Top-quark pair hadroproduction at next-to-next-to-leading order in QCD

We report on a new calculation of the next-to-next-to-leading order (NNLO) QCD radiative corrections to the inclusive production of top-quark pairs at hadron colliders. The calculation is performed by using the $q_T$ subtraction formalism to handle and cancel infrared singular contributions at intermediate stages of the computation. We present numerical results for the total cross section in $pp$ collisions at $\sqrt{s}=8$ TeV and $13$ TeV, and we compare them with those obtained by using the publicly available numerical program Top++. Our computation represents the first complete application of the $q_T$ subtraction formalism to the hadroproduction of a colourful high-mass system at NNLO.

The top quark (t) is the heaviest known elementary particle, and due to its large coupling to the Higgs boson it is expected to play a special role in electroweak symmetry breaking. Studies of top-quark production and decay are central in the LHC physics programme, allowing us to precisely test the Standard Model and, at the same time, offering a window on possible physics beyond the Standard Model. The LHC supplies a huge number of top-quark events, thereby offering an excellent environment for such studies.
Within the Standard Model the main source of top quarks in hadronic collisions is top-quark pair production. Studying the production of tt pairs at hadron colliders can not only shed light on the nature of the electroweak-symmetry breaking, but it also provides information on the backgrounds of many new-physics models.
Next-to-leading order (NLO) QCD corrections to the total cross section for this production process have been computed thirty years ago [1][2][3][4]. The calculation of the next-to-next-to-leading order (NNLO) QCD corrections to the tt total cross section was completed a few years ago [5][6][7][8]. Besides the total cross section, differential cross sections and more general kinematical distributions are of great importance for precision studies [9]. The tt asymmetry, which is non-vanishing starting from the NLO level, is known up to NNLO [10]. Other NNLO results on differential distributions are available [11][12][13].
At the partonic level, the NNLO calculation of tt production requires the evaluation of treelevel contributions with two additional partons in the final state, of one-loop contributions with one additional parton and of purely virtual contributions. The required tree-level and one-loop scattering amplitudes are known and they are the same as those entering the NLO calculation of the associated production of a tt pair and one jet [14,15]. The purely virtual contributions depend on the two-loop scattering amplitudes and on the square of one-loop scattering amplitudes. Partial results for the two-loop amplitude are available in analytic form [16][17][18][19], and the complete computation has been carried out numerically [20,21]. The square of the one-loop amplitudes is also known [22][23][24].
The implementation of the above contributions in a (fully differential) NNLO calculation is a highly non-trivial task because of the presence of infrared (IR) divergences at intermediate stages of the calculation. Various methods have been proposed and used to overcome these difficulties at the NNLO level (see e.g. Ref. [25] and references therein).
In this Letter we report on a new complete computation of tt production at NNLO based on the q T subtraction formalism [34]. The q T subtraction formalism is a method to handle and cancel the IR divergences in QCD computations at NLO and NNLO accuracy. The method uses IR subtraction counterterms that are constructed by considering the transverse-momentum (q T ) distribution of the produced high-mass system in the limit q T → 0. If the produced high-mass system is composed of non-QCD (colourless) partons (e.g. leptons, vector bosons or Higgs bosons), the behaviour of the q T distribution in the limit q T → 0 has a universal (process-independent) structure that is explicitly known up to NNLO through the formalism of transverse-momentum resummation [35]. These results on transverse-momentum resummation are sufficient to fully specify the q T subtraction formalism for this entire class of processes. In the case of heavy-quark production the transverse-momentum resummation formalism has been developed only recently [36][37][38]. Nonetheless, such information was already sufficient to apply the q T subtraction formalism to tt production and to obtain the complete NLO corrections and the NNLO contributions in all the flavour off-diagonal channels [39]. The NNLO computation in the flavour diagonal channels requires additional perturbative information (see below), and the ensuing results are presented here for the first time.
