Soft-collinear supersymmetry

Journal of High Energy Physics, Mar 2017

Abstract Soft-Collinear Effective Theory (SCET) is a framework for modeling the infrared structure of theories whose long distance behavior is dominated by soft and collinear divergences. This paper demonstrates that SCET can be made compatible with super-symmetry (SUSY). Explicitly, the effective Lagrangian for \( \mathcal{N}=1 \) SUSY Yang-Mills is cconstructed and shown to be a complete description for the infrared of this model. For contrast, we also construct the effective Lagrangian for chiral SUSY theories with Yukawa couplings, specifically the single flavor Wess-Zumino model. Only a subset of the infrared divergences are reproduced by the Lagrangian — to account for the complete low energy description requires the inclusion of local operators. SCET is formulated by expanding fields along a light-like direction and then subsequently integrating out degrees-of-freedom that are away from the light-cone. Defining the theory with respect to a specific frame obfuscates Lorentz invariance — given that SUSY is a space-time symmetry, this presents a possible obstruction. The cleanest language with which to expose the congruence be-tween SUSY and SCET requires exploring two novel formalisms: collinear fermions as two-component Weyl spinors, and SCET in light-cone gauge. By expressing SUSY Yang-Mills in “collinear superspace”, a slice of superspace derived by integrating out half the fermionic coordinates, the light-cone gauge SUSY SCET theory can be written in terms of superfields. As a byproduct, bootstrapping up to the full theory yields the first algorithmic approach for determining the SUSY Yang-Mills on-shell superspace action. This work paves the way toward discovering the effective theory for the collinear limit of \( \mathcal{N}=4 \) SUSY Yang-Mills.

A PDF file should load here. If you do not see its contents the file may be temporarily unavailable at the journal website or you do not have a PDF plug-in installed and enabled in your browser.

Alternatively, you can download the file locally and open with any standalone PDF reader:

https://link.springer.com/content/pdf/10.1007%2FJHEP03%282017%29017.pdf

Soft-collinear supersymmetry

Received: October Soft-collinear supersymmetry Timothy Cohen 0 4 Gilly Elor 0 2 4 Andrew J. Larkoski 0 1 3 Eugene 0 OR 0 U.S.A. 0 Cambridge 0 MA 0 U.S.A. 0 Portland 0 OR 0 U.S.A. 0 0 Open Access , c The Authors 1 Center for Fundamental Laws of Nature, Harvard University 2 Center for Theoretical Physics, Massachusetts Institute of Technology 3 Physics Department, Reed College 4 Institute of Theoretical Science, University of Oregon Soft-Collinear E ective Theory (SCET) is a framework for modeling the infrared structure of theories whose long distance behavior is dominated by soft and collinear divergences. This paper demonstrates that SCET can be made compatible with supersymmetry (SUSY). Explicitly, the e ective Lagrangian for N = 1 SUSY Yang-Mills is constructed and shown to be a complete description for the infrared of this model. For contrast, we also construct the e ective Lagrangian for chiral SUSY theories with Yukawa couplings, speci cally the single avor Wess-Zumino model. Only a subset of the infrared divergences are reproduced by the Lagrangian | to account for the complete low energy description requires the inclusion of local operators. SCET is formulated by expanding elds along a light-like direction and then subsequently integrating out degrees-of-freedom that are away from the light-cone. De ning the theory with respect to a speci c frame obfuscates Lorentz invariance | given that SUSY is a space-time symmetry, this presents a possible obstruction. The cleanest language with which to expose the congruence between SUSY and SCET requires exploring two novel formalisms: collinear fermions as two-component Weyl spinors, and SCET in light-cone gauge. By expressing SUSY YangMills in \collinear superspace", a slice of superspace derived by integrating out half the fermionic coordinates, the light-cone gauge SUSY SCET theory can be written in terms of super elds. As a byproduct, bootstrapping up to the full theory yields the rst algorithmic approach for determining the SUSY Yang-Mills on-shell superspace action. This work paves the way toward discovering the e ective theory for the collinear limit of N = 4 SUSY Yang-Mills. ArXiv ePrint: 1609.04430 E ective eld theories; Superspaces; Supersymmetric E ective Theories; Su- - persymmetric gauge theory 5 Collinear superspace Non-interacting chiral multiplet Abelian gauge theory 5.3 A candidate supercurrent 6 Soft-collinear SUSY Yang-Mills 6.1 SUSY Yang-Mills in collinear superspace 6.2 SUSY Yang-Mills in components 7 The collinear Wess-Zumino model Infrared divergences Collinear Lagrangian for the Wess-Zumino model 7.3 The Wess-Zumino model in collinear superspace 8 Outlook A Technical details Wilson lines and SUSY A.2 Collinear factor from SCET in covariant gauge A.3 Notation and identities Contents 1 Introduction and summary 2 SCET for two-component spinors Universal soft and collinear limits The SCET Lagrangian 2.3 RPI in two-components 3 SCET in light-cone gauge 4 Supersymmetry and SCET The Abelian theory in LCG Interactions and IR structure in LCG The non-Abelian theory in LCG Power counting the supersymmetry algebra Super-RPI; the