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

Abstract

A matrix-based approach to numerical integration of the DGLAP evolution equations is presented. The method arises naturally on discretisation of the Bjorken x variable, a necessary procedure for numerical integration. Owing to peculiar properties of the matrices involved, the resulting equations take on a particularly simple form and may be solved in closed analytical form in the variable t = ln(α 0 /α). Such an approach affords parametrisation via data x bins, rather than fixed functional forms. Thus, with the aid of the full correlation matrix, appraisal of the behaviour in different x regions is rendered more transparent and free of pollution from unphysical cross-correlations inherent to functional parametrisations. Computationally, the entire programme results in greater speed and stability; the matrix representation developed is extremely compact. Moreover, since the parameter dependence is linear, fitting is very stable and may be performed analytically in a single pass over the data values.

Resummation of the jet broadening in DIS

“…This therefore remains much more manageable than the O (n 3 ) factor that is relevant with a non-uniform grid, while preserving formal p th order accuracy at all values of x. We of course maintain the property that the evaluation of P ⊗ q requires O (n 2 /2) operations We point out that the approach used here differs significantly in philosophy from that in say [45] in that the evolution in Q 2 is performed with a Runge Kutta algorithm, rather than by a formal analytic solution to the evolution equations expressed in terms of a power series of P . This gives considerable simplicity because, for example, the inclusion of NNLL splitting functions and 3-loop running for α s requires no extra analytical calculations.…”

“…As has been exploited by [44,45], on a uniform grid, away from the edges of the integral, the property (F.8) allows a further simplification: 11) i.e. P ij is only a function of i − j.…”

“…In [44] this method has been implemented only for p = 1, while [45] has applied it to the general p case. For the general-p case some subtleties arise with grid edges, because eqs.…”

“…(F.6) and (F.7) taken together imply a sum over points outside one's grid. The approach in [45] is to sacrifice the formal accuracy of the approach at large x, so that for the integration region between grid points j − 1 and j, the order is min(p, j) th order.…”

“…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 .…”