Diphoton production at the LHC:
[0.2cm] a QCD study up to NNLO
Stefano Catani Leandro Cieri Daniel de Florian,
[0.2cm] Giancarlo Ferrera and Massimiliano Grazzini
INFN, Sezione di Firenze and Dipartimento di Fisica e Astronomia,
Università di Firenze, I-50019 Sesto Fiorentino, Florence, Italy
INFN, Sezione di Milano-Bicocca,
Piazza della Scienza 3, I-20126 Milano, Italy
Physik-Institut, Universität Zürich, CH-8057 Zurich, Switzerland
International Center for Advanced Studies (ICAS), ECyT-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
We consider the production of prompt-photon pairs at the LHC and we report on a study of QCD radiative corrections up to the next-to-next-to-leading order (NNLO). We present a detailed comparison of next-to-leading order (NLO) results obtained within the standard and smooth cone isolation criteria, by studying the dependence on the isolation parameters. We highlight the role of different partonic subprocesses within the two isolation criteria, and we show that they produce large radiative corrections for both criteria. Smooth cone isolation is a consistent procedure to compute QCD radiative corrections at NLO and beyond. If photon isolation is sufficiently tight, we show that the NLO results for the two isolation procedures are consistent with each other within their perturbative uncertainties. We then extend our study to NNLO by using smooth cone isolation. We discuss the impact of the NNLO corrections and the corresponding perturbative uncertainties for both fiducial cross sections and distributions, and we comment on the comparison with some LHC data. Throughout our study we remark the main features that are produced by the kinematical selection cuts that are applied to the photons. In particular, we examine soft-gluon singularities that appear in the perturbative computations of the invariant mass distribution of the photon pair, the transverse-momentum spectra of the photons, and the fiducial cross section with asymmetric and symmetric photon transverse-momentum cuts, and we present their behaviour in analytic form.
The production of photon pairs (diphotons) with high invariant mass at high-energy hadron colliders is a very relevant process in the context of both Standard Model (SM) studies and searches for new-physics signals.
Experimentally a pair of photons is a very clean final state, and photon energies and momenta can be measured with high precision in modern electromagnetic calorimeters. Since photons do not interact strongly with other final-state particles, prompt photons represent ideal probes to test the properties of the SM and corresponding theoretical predictions (see Refs. – for recent experimental analyses of Tevatron and LHC diphoton data). Measurements involving a pair of isolated photons have played a crucial role in the discovery at the LHC [9, 10] of a Higgs boson, whose properties are compatible with those of the SM one. Studies of the Higgs boson properties in the diphoton decay mode have been performed [11, 12]. Diphoton measurements (see, e.g., Refs. –) are also important in many new-physics scenarios, including searches for extra dimensions or supersymmetry. The relevance of LHC measurements of the diphoton invariant-mass spectrum is highlighted by the recent observation – of an excess of events with invariant mass of about 750 GeV that might have indicated the presence of resonances over the diphoton SM background. That observation raised a great deal of attention from December 2015 till the time of the 2016 Summer Conferences [24, 25].
Owing to its physics relevance, the study of diphoton production requires accurate theoretical calculations which, in particular, include QCD radiative corrections at high perturbative orders. In high-energy collisions, final-state prompt photons with high transverse momentum can be originated through direct production form hard-scattering subprocesses and through fragmentation subprocesses of QCD partons. The theoretical computation of fragmentation subprocesses requires non-perturbative information, in the form of parton fragmentation functions of the photon, which typically has large associated uncertainties. However, the effect of fragmentation contributions is significantly reduced by the photon isolation criteria that are necessarily applied in hadron collider experiments to suppress the very large reducible background of ‘non-prompt’ photons (e.g., photons that are faked by jets or produced by hadron decays). Two such criteria are the so-called ‘standard’ cone isolation and the ‘smooth’ cone isolation proposed by Frixione . The standard cone isolation is the criterion that is typically used by experimental analyses. This criterion can be experimentally implemented in a relatively straightforward manner, but it only suppresses part of the fragmentation contribution. By contrast, the smooth cone isolation (formally) eliminates the entire fragmentation contribution, although, due to the finite granularity of the detectors, it cannot be directly applied at the experimental level in its original form. Owing to the absence of fragmentation contributions, theoretical calculations are much simplified by using the smooth cone isolation, and it is relatively simpler to compute radiative corrections at high perturbative orders. Considering calculations at the next-to-leading order (NLO) in the QCD coupling , in Ref.  it was shown that, if the isolation is ‘tight enough’, the two isolation criteria lead to theoretical NLO results that are quantitatively very similar for various observables in diphoton production processes at high-energy hadron colliders.
In the present paper we deal with the diphoton production process , where is a colliding proton and denotes the inclusive final state that accompanies the pair. QCD radiative corrections to this process were first computed up to the NLO in Ref. . This is a complete NLO calculation of both the direct and fragmentation (specifically, single- and double-fragmentation) components of the cross section. The calculation is implemented at the fully-differential level in the numerical Monte Carlo code DIPHOX, which can be used to perform computations with any infrared and collinear safe isolation criteria (including the standard and smooth cone criteria). The DIPHOX calculation also includes the so-called box contribution  to the partonic channel (which is formally a contribution of next-to-next-to-leading order (NNLO) type), since this contribution can be quantitatively enhanced by the large gluon–gluon parton luminosity at high-energy hadron–hadron colliders. The next-order gluonic corrections to the box contribution (which are part of the NLO QCD corrections to diphoton production) were computed in Ref.  (and implemented in the numerical program GAMMA2MC) and found to have a moderate quantitative effect. An independent NLO calculation  is implemented in the code MCFM, which, however, includes the fragmentation component only at the leading order (LO), while the box contribution is treated according to the GAMMA2MC code. A complete calculation at the NNLO of both the direct and fragmentation components still nowadays remains computationally very challenging.