super-Poincare group on the light cone Representations of the light-cone super-Poincare algebra Non-interacting SUSY SCET Lagrangian Introduction and summary and collinear divergence structure of quantum eld theories; see [9, 10] for reviews. This cidate aspects of quantum eld theory | by demonstrating how SUSY can be maintained of results that exists for both SUSY eld theory and even supergravity will allow us to phase space of SUSY models. the interplay of SCET with SUSY theories. superspace" [57], which is the natural setting for SUSY SCET. elds of the (single avor) Wess-Zumino model only accounts for a subset of Higgs is emitted o a top quark. Identifying the local operators and their organization see [19, 58, 59]. that remain after taking the collinear limit of the theory. since it is both RPI and SUSY invariant. gure 1. A few technical appendices are also provided. SCET for two-component spinors Integrate out modes away from light cone Full theory + adjoint fermion Confirm SUSY Full theory light cone gauge SUSY Yang-Mills Integrate out coordinates away from light cone + adjoint fermion Integrate out SUSY Yang-Mills light cone EFT Light cone + adjoint fermion to an adjoint Weyl fermion. Section 3 details the gauge xing procedure and the corresponding analyzed in section 7. it will be relevant for our exploration of the Wess-Zumino model. We will work in Minkowski space with signature g = diag (+1; 1; 1; 1). The collinear direction is taken along the +z^ light-cone direction: n = (1; 0; 0; 1). Then the anti-collinear direction is de ned by n to make the explicit choice n expanded as p = or p = n p; n p; p~? : and in everything that follows, the \ " is a 4-vector dot-product. ? = p22. Here, The collinear limit is de ned by the momentum shells which scale like pn is some dimensionful scale, and 1 is the SCET power counting parameter.1 2 as the virtuality (or allowed distance from the light ; ). Depending on the process of interest there are also soft modes ps ( ; ; ) or ultra-soft modes pus ; 2). When relevant, we will use the ultrasoft in what follows, we will follow standard practice and work with units where = 1. Universal soft and collinear limits current. The tree-level amplitude for emission is p = xy_ (p) i (p + q) motion xy_ (p)( spinor) indices _ range from 1 to 2, and we work in the Weyl basis, see appendix A.3 the soft limit, while both terms are divergent in the collinear limit. To study the soft limit, we rescale the momentum q by 2: q q and insert it into eq. (2.2). We then have Eq. (2.2) ! (n p)(n q) n (n p)=2. In the limit that of the amplitude is ! 0, the rst term dominates, and so the leading soft limit inclusive rate for B ! Xs . In this case a large separation of scales is due to QCD=mb Note that the polarization vector does not scale with since the soft limit is isotropic. By ing factor: S(q) = so-called soft current and is gauge invariant by conservation of charge. gluon polarization vector is constrained. The Ward identity q (q) = 0 must be maintained as an expansion in so that when q is collinear, n (q) must be suppressed by a power of . Additionally, the collinear completeness relation2 (q) = = 0 : implies that 1. It follows that n . We must also use the equation of motion for xy_ (p) which, in the exactly collinear limit, reduces to Then, the leading collinear limit of the amplitude is (n p)(n q) M (p+q) : further discussed in sections 3.2 and 7.1. The SCET Lagrangian 2Only the perpendicular components of the metric exist at leading power in because the physical dominantly in the ? plane. of the IR divergences of QCD. The position-space Weyl equation for a left-handed two component eld u is L = i uy(x) Boosting u(x) along the light-cone in the z^ direction, i.e. n direction, yields:5 x(p) n = N = 1 SUSY. is implied. is a two-component spinor, we have inserted the appropriate factor of 0 to make scalings for the momentum, n p 1 and n p . We see that the upper component of For concreteness, one can construct the following combinations of Pauli matrices: _ = _ = u = (Pn + Pn) u = un + un: Comparing to eq. (2.10) yields the explicit forms Pn un = Pn un = 0 ; un = un ; Pn un = Pn un = 0 : un = un ; de ned for momenta pointing along the n direction. 5For example, see section 3.3 of [67]. L = i uyn;2_ n @ un;2 + i uyn;1_ ( @?)2_1 un;1 + i uyn:1_ n @ un;1; (2.14) where we have used for instance ( term in eq. (2.14) must scale as Ln 4 and the action must be unsuppressed. This xes6 un The scalings of the elds are summarized in table 1. 4 ; the collinear volume element scales as the classical equation of motion: un = propagating mode un = un = 4 will often suppress the subscript \n" and take u2 un;2, and similarly uyn;2_ for a free Weyl fermion: Lun = uyn i n @ whqere we have used in terms of its momentum space representation: ? = (x) = d4p e i p x ~(p) = tum space Lagrangian: = i 6The -scaling of the collinear elds is equivalent to the twist = dm s, where dm is the mass dimension. For the fermion un = 3=2 dimension is = 1 the twist is = 1 0 = 1. O( ), since its mass scale as O(1) which the interactions). For instance, for soft ps scales like O( 2). So the soft volume element d4xs scales as O( 1= 2 since ps xs + Lu + Lg)s, must xes the scalings of the soft and collinear elds (all dependence is moved to 1 and every component of gauge boson elds as = An + As : acting on