An analytic determination of the gluon distribution function from HERA combined data for the proton structure function $F_2^{\gamma p}(x,Q^2)$

We exploit a new method based on Laplace transforms to find an exact analytic solution of the leading-order (LO) gluon distribution function $G(x,Q^2)=xg(x,Q^2)$ directly from the proton structure function $F_2^{\gamma p}(x,Q^2)$. The method uses only the LO DGLAP evolution equation for $F_2^{\gamma p}(x,Q^2)$ without making any assumptions about the shape of the low $x$ gluon distributions. Using an accurate, new global parametrization of the complete set of low Bjorken-$x$ HERA data on $F_2^{\gamma p}(x,Q^2)$, including the data at very low $Q^2$, we determine the gluon distribution $G(x,Q^2)$ for massless quarks and compare our results to other published gluon distributions. An extension of the present method to determine the NLO corrections to the gluon distributions is also presented.