Real-virtual corrections for gluon scattering at NNLO

We use the antenna subtraction method to isolate the mixed real-virtual infrared singularities present in gluonic scattering amplitudes at next-to-next-to-leading order. In a previous paper, we derived the subtraction term that rendered the double real radiation tree-level process finite in the single and double unresolved regions of phase space. Here, we show how to construct the real-virtual subtraction term using antenna functions with both initial- and final-state partons which removes the explicit infrared poles present in the one-loop amplitude, as well as the implicit singularities that occur in the soft and collinear limits. As an explicit example, we write down the subtraction term that describes the single unresolved contributions from the five-gluon one-loop process. The infrared poles are explicitly and locally cancelled in all regions of phase space prior to integration, leaving a finite remainder that can be safely evaluated numerically in four-dimensions. We show numerically that the subtraction term correctly approximates the matrix elements in the various single unresolved configurations.


Introduction
In hadronic collisions, the most basic form of the strong interaction at short distances is the scattering of a coloured parton off another coloured parton. Experimentally, such scattering can be observed via the production of one or more jets of hadrons with large transverse energy. In QCD, the (renormalised and mass factorised) inclusive cross section has the form, where the probability of finding a parton of type i in the proton carrying a momentum fraction ξ is described by the parton distribution function f i (ξ, µ 2 F )dξ and the parton-level scattering cross section dσ ij for parton i to scatter off parton j normalised to the hadronhadron flux 1 is summed over the possible parton types i and j. As usual µ R and µ F are the renormalisation and factorisation scales.
The infrared-finite partonic cross section for a parton of type i scattering off parton of type j, dσ ij , has the perturbative expansion where the next-to-leading order (NLO) and next-to-next-to-leading order (NNLO) strong corrections are identified. Note that the leading order cross section is O(α 2 s ). The single-jet inclusive and dijet cross sections have been studied at NLO [1][2][3][4][5][6] and successfully compared with data from the high energy frontier at the TEVATRON [7][8][9]

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Suppressing the labels of the partons in the initial state of the hard scattering, the general form for the subtraction terms for an m-particle final state at NNLO is given by [21]: Note that because integration of the subtraction term dσ S N N LO gives contributions to both the (m+1)-and m-parton final states, we have explicitly decomposed the integrated double real subtraction term into a piece that is integrated over one unresolved particle phase space and a piece that is integrated over the phase space of two unresolved particles, In a previous paper [75], the subtraction term dσ S N N LO corresponding to the leading colour pure gluon contribution to dijet production at hadron colliders was derived. The

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subtraction term was shown to reproduce the singular behaviour present in dσ RR N N LO in all of the single and double unresolved limits.
It is the purpose of this paper to construct the appropriate subtraction term dσ T N N LO to render the leading colour five-gluon contribution dσ RV N N LO explicitly finite and numerically well behaved in all single unresolved limits. Our paper is organised in the following way. In section 2, the general structure of 1 dσ S,1 N N LO and dσ V S N N LO are discussed and analysed. The coupling constant renormalisation to remove the UV singularities and the mass factorisation of the initial-state singularities into the parton distributions are described in section 3. In section 4 we turn our attention to the specific process of gluon scattering at NNLO. Our notation for gluonic amplitudes is summarised in section 4.1 and the one-loop five-gluon amplitudes discussed in section 4.2.
There are two separate configurations relevant for gg → ggg scattering depending on whether the two initial state gluons are colour-connected or not. We denote the configuration where the two initial state gluons are colour-connected (i.e. adjacent) by IIFFF, while the configuration where the colour ordering allows one final state gluon to be sandwiched between the initial state gluons is denoted by IFIFF. Explicit forms for the real-virtual subtraction term dσ T N N LO for these two configurations are given in section 5 where the cancellation of explicit poles is made manifest. The validity of the subtraction term is tested numerically in section 6 by studying the subtracted one-loop matrix elements in all of the single unresolved limits. In particular, we show that in all cases the ratio of the finite part of the real-virtual and subtraction terms approaches unity. Finally, our findings are summarised in section 7.
Four appendices are also enclosed. Appendix A summarises the phase space mappings for the final-final, initial-final and initial-initial configurations. Appendix B gives a description of the tree-level antennae appearing in the subtraction terms in both their unintegrated form (appendix B.1) and after integration over the unresolved phase space (appendix B.2). The unintegrated one-loop antenna is given in appendix B.3. Appendix C contains a modified form for the wide angle soft subtraction terms present in dσ S N N LO , while formulae relating to the mass factorisation contribution are given in appendix D.

Real-virtual antenna subtraction at NNLO
In this paper, we focus on the kinematical situation of the scattering of two massless coloured partons to produce massless coloured partons, and particularly the production of jets from gluon scattering in hadronic collisions. To establish some notation, consider the leading-order parton-level contribution from the (m + 2)-parton processes to the m-jet cross section at LO in pp collisions,

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To make the subsequent discussion more general, we denote a generic tree-level (m + 2)parton colour ordered amplitude by the symbol M m+2 (1,2 . . . , m + 2), where1 and2 denote the initial state partons of momenta p 1 and p 2 while the m-momenta in the final state are labeled p 3 , . . . , p m+2 . For convenience, and where the order of momenta does not matter, we will often denote the set of (m + 2)-momenta {p 1 , . . . , p m+2 } by {p} m+2 . The symmetry factor S m accounts for the production of identical particles in the final-state. At leading colour, the colour summed squared matrix elements are determined by the squares of the individual colour ordered amplitudes where the sum runs over the various colour ordered amplitudes. 2 For gluonic amplitudes this is the sum over the group of non-cyclic permutations of n symbols. The normalisation factor N LO includes the hadronhadron flux-factor, spin and colour summed and averaging factors as well as the dependence on the renormalised QCD coupling constant α s . dΦ m is the 2 → m particle phase space: The jet function J (n) m ({p} n+2 ) defines the procedure for building m-jets from n final state partons. Any initial state momenta in the set {p} n+2 are simply ignored by the jet algorithm. The key property of J (n) m is that the jet observable is collinear and infrared safe. In a previous paper [75], we have discussed the NNLO contribution coming from processes where two additional partons are radiated -the double real contribution dσ RR N N LO and its subtraction term dσ S N N LO . dσ RR N N LO involves the (m + 4)-parton process at tree level and is given by, (2.4) In this paper, we are concerned with the NNLO contribution coming from single radiation at one-loop, i.e. the (m + 3)-parton process, dσ RV N N LO . In our notation, the one-loop (m+3)-parton contribution to m-jet final states at NNLO in hadron-hadron collisions is given by where we introduced a shorthand notation for the interference of one-loop and treeamplitudes,

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which explicitly captures the colour-ordering of the leading colour contributions. The subleading contributions in colour are implicitly included in (2.6) but will not be considered in detail in this paper. As usual, the sum is the appropriate combination of colour ordered amplitudes. For gluonic amplitudes, this is the sum over the group of non-cyclic permutations of n symbols. The normalisation factor N LO depends on the specific process and parton channel under consideration. Nevertheless, N RR N N LO and N RV N N LO are related both to each other and to N LO for any number of jets and for any partonic process by As expected, each power of the (bare) coupling is accompanied by a factor ofC(ǫ). In this paper, we are mainly concerned with the NNLO corrections to (2.1) when m = 2 and for the pure gluon channel. The normalisation factor N LO will be given for this special case in section 4. Eq. (2.5) contains two types of infrared singularities. The renormalised one-loop virtual correction M 1 m+3 to the (m + 3)-parton matrix element contains explicit infrared poles, which can be expressed using the infrared singularity operators defined in [76,77]. On the other hand, the requirement of building m-jets from (m + 1)-partons allows one of the final state partons to become unresolved, leading to implicit local infrared singularities which become explicit only after integration over the unresolved patch of the final state (m + 1)-parton phase space. The single unresolved infrared singularity structure of oneand two-loop amplitudes has been studied in [78][79][80][81][82][83][84][85][86][87][88][89][90].
As discussed in section 1, in order to carry out the numerical integration over the (m + 1)-parton phase, weighted by the appropriate jet function, we have to construct an infrared subtraction term 3 dσ T N N LO which (a) removes the explicit infrared poles of the virtual one-loop (m + 3)-parton matrix element.
(b) correctly describes the single unresolved limits of the virtual one-loop (m + 3)-parton matrix element.
3 Strictly speaking, dσ T N N LO is not a subtraction term since it adds back part of the the double radiation subtraction term dσ S N N LO integrated over the phase space of a single unresolved particle. Nevertheless, since it contains all the terms needed to render the (m + 1)-particle final state finite, it is convenient to call it the real-virtual subtraction term.

