Azimuthal asymmetries in QCD hard scattering: infrared safe but divergent
Azimuthal asymmetries in QCD hard scattering:
[0.2cm] infrared safe but divergent
Stefano Catani Massimiliano Grazzini and Hayk Sargsyan
INFN, Sezione di Firenze and Dipartimento di Fisica e Astronomia,
Università di Firenze, I-50019 Sesto Fiorentino, Florence, Italy
Physik-Institut, Universität Zürich, CH-8057 Zürich, Switzerland
We consider high-mass systems of two or more particles that are produced by QCD hard scattering in hadronic collisions. We examine the azimuthal correlations between the system and one of its particles. We point out that the perturbative QCD computation of such azimuthal correlations and asymmetries can lead to divergent results at fixed perturbative orders. The fixed-order divergences affect basic (and infrared safe) quantities such as the total cross section at fixed (and arbitrary) values of the azimuthal-correlation angle . Examples of processes with fixed-order divergences are heavy-quark pair production, associated production of vector bosons and jets, dijet and diboson production. A noticeable exception is the production of high-mass lepton pairs through the Drell–Yan mechanism of quark-antiquark annihilation. However, even in the Drell–Yan process, fixed-order divergences arise in the computation of QED radiative corrections. We specify general conditions that produce the divergences by discussing their physical origin in fixed-order computations. We show lowest-order illustrative results for asymmetries (with ) in top-quark pair production and associated production of a vector boson and a jet at the LHC. The divergences are removed by a proper all-order resummation procedure of the perturbative contributions. Resummation leads to azimuthal asymmetries that are finite and computable. We present first quantitative results of such a resummed computation for the asymmetry in top-quark pair production at the LHC.
Angular distributions of final-state particles produced by high-energy collisions are known to be relevant observables for the understanding of the underlying dynamics of their production mechanism. In this paper we deal with azimuthal-angle distributions and related asymmetries.
In the case of collisions that are produced by spin polarized particles, azimuthal angles with respect to the spin direction can be specified and measured: the analysis of the ensuing azimuthal correlations is a much studied topic and the subject of intense ongoing investigations (see, e.g., Ref.  and references therein).
Azimuthal correlations in spin unpolarized collisions are less studied. A relevant exception is the lepton azimuthal distribution  of high invariant mass lepton pairs that are produced in hadron collisions through the Drell–Yan (DY) mechanism of quark-antiquark annihilation. Angular distributions for the DY process are a well known topic that has been deeply studied at both the theoretical and experimental levels (see, e.g., Refs. [3, 4, 5] and references therein).
In the case of spin unpolarized collisions, much attention to azimuthal asymmetries has been devoted in recent years within the context of a QCD framework based on the introduction of non-perturbative transverse-momentum dependent (TMD) distributions of quarks and gluons inside the colliding unpolarized hadrons. In particular, the TMD distribution of linearly-polarized gluons  inside unpolarized hadrons plays a distinctive role since it leads to specific and modulations of the dependence on the azimuthal-correlation angle . TMD distribution studies of such modulations have been performed, for instance, for the production process of heavy quark-antiquark () pairs  and for the associated production of a virtual photon and a jet ( jet) . Many more related studies can be found in the list of references of Ref. .
In this paper we consider spin unpolarized collisions and, specifically, we focus our discussion on hadron–hadron collisions, although many features that we discuss are generalizable to (and, hence, valid for) lepton–hadron and lepton–lepton collisions. We consider the production of a system of two or more particles with a high value of the total invariant mass, and we examine the azimuthal correlations between the momenta of the system and of one of its particles. Roughly speaking, the relevant azimuthal angle is related to the difference between the azimuthal angles of the transverse momenta of the system and of the particle. Considering high values of the invariant mass of the system (additional kinematical cuts on the momentum of the produced particle can be applied or required), we select the hard-scattering regime of the production process. In this regime, azimuthal correlations are infrared and collinear safe observables , and they can be studied and computed by applying the standard QCD framework of factorization in terms of perturbative partonic cross sections and parton distribution functions (PDFs) of the colliding hadrons. Starting from the results in Refs. [10, 11] and, especially, from those in Ref.  and the analysis on the role of soft wide-angle radiation that is performed therein, we develop and present some general (process-independent) considerations on azimuthal correlations and asymmetries.
We point out with some generality that, in spite of the infrared and collinear safety of azimuthal-correlation observables, their computation at fixed perturbative orders leads to divergences. The divergences appear also in basic quantities such as the total cross section at fixed (and arbitrary) values of the azimuthal-correlation angle . Example of processes with fixed order (f.o.) divergences are heavy-quark pair () production, associated production of vector () or Higgs bosons and jets (e.g., ’ jet’ with ), dijet and diboson production. We support our theoretical discussion by performing the numerical calculation of modulations (for some values of ) at the lowest perturbative order for some specific processes. The numerical results are found to be consistent with the divergent behaviour if in the case of top-quark pair () production and if in the case of associated ’ jet’ production. A noticeable exception to the appearance of QCD divergences is the case of the DY process. However, even in the case of the DY process, we predict the presence of f.o. divergences in the perturbative computation of QED radiative corrections. Moreover, analogous divergences arise in the case of lepton–lepton collisions from the computation of radiative corrections in a pure QED context (e.g., in the QED inclusive-production process , or even in the simpler process ). In summary, processes that at the leading order (LO) are produced with an exactly vanishing value, , of the transverse momentum of the system tend to develop sooner or later (in the computation of QCD and QED radiative corrections at subsequent perturbative orders) f.o. divergences.
