ar X iv : h ep - p h / 98 10 22 3 v 1 2 O ct 1 99 8 . Q 2 evolution of parton distributions at low x . Soft initial conditions

We investigate the Q evolution of parton distributions at small x values, recently obtained in the case of soft initial conditions. The results are in excellent agreement with deep inelastic scattering experimental data from HERA. The measurements of the deep-inelastic scattering structure function F2 in HERA [1] have permitted the access to a very interesting kinematical range for testing the theoretical ideas on the behavior of quarks and gluons carrying a very low fraction of momentum of the proton, the so-called small x region. In this limit one expects that non-perturbative effects may give essential contributions. However, the resonable agreement between HERA data and the NLO approximation of perturbative QCD that has been observed for Q > 1GeV (see the recent review in [2]) indicates that perturbative QCD could describe the evolution of structure functions up to very low Q values, traditionally explained by soft processes. It is of fundamental importance to find out the kinematical region where the well-established perturbative QCD formalism can be safely applied at small x. The standard program to study the small x behavior of quarks and gluons is carried out by comparison of data with the numerical solution of the DGLAP equations 1 by fitting the parameters of the x profile of partons at some initial Q0 and the QCD energy scale Λ (see, for example, [3]). However, if one is interested in analyzing exclusively the small x region, there is the alternative of doing a simpler analysis by using some of the existing analytical solutions of DGLAP in the small x limit (see [2] for review). This was done so in Ref. [4]-[6] where it was pointed out that the HERA small x data can be interpreted in terms of the so called doubled asymptotic scaling phenomenon related to the asymptotic behaviour of the DGLAP evolution discovered in [7] many years ago. Here we will illustrate results obtained recently in [6]. 1. Thus, our purpose is to demonstrate the small x asymptotic form of parton distributions in the framework of the DGLAP equation starting at some Q0 with the flat function: fa(Q 2 0) = Aa (a = q, g), (1) 1 At small x there is another approach based on the BFKL equation, whose application is out of the scope of this work. where fa are the parton distributions multiplied by x and Aa are unknown parameters that have to be determined from data. Through this work at small x we neglect the non-singlet quark component. In [6] an effective method to reproduce the x-dependence of parton distributions has been developed. It is based on a separation of the singular and regular parts of the exact solution for the moments of parton distributions. The method allows in simplest way to reproduce the leading order (LO) results [4] and to construct the x-dependence of parton distributions at next-to-leading order (NLO): fa(x,Q ) = f a (x,Q ) + f a (x,Q ) and f a (x,Q ) = A−a (Q , Q0)exp(−d−(1)s−D−(1)p) + O(x) f g (x,Q ) = Ag (Q , Q0)I0(σ)exp(−d+(1)s−D+(1)p) + O(ρ) (2) f q (x,Q ) = Aq (Q , Q0) [ (1− d̄q±(1)α(Q ))ρI1(σ) + 20α(Q )I0(σ) ] ·exp(−d+(1)s−D+(1)p) +O(ρ) F2(x,Q ) = e · ( fq(x,Q ) + 2 3 fα(Q)fg(x,Q ) ) where Iν(σ) is modified Bessel function, s and p are given by s = ln(α(Q 2 0)/α(Q )), p = α(Q0)− α(Q ), and