Longitudinal structure function from logarithmic slopes of $F_{2}$ at low $x$

Using Laplace transform techniques, I calculate the longitudinal structure function $F_{L}(x,Q^{2})$ from the scaling violations of the proton structure function $F_{2}(x,Q^{2})$, and make a critical study of this relationship between the structure functions at leading order (LO) up to next-to-next-to leading order (NNLO) analysis at small $x$. Furthermore, I consider heavy quark contributions to the relation between the structure functions, which leads to compact formula for $N_{f}=3+\mathrm{Heavy}$. The nonlinear corrections to the longitudinal structure function at LO up to NNLO analysis are shown in the $N_{f}=4$ (light quark flavor) based on the nonlinear corrections at $R=2$ and $R=4 GeV^{-1}$. The results are compared to experimental data of the longitudinal proton structure function $F_{L}$ in the range of $6.5 \leq Q^{2} \leq 800 GeV^{2}$.

The inclusive deep inelastic scattering (DIS) measurements are of importance to understanding the gluonic substructure of proton at low values of Bjorken variable x. The reduced cross section is defined in the following form where Y + = 1+(1−y) 2 , y = Q 2 /xs is the inelasticity, s is the center-of-mass squared energy of incoming electrons and protons, F 2 (x, Q 2 ) and F L (x, Q 2 ) are the transverse and longitudinal structure functions, respectively. The structure functions describe the momentum distributions of partons in nucleon. A measurement of the proton structure functions (F 2 and F L ) at low values x * Electronic address: grboroun@gmail.com; boroun@razi.ac.ir is directly sensitive to the gluon density. This provides a sensitive test for perturbative QCD (pQCD). The longitudinal structure function is determined by measurements of differential cross sections at different values of √ s at HERA, where data on √ s for electron beam energies of E e ≃ 27.5GeV and for proton beam energies of E p = 920, 820, 575 and 460GeV are collected [1][2]. The experimental data for neutral current were also collected for 0.045 ≤ Q 2 ≤ 50000GeV 2 and 6E − 7 ≤ x ≤ 0.65 at values of the inelasticity 0.005 ≤ y ≤ 0.95. The contribution of F L to reduced cross section ( Eq.(1)) is significant only at high value of the inelasticity y, i.e., the kinematic region. The latter case corresponds to the low values of the Bjorken variable x, and the longitudinal structure function is related to the gluon density of the proton. As gluons are the most influencing at low x, therefore the quark contribution to the longitudinal structure function and singlet structure function is ignored in next step. The gluonic longitudinal structure function can be written as x −1 F L (x, Q 2 )= < e 2 > C L,g (α s , x) ⊗ g(x, Q 2 ), (2) where g(x, Q 2 ) represents the gluon density, and < e 2 > is the average squared charge (= 5/18 for even N f , where N f denotes number of active light flavours). The symbol ⊗ denotes the Mellin convolution according to the usual prescription.
The perturbative expansion of the gluon coefficient function can be written as where n is the order in running coupling constant. The reduced cross section for deep inelastic leptonnucleon scattering (Eq.1) is defined in terms of the proton structure function F 2 . At low values of x, the gluon contribution to the proton structure function F 2 dominates over the flavor singlet contribution. The DGLAP evolution equation for gluon dominating F 2 structure function is given by where F s 2 is the singlet distribution function. The splitting function P qg is the leading order (LO) up to highloop corrections to the QCD β-function as Several methods to relate F L and F 2 scaling violation to the gluon density at small x were suggested previously [3][4]. These methods were proposed to isolate the gluon distribution by its expansion around z = 1 2 . In LO analysis with N f = 4, the authors of Ref. [3] suggested an approximate relation between the gluon density at the point 2.5x and the longitudinal structure function F L at the point x in the following form Equation (6) was derived in an expansion of gluon distribution around z = 1 2 . A similar relation for derivation of F 2 (x, Q 2 ) with respect to lnQ 2 based on the expansion of the gluon distribution around z = 1 2 was found in [4], where the following result was obtained Combining Eqs. (6) and (7), one could calculate the longitudinal structure function by derivation of the structure function at a rescaled value ζ2 ζL x, where ζ 2 ≃ 0.5 and ζ L ≃ 0.4. The corresponding LO expression is where η ≃ 1.25. In addition, two different methods were suggested [5][6], the derivatives of the structure functions were based in the expansion of the gluon distribution around the arbitrary point z = α. The results were derived at an arbitrary point of expansion as follows and Eqs. (9) and (10) strongly depend on the momentum fraction carried