Transverse-momentum resummation for heavy-quark hadroproduction

We consider the production of a pair of heavy quarks ($Q{\bar Q}$) in hadronic collisions. When the transverse momentum $q_T$ of the heavy-quark pair is much smaller than its invariant mass, the QCD perturbative expansion is affected by large logarithmic terms that must be resummed to all-orders. This behavior is well known from the simpler case of hadroproduction of colourless high-mass systems, such as vector or Higgs boson(s). In the case of $Q{\bar Q}$ production, the final-state heavy quarks carry colour charge and are responsible for additional soft radiation (through direct emission and interferences with initial-state radiation) that complicates the evaluation of the logarithmically-enhanced terms in the small-$q_T$ region. We present the all-order resummation structure of the logarithmic contributions, which includes colour flow evolution factors due to soft wide-angle radiation. Resummation is performed at the completely differential level with respect to the kinematical variables of the produced heavy quarks. Soft-parton radiation produces azimuthal correlations that are fully taken into account by the resummation formalism. These azimuthal correlations are entangled with those that are produced by initial-state collinear radiation. We present explicit analytical results up to next-to-leading order and next-to-next-to-leading logarithmic accuracy.


Introduction
We consider the inclusive production of a QQ pair of heavy quarks (Q) in hadron-hadron collisions. The bulk of the cross section is produced in the kinematical region where the transverse momentum q T of the QQ pair is smaller than the mass m of the heavy quark. In this paper we are interested in the small-q T region, namely, the region where q T ≪ m (including the limit q T → 0). From the phenomenological point of view, the most relevant process is the production of a pair of topantitop (tt) quarks [1], because of its topical importance in the context of both Standard Model (SM) and beyond-SM physics. In our theoretical study at the formal level, we consider a generic pair of heavy quarks.
The q T cross section of the QQ pair is computable in QCD perturbation theory [2], provided m is much larger than the QCD scale Λ QCD . The cross section is obtained by convoluting the parton densities of the colliding hadrons with the partonic cross sections, which are evaluated as power series expansion in the QCD coupling α S . In the small-q T region the perturbative expansion is badly behaved, since the size of the perturbative coefficients is enhanced by powers of ln q T . A reliable theoretical calculation requires the all-order resummation of these logarithmicallyenhanced terms. This type of perturbative behaviour is well known [3,4,5] from the simpler case of hadroproduction of a high-mass lepton pair through the Drell-Yan (DY) mechanism. In the case of the DY process the all-order resummation of the ln q T terms is fully understood [3,4,5,6]. At the level of leading-logarithmic (LL) contributions, the extension of resummation from the DY process to the heavy-quark process is relatively straightforward, and it was first discussed long ago in Ref. [7] (related studies were presented in Ref. [8]). Beyond the LL level, the structure of ln q T terms for the heavy-quark process is definitely different (the main physical differences are discussed below) from that of the DY process, and this difference implies very relevant theoretical complications. The all-order resummation for the heavy-quark process has been discussed only very recently by H. X. Zhu et al. in Refs. [9,10]. The analysis of Refs. [9,10] is limited to the study of the q T cross section after integration over the azimuthal angles of the produced heavy quarks. In this paper we illustrate the results of our independent study of transversemomentum resummation for QQ production. We present our all-order resummation formalism for QQ production, and we perform the resummation up to the next-to-next-to-leading logarithmic (NNLL) level, by explicitly including all the contributions up to the next-to-leading order (NLO) in the perturbative expansion. Our formalism and results are valid at the fully-differential level with respect to the kinematics of the produced heavy quarks. In particular, we consider the explicit dependence on the azimuthal angles of the heavy quarks and we have full control, at the resummed level, of the ensuing azimuthal correlations in the small-q T region. In the case of the azimuthally-averaged q T cross section we find agreement with the NNLL results of Refs. [9,10].
The DY lepton-pair production is a specific process of a general class of hard-scattering processes in which the produced high-mass system F in the final state is formed by a set of colourless (i.e., non-strongly interacting) particles (e.g., F can be a lepton pair, or a photon pair, or one or more vector bosons or Higgs bosons). Transverse-momentum resummation for the q T distribution of F is fully understood for this entire class of processes (i.e., independently of the specific particle content of the system F ). Indeed, transverse-momentum resummation for these processes has an all-order universal (process-independent) structure [11,6,12,13,14], which has been explicitly worked out [14,15,16,17] at NNLL accuracy and the next-to-next-to-leading order (NNLO) in the perturbative expansion. This universality structure eventually originates from the underlying physical mechanism that produces the q T broadening of the system F at small q T : the transverse momentum of F is produced by (soft and collinear) QCD radiation from the initial-state colliding partons. The QQ production process definitely belongs to a different class of processes, since the produced final-state heavy quarks carry colour charge and, therefore, they act as additional source of QCD radiation. The q T of the QQ pair depends on initial-state radiation, on final-state radiation and on quantum (and colour flow) interferences between radiation from the initial and final states. These physical differences between QQ production and the production of a colourless system F lead to very relevant technical and conceptual complications in the context of transverse-momentum resummation for QQ production. An important issue regards the presence of possible contributions from factorization-breaking effects of collinear radiation [18,19,20,21]. Other complications, which already arise in the context of threshold resummation for the QQ total cross section [22,23,24,25,26], regard the effect of non-abelian colour correlations produced by initial-state and final-state interferences. Additional important complications and effects, which are specific of transverse-momentum resummation, regard the azimuthal-angle distribution of the QQ pair. In the case of the DY process, q T resummation has no effect on the azimuthal correlation between the produced leptons, since the q T broadening of the lepton pair is entirely due to QCD radiation from the initial-state (qq) partons. In contrast, the q T of the QQ pair is also due to radiation from Q andQ separately, and this leads to q T -dependent azimuthal correlations. The main features of QQ production that we have just highlighted will be briefly recalled in the presentation of our resummation results.
