Decoupling of the structure functions in momentum space based on the Laplace transformation

Using Laplace transform techniques, we describe the determination of the longitudinal structure function $F_{L}(x,Q^2)$, at the leading-order approximation in momentum space, from the structure function $F_{2}(x,Q^2)$ and its derivative with respect to ${\ln}Q^2$ in a kinematical region of low values of the Bjorken variable $x$. Since the $x$ dependence of $F_2(x,Q^2)$ and its evolution with $Q^2$ are determined much better by the data than $F_L(x,Q^2)$, this method provides both a direct check on $F_L(x,Q^2)$ where measured, and a way of extending $F_L(x,Q^2)$ into regions of $x$ and $Q^2$ where there are currently no data. In our calculations, we ultilize the Block-Durand-Ha parametrization for the structure function $F_{2}(x,Q^2)$ [M. M. Block, L. Durand and P. Ha, Phys.Rev.D {\bf89}, 094027 (2014)]. We find that the Laplace transform method in momentum space provides correct behaviors of the extracted longitudinal structure function $F_{L}(x,Q^2)$ and that our obtained results are in line with data from the H1 Collaboration and other results for $F_{L}(x,Q^2)$ obtained using Mellin transform method.


I. INTRODUCTION
Recently, evolution of the longitudinal and transversal structure functions in momentum space has been considered in [1].Structure functions measurable in deep inelastic scattering (DIS) are formulated in the momentum-space Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equations [2][3][4].Scheme independent evolution equations for the structure functions F i (x, Q 2 ) proposed some time ago in [5,6], as the physical observables, read where anomalous dimensions, P ij , are computable in perturbative QCD (pQCD).Determination of the longitudinal structure function in the nucleon from the proton structure function, based on a form of the deep inelastic lepton-hadron scattering structure function proposed by Block-Durand-Ha (BDH) in [7], is considered in [8][9][10][11].Parametrization of the proton structure function proposed in [7] describes the available experimental data on the reduced cross sections at low x and provides a behavior of the hadron-hadron cross sections ∼ln 2 s at large s in a full accordance with the Froissart predictions [12] (s is the Mandelstam variable denoting the square of the total invariant energy of the process).
Deep inelastic scattering (DIS) is characterized by structure functions F k that depend on kinematic variables Bjorken x and momentum transfer Q by the following form where C k,a (x, α s ) are the known Wilson coefficient functions in the order of the perturbation theory, α s is the strong coupling, and < e 2 > is the average of the charge e 2 for the active quark flavors, < e 2 >= n −1 f n f i=1 e 2 i with n f as the number of considered flavors.The symbol ⊗ denotes convolution according to the usual prescription and xf a=2,g (x, Q 2 ) are the singlet-quark and gluon densities respectively (the non-singlet quark distributions at small x become negligibly small in comparison with the singlet distributions).The DGLAP equations, which describe how the parton distribution functions (PDFs) vary as the energy scale of the scattering process changes, are important for understanding a wide range of high-energy processes, including DIS, hadron collisions, and deep-inelastic scattering of heavy ions at future colliders (Large Hadron electron Collider (LHeC) [13] and Electron-Ion Collider (EIC) [14,15]).
Numerical and analytical methods (which extract the PDFs from the experimental data) to solve the DGLAP evolution equations have been extensively studied in the literature [16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32].The solutions to these equations provide a theoretical prediction for the PDFs used in the interpretation and description of the Hadron-Electron Ring Accelerator (HERA) data on the total and diffractive cross-sections in deep inelastic electron-proton scattering.They serve as a mean to test our understanding of QCD and extract information about the structure of the proton.
In this paper, we extend the method using a Laplace-transform technique and obtain an analytical method for the solution of the momentum-space of the DGLAP equations for F L (x, Q 2 ) in terms of F 2 (x, Q 2 ) and known derivative dF 2 (x, Q 2 )/dlnQ 2 in the kinematical region of low values of the Bjorken variable x.The parameterization of the structure function F 2 (x, Q 2 ) in [7] is obtained from a combined fit to HERA data in a wide range of the kinematical variables x and Q 2 by the following explicit expression as where M is the effective mass and µ 2 is a scale factor defined by the Block-Halzen fit to the real photon-proton cross section [33] in Table I.In the following, we apply this parametrization function to test the consistency of the longitudinal structure function owing to the momentum-space of the DGLAP equations with HERA data on deep inelastic electron-proton scattering.The paper is organized as follows: in section II, we present the basics of the momentum-space of the DGLAP equations.Section III summarizes the Laplace transform method for obtaining an analytical solution for the longitudinal structure function.In section IV, the numerical results are obtained and compared with the available H1 Collaboration data [34][35][36] and the Large Hadron electron Collider (LHeC) [13] simulated errors.Conclusions are given in Sec.V. Some detailed calculations are relegated to Appendix.