collinear elds:7 Using An ( 2; 2; 2), it is straightforward to derive the leading i n Dn;s = i n @ + g n An + g n As in Dn = i n @ + g n An iD?;n = i @ respect to the collinear elds. The collinear eld strength can be written as g Fn = i Dn; Dn . This includes the As, which do not transfer enough momentum as to spoil the virtuality of the details, including a discussion of gauge xing, the gauge boson propagator and Feynman expanded as i g Fna ta = n n ) n @; n Dn + D?;n; D?;n 7Here D means the usual covariant derivative. This should not be confused with the common notational Wilson line. spectively. Note that upon contraction of Lorentz indices for (F )2, the leading order Lagrangian density scales as O( 4). This gives the two-component Lagrangian for QCD SCET: L = uyn i n Dn;s + i Lu + Lg : (2.23) two-components of a four-component fermion, D ! D;L = PL D. multiply Xn by a Wilson line for soft gluons to de ne a new eld Xen as The soft Wilson line Yn is de ned by Xen = Yn Xn : Yny Yn = 1 ; Ds Yn = 0 ; typically ignore the soft/ultrasoft gauge boson elds in what follows. Finally for scalars we can write (for a review of scalar SCET see [9]) free Lagrangian @ in powers of , yielding (for simplicity we omit possible gauge interactions):8 Ln = n2 n = Note that 2 O( 2) so the collinear scalar (as well as the soft and anti-collinear scalars) made for the anti-collinear and soft scalar. See table 1 for a summary. 8Note that if one performs a eld rede nition, n ! RPI in two-components invariance (RPI). Note that if all orders in were included, SCET must be equivalent to operators that are consistent with RPI. Practically, RPI can be characterized by noting that any choice of n and n These conditions are invariant under three di erent kinds of reparameterization: ? = n ? = n ? = n ? = 0. The RPI generators and algebra. The Poincare group, relevant to the full Lorentz invariant theory, is de ned by P ; P i = 0 ; ; P i = i g i = n break ve of the Lorentz generators, corresponding to the following: R1 = n M R2 = n M R3 = n n M RPI and the unbroken Lorentz subgroup closes [40]. RPI with the various components of P is: = 2 i P ; = 0 ; = 0 ; = 2 i P ; = 2 i n P ; = 0 : RPI scale: imbuing them with dependence. RPI-III is essentially a boost in the z^ direction, and under the three RPI transformations at every order in to ensure full Lorentz invariance when all-orders are included. The RPI transformations. Next, we motivate the RPI transformations of the EFT and n . The A component transformations can be determined by demanding that the di erent components of the gauge eld. Using the expression for un in eq. (2.15), its RPI-1 transformation is u = 1 1 + Mun ( the matrix Mun ( ?). Demanding that all terms proportional to ? sum to zero and eliminating those terms that are identically zero, we nd ) = been integrated out: of the fermion elds un un = un = must be invariant under RPI-III, and in particular the term Lu = i uyn n @ 2 spinor un does not transform under RPI-III.9 calculation here. SCET in light-cone gauge for a review. Additionally, once LCG has been xed, the gauge eld mode n An is nonsuch that the term in eq. (2.40) can be written as mation u2 ! e =2u2. O( ) parametrizes deviations away from the collinear n direction, and ? parametrizes deviations away from the anti-collinear n direction while maintaining the constraint components of the derivative (n @; @? ), gauge eld, and gauge covariant derivative Dn = i g An all have the same transformations. Note that there is some subtlety by what speci cally is denoted by un; the details given in the text should resolve any ambiguities. \n" on the components of the collinear gauge boson when it is clear by context. between these elds. However, note that gauge and Poincare invariance (and therefore the action of RPI on the LCG elds to future work. for previous work, see e.g. [8, 48]. To begin, we simply set n A = 0. Then we will solve for the equation of motion of the non-propagating gauge mode n A in order to ? = where we have introduce the notation p ? = p @1 + i @2 Wess-Zumino gauge. = complex conjugate r conjugation for A and @. The as is clear from eq. (3.4); the is equivalent to complex -matrix contracted with derivatives, gauge elds, and therefore covariant derivatives can be written in the following matrix form: @ = The Abelian theory in LCG As a warm up, we begin with the model of a free U(1) collinear gauge boson: LU(1) = @ A @ A = (@? A it is straightforward to expand eq. (3.6) in light-cone coordinates: 2LU(1) = A A ) + A where we have used @ ? = form is convenient for converting to the A scalars. We have the following identities: Similarly, we can write the light-cone scalar as Likewise for the transverse gauge covariant derivative:11 pD? = ?2 = ? = A )2 = @22 = ? (n @ n @A?) = A(n @ n @A ) + h.c. ; 11In the notation of Leibbrandt [64] AT where we have integrated by parts to combine factors. The Lagrangian becomes 2LU(1) = + h.c.i + h(@A )2 + (@ A) (@ A + @A )(n @ n A) + of freedom are A and A . Therefore, it is prudent to integrate out n A: LU(1) = 0 =) n A = If A can really be interpreted as a complex scalar, then LU(1) should reduce to the free this explicitly by plugging eq. (3.14) into eq. (3.13), which yields LU(1) = the kinetic term for a complex scalar. Interactions and IR structure in LCG expansion, the soft limits are identical to the full