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The subtraction term has three components; where 1 dσ S,1 N N LO is derived from the double real radiation subtraction term dσ S N N LO integrated over the phase space of one unresolved particle. Part of this contribution cancels the explicit poles in the virtual matrix element, while the real-virtual subtraction term dσ V S N N LO accounts for the single unresolved limits of the virtual matrix element.
In the following subsections we shall present the general structure of 1 dσ S,1 N N LO and dσ V S N N LO . The remaining poles are associated with the initial state collinear singularities and are absorbed by the mass factorisation counterterm dσ M F,1 N N LO which will be presented explicitly in section 3.
A key element of the antenna subtraction scheme is the factorisation of the matrix elements and phase space in the singular limits where one or more particles are unresolved. In determining the various contributions to dσ T N N LO , we shall therefore specify the unintegrated and/or integrated antennae and the reduced colour ordered matrix-element squared involved. The factorisation is guaranteed by the momentum mapping described in the appendix A. For conciseness, only the redefined hard radiator momenta will be specified in the functional dependence of the matrix element squared. The other momenta will simply be denoted by ellipsis.
In order to combine the subtraction terms and real-virtual matrix elements, it is convenient to slightly modify the phase space, such that (2.13) The integration over x 1 and x 2 reflects the fact that the subtraction terms contain contributions due to radiation from the initial state such that the parton momenta involved in the hard scattering carry only a fraction x i of the incoming momenta. In general, there are three regions: the soft (x 1 = x 2 = 1), collinear (x 1 = 1, x 2 = 1 and x 1 = 1, x 2 = 1) and hard (x 1 = 1, x 2 = 1). The real-virtual matrix elements only contribute in the soft region, as indicated by the two delta functions. In sections 2.1 and 2.2, we discuss the first two terms that contribute to dσ T N N LO given in eq. (2.12), namely 1 dσ S,1 N N LO and dσ V S N N LO . The final contribution dσ M F,1 N N LO is discussed in section 3.  Table 1. Type of contribution to the double real subtraction term dσ S N N LO , together with the integrated form of each term. The unintegrated antenna and soft functions are denoted as X 0 3 , X 0 4 and S while their integrated forms are X 0 3 , X 0 4 and S respectively. M 0 n denotes an n-particle tree-level colour ordered amplitude.
(c) Two unresolved partons that are not colour-connected but share a common radiator (almost colour-connected).
(d) Two unresolved partons that are well separated from each other in the colour chain (colour-unconnected).
(e) Compensation terms for the over subtraction of large angle soft emission.
Each type of contribution takes the form of antenna functions multiplied by colour ordered matrix elements. The various types of contributions are summarised in table 1. We see that the a, c and e types of subtraction term, as well as the b-type terms that are products of three-particle antenna, can be integrated over a single unresolved particle phase space and therefore contribute to the (m + 1)-particle final state so that, with, On the other hand, the double unresolved antenna functions X 0 4 in dσ S,b N N LO and the colour-unconnected X 0 3 X 0 3 terms in dσ S,d N N LO can immediately be integrated over the phase space of both unresolved particles and appear directly in the m-particle final state, These integrated contributions will be discussed elsewhere.
We now turn to a detailed discussion of each of the terms in eq. (2.14). In the antenna subtraction approach, the single unresolved configuration coming from the tree-level process with two additional particles, i.e. the double-real process involving (m + 4)-partons, is subtracted using a three-particle antenna function -two hard radiator

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partons emitting one unresolved parton. Once the unresolved phase space is integrated over, one recovers an (m + 3)-parton contribution that precisely cancels the explicit pole structure in the virtual one-loop (m + 3)-parton matrix element. The integrated subtraction term formally written as 1 dσ S,a N N LO is split into three different contributions, depending on whether the hard radiators are in the initial or final state.
When both hard radiators i and k are in the final state, then X ijk is a final-final (FF) antenna function that describes all singular configurations (for this colour-ordered amplitude) where parton j is unresolved. The subtraction term for this single unresolved configuration, summing over all possible positions of the unresolved parton, reads, dσ S,a(F F ) Besides the three parton antenna function X 0 ijk which depends only on p i , p j and p k , the subtraction term involves an (m + 3)-parton amplitude depending on the redefined onshell momenta p I and p K , whose definition in terms of the original momenta are given in appendix A.1. The (m+3)-parton amplitude also depends on the other final state momenta which, in the final-final map, are not redefined and on the two initial state momenta p 1 and p 2 . This dependence is manifest as ellipsis in (2.17). The jet function is applied to the (m + 1)-momenta that remain after the mapping, i.e. p 3 , . . . , p I , p K , . . . , p m+4 .
To perform the integration of the subtraction term in eq. (2.17) and make its infrared poles explicit, we exploit the following factorisation of the phase space, dΦ m+2 (p 3 , . . . , p m+4 ; p 1 , p 2 ) = dΦ m+1 (p 3 , . . . , p I , p K , . . . , p m+4 ; p 1 , p 2 ) · dΦ X ijk (p i , p j , p k ; p I + p K ) where we have simply relabeled the final-state momenta in the last step. In (2.18) the antenna phase space dΦ X ijk is proportional to the three-particle phase space relevant to a 1 → 3 decay, and one can define the integrated final-final antenna by where C(ǫ) is given in (2.10). The integrated single-unresolved contribution necessary to cancel the explicit infrared poles from virtual contributions in this final-final configuration then reads,

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where the sum runs over all colour-connected pairs of final state momenta (p i , p k ) and the final state momenta I, K have been relabelled as i, k. Expressions for the integrated final-final three-parton antennae are available in ref. [21]. When only one of the hard radiator partons is in the initial state, X i,jk is a initial-final (IF) antenna function that describes all singular configurations (for this colour-ordered amplitude) where parton j is unresolved between the initial state parton denoted byî (whereî =1 orî =2) and the final state parton k. The antenna only depends on these three parton momenta p i , p j and p k . The subtraction term for this single unresolved configuration, summing over all possible positions of the unresolved parton, reads, dσ S,a,(IF ) As in the previous final-final case, the reduced (m + 3)-parton matrix element squared involves the mapped momentaÎ and K which are defined in appendix A.2. Likewise, the jet algorithm acts on the (m + 1)-final state momenta that remain after the mapping has been applied. In this case, the phase space in (2.21) can be factorised into the convolution of an (m + 1)-particle phase space, involving only the redefined momenta, with a 2-particle phase space [71]. For the special case i = 1, it reads, dΦ m+2 (p 3 , . . . , p m+4 ; p 1 , p 2 ) = dΦ m+1 (p 3 , . . . , p K , . . . , p m+4 ; with Q 2 = −q 2 and q = p j + p k − p 1 and where we have also relabelled the final-state momenta in the last step. The quantityx 1 is defined in eq. (A.7).
Using this factorisation property one can carry out the integration over the unresolved phase space of the antenna function in (2.21) analytically. We define the integrated initialfinal antenna function by, where C(ǫ) is given in (2.10). Similar expressions are obtained when i = 2 via exchange of x 1 and x 2 ,

