A detail study of the LHC and TEVATRON hadron-hadron prompt-photon pair production experiments in the angular ordering constraint k_t-factorization approaches

TL;DR

This study analyzes prompt-photon pair production in high-energy hadron collisions using k_t-factorization with AOC-imposed UPDFs, comparing KMR and MRW approaches against experimental data and other models.

Contribution

It provides a detailed comparison of KMR and MRW unintegrated PDFs within the k_t-factorization framework for prompt-photon production, highlighting their energy-dependent performance and effects of higher-order corrections.

Findings

KMR framework performs better at higher energies.

LO-MRW scheme is more accurate at lower energies.

The models predict experimental features like shoulders and tails.

Abstract

In the present work, which is based on the k_-factorization framework, it is intended to make a detail study of the isolated prompt-photon pairs (IPPP) production in the high-energy inelastic hadron-hadron collisions differential cross section. The two scheme-dependent unintegrated parton distribution functions (UPDF) in which the angular ordering constraints (AOC) are imposed, namely the Kimber-Martin-Ryskin (KMR) and the Martin-Ryskin-Watt (MRW) approaches, in the leading and the next-to-leading orders (LO and NLO) are considered, respectively. These two prescriptions (KMR and MRW) utilize the phenomenological parton distribution functions (PDF) libraries of Martin et al, i.e. the MMHT2014. The computations are performed in accordance with the initial dynamics of latest existing experimental reports of the D0, CDF, CMS and ATLAS collaborations and the different experimental constraints. It is shown that above frameworks are capable of producing acceptable results, compared to the experimental data, the pQCD and some Monte Carlo calculations (i.e. 2\gamma-NNLO, SHERPA, DIPHOX and RESBOS). It is also concluded that the KMR framework produces better results in the higher center-of-mass energies, while the 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 higher pQCD effects become important. A clear prediction for the various shoulders and tails which were detected experimentally are observed and discussed in the present theoretical approaches. The possible double countings between 2 \rightarrow 2 and 2 \rightarrow 3 processes are studied.

Equations

$$xa(x,Q^2)\simeq {\int }^{Q^2}\frac{d{k}_t^2}{k_t^2}f(x,k_t^2,Q^2),$$

$$P_1=\frac{\sqrt{s}}{2}(1,0,0,1),P_2=\frac{\sqrt{s}}{2}(1,0,0,-1)$$

$$k_i=x_i{P}_i+k_{i,t},$$

$$P^{\mu }=(P^{+},\frac{M}{2P^{+}},0_t),$$

$$y=\frac{1}{2}ln\frac{P^{+}}{P^{-}}.$$

$$P^{\mu }=(\sqrt{\frac{M^2+{P_t}^2}{2}}{e}^y,\sqrt{\frac{M^2+{P_t}^2}{2}}{e}^{-y},P_t),$$

$${\text{k}}_{1,t}+{\text{k}}_{2,t}={\text{k}}_{3,t}+{\text{k}}_{4,t},$$

$$x_1=\left(|k_{3,t}|e^{y_3}+|k_{4,t}|e^{y_4}\right)/\sqrt{s},$$

$$x_2=\left(|k_{3,t}|e^{-y_3}+|k_{4,t}|e^{-y_4}\right)/\sqrt{s}.$$

$${\text{k}}_{1,t}+{\text{k}}_{2,t}={\text{k}}_{3,t}+{\text{k}}_{4,t}+{\text{k}}_{5,t},$$

$$x_1=\left(|k_{3,t}|e^{y_3}+|k_{4,t}|e^{y_4}+M_{5,t}{e}^{y_5}\right)/\sqrt{s},$$

$$x_2=\left(|k_{3,t}|e^{-y_3}+|k_{4,t}|e^{-y_4}+M_{5,t}{e}^{-y_5}\right)/\sqrt{s}.$$

$$M_{5,t}=\sqrt{m^2+{|k_{5,t}|}^2},$$

$$\mathcal{M}=e^2{e}_q^2{ϵ}_{\mu }(k_3){ϵ}_{\nu }(k_4)\bar{U}(k_2)\left({\gamma }^{\nu }\frac{/k_1-/k_3+m}{(k_1-k_3{)}^2-m^2}{\gamma }^{\mu }+{\gamma }^{\mu }\frac{/k_1-/k_4+m}{(k_1-k_4{)}^2-m^2}{\gamma }^{\nu }\right)U(k_1),$$

$$\sum _{i=3,4}{\epsilon }^{\mu }(k_i){\epsilon }^{\ast \upsilon }(k_i)=-g^{\mu \upsilon }.$$

$$\mathcal{M}=e^2{e}_q^2\sqrt{4\pi {\alpha }_s}{T}^{\alpha }{ϵ}_{\xi }(k_2){ϵ}_{\mu }(k_3){ϵ}_{\nu }(k_4)\bar{U}(k_5)[\sum _{i=1,6}{\mathcal{A}}_i^{\xi \mu \nu }]U(k_1),$$

$$T^{\alpha }=\frac{{\lambda }^{\alpha }}{2}.$$

$${ϵ}_{\xi }(k_2)=\frac{2k_{2,\xi }}{\sqrt{s}}.$$

$$\sum {\epsilon }^{\mu }(k_2){\epsilon }^{\ast \upsilon }(k_2)=\frac{k_{2,t}^{\mu }{k}_{2,t}^{\upsilon }}{k_{2,t}^2}.$$

$${\sigma }_{\gamma \gamma }=\sum _{i,j=q,g}\int {\hat{\sigma }}_{ij}(x_1,x_2,{\mu }^2)f_i(x_1,{\mu }^2)f_j(x_2,{\mu }^2)d{x}_{1}d{x}_2,$$

$${\sigma }_{\gamma \gamma }=\sum _{a_1,a_2=q,g}{\int }_0^1\frac{d{x}_1}{x_1}{\int }_0^1\frac{d{x}_2}{x_2}{\int }_0^{\infty }\frac{d{k}_{1,t}^2}{k_{1,t}^2}{\int }_0^{\infty }\frac{d{k}_{2,t}^2}{k_{2,t}^2}{\int }_0^{2\pi }\frac{d{\varphi }_1}{2\pi }{\int }_0^{2\pi }\frac{d{\varphi }_2}{2\pi }{f}_{a_1}(x_1,k_{1,t}^2,{\mu }_1^2)f_{a_2}(x_2,k_{2,t}^2,{\mu }_2^2)$$

$$\times {\hat{\sigma }}_{a_1{a}_2}(x_1,k_{1,t}^2,{\mu }_1^2;x_2,k_{2,t}^2,{\mu }_2^2),$$

$$f_a(x,k_t^2,{\mu }^2)=T_a(k_t^2,{\mu }^2)\sum _{b=q,g}\left[\frac{{\alpha }_S(k_t^2)}{2\pi }{\int }_x^{1-\Delta }dz{P}_{ab}^{(LO)}(z)b\left(\frac{x}{z},k_t^2\right)\right],$$