by gluons in Eqs. (2) and (4), and show the behavior of the gluonic structure functions based on the expansion of the gluon distribution around z = α. In this paper I introduce a method to calculate structure functions using the Laplace transform techniques. The paper is organized as follows. In section II, I find the relation between the structure functions at small x at LO analysis. In section III, I consider the high-order corrections to the relation between the structure functions. In section IV, I utilize obtained solution to calculate the nonlinear behavior of the longitudinal structure function at hot-spot point at LO analysis, and present an analytical analysis of the longitudinal structure function. Then I compare the obtained result with H1 experimental data. In section V, I study the high-order corrections to the nonlinear behavior of the longitudinal structure function. My conclusion is given in section VI. In Appendix A, I present the results for the splitting functions and coefficients in the inverse-Laplace transform method at some values of Q 2 . Appendix B includes the analytical expression for F γp 2 (x, Q 2 ). In Appendix C and D I present the high-order corrections and high-order ratios at NLO up to NNLO at small x. Appendix E deals with a technical detail including the inverse-Laplace transform of the nonlinear kernels at LO and high order corrections presented in Appendix F.

II. General Method
In pQCD, the evolution equations for proton and longitudinal structure functions are given by in terms of the non singlet, singlet and gluon coefficient functions. At small values of x the gluon contribution to the structure functions dominate over the flavour singlet and nonsinglet contribution. Therefore, in this research I consider the gluonic structure functions evolution equations. A: Four Flavours: One could write the LO equation for the evolution of the proton structure function at low values of x as The longitudinal structure function for gluon dominating is given by Considering the coordinate transformation as ν = ln(1/x) and ω = ln(1/y) [7], one could rewrite Eqs. (11) and (12) with respect to these variables and and F L (ν, Q 2 ) = 9π 20αs F L (ν, Q 2 ). Defining the Laplace transforms F 2 (s, (13) and (14), one obtains the structure functions in s-space as and In Eqs. (15) and (16) I used the fact that the Laplace transform of a convolution function is simply ordinary product of the Laplace transform of that function. Taking into account the gluon distribution, one could extend the Laplace transformation to the high-order corrections in the following form (17) and The leading-order splitting functions at Laplace s-space are given by: and I have, therefore, the derivative of the structure function in the form of the longitudinal structure function in sspace as or where h(s) = and Consequently, the general relation between the structure functions in x-space is given by Finally, one could write the leading-order relation for the longitudinal structure function for massless quarks in the form of the derivative of the structure function as B:Three flavours + Heavy: The heavy quark contribution (charm and bottom) to relation between F L and F 2 is define by fixed-order number scheme using m c = 1.5 GeV and m b = 4.5 GeV [8]. The mass of these heavy quarks satisfies m Q ≫Λ QCD , and provides a hard scale for pQCD calculations. One could consider the perturbative predictions for the longitudinal structure function. Eq. (2) can be rewritten as the convolution form where ⊗ in the N f = 3 for massless quarks u, d and s denotes the common convolution, and F c(b) L are heavy quark corrections to the longitudinal structure function at small x. These corrections in deep inelastic electron-proton scattering collisions serve as a test of pQCD and the heavy quark production is directly sensitive to the gluon density and heavy-quarks mass. One should write the individual longitudinal structure functions as At small x, where the gluon distribution is dominant, the heavy quark contributions F i k (x, Q 2 , m 2 i ) with i = b, c and k = 2, L in the proton structure function is written as where a = 1 + 4 m 2 i Q 2 , and the renormalization scale µ is assumed to be average < µ 2 >= 4m 2 i + Q 2 2 .
Using the Laplace transform method [7], one can rewrite the heavy structure functions in terms of the convolution integrals with respect to ν ′ and ω ′ variables at small x aŝ Here ν ′ = ln 1 ax , ω ′ = ln 1 ay and The Laplace transformation ofĤ i The convolution theorem for Laplace transforms allows one to rewrite the heavy distribution functions as a product of their Laplace transforms g(s, Q 2 ) and h i k (s). In this case one has Therefore the ratio of the heavy structure functions are independent of the gluon distribution function in s-space. This ratio can be written as If one takes the inverse Laplace transformation of Eq.
