Collinear Superspace

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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 ∂ ≡ −∂A − ∂A the appropriate kinetic terms, and must be necessarily ⊥ · An⊥ ; ð4Þ tracked when determining the order of a given operator. where ∂ and ∂ are also implicitly defined by this As discussed previously, studying the collinear fermion equation [37]. Then L ¼ − 1 F2 → A□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 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. Expressing the supercharges as differential operators in expanded when acting on collinear fields, and the nonlocal superspace, and expanding on the light cone yields; operator is defined in terms of its momentum eigenvalues, ∂ pffiffiffi see e.g., [23,24]. ¯2_ ¯1_ Q2 ¼ i − θ n¯ · ∂ − 2θ ∂ ; The gauge bosons of the full theory can simply be ∂θ2 μ μ μ expanded as A ¼ An þ An¯ , with a corresponding gauge ∂ pffiffiffi μν μν ¯1_ ¯2_ L − 1 2 − 1 2 Q1 ¼ i − θ n · ∂ − 2θ ∂ ; ð7Þ Lagrangian for each sector ¼ 4 ðFn Þ 4 ðFn¯ Þ . Note ∂θ1 that the gauge field is decomposed into components that scale as momentum along the collinear, anticollinear, and with analogous expressions for the conjugate charges. μν perpendicular directions. Thus the field strength igFn ¼ Therefore the scaling of the momentum operator, and the ½Dμ; Dν scales inhomogeneously with λ. However, after supercharges as given in (6), induce a nontrivial scaling of the

125013-2 COLLINEAR SUPERSPACE PHYSICAL REVIEW D 93, 125013 (2016) pffiffiffi TABLE I. Power counting for the superspace coordinates. 1 †1_ 1 †2_ 2 †1_ Vn ¼ −θ θ n · An − 2ðθ θ An þ θ θ AnÞ θ1 θ †1_ † θ2 −θ †2_ † 1 2 †2_ †1_ †2_ 2 Coordinate ¼ 2 θ ¼ θ ¼ 1 θ ¼ −θ 2iθ θ θ u − 2iθ θ θ u 2; 2_ 1_ þ n;2_ n; Scaling λ−1 λ−1 11 pffiffiffi −θ1θ†1_ − θ2θ†2_ ¯ − 2 θ1θ†2_ A θ2θ†1_ A Vn¯ ¼ n · An¯ n · An¯ ð n¯ þ n¯ Þ 1 2 †1_ †1_ †2_ 1 2iθ θ θ u − 2iθ θ θ u¯ 1; 11 superspace coordinates, see Table I. Hence, two out of the þ n;¯ 1_ n; ð Þ four N ¼ 1 superspace Grassmann coordinates have a high ¯ 0 virtuality and should not play a role in the EFT. where we have fixed the LCG condition n · An ¼ . In terms of yμ ¼ xμ þ iθσμθ†, the superspace derivatives The action for the Abelian theory is ¯ †α_ Z are Dα_ ¼ −∂=∂θ . Table I implies that they scale as ¯ ¯ 4 2θWαW D2_ ∼ Oð1Þ and D1_ ∼ OðλÞ. To leading order in λ, S ¼ d xd α þ H:c:; ð12Þ ¯ fD2; D2_ g¼−in¯ · ∂ ∼ Oð1Þ, while all other components of the anticommutator are suppressed. where Wα is a chiral superfield which in Wess-Zumino Chiral and antichiral SCET superfields are defined gauge is such that they obey the EFT chirality condition, † ¯ i D2Φ ¼ 0 ¼ D_ Φ. The physical degrees of freedom of ¯ ¯ 2 Wα ¼ − D DDαðVn þ Vn¯ Þ; ð13Þ the SCET LCG vector multiplet can be repackaged into a 4 chiral superfield. Enforcing the chirality condition in the α ¯ ¯ ¯ ¯ α_ Φ where DD ¼ D Dα and D D ¼ Dα_ D . EFT, the chiral superfield takes the form; The anticollinear vector superfield can be integrated out −iθ†2_ θ2 ¯ ∂ 2 Φ 2 n· A θ using the variation of the superspace action. This yields a ¼ e ð þ un;2Þ α superspace constraint equation, D Wα ¼ 0, which encodes 2 i †2_ 2 A θ − θ θ ¯ ∂A the equation of motion for Vn¯ ; ¼ þ un;2 2 n · ; ð8Þ μ μ −16□ 4 α σ ∂ ¯ α_ 0 where in the second line we have converted from y to x ð þ iD ð · Þαα_ D ÞðVn þ Vn¯ Þ¼ : ð14Þ coordinates, dropped terms that are subleading in λ, and