Factorization for the light-jet mass and hemisphere soft function

Many collider observables suffer from non-global logarithms not captured by standard resummation techniques. Classic examples are the light-jet mass event shape in the limit of small mass and the related hemisphere soft function. We derive factorization formulas for both of these and explicitly demonstrate that they capture all logarithms present at NNLO. These formulas achieve full scale separation and provide the basis for all-order resummations. A characteristic feature of non-global observables is that the soft radiation is driven by multi-Wilson-line operators, and the ones arising here map onto those relevant for the case of narrow-cone jet cross sections. Numerically, the contributions of non-global logarithms to resummed hemisphere-mass event shapes are sizeable.


Contents
1 Non-global logarithms in hemisphere-mass observables Perturbative corrections to observables which involve a hierarchy of scales are enhanced by logarithms of the scale ratios. Starting with the pioneering work of Sudakov [1], methods were developed to resum such logarithmically enhanced corrections to all orders. A crucial simplification is exponentiation, the statement that the leading logarithms can be obtained from exponentiating the leading-order correction to a process. Effective field theories provide a modern way to analyze multi-scale problems. In these theories exponentiation is a consequence of the renormalization group (RG). The logarithms are resummed by evolving Wilson coefficients, which encode the physics associated with high scales, down to lower scales and the leading-order solution of the RG equation is an exponential. Interestingly, this simple exponentiation property does not hold for all observables. For example, if one considers interjet energy flow, one finds that the relevant wide-angle soft radiation produces a very intricate pattern of leading logarithms [2]. Instead of a simple linear evolution equation, one needs to solve a complicated non-linear integral equation to obtain the leading logarithms, the Banfi-Marchesini-Smye (BMS) equation [3]. Interjet energy flow is an example of a non-global observable. Such observables are insensitive to radiation in certain regions of phase space (the inside of the jets, for the case of the interjet energy flow) and the same complicated pattern of "non-global" logarithms is present in all of them. Perhaps the simplest quantity which suffers from such logarithms is the hemisphere soft function, which is obtained by considering the radiation from two Wilson lines in opposite directions. Allowing for large energy in one hemisphere, but only a small amount in the other leads to non-global logarithms. This soft function is also relevant in the context of the light-jet mass event shape in e + e − collisions, in which the complicated pattern of logarithms was originally discovered [4].
The BMS equation makes crucial use of the simple form of strongly ordered gluonemission amplitudes. Beyond leading logarithmic accuracy these simplifications do not apply and it was therefore not clear how to generalize the BMS equation to higher accuracy.
In the past few years, the problem of non-global logarithms has received renewed interest, in particular in the context of Soft-Collinear Effective Theory (SCET) [5][6][7] (see [8] for a review). Several papers have computed hemisphere soft functions up to next-to-nextto-leading order (NNLO) to obtain full results for their non-global structure at this order [9][10][11][12]. Furthermore, by perturbatively expanding the BMS equation, the analytic form of the leading-logarithmic terms up to five-loop order was extracted [13,14]. Using an efficient new method to perform the angular integrations [15], this result has now been extended to 12 (!) loops [16]. In addition to these fixed-order considerations, a method to approximately resum the non-global logarithms was proposed [17,18]. At leading-logarithmic accuracy it reduces to an iterative solution of the BMS equation [19].
In the recent papers [20,21], two of us have analyzed cone-jet cross sections and have derived factorization theorems for the case where the outside energy is small. The characteristic feature of these theorems is the presence of multi-Wilson-line operators which describe the soft emissions from energetic partons inside jets. In our effective-field-theory framework, the non-global logarithms are obtained from an RG-evolution equation which generalizes the BMS equation to arbitrary logarithmic accuracy. The complicated structure arises because operators with an arbitrary number of soft Wilson lines are present in the factorization theorem. To obtain the large logarithms, one needs to exponentiate an infinite-dimensional anomalous-dimension matrix, which, at leading-logarithmic accuracy and large N c , is equivalent to solving the BMS equation. The exponentiation property mentioned earlier is thus present also for non-global logarithms, but takes a very complicated form. Our framework is closely related to the one proposed in [22] and involves the same anomalous dimension, which was computed to two-loop order in that reference and has recently even been derived at three-loop accuracy in the planar limit in N = 4 super Yang-Mills theory [15].
To make contact with the previous literature which has focused mostly on the hemisphere soft function, it is important to analyze this quantity using our framework. We do this in the present paper and at the same time also derive a factorization theorem for the light-jet mass event shape. To define this e + e − event shape, one first introduces the thrust axis n as the direction of maximum momentum flow. More precisely, the unit vector n is chosen to maximize the quantity i | n · p i |, where the sum runs over all particles in the final state. The event shape thrust is defined as this sum normalized to Q, where Q is the center-of-mass energy of the collision. The thrust axis splits each event into two hemispheres, which can arbitrarily be labelled as "left" and "right", and one can define additional event shapes by considering the invariant masses M L and M R of the particles in the hemispheres. Two commonly used event shapes are heavy-jet mass: light-jet mass: (1. 2) In the limit where the jet masses become small, perturbative corrections to these observables are logarithmically enhanced. For the heavy-jet mass these logarithms have been resummed up to next-to-next-to-next-to-leading logarithmic (N 3 LL) accuracy [23], while only NLL predictions are available for the light-jet mass ρ ℓ [4,24]. The reason for the poor accuracy for ρ ℓ was that it was not known how this non-global observable factorizes in the limit of small ρ ℓ , while the factorization is well known for the heavy-jet mass. Due to left-right symmetry, the three possible scale hierarchies for the hemisphere masses are a.) The relevant factorization theorem for case a.) has the form [25] dσ where σ 0 is the Born level cross section. The hard function H collects the virtual corrections to γ * → qq which are known to three loops [26,27]. The jet function J q is the usual inclusive jet function in SCET, which is known to two loops [28,29]. The hemisphere soft function S(ω L , ω R ) is a matrix element of Wilson lines along the two jet directions and is also known at NNLO [9,10,30]. This function measures the contribution of the soft radiation to the hemisphere mass in each hemisphere. Since the relevant anomalous dimensions are known for all ingredients in (1.3), one can solve their RG evolution equations to obtain N 3 LL resummation for hierarchy a.) which is the one relevant for the heavy-jet mass ρ h . However, the above theorem does not achieve resummation for case b.) since for ω L ≪ ω R the soft function S(ω L , ω R ) itself contains large logarithms of κ = ω L /ω R , which are examples of non-global logarithms. To be able to resum also these logarithms one must factorize the physics at the two different soft scales ω L and ω R . In the context of the function S(ω L , ω R ), we will refer to ω R as the hard scale and ω L the soft one. One of the main results of the present paper is that the hemisphere soft function factorizes in the limit κ → 0 as The hard functions H S m are the squared amplitudes for m-parton emissions from the two Wilson lines in the hemisphere soft function into the right hemisphere, integrated over their energies but at fixed directions {n} = {n 1 , . . . , n m }, where the n i 's are light-like vectors. The soft functions S m+1 consist of m + 2 Wilson lines along the directions {n} of the m hard partons and the two jets along n µ = (1, n) andn µ = (1, − n). Both of these are matrices in color space [32,33], and . . . indicates a sum over color indices. The symbol ⊗ indicates that one has to integrate over the m directions of the emissions into the right hemisphere. The form of the factorization theorem (1.4) is basically the same as the one for wide-angle cone-jet cross sections derived in [20]. To see the connection, one should view the right hemisphere as the inside of a jet which contains hard particles with momenta p µ ∼ ω R and the left hemisphere as the outside region where a veto on radiation is imposed which constrains the momenta to p µ ∼ ω L .