$$T_a(k_t^2,{\mu }^2)=exp\left(-{\int }_{k_t^2}^{{\mu }^2}\frac{{\alpha }_S(k^2)}{2\pi }\frac{d{k}^2}{k^2}\sum _{b=q,g}{\int }_0^{1-\Delta }d{z}^{\prime }{P}_{ab}^{(LO)}(z^{\prime })\right).$$

$$\Delta =\frac{k_t}{\mu +k_t}.$$

$$f_q^{LO}(x,k_t^2,{\mu }^2)=T_q(k_t^2,{\mu }^2)\frac{{\alpha }_S(k_t^2)}{2\pi }{\int }_x^{1}dz[P_{qq}^{(LO)}(z)\frac{x}{z}q\left(\frac{x}{z},k_t^2\right)\Theta \left(\frac{\mu }{\mu +k_t}-z\right)$$

$$+P_{qg}^{(LO)}(z)\frac{x}{z}g\left(\frac{x}{z},k_t^2\right)],$$

$$T_q(k_t^2,{\mu }^2)=exp\left(-{\int }_{k_t^2}^{{\mu }^2}\frac{{\alpha }_S(k^2)}{2\pi }\frac{d{k}^2}{k^2}{\int }_0^{z_{max}}d{z}^{\prime }{P}_{qq}^{(LO)}(z^{\prime })\right),$$

$$f_g^{LO}(x,k_t^2,{\mu }^2)=T_g(k_t^2,{\mu }^2)\frac{{\alpha }_S(k_t^2)}{2\pi }{\int }_x^{1}dz[P_{gq}^{(LO)}(z)\sum _q\frac{x}{z}q\left(\frac{x}{z},k_t^2\right)$$

$$+P_{gg}^{(LO)}(z)\frac{x}{z}g\left(\frac{x}{z},k_t^2\right)\Theta \left(\frac{\mu }{\mu +k_t}-z\right)],$$

Full text

Keywords: unintegrated parton distribution functions, isolated prompt-photon pair production, di-photon production, $k_{t}$-factorization, Guillet shoulder.

I 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 Higgs boson’s decay to diphotons, as well as (3) some theories, which extended beyond the standard model and should give some predictions regarding the new phenomena in the fundamental particle physics collins . Many experimental efforts at the LHC and TEVATRON colliders have been performed to explore the physics of these regions, e.g. the D0, CDF, CMS and ATLAS collaborations D010 ; D013 ; CDF11 ; CDF13 ; CMS11 ; CMS14 ; ATLAS10 ; ATLAS13 . These investigations are probing different channels and exploring different aspects of the above subjects, such as producing differential cross-section of the photon pair production as a function of the azimuthal separation angle between the photon pair in the laboratory frame ($\Delta\phi_{\gamma\gamma}$) and the transverse momenta of the photon pair ($p_{t,\gamma\gamma}$). They are particularly useful to study the higher order pQCD and the fragmentation effects ATLAS13 . Other observable, namely the photon pair invariant-mass ($M_{\gamma\gamma}$) and the polar angle of the highest photon-transfer-energy-momentum in the Collins-Soper isolated prompt photon pair (IPPP) rest frame ($cos\theta_{\gamma\gamma}^{*}$), are also powerful tools to investigate the spin of the photon pair resonances ATLAS13 . The experimental collaborations conventionally use some parton level Monte-Carlo programs, e.g. RESBOS, DIPHOX, $2\gamma$NNLO and SHERPA RESBOS ; DIPHOX ; NNLO ; sherpa , to test the pQCD theory against their data. Also, in the recent years, (2016), a new article based on MCFM program, (Monte Carlo for FeMtobarn), which uses the collinear factorization formalism, with NNLO accuracy was published that its result has good agreement with the available data new .

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 include the single photon fragmentation tomas as well. The DIPHOX Monte-Carlo event generator performs the IPPP production at the NLO pQCD level, in which the single and double fragmentation contributions tomas are also included. The $2\gamma$NNLO is developed to include the full next-to-next-to-leading order (NNLO) pQCD, without considering fragmentation contributions NNLO . Another choice is the SHERPA sherpa Monte Carlo event generator, that could simulate the high-energy reactions of particles in the hadron-hadron collisions.

To perform such a analysis one usually needs parton distribution functions (PDF), $a(x,Q^{2})$, or unintegrated PDF (UPDF), $f(x,k_{t},Q^{2}$), see the references DGLAP1 ; DGLAP2 ; DGLAP3 ; DGLAP4 ; BFKL1 ; BFKL2 ; BFKL3 ; BFKL4 ; BFKL5 ; CCFM1 ; CCFM2 ; CCFM3 ; CCFM4 ; CCFM5 ; Q-CCFMI ; Q-CCFMII and the section III. Note that $x$, $Q^{2}$ and $k_{t}$ are the Bjorken scale,the hard scale and the parton transverse 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 some features of these theories in the existing experimental data. Consequently, a detailed analysis over the experimental data is vital, to determine the exact contributions out of the standard model, in order to justify or reject such theories Lip0 .

Regarding the complication and the weakness of different prescriptions DGLAP1 ; DGLAP2 ; DGLAP3 ; DGLAP4 ; BFKL1 ; BFKL2 ; BFKL3 ; BFKL4 ; BFKL5 ; CCFM1 ; CCFM2 ; CCFM3 ; CCFM4 ; CCFM5 ; Q-CCFMI ; Q-CCFMII , Martin et al KMR ; MRW defined the UPDF in the $k_{t}$-factorization framework, in relation to the conventional PDF MMHT , through the identity,

[TABLE]

and developed the Kimber-Martin-Ryskin (KMR) and the Martin-Ryskin-Watt (MRW) approaches KMR ; MRW . These formalisms were analyzed thoroughly via the calculation of the proton structure functions ($F_{2}$ and $F_{L}$) in the references Mod1 ; Mod2 ; Mod3 ; Mod4 ; Mod5 ; Mod6 ; Mod7 ; FL . Also, the applications of KMR and MRW frameworks in the LO and the NLO levels were investigated against the existing experimental data in the references FL ; FL-dipole ; W/Z-NLO ; W/Z-LHCb ; Di-jet , and some successful results were achieved.

