$PATH/SSP/Output/MZpTBpGrid/SmallSpectrum.out", → "ENDOFPARAMETERFILE"]; SubData = MakeSubNum/@ AllLesHouchesInput; 100 Advances in High Energy Physics

耠 and we can use it, for instance, to make some more plots. For second lightest Higgs in the (tan 𝛽 ,𝑀푍耠 ) plane, we can use instance, to make a contour plot of the doublet fraction of the the following.

ListContourPlot[ Table[{ MINPAR[7], MINPAR[8], SCALARMIX[2, 1]^2 + SCALARMIX[2, 2]^2} /. SubData[[k]], {k, 1, Length[SubData]}], Frame -> True, Axes -> False,ContourLabels -> True, FrameLabel -> {"tan\[Beta]’", Subscript["M", "Z’"]}]

The obtained plot is shown in Figure 14. We can also apply some cuts and collect points with a neutralino mass below 500 GeV

MChi500=Select[SubData , (Abs[MASS[1000022]] < 500 ) /. # &];

and check for which values of 𝑀푍耠 this occurs.

MINPAR[8] /. Chi500 → {3000., 2944.44, 3000., 2888.89, 2944.44, 2888.89}

耠 Thus, this is only the case for heavy 𝑍 masses. Similarly, demanding that the lightest neutralino is lighter than the one can also extract all points with a neutralino LSP by lightest CP even sneutrino

ChiLSP = Select[ SubData , (Abs[MASS[1000022]] < Abs[MASS[1000012]] ) /. # &];

or check what is the lightest mass appearing for the second scalar.

LightestHiggs = Select[SubData , (Abs[MASS[35]] == Min[Table[MASS[35] /. SubData[[i]], {i, 1, → Length[SubData]}]]) /. # &];

The relevant information about this point is shown via the following.

{MINPAR[7], MINPAR[8], MASS[25], MASS[35], SCALARMIX[1, 1]^2 + SCALARMIX[1, 2]^2} /. LightestHiggs → {1.2, 2500., 123.863, 355.269, 0.999013} Advances in High Energy Physics 101

3000 is highly reduced compared to a home-brewed calculation. 0.0004 I hope that the detailed explanation of a specific example 0.0005 simplifies the first contact of interested users with the many 2900 different tools which are available today.

Appendices 2800 A. Some More Details

㰀 0.0007 z

M I could not address all interesting topics in the main part. 2700 Therefore, I give in this appendix a few more details to selected topics.

2600 0.0006 A.1. Flags in SARAH Model Files. There are different flags to enable or disable distinctive features which might be present 0.0008 in some models: 0.0009 2500 (i) AddDiracGauginos = True/False; 1.20 1.21 1.22 1.23 1.24 1.25 False 㰀 default: , it includes/excludes Dirac Gaugino tan 훽 mass terms, Figure 14: Example for contour plots with SSP and Mathematica: (ii) AddFIU1 = True/False; default: False,it the doublet fraction of the second lightest scalar. We used here EP1 includes/excludes Fayet-Iliopoulos 𝐷-terms, 𝛽耠 𝑀 and varied tan and 푍耠 . (iii) NoU1Mixing = True/False; default: False,disables effects from gauge-kinetic mixing, (iv) IgnoreGaugeFixing = True/False; Weseethatthereisnowaninfinitenumbersofpossibilitiesto default: False, it disables the calculation of the study data within Mathematica using the Select or also gauge fixing and ghost terms. Note that this is just other Mathematica commands. possible at tree-level. For loop calculations the ghosts are needed. 9. Summary Specific parts of the Lagrangian are turned off via the following: In the first part of this paper I have given an overview of what models SARAH can handle and what calculations it (i) AddTterms = True/False; it includes/excludes can do for these models. In addition, I have discussed to trilinear soft-breaking couplings, SARAH what other HEP tools the information derived by (ii) AddBterms = True/False; it includes/excludes canbelinked.InthesecondpartIhavediscussedingreat bilinear soft-breaking couplings, detail how all aspects of a SUSY model can be studied AddLterms = True/False; with SARAH andtherelatedtools.ForthispurposeIchoose (iii) it includes/excludes the B-L-SSM as an example. The implementation of the B- linear soft-breaking couplings, L-SSM in SARAH was explained, and it was shown what can (iv) AddSoftScalarMasses = True/False; it be done within Mathematica to gain some understanding includes/excludes soft-breaking scalar masses, SARAH about the model. Afterwards, I have explained how (v) AddSoftGauginoMasses = True/False; it SPheno in combination with is used to calculate the mass includes/excludes Majorana masses for gauginos, spectrum, decays, flavour and precision observables, and the AddSoftTerms = True/False; fine-tuning. The next step was to check parameter points (vi) it with HiggsBounds and HiggsSignals for their Higgs includes/excludes all soft-breaking terms, properties, with Vevacious for their vacuum stability, and (vii) AddDterms = True/False; it includes/excludes all with MicrOmegas for their dark matter relic density. I have 𝐷-terms, given two short examples for collider studies using SARAH (viii) AddFterms = True/False; it includes/excludes all model files. First, monojet events with WHIZARD were gen- 𝐹-terms. erated. Second, a simple dilepton analysis with MadGraph was done. Finally, I discussed possibilities how to perform By default all terms are included. In particular the last two parameter scans using either shell scripts or SSP. flags have to be used very carefully. This paper hopefully shows how helpful SARAH can be to study models beyond the SM or MSSM. Because of a A.2. Parts of the Lagrangian in SARAH. SARAH saves the dif- very high level of automatization the user can get quickly ferent parts of the Lagrangian which it has derived in different results with a precision which is otherwise just available for variables. This happens for all eigenstatesEIGENSTATES ($ ) theMSSM.Ofcourse,alsothepossibilitytomakemistakes and the user has access to this information: 102 Advances in High Energy Physics

(i) LagSV[$EIGENSTATES]: parts with scalars and vec- (vii) kinetic terms for scalars: KinScalar, tor bosons (i.e., kinetic terms for scalars), (viii) kinetic terms for fermions: KinFermion, (ii) LagFFV[$EIGENSTATES]: parts with fermions and (ix) D-terms: DTerms, vector bosons (i.e., kinetic terms of fermions), (x) interactions between gauginos and a scalar and a (iii) LagSSSS[$EIGENSTATES]: parts with only scalars fermion: FSGaugino, (i.e., scalar potential), (xi) trilinear self-interactions of gauge bosons: GaugeTri, LagFFS[ EIGENSTATES] (iv) $ : parts with fermions and (xii) quartic self-interactions of gauge bosons: scalars, GaugeQuad, (v) LagVVV[$EIGENSTATES]: parts with three vector (xiii) interactions between vector bosons and gauginos: bosons, BosonGaugino. (vi) LagVVVV[$EIGENSTATES]:partswithfourvector bosons, A.3. A More Precise Mass Calculation. In some cases a (vii) LagGGS[$EIGENSTATES]: parts with ghosts and numerical more precise calculation is needed to diagonalize scalars, mass matrices in SPheno. This is the case if the hierarchy in (viii) LagGGV[$EIGENSTATES]: parts with ghosts and vec- themassmatrixisverylarge.Inthatcasedoubleprecision tor bosons, with about 15 digits precision might not be sufficient. The best example is models with 𝑅-parity violation where neutrinos (ix) LagSSA[$EIGENSTATES]: parts with scalars and and neutralinos mix. Another example is seesaw type-I auxiliary fields. like models where TeV-scale right neutrinos mix with the In addition, the different steps to derive the Lagrangian of the left-neutrinos. In this case one has to go for quadruple gauge eigenstates are also saved in different variables: precision which gives a precision of about 32 digits. To enable quadruple precision for specific masses, two small changes (i) superpotential: Superpotential, are necessary. (1) In SPheno.m used to set up the SPheno out- (ii) fermion-scalar interactions coming from the put, one has to define for which particles the Wij superpotential: , higher precision is needed. This is done with the QuadruplePrecision (iii) F-terms: FTerms, variable which accepts a list of mass eigenstates. If we just want to have the masses SoftScalarMass (iv) scalar soft-breaking masses: , oftheneutrinos,whicharecalled Fv in the consid- (v) gaugino masses: SoftGauginoMass, ered model, with higher precision, the corresponding line reads as follows. (vi) soft-breaking couplings: SoftW,

SPheno.m QuadruplePrecision = {Fv};

-DONLYDOUBLE. This flags forces all calculations just (2) We must change the Makefile of SPheno located to be done with double precision. in the src directory and remove the compiler flag

Makeﬁle

1 PreDef = -DGENERATIONMIXING

cd $PATH/SPHENO make cleanall By doing that, the routines necessary for a higher precision get compiled. To make sure that everything make MODEL=$NAME is consistent, it might be a good idea to recompile the entire code after changing the Makefile. Advances in High Energy Physics 103

耠 A.4. More about Tadpole Equations and SPheno one wants not to use tan 𝛽 and 𝑀푍耠 as input but obtain 𝑥1 and 𝑥2 from the minimum conditions, an analytical solution A.4.1. Numerical Solutions. In the main part of this paper does not exist. To solve the equations numerically and to SPheno we solved the tadpole equations for the output with define the initialization used by the Broydn routines used for 𝜇 𝜇耠 𝐵 𝐵耠 respect to , , 휇,and 휇 for which an analytical solution that, SPheno.m has to contain the following lines. exists. This must not always be the case. For instance, if

1 ParametersToSolveTadpoles = {\[Mu],B[\[Mu]], x1, x2}; 2 NumericalSolutionTadpoleEquations = True; 3 InitializationTadpoleParameters = { \[Mu] -> m0, B[\[Mu]]-> m0^2, → x1->m0, x2->m0};

Thefirstlineisthesameasfortheanalyticalapproachand inside Mathematica areskipped.Thethirdlineassumes 2 defines that the tadpole equations have to be solved with that 𝜇, 𝑥1,and𝑥2 are 𝑂(𝑚0) and 𝐵휇 is 𝑂(𝑚0). These values are respect to 𝜇, 𝐵휇, 𝑥1,and𝑥2 this time. Without the other used in the numerical routines for initializing the calculation. two lines, Mathematica’s function Solve would try to Of course, other possible and reasonable choices would have findananalyticalsolutionbutitfails.SARAH would then been to relate 𝜇,휇 𝐵 with the running soft-breaking terms of 耠 stop the output with an error message. However, due to the Higgs, and 𝑥1 and 𝑥2 to 𝜇 which is now used as input. thesecondlinetheattemptstosolvethetadpoleequations

1 InitializationTadpoleParameters = { \[Mu] -> Sqrt[mHd2], B[\[Mu]]-> → mHd2, x1->MuP, x2->MuP};

Alsoconstantvaluescanbeusedasfollows.

1 InitializationTadpoleParameters = { \[Mu] -> 10^3, B[\[Mu]]-> 10^6, → x1->10^3, x2->10^3};

Usually, the time needed to find the solution changes only oneoratleastlong-lived.Thiscouldbedone,forinstance, slightly with the chosen initialization values as long as they with Vevacious. are not completely off. Note that all choices above would only find the solution which is the closest one to the initialization A.4.2. Assumptions and Fixed Solutions values. However, the equations are cubic in the VEVs and there will be in general many solutions. Thus, it would Assumptions. It is possible to define a list with replacements be necessary to check if the found vacuum is the global which are done by SARAH when it tries to solve the tadpole equations. For instance, to approximate some matrices as diagonal and to assume that all parameters are real, one could use the following.

AssumptionsTadpoleEquations = {Yx[a__]->Delta[a] Yx[a], T[Yx][a__]->Delta[a] T[Yx][a], conj[x_]->x};

Thathas,ofcourse,noimpactonourexamplebecausethese Fixed Solutions. There might be cases in which an analyt- matrices do not show up in the tadpole equations. However, ical solution exists when some approximations are made, in the 𝑅-parity violating case with sneutrinos VEVs this but Mathematica does not find this solution. Then, it might might help to find analytical solutions which do not exists in be useful to give the solutions as input in SPheno.m.Thiscan the most general case. be done via the following. 104 Advances in High Energy Physics

UseGivenTapdoleSolution=True;

SubSolutionsTadpolesTree = {x1 -> sol1Tree , x2 -> sol2Tree ,...}; SubSolutionsTadpolesLoop = {x1 -> sol1Loop , x2 -> sol2Loop , ....};

Note that the solutions have to be given for the tree-level and (ii) The superpotential and soft-couplings which involve loop-corrected tadpole equations. In the loop-corrected tad- the heavy states are set to zero when a threshold pole equations the one-loop contributions are parametrized is crossed. For instance, we take the Yukawa-like 푖푗 by Tad1Loop[i], where 𝑖 is an integer counting the equa- coupling 𝑌ΦΦ푖𝜙푗𝐻 which involves three generations tions. of the heavy field Φ.AtthethresholdofΦ푘,the𝑘th row of 𝑌Φ is set to zero. A.5. Thresholds in SUSY Models In addition, two assumptions have to be satisfied: (i) the difference between the masses of the scalar and fermionic Assumptions.Itispossibletoincludethresholdeffectsin component of the heavy superfield is negligible; that is, the SPheno the RGE evaluation with . I concentrate here on the masses coming from superpotential interactions are much simpler case where the gauge symmetry does not change. In larger than the soft-breaking term; (ii) these masses are a SARAH that case can derive the RGEs for all scales from the consequence of bilinear terms in the superpotential. Both RGEs of the highest scale above all thresholds as follows. assumptions are fulfilled, for instance, for very heavy vector- like particles or for singlets which have a large Majorana mass. (i) The numbers of generations of the fields which are supposed to be integrated out during the RGE eval- Procedure. There are two steps necessary to implement 𝑛 (Φ ) uation are parametrized by new variables gen 푖 . thresholds according to the above assumptions. First, small All gauge group constants like the Dynkin index changes in the model file of the considered model are 𝑆(𝑅) 𝑛 (Φ ) 𝑛 (Φ ) are expressed as function of gen 푖 . gen 푖 is necessary: the heavy states have to be “deleted” at the dynamically adjusted during the SPheno run when SUSY scale. This is done by adding the superfields to the the RGEs cross the different thresholds. array DeleteFields.

DeleteFields = {...};

This ensures that the heavy particles are not take into which will be known by SPheno. In addition, the fields account in the calculation of mass matrices, vertices, or loop must be stated which should be integrated out at that corrections at the SUSY scale. scale. Afterwards, the boundary conditions for all threshold The second step is to add the thresholds to SPheno.m. scales can be define via the arrays BoundaryConditionsUp Forthispurpose,thethresholdscaleshavetobedefinedvia and BoundaryConditionsDown. The conditions in the first Thresholds 𝑀 𝑀 the array . In this array, the numerical value array are applied during the evaluation from 푍 to GUT and ofeachthresholdscalehastobefixedbysomeparameter the conditions of the second array when running down from 𝑀 GUT.

