The Soft Quark Sudakov

There has been recent interest in understanding the all loop structure of the subleading power soft and collinear limits, with the goal of achieving a systematic resummation of subleading power infrared logarithms. Most of this work has focused on subleading power corrections to soft gluon emission, whose form is strongly constrained by symmetries. In this paper we initiate a study of the all loop structure of soft fermion emission. In $\mathcal{N}=1$ QCD we perform an operator based factorization and resummation of the associated infrared logarithms, and prove that they exponentiate into a Sudakov due to their relation to soft gluon emission. We verify this result through explicit calculation to $\mathcal{O}(\alpha_s^3)$. We show that in QCD, this simple Sudakov exponentiation is violated by endpoint contributions proportional to $(C_A-C_F)^n$ which contribute at leading logarithmic order. Combining our $\mathcal{N}=1$ result and our calculation of the endpoint contributions to $\mathcal{O}(\alpha_s^3)$, we conjecture a result for the soft quark Sudakov in QCD, a new all orders function first appearing at subleading power, and give evidence for its universality. Our result, which is expressed in terms of combinations of cusp anomalous dimensions in different color representations, takes an intriguingly simple form and also exhibits interesting similarities to results for large-x logarithms in the off diagonal splitting functions.


Introduction
One of the most powerful approaches to understanding the all orders perturbative structure of scattering amplitudes and cross sections is to consider simplifying kinematic limits, such as the soft, collinear, or Regge limits. While much is known about the leading power behavior of these particular limits, there has recently been considerable effort to understand the all orders structure of subleading power corrections , and ultimately formalize a systematic power expansion of amplitudes and cross sections about these limits.
Recently, using soft collinear effective theory (SCET) [45][46][47][48], subleading power infrared logarithms have been resummed to all orders both for the thrust event shape observable [1], which involves both soft and collinear real radiation, and for the case of threshold logarithms [33,49], which involve only soft real radiation (threshold logarithms were also resummed using diagramatic arguments in [41]). However, both these cases considered only subleading power corrections to gluon emission, which, as shown in [1], are related by symmetries to the leading power gluon emission, and therefore exhibit an exponentiation into a standard Sudakov exponential [50]. To understand more generally the structure of subleading power soft and collinear limits, it is necessary to understand cases that go beyond these corrections to soft gluon emission. The most interesting such case is the emission of soft quarks, which generically contribute at leading logarithm at subleading power. Soft quark emission has been used to calculate subleading power corrections in fixed order perturbation theory in Refs. [22,23,26,35,36,38,40,43], but its all orders structure has not been considered.
-1 -In this paper, we initiate a study of the all loop structure of subleading power contributions in the soft and collinear limits that are related to the emission of soft quarks (for example soft quark emission is related to the collinear limit of quark pairs). Through explicit fixed order calculations, we illustrate that the loop level structure of these limits is more non-trivial than for subleading power soft gluon emission, and in particular, that the naive Sudakov exponentiation is violated, even at leading logarithm, by terms proportional to the difference of color generators, C A − C F . By comparing the results of our fixed order calculation with the organization of the effective theory, we provide insight into the physical nature of the violation of naive Sudakov exponentiation. Interestingly, we show that these terms arise as endpoint divergences in convolution integrals.
To understand the all loop structure of soft quark emission in a simplified context, we consider N = 1 SUSY QCD (without matter multiplets 1 ), where fermions are present but these endpoint divergences are absent. Here we perform an operator level factorization and renormalization extending the treatment of subleading power resummation in pure gluodynamics presented in [1]. This involves the introduction of several new subleading power jet and soft functions involving soft quarks. Using this formalism, we are able to prove that in N = 1 QCD, subleading power logarithms associated with soft quarks exponentiate into a standard Sudakov.
Although we do not here solve the problem of endpoint divergences in QCD, based on our fixed order calculations, combined with intuition from the case of N = 1 QCD, we conjecture a form for the leading logarithmic "soft quark Sudakov" in QCD. Our result takes a simple, but interesting form with similarities to structures observed in the large-x resummation of off-diagonal splitting functions [51][52][53][54]. We also show that this soft quark Sudakov seems to have some degree of universality. Our paper therefore provides three new ingredients towards understanding factorization at subleading power • An explicit calculation of the subleading power collinear limits involving quarks for e + e − → 3 parton and H → 3 parton amplitudes at two loops.
• An operator level factorization and resummation of subleading power logarithms in N = 1 QCD, involving new jet and soft functions with soft quarks that go beyond those considered in [1].
• A conjecture for the full leading logarithmic soft quark Sudakov in QCD.
These provide a considerable step beyond the resummation of subleading power corrections to soft gluon emission that have been considered to date. In particular, we hope that by highlighting that endpoint divergences can have an effect even at leading logarithmic order, and by giving them a physical interpretation as arising when there are fields in different color representations, we provide some insight into an eventual complete field theory solution of subleading power resummation in QCD.
