ZU-TH 25/13

IFUM-1017-FT

Universality of transverse-momentum resummation

[0.3cm] and hard factors at the NNLO

Stefano Catani Leandro Cieri Daniel de Florian, Giancarlo Ferrera

and
Massimiliano Grazzini^{*}^{*}*On leave of absence from INFN, Sezione di Firenze, Sesto Fiorentino, Florence, Italy.

INFN, Sezione di Firenze and Dipartimento di Fisica e Astronomia,

Università di Firenze, I-50019 Sesto Fiorentino, Florence, Italy

Dipartimento di Fisica, Università di Roma “La Sapienza” and

INFN, Sezione di Roma, I-00185 Rome, Italy

Departamento de Física, FCEYN, Universidad de Buenos Aires,

(1428) Pabellón 1 Ciudad Universitaria, Capital Federal, Argentina

Dipartimento di Fisica, Università di Milano and

INFN, Sezione di Milano, I-20133 Milan, Italy

Institut für Theoretische Physik, Universität Zürich, CH-8057 Zürich, Switzerland

Abstract

We consider QCD radiative corrections to the production of colourless high-mass systems in hadron collisions. The logarithmically-enhanced contributions at small transverse momentum are treated to all perturbative orders by a universal resummation formula that depends on a single process-dependent hard factor. We show that the hard factor is directly related to the all-order virtual amplitude of the corresponding partonic process. The direct relation is universal (process independent), and it is expressed by an all-order factorization formula that we explicitly evaluate up to the next-to-next-to-leading order (NNLO) in QCD perturbation theory. Once the NNLO scattering amplitude is available, the corresponding hard factor is directly determined: it controls NNLO contributions in resummed calculations at full next-to-next-to-leading logarithmic accuracy, and it can be used in applications of the subtraction formalism to perform fully-exclusive perturbative calculations up to NNLO. The universality structure of the hard factor and its explicit NNLO form are also extended to the related formalism of threshold resummation.

November 2013

## 1 Introduction

The transverse-momentum distribution of systems with high invariant mass produced in hadron collisions is important for physics studies within and beyond the Standard Model (SM). This paper is devoted to a theoretical study of QCD radiative corrections to transverse-momentum distributions.

We consider the inclusive hard-scattering reaction

(1) |

where the collision of the two hadrons and with momenta and produces the triggered final state , and denotes the accompanying final-state radiation. The observed final state is a generic system of one or more colourless particles, such as lepton pairs (produced by Drell–Yan (DY) mechanism), photon pairs, vector bosons, Higgs boson(s), and so forth. The momenta of these final state particles are denoted by ,…. The system has total invariant mass , transverse momentum and rapidity . We use to denote the centre-of-mass energy of the colliding hadrons, which are treated in the massless approximation ().

The transverse-momentum cross section for the process in Eq. (1) is computable by using perturbative QCD. However, in the small- region (roughly, in the region where ) the convergence of the fixed-order perturbative expansion in powers of the QCD coupling is spoiled by the presence of large logarithmic terms of the type . The predictivity of perturbative QCD can be recovered through the summation of these logarithmically-enhanced contributions to all order in [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. As already stated, we shall limit ourselves to considering the production of systems of non-strongly interacting particles. The all-order analysis of the distribution of systems that involve coloured QCD partons has just started to be investigated, by considering [11] the specific case in which is formed by a pair.

In the case of a generic system of colourless particles, the large logarithmic contributions to the cross section can be systematically resummed to all perturbative orders, and the structure of the resummed calculation can be organized in a process-independent form [4, 6, 9, 10]. The all-order resummation formalism was first developed for the DY process [6] (and the kinematically-related process of two-particle correlations in annihilation [4]). The process-independent extension of the formalism has required two additional main steps: the understanding of the all-order process-independent structure of the Sudakov form factor (through the factorization of a single process-dependent hard factor) [9], and the complete generalization to processes that are initiated by the gluon fusion mechanism [10].

