A detail study of the LHC and TEVATRON hadron-hadron prompt-photon pair production experiments in the angular ordering constraint k_t-factorization approaches
M. Modarres, R. Aminzadeh Nik, R. Kord Valeshbadi, H. Hosseinkhani, N. Olanj
AA detail study of the LHC and TEVATRON hadron-hadronprompt-photon pair production experiments in the angularordering constraint k t -factorization approaches M. M odarres , ∗ R. Aminzadeh N ik , and
R. Kord V aleshbadi
Department of Physics, University of
T ehran , 1439955961,
T ehran , Iran.
H. Hosseinkhani
Plasma and Fusion Research School,Nuclear Science and Technology Research Institute, 14395-836 Tehran, Iran.
N.Olanj
Physics Department, Faculty of Science, Bu - AliSina
University, 65178,
Hamedan , Iran. a r X i v : . [ h e p - ph ] J a n bstract In the present work, which is based on the k t -factorization framework, it is intended to make adetail study of the isolated prompt-photon pairs (IPPP) production in the high-energy inelastichadron-hadron collisions differential cross section. The two scheme-dependent unintegrated partondistribution functions (UPDF) in which the angular ordering constraints (AOC) are imposed,namely the Kimber-Martin-Ryskin (KMR) and the Martin-Ryskin-Watt (MRW) approaches, inthe leading and the next-to-leading orders (LO and NLO) are considered, respectively. These twoprescriptions (KMR and MRW) utilize the phenomenological parton distribution functions (PDF)libraries of Martin et al, i.e. the MMHT2014. The computations are performed in accordancewith the initial dynamics of latest existing experimental reports of the D0, CDF, CMS and ATLAScollaborations and the different experimental constraints. It is shown that above frameworks arecapable of producing acceptable results, compared to the experimental data, the pQCD and someMonte Carlo calculations (i.e. 2 γ NNLO, SHERPA, DIPHOX and RESBOS). It is also concludedthat the KMR framework produces better results in the higher center-of-mass energies, whilethe same thing can be argued about the LO-MRW prescription in lower energies. Additionally,these two schemes show different behavior in the regions where the fragmentation and higherpQCD effects become important. A clear prediction for the various shoulders and tails which weredetected experimentally are observed and discussed in the present theoretical approaches. Thepossible double countings between 2 → → PACS numbers: 12.38.Bx, 13.85.Qk, 13.60.-r
Keywords: unintegrated parton distribution functions, isolated prompt-photon pair production, di-photonproduction, k t -factorization, Guillet shoulder. ∗ Corresponding author, Email: [email protected],Tel:+98-21-61118645, Fax:+98-21-88004781. . INTRODUCTION The study of photon pair production plays an important role in the investigation of: (1)the perturbative quantum chromodynamics (pQCD) and (2) better observation of the Higgsboson’s decay to diphotons, as well as (3) some theories, which extended beyond the standardmodel and should give some predictions regarding the new phenomena in the fundamentalparticle physics [1]. Many experimental efforts at the LHC and TEVATRON colliders havebeen performed to explore the physics of these regions, e.g. the D0, CDF, CMS and AT-LAS collaborations [2–9]. These investigations are probing different channels and exploringdifferent aspects of the above subjects, such as producing differential cross-section of thephoton pair production as a function of the azimuthal separation angle between the photonpair in the laboratory frame (∆ φ γγ ) and the transverse momenta of the photon pair ( p t,γγ ).They are particularly useful to study the higher order pQCD and the fragmentation effects[9]. Other observable, namely the photon pair invariant-mass ( M γγ ) and the polar angle ofthe highest photon-transfer-energy-momentum in the Collins-Soper isolated prompt photonpair (IPPP) rest frame ( cosθ ∗ γγ ), are also powerful tools to investigate the spin of the photonpair resonances [9]. The experimental collaborations conventionally use some parton levelMonte-Carlo programs, e.g. RESBOS, DIPHOX, 2 γ NNLO and SHERPA [10–13], to testthe pQCD theory against their data. Also, in the recent years, (2016), a new article basedon MCFM program, (Monte Carlo for FeMtobarn), which uses the collinear factorizationformalism, with NNLO accuracy was published that its result has good agreement with theavailable data [14].The RESBOS Monte-Carlo event generator, provides the next-to-leading order (NLO)level pQCD predictions for the IPPP with the soft gluon re-summation which can includethe single photon fragmentation [15] as well. The DIPHOX Monte-Carlo event generatorperforms the IPPP production at the NLO pQCD level, in which the single and doublefragmentation contributions [15] are also included. The 2 γ NNLO is developed to includethe full next-to-next-to-leading order (NNLO) pQCD, without considering fragmentationcontributions [12]. Another choice is the SHERPA [13] Monte Carlo event generator, thatcould simulate the high-energy reactions of particles in the hadron-hadron collisions.To perform such a analysis one usually needs parton distribution functions (PDF), a ( x, Q ), or unintegrated PDF (UPDF), f ( x, k t , Q ), see the references [16–31] and the3ection III. Note that x , Q and k t are the Bjorken scale,the hard scale and the partontransverse momentum.In the most of the recent theories, which explore the domain beyond the standard model,the photon is expected to be present at the final states. Therefore, we expect to see somefeatures of these theories in the existing experimental data. Consequently, a detailed analysisover the experimental data is vital, to determine the exact contributions out of the standardmodel, in order to justify or reject such theories [32].Regarding the complication and the weakness of different prescriptions [16–31], Martin etal [33, 34] defined the UPDF in the k t -factorization framework, in relation to the conventionalPDF [35], through the identity, xa ( x, Q ) (cid:39) (cid:90) Q dk t k t f ( x, k t , Q ) , (1)and developed the Kimber-Martin-Ryskin (KMR) and the Martin-Ryskin-Watt (MRW) ap-proaches [33, 34]. These formalisms were analyzed thoroughly via the calculation of theproton structure functions ( F and F L ) in the references [36–43]. Also, the applicationsof KMR and MRW frameworks in the LO and the NLO levels were investigated againstthe existing experimental data in the references [43–47], and some successful results wereachieved.In the present work, we intend to study the production of the IPPP in the high energyHadron-Hadron collisions, in the frameworks of KMR and MRW procedures. A primaryinvestigation, using the KMR k t -factorization approach, was performed in the reference [48],with some comparisons to the old data [2, 4, 6, 8], and some discrepancies especially inthe fragmentation regions, were observed. To investigate this problem, it is intended to usethree different procedure via the k t -factorization formalism, by utilizing the UPDF of KMR,LO-MRW and NLO-MRW frameworks. Then the extracted results are compared with thelatest, as well as old, experimental data of the D0, CDF, CMS and ATLAS collaborations intheir respective dynamical specifications [2–9] and other theoretical approaches [10–13, 15]discussed above. It will be shown that the k t -factorization framework is reasonably capableof describing of the high energy experiments data for the IPPP production. We also discussthe various advantages and disadvantages of the KMR and MRW prescriptions in connectionto each experiments conditions by presenting a detail comparison. One of the main goalsof our work is to observe and analyze the effect of imposing different visualizations of the4OC (embedded in different UPDF prescription schemes) in the partonic dynamics thatdepends on the deriving factors, (i.e. different experimental constraints), which are assumedin the existing experimental data, such that to cover the sensitive area to the fragmentationand the higher order pQCD effects (see also the sections V and VI). On the other hand,very recently there was some dispute about the application of the AOC and the cut off inthe KMR prescription [49] which is different in case of the MRW (note that for the KMRscheme the AOC is applied on both quark and gluon radiations but this is not the case inthe MRW approaches (see the section III)). As it was discussed in the reference [49], ourcalculations show that a qualitative agreements between the different schemes can be achieveat least in the calculation of differential cross sections [50]. Beside these, the ambiguity aboutthe fragmentation region [48] is considered by performing MRW-LO, which show a betteragreement with data at the fragmentation regions. On the other hand the Guillet shoulder[51] as well as new shoulders are observed (see the section V). In the reference [48], thevalence quarks were only considered in the case of q ∗ +¯ q ∗ for P ¯ P collision and the see-quarkscontributions (which is very small) were ignored. Beside these, in the PP collision the lattercontributions are sizable, in contrast to the reference [48] which again was not included.There are also some other essential points which will be discussed in the end of section II,IV and V.In the above calculation, one should evaluate the off-shell transition matrix elements.Various formalisms were introduced to calculate these off-shell matrix elements to insure thegauge invariance and the satisfactions of the Ward identities [52–59]. The off-shell matrixelement violates the gauge invariant which is necessary for the cross section calculation.In the references [55, 56, 60], it was shown that a suitable gauges for gluons and photonspolarizations lead to saving the gauge invariant of the off-shell gluons matrix elements. Theeikonal polarization is the result of using axial gauge which is considered in the current work(see the section II). But the problem of gauge invariance violation is still remained in allprocesses that the quarks are the incoming off-shell legs. However, in the small x limit andthe large transverse momentum, using the approximation made in the references [61, 62],the off-shell matrix element satisfies the gauge invariance requirements. In this work, withthe aforementioned constraints (the small x limit and the large transverse momentum),we check numerically the gauge invariant of each process individually. However there are(i) reggeization methods to evaluate the off-shell quark density matrix elements which are5nherently satisfy the gauge invariance in the all regions [57, 58] or (ii) the method developedin the references [52] , in which by modifying the vertexes and using auxiliary photons andquarks, an off-shell quark matrix elements are produced, which satisfy the Ward identity.To be insure about the above problems, in the section V we check our result against thoseof reference [57].The possible double countings between 2 → → k t -factorization, and individually the KMR, LO-MRWand NLO-MRW prescriptions, are presented in the section III. The section IV contains acomprehensive description about the methods and the tools for the calculation of the k t -dependent cross-section of the IPPP production in the various proton-proton (or proton-antiproton) inelastic collisions. The constraints of each experiment are discussed in theappendix A and finally, results, discussions and conclusions are given in the sections V andVI, respectively. II. THE THEORETICAL FRAMEWORK