(36), then one has where Here I used the following property for inverse Laplace transformation Then Eq. (37) becomes as where The analytical results for the parameters J i for a particular range of Q 2 under study are given in Appendix A.
In a similar manner, the longitudinal structure function can be determined at small x by considering the heavy corrections to the structure function. Thus, applying the convolution theorem, the analytical solution for the longitudinal structure function for N f = 3 + Heavy should be converted to usual (x, Q 2 ) space. Therefore one has one observes that the connection between the structure functions (in Eqs. (26) and (40)) are independent of the running coupling constant at LO analysis and gluon density behavior. To calculate the right hand side of these equations (Eqs. (26) and (40)) one has to have an expression for the proton structure function [9] and heavy quark structure [10] functions for massless and heavy quarks . The H1 Collaboration reported a measurement of inclusive ep cross sections at high Q 2 at √ s = 225 and 252 GeV . HERA provided the first measurements of F L in the region 120 ≤ Q 2 ≤ 800 GeV 2 and 6.5×10 −4 < x < 0.032 [1]. My results are compared with extracted longitudinal proton structure function F L in the range of 6.5 ≤ Q 2 ≤ 800 GeV 2 . In Fig.1 the determined longitudinal structure function F L is shown for Q 2 = 20 GeV 2 and 200 GeV 2 , respectively. In this figure, the longitudinal structure functions determined for four massless quarks at m 2 c < µ 2 and also to account for fixed N f = 3 flavor number scheme as the heavy flavor contributions to F L are taken as given by fixed order perturbation theory. The results for Q 2 = 20 and 200 GeV 2 are presented for N f = 4 and N f = 3 + Heavy, and are compared with H1 Collaboration data [1]. For heavy contributions to F L , the renormalization scale is < µ 2 >= 4m 2 H + Q 2 /2. These results are accompanied with errors due to fit parametriztions of ∂F 2 /∂lnQ 2 , as listed in Appendix B and Table I. It is seen from Fig.1 that the results are comparable with the experimental data as accompanied with total errors, although those are independent of the gluon behavior. I also present the Cooper-Prytz (CP) fit [3][4], which depends on expanding of the gluon distribution at z = 1/2, and the Gay Ducati-Boroun (BG) fit [5][6] which depends on expanding of the gluon distribution at z = α. In Fig.2 I present the longitudinal structure function F L for Q 2 = 45 and 500 GeV 2 without considering the heavy quark contributions in the same Fig.1. The longitudinal proton structure function F L (x, Q 2 ) compared by averaging F L data from Table 5 in Ref. [1] at the given values of Q 2 and x with total uncertainty on F L , shown in Fig.3. A reasonable agreement between the longitudinal structure function as extracted from the direct measurement of the derivative of F 2 with the experimental data is observed at moderate and high Q 2 values at low values of x. These results extend from the LO up to NNLO analysis with respect to the Laplace transform method at small x and I will try to compare our result with experimental data in the next section.

