Computer Algebra for Multi-Loop Feynman Integrals

@article{RISC6688,
author = {E.D. Ocansey and C. Schneider},
title = {{Representation of hypergeometric products of higher nesting depths in difference rings}},
language = {english},
journal = {J. Symb. Comput.},
volume = {120},
pages = {1--50},
isbn_issn = {ISSN: 0747-7171},
year = {2024},
note = {arXiv:2011.08775 [cs.SC]},
refereed = {yes},
length = {50},
url = {https://doi.org/10.1016/j.jsc.2023.03.002}
}

@article{RISC6710,
author = {Koustav Banerjee and Peter Paule and Cristian-Silviu Radu and Carsten Schneider},
title = {{Error bounds for the asymptotic expansion of the partition function}},
language = {english},
abstract = {Asymptotic study on the partition function $p(n)$ began with the work of Hardy and Ramanujan. Later Rademacher obtained a convergent series for $p(n)$ and an error bound was given by Lehmer. Despite having this, a full asymptotic expansion for $p(n)$ with an explicit error bound is not known. Recently O'Sullivan studied the asymptotic expansion of $p^{k}(n)$-partitions into $k$th powers, initiated by Wright, and consequently obtained an asymptotic expansion for $p(n)$ along with a concise description of the coefficients involved in the expansion but without any estimation of the error term. Here we consider a detailed and comprehensive analysis on an estimation of the error term obtained by truncating the asymptotic expansion for $p(n)$ at any positive integer $n$. This gives rise to an infinite family of inequalities for $p(n)$ which finally answers to a question proposed by Chen. Our error term estimation predominantly relies on applications of algorithmic methods from symbolic summation. },
journal = {Rocky Mt J Math },
volume = {to appear},
pages = {?--?},
isbn_issn = {ISSN: 357596},
year = {2023},
note = {arXiv:2209.07887 [math.NT]},
refereed = {yes},
keywords = {partition function, asymptotic expansion, error bounds, symbolic summation},
length = {43},
url = {https://doi.org/10.35011/risc.22-13}
}

@article{RISC6740,
author = {Johannes Bluemlein and Nikolai Fadeev and Carsten Schneider},
title = {{Computing Mellin representations and asymptotics of nested binomial sums in a symbolic way: the RICA package}},
language = {english},
abstract = {Nested binomial sums form a particular class of sums that arise in the context of particle physics computations at higher orders in perturbation theory within QCD and QED, but that are also mathematically relevant, e.g., in combinatorics. We present the package RICA (Rule Induced Convolutions for Asymptotics), which aims at calculating Mellin representations and asymptotic expansions at infinity of those objects. These representations are of particular interest to perform analytic continuations of such sums. },
journal = {ACM Communications in Computer Algebra},
volume = {57},
number = {2},
pages = {31--34},
isbn_issn = {ISSN:1932-2240},
year = {2023},
month = {June},
note = {arXiv:2308.06042 [hep-ph]},
refereed = {yes},
keywords = {Mellin transform, asymptotic expansions, nested sums, nested integrals, computer algebra},
length = {4},
url = {https://doi.org/10.1145/3614408.3614410}
}

@article{RISC6497,
author = {J. Ablinger and J. Blümlein and A. De Freitas and M. Saragnese and C. Schneider and K. Schönwald},
title = {{New 2– and 3–loop heavy flavor corrections to unpolarized and polarized deep-inelastic scattering}},
language = {english},
abstract = {A survey is given on the new 2-- and 3--loop results for the heavy flavor contributions to deep--inelastic scattering in the unpolarized and the polarized case. We also discuss related new mathematical aspectsapplied in these calculations.},
journal = {SciPost Phys. Proc.},
number = {8},
pages = {137.1--137.15},
isbn_issn = {ISSN 2666-4003},
year = {2022},
note = {DIS2021, arXiv:2107.09350 [hep-ph]},
refereed = {yes},
length = {15},
url = {https://www.doi.org/10.21468/SciPostPhysProc.8.137}
}

@article{RISC6527,
author = {J. Blümlein and P. Marquard and C. Schneider and K. Schönwald},
title = {{The Two-Loop Massless Off-Shell QCD Operator Matrix Elements to Finite Terms}},
language = {english},
abstract = {We calculate the unpolarized and polarized two--loop massless off--shell operator matrix elements in QCD to $O(ep)$ in the dimensional parameter in an automated way. Here we use the method of arbitrary high Mellin moments and difference ring theory, based on integration-by-parts relations. This method also constitutes one way to compute the QCD anomalous dimensions. The presented higher order contributions to these operator matrix elements occur as building blocks in the corresponding higher order calculations upto four--loop order. All contributing quantities can be expressed in terms of harmonic sums in Mellin--$N$ space or by harmonic polylogarithms in $z$--space. We also perform comparisons to the literature. },
journal = {Nuclear Physics B},
volume = {980},
pages = {1--131},
isbn_issn = {ISSN 0550-3213},
year = {2022},
note = {arXiv:2202.03216 [hep-ph]},
refereed = {yes},
keywords = {QCD, Operator Matrix Element, 2-loop Feynman diagrams, computer algebra, large moment method},
length = {131},
url = {https://www.doi.org/10.1016/j.nuclphysb.2022.115794}
}