Light fermionic NNLO QCD corrections to top-antitop production in the quark-antiquark channel

We present the NNLO corrections to top pair production in the quark-antiquark channel proportional to the number of light quark flavors $N_l$. While the double real corrections were derived previously, here we compute the real-virtual and virtual-virtual contributions in this partonic channel. Using the antenna subtraction formalism, we show that the subtraction terms correctly approximate the real-virtual contributions in all their infrared limits. Combined with the integrated forms of the double real and real-virtual subtraction terms, we show analytically that the explicit infrared poles cancel at the real-virtual and virtual-virtual levels respectively, thereby demonstrating the validity of the massive extension of the NNLO antenna formalism. These NNLO corrections are implemented in a Monte Carlo parton level generator providing full kinematical information on an event-by event basis. With this program, NNLO differential distributions in the form of binned histograms are obtained and presented here.


Introduction
With the large number of top quark pairs produced at the LHC, the study of the properties of top quarks is becoming precision physics. Recently, the ATLAS and CMS collaborations at CERN have reported first measurements of differential observables in top-quark pair production, such as the transverse momentum and rapidity of the top quark as well as the transverse momentum, rapidity, and invariant mass of the tt system [1][2][3]. These measurements will become more and more accurate with higher statistics and will allow for a much more detailed probe of the top quark production mechanism than what can obtained from the total cross section. To reliably interpret the data, those precise measurements have to be matched by equally accurate theoretical predictions. Those can be obtained by computing these hadron collider observables at the next-to-next-to leading order (NNLO) in perturbative QCD. At present, a fully differential NNLO calculation of the cross section for top pair production including all partonic channels is still missing. Intermediate results have recently become available in [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22], and new developments were achieved in [23][24][25][26][27]. Most notably, the total hadronic cross section for tt production has been presented in [28]. It has been obtained using a combination of two well-known and tested procedures: the NLO so-called FKS subtraction formalism [29] and the sector-decomposition approach [30,31], leading to a numerical extraction and cancellation of the infrared -poles.
In this paper we consider the hadro-production of a top-antitop pair and present those O(α 4 s ) corrections to the partonic process qq → tt which are proportional to the number of light quark flavours N l . While at O(α 3 s ) this partonic channel only contains a N l dependence in the finite part of the virtual contributions, at O(α 4 s ) we must consider three different ingredients: virtual-virtual, real-virtual and double real, with two, three and four particles in the final state respectively. Individually, these contributions can contain both ultraviolet and infrared singularities. While the ultraviolet divergencies are removed by ultraviolet renormalisation, the infrared singularities can be made explicit using a subtraction formalism in which subtraction terms that approximate the infrared behaviour of the real radiation matrix elements are added and subtracted. At NNLO, a systematic procedure to construct these terms is provided by the antenna formalism [5,7,8,[32][33][34][35][36][37].
The double real contributions for the process considered in this paper were treated in [5], where antenna subtraction terms were derived and their convergence to the real radiation matrix elements in all single and double unresolved limits was shown. In this paper we shall present the real-virtual and virtual-virtual contributions, employing the massive extension of the antenna formalism to construct their corresponding subtraction terms. We will show that the real-virtual subtraction terms correctly approximate the corresponding matrix elements in all singular limits, and that all explicit infrared singularities are cancelled analytically. Furthermore we also provide differential cross sections obtained with a fully differential parton-level event generator containing the real-virtual and virtual-virtual contributions presented in this paper together with the double real contributions of [5].
Employing a subtraction method, the NNLO partonic cross section to the tt production cross section in a given partonic channel reads, where dΦm represents the phase space integration for an m-parton final state. dσ S N N LO denotes the subtraction term for the four-parton final state which, by construction, behaves like the double real radiation contribution dσ RR N N LO in all single and double unresolved infrared limits. Likewise, dσ V S N N LO is the one-loop virtual subtraction term coinciding with the one-loop three-parton final state dσ RV N N LO in all single unresolved singular limits. dσ V V N N LO denotes the virtual-virtual two-parton final state contribution, where no particles can become unresolved and therefore no subtraction is needed. In addition, due to the presence of initial-state partons that can emit collinear radiation, there are also two mass factorisation counter-terms dσ M F,1 N N LO and dσ M F,2 N N LO , for the three and two-particle final state respectively. Both are constructed as convolutions of Altarelli-Parisi splitting kernels with lower order partonic cross sections: dσ M F,1 N N LO involves the next-to-leading order (NLO) real radiation cross sections, while dσ M F,2 N N LO contains the leading order (LO) and NLO virtual partonic cross sections. Since there are no NLO real contributions to tt production proportional to N l , in this paper we do not need to consider (M F, 1)-type counter-terms. As mentioned above, the quark-antiquark channel contains a piece proportional to N l in the NLO virtual contributions, and therefore dΦ 2 dσ M F,2 N N LO is present and will be derived below.
The numerical implementation of the NNLO cross section requires the rearrangement of the different terms in eq.(1.1) according to the multiplicity of the final states. After this rearrangement is performed we have where the terms in each of the square brackets is finite, well behaved in the infrared singular regions and can be implemented in a parton-level event generator computing the NNLO cross section and related differential distributions numerically. The double real subtraction term dσ S N N LO contains distinct pieces corresponding to different limits in different colour-ordered configurations. Some of these pieces ought to be integrated analytically over the unresolved phase space of one particle and combined with the three-parton final state integral, while the remaining terms are to be integrated over the unresolved phase space of two particles and combined with the two-parton contribution. This separation amounts to splitting the integrated form of dσ S N N LO as explained in [8,32,33,35,36]  After performing this separation, the counterterms for the partonic channel considered in this paper which must be added at the three and two-parton final states are While the construction of dσ S qq,N N LO,N l has been presented in [5], the explicit derivation of dσ T qq,N N LO,N l and dσ U qq,N N LO,N l will be given here for the first time. The NNLO antenna subtraction method, which we shall employ in this paper, can be viewed as belonging to the class of so-called classical subtraction methods in which the matrix elements, subtraction terms and especially the integrated subtraction terms are evaluated analytically. As a result, at the real-virtual and virtual-virtual levels one obtains an exact and analytic pole cancellation. The finite remainders at each of those levels can be implemented in a parton-level Monte Carlo event generator containing the full kinematical information on the final states on an event-by-event basis and enabling the description of differential NNLO observables.
The general structure of the NNLO antenna subtraction terms that are required for massless hadronic observables has been detailedly explained in [32,33,[35][36][37]. For the treatment of observables involving massive final states, further ingredients are needed, namely, massive antennae and different phase space factorisations and mappings [5,8]. The general structure of the subtraction terms, however, remains unchanged it will not be repeated in this paper. We shall instead restrict ourselves to presenting those ingredients of our computation which are new at the real-virtual and virtual-virtual levels focusing on challenging aspects such as the correct subtraction in the subleading colour contributions at each partonic level.
The plan of the paper is as follows: In section 2 we present the general structure of the individual NNLO contributions related to the process qq → tt. Section 3 deals with the double real contributions proportional to the colour factors N l N c and N l /N c , and it specifies how the integrated form of the double real subtraction term dσ S N N LO ought to be added back to the three and two-parton phase space integrals. In section 4, we derive the real-virtual contributions and explicitly construct their corresponding subtraction term dσ T N N LO , while in section 5 the two-parton final state contributions involving the two-loop matrix elements and the counterterm dσ U N N LO are derived. Section 4 also presents some numerical tests checking the validity of the real-virtual subtraction term dσ T N N LO , while in section 5 we show that the explicit poles present at the two-parton level cancel analytically. In section 6 we present differential distributions at NNLO in kinematical variables of the top quark and the top-antitop system. Finally, section 7 contains our conclusions and an outlook.
