# Computer Algebra for Multi-Loop Feynman Integrals

### Project Lead

### Project Duration

01/09/2021 - 31/08/2025### Project URL

Go to Website## Partners

### The Austrian Science Fund (FWF)

## Publications

### 2024

[de Freitas]

### The first-order factorizable contributions to the three-loop massive operator matrix elements $A_{Qg}^{(3)}$ and $Delta A_{Qg}^{(3)}$

#### J. Ablinger, A. Behring, J. Bluemlein, A. De Freitas, A. von Manteuffel, C. Schneider, K. Schoenwald

Nuclear Physics B 999(116427), pp. 1-42. 2024. ISSN 0550-3213. arXiv:2311.00644 [hep-ph]. [doi]@

author = {J. Ablinger and A. Behring and J. Bluemlein and A. De Freitas and A. von Manteuffel and C. Schneider and K. Schoenwald},

title = {{The first--order factorizable contributions to the three--loop massive operator matrix elements $A_{Qg}^{(3)}$ and $Delta A_{Qg}^{(3)}$}},

language = {english},

abstract = {The unpolarized and polarized massive operator matrix elements $A_{Qg}^{(3)}$ and $Delta A_{Qg}^{(3)}$contain first--order factorizable and non--first--order factorizable contributions in the determining difference or differential equations of their master integrals. We compute their first--order factorizable contributions in the single heavy mass case for all contributing Feynman diagrams. Moreover, we present the complete color--$zeta$ factors for the cases in which also non--first--order factorizable contributions emerge in the master integrals, but cancel in the final result as found by using the method of arbitrary high Mellin moments. Individual contributions depend also on generalized harmonic sums and on nested finite binomial and inverse binomial sums in Mellin $N$--space, and correspondingly, on Kummer--Poincar'e and square--root valued alphabets in Bjorken--$x$ space. We present a complete discussion of the possibilities of solving the present problem in $N$--space analytically and we also discuss the limitations in the present case to analytically continue the given $N$--space expressions to $N in mathbb{C}$ by strict methods. The representation through generating functions allows a well synchronized representation of the first--order factorizable results over a 17--letter alphabet. We finally obtain representations in terms of iterated integrals over the corresponding alphabet in $x$--space, also containing up to weight {sf w = 5} special constants, which can be rationalized to Kummer--Poincar'e iterated integrals at special arguments. The analytic $x$--space representation requires separate analyses for the intervals $x in [0,1/4], [1/4,1/2], [1/2,1]$ and $x > 1$. We also derive the small and large $x$ limits of the first--order factorizable contributions. Furthermore, we perform comparisons to a number of known Mellin moments, calculated by a different method for the corresponding subset of Feynman diagrams, and an independent high--precision numerical solution of the problems.},

journal = {Nuclear Physics B},

volume = {999},

number = {116427},

pages = {1--42},

isbn_issn = {ISSN 0550-3213},

year = {2024},

note = {arXiv:2311.00644 [hep-ph]},

refereed = {yes},

keywords = {Feynman diagram, massive operator matrix elements, computer algebra, differential equations, difference equations, coupled systems, nested integrals, nested sums},

length = {42},

url = {https://doi.org/10.1016/j.nuclphysb.2023.116427}

}

**article**{RISC6755,author = {J. Ablinger and A. Behring and J. Bluemlein and A. De Freitas and A. von Manteuffel and C. Schneider and K. Schoenwald},

title = {{The first--order factorizable contributions to the three--loop massive operator matrix elements $A_{Qg}^{(3)}$ and $Delta A_{Qg}^{(3)}$}},

