Importance of heavy quark longitudinal structure function measurements at future circular collider energies

In this article,we consider the ratio of structure functions for heavy quark pair production at low values of The importance of this ratio for charm and beauty pair production is examined according to the Hadron Electron Ring Accelerator(HERA) data.The behavior of these ratios is considered due to the hard pomeron behavior of the gluon distribution function.The results are in good agreement with the HERA data.Expanding this data to the range of new energies underscores the importance of these measurements for heavy quarks.The ratio of charm and beauty structure functions at the proposed Large Hadron electron Collider(LHeC) is considered as a function of invariant center-of-mass energy.For top pair production this ratio is extracted with known kinematics of the LHeC and Future Circular Collider electron-hadron(FCC-eh) colliders.Comparison of the results obtained for the ratio of top structure functions in LHeC and FCC-eh are proportional to the specified inelasticity y range.     

Analytic approach to the approximate solution of the independent DGLAP evolution equations with respect to the hard-Pomeron behavior

We show that it is possible to use hard-Pomeron behavior to the gluon distribution and singlet structure function at low x. We derive a second-order independent differential equation for the gluon distribution and the singlet structure function. In this approach, both singlet quarks and gluons have the same high-energy behavior at small x. These equations are derived from the next-to-leading order DGLAP evolution equations. All results can be consistently described in the framework of perturbative QCD, which shows an increase of gluon distribution and singlet structure functions as x decreases.     

Calculation of Gluon Distribution Functions in Leading Order by Regge-like Behavior at Low-x

An approximate analytical form of the gluon distribution function from the derivative F2 proton structure function data at low-x assuming the Regge-like behavior of the gluon distribution function at this limit is presented. In this method, we calculate λ and C parameters for each low-x value at several Q^2 values using the scaling violation of the F2 proton structure function. For low-x, A is found to be independent of x within the experimental accuracy. We make use of the leading order Altarelli-Parisi （A-P） evolution equations in our analysis. To test the validity of our new determined gluon distribution functions, we compare them with the QCD parton distribution functions at low-x region.     

Shadowing correction to the gluon distribution behavior at small x

We determined the saturation exponent of the gluon distribution using the solution of the QCD nonlinear Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (NLDGLAP) evolution equation at small x . The very small-x behavior of the gluon distribution is obtained by solving the Gribov, Levin, Ryskin, Mueller and Qiu (GLR-MQ) evolution equation with the nonlinear shadowing term incorporated. The form of the initial condition for the equation is determined. We find, with decreasing x , the emergence of a singular behavior and the eventual taming (at R = 5 GeV^-1) and the essential taming (at R = 2 GeV^-1) of this singular behavior by the shadowing term. The nonlinear gluon density functions are calculated and compared with the results for the integrated gluon density from the Balitsky-Kovchegov (BK) equation for the different values of Q². It is shown that the results for the gluon density function are comparable with the results obtained from the BK equation solution. Also we show that for each x , the Q²-dependence of the data is well described by gluon shadowing corrections to the GLR-MQ equation. The resulting analytic expression allows us to predict the logarithmic derivative \( {\frac{{\partial F^{s}_{2} (x,Q^{2})}}{{\partial \ln Q^{2}}}}\) and to compare the results with H1 data and a QCD analysis fit.     

The ratio of the charm structure functions F k c (k = 2, L) at low x in deep inelastic scattering with respect to the expansion method

We study the expansion method for the gluon distribution function at low x values and calculate the charm structure functions in the LO and NLO analysis. Our results provide a compact formula for the ratio R ^c = F _L ^c /F ₂ ^c , which is approximately independent of x and the details of the parton distribution function at low x values. This ratio could be a good probe of the charm structure function F ₂ ^c in the proton deduced from the reduced charm cross sections at DESY HERA. These results show that the charm structure functions obtained are in agreement with HERA experimental data and other theoretical models.     

Calculation of Gluon Distribution Functions in Leading Order by Regge-like Behavior at Low-x

An approximate analytical form of the gluon distribution function from the derivative F2 proton structure function data at low-x assuming the Regge-like behavior of the gluon distribution function at this limit is presented. In this method, we calculate λ and C parameters for each low-x value at several Q2 values using the scaling violation of the F2 proton structure function. For low-x, λ is found to be independent of x within the experimental accuracy. We make use of the leading order Altarelli-Parisi (A-P)evolution equations in our analysis. To test the validity of our new determined gluon distribution functions, we compare them with the QCD parton distribution functions at low-x region.     

Calculation of the Gluon Distribution Function Using Alternative Method for the Proton Structure Function

A calculation of the proton structure function F2(x,Q2) is reported with an approximation method that relates the reduced cross section derivative and the F2(x, Q2) scaling violation at low x by using quadratic form for the structure function. This quadratic form approximation method can be used to determine the structure function F2 (x, Q2)from the HERA reduced cross section data taken at low x. This new approach can determine the structure functions F2(x,Q2) with reasonable precision even for low x values which have not been investigated. We observe that the Q2 dependence is quadratic over the full kinematic covered range. To test the validity of our new determined structure functions, wefind the gluon distribution function in the leading order approximation with our new calculation for the structure functions and compare them with the QCD parton distribution functions.     

Calculation of the Longitudinal Structure Function from Regge-Like Behaviour of the Gluon Distribution Function in Leading Order Approximation at Low x

We present the calculations of FL longitudinal structure functions from DGLAP evolution equation in leading order （LO） at low-x, assuming the Regge-like behaviour of gluon distribution at this limit. The calculated results are compared with the H1 data and QCD fit. It is shown that the obtained results are very close to the mentioned methods. The proposed simple analytical relation for EL provides a t-evolution equation for the determination of the longitudinal structure function at low-x. All the results can consistently be described within the framework of perturbative QCD, which essentially shows increases as x decreases.     

Calculation of the Gluon Distribution Function Using Alternative Method for the Proton Structure Function

A calculation of the proton structure function F2(x,Q^2) is reported with an approximation method that relates the reduced cross section derivative and the F2(x, Q^2) scaling violation at low x by using quadratic form for the structure function. This quadratic form approximation method can be used to determine the structure function F2 (x, Q^2) from the HERA reduced cross section data taken at low x. This new approach can determine the structure functions F2(x,Q^2) with reasonable precision even for low x values which have not been investigated. We observe that the Q^2 dependence is quadratic over the full kinematic covered range. To test the validity of our new determined structure functions, we find the gluon distribution function in the leading order approximation with our new calculation for the structure functions and compare them with the QCD parton distribution functions.     

The nonsinglet spin-dependent structure function evolution by Laplace and characteristics methods

We evaluate the non-singlet spin-dependent structure function g₁^NS at leading order (LO) and next-to-leading order (NLO) by using the Laplace-transform technique and method of characteristics and also obtain its first moment at NLO. The polarized non-singlet structure function results are compared with the data from HERMES (A. Airapetian et al., Phys. Rev. D 75, 012007 (2007)) and E143 (K. Abe et al. (E143 Collab.), Phys. Rev. D 58, 112003 (1998)) at LO and NLO analyses and the first-moment the result at NLO is compared with the result of the NLO GRSV2000 fit. Considering the solution, this method is valid at low- and large-x regions.