In the study of photon production, there exists two possible categories; the prompt-photon and the non-prompt-photon. The first, includes the fragmentation and the directproduction of a photon while the second is created in the processes of hadronic decay. Inthis paper, we intend to base our calculations only on direct IPPP production. In order toset the kinematics of the pair photon production, we choose to work in the center-of-massframe of the initial protons. So, we can set their four-momenta as: P = √ s , , , , P = √ s , , , − s is the center-of-mass (CM) energy and P and P are the four-momenta of thecolliding protons. One can write the total cross-section for the production of the prompt-photons, summing over all the contributing partonic sub-processes, i.e. q ∗ + ¯ q ∗ → γ + γ , q ∗ (¯ q ∗ ) + g ∗ → γ + γ + q (¯ q ) and g ∗ + g ∗ → γ + γ [48]. Hence, it is required to write thefour-momenta of the incoming partons based on the four-momenta of the initial protons,using the Sudakov decomposition assumption as, k i = x i P i + k i,t , (2)where as we pointed out before, k i,t are the transverse momenta of the i th partons and x i arethe fraction of the longitudinal momentum of the protons that are inherited to that partons.Now, consider a particle of mass M that obtains a boost ψ from the rest-frame. Itsmomentum reads as, P µ = ( P + , M P + , t ) , with P + being the positive light-cone momentum of the particle. In general, one is able towrite its momentum based on the rapidity ( y ), which is defined as: y = 12 ln P + P − . The result is the following expression for the momentum of the particle: P µ = ( (cid:114) M + P t e y , (cid:114) M + P t e − y , P t ) , where (cid:112) M + P t is the so-called transverse energy of the particle, E t , [65]. Using the abovemethod and the conservation of energy-momentum, we can derive the following relations forthe subprocesses q ∗ + ¯ q ∗ −→ γ + γ and g ∗ + g ∗ −→ γ + γ : k ,t + k ,t = k ,t + k ,t ,x = ( | k ,t | e y + | k ,t | e y ) / √ s,x = (cid:0) | k ,t | e − y + | k ,t | e − y (cid:1) / √ s. (3)Similarly for the subprocess g ∗ + q ∗ (¯ q ∗ ) −→ γ + γ + q (¯ q ), we find: k ,t + k ,t = k ,t + k ,t + k ,t ,x = ( | k ,t | e y + | k ,t | e y + M ,t e y ) / √ s, = (cid:0) | k ,t | e − y + | k ,t | e − y + M ,t e − y (cid:1) / √ s. (4) k i,t and y i , i = 3 , M ,t is the transverse mass of produced quark or anti-quark with mass m thatis defined by: M ,t = (cid:113) m + | k ,t | , while y is its rapidity (the quarks masses are set equal to zero as it is stated above theequation (22)).In this work, we consider the simplest processes for the isolated prompt photon pairproduction. Therefore, the diagrams 2 → q ∗ + ¯ q ∗ or g ∗ + g ∗ → γ + γ ) and 2 → q ∗ (¯ q ∗ ) + g ∗ → γ + γ + q (¯ q )) are selected using the figure 1. The incoming legs in the figure1 can be the (anti)-quarks or gluons UPDF according to the k t -factorization procedure ofcorresponding differential cross section calculation. We also investigate the dependence ofthe differential cross section to the three prescriptions of UPDF (see the section III).Furthermore, it can be demonstrated that, the matrix element of all diagrams for the q ∗ + ¯ q ∗ → γ + γ sub-process is as follows: M = e e q (cid:15) µ ( k ) (cid:15) ν ( k ) ¯ U ( k ) (cid:18) γ ν /k − /k + m ( k − k ) − m γ µ + γ µ /k − /k + m ( k − k ) − m γ ν (cid:19) U ( k ) , where e and e q are the electron charge and the quark electric charge respectively and ε and ε are the polarization 4-vectors of the isolated prompt photons, that satisfy the co-variantequation: (cid:88) i =3 , ε µ ( k i ) ε ∗ υ ( k i ) = − g µυ . (5)Similarly for the g ∗ + q ∗ (¯ q ∗ ) → γ + γ + q (¯ q ) sub-process, the transition amplitude is: M = e e q √ πα s T α (cid:15) ξ ( k ) (cid:15) µ ( k ) (cid:15) ν ( k ) ¯ U ( k )[ (cid:88) i =1 , A ξµνi ] U ( k ) , (6)where A ξµνi are defined in the reference [48]. T α are the generators of the SU (3) color gauge group, as the color transition operators,that are defined in the relation with the Gell-Mann matrices ( λ α ), T α = λ α .ε ξ ( k ) is the polarization vector of the incoming off-shell gluon which should be modifiedwith the eikonal vertex (i.e the BFKL prescription, see the reference [66]). One choice is to8mpose the so called non-sense polarization conditions on ε ξ ( k ) which is not normalized toone [1, 66] (and it will not be used in the present work): (cid:15) ξ ( k ) = 2 k ,ξ √ s . But in the case of k t -factorization scheme and the off-shell gluons, the better choice is (cid:15) µ ( k ) = k µ ,t | k ,t | , which leads to the following identity and can be easily implemented in ourcalculation [1, 66]: (cid:88) ε µ ( k ) ε ∗ υ ( k ) = k µ ,t k υ ,t k ,t . (7)Finally, for the matrix element of the g ∗ + g ∗ → γ + γ sub-process, we use those which wascalculated before by Berger et al. [67], with this difference that the kinematics given in theequations (3) and (4) is imposed [48].So, generally the cross-section of IP P P production is: σ γγ = (cid:88) i,j = q,g (cid:90) ˆ σ ij ( x , x , µ ) f i ( x , µ ) f j ( x , µ ) dx dx , (8)where ˆ σ ij ( x , x , µ ) is the partonic cross-section and f j ( x i , µ ) (= xa j ( x i , µ )) are partondistribution function refer to the incoming parton i that depends on two variables, x and µ as the scales of the hard process. But in the high-energy domain, using the k t -factorizationtheory, we could rewrite the collinear cross-section, i.e., the equation (8), as, σ γγ = (cid:88) a ,a = q,g (cid:90) dx x (cid:90) dx x (cid:90) ∞ dk ,t k ,t (cid:90) ∞ dk ,t k ,t (cid:90) π dφ π (cid:90) π dφ π f a ( x , k ,t , µ ) f a ( x , k ,t , µ ) × ˆ σ a a ( x , k ,t , µ ; x , k ,t , µ ) , (9)where f ( x i , k t, , µ ) are the U P DF that depend on three parameters, i.e. x , k t and µ .The UPDF are directly obtained from the PDF by using different prescriptions (see thenext section). In this paper, we use the three approaches namely KMR [33], LO-MRW andNLO-MRW [34] to generate the UPDF from the PDF, to be inserted in the equation (9).In general, one should consider the KMR or MRW parton densities in the k t -factorizationcalculations correspond to non-normalized probability functions. They are used as the weightof the given transition amplitudes (the matrix elements in these cases). The transversemomentum dependence of the UPDF comes from considering all possible splittings up toand including the last splitting, see the references [33, 64, 68, 69], while the evolution up to9he hard scale without change in the k t , due to virtual contributions, is encapsulated in theSudakov-like survival form factor. Therefore, all splittings and real emissions of the partons,including the last emission, are factorized in the function f g ( x, k t , µ ) as its definition. Thelast emission from the definition of the produced UPDF can not be disassociated and to becount as the part of the 2 → q ∗ + ¯ q ∗ → γ + γ and g ∗ + q ∗ (¯ q ∗ ) → q (¯ q ) + γ + γ processes together in the k t -factorization approach in the present calculation [70]. On the other hand some authorsdo believe on the double counting. The argument goes as follows: in the region where thetransverse momentum of one of the parton is as large as the hard scale and the additionalparton is highly separated in the rapidity from the hard process (multi-Regge region), theadditional emission in the 2 → III. THE k t -FACTORIZATION FRAMEWORK In the equation (8) all partons are usually assumed to move in the plane of the incomingprotons. Therefore, they do not posses any transverse momenta. This is so called thecollinear approximation (see the appendix A). However, at high energies and in the small- x region, the transverse momentum, k t , of the incoming partons are expected to becomeimportant. Therefore, the cross-sections are factorized into the k t -dependent partonic cross-sections ˆ σ ( x, k t , µ ), where the incoming partons are treated as the off-shell particles. So,one should use the UPDF ( f ( x, k t , µ )) instead of the PDF in the equation (8), accordingto the equation (1) which leads to the equation (9).In the rest of this section, we briefly explained how to evaluate these UPPF in thesimplistic frameworks. 10 . The KMR prescription The KMR UPDF are generated through a procedure that was proposed by Kimber,Martin and Ryskin (KMR) [33]. In this method, the UPDF are generated such that thepartons developed from some starting parameterizations up to the scale k t according tothe DGLAP evolution equations. So the partons are evolved in the single evolution ladder(carrying only the k t dependency) and get convoluted with the second scale ( µ ) at the hardprocess. This is the last-step evolution approximation. Then the k t is forced to depend on thescale µ , without any real emission, and there is a summation over the virtual contributionsby imposing the Sudakov form factor ( T a ( k t , µ )). So, the general form of the KMR-UPDFare: f a ( x, k t , µ ) = T a ( k t , µ ) (cid:88) b = q,g (cid:20) α S ( k t )2 π (cid:90) − ∆ x dzP ( LO ) ab ( z ) b (cid:16) xz , k t (cid:17)(cid:21) , (10)where T a ( k t , µ ) are : T a ( k t , µ ) = exp (cid:32) − (cid:90) µ k t α S ( k )2 π dk k (cid:88) b = q,g (cid:90) − ∆0 dz (cid:48) P ( LO ) ab ( z (cid:48) ) (cid:33) . (11) T a are considered to be unity for k t > µ . In the above equation ∆ is proposed to preventthe soft gluon singularity, but this constraint is imposed on the quark radiations too. Theangular ordering constraint is imposed to determine ∆,. Angular ordering originates fromthe color coherence effects of the gluon radiations [33]. So ∆ is:∆ = k t µ + k t . The P ( LO ) ab ( z ) are the familiar LO splitting functions [1]. B. The LO-MRW prescription
The LO-MRW formalism, similar to the KMR scheme, was proposed by Martin, Ryskinand Watt (MRW) [34]. This formalism has the same general structure as the KMR, butonly with one significant difference that: the angular ordering constraint is correctly imposedonly on the on-shell radiated gluons, i.e. the diagonal splitting functions P qq ( z ) and P gg ( z )[34]. So, the LO-MRW prescription is written as: f LOq ( x, k t , µ ) = T q ( k t , µ ) α S ( k t )2 π (cid:90) x dz (cid:20) P ( LO ) qq ( z ) xz q (cid:16) xz , k t (cid:17) Θ (cid:18) µµ + k t − z (cid:19) P ( LO ) qg ( z ) xz g (cid:16) xz , k t (cid:17)(cid:105) , (12)with T q ( k t , µ ) = exp (cid:32) − (cid:90) µ k t α S ( k )2 π dk k (cid:90) z max dz (cid:48) P ( LO ) qq ( z (cid:48) ) (cid:33) , (13)for the quarks and f LOg ( x, k t , µ ) = T g ( k t , µ ) α S ( k t )2 π (cid:90) x dz (cid:34) P ( LO ) gq ( z ) (cid:88) q xz q (cid:16) xz , k t (cid:17) + P ( LO ) gg ( z ) xz g (cid:16) xz , k t (cid:17) Θ (cid:18) µµ + k t − z (cid:19)(cid:21) , (14)with T g ( k t , µ ) = exp (cid:32) − (cid:90) µ k t α S ( k )2 π dk k (cid:20)(cid:90) z max z min dz (cid:48) z (cid:48) P ( LO ) qq ( z (cid:48) ) + n f (cid:90) dz (cid:48) P ( LO ) qg ( z (cid:48) ) (cid:21)(cid:33) , (15)for the gluons. In the equations (13) and (15), z max = 1 − z min = µ/ ( µ + k t ) [68]. The UPDFof KMR and MRW to a good approximation, include the main kinematical effects involvedin the IS processes. Note that the particular choice of the AOC in the KMR formalismdespite being of the LO, includes some contributions from the NLO sector, hence in the caseof MRW framework, these contributions must be inserted separately. C. The NLO-MRW prescription