language = {english},

abstract = {The unpolarized and polarized massive operator matrix elements $A_{Qg}^{(3)}$ and $Delta A_{Qg}^{(3)}$contain first--order factorizable and non--first--order factorizable contributions in the determining difference or differential equations of their master integrals. We compute their first--order factorizable contributions in the single heavy mass case for all contributing Feynman diagrams. Moreover, we present the complete color--$zeta$ factors for the cases in which also non--first--order factorizable contributions emerge in the master integrals, but cancel in the final result as found by using the method of arbitrary high Mellin moments. Individual contributions depend also on generalized harmonic sums and on nested finite binomial and inverse binomial sums in Mellin $N$--space, and correspondingly, on Kummer--Poincar'e and square--root valued alphabets in Bjorken--$x$ space. We present a complete discussion of the possibilities of solving the present problem in $N$--space analytically and we also discuss the limitations in the present case to analytically continue the given $N$--space expressions to $N in mathbb{C}$ by strict methods. The representation through generating functions allows a well synchronized representation of the first--order factorizable results over a 17--letter alphabet. We finally obtain representations in terms of iterated integrals over the corresponding alphabet in $x$--space, also containing up to weight {sf w = 5} special constants, which can be rationalized to Kummer--Poincar'e iterated integrals at special arguments. The analytic $x$--space representation requires separate analyses for the intervals $x in [0,1/4], [1/4,1/2], [1/2,1]$ and $x > 1$. We also derive the small and large $x$ limits of the first--order factorizable contributions. Furthermore, we perform comparisons to a number of known Mellin moments, calculated by a different method for the corresponding subset of Feynman diagrams, and an independent high--precision numerical solution of the problems.},

journal = {Nuclear Physics B},

volume = {999},

number = {116427},

pages = {1--42},

isbn_issn = {ISSN 0550-3213},

year = {2024},

note = {arXiv:2311.00644 [hep-ph]},

refereed = {yes},

keywords = {Feynman diagram, massive operator matrix elements, computer algebra, differential equations, difference equations, coupled systems, nested integrals, nested sums},

length = {42},

url = {https://doi.org/10.1016/j.nuclphysb.2023.116427}

}

[de Freitas]

### The non-first-order-factorizable contributions to the three-loop single-mass operator matrix elements $A_{Qg}^{(3)}$ and $Delta A_{Qg}^{(3)}$

#### J. Ablinger, A. Behring, J. Bluemlein, A. De Freitas, A. von Manteuffel, C. Schneider, K. Schoenwald

Physics Letter B 854(138713), pp. 1-8. 2024. ISSN 0370-2693. arXiv:2403.00513 [[hep-ph]. [doi]@

author = {J. Ablinger and A. Behring and J. Bluemlein and A. De Freitas and A. von Manteuffel and C. Schneider and K. Schoenwald},

title = {{The non-first-order-factorizable contributions to the three-loop single-mass operator matrix elements $A_{Qg}^{(3)}$ and $Delta A_{Qg}^{(3)}$}},

language = {english},

abstract = {The non-first-order-factorizable contributions to the unpolarized and polarized massive operator matrix elements to three-loop order, $A_{Qg}^{(3)}$ and $Delta A_{Qg}^{(3)}$, are calculated in the single-mass case. For the $_2F_1$-related master integrals of the problem, we use a semi-analytic method basedon series expansions and utilize the first-order differential equations for the master integrals whichdoes not need a special basis of the master integrals. Due to the singularity structure of this basis a part of the integrals has to be computed to $O(ep^5)$ in the dimensional parameter. The solutions have to be matched at a series of thresholds and pseudo-thresholds in the region of the Bjorken variable $x in ]0,infty[$ using highly precise series expansions to obtain the imaginary part of the physical amplitude for $x in ]0,1]$ at a high relative accuracy. We compare the present results both with previous analytic results, the results for fixed Mellin moments, and a prediction in the small-$x$ region. We also derive expansions in the region of small and large values of $x$. With this paper, all three-loop single-mass unpolarized and polarized operator matrix elements are calculated.},