The paper is organized as follows. In Sect. 2 we introduce our notation and we illustrate our all-order resummation formalism. In Sect. 3 we present and discuss the explicit form of the resummation coefficients up to NLO and NNLL accuracy. Our results are summarized in Sect. 4.

All-order resummation
We consider the inclusive hard-scattering process where the collisions of the two hadrons h 1 and h 2 with momenta P 1 and P 2 produces the QQ pair, and X denotes the accompanying final-state radiation. The hadron momenta P 1 and P 2 are treated in the massless approximation (P 2 1 = P 2 2 ≃ 0). The heavy quarks have momenta p 3 and p 4 , and the total four-momentum of the QQ pair is q µ = p µ 3 + p µ 4 . In a reference frame where the colliding hadrons are back-to-back, the total momentum q µ is fully specified by its invariant mass M (M 2 = q 2 ), rapidity y (y = 1 2 ln q·P 2 q·P 1 ) and transverse-momentum vector q T . Analogously, the momentum p µ j (j = 3, 4) of the heavy quark is specified by the heavy-quark mass m (p 2 3 = p 2 4 = m 2 ), rapidity y j and transverse-momentum vector p Tj . The two-dimensional transverse-momentum vectors q T , p T 3 and p T 4 have azimuthal angles φ q , φ 3 and φ 4 .
The kinematics of the observed heavy quarks is fully determined by the their total momentum q and by two additional and independent kinematical variables that specify the angular distribution of Q andQ with respect to the momentum q of the QQ pair. These two additional kinematical variables are generically denoted as Ω = {Ω A , Ω B } (correspondingly, we define dΩ = dΩ A dΩ B ). For instance, we can use Ω = {y 3 , φ 3 } or any other equivalent pairs of kinematical variables (e.g., y 3 → y 3 − y, φ 3 → φ 4 and so forth). We thus consider the most general fully-differential cross section dσ(P 1 , P 2 ; q T , M, y, Ω) d 2 q T dM 2 dy dΩ (2) for the inclusive-production process in Eq. (1). Note that the cross section in Eq. (2) and the corresponding q T resummation formula can be straightforwardly integrated with respect to one or more of the final-state variables {Ω A , Ω B , y, φ q , M}, thus leading to results for observables that are more inclusive than the differential cross section in Eq. (2).
The hadronic cross section in Eq. (2) is computable within QCD by convoluting partonic cross sections with the scale-dependent parton distributions f a/h (x, µ 2 ) (a = q f ,q f , g is the label of the massless partons) of the colliding hadrons. The partonic cross sections are expressed as a power series expansion in α S . At the leading order (LO) in the perturbative expansion, the partonic cross sections are proportional to α 2 S and there are only two contributing partonic processes, namely, the quark-antiquark (qq) annihilation process q fqf → QQ and the gluon fusion process g g → QQ. In both LO processes, the q T dependence of the partonic cross section (and of the ensuing hadronic cross section) is simply proportional to δ (2) (q T ), because of transverse-momentum conservation. At higher perturbative orders, the partonic cross sections receive contributions from elastic (cc → QQ) and inelastic (a b → QQ + X) partonic processes. The q T dependence of the partonic cross section includes contributions that are 'singular' in the limit q T → 0: these singular contributions are proportional to α n+2 S δ (2) (q T ) or to logarithmic terms of the type α n+2 with k ≤ 2n − 1 (more precisely, the logarithmic terms are expressed in terms of singular, though integrable over q T , 'plus'-distributions). We thus decompose the cross section in Eq. (2) as follows: where the component dσ (sing) embodies all the singular terms in the limit q T → 0, whereas dσ (reg) includes the remaining non-singular terms. In this paper we deal with the all-order evaluation and resummation of the small-q T singular terms in dσ (sing) . At fixed value of q T the cross section depends on the mass scales M and m. We use M to set the scale of the ln q T terms, and the remaining dependence on the two mass scales is controlled by the dimensionless ratio 2m/M or, equivalently, by the relative velocity v of Q andQ, In our resummation treatment at small q T , the mass scales M and m are considered to be parametrically of the same order. In two particular regions, namely, the threshold region where 2m/M → 1 (or v → 0) and the high-mass region where 2m/M → 0 (or v → 1), the size of the coefficients of the ln q T terms can be enhanced, and accurate quantitative predictions may require additional resummation of the dependence on 2m/M (or v). Note, however, that our treatment of the small-q T dependence is valid in the entire region q T ≪ M (and not only in the subregion q T ≪ m). In other words, in our treatment of the small-q T region, the decomposition in Eq. We illustrate the method that we have used to derive our resummation results for dσ (sing) . More details and additional results will be presented in forthcoming studies. We carry out our analysis of the singular terms in the small-q T region by working in impact parameter (b) space and, thus, we first perform the Fourier transformation of dσ (sing) /d 2 q T with respect to q T at fixed b. The final results for dσ (sing) /d 2 q T are then eventually recovered by performing the inverse Fourier transformation from b space to q T space (see Eq. (5)). In b space the singular terms are proportional to power of ln(Mb) (q T ≪ M corresponds to bM ≪ 1). These ln(Mb) terms are produced by the radiation of soft and collinear partons (i.e., partons with low transverse momentum k T , say, with k T ≪ M) in the inclusive final state X of the inelastic partonic processes a b → QQ + X. Soft