theory and are thus trivially invariant on its own in the collinear limit, we expect to nd the exact same expression in LCG. For simplicity, we will show this agreement for SCET QED in LCG. First, we must construct the Feynman rules for the O(e) interaction vertex coupling the Lorentz vector gauge eld A rather then LC scalars so that the explicit dependence on ieA is the covariant derivative. Expanding the term in brackets yields e uyn(n A) = @ un;p0 + uyn;p tegrated by parts to get the second line. This gives a contribution to the u-uy-A which goes as n2 ; these are the same spin-dependent terms that appear in the standard QCD-SCET Feynman rule, see appendix A.2. = i ? = i n · fermion. The dashed wavy line is a light cone gauge boson. Next we integrate out the non-propagating mode n A: LU(1) = 0 =) n A = O(e) vertex comes from the fermion Lagrangian: e uyn(n A) un = e uyn Feynman rule is given in gure 2. To check that wpe reproduce the expected collinear factor, we compute the following p = i e xyn(p) 2 ? = 0, and we have absorbed the projection operator (n =2) into the de nition of M(p0). We can simplify the expression using the QED Ward identity: =q 0 ) q )(n q) = )(n q) ; (3.21) equation of motion xyn where we have used the LCG condition n n = 0, the amplitude simpli es to and the Weyl p = (n p)(n q) M(p + q); since it required including terms that result from integrating out n A. The non-Abelian theory in LCG gauge part of the Lagrangian, which is the same as YM theory: Lg = 1 F a F a = gf abcAb Ac (@ A a 1 g2f abcf adeAb Ac A dA e Lg;0 + Lg;1 + Lg;2 to terms involving di erent powers of g. Lu = i uy The rst step is to derive the equations of motion for n Aa. Integrating out n Working with a non-Abelian theory will not change the argun Ad = Making use of eq. (2.22) we nd: = g uy2 u2ta i tan(n @)2n A + 2 D? (n @A?)o ; In order to convert to LC elds and derivatives, note that Plugging this in and solving the above for n Aa yields: n Ab = bc(n @Ac) + rbc(n @A ) generators (te)bc = i f ebc and we expanded r bc = bc @ Lagrangian eq. (2.23). Deriving Lg;0 in LCG. To begin, we will derive the terms which come from the part of in eq. (3.29) for n Ab yields the total Lg;0 LCG Lagrangian: Lg;0 = 2 g2f abcf ebh + 2 g2f abcf ebh will nd for the Wess-Zumino model in section 7. Deriving Lg;1 in LCG. Starting with the Lg;1 Lagrangian: Lg;1 = g f abcAb Ac (@ A a we can expand these interactions in light-cone coordinates: light cone scalars: Lg;1 = @ A a + n Ab Ac? (n @Aa )i : ? ? ? eq. (4.37) with A candidate supercurrent to warrant presenting it here. is given by = n un n @ A It has the corresponding conserved charge Q = d(n x) d2x = i Using the light-cone canonical (anti-)commutation relations un(n x; x?); uyn(n y; y?)o = of the LCG collinear gauge eld: = i as expected from eq. (4.37) with while the power suppressed Q1 charges as generating the dynamical supersymmetry. Soft-collinear SUSY Yang-Mills in the context of deriving the UV generic way. SUSY Yang-Mills in collinear superspace section, we will keep the details to a minimum. The superspace action of the full theory is; S = i DDe V D eV is a matrix valued chiral super eld. We again decompose the DD e (Vn+Vn) D e(Vn+Vn) ; DD e (Vn+Vn) D _ e(Vn+Vn) : (6.2) where W =) a = W _ = Expanding the exponents and enforcing Wess-Zumino gauge yields DD D (Vn + Vn) + DD (Vn + Vn); D (Vn + Vn) DD D (Vn + Vn)a + g f abc DD (Vn + Vn)b D (Vn + Vn)c ; where the second line is expressed in the adjoint representation with W Next, we integrate out the anti-collinear super eld Vn. The variation of the action is = 2 a _ = 0 ; The constraint equation is then found by combining eqs. (6.3), (6.4), and (6.5): 0 = D DD D V e + ig f ace D DD V c + D ; V c DD D V a g2f acef adh D ; V c DDV dD V h : be organized into a collinear chiral super eld an(x) = p to construct the superspace Lagrangian of the EFT in LCG:19 L = generalization the LCG SUSY transformations given in eq. (4.37) with ! A, which are SUSY Yang-Mills in components To begin, we need identities such as cn = 2 A + ip2 2uy2aAb + 2 2 y2_ uy2aub2 Expanding in components and integrating over superspace yields g LO(g) = f abc By permutation of asymmetric indices the rst term becomes b(n @Aa) = f abc=n @(@A a) + A c(n @Aa) . With these substitutions, we nd elds), we nd that these terms reproduce the second line of eq. (3.40). The collinear Wess-Zumino model of these features could change in the presence of multiple avors. The massless Wess-Zumino model can be simply expressed in superspace as L = (y) = + 2F is a chiral super eld consisting of a complex scalar , a left-handed Weyl fermion , and an auxiliary eld F . Expanded in components and integrating out the auxiliary eld yields L = The rest of this section is devoted to exploring the IR of this Lagrangian. Infrared divergences single avor collinear Wess-Zumino EFT. The Yukawa coupling in eq. (7.2) leads to the following interaction vertex: p = ( i y)xy_ (p; s) = y xy_ (p; s) i(p + q) equation of motion xy(p)p _ 2 collinea!r divergence. Similarly, taking p collinear and q soft yields _ 2 We see that there are collinear but not soft divergences. _ 2 _ 2 ips the helicity. _ 2 limit. Then the scalings are This is most straightforward to do using eq. (7.4). We can write M = xy_ M , where the last equality is due to the fact that only xy2_ survives in the collinear = y 2 p q xy_ (p; s)xy _ (q; s)M(p + q) : Assuming that the momentum owing through the current p + q 1, there is no soft the current is small. scaling of the spinor product xy_ (p; s)xy _ (q; s) can be derived by noting that X tr xy(p)xy(q)x(q)x(p) = ) = (n p)(n q) ; which implies Hence, in the this limit xy(p)xy(q) xy(p; s)xy(q; s)M(p + q) demonstrating that the splitting of a scalar to collinear fermions is singular. We see that in both examples of 1 ! 