Before analyzing the factorization formula (1.4) in more detail and providing operator definitions for its ingredients, we now turn to the light-jet mass ρ ℓ . Due to left-right symmetry and its definition, ρ ℓ is directly related to the left-jet mass ρ Instead of the light-jet mass one can therefore equally well analyze the factorization for ρ L . If one only measures the left-jet mass, the mass of the right jet will typically be large, so that scale hierarchy c.) applies. We find that the cross section for the left-jet mass factorizes as Since the unobserved radiation in the right hemisphere is typically hard, such that p µ ∼ Q, we no longer encounter a jet function for this hemisphere, in contrast to the previous case (1.3). The hard functions also differ from the function H S m encountered for the hemisphere soft functions. Rather than Wilson-line matrix elements as in (1.4), the functions H i m in this case are given by squared QCD amplitudes with a single parton of flavor i in the left hemisphere propagating along then-direction and m partons in the right hemisphere. The subsequent branchings of the hard parton on the left are described by the jet functions J i . A graphical representation of the factorization theorems is shown in Figure 1.
Our paper is organized as follows. In the next section, we will flesh out the factorization formulas for the hemisphere soft function and for the light-jet mass event shape and discuss their derivation, which can be obtained following similar steps as in [20]. The soft functions in these theorems can be related to the coft functions computed in that reference so that the only new ingredients to our factorization formulas are the hard functions. After computing these in Section 3 up to O(α 2 s ), we verify that we reproduce the known NNLO result for the hemisphere soft function in the limit ω L → 0. Next, we analyze the light-jet mass distribution in Section 4 and compare to the numerical fixed-order result for this quantity. In Section 5 we use the known result for the leading non-global logarithms in the hemisphere soft function to obtain numerical results for the light-jet mass at NLL accuracy. In Section 6 we discuss the necessary steps to perform higher-order resummation for this event shape and conclude. Figure 1. Pictorial representation of the factorization theorems for the differential cross sections with respect to the hemisphere jet masses in the limit M L ≪ M R ≪ Q (left), and to the left-jet mass when M L ≪ M R ∼ Q (right). Blue lines correspond to collinear partons inside the jet functions, the red lines represent soft emissions. The green lines in the left picture correspond to the hard part of the hemisphere soft function, while the black lines in the right picture correspond to hard emission into the right hemisphere.

Factorization
The derivation of the factorization formula follows the same steps in both cases and is similar to the one relevant for wide-angle cone-jet cross sections presented in [20]. We will first sketch the derivations of the theorems and specify the ingredients. We then relate the soft functions to the ones which arise in the case of the narrow-cone jet cross sections. Due to this relation, we can use the results [20] for these and only the hard functions need to be computed.

Hemisphere soft function
The hemisphere soft function describes radiation originating from a quark and an antiquark along the directions n andn of the two jets. Their soft radiation is described by Wilson lines. The one generated by the outgoing quark along the n direction is 1) and the soft function is defined as where the trace is over color indices. We call the hemisphere which contains the thrust vector the right hemisphere. The right-moving particles therefore haven · p > n · p and P R(L) is the total momentum in the right (left) hemisphere. Usually, the function S(ω L , ω R ) is defined in terms of the soft gluon field in SCET. However, the soft SCET Lagrangian is equivalent to the full QCD one so for our discussion we will consider (2.2) as a matrix element in QCD. In the asymmetric case ω L ≪ ω R the function S(ω L , ω R ) develops large, non-global logarithms (NGLs) in the ratio κ ≡ ω L /ω R ≪ 1. It is these logarithms which we seek to resum using effective-field-theory methods.
Before constructing the appropriate effective theory, it is useful to study the structure of NGLs in the matrix element (2.2) perturbatively. Clearly, one method is to calculate the hemisphere soft function at a given order in perturbation theory, and then take the limit κ → 0 in the final result. This was the approach taken in the NNLO calculations of [9,10], and the obvious benefit of such a computation is that it provides the hemisphere soft function for any value of κ. On the other hand, if one is interested only in NGLs appearing in the limit κ → 0, it is much simpler to obtain results by expanding the phasespace integrals appearing in the hemisphere soft function using the method of regions [31]. Indeed, in a first step we have used this method to reproduce the NNLO fixed-order calculations in the non-global limit. The factorization results discussed below can be viewed as a translation of this diagrammatic approach into the language of effective field theory.