SPheno.m ... Thresholds = {{Scale1 , {HeavyFields1}}, {Scale2 , ... }}; ... BoundaryConditionsUp[[x]] = { ...}; BoundaryConditionsDown[[x]] = { ...}; Advances in High Energy Physics 105

Seesaw Type-I. To exemplify these steps, I will discuss briefly neutrino is fixed by a Majorana mass term 𝑀].Weassume 𝑀 ≫𝑀 the simplest model with a threshold scale: seesaw type- that ] SUSY; that is, the right neutrinos should be 𝑀 I. In this model, the MSSM particle content is extended integrated out and should not play any role in SUSY.This by three generations of right-handed neutrino superfields. will generate the Weinberg operator 𝑊] which couples in its In addition, a neutrino Yukawa coupling 𝑌] between the supersymmetric version two left-lepton superfields with two left and right neutrinos is present. The mass of the right up-Higgs superfields.

Seesaw1.m ... SuperFields[[8]] = {v, 3, vR, 0, 1, 1, RpM};

... SuperPotential = Yu u.q.Hu - Yd d.q.Hd -Ye e.l.Hd + \[Mu] Hu.Hd + Yv v.l.Hu + Mv/2 v.v +WOp/2 l.Hu.l.Hu; ... DeleteParticles={v};

Itmightlookabitoddthat𝑀] and 𝑊] show up in initialized when the right neutrinos are integrated out. In the the same superpotential. However, we will make sure that following lines in SPheno.m of the seesaw I take care of the during the numerical analysis the Weinberg operator just gets following.

SPheno.m Thresholds={ {Abs[MvIN[1,1]],{v[1]}}, {Abs[MvIN[2,2]],{v[2]}}, {Abs[MvIN[3,3]],{v[3]}} };

BoundaryConditionsUp=Table[{},{Length[Thresholds]}]; BoundaryConditionsDown=Table[{},{Length[Thresholds]}];

BoundaryConditionsDown[[1]]={ {WOp[index1 ,index2], WOp[index1 ,index2] - Yv[1,index1] Yv[1,index2]/MassOfv[1]} };

BoundaryConditionsDown[[2]]={ {WOp[index1 ,index2], WOp[index1 ,index2] - Yv[2,index1] Yv[2,index2]/MassOfv[2]} };

BoundaryConditionsDown[[3]]={ {WOp[index1 ,index2], - Yv[3,index1] Yv[3,index2]/MassOfv[3]} }; 106 Advances in High Energy Physics

Here, we defined three threshold scales which are given by 14 if 1, the running parameters at the mass scale of the diagonal entries of the input value of 𝑀] (MvIN[X,X]). At thedecayingparticlearecalculated.Otherwise,the each scale the corresponding generation of the right neutrino parameters at the standard renormalization scale are superfields is integrated out. Then, we initialize the three used; default is 1; boundary conditions for each threshold scale when running 15 up and down. While we need not define any boundary defines minimum value for a width to be included 10−30 condition when running up, we initialize the Weinberg in output; default is ; operator when running down. The shifts of the coefficients 31 positivevaluesareusedasGUTscale;otherwisea of the Weinberg operator at each threshold scale 𝐼 are given dynamicalGUTscalefulfillingthegivenconditionis by used; default is −1;

퐼푖 퐼푗 𝑌 𝑌 32 if 1, forcing strict unification, that is, 𝑔1 =𝑔2 = 𝛿𝑊푖푗 =− ] ] , ] 퐼 (A.1) 𝑔3; default is 0; 𝑚] 33 if set, a fixed renormalization scale is used; 퐼 where 𝑚 is the 𝐼th eigenvalue of the running matrix 𝑀] ] 34 sets the relative precision of the mass calculation; which in general is not diagonal. −4 default is 10 ; 35 B. Flags in SPheno Input File sets the maximal number of iterations in the calculation of the masses; default is 40; There are many options which can be used in the 36 sets the minimal number of iterations block SPhenoInput intheLesHouchesinputfileto before SPheno stops because of tachyon in the set up the calculations and the output done by SPheno: spectrum; default is 5; 1 sets the error level; default is 0; 37 defines if CKM matrix is taken to be in the up- (1) or down- (2) quark sector; default is 1; 2 if 1, the SPA conventions are used; default is 0; 38 sets the loop order of the RGEs: 1 or 2 can be 7 if 1, it skips two loop Higgs masses; default is 0; used; default is 2; 8 method to calculate two-loop corrections; default 39 if 1, writes output using SLHA1 format; default is 3; is 0;

41 sets the width of the 𝑍-boson Γ푍, default is 1: fully numerical method, 2.49 GeV;

2: semianalytical method, 42 sets the width of the 𝑊-boson Γ푊, default is 3: diagrammatic calculation, 2.06 GeV; 8/9: using results from the literature if 50 if 1, negative fermion masses are rotated to real available; 8 includes only 𝛼푆 corrections, ones by multiplying the rotation matrix with 𝑖; default is 1; 51 0 𝑌 ,𝑌 ,𝑇 ,𝑇 ,𝑚2,𝑚2 ,𝑚2 9 if 1, two-loop corrections are calculated in gauge- if , the parameters 푢 푑 푢 푑 푞 푑 푢 are less limit; default is 1; not rotated into the SCKM basis in the spectrum file; default is 0; 10 1 if ,safemodeisusedforthenumericalderiva- 52 1 tive in the two-loop Higgs calculations; default is 0; if ,anegativemasssquaredisalwaysignored and set 0; default is 0; 11 if 1, the branching ratios of the SUSY and Higgs 53 if 1, a negative mass squared at 𝑀푍 is always particles are calculated; default is 1; ignored and set 0; default is 0; 12 defines minimum value for a branching ratios to 54 1 10−4 if , the output is written even if there has been a be included in output; default is ; problem during the run; default is 0; 13 0 adjusts the three-body decays: : no three-body 55 if 0, the loop corrections to all masses are 1 decays are calculated; : only three-body decays of skipped; default is 1; fermions are calculated; 2: only three-body decays of scalars are calculated; 3: three-body decays of 57 if 0, the calculation of the low-energy observables fermions and scalars are calculated; default is 1; is skipped; default is 1; Advances in High Energy Physics 107

58 0 𝛿 510 1 SPheno if ,thecalculationof VB in the boundary if , writes solution of tadpole equa- conditions at the SUSY scale is skipped; default is 1; tions at tree-level; default is 1. This is needed Vevacious 60 if 0, possible effects from kinetic mixing are for ; neglected; default is 1; 515 if 1, SPheno writes all running values at the 61 if 0, the RGE running of SM parameters is GUT scale; default is 0; skipped in a low scale input; default is 1; 520 1 SPheno HiggsBounds 62 if 0, the RGE running of SUSY parameters to the if , writes blocks low scale is skipped for the calculation of the flavour (effective coupling ratios of Higgs particles to SM 1 and precision observables; default is 1; fields); default is ; 63 0 if , the RGE running of SM parameters to the 525 if 1, SPheno writesthesizeofalldifferent low scale is skipped for the calculation of the flavour contributions to the Higgs diphoton rate; default is 0; and precision observables; default is 1; 64 if 1, the running parameters at the scale 𝑄 = 160 530 if 1, the tree-level values of the tadpole equa- are written in the spectrum file; default is 0; tions appear in the output instead of the loop- corrected ones; default is 0; 65 can be used if several, independent solutions to 1 the tadpole equations exist; default is . An integer 550 if 0,thefine-tuningcalculationisskipped; is used to pick one solution default is 1; 75 1 WHIZARD if , a file containing all parameters in 551 1 𝑍 format is created; default is 1; if , one-loop corrections to -mass are included in fine-tuning calculation; default is 0; 76 if 1,inputfilesforHiggsBounds and HiggsSignals are written; default is 1; 999 if 1, debug information is printed on the screen; 86 sets the maximal width which is taken as default is 0. “invisible” in the output for HiggsBounds HiggsSignals 0 and ; default is ; C. Model Files for the B-L-SSM 88 sets a maximal mass of particles which are 16 included in loop calculations; default is 10 GeV. The full model file for the implementation ofthe B-L- Note that this option must be turned in SARAH first; SSM in SARAH is shown. In addition, I summarize all particles.m parameters.m 89 sets the maximal mass for scalars which is treated changes in and compared −8 as numerical zero; default is 10 GeV; to the MSSM. This includes definition of new parameters and changed properties of parameters already present in the 95 1 if , mass matrices at one-loop are forced to be MSSM.FortheparticlesIrestrictmyselftotheintermediate 0 real; default is ; states and the mass eigenstates after EWSB but skip the 400 fixes initial step-size in numerical derivative for gauge eigenstates. Finally, the additional input file to create the purely numerical method to calculate two-loop a SPheno version for the B-L-SSM is given in Appendix C.4. Higgs masses; default is 0.1; 401 fixes initial step-size in numerical derivative for C.1. Model File the semi-analytical method to calculate two-loop Higgs masses; default is 0.001; 108 Advances in High Energy Physics

B-L-SSM.m

1 Model ‘Name = "BLSSM"; 2 Model ‘NameLaTeX ="B-L-SSM"; 3 Model ‘Authors = "L.Basso , F.Staub"; 4 Model ‘Date = "2012-09-01"; 5 6 (* 2013-09-01: changing to new conventions for Superfields , Superpotential and → global symmetries *) 7 8 9 10 (*------*) 11 (* Particle Content*) 12 (*------*) 13 14 (* Global symmetries *) 15 16 Global[[1]] = {Z[2],MParity}; 17 MpM = {-1,-1,1}; 18 MpP = {1,1,-1}; 19 20 (* Vector Superfields *) 21 22 Gauge[[1]]={B, U[1], hypercharge , g1, False , MpM}; 23 Gauge[[2]]={WB, SU[2], left, g2, True, MpM}; 24 Gauge[[3]]={G, SU[3], color , g3, False , MpM}; 25 Gauge[[4]]={Bp, U[1], BminusL , gBL, False , MpM}; 26 27 (* Chiral Superfields *) 28 29 SuperFields[[1]] = {q, 3, {uL, dL}, 1/6, 2, 3, 1/6, MpM}; 30 SuperFields[[2]] = {l, 3, {vL, eL}, -1/2, 2, 1, -1/2, MpM}; 31 SuperFields[[3]] = {Hd,1, {Hd0, Hdm}, -1/2, 2, 1, 0, MpP}; 32 SuperFields[[4]] = {Hu,1, {Hup, Hu0}, 1/2, 2, 1, 0, MpP}; 33 34 SuperFields[[5]] = {d, 3, conj[dR], 1/3, 1, -3, -1/6, MpM}; 35 SuperFields[[6]] = {u, 3, conj[uR], -2/3, 1, -3, -1/6, MpM}; 36 SuperFields[[7]] = {e, 3, conj[eR], 1, 1, 1, 1/2, MpM}; 37 SuperFields[[8]] = {vR,3, conj[vR], 0, 1, 1, 1/2, MpM}; 38 39 SuperFields[[9]] = {C1, 1, C10, 0, 1, 1, -1, MpP}; 40 SuperFields[[10]] = {C2, 1, C20, 0, 1, 1, 1, MpP}; 41 42 43 (*------*) 44 (* Superpotential *) 45 (*------*) 46 47 SuperPotential = Yu u.q.Hu - Yd d.q.Hd - Ye e.l.Hd + \[Mu] Hu.Hd + Yv vR.l.Hu - MuP → C1.C2 + Yn vR.C1.vR; 48 49 50 (*------*) 51 (* ROTATIONS *) 52 (*------*) 53 54 NameOfStates={GaugeES , EWSB}; 55 56 (* Dirac Spinors for gauge eigenstates *) 57 DEFINITION[GaugeES][DiracSpinors]={ 58 Bino ->{fB, conj[fB]}, 59 Wino -> {fWB, conj[fWB]}, 60 Glu -> {fG, conj[fG]}, 61 H0 -> {FHd0, conj[FHu0]}, 62 HC -> {FHdm, conj[FHup]}, 63 Fd1 -> {FdL, 0}, 64 Fd2 -> {0, FdR}, Advances in High Energy Physics 109

65 Fu1 -> {FuL, 0}, 66 Fu2 -> {0, FuR}, 67 Fe1 -> {FeL, 0}, 68 Fe2 -> {0, FeR}, 69 Fv1 -> {FvL, 0}, 70 Fv2 -> {0, FvR}, 71 FC -> {FC10, conj[FC20]}, 72 FB -> {fBp, conj[fBp]} 73 }; 74 75 76 (*--- Gauge Sector ---- *) 77 DEFINITION[EWSB][GaugeSector] = 78 { 79 {{VB,VWB[3],VBp},{VP,VZ,VZp},ZZ}, 80 {{VWB[1],VWB[2]},{VWm,conj[VWm]},ZW}, 81 {{fWB[1],fWB[2],fWB[3]},{fWm,fWp,fW0},ZfW} 82 }; 83 84 85 86 (*--- VEVs ---- *) 87 DEFINITION[EWSB][VEVs]= 88 {{SHd0, {vd, 1/Sqrt[2]}, {sigmad , I/Sqrt[2]},{phid ,1/Sqrt[2]}}, 89 {SHu0, {vu, 1/Sqrt[2]}, {sigmau , I/Sqrt[2]},{phiu ,1/Sqrt[2]}}, 90 {SvL, {0, 0}, {sigmaL , I/Sqrt[2]},{phiL ,1/Sqrt[2]}}, 91 {SvR, {0, 0}, {sigmaR , I/Sqrt[2]},{phiR ,1/Sqrt[2]}}, 92 {SC10, {x1, 1/Sqrt[2]}, {sigma1 , I/Sqrt[2]},{phi1, 1/Sqrt[2]}}, 93 {SC20, {x2, 1/Sqrt[2]}, {sigma2 , I/Sqrt[2]},{phi2, 1/Sqrt[2]}} 94 }; 95 96 97 (*--- Matter Sector ---- *) 98 DEFINITION[EWSB][MatterSector]= 99 { {{SdL, SdR}, {Sd, ZD}}, 100 {{SuL, SuR}, {Su, ZU}}, 101 {{SeL, SeR}, {Se, ZE}}, 102 {{sigmaL ,sigmaR}, {SvIm, ZVI}}, 103 {{phiL,phiR}, {SvRe, ZVR}}, 104 {{phid, phiu,phi1, phi2}, {hh, ZH}}, 105 {{sigmad , sigmau ,sigma1 ,sigma2}, {Ah, ZA}}, 106 {{SHdm,conj[SHup]},{Hpm,ZP}}, 107 {{fB, fW0, FHd0, FHu0,fBp,FC10,FC20}, {L0, ZN}}, 108 {{{fWm, FHdm}, {fWp, FHup}}, {{Lm,UM}, {Lp,UP}}}, 109 {{FvL,conj[FvR]},{Fvm,UV}}, 110 {{{FeL},{conj[FeR]}},{{FEL,ZEL},{FER,ZER}}}, 111 {{{FdL},{conj[FdR]}},{{FDL,ZDL},{FDR,ZDR}}}, 112 {{{FuL},{conj[FuR]}},{{FUL,ZUL},{FUR,ZUR}}} \ 113 }; 114 115 116 (* Phases *) 117 DEFINITION[EWSB][Phases]= 118 { {fG, PhaseGlu} 119 }; 120 121 (* Dirac Spinors for Mass eigenstates *) 122 DEFINITION[EWSB][DiracSpinors]={ 123 Fd ->{ FDL, conj[FDR]}, 124 Fe ->{ FEL, conj[FER]}, 125 Fu ->{ FUL, conj[FUR]}, 126 Fv ->{ Fvm, conj[Fvm]}, 127 Chi ->{ L0, conj[L0]}, 128 Cha ->{ Lm, conj[Lp]}, 129 Glu ->{ fG, conj[fG]} 130 }; 110 Advances in High Energy Physics

C.2. Parameters Files (i) New gauge couplings.