Fragmentation contributions are absent by considering smooth cone isolation. In the context of smooth cone isolation, diphoton production at the LO (i.e. at ) emerges via the quark–antiquark annihilation subprocess . The NLO QCD corrections are due to quark–antiquark annihilation and to new partonic channels (via the subprocess and ) with an initial-state colliding gluon. At the NNLO the channel starts to contribute and it (the entire channel and not only its box contribution) can be fully consistently included in the perturbative QCD calculation. The large value of the gluon parton distribution function (PDF) of the colliding proton makes the gluon initiated channels important, especially at small and intermediate values of the diphoton invariant mass. The scattering amplitudes that are needed to evaluate the NNLO QCD corrections were computed in Ref. [29, 32, 33, 34], and they were used in Ref.  to perform the first NNLO calculation of diphoton production in hadronic collisions. The NNLO calculation at the fully-differential level is based on the subtraction method  and it was implemented  in the numerical code 2NNLO. More recently, an independent calculation of diphoton production at the NNLO has been performed in Ref.  by using the method of -jettiness subtraction [38, 39]. The NNLO calculation of Ref.  is implemented  in the MCFM program. An independent NNLO computation of diphoton production at the NNLO, based on the subtraction method, has been implemented in the general purpose NNLO generator MATRIX .
The transverse-momentum spectrum of the photon pair is sensitive to logarithmically-enhanced contributions at high perturbative orders. Transverse-momentum resummation at next-to-next-to-leading logarithmic (NNLL) accuracy for inclusive diphoton production was implemented  in the ResBos code, and more recently the complete NNLL calculation has been combined with the NNLO contributions and implemented in the numerical program 2Res .
Using smooth cone isolation, associated production of diphotons and jets has been computed up to NLO for various jet multiplicities, namely, jet [44, 45] (the fragmentation component at LO is included in the calculations of Refs. [45, 46]), jets [47, 48, 49], -jets  and jets . Owing to the computational simplifications of smooth cone isolation (with respect to standard cone isolation), some hadron collider processes with one final-state photon, such as associated [51, 52, 53] and  production and inclusive single-photon production , have also been computed up to the NNLO in QCD perturbation theory.
Lowest-order electroweak radiative corrections to diphoton production at LHC energies have been computed in Refs. [55, 56]. They produce small effects on NNLO QCD results for inclusive diphoton production. The effects can increase by selecting photons with transverse momenta in the TeV region.
In this paper we present studies of NLO and NNLO QCD radiative corrections to inclusive diphoton production at LHC energies. The results at the NNLO are based on the theoretical calculation of Ref. , as implemented in the fully-differential Monte Carlo program 2NNLO, and on the independent NNLO calculation implemented in MATRIX . The first version of 2NNLO, which was originally used in Ref. , had a numerical implementation bug that was subsequently corrected. A main result of Ref.  is that NNLO radiative corrections are substantial for diphoton kinematical configurations of interest at high-energy hadron colliders. The selected quantitative results that were presented in Ref.  are mostly related to diphoton kinematical configurations that are typically examined in Higgs boson searches and studies at the LHC. In this paper we consider more general kinematical configurations. In particular, we present NLO and NNLO studies related to photon isolation, and we discuss various detailed features of the theoretical results up to NNLO.
The results of the code 2NNLO have been used by experimental collaborations in some of their data/theory comparisons. The analyses performed at the Tevatron ( TeV) by the CDF  and D0  collaborations and at the LHC ( TeV and 8 TeV) by the ATLAS [4, 8] and CMS  collaborations show that the inclusion of the NNLO corrections greatly improves the description of diphoton production data. The NNLO predictions have also been used in data analyses related to new-physics searches in high-mass diphoton events at GeV . In summary, these measurements prove that the NNLO results are important to understand phenomenological aspects of diphoton production.
The paper is organized as follows. Section 2 is devoted to a comprehensive study of photon isolation. In particular, we perform a detailed comparison of the standard and smooth isolation criteria in the context of perturbative QCD results at LO and NLO. We discuss the role of different partonic subprocesses within the two different isolation criteria. We also remark the effects that are produced by the isolation parameters and by the kinematical selection cuts that are applied to the photons. Specifically, in Sect. 2.1 we introduce the photon isolation criteria and comment on some of their features. The QCD calculations that are used in our study are briefly described in Sect. 2.2. The quantitative results for total and differential cross sections are presented in Sect. 2.3. Section 2.3.1 is devoted to the LO and NLO results for total cross sections. Results for differential cross sections at LO and NLO are presented in Sects. 2.3.2 and 2.3.3, respectively. In Sect. 3 we present detailed NNLO results for diphoton production within the smooth isolation prescription. We study the perturbative stability of the results and we discuss related theoretical uncertainties by considering several observables that are relevant in diphoton production at hadronic colliders. We also discuss the comparison of the NNLO predictions to recent LHC data . Specifically, results for total cross sections and differential cross sections are presented in Sects. 3.1 and 3.2, respectively. In Sect. 3.3 we present some results on the dependence on the isolation parameters. The comparison with LHC data is discussed in Sect. 3.4. In Sect. 3.5 we discuss the effects of (asymmetric and symmetric) photon transverse-momentum cuts on total and differential cross sections and, in particular, we comment on related perturbative (soft-gluon) instabilities. Finally, in Sect. 4 we summarize our results.
