On the reduction of generalized polylogarithms to Li n and Li2,2 and on the evaluation thereof

Abstract

Abstract:We give expressions for all generalized polylogarithms up to weight four in terms of the functions log, Li n , and Li 2,2 , valid for arbitrary complex variables. Furthermore we provide algorithms for manipulation and numerical evaluation of Li n and Li 2,2 , and add codes in Mathematica and C++ implementing the results. With these results we calculate a number of previously unknown integrals, which we add in appendix C.

Two loop virtual corrections to $$b\rightarrow (d,s)\ell ^+\ell ^-$$ b → ( d , s )

“…The results are valid for arbitrary momentum transfer and masses, when the light mass is neglected. For the MIs we used the works of [13,14]. Our computation extends previous work on b → (d, s) + − transitions [8][9][10][11] and is new for c → u + − transitions.…”

“…m 2 i → m 2 i − iη. We write the HPLs as generalized/Goncharov polylogarithms (GPLs); see, e.g., [13] (and the references therein) and feed them into the computer package lieevaluate [14]. Other packages for the numerical evaluation of GPLs can be found in [23,24].…”

“…While evaluating the expressions with lieevaluate, we encounter the undefined function θ(0) and the function MyP [14] that is divergent for some arguments of GPLs [14]. We regulate the former by a perturbation of these arguments; see [14].…”

“…We regulate the former by a perturbation of these arguments; see [14]. We have numerically checked that the form factors are insensitive to the actual choice of such perturbations.…”

NNLO QCD corrections to γ + η c (η b ) exclusive production in electron-positron collision

“…Goncharov Polylogarithms can be numerically evaluated with the GINAC implementation [32,33], and can also be transformed into functions of ln, Li n and Li 22 up to weight four, by using the method described in ref. [34]. The numerical evaluation of integrals over polylogarithms and integrals over elliptic functions and polylogarithms can be simply performed by software MATHEMATICA.…”