and collinear radiation is treated by using the universal (process-independent) all-order factorization formulae [27,28,29,30,31,19,32] of QCD scattering amplitudes. Soft/collinear factorization at the amplitude (and squared amplitude) level is not spoiled by kinematical effects at the cross section level, since we are working in b space (in the small-q T limit, the kinematics of the q T cross section is exactly factorized [4] by the Fourier transformation to b space). Therefore, the ln(Mb) terms are explicitly computed by the phase space integration (in b space) of the soft/collinear factors. The application of the known explicit expressions [33,28,34,29,30,35] of soft/collinear factorization formulae allows us to compute the structure of dσ (sing) up to NNLO and NNLL accuracy. The method that we have just described is completely analogous (as applied in the NNLL+NLO computation of Ref. [36] and outlined to all orders in Ref. [14]) to the method that is applicable to transverse-momentum resummation for the production of a system F of colourless particles. The differences between the production of F and QQ production are due to the non-abelian colour charge of the produced heavy quarks. The complications that arise from these differences are basically related to soft radiation at wide angles with respect to the direction of the colliding partons. As a consequence, the structure of dσ (sing) for QQ production definitely differs (and the differences already appear at the NLO) from that of transverse-momentum resummation for the production of a colourless system F . Beyond the NNLL+NNLO level of perturbative accuracy, non-abelian soft wide-angle interactions of absorptive origin produce violation of strict factorization for space-like collinear radiation [19]. Therefore, the all-order formula of dσ (sing) that is presented below is based on some assumptions about possible contributions that can arise from factorization-breaking effects of collinear radiation [18,19,20,21]. In particular, we assume that infrared divergences produced by inclusive parton radiation at transverse momentum k T ≪ 1/b are either cancelled or customarily factorized in the parton distributions f a/h (x, 1/b 2 ) evolved up to the scale µ ∼ 1/b. Moreover, our resummed result for dσ (sing) includes only the possible soft/collinear correlation structures that we have explicitly uncovered up to NNLL+NNLO. These issues certainly deserve further and future investigations. We remark that we have full control of the all-order structure of dσ (sing) up to NNLL+NNLO accuracy. Possible additional structures are likely to be absent till very high perturbative orders [18,19,20,37].
In the following we use parton densities f a/h (x, µ 2 ) as defined in the MS factorization scheme. The running coupling α S (µ 2 ) denotes the renormalized QCD coupling in the MS renormalization scheme with decoupling of the heavy quark Q [38] (e.g., in the case of tt production, α S (µ 2 ) is the MS coupling in the 5-flavour scheme), and m is the renormalized pole mass of the heavy quark Q. Obviously our explicit results can be straightforwardly expressed in different factorization/renormalization schemes by applying the corresponding scheme transformation relations (e.g., the pole mass m can be replaced by the MS running mass m(µ 2 )). To present the resummation results for QQ production we closely follow the formulation of transverse-momentum resummation for the production of a colourless system F , and we use the same notation as in Refs. [13,14] (more details about the notation can be found therein). This presentation allows us to clearly identify and highlight the structural differences that arise in the context of QQ production.
Our results for the singular component dσ (sing) of the QQ production cross section are given by the following all-order resummation formula: . is the Euler number) is a numerical coefficient, and the kinematical variables x 1 and x 2 are The right-hand side of Eq. (5) involves the (inverse) Fourier transformation with respect to the impact parameter b and two convolutions over the longitudinal-momentum fractions z 1 and z 2 .
The parton densities f a i /h i (x, µ 2 ) of the colliding hadrons are evaluated at the scale µ = b 0 /b, which depends on the impact parameter. The factor that is denoted by the symbol dσ (0) cc refers to the partonic elastic-production process cc → QQ of the QQ pair, with where P i (i = 1, 2) are the momenta of the colliding hadrons (see Eq. (1)) and x i (i = 1, 2) are the momentum fractions in Eq. (6). Making the symbolic notation explicit, the symbol dσ (0) cc is related the LO cross section dσ (0) for QQ production by the partonic process in Eq. (7), and we have QCD radiative correction are embodied in the factors S c and [(H ∆) C 1 C 2 ] on the right-hand side of Eq. (5). The expression in Eq. (5) involves the sum of two types of contributions, which correspond to the LO partonic channels: the contribution of the qq annihilation channel (c = q,q) and the contribution of the gluon fusion channel (c = g). In each of these channels, the structure of Eq. (5) is apparently similar to the structure of transverse-momentum resummation for the production of a colourless system F [11,6,12,13,14] (see Eq. (6) of Ref. [14] for direct comparison). The important differences that occur in the case of QQ production are hidden in the symbolic notation of the factor [(H ∆) C 1 C 2 ] and, more specifically, they are due to the factor ∆ that is related to the accompanying soft-parton radiation in QQ production. In the case of production of a colourless system F , the factor ∆ is absent (i.e. ∆ = 1).
