Diphoton production at Tevatron and the LHC in the NLO* approximation of the Parton Reggeization Approach
DDiphoton production at Tevatron and the LHC in the NLO (cid:63) approximation of the Parton Reggeization Approach.
M.A. Nefedov ∗ Samara State University, Ac. Pavlov st., 1, 443011 Samara, Russia
V.A. Saleev † Samara State University, Ac. Pavlov st., 1, 443011 Samara, Russia andS.P. Korolyov Samara State Aerospace University,Moscow Highway, 34, 443086, Samara, Russia
Abstract
The hadroproduction of prompt isolated photon pairs at high energies is studied in the NLO (cid:63) framework of the Parton Reggeization Approach. The real part of the NLO corrections is computed,and the procedure for the subtraction of double counting between real parton emissions in the hard-scattering matrix element and unintegrated PDF is constructed for the amplitudes with Reggeizedquarks in the initial state. The matrix element of the important NNLO subprocess RR → γγ withfull dependence on the transverse momenta of the initial-state Reggeized gluons is obtained. Wecompare obtained numerical results with diphoton spectra measured at Tevatron and the LHC,and find a good agreement of our predictions with experimental data at the high values of diphotontransverse momentum, p T , and especially at the p T larger than the diphoton invariant mass, M .In this multi-Regge kinematics region, the NLO correction is strongly suppressed, demonstratingthe self consistency of the Parton Reggeization Approach. PACS numbers: 12.38.Bx, 12.39.St, 12.40.Nn, 13.87.Ce ∗ Electronic address: [email protected] † Electronic address: [email protected] a r X i v : . [ h e p - ph ] A ug . INTRODUCTION Nowadays, the inclusive hadroproduction of pairs of isolated prompt photons (diphotons)is a subject of intense experimental and theoretical studies. From the experimental pointof view, this process forms an irreducible background in the searches of heavy neutral res-onances in the diphoton decay channel, such as Standard Model Higgs boson [1] and it’sBeyond Standard Model counterparts [2]. As for the process itself, it allows us to define theset of inclusive differential cross sections over such variables as the invariant mass of the pair( M ), it’s transverse momentum ( p T ), azimuthal angle between transverse momenta of thephotons (∆ φ ), rapidity of the photon pair ( Y γγ ), Collins-Soper angle in the center of massframe of the photon pair ( θ ) and a few others [3]. Most of these spectra are measured withhigh precision both at Tevatron [3] and the LHC [4].On the theoretical side, providing the predictions for the above mentioned rich set ofdifferential spectra is a challenging task even for the state of the art techniques in perturba-tive Quantum Chromodynamics (pQCD). While, for the inclusive isolated prompt photonproduction, the p T -spectra from CDF [5], ATLAS [6] and CMS [7] are described withinexperimental uncertainties in the Next to Leading Order (NLO) of conventional CollinearParton Model (CPM) of the QCD [8]. Also, the notably good results where obtained forthese spectra already in the Leading Order (LO) of k T -factorization in the Ref. [9, 10].In contrast, existing NLO CPM calculations, implemented in the DIPHOX [11] Monte-Carloevent generator, provide very poor description of p T and ∆ φ distributions measured by AT-LAS [4]. In the CPM, the full NNLO accuracy is required to provide qualitatively reasonabledescription of all distributions [12].Part of these difficulties can be traced back to the shortcomings of the CPM approxi-mation, where the transverse momentum of initial state partons is integrated over in theParton Distribution Functions (PDFs), but neglected in the hard scattering part of theprocess. Such treatment is justified for the fully inclusive single scale observables, such asdeep inelastic scattering structure functions or p T -spectra of single prompt photons and jets,where the corrections breaking the collinear factorization are shown to be suppressed by apowers of the hard scale [13].For the multi-scale differential observables, there is no obvious reason why the fixed-ordercalculation in the CPM should be a good approximation. Usually the simple picture of fac-2orization of the cross section of the hard process into the convolution of hard-scatteringcoefficient and some PDF-like objects is kept, but kinematical approximations are relaxed.In the treatment of Initial State Radiation (ISR) corrections in the Soft Collinear EffectiveTheory (SCET) [14] or in the Transverse Momentum Dependent (TMD) factorization for-malism [13, 15, 16], the transverse-momentum of the initial-state parton is kept unintegratedon the kinematical level, but neglected in the hard-scattering part, which is justified e. g.when the p T of the exclusive final state is much smaller than it’s invariant mass, so that thefollowing hierarchy of the light-cone momentum components for the initial-state parton ispreserved: q ∓ (cid:28) | q T | (cid:28) q ± = x √ S .In the opposite limit, when q ∓ (cid:28) | q T | ∼ q ± = x √ S , the k T -factorization [17] is valid,and transverse momentum of the initial state parton can no longer be neglected in the hardscattering amplitudes. To obtain