1 Introduction

ICAS 02/15

TIF-UNIMI-2015-10

ZU-TH 24/15

Vector boson production at hadron colliders:

[0.15cm] transverse-momentum resummation

[0.15cm] and leptonic decay

Stefano Catani, Daniel de Florian,

Giancarlo Ferrera and Massimiliano Grazzini1

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

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

Departamento de Física, FCEYN, Universidad de Buenos Aires,

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

and International Center for Advanced Studies (ICAS),

UNSAM, Campus Miguelete,  25 de Mayo y Francia,

1650 Buenos Aires, Argentina

Dipartimento di Fisica, Università di Milano and

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

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

Abstract


We consider the transverse-momentum () distribution of Drell–Yan lepton pairs produced, via and decay, in hadronic collisions. At small values of , we resum the logarithmically-enhanced perturbative QCD contributions up to next-to-next-to-leading logarithmic accuracy. Resummed results are consistently combined with the known fixed-order results at intermediate and large values of . Our calculation includes the leptonic decay of the vector boson with the corresponding spin correlations, the finite-width effects and the full dependence on the final-state lepton(s) kinematics. The computation is encoded in the numerical program DYRes, which allows the user to apply arbitrary kinematical cuts on the final-state leptons and to compute the corresponding distributions in the form of bin histograms. We present a comparison of our results with some of the available LHC data. The inclusion of the leptonic decay in the resummed calculation requires a theoretical discussion on the recoil due to the transverse momentum of the produced vector boson. We present a recoil procedure that is directly applicable to resummed calculations for generic production processes of high-mass systems in hadron collisions.

July 2015

1 Introduction

The production of high-mass lepton pairs through the Drell–Yan (DY) mechanism [1] is a benchmark hard-scattering process at hadron colliders. It provides important tests of the Standard Model (SM) with both precise measurements of its fundamental parameters and, at the same time, stringent constraints on new physics.

It is thus a major task to achieve accurate theoretical predictions for the DY production cross section and related kinematical distributions. This requires, in particular, the evaluation of QCD radiative corrections, which can be perturbatively computed as power series expansion in the strong coupling . The total cross section [2] and the rapidity distribution [3] of the vector boson are known up to the next-to-next-to-leading order (NNLO) in perturbative QCD. Two independent fully exclusive NNLO calculations, which include the leptonic decay of the vector boson, have been performed [4, 5, 6]. Electroweak (EW) radiative corrections are also available for both [7] and [8] production. Mixed QCD-EW corrections have been considered in Refs. [9, 10, 11, 12].

A particularly relevant observable is the transverse-momentum () distribution of the vector boson. To obtain a precise measurement of the mass it is important to have accurate theoretical calculations of the and bosons spectra. In the large- region (), where the transverse momentum is of the order of the vector boson mass , QCD corrections are known up to [13, 14, 15], and these results were extended in Refs. [16, 17] with the inclusion of the dependence on the leptonic decay variables. Very recently the fully exclusive computation of vector boson production in association with a jet has been performed in Ref. [18] (in the case of production) and Ref. [19] (in the case of production).

The bulk of the vector boson cross section is produced in the small- region (), where the reliability of the fixed-order expansion is spoiled by the presence of large logarithmic corrections, (with ), of soft and collinear origin. To obtain reliable predictions, these logarithmically-enhanced terms have to be evaluated and systematically resummed to all orders in perturbation theory [20][29]. In recent years, the resummation of small- logarithms has been reformulated [30][38] by using Soft Collinear Effective Theory (SCET) methods and transverse-momentum dependent (TMD) factorization.

The resummed and fixed-order calculations, which are valid at small and large values of , respectively, can be consistently matched at intermediate values of to achieve a uniform theoretical accuracy for the entire range of transverse momenta.

In this paper we compute the vector boson transverse-momentum distribution [39, 40] by using the resummation formalism proposed in Refs. [25, 26, 27], which can be applied to a generic process in which a high-mass system of non strongly-interacting particles is produced in hadronic collisions [41, 26, 27], [42][52]. Other phenomenological studies of the DY distribution, which combine resummed and fixed-order perturbative results at different levels of theoretical accuracy, can be found in Refs. [53][69]. Within the studies in Refs. [53][69], the kinematical dependence on the momenta of the final-state leptons is considered only in the RESBOS calculation [55, 56, 68] and in the calculations of Refs. [57] and [67].