The expression of the symbolic factor [(H ∆) C 1 C 2 ] for the qq annihilation channel is whereas for the gluon fusion channel (c = g) we have The functions C ca and C µν ga are described below. The factors (H ∆) in Eqs. (10) and (11) depend on b, M and on the kinematical variables of the partonic process in Eq. (7) (this dependence is not explicitly denoted in Eqs. (10) and (11)). Equation (11) includes the sum over the repeated indices {µ i , ν i }, which refer to the Lorentz indices of the colliding gluons g(p i ) (i = 1, 2) in Eq. (7). In Eqs. (10) and (11) we use the shorthand notation (H ∆) for the contribution of the factors H and ∆, since these factors embody a non-trivial dependence on the colour structure (and colour indices) of the partonic process in Eq. (7). To take into account the colour dependence, we use the colour space formalism of Ref. [39]: the colour-index dependence of the scattering amplitude M of the process in Eq. (7) is represented by a vector | M in colour space, and colour matrices are represented by colour operators acting onto | M . Using the colour space formalism, we can write the explicit representation of (H ∆). In the case of the qq annihilation channel, we have where the 'hard-virtual' amplitude M cc→QQ is directly related to the infrared-finite part of the all-order (virtual) scattering amplitude M cc→QQ of the partonic process in Eq. (7), and M cc→QQ | 2 is the squared amplitude summed over the colours and spins of the partons c,c, Q,Q). The relation between M and M is given in Eq. (26). The analogue of Eq. (12) in the gluon fusion channel is where {µ ′ i , ν ′ i } (i = 1, 2) are exactly (see Eq. (26)) the gluon Lorentz indices of the scattering amplitude M gg→QQ (p 1 , p 2 ; p 3 , p 4 ), and d µν = d µν (p 1 , p 2 ) is the following polarization tensor, which projects onto the Lorentz indices in the transverse plane. The soft-parton factor ∆ depends on colour matrices, and it acts as a colour space operator in Eqs. (12) and (13). We can also introduce a colour space operator H through the definition α 2 S |M (0) | 2 H = | M M|. Therefore, according to Eqs. (12) and (13), the shorthand notation (H ∆) is equivalent to (H ∆) = Tr [H ∆], where 'Tr' exactly denotes the colour space trace of the colour operator H ∆.
We now illustrate the structural form of the resummation formulae in Eqs. (5), (10)- (13), and the differences between QQ production and the production of a colourless system F . The hard factor H is independent of the impact parameter b, and it depends on the scattering amplitude M cc→QQ . An analogous process-dependent hard factor (which depends on the scattering amplitude of the process cc → F ) [14] appears for the production of a colourless system F . The functions C c a [12] and C µ ν g a [13] in Eqs. (10) and (11) are universal (they are process independent and only depend on the parton indices), and they are computable as power series expansions in α These functions originate from initial-state collinear radiation of partons with typical transverse momentum k T ∼ 1/b. The function S c (M, b) in Eq. (5) is the Sudakov form factor [11], and it is also universal. Thus, for instance, the qq annihilation channel functions S q and C q a also contribute to transverse-momentum resummation for the DY process [6], whereas the gluon fusion channel functions S g and C µ ν g a also contribute in the case of Higgs boson production [13]. The Sudakov form factor S c (M, b) resums logarithmic terms α n S ln k (Mb), starting from the LL contributions (those with k = 2n) to the q T cross section. The Sudakov form factor is due to QCD radiation from the initial-state partons c andc in the process of Eq. (7) and, more precisely, it is produced by soft and flavour-conserving collinear radiation with typical transverse momentum k T in the range 1/b ∼ < k T ∼ < M. The factor ∆ in Eqs. (5), (10)-(13) is specific of QQ production (∆ = 1 for the production of a colourless system F ), and it is due to QCD radiation of soft noncollinear (at wide angles with respect to the direction of the initial-state partons) partons from the underlying subprocess cc → QQ. Therefore, ∆ embodies the effect of soft radiation from the QQ final state and from initial-state and final-state interferences. As in the case of the Sudakov form factor, the soft radiation contribution to ∆ involves the transverse-momentum range 1/b ∼ < k T ∼ < M. Therefore, ∆ resums additional logarithmic terms α n S ln k (Mb) (see Eq. (15)), although the dominant contributions to ∆ are of next-to-leading-logarithmic (NLL) type, since they are produced by non-collinear radiation. Moreover, soft-parton radiation at the scale k T ∼ 1/b has a 'special' physical role, since it is eventually responsible for azimuthal correlations (see Eqs. (15) and (18)).
The soft-parton factor ∆ depends on the impact parameter b, on M and on the kinematics of the partonic process in Eq. (7). To explicitly denote the kinematical dependence (which is in turn related to the two angular variables Ω of the q T cross section), we use the rapidity difference y 34 = y 3 − y 4 between Q(p 3 ) andQ(p 4 ) and the azimuthal angle φ 3 of the quark Q(p 3 ) (the dependence on 2m/M is not explicitly denoted in the following). The all-order structure of ∆ is where The colour operator (matrix) Γ t is the soft anomalous dimension matrix that is specific of transverse-momentum resummation for QQ production. This quantity is computable order-byorder in α S as in Eq. (17). The evolution factor V in Eq. (16) is obtained by the exponentiation of the integral of the soft anomalous dimension. The integral is performed over the transversemomentum scale q 2 of the the QCD running coupling, and the symbol P q in Eq. (16) denotes the anti path-ordering of the exponential matrix with respect to the integration variable q 2 . The evolution operator V explicitly resums logarithmic terms α n S (M 2 ) ln k (Mb) (with k ≤ n) through the integration over q 2 . Soft-parton radiation from the process cc → QQ produces non-abelian colour correlations that are embodied in the soft anomalous dimension matrix. The structure of V is typical of the resummation of soft-gluon logarithmic contributions in QCD multiparton hardscattering processes [40,41]. Operators that are analogous to V arise in the context of threshold resummation for the QQ total cross section [22,23,24,25,26]. The colour operator D in Eq. (15) is computable as a powers series expansion in α S (b 2 0 /b 2 ) (see Eq. (18)). This operator does not explicitly depend on the hard scale M 2 , and its dependence on the scale b 2 is due to the running coupling α S (b 2 0 /b 2 ). Therefore, the operator D effectively resums ln(Mb) contributions to ∆ by using the renormalization group evolution of α S (µ 2 ) to express α S (b 2 0 /b 2 ) in terms of α S (M 2 ) and ln (M 2 b 2 ).
