Infrared finiteness in the factorization of the dijet cross section in hadron-hadron collision near threshold

# Infrared finiteness in the factorization of the dijet cross section in hadron-hadron collision near threshold

Junegone Chay Department of Physics, Korea University, Seoul 136-713, Korea    Taewook Ha Department of Physics, Korea University, Seoul 136-713, Korea    Inchol Kim Department of Physics, Korea University, Seoul 136-713, Korea
###### Abstract

The factorization theorem for the dijet cross section is considered in hadron-hadron collisions near threshold with a cone-type jet algorithm. We focus on the infrared finiteness of the factorized parts by carefully distinguishing the ultraviolet and infrared divergences in dimensional regularization. The soft function, subject to a jet algorithm, shows a complicated divergence structure. It is shown that the soft function becomes infrared finite only after the emission in the beam directions is included. Among many partonic processes, we take as a specific example to consider the dijet cross section, and verify explicitly that each factorized part is infrared finite. We also compute the anomalous dimensions of the factorized components to next-to-leading logarithmic accuracy. The hard and the soft functions have nontrivial color structure, while the jet and the collinear distribution functions are diagonal in color space. The dependence of the soft anomalous dimension on the jet algorithm is color diagonal and is cancelled by that of the jet functions. The sum of the remaining anomalous dimensions also cancels, thus the dijet cross section is independent of the renormalization scale.

## I Introduction

The study of jet physics in high energy scattering has reached a sophisticated level. Many of the factorization theorems for inclusive scattering processes have been established both in QCD and in soft-collinear effective theory (SCET) Bauer:2000ew (); Bauer:2000yr (); Bauer:2001yt (). More differential quantities such as the transverse momentum dependence of the final-state particles or jets Stewart:2013faa (), and the jet substructures Dasgupta:2013ihk () have been studied. Though we probe more differential quantities, not all the factorization theorems are provided yet. But the factorization theorems should be proved since they offer a fundamental basis for theoretical predictive power.

In general, the factorization theorem states that the prediction for physical observables consists of the product or the convolution of the hard, the collinear and the soft parts. In proving the factorization theorem for various scattering processes, it is important to verify that each factorized part is infrared (IR) finite. Otherwise the dependence of the renormalization scale does not solely come from the ultraviolet (UV) divergence, which invalidates the scaling behavior of the factorized parts. If some components are not IR finite in the factorized form, the factorized parts should be reshuffled such that the redefined or rearranged quantities are IR finite. If the IR divergence remains even after the rearrangement, the quantity at hand is not physical.

A fully inclusive quantity is IR finite to all orders due to the Kinoshita-Lee-Nauenberg theorem Kinoshita:1962ur (); Lee:1964is () since the IR divergence from the virtual correction is cancelled by that of the real correction. It should hold true also for exclusive physical quantities such as the dijet cross section, in which the phase space for the real gluon emission is constrained by the jet algorithm.

The verification that each factorized part is IR finite in the dijet cross section from annihilation with various jet algorithms has been performed in Refs. Chay:2015ila (); Chay:2015dva (). On the other hand, here we take the dijet cross section in hadron-hadron collision near threshold with a cone-type jet algorithm Salam:2009jx () to show explicitly that each factorized part is IR finite by carefully dissecting the phase space and performing the corresponding computation. This process is more complicated due to the complex color structure and the existence of the beam hadrons, thus more illuminating to show how to disentangle the interwoven structure of divergence.

The dijet cross section in hadron-hadron collision near threshold is shown to be factorized into the hard, collinear and soft parts, which is schematically written as

 σJ∼tr(H⊗S)⊗fi/N1⊗fj/N2⊗J3⊗J4. (1)

A more rigorous expression will be derived in Sec. II. Here is the hard function depending only on the hard scales, and is the soft function which describes the soft interactions among the energetic particles. The hard and soft functions are matrices in color space because they arise from different color channels. Near threshold, the incoming partons are described by the collinear distribution functions , for the parton to be contained in the hadron , rather than the parton distribution functions (PDFs) . The distinction will be discussed later. And are the integrated jet functions describing the outgoing collinear particles prescribed by a jet algorithm in the final state.