Included are also different appendices: appendix A presents the double real subtraction term dσ S N N LO derived in [5]. In appendix B we define the massive and massless infrared singularity operators required to express the explicit infrared poles of several ingredients in our calculation. Appendix C summarises the infrared factorisation of tree-level and oneloop amplitudes, while appendix D presents the infrared limits of the one-loop antennae present in the real-virtual subtraction term. Finally, appendix E contains the expression for the integrated initial-final four-parton B-type massive antenna required in our calculation at the virtual-virtual level.
2 General structure of the NNLO contributions to top pair production in the qq channel At leading order (O(α 2 s )), the partonic cross section for top-antitop pair production via the quark-antiquark channel reads In this equation, dΦ 2 (p 3 , p 4 ; p 1 , p 2 ) is a 2 → 2 phase space with p 1 and p 2 denoting the momenta of the incoming massless quark and antiquark respectively, and p 3 and p 4 those of the top and antitop. J 2 (p 3 , p 4 ) represents the measurement function that constructs an experimental observable with a top-antitop quark pair in the final state. For the total cross section, we have J (2) 2 (p 3 , p 4 ) = 1 while for a generic differential distribution in a given variable X, which is defined in terms of the momenta p 3 and p 4 through the function X(p 3 , p 4 ) we take J (2) 2 (p 3 , p 4 ) = δ(X −X(p 3 , p 4 )). The tree-level matrix element M 0 4 (...) in eq.(2.1) is colour and coupling-stripped, and it is related to the full tree-level amplitude for q 1q2 → Q 3Q4 through the (trivial) colour decomposition The hats in1 q and2q indicate that p 1 and p 2 are incoming momenta. The normalisation factor N qq LO is given by where s is the energy squared in the hadronic center-of-mass frame. Included in this normalisation factor are the flux factor, as well as the sum and average over colour and spin. The constants C( ) andC( ) are defined as: providing the useful relation The NNLO corrections to the leading-order cross section in eq.(2.1) receive contributions from partonic processes at tree, one-loop and two-loop levels. The double real NNLO corrections that are proportional to the number of light quark flavors are due to the partonic process qq → QQq q at tree level. They read 2 (p 3 , p 4 , p 5 , p 6 ), (2.6) where dΦ 4 is the 2 → 4 phase space, and |M 0 q 1q2 →Q 3Q4 q 5q 6 | 2 is the square of the full coupling-stripped tree-level amplitude normalized to (N 2 c − 1). This factor is included in the overall normalisation N RR,qq N N LO , which is given by Given the fact that the measurement function J (4) 2 allows the massless final state fermions with momentum p 5 and p 6 to become unresolved, i.e. soft or collinear, eq.(2.6) contains infrared divergences. These divergences are implicit, in the sense that they only become explicit as poles in the dimensional regulator after the integration over the phase space is performed. The infrared behaviour of dσ RR qq,N N LO,N l can be captured with the antenna subtraction term presented in [5], which is also recalled in appendix A. A more detailed discussion on the structure of the double real contributions and their infrared behaviour will be presented in section 3.
The mixed real-virtual contributions are given by the phase space integral of the oneloop and tree-level amplitudes for the 2 → 3 process qq → QQg. They read with the normalisation factor N qq,RV N N LO given by The subscript N l on the interference of the tree-level and one-loop amplitudes indicates that only those terms proportional to N l are to be kept. The real-virtual contributions in eq.(2.8) contain explicit ultraviolet and infrared divergences as well as implicit infrared ones. The explicit poles in originate from the loop integration in M 1 q 1q2 →Q 3Q4 g 5 , whereas the implicit singularities are due to the phase space integration over regions where the matrix elements diverge: the soft limit p 5 → 0 and the collinear limits p 1 ||p 5 , p 2 ||p 5 . While the ultraviolet poles are cancelled upon renormalisation, we employ a subtraction term dσ T qq,N N LO,N l to deal with the infrared ones. This subtraction term has the twofold purpose of canceling the explicit poles, whose structure is well known [38], while simultaneously regularising the phase space integrand in the soft and collinear limits. We shall present the explicit construction of dσ T qq,N N LO,N l in section 4.
Finally, the double virtual contributions contain two terms: the interference of a twoloop 2 → 2 matrix element with its tree-level counterpart, and a one-loop 2 → 2 amplitude squared. Those can be written as, where the normalisation factor N qq,V V N N LO is given by After ultraviolet renormalisation, these double virtual contributions have explicit infrared poles coming from the loop integration, but no implicit poles, since the measurement function J (2) 2 , does not allow any final state particle to be unresolved. The construction of dσ U qq,N N LO,N l to be associated with these virtual-virtual contributions and dealing with the explicit infrared divergencies present in those will be presented in section 5.
In the expressions of the real-virtual and virtual-virtual contributions defined above as dσ RV qq,N N LO,N l and of dσ V V qq,N N LO,N l , we have introduced a dependence on x 1 and x 2 , the momentum fractions carried by the initial state quark-antiquark pair, through delta functions. This trivial dependence on x 1 and x 2 is introduced in order to facilitate the combination with the counterterms dσ T qq,N N LO,N l and dσ U qq,N N LO,N l respectively, whose dependance on these two variables is not trivial. As we will see in sections 4 and 5, this dependance on x 1 and x 2 is related to the fact that in the integrated subtraction terms and mass factorisation counterterms there are contributions where the partons that enter the hard scattering carry a fraction x i of the incoming momenta. In general, there are three regions: the soft (x 1 = x 2 = 1), the collinear (x 1 = 1, x 2 = 1 and x 1 = 1 x 2 = 1) and the hard (x 1 = 1 x 2 = 1). The two delta functions in the above equations imply that the real-virtual and the virtual-virtual corrections only contribute in the soft region. Their corresponding subtraction terms denoted respectively as dσ T qq,N N LO,N l and dσ U qq,N N LO,N l , on the other hand, will contribute in all three regions.
3 Real-real contributions to qq → tt: the N l part The double real contributions to qq → tt that are proportional to the number of light quark flavours and their corresponding subtraction terms were derived in [5]. It is the purpose of this section to identify the parts of the integrated form of the subtraction term that must be included in the counter-terms at the two and three parton levels.
The (S, b)-type subtraction term, denoted by dσ S,b qq,N N LO,N l , takes care of the double unresolved limits. Two different kinds of structures are involved in this subtraction term: dσ S,b,4 qq,N N LO,N l which has the form X 0 4 ×|M 0 4 | 2 , with X 0 4 being a general four parton tree-level antenna, and dσ S,b,3×3 qq,N N LO,N l which has the form X 0 3 × X 0 3 × |M 0 4 | 2 . The former subtracts the double unresolved limits while introducing spurious single unresolved singularities, whereas the latter removes these spurious limits ensuring that the four-parton antenna is only active in the double unresolved regions.
The integrated form of dσ S,b,4 qq,N N LO,N l is obtained by integrating the four-parton antennae X 0 4 inclusively over their corresponding antenna phase space. These integrated subtraction terms are added back at the two-parton level, and therefore, recalling eq.(1.5), for the present calculation we have On the other hand, the subtraction terms dσ S,a qq,N N LO,N l and dσ S,b,3×3 qq,N N LO,N l are added back in integrated form at the three-parton level. The integrated forms are obtained by integrating the three-parton antennae X 0 3 over the corresponding antenna phase space (in the case of dσ S,b,3×3 qq,N N LO,N l the "outer" antenna is integrated). We therefore have Before concluding this section we would like to make a few more remarks about the double real subtraction terms.
As was pointed out in [5], the construction of dσ S,b,4 qq,N N LO,N l , which is given in appendix A, requires three different B-type antennae: a massive final-final, a flavour-violating initialfinal, and an (massless) initial-initial antenna. While the leading-colour piece, that is, the part proportional to N l N c , only requires the initial-final antenna, a combination of all three of them is needed in the subleading-colour part, propotional to N l /N c . The form of this combination is non-trivial and it is derived in such a way that the subtraction term correctly matches the double soft limit of the double real radiation matrix elements. The integrated forms of the three different B-type antennae, which shall be used in section 5 for the construction of the double virtual counter-term, were derived in [6,11,39,40].
For the construction of dσ S,a qq,N N LO,N l , only one antenna function is needed: a massive final-final E-type antenna. The same antenna function is employed in dσ S,b,3×3 qq,N N LO,N l in order to subtract the single collinear limits of the B-type four-parton antennae. The unintegrated and integrated form of this massive E-type three-parton antenna has been derived in [4,41].
4 Real-virtual contributions to qq → tt: the N l part In this section we shall present the real-virtual contributions to qq → tt proportional to the number of light quark flavours together with their corresponding antenna subtraction terms. In other words we will construct the phase space integrand of the following threeparton contribution