An important point about the structure of the soft factor ∆ in Eq. (15) regards its dependence on the rapidity and azimuth kinematical variables of the QQ pair. Both Γ t and D depend on y 34 and this produces an ensuing dependence of the operators V and ∆. The azimuthal dependence is specific of transverse-momentum resummation. In particular, we remark that Γ t and, thus, the evolution operator V do not depend on azimuthal angles. In contrast, the operator D does depend on φ 3 and, more importantly, it depends on φ 3b = φ 3 − φ b , where φ b is the azimuth of the two-dimensional impact parameter vector b. Inserting this dependence on φ 3b in the resummation formula (5) and performing the inverse Fourier transformation from b space to q T space, we obtain an ensuing dependence of the q T cross section on φ 3 − φ q (where φ q is the azimuthal angle of q T ). In other words, the resummation formula (5) leads to q T -dependent azimuthal correlations of the produced QQ pair in the small-q T region. These azimuthal correlations are produced by the dynamics of soft-parton radiation, and they are entirely embodied in the soft-parton factor D of Eq. (15). The φ 3b dependence occurs in D, at the characteristic scale 1/b, and does not occur in the evolution operator V: this fact has a definite physical origin in the distinction between real and virtual radiative contributions. Virtual radiation involves soft partons with transverse momentum k T in the entire range k T ∼ < M, while real radiation is due to partons with k T ∼ < q T ∼ 1/b. The dynamics of V is essentially driven by soft virtual partons, which cannot produce azimuthal correlations. Real radiation plays a 'minimal' role in V: it simply produces the cancellation of virtual terms (and the ensuing infrared divergences) in the region k T ∼ < 1/b, thus leading to remaining contributions from the region 1/b ∼ < k T ∼ < M (see the limit of integrations over q ∼ k T in Eq. (16)). Azimuthal correlations are instead necessarily produced by real radiation, which first occur at scale k T ∼ q T ∼ 1/b: these correlations are thus 'trapped' in the soft factor D(α S (b 2 0 /b 2 )), at the corresponding scale 1/b.
As first pointed out in Ref. [12], the structure of transverse-momentum resummation is invariant under a class of renormalization group transformations, named resummation-scheme transformations. This symmetry permits a redefinition of the individual resummation factors in such a way that their total contribution to the q T cross section is left unchanged. In particular, we can consider a resummation-scheme transformation that changes (redefines) the separate factors H, V and D in such a way that (H ∆) (i.e., Eqs. (12) and (13)) is invariant. Such a transformation can introduce an arbitrary φ 3b dependence of the redefined factors H, V, D. Our key point about the structure of the azimuthal correlations in Eq. (15) is that there are necessarily schemes in which the dependence on φ b is absent from H and V, and it is entirely embodied in D. This key point eventually follows from our previous discussion on the physical origin of the soft-parton azimuthal correlations. In particular, we can define the factor D in Eq. (15) in such a way that it gives a trivial contribution after azimuthal average over b. Thus, the soft factor D can fulfils the property where the symbol . . . av. denotes the azimuthal average over the angle φ b of the impact parameter vector b.
We note that the transverse-momentum resummation formula (5) has an additional source of azimuthal correlations. These additional azimuthal correlations are due to the b dependence of the function C µ ν g a that contributes to Eq. (11). The two sources of azimuthal correlations have a definitely different physical origin. The azimuthal correlations produced by C µ ν g a originate from initial-state collinear radiation [13], while those produced by D originate from soft radiation in the processes, such as QQ production, with final-state coloured partons. This difference is manifest in the qq annihilation channel, where we find soft-parton azimuthal correlations (produced by D) without accompanying azimuthal correlations of collinear origin (see Eq. (10)).
The gluon collinear function C µ ν g a of Eq. (11) has the following all-order form [13]: where d µν is given in Eq. (14), and b µ = (0, b, 0) is the two-dimensional impact parameter vector in the four-dimensional notation , analogously to the collinear functions C q a and Cq a in Eq. (10), whereas the expansion of the gluonic function G g a starts at O(α S ). From Eq. (20) we see that the dependence of C µν g a on the azimuthal angle φ b of b is entirely embodied in the Lorentz tensor D µν of Eq. (21): therefore, this azimuthal dependence is uniquely specified at arbitrary perturbative orders in α S . This specific azimuthal dependence is a consequence [13] of the fact that gluonic collinear radiation is intrinsically spin-polarized and its spin-polarization structure is uniquely specified (see, e.g., Eq. (50) in Ref. [13]) by helicity conservation rules. The contribution of the gluon fusion channel is the sole source of azimuthal correlations [42,13] in transverse-momentum resummation for the production of a colorless system F . The azimuthal dependence of C µν g a produces a definite structure of azimuthal correlations with respect to the azimuthal angle φ q of the transverse momentum q T . As shown in Ref. [13], the small-q T resummed cross section for the production of a colourless system F through gluon fusion leads to azimuthal correlations that are expressed in terms of a linear combination of only four Fourier harmonics (cos(2φ q ), sin(2φ q ), cos(4φ q ), sin(4φ q )).