The configuration of the dijet production near threshold is schematically shown in Fig. 1. The incoming partons with momenta and take most of the momenta from the two hadrons with momenta and and participate in the hard collision. Then the partonic CM frame coincides with the hadronic CM frame. Jet algorithms in annihilation and in hadron-hadron scattering are different since the latter should preserve the boost invariance along the beam direction. However, near threshold, we can employ the same cone-type jet algorithm with the rescaling of the jet radius Hornig:2016ahz (). For central jets with the rapidity close to zero, the cone-type jet algorithms in both cases take the same form.

Since the partons are back to back in the hadronic CM frame near threshold, the resultant dijets are also produced back to back. Away from the threshold, the dijets do not have to be back to back, and the initial-state particles also form beam jets in the beam direction. In this case, the factorized form for the jet cross section retains almost the same form except that the collinear distribution functions are replaced by the beam functions Stewart:2009yx (), describing the emission of the collinear particles along the initial parton from the hadron. This case will be investigated later, and here we consider the dijets near threshold.

In every high-energy process, the IR finiteness should be guaranteed, but here we focus on the dijet cross section to illustrate how carefully the UV and IR divergences should be treated. We examine various radiative corrections by carefully separating the UV and IR divergences and show that each factorized part is indeed IR finite. This type of analysis should be applied to other various differential processes for the rigorous proof of the factorization theorems.

The claim that the factorization of the cross section is established is to verify that each factorized part should be free of IR divergence. Numerous types of computations have been performed to extract the UV divergence either by applying the dimensional regularization or by introducing some IR regulators. In many cases, the divergences appearing in the computation are regarded as the UV divergence based on the belief that the factorized parts are physical quantities, hence should be free of IR divergence. However, the mere separation of the long- and short-distance effects via the factorization guarantees in no way that each factorized part is IR finite. It should be rather explicitly checked than presumed. If each factorized part is IR free, the factorization theorem remains valid. If not, a new combination should be devised such that there is no IR divergence in the recombined parts. In fact, if the divergences are carefully distinguished, there may remain IR divergence, which endangers the validity of the factorization. This aspect will be investigated in detail here.

Only after the remaining divergence is guaranteed to be of the UV origin, we can safely apply the renormalization group equation to resum large logarithms. In this paper, we employ the pure dimensional regularization with the spacetime dimension and the scheme, in which we carefully distinguish the UV and the IR divergences in computing radiative corrections of the jet and soft functions with a jet algorithm. Especially in the soft function, there exist IR divergences if we naively apply the jet algorithm. However, we have found some additional contributions from the phase space pertaining to the beam directions. By including this contribution, the soft function becomes IR finite, and we have a firm status in proving the factorization theorem. It will be explained in detail in Sec. III.

The dijet cross section is described by processes at the parton level. We can analyze all the processes, but we rather choose the specific process to show how the extraction of the IR divergence works. This process involves the computation of the gluon jet function with the jet algorithm, while the quark jet function with the jet algorithm was calculated in Ref. Chay:2015ila (); Jouttenus:2009ns (); Cheung:2009sg (). And the structure of the hard and soft functions is interesting and complicated enough to seek the consistency in the relations among the anomalous dimensions.

This paper is organized as follows: The factorization of the dijet cross section in hadron-hadron collision is presented in Sec. II. We take a specific example of the partonic process to express the individual factorized components explicitly. In Sec. III, we discuss the main ideas and techniques, in which we explain the structure of the phase space for the soft function and the technical details in dimensional regularization to handle the aforementioned phase space. In Sec. IV, the soft function is computed with the jet algorithm. The cancellation of the IR divergence is the most nontrivial issue and it will be treated in detail. In Sec. V, the gluon jet function and its anomalous dimensions are computed with the cone-type algorithm at next-to-leading order. In Sec. VI, the collinear distribution function, and its anomalous dimension are computed at one loop. In Sec. VII, we collect the anomalous dimensions of the hard, soft, jet and collinear distribution functions at one loop. The independence of the renormalization scale in the dijet cross section is confirmed explicitly. In Sec. VIII, the conclusion and the perspective are presented. In Appendix A, the detailed computation is presented for the real gluon emission in the soft part from the beam-jet contributions. In Appendix B, the soft contribution along the beam direction is derived both from the beam-beam and the beam-jet contributions.