In the present work, we intend to study the production of the IPPP in the high energy Hadron-Hadron collisions, in the frameworks of KMR and MRW procedures. A primary investigation, using the KMR $k_{t}$-factorization approach, was performed in the reference LIPIPPP , with some comparisons to the old data D010 ; CDF11 ; CMS11 ; ATLAS10 , and some discrepancies especially in the fragmentation regions, were observed. To investigate this problem, it is intended to use three 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 the latest, as well as old, experimental data of the D0, CDF, CMS and ATLAS collaborations in their respective dynamical specifications D010 ; D013 ; CDF11 ; CDF13 ; CMS11 ; CMS14 ; ATLAS10 ; ATLAS13 and other theoretical approaches RESBOS ; sherpa ; NNLO ; tomas ; DIPHOX discussed above. It will be shown that the $k_{t}$-factorization framework is reasonably capable of describing of the high energy experiments data for the IPPP production. We also discuss the various advantages and disadvantages of the KMR and MRW prescriptions in connection to each experiments conditions by presenting a detail comparison. One of the main goals of our work is to observe and analyze the effect of imposing different visualizations of the AOC (embedded in different UPDF prescription schemes) in the partonic dynamics that depends on the deriving factors, (i.e. different experimental constraints), which are assumed in the existing experimental data, such that to cover the sensitive area to the fragmentation and 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 in the KMR prescription re which is different in case of the MRW (note that for the KMR scheme the AOC is applied on both quark and gluon radiations but this is not the case in the MRW approaches (see the section III)). As it was discussed in the reference re , our calculations show that a qualitative agreements between the different schemes can be achieve at least in the calculation of differential cross sections AMIN . Beside these, the ambiguity about the fragmentation region LIPIPPP is considered by performing MRW-LO, which show a better agreement with data at the fragmentation regions. On the other hand the Guillet shoulder Guilet binoth as well as new shoulders are observed (see the section V). In the reference LIPIPPP , the valence quarks were only considered in the case of $q^{*}$+$\bar{q}^{*}$ for $P\bar{P}$ collision and the see-quarks contributions (which is very small) were ignored. Beside these, in the PP collision the latter contributions are sizable, in contrast to the reference LIPIPPP 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 the gauge invariance and the satisfactions of the Ward identities KUTAK1 ; KUTAK2 ; KUTAK3 ; KUTAK4 ; 1p ; saleev ; saleev1 ; LIP2016 . The off-shell matrix element violates the gauge invariant which is necessary for the cross section calculation. In the references 1p ; KUTAK4 ; 2 , it was shown that a suitable gauges for gluons and photons polarizations lead to saving the gauge invariant of the off-shell gluons matrix elements. The eikonal 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 all processes that the quarks are the incoming off-shell legs. However, in the small x limit and the large transverse momentum, using the approximation made in the references lip31 ; SPL , the off-shell matrix element satisfies the gauge invariance requirements. In this work, with the 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 are inherently satisfy the gauge invariance in the all regions saleev ; saleev1 or (ii) the method developed in the references KUTAK1 , in which by modifying the vertexes and using auxiliary photons and quarks, 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 those of reference saleev1 .

The possible double countings between 2$\rightarrow$2 and 2$\rightarrow$3 processes, which were pointed out in the references saleev1 ; LIP2011 as well as our previous work W/Z-LHCb , will be discussed in the sections II, V and VI.

The others theoretical and the Monte Carlo calculation (as explained above) are also presented against our results. However, we should make this note that, as it was discussed in the reference WattWZ , the present approach should not be as good as pQCD approaches, on the other hand, it is more simplistic.

In what follows, the theoretical framework of the IPPP production is presented in the section II. A brief introduction to the $k_{t}$-factorization, and individually the KMR, LO-MRW and NLO-MRW prescriptions, are presented in the section III. The section IV contains a comprehensive 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 the appendix A and finally, results, discussions and conclusions are given in the sections V and VI, 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 direct production of a photon while the second is created in the processes of hadronic decay. In this paper, we intend to base our calculations only on direct IPPP production. In order to set the kinematics of the pair photon production, we choose to work in the center-of-mass frame of the initial protons. So, we can set their four-momenta as:

[TABLE]

where $s$ is the center-of-mass (CM) energy and $P_{1}$ and $P_{2}$ are the four-momenta of the colliding 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. $q^{*}+\bar{q}^{*}\to\gamma+\gamma$, $q^{*}(\bar{q}^{*})+g^{*}\to\gamma+\gamma+q(\bar{q})$ and $g^{*}+g^{*}\to\gamma+\gamma$ LIPIPPP . Hence, it is required to write the four-momenta of the incoming partons based on the four-momenta of the initial protons, using the Sudakov decomposition assumption as,

[TABLE]

where as we pointed out before, $k_{i,t}$ are the transverse momenta of the $i^{th}$ partons and $x_{i}$ are the 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 $\psi$ from the rest-frame. Its momentum reads as,

[TABLE]

with $P^{+}$ being the positive light-cone momentum of the particle. In general, one is able to write its momentum based on the rapidity ($y$), which is defined as:

[TABLE]

The result is the following expression for the momentum of the particle:

[TABLE]

where $\sqrt{M^{2}+{P_{t}}^{2}}$ is the so-called transverse energy of the particle, $E_{t}$, light-cone . Using the above method and the conservation of energy-momentum, we can derive the following relations for the subprocesses $q^{*}+\bar{q}^{*}\longrightarrow\gamma+\gamma$ and $g^{*}+g^{*}\longrightarrow\gamma+\gamma$ :

[TABLE]

[TABLE]

[TABLE]

Similarly for the subprocess $g^{*}+q^{*}(\bar{q}^{*})\longrightarrow\gamma+\gamma+q(\bar{q})$, we find:

[TABLE]

[TABLE]

[TABLE]

$k_{i,t}$ and $y_{i}$, $i=3,4$ are the transverse momentum and the rapidity of outgoing particles, respectively. $M_{5,t}$ is the transverse mass of produced quark or anti-quark with mass $m$ that is defined by:

[TABLE]

while $y_{5}$ is its rapidity (the quarks masses are set equal to zero as it is stated above the equation (22)).

In this work, we consider the simplest processes for the isolated prompt photon pair production. Therefore, the diagrams $2\rightarrow 2$ $(q^{*}+\bar{q}^{*}$ or $g^{*}+g^{*}\rightarrow\gamma+\gamma)$ and $2\rightarrow 3$ $(q^{*}(\bar{q}^{*})+g^{*}\rightarrow\gamma+\gamma+q(\bar{q}))$ are selected using the figure 1. The incoming legs in the figure 1 can be the (anti)-quarks or gluons UPDF according to the $k_{t}$-factorization procedure of corresponding differential cross section calculation. We also investigate the dependence of the 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 $q^{*}+\bar{q}^{*}\to\gamma+\gamma$ sub-process is as follows:

[TABLE]

where $e$ and $e_{q}$ are the electron charge and the quark electric charge respectively and $\varepsilon_{3}$ and $\varepsilon_{4}$ are the polarization 4-vectors of the isolated prompt photons, that satisfy the co-variant equation:

[TABLE]

Similarly for the $g^{*}+q^{*}(\bar{q}^{*})\to\gamma+\gamma+q(\bar{q})$ sub-process, the transition amplitude is:

[TABLE]

where $\mathcal{A}^{\xi\mu\nu}_{i}$ are defined in the reference LIPIPPP .