II. MOMENTUM SPACE
The DIS structure functions F 2 and F L at low x are defined [1] into the singlet and gluon distribution functions by the following forms and L,s − ln L,g − ln L,s ⊗P qg − ln where xg(x, µ 2 r ) and xΣ(x, µ 2 r )≡xf s (x, ] are the gluon and singlet distribution functions at the renormalization scale µ 2 r , respectively.In Eqs.( 5) and ( 6), C ij (i = 2, L; j = s, g) denote the scheme-dependent coefficient functions defined, at the first non-zero order in α s , by [1] with the color factors C A = 3, T R = 1/2 and C F = 4/3 associated with the color group SU(3).The authors in [1] inverted the leading non-zero order part of Eqs.( 5) and ( 6).As a result, the singlet and gluon densities in terms of the structure functions read Using the leading-order (LO) renormalization group equation and setting the renormalization scale equal to the momentum transfer, µ 2 r = Q 2 , the authors in [1] derived the evolution equation of the structure function F 2 (x, Q 2 ) at the first non-zero order in α s as where the plus function is defined as By writing we can rewrite the last integral in Eq. ( 10) as Substituting Eq. ( 13) into Eq.( 10), we get

III. LAPLACE TRANSFORMATION
In the following, we use the method developed in detail in [37][38][39][40] to obtain the longitudinal structure function into the proton structure function and its derivative using a Laplace-transform method.We now rewrite the momentumspace DGLAP evolution equation for the longitudinal structure function (i.e., Eq.( 14)) in terms of the variables υ = ln(1/x) and Q 2 instead of x and Q 2 .Using the notation F i (υ, Q 2 )≡F i (e −υ , Q 2 ) for structure functions, explicitly, from Eq.( 14), we find Introducing the notation that the Laplace transform of structure functions ; s] and using the fact that the Laplace transform of a convolution factors is simply the ordinary product of the Laplace transform of the factors by the following form where h(s)≡L[e −υ H(υ)], we find that the Laplace transform of Eq. ( 15) is given by where Here ψ(1) = −γ = −0.5772156... is Euler constant.The functions F i (0) (or exactly F i (+0)) are the boundary conditions due to the Laplace derivatives and defined Then, from Eq. ( 17), we find where 2π h2 (s)h −1 L (s) with h L (s) and h2 (s) given by The inverse Laplace transform of the coefficients h i in Eq. ( 18), defined by the kernels J i (υ)≡L −1 [h i (s); υ], is straightforward.We find that and where as shown in the Appendix.Transforming back in to x space, the longitudinal structure function F L (x, Q 2 ) is given by