The all-order process-independent form of the resummed calculation has a factorized structure, whose resummation factors are (see Sect. 2) the (quark and gluon) Sudakov form factor, process-independent collinear factors and a process-dependent hard or, more precisely (see Sect. 4), hard-virtual factor. The resummation of the logarithmic contributions is controlled by these factors or, equivalently, by a corresponding set of perturbative functions whose perturbative resummation coefficients are computable order-by-order in . The perturbative coefficients of the Sudakov form factor are known, since some time [5, 7, 12, 8, 13], up to the second order in , and the third-order coefficient (which is necessary to explicitly perform resummation up to the next-to-next-to-leading logarithmic (NNLL) accuracy) is also known [14]. The next-to-next-to-leading order (NNLO) QCD calculation of the cross section (in the small- region) has been explicitly carried out in analytic form for two benchmark processes, namely, SM Higgs boson production [15] and the DY process [16]. The results of Refs. [15, 16] provide us with the complete knowledge of the process-independent collinear resummation coefficients up to the second order in , and with the explicit expression of the hard coefficients for these two specific processes. The purpose of the present paper (see below) is to explicitly point out and derive the underlying universal (process-independent) structure of the process-dependent hard factor of the QCD all-order resummation formalism.

In Refs. [17, 18, 14, 19, 20, 21, 22], the resummation of small- logarithms has been reformulated in terms of factorization formulae that involve Soft Collinear Effective Theory operators and (process-dependent) hard matching coefficients. The formulation of Ref. [14] has been applied [23] to the DY process by explicitly computing the (process-independent) collinear quark–quark coefficients and the DY hard coefficient at the NNLO. The results of this calculation [23] agree with those obtained in Ref. [16]. Transverse-momentum cross sections can also be studied by using other approaches (which go beyond the customary QCD resummation formalism of the present paper) that use transverse-momentum dependent (TMD) factorization (see Refs. [24, 25, 26, 27] and references therein) and, consequently, -unintegrated parton densities and partonic cross sections that are both TMD quantities.

In this paper we study the process-dependent hard factor of the transverse-momentum resummation formula. We show that, for any process of the class in Eq. (1), the all-order hard factor has a universal structure that involves a minimal amount of process-dependent information. The process-dependent information is entirely given by the scattering amplitude of the Born-level partonic subprocess and its virtual radiative corrections. Knowing the scattering amplitude, the hard-virtual resummation factor is determined by a universal (process-independent) factorization formula. The universality structure of the factorization formula has a soft (and collinear) origin, and it is closely (though indirectly) related to the universal structure of the infrared divergences [28] of the scattering amplitude. This process-independent structure of the hard-virtual term, which generalizes the next-to-leading order (NLO) results of Ref. [13], is valid to all perturbative orders, and we explicitly determine the process-independent form of the hard-virtual term up to the NNLO. Using this general NNLO result, the hard-virtual resummation factor for each process of the class in Eq. (1) is straightforwardly computable up to its NNLO, provided the corresponding scattering amplitude is known.

In the final part of the paper, we consider the related formalism of threshold resummation [29, 30] for the total cross section. The process-independent formalism of threshold resummation also involves a corresponding process-dependent hard factor. We shall show that this factor has a universality structure that is analogous to the case of transverse-momentum resummation. In particular, we directly relate the process-dependent hard factors for transverse-momentum and threshold resummation in a form that is fully universal and completely independent of each specific process (e.g., independent of the corresponding scattering amplitude).

The knowledge of the NNLO hard-virtual term completes the resummation formalism in explicit form up to full NNLL+NNLO accuracy. This permits direct applications to NNLL+NNLO resummed calculations for any processes of the class in Eq. (1) (provided the corresponding NNLO amplitude is known), as already done for the cases of SM Higgs boson [31, 32, 33] and DY [34, 35] production.

The NNLO information of the resummation formalism is also relevant in the context of fixed order calculations. Indeed, it permits to carry out fully-exclusive NNLO calculations by applying the subtraction formalism of Ref. [36] (the subtraction counterterms of the formalism follow [36] from the fixed-order expansion of the resummation formula, as in Sect. 2.4 of Ref. [31]). The subtraction formalism has been applied to the NNLO computation of Higgs boson [36, 37] and vector boson production [38], associated production of the Higgs boson with a boson [39], diphoton production [40] and production [41]. The computations of Refs. [36, 37, 38, 39] were based on the specific calculation of the NNLO hard-virtual coefficients of the corresponding processes [15, 16]. The computations of Refs. [40, 41] used the NNLO hard-virtual coefficients that are determined by applying the universal form of the hard-virtual term that is derived and illustrated in the present paper.

