A validity check of the KATIE parton level event generator in the $k_t$-factorization and collinear frameworks

In the present paper, we check and study the validity of the \textsc{KaTie} parton level event generator, by calculating the inclusive electron-proton (ep) dijet and the Proton-proton (p-p) Drell-Yan electron-pair productions differential cross sections in the $k_t$- and collinear factorization frameworks. The Martin-Ryskin-Watt (MRW) unintegrated parton distribution functions (UPDFs) are used as the input UPDFs. The results are compared with those of ZEUS ep inclusive dijet and ATLAS p-p Drell-Yan electron-pair productions, experimental data. The \textsc{KaTie} parton level event generator can directly calculate the cross sections in the $k_t$-factorization framework. It is noticed that the lab to the Breit transformation in this generator is not correctly implemented by its author, so the produced output does not cover the ep ZEUS experimental data in which the mentioned transformation is applied. By fixing the above transformation in the \textsc{KaTie} generator code, we could appropriately produce the ep inclusive dijet differential cross section, in comparison with those of ZEUS data. It is also shown that the MRW at the NLO level, with the angular ordering constraint can successfully predict the ATLAS p-p Drell-Yan data. Finally, as it is expected, we conclude that the $k_t$-factorization is an appropriate tool for the small longitudinal parton momenta and high center of mass energies, with respect to the collinear one.


I. INTRODUCTION
Monte Carlo event generators are essential tools for experimentalists and theoreticians who attempt to simulate hadronic collisions. These generators are mostly based on the assumption of collinear factorization, in which the partons are assumed to behave collinearly in the hadrons. However, generally, this assumption is not correct and parton should be allowed to have transverse momenta (k t ). Hence the k t -factorization [1,2] comes into play, in which the transverse momenta of partons can be considered as an extra variable into the hard interaction calculations. This framework allows to calculate the differential cross sections for different processes by using unintegrated parton distribution functions (UPDFs).
One of the parton level event generators that can calculate the cross sections for arbitrary final state particles, in the above framework, is the KaTie [3]. The program of this generator can produce Les Houches Event File (LHEF) [4], where thereafter, this LHEF file can be passed to CASCADE3 [5] event generator to perform parton showering models. The KaTie due to its simplicity has this capability to put into use for calculating different differential cross sections with the various UPDFs. So, it is a suitable choice to perform phenomenological studies of these distributions, as well. Therefore, in order to obtain reliable results, one task is to test this generator for different processes and comparing the results to the experimental data. Hence, checking the validity of this parton level event generator at this early stage is crucially important for future studies of the k t -factorization framework.
The k t -factorization framework considers a more realistic picture of partons inside proton with respect to collinear factorization, and hence it is expected to give a better description of the experimental data, especially in high energies (much greater than few TeV) . Although, due to complications which arise when the transverse momentum of parton come into play, obtaining a fully transverse momentum dependent evolution equation is a challenging task.
For example the CCFM evolution equation [6][7][8][9] which is based on the angular ordering of soft gluon emissions is not defined for all quark flavors. However, recently, another approach which is based on the parton branching method [10,11], allows one to obtain the UPDFs naturally by solving the DGLAP evolution equations [12][13][14]. With this method one can also obtain the UPDFs for both quarks and gluon and it is shown to have successful predictions [15,16].
Recently, the KaTie generator is extended its application to the electron(positron)proton (ep) collision [35]. Therefore, for the first time we calculated the inclusive dijet production differential cross sections in the k t -factorization framework and compared our results to those of ZEUS collaboration data [36,37]. It was observed that our results [36] cannot describe the data in a satisfying way, and hence we found that the lab to Breit transformation was not appropriately implemented in this parton level generator.
In the present work, it is intended to study the prediction of the MRW formalism at the leading (LO) and next-to-leading order (NLO) levels for two different processes, i.e. the electon(positron)-proton (ep) inclusive dijet and proton-proton (p-p) Drell-Yan differential cross sections which can be compared with the ZEUS [37] and ATLAS [38] collaborations data, respectively. The KaTie parton level event generator is used, in order to check the validity of this generator by correcting the Breit transformation and also investigating the MRW method.
So, the paper is organized as follows: In the section II the theoretical framework is presented which includes the KaTie parton level event generator and the MRW formalism.
The results and discussions are given in the section III and the section IV is devoted to the conclusions.

II. THE THEORETICAL FRAMEWORK
In this section, it is intended to first explain the KaTie parton level event generator, and then discuss about the UPDFs which are used to obtain the differential cross sections for the ep inclusive dijet and the p-p Drell-Yan processes to be compared with those of the ZEUS and ATLAS collaboration data, respectively.