$T^{\alpha}$ 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 ($\lambda^{\alpha}$),

[TABLE]

$\varepsilon_{\xi}(k_{2})$ is the polarization vector of the incoming off-shell gluon which should be modified with the eikonal vertex (i.e the BFKL prescription, see the reference Deak1 ). One choice is to impose the so called non-sense polarization conditions on $\varepsilon_{\xi}(k_{2})$ which is not normalized to one Deak1 ; collins (and it will not be used in the present work):

[TABLE]

But in the case of $k_{t}$-factorization scheme and the off-shell gluons, the better choice is $\epsilon^{\mu}_{2}(k_{2})={k^{\mu}_{2,t}\over|k_{2,t}|}$, which leads to the following identity and can be easily implemented in our calculation Deak1 ; collins :

[TABLE]

Finally, for the matrix element of the $g^{*}+g^{*}\to\gamma+\gamma$ sub-process, we use those which was calculated before by Berger et al. berger , with this difference that the kinematics given in the equations (3) and (4) is imposed LIPIPPP .

So, generally the cross-section of $IPPP$ production is:

[TABLE]

where $\hat{\sigma}_{ij}(x_{1},x_{2},\mu^{2})$ is the partonic cross-section and $f_{j}(x_{i},\mu^{2})$ (= $xa_{j}(x_{i},\mu^{2})$) are parton distribution function refer to the incoming parton $i$ that depends on two variables, $x$ and $\mu^{2}$ as the scales of the hard process. But in the high-energy domain, using the $k_{t}$-factorization theory, we could rewrite the collinear cross-section, i.e., the equation (8), as,

[TABLE]

[TABLE]

where $f(x_{i},k_{t,1}^{2},\mu^{2})$ are the $UPDF$ that depend on three parameters, i.e. $x$, $k^{2}_{t}$ and $\mu^{2}$.

The UPDF are directly obtained from the PDF by using different prescriptions (see the next section). In this paper, we use the three approaches namely KMR KMR , LO-MRW and NLO-MRW MRW 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 $k_{t}$-factorization calculations correspond to non-normalized probability functions. They are used as the weight of the given transition amplitudes (the matrix elements in these cases). The transverse momentum dependence of the UPDF comes from considering all possible splittings up to and including the last splitting, see the references KMR ; WattWZ ; WATT ; KMR1 , while the evolution up to the hard scale without change in the $k_{t}$, due to virtual contributions, is encapsulated in the Sudakov-like survival form factor. Therefore, all splittings and real emissions of the partons, including the last emission, are factorized in the function $f_{g}(x,k_{t}^{2},\mu^{2})$ as its definition. The last emission from the definition of the produced UPDF can not be disassociated and to be count as the part of the $2\to 3$ diagrams. This point also discussed in the reference W/Z-LHCb and it is in contrast to the reference LIPIPPP ; LIP2011 i.e. there may not be any double counting by taking into account $q^{*}+{\bar{q}^{*}}\rightarrow\gamma+\gamma$ and $g^{*}+q^{*}({\bar{q}^{*}})\rightarrow q({\bar{q}})+\gamma+\gamma$ processes together in the $k_{t}$-factorization approach in the present calculation KIMBER . On the other hand some authors do believe on the double counting. The argument goes as follows: in the region where the transverse momentum of one of the parton is as large as the hard scale and the additional parton is highly separated in the rapidity from the hard process (multi-Regge region), the additional emission in the 2$\rightarrow$3 should be subtracted i.e. considering the definition of the UPDF. However we will check the above agrement in the sections IV and V, by modification of the UPDF to find out about the possible double counting. One should also not that the UPDF should satisfy the condition given in the equation (1). So any changes in the UPDF certainly affect the original PDF definitions.

III The $k_{t}$-factorization framework

In the equation (8) all partons are usually assumed to move in the plane of the incoming protons. Therefore, they do not posses any transverse momenta. This is so called the collinear 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 become important. Therefore, the cross-sections are factorized into the $k_{t}$-dependent partonic cross-sections $\hat{\sigma}(x,k_{t}^{2},\mu^{2})$, where the incoming partons are treated as the off-shell particles. So, one should use the UPDF ($f(x,k^{2}_{t},\mu^{2})$) instead of the PDF in the equation (8), according to the equation (1) which leads to the equation (9).

In the rest of this section, we briefly explained how to evaluate these UPPF in the simplistic frameworks.

III.1 The KMR prescription

The KMR UPDF are generated through a procedure that was proposed by Kimber, Martin and Ryskin (KMR) KMR . In this method, the UPDF are generated such that the partons developed from some starting parameterizations up to the scale $k_{t}$ according to the DGLAP evolution equations. So the partons are evolved in the single evolution ladder (carrying only the $k_{t}^{2}$ dependency) and get convoluted with the second scale ($\mu^{2}$) at the hard process. This is the last-step evolution approximation. Then the $k_{t}$ is forced to depend on the scale $\mu^{2}$, without any real emission, and there is a summation over the virtual contributions by imposing the Sudakov form factor ($T_{a}(k_{t}^{2},\mu^{2})$). So, the general form of the KMR-UPDF are:

[TABLE]

where $T_{a}(k_{t}^{2},\mu^{2})$ are :

[TABLE]

$T_{a}$ are considered to be unity for $k_{t}>\mu$. In the above equation $\Delta$ is proposed to prevent the soft gluon singularity, but this constraint is imposed on the quark radiations too. The angular ordering constraint is imposed to determine $\Delta$,. Angular ordering originates from the color coherence effects of the gluon radiations KMR . So $\Delta$ is:

[TABLE]

The $P_{ab}^{(LO)}(z)$ are the familiar LO splitting functions collins .

III.2 The LO-MRW prescription

The LO-MRW formalism, similar to the KMR scheme, was proposed by Martin, Ryskin and Watt (MRW) MRW . This formalism has the same general structure as the KMR, but only with one significant difference that: the angular ordering constraint is correctly imposed only on the on-shell radiated gluons, i.e. the diagonal splitting functions $P_{qq}(z)$ and $P_{gg}(z)$ MRW . So, the LO-MRW prescription is written as:

[TABLE]

[TABLE]

with

[TABLE]

for the quarks and

[TABLE]

[TABLE]

with

[TABLE]

for the gluons. In the equations (13) and (15), $z_{max}=1-z_{min}=\mu/(\mu+k_{t})$ WATT . The UPDF of KMR and MRW to a good approximation, include the main kinematical effects involved in the IS processes. Note that the particular choice of the AOC in the KMR formalism despite being of the LO, includes some contributions from the NLO sector, hence in the case of MRW framework, these contributions must be inserted separately.