the suitable hard scattering matrix element we will applythe hypothesis of parton Reggeization, which will be described below. In what follows wewill refer to the combination of k T -factorization with hard scattering matrix elements withReggeized partons in the initial state as the Parton Reggeization Approach (PRA). Thisapproach is mostly suitable for the study of the production of the final states with high p T and small invariant mass in the central rapidity region. At high energies √ S (cid:29) p T , suchfinal states are produced by the small- x partons, and the resummation of log(1 /x )-enhancedterms into the unintegrated PDF (unPDF) can be implemented [18]. Clearly, the regionsof applicability of the TMD and k T -factorization are overlapping, and they should matchwhen x → k T -factorization covers all available range of experimental data. Most of the cross sectioncomes from the region where the diphoton has small p T and photons fly nearly back-to-backin the transverse plane, so additional QCD radiation is kinematically constrained to be softand collinear, and the approach of SCET factorization will be preferable. On the contrary,at high p T and small ∆ φ , the k T -factorization will do a good job, as we will show below.The previous attempts to study the prompt diphoton production in k T -factorization [19,20] had their own problems. In the Ref. [19], only LO 2 → Q ¯ Q → γγ matrix element with off-shell initial state Reggeized quarks ( Q ), the matrix element for the RR → γγ with off-shell Reggeized gluons in the initial state was taken the same as in the3PM. In fact, this contribution was overestimated in Ref. [19] due to the erroneous overallfactor 4 in the partonic cross section of the subprocess gg → γγ presented in the Ref. [21],which have lead to the accidental agreement with the early Tevatron data [22]. This factorwas carefully checked against the results presented in the literature [23, 24], as well as byour independent calculations of the exact RR → γγ amplitude, described in the Sec. IV ofthe present paper.In the Ref. [20], the attempt to take into account the NLO 2 → → → q (cid:63) g (cid:63) → qγγ subprocess and the unPDF was not subtracted, which have lead to thequestionable conclusion, that no resummation of the effects of soft radiation is needed inthe small- p T region to describe the data.In view of above mentioned shortcomings of the previous calculations, the present studyhas two main goals. The first one is to calculate the real part of NLO corrections to theprocess under consideration in the PRA, and develop the technique of subtraction of doublecounting between real NLO corrections and unPDF in PRA. The second goal is to calculatethe matrix element of the quark-box subprocess RR → γγ in PRA, taking into account theexact dependence on the transverse momenta of initial-state Reggeized gluons.The present paper has the following structure, in Sec. II the relevant basics of the PRAformalism are outlined and the amplitude for the LO subprocess Q ¯ Q → γγ is derived. Inthe Sec. III the NLO 2 → RR → γγ is reviewed, and inthe Sec. V we compare our numerical results with the most recent CDF [3] and ATLAS [4]data. Our conclusions are summarized in the Sec. VI. II. BASIC FORMALISM AND LO CONTRIBUTION
As collinear factorization is based on the property of factorization of collinear singu-larities in QCD [25], the k T -factorization is based on the Balitsky-Fadin-Kuraev-Lipatov(BFKL) [26] (see [27, 28] for the review) factorization of QCD amplitudes in the Multi-ReggeKinematics (MRK), i. e. in the limit of the high scattering energy and fixed momentum4ransfers. For example, the amplitude for the subprocess q ( q ) + q ( q ) → q ( q ) + g ( q ) + q ( q )in the limit when s (cid:29) − t , s (cid:29) − t , where s ij = ( q i + q j ) , t ij = ( q i − q j ) , has the form of the amplitude with exchange of theeffective Reggeized particle in the t -channel: A c,µ = 2 s (¯ u ( q ) γ r u ( q )) · t (cid:18) s s (cid:19) ω ( t ) · Γ c,µr r ( q t , q t ) · t (cid:18) s s (cid:19) ω ( t ) · (¯ u ( q ) γ r u ( q )) , (1)where q t = q − q , q t = q − q , c, r , r are the color indices, γ r is the effective qqR vertex, Γ c,µr ,r ( q t , q t ) is the central gluon production vertex RRg , ω ( t ) is the gluon Reggetrajectory. The Slavnov-Taylor identity ( q t − q t ) µ Γ c,µr r ( q t , q t ) = 0 holds for the effectiveproduction vertex, which ensures the gauge invariance of the amplitude.The analogous form for the MRK asymptotics of the amplitude with quark exchange inthe t -channel was shown to hold in the Leading Logarithmic Approximation (LLA) in [29].For the review of modern status of the quark Reggeization in QCD see Ref. [30].The Regge factor s ω ( t ) resums the loop corrections enhanced by the log( s ) to all orders instrong coupling constant α s , and the dependence on the arbitrary scale s should be canceledby the analogous dependence of the effective vertices, taken in all orders of perturbationtheory. The Reggeized gluon in the t -channel is a scalar particle in the adjoint representationof the SU ( N c ). In the MRK limit, when all three particles in the final state are highlyseparated in rapidity, the light-cone