(ii) New gauge boson mass.

1 {MZp, {Description -> "Z’ mass", 2 LaTeX -> "M_{Z’}", 3 Real -> True, 4 LesHouches -> None, 5 OutputName -> MZp }},

(iii) New gaugino masses.

1 {MassBp , {Description -> "Bino’ Mass", 2 LaTeX -> "{M}_{BL}", 3 LesHouches -> {BL,31}, 4 OutputName -> MBp }}, 5 {MassBBp , {Description -> "Mixed Gaugino Mass 1", 6 LaTeX -> "{M}_{B B’}", 7 LesHouches -> {BL,32}, 8 OutputName -> MBBp }},

(iv) New gauge boson mixing angle.

(v) New angle to give ratio of VEVs.

1 {TBetaP , { LaTeX -> "\\tan(\\beta ’)", 2 Real -> True, 3 LesHouches -> None, 4 OutputName -> TBp }},

(vi) New superpotential and soft-breaking parameters.

1 {MuP, {Description -> "Mu’ Parameter", 2 LaTeX -> "{\\mu_{\\eta}}", 3 LesHouches -> {BL,1}, 4 OutputName -> MuP }}, 5 {B[MuP], {Description -> "B’ Parameter", 6 LaTeX -> "B_{\\eta}", 7 LesHouches -> {BL,2}, 8 OutputName -> BMuP}},

10 {Yv, {Description -> "Neutrino -Yukawa-Coupling", 11 LaTeX -> "Y_\\nu", 12 LesHouches -> Yv, 13 OutputName -> Yv}}, 14 {T[Yv], {Description -> "Trilinear -Neutrino -Coupling", 15 LaTeX -> "T_\\nu", 16 LesHouches -> Tv, 17 OutputName -> Tv}}, 18 {Yn, {Description -> "Neutrino -X-Yukawa -Coupling", 19 LaTeX -> "Y_x", 20 OutputName -> Yx, 21 LesHouches ->Yx }}, 22 {T[Yn], {Description -> "Trilinear -Neutrino -X-Coupling", 23 OutputName -> Tx, 24 LaTeX -> "T_x", 25 LesHouches ->TX}},

27 {mvR2, { Description -> "Softbreaking right Sneutrino Mass", 28 LaTeX -> "m_{\\nu}^2", 29 LesHouches -> mv2, 30 OutputName -> mv2 }}, 31 {mC12, {Description -> "Bilepton 1 Soft-Breaking mass", 32 LaTeX->"m_{\\eta}^2", 33 LesHouches -> {BL,11} , 34 OutputName -> mC12}}, 35 {mC22, {Description -> "Bilepton 2 Soft-Breaking mass", 36 LaTeX->"m_{\\bar{\\eta}}^2", 37 LesHouches -> {BL,12} , 38 OutputName -> mC22}}, 112 Advances in High Energy Physics

(vii) New VEVs.

1 {x1, { Description -> "Bilepton 1 VEV", 2 LaTeX -> "v_{\\eta}", 3 DependenceNum -> Sin[BetaP]*vX, 4 OutputName -> x1, 5 Real -> True, 6 LesHouches -> {BL,41} }}, 7 {x2, {Description -> "Bilepton 2 VEV", 8 LaTeX -> "v_{\\bar{\\eta}}", 9 DependenceNum -> Cos[BetaP]*vX, 10 OutputName -> x2, 11 Real -> True, 12 LesHouches -> {BL,42} }}, 13 {vX, {Description -> "Bilepton VEV", 14 LaTeX -> "x", 15 Dependence -> None, 16 OutputName -> vX, 17 DependenceSPheno -> Sqrt[x1^2 + x2^2], 18 Real -> True, 19 LesHouches -> {BL,43} }},

(viii) New rotation matrices in matter sector.

1 {ZVR, { LaTeX -> "Z^R", 2 OutputName -> ZVR, 3 LesHouches -> SNUMIXR }}, 4 {ZVI, { LaTeX -> "Z^I", 5 OutputName -> ZVI, 6 LesHouches -> SNUMIXI }},

8 {UV, {Description ->"Neutrino -Mixing-Matrix", 9 LaTeX -> "U^V", 10 LesHouches -> UVMIX, 11 OutputName -> UV }},

(ix) Modified rotation matrix in gauge sector.

(x) Modified rotation matrix in matter sector.

1 {ZH, { Description ->"Scalar -Mixing -Matrix",

2 Dependence ->None,

3 DependenceNum ->None,

4 DependenceOptional ->None}},

5 {ZA, { Description ->"Pseudo -Scalar -Mixing -Matrix",

6 Dependence ->None,

7 DependenceNum ->None,

8 DependenceOptional ->None}},

C.3. Particles Files (i) Intermediate states.

1 WeylFermionAndIndermediate = {

2 ...

3 (* Superfields *)

4 {vR, { Description -> "Right Neutrino Superfield" }},

5 {C1, { LaTeX -> "\\hat{\\eta}" }},

6 {C2, { LaTeX -> "\\hat{\\bar{\\eta}}" }},

8 (* Intermediate fermions *)

9 {FC10, { LaTeX -> "\\tilde{\\eta}" }},

10 {FC20, { LaTeX -> "\\tilde{\\bar{\\eta}}" }},

11 {fBp, { LaTeX -> "{\\tilde{B}{}’}"}},

13 (* Intermediate Scalars *)

14 {phi1, { LaTeX -> "\\phi_{\\eta}" }},

15 {phi2, { LaTeX -> "\\phi_{\\bar{\\eta}}" }},

17 {sigma1 , { LaTeX -> "\\sigma_{\\eta}"}},

18 {sigma2 , { LaTeX -> "\\sigma_{\\bar{\\eta}}" }},

20 {SC10, { LaTeX -> "\\eta" }},

21 {SC20, { LaTeX -> "\\bar{\\eta}" }},

23 };

for those is used in the different output for SPheno (ii) New Mass Eigenstates. More interesting are the andtheMC-tools.Webeginwiththenewstateswhich mass eigenstates. The additional information given are not present in the MSSM. 114 Advances in High Energy Physics

(c) ElectricCharge: ... (d) OutputName: ... Some comments are at place: (iii) Modified Mass Eigenstates. For other states only more PDG PDG.IX ... generations appear compared to the MSSM. There- (a) and : fore, it is only possible to extent the lists for the PDGs. (b) FeynArtsNr: ...

1 {hh , { Description -> "Higgs", 2 PDG -> {25,35,9900025, 9900035}, 3 PDG.IX->{101000001,101000002,101000003,101000004} }}, 4 {Ah , { Description -> "Pseudo -Scalar Higgs", 5 PDG -> {0,0,36,9900036}, 6 PDG.IX->{0,0,102000001,102000002} }}, 7 {Fv, { Description -> "Neutrinos", 8 Mass -> Automatic , 9 Width -> Automatic , 10 PDG ->{12,14,16,112,114,116}, 11 PDG.IX->{111000001,111000002,111000003, 12 111000004,111000005,111000006 } }}, 13 14 {Chi, { Description -> "Neutralinos", 15 PDG -> {1000022,1000023,1000025,1000035, 16 1000032,1000036,1000039}, 17 PDG.IX ->{211000001,211000002,211000003,211000004, 18 211000005,211000006,211000007 } }} Advances in High Energy Physics 115

C.4. SPheno File

SPheno.m

1 MINPAR={{1,m0}, 2 {2,m12}, 3 {3,TanBeta}, 4 {4,SignumMu}, 5 {5,Azero}, 6 {6,SignumMuP}, 7 {7,TBetaP}, 8 {8,MZp}}; 9 10 RealParameters = {TanBeta , TBetaP ,m0}; 11 ParametersToSolveTadpoles = {B[\[Mu]],B[MuP],\[Mu],MuP}; 12 13 RenormalizationScaleFirstGuess = m0^2 + 4 m12^2; 14 RenormalizationScale = MSu[1]*MSu[6]; 15 16 ConditionGUTscale = (g1*gBL-g1BL*gBL1)/Sqrt[gBL^2+gBL1^2] == g2; 17 18 BoundaryHighScale={ 19 {g1,(g1*gBL-g1BL*gBL1)/Sqrt[gBL^2+gBL1^2]}, 20 {g1,Sqrt[(g1^2+g2^2)/2]}, 21 {g2,g1}, 22 {gBL, g1}, 23 {g1BL ,0}, 24 {gBL1 ,0}, 25 {T[Ye], Azero*Ye}, 26 {T[Yd], Azero*Yd}, 27 {T[Yu], Azero*Yu}, 28 {T[Yv], Azero*Yv}, 29 {T[Yn], Azero*Yn}, 30 {mq2, DIAGONAL m0^2}, 31 {ml2, DIAGONAL m0^2}, 32 {md2, DIAGONAL m0^2}, 33 {mu2, DIAGONAL m0^2}, 34 {me2, DIAGONAL m0^2}, 35 {mvR2, DIAGONAL m0^2}, 36 {mHd2, m0^2}, 37 {mHu2, m0^2}, 38 {mC12, m0^2}, 39 {mC22, m0^2}, 40 {MassB , m12}, 41 {MassWB ,m12}, 42 {MassG ,m12}, 43 {MassBp ,m12}, 44 {MassBBp ,0} 45 }; 46

47 48 BoundarySUSYScale = { 49 {g1T,(g1*gBL-g1BL*gBL1)/Sqrt[gBL^2+gBL1^2]}, 50 {gBLT, Sqrt[gBL^2+gBL1^2]}, 51 {g1BLT ,(g1BL*gBL+gBL1*g1)/Sqrt[gBL^2+gBL1^2]}, 52 {g1, g1T}, 53 {gBL, gBLT}, 54 {g1BL, g1BLT}, 55 {gBL1 ,0}, 56 {vevP, MZp/gBL}, 116 Advances in High Energy Physics

57 {betaP ,ArcTan[TBetaP]}, 58 {x2,vevP*Cos[betaP]}, 59 {x1,vevP*Sin[betaP]}, 60 {Yv, LHInput[Yv]}, 61 {Yn, LHInput[Yn]} 62 }; 63 64 BoundaryEWSBScale = { 65 {g1T,(g1*gBL-g1BL*gBL1)/Sqrt[gBL^2+gBL1^2]}, 66 {gBLT, Sqrt[gBL^2+gBL1^2]}, 67 {g1BLT ,(g1BL*gBL+gBL1*g1)/Sqrt[gBL^2+gBL1^2]}, 68 {g1, g1T}, 69 {gBL, gBLT}, 70 {g1BL, g1BLT}, 71 {gBL1 ,0}, 72 {vevP, MZp/gBL}, 73 {betaP ,ArcTan[TBetaP]}, 74 {x2,vevP*Cos[betaP]}, 75 {x1,vevP*Sin[betaP]} 76 }; 77 78 InitializationValues = { 79 {gBL, 0.5}, 80 {g1BL, -0.06}, 81 {gBL1, -0.06} 82 } 83 84 BoundaryLowScaleInput={ 85 {vd,Sqrt[4 mz2/(g1^2+g2^2)]*Cos[ArcTan[TanBeta]]}, 86 {vu,Sqrt[4 mz2/(g1^2+g2^2)]*Sin[ArcTan[TanBeta]]} 87 }; 88 89 ListDecayParticles = Automatic; 90 ListDecayParticles3B =Automatic; 91 92 UseBoundarySUSYatEWSB = True; 93 94 (* Example for mSugra input values *) 95 DefaultInputValues = {m0 -> 1000, m12 -> 1500, TanBeta ->20, SignumMu ->1, Azero -> → -1500, SignumMuP -> 1, TBetaP -> 1.15, MZp -> 2500, Yn[1,1]->0.37, → Yn[2,2]->0.4, Yn[3,3]->0.4};

D. Models Included in SARAH (a) next-to-minimal supersymmetric standard model (NMSSM, NMSSM/NoFV, NMSSM/CPV, I show here the list of models which are included in the public and NMSSM/CKM), SARAH version of . Additional models created and provided (b) near-to-minimal supersymmetric standard byuserarealsofoundhere model (near-MSSM), https://sarah.hepforge.org/trac/wiki. (c) general singlet extended, supersymmetric standard model (SMSSM), DiracNMSSM D.1. Supersymmetric Models (d) Dirac NMSSM ( ), (e) next-to-minimal supersymmetric standard (i) Minimal supersymmetric standard model: model with inverse seesaw (inverse-seesaw- NMSSM). (a) with general flavour and CP structureMSSM ( ), (b) without flavour violation (MSSM/NoFV), (iii) Triplet extensions: (c) with explicit CP violation in the Higgs sector (MSSM/CPV), (a) triplet extended MSSM (TMSSM), (d) in SCKM basis (MSSM/CKM). (b) triplet extended NMSSM (TNMSSM).