Hadron collider experiments can directly measure only the decay products of vector bosons in finite kinematical regions. Therefore, it is important to include the vector boson leptonic decay in the theoretical calculations, by retaining the kinematics of the final-state leptons. In this way it is possible to obtain predictions for the transverse-momentum distribution of the measured leptons. This is specially relevant in the case of production where, because of the final-state neutrino, the transverse momentum of the vector boson can only be reconstructed through a measure of the hadronic recoil. Moreover, in both cases of and production, the inclusion of the leptonic decay allows one to apply kinematical selection cuts, thus providing a more realistic simulation of the actual experimental analysis.

In Ref. [39, 40] we have presented a resummed computation of the transverse-momentum spectrum for production at Tevatron energies. We have combined resummation at the next-to-next-to-leading logarithmic (NNLL) accuracy in the small- region with the fixed-order results at in the large- region. This leads to a calculation with uniform theoretical accuracy from small to intermediate values of . In particular, the integral over the range ( is a generic upper limit in the small- region) of the distribution includes the complete perturbative terms up to NNLO. Moreover, at large values of the calculation implements a unitarity constraint that guarantees to exactly reproduce the NNLO value of the total cross section after integration over . In this paper we extend the NNLL+NNLO calculation of Ref. [40] to boson production, and we include the leptonic decay of the vector boson with the corresponding spin correlations. The spin of the vector boson dynamically correlates the decaying lepton momenta with the transverse momentum acquired by the vector boson through its production mechanism. Therefore, the inclusion of the full dependence on the lepton decay variables in the resummed calculation requires a theoretical discussion on the treatment of the recoil due to the transverse momentum of the vector boson. We treat the recoil by introducing a general procedure that is directly applicable to resummed calculations for generic production processes of high-mass systems in hadron collisions. The calculation presented in this paper parallels the one performed in Ref. [43] for the case of SM Higgs boson production, with the non-trivial additional complication of dealing with the spin correlations that are absent in the Higgs boson case. Our vector boson computation is implemented in the numerical code DYRes, which allows the user to apply arbitrary kinematical cuts on the final-state leptons and to compute the corresponding relevant distributions in form of bin histograms. The code DYRes is publicly available and it can be downloaded from the URL address http://pcteserver.mi.infn.it/~ferrera/dyres.html .

The paper is organized as follows. In Sect. 2 we briefly review the resummation formalism of Refs. [25, 26, 27], and we discuss the main features of resummation for the DY process with full dependence on final-state lepton variables. In Sect. 3 we present our quantitative results for vector boson production at LHC energies. Section 3.1 is devoted to the spectrum of the vector boson after integration over the final-state leptons. We present results at different orders of logarithmic accuracy, we study the corresponding dependence on scale variations, and we briefly comment on uncertainties due to parton densities and on non-perturbative effects. In Sect. 3.2 we compare our numerical results for and production with some of the available LHC data, and we also study the impact of transverse-momentum resummation on lepton kinematical variables. In Sect. 4 we summarize our results. The Appendix presents a detailed discussion of recoil and of its implementation.

2 Transverse-momentum resummation

In this Section we briefly recall the main features of the transverse-momentum resummation formalism that we use in this paper. A more detailed discussion of the resummation formalism can be found in Refs. [25, 26, 27, 29]. In Ref. [40] we have considered NNLL resummation for the distribution of the vector boson after integration over the kinematical variables of the decaying leptons and the rapidity of the vector boson. In this paper we extend the results of Ref. [40] to include the entire kinematical dependence on the final-state leptons. The presentation in this Section parallels that of Sect. 2 in Ref. [40] and, in particular, we highlight the main differences that arise in the treatment of the rapidity of the vector boson and, especially, of the lepton kinematics.

We consider the inclusive hard-scattering process

(1)

where the collision of the two hadrons and with momenta and produces the vector boson ( and/or ) with total momentum , which subsequently decays in the lepton pair , and denotes the accompanying final-state radiation. We consider high values of the invariant mass of the lepton pair (in general, differs from the on-shell mass of the vector boson ), and we treat the colliding hadrons and the leptons in the massless approximation () throughout the paper. In a reference frame where the colliding hadrons are back-to-back, the momentum is fully specified by the invariant mass (), the two-dimensional transverse-momentum vector (with magnitude and azimuthal angle ) and the rapidity () of the vector boson. Analogously the momentum of the lepton () is specified by the lepton rapidity and transverse momentum .