momentum components carried by the Reggeons in t -channels obey the hierarchy q ∓ t (cid:28) | q t ⊥ | ∼ q ± t , so in the strict MRK limit, the “small”light-cone component is usually neglected.To go beyond the LLA in log( s ), one needs to consider the processes with a few clustersof particles in the final state, which are highly separated in rapidity, but keeping the exactkinematics within clusters. This is so called Quasi-Multi-Regge Kinematics(QMRK), andto obtain the amplitudes in this limit, the gauge invariant effective action for high energyprocesses in QCD was introduced in the Ref. [31]. Apart from the usual quark and gluonfields of QCD, which are supposed to live within a fixed rapidity interval, the fields ofReggeized gluons [31] and Reggeized quarks [32] are introduced to communicate betweenthe different rapidity intervals. To keep the t -channel factorized form of the amplitudes inthe QMRK limit, the Reggeon fields have to be gauge invariant, which leads to the specificform of their nonlocal interaction with the usual QCD fields, containing the Wilson lines.5auge invariance of Reggeon fields also ensures the gauge invariance of the effective emissionvertices, which describe the production of particles within the given interval of rapidity, asit was the case in (1). The Feynman Rules (FRs) of the effective theory are collected in therefs. [32, 33], but for the reader’s convenience, we also list the FRs relevant for the purposesof the present study in the Fig. 1. To compute the hard-scattering matrix elements in PRAone have to combine the FRs of the Fig. 1 with the usual FRs of QCD and QED, and usethe factors for the Reggeons in the initial-state of the hard subprocess, also defined in theFig. 1 to be compatible with the normalization of the unPDF described below.Recently, the new scheme to obtain gauge-invariant matrix elements for k T factorization,by exploiting the spinor-helicity representation and recursion relations for the tree-levelamplitudes was introduced [34, 35]. This technique is equivalent to the PRA for the tree-level amplitudes without internal Reggeon propagators, however, the construction of thesubtraction terms in the Sec. III requires the usage of the FRs of the refs. [32, 33].So far as, the form of the central production vertex, propagators of Reggeized gluonsand Regge trajectories, depends only on the quantum numbers of the Reggeon in the t -channel, and do not depend on what particles are in the initial state, the cross section ofthe production of particles to the central rapidity region in the inelastic pp collisions can bewritten in the form [17]: dσ = (cid:88) i,j (cid:90) dx x (cid:90) d q T π Φ i ( x , t , µ F ) (cid:90) dx x (cid:90) d q T π Φ j ( x , t , µ F ) d ˆ σ ij ( q , q ) , (2)where the sums are taken over the parton species, q , = x , P , + q T , are the momentaof the partons, q , = − q T , = − t , , P , are the four-momenta of the protons, 2 P P = S , d ˆ σ is the partonic cross section with Reggeized partons in the initial state. In what followswe will often use the Sudakov decomposition of the momenta: k = 12 (cid:0) n + k − + n − k + (cid:1) + k T , where k ± = k ± k , P = √ Sn − / P = √ Sn + /
2, ( n + ) = ( n − ) = 0, n + n − = 2.The unPDF Φ i ( x, q T , µ F ) is unintegrated over the transverse momentum q T , but stillintegrated over the ”small” light-cone component of momentum, so this light-cone compo-nent is neglected in the hard scattering part, which is hence formally in the QMRK withthe ISR and therefore it is gauge invariant. The exact kinematics will be restored by the6igher-order QMRK corrections. The factorization scale µ F is introduced to keep track ofthe position of the hard process on the axis of rapidity.The unPDF is normalized on the CPM number density PDF via: µ F (cid:90) dt Φ i ( x, t, µ F ) = xf i ( x, µ F ) . In the case of an inelastic scattering of objects with intrinsic hard scale, such as photonswith high center-of-mass energy and virtuality, the evolution of the unPDFs is governed bythe large log(1 /x ) and they satisfy the BFKL evolution equation [26]. In proton-proton colli-sions, the initial state does not provide us with intrinsic hard scale, therefore, the k T -orderedDokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) [36] evolution at small k T should bemerged with rapidity-ordered BFKL evolution at high- k T final steps of the ISR cascade.The last problem is highly nontrivial and equivalent to the complete resummation of thelog( k T )-enchanced terms in the BFKL kernel. A few phenomenological schemes to com-pute unPDFs of a proton where proposed, such as the Ciafaloni-Catani-Fiorani-Marchesini(CCFM) approach [18], the Bl¨umlein approach [37] and the Kimber-Martin-Ryskin ap-proach [38]. In the LO calculations in PRA, the definition of the hard-scattering coefficient d ˆ σ is independent on the approximations made in the unPDF, so any unPDF can be used,and spread between them gives the theoretical uncertainty. In fact, the recent studies [39, 40]show, that in the realistic kinematical conditions, the LO calculations with KMR and re-cent version of the CCFM unPDFs [41] give very close results. At the NLO, one shoulddevelop the proper matching