IV. RESULTS AND DISCUSSIONS
With the explicit form of the longitudinal structure function (i.e., Eq. ( 23)), we begin to extract the numerical results at small x in a wide range of Q 2 values, using the parametrization of F 2 (x, Q 2 ) given by Eq. ( 3).The QCD parameter Λ for four numbers of active flavor has been extracted [10] due to α s (M 2 z ) = 0.1166 with respect to the LO form of α s (Q 2 ) with Λ = 136.8MeV.In order to make the effect of production threshold for charm quark with m c = 1.29 +0.077 −0.053 GeV [42,43], the rescaling variable χ is defined by the form χ = x(1 + 4 where reduced to the Bjorken variable x at high Q 2 [43].In Fig. 1, we analyze the function f (υ) in a wide range of υ according to the expansion of the second term in Eq. ( 22) in the ranges k = 1 − 2, k = 1 − 10, and k = 1 − 50, respectively.We observe that the function f (υ) is very small for υ ≥ 2 as would be expected from the decreasing exponential factor in υ in Eq. (22).It is nearly independent of the cutoff in the expansion for υ ≥ 1, but the expansion must be carried to large k for υ small.In the following we choose the value of k = 50 in the numerical results.As seen in the figure, the result for f (υ) appears to converge well even for small υ > 0 for the maximum value of k sufficiently large.We have found that the choice k = 50 for the upper limit gives results sufficient accurate for our purposes.
In Fig. 3, we have separated our analysis of the longitudinal structure function at any fixed Q 2 and compare it with the results in [8] and [10] at the LO approximation as a function of x.The longitudinal structure function extracted at Q 2 values are in good agreement with experimental data in comparison with those in [8] at the LO approximation, as the mathematical structure of Eq. ( 10) in momentum space differs from the DGLAP equations for structure functions.
In Fig. 4, the longitudinal structure functions in momentum space at selected x and Q 2 are associated with the LHeC simulated uncertainties [13].We observe that the longitudinal structure functions (central values) are determined owing to Eq. ( 23) for the Q 2 values (4.5, 8.5, 18, and 35 GeV 2 ) and accompanied with the simulated uncertainties reported by the LHeC study group [13].The H1 collaboration data with total errors for Q 2 = 8.5 and 35 GeV 2 are shown in Fig. 4.
In Fig. 5, we show a comparison between the longitudinal structure functions in momentum space with the H1 Collaboration data at a fixed value of the invariant mass W (i.e.W = 230 GeV) at low values of x. Figure 5 clearly demonstrates that the Laplace transform method in momentum space provides correct behaviors of the extracted longitudinal structure function in comparison with the LO and NLO analysis reported in [10].As can be seen in this figure, the results are comparable with the H1 data and the NLO corrections to the Mellin transform method at all Q 2 values.
Indeed, the momentum-space DGLAP evolution equations for structure functions measurable in deeply inelastic scattering have some importance in contrast to the existing literature on the subject where the evolution has been written in Mellin space.In the momentum-space, there is no need to define a factorization scheme and also, the approach in terms of physical structure functions has the advantage of being more transparent in the parametrization of the initial conditions of the evolution.

V. CONCLUSIONS
We have presented a method based on the Laplace transform method to determine the longitudinal structure function at the LO approximation in momentum space.This method relies on the parametrization of the function F 2 (x, Q 2 ) and its derivative dF 2 (x, Q 2 )/dlnQ 2 within a kinematical region characterized by low values of the Bjorken variable x.The x dependence of F 2 (x, Q 2 ) and its evolution with Q 2 are determined much better by the data than F L (x, Q 2 ), so this method provides both a direct check on F L (x, Q 2 ) where measured, and a way of extending F L (x, Q 2 ) into regions of x and Q 2 where there are currently no data.We find that the Laplace transform method in momentum space provides correct behaviors of the extracted longitudinal structure function F L (x, Q 2 ) and that our results for F L (x, Q 2 ) demonstrate comparability with data from the H1 Collaboration and other results obtained using the Mellin transform method.
VI. ACKNOWLEDGMENTS G.R.Boroun thanks M.Klein and N.Armesto for allowing access to data related to simulated errors of the longitudinal structure function at the Large Hadron Electron Collider (LHeC).Phuoc Ha would like to thank Professor Loyal Durand for useful comments and invaluable support.

VII. APPENDIX
The inverse Laplace transform of the function can be evaluated analytically in terms of an infinite series, rapidly convergent except for υ near zero where it grows logarithmically.The denominator in f (s) has zeros at s 2 .which lead to simple poles in f (s) at those points.The function ψ(s + 1) has simple poles with residue −1 at s + 1 = 0, −1, −2, ... [44] .There are no other sigularities in the integrand which decreases exponentially rapidly for s → −∞ and Re(υ) > 0. We can therefore close the integration contour in the left-half s plane and evaluate the integral as the sum of the residues at the poles multiplied by 2πi by Cauchy's residue theorem.
The central values of the longitudinal structure function are plotted at low x as accompanied by the LHeC total errors [13].H1 Collaboration data [36,42] are collected at Q 2 = 8.5 and 35 GeV 2 with total errors.

FIG. 5 .For υ ≪ 1 ,
FIG.5.The extracted longitudinal structure function (solid brown curve) in momentum space at fixed value of the invariant mass W (W = 230 GeV) compared with the H1 Collaboration data[42] as accompanied with total errors and the results in Refs.[8] and[10] at the LO (dotted black curve) and NLO (dot-dashed green curve) approximation.