Nonperturbative contribution to the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi and Gribov-Levin-Ryskin equation

By studying the nonperturbative contribution to the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi and Gribov-Levin-Ryskin equation, it is found that (i) the nonperturbative contribution suppresses the evolution rate at the low $Q^2$, small-x region; (ii) the nonperturbative contribution weakens the shadowing effect. The method in this paper suggests a smooth transition from the low $Q^2$ ("soft"), where nonperturbative contribution dominates, to the large $Q^2$ ("hard") region, where the perturbative contribution dominates and the nonperturbative contribution can be neglected. PACS numbers:12.38.Aw, 13.60.Hb