suppressed a gauge index in the case of non-Abelian fields. It is instructive to see that the equations of motion for the There is only one complex fermionic degree of freedom component fields that are integrated out in the EFT, un¯ and n · An, are equivalent to this constraint equation. To isolate in Φ, and it obeys P u ¼ u 2, since the spin up state has n n n; the leading order fermionic components of the vector been projected out. Similarly, we have integrated out only ¯ one (spin-up) anticollinear fermionic degree of freedom; superfield expanded in (11), apply D2_ to the constraint this depends on the specific choice for n¯ μ. equation: The (on-shell) SUSY transformations of the component ¯ 2 ¯ 2_ ¯ 1 ¯ 2_ D2_ D ðσ · ∂Þ22_ D Vn¯ ¼ −D2_ D ðσ · ∂Þ12_ D Vn; fields in the EFT follow from the SCET expansion of the pffiffiffi charges in (7). Additionally, they are consistent with the 2∂ ⇒ u¯ 1 ¼ u 2; ð15Þ expected component transformations of a chiral superfield: n; n¯ · ∂ n; pffiffiffi pffiffiffi †2_ 2 which reproduces the expected equation of motion for the δηun;2 ¼ i 2η n¯ · ∂A; δηA ¼ 2η un;2; ð9Þ anticollinear gaugino, see (2). Additionally, it is straight- where we have used ðn · σÞ22_ ¼ 2. The collinear SCET forward to show that (14) integrates out the unphysical Lagrangian is invariant under these transformations [36]. gauge polarization n · An, thereby reproducing the LCG Now that we have explored some general aspects of Lagrangian. marrying SCET and SUSY, we will focus our attention on a This motivates an ansatz for the equation of motion of specific example. In the rest of this paper, we will apply the the anticollinear vector superfield: general algorithm presented above to the free Abelian 1 gauge theory. Then we will conclude by quoting the result − − ¯ ¯ Vn¯ ¼ Vn ¯ D2_ DDðD2_ D1VnÞ for non-Abelian gauge theory [36]. n¯ · ∂D2D1D1 Since SUSY is a good symmetry, the projection oper- 1 − ¯ ¯ ¯ ators acting on the gauginos of a vector multiplet imply that ¯ ¯ D2D DðD2D1_ VnÞ: ð16Þ n¯ · ∂D2_ D1D1_ the entire superfield obeys the decomposition:

† Both nontrivial terms are required to ensure the reality V ¼ V ¼ PnV þ Pn¯ V ¼ Vn þ Vn¯ ; ð10Þ † condition Vn¯ ¼ Vn¯ . Dividing by superspace derivatives is where the projection operators are defined in (1). Using well defined by taking a super-Fourier transform and un;1 ¼ 0 ¼ un;¯ 2, the collinear and anticollinear on-shell considering momentum and supermomentum eigenvalues. superfields are (16) satisfies the constraint equation (14). Furthermore, it

125013-3 COHEN, ELOR, and LARKOSKI PHYSICAL REVIEW D 93, 125013 (2016) ¯ reproduces the component equations of motion for unphys- which obeys the chirality constraint D2_ Φ ¼ 0; for the ical degrees of freedom. For example, projecting with antichiral multiplet, simply take the conjugate of (18). ¯ D2D1D1 reproduces (15). Therefore, the ansatz for integrating out the anticollinear In the LCG EFT the remaining physical degrees of modes (16) can be expressed in terms of the chiral and freedom un and A form a chiral superfield. This can be antichiral superfields. justified in superspace by taking projections on a vector AfterR some manipulations, the action (12) is superfield (11), for instance 4 4 ¯ α ¯ S ∝ d xd θD1_ D ðVn þ Vn¯ ÞD2_ DαðVn þ Vn¯ Þ. Using (16) ¯ to integrate out the anticollinear superfield yields the Φ ≡ _ 1 †1_ D2D1Vnjθ ¼0¼θ pffiffiffi pffiffiffi EFT action1 2A 2 θ2 2θ2θ†2_ ¯ ∂A ¼ þ i un;2 þ i n · ; ð17Þ