## Ii Factorization of the dijet cross section

 N1(P1)+N2(P2)→j3(p3)+j4(p4)+Xs, (2)

where and are incoming hadrons (protons in the case of LHC), and denote two back-to-back energetic collinear jets and represents soft particles. Near threshold, the momenta of the incoming partons and are close to the hadronic momenta and respectively. We choose the beam directions to be in the and directions with , , and . The jet directions are chosen to be , and with , , and . And we consider the dijets away from the beam direction, which can be stated as .

The dijet cross section near threshold in SCET is written as

 σ(N1N2→j3j4Xs) = 12S2∏X3,X4,Xs(2π)4δ(4)(Pμ1+Pμ2−pμ3−pμ4−pμXs) (3) × ∑IJCIC∗J⟨N1N2|O†J|X3,X4,Xs⟩⟨X3,X4,Xs|OI|N1N2⟩.

Here denotes the phase space for the final-state particles, and is the hadronic center-of-mass energy squared. The set of operators are the SCET operators for processes and are the Wilson coefficients obtained by matching SCET and full QCD Ellis:1985er (); Kelley:2010fn ().

The operators can be categorized by the partons participating in the hard scattering processes. If the initial- and final-state particles consist of quarks or antiquarks, the set of the SCET collinear operators for are given as

 O1=¯¯¯¯χ2Taγμχ1⋅¯¯¯¯χ4Taγμχ3,  O2=¯¯¯¯χ2γμχ1⋅¯¯¯¯χ4γμχ3, (4)

where the collinear fields are the collinear gauge-invariant combination with the collinear Wilson line . The SU(3) generators for the strong interaction are in the fundamental representation. These operators are responsible for the scattering of , , including different types of quarks, which are related by the appropriate crossing symmetry.

For the processes , , and , the relevant SCET collinear operators are given by111In Ref. Chiu:2009mg (), another independent set of operators are introduced: , , and . They are related by , , and .

 O1=¯¯¯¯χ2TaTbχ1Bμa⊥4Bb⊥μ3, O2=¯¯¯¯χ2TbTaχ1Bμa⊥4Bb⊥μ3, O3=¯¯¯¯χ2χ1Bμa⊥4Ba⊥μ3, (5)

where is the collinear gauge-invariant gluon field in the direction in SCET. For the process , there are 9 independent collinear SCET operators, which are of the form (), where indicates the helicity of the gluons. The explicit form can be found in Ref. Kelley:2010fn ().

The factorization procedure can be performed for any partonic processes, but it is illustrative to pick up one process and treat the factorization in detail. As a specific example, we consider the partonic process . The relevant operators with the redefinition of the collinear fields to decouple the soft interaction , are given by

 OI=(¯¯¯¯χα2χβ1Ba4⊥μBbμ3⊥)(Y†2Y†aa′4Ta′b′IYb′b3Y1)αβ, (6)

where , and . The indices , (, ) refer to the adjoint (fundamental) representation. The soft Wilson line associated with the -collinear fermion is given in the fundamental representation, while the soft Wilson line from the -collinear gluon is given in the adjoint representation.

The collinear matrix element for the operators in Eq. (6) is given by

 ∑X3,X4⟨N1N2|¯¯¯¯χρ1χσ2Bcν⊥3Bd⊥4ν|X3X4⟩⟨X3X4|¯¯¯¯χα2χβ1Ba⊥4μBbμ⊥3|N1N2⟩, (7)

and it can be expressed in terms of the gluon jet functions and the collinear distribution functions. The gluon jet functions in the and directions are defined as

 ∑X3⟨0|Bcν⊥3|X3⟩ΘJ⟨X3|Bbμ⊥3|0⟩ = −gμν⊥δbc∫d4p3(2π)3Jg(p23), ∑X4⟨0|Bdν⊥4|X4⟩ΘJ⟨X4|Baμ⊥4|0⟩ = −gμν⊥δad∫d4p4(2π)3Jg(p24), (8)

where denotes the jet algorithm to be employed. The jet functions are normalized to at tree level.