III.3 The NLO-MRW prescription

Finally, MRW MRW proposed a method for the promotion of the LO-MRW to the NLO-MRW prescription. Utilizing the NLO PDF and corresponding splitting functions from DGLAP evolution equations lead to the MRW-NLO formalism MRW . The general form of the NLO-MRW UPDF are:

[TABLE]

[TABLE]

with the ”extended” NLO splitting functions, $\tilde{P}_{ab}^{(i)}(z)$, being defined as,

[TABLE]

and

[TABLE]

where $i=0$ and $1$ stand for the LO and the NLO, respectively. The reader can find a comprehensive description of the NLO splitting functions in the references MRW ; PNLO . We must however emphasize that in contrary to the KMR and the LO-MRW frameworks, the AOC is being introduced into the NLO-MRW formalism via the $\Theta(z-(1-\Delta))$ constraint, in the ”extended” splitting function. Now $\Delta$ can be defined as:

[TABLE]

This framework are the collection of the NLO PDF, the NLO splitting functions and the constraint $\Theta\left(1-z-{k_{t}^{2}/\mu^{2}}\right)$ which impose the NLO corrections to this method. Nevertheless, it was shown that using only the LO part of the ”extended” splitting functions, instead of the complete definition of the equation (17), would result a reasonable accuracy in the computation of the NLO MRW UPDF MRW . Additionally, the Sudakov form factors in this framework are defined as:

[TABLE]

[TABLE]

Each of the KMR, the LO and the NLO MRW UPDF can be used to identify the probability of finding a parton of a given flavor, with the fraction $x$ of longitudinal momentum of the parent hadron and the transverse momentum $k_{t}$, in the scale $\mu^{2}$ at the semi-hard level of a particular IS process.

The modifications to the above KMR, LO-MRW and NLO-MRW UPDF are made in our calculation of IPPP production cross sections, in the section V, to investigate the possible double counting concerning the $2\rightarrow 3$ process, according to the reference LIP2011

IV the IPPP production and the technical prescription

For calculating the partonic cross-section, we need the matrix element squared ($|\mathcal{M}|^{2}$) of sub-processes. Since the incoming quarks and gluons are off-shell, the expression for such matrix element will be more complicated. Therefore, we use the BFKL prescription Deak1 for the gluons in the equation (7) and apply the method proposed in the references lip31 ; lip32 for the incoming quarks, for the small x region. In this method, it is assumed that the incoming quarks with 4-momenta ($p_{\mu}$) radiate a gluon (or a photon) and consequently become off-shell LIPIPPP ; W/Z-NLO . Therefore the extended $|\mathcal{M}|^{2}$ becomes,

[TABLE]

where $\tilde{T}^{\alpha}$ and $T_{\alpha}$ represent the rest of the matrix elements. Since, the expression presented between $T_{\alpha}$ and $\tilde{T}^{\alpha}$ 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 $m\to 0$ limit and imposing the Sudakov decomposition, $k=xp+k_{t}$, with $k^{2}=k^{2}_{t}=-\textbf{k}^{2}_{t}$ and some straightforward algebra we obtain W/Z-NLO ; Deak1 ; LIPIPPP :

[TABLE]

where $x\hat{p}$ represents the properly normalized off-shell spin density matrix.

Since the calculation of the $|\mathcal{M}|^{2}$ is a laborious task, we use the algebraic manipulation system $\mathtt{FORM}$ FORM . The above approximations, which are valid at small x region, force some limits on our kinematics range. So the resulted differential cross section may not cover the whole experimental data of Tevatron and LHC colliders (see our discussion in the section V) LIPIPPP ; KUTAK1 ; KUTAK2 ; KUTAK3 ; KUTAK4 ; LIP2016 .

In the section II, we defined the total cross-section for the IPPP production at hadronic collisions, $\sigma_{\gamma\gamma}$, as:

[TABLE]

$d\phi_{a_{1}a_{2}}$ and $F_{a_{1}a_{2}}$ are the multi-particle phase apace and the flux factor, respectively which can be defined according to the specifications of the partonic process,

[TABLE]

[TABLE]

where the $s$ is the center of mass energy squared,

[TABLE]

$d\phi_{a_{1}a_{2}}$ 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, $\varphi_{i}$,

[TABLE]

In the present work, ${\mathcal{M}_{a_{1}a_{2}}}$ in the equation (23), are the matrix elements of the partonic diagrams 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 for the total cross-section of the IPPP production in the framework of $k_{t}$-factorization. So the total cross-section for $q\bar{q}$ and $gg$ are:

[TABLE]

[TABLE]

and for $qg$ and $\bar{q}g$ are,

[TABLE]

[TABLE]

[TABLE]

Note that the integration boundaries for ${dk_{i,t}^{2}/k_{i,t}^{2}}$ are limited by the kinematics. So one can introduce an upper limit for these integration, say $k_{i,max}$, several times larger than the scale $\mu$. In addition, $k_{t,min}=\mu_{0}\sim 1$ $GeV$, is considered as the lower limit, that separates the non-perturbative and the perturbative regions, by assuming that,

[TABLE]

As a result of above formulation, the densities of partons are constant for $k_{t}<\mu_{0}$ at fix $x$ and $\mu$ MRW . For the above calculations, we use the LO-MMHT2014 PDF libraries for the KMR and the LO-MRW UPDF schemes, and the NLO-MMHT2014 PDF libraries for the NLO-MRW formalism.

The VEGAS algorithm is considered for performing the multidimensional integration of the total cross-section in the equations (27) and (28). Since the sea quarks become significant in the high energy limit, we calculate the cross-section of IPPP production, by considering four 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). Before we present our results, it is important to have the relations between the different channel parameters, i.e. $M_{\gamma\gamma}$, $p_{t,\gamma\gamma}$, $cos\theta^{*}_{\gamma\gamma}$, $\Delta\phi_{\gamma\gamma}$, $z_{\gamma\gamma}$ and $y_{\gamma\gamma}$, which is given in the references D010 ; D013 ; CDF11 ; CDF13 ; CMS11 ; CMS14 ; ATLAS10 ; ATLAS13 .

Some divergences appear because of the small $k_{t}$ ($<<\mu$) of the outgoing quark in the case of $q^{*}(\bar{q}^{*})+g^{*}\rightarrow\gamma+\gamma+q(\bar{q})$ process. But since this quark is in the direction of outgoing photons it is eliminated by excluding the above mentioned regions in our calculation and also implementing isolated and separated cone in this computation. In this work, we applied the same method as the reference DIPHOX and LIPIPPP for the phase space cut. To avoid the double counting and divergence, the invariant mass of the photon-quark subsystem is considered to be greater than 1 GeV. In order to be insure about the possible double counting in UPDF 9B ; 10B , we checked our result by modifying the UPDF according to reference LIPIPPP ; LIP2011 and suppressing the quarks splitting in the UPDF that will be discussed in the section V. Otherwise one should perform substraction procedure saleev1 ; saleev . We should point out here that the fragmentation contribution of the $q(\bar{q})\rightarrow\gamma$ or $g\rightarrow\gamma$ to the $2\rightarrow 3$ processes can be dramatically reduced by applying the same photon isolation and separation cone implemented by the experimental setup D010 ; D013 ; CDF11 ; CDF13 ; CMS11 ; CMS14 ; ATLAS10 ; ATLAS13 . Beside this restriction, as we stated before, we also choose the invariant mass of quark and photon subsystem to be greater than 1 GeV , in order to eliminate any divergence from our cross section calculation DIPHOX ; LIPIPPP .