A. The KATIE Parton Level Event Generator
This parton level event generator is mainly composed of four parts, i.e, the input file, the optimization stage, the event generation, and the histograms creation. In the input file of this histogram, one should write information about the sub-processes, the factorization and renormalization scales, the experimental cuts, the off-shellness or on-shellness of partons, the PDFs, the UPDFs, the order of non-QCD couplings and the energies of incoming particles. Furthermore, it is also possible to calculate the multi-parton scattering, instead of the single one. It is also worth to mention, that one can use desired UPDFs by providing grid files in columns of ln(x), ln(k 2 t ), ln(µ 2 ) and f a (x, k 2 t , µ 2 ) for each parton flavor. Additionally, it is possible to directly use UPDFs grid files of TMDLIB [39].
After providing an input file, an optimization of all sub-processes and event generation should be performed, which are not of interest for the end-user. Finally, one can obtain differential cross sections by a FORTRAN file with the name "create eventfile.f90". In this file, the histograms of interests are developed and the program can read the recorded events in a file which is called raw file and create the distributions of interests. Additionally, in this part one can produce LHEF file, which can itself with the help of CASCADE [40] to make showering and hadronisation. Therefore, this generator is a beneficial phenomenological tool to investigate the k tfactorization framework, and different UPDFs models. In the next subsection, we give an overview of the MRW UPDF models at the LO and NLO levels which will be used through this report.

B. The MRW UPDFs
The MRW UPDFs at the leading order level (LO-MRW UPDFs) are developed by Martin, et al [17,18], and are based on the DGLAP evolution equations. In this framework, by choosing the factorization scale of the DGLAP evolution equation to be the transverse momentum of the parton, a parton evolves collinearly in the proton, until it reaches the last evolution step. In this step, the parton which has the transverse momentum k t , emits a parton and evolves to the factorization scale µ without any further real emissions. Therefore, the LO-MRW UPDFs are defined as follows: where f LO b ( x z , k 2 t ) in the above equation is the momentum weighted parton density at the leading order (LO), and can be either for quark (anti-quark) or gluon, respectively. Additionally, T a (k 2 t , µ 2 ) is the Sudakov form factor and resums no real emissions probability from the scale k 2 t to µ 2 and which is: It should also be noted that the Sudakov form factor is defined for the k 2 t ≤ µ 2 and for the k t > µ, T a → 1.
One should note that the LO-MRW UPDFs, in the equations (1) and (2), which can be defined for all quark flavors f q (x, k 2 t , µ 2 ) and gluon, f g (x, k 2 t , µ 2 ), are divergent at z → 1 and ξ → 1 for the diagonal terms, i.e. P LO qq (z), P LO gg (z), P LO qq (ξ), P LO gg (ξ). To avoid such divergences which happen as a result of soft gluon emissions, one can either use the angular ordering of the soft gluon emissions or the strong ordering of partons, along the evolution ladder, to put a cutoff on z and ξ. The authors of this model, adopted the angular ordering of the soft gluon emissions, hence we use the same cutoff here, i.e,: The z max and ξ max are the maximum allowed values of z and ξ. Therefore one should put a Heaviside step functions for the diagonal terms of the equations (1) and (2), i.e, Θ(z M ax − z) and Θ(ξ M ax − ξ), respectively.
It should be noted that the LO-MRW formalism is limited to the k t ≥ µ 0 , and hence for defining such distribution in the limit k t < µ 0 , the density of the partons are assumed to be constant at fixed x and µ 2 in the reference [17,18], and satisfy the normalization condition.
Therefore one obtains the density of partons in the limit k t < µ 0 , as: 1 where in the above equation we fix µ 0 = 1 GeV .
The LO-MRW formalism is also extended to the next-to-leading order level, (NLO-MRW), in which instead of the scale k 2 t , the virtuality of the parton along the evolution ladder is . Additionally, the PDFs, the strong ordering coupling constant, and the splitting functions are at the NLO level. However, it is shown in [17] with the use of NLO strong ordering coupling constant along with the NLO-PDFs one can obtain results close to those use in the fully NLO case. For simplicity the first form is used, where the PDFs and strong ordering coupling constant are at the NLO level, but the splitting functions are at the LO level. Therefore one can write the NLO-MRW UPDFs as: where, in the above equation, we implicitly insert, once again, a Heaviside step functions to avoid the soft gluon divergence. There is also an additional cutoff Θ(µ 2 − k 2 ) in the above equation, which limits the virtuality in the region of k 2 < µ 2 . Because, the Sudakov form factor in this model is:

III. RESULTS AND DISCUSSION
In this section, we intend to calculate the differential cross sections in the k t -factorization frameworks with the LO and NLO-MRW UPDFs by utilizing the KaTie parton level event generator. In this calculation, we calculate and provide the grid files of our UPDFs for each quark flavor and gluon, using the MMHT2014lo68cl and MMHT2014nlo68cl PDFs [41,42] of LHAPDF6 library [43], as an input PDFs for our model in the equations (1) and (2). It should also be noted that the UPDFs are divided by k 2 t , due to the definition of differential cross section in the KaTie generator.