1 Which can be obtained from QCD by setting CF → CA and n f → CA.
-2 -An outline of this paper is as follows. In Sec. 2 we study the subleading power collinear limits of amplitudes at loop level, highlighting interesting effects that arise when considering collinear limits that do not exist at leading power. We also discuss the relation of these results to the violation of the most naive anticipated form of factorization at subleading power (Sudakov exponentiation). In Sec. 3 we perform an operator level factorization and resummation in N = 1 QCD, building on our work in [1]. In Sec. 4 we conjecture a form for the leading logarithmic soft quark Sudakov in QCD. We conclude in Sec. 5, and discuss some remaining open issues for achieving a complete understanding of leading logarithmic subleading power resummation in QCD.

Subleading Power Collinear Limits at Loop Level
In this section we explicitly compute the subleading power collinear limits of e + e − → 3 partons and H → 3 partons and study the behavior of the limits that do not have a leading power analogue. The matrix elements for e + e − → 3 partons and H → 3 partons were computed at two loops in [55][56][57], and we obtain our results by expanding these amplitudes in the collinear limits. Details of how this expansion is performed were given in [1]. The results of [55][56][57] provide the complete color structure, so that in addition to QCD, we can obtain the result in N = 1 SUSY with an adjoint gluino by setting C F → C A and n f → C A .
Note that there has recently been significant interest in Higgs form factors from the perspective of maximal transcendentality, and their relation to similar form factors in N = 4 [58][59][60][61][62], which have primarily focused on the C F → C A limit. Here we are interested in the structure of the maximally transcendental terms that persist in the C F → C A limit.

Physical Observables from Consistency Relations
We are ultimately interested in the behavior of subleading power corrections to infrared event shape observables. Such event shapes are complicated to directly compute perturbatively, since they involve both real and virtual corrections. An efficient way to obtain the leading logarithms is to use consistency relations derived in [22,26], which allow the leading logarithms to be expressed entirely in terms of virtual corrections that can computed efficiently using known amplitudes. Here we briefly review this approach to make clear why we focus on a particular set of virtual corrections. More details can be found in [22,26].
The N -loop fixed order result for an event shape τ at next-to-leading power (NLP), i.e. terms that scale like τ 0 modulo logarithms, can be written as [22,26] The subleading poles in are only relevant beyond LL order, and will not be considered here. The sum in Eq. (2.1) is over κ, which denotes different combinations of soft, collinear, or hard loops (we will often use the word "loops" generically, such that it includes both loop and -3 -phase space integrals). The power of τ appearing in the result, denoted m(κ) is determined by the number of soft, collinear or hard loops. As a concrete example, at one-loop we can have either For an infrared safe observable, the constraint that all infrared poles in cancel places strong constraints on the coefficients of different poles. It can be shown (see [22,26] for details) that the leading logarithmic result can be expressed entirely in terms of the hard-collinear contributions, i.e. those with one collinear loop, and N − 1 hard loops. Explicitly, Here c hc,2N −1 denotes the leading pole in with N − 1 hard loops correcting a single collinear splitting. Because of this, we can gain insight into the structure of the leading logarithms for infrared safe observables by studying the virtual corrections to the subleading power collinear limits of squared amplitudes. These must then be integrated over the simple two particle collinear phase space to obtain the result. This will enable us to compute subleading power corrections to event shape observables to O(α 3 s ) analytically, which is not feasible directly. Additionally, the study of these virtual corrections is interesting in its own right.

e + e − → 3 Partons
We begin by looking at the case of e + e − → qqg. Here we will be interested in the limit where the two quarks become collinear, which does not have a leading power analog. The analog of the analysis below where the gluon becomes collinear to a quark, has been considered in [1]. We label the kinematics as 3 Quark-antiquark collinear limit for e + e In this case, we write the collinear expansion as We define s = (p 1 + p 2 ) 2 in this section.