2 Photon isolation
2.1 Isolation criteria
Hadron collider experiments at the Tevatron and the LHC do not perform inclusive photon measurements. The background of secondary photons coming from the decays of , , etc. overwhelms the signal by several orders of magnitude and the experimental selection of prompt diphotons requires isolation cuts (or criteria) to reject this background. The standard cone isolation and the smooth cone isolation are two of these criteria. Both criteria consider the amount of hadronic (partonic) transverse energy‡‡‡For each four-momentum , the corresponding transverse momentum (), transverse energy (), rapidity () and azimuthal angle () are defined in the centre–of–mass frame of the colliding hadrons. Angular distances are defined in rapidity–azimuthal angle space (). inside a cone of radius around the direction of the photon momentum . Then the isolated photons are selected by limiting the value of .
The standard cone isolation criterion fixes the size of the radius of the isolation cone and it requires
where the isolation parameter can be either a fixed value of transverse energy or a function of the photon transverse momentum (i.e., with a fixed parameter ). A combination of these two options is also possible: for instance, GeV is used in the study of Refs. [19, 22].
Provided is finite (not vanishing) standard cone isolation leads to infrared-safe cross sections  in QCD perturbation theory. Parton radiation exactly collinear with the direction of the photon momentum is allowed by the constraint in Eq. (1) and, as a consequence, the treatment of standard cone isolation within perturbative QCD requires the introduction of parton to photon fragmentation functions. Decreasing the value of reduces and suppresses the effect of the fragmentation function (and of the corresponding partonic subprocesses).
with a suitable choice of the dependence of the isolation function . The two key properties  of the isolation function are: has to smoothly vanish as the cone radius vanishes (), and it has to fulfil the condition (in particular, must not vanish) for any finite (non-vanishing) value of (in practice, Eq. (2) is effective only if monotonically decreases as decreases).
The smooth cone isolation criterion implies that, closer to the photon, less hadronic activity is allowed. The amount of energy deposited by parton radiation at angular distance from the photon is required to be exactly equal to zero, and the fragmentation component (which has a purely collinear origin in perturbative QCD) of the cross section vanishes completely. The cancellation of perturbative QCD soft divergences still takes place as in ordinary infrared-safe cross sections, since parton radiation is not forbidden in any finite region of the phase space . It is also preferable to chose isolation functions with a sufficiently smooth dependence on over the entire range . In particular, large discontinuities of at finite values of are potential sources of instabilities  in fixed-order perturbative calculations. Small discontinuities of the function (such as those in the discretized version  of smooth cone isolation) are instead acceptable.
A customary choice of the isolation function is
where the value of the power is typically set to . We also consider the following isolation function:
Comparing the isolation requirements in Eqs. (1) and (2) by using the same values of and in both equations, we see that smooth cone isolation is more restrictive than standard cone isolation. Therefore, the following physical constraint applies:
where generically denotes total cross sections and differential cross sections with respect to photon kinematical variables, and the subscripts ‘smooth’ and ‘standard’ refer to smooth and standard isolation, respectively. Note that the isolation parameters and are set at the same values in the two isolated cross sections, and , that are compared in the inequality (5) (e.g., the inequality is not necessarily valid if smooth isolation at a given value of is compared with standard isolation at a different and smaller value of ). An analogous reasoning applies to the cross section dependence on the isolation parameters and , since the isolation requirement can become more or less restrictive by varying these parameters. Therefore, we have the following physical behaviour:
and the subscript ‘is’ equally applies to both isolation criteria (e.g., ‘is’=‘smooth’ or ‘is’=‘standard’). The relation (8) refers to the dependence on the power in the case of the isolation function in Eqs. (3) or (4) (a similar relation applies to the cross section dependence by considering two isolation functions and such that ).
The standard cone isolation criterion is simpler and, as stated in the Introduction, it is the criterion that is used in experimental analyses at hadron colliders (the actual experimental selection of isolated photons, including isolation requirements, is definitely much more involved than the relatively straightforward implementation of the criterion). The experimental implementation of smooth cone isolation (in its strict original form) is complicated§§§There is activity related to the experimental implementation [61, 62, 63, 64] of the discretized version of smooth cone isolation. An experimental implementation of the smooth isolation criterion was done by the OPAL collaboration  in a study of isolated-photon production in photon–photon collisions at LEP. A discretized version of smooth isolation is used in the QCD calculation of Ref. . especially by the finite granularity of the Tevatron and LHC detectors.
Independently of the intrinsic differences between the standard and smooth cone isolation criteria, Eq. (5) implies that a reliable (theoretical or experimental) cross section result with smooth cone isolation represents a lower bound on the corresponding (i.e., with the same values of and ) cross section result with standard cone isolation (this statement is valid within reliable estimates of related theoretical or experimental uncertainties). In particular, this also implies that a comparison between theoretical smooth isolation results and experimental standard isolation results leads to meaningful and valuable information.
The comparison between smooth and standard isolation can be sharpened by considering tight isolation requirements. As expected on general grounds, the differences between the two isolation criteria should be reduced in the case of tight isolation cuts. This expectation is confirmed by the diphoton studies in Refs. [27, 67], which show theoretical NLO results that are similar for the two isolation criteria if the isolation parameters are tight enough (e.g., and GeV or with as in Sect. 11 of Ref.  or Sect. 4.3.1 of Ref. ).
In Sect. 2.3 we present detailed results for smooth and standard isolation. We study the dependence on the isolation parameter and its effects on both total cross sections and differential cross sections in various kinematical regions.