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The explicit poles present in dσ RV N N LO (defined in (2.13)) associated with the colourconnected initial-final radiatorsî and k can therefore be removed with the following form, (2.25) In this expression, only the redefined momentum K has been relabelled (K → k). The rescaled initial state radiatorÎ(=ī) is not relabelled and appears in the functional dependence of the integrated antenna and in the matrix-element squared. Explicit forms for the integrated initial-final three-parton antennae are available in ref. [71].
If we consider the case where the two hard radiator partons i and k are in the initial state finally, then X ik,j is a initial-initial (II) antenna function that describes all singular configurations (for this colour-ordered amplitude) where parton j is unresolved. The subtraction term for this single unresolved configuration, summing over all possible positions of the unresolved parton, reads, dσ S,a,(II) where as usual we denote momenta in the initial state with a hat. The radiatorsî andk are replaced by new rescaled initial state partonsÎ andK and all other spectator momenta are Lorentz boosted to preserve momentum conservation as described in appendix A.3. For the initial-initial configuration the phase space in (2.26) factorises into the convolution of an (m + 1)-particle phase space, involving only redefined momenta, with the phase space of parton j [71] so that when i = 1 and k = 2, dΦ m+2 (p 3 , . . . , p m+4 ; p 1 , p 2 ) (2.27) = dΦ m+1 (p 3 , . . . ,p m+4 ; where the single particle phase space measure is [dp The only dependence on the original momenta lies in the antenna function X ik,j and the antenna phase space. One can therefore carry out the integration over the unresolved phase space analytically, to find the integrated antenna function, where C(ǫ) is given in eq. (2.10). Explicit forms for the integrated initial-initial threeparton antennae are available in ref. [71].

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We can therefore remove the explicit poles present in dσ RV N N LO (defined in (2.13)) associated with the colour-connected initial-state pairî andk with the subtraction term, 1 dσ S,a,(II) (2.29) As in the initial-final case, the redefined initial state momentaÎ =ī andK =k are not relabelled, neither in the functional dependence of the integrated antenna X 0 ik,j (sīk, x i , x k ) nor for the reduced matrix element squared |M m+3 (. . .Î,K, . . .)| 2 .
Each integrated antenna has an explicit dependence on the variables x 1 and x 2 as stated in eqs. (2.19), (2.23) and (2.28).
Summing over the different final-final, initial-final and initial-initial configurations, we find that the explicit poles in dσ RV N N LO are removed by 1 dσ S,a N N LO to yield an integrand free from explicit infrared poles over the whole region of integration. This is of course merely a consequence of the cancellation of infrared poles between a virtual contribution and integrated subtraction terms from real emission. However, as discussed later in section 2.2.2, dσ RV N N LO and 1 dσ S,a N N LO develop further infrared singularities in singly unresolved regions of the phase-space which do not coincide. Therefore, we have to introduce a further subtraction term, dσ V S N N LO , to ensure an integrand that has no explicit global ǫ-poles, and that does not have implicit singularities in single unresolved regions. As discussed earlier in section 2.1, contributions to the double real subtraction term due to colour-connected or almost colour-unconnected hard radiators that have the generic form X 3 ×X 3 must also be added back integrated over the phase space of the first (outer) antenna. The structure of these terms is very similar to the single unresolved contributions of the previous sub-section; each term with an "outer" final-final antenna present in dσ S,(b,c) N N LO produces a contribution given by, In this case, i and k represent the momenta in the set {p} m+3 . The set of momenta denoted by {p} m+2 is obtained from {p} m+3 set through a phase space mapping that is determined by the type of the unintegrated "inner" antenna X 0 3 which is fixed by the corresponding term in dσ S N N LO .

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When the outer antenna is of initial-final or initial-initial type, we find the integrated forms, Note that the (b, c)-type terms reflect different physical origins; dσ S,b N N LO is designed to account for the unresolved contributions from two colour-connected unresolved partons, while dσ S,c N N LO treats the singularities from two almost colour-connected unresolved partons.
The general structure of dσ S,b N N LO consists of groups of terms involving the difference between a four-parton antenna and products of three-parton antennae. The latter removes the single unresolved limits present in the former. There are two distinct types of four-parton antenna which, for sake of argument, we label X 0 where h 1 , h 2 represent the hard radiators and s 1 , s 2 the unresolved particles. By construction, there are no unresolved limits when h 1 or h 2 become unresolved. In both, the four partons are colour-connected. The key difference between X 0 In the case of X 0 4 , dσ S,b N N LO has the form, where the two X 0 3 × X 0 3 terms subtract the single unresolved limits present in X 0 4 . Most antennae fit into this class including, for example, the quark-antiquark antenna A 0 4 , the quark-gluon subantenna D 0 4,a [58] and the gluon-gluon subantenna F 0 4,a [75]. Upon integration, the X 0 3 × X 0 3 terms produce integrated antenna that cancel against the explicit poles present in X 1 3 that will be discussed in section 2.2.2. Similarly, one must also subtract the limits where s 1 and s 2 are unresolved fromX 0 4 , Antennae of this type include the quark-antiquark antennaÃ 0 4 , the quark-gluon subantenna D 0 4,b [58] and the gluon-gluon subantenna F 0 4,b [75]. As before, after integration the X 0 terms produce integrated antenna that cancel against the explicit poles present in X 1 3 . However, unless the single unresolved contribution dσ S,a N N LO has the form, 4 it is clear that eq. (2.34) oversubtracts the double unresolved limits. In this case, a correction term of the form These terms produce poles that cancel against the integrated wide angle soft terms discussed in section 2.1.3. and the real-virtual subtraction terms discussed in section 2.2.3. The presence of D 0 4,b and F 0 4,b antennae in dσ S N N LO is an indicator that there are contributions from wide angle soft radiation, see for example refs. [58,75].
Similarly, the almost colour-connected contributions in dσ S,c N N LO are produced by matrix elements of the type |M n+2 (. . . , a, s 1 , h 1 , s 2 , h 2 , . . .)| 2 and have the form [21,75], Note that unlike the contributions described in eq. (2.36), the radiators for the inner antenna are not both constructed from mapped momenta. Upon integration, these terms also produce poles that cancel against wide angle soft terms and will be further discussed in section 2.2.3.

Integration of large angle soft terms: 1 dσ S,e N N LO
For processes involving soft gluons the real-real channel has an additional subtraction contribution denoted by dσ S,e N N LO due to large angle soft radiation [58,59,62,75]. This term removes the remnant soft gluon behaviour associated with the phase space mappings of the iterated structures of the double-real subtraction contributions dσ S,(b,c) N N LO . In these contributions two successive mappings have been applied and the gluon emission can occur before or after the first mapping. The large angle soft subtraction term is constructed to account for that and the soft eikonal factors can involve either unmapped or mapped momenta.