LO splitting
At LP, the qq splitting function vanishes, The LO NLP splitting function is Note that there is both z 1 and (1 z) 1 poles in this splitting function, which means that we need to keep track of ln z and ln(1 z) to all orders in ✏ at higher order in ↵ s . 4) and the superscripts in P (i,j) qq are the power i in the collinear expansion, and the order j in the α s expansion and here we are always keeping the number of external particles fixed, hence higher orders in α s indicate higher loop corrections. Note that we have -4 -meaning that the leading power (LP) contribution to the limit where the two quarks become collinear vanishes at all orders in α s . Hence, as anticipated, this limit starts at next-toleading-power (NLP). At NLP at tree level, we write the result as Here the exponent (2, 0) labels that this is subleading power, and tree level. We now consider loop corrections to this result. We keep only the highest transcendental terms, which are obtained by assigning 1/ n , log(z) n and log(1 − z) n transcendental weight n and keeping only the weight 2 part of the result, and ignore weight 2 functions which are not logarithmic divergent in the z → 0 or z → 1 limit, such as Li 2 (z), since these will not contribute leading logarithms once integrated against the collinear phase space. At one-loop, we find where s = (p 1 + p 2 ) 2 is the invariant mass of the qq pair. Note that since p µ 1 and p µ 2 are collinear to the same direction, their invariant mass s is much smaller than the hard scale of the process, which we identify with Q. In formulae where τ is an infrared power counting parameter that we will ultimately identify with a physical observable. In terms of Casimirs we have the following expressions for the color generators (2.9) Note that, even though each term in Eq. (2.7) carries a 1/ 2 , only the terms in the first line of Eq. (2.7) are predicted by Catani's dipole formula [63] and its generalization [64][65][66][67]. The other terms in Eq. (2.7), after expanding in , will generate 1/ 2 and 1/ contributions that cancel against each other, leaving only finite terms that would belong to the remainder part of the amplitude. However, they contribute at leading logarithmic order once integrated over the collinear phase space. The fact that this can happen is generic and not a new finding of this paper. A more detailed discussion of this point can be found, for example, in Sec. 5 of [1]. When integrating Eq. (2.7) over collinear phase space, the leading result will come from the z → 0, or z → 1 limits. It is therefore interesting to study the behavior of this result in -5 -these limits. Since the result is symmetric in z ↔ 1 − z, we consider z → 0. We find This result is quite interesting. The naive intuition from leading power factorization is that the leading double logs (equivalently the leading poles in ) arise from soft gluons that view the two collinear quarks coherently, and are insensitive to their energy fractions. This would lead to a cusp like result [68,69] with color structure (T 1 + T 2 ) · T 3 (in this case an adjoint cusp), as illustrated schematically by qq (z, ) were non-singular as z → 0, then the second term in Eq. (2.10) could be expanded in , and would not contribute to the leading logarithm, and we would have exactly this picture. However, instead, we get a contribution from z → 0 that violates this naive expectation. We will refer to this as an endpoint contribution. Note that if T 1 = T 3 , as would occur in N = 1 SUSY QCD, then the naive factorization picture holds. We will discuss the interpretation in terms of factorization in SCET in Sec. 2.4. This analysis can be extended to two-loops 2 . Remarkably, we find that the one loop result simply exponentiates The O( 0 ln 2 ) of this result can be checked against the expansion of the analytic amplitudes (squared) for e + e − → 3 jets at NNLO [55]. Given the size of these O(α 3 s ) expressions, the extraction of the subleading collinear limit from them is non trivial. However, to extract 2 Note that P (2,n) qq is the NLP O(α n+1 s ) contribution to the amplitude squared in the collinear limit. Therefore, P (2,2) qq contains both the subleading two loop correction interfering with the tree level as well as the squared one loop subleading splitting function. Note also that since next-to-leading power corrections start with one real emission, P (i,2) qq is part of the O(α 3 s ) RVV terms contributing to color singlet decay.
-6 -only the leading transcendental part is quite simple. The matrix elements squared is written in terms of rational functions multiplying transcendental functions. We can simplify the procedure of taking the collinear limit of such expressions by performing the expansion of the transcendental functions at the symbol level. 3 Then the only functions that are needed beyond the leading collinear limit are the rational functions, whose expansion is straightforward. Alternatively, the result of [55] is expressed in terms of two-dimensional harmonic polylogarithms that can be directly expanded using [70].
Coming back to the result of Eq. (2.12), we see from explicit calculation that at least to two loops, loop corrections to the collinear limit behave as expected from naive factorization, with the exception of endpoint contributions. These endpoint contributions are proportional to (C A − C F ) n and therefore vanish in N = 1 SUSY. In Sec. 3, we will perform an operator level factorization analysis to show that in N = 1 QCD, the leading logarithms exponentiate to all orders. In QCD, we are not yet able to perform such an analysis due to these endpoint contributions. Nevertheless, we give a conjecture for the form of the soft quark Sudakov in Sec. 4.

H → 3 Partons
We can also consider the λHG µν G µν effective operator producing the decay of H → qqg, in the limit where the quark and gluon become collinear, which much like the collinear limit of two quarks in e + e − → qqg does not have a leading power limit. Here we will find an identical story to that for the qq collinear limit just considered, providing some evidence that it is generic.
