aa r X i v : . [ h e p - t h ] M a r Collinear Superspace
Timothy Cohen, Gilly Elor, and Andrew J. Larkoski Institute of Theoretical Science, University of Oregon, Eugene, OR 97403 Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139 Center for the Fundamental Laws of Nature, Harvard University, Cambridge, MA 02138
Abstract
This letter provides a superfield based approach to constructing a collinear slice of N = 1 su-perspace. The strategy is analogous to integrating out anti-collinear fermionic degrees-of-freedomas was developed in the context of soft-collinear effective theory. The resulting Lagrangian can beunderstood as an integral over collinear superspace, where half the supercoordinates have been inte-grated out. The application to N = 1 super Yang-Mills is presented. Collinear superspace providesthe foundation for future explorations of supersymmetric soft-collinear effective theory. Supersymmetry (SUSY) is a powerful framework forexploring the properties of quantum field theory. Thereare many examples of extraordinary results derivedfor SUSY models, for instance the exact NSVZ β -function [1], Seiberg duality [2], Seiberg-Witten the-ory [3, 4], and the finiteness of N = 4 SUSY Yang-Mills(SYM) [5]. Identifying models that manifest SUSY innon-trivial ways has yielded many fruitful developments,see [6–10] for recent examples. In this letter, we explorea new class of N = 1 SUSY effective field theory (EFT)models which live on a “collinear slice” of superspace;defining this collinear superspace is the subject of thisletter.The connection between collinear superspace andgauge theories becomes apparent in the infrared (IR),where the physics can be largely inferred from the pres-ence of soft and collinear divergences. There is a richhistory associated with the IR structure of gauge theo-ries. For example, a correspondence between the coeffi-cients of Sudakov logs in Yang-Mills theory and the cuspanomalous dimension of Wilson loops was discovered asearly as 1980 [11]. The importance of these IR effectshelped lead to the discovery of Soft-Collinear EffectiveTheory (SCET) [12–19], which is a powerful formalismdeveloped for resumming the IR divergences occurringfor processes that are dominated by soft (low momen-tum) and collinear degrees of freedom; see [20, 21] forreviews. There exists an ever growing literature explor-ing practical applications of SCET to heavy meson de-cays [18, 19, 22, 23], LHC collisions [24–28], and evenWIMP dark matter systems [29–31]. Our purpose hereis to lay the groundwork for supersymmetrizing SCET, inhopes of further illuminating non-trivial aspects of fieldtheory.SCET can be understood in terms of a mode expan-sion, where a power-counting parameter λ is used to sep-arate degrees-of-freedom that are “near” a light-like di-rection, thereby capturing the IR dynamics as an ex- pansion in λ , from the “far” modes. Integrating outthese “anti-collinear” degrees-of-freedom yields the effec-tive Lagrangian of SCET. Note that this procedure ob-scures the underlying Lorentz invariance of the theory,leaving behind the constraints known as reparameteriza-tion invariance (RPI) [32]. Given its spacetime nature, itis unclear that SUSY can be preserved in any meaningfulway. Our main result is to show how collinear superspacepackages a SCET Lagrangian in a language that makesthe SUSY of the EFT manifest.To derive the collinear limit for a fermion requires in-tegrating out the anti-collinear modes, which in prac-tice are half of the full theory fermion helicity degreesof freedom (the momenta of the EFT fields are also con-strained). This procedure guides the construction here:the EFT can be characterized in terms of half the su-percharges for N = 1 SUSY – the other half of the su-persymmetries are non-linearly realized. We refer to thisas “integrating out” half of superspace, which leave be-hind a collinear subsurface of superspace. Our procedurefor deriving collinear superspace, which should be gen-erally applicable to a wide class of SUSY EFTs, can bedescribed by the following algorithm: General Algorithm • Find projection operators that separate the superfieldinto collinear/anti-collinear superfields [ e.g.
Eq. (10)]. • Starting with the superspace action for the full theory,integrate out the entire anti-collinear superfield. Thiswill yield a constraint equation [ e.g.
Eq. (14)]. • Based on the constraint equation, guess an ansatz forthe equation of motion for the anti-collinear superfieldin terms of collinear degrees-of-freedom [ e.g.