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The wide angle soft terms are however not uniquely defined, only their soft behaviour is determined. They can be obtained after two successive mappings of the same kind have been applied. In other words, either two final-final, two initial-final or two initial-initial mappings are used. Those are presented in [75].
Alternatively, they can be constructed after the application of two successive mappings where the first of those is always a final-final mapping, (i, j, k) → (I, K). To achieve this we modified the large angle soft contribution of the double-real piece, given by eqs. (5.9), (5.13) and (5.17) of ref. [75], to the equally valid forms given in (C.1), (C.2) and (C.3) respectively. Following this latter choice, the integrated large angle soft subtraction terms can simply be obtained through integration over the final-final antenna phase space dΦ X ijk given in eq. (2.18).
The unintegrated soft factor terms are eikonal factors of the form where a, c denotes initial or final state external partons. The hard radiators, a and c, may be momenta present after the first mapping, or momenta produced by the second mapping, i.e.
Because j is eliminated in the first mapping, it is not a member of {p} m+3 or {p} m+2 . Nevertheless, in the latter case, the second mapping should also be applied to p j to produce a new soft momentum p j ′ . When this is a final-final or initial-final mapping, it is a trivial mapping p j → p j ′ ≡ p j . However, when the second mapping is of the initial-initial form, then p j → p j ′ ≡p j as given by eq. (A.8).
In general, and after factorisation of the phase space according to eq. (2.18), one always finds differences of terms of the form, where X 0 3 is the antenna corresponding to the second mapping, i.e. final-final, initial-final or initial-initial.
For the final-final kinematical configuration, where the second mapping transforms the momenta (I, l, K) → (I ′ , K ′ ), we find a contribution of the form, dσ S,e,(F F )

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We see that there are three pairs of soft antennae which correspond to inserting the soft gluon around the radiator gluons, in this case, between a and I ′ , between I ′ and K ′ and between K ′ and b.
Similarly, when the second mapping is of the initial-final type corresponding to the mapping (n, l, K) → (N , K ′ ), we have, dσ S,e,(IF ) Once again, there are three pairs of soft terms, corresponding to inserting the soft gluon in the three positions into the colour-connected gluons, . . . , a,N , K ′ , b, . . .. In this case, the last two terms simply cancel. This is because the radiator momenta appears in both the numerator and denominator of the eikonal factor, coupled with the fact that the initial momentum is simply scaled in the initial-final mapping, S ajn ≡ S ajN . Finally, when the second mapping is of the initial-initial type: (n,p, l, . . . , a, b, .
As anticipated, when the soft factor involves hard radiators produced by the initial-initial mapping, the soft momentum p j is replaced by the momentum pj obtained by applying the boost for the initial-initial mapping directly to momentum p j . When the unresolved momentum is unboosted and denoted by p j , the integrated form of the soft factor is given by, where a and c are arbitrary hard radiator partons. After relabelling the momenta such that I, K → i, k, then [58], where we have defined x ac,ik = s ac s ik (s ai + s ak )(s ci + s ck ) .
Since the soft eikonal factor is invariant under crossing of one or two partons from the final to the initial state the integrated soft factor is independent of where the external partons are situated. The integrated final-final soft factor involving a boosted unresolved momentum (denoted by pj in the soft eikonal factor is defined by, Since S abc is composed of Lorentz invariants, we can simply invert the Lorentz boost that mapped j →j (A.8) such that where a, c are also obtained by inverting the same Lorentz boost so that The r.h.s. has precisely the same form as eq. (2.44) with p a → p a and p c → p c , so that Furthermore, we can exploit the fact that S only depends on invariants and can apply the same Lorentz boost such that The integrated soft factors only contribute to the soft region (x 1 = x 2 = 1). Using the (m + 1) phase space factorisation as given in eq. (2.18) and inserting the integrated large angle subtraction factors into eqs. (2.41), (2.42) and (2.43) while relabelling I ′ → I, I → i, Note that the double ǫ-poles cancel within the square brackets, so that, for example, the leading pole for the square bracket in eq. (2.52) is and similarly for eqs. (2.53) and (2.54). Moreover, this combination of integrated soft factors does not have any single unresolved limit and therefore does not require further subtraction. Each subtraction term takes the form of a tree-or one-loop-antenna function multiplied by the one-loop or tree-colour ordered matrix elements respectively which we denote by dσ V S,a N N LO .
(b) Terms of the type X 0 3 X 0 3 that cancel the explicit poles introduced by one-loop matrix elements and one-loop antenna functions present in dσ V S,a N N LO . This term is named dσ V S,b N N LO .
(c) Terms of the type X 0 3 X 0 3 that compensate for any remaining poles, specifically those produced by dσ  Table 2. Type of contribution to the real-virtual subtraction term dσ V S N N LO , together with the integrated form of each term. The unintegrated antenna functions are denoted as X 0 3 and X 1 3 while their integrated forms are X 0 3 and X 1 3 respectively. |M 1 n | 2 denotes the interference of the tree-level and one-loop n-particle colour ordered amplitude while |M 0 n | 2 denotes the square of an n-particle tree-level colour ordered amplitude.  Figure 1. Illustration of NNLO antenna factorisation representing the factorisation of both the one-loop "squared" matrix elements (represented by the white blob) and the (m + 1)-particle phase space when the unresolved particles are colour-connected. The terms in square brackets represent both the three-particle tree-level antenna function X 0 ijk and the three-particle one-loop antenna function X 1 ijk and the antenna phase space.
The types of contributions present in each of these terms are summarised in table 2. Note that dσ V S N N LO is a subtraction term, and it must therefore be added back in integrated form to the m-parton final state. There it combines with the twice integrated double real subtraction term 2 dσ S,2 N N LO and the mass-factorisation counterterm denoted by dσ M F,2 N N LO according to eq. (1.5) to provide the singularity structure necessary to cancel the pole structure of the double virtual contribution dσ V V N N LO .

One-loop single-unresolved contributions: dσ V S,a N N LO
In single unresolved limits, the behaviour of (m+3)-parton one-loop amplitudes is described by the sum of two different contributions [78][79][80][81][82]: a single unresolved tree-level factor times a (m + 2)-parton one-loop amplitude and a single unresolved one-loop factor times a (m+2)-parton tree-level amplitude, as illustrated in figure 1. Accordingly, we construct the one-loop single unresolved subtraction term from products of tree-and one-loop antenna functions with one-loop and tree-amplitudes respectively. Analogously to the case of handling explicit infrared poles present in 1 dσ S,a N N LO or to deal with single unresolved limits of tree level amplitudes, we need to decompose this JHEP02(2012)141 contribution into three parts according to where the two hard radiators are situated. We note that the momentum mappings which implement momentum conservation away from the single unresolved limits of each configuration are the same as in the previous section, namely equations (A.4), (A.6) and (A.8).
In the final-final configuration, the subtraction term is given by, where the one-loop three-parton antenna function X 1 ijk depends only on the antenna momenta p i , p j , p k . X 1 ijk correctly describes all simple unresolved limits of the difference between an (m + 2)-parton one-loop corrected squared matrix element and the product of a tree-level antenna function with the m-parton one-loop corrected squared matrix element. It can therefore be constructed out of colour ordered and renormalised one-loop three-parton and two-parton matrix elements as where S ijk,IK denotes the symmetry factor associated with the antenna, which accounts both for potential identical particle symmetries and for the presence of more than one antenna in the basic two-parton process.
It should be noted that X 1 ijk is renormalised at a scale corresponding to the invariant mass of the antenna partons, s ijk , while the one-loop (m + 2)-parton matrix element is renormalised at a scale µ 2 . To ensure correct subtraction of terms arising from renormalisation, we have to substitute in (2.56). The terms arising from this substitution will in general be kept apart in the construction of the colour ordered subtraction terms, since they all share a common colour structure β 0 . Similar subtraction terms are appropriate in the initial-final and initial-initial configurations. In the initial-final case, we have,