A possible reaction to this state of affairs is to advocate non-perturbative strong-interactions dynamics and related non-perturbative QCD effects that can cure the f.o. divergences. This cure would spoil the common wisdom according to which non-perturbative contributions to infrared and collinear safe observables are (expected to be) power suppressed, namely, suppressed by some inverse power of the relevant hard-scattering scale (as set by the high invariant mass of the system ). Moreover, non-perturbative strong-interactions dynamics cannot be advocated in the case of f.o. divergences in a pure QED context.
We pursue a more conventional viewpoint within perturbation theory. We examine the origin of the singularities for azimuthal-correlation observables order by order in perturbation theory, and we proceed to resum the singular terms to all orders [11, 12].
The singularities originate from the kinematical region where . We identify two sources of singularities. One source [10, 11] is initial-state collinear radiation in the case of systems that can be produced with vanishing by gluon initiated partonic collisions. The other source  is soft wide-angle radiation in the case of systems that contain colour-charged particles (or electrically-charged particles for QED radiative corrections). Collinear radiation is, per se, responsible for singularities in (and ) harmonics with and , only . Soft radiation is responsible for singularities that, in principle, can affect harmonics with any (both even and odd) values of ().
At high perturbative orders collinear and soft radiation is mixed up, and the singularities are enhanced by logarithmic () contributions. We discuss in general terms how the all-order resummation of the logarithmic contributions leads to ( integrated) azimuthal asymmetries that are finite and computable. Such procedure is effective in both QCD and QED contexts. In the QCD case this does not imply that non-perturbative effects have a negligible quantitative role in the region of very-low values of , but these effects give small contributions to -integrated azimuthal-correlation observables. Within perturbative QCD, the most advanced theoretical treatment of singular azimuthal correlations that is available at present regards the process of heavy-quark pair production . We use the process of top-quark pair production in collisions at the LHC as a prototype to present a first illustration of the resummation of singular azimuthal asymmetries at the quantitative level.
A correspondence can be established [10, 11, 13] between the TMD distribution of linearly-polarized gluons and singular azimuthal correlations that are due to initial-state collinear radiation in gluon initiated partonic subprocesses. Therefore, both the framework used in Refs. [7, 8] and our perturbative QCD treatment lead to corresponding and azimuthal modulations. In this respect, the TMD distribution of linearly-polarized gluons can be regarded as the extension of specific features of QCD collinear dynamics from perturbative to non-perturbative transverse-momentum scales. Much discussion of the present paper is related to soft radiation effects that occur in processes with colour-charged final-state particles. Soft radiation produces azimuthal asymmetries (which are singular at f.o. and finite after resummation) for both quark (or antiquark) and gluon initiated subprocesses, and for harmonics with arbitrary even values ()  and also odd values of . These soft wide-angle radiation effects are unrelated to the TMD distribution of linearly-polarized gluons.
The paper is organized as follows. In Sect. 2 we start our discussion on azimuthal asymmetries and we specify the conditions that lead to divergences in f.o. computations. In Sect. 3 we discuss azimuthal asymmetries in two examples of hadron collider processes, i.e. the production of lepton pairs through the DY mechanism and the production of a pair, by contrasting the different behaviour of the corresponding azimuthal harmonics. In Sect. 4 we start our analysis of the small- limit: in Sect. 4.1 we focus on the small- behavior at f.o. and in Sect. 4.2 we recall the transverse-momentum resummation procedure for the case of azimuthally-averaged cross sections. In Sect. 5.1 we discuss the origin of singular azimuthal correlations and present illustrative lowest-order results for the jet production process. In Sect. 5.2 we outline the resummation procedure of singular terms in the case of azimuthally-correlated cross sections and we contrast the small- behavior expected for the resummed cross section with the known behavior of the azimuthally-averaged transverse-momentum cross section. In Sect. 5.3 we focus on the harmonic for production and we present first quantitative results of a resummed calculation at next-to-leading logarithmic accuracy. After the matching with the complete next-to-leading order (NLO) result, the resummed computation offers an effective ‘lowest-order’ prediction for the harmonic. We also comment on the possible role of non-singular terms. In Sect. 6 we summarize our results.
2 Azimuthal correlations and asymmetries in fixed-order perturbation theory
Our discussion on azimuthal correlations has a high generality. To simplify the illustration of the key points, we consider the simplest class of processes, in which the produced high-mass system in the final state is formed by only two ‘particles’ in generalized sense (point-like particles and/or jets).
We consider the inclusive hard-scattering hadroproduction process
in which 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 that is formed by two ‘particles’, and , with four momenta and , respectively. The two particles can be point-like particles or hadronic jets (), which are reconstructed by a suitable (infrared and collinear safe) jet algorithm. As for the case of point-like particles, the most topical process is the production of a high-mass lepton pair through the DY mechanism of quark–antiquark annihilation. We consider many other cases such as, for instance, the production of a photon pair (), a pair of top quark and antiquark () or a pair of vector bosons (), in addition to dijet production () and associated production processes such as vector boson plus jet (). The invariant masses of the two particles have a little role in the context of our discussion (and they do not affect any conceptual aspects of our discussion). The system has total invariant mass (), transverse momentum and rapidity (transverse momenta and rapidities are defined in the centre–of–mass frame of the colliding hadrons). We require that is large (, being the QCD scale), so that the process in Eq. (1) can be treated within the customary perturbative QCD framework. We use to denote the centre–of–mass energy of the colliding hadrons, which are treated in the massless approximation ().