scheme between the unPDF and corrections included into thehard-scattering kernel, which introduces a difference in treatment of different unPDFs.In the present paper we will work with the version of the KMR formula for the unPDFs,described in the Ref. [42]. The KMR prescription introduces the simplest possible scenario,where the k T -ordered DGLAP chain of the emissions is followed by exactly one emission,ordered in rapidity with the particles produced in the hard subprocess. Due to the strong k T -ordering of the DGLAP evolution, the transverse momentum of the parton in the initial-state of the hard subprocess is approximated to come completely from the last step of theevolution. With this approximations, one can obtain the unPDF from the conventionalcollinear PDF as follows: 7 i ( x, q T , µ ) = 1 q T (cid:90) x dzT i ( q , µ ) α s ( q )2 π (cid:88) j P ij ( z ) f j (cid:16) xz , q (cid:17) θ (cid:0) ∆ ij ( q T , µ ) − z (cid:1) , (3)where P ij ( z )- DGLAP splitting function, q = q T / (1 − z ) – virtuality of the parton in the t -channel, the Sudakov formfactor T i is defined as: T i ( q , µ ) = exp − µ (cid:90) q dk k α s ( k )2 π (cid:88) i,j (cid:90) dξξP ij ( ξ ) θ (cid:0) ∆ ij ( k (1 − ξ ) , µ ) − ξ (cid:1) , (4)and the ordering in rapidity between the last parton emission and the particles produced inthe hard subprocess is implemented via the following infrared cutoff [53]:∆ ij ( q T , µ ) = µµ + q T δ ij + (1 − δ ij ) . In the present study, we use the version of the KMR formula (3) with LO DGLAP splittingfunctions, but NLO PDFs as a collinear input, because, as it was shown in the Ref. [42] theusage of the NLO PDFs and the exact scale q are the most numerically important effectsdistinguishing the LO KMR distribution of the Ref. [38] and the NLO prescription of theRef. [42]. Also, as it will be shown in the Sec. III, the usage of the LO DGLAP splittingfunctions is compatible with the PRA, while at the NLO, the splitting functions should berecalculated using the effective theory of the refs. [31, 32].The effects of the Sudakov resummation are known to be dominant in the doubly-asymptotic region, q T (cid:28) µ and z → ∆, which is most important in pp collisions, thereforewe use the Sudakov form-factor in (4). The opposite limit, when z → q T (cid:28) µ , iscaptured by the Regge factor s ω ( t ) , but the proper procedure of matching of the double-logarithmic corrections, between Sudakov and Regge factors is also beyond the scope of thepresent study.Now we are at step to discuss the LO and NLO contributions to the prompt photon pairproduction in the PRA. There is only one LO ( O ( α α s )) subprocess: Q ( q ) + ¯ Q ( q ) → γ ( q ) + γ ( q ) (5)The set of Feynman diagrams for this subprocess is presented on the Fig. 2. The amplitudeof the process (5) obeys the Ward identity of Quantum Electrodynamics (QED), and the8mplitude squared and averaged over the spin and color quantum numbers of the initialstate, which was obtained in a first time in the Ref. [19], has the form: |A ( Q ¯ Q → γγ ) | = 323 π e q α x x a a b b S ˆ t ˆ u (cid:16) w + w S + w S + w S (cid:17) , (6)where a = q +3 / √ S , a = q +4 / √ S , b = q − / √ S , b = q − / √ S , ˆ s = ( q + q ) , ˆ t = ( q − q ) ,ˆ u = ( q − q ) , x = a + a , x = b + b , α = e / (4 π ), e q is the electric charge of the quarkin the units of the electron charge and the coefficients w i can be represented as follows: w = t t ( t + t ) − ˆ t ˆ u (ˆ t + ˆ u ) , − w = t t ( a − a )( b − b ) + t x ( b ˆ t + b ˆ u ) ++ t x ( a ˆ t + a ˆ u ) + ˆ t ˆ u ( a b + 2 a b + 2 a b + a b ) , − w = b b x t + a a x t + a b ˆ t ( x a + a b ) + a b ˆ u ( a b + a x ) , − w = a a b b (cid:16) a b (cid:16) ˆ t ˆ u (cid:17) + a b (cid:16) ˆ u ˆ t (cid:17)(cid:17) . Taking the collinear limit t , → q ¯ q → γγ : |A ( q ¯ q → γγ ) | = 323 π e q α (cid:18) ˆ t ˆ u + ˆ u ˆ t (cid:19) . In the next section we will discuss the tree-level NLO corrections.
III. REAL NLO CORRECTIONS
The tree-level NLO ( O ( α α s )) subprocesses are: Q ( q ) + R ( q ) → γ ( q ) + γ ( q ) + q ( q ) , (7) Q ( q ) + ¯ Q ( q ) → γ ( q ) + γ ( q ) + g ( q ) . (8)The sets of Feynman diagrams for them are presented in the Figs. 3 and 4. The FRsof the Fig. 1 where implemented as the model file for the FeynArts [45],
Mathematica based package, and the computation of the squared matrix elements was performed using9 eynArts , FeynCalc [46], and
FORM programs. It was checked analytically, that the ampli-tudes for the NLO subprocesses (7) and (8) obey the Ward (Slavnov-Taylor) identities withrespect to all final state photons (gluons) independently on the transverse momentum of theinitial state Reggeized partons. Unfortunately, the obtained expressions are too large andnon-informative to present them here.The squared matrix element of the subprocess (7) contains the collinear singularity, whenthe three-momentum of the quark becomes collinear to the three-momentum of one of thephotons. This collinear singularity can be absorbed into the nonperturbative parton-to-photon fragmentation function, and then, the theoretical cross section is represented as thesum of direct contribution, where the collinear singularity is subtracted, according to e. g.