In the case of q T resummation for QQ production, the azimuthal dependence is present in both the qq annihilation channel and the gluon fusion channel. In both channels, the φ b dependence of the resummation formula (5) is embodied in the resummation factors at scale b 2 0 /b 2 , which are (see Eqs. (10), (11) and (15)) where we have omitted the argument of the various factors to shorten the notation. As we have just recalled, the azimuthal dependence of the collinear function C µν g a is relatively simple and it is uniquely specified to all perturbative orders. In contrast, the φ b dependence of D is determined by the process-dependent dynamics of soft-parton radiation in QQ production: this dependence is definitely cumbersome already at the first perturbative order (see Eq. (36)), and it receives additional contributions to each subsequent order. Therefore, the ensuing azimuthal correlations of the q T cross section depend on Fourier harmonics of any degrees. In particular, in the gluon fusion channel (see Eq. (23)), the azimuthal dependence originating from soft-parton radiation is entangled with the azimuthal dependence of collinear origin: the complete azimuthal dependence is determined by a non-trivial interplay of colour (soft) and spin (collinear) correlations.
The resummation formula (5) can be straightforwardly averaged over the azimuth φ q of q T . The resummation formula for the azimuthally-averaged q T cross section is obtained from Eq. (5) through two simple replacements: the integrand factor e ib·q T is replaced by the 0-th order Bessel function J 0 (bq T ) and the factors in Eq. (22) and (23) are replaced by their azimuthal average over φ b . Performing the azimuthal average over φ b , we have Owing to the property in Eq. (19), the effect of the soft-parton factor D disappears from the right-hand side of Eq. (24): therefore, in the qq annihilation channel, soft wide-angle radiation contributes to the azimuthally-averaged q T cross section only through the evolution factor V † V from Eq. (15). Despite the property in Eq. (19), however, in the gluon fusion channel we have the inequality in Eq. (25) (owing to Eq. (19) and the fact that G g a = O(α S ), the inequality is due to contributions at O(α 2 S )). Therefore, the soft factor D still gives a non-trivial effect to the azimuthally-averaged q T cross section through the contribution of the gluon fusion channel. This effect is proportional to the factor D C µ 1 ν 1 g a 1 C µ 2 ν 2 g a 2 av. , which originates from the entangled soft/collinear azimuthal dependence of the q T resummation formula (5).
The contribution of the hard factor H to Eqs. (12) and (13) is independent of b, it depends on the hard scale M and it is entirely specified by the hard-virtual amplitude M cc→QQ . The auxiliary amplitude M cc→QQ is related to the scattering amplitude M cc→QQ by the following all-order factorization formula: where and µ R is the renormalization scale. The function I cc→QQ (α S , ǫ) also depends on the momenta p i (i ≤ 4), although this dependence is not explicitly denoted in its argument. The structure of Eq. (26) is analogous [14] to that of the hard-virtual amplitudes of transverse-momentum resummation for the production of colourless systems F . The main technical difference regards the colour treatment and, thus, the 'subtraction' operator I cc→QQ is a colour operator acting onto the colour vector |M cc→QQ .
The all-order (virtual) amplitude of the process cc → QQ has ultraviolet (UV) and infrared (IR) divergences. We consider their regularization by analytic continuation in d = 4 − 2ǫ spacetime dimensions, and we use the customary scheme of conventional dimensional regularization (CDR). The quantity M cc→QQ (p 1 , p 2 ; p 3 , p 4 ) ≡ M cc→QQ ({p i }) in the right-hand side of Eq. (26) is the renormalized on-shell scattering amplitude [43,44], and it has the perturbative expansion The perturbative expansion of M cc→QQ is completely analogous to that in Eq. (28), with M The renormalized virtual amplitude M cc→QQ still has IR divergences in the form of 1/ǫ poles. The subtraction operator I cc→QQ (α S , ǫ), which originates from real emission contributions to the q T cross section, also contains IR divergences. More precisely, it exactly includes the IR divergent terms that are necessary to cancel the IR divergences of the amplitude M cc→QQ , and it includes additional IR finite terms that are specific of the q T cross section in Eq. (5). Therefore, the hard-virtual amplitude M cc→QQ can be safely computed in the limit ǫ → 0. The expressions of (H ∆) in Eqs. (12) and (13) have to be evaluated by setting ǫ = 0 in M cc→QQ , although the fourdimensional limit ǫ → 0 is not explicitly denoted in the right-hand side of those equations. We note the the all-order factors M, I and, hence, M are renormalization-group invariant quantities (i.e., they are independent of µ R ). Their dependence on µ R only appears throughout the fixedorder truncation of the perturbative series in powers of α S (µ 2 R ) (see Eqs. (27), (28) and (29)). We also remark that the operator I cc→QQ is completely independent of the spin of the four external hard partons of the process cc → QQ. In particular, the gluon Lorentz indices {µ ′ i , ν ′ i } (i = 1, 2) of M gg→QQ in Eq. (13) are exactly those of the corresponding amplitude M gg→QQ in the right-hand side of Eq. (26).