The kinematics of the lepton pair is completely specified by six independent variables (e.g., the three-momenta of the two leptons). For our purposes, it is convenient to use the vector boson momentum to select four independent variables. Therefore, the final-state lepton kinematics is fully determined by the vector boson momentum and by two additional and independent variables that specify the angular distribution of the leptons with respect to the vector boson momentum . We generically denote these two additional kinematical variables as . These two independent variables can be chosen in different ways. For instance, we can use longitudinally boost invariant variables such as the rapidity difference and the azimuthal angle (or the azimuthal angle difference ) of the lepton and the vector boson in the hadronic back-to-back reference frame. Alternatively, we can use the polar and azimuthal angles of one lepton in a properly specified rest frame of the vector boson (such as, for instance, the Collins–Soper rest frame [70]). Independently of the actual specification of the variables , the most general fully-differential hadronic cross section is expressed in terms of the sixfold differential distribution

(2)

where is the square of the hadronic centre–of–mass energy. Obviously, the differential distribution also depends on the EW parameters (including the mass of the vector boson ): unless otherwise specified, this dependence is not explicitly denoted throughout the paper.

The differential hadronic cross section can be written as

(3)

where () are the parton densities of the colliding hadron at the factorization scale , are the differential partonic cross sections, is the square of the partonic centre–of–mass energy, is the vector boson rapidity with respect to the colliding partons, and is the renormalization scale. Note that the partonic cross sections do not have any explicit dependence on hadronic kinematical variables, since the leptonic variables are specified with respect to . The partonic cross section is computable in QCD perturbation theory as a power series expansion in the QCD coupling .

In the region where , the perturbative expansion of the partonic cross section starts at . In this region the value of the auxiliary scales and can be chosen to be of the order of , and the QCD perturbative series is controlled by a small expansion parameter . Therefore, fixed-order calculations of the partonic cross section are theoretically justified. The QCD radiative corrections are known analytically up to after integration over the lepton angular variables [14, 15] and with the inclusion of the full dependence on these angular variables [16, 17]. The numerical results at can be obtained also from the fully-exclusive calculations of Refs. [4, 5, 6]. Results at can be derived from the recent numerical computations of production [18] and production [19].

In the small region (), the perturbative computation of the partonic cross section starts at through the leading-order (LO) EW process of quark–antiquark annihilation. In this region, the QCD radiative corrections are known up to NNLO (i.e., ) in analytic form [74] by neglecting corrections of (these corrections can directly be extracted from Refs. [14, 15, 16, 17]). The complete (i.e., by including corrections of ) NNLO result can be obtained from the numerical computations of Refs. [4, 5]. However, in the small region the convergence of the fixed-order perturbative expansion is spoiled by the presence of powers of large logarithmic terms, (with ). In particular, these terms become singular in the limit . To obtain reliable predictions these terms have to be resummed to all orders.

Within our formalism, the resummation is performed at the level of the partonic cross section, which is decomposed as follows:

(4)

Here we have introduced a shorthand notation: the symbol denotes the multidifferential partonic cross section that appears as the last factor in the right-hand side of Eq. (3). The first term, , on the right-hand side of Eq. (4) is the resummed component. It contains all the logarithmically-enhanced contributions (at small ) that have to be resummed to all orders in . The second term, the finite component , is free of such contributions and thus it can be evaluated at fixed order in perturbation theory. Note that part of the non-singular (i.e., not logarithmically-enhanced) contributions can also be included in , and we comment later about this point.

The resummation of the logarithmic contributions has to be carried out in the impact parameter () space [20, 21, 22, 53, 23] to fulfil the important constraint of transverse-momentum conservation for inclusive multiparton radiation. The impact parameter is the conjugate variable to through a Fourier transformation. The small- region () corresponds to the large- region () and the logarithmic terms become large logarithmic contributions in space. The resummed component of the cross section is then obtained by performing the inverse Fourier transformation (or the Bessel transformation in Eq. (6)) from space to space. The resummed component of the partonic cross section in Eq. (4) can be expressed as