Real-virtual contributions
The real-virtual corrections to top pair production in the qq channel are obtained from the interference of the one-loop and tree-level amplitudes for the partonic process qq → ttg.
The colour decomposition of the one-loop matrix-element reads, where each of the sub-amplitudes has the following decomposition into primitives: For the tree-level amplitude, the colour decomposition reads, Interfering these one-loop and tree-level amplitudes given respectively in eq.(4.2) and eq.(4.4), combining the result with the phase space and the measurement function, and retaining only the terms multiplied by N l , we have where the overall factor N qq,RV N N LO was given in eq.(2.9). Furthermore, in eq.(4.5) we have defined the following amplitude where the gluon is photon-like.
The matrix elements in eq.(4.5) were not available in analytic form in the literature. We computed them ourselves with an in-house program based on Qgraf [42]. Only eight Feynman diagrams of the full one-loop amplitude M 1 q 1q2 →Q 3Q4 g 5 contribute to the N l piece, with topologies involving up to three massless loop propagators. Two sample Feynman diagrams are presented in figure 1. After the reduction of all tensor integrals to scalars, we find that all the one-loop amplitudes in dσ RV qq,N N LO,N l can be written entirely in terms of two massless bubbles with virtualities (p 1 + p 2 ) 2 and (p 3 + p 4 ) 2 .
These matrix elements contain explicit infrared and ultraviolet poles, and because of their singular behavior in soft and collinear limits, they also yield implicit infrared singularities upon phase space integration. The ultraviolet poles are removed after renormalisation, for which we follow the scheme of [14][15][16]22]: the heavy quark mass and wave function are renormalised on-shell, while for the strong coupling, the MS scheme is employed.
In general, the explicit infrared pole structure of real-virtual contributions can be written in terms of colour-ordered infrared singularity operators denoted as I (1) ij , with i, j the partons involved in this operators which can be either massless or massive. For completeness those are recalled in appendix B. In the context of this computation, it turns out to be more convenient to write the pole parts explicitly, since the I Qg,F ( , s ij ) and I (1) qg,F ( , s ij ), all of which yield the same explicit poles proportional to the colour-ordered renormalisation constant b 0,F = −1/3. For all colour orderings we have: With this, we can write the pole part of the real-virtual contributions as, The implicit infrared singularities in eq.(4.5) are related to the fact the final state gluon can become either soft or collinear to one of incoming particles. These implicit divergences as well as the explicit poles are captured by the real-virtual subtraction term dσ T qq,N N LO,N l which we shall construct below.