2 splittings, the IR singularities contained in non-trivial EFT couplings which appear in the SCET Lagrangian. from the 1 ! 3 process of a collinear fermion emitting a pair of collinear scalars: (p + q2) (p + q1 + q2) (p + q1 + q2)2 M(p + q) : We are interested in the properties of the 1 ! 3 splitting function S1!3 = (p + q2) (p + q1 + q2) S1!3 = n2 n (p + q2) + n2 n (p + q2) n (p + q2) n (p + q2) n2 n (p + q1 + q2) + n2 n (p + q1 + q2) + q1? (p + q1 + q2)2 In the collinear limit, the constraint on the spinor xy(p) is = 0 ; p is incoming. which eliminates terms in S1!3. Additionally, the ? matrix ips helicity, and so projects these constraints, the splitting function dramatically simpli es: S1!3 = 2 n (p + q2) n(p(+p +q1q+1 +q2q)22) ; 1 where we also have used (n we then nd, in the collinear limit, n (p + q2) (p + q1 + q2)2 M(p + q) : (7.15) The term in the square brackets scales like 0, and the propogator factor scales like interacting SCET to reproduce this divergence. Collinear Lagrangian for the Wess-Zumino model classical equation of motion for the anti-collinear eld un; = ( we arrive at the following collinear Lagrangian: Ln = that in the full theory Lagrangian, this cubic coupling has the form u u = clear that this term vanishes due to the anti-commuting nature of the spinors: = 0 of freedom to reproduce the IR of the full theory. EFT sector on its own for the reasons given previously. source J . Begin with the fermion Lagrangian L(J ) = L + un J + uyn J y ; motion for un and plugging it back into the Lagrangian yields with sources: Lu = Ln n (p + q) = y xyn ? (p + q)? M : in (p + q) n (p + q) of the full theory . n¯ @ so RPI is obscured as expec where the derivative onlyCoalcltinseoanrwltihmheeirtesfatscheteeomrdsetniornivsauthtgiegveessqtounsaolyrmeen¯abtchrtaisncgkoenmtn¯sto.hreeTlfihakicest:oirss fuinllythRe PsIquinavrearbiarnatckaentsd. Thi 12 n¯ n¯ n ✓ n in¯ @ 2 y@ 22 ◆ n¯ 4 ¯ nyi2n¯✓ @ y 222 ✓ @ 12 ◆yn¯2n¯in¯@¯2·¯@ ◆2n¯ (0) 2 (0) 2 · @ · (0) | |(0) 2 4 · | 2| · | 2| 4 | | · · 2 In this expression, vertiLcnes =from⇤nt@he nL+aLgunr†nan=ginia·n@ ⇤ne@q. n( 7?+.2u1†n) airnLen·uw@nr=ittLenn?⇤n|@w=nit|hni+n⇤nu@ut†nhne|nisn+nq|·uu@a†n|rue†ni|n ·? @ | ? un|nun ,u†n ·| · 2 for instance using the identity _ where. the derivative only acts on the fact _ _ the frame where p In eq. (7.4), making this choice leaves only the product q at leading power in the collinear limit, demonstrating that the two approaches agree. Infrared structure: 1 However, as illustrated in gure 4, the full theory Yukawa interaction generates the four Wess-Zumino theory collinear structure discussed in section 7.1. = xy (p)( i y ) n j j n (p + q ) (p + q + q ) 2 1 2 (p + q) ; (p + q) (7.23) from which we can read o the full theory. . This agrees with eq. (7.15), the explicit calculation in Reparameterization invariance. The key step in deriving the fermion terms in the into un and un by projecting with n , see eq. (2.13). Because the vector n breaks RPI transformations are the Lorentz generators that are broken when xing n , which in for gauge theories; we apply the same arguments here to the Wess-Zumino model. u = un; + un; = un; + ( transformations. For concreteness, we can consider RPI-I. A linear transformation implies that uyn ! T uyn ; n since it is a scalar. It is possible that the transformations of the two terms for this case the n vector does shift, so the argument is not as simple. T a] transforms linearly under the broken generators T a. Zumino Model,21 taking L= yn yields W = DD = 0 : term has a higher scaling and should be dropped. In terms of super elds we postulate the solution The Wess-Zumino model in collinear superspace this result to the similar interactions that arise in N 1 SUSY Yang-Mills theories, see e.g. eq. (6.8). The intuition we gain in the simple case at hand will be useful for future work. As in previous sections we begin by noting that supersymmetry requires = Therefore we expand the full theory Lagrangian as L = 2 W ( n; n) + h.c. : Taking the variation in superspace with respect to yn yields the constraint equation; DD = 0 ; n = 4 i (n @)D1 the Lagrangian, an interaction term is inherited from the Yukawa coupling: Ln = = i uy2 n @ u2 This reproduces the desired interaction that we found above in eq. (7.17). 