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where the one-loop antenna function X 1 i,jk is obtained from X 1 ijk by crossing parton i from the final state to the initial state.
The initial-initial subtraction term is with X 1 ik,j obtained by crossing partons i and k from X 1 ijk . We note that in order to fulfil overall momentum conservation, the initial-initial momentum mapping requires all the momenta in the arguments of the reduced matrix elements and the jet functions to be redefined.
The one-loop single unresolved subtraction terms, X 1 ijk , X 1 i,jk and X 1 ik,j can never be related to integrals of tree-level subtraction terms. Therefore, after integration over the three-parton antenna phase space, this component of the subtraction term must be added back in integrated form to the terms yielding the m-parton final state contribution. This can be accomplished using the techniques described in [91] and analytic expressions for all of the integrated three-parton one-loop antennae are available in refs. [21,72,74]. It is the purpose of this subsection to present a new term denoted by dσ V S,b N N LO which removes the explicit ǫ-poles present in dσ V S,a N N LO . To achieve this, we introduce further partially integrated subtraction terms built with products of an unintegrated and an integrated three parton antenna function times a reduced matrix element squared. For a given configuration, the integrated antenna will be of a given type (final-final, initial-final or initial-initial) multiplying a sum of three unintegrated antennae of each configuration type.

Cancellation of explicit infrared divergences in dσ
Let us consider first the explicit poles in the one-loop reduced matrix elements present in eqs. (2.56), (2.59) and (2.60). The pole structure of |M 1 m+2 ({p} m+2 )| 2 is well understood as a sum of I (1) operators [76]. These ǫ-poles can be simply subtracted using integrated X 0 3 antennae. To derive the relevant term, we simply add a contribution constructed from each I (1) operator present in the pole structure of |M 1 m+2 ({p} m+2 )| 2 according to the relation shown in table 3, (2.61) The particle type (x, y) fixes the flavour of antenna, while the momenta (a, b) determines whether the integrated antenna is final-final, initial-final or initial-initial. This is sum- marised by adding the contribution obtained by the replacement There are also explicit poles present in the one-loop antenna, X 1 ijk which are produced by the physical matrix elements making up the antenna (2.57). These can also be described by integrated antenna, however now the relevant mass scale is constructed from momenta before the mapping i.e. the momenta lying in the set {p} m+3 .
The first type of poles are associated with |M 1 ijk | 2 in eq. (2.57). The second contribution comes from the term proportional to |M 1 IK | 2 . For example, the explicit poles in X 1 ijk are removed by adding the contribution obtained by the replacement JHEP02(2012)141 2  1  2  3  3  2  1  0  3  3  2  1  0   M X  1  1  0  2  2  2  0  2  2  2  2  0  2   Table 4. Number of colour-connected pairs N X in the one-loop antenna X 1 3 , and the coefficient M X . from eqs. (2.33) and (2.34). This is another example of the cancellation of infrared poles between a virtual contribution and integrated subtraction terms from real emission. On the other hand, the last term must be introduced as a new subtraction term in dσ V S,b N N LO . To be more explicit, in the final-final configuration, dσ V S,b N N LO is given by, Here p I and p K are the momenta produced by the mapping for the ijk antenna, while ab are the pairs of colour-connected particles appearing in the matrix element M 0 m+2 ({p} m+2 ). Similar subtraction terms are appropriate in the initial-final and initial-initial configurations, where the invariant mass s ik is constructed from momenta in the set {p} m+3 . For each term like this, we introduce a subtraction term (with the opposite sign) where the appropriate invariant mass is constructed out of momenta in the set {p} m+2 i.e. the momenta produced by the mapping appropriate to the type of antenna X 0 3 ({p} m+3 ). That is, Taken together, we find that where the term in square brackets does not have a double pole in ǫ, but has a leading singularity of the form 1 2ǫ log s ik s IK + O(1).
Combinations of terms like this, plus similar contributions from the initial-final and initialinitial antenna, together with correction terms coming from the oversubtraction of the single unresolved limits from theX 0 4 antenna (see eq. 2.36) ultimately cancel against the integrated wide angle soft radiation term 1 dσ S,e N N LO .

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To make this cancellation more precise, consider the coefficients of terms proportional to X 0 iℓk |M m+2 (. . . , a, I, K, b, . . .)| 2 J (m) m ({p} m+2 ). (2.72) Here we consider final-final radiation, but the argument is general. The wide angle soft contribution is given by eq. (2.41). As discussed in section 2.1.3, the leading singularity from the wide angle soft terms is proportional to This pole cancels against combinations of the form (where the dependence on x 1 , x 2 is suppressed), . This can be understood as being due to repeated radiation where the hard radiators are particles i and k. The second and third terms have opposing signs (compared to the first term) and come from the almost colourconnected term 1 dσ S,c N N LO and occur when the integrated antenna describes unresolved radiation which took place between hard radiators a and i (b and k). For these three terms, the unresolved radiation is emitted from an outer X 0 3 antenna, while X 0 iℓk is the inner antenna. The final three terms form part of the real-virtual subtraction term dσ V S,c N N LO , which we can understand as corresponding to emitting an unresolved momentum between pairs in the colour-connected set . . . , a, I, K, b, . . . i.e. unresolved radiation from an inner antenna. In this case, X 0 iℓk plays the role of outer antenna. The signs are always fixed to be opposite of the partner term coming from 1 dσ S,(b,c) N N LO . As usual, the subtraction terms constituting dσ V S,c N N LO must be integrated over the unresolved phase space and added back in integrated form to the double virtual contribution. Expanding (2.73), we find that the leading pole is proportional to which cancels against the leading pole coming from the wide angle soft term.

Renormalisation and mass factorisation
We are concerned with m-jet production in the collision of two hadrons h 1 , h 2 carrying momenta Within the framework of QCD factorisation the cross section for this process is written in eq. (1.1) as an integral over the infrared and ultraviolet finite hard partonic scattering

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cross section for m-jet production from quarks and gluons, multiplied by parton distribution functions (PDF's) describing the momentum distribution of these partons inside the colliding hadrons. However, after combining the real and virtual contributions together with the antenna subtraction terms, the partonic cross section dσ ij contains both ultraviolet and initial-state collinear singularities. We remove the UV singularities through coupling constant renormalisation in the MS scheme and absorb the initial-state singularities into the PDFs using the MS factorisation scheme. For simplicity, we first set the factorisation and renormalisation scales equal to a common scale, It is straightforward to restore the dependence of the partonic cross section on these scales using the requirement that the hadronic cross section is independent of them.

Ultraviolet renormalisation
In terms of the bare (and dimensionful) coupling α b s , the unrenormalised cross section has the perturbative expansion, bearing in mind that the LO cross section is O (α b s ) m . Furthermore, each power of the bare coupling is accompanied by a factor ofC(ǫ) given by eq. (2.11).

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Since the m-jet cross section has a leading order behaviour proportional to α m s , we have, (3.10) Note that for the double real radiation contribution dσ RR N N LO and its associated subtraction term dσ S N N LO , renormalisation simply amounts to the replacementC(ǫ)α b s → α s . Likewise, the inverse powers ofC(ǫ) are immediately cancelled against the additional factors ofC(ǫ) present in dσ un,N LO ij and dσ un,N N LO ij .

