A new phenomenological Investigation of $KMR$ and $MRW$ $unintegrated$ parton distribution functions

The longitudinal proton structure function, $F_L(x,Q^2)$, from the $k_t$ factorization formalism by using the unintegrated parton distribution functions (UPDF) which are generated through the KMR and MRW procedures. The LO UPDF of the KMR prescription is extracted, by taking into account the PDF of Martin et al, i.e. MSTW2008-LO and MRST99-NLO and next, the NLO UPDF of the MRW scheme is generated through the set of MSTW2008-NLO PDF as the inputs. The different aspects of $F_L(x,Q^2)$ in the two approaches, as well as its perturbative and non-perturbative parts are calculated. Then the comparison of $F_L(x,Q^2)$ is made with the data given by the ZEUS and H1 collaborations. It is demonstrated that the extracted $F_L(x,Q^2)$ based on the UPDF of two schemes, are consistent to the experimental data, and by a good approximation, they are independent to the input PDF. But the one developed from the KMR prescription, have better agreement to the data with respect to that of MRW. As it has been suggested, by lowering the factorization scale or the Bjorken variable in the related experiments, it may be possible to analyze the present theoretical approaches more accurately.


I. INTRODUCTION
more compliance with the DGLAP evolution equations requisites, but it seems in the KMR case, the angular ordering constraint spreads the UP DF to whole transverse momentum region, and makes the results to sum up the leading DGLAP and Balitski-F adin-Kuraev-Lipatov (BF KL) logarithms [34][35][36][37][38].
Another important observable quantity in this connection is the longitudinal structure function, i.e. F L (x, µ 2 ), which is proportional to the cross section of the longitudinal polarized virtual photon with proton. Particulary at small x, it is directly sensitive to the gluon distributions i.e. g → qq process. Moreover its calculations in this region need the k t factorization formalism [39][40][41][42][43], which is beyond the standard collinear factorization procedure [44]. Recently, Golec − Biernat and Staśto [45,46] (GS) have used the k t and collinear factorizations [39][40][41][42][43] as well as the dipole approach to generate the longitudinal structure function, but by using the DGLAP /BF KL re-summation method, developed by Kwiecinski, Martin and Stasto (KMS) [47], for calculation of the unintegrated gluon density at small x. They have parameterized the input non-perturbative gluon distribution such that they could get the best fit to the experimental proton structure function data [47].
On the experimental side, the longitudinal structure function has been measured by both the H1 [48,49] and ZEUZ [50,51] collaborations at the DESY electron-proton collider HERA. The Q 2 ranges have been varied between 12 to 90 and 24 to 110 GeV 2 in each experiments, respectively.
As it was pointed out above, similar to our recent publication on F 2 (x, Q 2 ) [32], in the present paper, we intend to calculate F L (x, Q 2 ) by working in the the k t -factorization scheme.
But rather than KMS re-summation method pointed out above, the KMR and MRW [18][19][20][21][22] formalisms are used to predict the UP DF with the input P DF of the MRST 99-NLO [52], MST W 2008-LO [53] and MST W 2008-NLO [53] which covers wide range of (x, Q 2 ) plane. Then our results can be compared both with the experimental data as well as the theoretical KMS − GS presentation of F L (x, Q 2 ). So the paper is organized as follows: In the section II we give a belief review of the KMR and the MRW formalisms [18][19][20][21][22] for extraction of the UP DF form the phenomenological P DF [52,53]. The formulation of F L (x, Q 2 ) based on the k t -factorization scheme is given in the section III. Finally, the section IV is devoted to results, discussions and conclusions.

II. A BRIEF REVIEW OF THE KM R AND THE M RW FORMALISMS
The KMR and MRW [18][19][20][21][22][23] ideas for generating the UP DF work as follows: Using the given integrated P DF as the inputs, the KMR and MRW procedures produce the UP DF as their outputs. They are based on the DGLAP equations along with some modifications due to the separation of virtual and real parts of the evolutions, and the choice of the splitting functions at leading order (LO) and the next-to-leading order (NLO) levels, respectively: (i) In the KMR formalism [18,19], the UP DF , f a (x, k 2 t , µ 2 ) (a = q and g), are defined in terms of the quarks and the gluons P DF , i.e.: and respectively, where, P aa ′ (x), are the LO splitting functions, and the survival probability factors, T a (k t , µ), are evaluated from: The angular ordering condition (AOC) [54,55], which is a consequence of coherent emission of gluons, on the last step of the evolution process [23], is imposed. The AOC determined the cut off, ∆ = 1 − z max = kt µ+kt , to prevent z = 1 singularities in the splitting functions, which arises from the soft gluon emission. As it has been pointed out in the references [18,19], the KMR approach has several main characteristics. The important one, is the existence of the cut off at the upper limit of the integrals, that makes the distributions to spread smoothly to the region in which k t > µ i.e. the characteristic of the small x physics, which is principally governed by the BF KL evolution [34][35][36][37][38]. This feature of the KMR, leads to the UP DF with the behavior very similar to the unified BF KL+DGLAP formalism [18,19]. The UP DF based on the KMR formalism, have been widely used in the phenomenological calculations which depend on the transverse momentum [56][57][58][59][60][61][62][63][64][65][66][67].