Z 1 1 L 4θ 1 ¯ ¯ ¯ ¯ 1_ ¯ ¼ d ¯ D D DðD2D1_ VnÞ D DDðD2_ D1VnÞ n¯ · ∂D_ n¯ · ∂D1 Z 1 1 1 θ2 θ†2_ θ†1_ θ1 1 ¯ ¯ ¯ ¯ 1_ ¯ ¼ d d d d ¯ ðD D DðD2D1_ VnÞÞ 2 ðD DDðD2_ D1VnÞÞ D1D_ ðn¯ · ∂Þ Z 1 Z ¯ ¯ ¯ 2 _ □ ∂ 2 †2_ † D DD2D2DD 2 †2_ † i ⊥ ¼ dθ dθ Φn Φ ¼ dθ dθ Φn Φ ⊂ iu 2 n · ∂ þ u 2 þ A □A; ð18Þ ðn¯ · ∂Þ2 n n¯ · ∂ n n; n¯ · ∂ n; which reproduces the expected equation of motion in the expressions to (19) do exist in the literature for N ¼ 4 free theory. We conclude that integrating out the anticol- SYM [5,38], the present work simultaneously provides the linear fermion translates into integrating out two super- first application to collinear fields along with a general space coordinates, namely θ1 ∼ 1=λ and θ†1 ∼ 1=λ, while algorithm that can be used to derive the Lagrangian. θ2 ∼ 1 and θ†2 ∼ 1 remain in the EFT. Note that in the In conclusion, this paper has provided a framework for above calculation we can identify the various projections of studying SUSY in the collinear limit. A general algorithm Vn with a chiral superfield by (18) in the EFT. for deriving an EFT defined on collinear superspace was Finally for completeness, we quote the result for the proposed, and it was applied to the case of an N ¼ 1 collinear superspace LCG Lagrangian in N ¼ 1 SYM. Abelian superfield. We also provided the result for a non- This model is invariant under the SUSY transformations (9) Abelian theory. In a followup work [36], we will provide a and meets additional requirements such as RPI demonstrat- more complete treatment of the EFT perspective, including ing that it is a consistent collinear EFT [36]: a detailed discussion of the remaining symmetries of the EFT, and an explanation of how the Super-Poincare Z ⋆ 2 †2_ † □ † ∂ generators reduce to RPI. This will provide the groundwork L¼ dθ dθ Φ a Φa þ2g fabcΦaΦ b Φc þH:c: n¯ ·∂ n¯ ·∂ for many interesting extensions, including models with a 1 1 larger number of supercharges, and even perhaps theories 2 abc ade b ¯ †c †d e þ2g f f ðΦ D_ Φ Þ ðΦ D2Φ Þ : ð19Þ of collinear supergravity. n¯ ·∂ 2 n¯ ·∂ Note that the form of this expression is what one would ACKNOWLEDGMENTS have naively obtained by supersymmetrizing the pure LCG We are grateful to Marat Freytsis and Duff Neill and for Yang Mills Lagrangian [37]. In this sense, the on-shell helpful comments. T. C. is supported by an LHC Theory collinear EFT makes SUSY transparent. While similar Initiative Postdoctoral Fellowship, under the National – R Science Foundation Grant No. PHY 0969510. G. E. is 1 α Recall that dθ Dαð…Þ is a total derivative in real space, and supported by the U.S. Department of Energy, under grant therefore we can drop surface terms when using integration by Contract Numbers DE–SC00012567. A. L. is supported an parts if we assume that they vanish sufficiently fast at infinity. In LHC Theory Initiative Postdoctoral Fellowship, under the SCET, integration by parts is well defined for the inverse – derivative operator 1=n¯ · ∂ because it can be cast in terms of National Science Foundation Grant No. PHY 1419008. its momentum space representation. By analogy we extend this This work was in part initiated at the Aspen Center for argument and use integration by parts on 1=D operators in the Physics, which is supported by National Science following calculation. Foundation Grant No. PHY–1066293.

