Geometrical scaling in charm structure function ratios

By using a Laplace-transform technique, we solve the next-to-leading-order master equation for charm production and derive a compact formula for the ratio $R^c=\frac{F_L^{{}^{c\bar{c}}}}{F_2^{{}^{c\bar{c}}}}$, which is useful for extracting the charm structure function from the reduced charm cross section, in particular, at DESY HERA, at small x. Our results show that this ratio is independent of xat small x. In this method of determining the ratios, we apply geometrical scaling in charm production in deep inelastic scattering (DIS). Our analysis shows that the renormalization scales have a sizable impact on the ratio Rcat high $Q^2$. Our results for the ratio of the charm structure functions are in a goodagreement with some phenomenological models.