LHC production of forward-center and forward-forward di-jets in the $k_t$-factorization $unintegrated$ parton distribution frameworks

The present work is devoted to study the high-energy $QCD$ events, such as the di-jet productions from proton-proton inelastic collisions at the $LHC$ in the forward-center and the forward-forward configurations, using the $unintegrated$ parton distribution functions ($UPDF$) in the $k_t$-factorization framework. The $UPDF$ of $Kimber$ et. al. ($KMR$) and $Martin$ et.al. ($MRW$) are generated in the leading order ($LO$) and next-to-leading order ($NLO$), using the $Harland-Lang$ et al. ($MMHT2014$) $PDF$ libraries. While working in the forward-center and the forward-forward rapidity sectors, one can probe the parton densities at very low longitudinal momentum fractions ($x$). Therefore, such a computation can provide a valuable test-field for these $UPDF$. We find very good agreement with the corresponding di-jet production data available from $LHC$ experiments. On the other hand, as we have also stated in our previous works, (i.e. the protons longitudinal and transverse structure function as well as hadron-hadron $LHC$ $W/Z$ production), the present calculations based on the $KMR$ prescriptions show a better agreement with the corresponding experimental data. This conclusion is achieved, due to the particular visualization of the angular ordering constraint ($AOC$), despite the fact that the $LO-MRW$ and the $NLO-MRW$ formalisms both employ better theoretical descriptions of the $Dokshitzer$-$Gribov$-$Lipatov$ -$Altarelli$-$Parisi$ ($DGLAP$) evolution equation, and hence are expected to produce better results. The form of the $AOC$ in the $KMR$ prescription automatically includes the re-summation of the higher-order $ln({1/x})$ type contributions, i.e. the $Balitski$-$Fadin$-$Kuraev$-$Lipatov$ ($BFKL$) logarithms, in the $LO$-$DGLAP$ evolution equation.


I. INTRODUCTION
Analyzing the raw data, which comes pouring out of the LHC, presents a challenge of considerable proportions, given that the dynamics of the true players in the hadronic inelastic collisions, i.e. partons, are shadowed bye the laws of strong interactions. However, to understand the nature of our universe, it is paramount to enlighten the behavior of these fundamental substances. Amazingly, an answer came a few decades ago, in the form of the Dokshitzer-Gribov-Lipatov-Altarelli-P arisi (DGLAP ) evolution equations, [1][2][3][4], g(x, Q 2 ) and q(x, Q 2 ) as the solutions of the DGLAP evolution equations, are single-scale parton density functions (P DF ), corresponding respectively to gluons and quarks. They depend on the fraction of the longitudinal momentum of parent hadron (x) and an ultra-violet cutoff (Q 2 ), which denotes the virtuality of the particle that is being exchanged throughout the inelastic scattering (IS). P α S represents the LO running coupling constant of the strong interaction, conventionally approximated as: , where n f is the number of involving flavors in the given strong interaction and Λ QCD is the QCD fundamental low energy scale. The value of the Λ QCD can be effectively extracted from experiment, around 300 M eV . The terms on the right-hand side of the equation (1), correspond to the real emission and the virtual contributions, respectively.
The main postulation in the DGLAP evolution equation, i.e. the strong ordering hypothesis, is to neglect the transverse momenta of the partons along the evolution ladder, and to sum over the α S ln(Q 2 ) contributions. One finds out that neglecting the contributions that come from this transverse dependency may harm the precision of the calculations, particularly in the high-energy processes and in the small-x region [10][11][12][13][14][15].  [26].
Mathematically speaking, solving the CCF M equation is rather difficult, usually possible with the help of Monte Carlo event generators, references [27,28]. On the other hand, the main feature of the CCF M equation, i.e. the AOC, can be used only for the gluon evolution and therefore, producing convincing quark contributions in this framework is only a recent development, see the references [29][30][31]. Given these complexities, Martin et al, employed the idea of last − step evolution along the k t -factorization framework, [5][6][7][8][9][10], and developed the Kimber-M artin-Ryskin (KM R) and the M artin-Ryskin-W att (M RW ) approaches [11,12]. Both of these formalisms are constructed around the solutions of the LO DGLAP evolution equations and modified with different visualizations of the angular ordering constraint. Although the unintegrated parton distribution functions (U P DF ) of M RW in the leading order (LO) and next-to-leading order (N LO) have been defined to improve the compatibility of the KM R approach with the theory of the LO DGLAP and extend it to a higher order QCD, the recent work suggests that the KM R framework is more successful (or at least as successful) in describing experimental data, see for example the references [32][33][34][35][36][37][38][39][40]. Nevertheless, to utter a rigid statement on this matter, further investigation is required.