The strong coupling is $\Lambda_{QCD}$=200 MeV and $\alpha_{s}$ is chosen to be one and two loops in case of LO and NLO level, respectively collins . As it was pointed out before, the same photon isolation cuts are implemented as the one imposed in the related experiment D010 ; D013 ; CDF11 ; CDF13 ; CMS11 ; CMS14 ; ATLAS10 ; ATLAS13 . We should mention that, the factorization scale $\mu$ is chosen such that, the renormalization scale $\mu_{R}$ to be equal to the invariant mass of photon-photon sub-system $M_{\gamma\gamma}$.

V Results and discussions

In this section, we present our results, regarding the IPPP production according to the experimental specifications discussed in the appendix A. Note that the fragmentation effects enhanced (suppressed) when $p_{t,\gamma\gamma}>M_{\gamma\gamma}$ or $\Delta\Phi_{\gamma\gamma}<\pi/2$ or $\mid\cos\theta^{*}_{\gamma\gamma}\mid>0.6$ ($p_{t,\gamma\gamma}<M_{\gamma\gamma}$ or $\Delta\Phi_{\gamma\gamma}>\pi/2$ or $\mid\cos\theta^{*}_{\gamma\gamma}\mid<0.6$). Since our calculations show different behavior corresponding to the different CM energies, our results and discussions are given into two subsections:

V.1 $E_{CM}=1.96\;TeV$

The figures 2 and 3, illustrate our calculations regarding the IPPP production differential cross section in the $E_{CM}=1.96\;TeV$, in accordance to the experimental data of the D0 (D010 and DO13) and CDF (CDF11 and CDF13) collaborations D010 ; D013 ; CDF11 ; CDF13 , using the KMR, LO-MRW and NLO-MRW prescriptions, as a function of $p_{t,\gamma\gamma}$, $M_{\gamma\gamma}$ and $cos\theta^{*}_{\gamma\gamma}$, respectively. Note that the contribution of the individual sub-processes i.e. $q^{*}+\bar{q}^{*}\to\gamma+\gamma$, $q^{*}(\bar{q}^{*})+g^{*}\to\gamma+\gamma+q(\bar{q})$ and $g^{*}+g^{*}\to\gamma+\gamma$ are given only for the $KMR$ approach.

Considering the different panels of above figures, one readily finds that the $q^{*}+\bar{q}^{*}\to\gamma+\gamma$ contributions dominate. But for larger transverse momenta ($p_{t,\gamma\gamma}$) region, the effects of $q^{*}(\bar{q}^{*})+g^{*}\to\gamma+\gamma+q(\bar{q})$ sub-process become non-negligible. Interestingly, one can see the so-called Guillet shoulder Guilet binoth (note that a new channel opens beyond the leading order (NLO, NNLO etc) where the transverse momentum of pair photons is close to the pair photons transverse momentum cut (threshold) which makes this shoulder) is forming in the intermediate 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 D013 , the fragmentation effects are not fully suppressed, this ”shoulder” can be seen more clearly in the panel (b) of the figure 2. Such behavior can be seen in all of the regions where the fragmentation and the higher order (pQCD) effects become important.

On the other hand, in the panel (c) of the figure 3, one can obviously see the ”low-tail of the mass” (i.e. the small $M_{\gamma\gamma}$ region, where a raise in the differential cross section is acquired), appearing at the small-$M_{\gamma\gamma}$ region, which strongly sensitive to the choice of the mid-$p_{t,\gamma\gamma}$ and the low-$\Delta\phi_{\gamma\gamma}$ domain in the range of our and the others calculations NNLO . In the panel (e) of this figure, the reader should notice that in the $|cos\theta^{*}_{\gamma\gamma}|>0.6$ region, the fragmentation effects become important. However, such behavior is missing or negligible in the panel (f), since the fragmentation effects are being suppressed in these areas by the means of introducing the $p_{t,\gamma\gamma}<M_{\gamma\gamma}$ constraint CDF11 ; CDF13 .

Because of the small x approximation which was made in the section IV for the partial insurance of the gauge invariance of the partonic cross section within high-energy factorization, our result may not be accurate for large $M_{\gamma\gamma}$ as far as we are working in the small x region in which the incoming partons have large transverse momenta.

Similar comparisons are made regarding the double differential cross sections $d^{2}\sigma/dp_{t,\gamma\gamma}dM_{\gamma\gamma}$, $d^{2}\sigma/\Delta\phi_{\gamma\gamma}dM_{\gamma\gamma}$ and $d^{2}\sigma/\cos\theta^{*}_{\gamma\gamma}dM_{\gamma\gamma}$, in the figure 4, against the data of the D010 collaboration D010 . To be specific, what makes the difference in these calculations, is different cuts on the $M_{\gamma\gamma}$, which defers from $30\;GeV<M_{\gamma\gamma}<50\;GeV$, $50\;GeV<M_{\gamma\gamma}<80\;GeV$ and $80\;GeV<M_{\gamma\gamma}<350\;GeV$ in the figure 4 and they are coated in each panel. One notices that, the best predictions are being obtained in the $50\;GeV<M_{\gamma\gamma}<80\;GeV$ range. Since it corresponds to the intermediate transverse momentum regions, where (in the absence of strong fragmentation effects) we expect to achieve the best outcome. In the $80\;GeV<M_{\gamma\gamma}<350\;GeV$ range, the higher order pQCD effects are larger, hence our results are generally lower than the data.

At the $\Delta\Phi_{\gamma\gamma}\leq\pi/2$ or $p_{t,\gamma\gamma}>M_{\gamma\gamma}$ domain, the contribution of the fragmentation becomes utterly non-negligible, as it can be seen in the panels (a)-(e) of the figure 5, as well as in the panels (a), (d) and (g) of the figures 6, where the predictions of our simplistic framework are clearly insufficient to describe the experimental data from the D0 and CDF collaborations D013 ; CDF11 ; CDF13 (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 the our framework. Moreover, in the figure 6, the reader can also find the IPPP production rates as the functions of $y_{\gamma\gamma}$ and $z_{\gamma\gamma}$ parameters. The symmetric form of the $y_{\gamma\gamma}$ distributions are due to the fact that the $q^{*}(\bar{q}*)+g^{*}\to\gamma+\gamma+q(\bar{q})$ and $g^{*}+q^{*}(\bar{q}^{*})\to\gamma+\gamma+q(\bar{q})$ are the $2\to 3$ asymmetric processes, so their summation becomes symmetric around $y_{\gamma\gamma}=0$. Also, the $q^{*}(\bar{q}^{*})+g\to\gamma+\gamma+q(\bar{q})$ sub-process causes an interesting effect on the $z_{\gamma\gamma}$ parameter distributions, by adding a ”shoulder” which can be roughly detected in the CDF13 data as well (which we name it MAK shoulders). In the panels (f) and (g) of the figure 5 and the panels (a), (b) and (c) of the figure 6, we have compared our results to the experimental data, regarding the $\Delta\Phi_{\gamma\gamma}$ dependency of the IPPP production rates. The low-$\Delta\phi_{\gamma\gamma}$ tail can be clearly seen here.