In the region of very small values of q T , q T ∼ < Λ ( Λ is the QCD scale) or, equivalently, at very large values of b ( bΛ ∼ > 1), the perturbative computation of the q T cross section has to be supplemented with non-perturbative corrections. Non-perturbative contributions are embodied in transverse-momentum dependent (TMD) parton densities [11,45,46] that can be used to express the q T cross section in the small-q T region through TMD factorization (see Ref. [47] and references therein). In the context of TMD factorization, roughly speaking, the factor here C denotes the collinear functions in Eqs. (10) and (11), and the symbol '⊗' denotes the convolution with respect to the momentum fraction z) of the resummation formula (5) arises from the TMD parton density [15,48] in the region bΛ ∼ < 1. In the case of production of a colourless system F , the resummation formula (5) has no other b dependent factors. In the case of QQ production, the presence in Eq. (5) of one additional b dependent factor, the soft-parton factor ∆, is consistent with a breakdown (in weak form) [49] of TMD factorization. In the production processes of strongly interacting systems (such as QQ pairs), TMD parton densities have to be supplemented with additional and process dependent non-perturbative factors [50]. As we have previously discussed, the breakdown (in strong form) [18] of TMD factorization can have connection with high-order structures in transverse-momentum resummation.

Explicit results for the resummation coefficients
In this Section we present our explicit analytic results for the resummed cross section in Eq. (5) up to NLO and NNLL accuracy. To this purpose we can exploit the knowledge of the universal (process-independent) factors S c , C c a and C µν g a up to NNLL+NNLO. The Sudakov form factor S c (M, b) has an all-order representation [11] (see, e.g., Eq. (8) in Ref. [14]) that is fully specified by two perturbative functions A c (α S ) and B c (α S ). The corresponding perturbative coefficients A c [45,46], B (2) c [51,36] and A (3) c [15] are explicitly known, and they determine S c (M, b) up to NNLL accuracy. The partonic collinear functions C c a (c = q,q) and C µν g a in Eqs. (10) and (11) are known [52,53,16,17] up to O(α 2 S ) (i.e., NNLO). The two computations in Refs. [16] and [17] are fully independent and they lead to results in full agreement. As we have already recalled, the determination of the individual (separate) factors of the resummation formula (5) requires the specification of a resummation scheme [12]. The collinear functions of Ref. [17], which refer to transverse-momentum resummation according to the formulation of Ref. [15], are eventually related to our functions C c a and C µν g a [16] throughout a transformation of resummation scheme. In the following, to present our results, we explicitly consider the 'hard scheme' used in Ref. [14]. The expressions of the universal factors S c , C c a and C µν g a in the hard scheme can be found in Ref. [14]. The remaining perturbative ingredients of the QQ resummation formula (5) are the hard factor H (i.e, the subtraction operator I cc→QQ ), the soft evolution factor V (i.e., the soft anomalous dimension Γ t ) and the soft azimuthal-correlation factor D. We have computed I cc→QQ , Γ t and D at O(α S ), and we have determined Γ t at O(α 2 S ) by relating it to the O(α 2 S ) computation [54,55,56] of the IR anomalous dimension of the scattering amplitude M gg→QQ : these results complete the evaluation of the QQ resummation formula (5) up to NNLL+NLO. Using the hard scheme, the results of our computation are presented below (see Eqs. (30), (33), (36) and (40)).
The colour operators I cc→QQ , Γ t and D depend on the colour charges (T i ) a (a = 1, . . . , N 2 c − 1 is the colour index of the radiated gluon) of the four (i ≤ 4) radiating partons c,c, Q,Q. Using the colour space formalism of Ref. [39], the colour charge (T i ) a is a colour matrix in either the fundamental (if i is a quark) or adjoint (if i is a gluon) representation of SU(N c ) in QCD with N c colours. Note that the colour flow of the process cc → QQ is treated as 'outgoing', so that T 3 and T 4 are the colour charges of Q(p 3 ) andQ(p 4 ), while T 1 and T 2 are the colour charges of the anti-partonsc(−p 1 ) and c(−p 2 ) in Eq. (7). According to this notation, colour conservation implies . is a colour-singlet state vector, such as |M cc→QQ or | M cc→QQ . We also define T i · T j ≡ (T i ) a (T j ) a and, in particular, T 2 i is a c-number term (more precisely, T 2 i is a multiple of the unit matrix in colour space) given by the Casimir factor (C F or C A ) of the corresponding representation of SU(N c ). We have T 2 1 = T 2 2 = C F = (N 2 c − 1)/(2N c ) in the qq annihilation channel, T 2 1 = T 2 2 = C A = N c in the gluon fusion channel, whereas T 2 3 = T 2 4 = C F . Considering the kinematics of the process cc → QQ in Eq. (7), four-momentum conservation leads to the relations y 3 − y = y − y 4 = y 34 /2, p 2 T3 = p 2 T4 ≡ p 2 T and the heavy-quark transverse mass m T = m 2 + p 2 T is related to y 34 by using M = 2m T cosh(y 34 /2). Using these kinematical relations the operators I cc→QQ , Γ t and D can eventually be expressed in term of the two independent variables y 34 and 2m/M (or, equivalently, the relative velocity v in Eq. (4)). As already discussed, D additionally depends on the relative azimuthal angle φ 3b (or, equivalently, φ 4b ).
The first-order term I (1) of the subtraction operator I cc→QQ in Eqs. (26) and (27) has the following form: The flavour dependent coefficients γ c (c = q,q, g) originate from collinear radiation: the explicit values of these coefficients are γ q = γq = 3C F /2 and γ g = (11C A −2N f )/6, and N f is the number of flavours of massless quarks (e.g., N f = 5 in the case of tt production). The IR finite contribution F where the function L 34 is and Li 2 is the customary dilogarithm function, The colour operator Γ (1) t (y 34 ) in the right-hand side of Eq. (30) is exactly equal to the first-order term of the soft anomalous dimension in Eq. (17), and its explicit form is We note that the second term in the right-hand side of Eq. (33) can be rewritten as i=1,2 j=3,4 where we have simply used colour conservation and kinematical relations.