21Note that 3n and 3n vanish due to the de nition of these projected super elds. W = results derived for SUSY models, for instance the exact NSVZ -function [84], Seiberg of collinear superspace and its possible extensions. a judicious choice of R-dependent projection operators. In particular, take the following Yukawa theory with multiple avors, L = i uiy( @)ui + i vky ( ylik l vky ui + uiyvk + h.c. un;i = vn;i = which lead to tri-linear interactions such as avor Wess-Zumino model above, due to the fact that a collinear fermion antiavor structure in a non-trivial way. for Higgsstrahlung o a top quark. understanding of collinear EFTs and beyond. Acknowledgments ni cant improvements for our treatment of the Wess-Zumino model. We thank Marat Freytsis, Ian Low, Du Neill, Arvind Rajaraman, and David Shih for discussions. TC supported by National Science Foundation grant PHY-1066293. Technical details appendix A.3 contains notation and conventions used throughout this paper. Wilson lines and SUSY eq. (2.33) of [10]. The QCD Lagrangian can therefore be written as; Lun = uyn i n D + i Wn = In other words, the eld n An can be summed into a collinear Wilson line. This has is given by: yn un+uynn = Wn + p un + uynn (Wn) = p (iD?) = un + uynn For additional discussions regarding SUSY Wilson lines see [96] and [97]. Collinear factor from SCET in covariant gauge The SCET Lagrangian gives rise to fermion-gauge interactions [10]: (n p0)(n p) : (A.6) (n p)(n q) (n p)(n q) n (p + q) iM(p + q) (n p)(n q) M(p + q); projection operator (n matrix identity + 2g . This expression matches the amplitude computed in eq. (2.8) and eq. (3.22) as expected. We work in Minkowski space with metric signature g = diag (+1; 1; 1; 1). Throughthe -matrices take the following form; = 1; i , = 1; i , and the Pauli matrices are: 0 = 1 = 3 = The upper/lower spinor index convention is Spinor indices are raised and lowered using the anti-symmetric -matrix: 2 = = 2g ; = 2g = g = g (g and we contract pairs of spinors and anti-spinors follows the usual way: Sigma matrix identities. Throughout this work we make use of various sigma-matrix identities. Those that were used most often are For additional identities see [65] and [76]. Shorthand conventions for collinear To keep the notation from being too only drop the collinear subscript when dealing with components: Note we also drop the \dot" on the subscript of the conjugate eld. @22 = literature which relate @ terms of this notation 2 to the explicit component expression with the opposite sign. In D _ = of objects in LCG. For instance, @ = @ = Note the slight abuse of notation; p2@ = @1 A . These expressions are independent of the choice of n , and n direction. Note that throughout we include the Lorentz contraction in the de nitions of @2 , as this is convenient when working with LCG scalars; we make use of the following identities: D DDD V = D _ DDD _ V ; DDD _ DD = 0 = DDD DD ; D D D = 0 = D _ D _ D _ ; [DD; D _ ] = [DD; D ] = 4i( ) _ @ D _ ; D DDD = D _ DDD _ = DDDD + 4iD ( = 162 + 4iD ( and D1 = D2), e.g. eq. (A.20); [DD; D1_ ] = [DD; D2_ ] = 2@ D2) = 2@D1) = 4i(n @D2 2@ D1) ; 2@D2) ; LY = i D D D + h.c. : D = The Lagrangian becomes LY = ivy( @)v + iuy( y (vyu + uyv) + h.c. left and right handed Weyl spinors: u = (Pn;L + Pn;L) u = un + un ; v = (Pn;R + Pn;R) v = vn + vn ; which we made use of for the derivation of collinear superspace, e.g. DDD1D1_ DD / DDD1h(n @)D1 D2D1_ Vn / DDDD(n @) D2D1_ Vn Yukawa theory. The Yukawa theory Lagrangian for a four component fermion is: [1] C.W. Bauer, S. Fleming and M.E. Luke, Summing Sudakov logarithms in B ! X(s ) in e ective eld theory, Phys. Rev. D 63 (2000) 014006 [hep-ph/0005275] [INSPIRE]. and soft gluons: heavy to light decays, Phys. Rev. D 63 (2001) 114020 [hep-ph/0011336] [INSPIRE]. 516 (2001) 134 [hep-ph/0107001] [INSPIRE]. Phys. Rev. D 65 (2002) 054022 [hep-ph/0109045] [INSPIRE]. Pn;L = Pn;L = Pn;R = Pn;R = )un = 0 ; )un = 0 ; )vn = 0 ; )vn = 0 : Open Access. This article is distributed under the terms of the Creative Commons any medium, provided the original author(s) and source are credited. Phys. Rev. D 66 (2002) 054005 [hep-ph/0205289] [INSPIRE]. B 657 (2003) 229 [hep-ph/0211018] [INSPIRE]. [hep-ph/0206152] [INSPIRE]. Notes Phys. 896 (2015) pp.1{206 [arXiv:1410.1892] [INSPIRE]. Lett. 87 (2001) 201806 [hep-ph/0107002] [INSPIRE]. e decays, Nucl. Phys. B 649 (2003) 349 [hep-ph/0210091] [INSPIRE]. factorization from e ective eld theory, Phys. Rev. D 66 (2002) 014017 [hep-ph/0202088] [INSPIRE]. initial state jets, Phys. Rev. D 81 (2010) 094035 [arXiv:0910.0467] [INSPIRE]. distributions in soft-collinear e ective theory, Phys. Rev. D 81 (2010) 093007 [arXiv:0911.4135] [INSPIRE]. and the collinear anomaly, Eur. Phys. J. C 71 (2011) 1665 [arXiv:1007.4005] [INSPIRE]. coloured heavy (s)particles at hadron colliders, Nucl. Phys. B 842 (2011) 414 [arXiv:1007.5414] [INSPIRE]. e ective eld theory, Phys. Rev. Lett. 100 (2008) 021802 [arXiv:0709.2377] [INSPIRE]. processes using e ective eld theory, Phys. Rev. D 77 (2008) 053004 [arXiv:0712.0396] [INSPIRE]. [INSPIRE]. radiation, JHEP 09 (2016) 025 [arXiv:1601.07190] [INSPIRE]. interacting massive particles, Phys. Rev. Lett. 114 (2015) 211301 [arXiv:1409.4415] WIMP annihilation, JHEP 01 (2015) 099 [arXiv:1409.7392] [INSPIRE]. eld theory, Phys. Rev. Lett. 114 (2015) 211302 [arXiv:1409.8294] [INSPIRE]. light decays, Phys. Rev. D 67 (2003) 071502 [hep-ph/0211069] [INSPIRE]. factorization, Phys. Rev. Lett. 93 (2004) 252001 [hep-ph/0408249] [INSPIRE]. theory, Phys. Rev. D 76 (2007) 074002 [hep-ph/0605001] [INSPIRE]. Drell-Yan processes, arXiv:0802.2973 [INSPIRE]. theory, JHEP 07 (2011) 077 [arXiv:1010.1027] [INSPIRE]. [INSPIRE]. (2014) 105020 [arXiv:1403.6472] [INSPIRE]. factorization violation, JHEP 08 (2016) 025 [arXiv:1601.04695] [INSPIRE]. Nucl. Phys. B 522 (1998) 321 [hep-ph/9711391] [INSPIRE]. [arXiv:1111.2589] [INSPIRE]. radiation: a diagrammatic approach, JHEP 01 (2011) 141 [arXiv:1010.1860] [INSPIRE]. [arXiv:1404.5551] [INSPIRE]. graviton theorems, Phys. Rev. D 90 (2014) 065024 [arXiv:1406.6574] [INSPIRE]. from gauge invariance, Phys. Rev. D 90 (2014) 084035 [arXiv:1406.6987] [INSPIRE]. collinear operators, Phys. Lett. B 539 (2002) 59 [hep-ph/0204229] [INSPIRE]. 