(5)

where

(6)

and is the th-order Bessel function. The factor in the right-hand side of Eq. (5) is the Born level differential cross section for the partonic subprocess of quark–antiquark annihilation, where the quark flavours and can be either different (if ) or equal (if ). This factor is of purely EW origin, and it completely encodes the dependence on the lepton kinematical variables . We postpone more detailed comments on (see Eq. (12) and the discussion therein). The QCD radiative corrections and their associated dependence on are embodied in the resummed factor , which depends on the produced vector boson but it is independent of the decay leptons (in particular, it does not depend on ). The integrand in Eq. (6) depends on and the inverse Fourier transformation is recast in terms of the Bessel transformation through the integration over the azimuthal angle of . Note that the resummation factor depends on and it does not contain any dependence on the azimuthal angle of . This azimuthal independence is a feature of transverse-momentum resummation [23] for the production processes of colourless systems (such as vector bosons) through quark–antiquark annihilation. In contrast, logarithmically-enhanced azimuthal correlations enter transverse-momentum resummation for processes initiated by gluon-gluon fusion [28] (such as Higgs boson production) and for production of systems that carry colour charges (such as heavy quarks) [71] through either quark–antiquark annihilation or gluon-gluon fusion.

The all-order resummation structure of in Eq. (6) can be organized in exponential form [26, 27]. The exponentiated structure is directly evident by considering the ‘double’ Mellin moments of the function with respect to the variables and at fixed . We have 2

(7)

where the dependence on (and on the large logarithm ) is denoted by defining and introducing the logarithmic expansion parameter with ( is the Euler number). The scale , named resummation scale [41], which appears in the right-hand side of Eq. (7), parametrizes the arbitrariness in the resummation procedure. Although does not depend on when evaluated to all perturbative orders, its explicit dependence on occurs when it is computed by truncation of the resummed expression at some level of logarithmic accuracy (see Eq. (8)). Variations of around can thus be used to estimate the size of yet uncalculated higher-order logarithmic contributions.

The contribution in the right-hand side of Eq. (7) includes the Sudakov form factor and collinear-evolution terms. This contribution (which does not depend on the factorization scale ) is universal (i.e. process independent), namely, it is independent on the produced vector boson and, more generally, it occurs in transverse-momentum resummation for all the processes that are initiated by quark–antiquark annihilation at the LO level. The generalized form factor contains all the terms that order-by-order in are logarithmically divergent as (or, equivalently, as ). The all-order expression of the form factor can be systematically expanded in terms of functions of the resummation parameter (each function resums terms and it is defined such that ). The resummed logarithmic expansion of in powers of reads

(8)

where the term collects the leading logarithmic (LL) contributions, the function includes the next-to-leading logarithmic (NLL) contributions [24], controls the NNLL terms [72, 32] and so forth. The function depends on the specific process of vector boson production and it is due to hard-virtual and collinear contributions. This function does not depend on the impact parameter (it includes all the perturbative contributions to that behave as constants in the limit ) and, therefore, it can be expanded in powers of as

(9)

The next-to-leading order (NLO) term is known since a long time [73], and the NNLO term has been obtained more recently by two independent calculations in Refs. [74] and [75]. The explicit form of the functions and and, in particular, their dependence on the Mellin moment indices can be found in Ref. [26] and in Appendix A of Ref. [27].

Incidentally, we recall that the generalized form factor is known up to NNLL accuracy also for processes initiated by the gluon fusion mechanism [76, 77, 28, 32], and that the collinear coefficients (which contribute to the NNLO term in Eq. (9)) are also known for all possible partonic channels [78, 74, 75, 79, 29]. Owing to the universality structure of transverse-momentum resummation, these results and those for the annihilation channel (which contribute to vector boson production) can be directly implemented in resummed calculations for production processes of generic high-mass systems.

The finite component in Eq. (4) has to be evaluated starting from the usual fixed-order perturbative truncation of the partonic cross section and subtracting the expansion of the resummed part at the same perturbative order. We have

(10)

where the subscript f.o. denotes the perturbative truncation at the order f.o. (NLO, NNLO and so forth). The customary fixed-order component (and consequently also the finite component) definitely contains azimuthal correlations with respect to , although these are not logarithmically-enhanced in the small- region.

To obtain NLL+NLO accuracy we have to include the functions and in the generalized form factor of Eq. (8), the function in the hard/collinear factor of Eq. (9) and the finite component of Eq. (10) up to . To reach NNLL+NNLO accuracy we need to include also the functions , and the finite component up to 3. This matching procedure between resummed and finite contributions guarantees to achieve uniform theoretical accuracy over the entire range of transverse momenta. In particular, we remark that the inclusion of in the resummed component at the NNLL+NNLO level is essential to achieve NNLO accuracy in the small- region (considering a generic upper limit value , the integral over the range of the distribution at the NNLL+NNLO level includes the complete perturbative terms up to NNLO). An analogous remark applies to the inclusion of at the NLL+NLO level.