One extraordinary test-ground for the U P DF of the k t -factorization is the probe of the forward-center and forward-forward rapidity sectors in the hadronic collisions, given that it involves the dynamics of the small-x region, e.g. x ∼ 10 −4 − 10 −5 , where the gluon density dominates. Since the decisive difference between the U P DF of KM R and M RW is in the different manifestations of the AOC, one could argue that working in such phenomenological setups could potentially exploit this diversity and unveil the true capacities of the presumed frameworks. For this propose, we have calculated the process of production of di-jets in the inelastic proton-proton collisions from the forward-center and the forward-forward rapidity regions, utilizing the U P DF of KM R and M RW in the LO and the N LO. Comparing these results with each other, and the results of the similar calculations in other frameworks, namely the linear and non-linear KS formalisms, [41][42][43][44][45], and with the experimental data from the CM S collaboration [46,47], would provide an excellent opportunity to study the strength and the weaknesses of the U P DF in the k t -factorization framework.
The outlook of this paper is as follows: In the section II we present a brief introduction to the framework of k t -factorization and develop the required prescriptions for the KM R and the M RW U P DF , stressing their key differences regarding the involvement of the AOC in their definitions. The U P DF will be prepared in their proper k t -factorization schemes The section III contains a comprehensive description over the utilities and the means for the calculation of the k t -dependent cross-section of the di-jets production in the p-p IS processes. The necessary numerical analysis will be presented in the section IV, after which a thorough conclusion will follow in section the V.

II. THE U P DF CALCULATIONS IN THE k t -FACTORIZATION FRAMEWORK
During a high energy hadronic collision, the involving partons, i.e. the partons that appear at the top of their respective evolution ladders, carry some inherently induced transverse momentum, as the remnant of the successive (an potentially infinite) number of evolution steps. When working within the framework of collinear factorization, such transverse momentum dependency is conventionally neglected, due to the assumption of the strong ordering that is embedded in the LO DGLAP evolution equation, Avoiding such assumption, one can include the contributions coming from the transverse momentum distributions of the partons, using either the solutions of the CCF M evolution equation or unify the BF KL and the DGLAP single-scaled evolution equations to form a properly tuned k t -dependent framework, [49,50]. Utilizing these methods does not always come easy, since these frameworks are mathematically complex and in the case of CCF M , not enough to include all of the contributing sub-processes. Alternatively, the single-scaled P DF of the DGLAP evolution equation can be convoluted with the required k t -dependency during the last step of the evolution [14], postulating that: Consequently, one may use the defining identity of the k t -factorization, to define the U P DF , f a (x, k 2 t , µ 2 ), with a(x, µ 2 ) being the solutions of the DGLAP equation times x (i.e. xq(x, Q 2 ) and xg(x, Q 2 )). we should make this comment here that in the more precise definition, one should use the generalized U P DF [5][6][7][8][9][10], i.e. the double-U P DF (DU P DF ), such that they take into account both quarks and gluons. Then we should write (compare with equation (2)): However, in this work we continue our calculations by using the U P DF . Afterwards, one can easily derive the direct expressions for the U P DF of the k t -factorization, f a (x, k 2 t , µ 2 ). Furthermore, in order to avoid the soft-gluon singularities, it is necessary to impose some physical constraint into this definition in the form of the AOC. Naturally, imposing different visualizations of the AOC will from different formalisms for the U P DF .
The first choice is the so called the KM R prescription. Introducing the virtual (loop) contributions via the Sudakov form factor, and utilizing the LO splitting functions, P The LO splitting functions parameterize the probability of evolving from a scale k t to a higher scale µ without any parton emissions. Naturally, the N LO extensions of these functions would take more complicated forms, see the following equation (10) in relation to the M RW prescriptions. The infra-red cut-off ∆ = k t /(µ + k t ) represents a visualization of the AOC, which automatically excludes the x = x point from the range of z-integration blocking the soft gluon singularities that arise form the 1/(1 − z) terms in the splitting functions.
One immediately notes that throughout the above definition, the k t -dependency gets introduced into the U P DF , only at the last step of the evolution. In order to produce these U P DF , the single scaled b(x, k 2 t ) functions can be obtained from the M M HT 2014 library, [48], where the calculation of the single-scaled functions have been carried out using the IS data on the F 2 structure function of the proton. Additionally, using the constraint, and with the modified loop contributions and where z max = 1 − z min = µ/(µ + k t ) [52].
with i = 0 corresponding to the LO and i = 1 to the N LO levels (It has been argued that, applying the approximation P (LO+N LO) (z) ∼ P (LO) (z) will simplify the N LO prescription and have a negligible effect on the outcome [12], therefore we do not need to express the exact forms of the N LO splitting functions) . Consequently, the introduction of the AOC into the N LO M RW formalism is through the extended splitting functions and the Θ(z − (1 − ∆ )) constraint, with ∆ being defined as: Additionally, one have to cut off the tail of the probability into the k t > µ region by inserting a secondary AOC related term into the body of the real emission sector, The Sudakov form factors in this framework are formulated as: The reader can find a comprehensive description of the N LO splitting functions in the references [12,53].