(ii) Singlet extensions: (iv) Models with 𝑅-parity violation: Advances in High Energy Physics 117

(a) bilinear RpV (MSSM-RpV/Bi), Conflict of Interests (b) lepton number violation (MSSM-RpV/LnV), The author declares that there is no conflict of interests (c) only trilinear lepton number violation regarding the publication of this paper. (MSSM-RpV/TriLnV), (d) Baryon number violation (MSSM-RpV/BnV), (e) 𝜇]SSM (munuSSM). Acknowledgments (v) Additional 𝑈(1)’s: The author is very grateful to Mark D. Goodsell and Kilian Nickel for providing routines for the two-loop calculation (a) 𝑈(1)-extended MSSM (UMSSM), via the SARAH–SPheno interface. In particular the author (b) secluded MSSM (secluded-MSSM), thanks Werner Porod who raised his interest in supersym- (c) minimal 𝐵-𝐿 model (B-L-SSM), metry. This was the starting point of the entire development of SARAH.TheauthorisindebttoMartinHirsch,Avelino (d) minimal singlet-extended 𝐵-𝐿 model Vicente, Daniel Busbridge, James Scoville, Alexander Voigt, (N-B-L-SSM). Peter Athron, Roberto Ruiz de Austri Bazan, Moritz McGar- SARAH (vi) SUSY-scale seesaw extensions: rie, Lorenzo Basso, and many others for testing of , helpful suggestions, and also their bug reports. Finally, it (a) inverse seesaw (inverse-Seesaw), has been a pleasure for the author to work with Jose Eliel (b) linear seesaw (LinSeesaw), Camargo-Molina, Ben O’Leary, and again Werner Porod and Vevacious FlavorKit (c) singlet extended inverse seesaw Avelino on and .Theauthorthanks (inverse-Seesaw-NMSSM), Manuel Krauss, Lukas Mitzka, Tim Stefaniak, and Avelino Vicente for their remarks on the paper. (d) inverse seesaw with 𝐵-𝐿 gauge group (B-L-SSM-IS), 𝑈(1) × 𝑈(1) Endnotes (e) minimal 푅 퐵-퐿 modelwithinverse seesaw (BLRinvSeesaw). 1. See for instance [320] for an overview of SUSY searches. (vii) Models with Dirac gauginos: 2. Another method to deal with gauge-kinetic mixing was proposed in [321]. (a) MSSM/NMSSM with Dirac gauginos (DiracGauginos), 3. For simplicity, I’ll use here “Wilson coefficients” also for the coefficients of LFV operators which are more (b) minimal 𝑅-symmetric SSM (MRSSM), commonly called “form factors”. (c) minimal Dirac gaugino SSM (MDGSSM). 4. Models with a threshold scale where heavy superfields (viii) High-scale extensions: are integrated out are Seesaw1, Seesaw2, Seesaw3. 5. A model with threshold scales where the gauge group (a) seesaw 1–3 (SU(5) version) changes is the left-right symmetric model called Omega (Seesaw1, Seesaw2, Seesaw3), in SARAH. (b) left/right model (ΩLR) (Omega). 6. Gauge bosons get this name with a prefix V,gauginos (ix) Others: with a prefix f and ghosts with the prefix g.For instance, the gluon and gluino are called VG and fG by (a) MSSM with nonholomorphic soft-terms this definition and the Ghost gG. (NHSSM), MSSM6C 7. The charge indices of non-Abelian groups start with the (b) MSSM with colour sextets ( ). first three letters of the gauge group’s name followed by an integer, that is, col1 isusedforcolourindices. D.2. Nonsupersymmetric Models 8. Note, gBp is not possible because this is already the name of the ghost! (i) Standard model (SM) (SM), standard model in CKM SM/CKM 9. One has to be careful with the rotations of charged basis ( ). + vector bosons like 𝑊 .Here,themassmatrixinthebasis (ii) Inert Higgs doublet model (Inert). 1 2 (𝑊 ,𝑊 ) is diagonal and the calculated rotation matrix (iii) 𝐵-𝐿 extended SM (B-L-SM). from diagonalizing this matrix would be just the identity (iv) 𝐵-𝐿 extended SM with inverse seesaw (B-L-SM-IS). matrix. In that case it is inevitable to give the standard 𝑍푊 =1/√2( 11) (v) SM extended by a scalar colour octet (SM-8C). parametrization −푖 푖 in numerical studies as input. For neutral vector bosons such a problems (vi) Two Higgs doublet models (THDM, THDM-II, won’t show up. THDM-III, THDM-LS, THDM-Flipped). 10. “/” is interpreted by Mathematica as escape sequence. (vii) Singlet extended SM (SSM). Therefore, “/” has to be replaced by “//” in the LaTeX (viii) Triplet extended SM (TSM). commands. 118 Advances in High Energy Physics

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Review Article Vacuum Condensates as a Mechanism of Spontaneous Supersymmetry Breaking

Antonio Capolupo and Marco Di Mauro

Dipartimento di Fisica E.R.Caianiello, UniversitadiSalernoandINFNGruppoCollegatodiSalerno,84084Fisciano,Italy´

Correspondence should be addressed to Antonio Capolupo; [email protected]

Received 27 April 2015; Revised 23 June 2015; Accepted 1 July 2015

Academic Editor: Ignatios Antoniadis

Copyright © 2015 A. Capolupo and M. Di Mauro. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.

A possible mechanism for the spontaneous breaking of SUSY, based on the presence of vacuum condensates, is reviewed. Such a mechanism could occur in many physical examples, at both the fundamental and emergent levels, and would be formally analogous to spontaneous SUSY breaking at finite temperature in the TFDormalism,inwhichcaseitcanbeappliedaswell.Apossible f experimental setup for detecting such a breaking through measurement of the Anandan-Aharonov invariants associated with vacuum condensates in an optical lattice model is proposed.

1. Introduction Namely,itispossiblethatacondensedmattersystemmay display SUSY at low energies, which may or may not be spon- Supersymmetry (SUSY) [1, 2]hashadahugeimpacton taneously broken. In particular, relativistic supersymmetric contemporary physics, not only from the purely theoretical theories could be simulated with cold atom systems in optical andmathematicalpointsofview,butalsofromthephe- lattices [6]. In what follows, we will describe a mechanism for nomenological and experimental ones. This is despite the SUSY breaking, based on vacuum condensates, which may absence, up to now, of any clear experimental signature of its be valid both at a fundamental level and at an emergent level existence at the fundamental level. The main reason for this [7, 8]. The latter possibility also suggests ways to investigate is that to date SUSY provides the best available explanation this mechanism in table top experiments. of the gauge hierarchy problem of the Standard Model [3], as The idea is to exploit the formal analogy between thermal well as providing candidates for dark matter and improving field theory in its Thermofield Dynamics (TFD) formula- the situation of the dark energy issue (which however is still tion [9] and different physical phenomena characterized far from settled). In a few years, the situation may radically by vacuum condensates similar to those appearing in the change due to the results coming from the LHC, but it is thermal context [10–15]. As in the thermal case, SUSY afactthatifSUSYexistsatafundamentallevel,itmustbe is spontaneously broken (see below); we expect that this broken, either spontaneously or explicitly, since otherwise the happens in the same way also in these other phenomena. superpartners of the known particles would be degenerate A possible experiment involving the measurement of the with the latter and thus would have been observed long Anandan-Aharonov invariant associated with the vacuum ago. For this reason over the years there has been a lot of condensate is also described. activity concerning SUSY breaking, and in particular the Before explicitly stating our conjecture, let us briefly recall spontaneous breaking case (see, e.g., [4, 5] and references howSUSYisspontaneouslybrokeninTFD. therein). It is well known that SUSY is spontaneously broken Besides fundamental SUSY, an interesting possibility, at any finite temperature [16, 17], with the fundamental both on its own right and as a way to experimentally test ideas reason being the different statistical behavior of bosons and onSUSYanditsbreakinginthenearfuture,is emergent SUSY. fermions. Finite temperature physics can be formulated in 2 Advances in High Energy Physics a way which is equivalent to the standard ensemble based give a good qualitative understanding of the vacuum of more picture but which emphasizes the appearance of vacuum complicated systems. condensates. This formalism goes under the name of Ther- Considering the range of the phenomena described by mofield Dynamics [9]. In this formalism vacuum condensates this picture, this mechanism may occur at a fundamental in the thermal ground state are conveniently described by level, for example, triggered by particle mixing, as proposed means of Bogoliubov transformations, and thermal effects in [21, 22], or at an emergent level. The first possibility is areencodedintheappearanceofanewvacuumwhichis very interesting from a phenomenological point of view and unitarily inequivalent to the zero temperature one. Thermal maybeusedformodelbuilding,whilethelatterpossibility averages are then just vacuum expectation values with respect suggests, as said, the possibility of conceiving experimental to this new vacuum. In the standard picture [16, 17], SUSY measurements of the vacuum energy due to the condensates, breaking is due to the fact that it is not possible to write therefore corroborating our conjecture [7]. This will be also down thermal averages in a way consistent with SUSY, while the object of the present paper, in which the possibility in the TFD picture it is due to the fact that the new vacuum of probing thermal spontaneous SUSY breaking through acquires a nonvanishing energy density. This picture thus geometric invariants [23] will be explored. To be specific, links thermal breaking of SUSY to the standard description the relevant quantity is the Anandan-Aharonov invariant of SUSY breaking, whose order parameter is precisely the [24],whichhasbeenshowntobeafeatureofphenomena vacuum energy density [4, 5].Thislastfactisaswellknown characterized by vacuum condensates [25]. a straightforward consequence of the SUSY algebra: if the A few comments are in order. First of all, since vacuum vacuum is not invariant under SUSY transformations, that is, condensates are a genuine field theoretical and nonpertur- 𝑄𝛼|0⟩ ≠ 0, then (here 𝑄 is the supercharge that generates bative phenomenon, this kind of SUSY breaking can occur SUSY transformations, 𝐻 is the Hamiltonian of the theory, only in QFT, and it is nonperturbative in nature (consistently and 𝐶 is the charge conjugation matrix) withthefactthatifSUSYisunbrokenattreelevel,itcanonly 󵄨 󵄨 be broken at the nonperturbative level [4]).Second,whilein 1 󵄨 0 󵄨 ⟨0 |𝐻| 0⟩ = ⟨0 󵄨Tr (𝐶𝛾 [𝑄, 𝑄]+)󵄨 0⟩ ≠ 0, (1) whatfollowswewillgiveevidenceforourconjectureinasim- 8 ple case, we do not address the issue of the dynamical origin whileofcourseifthevacuumisinvariant,then⟨0|𝐻|0⟩=0. of that breaking or, which is the same, of the origin of the Physically, this is due to the fact that the zero point energies vacuum condensates, which depends on the specific details of fermions and bosons cancel out; schematically, of the phenomena under study and which in some cases such as mixing is to date unknown. The effective description 𝐻=𝐻𝜓 +𝐻𝐵 of condensates in terms of Bogoliubov transformations is instead universal (besides being technically straightforward), 𝜓 𝜓 1 𝐵 𝐵 1 (2) since the form of this transformation is always the same, ∼ ∑ {𝜔k,𝑖 (𝑁k,𝑖 − )+𝜔k,𝑖 (𝑁k,𝑖 + )} , k,𝑖 2 2 with the details of the specific case being encoded in the coefficients. This means that our discussion will be necessarily 𝜓 𝐵 and in a supersymmetric theory 𝜔k,𝑖 =𝜔k,𝑖 ≡𝜔k,𝑖.Inthe qualitative, while a more quantitative approach will need caseofTFD,thecondensateswhicharepresentinthethermal dealing with the complexities of the dynamics on various vacuum lift the vacuum energy. Such a lift is not canceled in cases. In particular, the computation of quantities such as a supersymmetric theory, thereby triggering the spontaneous the scale of the breaking and mass differences between breaking of SUSY. superpartners lies beyond the scope of the present paper. ThepointisthattheformalismofBogoliubovtrans- Also, we do not address the issue of the Goldstone fermion formations [18], on which this vacuum condensate based associated with the breaking. This issue, as well as the description of thermal physics is founded, is quite universal detailed study of some specific case, is left for some future and describes vacuum condensates in a host of different publication. quantum field theoretical (QFT) phenomena at various length scales, from fundamental to emergent models [19]. 2. Vacuum Condensate and SUSY Breaking Examples of such phenomena include quantum fields in external fields, such as Schwinger [10]andUnruh[11]effects As a model of the supersymmetric extension of any of the and examples from condensed matter physics such as the above systems, we consider a situation in which SUSY is BCS theory of superconductivity [12] and graphene [13], preserved at the Lagrangian level and study the vacuum mixing in particle physics [14, 15]. (In the case of mixing condensation effects. These are described by a Bogoliubov the situation is slightly different, in that the Bogoliubov transformation acting simultaneously, and with the same transformation is nested in a unitary transformation of the parameters, on the bosonic and on the fermionic degrees fields; however, this does not qualitatively change what we of freedom. This is required in order not to break SUSY will say.) This leads to the conjecture that, in all these cases, explicitly. We conjecture that, in such a situation, SUSY is when a supersymmetric extension is possible at the classical spontaneously broken by the appearance of vacuum conden- level, vacuum condensates lift the vacuum energy, thereby sates. In the present section, we collect some basic facts about spontaneously breaking SUSY [7, 8]. We give some evidence Bogoliubov transformations in QFT (see, e.g., [19]), and then for this conjecture by considering the free Wess-Zumino we prove in a simple case that vacuum condensates do shift model [20]. Despite its simplicity, this simple picture should the vacuum energy. Advances in High Energy Physics 3