We label the partons for the Higgs decay as

Quark-gluon collinear limit for H decay
We summarize the result for gluon gluon collinear limit for H(Q) ! q(p 1 )q(p 2 )g(p 3 ) in this section. We write the squared amplitude as The color algebra is Just as in e + e case. The LO NLP splitting function is And the NLO NLP splitting function is 6 Quark-anitquark collinear limit for Higgs decay The last configuration we consider is quark-antiquark collinear limit for Higgs decay. We . (2.13) The LO NLP splitting function is (2.14) At one loop, we find In terms of Casimirs we have We can now consider the behavior in the z → 1 limit, which gives rise to the leading logarithm when integrated over z. In the z → 1 limit, we have (2.17) Here we again see an identical structure as found in the case of e + e − decay. Away from z → 1, we can expand the second term in Eq. (2.17), and find the expected result from a cusp with color structure (T 1 + T 3 ) · T 2 . However, there is another contribution from the endpoint z → 1. Again, if T 3 = T 2 , as would occur in N = 1 QCD, then the second term vanishes and naive factorization holds. This result can also be extended to two loops. As with the case of the e + e − decay, we find that it exponentiates We conjecture that it exponentiates to all loops, however, we are only able to prove this in N = 1 QCD. Note that the O( 0 ) expansion of the result in Eq. (2.18) can be cross checked against the subleading power collinear expansion of the matrix elements squared constructed from the analytic result of the amplitudes for H → 3 partons at NNLO [57]. The procedure is analogous as the one we explained for the case of e + e − → 3 jets below Eq. (2.12).

Insights into Factorization and Endpoint Divergences
In this section we briefly discuss the interpretation of the above fixed order calculation in terms of subleading power factorization formulas in SCET. This section assumes significant familiarity with SCET at subleading power, which we do not attempt to review in this paper. Details building up the formalism for subleading power factorization used here can be found in [1,30,31,44,71]. Factorization formulas at subleading power generically involve convolutions in the label momentum of collinear partons, or in certain momenta of soft partons. A general issue is that these convolutions (at least when treated naively) can exhibit divergences. It is currently not understood when such divergences occur, and when they do occur how to overcome them to achieve subleading power resummation. While such divergent convolution integrals are expected to occur somewhat generically at next-to-leading logarithm, we believe that focusing on cases where they occur at leading logarithm may be useful for understanding their physical origin.
For the case of soft gluon emission at subleading power for dijet event shapes, it was shown that such divergences do not occur at leading logarithm due to reparametrization invariance. In the case of N = 1 SUSY, they also do not occur for soft quark emissions due to supersymmetry. This constrains such divergences to be proportional to (C A − C F ) n at leading logarithm in QCD. In this section, we briefly discuss the physical picture of such divergences, which we hope will lead to an understanding in the near future. In addition to this insight into the structure of the endpoint contributions, we also wish to emphasize that it is generally assumed that leading logarithms can be derived by considering anomalous dimensions that are diagonal in the convolution variables. This provides an example where this does not occur in this naive manner.
To understand the structure of the endpoint divergences appearing in our fixed order calculations, we consider how these limits are factorized in the EFT. The matching for the relevant operators was studied in detail for Higgs decay in [31] and for e + e − annihilation in [30]. We consider first the case of Higgs decay, where a quark and gluon become collinear. The full theory diagram that contributes to the matching, expanded to O(λ) in the power counting is given by Here ω 1 ≡n · p 1 is the large component of the n-collinear momentum p µ 1 and ω 2,3 ≡ n · p 2,3 are the large component of then-collinear momenta p µ 2 and p µ 3 . The large component of the momentum is usually referred as the label momentum of the particle. This is matched in the EFT onto an operator whose Feynman rule is given by The matching coefficient for this operator exhibits a divergence as ω 2 → 0. In this limit, the off-shell gluon in the diagram in Eq. (2.19) that is being integrated out becomes on-shell, and should no longer be incorporated in the hard matching coefficient, but rather in the long -9 -distance emission of the soft quark. In the effective theory, this overlap should be removed by the zero bin [72], however, the exact implementation of the zero bin in this case is not fully understood. This complication generally does not arise at leading power. At leading power both for Higgs decay and e + e − annihilation, we have only one field per collinear direction in the hard scattering operator and its label momentum, let's call it ω, is fixed by momentum conservation to be a hard scale of the process, ω = Q. The reason why here it is meaningful to talk about the singular behavior of the Wilson coefficient as a label momentum goes to zero is because at subleading power we can have multiple collinear fields in one collinear sector (as in the operator in Eq. (2.20) where both / B ⊥n,ω 3 and χn ,−ω 2 belong to the samen-collinear sector and have label momenta ω 3 and ω 2 respectively). Therefore, momentum conservation for the label components constrains only the sum of the labels in a collinear sector, hence in our example we have ω 2 + ω 3 = Q. The relative size of ω 2 and ω 3 can be arbitrary and it is necessary to integrate over all possible configurations, including the one where one label momentum vanishes as the other becomes closer and closer to the hard scale.