We have so far illustrated the resummation formalism for the most general sixfold differential partonic cross section (and for the corresponding hadronic cross section in Eq. (3)). Starting from and performing integrations over some kinematical variables, we can obtain resummed results for more inclusive -dependent distributions. For instance, integrating over the lepton kinematical variables , we obtain the cross section at fixed invariant mass and rapidity of the lepton pair. The corresponding resummed component of the partonic cross section, as obtained from Eq. (5), is

(11)

where is the Born level (EW) total cross section for the partonic subprocess . By performing an additional integration over the rapidity of the vector boson (lepton pair), we obtain and the corresponding resummed component of the partonic cross section simply involves the integration over of the resummed factor in Eqs. (5) and (11) (or, equivalently, the factor in Eq. (6)). After integration over , the ensuing resummed factor depends on and , and it can be conveniently expressed in exponentiated form [26] by considering ‘single’ Mellin moments with respect to the variable at fixed . The resummed expression for these ‘single’ moments is exactly obtained by simply setting in Eqs. (7)–(9). Our resummed calculation of was discussed in Ref. [39, 40], and it is implemented in the numerical code DYqT. In Refs. [39, 40] we presented detailed quantitative results for vector boson production at Tevatron energies. Results from DYqT at LHC energies are presented in the following Sect. 3.1.

Within our formalism the resummation of the large terms at small values of is achieved by first performing the Fourier transformation of the cross section (or, more precisely, of its singular behaviour in the small- region) from space to space (incidentally, the renormalization scale and the others auxiliary scales and are kept fixed and, especially, independent of in the integration over of the Fourier transformation). In space, the large logarithmic variable (whose dependence has to be resummed) is , at large values of . Note that in the context of the resummation approach, the parameter is formally considered to be of order unity. Therefore, the ratio of two successive terms in the expansion (8) is formally of (with no enhancement). In this respect the resummed logarithmic expansion in Eq. (8) is as systematic as any customary fixed-order expansion in powers of . Analogously to any perturbative expansions, the perturbative terms in Eq. (8) have an explicit logarithmic dependence on or (see, e.g., Eqs. (22) and (23) in Ref. [26]). Therefore, to avoid additional large logarithmic enhancements that would spoil the formal behaviour of the expansion in Eq. (8), the renormalization scale has to be set at a value of the order of . A completely analogous reasoning applies to the dependence of in the expansion of Eq. (9) and, therefore, we should set . In other words, once the enhanced perturbative dependence on (i.e., on the two different scales and ) is explicitly resummed (albeit at a definite logarithmic accuracy), we are effectively dealing with a single-scale observable at the hard scale and we can set in both the resummed and finite components of the cross section in Eq. (4).

We remark that setting (here generically denotes the auxiliary scales ) does not mean that the cross section is physically controlled by parton radiation with intensity that is proportional to . The resummed form factor in Eq. (7) (and the ensuing logarithmic expansion in Eq. (8)) is produced by multiparton radiation with intensity that is proportional to and is a dynamical scale that varies in the range (see, for instance, Eq. (19) in Ref. [26]), where can be physically identified with at small values of . Setting in Eqs. (7) and (8) corresponds, roughly speaking, to consider the scale range (it does not correspond to set ).