(ii) In the MRW formalism [20][21][22], the similar separation of real and virtual contributions to the DGLAP evolution is done, but the procedure is performed at the NLO level i.e., where P (0+1) ab In the equations (4) and (5) the P ab denote the LO and the NLO contributions of the splitting functions, respectively. It is obvious from equation (4) that in the MRW formalism, the UP DF are defined such that to ensure k 2 < µ 2 . Also, the survival probability factor, T a (k 2 , µ 2 ), are obtained as follows: where P (i) ab (which is singular in the z → 1) is given in the reference [68]. MRW have demonstrated that the sufficient accuracy can be obtained by keeping only the LO splitting functions together with the NLO integrated parton densities. So, by considering angular ordering, we can use P (0) instead of P (0+1) . As it is mentioned above unlike the KMR formalism, where the angular ordering is imposed to the all of terms of the equations (1) and (2), in the MRW formalism, the angular ordering is imposed to the terms in which the splitting functions are singular, i.e. the terms that include P qq and P gg .
The k t -factorization approach has been discussed in the several works i.e. references [3,39,42,69,70]. In the following equation [45,[71][72][73], the different terms i.e the perturbative and the non-perturbative contributions to the F L (x, Q 2 ) has been broken into the sum of gluons from the quark-box (the first term i.e. the k t factorization part), see figure 1 [22]), quarks (the second term) and the non-perturbative gluon (the third term) Parts: where the second term is (see [74,75] ): In the above equation, in which the graphical representations of k t and κ t have been introduced in the figure 1, the variable β is defined as the light-cone fraction of the photon momentum carried by the internal quark [70]. Also, the denominator factors are: Then by defining κ ′ t = κ t − (1 − β)k t , the variable y takes the following form: As in the reference [47], the scale µ which controls the unintegrated gluon and the QCD coupling constant α s , is chosen as follows: One should note that the coefficients used for quark and non-perturbative gluon contributions depend on the transverse momentum. As it has been briefly explained before, the main prescription for F L consists of three terms; the first term is the k t factorization which explains the contribution of the UP DF into the F L . This term is derived with the use of pure gluon contribution. However, it only counts the gluon contributions coming from the perturbative region, i.e. for k t > 1 GeV , and does not have anything to do with the non-perturbative contributions. In the reference [74], it has been shown that a proper non-perturbative term can be derived from the k t factorization term, compacting the k t dependence and the integration with the use of a variable-change, i.e. y, that carries the k t dependence. Nevertheless, there is a calculable quark contribution in the longitudinal structure function of the proton, which comes from the collinear factorization, i.e. the second term of the equation (7).
For the charm quark, m is taken to be m c = 1.4GeV , and u, d and s quarks masses are neglected. We also use the same approximation to save the computation time [19], the one we did for the calculation of F 2 (x, Q 2 ) [32] i.e the representative "average" value for φ, φ = π 4 for perturbative gluon contribution. This approximation has been checked in the reference [19] (page 83). The rest of φ angular integration can be performed analytically by using a series of integral identities given in the reference [76]. We will also verify this approximation in the next section. The unintegrated gluon distributions are not defined for k t and κ t < k 0 , i.e. the non-perturbative region. So, according to the reference [71], k 0 is chosen to be about one GeV which is around the charm mass in the present calculation, as it should be. On the other hand, one expects that the discrepancy between the k t -factorization calculation and the experimental data can be eliminated by using the P DF , which have been fitted to the same data for F 2 (x, Q 2 ) [77] with respect to the re-summation method of KMS respectively. Their total F L (x, Q 2 ) and the contributions from k t factorization scheme, the quarks and the no-perturbative parts (see the equation (7) are presented with different curve styles. The behavior of F L (x, Q 2 ) mostly comes from the k t factorization contribution especially as the Q 2 is increased and it is more sizable in case of MRW approach. By rising up the increases, which is a heritage of the parent DGLAP evolution.