2.2 Perturbative QCD calculations
In this paper we present quantitative results on QCD radiative corrections to diphoton production by using both smooth cone and standard cone isolations. We consider total cross sections (more precisely, fiducial cross sections) and differential cross sections as functions of various kinematical variables of the prompt photons. The kinematical variables that we use are the diphoton invariant mass , the photon polar angle in the Collins–Soper rest frame  of the diphoton system, the azimuthal angle separation between the two photons, the diphoton transverse momentum and the transverse momenta and () of the harder and softer photon of the diphoton pair. As previously specified, azimuthal angles and transverse momenta are defined in the centre–of–mass frame of the colliding hadrons.
The QCD results on smooth cone isolation are obtained by using the numerical programs 2NNLO  and MATRIX , which include perturbative QCD contributions up to NNLO. The NNLO calculations are based on the subtraction method , which is applicable to hadroproduction processes of generic high-mass systems of colourless particles and it has been already applied to explicit NNLO calculations of several specific processes (Higgs boson [36, 69] and vector boson [70, 71] production, associated production of a Higgs boson and a vector boson [72, 73], diboson production processes such as [51, 52], , [74, 75], [76, 77] and  production, Higgs boson pair production , associated production of a Higgs boson pair and a or a boson ). The subtraction method has recently been extended to heavy-quark production and applied to the NNLO calculation of top quark pair production .
In the case of diphoton production, the method combines the NLO cross section for the production of the photon pair plus one or two jets (partons) with an appropriate subtraction counterterm (its explicit expression is given in Ref. ) and a hard-virtual contribution (see Ref. ) for diphoton production with no additional final-state jets (partons). In the code 2NNLO the cross section is computed up to NLO by using the dipole-subtraction method  as implemented in the diphoton calculation of Ref. . In MATRIX the cross section is computed at NLO by using the fully automated implementation of the dipole-subtraction method within the Monte Carlo program MUNICH‡‡‡MUNICH is the abbreviation of “MUlti-chaNnel Integrator at Swiss (CH) precision” — an automated parton-level NLO generator by S. Kallweit., and all (spin- and colour-correlated) tree-level and one-loop amplitudes are obtained from OpenLoops . The combination of and its counterterm is numerically finite, although the two contributions are separately divergent in the limit of vanishing transverse momentum of the photon pair. In 2NNLO the cancellation of the separate divergences is numerically treated by introducing a lower limit on () and considering small values of (formally performing the limit ). Decreasing the value of reduces the size of systematical errors (due to the finite value of ) at the expenses of increasing computational time and resources to handle numerical instabilities. As a practical compromise, in our study we use a finite value of in the range GeV–0.2 GeV. The NNLO generator MATRIX implements instead a lower limit on the ratio () , and we use values in the range –.
Owing to the finite values of or , a systematical uncertainty of about affects all the NNLO results§§§At NLO the generator MATRIX can also use the dipole-subtraction method , which does not require the regularization parameter . Our diphoton NLO results have been quantitatively cross-checked by numerical comparisons of the calculations that use subtraction (2NNLO, MATRIX) and dipole subtraction (MATRIX). presented in this paper. The quoted systematical uncertainty is sufficient for the purpose of the studies that we present throughout the paper. It is substantially smaller than the size of additional (NLO) NNLO theoretical uncertainties that we find, for instance, by performing customary variations of the factorization and renormalization scales at (NLO) NNLO. We also note that, at fixed value of (), the NNLO systematical error on differential cross sections tends to be larger than the corresponding error on total cross sections. In particular, some specific observables (and, more precisely, their value in restricted kinematical regions) that are very sensitive to the detailed shape of the distribution in the limit can be affected by larger NNLO systematical errors due to the finite value of (): these observables may require refined numerical studies with smaller values of (). Nonetheless, these same specific observables are (expected to be) affected by increased theoretical uncertainties due to large higher-order perturbative corrections (instabilities). Examples of these observables are the differential cross section at , and the differential cross sections , and (or related integrated quantities) in soft-gluon sensitive kinematical regions (see Sect. 3.5).
We use 2NNLO and MATRIX to obtain results at LO, NLO and NNLO in QCD perturbation theory. At each order our calculation includes all the terms (and only those terms) that contribute to the total cross section at the corresponding perturbative order according to the formal expansion in powers of . Therefore (in the context of smooth cone isolation), only the initial-state partonic channel contributes at LO, the initial-state and channels start to contribute at NLO, and the initial-state channel starts to contribute at NNLO. In particular the box contribution  only enters at NNLO, together with all the partonic subprocesses (e.g., including the gluon initiated subprocess ) that contribute at the same order. We do not include the NLO correction  to the box contribution, since it is part of the complete (and still unknown) NLO corrections to inclusive diphoton production. All the partonic subprocesses are treated in the massless-quark framework with quark flavours. In particular, we do not include NNLO contributions with virtual top quarks since they are not yet fully known (e.g., the loop top quark contribution to the two-loop scattering amplitude has not yet been computed in complete form ). The effect of including the NLO correction  and the top quark correction to the (massless-quark) box contribution has been considered in the diphoton calculation of Ref. . The top quark correction to the box contribution is also studied in Ref. .