The real-virtual subtraction term: dσ T
The purpose of the real-virtual counter term dσ T qq,N N LO,N l is to cancel the explicitpoles of the real-virtual contributions dσ RV qq,N N LO,N l and to simultaneously subtract their infrared limits in such a way that the difference dσ RV qq,N N LO,N l − dσ T qq,N N LO,N l can be safely integrated numerically in four dimensions. The general structure of this subtraction term was developed for hadronic (massless) jet observables in [35], and it remains unchanged for processes involving massive fermions. For the leading-colour contributions to qq → tt it has been applied in [8]. We shall not repeat it here.
As shown in eq.(1.4), the real-virtual subtraction term for the present calculation contains singly integrated double real subtraction terms, as well as a pure virtual subtraction term dσ V S qq,N N LO,N l . The pieces of the integrated double real subtraction terms that must be added back in this three-parton final state are specified in eq. (3.4). Regarding the purely virtual subtraction term dσ V S qq,N N LO,N l , only two of the four contributions discussed in [35] are needed in the context of this paper: a term denoted as dσ V S,a qq,N N LO,N l , which approximates the behaviour of the real-virtual contributions in their infrared limits, and another term that we denote as dσ V S,d qq,N N LO,N l , which is related to the ultraviolet renormalisation of one-loop antennae. With all these ingredients put together, the real-virtual subtraction term that will be presented in the remainder of this section takes the following form

Explicit infrared singularity subtraction
We start the construction of dσ T qq,N N LO,N l by showing that the explicit poles of the realvirtual contributions are cancelled as The explicit poles of the real-virtual matrix elements were given in eq.(4.8). Those of the singly integrated subtraction term 1 dσ S,a qq,N N LO,N l can be obtained from its corresponding unintegrated form given in eq.(A.1) by replacing in that expression the unintegrated antennae by their integrated forms and relabeling the remapped momenta. We find (4.11) The integrated final-final massive three-parton E-type antenna function denoted by E 0 Qqq was derived in [4]. It is given by We omit the explicit functional dependence of the integrated antennae on ρ for conciseness. Combining eqs.(4.8), (4.11) and (4.12) we find that eq.(4.10) is satisfied.