Mass factorisation
After renormalisation, the physical cross section is given by where the bare PDF for a parton of type a carrying a momentum fraction ξ is denoted bỹ f a (ξ). For clarity, we have made the dependence of the partonic cross section dσ ij on the initial state parton momenta, ξ 1 H 1 and ξ 2 H 2 , explicit. The initial-state singularities present in dσ ij are removed using mass factorisation to produce the hadronic cross section where the finite partonic cross section multiplies the physical PDF f a (ξ, µ 2 F ) at the factorisation scale µ 2 F = µ 2 R = µ 2 . The physical PDF are related to the bare PDFf a (ξ) by the convolution, The kernel Γ ba has the (renormalised) perturbative expansion, where in the MS scheme, (3.15) where the p (n) ba are the standard four-dimensional LO and NLO Altarelli-Parisi kernels in the MS scheme given in refs. [97][98][99][100][101] and are collected in the appendix D for the gluonic channel.

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Eq. (3.12) can be systematically inverted such that the bare PDF is given by, where, Inserting eq. (3.16) and (3.17) into (3.11) and applying the rescaling where the infrared-finite mass factorised partonic cross section is Expanding in the strong coupling as in eq. (1.1), we find the connection between the infrared singular cross sections and the mass factorisation counterterms, Recalling that   (3.27) and will be discussed elsewhere.

Scale dependence of the partonic cross section
We start from the hadronic cross section that depends on µ F = µ R = µ through the strong coupling and the PDF's, The scale variation of the coupling constant α s (µ) is given by, (3.29) where,

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with β 0 , β 1 given in (3.5), (3.6). Similarly, the scale variation of the parton distribution function f i (x, µ 2 ) is determined by the DGLAP evolution equation, (3.31) where, Demanding the independence of the physical cross section in (3.28) on the unphysical scale µ 2 , we obtain the following scale variation equation for the partonic cross section, Bearing in mind that the leading order m-jet cross section depends on α m s , and solving eq. (3.34) order by order in α s , we see that the partonic cross section at scale µ 2 is related to that at scale µ 1 via, where L = ln(µ 2 2 /µ 2 1 ).

Real-virtual corrections for gluon scattering at NNLO
In this section we discuss the amplitudes that enter in the implementation of the mixed real-virtual correction. We focus on the pure gluon channel and describe the colour decomposition of the gluonic matrix elements at tree and loop-level. The remaining contributions

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that we use for subtraction, namely the three parton tree-level unintegrated and integrated antennae as well as one-loop three parton antennae have been derived in previous publications. For convenience and completeness we collect them in appendix B.

Gluonic amplitudes
The leading colour contribution to the m-gluon n-loop amplitude can be written as [78,[102][103][104][105], where the permutation sum, S m /Z m is the group of non-cyclic permutations of m symbols. We systematically extract a loop factor of C(ǫ)/2 per loop with C(ǫ) defined in (2.10). The helicity information is not relevant to the discussion of the subtraction terms and from now on, we will systematically suppress the helicity labels. The T a are fundamental representation SU(N ) colour matrices, normalised such that Tr(T a T b ) = δ ab /2. A n m (1, · · · , n) denotes the n-loop colour ordered partial amplitude. It is gauge invariant, as well as being invariant under cyclic permutations of the gluons. For simplicity, we will frequently denote the momentum p j of gluon j by j.
At leading colour, the tree-level (m + 2)-gluon contribution to the M -jet cross section is given by, For tree-processes involving four-and five-gluons, there are no sub-leading colour contributions. The normalisation factor N 0 m+2 includes the average over initial spins and colours and is given by, and where we have absorbed the factors of g 2 using the useful factors C(ǫ) (2.10) and C(ǫ) (2.11),

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The leading colour six-gluon real-real contribution to the NNLO dijet cross section is obtained by setting m = 4 in (4.4) and M = 2 in (4.2) such that in this case N RR N N LO appearing in section 2 in (2.4) is given by N RR N N LO = N 0 6 . For convenience, we introduce the additional notation for the one-loop "squared" matrix elements (4.6) so that the one-loop, (m + 2)-gluon contribution to the M -jet cross section is given by, As before, the normalisation factor N 1 m+2 includes the average over initial spins and colours and is given by, We will encounter both A 1 5 and A 1 4 when computing the real-virtual corrections relevant for the NNLO dijet cross section.
where the sums runs over the 3! permutations of the final state gluons. Therefore, depending on the position of the initial state gluons in the colour ordered matrix elements, we have two different topologies. These are labelled by the the colour ordering of initial and final state gluons. We denote the configurations where the two initial state gluons are colour-connected (i.e. adjacent) as IIFFF and those where the colour where, 2 (p 3 , . . . , p 5 ), (4.11) 2 (p 3 , . . . , p 5 ) . (4.12) The one-loop helicity amplitudes for gg → ggg have been available for some time [106]. We have cross checked our implementation of the one-loop helicity amplitudes of ref. [106] against the numerical package NGluon [107].
We note that the renormalised singularity structure of the contribution in (4.9) can be easily written in terms of the tree-level squared matrix elements multiplied by combinations of the colour ordered infrared singularity operator [76] Therefore the real-virtual correction contains only I (1) type of operators and in the gluonic approximation I (1) gg is the only operator that appears. The singular part of the renormalised colour ordered gluonic amplitude takes the form, A 1 5 (1 g ,2 g , i g , j g , k g ) = 2 I (1) gg (ǫ, s 12 ) + I (1) gg (ǫ, s 2i ) + I (1) gg (ǫ, s ij ) +I (1) gg (ǫ, s jk ) + I (1) gg (ǫ, s k1 ) A 0 5 (1 g ,2 g , i g , j g , k g ) + O(ǫ 0 ), In section 5 we will explicitly write down the counterterm that regularises the infrared divergences of the real-virtual correction for topology (4.11) and (4.12) separately.

Construction of the NNLO real-virtual subtraction term
As stated in the introduction, the aim of this paper is to construct the subtraction term for the real-virtual contribution such that the (m + 1)-parton contribution to the m-jet rate is free from explicit ǫ-poles over the whole of phase space and the subtracted integrand is well behaved in the single unresolved regions of phase space. As discussed earlier, this is

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achieved with the help of the antenna functions and, as will be explained here and in the subsequent section, the limit ǫ → 0 can be safely taken and the finite remainders evaluated numerically in four dimensions.
We start this section by recovering the general formula for the real-virtual channel which has to be integrated over the (m + 1)-parton final state phase space numerically. It reads, where 1 dσ S,1 N N LO contains the once integrated part of the integrated double real subtraction term discussed in section 2.1, dσ V S N N LO is the the real-virtual subtraction term discussed in section 2.2 and the mass factorisation term dσ M F,1 N N LO is given in eq. (3.26). The remaining contribution from the double real subtraction term denoted by 2 dσ S,2 N N LO must be integrated over two unresolved particles and contributes directly to the m-jet final states. It must therefore be added to the integrated real-virtual subtraction term 1 dσ V S N N LO , the two-loop matrix elements dσ V V N N LO together with the mass-factorisation counterterm denoted by dσ M F,2 N N LO . These all contribute to the m-parton final state and will be treated elsewhere. At the end of this section, however, we present the contribution from dσ V S N N LO which must be added back in integrated form to the m-parton final state. As discussed in section 2, the phase space for the (m + 1)-parton phase space can be written as an integral over the longitudinal momentum fractions x 1 , x 2 in the form dΦ m+1 (p 3 , . . . , p m+3 ; to account for initial state radiation.
In the next subsections we present separately the subtraction terms required for gluongluon scattering divided for the IIFFF and IFIFF topologies. Subtraction terms related to each topology are denoted with a superscript X 5 and Y 5 respectively such that, 6 When presenting the subtraction term we group the terms which are free of explicit ǫ-poles in square brackets. For conciseness we suppress the explicit x 1 , x 2 dependence of the integrated antennae appearing in the formulae below.