The results on standard cone isolation are obtained by using the DIPHOX calculation , which includes QCD radiative corrections up to the NLO. The NLO calculation that is implemented in DIPHOX is based on a combined use [28, 88] of the subtraction and phase-space slicing methods. The slicing approximation is applied  to the phase-space region where the minimum transverse-momentum of the three final-state partons at NLO is smaller than (formally considering the limit ). Our numerical results are obtained by using GeV, which is the default value of the slicing parameter (regulator) that is suggested in the DIPHOX program. Since we are interested in an order-by-order comparison with smooth cone isolation results, the box contribution is not included in the DIPHOX NLO results. Note, however, that at LO and NLO all the initial-state partonic channels () contribute in DIPHOX because of the presence of a non-vanishing fragmentation component (though the fragmentation component is quantitatively suppressed by the isolation procedure). Owing to the double-fragmentation component, even the initial-state channel is not vanishing at LO (at NLO the initial-state channel contributes also through the single-fragmentation component).
We note that (due to the limited number of final-state partons in fixed-order computations) LO calculations do not cover the entire kinematical region for inclusive diphoton production. LO calculations give non-vanishing cross sections only in limited LO kinematical subregions. Outside these subregions, only the NLO results start to give non-vanishing cross section contributions. Therefore, outside the LO kinematical subregions the NLO (NNLO) results effectively represent an LO (NLO) perturbative QCD prediction. In spite of the effective meaning of the results, we always use the default labels LO, NLO and NNLO according to the perturbative order in which the results contribute to the inclusive (total) cross section. For example, in the case of the distribution, the LO kinematical subregion has : the region of small values of receives contributions only from the NLO and NNLO results that represent effective LO and NLO predictions, respectively. We also note that the LO kinematical subregions can be different in the context of smooth cone and standard cone isolation. For instance, at LO and in the case of smooth cone isolation, whereas can be different from zero and in the case of standard cone isolation (non-vanishing values of are produced through the LO fragmentation component of the cross section). We also comment about the dependence on the isolation parameters and (the comment has similarities with our previous comment on LO kinematical subregions). The LO results are independent of the size of the isolation cone. The LO cross section depends on in the case of standard cone isolation (the dependence is due to the fragmentation component), while it is independent of the value of in the case of smooth cone isolation.
In Eqs. (5)–(8) we have listed some constraints on isolated photon cross sections. These are physical constraints (properties) in the sense that they apply to physical (‘positive definite’) events: if the isolation requirements are more (less) restrictive, the selected number of isolated photons decreases (increases). Such properties do not ‘a priori’ apply to theoretical calculations based on perturbative QCD (of course, if the constraints are not fulfilled, the perturbative QCD calculation is not a reliable approximation of the physical result) since, beyond the effective LO approximation, they involve negative-weight contributions (which are due to virtual radiative corrections and to subtraction contributions related to the factorization procedure of parton distribution functions and fragmentation functions).
As is well known, fixed-order perturbative results can show unphysical behaviour in specific kinematical regions. In the case of diphoton production, for instance, it is known  that the differential cross sections at small values of () or large values of () cannot be reliably computed in fixed-order perturbation theory: the disease is due to large logarithmic corrections that have to be resummed to all perturbative orders [42, 43] to recover the physical behaviour and predictivity.
In the presence of photon isolation cuts, perturbative computations of total cross sections can also show some misbehaviour. Isolated (with both smooth and standard isolation) photon cross sections fulfil the physical constraint , where is the inclusive (non-isolated) cross section. Nonetheless, in the case of standard cone isolation at NLO this constraint is violated  at very small values () of the radius of the isolation cone. The violation is due to large logarithmic () corrections, and the physical behaviour is recovered through a corresponding all-order resummation procedure . We expect a (qualitatively) similar dependence in the case of smooth cone isolation, though in the present paper we do not consider very small values of .
Violation of expected properties is not necessarily related to logarithmically-enhanced QCD corrections: it can be simply a consequence of a slowly convergent (toward a reliable estimate of the physical result) perturbative expansion. The LO, NLO and NNLO results obtained in Ref.  with smooth cone isolation certainly indicate that we are not dealing with a fastly convergent perturbative expansion in the case of diphoton production at high-energy hadron colliders. Additional complications can occur in direct comparisons (as in Eq. (5)) between calculations with smooth and standard isolation. The complications follow from the fact that such a comparison does not involve ingredients that are in one-to-one correspondence. Owing to the presence of the fragmentation component, at each perturbative order the standard isolation result depends on the photon fragmentation function, on the corresponding factorization scale and on related partonic subprocesses. The poorly known fragmentation function certainly affects the standard isolation results and, especially, the comparison with smooth isolation results (which have no analogue of the fragmentation function).
Throughout the paper we comment on these and additional points related to physical behaviour and perturbative QCD calculations
2.3 Quantitative results
In our theoretical study of standard and smooth isolation we consider isolated diphoton production in collisions at the centre–of–mass energy TeV. We apply the following kinematical cuts on photon transverse momenta and rapidities: GeV, GeV and the rapidity of both photons is limited in the range . The minimum angular distance between the two photons is .
These are basically the kinematical cuts used in the ATLAS Collaboration study of Ref. . The analysis of Ref.  is restricted to a smaller rapidity region since it excludes the rapidity interval , which is outside the acceptance of the electromagnetic calorimeter. For the sake of simplicity, in this section we do not consider such additional rapidity restriction; the rapidity restriction is instead applied in the results of the following Sect. 3.
In the perturbative calculation, the QED coupling constant is fixed at the value . We use the MMHT 2014 sets  of parton distribution functions (PDFs), with parton densities and evaluated at each corresponding perturbative order (i.e., we use the -loop running at NLO, with ). We use the NLO photon fragmentation functions of Ref. , and specifically the BFG set II (we have checked that BFG set I leads to very small quantitative differences). The central value of the renormalization scale (), PDF factorization scale () and fragmentation function scale () is set to be equal to the invariant mass of the diphoton system, . We compute scale variation uncertainties by considering the two asymmetric scale configurations with , and , .