The implicit infrared substraction term: dσ V S
As we mentioned before, the real-virtual subtraction term dσ V S qq,N N LO,N l receives two different contributions. It can be written as Since dσ V S,d qq,N N LO,N l does not have any explicit infrared poles, and no dσ M F,1 qq,N N LO,N l mass factorisation counterterms terms are present, a way to see that no (V S, b)-type subtraction terms are needed, unlike in the general case [35], is by showing that and that the implicit infrared singularities of 1 dσ S,a qq,N N LO,N l are compensated by

Construction of dσ V S,a
Following the general framework described in [35], in order to subtract the single unresolved limits of the real-virtual contributions given in eq.(4.5) we construct our subtraction terms of the type (V S, a) with tree-level and one-loop antennae multiplied by reduced matrix elements at one-loop and tree level respectively. For the process under consideration we find This subtraction term contains three different three-parton tree-level A-type antennae: a (massless) initial-initial A 0 3 (q, g,q), a massive flavour-violating initial-final A 0 3 (Q, g,q) and a massive final-final A 0 3 (Q, g,Q). All these antennae were derived and integrated in [4,41,43]. The one-loop antennae in eq.(4.16) are A 1,l 3 (q, g,q), A 1,l 3 (Q, g,q) and A 1,l 3 (Q, g,Q). Only the first of them has been already derived [44]. The remaining two are massive antennae which we present for the first time together with their singular limits and infrared structure in appendix D.
For the construction of dσ V S,a qq,N N LO,N l , the U(1)-like sub-amplitude squared |M [l] 5 (3 Q , 4Q,2q,1 q , 5 γ )| 2 in eq.(4.5) requires a special treatment, since the hard radiations cannot be simply identified as the "colour neighbours" of the unresolved gluon. It is crucial to notice here that the unresolved behaviour of the one-loop matrix elements M [l] 5 is "tree-level-like". This can be seen in the fact that the N l parts of the one-loop colourordered soft factor and splitting functions are proportional to their tree-level counterparts as shown in appendix C. Therefore, the subtraction terms for the one-loop matrix-element |M antennae, which can be found in appendix D. We find can be obtained from its unintegrated form by integrating the "outer" antennae in those terms in eq.(A.2) which contain a product of two three-parton antennae. It reads 2 ( p 3 , p 4 ) .

Renormalisation scale dependent subtraction
In general, one-loop antennae are evaluated at the renormalisation scale |s ijk | (see Appendix C) while all one-loop matrix elements are evaluated at scale µ 2 . To ensure the proper renormalisation of the real-virtual subtraction terms, a contribution denoted as dσ V S,d qq,N N LO,N l is required. It is given by 2 ( p 3 , p 4 ) .

The real-virtual subtraction term dσ T
Putting everything together, the real-virtual counterterm term dσ T qq,N N LO,N l takes the following form 2 (p 3 , p 4 , p 5 ) In the following section we shall show that the counter-term in eq.(4.20) approximates the real-virtual contributions dσ RV qq,N N LO,N l given in eq.(4.5) in all single soft and collinear limits.

Numerical tests of soft and collinear cancellations
The real-virtual contributions and their related subtraction terms presented above have been implemented in a Fortran code. In this section we show the results of a series of numerical tests devised in order to test how well the subtraction terms fulfill their purpose of approximating the real-virtual contributions in all single unresolved regions. Since the heavy quark mass regulates all final state collinear limits, we must consider only soft and initial state collinear limits.
The tests were performed by using RAMBO [45] to generate phase space points in the singular regions, with the exact distance between each event and the singularity parametrized    In this section we present the double virtual contributions to qq → tt proportional to N l . We shall focus in particular in the double-virtual counter-term dσ U qq,N N LO,N l with which we construct the two-parton contribution

The virtual-virtual contributions
Following the general structure given in eq.(2.10), the virtual-virtual contributions can be written as where we have abbreviated and the normalisation factor N V V N N LO was given in eq.(2.11). For the N l part of the two-loop matrix element in eq.(5.3) we employ the analytic results of [14]. The "one-loop squared" term has been computed analytically in [21]. We re-derived it ourselves, also analytically, and use our own result in our event generator. We further compared our one-loop amplitude squared with results provided by Roberto Bonciani and found full agreement.
In general, NNLO double virtual contributions contain poles of up to order four, all of which originate from the loop integrations. However, in the colour factors that we are considering in this paper, i.e. N l N c and N l /N c , the deepest poles are of order three. Since at this level no partons can become unresolved, the phase space integration of dσ V V qq,N N LO,N l does not yield any additional singularities. As we shall see below, all explicit poles of the double virtual contributions are captured and cancelled by those in the subtraction term dσ U qq,N N LO,N l , which, as shown in eq.(1.5), contains mass factorisation counter-terms as well as double real and real-virtual integrated subtraction terms.