M S scheme, and fragmentation contribution, which is equal to the convolution of the crosssection of the parton production in pQCD and the parton-to-photon fragmentation function.Experimental (hard-cone) isolation condition require the amount of hadronic energy withinthe photon isolation cone of the radius R to be smaller than the fixed value E ( ISO ) T ∼ O (1)GeV: E ( had ) T ( r < R ) < E ( ISO ) T , (9)where r = (cid:112) ∆ η + ∆ φ is the distance in the pseudorapidity–azimuthal angle plane, E ( had ) T ( r < R ) is the amount of the hadronic transverse energy within the isolation conearound the photon. This isolation condition strongly suppresses the fragmentation compo-nent, but at high energies, fragmentation is still non-negligible, constituting up to the 20%of the cross section [11].The proper treatment of the collinear singularity, considerably complicates the analyticalcomputations both in the NNLO of CPM [12, 47], and in the NLO of PRA. Since in the PRA,the part of transverse momentum is provided to the hard subprocess by the phenomenologicalunPDFs. To avoid this difficulties, one can define the direct part of the cross section in theinfrared-safe way, using the smooth-cone isolation condition [48]: r < R ⇒ E ( had ) T ( r ) < E ( ISO ) T χ ( r ) , (10)where χ ( r ) = (cid:16) − cos( r )1 − cos( R ) (cid:17) n , n ≥ /
2. The isolation condition (10) is easy to implement intothe process of Monte-Carlo integration, and it makes the cross section of the subprocess(7) finite, because the collinear singularities associated with the initial-state are regularizedby the unPDF. Applying the smooth-cone isolation to our calculation we are completely10liminating the need in the fragmentation component [47, 48], but of course this isolationdo not match to the experimental one. However, as it was shown in the Ref. [47], the crosssection obtained with the isolation condition (10) is a lower estimate for the direct plusfragmentation cross section, obtained with the hard-cone isolation. Numerically, for n = 1this estimate is very good, since it reproduces the NLO results with standard isolation withthe accuracy of O (1%) [47]. Having in mind, that we are going to discuss O (50% − q T (cid:28) µ faster than any positive power of q T and therefore regularizesthe collinear and soft singularities of the matrix element of the subprocess (8) in the limitof q T → α s in the hard-scattering coefficient. Therefore, the corresponding MRK asymptoticsshould be subtracted from the NLO QMRK contributions (7), and (8) to avoid the doublecounting, when the additional parton is highly separated in rapidity from the central region.The analogous procedure of the “localization in rapidity” of the QMRK contributions wasproposed in the refs. [43, 44]. To be compatible with our definition of the KMR unPDF (3),this subtraction term should interpolate smoothly between the strict MRK limit, when addi-tional parton goes deeply forward or backward in rapidity with fixed transverse-momentum,and collinear factorization limit, when the initial-state partons are nearly on-shell and ad-ditional parton has a small transverse momentum but it’s rapidity is arbitrary. Below, suchsubtraction term is constructed in close analogy with the High Energy Jets approach [49].The Feynman diagrams for the subtraction terms, required for the squared amplitudesof the subprocesses (7), and (8) are shown in the Fig. 5 and can be easily written accordingto the FRs of the Fig. 1. To extend the applicability of the subtraction terms outside of thestrict MRK limit, one have to implement the exact 2 → t -channel momentum in the propagator of the Reggeizedquark. In what follows, we will refer to the amplitudes with the Reggeon propagators andvertices, but without kinematical approximations, characteristic for the MRK, as modifiedMRK (mMRK) amplitudes. As it was checked explicitly, the implementation of the exact11inematics do not destroy the gauge invariance of the subtraction terms with the Reggeizedquarks in the ˆ t -channels, presented in the Fig. 5, as it was the case for the mMRK amplitudeswith the Reggeized gluons in the ˆ t -channels in the Ref. [49].The last ambiguity, which we have to fix in the definition of our mMRK amplitudes isthe position of the ˆ P ± -projector in the numerator of the propagator of the Reggeized quark.In the MRK limit, the “small” light-cone component of the Reggeon momentum can beneglected and the projectors ˆ P ± commute with ˆ q t under the sign of the trace, but outsideof this limit, this is not true anymore. To fix this ambiguity, let’s study the amplitudes ofthe mMRK subprocesses in the Fig. 6. Explicitly, they have the forms: A abµν ( g ¯ Q → ¯ qg ) = g s ¯ v ( q (cid:107) ) (cid:18) γ ν − ˆ q n + ν q +4 − ˆ q t