Production of three isolated photons in the parton Reggeization approach at high energies

We study a large-$p_T$ three-photon production in proton-proton collisions at the LHC. We use the leading order (LO) approximation of the parton Reggeization approach consistently merged with the next-to-leading order corrections originated from the emission of additional jet. For numerical calculations we use the parton-level generator KaTie and modified KMR-type unintegrated parton distribution functions which satisfy exact normalization conditions for arbitrary $x$. We compare our prediction with data from ATLAS collaboration at the center-of-mass energy $\sqrt{s}=8$ TeV. We find that the inclusion of the real next-to-leading-order corrections leads to a good agreement between our predictions and data with the same accuracy as for the next-to-next-to-leading calculations based on the collinear parton model of QCD. At higher energies ($\sqrt{s}=13$ and 27 TeV) parton Reggeization approach predicts larger cross sections, up to $\sim 15$ \% and $\sim 30$ \%, respectively.


I. INTRODUCTION
The recent experimental data for a large-p T multi-photon production at the Tevatron [1,2] and LHC [3][4][5] at the energy range from 1.96 TeV up to 8 TeV are extensively studied in the collinear parton model (CPM) of perturbative quantum chromodynamics (QCD) beyond the leading-order (LO) accuracy in strong-coupling constant, α S , i.e. at the next-to-leadingorder (NLO) [6][7][8] and even at next-to-next-to-leading-order (NNLO) [9][10][11][12].The high-order calculations for two-photon or three-photon production in CPM of QCD provide rather bad agreement with data at the level of NLO accuracy.For example, NLO QCD calculations strongly underestimate, by factor 2 or even more, recent data from ATLAS Collaboration at √ s = 8 TeV [5] for three-photon production.The inclusion of additional contribution from parton-shower mechanism and hadronization effects [7] to the NLO calculations increase theoretical prediction but they are being yet far from measured cross sections.
Inclusion of the NNLO QCD corrections for two-photon production [9] and three-photon production [11,12] eliminates the existing discrepancy with respect to NLO QCD predictions.However, for three-photon production the agreement with data is not so good as for two-photon production and it is achieved when hard scale parameter µ should be taken very small relatively usual used value [11].
In CPM of QCD we neglect the transverse momenta of initial-state partons in hardscattering amplitudes that is a correct assumption for the fully inclusive observables, such as p T -spectra of single prompt photons or jets, where their large transverse momentum defines single hard scale of the process, µ ∼ p T .The corrections breaking the collinear factorization are shown to be suppressed by powers of the hard scale [13].
The multi-photon large-p T production is multi-scale hard process in which using the simple collinear picture of initial state radiation may be a bad approximation.In the present paper, we calculate different multi-scale variables in three-photon production from a point of view of High-Energy Factorization (HEF) or k T −factorization, which initially has been introduced as a resummation tool for ln( √ s/µ)-enhanced corrections to the hardscattering coefficients in CPM, where invariants √ s referees to the total energy of process.
In the same manner of the PRA, we have studied previously one-photon production [18], two-photon production [19] and photon plus jet production [20] in proton-(anti)proton collisions at the Tevatron and the LHC.At the present paper we study a production or three isolated photons at the LHC.Preliminary, our predictions has been presented as short note at DIS2021 Conference, see Ref. [21].The similar study of three-photon production in the k T −factorization approach was published recently in Ref. [22], where authors compared predictions obtained with different unPDFs [23][24][25][26] that is compliment to our study in PRA [21].
The paper has the following structure, in Section II the relevant basics of the PRA formalism are outlined.In the Section III we overview Monte-Carlo (MC) parton-level event generator KaTie and the relation between PRA and KaTie MC calculations for treelevel amplitudes.In the Section IV we compare obtained in the PRA results with the recent ATLAS [5] data as well as with theoretical predictions obtained in NNLO calculations of CPM [11,12].Our conclusions are summarized in the Section V.