It is interesting to note that in these relatively low CM energies, the LO-MRW framework performs much better with respect to other schemes, specially in the D013 data. Additionally, one finds out that in the higher transverse momenta, i.e. where the higher-order pQCD effects become important, the KMR results behave similar to the NLO-MRW rather than its 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. Generally speaking, in the low-$\Delta\Phi_{\gamma\gamma}$ domain, the effect of the fragmentation and the higher-order contributions 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 are larger compared to the NLO-MRW ones. Because of the different AOC implementations on these prescriptions, the predictions get quite separated in their respective regions. We should point out that the MRW sub-processes in the above calculations behave roughly the same as those of KMR. However, some discrepancy in case of NLO-MRW is observed. There is not a sizable difference between the above schemes which use various AOC and cut off in the differential cross sections, which is in agreement to reference re . As one should expect, this is not the case on fragmentation domain.

V.2 $E_{CM}=7\;TeV$

We have performed another set of calculations, with the CM energy of $7\;TeV$, in accordance with the specifications of the ATLAS and the CMS reports, i.e. the references CMS11 ; CMS14 ; ATLAS10 ; ATLAS13 . Therefore, in the figures 7 through 9, the reader is presented with comprehensive comparisons regarding the dependency of the differential total cross-section of the IPPP production, as the functions of $p_{t,\gamma\gamma}$, $M_{\gamma\gamma}$, $\Delta\phi_{\gamma\gamma}$ and $cos\theta^{*}_{\gamma\gamma}$. The general behavior of the results are the same as in the $E_{CM}=1.96\;TeV$ case, with the exception that the contributions coming from the $q^{*}(\bar{q}^{*})+g^{*}\to\gamma+\gamma+q(\bar{q})$ sub-process is visibly greater than that of the $q^{*}+\bar{q}^{*}\to\gamma+\gamma$ sub-process, with some exceptions for the LO-MRW case (we do not present their data in order not to crude these figures). This happens, because the shares of the gluons and the sea-quarks become important, with the increase of the CM energy.

As a result of increasing the CM energy, the Guillet shoulder phenomena can be seen more clearly in our $7\;TeV$ calculations. Additionally, as in the $1.96\;TeV$ case, in the regions where the fragmentation effects become non-negligible, the low-tails of mass and the low-$\Delta\Phi_{\gamma\gamma}$ tails are visible and generally followed by the separation of the KMR, LO-MRW and NLO-MRW results. Generally speaking, in these areas the LO-MRW and the NLO-MRW results 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 of the IPPP production in the CMS14 data. So, we perform our calculations for $q^{*}(\bar{q}^{*})+g$ and $g^{*}+q^{*}(\bar{q}^{*})$ configurations of the $q^{*}(\bar{q}^{*})+g^{*}\to\gamma+\gamma+q(\bar{q})$ sub-process, separately. As a result of this asymmetric constraint, the production of the back-to-back photons are being suppressed in the transverse plane CMS14 . Therefore, the higher order contributions, e.g. the quark-gluon scattering, become more important. In the CMS14 measurement, despite our expectations, the quark annihilation has a significant contribution in the LO-MRW, while the KMR and the NLO-MRW results have the ”expected” behavior. Nevertheless, only the KMR prescription is somehow successful in describing the experimental data in ”these kinematic regions”.

Unlike the CMS14 measurement, the ATLAS12 experiment utilizes symmetric constraints ATLAS13 . Therefore as one expects, the lower order pQCD contributions should be enhanced these data. So the NLO-MRW should perform a better behavior for the prediction of the experimental outcome, see for example the figure 7. Therefore, we may conclude that by including suitable higher-order contributions, in accordance with the experiment conditions, i.e. the imposed kinematics constraints, the predictions of the NLO-MRW framework becomes better 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 Carlo event generator as well as the pQCD which were introduced in the introduction RESBOS ; DIPHOX ; NNLO ; sherpa . Furthermore, we make a careful scrutiny of our calculation by dividing our results in different frameworks (i.e. KMR, LO-MRW and NLO-MRW) to that of the corresponding experimental data. This can highlight the difference of our works over the experiments. The outcome of above comparisons are demonstrated in the figure 10 through figure 16. At the lower panels of these figures the red circles show the KMR ratio and the black triangles and the 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 compared with the SHERPA and the NNLO (or $2\gamma$NNLO) pQCD CDF13 ; D013 , as well as CDF13 and D013 data. It is observed that our KMR approach predicts the differential cross section data as a function of corresponding variable very well, but in the region where the fragmentation is not important. While in the regions where the fragmentation and higher order pQCD become dominant (for example, low $\Delta\phi$ region in the panel (g) of the figure 10), the SHERPA and NNLO methods produce better results. At these regions without considering higher order contributions 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 10 and 11 and the panels (b) and (h) of the figure 12. However, the SHERPA and $2\gamma$NNLO calculations are well behaved in whole regions and all of the panels.

The figure 13 compares the D010 data D010 with our results, as well as the RESBOS and DIPHOX calculations D010 . One can clearly observe that the KMR $k_{t}$-factorization approach predicts the acceptable result with respect to other theoretical methods that presented in this 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 high value of $M_{\gamma\gamma}$, the Monte Carlo calculation is more successful. The remaining panels of this figure which is related to the CMS collaborations CMS11 ; CMS14 , show that our results predict the data with higher accuracy with respect to those of DIPHOX calculation CMS11 ; CMS14 ; DIPHOX .

In the panels (a), (d) and (g) of the figure 15 similar to the figure 14, the KMR approach behaves as before, but the results of DIPHOX calculation are closer to the data, since the rapidity was increased. In rest of the panels of the figures 15 and the panels (a) to (e) of 16, our results are examined against DIPHOX and $2\gamma$NNLO, and their behavior are much similar. However, our KMR or MRW, as it was discussed before, are closer to the $2\gamma$NNLO calculation.

In order to check our results against those of reggization methods, saleev ; saleev1 , and also to give the uncertainty of present calculation (by multiplying the factorization scale by half and 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 saleev1 , except the small $p_{t,\gamma\gamma}$ and the large $\Delta\Phi_{\gamma\gamma}$ regions, where their results are off the data. However our uncertainty bounds are reasonably cover the data as well as the reggization and $2\gamma$NNLO calculations (see panels (a), (e), (f) and (g)). It is interesting that the $2\gamma$NNLO method, in which the fragmentation contribution has been also taken into account, is off the experimental data while 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 also modified our UPDF according to e.g. the reference LIP2011 and find less than 15 per cent effect, which still keeps our result inside the uncertainty bounds. But, as we pointed out in the introduction, the UPDF should satisfy the condition given in the equation (1), so any changes in the UPDF certainly affect the original PDF definitions or it may be in contrast with the original definition of UPDF KIMBER . 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 Martin et al WattWZ .