125013-4 COLLINEAR SUPERSPACE PHYSICAL REVIEW D 93, 125013 (2016) [1] V. A. Novikov, M. A. Shifman, A. I. Vainshtein, and V. I. [20] R. J. Hill and M. Neubert, Spectator interactions in soft Zakharov, Exact Gell-Mann-Low function of supersymmet- collinear effective theory, Nucl. Phys. B657, 229 (2003). ric Yang-Mills theories from instanton calculus, Nucl. Phys. [21] J. Chay and C. Kim, Collinear effective theory at subleading B229, 381 (1983). order and its application to heavy-light currents, Phys. Rev. [2] N. Seiberg, Electric-magnetic duality in supersymmetric D 65, 114016 (2002). non-Abelian gauge theories, Nucl. Phys. B435, 129 [22] M. Beneke, A. P. Chapovsky, M. Diehl, and T. Feldmann, (1995). Soft collinear effective theory and heavy to light currents [3] N. Seiberg and E. Witten, Monopoles, duality and chiral beyond leading power, Nucl. Phys. B643, 431 (2002). symmetry breaking in N ¼ 2 supersymmetric QCD, Nucl. [23] T. Becher, A. Broggio, and A. Ferroglia, Introduction to Phys. B431, 484 (1994). soft-collinear effective theory, arXiv:1410.1892. [4] N. Seiberg and E. Witten, Electric-magnetic duality, [24] I. W. Stewart, Lectures on the soft-collinear effective monopole condensation, and confinement in N ¼ 2 super- theory., http://ocw.mit.edu/courses/physics/8‑851‑effective‑ symmetric Yang-Mills theory, Nucl. Phys. B426, 19 (1994); field‑theory‑spring‑2013/lecture‑notes/MIT8_851S13_ [5] S. Mandelstam, Light cone superspace and the ultraviolet scetnotes.pdf. finiteness of the N ¼ 4 model, Nucl. Phys. B213, 149 [25] C. W. Bauer, D. Pirjol, and I. W. Stewart, A Proof of (1983). Factorization for B → Dπ, Phys. Rev. Lett. 87, 201806 [6] G. Dall’Agata and F. Farakos, Constrained superfields in (2001). supergravity, J. High Energy Phys. 02 (2016) 101. [26] E. Lunghi, D. Pirjol, and D. Wyler, Factorization in [7] S. Ferrara, R. Kallosh, and J. Thaler, Cosmology with leptonic radiative b → γeν decays, Nucl. Phys. B649, orthogonal nilpotent superfields, Phys. Rev. D 93, 043516 349 (2003). (2016). [27] C. W. Bauer, S. Fleming, D. Pirjol, I. Z. Rothstein, and I. W. [8] Y. Kahn, D. A. Roberts, and J. Thaler, The goldstone and Stewart, Hard scattering factorization from effective field goldstino of supersymmetric inflation, J. High Energy Phys. theory, Phys. Rev. D 66, 014017 (2002). 10 (2015) 001. [28] I. W. Stewart, F. J. Tackmann, and W. J. Waalewijn, Fac- [9] Z. Komargodski and N. Seiberg, From linear SUSY to torization at the LHC: From PDFs to initial state jets, Phys. constrained superfields, J. High Energy Phys. 09 (2009) Rev. D 81, 094035 (2010). 