The use of Mellin-Barnes Integrals for the Calculation of Two-loop massive Operator Matrix Elements

Abstract

We calculate the O ¡ α 2 s ¢ gluonic massive operator matrix elements for unpolarized and polarized heavy flavor production, at asymptotic values Q 2 £ m 2 . This calculation is done in Mellin space and without applying the integration-by-parts method, which was used in a former x-space calculation in Ref.[1], the results of which we confirm. However, our results present themselves in a more compact way, needing a basis of only 2 compared to 48 basic functions in the unpolarized case, and 24 in the polarized case. The results are obtained in two ways, by hypergeometric functions and by the use of Mellin-Barnes integrals, on which we put the emphasis in this paper. XI International Workshop on Advanced Computing and Analysis Techniques in Physics Research

The theory of deeply inelastic scattering

“…The results of [71] were confirmed in [270] and the O(a 2 s ε) corrections were derived in [271]. In the polarized case the NLO heavy flavor Wilson coefficients were computed for the asymptotic region only [71,72]. At LO and NLO also heavy flavor corrections for the charged current case have been calculated [272].…”

“…One problem in this context is that m 2 c /m 2 b ≈ 1/10 and one cannot consider charm quarks as massless at µ 2 = m 2 b . The matching conditions for the flavor non-singlet and singlet parton distributions have been given at NLO in [283] and NNLO in [67] and the transition functions in [67,71,72,283,284], with first general N results at NNLO in [194,195]. Different descriptions to interpolate between the heavy-quark threshold region and the asymptotic region are considered in the literature.…”

“…Heavy quark corrections are known to 2-loop order [65,66] and for larger scales a series of Mellin moments has been calculated to 3-loop order [67]. In the polarized case the level of 2-loop corrections has been reached for massless and massive corrections [68][69][70][71][72], with first results at 3-loop order [73]. The present massless and massive 3-and 4loop calculations require powerful computer algebra systems and packages like FORM [74], MATAD [75], and Sigma [76], cf.…”

Massive 3-loop ladder diagrams for quarkonic local operator matrix elements

“…[7,8]. The 2-loop corrections in the neutral current case have been calculated in [4,[9][10][11][12][13] and for the charged current case up to NLO for Q 2 ≫ m 2 in [14]. For the structure function F L (x, Q 2 ) the general result in the asymptotic case was obtained at 3-loop order in Ref.…”