Analytic derivation of the leading-order gluon distribution function $G(x,Q^2)=xg(x,Q^2)$ from the proton structure function $F_2^{\gamma p}(x,Q^2)$. II. Effect of heavy quarks

Abstract

We extend our previous derivation of an exact expression for the leading-order (LO) gluon distribution function $G(x,Q^2)=xg(x,Q^2)$ from the DGLAP evolution equation for the proton structure function $F_2^{\gamma p}(x,Q^2)$ for deep inelastic $\gamma^* p$ scattering to include the effects of heavy-quark masses. We derive the equation for $G(x,Q^2)$ in two different ways, first using our original differential-equation method, and then using a new method based on Laplace transforms. The results do not require the use of the gluon evolution equation, or, to good approximation, knowledge of the individual quark distributions. Given an analytic expression that successfully reproduces the known experimental data for $F_2^{\gamma p}(x,Q^2)$ in a domain ${\cal D}(x,Q^2)$--where $x_{\rm min}(Q^2) \le x \le x_{\rm max}(Q^2)$, $Q^2_{\rm min}\le Q^2\le Q^2_{\rm max}$ of the Bjorken variable $x$ and the virtuality $Q^2$--$G(x,Q^2)$ is uniquely determined in the same domain. As an application of the method, we construct a new global parametrization of the complete set of ZEUS data on $F_2^{\gamma p}(x,Q^2)$, and use this to determine the 5 quark gluon distribution, $G(x,Q^2)$, for massless $u, d, s$ and massive $c, b$ quarks and discuss the mass effects evident in the result. We compare these results to the gluon distributions for CTEQ6L, and in the domain ${\cal D}(x,Q^2)$ where they should agree, they do not; the discrepancy is due to the fact that the CTEQ6L results do not give an accurate description of the ZEUS $F_2^{\gamma p}(x,Q^2)$ experimental data. We emphasize that our method for obtaining the LO gluon distribution connects $G(x,Q^2)$ {\em directly} to the proton structure function without either the need for individual parton distributions or the gluon evolution equation.