Finally, MRW [34] proposed a method for the promotion of the LO-MRW to the NLO-MRW prescription. Utilizing the NLO PDF and corresponding splitting functions fromDGLAP evolution equations lead to the MRW-NLO formalism [34]. The general form ofthe NLO-MRW UPDF are: f NLOa ( x, k t , µ ) = (cid:90) x dzT a (cid:18) k = k t (1 − z ) , µ (cid:19) α S ( k )2 π (cid:88) b = q,g ˜ P ( LO + NLO ) ab ( z ) × b NLO (cid:16) xz , k (cid:17) Θ (cid:18) − z − k t µ (cid:19) , (16)with the ”extended” NLO splitting functions, ˜ P ( i ) ab ( z ), being defined as,˜ P ( LO + NLO ) ab ( z ) = ˜ P ( LO ) ab ( z ) + α S π ˜ P ( NLO ) ab ( z ) , (17)12nd ˜ P ( i ) ab ( z ) = P iab ( z ) − Θ( z − (1 − ∆)) δ ab F iab P ab ( z ) , (18)where i = 0 and 1 stand for the LO and the NLO, respectively. The reader can find acomprehensive description of the NLO splitting functions in the references [34, 71]. Wemust however emphasize that in contrary to the KMR and the LO-MRW frameworks, theAOC is being introduced into the NLO-MRW formalism via the Θ( z − (1 − ∆)) constraint,in the ”extended” splitting function. Now ∆ can be defined as:∆ = k √ − zk √ − z + µ . This framework are the collection of the NLO PDF, the NLO splitting functions and theconstraint Θ (1 − z − k t /µ ) which impose the NLO corrections to this method. Neverthe-less, it was shown that using only the LO part of the ”extended” splitting functions, insteadof the complete definition of the equation (17), would result a reasonable accuracy in thecomputation of the NLO MRW UPDF [34]. Additionally, the Sudakov form factors in thisframework are defined as: T q ( k , µ ) = exp (cid:32) − (cid:90) µ k α S ( q )2 π dq q (cid:90) dz (cid:48) z (cid:48) (cid:104) ˜ P (0+1) qq ( z (cid:48) ) + ˜ P (0+1) gq ( z (cid:48) ) (cid:105)(cid:33) , (19) T g ( k , µ ) = exp (cid:32) − (cid:90) µ k α S ( q )2 π dq q (cid:90) dz (cid:48) z (cid:48) (cid:104) ˜ P (0+1) gg ( z (cid:48) ) + 2 n f ˜ P (0+1) qg ( z (cid:48) ) (cid:105)(cid:33) . (20)Each of the KMR, the LO and the NLO MRW UPDF can be used to identify the probabilityof finding a parton of a given flavor, with the fraction x of longitudinal momentum of theparent hadron and the transverse momentum k t , in the scale µ at the semi-hard level of aparticular IS process.The modifications to the above KMR, LO-MRW and NLO-MRW UPDF are made in ourcalculation of IPPP production cross sections, in the section V, to investigate the possibledouble counting concerning the 2 → IV. THE IPPP PRODUCTION AND THE TECHNICAL PRESCRIPTION
For calculating the partonic cross-section, we need the matrix element squared ( |M| ) ofsub-processes. Since the incoming quarks and gluons are off-shell, the expression for suchmatrix element will be more complicated. Therefore, we use the BFKL prescription [66] for13he gluons in the equation (7) and apply the method proposed in the references [61, 72] forthe incoming quarks, for the small x region. In this method, it is assumed that the incomingquarks with 4-momenta ( p µ ) radiate a gluon (or a photon) and consequently become off-shell[45, 48]. Therefore the extended |M| becomes, |M| ∼ | ˜ T α /k + m ( k ) − m γ β U ( p ) ¯ U ( p ) γ β /k + m ( k ) − m T α | , (21)where ˜ T α and T α represent the rest of the matrix elements. Since, the expression presentedbetween T α and ˜ T α is considered to be the off-shell quark spin density matrix elements,then by using the on-shell identity, performing some Dirac algebra at the m → k = xp + k t , with k = k t = − k t and somestraightforward algebra we obtain [45, 48, 66]: |M| ∼ xk tr [ T µ x ˆ p ˜ T µ ] . (22)where x ˆ p represents the properly normalized off-shell spin density matrix.Since the calculation of the |M| is a laborious task, we use the algebraic manipulationsystem FORM [73]. The above approximations, which are valid at small x region, force somelimits on our kinematics range. So the resulted differential cross section may not cover thewhole experimental data of Tevatron and LHC colliders (see our discussion in the sectionV) [48, 52–55, 59].In the section II, we defined the total cross-section for the IPPP production at hadroniccollisions, σ γγ , as: d ˆ σ a a = dφ a a F a a |M a a | . (23) dφ a a and F a a are the multi-particle phase apace and the flux factor, respectively whichcan be defined according to the specifications of the partonic process, dφ a a = (cid:89) i d p i E i δ (4) (cid:16)(cid:88) p in − (cid:88) p out (cid:17) , (24) F a a = x x s, (25)where the s is the center of mass energy squared, s = ( P + P ) = 2 P .P . φ a a can be characterized in terms of the transverse momenta of the product particles p i,t ,their rapidities, y i , and the azimuthal angles of the emissions, ϕ i , d p i E i = π dp i,t dy i dϕ i π . (26)In the present work, M a a in the equation (23), are the matrix elements of the partonicdiagrams which are involved in the production of the final results (see the section II).By using the kinematics given in the section II, we can derive the following equations forthe total cross-section of the IPPP production in the framework of k t -factorization. So thetotal cross-section for q ¯ q and gg are: σ ( P + ¯ P → γ + γ ) = (cid:88) a i ,b i = q,g (cid:90) dk a ,t k a ,t dk a ,t k a ,t dp γ ,t dy γ dy γ dϕ a π dϕ a π dϕ γ π ×|M ( a + a → γ + γ ) | π ( x x s ) f a ( x , k a ,t , µ ) f a ( x , k a ,t , µ ) , (27)and for qg and ¯ qg are, σ ( P + ¯ P → γ + γ + X ) = (cid:88) a i ,b i = q,g (cid:90) dk a ,t k a ,t dk a ,t k a ,t dp γ ,t dp γ ,t dy γ dy γ dy × dϕ a π dϕ a π dϕ γ π dϕ γ π ×|M ( a + a → γ + γ + X ) | π ( x x s ) f a ( x , k a ,t , µ ) f a ( x , k a ,t , µ ) , (28)Note that the integration boundaries for dk i,t /k i,t are limited by the kinematics. So one canintroduce an upper limit for these integration, say k i,max , several times larger than the scale µ . In addition, k t,min = µ ∼ GeV , is considered as the lower limit, that separates thenon-perturbative and the perturbative regions, by assuming that,1 k t f a ( x, k t , µ ) | k t <µ = 1 µ a ( x, µ ) T a ( µ , µ ) . (29)As a result of above formulation, the densities of partons are constant for k t < µ at fix x and µ [34]. For the above calculations, we use the LO-MMHT2014 PDF libraries for theKMR and the LO-MRW UPDF schemes, and the NLO-MMHT2014 PDF libraries for theNLO-MRW formalism.The VEGAS algorithm is considered for performing the multidimensional integration ofthe total cross-section in the equations (27) and (28). Since the sea quarks become significant15n the high energy limit, we calculate the cross-section of IPPP production, by consideringfour flavors (i.e. the up, down, charm and strange flavors) for CM energy of 1.960 TeV(D0 and CDF) and add bottom flavor for CM energy of 7 TeV (CMS and ATLAS). Beforewe present our results, it is important to have the relations between the different channelparameters, i.e. M γγ , p t,γγ , cosθ ∗ γγ , ∆ φ γγ , z γγ and y γγ , which is given in the references [2–9].Some divergences appear because of the small k t ( << µ ) of the outgoing quark in thecase of q ∗ (¯ q ∗ ) + g ∗ → γ + γ + q (¯ q ) process. But since this quark is in the direction of outgoingphotons it is eliminated by excluding the above mentioned regions in our calculation andalso implementing isolated and separated cone in this computation. In this work, we appliedthe same method as the reference [11] and [48] for the phase space cut. To avoid the doublecounting and divergence, the invariant mass of the photon-quark subsystem is considered tobe greater than 1 GeV. In order to be insure about the possible double counting in UPDF[74, 75], we checked our result by modifying the UPDF according to reference [48, 63]and suppressing the quarks splitting in the UPDF that will be discussed in the section V.Otherwise one should perform substraction procedure [57, 58]. We should point out herethat the fragmentation contribution of the q (¯ q ) → γ or g → γ to the 2 → QCD =200 MeV and α s is chosen to be one and two loops incase of LO and NLO level, respectively [1]. As it was pointed out before, the same photonisolation cuts are implemented as the one imposed in the related experiment [2–9]. Weshould mention that, the factorization scale µ is chosen such that, the renormalization scale µ R to be equal to the invariant mass of photon-photon sub-system M γγ . V. RESULTS AND DISCUSSIONS
In this section, we present our results, regarding the IPPP production according to theexperimental specifications discussed in the appendix A. Note that the fragmentation effectsenhanced (suppressed) when p t,γγ > M γγ or ∆Φ γγ < π/ | cos θ ∗ γγ | > . p t,γγ
The figures 2 and 3, illustrate our calculations regarding the IPPP production differentialcross section in the E CM = 1 . T eV , in accordance to the experimental data of the D0(D010 and DO13) and CDF (CDF11 and CDF13) collaborations [2–5], using the KMR,LO-MRW and NLO-MRW prescriptions, as a function of p t,γγ , M γγ and cosθ ∗ γγ , respectively.Note that the contribution of the individual sub-processes i.e. q ∗ + ¯ q ∗ → γ + γ , q ∗ (¯ q ∗ ) + g ∗ → γ + γ + q (¯ q ) and g ∗ + g ∗ → γ + γ are given only for the KM R approach.Considering the different panels of above figures, one readily finds that the q ∗ + ¯ q ∗ → γ + γ contributions dominate. But for larger transverse momenta ( p t,γγ ) region, the effects of q ∗ (¯ q ∗ ) + g ∗ → γ + γ + q (¯ q ) sub-process become non-negligible. Interestingly, one can seethe so-called Guillet shoulder [51] (note that a new channel opens beyond the leading order(NLO, NNLO etc) where the transverse momentum of pair photons is close to the pairphotons transverse momentum cut (threshold) which makes this shoulder) is forming in theintermediate transverse momentum range (see the panels (a)-(c) of figure 2 and the panels(a)-(b) of figure 3). Additionally, since in the D013 report [3], the fragmentation effects arenot fully suppressed, this ”shoulder” can be seen more clearly in the panel (b) of the figure2. Such behavior can be seen in all of the regions where the fragmentation and the higherorder (pQCD) effects become important.On the other hand, in the panel (c) of the figure 3, one can obviously see the ”low-tailof the mass” (i.e. the small M γγ region, where a raise in the differential cross section isacquired), appearing at the small- M γγ region, which strongly sensitive to the choice of themid- p t,γγ and the low-∆ φ γγ domain in the range of our and the others calculations [12]. Inthe panel (e) of this figure, the reader should notice that in the | cosθ ∗ γγ | > . p t,γγ < M γγ constraint [4, 5].Because of the small x approximation which was made in the section IV for the partialinsurance of the gauge invariance of the partonic cross section within high-energy factoriza-17ion, our result may not be accurate for large M γγ as far as we are working in the small xregion in which the incoming partons have large transverse momenta.Similar comparisons are made regarding the double differential cross sections d σ/dp t,γγ dM γγ , d σ/ ∆ φ γγ dM γγ and d σ/ cos θ ∗ γγ dM γγ , in the figure 4, against the dataof the D010 collaboration [2]. To be specific, what makes the difference in these cal-culations, is different cuts on the M γγ , which defers from 30 GeV < M γγ < GeV ,50
