The qT\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_T$$\end{document} subtraction method for top-quark production at hadron colliders

We consider QCD radiative corrections to top-quark pair production at hadron colliders. We use the qT\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_T$$\end{document} subtraction formalism to perform a fully differential computation for this process. Our calculation is accurate up to the next-to-leading order in QCD perturbation theory and it includes all the flavour off-diagonal partonic channels at the next-to-next-to-leading order. We present a comparison of our numerical results with those obtained with the publicly available numerical programs MCFM and Top++.

sions in analytic form of the total partonic cross section 1 at NLO were obtained in Ref. [11]. Recently the calculation of the next-to-next-to-leading order (NNLO) QCD corrections to the tt total cross section was completed [12][13][14][15]. Besides the total cross section, differential cross sections and more general kinematical distributions are of great importance for precision studies. For instance, the tt (forward-backward and charge) asymmetry has received much attention in recent years (see, e.g., Ref. [16]). The tt asymmetry, which is nonvanishing starting from the NLO level [17,18], has recently been computed up to the NNLO level [19]. Other NNLO results on differential distributions are starting to appear [20][21][22][23].
This letter is devoted to the NNLO (and NLO) QCD calculation of tt production. In particular, we present the results of the first NNLO application of the q T subtraction formalism [24] to the process of tt production in hadron collisions.
At the partonic level, the NNLO calculation of tt production requires the evaluation of tree-level contributions with two additional unresolved partons in the final state, of oneloop contributions with one unresolved parton and of purely virtual contributions. The required tree-level and one-loop scattering amplitudes are known and they are the same amplitudes that enter the NLO calculation of tt + jet [25,26], the associated production of a tt pair and one jet. The purely virtual contributions depend on the two-loop scattering amplitudes and on the square of one-loop scattering amplitudes. The two-loop amplitude for tt production is partly known in analytic form [27][28][29][30] and its complete computation has been carried out numerically [31,32]. The square of one-loop scattering amplitudes is known [33][34][35].
The implementation of the various scattering amplitudes in a complete NNLO calculation at the fully differential (exclusive) level is a highly non-trivial task because of the presence of infrared (IR) divergences at intermediate stages of the calculation. In particular, these divergences do not permit a straightforward implementation of numerical techniques. Various methods have been proposed and used to overcome these difficulties at the NNLO level. The formalisms of antenna subtraction [36][37][38][39][40][41] and colourful subtraction [42,43] are more related to NNLO extensions of established NLO formulations [44][45][46][47][48] of the subtraction method. The Stripper formalism [49][50][51] is a combination of the subtraction method with numerical techniques based on sector decomposition [52,53]. Variants of the subtraction methods are the q T subtraction formalism [24] and the recently proposed N -jettiness subtraction [54][55][56].
The NNLO computations in Refs. [12][13][14][15]19] for tt production have been performed by using the Stripper method [49]. Parallely, an ongoing effort is being made by using the antenna subtraction method [39,57], which led to the NNLO fully differential computation of tt production in the qq channel [21,22,58] at leading colour and including the light-quark contributions.
The q T subtraction formalism [24] is a method to handle and cancel the IR divergences at the NLO and NNLO level. The method has been successfully applied to the fully differential computation of NNLO QCD corrections to several hard-scattering processes [24,[59][60][61][62][63][64][65][66][67]. The method uses IR subtraction counterterms that are constructed by considering and explicitly computing the transverse-momentum (q T ) distribution of the produced final-state system in the limit q T → 0. If the produced final-state 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 the NNLO level through the formalism of transverse-momentum resummation [68]. These results on transverse-momentum resummation are sufficient to fully specify the q T subtraction formalism for this entire class of processes. Therefore, up to now, the applications of the q T subtraction formalism have been limited to the production of colourless high-mass systems in hadron collisions. In this Letter we present first results on the application of the q T subtraction method to the NNLO computation of heavyquark production in hadron collisions. To this purpose, we use the recent progress on transverse-momentum resummation for heavy-quark production [69][70][71]. We exploit the formulation of transverse-momentum resummation in Ref. [71] that includes the complete dependence on the kinematics of the heavy-quark pair. This dependence and, in particular, the complete control on the heavy-quark azimuthal correlations are essential (see below) to extract all the NNLO counterterms of the q T subtraction method. Although the structure of transverse-momentum resummation for heavyquark production is fully worked out up to the NNLO level, the explicit NNLO results for the hard-virtual factors [71] in the flavour diagonal partonic channels qq → tt + X and gg → tt + X (X denotes the unobserved inclusive final state) are not yet known. Therefore, in the NNLO calculation of this paper we present numerical results for all the flavour off-diagonal channels ab → tt + X , with ab = qg(qg), qq(qq), qq (qq ), qq (qq ) (q and q denote quarks with different flavour).