The paper is organized as follows. In Sect. 2 we recall the transverse-momentum resummation formalism in impact parameter space, and we introduce our notation. In Sect. 3 we present the explicit expressions of the process-independent resummation coefficients up to NNLO. Section 4 is devoted to the process-dependent hard coefficients. We discuss and illustrate the universal all-order form of the hard-virtual coefficients by relating them to the process-dependent scattering amplitudes, through the introduction of suitably subtracted hard-virtual matrix elements. The process-independent structure of the hard-virtual coefficients is explicitly computed up to the NNLO. In Sect. 5 we extend our discussion and results on the universal structure of the hard-virtual coefficients to the case of threshold resummation. In Sect. 6 we summarize our results. In Appendix A we report the explicit expressions of the NLO and NNLO hard-virtual coefficients for DY, Higgs boson and diphoton production.

## 2 Small- resummation

In this section we briefly recall the formalism of transverse-momentum resummation in impact parameter space [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. We closely follow the notation of Ref. [10] (more details about our notation can be found therein).

We consider the inclusive-production process in Eq. (1), and we introduce the corresponding fully differential cross section

(2) |

which depends on the total momentum of the system (i.e. on the variables ). The cross section also depends on the set of additional variables that controls the kinematics of the particles in the system . In Eq. (2) these additional variables are generically denoted as (correspondingly, we define . To be general, we do not explicitly specify these variables, and we only require that the kinematical variables are independent of and and that the set of variables completely determines the kinematical configuration (i.e., the momenta ) of the particles in the system . For instance, if the system is formed by two particles, there are only two variables in the set , and they can be the rapidity and the azimuthal angle of one of the two particles (the particle with momentum ). Note that the cross section in Eq. (2) and the corresponding resummation formula can be straightforwardly integrated with respect to some of the final-state variables , thus leading to results for observables that are more inclusive than the differential cross section in Eq. (2).

We also recall that we are considering the production of a system of colourless particles or, more precisely, a system of non-strongly interacting particles (i.e., cannot include QCD partons and their fragmentation products). Therefore, at the Born (lowest-order) level, the cross section in Eq. (2) is controlled by the partonic subprocesses of quark–antiquark () annihilation,

(3) |

and gluon fusion,

(4) |

Owing to colour conservation, no other partonic subprocesses can occur at the Born level. More importantly (see below), the distinction between annihilation and gluon fusion leads to relevant (and physical) differences [42, 10] in the context of small- resummation.

To study the dependence of the differential cross section in Eq. (2) within QCD perturbation theory, we introduce the following decomposition:

(5) |

Both terms in the right-hand side are obtained through convolutions of partonic cross sections and the scale-dependent parton distributions ( is the parton label) of the colliding hadrons. We use parton densities as defined in the factorization scheme, and is the QCD running coupling in the renormalization scheme. The partonic cross sections that enter the singular component (the first term in the right-hand side of Eq. (5)) contain all the contributions that are enhanced (or ‘singular’) at small . These contributions are proportional to or to large logarithms of the type . On the contrary, the partonic cross sections of the second term in the right-hand side of Eq. (5) are regular (i.e. free of logarithmic terms) order-by-order in perturbation theory as . More precisely, the integration of over the range leads to a finite result that, at each fixed order in , vanishes in the limit .

The regular component of the cross section depends on the specific process in Eq. (1) that we are considering. In the following we focus on the singular component, , which has a universal all-order structure. The corresponding resummation formula is written as [6, 9, 10]

(6) |

where ( is the Euler number) is a numerical coefficient, and the kinematical variables and are

(7) |

The right-hand side of Eq. (2) involves the Fourier transformation with respect to the impact parameter and two convolutions over the longitudinal-momentum fractions and . The parton densities of the colliding hadrons are evaluated at the scale , which depends on the impact parameter. The function is the Sudakov form factor. This factor, which only depends on the type ( or ) of colliding partons, is universal (process independent) [9], and it resums the logarithmically-enhanced contributions of the form (the region corresponds to in impact parameter space). The all-order expression of is [6]

(8) |

where and are perturbative series in ,

(9) |

The perturbative coefficients [5, 7], [12, 8, 13] and [14] are explicitly known.

