Decoupling the NLO coupled DGLAP evolution equations: an analytic solution to pQCD

Abstract

Using repeated Laplace transform techniques, along with newly-developed accurate numerical inverse Laplace transform algorithms [1,2], we transform the coupled, integral-differential NLO singlet DGLAP equations first into coupled differential equations, then into coupled algebraic equations, which we can solve iteratively. After Laplace inverting the algebraic solution analytically, we numerically invert the solutions of the decoupled differential equations. Finally, we arrive at the decoupled NLO evolved solutionswhere Fs and G are known functions-determined using the DGLAP splitting functions up to NLO in the strong coupling constant αs(Q 2 ). The functions Fs0(x) ≡ Fs(x, Q 2 0 ) and G0(x) ≡ G(x, Q 2 0 ) are the starting functions for the evolution at Q 2 = Q 2 0 . This approach furnishes us with a new tool for readily obtaining, independently, the effects of the starting functions on either the evolved gluon or singlet structure functions, as a function of both Q 2 and Q 2 0 . It is not necessary to evolve coupled integral-differential equations numerically on a two-dimensional grid, as is currently done. The same approach can be used for NLO non-singlet distributions where it is simpler, only requiring one Laplace transform. We make successful NLO numerical comparisons to two non-singlet distributions, using NLO quark distributions published by the MSTW collaboration [3], over a large range of x and Q 2 . Our method is readily generalized to higher orders in the strong coupling constant αs(Q 2 ).

Applications of the leading-order Dokshitzer-Gribov-Lipatov-Altarelli-Parisi evolution equations to the combined HERA data on deep inelastic scattering

“…(5) and (6) providedF s (v, Q 2 ) andĜ(v, Q 2 ) vanish sufficiently rapidly for v → 0 that the products in Eqs. (2) and (3) vanish as a power of 1/s for s → ∞. These conditions are satisfied in practice.…”

“…In the present paper, we use the method developed in detail in [1,2] to solve the coupled DGLAP evolution equations for F s and G. We will not give the details here, but note that our method is based on Laplace transforms. We first rewrite the evolution equations in terms of the variables v = ln (1/x) and Q 2 instead of x and Q 2 .…”

“…We have discussed the generalization of these results to next-to-leading order in [2]. The decoupling of the evolution equations in that case requires a double Laplace transform and is considerably more complicated in detail, but can still be carried through analytically.…”

“…In recent papers [1,2], we showed that it is possible to solve the coupled leading order (LO) Dokshitzer-GribovLipatov-Altarelli-Parisi (DGLAP) evolution equations [3][4][5] for the singlet quark structure function F s (x, Q 2 ) = i x[q i (x, Q 2 ) +q i (x, Q 2 )] and the gluon distribution G(x, Q 2 ) = xg(x, Q 2 ) directly using a method based on Laplace transforms. While the method is formally equivalent through the known connection between Laplace and Mellin transforms [6] to methods based on the latter -see, e.g.…”

“…These do not require that we work on a pre-assigned numerical grid, and make the solution of the evolution equations at arbitrary values x and Q 2 straightforward on desktop or laptop computers. We have extended the Laplace method elsewhere [2] to next-to-leading order (NLO) in α s , including to non-singlet distributions, but will not pursue that extension here.…”