II. PARTON REGGEIZATION APPROACH
The PRA is based on high-energy factorization for hard processes in the Multi-Regge kinematics.The basic ingredients of PRA are k T −dependent factorization formula, unintegrated parton distribution functions (unPDF's) and gauge-invariant amplitudes with offshell initial-state partons.The first one is proved in the leading-logarithmic-approximation of high-energy QCD [27,28], the second one is constructed in the same manner as it was suggested by Kimber, Martin, Ryskin and Watt [23,24], but with sufficient revision [29].
The off-shell amplitudes are derived using the Lipatov's Effective Field Theory (EFT) of Reggeized gluons [30] and Reggeized quarks [31].The brief description of LO in α S approximation of PRA is presented below.More details can be found in Refs.[32,33], the inclusion of real NLO corrections in the PRA was studied in Ref. [33], the development of PRA in the full one-loop NLO approximation was further discussed in [34][35][36].
Factorization formula of PRA in LO approximation for the process p + p → Y + X, can be obtained from factorization formula of the CPM for the auxiliary hard subprocess like g + g → q + Y + q.For discussed here process of three-photon production, Y = γγγ.In the Ref. [33] the modified Multi-Regge Kinematics (mMRK) approximation for the auxiliary amplitude has been constructed, which correctly reproduces the Multi-Regge and collinear limits of corresponding QCD amplitude.This mMRK-amplitude has t-channel factorized form, which allows one to rewrite the cross-section of auxiliary subprocess in a k T -factorized form: where t 1,2 = −q 2 T 1,2 , the off-shell partonic cross-section σPRA in PRA is determined by squared PRA amplitude, |A P RA | 2 .Despite the fact that four-momenta (q 1,2 ) of partons in the initial state of amplitude A P RA are off-shell (q 2  1,2 = −t 1,2 < 0), the PRA hardscattering amplitude is gauge-invariant because the initial-state off-shell partons are treated as Reggeized partons of gauge-invariant EFT for QCD processes in Multi-Regge Kinematics (MRK), introduced by L.N. Lipatov in [30,31].The Feynman rules of this EFT are written down in Refs.[31,37].
The tree-level unPDFs Φi (x 1,2 , t 1,2 , µ 2 ) in Eq. ( 1) are equal to the convolution of the collinear PDFs f i (x, µ 2 ) and Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) splitting where Here and above we put µ F = µ R = µ.Consequently, the cross-section (1) with such unPDFs contains the collinear divergence at t 1,2 → 0 and infrared To resolve collinear divergence problem of Φi (x, t, µ) we require that modified unPDF Φ i (x, t, µ) should be satisfied exact normalization condition: which is equivalent to: where T i (t, µ 2 , x) is usually referred to as Sudakov form-factor, satisfying the boundary conditions T i (t = 0, µ 2 , x) = 0 and T i (t = µ 2 , µ 2 , x) = 1.Such a way, modified unPDF can be written as follows from KMR model: Here, we resolved also IR divergence taking into account observation that the mMRK expression gives a reasonable approximation for the exact matrix element only in the rapidityordered part of the phase-space.From this requirement, the following cutoff on z 1,2 can be derived: function [23].
The solution for Sudakov form-factor in Eq. ( 4) has been obtained in Ref. [29]: with Let us summarize important differences between the Sudakov form-factor obtained in our mMRK approach (6) and in the KMR approach [23].At first, the Sudakov form-factor (6) contains the x−depended ∆τ i -term in the exponent which is needed to preserve exact normalization condition for arbitrary x and µ.The second one is a numerically-important difference that in our mMRK approach the rapidity-ordering condition is imposed both on quarks and gluons, while in KMR approach it is imposed only on gluons.
To illustrate differences between unPDFs at large x, obtained in original KMR [23,24] model and in our modified approach [29], we plot ratios for integrated over transverse momentum unPDFs to parent collinear PDFs for gluon and u−quark as function of x at different choice of hard scale µ in Figures 3 and 4, correspondingly.
In contrast to most of studies in the k T -factorization, the gauge-invariant matrix elements with off-shell initial-state partons (Reggeized quarks and gluons) from Lipatov's EFT [30,31] allow one to study arbitrary processes involving non-Abelian structure of QCD without violation of Slavnov-Taylor identities due to the nonzero virtuality of initial-state partons.
This approach, together with KMR-type unPDFs gives stable and consistent results in a wide range of phenomenological applications, which include the description of the angular correlations of dijets [32], charmed [39,40] and bottom-flavored [33,38] mesons, charmonia [41,42] as well as some other examples.

