The general behavior of $NLO$ unintegrated parton distributions based on the single-scale evolution and the angular ordering constraint

To overcome the complexity of generalized two hard scale ($k_t$,$\mu$) evolution equation, well known as the $Ciafaloni$, $Catani$, $Fiorani$ and $Marchsini$ ($CCFM$) evolution equations, and calculate the unintegrated parton distribution functions ($UPDF$), $Kimber$, $Martin$ and $Ryskin$ ($KMR$) proposed a procedure based on ($i$) the inclusion of single-scale ($\mu$) only at the last step of evolution and ($ii$) the angular ordering constraint ($AOC$) on the $DGLAP$ terms (the $DGLAP$ collinear approximation), to bring the second scale, $k_t$ into the $UPDF$ evolution equations. In this work we intend to use the $MSTW 2008$ (Martin et al) parton distribution functions (PDF) and try to calculate $UPDF$ for various values of $x$ (the longitudinal fraction of parton momentum), $\mu$ (the probe scale) and $k_t$ (the parton transverse momentum) to see the general behavior of three dimensional $UPDF$ at the $NLO$ level up to the $LHC$ working energy scales ($\mu^2)$. It is shown that there exits some pronounced peaks for the three dimensional $UPDF$ $(f_a(x,k_t))$ with respect to the two variables $x$ and $k_t$ at various energies ($\mu$). These peaks get larger and move to larger values of $k_t$, as the energy ($\mu$) is increased. We hope these peaks could be detected in the $LHC$ experiments at $CERN$ and other laboratories in the less exclusive processes.


I. INTRODUCTION
To understand the event structure observed in different laboratories i.e. SLAC, HERA, DESY etc, and especially the one would be expected in LHC (CERN), the theoretical formalisms which describe the small x (x is Bejorken variable) region are vital. The main unknown parameters in these models are the unintegrated parton distribution functions (UP DF ) [1][2][3][4]. The UP DF are two-scales dependent distributions which are functions of x (longitudinal momentum fraction of the parent hadron) and the scales k 2 t and µ 2 , the squared transverse momentum of the parton and the factorization scale, respectively. As we pointed out these distributions are the essential ingredients for the less exclusive phenomenological computations in the high energy collisions of particle physics.
It is well known that in the region of high energy and moderate momentum transfer i.e. small x, the collinear factorization theorem i.e. Dokshitzer-Gribov-Lipatov-Altarelli-P arisi (DGLAP ) [5][6][7][8] evolution, breaks down. This happens because of the large increase of the phase space available for the gluon emissions (i.e. a rapid rise in the gluon density), which makes the quantum chromodynamics (QCD) perturbative expansions unjustified and one can not obtain the UP DF . On the other hand, at above high energy limit, the cross section can be predicted by using the k t factorization and the Balitsky-F adin-Kuraev-Liptov (BF KL) [9][10][11] evolution. But the precision of k t factorization is not good e.g. the next-to-leading order (NLO) corrections to BF KL are very large [12][13][14][15]. Another approach to derive the UP DF is the Ciaf aloni-Catani-F iorani-Marchesini (CCF M) equations [16][17][18][19][20]. Although the CCF M equations describe the evolution of the UP DF correctly, but working in this framework is a complicated task, so practically they are used only in the Monte Carlo event generators [21][22][23][24][25]. On the other hand, up to now, there is not a complete quark version for these kind of equations [16][17][18][19][20]26], since the enhanced terms that are resumed by CCF M come from gluon evolution. However, to over come this problem, it has been shown that the CCF M equation can be reformulated (the linked dipole chain model) by reducing the division between the initial and the final state radiation diagrams using the colour dipole cascade model [27][28][29].
In equation (1) the symbol ⊗ denotes a convolution as, In this approach under the certain approximation the UP DF are obtained from the PDF by introducing the scale µ only in the last step of evolution with the inclusion of angular ordering constraint (AOC). It has been shown that the KMR prescription gives the same results both for the DGLAP and the unified BF KL-DGLAP equations [31] and the AOC is applicable to all orders as in the CCF M formalism, i.e. all the loops contributions via the chain of evolution which are restricted by AOC, are resumed.
In this work, along the lines of our recent calculations [33,34], we intend to use the KMR prescription with MST W 2008 [32] P DF to produce three dimensional plots of UP DF at different energies (µ) and discussed the various behavior of UP DF i.e. f (x, k t , µ) . So the paper is organized as follows: In section II we briefly introduce the KMR formalism and finally, section III is devoted to the results and the discussions concerning the three dimensional (3D) graphs of the UP DF produced via this approach.