n − ν q − (cid:19) T b ˆ P + ˆ q t q t (cid:18) γ µ − ˆ q t n + µ q +1 (cid:19) T a v ( q ) , (11) A abµν ( q ¯ Q → gg ) = g s ¯ v ( q (cid:107) ) (cid:18) γ ν − ˆ q n + ν q +4 − ˆ q t n − ν q − (cid:19) T b ˆ P + ˆ q t q t (cid:18) γ µ + ˆ q t n + µ q +3 (cid:19) T a u ( q ) , (12) A aµ ( Q ¯ Q → g ) = g s ¯ v ( q (cid:107) ) (cid:18) γ ν − ˆ q n + µ q +4 − ˆ q t n − µ q − (cid:19) T a u ( q (cid:107) t ) . (13)Taking the squared modulus of these amplitudes and averaging them over the initial-statespin and color quantum numbers, we get: |A ( g ¯ Q → ¯ qg ) | = g s ˆ s + t ˆ s + t + ˆ t P qg ( z ) z ˆ t |A ( Q ¯ Q → g ) | , (14) |A ( q ¯ Q → gg ) | = g s ˆ s + t ˆ s + t + ˆ t P qq ( z ) z ˆ t |A ( Q ¯ Q → g ) | , (15) |A ( Q ¯ Q → g ) | = g s C A C F N c ( q t ⊥ + t ) , (16)where we have taken the limit q = 0 to facilitate the study of the collinear singularity, q = − t , the invariants ˆ s , ˆ t , ˆ u are defined after the Eq. (6), q t ⊥ = ( q T − q T ), z = 1 − q +3 /q +1 and P qg ( z ) = ( z + (1 − z ) ), P qq ( z ) = C F z − z are the LO DGLAP splitting functions.When z (cid:28) t -fixed, the partons 3 and 4 are in the MRK. In the opposite (collinear)limit ˆ t →
0, the squared amplitudes (14), and (15) factorize into the collinear singularitywith the corresponding DGLAP splitting function and the squared amplitude (16). Fromthis example one can conclude, that the factor ˆ q t should be taken together with the vertexof the MRK-emission to correctly reproduce the collinear singularity of the amplitude. Thisprescription is denoted by the crosses on the quark propagators in the Figs. 1, 5 and 6.The squared amplitudes (14, 15) can also be used to explain the structure of the fac-torization formula (2) and the unPDF (3). The presence of the exact DGLAP splitting12unctions in (14), and (15) corresponds to the usage of the exact splitting functions in theunPDF (3). Factor z in the denominators of (14, 15) is nothing but a flux factor of the ˆ t -channel partons, which tells us, that for the Reggeized partons one should use the same fluxfactor I = 2 Sx x as for the CPM partons. After the integration over the “small” light-conecomponent in the definition of the unPDF, the additional factor 1 / (1 − z ) appears, whichconverts ˆ t into ˆ t (1 − z ) = q t ⊥ , that’s why the q T and not q stands in the denominator of(3).The rapidity ordering conditions are imposed in the subtraction terms for the subprocess(8) (see Fig. 5, lower panel), while for the case of the subprocess (7) the rapidity of thequark in the final state is unconstrained (Fig. 5, upper panel). This corresponds to thefact, that in the KMR unPDF, the radiation of the gluon is ordered in rapidity with theparticles, produced in the hard subprocess, while for the quark it is not the case. So, themMRK terms constructed according to the Feynman diagrams in the Fig. 5 are completelywell-defined and correspond to the definition of the unPDF (3). IV. THE QUARK-BOX CONTRIBUTION
The subprocess: R ( q ) + R ( q ) → γ ( q ) + γ ( q ) , (17)is described by the quark-box amplitude, and is formally NNLO ( O ( α α s )), but it it’scontribution to the total cross section is expected to be comparable with NLO contributions,due to the enhancement by two gluon unPDFs. The helicity amplitudes for the subprocess(17) could be written as: A ( RR, λ λ ) = q +1 q − √ t t n − µ n + µ ε (cid:63)µ ( λ ) ε (cid:63)µ ( − λ ) M µ µ µ µ , (18)where λ , λ are the helicities of the photons in the final state and the fourth-rank vacuumpolarization tensor has the form: M µ µ µ µ = 2 (cid:90) d q (cid:26) tr [(ˆ q − ˆ q ) γ µ (ˆ q + ˆ q − ˆ q ) γ µ (ˆ q + ˆ q ) γ µ ˆ qγ µ ]( q − q ) ( q + q − q ) ( q + q ) q ++ ( q ↔ q , µ ↔ µ ) + ( q ↔ − q , µ ↔ µ ) } , (19)13here the factor 2 takes into account the diagrams with the opposite direction of the fermionnumber flow. The following overall factor is taken out from the amplitude (18): e g s (2 π ) δ ab (cid:32)(cid:88) q e q (cid:33) . We take the polarization vectors for the final-state photons in the form: ε µ ( λ ) = 1 √ (cid:0) n µx + iλn µy (cid:1) , where n µx = 1∆ (( q q ) q µ − ( q q ) q µ − ( q q ) q µ ) ,n µy = − (cid:15) µq q q , and ∆ = (cid:112) ˆ s ˆ t ˆ u − ˆ st t / γR → γg , and explicitly demonstrate the cancellation of the spurious collinearsingularity 1 / ( t t ) in the squared amplitude. For the process (17) it turns out to be im-possible to obtain the reasonably compact results, and the task is actually to obtain therepresentation for the helicity amplitudes which will be feasible for the numerical evaluationat all. To do this, we observe, that exploiting the Ward identity q µ , , M µ µ µ µ = 0 for thetensor (19), one can make the following substitutions in (17): q +1 q − √ t t ( n − ) µ ( n + ) µ → n µ T n