journal = {Physics Letter B},

volume = {854},

number = {138713},

pages = {1--8},

isbn_issn = {ISSN 0370-2693},

year = {2024},

note = {arXiv:2403.00513 [[hep-ph]},

refereed = {yes},

keywords = {Feynman diagram, massive operator matrix elements, computer algebra, differential equations, difference equations, coupled systems, numerics},

length = {8},

url = {https://doi.org/10.1016/j.physletb.2024.138713}

}

**article**{RISC7058,author = {J. Ablinger and A. Behring and J. Bluemlein and A. De Freitas and A. von Manteuffel and C. Schneider and K. Schoenwald},

title = {{The non-first-order-factorizable contributions to the three-loop single-mass operator matrix elements $A_{Qg}^{(3)}$ and $Delta A_{Qg}^{(3)}$}},

language = {english},

abstract = {The non-first-order-factorizable contributions to the unpolarized and polarized massive operator matrix elements to three-loop order, $A_{Qg}^{(3)}$ and $Delta A_{Qg}^{(3)}$, are calculated in the single-mass case. For the $_2F_1$-related master integrals of the problem, we use a semi-analytic method basedon series expansions and utilize the first-order differential equations for the master integrals whichdoes not need a special basis of the master integrals. Due to the singularity structure of this basis a part of the integrals has to be computed to $O(ep^5)$ in the dimensional parameter. The solutions have to be matched at a series of thresholds and pseudo-thresholds in the region of the Bjorken variable $x in ]0,infty[$ using highly precise series expansions to obtain the imaginary part of the physical amplitude for $x in ]0,1]$ at a high relative accuracy. We compare the present results both with previous analytic results, the results for fixed Mellin moments, and a prediction in the small-$x$ region. We also derive expansions in the region of small and large values of $x$. With this paper, all three-loop single-mass unpolarized and polarized operator matrix elements are calculated.},

journal = {Physics Letter B},

volume = {854},

number = {138713},

pages = {1--8},

isbn_issn = {ISSN 0370-2693},

year = {2024},

note = {arXiv:2403.00513 [[hep-ph]},

refereed = {yes},

keywords = {Feynman diagram, massive operator matrix elements, computer algebra, differential equations, difference equations, coupled systems, numerics},

length = {8},

url = {https://doi.org/10.1016/j.physletb.2024.138713}

}

[de Freitas]

### The three-loop single-mass heavy flavor corrections to deep-inelastic scattering

#### J. Ablinger, A. Behring, J. Bluemlein, A. De Freitas, A. von Manteuffel, C. Schneider, K. Schoenwald

Technical report no. 24-04 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). July 2024. arXiv:2407.02006 [hep-ph]. Licensed under CC BY 4.0 International. [doi] [pdf]@

author = {J. Ablinger and A. Behring and J. Bluemlein and A. De Freitas and A. von Manteuffel and C. Schneider and K. Schoenwald},

title = {{The three-loop single-mass heavy flavor corrections to deep-inelastic scattering}},

language = {english},

abstract = {We report on the status of the calculation of the massive Wilson coefficients and operator matrix elements for deep-inelastic scatterung to three-loop order. We discuss both the unpolarized and the polarized case, for which all the single-mass and nearly all two-mass contributions have been calculated. Numerical results on the structure function $F_2(x,Q^2)$ are presented. In the polarized case, we work in the Larinscheme and refer to parton distribution functions in this scheme. Furthermore, results on the three-loop variable flavor number scheme are presented.},

number = {24-04},

year = {2024},

month = {July},

note = {arXiv:2407.02006 [hep-ph]},

keywords = {Feynman integrals, deep-inelastic scattering, numerical results},

length = {12},

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**{RISC7059,author = {J. Ablinger and A. Behring and J. Bluemlein and A. De Freitas and A. von Manteuffel and C. Schneider and K. Schoenwald},

title = {{The three-loop single-mass heavy flavor corrections to deep-inelastic scattering}},