The dynamics of the production process in Eq. (1) can be described in
terms of five kinematical variables: the total mass , transverse momentum
and rapidity of the system and two independent angular variables
that specify the kinematics of the two particles and with respect to
the total momentum of . These two angular variables are a
polar-angle variable and an azimuthal-angle variable. To be definite and to avoid
the use of ‘exotic’ variables, we refer to a widely-used set of angular variables
and we use the polar angle and the azimuthal angle (of particle
) in the Collins–Soper (CS) rest frame
The variable is the relevant variable for our discussion of azimuthal correlations. We remark that specifies the azimuth of one of the two particles in the system with respect to the total momentum of the system. In particular, we also remark that we are not considering the relative azimuthal separation (, with , is the azimuthal angle of the transverse-momentum vector in the centre–of–mass frame of the colliding hadrons) between the two particles. However, we can anticipate (we postpone comments on this point) that our main findings are not specific of the CS frame, and they are equally valid for other azimuthal variables with respect to the system (for instance, we can consider the azimuthal angle in a different rest frame of or, simply, the azimuthal difference , where is the azimuthal angle of in the centre–of–mass frame of the colliding hadrons).
Using the kinematical variables in the CS frame, we can consider azimuthal distributions for the process in Eq. (1). The most elementary azimuthal-dependent observable is the azimuthal cross section at fixed invariant mass,
All these quantities are related (from the less inclusive to the more inclusive case) through integration of kinematical variables (for instance, the azimuthal cross section in Eq. (2) is obtained by integrating Eq. (3) over ) and, in particular, the azimuthal integration of Eq. (2) gives the total cross section (at fixed invariant mass) of the process:
Obviously, we can also consider differential cross sections that are integrated over a certain range of values of the invariant mass.
Since all the cross sections that we have just mentioned are infrared and collinear safe quantities , they can be computed perturbatively within the customary QCD factorization framework (see Ref.  and references therein). The only non-perturbative (strictly speaking) input is the set of PDFs of the colliding hadrons. The PDFs are convoluted with corresponding partonic differential cross sections that can be evaluated as a power series expansion in the QCD running coupling .
This perturbative QCD framework is applicable at any finite (and arbitrary) fixed perturbative order. Despite this statement, the first main observation that we want to make is that the f.o. perturbative calculation of the azimuthal distributions can lead to divergent (and, hence, unphysical and useless) results. More specifically, the f.o. calculation of the azimuthal cross section of Eq. (2) gives the following results:
We mean that the perturbative computation of for the DY
process gives a finite result order–by–order in QCD perturbation theory, while
in most of the other cases (some of them are listed in the right-hand side of
Eq. (6)) the computation gives a divergent (meaningless) result for
any values of the azimuthal angle starting from some
We note that the integration over of gives the
total cross section (see Eq. (5)), which is known to be finite at any
Therefore, the divergent behaviour that is highlighted in Eq. (6)
originates from the azimuthal-correlation
where the notation denotes the azimuthal average and is the total cross section in Eq. (5).
To understand the origin of the divergent behaviour in Eq. (6), we first comment about kinematics. If the system has vanishing transverse momentum (), the rest frame of is obtained by simply applying a longitudinal boost to the centre–of–mass frame of the colliding hadrons. Any additional rotation in the transverse plane of the collision leaves the system at rest and makes the particle azimuthal angle ambiguously defined. Owing to the azimuthal symmetry of the collision process, if there is no preferred direction to define . In other words, considering the azimuthal angle (as defined in the CS frame or any other rest frame of ) and performing the limit , we have
where and are the azimuthal angles of the corresponding transverse-momentum vectors in the centre–of–mass frame of the colliding hadrons. If , is not defined and, consequently, is not (unambiguously) defined. At the strictly formal level, to define the azimuthal correlation we have to exclude the phase space point at . If we want to consider integrated azimuthal correlations (such as, for instance, the cross section in Eq. (7)), we have to introduce a minimum value of () and eventually perform the limit . The divergences that are highlighted in Eq. (6) do not appear by considering azimuthal correlations at fixed and finite values of (e.g., the azimuthal-dependent cross sections in Eq. (3) and (4)). These divergences are related to the limit after integration of the azimuthal correlations over .
Our discussion on the limit reconciles the appearance of divergences in Eq. (6) with the perturbative criterion of infrared and collinear safety at the formal level: no divergence occurs provided , namely, provided the azimuthal-correlation observable is well (unambiguously) defined. However, a single phase space point at is physically harmless. Therefore, the integrated azimuthal correlations (i.e., their limiting behaviour as ) are finite and measurable quantities. Moreover, they are also finite and measurable if is non-vanishing, or, equivalently if is not vanishing and fixed at an arbitrarily small value. The divergence in Eq. (6) implies that the dependent azimuthal correlations become singular at small values of , and that this singular behaviour is not integrable over in the limit . This unphysical behaviour of f.o. perturbative QCD at small values of and the divergence of the integrated azimuthal correlations certainly require some deeper understanding to have a QCD theory of physically measurable azimuthal correlations. As a possible shortcut, one can still use f.o. perturbative QCD but avoid the region of small values of . In this case one still needs some understanding of the phenomenon in order to assess the extent of the dangerous small- region where the f.o. predictions are ‘unphysical’ or, anyhow, not reliable at the quantitative level.
We note that the kinematical relation in Eq. (8) implies that the azimuthal angle in the CS frame has no privileged role in the context of our discussion of azimuthal correlations in the small- region. The main features of the small- behaviour of azimuthal correlations that are discussed in this paper are basically unaffected by using azimuthal angles as defined in other rest frames of the system , or, by using other related definitions of azimuthal angles. For instance, one can simply replace the CS frame angle with one of the relative azimuthal angles () in the centre–of–mass frame of the colliding hadrons. Alternatively, one can use the difference between the azimuthal angles (in the centre–of–mass frame of the colliding hadrons) of the two transverse-momentum vectors and . All these definitions of the relevant (for our purposes) azimuthal-angle variable turn out to be equivalent (because of Eq. (8)) in the limit (or, at very small values of ). In the following we continue to mainly refer ourselves to the azimuthal angle in the CS frame, although all the basic features that we discuss are unchanged by considering the other definitions of the azimuthal angle.