The differential cross section dσ tt for the inclusive production process pp( pp) → tt + X is computable by convoluting the corresponding partonic cross sections dσ tt ab of the various partonic channels with the parton distribution functions (PDFs) of the colliding hadrons pp ( pp). According to the q T subtraction method [24], the (N)NLO partonic cross section dσ tt (N)NLO can be written as where dσ tt+jet (N)LO is the tt+jet cross section at (N)LO accuracy. Applying Eq. (1) at NLO, the leading-order (LO) cross section dσ tt+jet LO can be directly obtained by integrating the corresponding tree-level scattering amplitudes. Applying Eq. (1) at NNLO, dσ tt+jet NLO can be evaluated by using any available NLO method (e.g., Refs. [44][45][46][47][48]) to handle and cancel the corresponding IR divergences. Therefore, dσ tt +jet The square bracket term of Eq. (1) is IR finite in the limit q T → 0, but its individual contributions, dσ tt+jet (N)LO and dσ tt, CT (N)NLO , are separately divergent. The IR subtraction counterterm dσ tt, CT (N)NLO is obtained from the (N)NLO perturbative expansion (see, e.g., Refs. [72,73]) of the resummation formula of the logarithmically enhanced contributions to the q T distribution of the tt pair [69][70][71]: the explicit form of dσ tt, CT (N)NLO can be completely worked out up to NNLO accuracy. For example, at the NLO, the explicit expression of dσ tt, CT NLO in the partonic channel ab → tt + X is where α S is the QCD coupling, M is the invariant mass of the produced tt pair and dσ tt LO ab is the LO partonic cross section. The expression (2) involves convolutions (which are denoted by the symbol ⊗) with respect to the longitudinal-momentum fractions z 1 and z 2 of the colliding partons c andc in dσ tt LO cc . The integration variable q T in Eq. (2) corresponds, in the limit q T → 0, to the transverse momentum of the produced tt pair in the cross section dσ tt+jet LO on the right-hand side of Eq. (1). The function (1) cc←ab enters into the q T subtraction method [24] for hard-scattering production of a generic final-state system. Its explicit form is [72,73] (1) and it derives from the small-q T singular behavior of the q T cross section for the production of a colourless system in the partonic cc production channel. The coefficients A (1) c and B (1) c are the first-order resummation coefficients for transverse-momentum resummation ( A ab (z) are the lowest-order DGLAP kernels (the overall normalizazion is specified according to the notation in Eq. (41) of Ref. [72]). The functionsĨ k (q T /M) (k = 1, 2), which appear in Eq. (3), encapsulate the singular behavior at small q T , and they are explicitly given in Appendix B of Ref. [72]. The other function (1)

tt−new cc←ab
in the round-bracket factor of Eq. (2) is due to soft radiation and it is an additional term that is specific of the q T subtraction method for the case of heavy-quark pair production. This function reads where (1) t is the first-order term of the soft anomalous dimension for transverse-momentum resummation in heavyquark production and its explicit expression is given in Eq. (33) of Ref. [71]. This soft anomalous dimension is a colour space matrix that acts onto the colour indices of the four partons {c,c, t,t} in the Born level scattering amplitude |M cc→tt of the partonic process cc → tt. The colour space notation is specified in Ref. [71] and, in particular, |M cc→tt | 2 = M cc→tt |M cc→tt denotes the colour summed square amplitude that contributes to dσ tt L O cc , whereas the factor M cc→tt |( (1) t + (1) † t )|M cc→tt embodies colour correlation terms with a definite kinematical dependence.
The first-order hard-collinear (IR finite) counterterms H tt NLO are also completely known [69][70][71] for all the partonic channels. The second-order (IR finite) counterterms H tt NNLO are not yet fully known. However, H tt NNLO can be explicitly determined for all the flavour off-diagonal partonic channels. In these off-diagonal channels, H tt NNLO embodies process-dependent and process-independent contributions. The process-dependent contributions to H tt NNLO derive 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 [71] (see, in particular, Eq. (25) in Ref. [71] and accompanying comments). The process-independent contributions to H tt NNLO are analogous to those that contribute to Higgs boson [24] and vector boson [60] production, and they are explicitly known [68,[74][75][76][77][78].