language = {english},

abstract = {We report on the status of the calculation of the massive Wilson coefficients and operator matrix elements for deep-inelastic scatterung to three-loop order. We discuss both the unpolarized and the polarized case, for which all the single-mass and nearly all two-mass contributions have been calculated. Numerical results on the structure function $F_2(x,Q^2)$ are presented. In the polarized case, we work in the Larinscheme and refer to parton distribution functions in this scheme. Furthermore, results on the three-loop variable flavor number scheme are presented.},

number = {24-04},

year = {2024},

month = {July},

note = {arXiv:2407.02006 [hep-ph]},

keywords = {Feynman integrals, deep-inelastic scattering, numerical results},

length = {12},

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)}

}

[de Freitas]

### Challenges for analytic calculations of the massive three-loop form factors

#### J Bluemlein, A. De Freitas, P. Marquard, C. Schneider

Technical report no. 24-05 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). August 2024. arXiv:2408.07046 [hep-ph]. Licensed under CC BY 4.0 International. [doi] [pdf]@

author = {J Bluemlein and A. De Freitas and P. Marquard and C. Schneider},

title = {{Challenges for analytic calculations of the massive three-loop form factors}},

language = {english},

abstract = {The calculation of massive three-loop QCD form factors using in particular the large moments method has been successfully applied to quarkonic contributions in [1]. We give a brief review of the different steps of the calculation and report on improvements of our methods that enabled us to push forward the calculations of the gluonic contributions to the form factors.},

number = {24-05},

year = {2024},

month = {August},

note = { arXiv:2408.07046 [hep-ph]},

keywords = {Form factor; computer algebra, coupled systems, differential equations, recurrences, analytic continuation, holonomic functions},

length = {18},

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**{RISC7062,author = {J Bluemlein and A. De Freitas and P. Marquard and C. Schneider},

title = {{Challenges for analytic calculations of the massive three-loop form factors}},

language = {english},

abstract = {The calculation of massive three-loop QCD form factors using in particular the large moments method has been successfully applied to quarkonic contributions in [1]. We give a brief review of the different steps of the calculation and report on improvements of our methods that enabled us to push forward the calculations of the gluonic contributions to the form factors.},

number = {24-05},

year = {2024},

month = {August},

note = { arXiv:2408.07046 [hep-ph]},

keywords = {Form factor; computer algebra, coupled systems, differential equations, recurrences, analytic continuation, holonomic functions},

length = {18},

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. 120, pp. 1-50. 2024. 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 = {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**{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}

}

[Schneider]

### Creative Telescoping for Hypergeometric Double Sums

#### P. Paule, C. Schneider

Technical report no. 24-01 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). January 2024. arXiv:2401.16314 [cs.SC]. Licensed under CC BY 4.0 International. [doi] [pdf]@

author = {P. Paule and C. Schneider},

title = {{Creative Telescoping for Hypergeometric Double Sums}},

language = {english},

abstract = {We present efficient methods for calculating linear recurrences of hypergeometric double sums and, more generally, of multiple sums. In particular, we supplement this approach with the algorithmic theory of contiguous relations, which guarantees the applicability of our method for many input sums. In addition, we elaborate new techniques to optimize the underlying key task of our method to compute rational solutions of parameterized linear recurrences.},

number = {24-01},

year = {2024},

month = {January},

note = {arXiv:2401.16314 [cs.SC]},

keywords = {creative telescoping; symbolic summation, hypergeometric multi-sums, contiguous relations, parameterized recurrences, rational solutions},

length = {26},

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**{RISC6894,author = {P. Paule and C. Schneider},

title = {{Creative Telescoping for Hypergeometric Double Sums}},

language = {english},

abstract = {We present efficient methods for calculating linear recurrences of hypergeometric double sums and, more generally, of multiple sums. In particular, we supplement this approach with the algorithmic theory of contiguous relations, which guarantees the applicability of our method for many input sums. In addition, we elaborate new techniques to optimize the underlying key task of our method to compute rational solutions of parameterized linear recurrences.},

number = {24-01},

year = {2024},

month = {January},

note = {arXiv:2401.16314 [cs.SC]},

keywords = {creative telescoping; symbolic summation, hypergeometric multi-sums, contiguous relations, parameterized recurrences, rational solutions},

length = {26},

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)}

}

### 2023

[de Freitas]