Eq. (16)]. • Plug the ansatz into the full theory action to yield thesuperspace action of the effective theory [ e.g.
Eq. (19)].In what follows, we will apply this procedure to the ex-plicit case of N = 1 SYM.To begin, we will provide some conventions. TheSUSY EFT is defined in Minkowski space with signa-ture g µν = diag (+1 , − , − , − z light-cone direction: n µ = (1 , , , n = 0 = ¯ n and n · ¯ n = 2. It is usually convenient to make theexplicit choice ¯ n µ = (1 , , , − p µ = ( n · p, ¯ n · p, ~p ⊥ ), where “ ⊥ ” refers to thetwo directions perpendicular to both n and ¯ n . A state iscollinear to the light-cone when it lives within a momen-tum shell which scales as p µn ∼ ( λ , , λ ), where λ ≪ p ∼ λ can beinterpreted as closeness to the light cone. Similarly, ananti-collinear momenta scales as p µ ¯ n ∼ (1 , λ , λ ). Fieldsalso scale as powers of λ ; the power counting rules canbe inferred from the appropriate kinetic terms, and mustbe necessarily tracked when determining the order of agiven operator.As discussed previously, studying the collinear fermionEFT will provide insight for the derivation of collinearsuperspace. A two-component left-handed Weylspinor can be decomposed into collinear and anti-collinear momentum modes using projection operators; u = ( P n + P ¯ n ) u = u n + u ¯ n , where P n = n · σ n · ¯ σ P ¯ n = ¯ n · σ n · ¯ σ . (1)These also correspond to chiral projection operators thatdistinguish the fermion’s spin states in the collinearlimit (a detailed discussion of two-component collinearfermions will be given in a forthcoming paper [33]). Theanti-collinear modes u ¯ n , which scale as O ( λ ), are powersuppressed relative to the collinear ones u n ∼ O ( λ ).Therefore, u ¯ n should be integrated out using the clas-sical equation of motion: u ¯ n = − ¯ n · σ n · D (cid:0) ¯ σ · D ⊥ (cid:1) u n , (2)yielding the following Lagrangian for a charged collinearfermion L u = i u † n (cid:18) n · D + ¯ σ · D ⊥ n · D σ · D ⊥ (cid:19) ¯ n · ¯ σ u n , (3)where D is the covariant derivative appropriately powerexpanded when acting on collinear fields, and the non-local operator is defined in terms of its momentum eigen-values, see e.g. [20, 21].The gauge bosons of the full theory can simply be ex-panded as A µ = A µn + A µ ¯ n , with a corresponding gauge La-grangian for each sector L = − ( F µνn ) − ( F µν ¯ n ) . Notethat the gauge field is decomposed into components thatscale as momentum along the collinear, anti-collinear, and perpendicular directions. Thus the field strength i g F µνn = (cid:2) D µ , D ν (cid:3) scales inhomogeneously with λ . How-ever, after contractions the gauge boson Lagrangian den-sity does scale homogeneously: F ∼ λ . In what follows,we focus on the collinear modes, as the soft modes can bedecoupled at leading power by a field redefinition [20, 21].Collinear superspace is on-shell, i.e. , only physicaldegrees-of-freedom will be present in the Lagrangian.To this end, it is convenient to work in Light ConeGauge (LCG) which corresponds to the non-(space-time)-covariant gauge choice ¯ n · A = 0, see e.g. [34] for areview. Additionally, the mode n · A is non-propagatingin this gauge (with respect to light-cone time) – it can beintegrated out by solving the classical equation of motion.The two remaining bosonic physical degrees of freedom,the transverse components of the gauge field, can be re-cast as a complex scalar A , defined by ∂ ⊥ · A n ⊥ ≡ − ∂ ∗ A − ∂ A ∗ , (4)where ∂ and ∂ ∗ are also implicitly defined by this equa-tion [34]. Then L = − F n → A ∗ ✷ A + L int .For concreteness, our focus here is on-shell N = 1SYM. The fermionic degree of freedom u n (the singleremaining spin state in the EFT after the SCET projec-tion) is the collinear gaugino whose superpartner is thebosonic light cone scalar A . In [33], we will provide adetailed derivation of the corresponding collinear SCETLagrangian along with a demonstration that it passeschecks necessary for EFT consistency, e.g. RPI.The N = 1 supercharges are defined by the gradedalgebra (cid:8) Q α , Q † ˙ α (cid:9) = 2 σ µα ˙ α P µ , where the spinor and anti-spinor indices run over α, ˙ α = 1 ,
2. Power counting thegenerator of translations P µ = i ∂ µ as appropriate forcollinear momenta, yields the scaling of the algebra inthe EFT: n Q α , Q † ˙ α o = 2 i (cid:20) n · ∂ √ ∂ ∗ √ ∂ ¯ n · ∂ (cid:21) α ˙ α ∼ (cid:20) O ( λ ) O ( λ ) O ( λ ) O (1) (cid:21) , (5)from which we can infer Q ∼ O (1) , Q ∼ O ( λ ) ,Q † ˙2 ∼ O (1) , Q † ˙1 ∼ O ( λ ) . (6)To leading power, only one supercharge ( Q ) is presentin the EFT. Expressing the supercharges as differentialoperators in superspace, and expanding on the light coneyields; Q = (cid:16) i ∂∂θ − ¯ θ ˙2 ¯ n · ∂ − √ θ ˙1 ∂ (cid:17) ,Q = (cid:16) i ∂∂θ − ¯ θ ˙1 n · ∂ − √ θ ˙2 ∂ ∗ (cid:17) , (7)with analogous expressions for the conjugate charges.Coordinate θ = θ θ † ˙1 = θ † ˙2 θ = − θ θ † ˙2 = − θ † ˙1 Scaling λ − λ − TABLE I. Power counting for the superspace coordinates.
Therefore the scaling of the momentum operator, andthe supercharges as given in Eq. (6), induce a non-trivialscaling of the superspace coordinates, see Table I. Hence,two out of the four N = 1 superspace Grassmann coor-dinates have a high virtuality and should not play a rolein the EFT.In terms of y µ = x µ + i θσ µ θ † , the superspace deriva-tives are ¯ D ˙ α = − ∂/∂θ † ˙ α . Table I implies that they scaleas ¯ D ˙2 ∼ O (1) and ¯ D ˙1 ∼ O ( λ ). To leading order in λ , { D , ¯ D ˙2 } = − i ¯ n · ∂ ∼ O (1), while all other componentsof the anti-commutator are suppressed.Chiral and anti-chiral SCET superfields are definedsuch that they obey the EFT chirality condition, D Φ † =0 = ¯ D ˙2 Φ. The physical degrees of freedom of the SCETLCG vector multiplet can be repackaged into a chiral su-perfield. Enforcing the chirality condition in the EFT,the chiral superfield Φ takes the form;Φ = e − i θ † ˙2 θ ¯ n · ∂ (cid:0) A ∗ + θ u ∗ n, (cid:1) = A ∗ + θ u ∗ n, − i θ † ˙2 θ ¯ n · ∂ A ∗ , (8)where in the second line we have converted from y µ to x µ coordinates, dropped terms that are subleading in λ ,and suppressed a gauge index in the case of non-Abelianfields. There is only one complex fermionic degree offreedom in Φ, and it obeys P n u ∗ n = u ∗ n, , since the spinup state has been projected out. Similarly, we have in-tegrated out only one (spin-up) anti-collinear fermionicdegree of freedom; this depends on the specific choice for¯ n µ .The (on-shell) SUSY transformations of the compo-nent fields in the EFT follow from the SCET expansionof the charges in Eq. (7). Additionally, they are consis-tent with the expected component transformations of achiral superfield: δ η u n, = i √ η † ˙2 ¯ n · ∂ A , δ η A = √ η u n, , (9)where we have used ( n · σ ) = 2. The collinear SCETLagrangian is invariant under these transformations [33].Now that we have explored some general aspects ofmarrying SCET and SUSY, we will focus our attentionon a specific example. In the rest of this letter, we willapply the general algorithm presented above to the freeAbelian gauge theory. Then we