GeV < M γγ < GeV and 80
GeV < M γγ < GeV in the figure 4 and theyare coated in each panel. One notices that, the best predictions are being obtained in the50
GeV < M γγ < GeV range. Since it corresponds to the intermediate transverse mo-mentum regions, where (in the absence of strong fragmentation effects) we expect to achievethe best outcome. In the 80
GeV < M γγ < GeV range, the higher order pQCD effectsare larger, hence our results are generally lower than the data.At the ∆Φ γγ ≤ π/ p t,γγ > M γγ domain, the contribution of the fragmentation becomesutterly non-negligible, as it can be seen in the panels (a)-(e) of the figure 5, as well as in thepanels (a), (d) and (g) of the figures 6, where the predictions of our simplistic framework areclearly insufficient to describe the experimental data from the D0 and CDF collaborations[3–5] (the constraints are presented on each panel). To account for the missing contributions,one has to incorporate the fragmentation and the higher-order pQCD corrections into theour framework. Moreover, in the figure 6, the reader can also find the IPPP production ratesas the functions of y γγ and z γγ parameters. The symmetric form of the y γγ distributions aredue to the fact that the q ∗ (¯ q ∗ ) + g ∗ → γ + γ + q (¯ q ) and g ∗ + q ∗ (¯ q ∗ ) → γ + γ + q (¯ q ) are the2 → y γγ = 0. Also,the q ∗ (¯ q ∗ ) + g → γ + γ + q (¯ q ) sub-process causes an interesting effect on the z γγ parameterdistributions, by adding a ”shoulder” which can be roughly detected in the CDF13 data aswell (which we name it MAK shoulders). In the panels (f) and (g) of the figure 5 and thepanels (a), (b) and (c) of the figure 6, we have compared our results to the experimentaldata, regarding the ∆Φ γγ dependency of the IPPP production rates. The low-∆ φ γγ tail canbe clearly seen here.It is interesting to note that in these relatively low CM energies, the LO-MRW frameworkperforms much better with respect to other schemes, specially in the D013 data. Addition-ally, one finds out that in the higher transverse momenta, i.e. where the higher-order pQCDeffects become important, the KMR results behave similar to the NLO-MRW rather than18ts LO counterpart and the KMR and MRW-NLO results are below the experimental data.So their behaviors are the same, while MRW-LO approximately cover the data. Generallyspeaking, in the low-∆Φ γγ domain, the effect of the fragmentation and the higher-ordercontributions are large (see the panels (b), (d) and (e) of the figure 5 and the panels (a),(d) and (g) of the figure 6). Hence as a clear pattern, the KMR and LO-MRW results arelarger compared to the NLO-MRW ones. Because of the different AOC implementationson these prescriptions, the predictions get quite separated in their respective regions. Weshould point out that the MRW sub-processes in the above calculations behave roughly thesame as those of KMR. However, some discrepancy in case of NLO-MRW is observed. Thereis not a sizable difference between the above schemes which use various AOC and cut off inthe differential cross sections, which is in agreement to reference [49]. As one should expect,this is not the case on fragmentation domain. B. E CM = 7 T eV
We have performed another set of calculations, with the CM energy of 7
T eV , in accor-dance with the specifications of the ATLAS and the CMS reports, i.e. the references [6–9].Therefore, in the figures 7 through 9, the reader is presented with comprehensive compar-isons regarding the dependency of the differential total cross-section of the IPPP production,as the functions of p t,γγ , M γγ , ∆ φ γγ and cosθ ∗ γγ . The general behavior of the results are thesame as in the E CM = 1 . T eV case, with the exception that the contributions coming fromthe q ∗ (¯ q ∗ ) + g ∗ → γ + γ + q (¯ q ) sub-process is visibly greater than that of the q ∗ + ¯ q ∗ → γ + γ sub-process, with some exceptions for the LO-MRW case (we do not present their data inorder not to crude these figures). This happens, because the shares of the gluons and thesea-quarks become important, with the increase of the CM energy.As a result of increasing the CM energy, the Guillet shoulder phenomena can be seenmore clearly in our 7 T eV calculations. Additionally, as in the 1 . T eV case, in the regionswhere the fragmentation effects become non-negligible, the low-tails of mass and the low-∆Φ γγ tails are visible and generally followed by the separation of the KMR, LO-MRW andNLO-MRW results. Generally speaking, in these areas the LO-MRW and the NLO-MRWresults are the lower and the upper bounds relative to the KMR diagrams, respectively.One should note that, the asymmetric constraint is applied on the transverse energies of19he IPPP production in the CMS14 data. So, we perform our calculations for q ∗ (¯ q ∗ ) + g and g ∗ + q ∗ (¯ q ∗ ) configurations of the q ∗ (¯ q ∗ ) + g ∗ → γ + γ + q (¯ q ) sub-process, separately. As aresult of this asymmetric constraint, the production of the back-to-back photons are beingsuppressed in the transverse plane [7]. Therefore, the higher order contributions, e.g. thequark-gluon scattering, become more important. In the CMS14 measurement, despite ourexpectations, the quark annihilation has a significant contribution in the LO-MRW, whilethe KMR and the NLO-MRW results have the ”expected” behavior. Nevertheless, onlythe KMR prescription is somehow successful in describing the experimental data in ”thesekinematic regions”.Unlike the CMS14 measurement, the ATLAS12 experiment utilizes symmetric constraints[9]. Therefore as one expects, the lower order pQCD contributions should be enhanced thesedata. So the NLO-MRW should perform a better behavior for the prediction of the experi-mental outcome, see for example the figure 7. Therefore, we may conclude that by includingsuitable higher-order contributions, in accordance with the experiment conditions, i.e. theimposed kinematics constraints, the predictions of the NLO-MRW framework becomes bet-ter and may be more consistent with respect to two other k t -factorization approaches.Finally, we would like to present a comparison between our results and the Monte Carloevent generator as well as the pQCD which were introduced in the introduction [10–13].Furthermore, we make a careful scrutiny of our calculation by dividing our results in dif-ferent frameworks (i.e. KMR, LO-MRW and NLO-MRW) to that of the correspondingexperimental data. This can highlight the difference of our works over the experiments. Theoutcome of above comparisons are demonstrated in the figure 10 through figure 16. At thelower panels of these figures the red circles show the KMR ratio and the black triangles andthe blue squares are presenting the LO-MRW and NLO-MRW ratios as explained above,respectively.In the figures 10, 11 and 12, our KMR ,LO-MRW and NLO-MRW results are comparedwith the SHERPA and the NNLO (or 2 γ NNLO) pQCD [3, 5], as well as CDF13 and D013data. It is observed that our KMR approach predicts the differential cross section data as afunction of corresponding variable very well, but in the region where the fragmentation is notimportant. While in the regions where the fragmentation and higher order pQCD becomedominant (for example, low ∆ φ region in the panel (g) of the figure 10), the SHERPA andNNLO methods produce better results. At these regions without considering higher order20ontributions and fragmentation, only the MRW can predict experimental data correctly,especially this well behavior can be observed in the panels (c), (f) and (i) of the figures 10and 11 and the panels (b) and (h) of the figure 12. However, the SHERPA and 2 γ NNLOcalculations are well behaved in whole regions and all of the panels.The figure 13 compares the D010 data [2] with our results, as well as the RESBOS andDIPHOX calculations [2]. One can clearly observe that the KMR k t -factorization approachpredicts the acceptable result with respect to other theoretical methods that presented inthis figure, especially for all of the double differential cross section channels (i.e. the panels(b) to (i)). On the other hand, in the panels (a), (d) and (g) of the figure 14 and in the highvalue of M γγ , the Monte Carlo calculation is more successful. The remaining panels of thisfigure which is related to the CMS collaborations [6, 7], show that our results predict thedata with higher accuracy with respect to those of DIPHOX calculation [6, 7, 11].In the panels (a), (d) and (g) of the figure 15 similar to the figure 14, the KMR approachbehaves as before, but the results of DIPHOX calculation are closer to the data, since therapidity was increased. In rest of the panels of the figures 15 and the panels (a) to (e) of16, our results are examined against DIPHOX and 2 γ NNLO, and their behavior are muchsimilar. However, our KMR or MRW, as it was discussed before, are closer to the 2 γ NNLOcalculation.In order to check our results against those of reggization methods, [57, 58], and also togive the uncertainty of present calculation (by multiplying the factorization scale by halfand two) , the panels (a) and (e) are repeated in the panels (f) and (g) of the figure 15.It is seen that the data are very close to those of reference [57], except the small p t,γγ and the large ∆Φ γγ regions, where their results are off the data. However our uncertaintybounds are reasonably cover the data as well as the reggization and 2 γ NNLO calculations(see panels (a), (e), (f) and (g)). It is interesting that the 2 γ NNLO method, in which thefragmentation contribution has been also taken into account, is off the experimental datawhile the reggization method cover them. As we pointed out in the end of section IV,beside the separation and isolation cone conditions for possible double counting, we alsomodified our UPDF according to e.g. the reference [63] and find less than 15 per centeffect, which still keeps our result inside the uncertainty bounds. But, as we pointed out inthe introduction, the UPDF should satisfy the condition given in the equation (1), so anychanges in the UPDF certainly affect the original PDF definitions or it may be in contrast21ith the original definition of UPDF [70]. On the other hand, there is no grantee that the k t factorization method produces results better than those of pQCD as it is stated by Martinet al [64].These comparisons show that one of the places in which the effect of k t factorizationframework obviously becomes important, with respect to its counterpart, would be theregions of large ∆ φ γγ and small p t,γγ . In these two regions, the predictions of collinearmatrix element method are overestimated the data, as it could be seen by DIPHOX and2 γ NNLO calculations. However, the prediction of RESBOS, due to the NLL (next leadinglogarithmic) resumption of soft initial state gluon radiation, is better than those of DIPHOXand 2 γ NNLO. But in our methods, the natural gluon resumption automatically is done inall orders [62], because of the Sudakov form factor, so this problem would not exist.