### Analytic results on the massive three-loop form factors: quarkonic contributions

#### J. Bluemlein, A. De Freitas, P. Marquard, N. Rana, C. Schneider

Physical Review D 108(094003), pp. 1-73. 2023. ISSN 2470-0029. arXiv:2307.02983 [hep-ph]. [doi]@

author = {J. Bluemlein and A. De Freitas and P. Marquard and N. Rana and C. Schneider},

title = {{Analytic results on the massive three-loop form factors: quarkonic contributions}},

language = {english},

abstract = {The quarkonic contributions to the three--loop heavy-quark form factors for vector, axial-vector, scalar and pseudoscalar currents are described by closed form difference equations for the expansion coefficients in the limit of small virtualities $q^2/m^2$. A part of the contributions can be solved analytically and expressed in terms of harmonic and cyclotomic harmonic polylogarithms and square-root valued iterated integrals. Other contributions obey equations which are not first--order factorizable. For them still infinite series expansions around the singularities of the form factors can be obtained by matching the expansions at intermediate points and using differential equations which are obeyed directly by the form factors and are derived by guessing algorithms. One may determine all expansion coefficients for $q^2/m^2 rightarrow infty$ analytically in terms of multiple zeta values. By expanding around the threshold and pseudo--threshold, the corresponding constants are multiple zeta values supplemented by a finite amount of new constants, which can be computed at high precision. For a part of these coefficients, the infinite series in front of these constants may be even resummed into harmonic polylogarithms. In this way, one obtains a deeper analytic description of the massive form factors, beyond their pure numerical evaluation. The calculations of these analytic results are based on sophisticated computer algebra techniques. We also compare our results with numerical results in the literature.},

journal = {Physical Review D},

volume = {108},

number = {094003},

pages = {1--73},

isbn_issn = {ISSN 2470-0029},

year = {2023},

note = {arXiv:2307.02983 [hep-ph]},

refereed = {yes},

keywords = {form factor, Feynman diagram, computer algebra, holonomic properties, difference equations, differential equations, symbolic summation, numerical matching, analytic continuation, guessing, PSLQ},

length = {92},

url = {https://www.doi.org/10.1103/PhysRevD.108.094003}

}

**article**{RISC6742,author = {J. Bluemlein and A. De Freitas and P. Marquard and N. Rana and C. Schneider},

title = {{Analytic results on the massive three-loop form factors: quarkonic contributions}},

language = {english},

abstract = {The quarkonic contributions to the three--loop heavy-quark form factors for vector, axial-vector, scalar and pseudoscalar currents are described by closed form difference equations for the expansion coefficients in the limit of small virtualities $q^2/m^2$. A part of the contributions can be solved analytically and expressed in terms of harmonic and cyclotomic harmonic polylogarithms and square-root valued iterated integrals. Other contributions obey equations which are not first--order factorizable. For them still infinite series expansions around the singularities of the form factors can be obtained by matching the expansions at intermediate points and using differential equations which are obeyed directly by the form factors and are derived by guessing algorithms. One may determine all expansion coefficients for $q^2/m^2 rightarrow infty$ analytically in terms of multiple zeta values. By expanding around the threshold and pseudo--threshold, the corresponding constants are multiple zeta values supplemented by a finite amount of new constants, which can be computed at high precision. For a part of these coefficients, the infinite series in front of these constants may be even resummed into harmonic polylogarithms. In this way, one obtains a deeper analytic description of the massive form factors, beyond their pure numerical evaluation. The calculations of these analytic results are based on sophisticated computer algebra techniques. We also compare our results with numerical results in the literature.},

journal = {Physical Review D},

volume = {108},

number = {094003},

pages = {1--73},

isbn_issn = {ISSN 2470-0029},

year = {2023},

note = {arXiv:2307.02983 [hep-ph]},

refereed = {yes},

keywords = {form factor, Feynman diagram, computer algebra, holonomic properties, difference equations, differential equations, symbolic summation, numerical matching, analytic continuation, guessing, PSLQ},

length = {92},

url = {https://www.doi.org/10.1103/PhysRevD.108.094003}

}

[de Freitas]