We also note that the discussion that we have presented so far about azimuthal correlations can be generalized in a straightforward way to consider the case in which the system is formed by more than two particles. In the multiparticle case, we can simply examine azimuthal correlations that are defined by using the azimuthal angles of the various particles in in a specified rest frame of (such as the CS frame), by simply using the various relative azimuthal angles in the centre–of–mass frame of the colliding hadrons, or by using some other related (i.e., equivalent in the limit ) azimuthal variables. Since the multiparticle cases do not present any additional conceptual issues with respect to the two-particle case, in the following (for the sake of technical simplicity) we continue to mostly consider the case of systems that are formed by two particles.
The f.o. perturbative divergences of the azimuthal correlations originate from QCD radiative corrections due to inclusive emission in the final state of partons with low transverse momentum (soft and collinear partons). The origin is discussed in more detail in the following Sections of the paper (see, in particular, Sect. 5.1). Since the azimuthal correlations behave differently in different processes (as stated in Eq. (6)), we anticipate here the conditions that produce the divergent behaviour.
Azimuthal correlations can have f.o. divergences if the final-state system can be produced by the partonic subprocess (see also Eq. (15) and accompanying comments) where
The conditions in (9) and (10) follow from a generalization of the results in Refs. [11, 12]. The divergences arise from the computation of the QCD radiative correction to the partonic process (they do not arise in the computation of that partonic subprocess itself!). Specifically, the f.o. divergences originate from collinear-parton radiation  in the case of the condition (9) and from soft-parton radiation  in the case of the condition (10). We remark that one of the conditions in (9) and (10) is sufficient to produce the f.o. divergences. In particular, the condition (10) necessarily produces f.o. divergences, while the condition (9) produces divergences with some ‘exception’ (words of caution) in few specific cases (see below). Having discussed the source of f.o. divergences in general terms, we can comment on some specific processes.
The production of heavy-quark pairs such as, for instance, can occur through the partonic subprocesses of quark-antiquark annihilation () and gluon fusion (). One of these subprocesses has initial-state gluons and, moreover, both and have QCD colour charge. Therefore, if both conditions (9) and (10) are fulfilled and the corresponding azimuthal correlations have f.o. divergences (as stated in Eq. (6)). The same reasoning and conclusions apply to the production of and (for instance, can be produced through , where at the lowest-order carries QCD colour charge). In the case of production the final-state photons do not carry QCD colour charge. However, diphoton production at the next-to-next-to-leading order (NNLO) can occur through the subprocess (the interaction is mediated by a quark loop): therefore, due to the condition (9), the azimuthal correlations for diphoton production diverge starting from the NLO computation. The cases and behave similarly to , and they also lead to azimuthal correlations that have f.o. divergences.
We consider the DY process, where and the high-mass lepton pair originates from the decay of a vector boson (). The final-state leptons have no QCD colour charge and, therefore, the condition (10) is not fulfilled. The partonic subprocesses and are forbidden by colour conservation, and the partonic subprocess is forbidden by the spin 1 of the vector boson. The DY lepton pair can be produced through , but this partonic subprocess has no initial-state gluons. Therefore, also the condition (9) is not fulfilled and the azimuthal correlation for the DY process have no f.o. divergences in the computation of QCD radiative corrections (as it is well known and recalled in Eq. (6)). However, we remark that f.o. divergences do appear in the computation of QED (or, generally, electroweak) radiative corrections to azimuthal correlations for the DY process (this important point is discussed in more detail in Sect. 3).
We can comment on the production of a system of colourless particles, such as or , in the specific case (or, better, within the approximation) in which those particles originate from the decay of a spin-0 boson, such as the Standard Model (SM) Higgs boson (e.g., or ). In this case the production mechanism (followed by the decay) is allowed. However, due to the spin-0 nature of , the decay dynamically decouples from the production mechanism: the angular distribution of the final-state particles ( or ) with momenta and is dynamically flat (it has no dynamical dependence on the decay angles, since it can only depend on the Lorentz invariant , namely, on the invariant mass of the produced pair of particles). As a consequence, although the condition (9) is fulfilled, there are no azimuthal correlations in this specific case and, hence, there are no accompanying f.o. divergences. The absence of azimuthal correlations follows from the requirement that the two final-state particles are due to the decay of a spin-0 boson. As a matter of principle at the conceptual level we note that, considering the final-state system or , the various subprocesses that contribute to the production mechanism (subprocesses with and without an intermediate , and corresponding interferences) are not physically distinguishable (this is, strictly speaking, correct although the applied kinematical cuts can sizeably affect the relative size of the various contributing subprocesses). As we have previously discussed, the production of such systems without the intermediate decay of a spin-0 boson leads to azimuthal correlations with f.o. perturbative divergences. Therefore, if the condition (9) is fulfilled, we can conclude that sooner or later (in the computation of subsequent perturbative orders) QCD radiative corrections produce non-vanishing azimuthal correlations and ensuing f.o. divergences.
