# Computer Algebra for Multi-Loop Feynman Integrals

### Project Lead

### Project Duration

01/09/2021 - 31/08/2025## Partners

### The Austrian Science Fund (FWF)

## Publications

### 2023

[Kauers]

### Order bounds for $C^2$-finite sequences

#### M. Kauers, P. Nuspl, V. Pillwein

Technical report no. 23-03 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). February 2023. Licensed under CC BY 4.0 International. [doi] [pdf]@

author = {M. Kauers and P. Nuspl and V. Pillwein},

title = {{Order bounds for $C^2$-finite sequences}},

language = {english},

abstract = {A sequence is called $C$-finite if it satisfies a linear recurrence with constant coefficients. We study sequences which satisfy a linear recurrence with $C$-finite coefficients. Recently, it was shown that such $C^2$-finite sequences satisfy similar closure properties as $C$-finite sequences. In particular, they form a difference ring. In this paper we present new techniques for performing these closure properties of $C^2$-finite sequences. These methods also allow us to derive order bounds which were not known before. Additionally, they provide more insight in the effectiveness of these computations. The results are based on the exponent lattice of algebraic numbers. We present an iterative algorithm which can be used to compute bases of such lattices.},

number = {23-03},

year = {2023},

month = {February},

keywords = {Difference equations, holonomic sequences, closure properties, algorithms},

length = {16},

license = {CC BY 4.0 International},

type = {RISC Report Series},

institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},

address = {Altenberger Straße 69, 4040 Linz, Austria},

issn = {2791-4267 (online)}

}

**techreport**{RISC6683,author = {M. Kauers and P. Nuspl and V. Pillwein},

title = {{Order bounds for $C^2$-finite sequences}},

language = {english},

abstract = {A sequence is called $C$-finite if it satisfies a linear recurrence with constant coefficients. We study sequences which satisfy a linear recurrence with $C$-finite coefficients. Recently, it was shown that such $C^2$-finite sequences satisfy similar closure properties as $C$-finite sequences. In particular, they form a difference ring. In this paper we present new techniques for performing these closure properties of $C^2$-finite sequences. These methods also allow us to derive order bounds which were not known before. Additionally, they provide more insight in the effectiveness of these computations. The results are based on the exponent lattice of algebraic numbers. We present an iterative algorithm which can be used to compute bases of such lattices.},

number = {23-03},

year = {2023},

month = {February},

keywords = {Difference equations, holonomic sequences, closure properties, algorithms},

length = {16},

license = {CC BY 4.0 International},

type = {RISC Report Series},

institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},

address = {Altenberger Straße 69, 4040 Linz, Austria},

issn = {2791-4267 (online)}

}

[Schneider]

### Representation of hypergeometric products of higher nesting depths in difference rings

#### E.D. Ocansey, C. Schneider

J. Symb. Comput. to appear, pp. ?-?. 2023. ISSN: 0747-7171. arXiv:2011.08775 [cs.SC]. [doi]@

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 = {to appear},

number = {?},

pages = {?--?},

isbn_issn = {ISSN: 0747-7171},

year = {2023},

note = {arXiv:2011.08775 [cs.SC]},

refereed = {yes},

length = {48},

url = {https://doi.org/10.1016/j.jsc.2023.03.002}

}

**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 = {to appear},

number = {?},

pages = {?--?},

isbn_issn = {ISSN: 0747-7171},

year = {2023},

note = {arXiv:2011.08775 [cs.SC]},

refereed = {yes},

length = {48},

url = {https://doi.org/10.1016/j.jsc.2023.03.002}

}

### 2022

[Schneider]

### New 2– and 3–loop heavy flavor corrections to unpolarized and polarized deep-inelastic scattering

#### J. Ablinger, J. Blümlein, A. De Freitas, M. Saragnese, C. Schneider, K. Schönwald

SciPost Phys. Proc.(8), pp. 137.1-137.15. 2022. ISSN 2666-4003. DIS2021, arXiv:2107.09350 [hep-ph]. [doi]@

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**{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}

}

[Schneider]

### The Two-Loop Massless Off-Shell QCD Operator Matrix Elements to Finite Terms

#### J. Blümlein, P. Marquard, C. Schneider, K. Schönwald

Nuclear Physics B 980, pp. 1-131. 2022. ISSN 0550-3213. arXiv:2202.03216 [hep-ph]. [doi]@

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}

}

**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}

}

[Schneider]

### The 3-loop anomalous dimensions from off-shell operator matrix elements

#### J. Blümlein, P. Marquard, C. Schneider, K. Schönwald

In: Proceedings of Loops and Legs in Quantum Field Theory, PoS(LL2022) 48, P. Marquard, M. Steinhauser (ed.)416, pp. 1-12. July 2022. ISSN 1824-8039. arXiv:2207.07943 [hep-ph]. [doi]@

author = {J. Blümlein and P. Marquard and C. Schneider and K. Schönwald},

title = {{The 3-loop anomalous dimensions from off-shell operator matrix elements}},

booktitle = {{Proceedings of Loops and Legs in Quantum Field Theory, PoS(LL2022) 48}},

language = {english},

abstract = {We report on the calculation of the three--loop polarized and unpolarized flavor non--singlet and the polarized singlet anomalous dimensions using massless off--shell operator matrix elements in a gauge--variant framework. We also reconsider the unpolarized two--loop singlet anomalous dimensions and correct errors in the foregoing literature.},

