Collinear superspace The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Cohen, Timothy; Elor, Gilly and Larkoski, Andrew J. "Collinear superspace." Physical Review D 93, 125013 (June 2016): 1-5 © 2016 American Physical Society As Published http://dx.doi.org/10.1103/PhysRevD.93.125013 Publisher American Physical Society Version Final published version Citable link http://hdl.handle.net/1721.1/110594 Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. PHYSICAL REVIEW D 93, 125013 (2016) Collinear superspace Timothy Cohen,1 Gilly Elor,2 and Andrew J. Larkoski3 1Institute of Theoretical Science, University of Oregon, Eugene, Oregon 97403, USA 2Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 3Center for the Fundamental Laws of Nature, Harvard University, Cambridge, Massachusetts 02138, USA (Received 6 May 2016; published 10 June 2016) This paper provides a superfield based approach to constructing a collinear slice of N ¼ 1 superspace. The strategy is analogous to integrating out anticollinear fermionic degrees-of-freedom as was developed in the context of soft-collinear effective theory. The resulting Lagrangian can be understood as an integral over collinear superspace, where half the supercoordinates have been integrated out. The application to N ¼ 1 super Yang-Mills is presented. Collinear superspace provides the foundation for future explorations of supersymmetric soft-collinear effective theory. DOI: 10.1103/PhysRevD.93.125013 Supersymmetry (SUSY) is a powerful framework for SCET. Note that this procedure obscures the underlying exploring the properties of quantum field theory. There Lorentz invariance of the theory, leaving behind the are many examples of extraordinary results derived for constraints known as reparameterization invariance (RPI) SUSY models, for instance the exact NSVZ β-function [1], [35]. Given its spacetime nature, it is unclear that SUSY Seiberg duality [2], Seiberg-Witten theory [3,4], and can be preserved in any meaningful way. Our main result the finiteness of N ¼ 4 SUSY Yang-Mills (SYM) [5]. is to show how collinear superspace packages a SCET Identifying models that manifest SUSY in nontrivial ways Lagrangian in a language that makes the SUSY of the EFT has yielded many fruitful developments, see [6–13] for manifest. recent examples. In this paper, we explore a new class of To derive the collinear limit for a fermion requires N ¼ 1 SUSY effective field theory (EFT) models which integrating out the anticollinear modes, which in practice live on a “collinear slice” of superspace; defining this are half of the full theory fermion helicity degrees of collinear superspace is the subject of this paper. freedom (the momenta of the EFT fields are also con- The connection between collinear superspace and gauge strained). This procedure guides the construction here: the theories becomes apparent in the infrared (IR), where the EFT can be characterized in terms of half the supercharges physics can be largely inferred from the presence of soft for N ¼ 1 SUSY—the other half of the supersymmetries and collinear divergences. There is a rich history associated are nonlinearly realized. We refer to this as “integrating with the IR structure of gauge theories. For example, a out” half of superspace, which leave behind a collinear correspondence between the coefficients of Sudakov logs subsurface of superspace. Our procedure for deriving in Yang-Mills theory and the cusp anomalous dimension of collinear superspace, which should be generally applicable Wilson loops was discovered as early as 1980 [14]. The to a wide class of SUSY EFTs, can be described by the importance of these IR effects helped lead to the discovery following algorithm: of soft-collinear effective theory (SCET) [15–22], which is General Algorithm a powerful formalism developed for resumming the IR (i) Find projection operators that separate the divergences occurring for processes that are dominated by superfield into collinear/anti-collinear superfields soft (low momentum) and collinear degrees of freedom; see [e.g., Eq. (10)]. [23,24] for reviews. There exists an ever growing literature (ii) Starting with the superspace action for the full exploring practical applications of SCET to heavy meson theory, integrate out the entire anticollinear decays [21,22,25,26], LHC collisions [27–31], and even superfield. This will yield a constraint equation WIMP dark matter systems [32–34]. Our purpose here is to [e.g., Eq. (14)]. lay the groundwork for supersymmetrizing SCET, in hopes (iii) Based on the constraint equation, guess an ansatz of further illuminating nontrivial aspects of field theory. for the equation of motion for the