III. DETAILS OF NUMERICAL CALCULATIONS
The first step of calculations in PRA is generation of amplitudes of relevant off-mass shell partonic processes by Feynman rules of Lipatov's EFT.It can be done using a model file ReggeQCD [43] for FeynArts tool [44].In the Fig. 1, the total set of 13 Feynman diagrams for LO process obtained with ReggeQCD is shown.The number of EFT diagrams for NLO in α S involved in our study process is getting too large for analytical calculation.In Fig. 2, the full gauge invariant set of 40 Feynman diagrams is shown.To proceed next step, we should analytically calculate squared off-shell amplitudes and perform a numerical calculation using factorization formula (1) with modified unPDFs (5).At present, we can do it with required numerical accuracy only for 2 → 2 and 2 → 3 off-shell parton processes.To calculate contributions from 2 → 4 processes with initial Reggeized partons we use parton-level generator KaTie [45].
A few years ago, a new approach to derive gauge-invariant scattering amplitudes with offshell initial-state partons for high-energy scattering, using the spinor-helicity techniques and BCFW-like recursion relations for such amplitudes has been introduced in the Refs.[46,47].
Some time later the MC parton-level event generator KaTie [45] has been developed to provide calculations for hadron scattering processes that can deal with partonic initial-state momenta with an explicit transverse momentum dependence causing them to be space-like.
The formalism [46,47] for numerical generation of off-shell amplitudes is equivalent to the results of Lipatov's EFT at the tree level [32,33,48].We should note here, that for the generalization of the formalism to full NLO level [34,35], the use of explicit Feynman rules and the structure of EFT is more convenient.
Taking in mind above mentioned discussion, the LO contribution of subprocess (7) has been calculated for crosscheck both with KaTie MC generator and using direct integration of analytical squared amplitudes obtained with the help of Feynman rules of Lipatov EFT.
All final calculations have been done using MC event generator KaTie [45].
We will neglect NLO contribution in quark-antiquark annihilation channel from subprocesses with additional final gluon which should be negligibly small in comparison with main others as in the similar case of NLO CPM calculations.First off all, because the relevant values of involving longitudinal parton momenta are very small (x < 10 −2 ) at the energy range of the LHC and the gluon density is much larger than the quark (antiquark) ones.Such a way, we avoid difficulties in a calculation of the process (9), which follow from an infra-red divergence, which should be regularized by a contribution from loop correction to the LO process (7) and from double counting between LO (7) and NLO (9) diagrams with emission of an additional gluon.

The technique of NLO calculations is still under development in PRA, see discussions in
Refs. [34][35][36].
The next important issue is that a calculation for the process (8) doesn't contain infra-red and collinear singularities in PRA, after taking in consideration isolation-cone conditions for final photons and partons.Numerical accuracy of total cross section calculations with MC generator KaTie by default is 0.1 % .