Near the threshold, the collinear distribution functions are defined as

 ⟨N1|¯¯¯¯χρ1kδ(ξ1−¯¯¯n1⋅P¯¯¯n1⋅P1)χβ1j|N1⟩ = ¯¯¯n1⋅P12N(/n12)jkδρβfq/N1(ξ1), ⟨N2|χσ2lδ(ξ2−¯¯¯n2⋅P¯¯¯n2⋅P2)¯¯¯¯χα2i|N2⟩ = ¯¯¯n2⋅P22N(/n22)liδσαf¯q/N2(ξ2), (9)

where the subscripts are the Dirac indices, and is the number of colors. And is the operator extracting the label momentum. The collinear distribution function is normalized as at tree level.

Then the dijet cross section near threshold is factorized as222To be rigorous, the effect of the Glauber gluons should be implemented to prove factorization.

 σ=116πN2S∑IJHIJ(μ)SJI(μ)∫dξ1fq/N1(ξ1,μ)∫dξ2f¯q/N2(ξ2,μ)Jg3Jg4+(q↔¯¯¯q), (10)

where the soft function is defined as

 SJI=∑Xstr⟨0|Y†1(Y†3T†JY4)baY2|Xs⟩ΘJ⟨Xs|Y†2(Y†4TIY3)abY1|0⟩, (11)

with the appropriate jet algorithm denoted by . The integrated jet function is defined as

 Jgi=∫dp2iJg(p2i), (i=3,4). (12)

If we are interested in the dijet invariant mass distribution , , the differential cross section with respect to the invariant jet masses is given by

 dσdm2j3dm2j4 = 116πN2S∫dξ1fq/N1(ξ1,μ)∫dξ2f¯q/N2(ξ2,μ) (13) × ∫dl+dl−Jg(m2j3−¯¯¯n⋅p3l−)Jg(m2j4−¯¯¯n4⋅p4l+)∑IJHIJ~SJI(l+,l−),

with and . The differential soft function is defined as

 ~SJI(l+,l−)=tr⟨0|Y†1(Y†3T†JY4)baY2δ(l++n3⋅Ps)ΘJδ(l−+n4⋅Ps)Y†2(Y†4TIY3)abY1|0⟩. (14)

From now on, we concentrate on the integrated jet and soft functions at next-to-leading order. In probing the structure of divergence, it is necessary to analyze the phase space for the soft function in detail, and to discuss the calculational scheme in using the dimensional regularization.

## Iii Phase space and divergence of the soft functions

In disentangling the divergence structure, the soft function is the most sophisticated part to be treated with care. The phase space, in which the UV or IR divergence arises, depends on which Wilson lines are involved in the soft part. And there is some phase space missing due to the existence of the beam particles. On the technical side, we employ the dimensional regularization both for the UV and IR divergences, and we explain how we proceed especially when there are double poles. This section constitutes the main idea and technique in this paper.

### iii.1 Phase space for the soft function

At the parton level, the dijet production from hadron-hadron scattering and the process jets are similar since they are related by the crossing symmetry. But in hadron-hadron scattering, only the final-state partons are organized by the jet algorithm, while all the final-state partons in jets are scrutinized by the jet algorithm. This affects the soft function and its anomalous dimension, which depend on the jet cone size. However, it turns out that the anomalous dimension of the soft function depending on the jet cone size is diagonal in color basis, which cancels the cone size dependence of the anomalous dimension in the jet function.

We consider the cone-type jet algorithm at next-to-leading order, in which there are at most two particles inside a jet. At this order, we choose the jet axis in the direction. The jet axis may be chosen as the thrust axis, or the weighted average of the rapidity and the azimuthal angle over the transverse energy. Then the particles inside a jet should satisfy the condition . Here is the angle of the -th particle with respect to the jet axis, and is the jet cone size.