will conclude by quotingthe result for non-Abelian gauge theory [33].Since SUSY is a good symmetry, the projection oper-ators acting on the gauginos of a vector multiplet implythat the entire superfield obeys the decomposition: V = V † = P n V + P ¯ n V = V n + V ¯ n , (10)where the projection operators are defined in Eq. (1).Using u n, = 0 = u ¯ n, , the collinear and anti-collinearon-shell superfields are V n = − θ θ † ˙1 n · A n − √ (cid:16) θ θ † ˙2 A ∗ n + θ θ † ˙1 A n (cid:17) + 2 i θ θ θ † ˙2 u ∗ n, ˙2 − i θ † ˙1 θ † ˙2 θ u n, ,V ¯ n = − θ θ † ˙1 n · A ¯ n − θ θ † ˙2 ¯ n · A ¯ n − √ (cid:16) θ θ † ˙2 A ∗ ¯ n + θ θ † ˙1 A ¯ n (cid:17) + 2 i θ θ θ † ˙1 u ∗ ¯ n, ˙1 − i θ † ˙1 θ † ˙2 θ u ¯ n, , (11)where we have fixed the LCG condition ¯ n · A n = 0.The action for the Abelian theory is S = Z d x d θ W α W α + h.c. , (12)where W α is a chiral superfield which in Wess-Zuminogauge is W α = − i D ¯ D D α (cid:0) V n + V ¯ n (cid:1) , (13)where DD = D α D α and ¯ D ¯ D = ¯ D ˙ α ¯ D ˙ α .The anti-collinear vector superfield can be integratedout using the variation of the superspace action. This yields a superspace constraint equation, D α W α = 0,which encodes the equation of motion for V ¯ n ; (cid:16) − ✷ + 4 i D α (cid:0) σ · ∂ (cid:1) α ˙ α ¯ D ˙ α (cid:17)(cid:0) V n + V ¯ n (cid:1) = 0 . (14)It is instructive to see that the equations of motion forthe component fields that are integrated out in the EFT, u ¯ n and n · A n , are equivalent to this constraint equation.To isolate the leading order fermionic components of thevector superfield expanded in Eq. (11), apply ¯ D ˙2 to theconstraint equation:¯ D ˙2 D (cid:0) σ · ∂ (cid:1) ¯ D ˙2 V ¯ n = − ¯ D ˙2 D (cid:0) σ · ∂ (cid:1) ¯ D ˙2 V n ;= ⇒ u ¯ n, = √ ∂ ∗ ¯ n · ∂ u n, , (15)which reproduces the expected equation of motion forthe anti-collinear gaugino, see Eq. (2). Additionally, it isstraightforward to show that Eq. (14) integrates out theunphysical gauge polarization n · A n , thereby reproducingthe LCG Lagrangian.This motivates an ansatz for the equation of motion ofthe anti-collinear vector superfield: V ¯ n = − V n − n · ∂D ¯ D D ¯ D ˙2 DD (cid:16) ¯ D ˙2 D V n (cid:17) − n · ∂ ¯ D ˙2 D ¯ D ˙1 D ¯ D ¯ D (cid:16) D ¯ D ˙1 V n (cid:17) . (16)Both nontrivial terms are required to ensure the realitycondition V ¯ n = V † ¯ n . Dividing by superspace derivativesis well-defined by taking a super-Fourier transform and considering momentum and super-momentum eigenval-ues. Eq. (16) satisfies the constraint equation Eq. (14).Furthermore, it reproduces the component equations ofmotion for unphysical degrees of freedom. For example,projecting with D ¯ D D reproduces Eq. (15).In the LCG EFT the remaining physical degrees offreedom u n and A form a chiral superfield. This can bejustified in superspace by taking projections on a vectorsuperfield Eq. (11), for instanceΦ ≡ ¯ D ˙2 D V n (cid:12)(cid:12)(cid:12) θ =0= θ † ˙1 = √ A ∗ + 2 i θ u ∗ n, + i √ θ θ † ˙2 ¯ n · ∂ A ∗ , (17)which obeys the chirality constraint ¯ D ˙2 Φ = 0; forthe anti-chiral multiplet, simply take the conjugate ofEq. (17). Therefore, the ansatz for integrating out theanti-collinear modes Eq. (16) can be expressed in termsof the chiral and anti-chiral superfields.After some manipulations, the action Eq. (12) is S ∝ R d x d θ ¯ D ˙1 D α (cid:0) V n + V ¯ n (cid:1) ¯ D ˙2 D α (cid:0) V n + V ¯ n (cid:1) . Us-ing Eq. (16) to integrate out the anti-collinear superfieldyields the EFT action L = Z d θ (cid:18) n · ∂ ¯ D ˙1 D ¯ D ¯ D (cid:16) D ¯ D ˙1 V n (cid:17)(cid:19) (cid:18) n · ∂ D ¯ D ˙1 D D (cid:16) ¯ D ˙2 D V n (cid:17)(cid:19) = Z d θ d θ † ˙2 d θ † ˙1 d θ D ¯ D ˙1 (cid:16) D ¯ D ¯ D (cid:16) D ¯ D ˙1 V n (cid:17)(cid:17) n · ∂ ) (cid:16) ¯ D ˙1 DD (cid:16) ¯ D ˙2 D V n (cid:17)(cid:17) = Z d θ d θ † ˙2 Φ † n ¯ D ¯ DD ¯ D ˙2 DD (¯ n · ∂ ) Φ n = Z d θ d θ † ˙2 Φ † n i ✷ ¯ n · ∂ Φ n ⊂ i u ∗ n, (cid:18) n · ∂ + ∂ ⊥ ¯ n · ∂ (cid:19) u n, + A ∗ ✷ A , (18)which reproduces the expected equation of motion in thefree theory. We conclude that integrating out the anti-collinear fermion translates into integrating out two su-perspace coordinates, namely θ ∼ /λ and θ † ∼ /λ ,while θ ∼ θ † ∼ V n with a chiral superfield by Eq. (17) in theEFT.Finally for completeness, we quote the result for thecollinear superspace LCG Lagrangian in N = 1 SYM.This model is invariant under the SUSY transformationsEq. (9) and meets additional requirements such as RPIdemonstrating that it is a consistent collinear EFT [33]: L = Z d θ d θ † ˙2 (cid:20) Φ † a ✷ ¯ n · ∂ Φ a +2 g (cid:18) f abc Φ a Φ † b ∂ ⋆ ¯ n · ∂ Φ c + h.c. (cid:19) +2 g f abc f ade n · ∂ (cid:16) Φ b ¯ D ˙2 Φ † c (cid:17) n · ∂ (cid:16) Φ † d D Φ e (cid:17)(cid:21) . (19) Recall that R d θ α D α (. . . ) is a total derivative in real space, andtherefore we can drop surface terms when using integration byparts if we assume that they vanish sufficiently fast at infinity.In SCET, integration by parts is well defined for the inversederivative operator 1 / ¯ n · ∂ because it can be cast in terms of its momentum space representation. By analogy we extend thisargument and use integration by parts on 1 /D operators in thefollowing calculation. Note that the form of this expression is what one wouldhave naively obtained by supersymmetrizing the pureLCG Yang Mills Lagrangian [34]. In this sense, the on-shell collinear EFT makes SUSY transparent. While sim-ilar expressions to Eq. (19) do exist in the literature for N = 4 SYM [5, 35], the present work simultaneously pro-vides the first application to collinear fields along witha general algorithm that can be used to derive the La-grangian.In conclusion, this letter has provided a framework forstudying SUSY in the collinear limit. A general algo-rithm for deriving an EFT defined on collinear super-space was proposed, and it was applied to the case of an N = 1 Abelian superfield. We also provided the resultfor a non-Abelian theory. In a followup work [33], wewill provide a more complete treatment of the EFT per-spective, including a detailed discussion of the remain- ing symmetries of the EFT, and an explanation of howthe Super-Poincare generators reduce to RPI. This willprovide the groundwork for many interesting extensions,including models with a larger number of supercharges,and even perhaps theories of collinear supergravity. ACKNOWLEDGMENTS
We are grateful to Marat Freytsis and Duff Neill andfor helpful comments. TC is supported by an LHC The-ory Initiative Postdoctoral Fellowship, under the Na-tional Science Foundation grant PHY–0969510. GE issupported by the U.S. Department of Energy, undergrant Contract Numbers DE–SC00012567. AL is sup-ported an LHC Theory Initiative Postdoctoral Fellow-ship, under the National Science Foundation grant PHY–1419008. This work was in part initiated at the AspenCenter for Physics, which is supported by National Sci-ence Foundation grant PHY–1066293. [1] V. A. Novikov, M. A. Shifman, A. I. Vainshtein, andV. I. Zakharov, “Exact Gell-Mann-Low Function ofSupersymmetric Yang-Mills Theories from InstantonCalculus,”
Nucl. Phys.