In the figure 1, the U P DF of the k t -factorization are plotted against the fractional longitudinal momentum of the parent hadron (x) and the transverse momentum of the parton, appearing on the top of the evolution ladder (k t ). The obvious difference in the behavior of the U P DF in different frameworks is a direct consequence of employing different manifestations of the AOC in their respective definitions.

III. THE DI-JET PRODUCTION IN THE P-P COLLISIONS AT THE LHC
Generally speaking, the main contributions into the hadronic cross-section of the di-jet productions at the LHC, i.e., are the LO partonic sub-processes: Since we are considering the forward sector for the partons that are produced in the k tfactorization, the stared partons in the equation (15), one can safely neglect the qq and qq sub-processes. In the collinear factorization framework, the cross-section of a hadronic IS can be written as a sum over all of the involving partonic cross-sections, times the probability of appearing the particular partonic configuration at top of the evolution ladder of the individual hadrons, i.e., whereσ a 1 −a 2 denotes the cross-section of the incoming partons a 1 and a 2 , respectively with the longitudinal momentum fractions x 1 and x 2 , the hard scales µ 1 and µ 2 and neglected transverse momenta.σ a 1 −a 2 may be defined as follows: with the multi-particle phase space dφ a 1 a 2 , and the flux factor F a 1 a 2 , s is the center of mass energy squared, with P 1 and P 2 being the 4-momenta of the incoming hadrons, where we have neglected the mass of the proton, while working in the infinite momentum frame. M a 1 a 2 in the equation (17) are the matrix elements of the partonic sub-processes, the equations (15). To calculate these quantities, one must first understand the exact kinematics that rule over the corresponding partonic sub-processes.
To include the contributions coming from the transverse momentum dependency of the probability functions, one can use the definition of the U P DF in the framework of k tfactorization, the equation (2) and rewrite the equation (16) as follows: Now, it is convenient to characterize dφ a 1 a 2 in term of the transverse momenta of the product particles, p i,t , their rapidities, y i , and the azimuthal angles of the emissions, ϕ i , Working in the proton-proton center of mass frame, one may use below kinematics, where the k i are the 4-momenta of the partons that enter the semi-hard process. Then, for each partonic sub-process, the conservation of the transverse momentum reads as, Afterwards, one can simply define, The figure 2 illustrates the schematics for a proton-proton deep inelastic collision in the forward-center (or the forward-forward) rapidity sector in a particular partonic sub-process, i.e. g * + g → q +q. Working within the boundaries of the forward-center or the forwardforward rapidity sector, without damaging the main assumptions, one can assume that x 1 ∼ 1 and x 2 1. In the direct consequent of a such approximation, we can safely neglect the transverse momentum dependency of the first parton entering the hard process (shift it to the collinear domain), and rewrite the equation (20) as, with the k t being defined as, and ∆ϕ = ϕ 1 − ϕ 2 .
After determining the kinematics of the involving processes, it is possible to calculate their matrix elements, i.e. M a 1 a 2 . To this end, one have to sum over the dk 2 i,t /k 2 i,t terms only from the ladder-type diagrams, and somehow systematically dispose the interference (the non-ladder) diagrams, e.g. by using a physical gauge for the gluons, Note that n = x 1 P 1 + x 2 P 2 is the gauge-fixing vector. One might expect that neglecting the contributions coming from the non-ladder diagrams, i.e. the diagrams where the production of the jets is a by-product of the hadronic collision (see the reference [40,54]), would have a numerical effect on the results. Hence, using the equation (27) as our choice for the axial gauge for the gluons, we can safely subtract the "unfactorizable" contributions coming from the non-ladder type diagrams. Thus, using the regular Feynman rules, inserting the "non-sense" polarization for the incoming gluons and imposing the "eikonal" approximation to justify the use of an on-shell prescription for the off-shell particles (via neglecting the exchanged momenta in the quark-gluon vertices and preserving the spin of the gluons, see the references [40,54,55]), one can manage to extract the matrix element, corresponding to the processes of the equation (15), see the appendix A. Now, using the above equations, one can derive the master equation for the total crosssection of the production of di-jets in the framework of k t -factorization, The term 1/(1 + δ cd ) restrains the over-counting indices. Note that, the existence of the term k −2 t in the equation (30) is the remnant of the re-summation factor, dk 2 t /k 2 t , from the equation (2) and since we are interested to look for the transverse momentum dependent jets with p i,t > 20 GeV , the presence of such denominator would not cause any complication in the master equation. Additionally, we have to decide how to validate our U P DF in the non-perturbative region. i.e. where k t < µ 0 with µ 0 = 1 GeV . A natural option would be to fulfill the requirement that: and therefore, one can safely choose the following approximation for the non-perturbative region: In the next section, we will introduce some of the numerical methods that have been used for the calculation of the cross-section of the production of di-jets, using the U P DF of KM R and M RW .