The mass factorisation counterterm dσ M F,2
We start the construction of the virtual-virtual subtraction term dσ U qq,N N LO,N l by deriving the mass factorisation counter-term dσ M F,2 qq,N N LO,N l . In general, for a given partonic process initiated by two partons i and j with momenta p 1 and p 2 , the mass factorisation counterterm dσ M F,2 ij,N N LO to be included at NNLO together with the double virtual matrix elements contributions reads [33,35,36], with the kernels Γ (1) ij,kl and Γ (2) ij,kl given by To simplify the construction of the double virtual subtraction term dσ U N N LO , the splitting kernel Γ (2) ij,kl can be written as suggested in [33] as such that, Γ ij;kl (x 1 , The kernel Γ (2) ij (z) is related to the usual Altarelli-Parisi splitting functions [46] as For the partonic process qq → tt, the part of the mass factorisation counter-term dσ M F,2 qq,N N LO that is proportional to N l reads where Γ (2),[N l ] qq;qq and dσ V qq,N LO,N l denote the pieces proportional to N l in Γ (2) qq;qq and dσ V qq,N LO respectively. Moreover, dσ qq,LO is the leading order partonic cross section for the process qq → tt. Trading Γ (2) qq;qq for Γ (2) qq;qq in eq.(5.7), dσ M F,2 qq,N N LO can be compactly written as the sum of qq;qq (x 1 , x 2 )dσ qq,LO (x 1 p 1 , x 2 p 2 ), (5.12) with the kernels Γ (1) qq;qq and Γ (2),[N l ] qq;qq constructed as in eqs.(5.5-5.7) using the expressions given in the appendix of [33].
Note finally that the mass factorisation counter-term denoted in [33] as dσ M F,2b , which is necessary in the most general cases, is absent here. This is due to the fact that the oneloop kernel Γ (1) qq;qq does not contain any terms proportional to N l .

The virtual-virtual subtraction term dσ U
In addition to the mass factorisation counter-term derived above, the virtual-virtual subtraction term dσ U N N LO contains in general contributions originated from the integrated double real and real-virtual subtraction terms. In the present case, it is given by Similarly, the integrated real-virtual subtraction terms 1 dσ V S,a qq,N N LO,N l and 1 dσ V S,d qq,N N LO,N l are obtained from eqs. (4.16) and (4.19) by integrating the tree-level and one-loop three parton antennae over their corresponding phase space. We obtain while for the integrated form of renormalisation scale dependent subtraction term we get 2 (p 3 , p 4 ). demonstrating that we have correctly implemented all the subtractions terms at real-real, real-virtual and virtual-virtual levels in leading (N l N c ) and subleading-colour (N l /N c ) contributions for the parton-level process qq → tt.
This pole cancellation provides furthermore a crucial check on the integrated forms of the massless and massive four-parton B-type antennae derived in [6,11,39] and of the correct implementation of the one-and two-loop massive amplitudes. Finally, it also provides a proof that the massive extension of the NNLO antenna formalism can be used to extract and cancel the infrared singularities of the top-antitop pair production cross section evaluated at the NNLO level, enabling the construction of a fully differential event generator.

Numerical results: NNLO differential distributions
With the double real counter-terms derived in [5] and recalled in appendix A together with the real-virtual and virtual-virtual subtraction terms presented above in sections 4 and 5, we developed a Monte Carlo parton-level event generator based on the set up of eq.(1.2). This program, written in Fortran following the structure of the program developed in [47] for the calculation of the di-jet hadronic cross section at NNLO, allows us to produce the first differential distributions for top pair production including exact NNLO results, namely the O(α 4 s ) corrections to the qq → tt process proportional to the number of light quark flavors N l .
In this section we present differential distributions in the top quark transverse momentum, the top quark rapidity, as well as in the rapidity and invariants mass of the top-antitop system. We consider the qq initiated process only, convoluted with parton distribution functions (PDFs) corresponding to proton-proton collisions, at leading order, next-to-leading order, and next-to-next-to-leading order. The next-to-next-to leading order results denoted as NNLO (N l ) in the figures 4-7 presented in this section are the sum of the full NLO corrections and the purely fermionic NNLO corrections to qq → tt discussed in this paper.
We employ a hadronic center-of-mass energy √ s = 7 TeV, a value of the top quark mass given by m t = 173.5 GeV, and the PDF sets MSTW2008LO90CL, MSTW2008NLO90CL and MSTW2008NNLO90CL for the computations at LO, NLO and NNLO respectively [48]. We set the factorisation and renormalisation scales to be equal to the top quark mass µ R = µ F = m t . In figure 4 we present the inclusive tt cross section as a function of the top quark transverse momentum p t T at LO, NLO, and NNLO together with the corresponding kfactors. From the NNLO/NLO ratio we see that the NNLO contributions proportional to N l decrease the cross section by 6−7% depending on the value of the transverse momentum p t T compared to the NLO result. In figure 5 we show the inclusive tt cross section as a function of the top quark rapidity y t at LO, NLO, and NNLO with the corresponding k-factors in the lower panel. The NNLO/NLO ratio shows that the NNLO contributions proportional to N l decrease the cross section by 8% in the central region and 5% in the forward and backward regions. Figure 6 contains the inclusive tt cross section as a function of the rapidity of the topantitop system y tt at LO, NLO and NNLO together with their corresponding k-factors. Once again we note an overall decrease in the cross section at NNLO, ranging from 2.5 to 8.5% with respect to the NLO result depending on the rapidity region considered. In figure 7 we present the cross section as a function of the invariant mass of the tt system at LO, NLO and NNLO, with all three ratios in the lower panel. As can be seen from the NNLO/NLO ratio, the NNLO corrections proportional to the number of light quark flavours N l decrease the cross section over the entire m tt spectrum considered. The decrease ranges from 6.5% in the low invariant mass region to almost 10% in the high invariant mass region.
Naïve power counting suggests that the leading-colour NNLO contributions considered in this paper, i.e. the pieces proportional to N l N c should approximately account for 90% of the NNLO corrections, with the remainder being given by the subleading-colour contributions, proportional to N l /N c . At the level of the total cross section we find that the subleading-colour pieces are opposite in sign to the leading-colour and represent 13% of the full result, in agreement with this naïve power counting expectation.
In order to assess the impact of the subleading-colour contributions, we studied the NNLO corrections proportional to N l N c and N l /N c separately as a function of the top quark transverse momentum and rapidity. The results of these studies are shown in figures 8(a) and 8(b). We find that the subleading-colour piece has a different shape and sign than -25 - the leading-colour part, and contributes more significantly in the forward and backwards regions and less when the top quark is more centrally produced. In particular, in the p t T spectrum of figure 8(a) we find that at low transverse momenta, the size of the N l /N c colour factor is of approximately 30% of the full colour result, while at high transverse momenta its contribution is negligible. As a check of our numerical results, we compared the value of the total NNLO correction proportional to N l with the result presented in [10]. In order to perform this check we had to incorporate to our calculation the NNLO contributions proportional to N h N l , with N h the number of heavy quark flavours, since those are included in the coefficient function F 1 (β) of [10]. These contributions are rather simple, as they only enter at the double virtual level, and they are finite. We employed the two-loop matrix elements derived in [14], and computed the square of the one-loop matrix elements ourselves. With the same input parameters and PDF sets used throughout the present section, we find that the total NNLO correction to top pair production proportional to N l in the qq channel is −0.5780 pb, whereas with the coefficient function F 1 (β) of [10] we obtain −0.5822 pb. These two results agree within less that 1%, which represents a very strong check of both completely independent calculations.