A. KATIE results for the ZEUS inclusive dijet production data
The ZEUS collaboration data [37] are collected from the collisions of the protons with energy 920 GeV and electrons or positrons with energy 27.5 GeV . The photon virtuality Q 2 is between 125 GeV < Q 2 < 20000 GeV . The minimum transverse energy of the two jets in the Breit frame should be larger than E T,B > 8 GeV and the invariant mass of the two jets required to be greater than M jj > 20 GeV . The inelasticity is between 0.2 < y < 0.6. Additionally, at least two jets are required to have the pseudo-rapidity in the range −1 < η lab < 2.5, in the lab frame.
We can simply perform such calculation with the help of the KaTie parton level generator [3]. For calculation of the dijet production, we considered the sub-processes γ * + q → q + g and γ * +g → q +q with the LO-MRW and NLO-MRW UPDFs. We also set the factorization and the renormalization scales µ F = µ R = Q, with n f = 5.
In the left panel of figure 1, the differential cross section with respect to the mean transverse jet energy in the Breit frame is obtained and compared to the experimental data.
As it can be seen in this figure, the result of the k t -factorization with the LO-MRW and NLO-MRW UPDFs are strangely off the data. Because, in the previous works, It was shown that the MRW UPDFs can produce the corresponding data reasonably [26][27][28][29][30][31][32][33][34], so to be sure what makes our calculation to get worse, we evaluated the differential cross section within the collinear factorization framework, using the MMHT2014nlo68cl PDFs. The prediction of the collinear factorization, as depicted in the left panel of figure 1, convinces us that there should be something wrong with this parton level event generator [3].
In order to diagnose the reason for such results, we decided to visualize the three dimensional plot with respect to the momenta p x , p y , and p z of 100 events generated by KaTie in the Briet frame, see the figure 2. In this figure, the three dimensional momenta of initial quark, virtual photon, final quark and gluon of the sub-process, i.e., γ * + q → q + g, in the Breit frame within the collinear factorization are shown. In the collinear factorization framework, the virtual photon should collide head to head along the z direction with the initial parton, according to the Breit frame transformation, which does not happen as expected.
According to the definition, in the Breit frame, the virtual photon has no energy component and only has momentum along the −z direction, q = (0, 0, 0, −Q), and collinearly scatter head to head with quark in the +z direction [37] which has momentum xP (P is the proton momentum). Therefore, it is obvious that there is problem due to the wrong choice of lab to Breit frame transformation and it should be fixed [44].
This lab to Breit transformation can be simply fixed for example according to the reference [45] (note that the minimum requirement is q + 2xP = 0). If we make such corrections in the implementation of the KaTie, one can see that events are now behaving according to our expectation, see the figure 3. Now, the virtual photon has a head to head collision with the initial quark along the z direction. is mostly due to the large UPDFs at large parton transverse momenta, see the reference [15] for detail. Additionally, the NLO-MRW UPDFs prediction, underestimate the data due to this fact that the choice of k 2 as a scale in this limit of energy, leads to the much larger scale than k 2 t in the LO-MRW formalism. As a result of this fact and also the additional cutoff Θ(µ 2 − k 2 ) in this method, one could expect worse estimation of cross section for the NLO-MRW UPDFs at this range of energy, see the top panels of figure 4 to gain an insight about these two different UPDFs for the up quark and the gluon at x = 0.3 and µ 2 = 400. However one should note that the k t -factorization works much more better at small x ≤ 0.01 and very high center of mass energy ≥ 1 T eV which could be very useful for saving computer time [31] with respect to NNLO collinear calculation in future LHeC [46].

B. KATIE results for the ATLAS p-p Drell-Yan electron-pair production data
In this section we investigate the normalized differential cross section with respect to the transverse momentum of the Drell-Yan electron-pair production in the k t -factorization framework in comparison with the ATLAS collaboration data [38]. The data is related to the collision of proton-proton with center of mass energy 13 T eV . The electron (positron) To make a proper comparison, the collinear calculation with the KaTie for the above two processes is also plotted in the figure 5 which gives worst results with respect to the k t -factorization formalisms, i.e., LO and NLO-MRW. One should note that the raise of the differential cross section at small p e − e + T is due to incomplete soft gluon resummation technique which should be used to make QCD predictions at low p e − e + Finally, in contrast to our dijet calculations, one can find that the application of k tfactorization in the KaTie parton level event generator, produce the differential cross section consistent with the Drell-Yan experimental data.

IV. CONCLUSION
In this work, we investigated the KaTie parton level event generator for calculating the differential cross section of ep inclusive dijet and p-p Drell-Yan productions within the k t -factorization framework in two energy ranges of the ZEUS and ATLAS collaboration data, respectively. In order to perform such calculation, we used the LO-MRW and NLO-MRW UPDFs with angular ordering constraint. For the ep dijet differential cross section, it is noticed that the lab to Breit transformation is performed not correctly, in this parton level event generator, and should be fixed. After correcting this problem, we calculated the differential cross section and obtained acceptable results. We could obtain satisfactory results, when the p-p Drell-Yan electron-pair production differential cross section in the above framework was calculated and compared with those of ATLAS collaboration data.
We understood that the NLO-MRW although underestimates the ZEUS collaboration data, but can in fact predict the data of the ATLAS accurately. However, we observed that the LO-MRW overshoot the data in both the ZEUS and ATLAS collaboration data at large transverse momenta. It was found, that the k t -factorization is very useful tools for the small x ≤ 0.01 and very high center of mass energy ≥ 1 T eV and could save much computer time [31] with respect to NNLO collinear calculation, especially in the future LHeC [46] and the present p-p LHC.