In the threshold region the jet algorithm can be expressed in terms of the lightcone momenta as follows Chay:2015ila (); Cheung:2009sg (); Ellis:2010rwa ():

 n3⋅ln3⋅l+¯¯¯n3⋅l<δ2,  n3 jet, n4⋅ln4⋅l+¯¯¯n4⋅l<δ2,  n4 jet, n3⋅l+n4⋅l<2βQ,  jet veto, (15)

where . We retain this form with the understanding that is actually replaced by , where is the cone size in the pseudorapidity-azimuthal angle space, and is the pseudorapidity of the jet in hadron-hadron scattering Hornig:2016ahz (). And is the fraction of the energy outside the jets acting as a jet veto, and . The jet veto is needed to guarantee that the final states form a dijet event. The constraint due to this jet algorithm for the final-state particles is shown in Fig. 2 (a). Eq. (III.1) as a whole incorporates the jet algorithm. But when there is no confusion, we sometimes refer to the first two equations of Eq. (III.1) as the jet algorithm since they are the conditions for the particles to be inside the jet, and refer to the third equation as the jet veto.

For power counting in SCET, the -collinear momentum scales as , where is the small parameter. Then the soft momentum scales as . And we also take and for definiteness. We may need other degrees of freedom if we are interested in the small resummation Chien:2015cka (). But this topic will be deferred.

Though the phase space is constrained by the jet algorithm as shown in Fig. 2, the structure of the divergence depends on which soft Wilson lines participate in the soft function. The integrand in computing the real gluon emission can be obtained by expanding the corresponding soft Wilson lines, and is of the form

 Iij=ni⋅njni⋅lnj⋅lδ(l2), i,j=1,⋯,4. (16)

The characteristics of are categorized into three classes: the jet-jet contribution , the beam-beam contribution , and the beam-jet contributions for the remaining combinations of . The three classes have different structure of divergence.

For the jet-jet contribution , the denominator of becomes singular when the momentum approaches zero or collinear to or . Therefore the UV or IR divergences arise in each of the phase space specified in Fig. 2 (a). However, when the virtual corrections are added, the soft part in the jet-jet contribution becomes IR finite Chay:2015ila ().

For the beam-beam contribution , the denominator never becomes singular unless approaches zero since the gluon momentum should be close to the jet directions , , and away from the beam directions , . Due to this fact, the contributions from the first two conditions of the jet algorithm in Eq. (III.1) (blue region) do not induce any (collinear) IR divergence and the contribution from that phase space is suppressed by . The detailed calculation of the explicit contribution to order is presented in Appendix A. Therefore the only divergence comes from the region specified by the jet veto, as illustrated in Fig. 2 (b), in which we show only the region of the phase space which yields the IR divergence. Here it is hard to visualize the and lines in Fig. 2 (b) and (c), so they are expressed as dashed lines for convenience. The contribution from the gluons in the and directions is critical in treating the IR divergence. This point will be discussed in detail.

For the beam-jet contributions, for example, or , as shown in Fig. 2 (c), the denominator becomes singular when the momentum approaches zero or is collinear to . When the momentum is collinear to , the contribution is finite and suppressed by for the same reason as in the beam-beam contribution. Therefore the divergence occurs in the phase space, as illustrated in Fig. 2 (c). We can also consider the cases with or , and the phase space which results in the IR divergence is given by the blue region near the axis instead along with the jet veto.

Considering the fact that the soft function from the jet-jet contributions with the virtual corrections is IR finite, we can see that the soft functions from the beam-jet and the beam-beam contributions should contain IR divergence because the IR divergence from the jet algorithms for the or jets or both is missing. This presents a predicament since the IR-divergent soft function loses its physical meaning. There must be another source for the IR divergence which cancels the existing divergence as mentioned above. That contribution is related to the emission of gluons exactly in the beam directions.

When a soft gluon is emitted outside the jet, its energy should be less than according to the jet veto. The beam directions also belong to the region outside the jet. However, when a soft gluon is emitted exactly in the beam directions, that is, in the or directions, the jet topology also corresponds to a dijet event. In this case, the energy of the soft gluon can be larger than the jet veto, and this contribution should be added in calculating the soft function. In contrast, in annihilation, there is no such contribution since there are no incoming hadrons in the beam direction. One of the main themes in this paper is to extract these contributions in order to show that the soft function becomes IR finite with this additional contribution.