VI. CONCLUSIONS
Throughout this work, we calculated the rate of the production of the isolated prompt-photon pairs, in the k t -factorization framework, using the UPDF of the KMR and theMRW prescriptions and compared our results to the existing experimental data from theD0, CDF, CMS and ATLAS collaborations. According to our discussions and observationsin the present work, the LO-MRW approach is the best suitable scheme for the predictionof the IPPP production rates in the lower CM energies, since this approach can predictthe experimental data within the regions where the fragmentation effects become impor-tant, without any additional manipulations in our calculations. In contrary, the LO-MRWformalism is not perfect for the higher CM energies in these kinematics. While the KMRapproach is able to accurately predict the experimental data in the 7 TeV center of massenergy. The main difference between these approaches arises due to the implementation ofdifferent visualizations of the AOC, which can be seen, specially in the regions where thefragmentation and the higher-order contributions become important, i.e. when the quark-radiation terms are enhanced. In these areas, we expect that the three approaches behavewell-separated. On the other hand it was shown that the application of different AOC andcut off, using the KMR [49] and MRW prescriptions do not show serious discrepancies anda qualitative agreements between different schemes can be achieved.We realized that the Guillet shoulder phenomena is more sensitive to the low- M γγ vari-22tions, compared to the low-∆ φ γγ and the cosθ ∗ γγ regions. Although, our predictions viaour simplistic calculations describe the experimental data well, one can improve the pre-cision of these results by including higher-order contributions and taking into account thefragmentation effects. We hope that in our future works, we can investigate these phenom-ena. A comparison was also made with the different theoretical methods such as DIPHOX, 2 γ NNLO, RESBOS and SHERPA and an overall agreement was found.It was shown that the possible double-counting can be removed by considering the phasespace cuts as well as modification in the UPDF. However by imposing the uncertainty of thefactorization scale in the resulted differential cross section, this issue may not be important.In this work we used the small x approximation, however as we stated in the introduction,we can use the effective action approaches for the off shell partons. We hope to investigatethis approximation in our future works [52–55, 59] as well as the gauge invariance andpossible double counting.
Acknowledgments
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D F g g ‡ p ⁄ - 1 d s /d|cos q* gg |(pb) K M R L O - M R W N L O - M R W q + q ( K M R ) q + g ( K M R ) g + g ( K M R ) • D 0 1 3 - 5 - 4 - 3 - 2 - 1 d s / dpT gg (pb/GeV) K M R L O - M R W N L O - M R W q + q ( K M R ) q + g ( K M R ) g + g ( K M R ) • D 0 1 3 - 3 - 2 - 1 d s / dpT, gg (pb/GeV) K M R L O - M R W N L O - M R W q + q ( K M R ) q + g ( K M R ) g + g ( K M R ) • D 0 1 0 - 1 d s /d|Cos gg *|(pb) K M R L O - M R W N L O - M R W q + q ( K M R ) q + g ( K M R ) g + g ( K M R ) • D 0 1 0 - 5 - 4 - 3 - 2 - 1 d s /dM gg (pb/GeV) K M R L O - M R W N L O - M R W q + q ( K M R ) q + g ( K M R ) g + g ( K M R ) • D 0 1 0 - 7 - 6 - 5 - 4 - 3 - 2 - 1 d s /dM gg (pb/GeV) M g g ( G e V ) K M R L O - M R W N L O - M R W q + q ( K M R ) q + g ( K M R ) g + g ( K M R ) • D 0 1 3
D F g g ‡ p ⁄ - 7 - 6 - 5 - 4 - 3 - 2 - 1 d s / dpT , gg (pb/GeV) p T ,g g ( G e V ) K M R L O - M R W N L O - M R W q + q ( K M R ) q + g ( K M R ) g + g ( K M R ) • D 0 1 3
D F g g ‡ p ⁄ - 5 - 4 - 3 - 2 - 1 d s /dM gg (pb/GeV) K M R L O - M R W N L O - M R W q + q ( K M R ) q + g ( K M R ) g + g ( K M R ) • D 0 1 3 ( a )( b )( c ) ( d )( e )( f ) ( g )( h )( i )F I G 2
FIG. 2: The differential cross section of the production of
IP P P as functions of the transversemomentum ( p t,γγ in the panels (a), (b) and (c)), the photon invariant mass ( M γγ in the panels (d),(e) and (f)) and cosθ ∗ γγ (in the panels (g), (h) and (i)) at E CM = 1960 GeV . The experimentaldata are from the D - 2 - 1 C D F 1 3 C D F 1 1 d s / dCos q * gg (pb) K M R L O - M R W N L O - M R W q + q ( K M R ) q + g ( K M R ) g + g ( K M R ) - 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 01 0 - 2 - 1 d s / dCos qgg *(pb) C o s q *g g C D F 1 3C D F 1 1
K M R L O - M R W N L O - M R W q + q ( K M R ) q + g ( K M R ) g + g ( K M R )
P t , g g < M g g - 4 - 3 - 2 - 1 C D F 1 3C D F 1 1 d s /dM gg [pb/GeV] M g g [ G e V ] K M R L O - M R W N L O - M R W q + q ( K M R ) q + g ( K M R ) g + g ( K M R )P t , g g < M g g - 5 - 4 - 3 - 2 - 1 d s /dM gg (pb/GeV) C D F 1 3C D F 1 1 K M R L O - M R W N L O - M R W q + q ( K M R ) q + g ( K M R ) g + g ( K M R ) - 6 - 5 - 4 - 3 - 2 - 1 d s /dpT gg (pb/GeV) p T g g ( G e V ) C D F 1 1
C D F 1 3 K M R L O - M R W N L O - M R W q + q ( K M R ) q + g ( K M R ) g + g ( K M R )P t , g g < M g g - 4 - 3 - 2 - 1 C D F 1 3C D F 1 1 d s / dpT gg (pb/GeV) K M R
L O - M R W N L O - M R W q + q ( K M R ) q + g ( K M R ) g + g ( K M R ) ( a )( b ) ( c )( d ) ( e )( f )F I G 3
FIG. 3: The differential cross section of the production of
IP P P as functions of the transversemomentum ( p t,γγ in the panels (a) and (b) ), photon invariant mass ( M γγ in the panels (c) and(d)) and cosθ ∗ γγ (in the panels (e) and (f)) at E CM = 1960 GeV . The experimental data are fromthe CDF collaboration [4, 5]. Note that the sub-processes are only given for the KMR approach. - 6 - 5 - 4 - 3 - 2 d s / dM gg dpT, gg (pb/GeV) K M R L O - M R W N L O - M R W q + q ( K M R ) q + g ( K M R ) g + g ( K M R ) • D 0 1 0 g g ( G e V ) < 8 0 - 6 - 5 - 4
K M R L O - M R W N L O - M R W q + q ( K M R ) q + g ( K M R ) g + g ( K M R ) • D 0 1 0 d s / dM gg dpT, gg (pb/GeV) p T ,g g ( G e V ) g g ( G e V ) < 3 5 0 - 5 - 4 - 3 - 2 - 1 d2 s /dM gg Dfgg (pb/rad)
D f g g ( r a d ) K M R L O - M R W N L O - M R W q + q ( K M R ) q + g ( K M R ) g + g ( K M R ) • D 0 1 0 g g ( G e V ) < 3 5 0 - 2 - 1 d2 s /dM gg Dfgg (pb/rad)
K M R L O - M R W N L O - M R W q + q ( K M R ) q + g ( K M R ) g + g ( K M R ) • D 0 1 0 g g ( G e V ) < 8 0 - 3 - 2 - 1 d2 s /dM gg Dfgg (pb/rad)
K M R L O - M R W N L O - M R W q + q ( K M R ) q + g ( K M R ) g + g ( K M R ) • D 0 1 0 g g ( G e V ) < 5 0 - 2 - 1 d s / dM gg d|cos q* gg |(pb) K M R L O - M R W N L O - M R W q + q ( K M R ) q + g ( K M R ) g + g ( K M R ) • D 0 1 0 g g ( G e V ) < 5 0 - 2 - 1 d s / dM gg d|cos q* gg |(pb) K M R L O - M R W N L O - M R W q + q ( K M R ) q + g ( K M R ) g + g ( K M R ) • D 0 1 0 g g ( G e V ) < 8 0 - 3 - 2 d s / dM gg d|cos q* gg |(pb) | c o s q *g g | K M R L O - M R W N L O - M R W q + q ( K M R ) q + g ( K M R ) g + g ( K M R ) • D 0 1 08 0 < M g g ( G e V ) < 3 5 0 - 5 - 4 - 3 - 2 d s / dM gg dpT, gg (pb/GeV) K M R L O - M R W N L O - M R W q + q ( K M R ) q + g ( K M R ) g + g ( K M R ) • D 0 1 0 g g ( G e V ) < 5 0 ( a )( b )( c ) ( d )( e )( f ) ( g )( h )( i )F I G 4
FIG. 4: The double-differential cross section of the production of