Having discussed the content of Eq. (1), we are in a position to apply it to tt production and to obtain the complete NLO results plus the NNLO corrections in all the flavour off-diagonal partonic channels. Our NLO implementation of the calculation has the main purpose of illustrating the applicability of the q T subtraction method to heavy-quark production and, in particular, of cross-checking the q T subtraction methodology by numerical comparisons with NLO calculations performed by using more established NLO methods. Our NNLO results on tt production represent a first step (due to the missing flavour diagonal partonic channels) towards the complete NNLO calculation for this production process. Up to NLO accuracy our numerical implementation is based on the scattering amplitudes and phase space generation of the MCFM program [79], suitably modified for q T subtraction along the lines of the corresponding numerical programs for Higgs boson [24] and vector boson [60] production. At NNLO accuracy the tt+jet cross section is evaluated by using the Munich code [80], which provides a fully automated implementation of the NLO dipole subtraction formalism [46][47][48] as well as an interface to the one-loop generator OpenLoops [81] to obtain all the required (spinand colour-correlated) tree-level and one-loop amplitudes. For the evaluation of tensor integrals we rely on the Collier library [82], which is based on the Denner-Dittmaier reduction techniques [83,84] of tensor integrals and on the scalar integrals of Ref. [85]. In OpenLoops problematic phase space points are addressed with a rescue system that uses the quadruple-precision implementation of the OPP method in CutTools [86] with scalar integrals from OneLOop [87].
We start the presentation of our results by considering pp collisions at √ s = 8 TeV. We use the MSTW2008 [88] PDFs with the QCD running coupling α S evaluated at each corresponding order (i.e., we use (n + 1)-loop α S at N n LO, with n = 1, 2). The pole mass of the top quark is m t = 173.3 GeV. The renormalisation and factorisation scales, μ R and μ F , are fixed at μ R = μ F = m t .
In Figs. 1 and 2 we compare the NLO differential distributions obtained by using MCFM (which implements the dipole subtraction method [46][47][48]) with those obtained by using our numerical program. 2 In particular in Fig. 1 we consider the invariant mass (m tt ) distribution (left) and the rapidity (y tt ) distribution (right) of the tt pair. In Fig. 2 we consider the rapidity (y t ) distribution (left) and the transverse-momentum ( p T,t ) distribution (right) of the top quark. We clearly see that the distributions obtained with q T subtraction are in excellent agreement with those obtained with MCFM. We have checked that the agreement persists also for different choices of μ R and μ F .
We now move to consider the NNLO contributions and, in particular, we compute the total cross section for tt production. In Table 1 we report our results and we compare them with the corresponding results from the numerical program Top++ [10], which implements the NNLO calculation of Refs. [12][13][14][15]. Specifically, we report the complete NLO results and the O(α 4 S ) contributions to the NNLO cross section from the flavour off-diagonal partonic channels ab → tt + X . The contribution from all the channels with ab = qg,qg is labelled by the subscript qg, and the contribution from all the channels with ab = qq,qq, qq ,qq , qq ,qq is labelled by the subscript q(q)q . From Table 1 we see that the results obtained by using q T subtraction are in agreement with those from Top++. We note that the numerical uncertainties of our O(α 4 S ) results are at the 2 % level. This is due to the fact that in both the qg and the q(q)q channels at √ s = 8 TeV there is a strong quantitative cancellation between the contributions of the two terms in the right-hand side of Eq. (1) (the term that is proportional to H tt NNLO and the term in the square bracket). The numerical uncertainties of our O(α 4 S ) calculation can be reduced by considering different centre-of-mass energies. In particular, the numerical cancellation that we have mentioned is reduced by decreasing the centre-of-mass energy of the collision.
We have computed the total cross section for pp collisions at √ s = 2 TeV and we report the corresponding results in Table 2. We note that the numerical agreement between our calculation and the Top++ result is still satisfactory, and the numerical uncertainties of our O(α 4 S ) results are reduced below the 1 % level.
Using Top++ we can compute the complete NNLO result and we note that the flavour off-diagonal partonic channels contribute to about 10% of the total result at O(α 4 S ) for both collision energies considered in Tables 1 and 2. We have presented the first application of the q T subtraction method to top-quark pair production at hadron colliders. Our implementation is based on the formulation of transverse-momentum resummation for heavy-quark production of Ref. [71], which includes the complete depen-dence on the kinematics of the heavy-quark pair. Our computation is accurate at NLO in QCD perturbation theory and it includes all the off-diagonal partonic channels at NNLO accuracy. At NLO we have compared our results for various kinematical distributions with those obtained by using the MCFM program, and we find good agreement. At NNLO our results for the tt total cross section agree with the corresponding results obtained by using the Top++ program. The extension of our NNLO computation to include the missing qq → tt + X and gg → tt + X channels requires the evaluation of the second-order hard-collinear functions H tt NNLO [71], and an implementation of the two-loop virtual amplitudes, which, at present, are known only in numerical form [31,32].
The computation that we have performed in this paper can straightforwardly be extended to the production of massivequark pairs of different flavour (e.g. bottom-quark pair). The extension of the method to production processes with massless coloured particles in the final state (e.g. inclusive dijet production) is definitely non-trivial and it would require additional theoretical advancements.