In order to analyze the above Q 2 dependent more clearly, in the figure 6, the longitudinal proton structure functions are plotted against Q 2 for two different values of x = 0.001 and 0.0001. Note, that for large Q 2 , especially the MRW approach, needs large computation time. So we have stopped at Q 2 = 100 GeV 2 for this procedure. There are sizable differences between the two approaches and results coming from the two different input P DF . But this should not be very important regarding the experimental data, that we will discuss later on.
In the figure 7, a comparison figure 3). Very similar behavior is observed especially between the k t factorization approaches.
In the figures 8, 9 and 10, we present our results in the range of energy available in the H1 and ZEUS data [31], respectively. Note that for Q 2 ≥ 80 GeV 2 , because of large computation time, we have only given four points (filled squares) for the MRW case. Very good agreements is observed between our result and those of experimental data at different Q 2 and x values. It seems with present existed data the UP DF of gluons generated with different input P DF and constraints procedures, one can reasonably explain the H1 and the ZEUS experimental data. It looks that even at low energies and small x values (see the figure 8); we find good agreement between our calculation and available data. However, as we mentioned before and it has been stated by several authors, the F L is mainly driven through the gluons distributions, especially at low values of x. The fact that F 2 is not accurately fit the data (see our previous work [32]), but we get good agreement between the F L calculations and H1 and ZEUS data, could be caused of the quark-quark contributions which has more contribution to F 2 . Since F L is more sensitive to the gluons UP DF with respect to F 2 . So one can conclude that present calculation can confirm that the KMR and MRW procedures (for generating the gluon UP DF ) and the k t -factorization scheme can reproduce reasonable F 2 (considering our previous work [32]) and present F L . On the other hand, as we stated previously:(1)Present results also shows good agreement with the theoretical calculations of GS, which have used more complicated approach such as KMS.
(2)It is interesting that the KMR and MRW UP DF can generate reasonable F L without using any free parameter in the (x, Q 2 )-plane even at low Q 2 (regarding figure 8), especially the UP DF generated for gluons.
Finally, the verification of the fact that the φ integration of perturbative gluon contribution can be averaged by setting < φ >= π/4, which was discussed in the end of previous section, is presented in the figure 11, for four values of Q 2 = 3.5, 12, 60 and 110 Gev 2 by using the KMR formalism and the MRST 99. It is clearly seen that the above approximation does work properly and one can save much computation time.
In conclusion, the longitudinal proton structure functions, F L (x, Q 2 ), were calculated based on the k t factorization formalism, by using the UP DF which are generated through the KMR and MRW procedures. The LO UP DF of the KMR prescription is extracted, by taking into account the P DF of MST W 2008-LO and MRST 99-NLO and also, the NLO UP DF of the MRW scheme is generated through the set of MST W 2008-NLO P DF as the inputs. The different aspects of the F L (x, Q 2 ) in the two approaches, as well as its perturbative and non-perturbative parts were calculated and discussed. It was shown that our approaches are in agreement with those given GS. Then the comparison of F L (x, Q 2 ) was made with the data given by the ZEUS and H1 collaborations at HERA. It was demonstrated that the extracted longitudinal proton structure functions based on the UP DF of above two schemes, were consistent with the experimental data, and by a good approximation, they are independent to the input P DF . But as it was pointed out in our previous work [32], the one developed from the KMR prescription, have better agreement to the data with respect to that of MRW . Although the MRW formalism is in more compliance with the DGLAP evolution equations requisites, but it seems in the KMR case, the angular ordering constraint spreads the UP DF to whole transverse momentum region, and makes the results to sum up the leading DGLAP and BF KL logarithms. At first, it seems that there should be a theoretical support for applying the angular ordering condition only to the diagonal splitting functions, in accordance with reference [22]. But as it has been mentioned in the references [32,33], this phenomenological modifications of the KMR approach (including the application of the AOC to all splitting functions) works as an "effective model" that spreads the UP DF to the k t > µ (a characteristic of low x physics) which enables it to represent a good level of agreement with the data. Beside this in our new work [33] in which we have calculated the F L in the dipole approximation according to the LO prescription of reference [22], it is shown that there is not much difference if one applies the AOC to the all splitting functions i.e. to use the KMR UP DF instead of using LO prescription of reference [22]. On the other hand, in this paper we have focused on comparison of the LO and the NLO calculation of F L and since the calculations are very time consuming we restricted the results to the LO − KMR and NLO − MRW .
As it has been suggested in the reference [45], by lowering the factorization scale or the Bjorken variable in the experimental measurements, it may be possible to analyze the present theoretical approaches more accurately.