According to the q T subtraction method [34], the NNLO differential cross section dσ tt NNLO for the inclusive production process pp → tt + X can be written as where dσ tt+jet NLO is the tt+jet cross section at NLO accuracy. The square bracket term of Eq. (1) is IR finite in the limit q T → 0, but its individual contributions, dσ tt+jet NLO and dσ tt, CT NNLO , are separately divergent. The contribution dσ tt+jet NLO can be evaluated with any available NLO method to handle and cancel IR divergences. The IR subtraction counterterm dσ tt, CT NNLO is obtained from the NNLO perturbative expansion (see e.g. Refs. [39][40][41]) of the resummation formula of the logarithmicallyenhanced contributions to the q T distribution of the tt pair [36][37][38]: the explicit form of dσ tt, CT NNLO is fully known.
To complete the NNLO calculation, the second-order functions H tt NNLO in Eq. (1) are needed. These functions embody process-independent and process-dependent contributions. The processindependent contributions to H tt NNLO are analogous to those entering Higgs boson [34] and vector boson [42] production, and they are explicitly known [43][44][45][46]. In the flavour off-diagonal channels the process-dependent contributions to H tt NNLO originate from the knowledge of the one-loop virtual amplitudes of the partonic processes qq → tt and gg → tt, and from the explicit results on the NLO azimuthal correlation terms in the transverse-momentum resummation formalism [38]. The computation of H tt NNLO in the diagonal qq and gg channels additionally requires the two-loop amplitudes for qq → tt and gg → tt and the evaluation of new contributions of purely soft origin. The two-loop amplitudes are available in numerical form [21], whereas the computation of the additional soft contributions has been completed by some of us [47]. † Therefore we are now in a position to complete the calculation of Ref. [39] and to obtain the full NNLO cross section.
Before presenting our results we briefly describe our implementation. The NNLO cross section can be expressed as The NLO contribution σ NLO is evaluated by using the Munich code [49], which provides a fully automated implementation of the NLO dipole subtraction formalism [50][51][52]. We use Eq. (1) to compute the NNLO correction ∆σ NNLO . The NLO cross section dσ tt+jet NLO is computed by using Munich. The subtraction counterterm dσ tt, CT NNLO is also implemented in Munich, whereas the contribution proportional to H tt NNLO is evaluated with an extension of the numerical programs developed for Higgs boson [34] and vector-boson [42] production. All the required (spin-and colour-correlated) tree-level and one-loop amplitudes are obtained by using OpenLoops [53], † An independent computation of these soft contributions has recently been presented in Ref. [48].  Table 1: Total cross section for tt production in pp collisions. The quoted uncertainties are obtained through scale variations as described in the text. Numerical uncertainties on the last digit are stated in brackets (and include the r cut → 0 extrapolation uncertainties).
except for the four-parton tree-level colour correlations that we obtain through an analytic implementation. OpenLoops relies on the fast and stable tensor reduction of Collier [54,55], supported by a rescue system based on quad-precision CutTools [56] with OneLOop [57] to deal with exceptional phase-space points. To the purpose of validating our results for the realvirtual contribution, we have used also the new on-the-fly reduction of OpenLoops 2 [58,59] and the independent matrix-element generator Recola [60], finding complete agreement.
The contribution in the square bracket in Eq. (1) is formally finite in the limit q T → 0, but both dσ tt+jet NLO and dσ tt, CT NNLO are separately divergent. In practice, the computation is carried out by introducing a cut-off r cut on the dimensionless variable r = q T /M , where M is the invariant mass of the tt pair. The final result is obtained by performing the limit r cut → 0. To do so, the cross section is computed at fixed values of r cut in the interval [0.01%, r max ]. Quadratic least χ 2 fits are performed by varying r max from 0.5% to 1%. The result with the lowest χ 2 /degrees-of-freedom value is kept as the best fit. The extrapolation uncertainty is determined by comparing the result of the best fit with the results of the other fits. This procedure is the same as implemented in Matrix [61], and it has been shown to provide a conservative estimate of the systematic uncertainty in the q T subtraction procedure for various processes (see Sec. 7 in Ref. [61]).
