$KMR$ $k_t$-factorization procedure for the description of the $LHCb$ forward hadron-hadron $Z^0$ production at $\sqrt{s}=13\;TeV$

Quit recently, two sets of new experimental data from the $LHCb$ and the $CMS$ collaborations have been published, concerning the production of the $Z^0$ vector boson in hadron-hadron collisions with the center-of-mass energy $E_{CM}= \sqrt{s}=13\;TeV$. On the other hand, in our recent work, we have conducted a set of $NLO$ calculations for the production of the electroweak gauge vector bosons, utilizing the unintegrated parton distribution functions ($UPDF$) in the frameworks of $Kimber$-$Martin$-$Ryskin$ ($KMR$) or $Martin$-$Ryskin$-$Watt$ ($MRW$) and the $k_t$-factorization formalism, concluding that the results of the $KMR$ scheme are arguably better in describing the existing experimental data, coming from $D0$, $CDF$, $CMS$ and $ATLAS$ collaborations. In the present work, we intend to follow the same $NLO$ formalism and calculate the rate of the production of the $Z^0$ vector boson, utilizing the $UPDF$ of $KMR$ within the dynamics of the recent data. It will be shown that our results are in good agreement with the new measurements of the $LHCb$ and the $CMS$ collaborations.


I. INTRODUCTION
Traditionally, the production of the electroweak gauge vector bosons is considered as a benchmark for understanding the dynamics of the strong and the electroweak interactions in the Standard Model. It is also an important test to assess the validity of collider data.
Many collaborations have reported numerous sets of measurements, probing different events in variant dynamical regions, in direct or indirect relation with such processes, to count a few see the references [1][2][3][4][5][6][7][8][9][10]. Among the most recent of these reports are the measurements of the production of Z 0 bosons at the LHCb and CM S collaborations, for proton-proton collisions at the LHC for √ s = 13T eV , with different kinematical regions [11,12]. The LHCb data are in the forward pseudorapidity region (2 < |η| < 4.5) while the CM S measurements are in th central domain (0 < |η| < 2.4).
In our previous work [13], we have successfully utilized the transverse momentum dependent (T M D) unintegrated parton distribution functions (U P DF ) of the k t -factorization (the references [14,15]), namely the Kimber-M artin-Ryskin (KM R) and M artin-Ryskin-W att (M RW ) formalisms in the leading order (LO) and the next-to-leading order (N LO) to calculate the inclusive production of the W ± and the Z 0 gauge vector bosons, in the proton-proton and the proton-antiproton inelastic collisions In order to increase the precision of the calculations, we have used a complete set of 2 → 3 N LO partonic sub-processes, i.e.
where V represents the produced gauge vector boson. k i and p i , i = 1, 2 are the 4-momenta of the incoming and the out-going partons. The results underwent comprehensive and rather lengthy comparisons and it was concluded that the calculations in the KM R formalism are more successful in describing the existing experimental data (with the center-of-mass energies of 1.8 and 8 TeV) from the D0, CDF , AT LAS and CM S collaborations [8,10,[16][17][18][19][20][21][22]. The success of the KM R scheme (despite being of the LO and suffering from some misalignment with its theory of origin, i.e. the Dokshitzer-Gribov-Lipatov-Altarelli-P arisi (DGLAP ) evolution equations, [23][24][25][26]) can be traced back to the particular physical constraints that rule its kinematics. To find extensive discussions regarding the structure and the applications of the U P DF of k t -factorization, the reader may refer to the references [27][28][29][30][31][32][33][34].
Meanwhile, arriving the new data from the LHCb and CM S collaborations, the references [11,12], gives rise to the necessity of repeating our calculations at the E CM = 13 T eV . This In the present work, we intend to calculate the transverse momentum and the rapidity distributions of the cross-section of production of the Z 0 boson using our N LO level diagrams (from the reference [13]) and the U P DF of the KM R formalism. The U P DF will be prepared using the P DF of M M HT 2014 − LO, [36]. In the following section, the reader will be presented with a brief introduction to the N LO ⊗ LO framework (i.e. N LO QCD matrix elements and LO U P DF ) that is utilized to perform these computations. The section II also includes the main description of the KM R formalism in the k t -factorization procedure. Finally, the section III is devoted to results, discussions and a thoroughgoing conclusion.