We can consider a more general case away from the threshold, in which there are two central jets and two beam jets. Then the emission of soft gluons in the beam direction does not have to be exactly collinear to the incoming particles, and can be smeared to be inside the beam jets. Then it requires a beam algorithm analogous to the jet algorithm to treat the soft gluon in the beam directions. In this case, the anomalous dimensions may depend also on the size of the beam jets, and there may be nonglobal logarithms Dasgupta:2001sh () related to the different sizes of the central jets and the beam jets. This may be interesting, but we will not consider this case here, and it is the topic to be pursued later. In a sense, the collinear distribution function in the dijet event near threshold corresponds to the beam function with the zero beam size.

### iii.2 Divergence structure and the dimensional regularization

Before we compute the soft functions, there are two important points which should be handled with care in using the dimensional regularization to separate the UV and the IR divergences. First, the ingredient to separate the UV and IR divergences is based on the fact that

 μϵ∫∞0duu−1−ϵ=1ϵUV−1ϵIR, (17)

where has the dimension of a momentum. In the naive dimensional regularization in which the UV and IR divergences are not distinguished, the integral vanishes since it is a scaleless integral. The UV divergence is extracted as a pole in by inserting an arbitrary IR regulator. Using the fact that the overall integral is zero, the IR divergence as a pole in is obtained as in Eq. (17).

Secondly, in the virtual corrections of the soft functions, there appears an integral of the form

 Vij = (μ2eγ4π)ϵ∫dDl(2π)Dni⋅njl2ni⋅lnj⋅l (18) = −i8π2Γ(1+ϵ)(ni⋅nj2)−ϵ(μ2eγ)ϵ∫∞0duu−1−ϵ∫∞0dvv−1−ϵ.

The only point we can claim about the integral is that it vanishes in the naive dimensional regularization, hence the integral is proportional to . Other than this observation, the result of the integration is ambiguous. To see why, let us consider two possible ways to compute this integral.

The first method is to treat the variables separately and it can be written as

 μ2ϵ∫∞0duu−1−ϵ∫∞0dvv−1−ϵ=(μϵ∫∞0duu−1−ϵ)2=(1ϵUV−1ϵIR)2. (19)

Then is given by

 Vij=−i8π2(1ϵUV−1ϵIR)2. (20)

On the other hand, the integral can also be performed by introducing the polar coordinates , , as

 μ2ϵ∫∞0duu−1−ϵ∫∞0dvv−1−ϵ = μ2ϵ∫∞0drr−1−2ϵ∫π/20dθ(sinθcosθ)−1−ϵ (21) = −1ϵIR(1ϵUV−1ϵIR).

The integral on yields both the UV and IR divergences, while the angular integral produces only the IR divergence. Therefore is given by

 Vij=i8π2(1ϵUV−1ϵIR)(1ϵIR−lnni⋅nj2). (22)

Comparing Eqs. (19) and (21), the integrals are indeed proportional to , but they are different.

Now the question is which one should be employed in actual computations. We emphasize that the treatment of the virtual correction is not determined by the virtual correction alone, but is determined by how the real gluon emission is computed. The ambiguity of the divergences in the soft function was also pointed out in Refs. Idilbi:2007ff (); Hornig:2009kv (). The point is that the method in computing the virtual and the real contributions should be performed in the same way. Usually the real gluon emission is complicated when the jet algorithm is implemented. The phase space for the virtual correction is simpler than that of the real contribution, hence the computation is more versatile in the virtual correction. Thus we calculate the virtual correction in the same way as we compute the real gluon emission.

The argument can be justified as follows: If there is no constraint for the soft gluon emission, the whole phase space is covered, and the contribution is proportional to . The virtual correction has the same form as the real gluon emission except the sign. The whole soft contribution at one loop, as well as all the divergences, vanishes as long as we maintain the same method of calculation either Eq. (19) or Eq. (21). Actually the inclusive soft function for the real gluon emission receives no radiative corrections to all orders. If we employ different methods for the virtual and real corrections, the result cannot be zero. This argument also holds for the computation of the soft function with the jet algorithm. The correct divergence structure is obtained only when the same method is applied consistently to the virtual and the real contributions.