The modes of any boson (fermion) field are described Bogoliubov transformations on the fermion and on the by a set of ladder operators 𝑎k, whose canonical (anti)com- bosons: [𝑎 ,𝑎†] =𝛿3( − ) − mutation relations (CCRs) are k p ± k p ,with for 𝛼̃𝑟 (𝜉,) 𝑡 =𝑈𝜓 (𝜉,) 𝑡 𝛼𝑟 (𝑡) +𝑉𝜓 (𝜉,) 𝑡 𝛼𝑟† (𝑡) , bosons and + for fermions and all other (anti)commutators k k k −k −k vanishing. The vacuum |0⟩ is defined by 𝑎k|0⟩,andaFock ̃ 𝑆 𝑆 † 𝑏k (𝜂, 𝑡) =𝑈 (𝜂, 𝑡)k 𝑏 (𝑡) −𝑉 (𝜂, 𝑡) 𝑏 (𝑡) , (6) space is built out of it by acting with the creation operators k −k −k 𝑎† 𝑃 𝑃 † k. 𝑐̃k (𝜂, 𝑡)k =𝑈 (𝜂, 𝑡)k 𝑐 (𝑡) −𝑉−k (𝜂, 𝑡)− 𝑐 k (𝑡) . A generic Bogoliubov transformation has the following form: The Bogoliubov coefficients of scalar and pseudoscalar 𝑆 𝑃 𝑆 𝑃 bosons are equal to each other: 𝑈k =𝑈k and 𝑉k =𝑉k . 𝑎̃ (𝜉) =𝑈𝑎 −𝑉𝑎†, 𝐵 𝐵 k k k k k (3) We thus denote such quantities as 𝑈k and 𝑉k ,respectively. For fermions and for bosons, the Bogoliubov coefficients have 2 2 𝜓 𝜓 |𝑈 | ±|𝑉| = − 𝑖𝜙1k 𝑖𝜙2k with the condition k k 1, with for bosons and + the general form: 𝑈k =𝑒 cos 𝜉k(𝜁), 𝑉k =𝑒 sin 𝜉k(𝜁), for fermions, ensuring the canonicity of the transformation. 𝐵 𝑖𝛾1k 𝐵 𝑖𝛾2k 𝑈 =𝑒 cosh 𝜂k(𝜁),and𝑉 =𝑒 sinh 𝜂k(𝜁),respectively, 𝑎̃ (𝜉) = k k The transformation (3) is conveniently rewritten as k where 𝜁 represents the relevant parameter which controls 𝐽−1(𝜉)𝑎 𝐽(𝜉) 𝐽(𝜉) k ,where is the generator which has the the physics underlying the Bogoliubov transformation. For 𝐽−1(𝜉) = 𝐽(−𝜉) 𝑎̃ (𝜉) property .Thetransformedoperators k example, 𝜁 is related to the temperature 𝑇 in Thermofield |̃(𝜉)⟩ 𝑎̃ (𝜉)|̃(𝜉)⟩ = define a state 0 through k 0 0, which is related Dynamics and to the acceleration of the observer in Unruh | ⟩ |̃(𝜉)⟩ =− 𝐽 1(𝜉)| ⟩. to the vacuum 0 by 0 0 Suchastateisanew effect case. Since the phases 𝜙𝑖k and 𝛾𝑖k,with𝑖=1, 2, are vacuum of the system, for the following reason: the above irrelevant, we neglect them. transformation is a unitary operation if k assumes a discrete The transformations (6) can be written at any time 𝑡 in range of values, that is, if there is a finite or denumerably terms of a generator 𝐽(𝜉, 𝜂, 𝑡); for example, for fermions we infinite number of CCRs. Then, the Fock spaces built on the have two vacua |0⟩ and |0̃(𝜉)⟩ areequivalent.Ifontheotherhand 𝑟 −1 𝑟 we assume that k has continuous infinity of values, which 𝛼̃k (𝜉,) 𝑡 =𝐽 (𝜉,𝜂,𝑡)𝛼k (𝑡) 𝐽(𝜉,𝜂,𝑡), (7) isthesituationwearereallyinterestedin,wefindthatthe −1 transformation |0̃(𝜉)⟩ = 𝐽 (𝜉)|0⟩ is not unitary any more. with similar relations holding for the bosonic annihilation and creation operators; in all of them, the generator is This means that the two vacua and thus the two Fock spaces 𝐽(𝜉, 𝜂, 𝑡) =𝐽 (𝜉, 𝑡)𝐽 (𝜂, 𝑡)𝐽 (𝜂, 𝑡) 𝐽 𝐽 𝐽 built over them are unitarily inequivalent. We thus have a 𝜓 𝑆 𝑃 ,where 𝜓, 𝑆,and 𝑃 are the generator of the Bogoliubov transformations for fermion, family of states |0̃(𝜉)⟩, each of which represents in principle scalar, and pseudoscalar fields [7]. a physical vacuum state for the theory. Of course, for these ̃ ̃ ̃ ̃ The new vacuum is |0(𝑡)⟩ =0 | (𝑡)⟩𝜓 ⊗|0(𝑡)⟩𝑆 ⊗|0(𝑡)⟩𝑃, states to be true vacua of the system, the issue of stability |̃(𝑡)⟩ 𝛼=𝜓,𝑆,𝑃 should be addressed, but this depends on the specific system where the states 0 𝛼,with , are related to the | ⟩ |̃(𝑡)⟩ =𝐽−1(𝜉, 𝑡)| ⟩ andisbeyondthescopeofthissimple,freemodel. original ones 0 𝛼 by the relations 0 𝜓 𝜓 0 𝜓, ̃ −1 ̃ −1 Now, as announced, we will perform a Bogoliubov trans- |0(𝑡)⟩𝑆 =𝐽𝑆 (𝜂, 𝑡)|0⟩𝑆,and|0(𝑡)⟩𝑃 =𝐽𝑃 (𝜂, 𝑡)|0⟩𝑃,respec- formation on the free Wess-Zumino model and study its tively; therefore, the full vacua are related by effects. The Lagrangian is given by (we adopt the notational 󵄨 󵄨̃ −1 conventions of [20]) 󵄨0 (𝑡)⟩=𝐽 (𝜉,𝜂,𝑡)|0⟩ . (8)

𝑖 𝜇 1 𝜇 1 𝜇 𝑚 |̃(𝑡)⟩ L = 𝜓𝛾𝜇𝜕 𝜓+ 𝜕𝜇𝑆𝜕 𝑆+ 𝜕𝜇𝑃𝜕 𝑃− 𝜓𝜓 We notice that 0 has the structure of a condensate of 2 2 2 2 particles, and indeed we have 2 (4) 𝑚 󵄨 󵄨 󵄨 󵄨2 − (𝑆2 +𝑃2), ⟨̃ (𝑡) 󵄨𝛼𝑟†𝛼𝑟 󵄨 ̃ (𝑡)⟩=󵄨𝑉𝜓 (𝜉,) 𝑡 󵄨 ; 0 󵄨 k k󵄨 0 󵄨 k 󵄨 2 (9) 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨2 ⟨̃ (𝑡) 󵄨𝑏†𝑏 󵄨 ̃ (𝑡)⟩=⟨̃ (𝑡) 󵄨𝑐†𝑐 󵄨 ̃ (𝑡)⟩=󵄨𝑉𝐵 (𝜂, 𝑡)󵄨 . where 𝜓 isaMajoranaspinorfield,𝑆 is a scalar field, and 𝑃 0 󵄨 k k󵄨 0 0 󵄨 k k󵄨 0 󵄨 k 󵄨 is a pseudoscalar field. This Lagrangian is invariant under the SUSY transformations: Such a structure leads to an energy density different from zero for |0̃(𝑡)⟩. To see this explicitly, we must compute the 𝛿𝑆 =𝑖𝜅𝜓, expectation value of the Hamiltonian 𝐻 corresponding to the Lagrangian in (4),whichhastheform𝐻=𝐻𝜓 +𝐻𝐵 (where 𝛿𝑃 =𝑖𝜅𝛾5𝜓, (5) ̃ 𝐻𝐵 =𝐻𝑆 +𝐻𝑃), on |0(𝑡)⟩. The results for the two pieces of 𝐻 𝜇 𝛿𝜓𝜇 =𝜕 (𝑆 −5 𝛾 𝑃) 𝛾 𝜅−𝑚(𝑆+𝛾5𝑃) 𝜅, are given by 󵄨 󵄨 󵄨 󵄨2 ⟨̃ (𝑡) 󵄨𝐻 󵄨 ̃ (𝑡)⟩=−∫ 𝑑3 𝜔 ( − 󵄨𝑉𝜓 (𝜉,) 𝑡 󵄨 ), where 𝜅 is a Grassmann valued spinorial parameter. 0 󵄨 𝜓󵄨 0 k k 1 2 󵄨 k 󵄨 (10) 𝑟 We denote by 𝛼k, 𝑏k,and𝑐k the annihilators for the 𝜓 𝑆 𝑃 󵄨 󵄨 󵄨 󵄨2 fields , ,and , respectively, which annihilate the vacuum ⟨̃ (𝑡) 󵄨𝐻 󵄨 ̃ (𝑡)⟩=∫ 𝑑3 𝜔 ( + 󵄨𝑉𝐵 (𝜂, 𝑡)󵄨 ), 𝜓 𝑆 𝑃 0 󵄨 𝐵󵄨 0 k k 1 2 󵄨 k 󵄨 (11) |0⟩=|0⟩ ⊗|0⟩ ⊗|0⟩ and we perform simultaneous 4 Advances in High Energy Physics respectively.Wethusobtainthefinalresult (We notice that the particle mixing phenomenon is peculiar for the following reason. Although also in this case SUSY is ⟨0̃ (𝑡) |𝐻| 0̃ (𝑡)⟩ spontaneously broken by condensate [21, 22], in this case the AAI arises mainly as an effect of the mixing of fields with 󵄨 󵄨2 󵄨 󵄨2 (12) = ∫ 𝑑3 𝜔 (󵄨𝑉𝜓 (𝜉,) 𝑡 󵄨 + 󵄨𝑉𝐵 (𝜂, 𝑡)󵄨 ) only a small contribution due to the condensate structure 2 k k 󵄨 k 󵄨 󵄨 k 󵄨 [26, 27]. Therefore, in this case the presence of the AAI is not directly linked with the presence of the condensate. In all the which is different from zero and positive unless we are in the 𝜓 2 𝐵 2 other cases instead the AAI is entirely due to the condensate trivial case |𝑉 | =|𝑉 | = 0. k k contribution.) The above computation clearly shows that the nonzero Now we study the specific case of thermal states and vacuum condensate energy, and thus the breaking of SUSY, is propose a possible experiment to detect thermal SUSY due to the fact that both the fermion and boson contributions violation by measuring nonvanishing AAIs. As is clear from to the condensate lift the vacuum energy by a positive all we said, in the TFD formalism [9], the thermal vacuum is amount,incontrastwiththezeropointenergieswhichcancel a condensate generated through Bogoliubov transformations each other. whose parameter is related to temperature. The Bogoliubov coefficients 𝑈 and 𝑉 have the general form [9] 𝑈k = √ 𝛽𝜔k 𝛽𝜔k √ 𝛽𝜔k 3. SUSY Breaking and 𝑒 /(𝑒 ± 1) and 𝑉k = 1/(𝑒 ± 1),with− for bosons the Anandan-Aharonov Invariant and + for fermions, and 𝛽=1/𝑘𝐵𝑇. The energy variances of a temperature dependent single It has been shown that the presence of the Anandan- particle state are given by Aharonov invariant (AAI) [24], describing the time-energy uncertainty, characterizes the time evolution of the systems 𝛽𝜔k/2 𝑆 𝑃 √ 𝐵 𝐵 √ 𝑒 in which the vacuum condensate is physically relevant [25]. Δ𝐸k =Δ𝐸k = 2𝜔k𝑈k 𝑉k = 2𝜔k , (16) (𝑒𝛽ℏ𝜔k − ) Then, AAIs could be used as a tool to reveal the SUSY break- 1 down [23]. The AAI appears in the evolution of any quantum for the bosonic states, and state |𝜒k(𝑡)⟩ which is not stationary; that is, its energy 2 2 2 𝜓 uncertainty Δ𝐸k(𝑡) = k⟨𝜒 (𝑡)|𝐻 |𝜒k(𝑡)⟩ − (⟨𝜒k(𝑡)|𝐻|𝜒k(𝑡)⟩) 𝛽𝜔 /2 𝜓 𝜓 𝜓 𝜓 𝜓 𝑒 k must be nonzero. This is the case in the above listed instances Δ𝐸 =𝜔 𝑈 𝑉 =ℏ𝜔 , k k k k k 𝛽𝜔𝜓 (17) [11–15]. When this condition is met, the AAI is defined as (we (𝑒 k + 1) 𝑡 ℏ 𝑆 =( /ℏ) ∫ Δ𝐸 (𝑡󸀠)𝑑𝑡󸀠. temporarily restore ) k 2 0 k This invariant is analogous to the geometric phase (but for the fermionic state. The corresponding AAIs are it is defined for noncyclic and nonadiabatic evolution) and 𝛽ℏ𝜔 /2 𝑒 k represents a time-energy uncertainty principle. It can be 𝑆𝑆 =𝑆𝑃 = √ 𝜔 𝑡 , k k 2 2 k 𝛽ℏ𝜔 measured by studying interference of particles or by the 𝑒 k − 1 (18) analysis of the uncertainty on the outcome of measurements. 𝛽ℏ𝜔 /2 𝜓 𝑒 k Weconsiderthesingleparticlestates: 𝑆k = 2𝜔k𝑡 . 𝑒𝛽ℏ𝜔k + 󵄨 1 󵄨 ̃ 𝑟† 󵄨̃ −1 󵄨 󵄨𝜓k (𝜉,) 𝑡 ⟩=𝛼̃k (𝜉,) 𝑡 󵄨0 (𝜉,) 𝑡 ⟩ =𝐽𝜓 (𝜉,) 𝑡 󵄨𝜓k ⟩, 󵄨 𝜓 In a supersymmetric model, at 𝑇 ≠ 0, the above invariants 󵄨 󵄨 󵄨 󵄨𝑆̃ (𝜂, 𝑡)⟩ = ̃𝑏† (𝜉,) 𝑡 󵄨̃ (𝜉,) 𝑡 ⟩ =𝐽−1 (𝜂, 𝑡) 󵄨𝑆 ⟩, are different from zero. 󵄨 k k 󵄨0 𝑆 𝑆 󵄨 k (13) 󵄨 󵄨 󵄨 󵄨𝑃̃ (𝜂, 𝑡)⟩ = 𝑐̃† (𝜂, 𝑡) 󵄨̃ (𝜂, 𝑡)⟩ =𝐽−1 (𝜂, 𝑡) 󵄨𝑃 ⟩. 4. Experimental Realization 󵄨 k k 󵄨0 𝑃 𝑃 󵄨 k

𝐵 The presence of the AAIs and then the SUSY violation The energy variances of these states are Δ𝐸 (𝑡) = k couldbetestedbyemployingamixtureofcoldfermion √ 𝜔 |𝑈𝐵(𝜂, 𝑡)||𝑉𝐵(𝜂, 𝑡)| Δ𝐸𝜓(𝑡) = 𝜔 |𝑈𝜓(𝜂, 𝑡)||𝑉𝜓(𝜂, 𝑡)| 2 k k k and k k k k , atoms and diatomic molecules trapped in two dimensional respectively. Then, the corresponding AAIs are given by optical lattices [6], in which the Wess-Zumino model in + 𝑡 󵄨 󵄨 󵄨 󵄨 2 1 dimensions can emerge at low energies. Such a system 𝑆𝑆 (𝑡) =𝑆𝑃 (𝑡) = √ ∫ 𝜔 󵄨𝑈𝐵 (𝜂,󸀠 𝑡 )󵄨 󵄨𝑉𝐵 (𝜂,󸀠 𝑡 )󵄨 𝑑𝑡󸀠 k k 2 2 k 󵄨 k 󵄨 󵄨 k 󵄨 (14) displays Dirac points in the Brillouin zone; therefore, the 0 excitations will have relativistic dispersion relations and for scalar and pseudoscalar bosons and SUSY will be described by the super-Poincare´ algebra, in contrast with other setups proposed in the literature, which 𝑡 󵄨 󵄨 󵄨 󵄨 display a nonrelativistic version of SUSY. The superpartner of 𝑆𝜓 (𝑡) = ∫ 𝜔 󵄨𝑈𝜓 (𝜉,󸀠 𝑡 )󵄨 󵄨𝑉𝜓 (𝜉,󸀠 𝑡 )󵄨 𝑑𝑡󸀠, k 2 k 󵄨 k 󵄨 󵄨 k 󵄨 (15) 0 the fermionic atom is a bosonic diatomic molecule. The setup allows simulating both the massless and the massive models, for the Majorana fermion field. Such invariants signal the with the latter being attained by putting a Bose-Einstein presence of the condensate, since their values are con- condensate of dimolecules nearby, allowing exchange of pairs trolled by the Bogoliubov coefficients and they vanish as the of molecules with the mixture through Josephson tunneling condensates disappear, that is, when 𝑈k and 𝑉k are zero. [6]. Advances in High Energy Physics 5

temperatures of the order of (20–200) nK, atomic excitation 1.0 frequencies characteristic of Bose-Einstein condensates, that 4 −1 5 −1 is, 𝜔 of order of 2 × 10 s –10 s and time intervals of 0.8 order of 𝑡=1/𝜔 [23]. The values of the AAI we found are in principle detectable.