The factor that is symbolically denoted by in Eq. (2) is the Born-level cross section (i.e., the cross section at its corresponding lowest order in ) of the partonic subprocesses in Eqs. (3) and (4) (in the case of the annihilation channel, the quark and antiquark can actually have different flavours). Making the symbolic notation explicit, we have

(10) |

where () is the momentum of the parton (). In Eq. (2), we have included the contribution of both the annihilation channel () and the gluon fusion channel (); one of these two contributing channels may be absent (i.e. in that channel), depending on the specific final-state system .

The Born level factor is obviously process dependent, although its process dependence is elementary (it is simply due to the Born level scattering amplitude of the partonic process ). The remaining process dependence of Eq. (2) is embodied in the ‘hard-collinear’ factor . This factor includes a process-independent part and a process-dependent part. The structure of the process-dependent part is the main subject of the present paper.

In the case of processes that are initiated at the Born level by the annihilation channel (), the symbolic factor in Eq. (2) has the following explicit form [9]

(11) |

and the functions and have the perturbative expansion

(12) | |||

(13) |

The function is process dependent, whereas the functions are universal (they only depend on the parton indices). We add an important remark on the factorized structure in the right-hand side of Eq. (11): the scale of is in the case of , whereas the scale is in the case of . The appearance of these two different scales is essential [9] to disentangle the process dependence of from the process-independent Sudakov form factor () and collinear functions ().

In the case of processes that are initiated at the Born level by the gluon fusion channel (), the physics of the small- cross section has a richer structure, which is the consequence of collinear correlations [10] that are produced by the evolution of the colliding hadrons into gluon partonic states. In this case, the resummation formula (2) and, specifically, its factor are more involved than those for the channel, since collinear radiation from the colliding gluons leads to spin and azimuthal correlations [42, 10]. The symbolic factor in Eq. (2) has the following explicit form [10]:

(14) |

where the function has the perturbative expansion

(15) |

and the following lowest-order normalization:

(16) |

Analogously to Eq. (11), in Eq. (2) the function is process dependent (and it is controlled by at the scale ) and the partonic functions are process independent (and they are controlled by at the scale ). At variance with Eq. (11) (where the factorization structure in the right-hand side is independent of the degrees of freedom of the colliding quark and antiquarks), in Eq. (2) the process-dependent function depends on the Lorentz indices (and, thus, on the spins) of the colliding gluons with momenta and this dependence is coupled to (and correlated with) a corresponding dependence of the partonic functions . The Lorentz tensor coefficients in Eq. (2) depend on (through the scale of ) and, moreover, they also depend on the direction (i.e., the azimuthal angle) of the impact parameter vector in the transverse plane. The structure of the partonic tensor is [10]

(17) |

where

(18) |

(19) |

and is the two-dimensional impact parameter vector in the four-dimensional notation . The gluonic coefficient function () in the right-hand side of Eq. (17) has the same perturbative structure as in Eq. (13), and it reads

(20) |

In contrast, the perturbative expansion of the coefficient functions , which are specific to gluon-initiated processes, starts at , and we write

(21) |

We recall [10] an important physical consequence of the different small- resummation structure between the annihilation and gluon fusion channels: the absence of azimuthal correlations with respect to in the annihilation channel, and the presence of correlations with a definite predictable azimuthal dependence in the gluon fusion channel. Indeed, in the case of annihilation, all the factors in the integrand of the Fourier transformation on the right-hand side of the resummation formula (2) are functions of , with no dependence on the azimuthal angle of . Therefore, the integration over in Eq. (2) can be straightforwardly carried out, and it leads [4, 6] to a one-dimensional Bessel transformation that involves the th-order Bessel function . This implies that the right-hand side of Eq. (2) and, hence, the singular part of the differential cross section depend only on , with no additional dependence on the azimuthal angle of . Unlike the case of annihilation, the gluon fusion factor in Eqs. (2) and (2) does depend on the azimuthal angle of the impact parameter . Therefore, the integration over in the Fourier transformation of Eq. (2) is more complicated. It leads to one-dimensional Bessel transformations that involve and higher-order Bessel functions, such as the 2-nd order and 4th-order functions and . More importantly, it leads to a definite structure of azimuthal correlations with respect to the azimuthal angle of the transverse momentum . The small- cross section in Eq. (2) can be expressed [10] in terms of a contribution that does not depend on plus a contribution that is given by a linear combination of the four angular functions and . No other functional dependence on is allowed by the resummation formula (2) in the gluon fusion channel.