IV. RESULTS
First of all, we review setup of ATLAS measurements at √ s = 8 TeV [5]: • Photon transverse energies (transverse momenta) (1 is leading photon, 2 is sub-leading photon and 3 is sub-sub-leading photon) GeV.
• Photon-photon isolation conditions are ∆R ij > R γγ = 0.45, where ∆R ij = • Photon-quark isolation conditions are ∆R iq > R 0 = 0.40 To take into account a fragmentation contribution, we use the Frixione smooth photon isolation [49].For any angular difference ∆R iq from each photon, when ∆R iq ≤ R 0 , it is required where E max T = 10 GeV, E iso T = E T q .We test dependence of predicted cross section on choice of factorization (µ F )and renormalization (µ R ) scales, which we take equal to each other, µ F = µ R = µ.In the Table IV we compare predictions obtained with µ = M 3γ -an invariant mass of the three-photon system and µ = E T, = E T,1γ + E T,2γ + E T,3γ -a sum of transverse momenta (transverse energies) moduli of photons.Errors indicate upper and lower limits of the cross section obtained due to variation of the hard scale µ by a factor ξ = 2 or ξ = 1/2 around the central value of the hard scale.
As we see in Table IV, where the total cross sections of three-photon production are presented, relative contribution of LO subprocesses grows with increase of the hard scale µ and contribution of NLO subprocesses oppositely falls down, however their sum changes only a little.Predicted absolute values of cross-section are in a quite well agreement with the experimental data [5] as well as with the NNLO CPM results [11,12] taking in mind the level of accuracy, which is originated from scale variation.approximately in 15 and 30 %, correspondingly.In the PRA we obtain also a strong decreasing of scale uncertainty in the NLO approximation instead of the LO one as it is estimated from general properties of perturbative QCD.In fact, one has LO scale uncertainty is about 25-30 %, but at NLO level of calculation it is only 4-5 % at different energies.Let us note that in NNLO CPM calculation of three-photon production [11,12] such uncertainty is still about 10 %.
Differential spectra, which demonstrate different kinematics correlations between final photons, are shown in Figures ( 5) - (9).There are no kinematics regions in invariant masses, pseudo-rapidities, azimuthal angles or transverse momenta where one of the relevant contributions can been considered as an absolutely dominant one.To describe the data, only both should be taken.The NLO contribution in α s (8) is enhanced evidently because it is proportional to a quark-gluon luminosity instead of a quark-antiquark luminosity in case of LO production (7) in proton-proton collision.

V. CONCLUSIONS
We obtain a quite satisfactory description for cross section and spectra for the threephoton production in the LO PRA with a matching of a real NLO correction from partonic subprocess (8) at the √ s = 8 TeV.We demonstrate an applicability of the new KMRtype quark and gluon unPDFs to use in high-energy factorization calculations.It has been shown that, as in our previous studies of hard processes in the PRA, obtained results in LO approximation coincide with full NLO predictions of the CPM and, respectively, NLO calculations in the PRA roughly reproduce NNLO predictions of the CPM.However, at higher energies (13 and 27 TeV) the PRA predicts larger cross sections, up to ∼ 15 % and ∼ 30 %, with respect to predictions of the NNLO CPM.The last fact can be used for a discrimination between the high-energy factorization and the collinear factorization for hard processes at high energies.

1 FIG. 3 : 1 FIG. 4 :
FIG.3: Ratio for integrated over transverse momentum unPDF to parent collinear PDF for gluon as function of x at different choice of hard scale µ 2 = 10 4 , 6 × 10 4 , 10 5 GeV 2 which correspond to dashed, solid and dotted-dashed lines.Blue lines are obtained in original KMR model[23,24] and red lines are obtained in our modified approach[29].

FIG. 5 :FIG. 6 :FIG. 7 :FIG. 8 :FIG. 9 :
FIG. 5: The differential cross sections for the production of three isolated photons as functions of M 123 = M 3γ .The hard scale in PRA calculations are taken as invariant mass of three-photon system, µ 0 = M 3γ .The green histogram corresponds LO contribution from Q Q → γγγ subprocess.The blue histogram corresponds NLO contribution from QR → qγγγ subprocess.The red histogram is their sum..

TABLE I :
PRA predictions for p + p → γγγ + X total cross section at √ s = 8 TeV for the different choice of factorization/renormalization scale (µ = µ F = µ R ).Errors indicate upper and lower limits of the cross section due to scale uncertainty.

TABLE II :
Predictions for p + p → γγγ + X total cross section at the different center-of-mass energies, √ s.Hard scale is taken as µ = M 3γ .Numerical error of total cross section calculation is equal to 0.1%.TeV, the PRA predicts larger cross sections in comparing with the NNLO CPM calculations, see TableIV.We estimate excess