µ T , where n T , = q T , / √ t , .To get rid of the (cid:15) -tensors and directly pass to the Passarino-Veltman reduction for theFeynman integrals with the scalar products in the numerator, we exploit the same trick asin Ref. [52]. We decompose the four-vectors n T , as follows: n T = β (1)0 q + β (1)3 q + β (1)4 q + γ n y , (20) n T = β (2)0 q + β (2)3 q + β (2)4 q + γ n y , (21)and the vector n y is introduced via it’s scalar products: n y = − n y q = n y q = n y q = 0.The coefficients of this decomposition can be straightforwardly expressed through theMandelstam invariants, transverse momenta of particles and azimuthal angles. After the14assarino-Veltman reduction, the helicity amplitudes where represented as a linear combi-nations of two, three and four-point scalar one loop integrals, and the cancellation of theUltra-Violet (UV) and Infra-Red (IR) divergences was checked both analytically and nu-merically. The coefficients of this decomposition depends on 5 invariants ˆ s, ˆ t, ˆ u, t , t and8 coefficients β ( j ) i , γ , , i.e. 13 parameters in total. They can be represented as rationalfunctions with tens of thousands terms in the numerators. It turns out, that just to reliablycheck the cancellation of the UV and IR divergences numerically, one have to compute thiscoefficients with 30 digits of accuracy at least.Also, it was checked, both analytically and numerically, that the collinear limit for thesquared helicity amplitudes (18), defined as: π (cid:90) dφ dφ (2 π ) lim t , → |A ( RR, λ λ ) | = 14 (cid:88) λ , = ± |A CP M ( λ λ , λ λ ) | , holds, where the |A CP M ( λ λ , λ λ ) | is the squared helicity amplitude of the process gg → γγ in the CPM. The numerical check of the collinear limit was performed by the techniquedescribed in the Ref. [52]. The numerical results for the subprocess (17) will be presentedin the next section, and the FORTRAN code for the calculation of the helicity amplitudes anddifferential cross sections of the process (17), as well as for the 2 → → V. NUMERICAL RESULTS
The differential cross section of the 2 → dσdq T dq T d ∆ φdy dy = 12! (cid:90) dt π (cid:90) dφ (cid:88) ij Φ i ( x , t , µ F )Φ j ( x , t , µ F ) q T q T |A ij | π ) ( Sx x ) , (22)where the factor 1 /
2! takes into account the identical nature of the photons, q T i = | q T i | , y i are the rapidities of the final-state particles, ∆ φ is the azimuthal angle between transversemomenta of the photons, φ is the azimuthal angle between q T and q T , t = ( q T + q T − q T ) , x , = ( q T e ± y + q T e ± y ) / √ S . The spectra differential in the diphoton invariantmass ( M ) and diphoton transverse momentum p T = (cid:112) q T + q T , could be obtained using15he following substitutions: dM = q T CM dq T , (23) dp T = Dp T dq T , (24)where C = cosh( y − y ) − cos(∆ φ ), D = (cid:112) p T − q T sin (∆ φ ), q T = M / (2 q T C ) for thecase of (23) and q T = D − q T cos(∆ φ ) for the (24).For the 2 → dσdq T dq T d ∆ φdy dy dy = 12! (cid:90) dt π (cid:90) dφ (cid:90) dt π (cid:90) dφ ×× (cid:88) ij Φ i ( x , t , µ F )Φ j ( x , t , µ F ) q T q T |A ij | π ) ( Sx x ) , (25)and the differential cross sections over the p T and M could be obtained using the substitu-tions (23), and (24) as in the 2 → α = 1 / .
036 was used in the calculations together with the NLO formula for the α s with α s ( M Z ) = 0 . m c = 1 . m b = 4 .
75 GeV. Thechoice of the factorization and renormalization scales µ R = µ F = ξM commonly used inthe literature [3, 4, 11, 12] was adopted, where the default value for ξ = 1 and the values ξ = 2 ± where used to estimate the scale uncertainty of the calculation, which is indicatedin the Figs. 8 – 12 as the gray band. The numerical computations where performed mostlyusing the Suave adaptive Monte-Carlo integration algorithm with the cross-checks againstthe results of
Vegas and
Divonne algorithms implemented in the
CUBA library [51].Before the presentation of the comparison of our calculations with experimental data,let us discuss the contribution of 2 → p T -spectra for the NLOcontributions (7, 8) in the ATLAS-2013 kinematical conditions (see the second column ofthe table I), is presented together with the corresponding mMRK subtraction contribution.For the CDF-2012 kinematics of the Ref. [3] (first column of the table I) the qualitativepicture is the same.From the Fig. 7 one can observe, that mMRK subtraction term reproduces the exactcontribution of the NLO subprocess (8) with the 10% accuracy and constitutes more than16 ¯ p , CDF-2012 [3] pp , ATLAS-2013 [4] √ S = 1960 GeV √ S = 7000 GeV q T , ≥ ,
17 GeV q T , ≥ ,
25 GeV | y , | ≤ . | y , | ≤ .
37, 1 . ≤ | y , | ≤ . R = 0 . E ( ISO ) T = 2 GeV R = 0 . E ( ISO ) T = 4 GeVTABLE I: Kinematical conditions for the CDF and ATLAS datasets.