Conclusions
In this paper we presented the O(α 4 s ) light fermionic corrections to the partonic process qq → tt. Those corrections are of three types: virtual-virtual, real-virtual and double real, with two, three and four particles in the final state respectively. While the ultraviolet divergencies are removed by renormalisation, the infrared singularities are treated using the massive extension of the NNLO antenna formalism.
The double real corrections had been derived in [5]. In this paper we presented the real-virtual and virtual-virtual contributions and derived their related subtraction terms explicitly. We showed that the real-virtual subtraction terms correctly approximate the real-virtual matrix elements in all their singular limits. Furthermore, combined with the integrated forms of the subtraction terms, we show analytically that all explicit infrared singularities are cancelled both at real-virtual and at the virtual-virtual levels. This analytic pole cancellation provides us with a very strong check on the correctness of our result and on the applicability of the antenna subtraction method to reactions involving massive final states.
The NNLO results of this paper are implemented in a Monte Carlo parton-level generator providing full kinematical information on an event-by event basis. This program, written in Fortran, allowed us to produce NNLO differential distributions for top pair production in the partonic channel under consideration for the first time. We studied the impact of the NNLO corrections proportional to N l for distributions in p T and rapidity of the top quark as well as in the rapidity and invariant mass of the tt system and found that those corrections are substantial. They reduce the cross section compared to the NLO results, with the reduction varying considerably over the phase space. We also assessed the separate sizes of the leading and subleading-colour N l contributions finding that they have opposite signs, and that, at the level of the total cross section, the subleading-colour part contributes with 13%. Its importance, however, varies substantially over different regions of phase space. It can be as high as 30% for low transverse momenta of the top quark, and negligible for high transverse momenta.
The results presented in this paper can be regarded as a major step towards the computation of the complete cross section of top pair production including all partonic channels in fully differential form. For the first time, we have presented exact NNLO (N l ) corrections to one partonic channel in this process and produced NNLO differential distributions. Further work will include the computation of all remaining partonic channels.
A The double real subtraction term for qq → tt: the N l part The double real corrections to the process qq → tt proportional to the number of light quark flavours are due to the tree-level partonic process qq → ttq q . We derived the corresponding subtraction terms in [5] and found that only two pieces are needed, namely 2 (p 35 , p 4 , p 56 ) 2 (p 1 , p 456 ) , where the pieces containing B 0 4 antennae subtract the double soft and triple collinear limits, while those of the form E 0 3 × A 0 3 subtract the single collinear limits of the B-type antennae, as well as spurious double unresolved limits of the dσ S,a qq,N N LO,N l subtraction term.

B Colour-ordered infrared singularity operators
The explicit pole structure of colour-ordered one-loop matrix elements can be written in terms of colour-ordered infrared singularity operators I (1) ij . Within the antenna subtraction method, the pole part of one-loop antennae as well as that of integrated three-parton antennae can be also captured by these operators.
If only massless particles are involved, the following set of operators is sufficient (in addition to the one-loop splitting kernels Γ (1) ij (x)) to express the pole structure of a QCD one-loop amplitude as well as that of a one-particle inclusive integral of a tree-level amplitude [34,43]: When massive fermions are involved the following operators must also be considered [4] I (1) (B.10) C Infrared properties of tree-level and one-loop amplitudes In this section we collect the single unresolved tree and one-loop factors needed in the context of the computation presented in this paper.

C.1 Infrared limits of tree-level amplitudes
The factorisation properties of colour-ordered tree-level amplitudes in single soft and collinear limits are well known [49,50]. They involve universal unresolved factors that are also encountered in the single unresolved limits of tree-level antennae. In the following we shall recall the general factorisation of tree-level matrix elements in single soft and collinear limits and present only those unresolved factors that are needed in the context of this paper.