114023 [arXiv:0908.4559] [INSPIRE]. [INSPIRE]. [INSPIRE]. [INSPIRE]. [INSPIRE]. e ective theory, Phys. Rev. D 69 (2004) 054017 [hep-ph/0308122] [INSPIRE]. Rev. Lett. 108 (2012) 151601 [arXiv:1104.0881] [INSPIRE]. Lett. B 704 (2011) 276 [arXiv:1104.4108] [INSPIRE]. rapidity logarithms in quantum eld theory, JHEP 05 (2012) 084 [arXiv:1202.0814] nonAbelian gauge symmetry, Phys. Lett. B 553 (2003) 267 [hep-ph/0211358] [INSPIRE]. lines, Phys. Rev. D 84 (2011) 011502 [arXiv:1104.0686] [INSPIRE]. the scattering equations, JHEP 02 (2017) 038 [arXiv:1608.04730] [INSPIRE]. theory, Phys. Rev. D 83 (2011) 016015 [arXiv:1011.6145] [INSPIRE]. Phys. Rev. Lett. 113 (2014) 261604 [arXiv:1405.6350] [INSPIRE]. hadronization, Nucl. Phys. B 513 (1998) 269 [hep-ph/9707338] [INSPIRE]. [INSPIRE]. arXiv:1611.00788 [INSPIRE]. e ective eld theories, Phys. Lett. B 286 (1992) 348 [hep-ph/9205228] [INSPIRE]. [INSPIRE]. eld theory and supersymmetry, Phys. Rept. 494 (2010) 1 [66] R.K. Ellis, W.J. Stirling and B.R. Webber, QCD and collider physics, Cambridge Press, Cambridge U.K. (1996). U.S.A. (1995). Cambridge U.K. (2014). [INSPIRE]. Princeton U.S.A. (1992). Oxford U.K. (2006). Oxford U.K. (2006). scattering from an external eld, Phys. Rev. D 3 (1971) 1382 [INSPIRE]. Mathematics and Physics, September 24{27, Kiev, Ukraine (2000). (1983) 381 [INSPIRE]. Phys. B 435 (1995) 129 [hep-th/9411149] [INSPIRE]. in ation, JHEP 10 (2015) 001 [arXiv:1504.05958] [INSPIRE]. models, JHEP 05 (2016) 082 [arXiv:1603.02661] [INSPIRE]. supersymmetry, in general, JHEP 04 (2016) 065 [arXiv:1603.02653] [INSPIRE]. [INSPIRE]. [97] V. Pestun, Wilson loops in supersymmetric gauge theories, Ph.D. thesis, Princeton [6] R.J. Hill and M. Neubert , Spectator interactions in soft collinear e ective theory, Nucl . Phys. [7] J. Chay and C. Kim , Collinear e ective theory at subleading order and its application to heavy-light currents , Phys. Rev. D 65 ( 2002 ) 114016 [hep-ph/0201197] [INSPIRE]. [8] M. Beneke , A.P. Chapovsky , M. Diehl and T. Feldmann , Soft collinear e ective theory and heavy to light currents beyond leading power, Nucl . Phys . B 643 ( 2002 ) 431 [9] T. Becher , A. Broggio and A. Ferroglia , Introduction to soft-collinear e ective theory , Lect. [10] I.W. Stewart , Lectures on the soft-collinear e ective theory , lecture notes ( 2013 ). [11] C.W. Bauer , D. Pirjol and I.W. Stewart , A proof of factorization for B ! D , Phys. Rev. [12] E. Lunghi, D. Pirjol and D. Wyler, Factorization in leptonic radiative B ! [13] C.W. Bauer, S. Fleming, D. Pirjol, I.Z. Rothstein and I.W. Stewart, Hard scattering [14] I.W. Stewart, F.J. Tackmann and W.J. Waalewijn, Factorization at the LHC: From PDFs to [15] S. Mantry and F. Petriello, Factorization and resummation of Higgs boson di erential [16] T. Becher and M. Neubert, Drell-Yan production at small qT , transverse parton distributions [17] M. Beneke, P. Falgari and C. Schwinn, Threshold resummation for pair production of [18] J.-y. Chiu, F. Golf, R. Kelley and A.V. Manohar, Electroweak Sudakov corrections using [19] J.-y. Chiu, F. Golf, R. Kelley and A.V. Manohar, Electroweak corrections in high energy [20] C.W. Bauer and N. Ferland, Resummation of electroweak Sudakov logarithms for real [21] M. Baumgart, I.Z. Rothstein and V. Vaidya, Calculating the annihilation rate of weakly [22] M. Bauer, T. Cohen, R.J. Hill and M.P. Solon, Soft collinear e ective theory for heavy [24] C.W. Bauer, D. Pirjol and I.W. Stewart, Factorization and endpoint singularities in heavy to [25] J.C. Collins and A. Metz, Universality of soft and collinear factors in hard-scattering [26] A.V. Manohar and I.W. Stewart, The zero-bin and mode factorization in quantum eld [27] F. Liu and J.P. Ma, Glauber gluons in soft collinear e ective theory and factorization of [28] C.W. Bauer, B.O. Lange and G. Ovanesyan, On Glauber modes in soft-collinear e ective [29] S. Fleming, The role of Glauber exchange in soft collinear e ective theory and the Balitsky-Fadin-Kuraev-Lipatov equation, Phys. Lett. B 735 (2014) 266 [arXiv:1404.5672] [30] I. Feige and M.D. Schwartz, Hard-soft-collinear factorization to all orders, Phys. Rev. D 90 [31] I.Z. Rothstein and I.W. Stewart, An e ective eld theory for forward scattering and [32] M. Beneke and V.A. Smirnov, Asymptotic expansion of Feynman integrals near threshold, [33] B. Jantzen, Foundation and generalization of the expansion by regions, JHEP 12 (2011) 076 [34] E. Laenen, L. Magnea, G. Stavenga and C.D. White, Next-to-eikonal corrections to soft gluon [35] E. Casali, Soft sub-leading divergences in Yang-Mills amplitudes, JHEP 08 (2014) 077 [36] J. Broedel, M. de Leeuw, J. Plefka and M. Rosso, Constraining subleading soft gluon and [37] Z. Bern, S. Davies, P. Di Vecchia and J. Nohle, Low-energy behavior of gluons and gravitons [38] A.J. Larkoski, D. Neill and I.W. Stewart, Soft theorems from e ective eld theory, JHEP 06 [41] J.F. Donoghue and D. Wyler, On Regge kinematics in SCET, Phys. Rev. D 81 (2010) [44] J.