IFIFF topology
The one-loop single unresolved subtraction term for the IFIFF topology is: +S(s2j, sjk, x2j ,jk ) − S(s1 k , s jk , x1 k,jk ) + S(s1k, sjk, x1k ,jk ) 2 (p i ,p kj ) For this topology, the contribution to the hard region (x 1 , x 2 = 1) turns out to be identically zero. This is because the only terms that contribute in the hard region are integrated initial-initial antennae. This configuration (4.12) does not involve colour-connected initial state gluons and therefore does not contain any integrated initial-initial antennae. 7

Infrared structure
With the explicit expressions for the integrated antenna functions given in appendix B, the integrated large angle soft terms given in section 2.1.3 and the pole structure of the one-loop matrix elements, it is straightforward to check that the explicit ǫ poles analytically cancel in each and every one of the groups of terms in square brackets in eqs. In summary, we have shown that, within the antenna subtraction formalism, the realvirtual corrections to gluon-gluon scattering are locally free of explicit ǫ-poles providing us with a stringent check on the construction of the necessary subtraction terms. This is in direct contradiction to the statement made in ref. [32].

Contributions to the m-jet final state
In this subsection we identify the contributions from the real-virtual channel that we have subtracted in unintegrated form and which therefore must be added back in integrated form in the double virtual (m + 2)-parton channel. As expected contributions from the X 5 topology collapse in integrated form to the X 4 topology of the virtual-virtual contribution while contributions from the Y 5 topology contribute to both X 4 and Y 4 . The contributions of dσ V S N N LO , which when integrated over the antenna phase space become proportional to the X 4 topology are denoted by dσ V S N N LO | X 0 4 and are given by,

Numerical results
In this section we will test how well the real-virtual subtraction term dσ T N N LO derived in the previous section approaches the real-virtual contribution dσ RV N N LO in all single unresolved regions of the phase space so that their difference can be integrated numerically in four dimensions. We will do this by generating a series of phase space points using RAMBO [108] that approach a given single unresolved limit. For each generated point we compute the ratio of the finite parts of dσ RV N N LO and dσ T N N LO , .
Here dσ RV N N LO is the interference between the one-loop and tree-level five gluon matrix elements given by eq. (4.9), and dσ T N N LO is the real virtual subtraction term given by eqs. (5.3) and (5.4). The ratio R should approach unity as we get closer to any singularity showing that the subtraction captures the infrared singularity structure of the real virtual contribution.
For each unresolved configuration, we will define a variable that controls how we approach the singularity subject to the requirement that there are at least two jets in the final state with p T > 50 GeV where the jets have been clustered with the anti-k t jet algorithm [109,110] with radius R=0.4. The partonic center-of-mass energy √ s is fixed to be 1000 GeV.

Soft limit
To probe the soft regions of the phase space, we generate an event configuration with a soft final state gluon k by making the invariant s ij close to the full center of mass energy s 12 . This kinematic configuration is pictured in figure 2(a). We define the small parameter x = (s − s ij )/s and show the distributions of the ratio between the real-virtual matrix element and the subtraction term for x = 10 −5 (green), x = 10 −6 (blue) and x = 10 −7 (red) in figure 2(b) using 10000 phase space points. The plot also shows the number of points that lie outside the range of the histogram. We see that the subtraction term rapidly converges to the matrix element as we approach the single soft limit.   JHEP02(2012)141

Collinear limit
Next we probe the final and initial state single collinear regions of the phase space. These event topologies are depicted in figures 3(a) when gluons i and j become collinear, and 4(a) where gluon i becomes collinear with the incoming gluon 1.
For the final-final collinear singularity, we introduce the small parameter x = s ij /s 12 . Figure 3(b) shows the distribution in R obtained for 10000 phase space points for x = 10 −7 (green), x = 10 −8 (blue) and x = 10 −9 (red).
Similarly in the initial-final collinear limit, the small parameter is x = s 1i /s 12 and figure 4(b) shows the distributions of R for the same x-values of x = 10 −7 (green), x = 10 −8 (blue) and x = 10 −9 (red).
As the small parameter x gets smaller, we see a systematic improvement in the convergence of the real-virtual matrix elements and the subtraction term. This is in contrast with the collinear limit of the double real all gluon subtraction term [75,111], but not surprising due to the simplicity of the final state where the partons are fixed to be in back-to-back pairs as shown in figures 3(a) and 4(a). Nevertheless, figures 3(b) and 4(b) show that the subtraction term does not approximate the real-virtual matrix element as well as in the soft limit ( figure 2(b)). This is due to the presence of angular correlations in the matrix elements stemming from gluon splittings g → gg. The collinear limits of tree and one-loop matrix elements are controlled by the unpolarised Altarelli-Parisi splitting functions which explicitly depend on the transverse momentum k ⊥ of the collinear gluons with respect to the collinear direction and on the helicity of the parent parton. As a result of this, the splitting functions produce spin correlations with respect to the directions of other momenta in the matrix element besides the momenta becoming collinear. These azimuthal terms coming from the single collinear limits vanish after integration over the azimuthal angle of the collinear system. This occurs globally after an azimuthal integration over the unresolved phase space. Here we are performing a point-by-point analysis on the integrand defined by the real-virtual matrix element and the subtraction term and because we use spin-averaged antenna functions to subtract the collinear singularities, the azimuthal angular terms produced by the spin correlations are simply not accounted for in the antenna subtraction procedure.
To improve on this, several approaches have been discussed in the past. One possible strategy discussed in [21] is to proceed with a tensorial reconstruction of the angular terms within the antenna subtraction terms. A second approach is to cancel the angular terms by combining phase space points which are related by rotating the collinear partons by an angle of π/2 around the collinear parton direction [58,112]. In this case, the azimuthal correlations present in the matrix element at the rotated point cancel precisely the azimuthal correlations of the un-rotated point.
This second procedure was demonstrated to be extremely powerful in improving the convergence of the double-real radiation subtraction contribution to dijet production in [75,111] in the pure gluonic channel. The strategy of combining pairs of phase space points related by a π/2 rotation eliminated the correlations from both: the real-radiation and its subtraction term. In the latter case, the four-parton antennae are responsible for angular correlations. In the real-virtual contribution discussed in this paper, the correlations can arise in the real-virtual matrix elements dσ V,1 N N LO and in the tree-level five gluon matrix elements present in the subtraction term. There is no contribution from the three-parton antennae. Therefore, the azimuthal effect is expected to be smaller than in the doublereal case [75]. Looking at the distributions shown in figure 3(b) for x = 10 −7 (green), x = 10 −8 (blue) and x = 10 −9 (red) and figure 4(b) for x = 10 −6 (green), x = 10 −7 (blue) and x = 10 −8 (red), we see that the correlations are clearly visible, but are indeed relatively small.
Nevertheless, to eliminate the remaining azimuthal correlations, we show the effect of combining related phase space points discussed above in figure 5(a) for the final-state collinear singularity and figure 5(b) for the initial-state collinear singularity for the same values of the small parameter as in figures 3(b) and 4(b) respectively. We observe a significant improvement in the convergence of the subtraction term, particularly in the case of the initial-final collinear limit. The conclusion is that by combining azimuthally correlated phase space points, the antenna subtraction term correctly subtracts the azimuthally enhanced terms in a point-by-point manner.