We find that two momentum regions are needed for the leading-power diagrammatic expansion in the limit κ → 0. Defining the light-cone components of an arbitrary vector p as (n · p,n · p, p ⊥ ), these regions are specified by the scalings The homogeneous scaling of the momentum components arises because the soft and hard radiation covers a wide angular range so that no specific direction is singled out. The expansion of individual diagrams also receives contributions from a left-collinear mode scaling as ω R (1, κ, √ κ). However, in the sum of all diagrams these collinear contributions vanish, and in Appendix A we present an all-orders proof of this result, based on the invariance of Wilson lines under rescalings of the reference vector. A non-trivial interplay between contributions of the two regions is responsible for the structure of NGLs in the hemisphere soft function. By NGLs, we mean contributions which cannot be written as a naive product of two component functions depending on ω L and ω R only. 1 An NLO analysis does not reveal the presence of NGLs, since the NLO result is the sum of the identical contributions of a single hard emission into the right hemisphere and a single soft emission into the left hemisphere, which can always be written as the product of identical one-scale functions for the hard and soft regions. At NNLO, on the other hand, it is possible for a virtual gluon to split into two particles flying into different hemispheres, and it is obvious that a simple product structure is insufficient to describe these contributions since they have a different color structure. Two types of opposite-hemisphere configurations are relevant. The first involves a soft gluon in the left hemisphere and a hard gluon in the right hemisphere and gives rise to double and single NGLs. The second involves one soft gluon in each hemisphere. Such a configuration is not possible for hard radiation, because a hard emission into the left hemisphere would violate the scaling ω L ≪ ω R . This asymmetry between double-hard and double-soft contributions generates the remaining single NGLs needed to reproduce the known NNLO result in the κ → 0 limit.
The effective field theory appropriate for describing the situation above has recently been developed in [20,21]. The basic observation of these papers was that each of the hard partons generates a soft Wilson line along its direction, so even though hard and soft contributions factorize in (1.4), new hard and soft functions appear at each order in perturbation theory. To obtain the operators in the low-energy effective theory, one therefore first considers a kinematic configuration with m hard partons along fixed directions and then introduces a soft Wilson line for each of them. The amplitudes for the emissions of m hard partons with momenta {p} = {p 1 , · · · , p m } from the two Wilson lines in (2.2) take the form Note that on the left-hand side of the above equation we use the color-space formalism of [32,33] in which the amplitude |M S m ({p)} is a vector in the color space of the m partons. However, on the right-hand side the color indices of the m partons are suppressed and the bra-ket notation denotes states in the Hilbert space. The superscript S indicates that the amplitude M S m is obtained from the Wilson line matrix element. A general soft Wilson line along the light-like direction n i ∝ p i is defined in analogy with (2.1) as where the color matrices for the representation of the underlying particle i are denoted by T a i . On the amplitude level, the soft radiation from the two original Wilson lines and the additional hard partons is obtained from the Wilson-line operator where S a (n) and S b (n) are the anti-quark and quark Wilson lines present in the original definition (2.2). A derivation of the formula (2.6) from SCET was given in [20].
To obtain the factorized result for the cross section we need to square the factorized amplitude (2.6), integrate over the energies and directions of the hard partons, and add up the contributions from different multiplicities of hard partons. Doing so, we obtain the factorization formula (1.4) for the hemisphere soft function in the limit κ → 0. The definitions of the hard functions in this formula read where d is the number of spacetime dimensions. The theta function Θ R ensures that all hard partons are inside the right hemisphere so that P R is simply the total hard momentum. Note that the directions of the hard partons are fixed. The integral over the directions is performed after multiplication with the soft function, which for m additional hard partons is obtained from squaring the Wilson-line operator matrix elements Note that the soft partons can be in either hemisphere. The ones in the left hemisphere contribute to ω L , but the ones in the right hemisphere are not constrained because their contribution to ω R is negligible compared to the hard partons. The strict expansion of the phase-space measure is crucial to achieve the desired factorization of scales and to avoid double counting of the contributions from different momentum regions.

Left-jet mass
The factorization for the left-jet mass distribution is rather similar to that for the hemisphere soft function, but the expansion parameter is λ = ω L /Q and the relevant momentum scalings are hard: (2.9) To derive the factorization theorem (1.6) and obtain the hard functions H i m ({n}, Q), one can first match onto a version of SCET with a collinear field along then-direction as well as m additional collinear fields along directions in the right hemisphere. Then one performs the usual decoupling transformation on the collinear fields [6], which gives rise to the relevant soft multi-Wilson-line operator. Finally one takes the matrix element where there is a single hard parton along each of the m directions in the right hemisphere, and a jet of partons along then-direction on the left. This yields the hard functions H i m ({n}, Q) together with the jet function J i . We refrain from going over this derivation in more detail since it involves, up to obvious modifications, exactly the same steps as the ones detailed for the wide-angle jet cross section in [20].
The explicit definition of the hard functions for the the decay of a virtual photon into a final state with m particles in the right hemisphere is where p µ 0 = Qn µ /2 is the momentum of the single hard parton of flavor i ∈ {q,q, g} in the left hemisphere, and the amplitudes |M i m+1 ({p 0 , p}) are standard QCD amplitudes for the decay of the virtual photon into (m + 1) partons. The associated soft function is This is exactly the same matrix element as (2.8) up to the fact that only the direction of the first Wilson line is fixed, as opposed to the case of the hemisphere soft function, where the first two, along then and n directions, are kept fixed. We can thus get the one in (2.8) by taking the result for (2.11) and setting the reference vector of the second Wilson line to the n direction. Furthermore, almost the same matrix element as (2.11) has arisen in the context of narrow-cone jet cross sections. In that case, the Wilson line structure is associated with coft emissions which are simultaneously collinear and soft. Rather than a hemisphere constraint, the coft functions involve a constraint on out-of-jet radiation of the form Qβ >n · p out and a particle is outside the right jet if n · p > δ 2n · p. If we set δ = 1 and replace Qβ → ω L , the coft functions are mapped onto the left hemisphere (up to the fact that we impose the constraint as a δ-function instead of an upper limit). Since Wilson lines are invariant under a rescaling of the reference vector, the transformation maps the coft Wilson line matrix elements directly onto the soft functions (2.11) and we can use the results of [20,21].

Hemisphere soft function at NNLO
In this section we demonstrate how our factorization formula can be used to reproduce the results for the hemisphere soft function at NNLO in perturbation theory in the asymmetric limit ω L ≪ ω R . In the following, it will be convenient to work in Laplace space, where the convolutions in the factorization formulas (1.3) and (1.6) turn into products. We define the renormalized, Laplace-transformed soft function as Whereas the soft function is a distribution in the arguments ω L,R , the Laplace-transformed soft function is a regular function of its arguments. The renormalized soft function in Laplace space is obtained from the bare one through multiplication by a UV renormalization factor. We write the relation between the bare and renormalized functions as The notation, used throughout the paper, is such that bare and renormalized functions are distinguished through their last argument, which is µ for renormalized functions and ǫ for bare ones, where the dimensional regulator is ǫ = (4 − d)/2. On the other hand, in generic expressions such as (1.4), we drop the dependence on µ or ǫ to indicate that the equations can refer equally well to bare or renormalized quantities. The form and explicit results for the renormalization factorZ S are well known -we collect some of the expressions we need in the analysis below in Appendix B. We now show how to reproduce the NNLO results of [9,10] for the hemisphere soft function using the factorization formalism from the previous section. We first define the Laplace-transformed component functions as The functions with different numbers of hard partons mix under renormalization. Following [20], we define the renormalized hard functions according to This equation states that lower-multiplicity hard functions absorb some of the divergences of the higher-point functions. This is familiar from fixed-order computations, where virtual corrections to lower-point amplitudes need to be combined with real-emission contributions.