IP P P as functions of the trans-verse momentum ( p t,γγ ) and M γγ in the panels (a), (b) and (c), cosθ ∗ γγ and M γγ (in the panels(d), (e) and (f)) and ∆ φ γγ and ( M γγ (in the panels (g), (h) and (i)) at E CM = 1960 GeV . Theexperimental data are from the D KM R approach. . 6 1 . 8 2 . 0 2 . 2 2 . 4 2 . 6 2 . 8 3 . 01 0 - 1 D f g g d s /d Dfgg (pb/rad)
K M R L O - M R W N L O - M R W q + q ( K M R ) q + g ( K M R ) g + g ( K M R ) • D 0 1 0 - 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 01 0 - 1 d s / dCos q* gg (pb) C o s q *g g C D F 1 1
C D F 1 3 M g g < P t , g g
K M R L O - M R W N L O - M R W q + q ( K M R ) q + g ( K M R ) g + g ( K M R ) ( g ) ( e ) - 6 - 5 - 4 - 3 - 2 - 1 d s /dpT gg (pb/GeV) p T g g ( G e V ) C D F 1 1
C D F 1 3M g g < P t , g g
K M R L O - M R W N L O - M R W q + q ( K M R ) q + g ( K M R ) g + g ( K M R ) - 7 - 6 - 5 - 4 - 3 - 2 - 1 d s /dM gg (pb/GeV) M g g ( G e V ) C D F 1 1
C D F 1 3 K M R L O - M R W N L O - M R W q + q ( K M R ) q + g ( K M R ) g + g ( K M R ) M g g < P t , g g - 2 - 1 d s /d Dfgg (pb/rad)
D f g g ( r a d ) K M R L O - M R W N L O - M R W q + q ( K M R ) q + g ( K M R ) g + g ( K M R ) • D 0 1 3 - 5 - 4 - 3 - 2 - 1 d s / dpT , gg (pb/GeV) K M R L O - M R W N L O - M R W q + q ( K M R ) q + g ( K M R ) g + g ( K M R ) • D 0 1 3
D F g g £ p ⁄ - 6 - 5 - 4 - 3 - 2 d s / dM gg (pb/GeV) K M R L O - M R W N L O - M R W q + q ( K M R ) q + g ( K M R ) g + g ( K M R ) • D 0 1 3
D F g g £ p ⁄ ( a )( b )( f ) ( c )( d ) F I G 5 .. . FIG. 5: The differential cross section of the production of
IP P P as functions of the transversemomentum ( p t,γγ ) in the panels (a) and (b)), photon invariant mass ( M γγ in panels (c) and(d)), cosθ ∗ γγ (in the panel (e)) and ∆ φ γγ (in the panels (f) and (g)) at E CM = 1960 GeV . Theexperimental data are from the D - 3 - 2 - 1 C D F 1 1
C D F 1 3 K M R L O - M R W N L O - M R W q + q ( K M R ) q + g ( K M R ) g + g ( K M R ) d s /dy gg (pb) M g g < P t , g g - 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 01 0 - 2 - 1 d s /dy gg (pb) y g g C D F 1 1
C D F 1 3P t , g g < M g g
K M R L O - M R W N L O - M R W q + q ( K M R ) q + g ( K M R ) g + g ( K M R ) - 1 d s /dy gg (pb) C D F 1 3C D F 1 1 K M R L O - M R W N L O - M R W q + q ( K M R ) q + g ( K M R ) g + g ( K M R ) - 4 - 3 - 2 - 1 d s / d Dfgg (pb/rad)
D f g g ( r a d )
C D F 1 1
C D F 1 3 K M R L O - M R W N L O - M R W q + q ( K M R ) q + g ( K M R ) g + g ( K M R )P t , g g < M g g - 3 - 2 - 1 d s / dz gg (pb) C D F 1 3C D F 1 1 K M R L O - M R W N L O - M R W q + q ( K M R ) q + g ( K M R ) g + g ( K M R ) - 4 - 3 - 2 - 1 d s / dz gg (pb) z g g C D F 1 1
C D F 1 3 K M R L O - M R W N L O - M R W q + q ( K M R ) q + g ( K M R ) g + g ( K M R ) P t , g g < M g g - 1 d s /d Dfgg (pb/rad)
C D F 1 3C D F 1 1 K M R L O - M R W N L O - M R W q + q ( K M R ) q + g ( K M R ) g + g ( K M R ) - 2 - 1 d s / dz gg (pb) C D F 1 1
C D F 1 3 K M R L O - M R W N L O - M R W q + q ( K M R ) q + g ( K M R ) g + g ( K M R ) • D 0 1 0M g g < P t , g g - 3 - 2 - 1 d s / d Dfgg (pb/rad)
C D F 1 1
C D F 1 3 K M R L O - M R W N L O - M R W q + q ( K M R ) q + g ( K M R ) g + g ( K M R ) M g g < P t , g g ( a )( b )( c ) ( d )( e )( f ) ( g )( h )( i )F I G 6 FIG. 6: The differential cross section of the production of
IP P P as functions of the ( φ γγ in thepanels (a), (b) and (c)), ( y γγ in the panels (d), (e) and (f)) and Z γγ (in the panels (g), (h) and (i))at E CM = 1960 GeV . The experimental data are from the
CDF collaboration, [4, 5]. Note thatthe sub-processes are only given for the KMR approach. . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 01 0 d s / d Dfgg (pb/rad)
K M R L O - M R W N L O - M R W q + q ( K M R ) q + g ( K M R ) g + g ( K M R ) • A T L A S 1 2 - 4 - 3 - 2 - 1 d s /dM gg (pb/GeV) M g g ( G e V ) K M R L O - M R W N L O - M R W q + q ( K M R ) q + g ( K M R ) g + g ( K M R ) • A T L A S 1 3 - 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 01 0 d s /dCos q* gg (pb) C o s q *g g K M R L O - M R W N L O - M R W q + q ( K M R ) q + g ( K M R ) g + g ( K M R ) • A T L A S 1 3 - 5 - 4 - 3 - 2 - 1 d s / dpT, gg (pb/GeV) p T , g g ( G e V )
K M R L O - M R W N L O - M R W q + q ( K M R ) q + g ( K M R ) g + g ( K M R ) • A T L A S 1 3 d s / d Dfgg (pb/rad)
D f g g ( r a d )
K M R L O - M R W N L O - M R W q + q ( K M R ) q + g ( K M R ) g + g ( K M R ) • A T L A S 1 3 - 3 - 2 - 1 d s / dpT, gg (pb/GeV) K M R L O - M R W N L O - M R W q + q ( K M R ) q + g ( K M R ) g + g ( K M R ) • A T L A S 1 2 - 2 - 1 d s / dM gg ( pb/GeV) K M R L O - M R W N L O - M R W q + q ( K M R ) q + g ( K M R ) g + g ( K M R ) • A T L A S 1 2 ( a )( b ) ( c )( d )( g ) ( e )( f )F I G 7
FIG. 7: The differential cross section of the production of
IP P P as functions of the transversemomentum ( p t,γγ in the panels (a) and (b)), the photon invariant mass ( M γγ in panels (c) and (d)), φ γγ (in the panels (e) and (f) ) and cosθ ∗ t,γγ (in the panel (g)) at E CM = 7 T eV . The experimentaldata are from the
AT LAS collaboration, [8, 9]. Note that the sub-processes are only given for theKMR approach. . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 011 01 0 0 d s / d|cos q* gg | ( pb) | c o s q *g g | K M R L O - M R W N L O - M R W q + q ( K M R ) q + g ( K M R ) g + g ( K M R ) • C M S 1 2 | h | < 1 . 4 4 - 3 - 2 - 1 d s / dM gg (pb/GeV) M g g ( G e V ) K M R L O - M R W N L O - M R W q + q ( K M R ) q + g ( K M R ) g + g ( K M R ) • C M S 1 2 | h | < 1 . 4 4 - 3 - 2 - 1 d s / dpT gg (pb/GeV) p T g g ( G e V ) K M R L O - M R W N L O - M R W q + q ( K M R ) q + g ( K M R ) g + g ( K M R ) • C M S 1 2 | h | < 1 . 4 4 d s / d|cos q* gg | (pb) K M R L O - M R W N L O - M R W q + q ( K M R ) q + g ( K M R ) g + g ( K M R ) • C M S 1 2 | h | < 2 . 5 - 3 - 2 - 1 d s /dM gg (pb/GeV) K M R L O - M R W N L O - M R W q + q ( K M R ) q + g ( K M R ) g + g ( K M R ) • C M S 1 2 | h | < 2 . 5 - 2 - 1 d s / dpT , gg (pb/GeV) K M R L O - M R W N L O - M R W q + q ( K M R ) q + g ( K M R ) g + g ( K M R ) • C M S 1 2 | h | < 2 . 5 ( a )( b ) ( c )( d ) ( e )( f )F I G 8 FIG. 8: The differential cross section of the production of
IP P P as functions of the transversemomentum ( p t,γγ in the panels (a) and (b)), the photon invariant mass ( M γγ in the panels (c) and(d)) and cosθ ∗ t,γγ (in the panels (e) and (f)) at E CM = 7 T eV . The experimental data are fromthe CMS collaboration, [6]. Note that the sub-processes are only given for the KMR approach. . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 01 0 d s /d|cos qgg *|(pb) | c o s q g g * | K M R L O - M R W N L O - M R W q + q ( K M R ) q + g ( K M R ) g + g ( K M R ) • C M S 1 4 - 4 - 3 - 2 - 1 d s /dM gg (pb/GeV) M g g ( G e V ) K M R L O - M R W N L O - M R W q + q ( K M R ) q + g ( K M R ) g + g ( K M R ) • C M S 1 4 - 3 - 2 - 1 d s / dpT, gg (pb/GeV) p T , g g ( G e V )
K M R L O - M R W N L O - M R W q + q ( K M R ) q + g ( K M R ) g + g ( K M R ) • C M S 1 4 - 1 d s / d Dfgg (pb/rad)
D f g g ( r a d )
K M R L O - M R W N L O - M R W q + q ( K M R ) q + g ( K M R ) g + g ( K M R ) • C M S 1 4 d s / d Dfgg (pb)
