MT: A Mathematica package to compute convolutions

Abstract

We introduce the Mathematica package MT which can be used to compute, both analytically and numerically, convolutions involving harmonic polylogarithms, polynomials or generalized functions. As applications contributions to next-to-next-to-next-to leading order Higgs boson production and the Drell-Yan process are discussed. * Corresponding author MT can be used to compute such convolutions. Method of solution:MT is implemented in Mathematica and we provide several functions in order to perform transformations to Mellin space, manipulations of the expressions, and inverse Mellin transformations. Restrictions on the complexity of the problem:In case the weight of the input quantities is too high the tables for the (inverse) Mellin transforms have to be extended. In the current implementation the tables contain expressions up to weight eight, code for the generation of tables of even higher weight is provided, too.MT can only handle convolutions of expressions involving harmonic polylogarithms, plus distributions and polynomials in the partonic variable x. Typical running time:In general the run time for the individual operations is at most of the order of a few minutes (depending on the speed and memory of the computer).

Scale dependence and collinear subtraction terms for Higgs production in gluon fusion at N3LO

“…They agree with the ones given in [41] after compensating for the factor of 1/z that was not included in that publication.…”

“…Let us remark that our method is different from the technique used in [41], where the convolutions were calculated in Mellin space (where convolutions turn into ordinary multiplications) and the problem was essentially the calculation of the inverse Mellin-transform.…”

“…We are not aware of results in [41] that involve the flavour-dependent quark-quark kernels. We close this section by giving the expressions for the four LO splitting kernels.…”

“…Since we chose to expand all our results in a = αs π as opposed to αs 4π as in [51,52], our kernels P (n) ij differ from the ones in the reference by a factor of 1 Also, since by P (n) qg we mean the emission of a single quark of a given flavour, we differ from the expression in [51,52] which parametrises the emission of any quark, by a factor of 1 2N F . Furthermore, there is also a conventional difference to the splitting kernels used in [41]. The authors of that publication use the quark-quark splitting kernel as defined in eq.…”

“…In [39,40], the real-virtual and double-real master integrals of the NNLO cross section have been calculated to higher orders in ǫ. In [41], the convolutions of collinear splitting kernels with lower-order partonic cross sections have been computed, which is also an ingredient for our result and has been re-derived in this work. Very recently, the soft limits of all master integrals appearing in triple-real radiation corrections (i.e.…”