This presence of a singularity in the matching coefficient is completely generic, and occurs also for e + e − annihilation. For the case that the two quarks become collinear, we have  To resum subleading power logarithms, one must renormalize these operators involving multiple collinear fields in each collinear sector. It is generically assumed that at leading logarithm, this renormalization is diagonal in the momentum fractions of the collinear fields, since it should arise from diagonal radiation which sees the two collinear fields coherently, as was illustrated in Eq. (2.11). Looking at our results from the fixed order calculation in this light (and recalling that z = ω 1 Q, (1 − z) = ω 2 Q), The first term in square brackets is proportional to the sum of the color charges of the fields in the collinear sector, as expected for the usual diagonal terms. However the second term is not, and due to the 1/z behavior of P (2,0) qq it contributes a leading logarithmic contribution from z → 0.
-10 - To understand physically what is occurring, it is instructive to look at the structure of the cusp at the hard scattering vertex for ω 2 > 0 and as ω 2 → 0. For ω 2 > 0, in the Higgs case, we have a fundamental cusp whereas, as ω → 0, for the long distance contribution, we get an adjoint cusp Therefore, there is a non-smooth transition in the color representation of the cusp as we approach the limit ω 2 → 0. Note that if the quarks are in the adjoint, as occurs in N = 1 SUSY, one has a fundamental cusp in both cases, and so the limit is smooth.
In the cases where factorization works naively, namely for subleading corrections to soft gluon emission, or soft quark emission in N = 1 SUSY, one does not have this issue of a nonsmooth limit for the representation of the cusp. We believe that this clearly demonstrates that this issue is related to the treatment of the zero bin, which apparently must have a non-trivial structure related to the changing of the color representations of the cusp. An interesting upshot of our analysis is that we have taken a step towards better understanding the nature of endpoint singularities by showing that at LL, they must be proportional to powers of C A − C F . This physical picture also strongly suggests that the result is still related at LL to the cusp anomalous dimension, albeit perhaps in different representations. While we do not at this stage have a complete understanding of the treatment of the zero-bin overlap region, we hope that this analysis provides an insight into its structure. We have also highlighted a particularly clear example where it is important even at LL, violating the standard lore.
Although we will restrict our attention in this paper to NLP, the loop level structure of the collinear limits given in Secs. 2.2 and 2.3 persists to all powers, with only a modification of the tree level coefficient. This suggests that at LL, endpoint contributions in N = 1 QCD are absent to all powers, and more generally, that the only difference between N = 1 and QCD at LL to all powers is endpoint contributions proportional to (C A − C F ) n . We are therefore of the belief that once the endpoint contributions are understood, the operator level resummation discussed in this paper can straightforwardly be extended to any order in the power expansion without additional conceptual complications.
-11 -In this section, as an example and extension of the formalism developed in [1], we perform an operator level factorization and renormalization in N = 1 QCD to resum subleading power infrared logarithms. Since many of the relevant details have been worked out in detail in [1], here we focus on the structure of the operators and the structure of the renormalization group mixing. This analysis extends beyond the case of pure gluo-dynamics considered in [1] through the inclusions of a number of new soft and jet functions involving quarks. We believe that this provides one more step towards a complete understanding of subleading power factorization at leading logarithm. In particular from our explicit calculations, in Sec. 2, it appears that at leading logarithm the only difference between N = 1 and full QCD is the presence of endpoint divergences.
The underlying reason for the simplification in N = 1 is that supersymmetry relates quark and gluon emissions. This has been studied in detail in the abelian case in [73], and it would be interesting to understand it in more detail at an operator level in SCET. However, we leave this to future work.
For concreteness, in this section we will consider the event shape thrust [74], although the identical set of operators and renormalization group structure would also apply for any other SCET I dijet event shape. Here we define the dimensionful thrust observable as T , and the dimensionless version as τ = T /Q, where Q is the center of mass energy of the scattering. In the τ → 0 limit, the thrust cross section can be expanded in powers of τ as Here the cross section dσ (n) /dτ describes all terms in α s that scale like τ n/2−1 . For thrust, dσ (n) /dτ = 0 for n odd. Here we will focus on n = 2, which is also known as the next-toleading power (NLP) cross section. The logarithmic structure of the NLP cross section is given by In this section we will extend our analysis in [44] to derive the structure of the leading logarithms in the NLP expression for e + e − annihilation and Higgs decay in N = 1 QCD. In particular, we will show that they exponentiate into a simple Sudakov, similarly to leading power.