More precisely, we have considered independent scale variations by a factor of two up and down with respect to the central value . We find a common overall behaviour of the cross sections: they increase by decreasing and decrease by decreasing or . Therefore the two asymmetric scale configurations are those that maximize the scale dependence within the considered scale variation range. The sole exception regards the invariant mass cross section at large values of ( GeV): this kinematical region is sensitive to larger values of the parton momentum fraction in the PDFs and, as a consequence, it turns out that the cross section decreases by increasing .
The radius of the isolation cone is fixed at . We study the isolation parameter dependence by considering values of in the range between 2 GeV and 10 GeV (in this section we mainly show results at the extreme values in this range). In the case of smooth cone isolation, we use the isolation function in Eq. (4) and we examine the cross section dependence on the power that controls the shape of . In Fig. 1 we show the dependence of the isolation function for some selected values of the power and the fixed value of the radius of the isolation cone. We note that the value is sufficiently small so that the two isolation functions in Eqs. (3) and (4) quantitatively coincides at the percent level in the case with .
We present perturbative QCD results for both standard and smooth isolation and, in particular, we perform comparisons by using the same value of the isolation parameter for both criteria. The comparison between the two criteria can also be performed differently. For instance, having fixed the values of and for standard isolation, one can use different values of isolation parameters ( and ) for smooth isolation to the purpose of trying to obtain similar quantitative results for the two criteria, as a pragmatic approach to mimic the standard cone isolation that is used in experimental conditions. We think that our comparison with the same value of (and ) is more informative to investigate and understand differences and similarities between perturbative QCD results for the two criteria.
The QCD results on standard cone isolation depend on the parton-to-photon fragmentation function , being the photon momentum fraction with respect to the momentum of the fragmenting parton . Owing to the isolation procedure, the value of is bounded by a minimum value (), and this leads to a quantitative suppression of the fragmentation component of the diphoton cross section. The typical value of is , being the transverse momentum of the photon that is involved in the fragmentation process. In our quantitative study we use relatively-large values of (i.e., typically, GeV) and relatively-small values of . Therefore, is always large ( at GeV, and still at GeV), and the suppression factor ‡‡‡At the formal level is the order of magnitude of the ratio between the fragmentation component and the direct component. due to is sizeable (roughly one order of magnitude or more, depending on ) . We note that at such high values of the quark (or antiquark) fragmentation function (or ) is much larger (roughly by more than a factor of ten) than the gluon fragmentation function . In our calculation we consistently (according to the formal perturbative expansion) include all the fragmentation functions. However, due to the dominance of and , in all our qualitative (or semi-quantitative) comments we neglect the effect of (i.e., we can assume that only and contribute). We also note that, because of QCD scaling violation, at high values of , increases (although weakly) by increasing .
2.3.1 Total cross sections at LO and NLO
We begin the presentation of our quantitative results by considering the total cross section (namely, the fiducial cross section for the applied kinematical cuts on the photons). Values of total cross sections at LO and NLO for both smooth and standard isolation are reported in Table 1.
Using smooth cone isolation, the total cross section at LO is (scale), where the percentage variation refers to the scale dependence of the result§§§Throughout the paper, any quantitative statements about scale dependence refer to scale variation effects that are computed as described in the text at the beginning of Sects. 2.3 and 3, respectively.. We note that is independent of and of the isolation function . The cross section is produced only by the initial-state channel through the partonic subprocess , and the scale dependence of (which is entirely due to variations of in the quark and antiquark PDFs) is quite small. All these features are a very crude approximation of the diphoton production dynamics.
Using standard cone isolation, the LO cross section depends on and it also depends on all the three scales and ( and enter through the fragmentation component). The scale dependence of is relatively similar to that of : we find (scale) and (scale) with GeV and GeV, respectively. The dependence of is instead more ‘surprising’. Considering GeV (very tight isolation), is slightly larger, by about 15%, than (actually the two cross sections are very similar within the corresponding scale dependence). However, increases by a factor of about 1.6 in going from GeV to GeV and, at GeV, is roughly a factor of 2 larger than .
Since does not depend on the isolation parameters (), there is obviously no way to approximate (or, mimic) the quantitative value of at GeV by using smooth cone isolation with different isolation parameters (e.g., a value of larger than 10 GeV).
The value GeV is not particularly large and it cannot be regarded as a very loose isolation parameter. Therefore, on physical grounds, we do not expect two actual features of the LO result: the large difference between and at GeV, and the large dependence of in going from GeV to GeV. We mean that physical events are not expected to have these features, since such features cannot be regarded as a physical consequence of the hadronic activity inside the photon isolation cones and of its detailed structure (which leads to an ensuing sensitivity to the isolation criteria). These observed LO features require some explanation, which we are going to discuss.
At the LO, in the context of standard isolation, the distinction between direct and fragmentation components is unambiguous, and the direct component exactly coincides with the entire contribution to the smooth isolation result. The double-fragmentation component always gives a small contribution (few percent at GeV and few permill at GeV) to . Therefore, the dependence of is due to the single-fragmentation component, whose contribution to is small (of the order of 10%) at GeV (because of the suppression due to the small value of ) and sizeable at GeV. At the larger value of the direct and single-fragmentation components have the same size, but this is not due to the fact that the fragmentation function is particularly large. At the LO the direct component only involves the partonic process
whereas the (single) fragmentation component also involves the partonic process
where the notation denotes the fragmentation of the final-state quark into a photon through (a similar notation is used in subsequent equations for partonic subprocesses). The PDF luminosity (the detailed definition of PDF luminosities is presented in Eq. (17)) of the initial-state direct subprocess is sizeably smaller than the luminosity of the initial-state fragmentation subprocess (this follows from the smaller size of the antiquark PDF with respect to the gluon PDF at small values, such as , of parton momentum fraction in the PDFs), and the suppression due to the isolated fragmentation function is compensated by the increased size of with respect to (the suppression from is much stronger by decreasing , because and increases by decreasing ).