volume = {416},

pages = {1--12},

isbn_issn = {ISSN 1824-8039},

year = {2022},

month = {July},

note = { arXiv:2207.07943 [hep-ph]},

editor = {P. Marquard and M. Steinhauser},

refereed = {no},

keywords = {hypergeometric series, Appell-series, symbolic summation, coupled systems, partial linear difference equations},

length = {12},

url = {https://doi.org/10.22323/1.416.0048 }

}

**inproceedings**{RISC6528,author = {J. Blümlein and P. Marquard and C. Schneider and K. Schönwald},

title = {{The 3-loop anomalous dimensions from off-shell operator matrix elements}},

booktitle = {{Proceedings of Loops and Legs in Quantum Field Theory, PoS(LL2022) 48}},

language = {english},

abstract = {We report on the calculation of the three--loop polarized and unpolarized flavor non--singlet and the polarized singlet anomalous dimensions using massless off--shell operator matrix elements in a gauge--variant framework. We also reconsider the unpolarized two--loop singlet anomalous dimensions and correct errors in the foregoing literature.},

volume = {416},

pages = {1--12},

isbn_issn = {ISSN 1824-8039},

year = {2022},

month = {July},

note = { arXiv:2207.07943 [hep-ph]},

editor = {P. Marquard and M. Steinhauser},

refereed = {no},

keywords = {hypergeometric series, Appell-series, symbolic summation, coupled systems, partial linear difference equations},

length = {12},

url = {https://doi.org/10.22323/1.416.0048 }

}

[Schneider]

### Computer Algebra and Hypergeometric Structures for Feynman Integrals

#### J. Bluemlein, M. Saragnese, C. Schneider

In: Proceedings of Loops and Legs in Quantum Field Theory, PoS(LL2022) 041, P. Marquard, M. Steinhauser (ed.)416, pp. 1-11. 2022. ISSN 1824-8039. arXiv:2207.08524 [math-ph]. [doi]@

author = {J. Bluemlein and M. Saragnese and C. Schneider},

title = {{Computer Algebra and Hypergeometric Structures for Feynman Integrals}},

booktitle = {{Proceedings of Loops and Legs in Quantum Field Theory, PoS(LL2022) 041}},

language = {english},

abstract = {We present recent computer algebra methods that support the calculations of (multivariate) series solutions for (certain coupled systems of partial) linear differential equations. The summand of the series solutions may be built by hypergeometric products and more generally by indefinite nested sums defined over such products. Special cases are hypergeometric structures such as Appell-functions or generalizations of them that arise frequently when dealing with parameter Feynman integrals.},

volume = {416},

pages = {1--11},

isbn_issn = {ISSN 1824-8039},

year = {2022},

note = { arXiv:2207.08524 [math-ph]},

editor = {P. Marquard and M. Steinhauser},

refereed = {no},

keywords = {hypergeometric series, Appell-series, symbolic summation, coupled systems, partial linear difference equations},

length = {11},

url = {https://doi.org/10.22323/1.416.0041 }

}

**inproceedings**{RISC6619,author = {J. Bluemlein and M. Saragnese and C. Schneider},

title = {{Computer Algebra and Hypergeometric Structures for Feynman Integrals}},

booktitle = {{Proceedings of Loops and Legs in Quantum Field Theory, PoS(LL2022) 041}},

language = {english},

abstract = {We present recent computer algebra methods that support the calculations of (multivariate) series solutions for (certain coupled systems of partial) linear differential equations. The summand of the series solutions may be built by hypergeometric products and more generally by indefinite nested sums defined over such products. Special cases are hypergeometric structures such as Appell-functions or generalizations of them that arise frequently when dealing with parameter Feynman integrals.},

volume = {416},

pages = {1--11},

isbn_issn = {ISSN 1824-8039},

year = {2022},

note = { arXiv:2207.08524 [math-ph]},

editor = {P. Marquard and M. Steinhauser},

refereed = {no},

keywords = {hypergeometric series, Appell-series, symbolic summation, coupled systems, partial linear difference equations},

length = {11},

url = {https://doi.org/10.22323/1.416.0041 }

}

[Schneider]

### Error bounds for the asymptotic expansion of the partition function

#### Koustav Banerjee, Peter Paule, Cristian-Silviu Radu, Carsten Schneider

Technical report no. 22-13 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). September 2022. arXiv:2209.07887 [math.NT]. Licensed under CC BY 4.0 International. [doi] [pdf]@

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. },

number = {22-13},

year = {2022},

month = {September},

note = {arXiv:2209.07887 [math.NT]},

keywords = {partition function, asymptotic expansion, error bounds, symbolic summation},

length = {43},

license = {CC BY 4.0 International},

type = {RISC Report Series},

institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},

address = {Altenberger Straße 69, 4040 Linz, Austria},

issn = {2791-4267 (online)}

}

**techreport**{RISC6620,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. },

number = {22-13},

year = {2022},

month = {September},

note = {arXiv:2209.07887 [math.NT]},

keywords = {partition function, asymptotic expansion, error bounds, symbolic summation},

length = {43},

license = {CC BY 4.0 International},

type = {RISC Report Series},

institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},

address = {Altenberger Straße 69, 4040 Linz, Austria},

issn = {2791-4267 (online)}

}