General Structure of Factorization and Consistency Relations
The complete set of contributions required to perform subleading power resummation involve kinematic corrections, subleading power hard scattering operators, and radiative corrections arising from the subleading Lagrangians in the effective theory. The treatment of these different contributions was worked out in detail in [44], and operator bases for both Higgs decays -12 -and e + e − annihilation were derived in [30][31][32]. The resummation of the leading logarithms for thrust in pure gluo-dynamics considered in [1] used only a subset of these operators. Here we are able to illustrate the complete set of operators required for the resummation of leading logarithms with both quarks and gluons. We believe that describing their structure all together illustrates a rigid and remarkably simple structure imposed by consistency. Furthermore, as has been emphasized above, apart from endpoint contributions, the same set of operators, renormalization and mixings are also sufficient in QCD. To simplify the discussion, we ignore in our presentation contributions arising from kinematic corrections, which were treated in [1], and focus only on those involving non-trivial additional fields (However, contributions from the kinematic corrections are reinstated in the final answer). Subleading power operators in SCET involve additional quark or gluon fields. Renormalization group consistency implies that for every operator structure with an additional collinear field, there is an analogous operator with an additional soft field, and so terms in the factorization necessarily appear in pairs. This splitting into two contributions arises simply from the fact that in SCET one has factorized into separate fields describing the soft and collinear dynamics, i.e. there are separate soft and collinear quark fields, and separate soft and collinear gluon fields. Indeed, it is apparent from the structure of the matrix elements involved that these two components are describing respectively the soft and collinear limits of the same full theory diagrams. This distinction works somewhat differently at leading power for hard scattering processes, since soft quark and gluon fields do not appear explicitly (soft gluons arise only through Wilson lines from BPS decoupling).
The complete description of all operators requires the introduction of a fair amount of technical machinery and notation. 4 Instead of distracting the reader with these details, here we choose to describe the basic structure of the factorization pictorially, since it in fact takes a simple and intuitive structure. For those familiar with SCET, it should be a simple exercise to translate these beautiful pictures into equations. We will describe the renormalization of the operators in Sec. 3.2.
At LL, renormalization group consistency is separately obeyed for the quark and gluon corrections, and we will indicate these distinct groups of corrections by square brackets. Each square bracket contains one term from a collinear correction, and one term from a soft correction. Furthermore, the structure of these pairs is effectively identical for the Higgs decays and e + e − annihilation, up to trivial exchanges. This gives the factorization formula at subleading a fairly transparent structure, consisting of pairs of operators correcting either the gluon or quark emission.
For the case of e + e − annihilation, the factorization can be depicted as In each line, we show from left to right the hard function, then the jet functions, and finally the soft function. Convolutions in the observable, τ are suppressed, and are incorporated in the ⊗ symbols. Convolution variables in the field momenta, which appear in subleading power factorization are made explicit. This factorization involves two separate RG invariant categories, which are indicated by the square bracket. The first category describes the corrections to the emission of a soft gluon, which are incorporated through the addition of a gauge invariant soft gluon field, or a gauge invariant collinear field. In either case, there is a single variable convolution appearing in the factorization, which describes a light cone momentum of the given field. These appear as the interference of an O(λ 2 ) operator or Lagrangian insertion with a O(λ 0 ) operator. This is forced by the LBK theorem [75,76], which guarantees that there is no O(λ) correction for soft gluon emission at amplitude level. The second category describes the corrections to quark emission, which are incorporated through the addition of a gauge invariant soft quark field, or a gauge invariant collinear quark field. Due to fermion number conservation, these fields must be inserted on either side of the cut. From a technical perspective, they therefore appear as the interference of two O(λ) operators, or two L (1) Lagrangian insertions in the effective theory. This leads to a factorization structure with two -14 -integration variables. For the case of Higgs decay, the factorization can be depicted as This takes an essentially identical structure to the case of e + e − annihilation, up to the exchange of some quark and gluon fields. Indeed, in N = 1 one finds an identical result in the two cases. While it may naively seem like there are a large number of operators involved in this factorization, we see that it ultimately has a simple structure: the four different contributions are understood as corrections to quarks and gluons that are then split between soft and collinear fields. With this structure understood, and using the formalism for the renormalization of subleading power operators that was introduced in [1], and will be reviewed in Sec. 3.2, LL resummation at subleading power in the N = 1 theory can be performed almost as easily as at leading power.

Renormalization and Leading Logarithmic Resummation
In [1], an understanding of the renormalization of subleading power operators was developed (working in a case where no endpoint divergences are present, as is the case at LL in N = 1 SUSY). Renormalization is conceptually non-trivial for subleading power soft and jet functions that involve additional fields (for example the soft functions involving an additional soft quark or gluon field, or the jet functions involving an additional collinear quark or gluon field that were shown in Eqs. (3.3) and (3.4)). The renormalization of these operators is described by mixing into a new set of subleading power operators, which were termed θ-jet and θsoft functions, that are not seen in the amplitude level matching. In [1] it was shown using renormalization group consistency that under general assumptions related to the convergence of convolution integrals, the matrix elements of these new subleading power operators are given by moments of the leading power functions.