We recall that, due to its specific factorization structure, the resummation formula in Eq. (2) is invariant under the following renormalization-group transformation [9]

(22) | ||||

(23) | ||||

(24) |

where is an arbitrary perturbative function (with ). More precisely, in the case of gluon-initiated processes, Eq. (24) becomes

(25) |

In the right-hand side of Eq. (23), denotes the QCD -function:

(26) |

(27) |

(28) |

where is the number of quark flavours, is the number of colours, and the colour factors are and in QCD. As a consequence of the renormalization-group symmetry in Eqs. (22)–(25), the resummation factors , , and are not separately defined (and, thus, computable) in an unambiguous way. Equivalently, each of these separate factors can be precisely defined only by specifying a resummation scheme [9].

To present the main results of this paper in the following sections, we find it convenient to specify a resummation scheme. Therefore, in the rest of this paper we work in the scheme, dubbed hard scheme, that is defined as follows. The flavour off-diagonal coefficients , with , are ‘regular’ functions of as . The dependence of the flavour diagonal coefficients and in Eqs. (13) and (20) is instead due to both ‘regular’ functions and ‘singular’ distributions in the limit . The ’singular’ distributions are and the customary plus-distributions of the form (). The hard scheme is the scheme in which, order-by-order in perturbation theory, the coefficients with do not contain any term. We remark (see also Sect. 4) that this definition directly implies that all the process-dependent virtual corrections to the Born level subprocesses in Eqs. (3) and (4) are embodied in the resummation coefficient .

We note that the specification of the hard scheme (or any other scheme) has sole practical purposes of presentation (theoretical results can be equivalently presented, as actually done in Refs. [15] and [16], by explicitly parametrizing the resummation-scheme dependence of the resummation factors). Having presented explicit results in the hard scheme, they can be translated in other schemes by properly choosing the functions and applying the transformation in Eqs. (22)–(25). Moreover, and more importantly, the cross section, its all-order resummation formula (2) and any consistent perturbative truncation (either order-by-order in or in classes of logarithmic terms) of the latter [9, 31] are completely independent of the resummation scheme.

## 3 Process-independent coefficients

Before discussing the general structure of the resummation coefficients , in this section we present the expressions of the process-independent resummation coefficients in the hard scheme, which is defined in Sec. 2.

The partonic functions and in Eqs. (13), (20) and (21) depend on the parton indices. Owing to charge conjugation invariance and flavour symmetry of QCD, the dependence on the parton indeces is fully specified by the five independent quark functions [16] ( and denote quarks with different flavour) and the four independent gluon functions [15].

The first-order coefficients are explicitly known [12, 43, 8, 13]. Their expressions in the hard scheme can be obtained from their corresponding expression in an arbitrary scheme by simply setting the coefficient of the term to zero. We get

(29) | ||||

(30) | ||||

(31) | ||||

(32) |

The first-order coefficients are resummation-scheme independent, and they read [10]

(33) |

where is the Casimir colour coefficient of the parton with and .

According to Eq. (23), the coefficients with of the Sudakov form factor do depend [9] on the resummation scheme. The second-order process-independent coefficient in Eq. (9) is known [12, 13]. In the hard scheme, its value reads

(34) |

where () are the coefficients of the term in the NLO quark and gluon splitting functions [44, 45], which read

(35) |

(36) |

and is the Riemann zeta-function ().

The second-order process-independent collinear coefficients of Eqs. (13) and (20) have been computed in Refs. [36, 38, 15, 16]. The quark–quark coefficient has been independently computed in Ref. [23]. The expressions of these coefficients in the hard scheme can be straightforwardly obtained from the results of Refs. [15, 16] and are explicitly reported below.