B229 (1983) 381.[2] N. Seiberg, “Electric - magnetic duality insupersymmetric nonAbelian gauge theories,”
Nucl. Phys.
B435 (1995) 129–146, arXiv:hep-th/9411149 [hep-th] .[3] N. Seiberg and E. Witten, “Monopoles, duality andchiral symmetry breaking in N=2 supersymmetricQCD,”
Nucl. Phys.
B431 (1994) 484–550, arXiv:hep-th/9408099 [hep-th] .[4] N. Seiberg and E. Witten, “Electric - magnetic duality,monopole condensation, and confinement in N=2supersymmetric Yang-Mills theory,”
Nucl. Phys.
B426 (1994) 19–52, arXiv:hep-th/9407087 [hep-th] . [Erratum: Nucl.Phys.B430,485(1994)].[5] S. Mandelstam, “Light Cone Superspace and theUltraviolet Finiteness of the N=4 Model,”
Nucl. Phys.
B213 (1983) 149–168.[6] Z. Komargodski and N. Seiberg, “From Linear SUSY toConstrained Superfields,”
JHEP (2009) 066, arXiv:0907.2441 [hep-th] .[7] G. Festuccia and N. Seiberg, “Rigid SupersymmetricTheories in Curved Superspace,” JHEP (2011) 114, arXiv:1105.0689 [hep-th] .[8] R. Kallosh, A. Karlsson, B. Mosk, and D. Murli,“Orthogonal Nilpotent Superfields from LinearModels,” arXiv:1603.02661 [hep-th] .[9] S. Ferrara, R. Kallosh, A. Van Proeyen, and T. Wrase,“Linear Versus Non-linear Supersymmetry, in General,” arXiv:1603.02653 [hep-th] .[10] G. Dall’Agata, E. Dudas, and F. Farakos, “On theorigin of constrained superfields,” arXiv:1603.03416 [hep-th] .[11] A. M. Polyakov, “Gauge Fields as Rings of Glue,” Nucl. Phys.
B164 (1980) 171–188. [12] C. W. Bauer, S. Fleming, and M. E. Luke, “SummingSudakov logarithms in B → X ( sγ ) in effective fieldtheory,” Phys. Rev.
D63 (2000) 014006, arXiv:hep-ph/0005275 [hep-ph] .[13] C. W. Bauer, S. Fleming, D. Pirjol, and I. W. Stewart,“An Effective field theory for collinear and soft gluons:Heavy to light decays,”
Phys. Rev.
D63 (2001) 114020, arXiv:hep-ph/0011336 [hep-ph] .[14] C. W. Bauer and I. W. Stewart, “Invariant operators incollinear effective theory,”
Phys. Lett.
B516 (2001) 134–142, arXiv:hep-ph/0107001 [hep-ph] .[15] C. W. Bauer, D. Pirjol, and I. W. Stewart, “Softcollinear factorization in effective field theory,”
Phys. Rev.
D65 (2002) 054022, arXiv:hep-ph/0109045 [hep-ph] .[16] C. W. Bauer, D. Pirjol, and I. W. Stewart, “Powercounting in the soft collinear effective theory,”
Phys. Rev.
