The study of heavy quarks pair production at the LHeC and FCC-eh in the collinear generalized double asymptotic scaling (DAS) approach

Abstract

In this paper, the main idea is the prediction of the linear and nonlinear behavior of heavy quarks pair production cross section at the deep inelastic scattering leptoproduction at the future electron–proton colliders, i.e., the Large Hadron electron Collider (LHeC) and the Future Circular Collider electron-hadron (FCC-eh) at the leading order (LO) and the next-to-leading order (NLO) approximation in perturbative quantum chromodynamics (pQCD) at small x. By considering the nonlinear corrections to the gluon distribution function in the framework of the nonlinear GLR-MQ-ZRS evolution equation and using the Laplace transform technique, based on the parametrization method of structure function $F_{2}(x,Q^{2})$ and its derivatives, we determine the linear and nonlinear evolution of production cross section and structure functions $F_{2}(x,Q^{2})$ and $F_{L}(x,Q^{2})$ and their ratio with respect to variables x and $Q^{2}$ for heavy quarks of charm and beauty. To study the heavy quark production processes, we use the collinear generalized double asymptotic scaling (DAS) approach in kinematics range of the LHeC and FCC-eh colliders. The computed results are compared with experimental data from HERA and the results of NNPDF4.0 and CT18 Collaborations.

Appendices

Appendix A

The NLO collinear coefficient function of a heavy quark production process are rather lengthy and not published in print, they are only available as computer codes [95]. For the purpose of this paper, it is sufficient to work in the high energy regime at $x\ll 1$, the coefficient functions in the LO and NLO orders of the perturbation theory have the compact form [44]

$$\begin{aligned} B_{2,g}^{(0)}(x,\xi )&=-2x\beta [(1-4x(2-\xi )(1-x)-(1-2x(1-2\xi )+2x^{2}(1-6\xi -4\xi ^{2}))L(\beta )], \nonumber \\ B_{2,L}^{(0)}(x,\xi )&=8x^{2}\beta [(1-x)-2x\xi L(\beta )], \nonumber \\ B_{k,g}^{(1)}(x,\xi )&=\beta [R_{k,g}^{(1)}(1,\xi )+4C_{A} B_{k,g}^{(0)}(1,\xi )L_{\mu }], \quad L_{\mu }=\ln (\frac{4m^{2}}{\mu ^{2}}), \end{aligned}$$

(29)

with

$$\begin{aligned} R_{2,g}^{(1)}(1,\xi )&=\frac{8}{9}C_{A}[5+(13-10\xi )J(\xi )+6(1-\xi )I(\xi )], \nonumber \\ R_{L,g}^{(1)}(1,\xi )&=-\frac{16}{9}C_{A}x_{1}[1-12\xi -(3+4\xi (1-6\xi ))J(\xi )+12\xi (1+3\xi )I(\xi )], \nonumber \\ B_{2,g}^{(0)}(1,\xi )&=\frac{2}{3}[1+2(1-\xi )J(\xi )], \nonumber \\ B_{L,g}^{(0)}(1,\xi )&=\frac{4}{3}x_{1}[1+6\xi -4\xi (1+3\xi )J(\xi )], \end{aligned}$$

(30)

where

$$\begin{aligned} I(\xi )&=-\sqrt{x_{1}}[\zeta _{2}+\frac{1}{2}\ln ^{2}(t)-\ln (\xi x_{1}) \ln (t)+2Li_{2}(-t)], \nonumber \\ J(\xi )&=-\sqrt{x_{1}}\ln (t), \quad t= \frac{1-\sqrt{x_{1}}}{1+\sqrt{x_{1}}}, \nonumber \\ \beta ^{2}&=1-\frac{4\xi x}{1-x}, \nonumber \\ L(\beta )&=\frac{1}{\beta }\ln \frac{1+\beta }{1-\beta } \nonumber \\ Li_{2}(x)&=-\int _{0}^{1}\frac{dy}{y}\ln (1-xy), \end{aligned}$$

(31)

where $Li_{2}(x)$ is the dilogarithmic function.

Appendix B

The effective parameters in parametrization of the proton structure function $F_{2}(x,Q^{2})$ are defined by the following forms

$$\begin{aligned} D(Q^{2})&=\frac{Q^{2}(Q^{2}+\lambda M^{2})}{(Q^{2}+ M^{2})^{2}}, \nonumber \\ A_{0}(Q^{2})&=a_{00}+a_{01}L_{2},\quad A_{i}(Q^{2})=\sum _{k=0}^{2}a_{ik}L_{2}^{k},i=(1,2), \end{aligned}$$

(32)

with the logarithmic terms L as

$$\begin{aligned} L=\ln \left( \frac{1}{x}\right) +L_{1},\quad L_{1}=\ln \left( \frac{Q^{2}}{Q^{2}+\mu ^{2}}\right) , \quad L_{2}=\ln \left( \frac{Q^{2}+\mu ^{2}}{\mu ^{2}}\right) , \end{aligned}$$

(33)

where the effective parameters M and $\mu ^{2}$ are the effective mass and a scale factor, respectively. The values of effective parameters from fit of experimental data [36,37,38] are given by

$$\begin{aligned} \mu ^{2}&=2.82\pm 0.29\, \textrm{GeV}^{2}, \quad M^{2}=0.753\pm 0.008\,\textrm{GeV}^{2}, \nonumber \\ n&=11.49\pm 0.99, \qquad \lambda =2.430\pm 0.153, \end{aligned}$$

(34)

and

$$\begin{aligned} a_{00}&=0.255\pm 0.016, \qquad a_{01}=0.1475\pm 0.03025, \nonumber \\ a_{10}&=8.205\times 10^{-4}\pm 4.62\times 10^{-4}, \quad a_{11} =-5.148\times 10^{-2}\pm 8.19\times 10^{-3},\nonumber \\ a_{12}&=-4.725\times 10^{-3}\pm 1.01\times 10^{-3}, \quad a_{20} =2.170\times 10^{-3}\pm 1.42\times 10^{-4}, \nonumber \\ a_{21}&=1.244\times 10^{-2}\pm 8.56\times 10^{-4}, \quad a_{22} =5.958\times 10^{-4}\pm 2.32\times 10^{-4}. \end{aligned}$$

(35)