We recall [26] a feature of our resummation formalism. The small- singular contributions that are resummed in Eqs. (5) (or Eq. (11)) are controlled by the large logarithmic parameter , which corresponds to (with ) in space at . In our resummation formula (7), we actually use the logarithmic parameter [41]. The motivations to use the logarithmic parameter are detailed in Ref. [26] (see, in particular, the Appendix B and the comments that accompany Eqs. (16)-(18) and Eqs. (74)-(75) in Ref. [26]), and here we simply limit ourselves to recalling some aspects. In the relevant resummation region , we have and, therefore, and are fully equivalent to arbitrary logarithmic accuracy (in other words, the replacement simply modifies the partition of small- non-singular contributions between the two components in the right-hand side of Eq. (4)). However, and have a very different behaviour as (and, thus, they differently affect the cross section in the large- region4). When , we have and, therefore, the replacement in Eq. (7) would produce the resummation of large and unjustified perturbative contributions in the large- region (strictly speaking, the replacement leads to a cross section that is even not integrable over when : see, in particular, Eqs. (131) and (132) of the arXiv version of Ref. [26] and related accompanying comments). In contrast, when we have and . Therefore, the use of reduces the impact of unjustified large contributions that can be introduced in the small- region through the resummation procedure. Moreover, the behaviour of the form factor at is related to the integral over of the -dependent cross section and, since we have at , our resummation formalism fulfils a perturbative unitarity constraint [26]: after inclusion of the finite component as in Eq. (10), the integration over of our resummed cross sections recovers the fixed-order predictions for the total cross sections. Specifically, the integral over of and at the NNLL+NNLO (NLL+NLO) accuracy completely and exactly (i.e., with no additional higher-order contributions) agrees with the rapidity distribution and the total cross section at NNLO (NLO) accuracy, respectively. In summary, the expressions (7) and (8) in terms of the logarithmic parameter correctly resum the large parametric dependence on at large values of and they introduce parametrically-small perturbative contributions at intermediate or small values of (the coefficients of the perturbative corrections are proportional to powers of with if or if ). After having combined the resummed calculation at NLL accuracy with the complete NLO calculation, as in Eqs. (4) and (10), these parametrically-small corrections produce residual terms that start to contribute at the NLO level. Therefore, the use of has the purpose of reducing the impact of unjustified and large higher-order (i.e., beyond the NLO level) contributions that can be possibly introduced at intermediate and large values of through the resummation of the logarithmic perturbative behaviour at small values of . In particular, no residual higher-order contributions are introduced in the case of the total (integrated over ) cross section (which is the most basic quantity that is not affected by logarithmically-enhanced perturbative corrections).

We add some relevant comments about the dependence of the resummed cross section on the kinematical variables that specify the angular distribution of the leptons with respect to the vector boson. By direct inspection of Eqs. (5) and (11) we see that they involve exactly the same resummation factor . The only difference between the right-hand side of these equations arises form the Born level factors and , which are related as follows through the integration over :

(12)

with the normalization condition

(13)

Although both factors depend on EW parameters (EW couplings, mass and width of the vector boson), they have a different dependence on the relevant kinematical variables. The vector boson distribution (and, analogously, ) involves the Born level total cross section , which depends on , whereas the less inclusive leptonic distribution involves the Born level differential cross section that additionally depends on and also on the transverse momentum of the lepton pair (see the function in the right-hand side of Eq. (12)). To our knowledge the dependence of has not received much attention in the previous literature on transverse-momentum resummation and, therefore, we discuss this issue with some details in Appendix A. Physically, this dependence is a necessary consequence of transverse-momentum conservation and it arises as a -recoil effect in transverse-momentum resummation. At the LO in perturbation theory the lepton angular distribution is determined by the Born level production and decay process of the vector boson, which carries a vanishing transverse momentum. Through the resummation procedure at fixed lepton momenta, higher-order contributions due to soft and collinear multiparton radiation dynamically produce a finite value of the transverse momentum of the lepton pair, and this finite value of has to be distributed between the two lepton momenta by affecting the lepton angular distribution. This -recoil effect on the Born level angular distribution is a non-singular contribution to the cross section at small values of and, therefore, it is not directly and unambiguously computable through transverse-momentum resummation. In other words, the Born level function in Eq. (12) has the form

(14)

where is uniquely determined, whereas the small- corrections of has to be properly specified. In any physical computations of lepton observables (i.e., in any computations that avoid possible unphysical behaviour due to violation of momentum conservation for the decay process ) through transverse-momentum resummation, a consistent -recoil prescription has to be actually (either explicitly or implicitly) implemented5. Note that, after having combined the resummed and finite components as in Eqs. (4) and (10), the recoil effects lead to contributions that start at (i.e., NLO) in the case of our resummed calculation at NNLL+NNLO accuracy (correspondingly, these contributions start at in the case of NLL+NLO accuracy). Obviously there are infinite ways of implementing the -recoil effect, and in Appendix A we explicitly describe a very general and consistent procedure6. Note that the -recoil effect completely cancels after integration over the leptonic variables (see Eq. (13)).