IV. THE NUMERICAL ANALYSIS
We perform the 5-fold integration of the master equation (30), using the VEGAS algorithm in Monte-Carlo integration. To do this, we have selected the hard-scale of the U P DF as the share of each of the parent hadrons from the total energy of the center-of-mass frame: Variating this normalization value around a factor of 2, will provide each framework with a decent uncertainty bound. One would also set the upper boundaries on the transverse momentum integrations to p i,max = 4µ, noting that increasing this upper value does not have any effect on the outcome.
The forward rapidity sectors is conventionally defined as, where η denotes the pseudorapidity of a produced particle, with θ being the angle between the propagation axis and the momentum of the particle.
Alternatively, to work in the central rapidity sector, one have to choose, Therefore, while working in the infinite momentum frame i.e. where η y, to perform our calculations in the forward-center region, we set: Trivially, the choice marks the forward-forward region. Such framework should be ideal to describe the inclusive CM S data regarding the forward-center di-jet measurements for p i,t > 35 GeV . After confirming that, one can go further, producing predictions in the framework of forwardforward di-jet production for the LHC.
Moreover, as a consequence of employing the inclusive scenario (i.e. p i,t > 35 GeV and limiting the rapidity integrations to the forward or central regions), one must assure that the produced jets must lie within this specific region. Thus, in order to cut-off the collinear and the soft singularities, it is conventional to use the anti-k t algorithm [56], with radius R = 1/2, bounding the jets to this particular initial setup, through inserting a constraint on the y − ϕ plane: Introducing the anti-k t jet constraint ensures the production of 2 separated jets and rejects any single-jet scenarios.

V. RESULTS, DISCUSSIONS AND CONCLUSIONS
Having in mind the theory and the notions of the previous sections, we are able to calculate the production rates belonging to the di-jets in the forward-center and the forwardforward rapidity sectors, from the perspective of the k t -factorization framework, utilizing the i.e., the equations (5), (7) and (12). Additionally, they are fit to be used as the solutions of the DGLAP , the P DF of the collinear factorization, directly in the master equation (30).
We tend to perform the above calculations in any of our presumed frameworks, the KM R,  (15), corresponding to the g * + g → g + g, g * + g → q +q and g * +q → g +q processes respectively. The black-horizontally stripped pattern represents the sum of the sub-contributions. The calculations have been compared against the experimental data of the CM S collaboration, the reference [46]. One immediately notices that the share of the g * + g → g + g sub-process dominates, relative to the negligible shares of the remaining two sub-processes. Although all of these frameworks are relatively successful in describing the experimental data, see the figure 6, it is interesting to find that the U P DF where M ax(p 1,t , p 2,t ) returns the higher value between the transverse momenta of the produced jets. To save computation time, we only considered the contributions coming form the dominant g * + g → g + g sub-processes. The choice a, which have been used in the similar calculations (e.g., the references [41][42][43][44][45]  After proving the success of our formalism in describing the experimental data for the production of di-jets in the forward-center rapidity region, we can move forward with the prediction of a similar event, in the forward-forward sector, i.e. by choosing the rapidity of the produced jets (y 1 and y 2 ) to be both in the boundaries that where specified within the equation (33). Therefore, in the figure 9 the reader is presented with our predictions regarding the dependency of the differential cross-section of the forward-forward di-jet production (dσ f /dp f t ) to the transverse momenta of the produced jets (p t ), in the framework of general behavior as in the forward-central case, in spite of the fact that the measured contribution for the g * + g → g + g and the g * + q → g + q sub-processes are closer, compared to their counterparts from the forward-center region, In the absence of any experimental data, we refrain ourselves from any assessments regarding these results. Nevertheless, the predictions of the KM R scheme (because of its previous success) may provide a base line for a sound comparison. Also, the singular behavior of the N LO M RW results may appear undesirable.
[59] Y. V. Kovchegov, Phys. Rev. D 60 (1999) 034008.  diagram shows the g * + g → q +q sub-process, assuming that one of the quarks is being produced in the forward sector (bounded by 3.2 < |η f | < 4.7) and the other in the center sector (bounded by |η c | < 2.8). The parton density related to the first proton is being described with the integrated P DF while the second parton is prepared using the U P DF in one of our presumed frameworks.              FIG. 11: The calculated predictions regarding the dependency of the differential cross-section for the production of forward-forward di-jets to rapidity of the produced jets, using the U P DF of k t -factorization for E CM = 7 T eV . The notion on the diagrams are as in the figure 9. In the panel (d), we have compared our results with the predictions made using the rcBK and KS T M D P DF from the reference [45].