According to our discussion, the presence of the fragmentation process of Eq. (10) in the LO standard cone isolation explains the quantitative dependence of on and the quantitative differences between and . At the same time, our discussion is useful to anticipate expected features of the NLO results. The NLO calculation within smooth cone isolation includes the partonic process
In the kinematical configuration where the final-state quark is inside the isolation cone of one photon, the process in Eq. (11) roughly corresponds to the LO fragmentation process of Eq. (10): therefore we expect that the NLO smooth isolation result receives a large NLO correction from this kinematical configuration of this process, in such a way to reduce the observed LO ‘deficit’ with respect to standard cone isolation (in other words, is suppressed with respect to by an extra power of , but this suppression is compensated by the increased luminosity of the partonic initial state). Moreover, the kinematical configuration where the final-state quark is outside the isolation cones of both photons contributes through the process of Eq. (11) to the NLO calculation of both smooth and standard isolations. This is not a very limited kinematical region (the size of the isolation cones is not large) and, due to the large value of the luminosity, it gives a sizeable and independent NLO contribution to both isolation prescriptions. It follows that, for both isolation criteria, we expect large NLO corrections and a much reduced dependence with respect to the LO result. As we are going to show shortly, the expectations of our discussion are confirmed by the actual NLO results.
We note that the LO results that we have presented are obtained by using LO PDFs and the NLO BFG fragmentation functions, since LO fragmentation functions are not readily available in the default setup of the DIPHOX code. This mismatch (at the strictly formal level) of perturbative order in the fragmentation functions should not strongly affect the main features of the LO standard isolation results. More generally, standard cone isolation results are certainly affected by an additional uncertainty, which is due to the poorly known fragmentation functions, that is difficult to be estimated at the quantitative level. The recent Ref.  presents a very brief overview on prospects for improving the determination of the photon fragmentation functions.
The value of the NLO total cross section , including its corresponding scale variation dependence, is reported in Table 1 and Fig. 2 for two different values of : GeV (Fig. 2-left) and GeV (Fig. 2-right).
In the case of smooth cone isolation, the NLO result depends on the power of the isolation function . We postpone the discussion of the dependence and we consider the case with . The values of the cross sections are (scale) with GeV and (scale) with GeV. The standard isolation cross section is very similar to at both values of (see Table 1 and Fig. 2): the differences are at most at the level of 2–3% and they are sizeably smaller than the scale dependence of . The dependence is small: the NLO cross section for smooth (standard) isolation increases by a factor of 1.06 (1.07) in going from GeV to GeV. These features are in qualitative agreement with physical expectations.
The NLO radiative corrections are large (as expected from our previous discussion). Considering the ratio , the value of is roughly 3 and the values of are approximately 2.6 ( GeV) and 1.7 ( GeV). In the case of the smooth isolation criterion, a sizeable part (roughly 50%) of the NLO total cross section is due to the initial-state partonic channel (which is absent at the LO), and also the LO channel receives sizeable (roughly 50%) NLO corrections that increase the total cross section. In the case of standard cone isolation, at the NLO the distinction between direct and fragmentation components is no longer unambiguous (‘physical’): it depends on the factorization scheme and on the fragmentation scale . Within the factorization scheme and the scale variation range that we use, the direct component contributes about 45–65% (85–90%) of with GeV ( GeV). Since and are very similar, the similarity is the consequence of a non-trivial interplay between the direct and fragmentation contributions to (especially in the case with GeV) and, in particular, the LO equivalence between the smooth cone result and the direct component of the standard cone result is definitely lost at the NLO.
We note that the scale dependence of the total cross section has a similar size at LO and at NLO, and it is much smaller than the size of the NLO corrections. This implies that the scale dependence of cannot be consistently regarded as a reliable estimate of uncalculated higher-order radiative corrections to : the ‘true’ theoretical uncertainty of is certainly larger than the NLO scale dependence that we have computed. A similar comment applies to the scale dependence of the results for the differential cross sections that we present in the following.
We now discuss the dependence of the smooth isolation cross section on the power of the isolation function (see Eq. (4)). The NLO results of the total cross section for some selected values of in the range are reported in Fig. 2. We note that the dependence of is small (in particular, it is smaller than the scale dependence) within this range of values of . Specifically, considering the interval , the central value of varies by about 3% (10%) if GeV ( GeV). Most of the qualitative features of the results in Fig.2 are consistent with physical expectations. The photons are more isolated by increasing at fixed and, consequently, monotonically decreases (in agreement with the physical behaviour in Eq. (8)). Moreover, by decreasing the photons are more isolated and consequently the total cross section is less sensitive to variations of the power . Nonetheless, we note that, by decreasing the value of , tends to become larger than , thus violating the physical constraint in Eq. (5). This feature deserves some comments, which are presented below.
The perturbative dependence of at very small () or very large () values of can be understood in relatively-simple terms. If is very small, the isolation function is approximately equal to unity with the exception of the angular region of very small values of : therefore, the dependence of is very sensitive to radiation of partons that are collinear to the photon direction. If is very large, the isolation function very strongly suppresses parton radiation inside the photon isolation cone: therefore, the dependence of is very sensitive to soft parton radiation. The dominant effects of soft and collinear radiation can be easily computed at NLO (see Refs. [26, 57, 93]). We consider the NLO correction , and we limit ourselves to sketch the dominant dependence of the soft () and collinear () contribution to within smooth isolation. We have
where is the typical hard scale of the cross section (the scale is of the order of the minimum value of ) and we have considered small values of (by neglecting relative corrections of ).