To present our quantitative results, we consider pp collisions at √ s = 8 TeV and 13 TeV, and we use the NNPDF31 [62] NNLO parton distribution functions throughout. The QCD running of α S is evaluated at three-loop order with α S (m Z ) = 0.118, and the pole mass of the top quark is fixed to m t = 173.3 GeV. The central values of the renormalization (µ R ) and factorization (µ F ) scales are fixed to µ R = µ F = m t . We start the presentation by considering the complete NNLO cross sections. In Table 1 our results at √ s = 8 TeV and 13 TeV are compared with the corresponding results obtained with the numerical program Top++ [63] ‡ , which implements the NNLO calculation of Refs. [5][6][7][8] (at NLO Top++ uses the parametrization of Refs. [64,65] of the analytic result of Ref. [66]). In Table 1 the NNLO cross sections are reported with their perturbative uncertainty, which is estimated through the customary procedure of independently varying µ R and µ F by a factor of two around their central value with the constraint 0 The program Top++ gives results without an associated numerical error. Our results are given with an uncertainty that is obtained by combining statistical errors from the Monte Carlo integration and the systematic uncertainty associated to the r cut → 0 extrapolation. Such combined uncertainty turns out to be at the per mille level, and our results are consistent with those of Top++ for all the considered values of µ R and µ F . ‡ The program Top++ is used with the input parameter Precision=3.  Table 2: NNLO corrections ∆σ NNLO , split into the different production channels, for µ R = µ F = m t . Numerical integration errors on the last digits are stated in brackets.
In Table 2 the NNLO corrections ∆σ NNLO in the various partonic channels ab → tt + X computed with q T subtraction are compared to the corresponding results obtained with Top++. The contribution from all the channels with ab = qg,qg is labelled as qg, and the contribution from all the channels with ab = qq,qq, qq ,qq , qq ,qq is labelled as q(q)q .
We see that the numerical uncertainties of our NNLO corrections are at the percent level or smaller, except for the qg contribution at √ s = 13 TeV, for which there is a large cancellation between the two terms in Eq. (1) (the term that is proportional to H tt NNLO and the term in the square bracket). Similar effects were already observed in Ref. [39]. Comparing our 8 TeV results for ∆σ NNLO with those obtained by using Top++, we see that they are fully compatible within 1σ. At 13 TeV we also find agreement at the 1σ level apart from the qq channel that exhibits a 1.6σ difference, which corresponds to about 0.5% of ∆σ NNLO in this channel. Considering the partly statistical nature of our error estimate and the fact that the uncertainties from Top++ were completely neglected in this discussion, we can state that our results are in agreement with the Top++ results throughout.
The quality of the r cut → 0 extrapolation can be assessed by investigating the behaviour of the cross section at fixed values of r cut . In Fig. 1 we study this behaviour in the different partonic channels. We see that the r cut dependence is larger than what is observed in the case of the production of a colourless system (see Sec. 7 of Ref. [61]), where the powerlike dependence of the total cross section on r cut is known to be quadratic (modulo logarithmic enhancements). In the case of tt production, due to the additional contribution of soft radiation from the heavy quarks, the r cut dependence of the cross section is expected to be linear [67], thereby implying a stronger sensitivity to the parameter r cut . The exception is the q(q)q channel, where the expected r cut dependence should be quadratic, since this channel does not receive contributions from soft radiation at NNLO.
To conclude, we have reported on a new complete computation of the tt cross section in hadron collisions at NNLO in QCD perturbation theory. The computation is performed by combining treelevel and one-loop QCD amplitudes, as obtained from OpenLoops, with two-loop contributions available from the literature. The results are obtained by using the q T subtraction formalism to handle and cancel IR singularities. The contributions needed to apply q T subtraction to this process that were previously unknown have been computed by some of us, and they will be reported q(q)q ′ channel     in a separate publication. We have presented numerical results in pp collisions at 8 TeV and 13 TeV and compared them to the corresponding results obtained with the numerical program Top++. We find good agreement within the numerical uncertainties. Our computation represents the first complete application of the q T subtraction formalism to the hadroproduction of a colourful highmass system at NNLO. The computation can be naturally extended to differential distributions and by applying arbitrary IR safe cuts on the tt pair and the associated QCD radiation. More details on the calculation and additional results will be presented elsewhere.