D f g g
K M R L O - M R W N L O - M R W q + q ( K M R ) q + g ( K M R ) g + g ( K M R ) • C M S 1 2 | h | < 2 . 5 d s / d Dfgg (pb)
D f g g
K M R L O - M R W N L O - M R W q + q ( K M R ) q + g ( K M R ) g + g ( K M R ) • C M S 1 2 | h | < 1 . 4 4 ( a )( b ) ( c )( d ) ( e )( f )F I G 9 FIG. 9: The differential cross section of the production of
IP P P as functions of the transversemomentum ( p t,γγ in the panel (a)), the photon invariant mass ( M γγ in the panel (c) ), cosθ ∗ t,γγ (inthe panel (e)) and ∆ φ γγ (in the panels (b), (d) and (f) ) at E CM = 7 T eV . The experimental dataare from the
CM S collaboration, [6, 7]. Note that the sub-processes are only given for the KMRapproach. . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 00 . 51 . 01 . 5 D f g g ( r a d )
Ratio
Ratio M g g ( G e V ) - 3 - 2 - 1 d s / d Dfgg (pb/rad)
C D F 1 3 K M R L O - M R W N L O - M R W S H E R P A
N N L O M g g < P t , g g - 4 - 3 - 2 - 1 d s / d Dfgg (pb/rad)
C D F 1 3
K M R
L O - M R W
N L O - M R W
S H E R P A N N L OP t , g g < M g g
Ratio - 7 - 6 - 5 - 4 - 3 - 2 - 1 d s /dM gg (pb/GeV) C D F 1 3 K M R L O - M R W N L O - M R W S H E R P A
N N L O M g g < P t , g g Ratio - 4 - 3 - 2 - 1 C D F 1 3 d s /dM gg [pb/GeV] K M R L O - M R W N L O - M R W S H E R P A
N N L OP t , g g < M g g
Ratio - 5 - 4 - 3 - 2 - 1 d s /dM gg (pb/GeV) C D F 1 3 K M R L O - M R W N L O - M R W S H E R P A
N N L O - 6 - 5 - 4 - 3 - 2 - 1 d s /dpT gg (pb/GeV) C D F 1 3M g g < P t , g g
K M R L O - M R W N L O - M R W S H E R P A
N N L O
Ratio p T g g ( G e V ) ( a )( b )( c ) ( d )( e )( f ) ( g )( h )( i )F I G 1 0 - 2 - 1 d s /d Dfgg (pb/rad)
C D F 1 3 K M R L O - M R W N L O - M R W S H E R P A
N N L O
Ratio - 6 - 5 - 4 - 3 - 2 - 1 d s /dpT gg (pb/GeV) C D F 1 3 K M R L O - M R W N L O - M R W S H E R P A
N N L OP t , g g < M g g
Ratio - 4 - 3 - 2 - 1 C D F 1 3 d s / dpT gg (pb/GeV) K M R
L O - M R W N L O - M R W
S H E R P A
N N L O
Ratio
FIG. 10: The comparison of
KM R and
M RW IP P P differential cross section (as function of p t,γγ , M γγ and ∆ φ γγ at E CM = 1960 GeV ) with the
SHERP A and
N N LO pQCD (2 γN N LO )calculations base on the CDF
13 experimental data [5]. The lower panels present the ratio of ourcomputation to that of the corresponding experimental data (the red circles, the black trianglesand the blue squares, show the ratio of KMR, LO-MRW and NLO-MRW, respectively). Ratio y g g - 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 00 . 51 . 01 . 5 Ratio - 1 d s /dy gg (pb) C D F 1 3 K M R L O - M R W N L O - M R W S H E R P A
N N L O - 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 00 . 51 . 01 . 5
Ratio - 2 - 1 C D F 1 3 d s /dy gg (pb) P t , g g < M g g K M R L O - M R W N L O - M R W S H E R P A N N L O - 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 00 . 51 . 01 . 5
Ratio - 3 - 2 - 1 C D F 1 3 K M R L O - M R W N L O - M R W S H E R P A
N N L O d s /dy gg (pb) M g g < P t , g g - 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 012 Ratio - 2 - 1 d s / dz gg (pb) C D F 1 3 K M R L O - M R W N L O - M R W S H E R P A
N N L OM g g < P t , g g z g g Ratio - 4 - 3 - 2 - 1 d s / dz gg (pb) C D F 1 3 K M R L O - M R W N L O - M R W S H E R P A
N N L O P t , g g < M g g Ratio - 3 - 2 - 1 d s / dz gg (pb) C D F 1 3 K M R L O - M R W N L O - M R W S H E R P A
N N L O
Ratio - 1 d s / dCos q* gg (pb) C D F 1 3 M g g < P t , g g
K M R L O - M R W N L O - M R W S H E R P A
N N L O - 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 00 . 51 . 01 . 5
Ratio
C o s q *g g - 2 - 1 d s / dCos qgg *(pb) C D F 1 3
K M R L O - M R W N L O - M R W S H E R P A
N N L O
P t , g g < M g g - 2 - 1 C D F 1 3 d s / dCos q * gg (pb) K M R L O - M R W N L O - M R W S H E R P A
N N L O ( a )( b )( c ) ( d )( e )( f ) ( g )( h )( i )F I G 1 1
FIG. 11: The same as the figure 11, but for the cosθ ∗ t,γγ , y γγ and Z γγ variables. Ratio
Ratio
Ratio M g g - 6 - 5 - 4 - 3 - 2 d s / dM gg (pb/GeV) K M R L O - M R W N L O - M R W S H E R P A g N N L O • D 0 1 3
D F g g < p ⁄ - 5 - 4 - 3 - 2 - 1 d s /dM gg (pb/GeV) K M R L O - M R W N L O - M R W S H E R P A 2 g N N L O • D 0 1 3
Ratio - 7 - 6 - 5 - 4 - 3 - 2 - 1 d s / dpT , gg (pb/GeV) K M R L O - M R W N L O - M R W S H E R P A 2 g N N L O • D 0 1 3
D F g g ‡ p ⁄ Ratio p T ,g g ( G e V ) Ratio | c o s q *g g | - 1 d s / d|cos q* gg |(pb) K M R L O - M R W N L O - M R W S H E R P A 2 g N N L O • D 0 1 3
D F g g ‡ p ⁄ Ratio d s /d|cos q* gg |(pb) K M R L O - M R W N L O - M R W S H E R P A 2 g N N L O • D 0 1 3 - 5 - 4 - 3 - 2 - 1 d s / dpT , gg (pb/GeV) K M R L O - M R W N L O - M R W S H E R P A g N N L O • D 0 1 3
D F g g < p ⁄ ( a )( b )( c ) ( d )( e )( f ) ( g )( h )( i )F I G 1 2 Ratio - 5 - 4 - 3 - 2 - 1 d s / dpT gg (pb/GeV) K M R L O - M R W N L O - M R W S H E R P A 2 g N N L O • D 0 1 3 - 6 - 5 - 4 - 3 - 2 - 1 d s /dM gg (pb/GeV) K M R L O - M R W N L O - M R W
S H E R P A g N N L O • D 0 1 3
D F g g ‡ p ⁄ - 2 - 1 d s /d Dfgg (pb/rad)
K M R L O - M R W N L O - M R W S H E R P A g N N L O • D 0 1 3
Ratio
D f g g ( r a d ) FIG. 12: The same as the figure 11 but for the D
013 experiment and extra cosθ ∗ t,γγ variable. - 2 - 1 d2 s /dM gg Dfgg (pb/rad)
K M R L O - M R W N L O - M R W R E S B O S
D I P H O X • D 0 1 0 g g ( G e V ) < 8 0 - 2 - 1 d s / dM gg d|cos q* gg |(pb) K M R L O - M R W N L O - M R W R E S B O S
D I P H O X • D 0 1 0 g g ( G e V ) < 8 0
Ratio
Ratio - 6 - 5 - 4 - 3 - 2 d s / dM gg dpT, gg (pb/GeV) K M R L O - M R W N L O - M R W R E S B O S
D I P H O X • D 0 1 0 g g ( G e V ) < 8 0 - 5 - 4 - 3 - 2 d s / dM gg dpT, gg (pb/GeV) K M R L O - M R W N L O - M R W R E S B O S
D I P H O X • D 0 1 0 g g ( G e V ) < 5 0
Ratio - 3 - 2 - 1 d s / dpT, gg (pb/GeV) K M R L O - M R W N L O - M R W R E S B O S D I P H O X • D 0 1 0
Ratio
D f g g ( r a d ) - 1 d s /d Dfgg (pb/rad)
K M R L O - M R W N L O - M R W R E S B O S D I P H O X • D 0 1 0 ( a )( b )( c ) ( d )( e )( f ) ( g )( h )( i )
Ratio p T ,g g ( G e V ) Ratio c o s q * g g
F I G 1 3 d s /d|cos qgg *|(pb) K M R L O - M R W N L O - M R W R E S B O S D I P H O X • D 0 1 0
Ratio - 2 - 1 d s / dM gg d|cos q* gg |(pb) K M R L O - M R W N L O - M R W R E S B O S
D I P H O X • D 0 1 0 g g ( G e V ) < 5 0 - 3 - 2 - 1 d s /dM gg d Dfgg (pb/rad)
K M R L O - M R W N L O - M R W R E S B O S
D I P H O X • D 0 1 0 g g ( G e V ) < 5 0
Ratio
Ratio
FIG. 13: The same as the figure 11 but with the RESBOS and DIPHOX results and p t,γγ , ∆ φ γγ and cosθ ∗ t,γγ , variables. . 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 70 . 51 . 01 . 5 Ratio | c o s q *g g | Ratio M g g ( G e V ) 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 00 . 51 . 01 . 5
Ratio
Ratio
D f g g - 3 - 2 d s / dM gg d|cos q* gg |(pb) K M R L O - M R W N L O - M R W R E S B O S
D I P H O X • D 0 1 0 g g ( G e V ) < 3 5 0 - 5 - 4 - 3 - 2 - 1 d s /dM gg d Dfgg (pb/rad)
K M R L O - M R W N L O - M R W R E S B O S
D I P H O X • D 0 1 0 g g ( G e V ) < 3 5 0
Ratio
D f g g ( r a d ) - 6 - 5 - 4 K M R L O - M R W N L O - M R W R E S B O S