The θ-soft function is defined as while the θ-jet functions for quarks and gluons are defined as These functions obtain their power suppression from the θ function, which is power suppressed relative to the standard δ function that appears in jet and soft functions. They obey the RG equations g,θ (s, µ) = ds γ J g (s − s )J (2) g,θ (s , µ) , q,θ (s , µ) . (3.9) Here γ S g , γ J g and γ J q are the anomalous dimensions of the leading power soft and jet functions, which can be found in [1] .
With this understanding of the structure of the renormalization and factorization at subleading power, it is now a technically straightforward exercise to derive the LL resummed result at subleading power. Using renormalization group consistency, we can choose to run all operators to the soft scale. This eliminates the need to consider the soft functions involving radiative quarks or gluons. We therefore must simply renormalize the hard functions involving two collinear quark fields, or a quark and gluon collinear field, as well as the subleading power jet functions.
-16 -Each of the jet functions involving multiple fields (which we denote generically as J (2) j ) obeys a two-by-two mixing equation, in which they mix with an adjoint θ-jet function (either quark or gluon). This mixing takes the form , (3.10) at leading logarithm. Note that we have supressed the dependence on the momentum labels of the fields, since at leading log the renormalization is diagonal in these momentum labels. This equation was solved in [1] with and without a running coupling. Without running coupling, it leads to a standard Sudakov exponentiation. Since the running coupling provides no additional insight to the structure at subleading power, we will not bother to consider it here. From explicit calculation, at LL each of the different hard functions in N=1 SUSY QCD satisfy a multiplicative RG equation determined by the standard cusp anomalous dimensions where the ellipses denote non-cusp terms that do not contribute to the leading logarithmic result. Here the subscript j is supposed to denote the different field configurations appearing in the hard functions in Eqs. (3.3) and (3.4). Note that since we are in N = 1, the different cusp anomalous dimensions are all the same, Γ g cusp = Γ q cusp , which enables us to treat all the anomalous dimensions on the same footing.
Once it is shown that the renormalization of the operators leads to a standard Sudakov exponentiation, it is then a simple combinatorial exercise to add up the contributions from different operators. We spare the reader these details, and just give the final result. We find that the NLP results for both Higgs decay and e + e − give an identical (as expected in N = 1) and simple result Note that this agrees with the fixed order results given in [22,26] upon taking C F → C A and n f → C A to obtain N = 1 from QCD. We believe that this result is interesting in its own right, since it shows that resummation of subleading power corrections at leading logarithm is understood in N = 1 (as well as for theories with more supersymmetry), and indeed has a simple form that, once understood, is technically no more difficult than LL resummation at leading power. In particular, we have illustrated that the mixing with θ function operators first introduced in [1] applies much more generally. We hope that the fact that factorization works in this manner in N = 1 may provide clues for how it can be extended to QCD. In particular, it seems to indicate that the only issue in QCD arises from the z → 0 or z → 1 limits of the convolution.

A Conjecture for the Soft Quark Sudakov in QCD
Although we are not yet in a position to derive the all loop result for soft quark emission in QCD from an operator level factorization, our explicit calculation to O(α 3 s ), combined with insight from the exponentiation in N = 1 QCD allows us to make a conjecture for the result. It will be interesting to derive this result (or prove that it is incorrect) from an operator level renormalization and factorization analysis in the future.
We arrive at our conjecture for the LL Sudakov in QCD by assuming that the radiative corrections to the squared amplitude exponentiate (at least for the highest order poles in ). Namely, that where K(z, ) describes the leading infrared poles in This result was illustrated in Sec. 2 for different partonic channels. Note that we have explicitly checked this for n = 1, 2 by direct calculation, and we know from our factorization and renormalization analysis that it holds for the cusp contributions, but we do not strictly speaking know that it holds for the endpoint contributions. This exponentiated conjecture for the splitting function can then be integrated against the measurement function, as was done in [1] to derive the result for a physical observable using the consistency equations in Sec. 2.1.
One of the reasons we make a conjecture for the result, is that under this assumption for the exponentiation of the splitting function, we find after integration that the color structure at any order is given by This is not at all apparent before integration. Intriguingly, identical color structures have appeared in the small x resummation of the off diagonal splitting functions [51][52][53][54]. We believe that this is not a coincidence, and we believe that it provides some non-trivial evidence for our conjecture. Alternatively, if one can argue for the color structure Eq. (4.3), as was done in [51][52][53][54] for the off-diagonal splitting functions, then one can extend the result in N = 1, where it can be proven to the case of QCD by adding terms proportional to (C A − C F ) n . Using either line of reasoning, both of which are conjectural, we can make an ansatz for the full result for the subleading power leading logarithmic result for thrust in e + e − in full Here the first term arises from the subleading power corrections to soft gluon emission, which take the form of a simple Sudakov, as shown in [1], and the second term gives the conjectured "soft quark Sudakov", which takes a more non-trivial form. Despite this more unusual structure, the result still seems to be controlled by a difference of cusp anomalous dimensions, as expected from the physical picture developed in Sec. 2.4. This suggests that understanding the endpoint divergences is a conceptual rather than technical difficulty. We can write this soft quark Sudakov in a more transparent form, as a modification of the standard Sudakov by (4.5) Here the term in square brackets becomes 1 in the C F → C A limit. It can be viewed as dressing the standard Sudakov that appears in the N = 1 theory. It represents a new all orders structure first appearing at subleading power. Functions of (C A − C F ) have also been observed in the high-energy limit for mass suppressed amplitudes in [77,78]. It would be interesting to understand the connections between these results in more detail.