) 0.6

𝜋 ≈

( At temperatures above 200 nK, the condensate (and thus S,P k the AAIs) is expected to disappear. As a final comment, S 0.4 we notice that as happens for any system which presents a 0.2 condensate structure [25], also in the present context, AAIs areunaffectedbythepresenceofnoise. 0.0 In conclusion, we have shown that, in the free Wess- 0.0 5.0 × 10−8 1.0 × 10−7 1.5 × 10−7 2.0 × 10−7 Zumino model, all the phenomena characterized by the Temperature (K) presence of the vacuum condensate generate spontaneous 𝑆𝑆,𝑃 SUSY breaking due to the nonzero vacuum energy. Indeed, Figure 1: Plots of AAI for bosons, k as a function of temperature bosons and fermion condensates both lift the vacuum energy 𝑇, for a time interval 𝑡=1/𝜔 and for sample values of 𝜔∈[2 × 4 −1 5 −1 4 −1 4 −1 byapositiveamount.Suchabreakingcouldbedetected 10 s , 10 s ], 𝜔=2 × 10 s (gray solid line), 𝜔=4 × 10 s 4 −1 (black dotted line), 𝜔=6 × 10 s (red dot dashed line), 𝜔=8 × by measuring the AAIs generated by the condensates in 4 −1 5 −1 10 s (blue dashed line), and 𝜔=10 s (brown solid line). a thermal bath in an optical lattice simulating the Wess- Zumino model.

0.30 Conflict of Interests 0.25 The authors declare that there is no conflict of interests 0.20 regarding the publication of this paper. )

𝜋 0.15 ( 𝜓 k

S Acknowledgments 0.10 Partial financial support from MIUR and INFN is acknowl- 0.05 edged.

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Research Article A Chargeless Complex Vector Matter Field in Supersymmetric Scenario

L. P. Colatto1 andA.L.A.Penna2,3 1 CEFET/RJ UnED Petropolis,´ 25620-003 Petropolis,´ RJ, Brazil 2Instituto de F´ısica, University of Bras´ılia, Bras´ılia, DF, Brazil 3International Center for Condensed Matter Physics, University of Bras´ılia, CP 04513, 70919-970 Bras´ılia, DF, Brazil

Correspondence should be addressed to L. P. Colatto; [email protected]

Received 25 April 2015; Revised 25 June 2015; Accepted 26 July 2015

Academic Editor: Shaaban Khalil

Copyright © 2015 L. P. Colatto and A. L. A. Penna. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.

We construct and study a formulation of a chargeless complex vector matter field in a supersymmetric framework. To this aim we combine two nochiral scalar superfields in order to take the vector component field to build the chargeless complex vector superpartner where the respective field strength transforms into matter fields by a global 𝑈(1) gauge symmetry. For the aim of dealing with consistent terms without breaking the global 𝑈(1) symmetry we imposes a choice to the complex combination revealing a kind of symmetry between the choices and eliminates the extra degrees of freedom which is consistent with the supersymmetry. As the usual case the mass supersymmetric sector contributes as a complement to dynamics of the model. We obtain the equations of motion of the Proca’s type field for the chiral spinor fields and for the scalar field on the mass-shell which showthesamemassasexpected.Thisworkestablishesthefirststepstoextendtheanalysisofchargedmassivevectorfieldina supersymmetric scenario.

1. Introduction Due to the supersymmetry which plays a fundamental role on strings theory fitting together quantum theories of Matter field dynamics was firstly established by Dirac ina the gravitational interactions, electroweak and strong forces, consistent relativistic framework. He has studied the free the studies on supersymmetric theories are of great interest to electron dynamics where its interaction yields the first steps high energy physicists such as applications in particle physics on QED, which was further developed by Feynmann and [4] and supersymmetry breaking [5] and in the treatment of others [1, 2]. These studies were very important for the classicalsupergravity[6] with inclusion of topological Chern- formulation and the understanding of QFT, standard model, Simons terms [7]. Furthermore, supersymmetry is required and also the strings theory. Indeed it has been the basement to understand the thermodynamics of quantum gravity [8], to of all theoretical analysis of any dynamics which is concerned build new scenarios for the electroweak baryogenesis in high with integer or half-integer spin particles. In fundamental energies [9, 10] as well as establishing superstring theories quantum theory we have classified into two types: boson and correctly [11–16]. Supersymmetry deals with graded Lie fermions, respectively. Fermions usually are the constituent algebra in the unique reliable algebra extension which holds of the matter and bosons are the interaction particle [3]. Nev- to be consistent with the S-matrix in relativistic quantum ertheless if we are treating to the weak force we have to deal field theory [17–20]. Recalling that this special symmetry with charged (or not) massive vector (boson) fields which are correlates fermionic and bosonic fields, called superpartners, the intermediate between the protons and neutrons. So we which puts them together in a superfield formulation. It could interpret as charged vector matter fields. remarks the important role played by the study of matter-like 2 Advances in High Energy Physics vector fields to construct appropriated supersymmetric mod- the Lorentz group. In order to build more ahead a complex els [21, 22]. Moreover supersymmetric models with chiral extension we introduce two chargeless real scalar superfields superfields and global gauge invariance involving matter doubling the number of degrees of freedom, which are fields are elegantly constructed [16, 20, 22]. Thus quarks, written as leptons, and vector bosons which participate in usual gauge 𝑎̇ Φ(𝑥𝜇,𝜃 , 𝜃 )=𝐶(𝑥) +𝜃𝑎𝜑 (𝑥) + 𝜃 𝜑 (𝑥) theories, as electroweak theory and chromodynamics, in a 𝑎 𝑎̇ 𝑎 𝑎̇ supersymmetric extension coexist along their superpartners: +𝜃2 [𝑚 (𝑥) +𝑖𝑛(𝑥)] squarks, sleptons, and the fermionic partner of the vector bosonswhichareatypeofvectormatterfield.Forinstance, 2 + 𝜃 [𝑚 (𝑥) −𝑖𝑛(𝑥)] the supersymmetric version of quantum electrodynamics involves a vector supermultiplet whose contents are a mass- 2 𝑎̇ 𝑖 𝜇 𝑎 less photon and its spin-1/2 superpartner, the photino [16, +𝜃 𝜃 [𝜆𝑎̇ (𝑥) − 𝜎 𝑎𝑎̇𝜕𝜇𝜑 (𝑥)] 20]. 2 Indeed theoretical formulation of supersymmetric gauge 2 𝑎 𝑖 𝜇 𝑎̇ + 𝜃 𝜃 [𝜆𝑎 (𝑥) − 𝜎 𝑎𝑎̇𝜕𝜇𝜑 ] vectorfieldhasbeenlargelystudied[16, 22–24]. It was shown 2 that gauge vector field component emerges from nonchiral 𝑎 𝜇 𝑎̇ scalar superfields when one uses some suitable constraint +𝜃 𝜎 𝑎𝑎̇𝜃 𝑋𝜇 (𝑥) (Wess-Zumino) to remove exceeding nonphysical compo- 2 nents fields [20, 22]. Nevertheless, there is a lack of studies +𝜃2𝜃 [𝐷 (𝑥) − 1◻𝐶 (𝑥)], on models that describe supersymmetric vector matter fields. 4 (1) Therefore,oneoftheaimsofthisworkistheattemptto 𝑎̇ Λ(𝑥𝜇,𝜃 , 𝜃 )=𝐴(𝑥) +𝜃𝑎𝜒 (𝑥) + 𝜃 𝜒 (𝑥) address this lack in order to further study the interactions 𝑎 𝑎̇ 𝑎 𝑎̇ which can be involved. To this purpose we construct a +𝜃2 [𝜌 (𝑥) +𝑖𝜏(𝑥)] formulationinwhichthevectorfield𝐵𝜇 is complex and massive, but with no local charge, which we simply called 2 “chargeless.” Indeed as a matterfieldittransformsbythe + 𝜃 [𝜌 (𝑥) −𝑖𝜏(𝑥)] global 𝑈(1) group [3];thatis,wewouldemphasizethat𝐵𝜇 2 𝑎̇ 𝑖 𝜇 𝑎 is not a local gauge field but a free Proca-type one. Such +𝜃 𝜃 [𝜁𝑎̇ (𝑥) − 𝜎 𝑎𝑎̇𝜕𝜇𝜒 (𝑥)] models are interesting in order to contribute to the under- 2 standing of the supersymmetric model of electroweak theory 2 𝑎 𝑖 𝜇 𝑎̇ + 𝜃 𝜃 [𝜁𝑎 (𝑥) − 𝜎 𝑎𝑎̇𝜕𝜇𝜒 (𝑥)] though it contains charged vector particles. Furthermore it 2 can improve our knowledge of the form of nuclear atomic 𝑎 𝜇 𝑎̇ structure and its interaction at high energy. Further, taking +𝜃 𝜎 𝑎𝑎̇𝜃 𝑌𝜇 (𝑥) this model in a fundamental scenario of the strings theory 2 [25], complex vector fields with matter symmetry are relevant +𝜃2𝜃 [𝑆 (𝑥) − 1◻𝐴 (𝑥)], to the vacuum polarization theory that can be connected 4 to models which deal with Lorentz symmetry violation in where the superfields Φ and Λ are particular constructions of high energy physics [26–46]. Another aim of this work is to matter vector supermultiplet which include real vector fields obtain the supermultiplet that will accommodate a charged 𝑋𝜇(𝑥) and 𝑌𝜇(𝑥) with helicity ±1; the fields 𝜑𝑎(𝑥), 𝜆𝑎(𝑥), matter vector field and its supersymmetric partners and also 𝜒𝑎(𝑥),and𝜁𝑎(𝑥) are two-component Weyl fermions with to get the most appropriate supersymmetric action for this helicity ±1/2; and the fields 𝐶(𝑥), 𝐷(𝑥), 𝐴(𝑥), 𝑆(𝑥), 𝑚(𝑥), fieldwhichwillbethesubjectofaforthcomingwork.To 𝑛(𝑥), 𝜌(𝑥),and𝜏(𝑥) are real scalar fields with spin-0. It is this purpose we are going to formulate a supersymmetric easy to verify that to both superfields the number of bosonic Lagrangian starting from chargeless nonchiral superfield and fermionic degrees of freedom is the same. We stress that which contains the vector (matter) field. The present paper we only have applied the reality condition on the superfields is outlined as follows: in Section 2,wepresentamodelthat which does not spoil the matter structure of these multiplets. acommodates two real vector matter fields; in Section 3 we Therefore the dynamics to chargeless supersymmetric vector compose the previous model in a complex form and we fields can be obtained through suitable field-strengths which present the Dirac superspinor field Ψ;inSection 4 we present accommodate the real superfields Φ and Λ. a general conclusion. In order to construct the supersymmetric field-strengths for the real superfields Φ and Λ, which we call chargeless supersymmetric field-strengths, we are going to apply 2. Two Chargeless Vector Matter supersymmetric covariant derivatives on the above scalar Superfields Model superfields which result in chiral superfields, in such way that We are going to present a chargeless (real) formulation for 1 𝜇 𝑊𝑎 =− 𝐷 𝐷𝐷𝑎Φ(𝑥 ,𝜃𝑎, 𝜃𝑎̇), vector matter field. To this aim we start from a general 4 (2) nonchiral scalar superfield which includes in the matter 1 𝜇 𝑊𝑎̇ =− 𝐷𝐷𝐷𝑎̇Φ(𝑥 ,𝜃𝑎, 𝜃𝑎̇), multiplet a vector field as irreducible representation of 4 Advances in High Energy Physics 3 and, by similarity for Λ,wehavethat thatcorrespondstotheunderlinedfieldtheory,wecanalso consider the supersymmetric mass term, given by 1 𝜇 Ω𝑎 =− 𝐷 𝐷𝐷𝑎Λ(𝑥 ,𝜃𝑎, 𝜃𝑎̇), 4 2 2 2 2 4 2 4 𝑆𝑚 = ∫ 𝑑 𝑥𝑑 𝜃𝛼 [Φ +Λ ]=∫ 𝑑 𝑥𝛼 [𝐶 (𝑥) 𝐷 (𝑥) (3) 1 𝜇 Ω𝑎̇ =− 𝐷𝐷𝐷𝑎̇Λ(𝑥 ,𝜃𝑎, 𝜃𝑎̇), 1 𝑎 𝑎̇ 4 − 𝐶 (𝑥) ◻𝐶 (𝑥) +𝜑𝑎 (𝑥) 𝜆 (𝑥) + 𝜑𝑎̇ (𝑥) 𝜆 (𝑥) 4 𝑊 Ω the 𝑎 and 𝑎 are chiral spinor superfields. 𝑎̇ 𝜇 𝑏 2 2 1 𝜇 We can redefine the superfields in the chiral superspace −𝑖𝜑 𝜎 𝑎𝑏̇ 𝜕𝜇𝜑 +𝑀(𝑥) +𝑁(𝑥) + 𝑋 𝑋𝜇 𝜇 𝜇 𝜇 4 coordinates, namely, Φ(𝑦 ,𝜃𝑎), Φ(𝑧 , 𝜃𝑎̇), Λ(𝑦 ,𝜃𝑎),and (8) Λ(𝑧𝜇, 𝜃 ) 𝑦𝜇 =𝑥𝜇 + 𝑖𝜃𝜎𝜇𝜃 𝑧𝜇 =𝑥𝜇 − 𝑖𝜃𝜎𝜇𝜃. 1 𝜇 1 𝑎̇ ,suchthat and + 𝑌 𝑌𝜇 +𝐴(𝑥) 𝑆 (𝑥) − 𝐴 (𝑥) ◻𝐴 (𝑥) Hence the supersymmetric covariant derivatives are defined 4 4 as 𝑎 𝑎̇ 𝑎̇ 𝜇 𝑏 +𝜒𝑎 (𝑥) 𝜁 (𝑥) + 𝜒𝑎̇ (𝑥) 𝜁 (𝑥) −𝑖𝜒 𝜎 𝑎𝑏̇ 𝜕𝜇𝜒 𝜕 𝑎̇ 𝜕 𝐷 = + 𝑖𝜎𝜇 𝜃 , 𝑎 2 𝑎𝑎̇ 𝜇 2 2 𝜕𝜃𝑎 𝜕𝑦 +𝜌(𝑥) +𝜏(𝑥) ], (4) 𝜕 𝜕 2 𝐷 =− − 𝑖𝜃𝑎𝜎𝜇 . where 𝛼 is the mass parameter. As usual, the “mass” part 𝑎̇ 2 𝑎𝑎̇ 𝜕𝑧𝜇 𝜕𝜃𝑎̇ of the action presents kinetic terms, beyond the usual mass terms, which were eliminated by spinor chirality property of According to these definitions we can compute the field- superfields 𝑊𝑎 and Ω𝑎.Furthermore,wecouldinferthatthe 𝑊 Ω strengths 𝑎 and 𝑎,andwehave mass term in action (8) arises as a dynamical complement to the supersymmetric vector matter fields. Indeed supersym- 𝑊 (𝑦, 𝜃) =𝜆 (𝑦) + 𝜃 𝐷(𝑦)+(𝜎𝜇]𝜃) 𝑋 (𝑦) 𝑎 𝑎 2 𝑎 𝑎 𝜇] metric matter-like fields are formulated with chiral spinor ̇ superfields. 2 𝜇 𝑏 −𝑖𝜃 𝜎 𝑎𝑏̇𝜕𝜇𝜆 (𝑦) , 𝜇] (5) 3. The Chargeless Complex Vector Ω𝑎 (𝑦, 𝜃)𝑎 =𝜁 (𝑦) + 2𝜃𝑎𝑆(𝑦) + (𝜎 𝜃) 𝑌𝜇] (𝑦) 𝑎 Matter Superfield Model ̇ 2 𝜇 𝑏 −𝑖𝜃 𝜎 𝑎𝑏̇𝜕𝜇𝜁 (𝑦) , We know that the supersymmetric action for two free vector matter fields might be built through nonchiral scalar super- andinasimilarwaywecancomputethefield-strengths fields [22]. Moreover, we can see that the degrees of freedom 𝑊𝑎̇ and Ω𝑎̇. So we are in conditions to construct the of this model are compatible with the dynamical free fields C supersymmetric model in terms of the superfields 𝑊𝑎 and in complex space .Inthissectionouraimistoderivethe Ω𝑎 wherechargelessvectormatterfieldispresent.Thekinetic appropriated complex superfield to describe supersymmetric part can be written as complex vector fields. To this aim we need to strongly define two complex nonchiral scalar superfields, defined as 4 2 𝑎 𝑎 𝑎̇ 𝑆kin = ∫ 𝑑 𝑥{𝑑 𝜃(𝑊𝑎𝑊 +Ω𝑎Ω ) 𝑎 2 Σ(𝑥𝜇,𝜃𝑎, 𝜃𝑎̇)=𝑘(𝑥) +𝜃 𝜉𝑎 (𝑥) + 𝜃 𝐶𝑎̇ (𝑥) +𝜃 𝑙 (𝑥) (6) 2 𝑎̇ 𝑎̇ 2 +𝑑 𝜃(𝑊𝑎̇𝑊 + Ω𝑎̇Ω )} . + 𝜃 𝑓 (𝑥)