Starting from the quark channel, the coefficient can be obtained from Eq. (34) of Ref. [16], and we have

(37) |

where is obtained from the right-hand side of Eq. (23) of Ref. [16] by setting the coefficient of the term to zero. Analogously, the coefficient can be obtained from Eq. (32) of Ref. [16] as

(38) |

where is given in Eq. (27) of Ref. [16]. The flavour off-diagonal quark coefficients , , are scheme independent and are presented in Eq. (35) of Ref. [16]. Moving to the gluon channel, the coefficient can be obtained from Eq. (32) of Ref. [15], and we have

(39) |

where is obtained from the right-hand side Eq. (24) of Ref. [15] by setting the coefficient of the term to zero. Finally, the coefficient can be obtained from Eq. (30) of Ref. [15], and we have

(40) |

where is given in Eq. (23) of Ref. [15].

The second-order gluon collinear coefficients of Eq. (21) are not yet known. We can comment on the role of in practical terms. In the specific and important case of Higgs boson production by gluon fusion, the coefficient does not contribute to the cross section at the NNLO (and NNLL accuracy). The Higgs boson cross section is discussed in detail in Ref. [10]: by direct inspection of Eq. (45) of Ref. [10], we can see that starts to contribute at the NLO. In most of the other processes (e.g., ), the system can be produced by both annihilation and gluon fusion. In these case, due to the absence of direct coupling of the gluons to the colourless particles in the system , the production channel is suppressed by some powers of with respect to the channel . Therefore, also in these cases the coefficient does not contribute to the NNLO cross section. This formal conclusion (based on counting the powers of ) has a caveat, since the channel can receive a quantitative enhancement from the possibly large luminosity of the gluon parton densities. However, the knowledge of the first-order coefficients and should be sufficient to compute the contribution from the channel to a quantitative level that is comparable to that of the contribution from the channel (whose collinear coefficients are fully known up to the second order). In summary, we conclude that the effect of rarely contributes in actual (practical) computations of the cross section at the NNLO or NNLL accuracy.

## 4 Hard-virtual coefficients

In this section we focus on the process-dependent coefficient . In the hard scheme that we are using, this coefficient contains all the information on the process-dependent virtual corrections, and, therefore, we can show that can be related in a process-independent (universal) way to the multiloop virtual amplitude of the partonic process . In the following we first specify the notation that we use to denote the all-loop virtual amplitude . Then we introduce an auxiliary (hard-virtual) amplitude that is directly obtained from by using a process-independent relation. Finally, we use the hard-virtual amplitude to present the explicit expression of the hard-virtual coefficient up to the NNLO.

We consider the partonic elastic-production process

(41) |

where the two colliding partons with momenta and
are either or (we do not explicitly denote the flavour of the
quark , although in the case with , the quark and the
antiquark can have different flavours), and is the triggered
final-state system in Eq. (1).
The loop scattering amplitude of the process in Eq. (41)
contains ultraviolet (UV) and infrared (IR) singularities,
which are regularized in space-time dimensions by using the customary scheme of
conventional dimensional regularization (CDR)^{†}^{†}†The relation between the
CDR scheme and other variants of dimensional regularization is explicitly known
[46] up to the two-loop level..
Before performing renormalization, the multiloop QCD amplitude
has a perturbative dependence on
powers of , where is the bare coupling and is
the dimensional-regularization scale. In the following we work with the renormalized
on-shell scattering amplitude
that
is obtained from the corresponding unrenormalized amplitude by just expressing the bare coupling
in terms of the running coupling according to the scheme
relation

(42) |

where is the renormalization scale, and are the first two coefficients of the QCD -function in Eq. (28) and the factor is

(43) |

The renormalized all-loop amplitude of the process in Eq. (41) is denoted by , and it has the perturbative (loop) expansion

(44) |

where the value of the overall power of depends on the specific process (for instance, in the case of the vector boson production process , and in the case of the Higgs boson production process through a heavy-quark loop). Note also that the lowest-order perturbative term is not necessarily a tree-level amplitude (for instance, it involves a quark loop in the cases and ). The perturbative terms are UV finite, but they still depend on (although this dependence is not explicitly denoted in Eq. (44)). In particular, the amplitude at the -th perturbative order is IR divergent as , and it behaves as

(45) |

where the dots stand for -poles of lower order. The IR divergent contributions to the scattering amplitude have a universal structure [28], which is explicitly known at the one-loop [47, 28], two-loop [28, 48] and three-loop [49, 50] level for the class of processes in Eq. (41).

The explicit calculations and the results of Ref. [13] show that the NLO hard-virtual coefficient is explicitly related in a process-independent form to the leading-order (LO) amplitude