Combined with the fact that the UV divergences for the hemisphere soft function are removed by the renormalization factorZ S , the renormalized soft functions can be written as The peculiar index structure arises because in the factorization theorem (1.4) for the hemisphere soft function, the hard function H S m multiplies S m+1 . This relation has several non-trivial features. First of all, it implies that higher-multiplicity soft functions enter the renormalization of lower-multiplicity ones. The higher-m functions depend on additional directions which need to be integrated over. This integral over unresolved directions is indicated by the symbol⊗. BothZ S and the Z lm depend on the hard scale τ R . It is a non-trivial cross check on our results that the renormalized soft function depends only on τ L , as it must.
The Laplace-transformed hemisphere soft function satisfies a factorization formula of the same form as (1.4). In order to verify it to NNLO, we first define expansion coefficients of the bare and renormalized functions as Writing out the contributions to the factorization theorem (1.4) to first order, we obtains (1) where we have made explicit that the two terms on the first line have no angular dependence, so that the convolution of functions reduces to simple product. Higher-multiplicity terms do not arise since the hard functions are suppressed, H S m ∼ α m s . The formula simplifies further after noting that perturbative corrections to the zero-emission hard function are scaleless and vanish in dimensional regularization, so that H Applying the same simplifications, the NNLO coefficient reads In the following, we give explicit results for the ingredients in these two formulas. We can evaluate equations (3.11) and (3.12) using bare ingredients or renormalized ones. In the main text, we will work with renormalized quantities, but in Appendix B we repeat the computation using bare ones.

Soft functions
As we stressed at the end of Section 2, the soft functions are trivially related to the coft functions U m relevant for narrow-jet cross sections defined in [20,21]. Indeed, after setting the cone-angle parameter δ = 1, the soft function for the left-jet mass (2.11) is identical to the coft function As discussed after (2.11), for the case of the hemisphere soft function the first reference vector must be set equal to n µ , see (2.8), because the Wilson line along the n-direction is present in the original hemisphere soft function (2.2) and only the remaining (m − 1) Wilson lines arise from hard partons. To be able to use our results in both cases, we will give results for the left-jet mass case. The one-loop soft function is a sum over dipoles where the summation of (ij) goes over all unordered pairs, and we can restrict the soft emission to the left hemisphere because the contribution from the right hemisphere is a scaleless integral. It is useful to separate out the dipoles involving the left-Wilson line S 0 (n) and write the one-loop coefficient of the function in Laplace space in the general form where the summation of [ij] goes over all unordered pairs with i, j = 0. Here φ i is the angle of the n i in the plane transverse to the thrust direction and parameterizes the angle with respect to the thrust axis. Since the terms in the first sum depend only on a single reference vector n i , the coefficient u(θ i , τ L ) is a function of the corresponding angle. The result for the renormalized coefficient functions can be obtained from the results for the coft function U 2 given in [20]. We find The function u involves double logarithms due to a collinear singularity from the region where the emission is collinear ton. The function v on the other hand, describes an exchange between Wilson lines in the right hemisphere. Since the gluon is emitted to the left, this function does not suffer from a collinear singularity. The auxiliary functions f 0 and g 0 were given in [20] and read For the function S 2 , it is sufficient to consider the case ∆φ = π due to transverse momentum conservation in the hard function H S 2 . For the hemisphere soft function in (1.4), we set n 1 = n so that we only need (3.20) To evaluate the color structure for the soft function with three legs explicitly, one can use the relation which follows from color conservation 2 i=0 T i = 0 together with T 2 i = C i 1, where C i is the quadratic Casimir of the relevant representation, C q = C F and C g = C A .
For S 1 in the left-jet case, we can set n 1 = n (θ 1 = 0) since the hard function will enforce that the single hard parton must fly along the thrust axis. For completeness, we reproduce the two-loop result for this function given in [20]. Using relation (3.13) we have 22) where L L = ln(τ L /µ) and (3.23) The renormalization of the soft function is quite non-trivial since higher-multiplicity function mix into lower ones, see (3.6). It is therefore interesting to test that the renormalization factor, obtained from absorbing the divergences of the hard functions, indeed renders the soft functions finite. For the case of narrow-jet cross sections, this was verified in [20].
Since we work with different hard functions in the present case, it is an important but somewhat tedious exercise to show that one recovers the same soft function after performing the renormalization. We have checked that this is the case -the details can be found in Appendix C.

Hard functions
Since H S 0 (τ R , ǫ) = 1 is trivial, the first nontrivial hard function is H S 1 ({n 1 }, τ R , ǫ), which arises from the emission of a single hard gluon from the Wilson-line operator in (2.4). The leading contribution to this hard function is given by The light-cone vector n 1 appearing as an argument in the hard function is related to the gluon momentum according to p µ 1 = E 1 n µ 1 . We parameterize this vector in d-dimensions as n 1 = (1, 0, . . . , cos θ 1 ), so that the theta-function constraint in (3.24) gives support to the hard function only in the region 0 < cos θ 1 < 1, that is, when the gluon is in the right hemisphere. After integrating over E 1 and ω R and performing the trivial angular integrations, we are left with an angular convolution in θ 1 . It is convenient to instead use the angular variableθ 1 defined in (3.16) and write where we have absorbed the trivial part of the angular integration into H S(1) 1 (θ 1 , τ R , ǫ). For the bare hard function at NLO, we obtain the simple result The hard function is thus a distribution in the angleθ 1 , in contrast the soft function which is regular forθ 1 → 0 . To obtain the renormalized hard function, one uses the identitŷ The renormalized one-loop function is given by At this order, renormalization is equivalent to dropping the divergences in the bare function. Doing so leaves the finite result with L R = ln(τ R /µ). . Both of these are O(α 2 s ) corrections. Rather than computing the full functions, it is sufficient to obtain the angular convolution of these functions with the trivial leading-order soft functions. The bare results for these can be extracted from the computations in [9,10] and are given in Appendix B. After renormalization one obtains as is shown in Appendix C.