2 𝑎̇ 𝑖 𝜇 𝑎 We have adopted the usual conventions for the spinor alge- +𝜃 𝜃 [𝐺𝑎̇ (𝑥) − 𝜎 𝑎𝑎̇𝜕𝜇𝜉 (𝑥)] bra, for the superspace parametrization, and the translation 2 (9) invariance of the integral in the chiral coordinates [16, 2 𝑎 𝑖 𝜇 𝑎̇ + 𝜃 𝜃 [𝑅𝑎 (𝑥) − 𝜎 𝑎𝑎̇𝜕𝜇𝐶 ] 22].Thenweobtainthatexpression(6) has the following 2 component expansion: 𝑎 𝜇 𝑎̇ +𝜃 𝜎 𝑎𝑎̇𝜃 𝐵𝜇 (𝑥) 𝑆 = ∫ 𝑑4𝑥{−𝑋 (𝑥) 𝑋𝜇] (𝑥) −𝑌 (𝑥) 𝑌𝜇] (𝑥) kin 𝜇] 𝜇] 2 +𝜃2𝜃 [𝑑 (𝑥) − 1◻𝑘 (𝑥)]. ̇ ̇ 4 𝑎 𝜇 𝑏 𝑎 𝜇 𝑏(𝑥) − 4𝑖𝜆 (𝑥) 𝜎 𝑎𝑏̇𝜕𝜇𝜆 (𝑥) − 4𝑖𝜁 (𝑥) 𝜎 𝑎𝑏̇𝜕𝜇𝜁 (7) 𝑎 𝑎̇ 𝐾(𝑥𝜇,𝜃𝑎, 𝜃𝑎̇)=𝑎(𝑥) +𝜃 𝑇𝑎 (𝑥) + 𝜃 𝐻𝑎̇ (𝑥) 2 2 + 8𝐷 (𝑥) + 8𝑆 (𝑥)}. 2 +𝜃2𝑗 (𝑥) + 𝜃 𝐸 (𝑥)

Action (7) describes the kinetic part of supersymmetric 2 𝑎̇ 𝑖 𝜇 𝑎 +𝜃 𝜃 [𝑄𝑎̇ (𝑥) − 𝜎 𝑎𝑎̇𝜕𝜇𝑇 (𝑥)] chargeless vector field. However, to write the full action 2 4 Advances in High Energy Physics

2 𝑎 𝑖 𝜇 𝑎̇ 𝑓 (𝑥) =𝑚(𝑥) +𝑛(𝑥) −𝑖(𝜌(𝑥) +𝜏(𝑥)), + 𝜃 𝜃 [𝑈𝑎 (𝑥) − 𝜎 𝑎𝑎̇𝜕𝜇𝐻 (𝑥)] 2 𝑅𝑎 (𝑥) =𝜆𝑎 (𝑥) +𝑖𝜁𝑎 (𝑥) , 𝑎 𝜇 𝑎̇ +𝜃 𝜎 𝑎𝑎̇𝜃 𝑍𝜇 (𝑥) 𝐵𝜇 (𝑥) =𝑋𝜇 (𝑥) +𝑖𝑌𝜇 (𝑥) , 2 +𝜃2𝜃 [V (𝑥) − 1◻𝑎 (𝑥)], 𝑑 (𝑥) =𝐷(𝑥) +𝑖𝑆(𝑥) . 4 (10) (13) To describe the dynamics of the supersymmetric complex andinasimilarwaywecandefinethecomplexconjugated vectorfieldswithmattersymmetryweneedtoconstructan superfields. appropriated complex supersymmetric field-strength model Superfields (9) and (10) present multiplets with complex in order to accommodate superfields Σ and 𝐾.Thiscanbe vector fields 𝐵𝜇(𝑥) and 𝑍𝜇(𝑥) and spin-1; the 𝜉𝑎(𝑥), 𝐶𝑎̇(𝑥), reached starting from the following definitions: 𝑅𝑎(𝑥), 𝐺𝑎̇(𝑥), 𝑇𝑎(𝑥), 𝐻𝑎̇(𝑥), 𝑈𝑎(𝑥),and𝑄𝑎̇(𝑥) are Weyl fermion fields with spin-1/2; and the 𝑘(𝑥), 𝑑(𝑥), 𝑎(𝑥),and 1 Υ𝑎 =− 𝐷 𝐷𝐷𝑎Σ(𝑥𝜇,𝜃𝑎, 𝜃𝑎̇), V(𝑥) are complex scalar fields with spin-0. So the rule that 4 implies an invariant mechanism is Υ =−1𝐷𝐷𝐷 Σ† (𝑥 ,𝜃 , 𝜃 ), † 𝑎̇ 𝑎̇ 𝜇 𝑎 𝑎̇ 𝐾=𝑖Σ. (11) 4 (14) 1 We observe that transformation rule (11) guarantees writing Γ𝑎 =− 𝐷 𝐷𝐷𝑎𝐾(𝑥𝜇,𝜃𝑎, 𝜃𝑎̇), 4 a consistent kinetic term for the complex vector field without breaking the global 𝑈(1) gauge symmetry. Another advantage 1 † Γ𝑎̇ =− 𝐷𝐷𝐷𝑎̇𝐾 (𝑥𝜇,𝜃𝑎, 𝜃𝑎̇), that came to light is that transformations (11) eliminate 4 the exceeding fields which does not contribute for the Υ Γ supersymmetric action, which allows bosons and fermions where 𝑎 and 𝑎 are charged spinor superfields. As a conse- to have the same physical degrees of freedom. Indeed the quence of the complex extension procedure we must relate Ω 𝑊 constraint relation to the superfields implies the relations of chargeless spinor superfields 𝑎 and 𝑎 with the complex the component fields as follows: definitions (14) which, in the simplest way, is

𝑎 (𝑥) =𝑖𝑘∗ (𝑥) , Υ𝑎 =𝑊𝑎 +𝑖Ω𝑎,

𝑇𝑎 (𝑥) =𝑖𝐶𝑎 (𝑥) , Υ𝑎̇ = 𝑊𝑎̇ −𝑖Ω𝑎̇, ∗ (15) V (𝑥) =𝑖𝑑 (𝑥) , Γ𝑎 =Ω𝑎 +𝑖𝑊𝑎, 𝐻 (𝑥) =𝑖𝜉 (𝑥) , 𝑎 𝑎 Γ𝑎̇ = Ω𝑎̇ −𝑖𝑊𝑎̇, 𝑗 (𝑥) =𝑖𝑓∗ (𝑥) , 𝑎 𝑎 (12) and by assuming the spinor identities 𝑊𝑎Ω =Ω𝑎𝑊 and 𝑎̇ 𝑎̇ 𝑊 Ω = Ω 𝑊 𝑈𝑎 (𝑥) =𝑖𝐺𝑎 (𝑥) , 𝑎̇ 𝑎̇ we can find the kinetic supersymmetric Lagrangian for the complex vector fields: 𝐸 (𝑥) =𝑖𝑙∗ (𝑥) , 𝑎̇ L =𝑖(Υ Γ −Υ Γ𝑎) 𝑄 (𝑥) =𝑖𝑅 (𝑥) , 𝑘 𝑎̇ 𝑎 𝑎 𝑎 (16) ∗ 𝑎 𝑎̇ 𝑎 𝑎̇ 𝑍𝜇 (𝑥) =𝑖𝐵𝜇 (𝑥) . =𝑊𝑎𝑊 + 𝑊𝑎̇𝑊 +Ω𝑎Ω + Ω𝑎̇Ω .

So, we can adjust the complex extension of the chargeless We can observe that the left-hand side of the latter superfields Φ and Λ by assuming the equation Σ=Φ+𝑖Λ, equation is the complex extension of chargeless Lagrangian wherewefindthefollowingrelationoffields: (7) that was written in terms of charged spinor superfields. Bearing this in mind, we can then redefine the kinetic 𝑘 (𝑥) =𝐶(𝑥) +𝑖𝐴(𝑥) , Lagrangian (16) simply by combining the charged spinor superfields Υ𝑎 and Γ𝑎 as a “Dirac superspinor” Ψ,suchthat 𝜉𝑎 (𝑥) =𝜑𝑎 (𝑥) +𝑖𝜒𝑎 (𝑥) , Υ 𝜇 𝑎 𝐶 (𝑥) = 𝜑 (𝑥) +𝑖𝜒 (𝑥) , Ψ(𝑥 ,𝜃𝑎, 𝜃𝑎̇)=( 𝑎̇), (17) 𝑎 𝑎 𝑎 Γ 𝑙 (𝑥) =𝑚(𝑥) +𝑛(𝑥) +𝑖(𝜌(𝑥) +𝜏(𝑥)), and also we assume Ψ to be the adjoint Dirac superspinor 𝐺𝑎 (𝑥) = 𝜆𝑎 (𝑥) +𝑖𝜁𝑎 (𝑥) , representation. Advances in High Energy Physics 5

† 0 𝑎 In this case we have that Ψ=Ψ𝛾 =(ΓΥ𝑎̇),andso However, it does not correspond to the conventional kinetic the supersymmetric action from the kinetic Lagrangian (16) term for the matter vector field, and action (18) shows more is now given by degrees of freedom than necessary. In order to get rid of such fields we must assume the rule of transformation (11) which 4 2 2 𝑆𝑘 = ∫ 𝑑 𝑥𝑑 𝜃𝑑 𝜃(𝑖ΨΨ) is a constraint of half of the degrees and consequently action (18) reaches the correct number of component fields. (18) 4 2 2 𝑎̇ 𝑎 Applying condition (11) in action (22) we can reach the =𝑖∫ 𝑑 𝑥𝑑 𝜃𝑑 𝜃(Υ𝑎̇Γ −Υ𝑎Γ ). usual dynamical matter field strength term, or

𝑖 𝜇] 𝑖 ∗ ∗𝜇] ∗ 𝜇] We can note that the product of Dirac superspinors ΨΨ obeys 𝐹𝜇]𝑍 − 𝐹 𝜇]𝑍 󳨐⇒ − 𝐹 𝜇]𝐹 , (23) matter symmetry and it presents an interesting analogy to 2 2 † 𝜇] charged scalar superfield product 𝑆 𝑆.Inthissenseweverify andsothe𝑍 tensor field is reabsorbed in this action. that Ψ and Ψ represent two chiral supersymmetric extensions Likewiseandwithoutlossofgenerality,wecouldhavechosen † 𝜇] for the matter vector field which can be transformed under the inverse relation Σ=𝑖𝐾 what implies reabsorbing the 𝐹 𝑈(1) global gauge group in the following way: tensor field. Then by using the whole relation (12) in action