-y. Chiu, A. Jain, D. Neill and I.Z. Rothstein, The rapidity renormalization group, Phys. [45] T. Becher , M. Neubert and D. Wilhelm , Electroweak gauge-boson production at small qT : infrared safety from the collinear anomaly , JHEP 02 ( 2012 ) 124 [arXiv:1109.6027] [46] T. Becher , G. Bell and M. Neubert , Factorization and resummation for jet broadening , Phys. [47] J.-Y. Chiu , A. Jain , D. Neill and I.Z. Rothstein , A formalism for the systematic treatment of [48] M. Beneke and T. Feldmann , Multipole expanded soft collinear e ective theory with [49] A. Idilbi and I. Scimemi , Singular and regular gauges in soft collinear e ective theory: the introduction of the new Wilson line T, Phys. Lett . B 695 ( 2011 ) 463 [arXiv:1009.2776] [50] M. Garcia-Echevarria , A. Idilbi and I. Scimemi , SCET, light-cone gauge and the T-Wilson [51] M. Beneke and G. Kirilin , Soft-collinear gravity, JHEP 09 ( 2012 ) 066 [arXiv:1207.4926] [52] D. Nandan , J. Plefka and W. Wormsbecher , Collinear limits beyond the leading order from [53] J. Chay and J.Y. Lee , N = 4 supersymmetric Yang-Mills theory in soft-collinear e ective [54] B. Basso , A. Sever and P. Vieira , Collinear limit of scattering amplitudes at strong coupling , [55] F.A. Berends and H. Kuijf , Jets at the LHC , Nucl. Phys . B 353 ( 1991 ) 59 [INSPIRE]. [56] M.H. Seymour , Jet shapes in hadron collisions: higher orders , resummation and [57] T. Cohen , G. Elor and A.J. Larkoski , Collinear superspace, Phys. Rev. D 93 ( 2016 ) 125013 [58] J.-y . Chiu, R. Kelley and A.V. Manohar , Electroweak corrections using e ective eld theory: applications to the LHC , Phys. Rev . D 78 ( 2008 ) 073006 [arXiv:0806.1240] [INSPIRE]. [59] A. Manohar , B. Shotwell , C. Bauer and S. Turczyk , Non-cancellation of electroweak logarithms in high-energy scattering , Phys. Lett . B 740 ( 2015 ) 179 [arXiv:1409. 1918 ] [60] J. Chen , T. Han and B. Tweedie , Electroweak splitting functions and high energy showering , [61] M.E. Luke and A.V. Manohar , Reparametrization invariance constraints on heavy particle [62] J. Heinonen , R.J. Hill and M.P. Solon , Lorentz invariance in heavy particle e ective theories , Phys. Rev. D 86 ( 2012 ) 094020 [arXiv:1208.0601] [INSPIRE]. [63] S. Mandelstam , Light cone superspace and the ultraviolet niteness of the N = 4 model , [64] G. Leibbrandt , Introduction to noncovariant gauges, Rev. Mod. Phys . 59 ( 1987 ) 1067 [65] H.K. Dreiner , H.E. Haber and S.P. Martin , Two-component spinor techniques and Feynman [67] M.E. Peskin and D.V. Schroeder , An introduction to quantum eld theory , Westview Press, [68] J. Collins , Foundations of perturbative QCD , Cambridge University Press, Cambridge U.K. [69] M.D. Schwartz , Quantum eld theory and the standard model , Cambridge University Press, [70] J.B. Kogut and D.E. Soper , Quantum electrodynamics in the in nite momentum frame , [71] J.D. Bjorken , J.B. Kogut and D.E. Soper , Quantum electrodynamics at in nite momentum: [72] E. Tomboulis , Quantization of the Yang-Mills eld in the null-plane frame , Phys. Rev. D 8 [73] G. Leibbrandt , The light cone gauge in Yang-Mills theory , Phys. Rev. D 29 (1984) 1699 [74] J. Wess and J. Bagger , Supersymmetry and supergravity, Princeton University Press, [75] J. Terning , Modern supersymmetry: dynamics and duality , Oxford Science Publications, [76] P. Binetruy , Supersymmetry: theory, experiment and cosmology, Oxford University Press, [77] S. Duplij and J. Wess , Noncommutative structures in mathematics and physics , in the [80] S.J. Gates Jr., and W. Siegel , Variant super eld representations, Nucl . Phys . B 187 ( 1981 ) [81] R. Kallosh , N = 8 supergravity on the light cone , Phys. Rev. D 80 (2009) 105022 [82] R. Kallosh , On a possibility of a UV nite N = 8 supergravity , arXiv:0808.2310 [INSPIRE]. [83] A.V. Belitsky , S.E. Derkachov , G.P. Korchemsky and A.N. Manashov , Quantum integrability in super Yang-Mills theory on the light cone , Phys. Lett . B 594 ( 2004 ) 385 [84] V.A. Novikov , M.A. Shifman , A.I. Vainshtein and V.I. Zakharov , Exact Gell-Mann -Low [85] N. Seiberg , Electric-magnetic duality in supersymmetric nonAbelian gauge theories , Nucl. [86] N. Seiberg and E. Witten , Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD, Nucl . Phys . B 431 ( 1994 ) 484 [hep-th /9408099] [INSPIRE]. [87] N. Seiberg and E. Witten , Electric-magnetic duality, monopole condensation and con nement in N = 2 supersymmetric Yang-Mills theory, Nucl . Phys . B 426 ( 1994 ) 19 [Erratum ibid . B 430 ( 1994 ) 485] [hep-th/9407087] [INSPIRE]. [88] G. Dall'Agata and F. Farakos , Constrained super elds in supergravity , JHEP 02 ( 2016 ) 101 [89] S. Ferrara , R. Kallosh and J. Thaler , Cosmology with orthogonal nilpotent super elds , Phys. [90] Y. Kahn , D.A. Roberts and J. Thaler , The Goldstone and Goldstino of supersymmetric [91] Z. Komargodski and N. Seiberg , From Linear SUSY to Constrained Super elds , JHEP 09 [93] R. Kallosh , A. Karlsson , B. Mosk and D. Murli , Orthogonal nilpotent super elds from linear [94] S. Ferrara , R. Kallosh , A. Van Proeyen and T. Wrase , Linear versus non-linear [95] G. Dall'Agata , E. Dudas and F. Farakos , On the origin of constrained super elds , JHEP 05 [96] K. Zarembo , Supersymmetric Wilson loops, Nucl. Phys . B 643 ( 2002 ) 157 [hep-th/0205160]


This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1007%2FJHEP03%282017%29017.pdf

Timothy Cohen, Gilly Elor, Andrew J. Larkoski. Soft-collinear supersymmetry, Journal of High Energy Physics, 2017, 17, DOI: 10.1007/JHEP03(2017)017