III. High-order corrections
An analytical solution based on the Laplace transformation for the relation between the longitudinal structure function in terms of a convolution of the derivative of F 2 obtained at LO accuracy in perturbative QCD in section II. Some analytical solutions of the DGLAP evolution equation in next-to-leading order (NLO) analysis using the Laplace transform method have been presented in Ref. [11]. In Refs. [12][13], the authors have been reported the complete two and three-order coefficient functions for the longitudinal structure functions in deepinelastic scattering (DIS). Now, a detailed analysis has been performed in order to find an analytical solutions of the longitudinal structure function into the derivative of the proton structure function with respect to lnQ 2 , us-ing the repeated Laplace transform, at NLO up to NNLO approximation. In s-space, one can rewrite the gluonic structure functions equations in terms of the convolution integrals up to NNLO analysis. The Laplace transform of these equations converted to an ordinary first order differential equations in s-space as one has and where the running coupling constants have the following forms in NLO and NNLO analysis respectively as and where β 0 = 1 3 (33−2n f ), β 1 = 102− 38 3 n f and β 2 = 2857 6 − 6673 18 n f + 325 54 n 2 f are the one-loop,two-loop and three-loop corrections to the QCD β-function and Λ is the QCD cutoff parameter. The Λ QCD parameter usually defined at NLO and NNLO analysis as Λ [12][13], respectively. The N n LO expansion coefficients are defined in Ref. [13] in Mellin-space and one should present these splitting functions and coefficient functions in Appendix C. In detail the shape of the structure functions are dominated by the gluon density at low values of x. Therefore one would find where DF 2 ≡ ∂F2 ∂lnQ 2 . The high-order H(ν, Q 2 ) for four Q 2 values presented in Appendix D. One can easily determine these high order corrections to the gluonic longitudinal structure function based on the derivative of the proton structure function with respect to ln Q 2 at low x. Now considering the terms from NLO up to NNLO, the gluonic longitudinal structure function takes the following form for a given Q 2 value as For my numerical investigation, the high order corrections to F L (x, Q 2 ) are shown in Fig.4 and compared with H1 data [1] for Q 2 = 20, 45, 200 and 500 GeV 2 . In this figure the straight and dash lines represent the gluonic longitudinal structure function solutions at NLO and NNLO respectively. These results are obtained with respect to the Laplace transform technique as described in Appendix D. In this figure, the circles represent the longitudinal structure functions from Ref. [1] as accompanied with total errors. These results are in agrement with F L (x, Q 2 ) predicted from the global fit at LO, NLO and NNLO in Ref. [12]. However it is a reflection of the behavior of the deep inelastic structure function and the coefficient functions at low values of x.
In the next sections, the recombination processes between gluons in a dense system have to be taken into account. Therefore the gluonic longitudinal structure function behavior has to be tamed by screening effects.

IV. Nonlinear behavior
The screening effects are provided by a multiple gluon interaction which leads to the nonlinear terms in the derivation of the linear DGLAP evolution equations. Therefore the standard linear DGLAP evolution equations will have to be modified in order to take the nonlinear corrections into account. Gribov, Levin, Ryskin, Mueller and Qiu (GLR-MQ) [14] performed a detailed study of these recombination processes. This widely known as the GLR-MQ equation and involves the two-gluon distribution per unit area of the hadron. This equation predicts a saturation behavior of the gluon distribution at very small x [15][16]. A closer examination of the small x scattering is resummation powers of α s ln(1/x) where leads to the k T -factorization form [17]. In the k T -factorization approach the large logarithms ln(1/x) are relevant for the unintegrated gluon density in a nonlinear equation. Solution of this equation develops a saturation scale where tame the gluon density behavior at low values of x and this is an intrinsic characteristic of a dense gluon system. Therefore one should consider the low-x behavior of the singlet distribution using the nonlinear GLR-MQ evolution equation. The shadowing correction to the evolution of the singlet quark distribution can be written as [13][14]18] ∂xq(x, Q 2 ) Eq. (47) can be rewrite in a convenient form as The first term is the standard DGLAP evolution equation (Eq. 11) and the value of R is the correlation radius between two interacting gluons. It will be of the order of the proton radius (R ≃ 5 GeV −1 ), if the gluons are distributed through the whole of proton, or much smaller (R ≃ 2 GeV −1 ) if gluons are concentrated in hot-spot within the proton. One would find Eq. (48) at LO analysis in s-space as The longitudinal structure function in s-space is given in the following form Combining Eqs. (49) and (50), one could calculate the nonlinear relation between the derivative of the structure function and longitudinal structure function in s-space as I have where ζ = 243π 2 25600R 2 Q 2 . At ζ→ 0, Eq. (51) leads to the linear relation between the structure functions (i.e., Eq. 21). Eq. (51) yields the gluonic longitudinal structure function with nonlinear effects as It is tempting, however, one of the roots of Eq. (52) can be discarded. Solution of Eq.