Conclusions
In this paper, we have generalised the antenna subtraction method for the calculation of higher order QCD corrections to derive the real-virtual subtraction term for exclusive collider observables for situations with partons in the initial state to NNLO. We focussed particular attention on the application of the antenna subtraction formalism to construct the subtraction term relevant for the gluonic real-virtual contribution to dijet production. The gluon scattering channel is expected to be the dominant contribution at NNLO. The subtraction term includes a mixture of integrated and unintegrated tree-and one-loop three-parton antennae functions in final-final, initial-final and initial-initial configurations.

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We note that the subtraction terms for processes involving quarks, as required for dijet or vector boson plus jet processes, will make use of the same types of antenna building blocks as those discussed here.
By construction the counterterm removes the explicit infrared poles present on the one-loop amplitude, as well as the implicit singularities that occur in the soft and collinear limits. The ǫ-poles present in the real-virtual contribution are analytically cancelled by the ǫ-poles in the subtraction term rendering the real-virtual contribution locally finite over the whole of phase space.
We tested that our numerical implementation of the antenna subtraction term behaves in the expected way by comparing the behaviour of the finite parts of the one-loop real-virtual contribution dσ RV N N LO with the finite part of the real-virtual subtraction term dσ T N N LO for the five-gluon process in the regions of phase space where one particle is unresolved. The numerical convergence of these terms gives confidence that the infrared structure of the real-virtual matrix element is captured by the antenna subtraction method in a systematic and accurate manner.
The real-virtual subtraction terms presented here provide a major step towards the NNLO evaluation of the dijet observables at hadron colliders. Future steps include; (i) completion of the analytic integration of the initial-initial four-parton antennae.
(ii) analytic cancellation of infrared poles between the analytically integrated antennae present in the subtraction terms and the two-loop four-gluon matrix elements.
(iii) full parton-level Monte Carlo implementation of the finite four-, five-and six-gluon channels.
(iv) the construction of similar subtraction terms, etc., for processes involving quarks.
The final goal is the construction of a numerical program to compute the NNLO QCD corrections to dijet production in hadron-hadron collisions.

A Momentum mappings
The NNLO corrections to an m-jet final state receive contributions from processes with different numbers of final state particles. In the antenna subtraction scheme, one is replacing antennae consisting of two hard radiators plus unresolved particles with two new hard radiators. A key element of the antenna subtraction scheme is the factorisation of the matrix elements and phase space in the singular limits where one or more particles are unresolved. This factorisation is guaranteed by the momentum mapping.
In this section we denote the set of momenta for the M -particle process by {p} M . In order to subtract a particular singular configuration in a given process, we derive subtraction terms which reproduce the exact singular behaviour of the matrix element in the unresolved configuration and employ a momentum mapping to implement momentum conservation away from the unresolved limit. This has the consequence of mapping a singular configuration in an M -particle process to an (M −1) or (M −2)-particle process, depending whether the given singular configuration involves a single or a double unresolved limit. In integrated form these subtraction terms have explicit ǫ-poles and contribute to final sates with fewer particles. The consistent momentum maps we require are, Let us consider the single unresolved emission that is relevant in this paper -either as part of the integrated single unresolved subtraction term that cascades down from the double real emission process (A.1) or in the single unresolved limit of the real-virtual contribution (A.3). If the antenna consists of an unresolved particle j colour linked to two hard radiators i and k, then the mapping must produce two new hard radiators I and K. Each mapping must conserve four-momentum and maintain the on-shellness of the particles involved. There are three distinct cases, where, as usual, initial state particles are denoted by a hat. In principle, the momenta not involved in the antenna are also affected by the mapping. For the final-final and initial-final maps, this is trivial. Only in the initial-initial case are the spectator momenta actually modified. The momentum transformations for these three mappings are described in refs. [71,113,114] and will be recalled below.

A.1 Final-final mapping
The final-final mapping is given in [113] and reads p µ I ≡ p µ (ij) = x p µ i + r p µ j + z p µ k p µ K ≡ p µ

Initial-initial emitters
The initial-initial gluon-gluon-gluon antenna is obtained by crossing symmetry from the corresponding initial-final antenna function (B.4), with the replacements s 12 → (p 1 + p 2 ) 2 , s 13 → (p 1 − p 3 ) 2 , s 23 → (p 2 − p 3 ) 2 and Q 2 = s 12 + s 13 + s 23 . It reads [71], F 0 3 (1 g , 3 g ,2 g ) = where the hat identifies the gluons crossed to the initial state. Only the final state gluon j may be soft, and it can also be collinear with the initial state gluonsî ork. Having well defined hard radiators, F 0 3 (î, j,k) does not need to be further decomposed. The full antenna can be used with a single initial-initial mapping, (î, j,k) → (Î,K) [71] of the type given in eq. (A.8).

B.2 Integrated tree-level three-parton antennae
In this subsection we give the expressions for the integrated forms of the antennae in (B.1), (B.4) and (B.7). The integrated three-parton antennae contain explicit ǫ-poles from the integration over the antenna phase space for one unresolved emission and finite remainders. They appear in the real virtual channel as single analytic integration of subtractions in the (m + 2)-parton channel, corresponding to the double real emission [75], as well as in the form of genuine subtraction terms to compensate for oversubtracted poles as discussed in section 2.2.2. Note that the full ǫ dependence in F 0 3 is retained during integration over the antenna phase space.

B.2.1 Final-final emitters
For final-final kinematics the integrated antenna was computed in [21] and reads, where the colour-ordered infrared singularity operator I (1) gg was defined in eq. (4.13).

B.2.2 Initial-final emitters
The full set of integrated initial-final three-parton tree-level antennae were computed in [71]. The pure gluon antenna reads,

JHEP02(2012)141
The splitting kernels, p 0 gg (x), and distributions D n (x 1 ) that appear above are defined in eqs. (D.1), and (D.4) respectively. The function H(m 1 , . . . , m w ; y) denotes the harmonic polylogarithms and their notation is also described in section D.

B.3 One-loop three-parton antennae
The one-loop antenna functions are obtained from the colour-ordered renormalised one-loop three-parton matrix elements according to eq. (2.57) [21]. These contain explicit poles from the loop integration. The integrated forms for the final-final, initial-final and initial-initial cases are available in [21,72] and [74] respectively.

B.3.2 Initial-final emitters
The initial-final one-loop three-gluon antenna function can be obtained from its final-final counterpart (B.11) by the appropriate crossing of one of the particles from the final to the initial state, i.e. by making the replacements, s 23 → (p 2 + p 3 ) 2 > 0, s 12 → (p 1 − p 2 ) 2 < 0, s 13 → (p 1 − p 3 ) 2 < 0 and s 123 → q 2 = s 12 + s 13 + s 23 < 0. Some care needs to be taken in the continuation, since the final-final antenna function is renormalised at µ 2 = q 2 = s 123 , while the initial-final antenna function is renormalised at µ 2 = −q 2 = −s 123 [72]. With this in mind, and defining again the y ij = s ij /s 123 , eq. (B.11) Since the hard radiators are uniquely identified with the initial-state partons, no further decomposition is necessary. In this appendix, we give explicit forms for the four-dimensional space-like splitting kernels used in the paper. Note that we systematically extract a factor of N from the splitting kernels, and furthermore, retain only the leading colour contribution. The relevant gluonic splitting kernels for the purpose of this paper read [115], We use the CHAPLIN Fortran library [116] to evaluate the harmonic polylogarithms up to weight four numerically. CHAPLIN is based on a reduction of harmonic polylogarithms to a minimal set of basis functions that are computed numerically using series expansions and provide fast and reliable numerical results.