D I P H O X • D 0 1 0 d s / dM gg dpT, gg (pb/GeV) g g ( G e V ) < 3 5 0 p T , g g ( G e V )
Ratio - 4 - 3 - 2 - 1 d s /dM gg (pb/GeV) K M R L O - M R W N L O - M R W R E S B O S D I P H O X • D 0 1 0
11 01 0 0 d s / d|cos q* gg | ( pb) K M R L O - M R W N L O - M R W D I P H O X • C M S 1 2 | h | < 1 . 4 4 d s / d Dfgg (pb)
K M R L O - M R W N L O - M R W D I P H O X • C M S 1 2 | h | < 1 . 4 4 ( a )( b )( c ) ( d )( e )( f ) ( g )( h )( i )F I G 1 4 Ratio
D f g g
Ratio
Ratio - 3 - 2 - 1 d s / dpT gg (pb/GeV) K M R L O - M R W N L O - M R W D I P H O X • C M S 1 2 | h | < 1 . 4 4 - 3 - 2 - 1 d s / dM gg (pb/GeV) K M R L O - M R W N L O - M R W D I P H O X • C M S 1 2 | h | < 1 . 4 4 d s / d Dfgg (pb)
K M R L O - M R W N L O - M R W D I P H O X • C M S 1 2 | h | < 2 . 5 FIG. 14: As the figure 14 but for the D010 ( E CM = 1960 GeV )and CMS12 ( E CM = 7 T eV )experiments and extra M γγ . E S B O S
Ratio
D f g g ( r a d ) d s / d Dfgg (pb/rad)
K M R L O - M R W N L O - M R W D I P H O X
R e s B o s • A T L A S 1 2 M g g Ratio - 2 - 1 d s / dM gg ( pb/GeV) K M R L O - M R W N L O - M R W D I P H O X
R e s B o s • A T L A S 1 2 p T , g g ( G e V )
Ratio - 3 - 2 - 1 d s / dpT, gg (pb/GeV) K M R L O - M R W N L O - M R W D I P H O X
R e s B o s • A T L A S 1 2
Ratio - 1 d s / d Dfgg (pb/rad)
K M R L O - M R W N L O - M R W D I P H O X g N N L O • C M S 1 4
Ratio
Ratio | c o s q *g g | d s / d|cos q* gg | (pb) K M R L O - M R W N L O - M R W D I P H O X • C M S 1 2 | h | < 2 . 5 - 2 - 1 d s / dpT , gg (pb/GeV) K M R L O - M R W N L O - M R W D I P H O X • C M S 1 2 | h | < 2 . 5 ( a )( b )( c ) ( d )( e )( f ) ( g )( h )( i )F I G 1 5 - 3 - 2 - 1 d s /dM gg (pb/GeV) K M R L O - M R W N L O - M R W D I P H O X • C M S 1 2 | h | < 2 . 5 M g g ( G e V ) Ratio
R E S B O SR E S B O S R E S B O S - 4 - 3 - 2 - 1 d s /dM gg (pb/GeV) K M R L O - M R W N L O - M R W D I P H O X g N N L O • C M S 1 4
Ratio
Ratio p T ,g g ( G e V ) - 3 - 2 - 1 d s / dpT, gg (pb/GeV) K M R L O - M R W N L O - M R W D I P H O X g N N L O • C M S 1 4
FIG. 15: As the figure 15 but for the CMS12, CMS14 and ATLAS12 ( E CM = 7 T eV ) and withthe 2 γ NNLO,
RESBOS and
DIP HOX results. . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 01 0 U n c e r t a i n t y
D f g g ( r a d ) d s / d Dfgg (pb/rad)
K M R L O - M R W N L O - M R W • A T L A S 1 3
R e g g e i z a t i o n - 5 - 4 - 3 - 2 - 1 p T , g g ( G e V )
U n c e r t a i n t y • A T L A S 1 3 d s / dpT, gg (pb/GeV) K M R L O - M R W N L O - M R W
R e g g e i z a t i o n
Ratio | c o s q g g * | - 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 00 . 51 . 01 . 5 Ratio
C o s q *g g ( e ) ( f ) D I P H O X 2 g N N L O A T L A S 1 3 d s /dCos q* gg (pb) K M R L O - M R W N L O - M R W
Ratio p T , g g ( G e V ) ( b ) - 4 - 3 - 2 - 1 d s /dM gg (pb/GeV) K M R L O - M R W N L O - M R W (cid:190)
D I P H O X (cid:190) g N N L O • A T L A S 1 3 d s /d|cos qgg *|(pb) K M R L O - M R W N L O - M R W D I P H O X g N N L O • C M S 1 4
Ratio
D f g g ( r a d ) d s / d Dfgg (pb/rad)
K M R L O - M R W N L O - M R W D I P H O X g N N L O • A T L A S 1 3
Ratio M g g ( G e V ) - 5 - 4 - 3 - 2 - 1 d s / dpT, gg (pb/GeV) K M R L O - M R W N L O - M R W D I P H O X g N N L O • A T L A S 1 3 ( a ) ( c )( d ) ( e )F I G 1 6
FIG. 16: As the figure 15 but for the
CM S
14 and
AT LAS
13 ( E CM = 7 T eV ) experiments andthe 2 γN N LO and
DIP HOX results . The panels (f) and (g) are similar to those of (a) and (e)but with the reggeization method of reference [57] and the uncertainty bands for our calculation.
Appendix A: The constraints of various experiments
We provide our results, by considering all constraints that are imposed in each experiment,among which two of them are considered generally, i.e. the isolated-cone and the separation-42one constraints. The isolated-cone is responsible for distinguishing the ”non-prompt decayphotons” from the prompt-photons. This constraint requires the transverse energy E isot (ina cone with the angular radius R = (cid:112) ( η − η γ ) + ( φ − φ γ ) < .
4) to be less than a fewGeV according to each experiment. To avoid, the overlap between the two photons, theseparation-cone constraint is imposed as,∆
R > . . Other constraints such as the p t -threshold of prompt-photons, the pseudo-rapidity regions,etc, are imposed according to the settings of the individual experiments. Obviously, thedifferent settings probe the various regions of pQCD. In what follows, we briefly present thereader with the specifications of the measurements that we intend to analyze throughoutthis work in each experiments.
1. The D0 collaboration
The D0 experiment was performed at the center of mass energy of 1 .
960 TeV. The twosets of D0 data, related to the IPPP production were investigated in the references [2, 3],i.e. D010 and D013, respectively. The constraints in the D010 experiment [2] are p t >
20, 21GeV (the transverse momentum of outgoing photons), | η | < . M γγ > p t,γγ which are applied to the IPPP production [2], suppressing the fragmentationeffects and some higher order contributions. In the D013 report [3], the constraints are p t >
17, 18
GeV and | η | < .
9. Also three regions are probed in the D013 report, i.e the regionI with the ∆ φ γγ ≥ π constraint which is suitable for the study of non-higher-order pQCD.By applying the ∆ φ γγ < π constraint in the region II, the fragmentation effects becomeimportant and the last region is without any extra constraint on ∆ φ γγ [2, 3].
2. The CDF collaboration
Similar to the D0 collaboration, the CDF experiment provides the two sets of data, thatare related to the IPPP production [4, 5] at the center of mass energy of 1 .
960 TeV. Theconstraints in the CDF13 [5] are the same as the CDF11 [4] reports. However, the luminosityis improved in the new sets of data (CDF13) [5] . These constraints are p t >
15, 17 GeV43nd | η | <
1. The
CDF collaboration explored three regions in their works: the regionI, via applying the p t,γγ > M γγ constraint, suitable for the study of higher-order pQCD.The region II, by applying the p t,γγ < M γγ constraint, which undermines the fragmentationeffects and emphasizes on the quark-antiquark annihilation. The last region is without anyextra constrains [4, 5].
3. The CMS collaboration
The CMS collaboration is presented at the 7 TeV CM energy [6, 7]. The constraints inthe CMS12 experiment [6] are p t >
20, 23 GeV and | η | < .
5, excluding the 1 . < | η | < . | η | < .
44 is separately canalized. In the reference [7] (
CM S
14) theasymmetric transverse momentum ( p t >
40, 25 GeV ) for the IPPP production is selectedin the regions of 1 . < | η | < . | η | < .
44. As a result, the higher order pQCDcontributions become dominant in these experiments [6, 7].
4. The ATLAS collaboration
Another experimental data at the 7 TeV CM energy is provided by the ATLAS collabo-ration [8, 9] (ATLAS12 and ATLAS13). In the reference [8], the data is sorted according to p t >
16 (16) GeV and | η | < .
5, excluding the 1 . < | η | < .
52 region. Similarly, ATLAS13[9] has the same pseudo-rapidity region, although the transverse momentum threshold ischanged to P t >>