Renormalized results to NNLO
Using (3.22) and (3.30), we immediately obtain the renormalized hemisphere soft function at NLO, which is given bỹ We observe that after the substitution τ R → τ L , the hard function contribution, given by the coefficient of the delta-function term in (3.30), agrees with the soft function contribution given in (3.22). This is easily understood since both arise from the same Wilson line matrix element and the single emission is always left for the soft function and right in the case of the hard function. This simple symmetry is no longer present at the two-loop level, since soft gluons can radiate to the right, while hard partons cannot enter the left hemisphere.
To obtain the NNLO result, we also need the convolution of H S 1 with the one-loop soft function. It is easy to show that With the final ingredient in place, we can now evaluate (3.12) by adding (3.22), (3.31) and (3.33). Explicitly, we havẽ This result is equivalent to a result for the integrated soft function given in [9], and to a position-space expression given in [10]. In those references the full hemisphere soft function was evaluated, while we directly obtain the function in the limit τ L ≪ τ R . The agreement provides a nontrivial check on our factorization formula (1.4). We have performed similar two-loop checks in our earlier work on jet cross sections. However, in that case we could only compare against numerical results from fixed-order event generators. The present case has the advantage that we can compare against the analytical results from [9,10].
In earlier work on the hemisphere soft function [9,10,34], the result was typically written in the form s(τ L , τ R , µ) = s µ (τ L , µ) s µ (τ R , µ) s ng (r) . (3.35) The non-global remainder s ng (r) is µ-independent but contains logarithms of the small ratio r = τ L /τ R ≪ 1. As it stands, the definition of the non-global piece in (3.35) is not unique. One way to fully specify it is to set s µ (τ, µ) = s(τ, τ, µ). Dividing out the global pieces from our result, we are then left with where Equally well, we could have defined the global part s µ (τ, µ) as the square root of the thrust soft function or the solution of the RG equation for s µ (τ, µ) with trivial boundary condition s µ (τ, τ ) = 1. With the latter two definitions, the non-global piece would involve constant terms.
The reasoning for splitting the soft function into global and non-global parts was that the global piece follows from the RG evolution of the soft function s(τ L , τ R , µ), while the logarithms in the non-global part do not. However, we have completely factorized this soft function in (1.4). Our factorization theorem splits the function into contributions from H S m , which live at the scale τ R , and contributions from S m , which live at the low scale τ L . The RG equations for these functions simultaneously resum all logarithms in the hemisphere soft function. So from the point of view of our effective theory, the splitting into global and non-global logarithms is artificial. The intricate structure of the logarithms is simply a reflection of the complicated operator structure in the effective theory.

Logarithmic corrections to the light-jet mass distribution at NNLO
We can obtain the logarithmic corrections to the light-jet mass distribution from those for the heavy-jet and left-jet mass distributions using (1.5). Since the NNLO corrections to the heavy-jet distribution are known, we first give new results for the NNLO corrections to the left-jet mass, before converting them into results for the light-jet mass and comparing with numerical results from event generators at the end of the section.
The factorization theorem for the left-jet mass distribution was given in (1.6). It is again convenient to work in Laplace space since the convolution with the jet function turns into an ordinary product. Introducing the Laplace transformation as in (3.1) the cross section becomesσ The Laplace-transformed jet functionsj i are the standard inclusive jet functions, which are well known. The soft functions are the same as the ones for the hemisphere soft case and were given in Section 3.1. This leaves us with a computation of the relevant hard functions and the evaluation of the angular integrals over the directions of the reference vectors.
The definition of the hard functions H i m for the left-jet mass, given in (2.10), involves matrix elements with a single hard parton of flavor i = q,q, g on the left and m hard partons on the right. The m = 1 hard functions have the form where σ 0 is the Born cross section for γ * → qq decay, given in d-dimensions by with α = e 2 /(4π) the fine structure constant and Q f the charge of the quark flavor q. Moreover, H(Q 2 , µ) is the standard dijet hard function present also in (1.3), and the δfunction in the angle arises because momentum conservation enforces that n 1 = n. The factor 1/2 is present because it is arbitrary whether we label the quark or anti-quark as being in the left hemisphere, so the two situations are averaged over. We also need the hard functions for the case of two hard partons in the right hemisphere. For the case of a quark-jet in the left hemisphere, we have whereθ 1 is the anti-quark angle andθ 2 the one of the gluon. Momentum conservation enforces ∆φ = φ 2 − φ 1 = π, which is why we only computed the soft function for this configuration. The thrust-axis constraint imposes the conditions 1 +θ 2 1 >θ 1 +θ 2 , 1 +θ 2 2 >θ 1 +θ 2 , (4.5) on the angular integration region, which can be added as θ functions to (4.4). This constraint implies in particular that the smaller of the two anglesθ 1 andθ 2 must be less than 1/ √ 3, which corresponds to a 60 • angle from the thrust axis. When the limit is reached the three partons are in a symmetric configuration and have all the same energy. If the angle becomes larger the thrust axis flips, since it always points in the direction of the most energetic parton in a three-parton configuration. For ǫ → 0, the function H q 2 has overlapping divergences when the anglesθ 1 andθ 2 go to zero simultaneously. To treat these, one splits the angular integration into two sectorsθ 1 <θ 2 andθ 1 >θ 2 and then parametrizesθ 1 = uθ 2 with u = 0 . . . 1 in the first sector and conversely in the second one. Once the divergences are separated one can expand both functions in ǫ using the identity (3.27) in the appropriate variables. At the one-loop level the renormalized expressions can be obtained by simply dropping the divergences which arise in this expansion.
The second configuration which is relevant is the one where we have a gluon jet on the left and a hard qq pair on the right. The hard function for this case reads and is subject to the same angular constraints (4.5). This hard function does not suffer from divergences when the angles go to zero, so we can immediately set ǫ → 0.
