The predictions of the charm structure function exponents behaviour at low x in deep inelastic scattering

We use the hard (Lipatov) pomeron for the low-x gluon distribution and provide a compact formula for the ratio $R^{c} =F_{L}^{c}/F_{2}^{c}$ that is useful to extract the charm structure function from the reduced charm cross-section, in particular at DESY HERA. Our results show that this ratio is independent of x and independent of the DGLAP evolution of the gluon PDF. As a result, we show that the charm structure function and the reduced charm cross-section exponents do not have the same behaviour at very low x. This difference is independent of the input gluon distribution functions and predicts the non-linear effects and some evidence for shadowing and antishadowing at HERA and RHIC.