These comparisons show that one of the places in which the effect of $k_{t}$ factorization framework obviously becomes important, with respect to its counterpart, would be the regions of large $\Delta\phi_{\gamma\gamma}$ and small $p_{t,\gamma\gamma}$. In these two regions, the predictions of collinear matrix element method are overestimated the data, as it could be seen by DIPHOX and $2\gamma$NNLO calculations. However, the prediction of RESBOS, due to the NLL (next leading logarithmic) resumption of soft initial state gluon radiation, is better than those of DIPHOX and $2\gamma$NNLO. But in our methods, the natural gluon resumption automatically is done in all orders SPL , 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 the MRW prescriptions and compared our results to the existing experimental data from the D0, CDF, CMS and ATLAS collaborations. According to our discussions and observations in the present work, the LO-MRW approach is the best suitable scheme for the prediction of the IPPP production rates in the lower CM energies, since this approach can predict the experimental data within the regions where the fragmentation effects become important, without any additional manipulations in our calculations. In contrary, the LO-MRW formalism is not perfect for the higher CM energies in these kinematics. While the KMR approach is able to accurately predict the experimental data in the 7 TeV center of mass energy. The main difference between these approaches arises due to the implementation of different visualizations of the AOC, which can be seen, specially in the regions where the fragmentation 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 behave well-separated. On the other hand it was shown that the application of different AOC and cut off, using the KMR re and MRW prescriptions do not show serious discrepancies and a qualitative agreements between different schemes can be achieved.

We realized that the Guillet shoulder phenomena is more sensitive to the low-$M_{\gamma\gamma}$ variations, compared to the low-$\Delta\phi_{\gamma\gamma}$ and the $cos\theta^{*}_{\gamma\gamma}$ regions. Although, our predictions via our simplistic calculations describe the experimental data well, one can improve the precision of these results by including higher-order contributions and taking into account the fragmentation effects. We hope that in our future works, we can investigate these phenomena. A comparison was also made with the different theoretical methods such as DIPHOX , $2\gamma$NNLO, RESBOS and SHERPA and an overall agreement was found.

It was shown that the possible double-counting can be removed by considering the phase space cuts as well as modification in the UPDF. However by imposing the uncertainty of the factorization 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 investigate this approximation in our future works KUTAK1 ; KUTAK2 ; KUTAK3 ; KUTAK4 ; LIP2016 as well as the gauge invariance and possible double counting.

Acknowledgements.

MM would like to acknowledge the Research Council of University of Tehran and the Institute for Research and Planning in Higher Education for the grants provided for him. RAN and MRM sincerely thank M. Kimber and A. Lipatov for their valuable discussions and comments.

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-cone constraints. The isolated-cone is responsible for distinguishing the ”non-prompt decay photons” from the prompt-photons. This constraint requires the transverse energy $E_{t}^{iso}$ (in a cone with the angular radius $R=\sqrt{(\eta-\eta^{\gamma})\;+\;(\phi-\phi_{\gamma})}\;\;<\;0.4$) to be less than a few GeV according to each experiment. To avoid, the overlap between the two photons, the separation-cone constraint is imposed as,

[TABLE]

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, the different settings probe the various regions of pQCD. In what follows, we briefly present the reader with the specifications of the measurements that we intend to analyze throughout this work in each experiments.

A.1 The D0 collaboration

The D0 experiment was performed at the center of mass energy of $1.960$ TeV. The two sets of D0 data, related to the IPPP production were investigated in the references D010 ; D013 , i.e. D010 and D013, respectively. The constraints in the D010 experiment D010 are $p_{t}$ $>$ 20, 21  GeV (the transverse momentum of outgoing photons), $|\eta|<0.9$ (the pseudo-rapidity) and $M_{\gamma\gamma}\;>\;p_{t,\gamma\gamma}$ which are applied to the IPPP production D010 , suppressing the fragmentation effects and some higher order contributions. In the D013 report D013 , the constraints are $p_{t}$ $>$ 17, 18 $GeV$ and $|\eta|<0.9$. Also three regions are probed in the D013 report, i.e the region I with the $\Delta\phi_{\gamma\gamma}\geq{\pi\over 2}$ constraint which is suitable for the study of non-higher-order pQCD. By applying the $\Delta\phi_{\gamma\gamma}<{\pi\over 2}$ constraint in the region II, the fragmentation effects become important and the last region is without any extra constraint on $\Delta\phi_{\gamma\gamma}$ D010 ; D013 .

A.2 The CDF collaboration

Similar to the D0 collaboration, the CDF experiment provides the two sets of data, that are related to the IPPP production CDF11 ; CDF13 at the center of mass energy of $1.960$ TeV. The constraints in the CDF13 CDF13 are the same as the CDF11 CDF11 reports. However, the luminosity is improved in the new sets of data (CDF13) CDF13 . These constraints are $p_{t}$ $>$ 15, 17 GeV and $|\eta|<1$. The $CDF$ collaboration explored three regions in their works: the region I, via applying the $p_{t,\gamma\gamma}>M_{\gamma\gamma}$ constraint, suitable for the study of higher-order pQCD. The region II, by applying the $p_{t,\gamma\gamma}<M_{\gamma\gamma}$ constraint, which undermines the fragmentation effects and emphasizes on the quark-antiquark annihilation. The last region is without any extra constrains CDF11 ; CDF13 .

A.3 The CMS collaboration

The CMS collaboration is presented at the 7 TeV CM energy CMS11 ; CMS14 . The constraints in the CMS12 experiment CMS11 are $p_{t}$ $>$ 20, 23 GeV and $|\eta|<2.5$, excluding the $1.44<|\eta|<1.57$ region. Also, the region $|\eta|<1.44$ is separately canalized. In the reference CMS14 ($CMS14$) the asymmetric transverse momentum ($p_{t}$ $>$ 40, 25 GeV ) for the IPPP production is selected in the regions of $1.57<|\eta|<2.5$ and $|\eta|<1.44$. As a result, the higher order pQCD contributions become dominant in these experiments CMS11 ; CMS14 .

A.4 The ATLAS collaboration

Another experimental data at the 7 TeV CM energy is provided by the ATLAS collaboration ATLAS10 ; ATLAS13 (ATLAS12 and ATLAS13). In the reference ATLAS10 , the data is sorted according to $p_{t}$ $>$16 (16) GeV and $|\eta|<2.5$, excluding the $1.37<|\eta|<1.52$ region. Similarly, ATLAS13 ATLAS13 has the same pseudo-rapidity region, although the transverse momentum threshold is changed to $P_{t}$ $>$25 (22) GeV.