To the purpose of studying azimuthal correlations and presenting corresponding quantitative results, we find it convenient to introduce harmonic components of azimuthal-dependent cross sections. In particular, we define the -th harmonic:
where is a positive integer . Note that, in view of our previous discussion on the origin of the divergences that are mentioned in Eq. (6), we have introduced a minimum value of (). We usually consider values of that are proportional to the invariant mass of the system (, with a fixed parameter ), although also fixed values of can be used. One can also consider -th harmonics of more differential cross sections (e.g., differential with respect to and as in Eq. (4)). The -th harmonic in Eq. (11) is defined by using the weight function ; -th harmonics with respect to the sine function can also be considered by simply replacing in the integrand of Eq. (11). In particular, the knowledge of the harmonics with respect to both and for all integer values of is equivalent to the complete knowledge of the azimuthal-correlation cross section in Eq. (7). Note that the -th harmonic in Eq. (11) is not a positive definite quantity. Although the physical azimuthal cross section is positive definite, the weight function (or ) has no definite sign.
The -th harmonic with weight (or ) gives a direct measurement of the (or ) asymmetry of the azimuthal-dependent cross section. The QCD computation of the -th harmonic gives a finite result provided is not vanishing. On the basis of Eq. (6), in the limit some harmonics (asymmetries) can be divergent if computed at some f.o. in QCD perturbation theory. We cannot draw general conclusions on harmonics with odd values of (), but from the results in Refs. [11, 12] we know that harmonics with even values of () can have f.o. divergences. Specifically, if the condition (9) is fulfilled the harmonics with and have f.o. divergences , and if the condition (10) is fulfilled all the -even ( and so forth) asymmetries have f.o. divergences . In Sect. 3 we explicitly show quantitative results on f.o. computations of harmonics for the DY and production processes. In Sect. 5.1 we also show f.o. results for production and we further comments on -odd harmonics.
As mentioned in the Introduction and discussed in Sect. 5.2, the f.o. divergences of the azimuthal asymmetries can be cured by a proper all-order perturbative resummation of QCD radiative corrections. The resummed computation leads to azimuthal correlations and asymmetries that are finite, as it is the case of the corresponding physical (and measurable) quantities.
The lepton angular distribution of the DY process is a much studied subject both experimentally and theoretically. Recent measurements of the lepton angular distribution for production at LHC energies ( TeV) are presented in Refs. [3, 4], together with corresponding QCD predictions from Monte Carlo event generators and f.o. calculations. A very recent phenomenological study of the DY lepton angular distribution is performed in Ref. . As for previous literature on the subject, we mainly refer the reader to the list of references in Refs. [3, 4, 5].
The QCD structure of the DY lepton angular distribution is well known, and it is a consequence of the spin-1 nature of the vector bosons involved in the DY mechanism. To be definite we consider the DY process at the Born level with respect to electroweak (EW) interactions. Within this Born level framework, the DY differential cross section is expressed in terms of a leptonic tensor (for the EW leptonic decay of the vector boson) and a hadronic tensor (for the hadroproduction of the vector boson). The QCD production dynamics is embodied in the hadronic tensor and the two tensors are coupled through the spin (helicity) correlations of the vector boson. Owing to quantum interferences, we are dealing with 9 helicity components of the five-fold differential cross section in Eq. (4) (the hadronic tensor is a helicity polarization matrix, due to the 3 polarization states of the vector boson), which, therefore, can be expressed  as a linear combination of 9 spherical harmonics or, better, harmonic polynomials (the leptonic tensor has a polynomial dependence of rank 2 on the lepton momenta) of the lepton angles and (we specifically use the CS frame). For the illustrative purposes of our discussion of azimuthal correlations, it is sufficient to consider the four-fold differential cross section , which is obtained by integrating Eq. (4) over . We write
where is the DY differential cross section integrated over , and the shorthand notation denotes differential cross section components that only depend on . The azimuthal dependence of Eq. (12) is entirely given by the harmonics that are explicitly written in the right-hand side of Eq. (12). The subscript in is defined according to a customary notation in the literature [16, 15, 3, 4], such that , where the functions are known as ‘DY angular coefficients’ and are normalization factors (specifically, we have ). A point that we would like to remark is that the QCD dependence of the azimuthal correlations of the DY cross section involves only four harmonics. The five-fold cross section in Eq. (4) for the DY process is expressed in terms of 9 harmonic polynomials of and : however, their dependence on is given only in terms of the four harmonics that also appear in Eq. (12).
At LO in QCD perturbation theory, the DY cross section
is due to the
partonic subprocess , which is of
. At this order in , there is no azimuthal dependence.
Azimuthal correlations start to appear at the
We recall that the cross section components and in Eq. (12) are parity conserving, while and are parity violating. Moreover, vanishes in the limit of vanishing axial coupling of to either quarks or leptons. Therefore, if , only the harmonic contributes in Eq. (12), while in the case of the five-fold differential cross section of Eq. (4) there is an additional non-vanishing azimuthal contribution that is proportional to and it is due to .
All the quantitative results that are presented in this paper refer to collisions at the LHC energy TeV. In the QCD calculations we use the set MSTW2008  of PDFs at NLO.
To present some illustrative results for the DY process we consider on-shell production and its leptonic decay in an electron–positron pair. Our QCD calculation is performed by using the numerical Monte Carlo program DYNNLO . The EW parameters are specified in the scheme and we use the following values: the Fermi constant is GeV, the mass of the boson is GeV and the mass of the boson is GeV. We use equal values of the factorization () and renormalization () scales and we set .
We specifically consider the numerical calculation of various asymmetries (see Eq. (11)), and is the azimuthal angle of the electron in the CS frame. We evaluate the asymmetries at their non-trivial lowest order in f.o. perturbation theory and, therefore, we compute all the NLO tree-level partonic processes whose final state is ’ parton’. Our numerical results are presented in Fig. 1-left.