The question remains whether the final finite answer could be the same irrespective of the different treatment for . The jet-jet contribution has been computed using Eq. (19) in Ref. Chay:2015ila (), and we will show here that the same answer is obtained using Eq. (21) with the angular integral in the real gluon emission. Note also that the different divergence structure in Eqs. (19) and (21) arises only when the whole phase space is covered. In other cases in which the whole phase space is not involved, for example, in the contribution from the jet veto as shown in Fig. 2 (b), the integration of Eq. (18) is the same in both approaches producing only the IR divergence.

## Iv Soft function

The soft function, defined in Eq. (11), is given again by

 SJI=∑Xstr⟨0|Y†1(Y†3T†JY4)baY2|Xs⟩ΘJ⟨Xs|Y†2(Y†4TIY3)abY1|0⟩. (23)

The Wilson lines and correctly describe the soft gluon emission from the antiquark and the quark in the initial state. But, to be exact, the Wilson lines and should be and , describing the soft gluon emission from the outgoing particles Chay:2004zn (). However, we keep the forms as they are for notational simplicity. The soft Wilson lines are given as

 Y1=∑perm.exp[1n1⋅R+i0(−gn1⋅As)], Y†2=∑perm.exp[−gn2⋅As1n2⋅R†−i0], (24)

where is the operator extracting the soft momentum.

The Feynman diagrams for the soft function at one loop is shown in Fig. 3. Figs. 3 (a) and (b) represent the virtual and the real corrections respectively. The vertical dashed lines are the unitarity cuts, and the hermitian conjugates are not shown. The contributions for the soft function can be expressed as the sum of the two parts, one from the jet veto and the other from the jet algorithm. These are given as

 Mvetoij = 2πg2(μ2eγ4π)ϵ∫dDl(2π)Dni⋅njni⋅lnj⋅lδ(l2)Θ(0βQ)ΘJ, (25)

apart from the group theory factors Chiu:2009mg (); Catani:1996jh (); Catani:1996vz (). The jet algorithm represents either of the first two equations in Eq. (III.1) depending on whether the jet in the or direction is considered. () is the contribution from the phase space colored as red (blue) in Fig. 2.

Since will be computed using the angular integral, the virtual correction with Eq. (21) is given by

 (26)

where .

### iv.1 Jet-jet contribution

This is the contribution from the case , where the soft gluons are emitted from the soft Wilson lines associated with the final-state partons. The results are given by

 Mveto34 = αs2π[1ϵ2IR+1ϵIR(2lnμ2βQ−lnn342)+2ln2μ2βQ Mjet34 = αs2π(−12ϵUVϵIR−1ϵIRlnμ2βQ+1ϵUVlnδ−ln2μ2βQδ+π224), (27)

where is kept explicitly for comparison with other cases. Note that the computation of each term in Eq. (IV.1) is performed using the angular integral in the phase space based on Eq. (21). The detailed calculation is presented in Appendix A. Since the phase space for the jet veto depends on the directions of and ( and in this case) in general, depends on . However, the jet contribution collects the contribution in a single jet direction, hence it depends not on but only on , and the jet veto parameter .

The soft contribution in the jet-jet contribution is given by

 M34 = Mveto34+2Mjet34+MV34 (28) = = αs2π[2(1ϵUV+2lnμ2βQ)lnδ−2ln2δ−π26],

where the last line is obtained by putting . The soft contributions to order at order are computed in detail in Appendix A. For the jet-jet contribution, it is given by

 M34(δ2)=αsπδ2. (29)

The same result is obtained by computing each term based on Eq. (19) Chay:2015ila (). This verifies the point that the divergences as well as the finite terms are the same in the two approaches as long as a consistent method is employed in the virtual and real contributions. The finite soft contribution at order is given as

 M(1)34=αs2π(4lnμ2βQlnδ−2ln2δ−π26). (30)

### iv.2 Beam-beam contribution

The beam-beam contribution corresponds to the case . The jet contribution is finite and suppressed by , hence neglected. However, it is explicitly computed in Appendix A . The contribution is given by

 Mveto12 = αs2π[1ϵ2IR+1ϵIR(2lnμ2βQ−lnn122)+2ln2μ2βQ (31)

Here the dependence of is retained but the final result is obtained by putting . Obviously, this is the same as .