Our resummed calculation of the sixfold differential distribution in Eq. (2) is implemented in the numerical code DYRes, which allows the user to apply arbitrary kinematical cuts on the momenta of the final-state leptons and to compute the corresponding relevant distributions in form of bin histograms. We add some comments on the numerical implementation of our calculation. In Eqs. (7)–(9) we have illustrated the structure of the resummed component in the double Mellin space. Through the inverse Mellin transformation, this structure can equivalently be expressed in terms of convolutions with respect to longitudinal momentum fractions and (see Eq. (3)). In the DYRes code, the Mellin inversion is carried out numerically. The results for the NNLO term in Eq. (9) are presented in Ref. [74] in analytic form directly in space. These results have to be transformed in Mellin space. Then, the Mellin inversion requires the numerical evaluation of some basic -moment functions that appear in the expression of : this evaluation has to be performed for complex values of , and we use the numerical results of Ref. [82]. This implementation of the resummed component is completely analogous to that of the DYqT code [39, 40] and of other previous computations [27]. Nonetheless, the efficient generation of ‘vector boson events’ according to the multidifferential distribution of Eq. (4) and the inclusion of the leptonic decay are technically non trivial, and this requires substantial improvements in the computational speed of the numerical code that evaluates the resummed component of the cross section. The fixed-order (NLO and NNLO) cross section in Eq. (10) and then the finite component of the cross section in Eq. (4) are evaluated through an appropriate modification of the DYNNLO code [5]: DYNNLO is particularly suitable to this purpose, since it is based on the subtraction formalism [83], which uses the transverse-momentum resummation formalism to construct the subtraction counterterms.

Using the resummation expansion parameter in Eq. (7) and the matching procedure (which implements the perturbative unitarity constraint on the total cross section) with the complete fixed-order calculation, our resummation formalism [26] formally achieves a uniform theoretical accuracy in the region of small and intermediate values of , and it avoids the introduction of large unjustified higher-order contributions in the large- region. In the large- region, the results of the resummed calculation are consistent with the customary fixed-order results and, typically [26, 40], show larger theoretical uncertainties (e.g., larger dependence with respect to auxiliary-scale variations) with respect to the corresponding fixed-order results. This feature is not unexpected, since the theoretical knowledge (and the ensuing resummation) of large logarithmic contributions at small cannot improve the theoretical predictions at large values of . In the large- region, where the resummed calculation shows ‘unjustified’ large uncertainties and ensuing loss of predictivity with respect to the fixed-order calculation, the reliability of the resummed calculation is superseded by that of the fixed-order calculation. In this large- region, we can simply use the theoretical results of the fixed-order calculation. In the computation of quantities that directly and explicitly depend on (e.g., the transverse-momentum spectrum of the vector boson), it is relatively straightforward to identify and select ‘a posteriori’ the large- region where the resummed calculation is superseded by the fixed-order calculation. In the present work, however, we are also interested in studying kinematical distributions of the vector boson decay products: our goal is thus to generate the full kinematics of the vector boson and its (leptonic) decay, to apply the required acceptance cuts, and to compute the relevant distributions of the lepton kinematical variables. In this framework, the actual results can become sensitive to the large- region in which the resummed calculation cannot improve the accuracy of the fixed-order calculation. To reduce this sensitivity, in the DYRes implementation of the resummed calculation we thus introduce a smooth switching procedure at large value of by replacing the resummed cross section in Eq. (4) as follows:

(15)

where the function is defined as

(16)

and the function is chosen as

(17)

We have quantitatively checked that the value of the parameter can be ‘suitably’ chosen in the large- region, and that both parameters and can be consistently chosen so as not to spoil our unitarity constraint (in Sect. 3.1 we show that the integral over of our NLL+NLO and NNLL+NNLO resummed results still reproduces well the NLO and NNLO total cross sections). We note that we do not introduce any switching procedure in the DYqT calculation (though, its introduction is feasible) since, as previously mentioned, the identification of the large- region is straightforward in the computation of .

We recall [26] that the resummed form factor of Eq. (7) is singular at very large values of . The singularity occurs in the region , where is the momentum scale of the Landau pole of the perturbative running coupling . This singularity is the ‘perturbative’ signal of the onset of non-perturbative (NP) phenomena at very large values of (which practically affect the region of very small transverse momenta). In this region NP effects cannot any longer be regarded as small quantitative corrections and they have to be taken into account in QCD calculations. A simple and customary procedure to include NP effects is as follows. The singular behaviour of the perturbative form factor is removed by using a regularization procedure7 and the resummed expression in Eq. (7) is then multiplied by a NP form factor and it is inserted as integrand of the space integral in Eq. (6). The regularization procedure that was used in the DYqT calculation [40] is the ‘minimal prescription’ of Ref. [84,