The -th harmonics are integrated over the lepton polar angle and over the rapidity and transverse-momentum of the pair, and they are computed as a function of , where is the minimum value of (). The results in Fig. 1 are obtained for very small values of in the range . As we have already recalled, the azimuthal asymmetries for the DY process have no f.o. divergences. Indeed, the results for that are presented in Fig. 1-left show that the corresponding asymmetries are basically independent of for very small values of and finite (the results in Fig. 1-left practically coincide with the numerical evaluation at ). The result for the asymmetry (blue line) in Fig. 1-left is consistent with a vanishing value, in agreement with the general expression in Eq. (12) (the very small deviations that are observed for in Fig. 1-left give an idea of the numerical uncertainties in our calculation of the various asymmetries). The result for the asymmetry (black line) gives a non-vanishing value (it corresponds to the integral of the differential cross section component in Eq. (12)). A non-vanishing value is obtained also for (red line), and it corresponds to the computation of the cross section component in Eq. (12). Note that the result reported in Fig. 1-left is rescaled by a factor of 0.1. Therefore, by inspection of Fig. 1-left we can see that the asymmetry is approximately a factor of 5 larger than the asymmetry.
In Fig. 1-left we also report the result for the harmonic (dashed line) of ’ parton’ production. The harmonic corresponds to the total (azimuthally-integrated) cross section, and it receives contributions from both real and virtual emission subprocesses. Real and virtual terms are separately divergent and their divergences cancel in the total contribution. In Fig. 1-left we report the result of the harmonic computed exactly as specified in Eq. (11), namely, by applying a non-vanishing lower limit on . Therefore, our computation selects only the real-emission term due to the ’ parton’ subprocesses. As is well known (see also Sect. 4.1), this real-emission term diverges in the limit and the divergent behaviour is proportional to : in Fig. 1-left we clearly see the increasing behaviour of the harmonic as . In the actual computation of the total cross section, the result in Fig. 1-left has to be combined with the NLO real-emission term at still smaller values of and with the LO and NLO virtual terms, thus leading to a total finite result. For the sake of completeness, we report the value (including the numerical error of the Monte Carlo integration) of the NLO total cross section that we obtain by using the same parameters as used in the results of Fig. 1-left: it is pb. We note that is roughly 100 times larger than the value of the asymmetry in Fig. 1-left.
The DY process is quite ‘special’ with respect to azimuthal correlations: the azimuthal correlations are finite order-by-order in QCD perturbation theory and their general QCD structure involves only 4 azimuthal harmonics (as shown in Eq. (12)). For most of the other hard-scattering processes (with few possible exceptions such as production, as discussed in Sect. 2) azimuthal correlations behave differently: usually they have an azimuthal dependence that involves an infinite set of harmonics (all values of ) and in many cases f.o. QCD computations lead to divergences.
We consider the production of heavy-quark pairs and we treat the heavy quark and antiquark as on-shell particles. Our discussion equally applies to any heavy quark, but we specifically consider top quarks since the on-shell treatment is more suitable in this case. In Fig. 1-right we present results for production that are obtained analogously to the DY results presented in Fig. 1-left.
Our QCD computation of production is performed by using the numerical Monte Carlo program of Ref. , which includes QCD radiative corrections at NLO (it uses the NLO scattering amplitudes of the MCFM program ) and part of the NNLO contributions. We set and we use the value GeV for the mass of the top quark. We consider the azimuthal angle of the top quark in the CS frame, and in Fig. 1-right we present the numerical results of various asymmetries (see Eq. (11)) after their integration over the invariant mass of the pair. As in the case of the results in Fig. 1-left, the results are integrated over the polar angle of the top quark and over the rapidity and transverse momentum () of the pair. The azimuthal asymmetries are evaluated (as a function of ) at their non-trivial lowest order (i.e., ) in f.o. perturbation theory and, therefore, we compute all the NLO tree-level partonic processes whose final state is ’ parton’.
Before commenting on the results in Fig. 1-right, we recall some of the results in Ref. . In Ref.  azimuthal correlations for production are computed in analytic form in the small- limit. At the result of Ref.  shows that the -dependent azimuthal-correlation cross section behaves proportionally to in the limit and, hence, it is not integrable over down to the region where . The behaviour is proportional to a non-polynomial function of that, therefore, leads to divergent harmonics for even values of . The spectra of harmonics with odd values of have instead a less singular behaviour at and they are integrable over in the limit .
The numerical results in Fig. 1-right are consistent with the convergent or divergent behaviour predicted in Ref. . The harmonic with (black line) is not vanishing and basically independent of for very small values of . The sign of the harmonic is negative (the harmonic of would be positive, analogously to the corresponding harmonic of the electron in the DY case of Fig. 1-left), and its absolute size (note that it is rescaled by a factor of 0.1 in Fig. 1-right) is roughly a factor of two smaller than the size of the harmonic for production (Fig. 1-left). The harmonics with (red line), (blue line) and (magenta line) in Fig. 1-right have instead an increasing (in absolute value) size for small and decreasing values of : this behaviour is consistent with a dependence, as expected from the analytical results in Ref. . The results for and 6 in Fig. 1-right have no straightforward quantitative implications for physical azimuthal asymmetries since they refer to small values of (and they eventually diverge in the limit ). Nonetheless we observe that the absolute magnitude of the -even asymmetries decreases as increases. As in the case of Fig. 1-left for the DY process, in Fig. 1-right we also present the result of the harmonic (dashed line) for the real-emission process ’ parton’. Analogously to the DY process, the ’ parton’ contribution to the harmonic diverges in the limit , and its dominant behaviour at small is proportional to . At small values of the shape of the -dependence of the result is thus steeper than that of the results for the harmonics with and 6. After combining real and virtual contributions, the harmonic gives the total cross section. The value (including the numerical error of the Monte Carlo integration) of the NLO total cross section that we find is pb. We note that is roughly 200 times larger than the absolute value of the asymmetry () in Fig. 1-right.
