A semianalytical method to evolve parton distributions

Abstract

We present a new method to solve in a semianalytical way the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi evolution equations at NLO order in the x-space. The method allows to construct an evolution operator expressed in form of a rapidly convergent series of matrices, depending only on the splitting functions. This operator, acting on a generic initial distribution, provides a very accurate solution in a short computer time (only a few hundredth of second). As an example, we apply the method, useful to solve a wide class of systems of integrodifferential equations, to the polarized parton distributions.

Matrix approach to a numerical solution of the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi evolution equations

“…With regard to the large value of t chosen for testing, we now comment on the method proposed in [8]: there the truncation of the power series in t at no more than 12 terms is justified on the basis of rapid convergence. Direct examination of the full finite expansion calculated here suggests this might be illusory: some coefficients do not become truly small until much later in the series and thus it may be that convergence is only achieved in [8] thanks to the not large values of t considered there.…”

“…Fortunately, the compact form of the matrices permits cheap computer storage of a large number of terms. We note here that the natural expansion is a power series in s and not t. Indeed, while for small t the two are essentially equivalent, for large values (where s → 0) the expansion in t contains only a part of the series expressed in terms of s. In the following section we shall comment further on this point with regard to the approach of [8].…”

“…The evolution equation now becomeṡ 8) where the sum runs up to N, the order of the approximation. Since the matrices P (n) are still all of the banded triangular type (i.e., commuting), an exponential solution of the form of eq.…”

“…Before proceeding it should be pointed out that an approach related to that proposed here has been presented by Santorelli and Scrimieri [8]. Rather than comment immediately on the similarities and differences, it is probably more convenient to highlight such in the following, as and when appropriate.…”

Resummation of the jet broadening in DIS

“…The parton distribution at arbitrary x is then defined to be equal to a linear combination of the parton distribution on neighbouring grid points Approaches of this kind have been adopted by many people, for example [42][43][44][45], and indeed are a standard numerical method [46]. In the algorithms of [42,43] the grid is non-uniform in ln x and there are roughly n 2 /2 elements of the P ij . Their evaluation leads to considerable overheads (analytical or numerical depending on the approach), while the requirement that they be held in memory during program execution (rather than on disk) places an upper limit on the value of n that can be used.…”