D66 (2002) 054005, arXiv:hep-ph/0205289 [hep-ph] .[17] R. J. Hill and M. Neubert, “Spectator interactions insoft collinear effective theory,”
Nucl. Phys.
B657 (2003) 229–256, arXiv:hep-ph/0211018 [hep-ph] .[18] J. Chay and C. Kim, “Collinear effective theory atsubleading order and its application to heavy - lightcurrents,”
Phys. Rev.
D65 (2002) 114016, arXiv:hep-ph/0201197 [hep-ph] .[19] M. Beneke, A. P. Chapovsky, M. Diehl, andT. Feldmann, “Soft collinear effective theory and heavyto light currents beyond leading power,”
Nucl. Phys.
B643 (2002) 431–476, arXiv:hep-ph/0206152 [hep-ph] .[20] T. Becher, A. Broggio, and A. Ferroglia, “Introductionto Soft-Collinear Effective Theory,” arXiv:1410.1892 [hep-ph] .[21] I. W. Stewart, “Lectures on the soft-collinear effectivetheory.” . [22] C. W. Bauer, D. Pirjol, and I. W. Stewart, “A Proof offactorization for B → Dπ ,” Phys. Rev. Lett. (2001) 201806, arXiv:hep-ph/0107002 [hep-ph] .[23] E. Lunghi, D. Pirjol, and D. Wyler, “Factorization inleptonic radiative b → γeν decays,” Nucl. Phys.
B649 (2003) 349–364, arXiv:hep-ph/0210091 [hep-ph] .[24] C. W. Bauer, S. Fleming, D. Pirjol, I. Z. Rothstein, andI. W. Stewart, “Hard scattering factorization fromeffective field theory,”
Phys. Rev.
D66 (2002) 014017, arXiv:hep-ph/0202088 [hep-ph] .[25] I. W. Stewart, F. J. Tackmann, and W. J. Waalewijn,“Factorization at the LHC: From PDFs to Initial StateJets,”
Phys. Rev.
D81 (2010) 094035, arXiv:0910.0467 [hep-ph] .[26] S. Mantry and F. Petriello, “Factorization andResummation of Higgs Boson Differential Distributionsin Soft-Collinear Effective Theory,”
Phys. Rev.
D81 (2010) 093007, arXiv:0911.4135 [hep-ph] .[27] T. Becher and M. Neubert, “Drell-Yan Production atSmall q T , Transverse Parton Distributions and theCollinear Anomaly,” Eur. Phys. J.
C71 (2011) 1665, arXiv:1007.4005 [hep-ph] .[28] M. Beneke, P. Falgari, and C. Schwinn, “Thresholdresummation for pair production of coloured heavy(s)particles at hadron colliders,”
Nucl. Phys.
B842 (2011) 414–474, arXiv:1007.5414 [hep-ph] .[29] M. Baumgart, I. Z. Rothstein, and V. Vaidya,“Calculating the Annihilation Rate of WeaklyInteracting Massive Particles,”
Phys. Rev. Lett. (2015) 211301, arXiv:1409.4415 [hep-ph] .[30] M. Bauer, T. Cohen, R. J. Hill, and M. P. Solon, “SoftCollinear Effective Theory for Heavy WIMPAnnihilation,”
JHEP (2015) 099, arXiv:1409.7392 [hep-ph] .[31] G. Ovanesyan, T. R. Slatyer, and I. W. Stewart,“Heavy Dark Matter Annihilation from Effective FieldTheory,” Phys. Rev. Lett. (2015) no. 21, 211302, arXiv:1409.8294 [hep-ph] .[32] C. Marcantonini and I. W. Stewart,“Reparameterization Invariant Collinear Operators,”
Phys. Rev.
D79 (2009) 065028, arXiv:0809.1093 [hep-ph] .[33] T. Cohen, G. Elor, and A. J. Larkoski, “Soft-CollinearSupersymmetry,”
In-Preparation .[34] G. Leibbrandt, “Introduction to Noncovariant Gauges,”
Rev. Mod. Phys. (1987) 1067.[35] L. Brink, O. Lindgren, and B. E. W. Nilsson, “N=4Yang-Mills Theory on the Light Cone,” Nucl. Phys.