As we have anticipated in Sect. 2 with a brief sentence, we expect (and predict) the appearance of f.o. perturbative divergences in the computation of QED radiative corrections to azimuthal correlations for the DY process. To explicitly explain this point we consider some analogies of the DY and production processes. We have illustrated the divergences of the -even asymmetries for production. Part of these divergences arise from the computation of the partonic subprocess and, specifically, they arise from the kinematical region where the radiated final-state gluon is soft and at wide angle with respect to the direction of the initial-state and . From this kinematical region the spectrum receives an azimuthal-correlation contribution that behaves as and that depends on the QCD colour charges of the final-state and . Specifically, this contribution is proportional to the Fourier transformation of the function in Eq. (36) of Ref. . The first-order QED radiative corrections to the DY process involve analogous partonic processes, such as , with a soft and wide-angle photon that is radiated in the final state. These photon radiative corrections produce f.o. QED divergences to azimuthal asymmetries for the DY process. The DY analogue of the function in Eq. (36) of Ref.  is simply obtained by replacing the QCD colour charges of and with the QED electric charges of the DY final-state leptons (by simple inspection of Eq. (36) in Ref. , such replacement leads to a non-vanishing result).
More generally, we remark that the expression in Eq. (12) for the DY process is valid only at the Born level with respect to the EW interactions (although the expression is valid at arbitrary orders in QCD perturbation theory). Including QED radiative corrections, we conclude that the azimuthal-correlation cross section for the DY process receives contributions from and harmonics with arbitrary values of (not only as in Eq. (12)). Moreover, the lowest-order perturbative QED computation of the contributions with -even already leads to a singular (not integrable) behaviour in the limit and to ensuing divergent azimuthal asymmetries upon integration over . The divergences are switched on by QED radiative corrections and can receive additional contributions from powers of in the context of mixed QEDQCD radiative corrections. Obviously, similar divergences arise also in a pure QED context such as, for instance, in the QED computation of the process . Eventually all these divergences originate from the QED analogue of the condition (10): the QED divergences arise if ’at least one the final-state particles with momenta and carries non-vanishing electric charge’.
We note that lepton pairs with high invariant mass can also be produced
throughout the partonic subprocess , in which the
initial-state colliding photons arise from the photon PDF of the colliding
Strictly speaking, lepton pairs that are produced in this way are not physically
To cure the f.o. perturbative divergences of azimuthal correlations one may advocate non-perturbative strong-interactions dynamics and related non-perturbative QCD effects, which can be sizeable in the small- region. However, in the case of -integrated azimuthal correlations (see Eq. (7)) the non-perturbative QCD dynamics should cancel divergent terms proportional to some powers of , and this would imply that non-perturbative QCD effects scale logarithmically with (i.e., these effects would not be suppressed by some power of in the hard-scattering regime ), thus spoiling not only the finiteness but also the infrared safety of the azimuthal correlations. Moreover such non-perturbative cure of the divergent behaviour cannot be effective in the case of QED radiative corrections and, in particular, it cannot be conceptually at work in a pure QED context since QED is well-behaved in the infrared (small-) region. In the following Sections we show that the problem of f.o. divergences in azimuthal correlations has a satisfactory solution entirely within the context of perturbation theory. Namely, the resummation of perturbative corrections to all orders leads to ( integrated) azimuthal asymmetries that are finite and computable. Such solution is effective in both QCD and QED contexts, although in the QCD case this does not imply that the non-perturbative effects have a negligible quantitative role in the small- region.
4 The small- region
The f.o. divergences of azimuthal correlations arise from the small- region and, eventually, from the behaviour at the phase space point where . From a physical viewpoint, however, a single (isolated) phase space point is harmless and, in particular, non-vanishing azimuthal correlations (which are defined with respect to the direction of ) cannot be measured if . These physical requirements imply that the dependence of azimuthal-correlation observables has to be sufficiently smooth in the small- region and, in particular, in the limit .
We specify more formally these smoothness requirements. Since physical measurements cannot be performed at a single phase space point, we split the integration region in the two intervals where (the lower- bin) and (the higher- region). We then consider a generic azimuthal-correlation observable (e.g., multidifferential cross sections as in Eqs. (3) and (4), or -th harmonics as in Eq. (11)) and we denote by and the ‘cross sections’ that are obtained by integration of the observable over the lower- bin and the higher- region, respectively. The smoothness requirements are
where denotes the ‘total’ cross section (i.e., the result obtained by integrating the observable over the entire region of ). These requirements specify azimuthal correlations that are physically well behaved. In particular, Eq. (13) implies that non-vanishing azimuthal correlations cannot be physically observed if , and Eq. (14) implies that the total azimuthal correlation is physically observable. Note, however, that Eqs. (13) and (14) do not specify how exactly behaves in the limit . Nonetheless, Eqs. (13) and (14) imply that the sole phase space point at (where azimuthal-correlation angles are not defined) has no relevant physical role (the lower- bin has an analogous harmless role if is sufficiently small).
The f.o. divergences of azimuthal-correlation observables are due to the fact that does not have a (‘sufficiently’) smooth dependence on at small values of if this quantity is computed order-by-order in perturbation theory. In particular, diverges in the limit and, consequently, is not computable, whereas is definitely not computable (divergent) even if has a finite value.