50% of the cross section of the subprocess (7) for the p T >
50 GeV. As the right panel of theFig. 7 shows, for the subprocess (7) the significant deviation from the mMRK asymptoticsstarts only for ∆ y = y − Y γγ < . y , the the QMRK 2 → → y . Consequently,more than 50% of the cross section of the subprocess (7) and almost all contribution of thesubprocess (8) will be canceled by the subtraction term. Having this in mind we do notinclude the contribution of the subprocess (8) in the further calculations.The squared amplitude for the subprocess (7) can be safely integrated from q T = 0 in(25). The cross section for the subprocess (8) is also finite, but for the small values of q T ,most of the cross section is accumulated at t , ∼ , which is nothing else than themanifestation of the usual infrared singularity for the radiation of the soft gluon. For thisreason the cutoff q T > t behavior of the unPDF is unphysical and willbe canceled away by the NLO real-virtual interference contribution.Now we are in a position to compare the predictions of our model with the experimentaldata of the refs. [3, 4]. For the comparison, we choose three main observables, dσ/dp T , dσ/d ∆ φ , as ones, where the fixed-order calculations in the CPM are experiencing the great-est difficulties and dσ/dM as the benchmark CPM observable, for which the multi-scalenature of the process under consideration is less important. The comparison for the otherobservables will be discussed elsewhere.In the Fig. 8 the p T -spectra of the photon pair, measured by the CDF Collaboration ispresented. For this dataset, three data samples are provided, the inclusive one and two data17amples with the additional kinematical constraint p T < M or p T > M imposed. One cannote, that the inclusive data and data for the p T < M are well described for the p T > p T > M data are well described by our prediction for all values of p T ,and the NLO contribution is, in fact, negligible. The contribution of the box subprocess(17) is only about 15% of the cross section predicted at small p T , and decreases with p T veryfast, contributing significantly only for the p T <
30 GeV.For the p T <
25 GeV one can observe the deficit of the predicted cross section whichreaches up to a factor of 5 at the p T = 7 GeV. The region of small- p T corresponds to thekinematics of CPM, where the radiation of soft gluons and virtual corrections are dominating.We expect, that computation of the NLO real-virtual interference correction in the PRA willsignificantly reduce this gap. One of the advantages of PRA is that at NLO this correction isfinite and can be considered separately from the real NLO corrections, which are the subjectof the present study.The good description of the data for the p T > M region supports the self-consistencyof our approach, since the NLO correction in this region is almost canceled by the mMRKsubtraction terms and the contribution of the real-virtual NLO correction is expected to besmall here.For the reader’s convinience, in the figs. 8 – 12, we also have plotted the correspondingNLO CPM predictions. Data for these plots correspond to the Diphox predictions [11],presented in the CDF [3] and ATLAS [4] experimental papers. The contribution of the gg → γγ subprocess is also included into this predictions, via GAMMA2MC program [24].Comparing the NLO CPM and NLO (cid:63)
PRA predictions in the figs. 8 – 12 one can conclude,that, NLO (cid:63) approximation in PRA can not describe the data on the dσ/dM -distributiondue to the absence of the loop correction, which contributes mostly in the back-to-backCPM-like kinematics. But for the configurations far away from the CPM kinematics, NLO (cid:63)
PRA describes data substantially better than NLO CPM, especially at the LHC. Moreover,in this region, the NLO (cid:63)
PRA prediction is dominated by the LO term, which demonstratesthe better stability of PRA predictions for the kinematics far away from CPM one. Inclusionof full NLO corrections should also improve the agreement in the CPM region.In the left panel of the Fig. 9 one can observe the same qualitative features as in theleft panel of the Fig. 8, despite the fact, that we have moved from Tevatron to the LHC18ith it’s 3 . pp collisions instead of p ¯ p ones. The NLOsubprocess (7) is more important at the LHC than at the Tevatron, contributing significantlyup to p T = 200 GeV.In the Fig. 10 and right panel of the Fig. 9 the ∆ φ -spectra for the Tevatron and LHCare presented. In the both figures one can observe a good agreement of our predictions withdata for ∆ φ < . p T -spectrum. The good description of the Tevatron data for the p T > M case is alsothere, as well, as the deficit of the predicted cross section for the back-to-back kinematics.As for the M -spectra of the Figs. 11 and 12, one certainly expects the deficit of thecalculated cross section for the most values of M due to the deficit of the cross section forthe CPM kinematics, observed earlier, since most of the total cross section is accumulatednear the CPM configurations. However, in the region of M below the peak, the data arewell described, demonstrating that PRA is suitable for the description of the effects ofkinematical cuts. Once again, we observe the good description of the M -spectrum for the p T > M subset of the Tevatron data.The contribution of the quark-box subprocess 17 to the M -spectra is found to be onlyabout 8% of the observed cross section in the peak, and 18% of the predicted cross section,both for the CDF-2012 and ATLAS-2013 kinematics. This result is 20 −
30% smaller than theusual CPM estimate [12], which is in accordance with the findings of the Ref. [52] where itwas shown, that the space-like virtuality of the initial-state partons suppresses the γR → γg contribution with respect to CPM expectation. VI. CONCLUSIONS
In the present study, the pair hadroproduction of prompt photons is considered in theframework of PRA with tree-level NLO corrections (7, 8) and NNLO quark-box subprocess(17) taken into account. The procedure of localization in rapidity of the tree-level NLOcorrections to avoid the double counting the real emissions between the hard-scattering partof the cross section and unPDF is proposed in the Sec. III. As a consequence of this pro-cedure, the real NLO corrections are put under quantitative control, and their contributionwas found to be numerically small at high p T or in the kinematical region p T > M . The19inematical region p T > M is interesting for the further theoretical and experimental study,as an ideal testing site for the PRA, where the MRK between the ISR and the hard sub-process is dominating. The contribution of the quark-box subprocess (17) was found to beabout 8% of the observed cross section in the peak of the dσ/dM distribution, which is abit smaller than the CPM estimate [12] due to the space-like virtuality of the initial-statepartons, similarly to the results of the Ref. [52]. Acknowledgements
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