The proportionality factor that is not explicitly denoted on the right-hand side of Eq. (12) depends on the LO cross section for the partonic process . The soft contribution in Eq. (12) is negative. It is due to a strong kinematical mismatch between (negative) soft virtual radiation (one-loop corrections in the subprocess ), which is not affected by isolation, and (positive) soft real radiation (the subprocess at the tree level), which is strongly suppressed by isolation. We note that is proportional to , so that eventually diverges to ‘’ in the limit . This NLO divergence is the perturbative signal of the infrared unsafety of the isolated cross section in the limit of completely isolated photons (no accompanying transverse energy inside the isolation cone). We observe that is proportional to , so that the soft contribution is strongly suppressed if the photon isolation cone has a small radius . We also note that is due to subprocesses with a initial state. The subprocess is formally subdominant in the soft limit (), but it represents a sizeable quantitative correction to because of the increased PDF luminosity of the initial state. The -suppressed dependence of and the large size of the (positive) correction to it from the initiated subprocess explain why the results for in Fig. 2 have a small dependence on at relatively-large values of (e.g., ).
The collinear contribution in Eq. (13) is relevant to discuss the dependence of the results in Fig. 2 at small values of . We note that the NLO contributions from the initial-state channel (e.g., ) are subdominant in the limit . The contribution in Eq. (13) is due to real radiation of a collinear quark or antiquark inside the photon isolation cone through the partonic processes and (the proportionality factor that is not explicitly denoted in the right-hand side of Eq. (13) depends on the LO cross section for the partonic processes and ). Therefore, is positive and independent of . Moreover, is proportional to , so that its induced dependence is reduced by decreasing (in agreement with the results in Fig. 2 at small ). Owing to its dependence on , sizeably increases by decreasing at fixed and, eventually, (and, consequently, ) diverges to ‘’ in the limit . Since becomes arbitrarily large by decreasing , it is obvious that at sufficiently small values of the physical requirement (see Eq. (5)) is unavoidably violated. This misbehaviour of at small values of implies that beyond-NLO contributions are relevant. Indeed, each higher-order contribution is equally misbehaved at : the NLO correction is proportional to (because of multiple collinear radiation inside the photon isolation cone) an it cannot be regarded as a small correction if (i.e., ). In principle, the perturbative treatment at small values of can be improved by a proper all-order resummation of these collinear contributions of . However, such resummation treatment cannot be pursued for arbitrarily small values of since it unavoidably fails in the limit , because of non-perturbative photon fragmentation effects (smooth isolation with requires photon fragmentation functions since it is equivalent to standard isolation).
We can draw some conclusions from our discussion on small values of . Owing to the physical requirements in Eqs. (5) and (8), in principle cross sections for standard and smooth isolation tend to agree at very small values of . However, fixed-order QCD computations for smooth isolation are not reliable if (they are affected by large higher-order corrections) and, in particular, they can violate the physical constraint in Eq. (5). In practice, to the purpose of approximating the standard isolation criterion within fixed-order QCD calculations it is more appropriate to consider smooth isolation with values of that are not too small. From the results in Fig. 2, we can conclude that the total cross sections and quantitatively agree if .
In the following we consider differential cross sections with respect to various kinematical variables and we limit ourselves to present smooth isolation results with . We have checked that the shape of the various NLO kinematical distributions is very little affected by variations of within the range . At the NLO, variations of basically produce overall normalization effects, whose size corresponds to the dependence of that is observed in Fig. 2.
2.3.2 Differential cross sections at the LO
In Fig. 3 we present the LO and NLO results (including their scale variation dependence) for the differential cross section with respect to the diphoton invariant mass . We consider two different values, 2 GeV (Fig. 3-left) and 10 GeV (Fig. 3-right), of and we use both standard and smooth isolation. In Fig. 4 we present the analogous results for the differential cross section with respect to the angular variable in the Collins–Soper rest frame. The results in Figs. 3 and 4 are obtained by numerical integration over small bins in and , respectively: we use bins of constant size equal to 2 GeV for and 0.08 for . In the following we discuss LO and NLO differential cross sections in turn. At the LO, we preliminarily note that standard and smooth isolation results for the differential cross sections have qualitatively similar shapes, with differences of the overall normalization that are quite similar to the quantitative differences (which we have previously discussed) of the corresponding LO total cross sections.
We first consider smooth cone isolation. Owing to transverse-momentum conservation in the corresponding LO partonic process , the diphoton azimuthal separation is and the transverse momentum of the diphoton system is . The corresponding differential cross sections , with or , are simply proportional to a -function ( or ) and the proportionality factor is the LO total cross section . The double differential cross section with respect to and the scattering angle is
where is the centre–of–mass energy of the hadronic collision (e.g., TeV as in Fig. 3), and the photon scattering angle is defined in the centre–of–mass frame of the LO partonic collision, and it is related to the diphoton rapidity separation :
We remark that, in the case of the LO smooth isolation cross section in Eq. (14), and the Collins–Soper polar angle actually coincides (), since . In the right-hand side of Eq. (14) we have not denoted an overall proportionality factor of order unity, which is independent of the kinematical variables and . The angular dependent function is
and it is specific of the angular distribution that is dynamically produced by the partonic process . The function is the PDF luminosity and is the quark electric charge in units of the positron charge (