### Recent 3-Loop Heavy Flavor Corrections to Deep-Inelastic Scattering

#### J. Ablinger, A. Behring, J. Bluemlein, A. De Freitas, A. Goedicke, A. von Manteuffel, C. Schneider, K. Schoenwald

In: Proc. of the 16th International Symposium on Radiative Corrections: Applications of Quantum Field Theory to Phenomenology , Giulio Falcioni (ed.), PoS RADCOR2023046, pp. 1-7. June 2023. ISSN 1824-8039. arXiv:2306.16550 [hep-ph]. [doi]@

author = {J. Ablinger and A. Behring and J. Bluemlein and A. De Freitas and A. Goedicke and A. von Manteuffel and C. Schneider and K. Schoenwald},

title = {{Recent 3-Loop Heavy Flavor Corrections to Deep-Inelastic Scattering}},

booktitle = {{Proc. of the 16th International Symposium on Radiative Corrections: Applications of Quantum Field Theory to Phenomenology }},

language = {english},

abstract = {We report on recent progress in calculating the three loop QCD corrections of the heavy flavor contributions in deep--inelastic scattering and the massive operator matrix elements of the variable flavor number scheme. Notably we deal with the operator matrix elements $A_{gg,Q}^{(3)}$ and $A_{Qg}^{(3)}$ and technical steps to their calculation. In particular, a new method to obtain the inverse Mellin transform without computing the corresponding $N$--space expressions is discussed.},

series = {PoS},

volume = {RADCOR2023},

number = {046},

pages = {1--7},

isbn_issn = {ISSN 1824-8039},

year = {2023},

month = {June},

note = {arXiv:2306.16550 [hep-ph]},

editor = {Giulio Falcioni},

refereed = {no},

keywords = {deep-inelastic scattering, 3-loop Feynman diagrams, (inverse) Mellin transform, binomial sums},

length = {7},

url = { https://doi.org/10.22323/1.432.0046 }

}

**inproceedings**{RISC6748,author = {J. Ablinger and A. Behring and J. Bluemlein and A. De Freitas and A. Goedicke and A. von Manteuffel and C. Schneider and K. Schoenwald},

title = {{Recent 3-Loop Heavy Flavor Corrections to Deep-Inelastic Scattering}},

booktitle = {{Proc. of the 16th International Symposium on Radiative Corrections: Applications of Quantum Field Theory to Phenomenology }},

language = {english},

abstract = {We report on recent progress in calculating the three loop QCD corrections of the heavy flavor contributions in deep--inelastic scattering and the massive operator matrix elements of the variable flavor number scheme. Notably we deal with the operator matrix elements $A_{gg,Q}^{(3)}$ and $A_{Qg}^{(3)}$ and technical steps to their calculation. In particular, a new method to obtain the inverse Mellin transform without computing the corresponding $N$--space expressions is discussed.},

series = {PoS},

volume = {RADCOR2023},

number = {046},

pages = {1--7},

isbn_issn = {ISSN 1824-8039},

year = {2023},

month = {June},

note = {arXiv:2306.16550 [hep-ph]},

editor = {Giulio Falcioni},

refereed = {no},

keywords = {deep-inelastic scattering, 3-loop Feynman diagrams, (inverse) Mellin transform, binomial sums},

length = {7},

url = { https://doi.org/10.22323/1.432.0046 }

}

[Kauers]