066. [29] S. Mantry and F. Petriello, Factorization and resummation [10] G. Festuccia and N. Seiberg, Rigid supersymmetric theories of Higgs boson differential distributions in soft-collinear in curved superspace, J. High Energy Phys. 06 (2011) 114. effective theory, Phys. Rev. D 81, 093007 (2010). [11] R. Kallosh, A. Karlsson, B. Mosk, and D. Murli, Orthogonal [30] T. Becher and M. Neubert, Drell-Yan production at small nilpotent superfields from linear models, J. High Energy qT , transverse parton distributions and the collinear Phys. 05 (2016) 082. anomaly, Eur. Phys. J. C 71, 1665 (2011). [12] S. Ferrara, R. Kallosh, A. Van Proeyen, and T. Wrase, [31] M. Beneke, P. Falgari, and C. Schwinn, Threshold resum- Linear versus non-linear supersymmetry, in general, J. High mation for pair production of coloured heavy (s)particles at Energy Phys. 04 (2016) 065. hadron colliders, Nucl. Phys. B842, 414 (2011). [13] G. Dall’Agata, E. Dudas, and F. Farakos, On the origin of [32] M. Baumgart, I. Z. Rothstein, and V. Vaidya, Calculating the constrained superfields, J. High Energy Phys. 05 (2016) 041. Annihilation Rate of Weakly Interacting Massive Particles, [14] A. M. Polyakov, Gauge fields as rings of glue, Nucl. Phys. Phys. Rev. Lett. 114, 211301 (2015). B164, 171 (1980). [33] M. Bauer, T. Cohen, R. J. Hill, and M. P. Solon, Soft [15] C. W. Bauer, S. Fleming, and M. E. Luke, Summing collinear effective theory for heavy WIMP annihilation, Sudakov logarithms in B → XðsγÞ in effective field theory, J. High Energy Phys. 01 (2015) 099. Phys. Rev. D 63, 014006 (2000). [34] G. Ovanesyan, T. R. Slatyer, and I. W. Stewart, Heavy Dark [16] C. W. Bauer, S. Fleming, D. Pirjol, and I. W. Stewart, An Matter Annihilation from Effective Field Theory, Phys. Rev. effective field theory for collinear and soft gluons: Heavy to Lett. 114, 211302 (2015). light decays, Phys. Rev. D 63, 114020 (2001). [35] C. Marcantonini and I. W. Stewart, Reparameterization invari- [17] C. W. Bauer and I. W. Stewart, Invariant operators in ant collinear operators, Phys. Rev. D 79, 065028 (2009). collinear effective theory, Phys. Lett. B 516, 134 (2001). [36] T. Cohen, G. Elor, and A. J. Larkoski, Soft-collinear [18] C. W. Bauer, D. Pirjol, and I. W. Stewart, Soft collinear supersymmetry (to be published). factorization in effective field theory, Phys. Rev. D 65, [37] G. Leibbrandt, Introduction to noncovariant gauges, Rev. 054022 (2002). Mod. Phys. 59, 1067 (1987). [19] C. W. Bauer, D. Pirjol, and I. W. Stewart, Power counting in [38] L. Brink, O. Lindgren, and B. E. W. Nilsson, N ¼ 4 the soft collinear effective theory, Phys. Rev. D 66, 054005 Yang-Mills theory on the light cone, Nucl. Phys. B212, (2002). 401 (1983).

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