(52) then leads us to a solution for the nonlinear gluonic longitudinal structure function. This equation can be solved by Taylor series expansion method around a particular choice of point of expansion. Since (ζR 2 Q 2 ) n <1, so this series is convergent when n→∞. This parameter decreases with increasing n, as seen from Table II. For the longitudinal structure function in s-space, one has Eq. (53) covers the whole range of expanding as it is shown in Table II. The contribution from the fourth term to the second term (such that Fourth term Second term ∝ ζ 3 ζ = ζ 2 ) is around the order of O(∼ 10 −2 ). To make a rough estimate of the accuracy in expansion method I find the longitudinal structure function until four order approximation with respect to the ζ expansion and neglecting the high order terms O(ζ 3 ) in Eq. (53). For this evolution, I retain the second order term into ζ. Therefore the gluonic longitudinal structure function in ν-space is defined as . The inverse-Laplace transform of kernels can be found in Appendix E. Applying the properties of Dirac delta function, finally I have the nonlinear gluonic longitudinal structure function in x-space by the following form where W LO 1 is independent of the values of Q 2 , but W LO 2 and W LO 3 are depend on Q 2 values. Thus I obtained an expression for the gluonic longitudinal structure function F L (x, Q 2 ) in leading order by solving the nonlinear GLR-MQ evolution equation. Eq. (56) shows that it is independent of the gluon behavior, the running coupling constant, and also the QCD cut off parameter in the LO approximation. One can easily solve this equation (i.e., Eq. 56), and extract the nonlinear gluonic longitudinal structure function. The nonlinear behavior of F L (x, Q 2 ) is shown in Fig.5 for values of Q 2 = 6.5 and 20 GeV 2 . It would appear that the effect of nonlinearity at low-x values should observe for moderate Q 2 values when compared with H1 data. In this figure, the nonlinear effect investigated at hot-spot point (R = 2 GeV −1 ). It is shown that the obtained results from present analysis based on Laplace transform are in good agreements with the ones obtained by H1 Collaboration [1]. The saturation of the gluon density at small x indirectly is significant for understanding the nonlinear effects in Eq. (56) and also high-order corrections. In the next section I apply high order corrections to the nonlinear behavior and compared with H1 data.

V. High-order corrections to the nonlinear behavior
Using the formalism given in the previous section, I calculate the high-order corrections to the nonlinear behavior of the gluonic longitudinal structure function at low x region. In terms of the derivative of proton structure function with respect to ln Q 2 , the GLR-MQ evaluation equation can be written in the high order correction in s-space as where Θ 2 (s, (s). One should consider the same method introduce in the previous section, I find the high order corrections to the gluonic longitudinal structure function by the following form Eq.(58) can be solved simultaneously to get the desired nonlinear equation for longitudinal structure function. Using the inverse Laplace transform to back from s-space to x-space, the simplified solution of the above equation at high-order corrections can be obtained by Indeed, comparison of the NNLO+NLCs with the NNLO calculations shows a turnover of the gluonic longitudinal structure function at Q 2 = 6.5 and 20 GeV 2 . This is due to the effect of the gluonic coefficient function to the gluonic splitting function ratio , which decreases the limit NNLO corrections when tamed with respect to the nonlinear saturation effect. Since gluon recombination introduces a negative correction to the NNLO linear behavior, the signal of its presence is a decrease of the scaling violation and this is strongly dependence to the correlation radius (i.e., R). In Fig.7, the effect of the nonlinearity in NNLO results for R = 4 GeV −1 at Q 2 = 6.5 and 20 GeV 2 investigated. It can be observed (in Figs.6 and 7) that NNLO results are very sensitive to R as x decreases. Indeed the effect third-order corrections to the coefficients functions and splitting functions in hot-spots point decrease the gluonic longitudinal structure function to the negative values as x decreases. This behavior is comparable when R increase throughout the entire proton at NNLO approximation.
At least there is another mechanism to prevent generation of the high density gluon states, as this is well known the vacuum color screening [19]. There is a transition between the nonperturbative and perturbative domains. In the QCD vacuum, the non-perturbative fields form structures with sizes ∼ R c which it is smaller than Λ QCD . The short propagation length for perturbative The gluon fusion effect in non-linear regime controlled by the new dimensionless parameter ∼ R 2 c 8B where B is the characteristic size of the interaction region as this parameter can be defined by ln(x 0 /x) and r where r 2 ∼ Q −2 . In all figures one should observe that the nonlinear effects are small even at lowest x values. This behavior is in accordance with the smallness of the ratio R 2 c 8B . It is interesting to look at the nonlinear limit where decreases as Q 2 increases. From [19], the nonlinear effects leads to the logarithmically ratio as the nonlinear/linear effects are proportional to R 2 c /8B(ln(x 0 /x), r 2 ) ln(Q 2 R 2 c ). Figs.6 are shown that high order corrections to the nonlinear behavior are very small at high Q 2 values and at lowest available Bjorken x.