II. THE KM R FORMALISM [30]
The KMR prescription [30] works as a machine that by taking a defined PDF as inputs, generates UP DF , as outputs. Using the leading order (LO) splitting functions, P aa ′ , the DGLAP equations can be written in a modified form as [30], where ∆ is a cutoff to prevent z = 1 singularities in the splitting functions arising from the soft gluon emission. In the conventional DGLAP formalism, ∆ = 0 and the singularities are canceled by the virtual terms. The value of ∆ can be determined by imposing an appropriate dynamical condition which is replaced by the angular ordering constraint arising from the coherency of the gluon emissions [35,36], where θ's are the radiation angels. This condition, at the final step of evolution, leads to [16][17][18][19]31], The first part of the equation (3), shows the contribution of real emissions, that can change the transverse momentum k t . The second term expresses the evolutions due to the virtual effects without changing the k t . The latter can be re-summed, to obtain a survival probability factor, Now, similar to the Sudakov form factor, the above survival probability, equation (6), is imposed into the equation (1), and by using equation (2), we find the equation which describes the UP DF , More explicit forms of the above equation for the gluon g and the different quark flavors q = u, d, s, ... are as follows, and The key observation here is the dependency on the scale µ 2 , which appears at the last step of the evolution. Another point is that, the Sudakov form factor which arises from the resumption of virtual effects, can be used at every order of approximation. Although the splitting functions must be used at the NLO level, but as it is shown in [37], the The primary computations based on this kind of approach to evaluate the UP DF , show very good agreement with the experimental data for F 2 [30]. Also, in recent years, the KMR prescription have been widely used for phenomenological calculations (see [33] and the references therein). Recently the stability and the reliability of the KMR UP DF have been investigated in [33,34].
Finally, we should mention here that, the key property of the CCF M approach (as given in their publications [16][17][18][19][20][21][22]) is the AOC, which in turn has root in the coherency of gluon radiation along the evolution chain, that is valid for whole range of x values. In the conventional DGLAP formalism, the strong ordering constraint on the transverse momenta, restricts the domain of study to the large and moderate values of x: where and So to obtain the DGLAP equations with ln(ŝ 4 ) ≃ ln(Q 2 ), x should not be very low. In the KMR prescription the AOC property of the CCF M formalism is applied to modified DGLAP evolution as a cut off on the integrals. Therefore, the results of these modifications show that the effect of application of AOC is even more important than the inclusion of the conventional low x effects in the BF KL approach [30].

III. RESULTS AND DISCUSSION
As we stated in the section II, by using the equations (8) and (9), the UP DF are generated via the KMR procedure. For the input P DF , the MST W 2008 [32] set of partons at the NLO level are used [40]. Since the generated UP DF (f a (x, k 2 t , µ 2 )) are three variable functions, by fixing the scale µ 2 , their values versus x and k 2 t are plotted in the various panels of figures 1, 2, 3  t , µ 2 )). For example for gluons, at x = 0.01 and µ 2 = 100 we get 6.7 whereas MST W 2008 gives the value of 6.5 i.e. 3% off. Situations are the same for other points and parton distributions. It is worth to say that in the original KMR work, they get 25% discrepancies [30] for above comparison. This is also evident by comparison of figure   5 with those of 1 to 4 i.e. the UP DF are decreasing by increasing x. On the other hand, as have been discussed in the KMR and other related works, because of the imposition of angular ordering, the UP DF have values for k 2 t ≥ µ 2 as x decreases. But this will not affect the above integration too much. The figures 1 to 4 also show that, for low scales (µ 2 ≃ 10 GeV 2 ) the UP DF become negative when x becomes close to one. This reflects the negative values of MST W 2008 gluon distributions at the NLO level and beyond that. So the negative values of UP DF have root in the parent integrated gluon distributions which in turn are the result of MST W 2008 assumptions [32]. As it was pointed, in the MST W 2008 [32], for better data fitting it is allowed that, the gluon distribution takes negative values, because there is no theorem that imposes positivity condition on P DF beyond the LO approximation. So they become negative in order to fit the data (in other words they can be traced to the slow evolution of F 2 at small x and Q 2 i.e. a positive gluon would give too rapid evolution to fit the dF 2 /d ln(Q 2 ) data. Then in the KMR integrals, the evaluation of input g(x, k 2 t ) at small x and k t (as a scale, instead of Q 2 in g(x, Q 2 )) leads to the negative values for the output UP DF . Finally, (i) the comparison of UP DF produced from different P DF sets have been made in our former works [33,34] . The different parameterizations procedures lead to different P DF , and a discussion about these procedures is presented in [33,34] and references therein. (ii) The differences between the LO and the NLO P DF are parameterizations dependent. In the MST W 2008 this is noticeable, but in some other parameterizations sets based on different assumptions and procedures it can be less (e.g GRV sets [33,34]), but as we have showed in [33,34] (by investigating the ratios of KMR UP DF compared to the corresponding ratios of input P DF ) the relative differences are less in the output UP DF and the KMR prescription suppresses these discrepancies. To show this point more transparently, in figure 6 we have plotted the gluon UP DF with three different input P DF , namely the original KMR [30] with MRST 99 [38] P DF , our recent works [33,34] with GJR08 P DF [39] and present calculation (MRST 2008) at µ 2 = 100 GeV 2 and x = 0.1, 0.01, 0.001 and 0.0001 in terms of k 2 t . It is clearly seen that different input P DF give very similar UP DF . (iii) In fact a complete prescription for producing the NLO UP DF needs to include both the P DF and the splitting functions at the NLO level.
This prescription is presented in [37], but as it is shown in this reference [37], inclusion of the NLO splitting functions have very low effect comparing to the contribution of the NLO P DF . Therefore, ignoring the corrections due to the NLO splitting functions do not affect our analysis of the general behavior of the NLO UP DF . (iv) There is no restriction on the k t dependency. As the orders of the approximation are in terms of orders of α s (k 2 t ), the NLO accuracy is contained in the NLO PDF and splitting functions that discussed in the former comments. Hence at scales k 2 t ≥ Q 2 0 ,where Q 2 0 is the scale that upper than it, the perturbative QCD is still applicable, these results are valid. and GJR08 (full curve). See the text for details.