To obtain the full NNLO result for the left-hemisphere cross section, we would need also the one-loop corrections to H i 2 ⊗ 1 and the three-parton functions H i 3 ⊗ 1. However, if we are only interested in the logarithmic terms, we can avoid their computation by setting µ = Q. For this scale choice these functions do not contain any logarithms and we can therefore recover the logarithmic part of the NNLO cross section from where the factor 2 in the first line accounts for the identical contribution when the antiquark is in the left hemisphere. The two-loop result for the soft function S 1 was given in the previous section in (3.22). The dijet hard function (4.2) and the Laplace-space quark jet functionj q are well known. Explicit two-loop results for both quantities can be found in Appendix B of [35]. We can thus immediately evaluate the first line of (4.7) and what remains is the convolution on the second line. Since the functions H i 2 start at O(α s ), we need the gluon jet functionj g and the soft function S 2 only to one-loop order.
We have obtained analytical results for the convolutions of the two-parton functions with the trivial leading-order soft functions where L Q = ln (Q/µ). The appearance of logarithms and polylogarithms in addition to the usual ζ-values is a result of the phase-space constraint (4.5). The result in (4.9) agrees with the quantity r 3 obtained in [24], see (22) in [4]. Putting (4.8) together with the other one-loop ingredients we obtain agreement with the result of [4] also in the quark channel. For the NNLO cross section we need results for the convolutions with the NLO soft function (3.14), which have the form (4.11) The expressions for the coefficients M Putting everything together and inverting the Laplace transformation we then obtain all logarithmic terms in the left-jet mass distribution. The inverse Laplace transformation can be obtained using the simple substitution rules (4.12) Using relation (1.5) together with the known result for the logarithmic terms in the heavyjet mass distribution [23] we then obtain the light-jet mass distribution. Up to NNLO, it has the general form Note that at NLO, the distribution is a δ-function since the lighter jet contains only a single parton. A nontrivial light-jet mass distribution first arises from four-particle configurations at NNLO in which each hemisphere contains two partons. The logarithmic terms from these configurations are encoded in the function B + (ρ ℓ ), for which we obtain B + (ρ) = C 2 F − 4 ln 3 ρ − 9 ln 2 ρ + −   Figure 2. Comparison of our analytic results (solid lines) for the coefficients of the three color structures in the two-loop coefficient B + (ρ l ) for the light-jet mass distribution with numerical results (points with invisibly small error bars) obtained using the Event2 event generator [33]. The two results must agree for small ρ ℓ . The lower panel shows the relative difference in per cent. (4.14) Due to the uncalculated two-loop constant terms in the hard functions H 2 and H 3 , we cannot give the two-loop coefficient B δ , but the δ-function terms do not contribute to the logarithmic corrections to the light-jet mass distribution. We have verified that the terms involving powers of ln ρ in (4.14) are in agreement with those implied by the results of [4,24]. The remaining pieces, on the other hand, are new. As a further check, we have repeated the computation of the logarithmic terms in the cross section using bare instead of renormalized quantities. The logarithms are related to divergences in the individual ingredients in the factorization theorem (1.6). To obtain the logarithmic terms in the cross section we thus insert the divergent bare ingredients together with their associated logarithmic terms into the Laplace-transformed version of (1.6). The divergences cancel and we are left with a logarithmic structure which agrees with (4.14). The details of this computation can be found in Appendix D.
In contrast to the hemisphere soft function, the full analytical result for the light-jet mass distribution is not known, but our result for the coefficient B + (ρ l ) can be compared to numerical results obtained from running a fixed-order event generator. Since our results are the leading term in the limit ρ ℓ → 0, we need to run the fixed-order code for very small values of ρ ℓ to suppress higher-power contributions, which makes the numerics delicate. For our comparison, we use Event2 [33], which is well suited to study the region of small ρ ℓ since the phase-space generation can be tuned to focus on this region. We note that the fixed-order result is known even one order higher [36][37][38] and available in the form of a public code eerad3 [39]. In order to ensure that the power-suppressed terms are small, we run down to values of ln ρ ℓ = −16. To ensure numerical stability, Event2 imposes a cutoff on the invariant mass of parton pairs, and we run the code in quadruple precision to be able to lower the cutoff enough to avoid cutoff effects. Figure 2 shows the Event2 result in blue, compared to our analytic result shown as red lines. The statistical error bars on the Event2 results are barely visible, since we have generated 300 billion events. The upper panels show that the numerical results indeed approach the leading-power analytic results as the value of ρ ℓ is lowered. In the lower panel, we show the difference between Event2 and the analytic result in per cent, and the two agree to better than half a per cent for low values of ρ ℓ . However, our statistical uncertainties are even smaller than this and we find residual deviations in all color channels which are larger than the uncertainties. As a cross check, we have performed the same comparison against the well-known analytical result for the heavy-jet mass [23] and find deviations of similar size. Indeed, earlier papers have identified similar numerical issues in several variables [23,40,41], so we believe that the remaining deviations are not indicative of a problem in our analytic computation. We have also compared with the results from eerad3 and from the CoLoRFulNNLO framework [38] but were not able to achieve small enough statistical uncertainties to resolve the difference between Event2 and the analytic result.

NLL resummation
Our focus has been on the factorization properties of the hemisphere soft function and the light-jet mass distribution. The factorization theorems we derived are important because they enable the resummation of the large logarithms. In our framework, this resummation is achieved by solving the RG evolution equations for the ingredients of the factorization theorem and evolving them to a common reference scale. To perform NLL resummation, which resums the leading non-global logarithms, one needs to evaluate the hard, jet and soft functions at tree level and evolve them using one-loop regular anomalous dimensions, together with the two-loop cusp anomalous dimension. The global part of the light-jet mass distribution at NLL was presented in [24] and the non-global part in the large-N c limit was computed in [4], but as far as we are aware a numerical result for the NLL resummed single-hemisphere mass distribution including NGLs was never presented in the literature.
The simplest way to obtain the NLL result for the left-jet mass distribution is to choose the factorization scale as µ = µ h ∼ Q. With this choice, the hard functions do not suffer from large logarithms and at NLL the factorization theorem (1.6) simplifies to We have used that all higher-order hard functions are suppressed by powers of α s (µ h ) and can be neglected at NLL. To obtain the cross section we thus need two ingredients: the resummed quark jet function and the soft function S 1 ({n}, ω, µ h ) evolved to the hard scale µ h . This soft function is the same as the NLL resummed result for the hemisphere soft function. Indeed, choosing µ = µ h and integrating ω R up to a large value Q ∼ µ h the factorization theorem (1.4) for this quantity at NLL accuracy reduces to This fact is of course well known and it is for this reason that the non-global logarithms in the light-jet mass are usually studied using the hemisphere soft function. Beyond NLL this simple relationship is no longer valid, because the left-jet mass receives contributions from hard radiation in the right hemisphere. Before analyzing the soft function further, let us quote the resummed result for the jet function at NLL. Using the Laplace-space technique of [42], one obtains where η J = 2A Γ (µ j , µ h ). Explicitly, the Sudakov exponent S(µ j , µ h ) and the single loga- 4) where r = α s (µ)/α s (µ j ). The result for A γ J is obtained by replacing Γ 0 → γ J 0 in A Γ (µ j , µ). The relevant expansion coefficients of the anomalous dimensions and the β-function can be found at the end of Appendix B.