For the case of thrust in Higgs decays, one can conjecture a similar result Note that setting C F , n f → C A in either Eq. (4.4) or Eq. (4.6) reproduces the N = 1 formula of Eq. (3.12). It is worth emphasizing that since this result follows from consistency relations, combined with the conjectural structure of the amplitudes, it will have considerable universality. In particular, it should hold for any SCET I type dijet observable (recoil free observable) up to trivial color factor changes. For this reason we refer to it as "the" soft quark Sudakov.
One numerically interesting feature of this result is that for a process that has a Sudakov in a given representation at leading power, at subleading power, both representations appear. Since the Sudakov with a C A dies off more rapidly, in e + e − , these corrections have a small -19 -effect. On the other hand, the effect for Higgs decays is much larger, since there is now a Sudakov with a C F , which dies off more slowly.
It is also interesting that while at leading power and leading logarithm, the results for N = 1 QCD and QCD are identical up to a color factor, at subleading power they have a significantly different form. Interesting simplifications in the infrared limits of N = 1 were also observed in collinear limits in [79] and at two loop level for the 2 → 2 amplitude in N = 2 in [80]. This suggests that further understanding at subleading powers may lead to a better understanding of the differences in their infrared and transcendental structure.

Conclusions
In this paper we have initiated a study of the all order loop level structure of soft fermion emission, which generically contributes leading infrared logarithms at subleading power. This supplements studies of subleading power corrections to soft gluon emission which have been resummed using the formalism introduced in [1].
We computed the subleading power collinear limits of e + e − → 3 parton and H → 3 parton amplitudes at two loops, focusing in particular on limits that do not have a leading power analogue. We found that in the case of QCD, Sudakov exponentiation is violated by endpoint divergences. These endpoint contributions were shown to be proportional to (C A − C F ) n , and hence vanish in N ≥ 1 QCD. We also gave a physical picture for these endpoint contributions, which we believe are related to a non-trivial zero bin when C A = C F . We leave finding a operator renormalization based understanding of the structure of these terms to future work.
We performed an operator level factorization and resummation of the infrared logarithms associated with soft quark emission in N = 1 QCD. This extends our analysis in [1] to include new subleading power jet and soft functions involving quarks. The renormalization of these operators involves mixing into θ function operators as was found for the pure gluon operators considered in [1]. Using this analysis, we showed that radiative contributions associated with soft quark emissions exponentiate into a Sudakov exponential. This also shows that leading logarithmic resummation at subleading powers is now understood in N ≥ 1 QCD.
Using the results from our fixed order analysis, as well as the structure of the exponentiation in N = 1 QCD, we made a conjecture for the all loop structure of the soft quark Sudakov in QCD. We believe that this structure will govern all SCET I dijet event shapes, and we have provided some non-trivial evidence of this universality by considering the event shape thrust for both Higgs decay and e + e − annihilation. It would be extremely interesting to prove (or disprove) this formula by better understanding the endpoint contributions, as well as to understand its relation to the off-diagonal splitting functions.
An interesting direction that deserves further study is to extend this work to the case of perpendicular momentum sensitive observables. These observables are affected by rapidity divergences and obey a set of additional rapidity renormalization group equations that are well understood at leading power [81]. The study of rapidity divergences at subleading power has been initiated at fixed order in [36]. While we believe that most of the structures we found in this current work will be present also for observables with rapidity divergences, the structure of the rapidity renormalization group at subleading power remains to be understood.
We believe it is of utmost interest to understand the structure of endpoint divergences in subleading power factorization, as this is now the primary obstacle to a complete understanding. The fact that the only difference in the LL subleading power resummation between N = 1 SUSY and QCD resides in endpoint contributions indicates that they are not simply a technical detail, but describe the interesting physical effect of having particles in different color representations. We find it interesting that the subleading power limit is so sensitive to the particular representations and matter content of the theory, and we therefore hope that better understanding subleading power limits can lead to an improved understanding of the infrared behavior of amplitudes and cross sections in gauge theories.