Note that the sum of and the virtual correction is still IR divergent, which is given by

 Mveto12+MV=αs2π(1ϵUVϵIR+2ϵIRlnμ2βQ+2ln2μ2βQ−π24). (32)

As explained in Sec. III.1, we should add the contribution from the real gluon emission in the beam direction with . It is given by

 Mbeam=αs2π(−12ϵUVϵIR−1ϵIRlnμ2βQ−ln2μ2βQ+π224). (33)

The detailed derivation of is presented in Appendix B.1.

The beam-beam contribution for the soft part at one loop can be obtained by setting , and it is given by

 M12 = Mveto12+MV12+2Mbeam (34) = αs2π[−(1ϵUV+2lnμ2βQ)lnn122−Li2(1−2n12)−π26]=αs2π(−π26).

Since it is IR finite, gives the finite soft contribution at one loop as

 (35)

In addition, the contribution at order and is given by

 (36)

### iv.3 Beam-jet contribution

This corresponds to the cases with and . The results are given by

 Mvetoij = αs2π[1ϵ2IR+1ϵIR(2lnμ2βQ−lnnij2)+2ln2μ2βQ Mjetij = αs2π(−12ϵUVϵIR−1ϵIRlnμ2βQ+1ϵUVlnδ−ln2μ2βQδ+π224). (37)

The contribution can be computed exactly in the same way as in Ref. Ellis:2010rwa () by replacing the pole by . This is because contains only the IR divergence due to the phase space constraint. In computing , care must be taken to separate the IR and UV divergences, and the detailed computation is presented in Appendix A.

Without the soft contribution in the beam direction, the soft part still remains IR divergent. This intermediate soft contribution is given by

 Mvetoij+Mjetij+MVij = αs2π[12ϵUVϵIR+1ϵIRlnμ2βQ+ln2μ2βQ+1ϵUVln2δnij (38) +2lnμ2βQln2δnij−ln2δ−Li2(1−2nij)−524π2].

It turns out that the beam contribution is the same both in the beam-beam and in the beam-jet contributions. This is explained in detail in Appendix B. Finally the soft part in the beam-jet contribution including the beam contribution is given by

 Mij = Mvetoij+Mjetij+MVij+Mbeam (39) =

which is IR finite.

The soft contribution for the beam-jet case at one loop is given by

 M(1)ij=αs2π[2ln2δnijlnμ2βQ−ln2δ−Li2(1−2nij)−π26]. (40)

And the suppressed term proportional to , using the jet algorithm for the -jet at order is given as

 (41)

## V Gluon jet function

The inclusive gluon jet function has been computed to one-loop order Becher:2009th (), and two-loop order Becher:2010pd (). Here we compute the gluon jet function with the cone jet algorithm at one loop. The cone jet algorithm at next-to-leading order involves at most two particles inside a jet. In computing the jet function, the matrix element squared is schematically shown in Fig. 4. The loop includes other particles. (See Fig. 6.) If we make a unitarity cut in any of the gluon lines, the cut gluon lines correspond to the final-state particles. For example, if a single leg is cut, it represents a single final-state particle with the virtual correction. If the loop is cut, it represents two final-state particles, to which the jet algorithm is applied.

The momentum of the jet is given by , and the momenta of the two gluons are labeled as and respectively. Suppose that the jet is collinear in the lightcone direction. Then the momenta of the gluons can be written as

 p1=(l−,l⊥,l+), p2=(Q−l−,−l⊥,p2/Q−l+), (42)

where . The energies of the gluons and the invariant mass squared of the jet are given by

 E1=12(l−+l+),  E2=12(Q−l−+p2Q−l+), p2=Ql+1−l−/Q. (43)

The cone jet algorithm for the -collinear jet requires and , where is the jet cone size, and is the angle of the gluon with respect to the jet axis, which is chosen to be in the direction. This jet algorithm for the collinear part can be written as Chay:2015ila (); Cheung:2009sg (); Ellis:2010rwa ()

 ΘJ=Θ(δ2>