The resummed soft function S 1 ({n}, ω L , µ h ) can be obtained by solving the RG equation for the soft functions, which in Laplace space takes the form Due to the factorization theorem (4.1), the anomalous dimension matrix must take the form The cusp piece is diagonal since the τ dependence of the anomalous dimension Γ S lm must cancel against that of the jet functionj q in (4.1). We can thus split the soft functions into a product where the global function fulfills the simple RG equation for the cusp part with trivial initial conditionS G (τ, τ ) = 1. In Laplace space this RG equation has the same form as for the jet function and is easily solved. Inverting the Laplace transformation, we obtain The remaining pieceŜ l ({n}, τ, µ) in (5.7) has a single logarithmic evolution driven byΓ lm ({n}), which can be derived from results given in Appendix C of [20]. This piece captures the non-global logarithms, through the formal solution where in the second line we usedŜ m ({n}, τ, µ s ) = 1 + O(α s ), and made explicit that at NLL the quantity S N G (µ s , µ h ) is thus a function of µ h and µ s only. The evolution matrix U S 1m evolves the soft function from the low scale µ s to the high scale µ h . It is obtained at NLL by exponentiating the one-loop anomalous dimension matrix but due to the angular convolutions and the color structure of the anomalous dimension matrix, deriving an explicit form for the evolution matrix is highly nontrivial. In our paper [20] we demonstrated that in the large-N c limit the exponentiation of the one-loop anomalous dimension matrix is equivalent to solving the BMS equation. The RG evolution equation (5.5) is also equivalent to a parton-shower equation and this is the way the resummation of the hemisphere soft function was performed in the original paper of Dasgupta and Salam [4], who presented a simple, accurate parameterization of their result. In the future, it will be very interesting to generalize this to higher logarithmic accuracy but for the moment we will simply use their result to obtain a resummed result for the left-jet mass and investigate the size of the leading non-global logarithms in this observable. The parameterization of Dasgupta and Salam has the form where the constants a = 0.85 C A , b = 0.86 C A , and c = 1.33 were determined by fitting to the parton-shower result. The resummed result for the soft function in momentum space is then simply the product of the global function with the non-global evolution factor, where η = η J + η S = 2A Γ (µ j , µ s ). The integrated left-jet distribution is then obtained as where we need to choose µ s ∼ ρ L Q and µ h ∼ Q. The quantity Σ q plays an important role in the coherent branching formalism [43][44][45], where it arises as an integral over the jet function. We verified that (5.14) indeed reproduces the result for this quantity given in [24] after setting the scales to the default values µ 2 j = ρ L Q 2 and µ s = ρ L Q. Formula (5.14) shows that the jet function in the coherent-branching formalism also includes the global part of the soft radiation. Our final resummed result (5.15) is therefore fully equivalent to that presented in [4]. Squaring Σ q , one obtains the integrated heavy-jet mass at NLL: (5.16) We have checked that using (5.14) in the above result reproduces the resummed result of [23]. Below we will use the result for R(ρ h ) together with relation (1.5) to obtain the light-jet mass from the left-jet mass distribution (5.15). The result for the resummed left-jet mass distribution (5.1) is shown in Figure 3. For our plots, we choose Q = M Z and α s (M Z ) = 0.1181 [46]. The red line shows the result for the default scale choices, and to estimate its uncertainty, we perform two different scale variations. In particular, we separately vary the hard scale µ h and the jet scale µ j by factors of two around the default choices µ 2 h = Q 2 and µ 2 j = ρ L Q 2 , and show in the plots the envelope of the two variations. At very low values of ρ L the spectrum ends because µ s = ρ L Q hits the Landau pole. One could also vary the soft scale, which would shift this end-point and thus generate a larger uncertainty band. The green line in the plot shows the global part of the left jet mass, i.e. the result without including S NG (µ s , µ h ). The difference between the two curves demonstrates that the non-global pieces have an important effect on the distribution. Note that the distributions shown in the plot are obtained from taking the derivative of the resummed cumulant R(ρ L ) in (5.15) with respect to ρ L . For fixed scales, integrating and differentiating would commute, but we choose the values of the scales in the cumulant and then take the derivative, which is advantageous, as explained in [47]. One benefit is that the spectrum is automatically normalized since R(ρ L ) → 1 for ρ L = 1 (the true upper limit of the spectrum is at a lower value and one often modifies the resummation prescription such that the result vanishes beyond the kinematical limit; for simplicity we will not do this here).
Our plots also include experimental results from the Aleph collaboration [48]. The LEP experiments have measured the light-jet and heavy-jet mass distributions and we have used relation (1.5) to convert their measurements into a result for the left-jet mass, naively adding the uncertainties on the two distributions in quadrature. It is obviously better to directly compare to the experimental result for the individual measurements, which is done in Figure 4. The comparison shows that non-perturbative effects, which will shift the peak to the right, are important at low values of ρ L , where the distribution is large. This is expected since the soft scale is µ s ∼ ρ L Q and takes non-perturbative values near the peak, especially for the light-jet mass. To reproduce the data, one would have to include such non-perturbative effects, and should also match to the fixed-order results to get a better description at higher values of ρ ℓ and ρ h . For the moment, we will not pursue these issues further. Our goal was to assess whether non-global effects are phenomenologically relevant and our results clearly show that this is indeed the case for the non-global hemisphere event shapes.

Conclusions and outlook
We have studied the factorization of large logarithmic corrections appearing in non-global hemisphere-mass observables at e + e − colliders. We focused our analysis on two particular cases: i.) the double differential cross section with respect to the left and right hemisphere masses M L and M R in the limit where M L ≪ M R ≪ Q, and ii.) the left-jet mass distribution in the limit where M L ≪ Q. Our main result in the first case was the derivation of a factorization formula for the hemisphere soft function S(ω L , ω R ) in the limit ω L ≪ ω R , while in the second case we presented a novel factorization formula for the differential cross section itself.