anticollinear SCET can be understood in terms of a mode expansion, superfield in terms of collinear degrees-of-freedom where a power-counting parameter λ is used to separate [e.g., Eq. (16)]. degrees-of-freedom that are “near” a lightlike direction, (iv) Plug the ansatz into the full theory action to yield the thereby capturing the IR dynamics as an expansion in λ, superspace action of the effective theory [e.g., (19)]. from the “far” modes. Integrating out these “anticollinear” In what follows, we will apply this procedure to the explicit degrees-of-freedom yields the effective Lagrangian of case of N ¼ 1 SYM. 2470-0010=2016=93(12)=125013(5) 125013-1 © 2016 American Physical Society COHEN, ELOR, and LARKOSKI PHYSICAL REVIEW D 93, 125013 (2016) To begin, we will provide some conventions. The contractions the gauge boson Lagrangian density does scale SUSY EFT is defined in Minkowski space with signature homogeneously: F2 ∼ λ4. In what follows, we focus on the gμν ¼ diagðþ1; −1; −1; −1Þ. The collinear direction is collinear modes, as the soft modes can be decoupled at taken along the zˆ light-cone direction: nμ ¼ð1; 0; 0; 1Þ. leading power by a field redefinition [23,24]. The anticollinear direction is defined by n2 ¼ 0 ¼ n¯ 2 and Collinear superspace is on-shell, i.e., only physical n · n¯ ¼ 2. It is usually convenient to make the explicit degrees-of-freedom will be present in the Lagrangian. To choice n¯ μ ¼ð1; 0; 0; −1Þ. Four vectors are expanded as this end, it is convenient to work in light cone gauge (LCG) μ p ¼ðn · p; n¯ · p; p~ ⊥Þ, where “⊥” refers to the two direc- which corresponds to the non-(space-time)-covariant gauge tions perpendicular to both n and n¯. A state is collinear to choice n¯ · A ¼ 0, see e.g., [37] for a review. Additionally, the light-cone when it lives within a momentum shell which the mode n · A is nonpropagating in this gauge (with μ 2 scales as pn ∼ ðλ ; 1; λÞ, where λ ≪ 1 is the SCET power respect to light-cone time)—it can be integrated out by counting parameter. The virtuality for the collinear modes solving the classical equation of motion. The two remain- in the effective theory p2 ∼ λ2 can be interpreted as ing bosonic physical degrees of freedom, the transverse closeness to the light cone. Similarly, an anticollinear components of the gauge field, can be recast as a complex μ ∼ 1 λ2 λ scalar A, defined by momenta scales as pn¯ ð ; ; Þ. Fields also scale as powers of λ; the power counting rules can be inferred from ∂ ≡ −∂ÃA − ∂Aà the appropriate kinetic terms, and must be necessarily ⊥ · An⊥ ; ð4Þ tracked when determining the order of a given operator. where ∂ and ∂à are also implicitly defined by this As discussed previously, studying the collinear fermion equation [37]. Then L ¼ − 1 F2 → AÃ□A þ L . EFT will provide insight for the derivation of collinear 4 n int For concreteness, our focus here is on-shell N ¼ 1 superspace. A two-component left-handed Weyl spinor can SYM. The fermionic degree of freedom u (the single be decomposed into collinear and anticollinear momentum n remaining spin state in the EFT after the SCET projection) modes using projection operators; u ¼ðP þ P¯ Þu ¼ n n is the collinear gaugino whose superpartner is the bosonic u þ u¯ , where n n light cone scalar A.In[36], we will provide a detailed n · σ n¯ · σ¯ n¯ · σ n · σ¯ derivation of the corresponding collinear SCET Lagrangian Pn ¼ 2 2 Pn¯ ¼ 2 2 : ð1Þ along with a demonstration that it passes checks necessary for EFT consistency, e.g., RPI. These also correspond to chiral projection operators that The N ¼ 1 supercharges are defined by the graded distinguish the fermion’s spin states in the collinear limit † μ algebra Qα;Q 2σ Pμ, where the spinor and anti- (a detailed discussion of two-component collinear fermions f α_ g¼ αα_ α α_ 1 will be given in a forthcoming paper [36]). The anticol- spinor indices run over ; ¼ , 2. Power counting the 2 generator of translations Pμ ¼ i∂μ as appropriate for collin- linear modes u¯ , which scale as Oðλ Þ, are power sup- n ear momenta, yields the scaling of the algebra in the EFT: pressed relative to the collinear ones un ∼ OðλÞ. Therefore, pffiffiffi u¯ should be integrated out using the classical equation of à 2 n † n · ∂ 2∂ Oðλ Þ OðλÞ motion: fQα;Qα_ g¼2i pffiffiffi ∼ ; 2∂ n¯ · ∂ αα_ OðλÞ Oð1Þ n¯ · σ 1 u¯ ¼ − ðσ¯ · D⊥Þu ; ð2Þ ð5Þ n 2 n¯ · D n yielding the following Lagrangian for a charged collinear from which we can infer fermion Q2 ∼ Oð1Þ;Q1 ∼ OðλÞ; 1 ¯ σ¯ † † † n · Q_ ∼ Oð1Þ;Q_ ∼ OðλÞ: ð6Þ L ¼ iu n · D þ σ¯ · D⊥ σ · D⊥ u ; ð3Þ 2 1 u n n¯ · D 2 n To leading power, only one supercharge (Q2) is present in the where D is the covariant derivative appropriately power EFT.