VI. Conclusion
In this paper I have estimated an analytical solution for the linear and nonlinear behaviors of the longitudinal structure function with respect to the derivative of the proton structure function inside the proton. This solution is independent of the gluon model and the running coupling constant at leading order analysis and it is free of any point expanding model for the gluon distribution behavior. The ratio of splitting functions applying the Laplace transform technique are calculated. I have used the heavy coefficient functions for heavyflavour production in DIS in the fixed-flavour-number scheme (FFNS) with N f = 3. In the present calcula-tions the high-order corrections (NLO and NNLO) for structure functions at low x values, arising from the coefficient functions and the splitting functions, are obtained. I have therefore used from these results for the gluonic longitudinal structure function at moderated and high values of Q 2 . The nonlinear GLR-MQ evolution equation predicted by considering the general Laplace transform method and studied the effects of adding the nonlinear corrections to the linear longitudinal structure function at hot-spot point (R = 2 GeV −1 ) with N f = 4. For the gluonic longitudinal structure function the nonlinear effects are found to play an increasingly important role at x ≤ 10 −3 . I have incorporated high-order corrections to the nonlinear behavior in the kinematic range of moderate-Q 2 and obtained the nonlinear longitudinal structure function at low x at NLO and NNLO approximation. It is interesting to see that the NNLO analysis at moderate Q 2 is dependence to the proton radius as the nonlinear behavior increase as R increases. This is due to the contribution from the NNLO terms in the ratio of coefficient function to the splitting function. It can be observed that with decreasing x, the taming of F L (x, Q 2 ) is apparently observed in NLO approximation at R = 2 GeV −1 and in NNLO approximation at R = 4 GeV −1 . This method presented in this analysis enable us to achieve strictly analytical linear and nonlinear solutions at LO up to NNLO approximation for the gluonic longitudinal structure function in terms of the derivative of the proton structure function with respect to the lnQ 2 at low values of x. The nonlinear effects are shown to be small at large Q 2 , even at lowest Bjorken values of x.

Acknowledgment
Author thanks Urs A.Wiedemann for discussions which completed this study and the Department of Physics of the CERN-TH for their warm hospitality. Also I would like to thank M.Tabrizi for his careful editorial revising of the paper.

Appendix. A
The kernels at leading order analysis are as follows: and The parameters J c (υ) and J b (υ) are given by the follow- Appendix. B The proton structure function parameterized with a global fit function [9] to the HERA combined data for F γp 2 (x, Q 2 ) for 0.85 < Q 2 < 3000 GeV 2 and x < 0.1, which ensures that the saturated Froissart ln 2 (1/x) behavior dominates at small-x. This global fit takes the where A(Q 2 ) = a 0 + a 1 lnQ 2 + a 2 ln 2 Q 2 , The fitted parameters are tabulated in Table I. At small x ( or large ν = ln(1/x)), the global fit becomes a quadratic polynomial in ν as where the coefficient functions are defined in Ref. [9].

Appendix. C
At small x the one-loop up to three-loop splitting functions for N f = 4 read where E qg 1 ≃ − 298.667N f and E qg 2 ≃ − 1268.28N f + 4.57613N 2 f . The gluonic longitudinal coefficient functions up to NNLO analysis at small x can be written as x . (66) Appendix. D Appendix. E The inverse-Laplace of the nonlinear kernels are as follows: where W , s are inverse Laplace transform of all coefficients at LO up to NNLO analysis accordance with results expanded in nonlinear behavior.    4)), Down triangle (N f = 3 + Heavy)) compared by H1 [1](circles) at the given values of Q 2 accompanied with total uncertainties. The determined error bars represent the derivative of F2(x, Q 2 ) uncertainties. The curves represent the prediction from the expanding of gluon behavior [3][4][5][6].  The longitudinal structure function FL compared with H1 data averaged over x in the region 6.5 ≤ Q 2 ≤ 800GeV 2 (solid points). The error bars represent the full errors as obtained by the Monte Carlo procedure described in the Ref. [1]. For each Q 2 the average value of x is given above each data point.