While the specifics of the two cases are slightly different, the ideas behind them are rather general, and indeed for the most part could be adapted from the analysis of cone-jet cross sections in [20]. In particular, the key feature of factorization formulas for such nonglobal observables is that additional wide-angle emissions of hard partons at each order in perturbation theory build up a tower of multi-Wilson-line operators in the effective field theory. The matrix elements of these operators define multi-Wilson-line soft functions, which appear in angular convolution integrals with their (distribution valued) Wilson coefficients, referred to as multi-parton hard functions.
We confirmed the validity of our factorization formulas through explicit NNLO calculations. For the hemisphere soft function, we showed that our results reproduce the known analytic ones from [9,10], including all constant and logarithmic pieces appearing in the limit ω L ≪ ω R . For the light-jet mass, on the other hand, we obtained only the logarithmically enhanced NNLO corrections, and validated them through numerical comparisons with event generators. In both cases, the main new perturbative results presented here were those for the multi-parton hard functions, since other ingredients appearing in the factorization could be taken from the literature. We calculated these to NLO in the case of the left-jet mass, and to NNLO in the case of the hemisphere soft function, thus providing a non-trivial example at NNLO of the renormalization procedure involving mixing of multi-Wilson-line operators characteristic of non-global observables.
The factorization formulas derived here provide the basis for all-orders resummation of non-global logarithms for these observables. To get an idea of the size of the effects, we have used the known result for the leading non-global logarithms in the hemisphere soft function to obtain the left-jet mass distribution at NLL. We find that the non-global effects, evaluated in the large-N c limit, are of the same magnitude as other NLL effects. For precision predictions of non-global observables, it would be important to include also higher-logarithmic effects. The necessary ingredients are available: we have computed the one-loop soft functions and hard functions and the relevant two-loop anomalous dimen-sions can be extracted from the work of [22]. Since one has to exponentiate an infinitedimensional anomalous dimension matrix, it is not possible to obtain analytic results and the resummation has to be performed numerically. One approach is to incorporate the corrections into the parton-shower framework used to compute the leading logarithmic corrections. It will be interesting to analyze how this can be done in an efficient way and to use our framework to produce precision predictions for non-global observables.
A Absence of leading-power collinear contributions to S(ω L , ω R ) One might expect that left-collinear modes with scaling could contribute to the hemisphere soft function, since they haven · p ∼ ω L , as required. The operator definition for the associated leading-power jet function has the form where the Wilson lines W n are built from collinear fields and are invariant under rescaling of the reference vector. The multipole expansion ensures that the left-collinear fields are always in the left hemisphere and for this reason, the collinear particles do not contribute to ω R . According to its definition the jet function transforms as J → J/α under the transformationn → αn , ω L → α ω L , (A. 3) or equivalently The n-loop corrections to J(α ω L ) scale as ω −1−2nǫ L and are thus incompatible with this scaling relation (A.4). We conclude that they must all vanish so that J(ω L ) = δ(ω L ) to all orders. The leading-power jet function is thus trivial and can be omitted. We note that power corrections do involve nontrivial collinear contributions, as can be checked through an explicit computation of the hemisphere soft function using the method of regions.

B Bare ingredients for the hemisphere soft function
In the main text, we have mostly presented renormalized results and have reconstructed the renormalized hemisphere soft function by combining renormalized ingredients. For completeness, we list here also the bare functions. These can be extracted from the results in [9,10] and they are used in Appendix C to derive the renormalized expressions.
The renormalization is interesting from an effective theory point of view and key to perform the resummation. However, to obtain the fixed-order result one can also combine the bare ingredients given in this appendix to recover the bare hemisphere soft function. The bare ingredients are also what is obtained when performing the method of regions computation. At NNLO, the regions computation yields three terms: i) purely hard contributions, ii) purely soft ones, and iii) a mixed contribution with one hard gluon on the right and a soft one on the left. We now list these in turn.
Let us first give the result for the purely hard corrections. They consist of a doublereal emission part and a virtual correction to single gluon emission. In the effective theory language they are When integrating also over the angles to compute these terms, one recovers the standard phase-space integration and the evaluation of these contributions simply amounts to computing the corrections to the Wilson line matrix element (2.4) in which all particles fly into the right hemisphere. This computation was performed in [9,10] and we can extract the coefficients h F , h A , h f and v A from those papers. The results are The differences quoted above are due to opposite-side emissions only and they contribute to subleading NGLs. These opposite-side contributions can be obtained from the computations in [9,10] by sending the right hemisphere energy ω R to infinity because ω R is much larger than the momentum components of the soft radiation. We have verified that a direct computation of the corresponding diagrams gives the same result. The final NNLO contribution is the convolution of NLO terms: It is worth noting that the product coefficient p A induced through the convolution of NLO functions is reproduced by the regions calculation of opposite-side gluon contributions, one with a hard scaling and the other with a soft scaling. This type of contribution is responsible for the leading NGLs, as well as part of the subleading ones. Evaluating the full NNLO expression according to (3.12) then yields where the same-side contributions are in the first line, and the opposite-side contributions in the second and third.
To obtain the renormalized function, we need to multiply by the renormalization factor Z S introduced in (3.2). Given the product structure of the factorization theorem (1.3) in Laplace space, it must have a factorized form Z S (τ L , τ R , ǫ, µ) =z s (τ L , ǫ, µ)z s (τ R , ǫ, µ) , (B.9) wherez s satisfies the RG equation Solving this equation perturbatively gives ln(z s (τ, ǫ, µ)) = α s 4π where L = ln(τ /µ). For convenience we give the necessary anomalous dimension in the above expression. The expansion of the anomalous dimensions in the strong coupling constant reads and To perform the NLL resummation in Section 5 we also need the anomalous dimensions     Here [. . . ] (2) and Z (2) lm refer to the second-order coefficients in the renormalized coupling, while S (2) 1 (ǫ) denotes the second order coefficient of the bare coupling. Notice that S (1) 2 is a regular function in its arguments, so the equations above imply that Z (1) 12 and Z (1) 11 are also regular functions and not distributions. It follows that the renormalized NLO functions are simply obtained from the bare functions by dropping the poles. Moreover, the following linear combinations of renormalization factors are immediately obtained For the renormalized soft function we obtain the result in (3.22). Because only the linear combinations of renormalization factors listed in (C.3) above is determined, and because we have the bare functions only after integrating over angles, we can only determine