󸀠 −2𝑖𝑞𝛽 (18) we find that the complex supersymmetric model for the Ψ =𝑒 Ψ, matter vector field can be written as (19) 󸀠 † Ψ = Ψ𝑒2𝑖𝑞𝛽 , 4 ∗ 𝜇] 𝑆𝑘 = ∫ 𝑑 𝑥{−𝐹 𝜇] (𝑥) 𝐹 (𝑥)

𝛽 𝑈( ) 𝑞 ̇ ̇ where is a global 1 gauge parameter and isthechargeof 𝑎 𝜇 𝑏 𝑎 𝜇 𝑏 the global symmetry. We can emphasize that the expressions − 2𝑖𝑅 (𝑥) 𝜎 𝑎𝑏̇𝜕𝜇𝑅 (𝑥) − 2𝑖𝐺 (𝑥) 𝜎 𝑎𝑏̇𝜕𝜇𝐺 (𝑥) (24) (19) representthateachofthecomponentsofmultiplet(17) ∗ has the same symmetry. So action (18) is then invariant + 8𝑑 (𝑥) 𝑑 (𝑥)}, under transformations (19). In order to obtain the component Lagrangian we can expand the product ΨΨ by considering ∗ 𝜇] where expression −𝐹 𝜇](𝑥)𝐹 (𝑥) represents the usual kinetic that term of the vector matter field while the terms represent 𝑎 𝑎 𝜇] with the components 𝑅 and 𝐺 the fermionic sector, and Υ𝑎 (𝑦, 𝜃)𝑎 =𝑅 (𝑦) + 2𝜃𝑎𝑑(𝑦)+(𝜎 𝜃)𝑎 𝐹𝜇] (𝑦) the last term corresponds to the auxiliary field 𝑑 term. To 𝑏̇ −𝑖𝜃2𝜎𝜇 𝜕 𝐺 (𝑦) , completeness we are going to introduce the massive action 𝑎𝑏̇ 𝜇 term in the model. Observing the symmetries of nonchiral (20) 𝜇] fields Σ and 𝐾 the massive supersymmetric term can be Γ𝑎 (𝑦, 𝜃)𝑎 =𝑈 (𝑦) + 2𝜃𝑎V (𝑦) + (𝜎 𝜃) 𝑍𝜇] (𝑦) 𝑎 suitable defined as ̇ 2 𝜇 𝑏 𝛼2 −𝑖𝜃 𝜎 𝑎𝑏̇𝜕𝜇𝑄 (𝑦) , 4 2 2 † † 𝑆𝑚 = ∫ 𝑑 𝑥𝑑 𝜃𝑑 𝜃[Σ Σ+𝐾 𝐾] , (25) 2 and similarly for Υ𝑎̇ and Γ𝑎̇. 𝛼2 Σ We note the presence of the complex matter field- where is a mass parameter. From nonchiral superfields and 𝐾 wecanobtainthemassivevectormatterfieldterm strengths; namely, ∗ 𝜇 𝐵 𝜇𝐵 as well as their supersymmetric partners. In order to 𝐹𝜇] =𝜕𝜇𝐵] −𝜕]𝐵𝜇; perform it we are going to compute action (25) by employing † † † (21) condition 𝐾=𝑖Σ where one has that (1/2)Σ Σ+(1/2)𝐾 𝐾= † 𝑍𝜇] =𝜕𝜇𝑍] −𝜕]𝑍𝜇 Σ Σ, and by applying definition (10) the full supersymmetric matter vector field model can be then obtained from Dirac hence action (18) can be expanded and we obtain superspinor field Ψ associated to the nonchiral scalar fields Σ and 𝐾 in the following form: 4 𝑖 𝜇] 𝑖 ∗ ∗𝜇] 𝑆𝑘 = ∫ 𝑑 𝑥{ 𝐹𝜇] (𝑥) 𝑍 (𝑥) − 𝐹 𝜇] (𝑥) 𝑍 (𝑥) 2 2 4 2 2 2 † 𝑆=𝑆𝑘 +𝑆𝑚 = ∫ 𝑑 𝑥𝑑 𝜃𝑑 𝜃(𝑖ΨΨ + 𝛼 Σ Σ) , (26) ̇ ̇ 𝑎 𝜇 𝑏 𝑎 𝜇 𝑏 −𝑅 (𝑥) 𝜎 𝑎𝑏̇𝜕𝜇𝑄 (𝑥) −𝑈 (𝑥) 𝜎 𝑎𝑏̇𝜕𝜇𝐺 (𝑥) (22) wherethemasspartofactioncanbeobtainedincomponent 𝑎̇ 𝜇 𝑏 𝑎̇ 𝜇 𝑏 + 𝑅 (𝑥) 𝜎 𝑎𝑏̇ 𝜕𝜇𝑄 (𝑥) + 𝑈 (𝑥) 𝜎 𝑎𝑏̇ 𝜕𝜇𝐺 (𝑥) fields as 2 4 ∗ ∗ ∗ ∗ 𝑆 =𝛼 ∫ 𝑑 𝑥(𝑑 (𝑥) 𝑘 (𝑥) +𝑑(𝑥) 𝑘 (𝑥) + 4𝑖V (𝑥) 𝑑 (𝑥) − 4𝑖V (𝑥) 𝑑 (𝑥)}. 𝑚

1 ∗ 1 ∗ 𝑎 In this format we can recognize the dynamical term − 𝑘 (𝑥) ◻𝑘 (𝑥) − 𝑘 (𝑥) ◻𝑘 (𝑥) +𝜉𝑎 (𝑥) 𝐺 (𝑥) 𝜇] 4 4 that describes the matter vector field as (𝑖/2)𝐹𝜇]𝑍 − ∗ ∗𝜇] 𝑎̇ 𝑎 𝑎̇ (𝑖/2)𝐹 𝜇]𝑍 .Itinvolvesboth𝐹𝜇] and 𝑍𝜇] matter tensors. + 𝜉𝑎̇ (𝑥) 𝐺 (𝑥) +𝐶𝑎 (𝑥) 𝑅 (𝑥) + 𝐶𝑎̇ (𝑥) 𝑅 (𝑥) 6 Advances in High Energy Physics

𝑎̇ 𝜇 𝑏 𝑎̇ 𝜇 𝑏 𝑑(𝑥) 𝑓(𝑥) 𝑙(𝑥) − 2𝑖𝜉 (𝑥) 𝜎 𝑎𝑏̇ 𝜕𝜇𝜉 (𝑥) − 2𝑖𝐶 (𝑥) 𝜎 𝑎𝑏̇ 𝜕𝜇𝐶 (𝑥) so the complex scalar fields , ,and have no dynamics and arise as auxiliary fields. Thus assuming the +𝐵∗ 𝐵𝜇 +𝑓∗ (𝑥) 𝑓 (𝑥) +𝑙∗ (𝑥) 𝑙 (𝑥)). action (26) to be the sum of the redefined actions (29) and 𝜇 (30) and rearranging (we adopt the Weyl representation to (27) the gamma matrices) the (Dirac) spinor fields Θ and Π,The sum of the action (29) and the action (30) results in an off- 𝜇 Here we observe the mass term to 𝐵 , 𝑓(𝑥),and𝑙(𝑥) fields. shell action 𝑆os writteninthefollowingform As in the usual supersymmetric models we note that mass action (27) also contributes to kinetic structure, namely, 4 1 ∗ 𝜇] 1 ∗ 𝜇 𝑆os = ∫ 𝑑 𝑥{− 𝐹 𝜇] (𝑥) 𝐹 (𝑥) + 𝜕𝜇𝑘 (𝑥) 𝜕 𝑘 (𝑥) ∗ 𝑎̇ 𝜇 𝑏 2 4 with the terms −(1/4)𝑘 (𝑥)◻𝑘(𝑥), −2𝑖𝜉 𝜎 𝑎𝑏̇ 𝜕𝜇𝜉 ,and 𝑎̇ 𝜇 𝑏 −𝑖Θ (𝑥) 𝛾𝜇𝜕 Θ (𝑥) −𝑖Π (𝑥) 𝛾𝜇𝜕 Π (𝑥) −2𝑖𝐶 𝜎 𝑎𝑏̇ 𝜕𝜇𝐶 .Theactionalsoshowsmixingmassscalar 𝜇 𝜇 ∗ ∗ and fermionic terms, namely, 𝑑 (𝑥)𝑘(𝑥), 𝑑(𝑥)𝑘 (𝑥), +𝛼Θ (𝑥) 𝛾5Θ (𝑥) +𝛼Π (𝑥) 𝛾5Π (𝑥) + 𝑑∗ (𝑥) 𝑑 (𝑥) 𝑎 𝑎̇ 𝑎 𝑎̇ 4 (31) 𝜉𝑎(𝑥)𝐺 (𝑥), 𝜉𝑎̇(𝑥)𝐺 (𝑥), 𝐶𝑎(𝑥)𝑅 (𝑥),and𝐶𝑎̇(𝑥)𝑅 (𝑥).By verifying the presence of extra kinetic terms in (27) what 1 ∗ 1 ∗ 2 ∗ 𝜇 + 𝑓 (𝑥) 𝑓 (𝑥) + 𝑙 (𝑥) 𝑙 (𝑥) +𝛼 𝐵 𝜇𝐵 can suggest that when we particularly treat supersymmetric 2 2 matter vector fields the mass action contributes with a “dynamic complement” to the kinetic action (24). +𝛼𝑑∗ (𝑥) 𝑘 (𝑥) +𝛼𝑑(𝑥) 𝑘∗ (𝑥)}, Furthermore, we remark that mass action (27) is important to match the number of bosonic and fermionic degrees of where we denote the Dirac spinors of mass 𝛼 as freedom of the supersymmetric matter vector action (26) for theconsistencyofthemodel.Wecanredefinethecomponent 𝜉 fields absorbing the mass parameter as follows: 𝑎 Θ (𝑥) =( 𝑎̇), 𝐺 𝑘 (𝑥) 󳨀→ 1 𝑘 (𝑥) , (32) 𝛼 𝐶𝑎 Π (𝑥) =( 𝑎̇). 𝜉 (𝑥) 󳨀→ 1 𝜉 (𝑥) , 𝑅 𝑎 𝛼 𝑎 1 For action (31) we have obtained chiral spinor mass terms 𝐶 (𝑥) 󳨀→ 𝐶 (𝑥) , 5 5 𝑎 𝛼 𝑎 (28) given by 𝛼Θ(𝑥)𝛾 Θ(𝑥) and 𝛼Π(𝑥)𝛾 Π(𝑥).Fromtheaction (31) we can obtain the equation of motion for the fields: 𝑓 (𝑥) 󳨀→ 1 𝑓 (𝑥) , 𝛼 𝜇] 2 ] 𝜕𝜇𝐹 (𝑥) +𝛼 𝐵 (𝑥) = 0, 𝑙 (𝑥) 󳨀→ 1 𝑙 (𝑥) . 𝜇 5 𝛼 𝛾 𝜕𝜇Θ (𝑥) −𝛼𝛾 Θ (𝑥) = 0, (33) 𝜇 5 Sotheactioncanberewrittenas 𝛾 𝜕𝜇Π (𝑥) −𝛼𝛾 Π (𝑥) = 0,

1 4 ∗ 𝜇] 2 𝑆ek = ∫ 𝑑 𝑥(−𝐹 𝜇] (𝑥) 𝐹 (𝑥) ◻𝑘 (𝑥) +𝛼 𝑘 (𝑥) = 0, 2 𝑏̇ 𝑏̇ with 𝑓(𝑥) = 𝑙(𝑥) = 0. Taking off-shell action (31) we − 𝑖𝑅𝑎 (𝑥) 𝜎𝜇 𝜕 𝑅 (𝑥) − 𝑖𝐺𝑎 (𝑥) 𝜎𝜇 𝜕 𝐺 (𝑥) 2 𝑎𝑏̇ 𝜇 2 𝑎𝑏̇ 𝜇 note that it has 16 bosonic degrees of freedom concerning the matter fields 𝐵𝜇(𝑥), 𝑘(𝑥), 𝑑(𝑥), 𝑓(𝑥),and𝑙(𝑥) and their 1 ∗ 1 ∗ − 𝑘 (𝑥) ◻𝑘 (𝑥) − 𝑘 (𝑥) ◻𝑘 (𝑥) (29) complex conjugated ones, as well as 16 fermionic degrees 4 4 of freedom for the Dirac spinor fields Θ(𝑥) and Π(𝑥) and 𝑎̇ 𝜇 𝑏 𝑎̇ 𝜇 𝑏 their conjugated complex ones, which is consistent with the +−2𝑖𝜉 (𝑥) 𝜎 𝑎𝑏̇ 𝜕𝜇𝜉 (𝑥) − 2𝑖𝐶 (𝑥) 𝜎 𝑎𝑏̇ 𝜕𝜇𝐶 (𝑥) supersymmetry. From equations of motion (33) we note that there are three auxiliary complex scalar fields 𝑑(𝑥), 𝑓(𝑥), + 𝑑∗ (𝑥) 𝑑 (𝑥) +𝑓∗ (𝑥) 𝑓 (𝑥) +𝑙∗ (𝑥) 𝑙 (𝑥)); 8 and 𝑙(𝑥) and massive dynamical complex scalar field 𝑘(𝑥). Moreover, as expected we have obtained a matter Proca-type analogously the mass action is now given by equation for field 𝐵𝜇. In this context, it is interesting to note that from the on-shell action we can easily extract from action 4 2 ∗ 𝜇 ∗ 𝑆em = ∫ 𝑑 𝑥{𝛼 𝐵 𝜇𝐵 +𝛼𝑑 (𝑥) 𝑘 (𝑥) (31) the supersymmetric generalization of matter vector field; it is only possible if we include two dynamical Dirac chiral 𝑎̇ spinor fields Θ(𝑥) and Π(𝑥) along a massive scalar field 𝑘(𝑥). +𝛼𝑑(𝑥) 𝑘∗ (𝑥) +𝛼𝜉 (𝑥) 𝐺𝑎 (𝑥) +𝛼𝜉 (𝑥) 𝐺 (𝑥) (30) 𝑎 𝑎̇ Furthermore, a peculiar aspect of the spinor fields Θ(𝑥) and 𝑎̇ Π(𝑥), in the present case, is that their mass terms (33) arise 𝑎 5 +𝛼𝐶𝑎 (𝑥) 𝑅 (𝑥) +𝛼𝐶𝑎̇ (𝑥) 𝑅 (𝑥)}, with chiral structure due to the presence of the matrix 𝛾 . Advances in High Energy Physics 7

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