The expression of I cc→QQ in Eq. (30) contains IR divergent terms in the form of double and single poles 1/ǫ 2 and 1/ǫ. We have explicitly checked that these IR divergent terms are exactly those that control the factorized IR structure [57] of general one-loop scattering amplitudes with massive external partons. This directly proves that the one-loop hard-virtual amplitude M (1) cc→QQ in Eq. (29) is IR finite in the limit ǫ → 0.
The soft-parton operator D in Eq. (18) also depends on the relative azimuthal angle φ 3b (or, equivalently, φ 4b ). The expression of the first-order term D (1) is quite involved. To shorten the notation we define the auxiliary variable c 3b , divergences due to soft wide-angle radiation in the process cc → QQ. This origin of Γ t remains valid at higher perturbative orders, and it leads to the contribution Γ sub. cc→QQ in Eq. (40). The subtracted anomalous dimension Γ sub. cc→QQ is given by the following relation: where the terms on the right-hand side are written by exactly using the notation of Eq. (5) of Ref. [56]. The term Γ(µ) is the anomalous-dimension matrix that controls the IR divergences of the scattering amplitude M cc→QQ , while the square-bracket term on the right-hand side of Eq. (41) is the corresponding expression of Γ(µ) for a generic process cc → F (where the system F is colorless). The square-bracket term is the contribution of soft and collinear radiation from the colliding partons c andc. In Eq. (41), this contribution is subtracted from Γ(µ), so that Γ sub.
cc→QQ embodies the remaining IR effects due to soft wide-angle radiation in the process cc → QQ.
We note that the subtraction in Eq. (41) exactly corresponds to the splitting procedure used in Eq. (57) of Ref. [10] to introduce the anomalous dimension γ h iī : therefore, we have Γ sub. cc→QQ = γ h cc . The expression of Γ(µ) at O(α 2 S ) is computed and explicitly given in Ref. [56]. This expression (which is too long to be reported here) straightforwardly leads to the O(α 2 S ) term of Γ sub. Transverse-momentum resummation for QQ production has been studied in Refs. [9,10]. The framework developed in Refs. [9,10] is an extension of the SCET formulation of q T resummation that was presented in Ref. [15] for the cases of DY and Higgs boson production. The authors of Refs. [9,10] consider the azimuthally-averaged q T cross section and present results at the NLO and NNLL accuracy. We have performed a comparison between those results and our results, and we find full agreement. The comparison poses no difficulties since, as we have discussed, we can straightforwardly obtain the azimuthally-averaged q T cross section by integrating the resummation formula (5). In particular, at NLO and NNLL accuracy, we can simply set ∆ = V † V (i.e., D = 1) in Eq. (5) (this follows from Eq. (24) and from the fact the inequality in Eq. (25) is due to terms of O(α 2 S ), which start to contribute at the NNLO and beyond NNLL accuracy). We note that the various (hard, soft, collinear) resummation factors in our Eq. (5) and those in Ref. [10] are separately different, since they correspond to the use of different resummation schemes.

Summary
In this paper we have considered the transverse-momentum distribution of a heavy-quark pair produced in hadronic collisions. As in the case of simpler processes, such as the hadroproduction of a system of non-strongly interacting particles, the perturbative QCD computation of the q T cross section is affected by large logarithmic terms that need be resummed to all perturbative orders. We have discussed the new issues that arise in the case of heavy-quark production, and we have presented our all-order resummation formula (see Eq. (5)) for the logarithmically-enhanced contributions. The main differences with respect to the production of colourless systems is the appearance of the soft factor ∆ (see Eq. (15)) that is due to soft-parton radiation at large angles with respect to the direction of the colliding hadrons (partons). The factor ∆ embodies the effect of soft radiation from the heavy-quark final state and from initial-state and final-state interferences. The dynamics of soft-parton radiation produces colour-dependent azimuthal correlations in the small-q T region. This azimuthal dependence is fully taken into account by the resummation formula and it is embodied in the soft-parton factor ∆: the dependence is controlled by the colour operator D and it is factorized with respect to the colour (soft) evolution factor V (see Eq. (15)). We have shown how the azimuthal correlations of soft-parton origin are entangled with the azimuthal dependence due to gluonic collinear radiation (see Eq. (23)), and we have discussed the ensuing effect on the azimuthally-averaged q T cross section. We have presented the explicit results of the perturbative coefficients of the resummation formula up to NLO and NNLL accuracy (see Eqs. (30), (33), (36) and (40)).
Transverse-momentum resummation for heavy-quark production is important for phenomenological applications through resummed calculations [10], especially for the production of top-quark pairs. Given the huge amount of top-quark pairs that have been produced at the LHC in its first run, and the even higher number of tt events that are expected at √ s = 13 (14) TeV, the possibility of relying on accurate computations of the transverse-momentum spectrum of the tt pair down to the low-q T region is very relevant for physics studies within and beyond the SM.
We point out that the q T resummation formalism for QQ production has implications not only for resummed calculations but also for fixed-order computations up to NNLO. The q T subtraction formalism [52] is an efficient method to perform fully-exclusive NNLO computations of hardscattering processes, and it is based on the knowledge of the small-q T limit of the transversemomentum cross section of the corresponding process. In the case of the production of colourless systems, thanks to the complete understanding of the all-order structure of the large logarithmic terms, the method is fully developed up to NNLO. The resummation formula presented in Eq. (5) makes possible to apply the q T subtraction formalism also to heavy-quark production at NNLO, once the explicit results of the resummation factors at the corresponding order will be available.