2 00 1 A fast and precise method to solve the Altarelli - Parisi equations in x space

A numerical method to solve linear integro-differential equations is presented. This method has been used to solve the QCD Altarelli-Parisi evolution equations within the H1 Collaboration at DESY-Hamburg. Mathematical aspects and numerical approximations are described. The precision of the method is discussed. This article is an extended version of an unpublished note [25]. In a recent publication [26], P. Ratcliffe proposed a numerical method similar to the one described in ref. [25]. In addition, he pointed out the problem of non commutativity of the Next-to-Leading-Logarithmic-Approximation (NLLA) Altarelli-Parisi kernels and the fact that we did not account for in ref. [25]. Our present extension of [25] therefore concerns the account for non commutativity. After having given the proper modification of the numerical method (section 2.1) we explicitly show that non commutativity effects can safely be neglected provided the Q evolution is performed, as usual, from points to points on a grid (section 2.2). The rest of the paper is untouched, in particular the references are not updated. Introduction Inclusive Deep Inelastic lepton-hadron Scattering (DIS)cross section measurements offer a powerful test of perturbative Quantum Chromo Dynamics (pQCD) [1]. The DIS process l(k)h(P ) → l(k)X (here we shall only consider the case of charged leptons l in order to simplify the discussion) kinematic is described by two Lorentz invariants. One usually chooses the transferred momentum squared Q = −q ≡ (k−k) and the Bjorken variable x = Q/(2P.q). In terms of these two kinematic variables, the internal dynamics of the 1 struck hadron enters the cross section via three structure functions: F1(x,Q ), F2(x,Q ) and F3(x,Q ). Because only F2(x,Q ) contributes significantly to the cross section for Q < M Z0c , we shall concentrate on this structure function in the present paper. Within the framework of pQCD, F2(x,Q ) is given by the convolution in x of the well known Wilson coefficients [2] and of the parton densities inside the hadron (we shall work in the MS factorisation and renormalisation scheme). The densities of partons, consisting of quarks and gluons, are computed from the solution of the Altarelli-Parisi (AP) equations [1]. According to [2], it is useful to define the gluon g, a non-singlet qNS and a singlet Σ quark combination densities. They are the solution of the set of AP integro-differential equations: ∂qNS ∂t = ∫ 1 x dw w PNS(w, t)qNS( x w , t) (1)