II. N LO ⊗ LO FRAMEWORK, KM R U P DF AND NUMERICAL ANALYSIS
Generally speaking, the total cross-section for an inelastic collision between two hadrons (σ Hadron−Hadron ) can be expressed as a sum over all possible partonic cross-sections in every possible momentum configuration: In the equation (3), x i and k i,t respectfully represent the longitudinal fraction and the are the density functions of the i th parton. The second scale, µ i , are the ultra-violet cutoffs related to the virtuality of the exchanged particle (or particles) during the inelastic scattering.σ a 1 a 2 are the partonic cross-sections of the given particles. For the production of the Z 0 boson, the equation (3) comes down to (for a detailed description see the reference [13]) y i are the rapidities of the produced particles (since y i η i in the infinite momentum frame, i.e. p 2 i m 2 i ). ϕ i are the azimuthal angles of the incoming and the out-going partons at the partonic cross-sections. |M| 2 represent the matrix elements of the partonic sub-processes in the given configurations. The reader can find a number of comprehensive discussions over the means and the methods of deriving analytical prescriptions of these quantities in the references [13,[37][38][39][40]. s is the center of mass energy squared. Additionally, in the protonproton center of mass frame, one can utilize the following definitions for the kinematic variables: Defining the transverse mass of the produced particles, m i,t = m 2 i + p 2 i , we can write, Furthermore, the density functions of the incoming partons, f a (x, k 2 t , µ 2 ) (which represent the probability of finding a parton at the semi-hard process of the partonic scattering, with the longitudinal fraction x of the parent hadron, the transverse momentum k t and the hardscale µ) can be defined in the framework of k t -factorization, through the KM R formalism: The Sudakov form factor, T a (k 2 t , µ 2 ), factors over the virtual contributions from the LO DGLAP equations, by defining a virtual (loop) contributions as: with T a (µ 2 , µ 2 ) = 1. α S is the LO QCD running coupling constant, P One can easily confirm that since the U P DF of KM R quickly vanish in the k t µ domain, further domain have no contribution into our results. Also we limit the rapidity integrations to [−8, 8], since 0 ≤ x ≤ 1 and according to the equation (6), further domain has no contribution into our results. The choice of above hard scale is reasonable for the production of the Z bosons, as has been discussed in the reference [40].
Finally, we choose to define the density of the incoming partons in the non-perturbative region, i.e. k t < µ 0 with µ 0 = 1 GeV . This appears to be a natural choice, since (see the references [13,43])  (4) to construct the Z cross-sections in the framework of k t -factorization. One must note that the experimental data of the LHCb collaboration, [11], and the preliminary data of the CM S collaboration, [12], are produced in different dynamical setups; the LHCb data are in the forward rapidity region, 2 < |y Z | < 4.5, while CM S data are in a central rapidity sector, i.e. 0 < |y Z | < 2.4. We have imposed the same restrictions in our calculations.
Thus, in the figure 1 we present the reader with a comparison between the different contributions into the differential cross-sections of the production of Z 0 , (dσ Z /dp t ), as a function of the transverse momentum (p t ) of the produced particles, in the KM R scheme.
One readily notices that the contributions from the g * + g * → Z 0 + q +q (the so-called gluon-gluon fusion process) dominate the the production. The share of other production vertices is small (but not entirely negligible) compared to these main contributions. This is to extent different from our observations in the smaller center-of-mass energies (see the section V of the reference [13]). Also, differential cross-sections are considerably larger at the central rapidity region compared to the results in the forward sector.
The total differential cross-section of the production of Z 0 vector boson is calculated within the figure 2, as the sum of the constituting partonic sub-processes (see the relation (2)). The calculations are carried out for the center-of-mass energy E CM = 13 T eV and plotted as a function of the transverse momentum of the produced particle. In the panels means of manipulating the hard-scale, µ, of the U P DF by a factor of 2, since this is the only free parameter in our framework. Also, as expected for the both regions, the contributions from the g * + g * → Z 0 + q +q sub-process dominate, The figure 3 presents the differential cross-section of the production of Z 0 vector boson, Overall, it appears that our N LO ⊗ LO framework is generally successful in describing the corresponding experimental measurements in the explored energy range. This success if by part owed to the U P DF of KM R, which as an effective model, has been very successful in producing a realistic theory in order to describe the experiment, see the references [13,[27][28][29][30][31][32][33][34]. One however should note that having a semi-successful prediction from the framework of k t -factorization by itself is a success, since our calculations utilizing these U P DF have inherently a considerably larger error compared to those from the N N LO QCD or even the N LO QCD, presented here by the relatively large uncertainty region. This is because we are incorporating the single-scaled P DF (with their already included uncertainties) to form double-scaled U P DF with additional approximations and further uncertainties. Being able to provide predictions with a desirable accuracy would require a thorough universal fit for these frameworks, see the reference [43]. Nevertheless, the k t -factorization framework, despite its simplicity and its computational advantages, see the reference [34,43], can provide us with a valuable insight regarding the transverse momentum dependency of various highenergy QCD events.
In summary, throughout the present work, we have calculated the production rate of the Z 0 gauge vector boson in the framework of k t -factorization, using a N LO ⊗ LO framework and the U P DF of the KM R formalism. The calculations have been compared with the experimental data of the LHCb and the CM S collaborations. Our calculation, within its uncertainty bounds, are in good agreement with the experimental measurements. We also reconfirm that the KM R prescription, despite its theoretical disadvantages and its simplistic computational approach, has a remarkable behavior toward describing the experiment.