C.1.1 Collinear limits
When two massless colour-connected partons i and j become collinear and cluster into a parent particle k, a sub-amplitude squared factorises as The spin averaged Altarelli-Parisi splitting function P ij→k (z) is different for the different parton-parton splittings. The definition of the momentum fraction z in eq.(C.1) depends on whether both collinear particles are in the final state, or if one of them is in the initial state. In the final-final case we have whereas if p i is in the initial state and p j in the final state, we have In the calculation presented in this paper, only the collinear limit of an initial state (anti)quark and a final state gluon must be considered. The corresponding splitting function is given by

C.1.2 Soft limits
Colour-ordered tree-level amplitudes develop an implicit soft singularity when the fourmomentum of a final state gluon vanishes. When a gluon with momentum p j becomes soft in a colour-ordered amplitude where it is colour-connected to two hard particles with momenta p i and p k , the amplitude factorises as, where µ (p j , λ) is the soft gluon's polarisation vector, and the soft currents are given by After squaring eq.(C.5) we obtain the well-known formula with the massive soft eikonal factor given by In multi-parton processes one often encounters certain subleading-colour contributions that, in addition to colour-ordered amplitudes squared, contain interferences of different colour-ordered amplitudes. Using eq.(C.5), we find that in their soft gluon limits, these types of interferences factorise as In some cases we can employ decoupling identities and replace the subleading-colour interference terms by sub-amplitudes squared in which a gluon is U (1)-like, and does not have any non-abelian couplings. In these cases, the factorisation in the soft limit at the amplitude level is analogous to the soft photon factorisation of QED matrix elements. It reads (C.11) where {q} is the set of all final state quarks and initial state antiquarks in the process, and {q} stands for all final state antiquarks and initial state quarks. Squaring eq.(C.11) and rearranging the result, we find the following factorisation for the amplitude squared: We have employed this factorisation in the construction of the real-virtual subtraction term for the colour-ordered amplitude squared |M [l] 5 (3 Q , 4Q,2q,1 q , 5 γ )| 2 in eq.(4.5).

C.2 Infrared limits of one-loop amplitudes
Like at tree level, in their infrared limits the one-loop colour-ordered amplitudes undergo well-known factorisations involving universal one-loop and tree-level soft and collinear factors [13,[51][52][53][54][55][56][57][58][59][60][61][62]. Those singular factors are also found in the single unresolved limits of one-loop antennae. Introducing the following notation where Sing 1 is a one-loop single unresolved factor and |M 1 m−1 | 2 is a one-loop reduced subamplitude squared. Following the decomposition of the one-loop colour-ordered amplitudes into primitives, the one-loop unresolved factor can be decomposed as In subleading-colour contributions involving interferences of one-loop and tree-level amplitudes with different colour-orderings the factorisation in eq.(C.14) does not hold for the soft limit. As was the case at tree level, multiple soft factors arise. In the following, we shall explicitly present the one-loop singular factors that we encountered in the calculation presented in this paper.

C.2.1 One-loop collinear factors
In single collinear limits, the interference of a one-loop and a tree-level colour-ordered matrix element factorises as in eq.(C.14), with Sing and to ensure that the antennae and real-virtual matrix elements are computed at the same renormalisation scale, for the N l contribution we must substitute As explained in section 4, this substitution gives rise to the real-virtual subtraction term of type dσ V S,(d) .
After renormalisation, one-loop antennae contain explicit and implicit infrared singularities. The structure of the former can be entirely captured by colour-ordered infrared singularity operators, whereas the latter occur when massless partons in the antenna become soft or collinear.
The one-loop three parton antennae encountered in the context of this paper are of A-type. They involve either no masses, as the initial-initial antenna A 1,l 3 (1 q , 3 g ,2q), one mass as the initial-final antenna A 1,l 3 (1 Q , 3 g ,2 q ), or two masses as the final-final antenna A 1,l 3 (1 Q , 3 g , 2Q).
D.2 Explicit infrared pole structure of A 1,l 3 one-loop antennae In general, the explicit infrared poles of one-loop antennae can be written in terms of massless and massive I (1) ij operators, which were all recalled in the appendix B. The poles of the one-loop antennae required in the calculation presented in this paper are proportional to their tree-level counterparts as Poles A 1,l 3 (1 Q , 3 g ,2 q ) = 2b 0,F A 0 3 (1 Q , 3 g ,2 q ) (D.6) Poles A 1,l 3 (1 q , 3 g ,2q) = 2 b 0,F A 0 3 (1 q , 3 g ,2q). (D.7) D.3 Infrared limits of A 1,l 3 one-loop antennae We conclude this section on one-loop antennae by listing the infrared limits of the three different A 1,l 3 antenna functions employed in this paper. The infrared limits of of the one-loop initial-initial antenna A 1,l 3 (1 q , 3 g ,2q) are given by E The integrated massive initial-final antenna B q,Qq q In [6] we have derived the integrated tree-level four-parton massive antenna B q,Qq q expressing our results in terms of the following variables with E 2 cm = y/(1 − y − z)Q 2 . In the present paper, this integrated antenna is more easily employed and combined with other integrated subtraction terms at the virtual-virtual level if it is expressed in terms of In terms of those variables, the pole part of the integrated four-parton antenna function denoted by B q,Qq q reads where the functions denoted as G are two dimensional harmonic polylogarithms [64], and, as usual,