As we have observed in Sect. 3, also the customary azimuthally-integrated (or azimuthally-averaged) cross section (i.e., the harmonic with in Eq. (11) or Fig. 1) does not fulfil the requirements in Eqs. (13) and (14) (both and can separately be divergent in the limit ). Therefore, the f.o. calculation of azimuthally-integrated cross section can have difficulties (as is well known) in describing the detailed shape in the small- region. In contrast to azimuthal-correlations observables, however, both and are computable for finite values of . In particular, the total cross section is always finite and computable (for arbitrary non-vanishing values of ) within f.o. perturbation theory.
In the following sections we discuss the small- behaviour of cross sections in f.o. perturbation theory and after all-order resummation.
4.1 Perturbative expansion with azimuthal-correlation terms
Among all the hadroproduction processes of the type in Eq. (1), we consider those that at the LO in perturbative QCD are produced by the following partonic subprocesses:
where and are the initial-state massless
colliding partons of the two hadrons and . In the case in which one or
both particles (with momenta ) in is a jet, the notation in
Eq. (15) means that the jet is replaced by a corresponding QCD parton.
Note that the LO process in Eq. (15) is an ‘elastic’ production process,
in the sense that is not accompanied by any additional final-state
The small- behaviour of azimuthal correlations for the processes that we have just specified is partly related to the behaviour of differential cross sections that are integrated over the entire azimuthal-angle region (or, equivalently, that are azimuthally averaged). The behaviour in the azimuthally-integrated (averaged) case is well known (starting from some of the ‘classical’ studies for the DY process [24, 25, 26, 27, 28]), and in our presentation we contrast it with the cases with divergent azimuthal correlations. To remark the differences between the two cases in general terms, we use a shorthand notation, which can be applied to final-state systems with two or more particles (a more refined kinematical notation can be found in Refs. [11, 12]). A generic multidifferential cross section (e.g., Eq. (3), Eq. (4) or related observables) with dependence on and on the azimuthal-correlation angle is simply denoted by , where is the transverse momentum of . Additional kinematical variables that are possibly not integrated (such as rapidities and polar angles) are not explicitly denoted. The dependence on a generic (as discussed in Sect. 2) azimuthal-correlation angle with respect to is denoted through the dependence on the direction of the two-dimensional vector . In practice such dependence will occur through functions of scalar quantities such as, for instance, . In particular, independent of means absence of azimuthal correlations, and the azimuthally-integrated (averaged) cross section corresponds to the azimuthal integration (average) with respect to the direction of .
Owing to transverse-momentum conservation in the process of Eq. (15), at the LO level is produced with vanishing . Its corresponding LO cross section is proportional to ,
and the proportionality factor is simply the LO total ( integrated) cross section. In the presence of the LO sharp behaviour of Eq. (16), NLO QCD radiative corrections are dynamically enhanced in the small- region. The dynamical enhancement is due to QCD radiation of low transverse-momentum partons (soft gluons or QCD partons that are collinear to the initial-state colliding partons), and it has the following general form:
where the dots on the right-hand side denote contributions that are less enhanced (‘non-singular’) in the limit . The ‘coefficients’ () and in Eq. (17) depend on the process and they are independent of (we mean they are independent of the magnitude of the vector ). The important point (see below) is that does depend on the direction of , whereas () do not depend on it. All these coefficients () can depend on the other kinematical variables (e.g., rapidities and polar angles). The QCD running coupling in Eq. (17) and in all the subsequent formulae is evaluated at a scale of the order of (e.g., we can simply assume ).
The terms that we have explicitly written in the right-hand side of Eq. (17) scale as (modulo logs) in the limit (i.e., they scale as under the replacement ). The non-singular terms (which are simply denoted by the dots) represent subdominant (‘power-correction’) contributions in the limit . These are, for instance, terms of the type or, more generally, terms that are relatively suppressed by some powers (modulo logs) of . Independently of their specific form and of their azimuthal dependence, these non-singular terms have an integrable and smooth behaviour in the limit . Because of these features, the non-singular terms and the ensuing azimuthal-correlation effects are well behaved in the small- region. Note, however, that the non-singular terms produce azimuthal effects whose actual size (and behaviour) depends on the specific definition of the azimuthal-correlation angle (see Eq. (8) and related comments in Sect. 2).
The expression in Eq. (17) is the master formula that we use for our discussion of the small- behaviour of azimuthal correlations at NLO and higher orders. Considering the ‘singular’ terms (the NLO terms that scale as , modulo logs, in the limit ), the azimuthal dependence is entirely embodied in . More precisely, the azimuthal-correlation dependence has been separated (as in Eq. (7)) and embodied in that, therefore, gives a vanishing result after azimuthal integration. Using the notation in Eq. (7) we have
This azimuthal-correlation term is absent for the DY process (i.e., in this case), and in all the other processes it has no effect by considering dependent but azimuthally-integrated observables. The presence of such term produces a behaviour that definitely differs from the behaviour studied in the ‘classical’ literature [24, 25, 26, 27, 28] on the small- region and on QCD transverse-momentum resummation.
The other (‘classical’) terms in Eq. (17) are proportional to the coefficients (), and the symbol denotes the (‘singular’) ‘plus’-distribution, which is defined by its action onto any smooth function of . The definition is
where () in Eq. (17), and is a scale of the order of (at the formal level, varying the value of changes the plus-distributions and the coefficient , so that the right-hand side of Eq. (17) is unchanged). The plus-distribution in Eq. (19) is equivalently defined through a limit procedure as follows:
Note that the point at is at the border of the phase space, but, at the
strictly formal level, it is inside the phase space (at variance with the
case of azimuthal correlations, in which it is formally outside the phase
space). This is essential to make sense of the LO differential cross
section in Eq. (16) and of the corresponding LO total cross
section. This is also essential (see Eqs. (19) and (20))
to transform the non-integrable behaviour of
into an integrable plus-distribution over the