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

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

In: Proceedings of the 2023 International Symposium on Symbolic and Algebraic Computation, A.Dickenstein, E. Tsigaridas and G. Jeronimo (ed.), ISSAC '23, Tromso{}, Norway , pp. 389-397. July 2023. Association for Computing Machinery, New York, NY, USA, 9798400700392}. [doi]@

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

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

booktitle = {{Proceedings of the 2023 International Symposium on Symbolic and Algebraic Computation}},

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.},

series = {ISSAC '23, Tromso{}, Norway},

pages = {389--397},

publisher = {Association for Computing Machinery},

address = {New York, NY, USA},

isbn_issn = {9798400700392}},

year = {2023},

month = {July},

editor = {A.Dickenstein and E. Tsigaridas and G. Jeronimo},

refereed = {no},

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

length = {9},

url = {https://doi.org/10.35011/risc.23-03}

}

**inproceedings**{RISC6751,author = {M. Kauers and P. Nuspl and V. Pillwein},

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

booktitle = {{Proceedings of the 2023 International Symposium on Symbolic and Algebraic Computation}},

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.},

series = {ISSAC '23, Tromso{}, Norway},

pages = {389--397},

publisher = {Association for Computing Machinery},

address = {New York, NY, USA},

isbn_issn = {9798400700392}},

year = {2023},

month = {July},

editor = {A.Dickenstein and E. Tsigaridas and G. Jeronimo},

refereed = {no},

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

length = {9},

url = {https://doi.org/10.35011/risc.23-03}

}

[Schneider]

### Hypergeometric Structures in Feynman Integrals

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

Annals of Mathematics and Artificial Intelligence 91(5), pp. 591-649. 2023. ISSN 1573-7470. arXiv:2111.15501 [math-ph]. [doi]@

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

title = {{Hypergeometric Structures in Feynman Integrals}},

language = {english},

abstract = {Hypergeometric structures in single and multiscale Feynman integrals emerge in a wide class of topologies. Using integration-by-parts relations, associated master or scalar integrals have to be calculated. For this purpose it appears useful to devise an automated method which recognizes the respective (partial) differential equations related to the corresponding higher transcendental functions. We solve these equations through associated recursions of the expansion coefficient of the multivalued formal Taylor series. The expansion coefficients can be determined using either the package {tt Sigma} in the case of linear difference equations or by applying heuristic methods in the case of partial linear difference equations. In the present context a new type of sums occurs, the Hurwitz harmonic sums, and generalized versions of them. The code {tt HypSeries} transforming classes of differential equations into analytic series expansions is described. Also partial difference equations having rational solutions and rational function solutions of Pochhammer symbols are considered, for which the code {tt solvePartialLDE} is designed. Generalized hypergeometric functions, Appell-,~Kamp'e de F'eriet-, Horn-, Lauricella-Saran-, Srivasta-, and Exton--type functions are considered. We illustrate the algorithms by examples.},

journal = {Annals of Mathematics and Artificial Intelligence},

volume = { 91},

number = {5},

pages = {591--649},

isbn_issn = {ISSN 1573-7470},

year = {2023},

note = {arXiv:2111.15501 [math-ph]},

refereed = {yes},

keywords = {hypergeometric functions, symbolic summation, expansion, partial linear difference equations, partial linear differential equations},

length = {59},

url = {https://doi.org/10.1007/s10472-023-09831-8}

}

**article**{RISC6643,author = {J. Blümlein and C. Schneider and M. Saragnese},

title = {{Hypergeometric Structures in Feynman Integrals}},

language = {english},

abstract = {Hypergeometric structures in single and multiscale Feynman integrals emerge in a wide class of topologies. Using integration-by-parts relations, associated master or scalar integrals have to be calculated. For this purpose it appears useful to devise an automated method which recognizes the respective (partial) differential equations related to the corresponding higher transcendental functions. We solve these equations through associated recursions of the expansion coefficient of the multivalued formal Taylor series. The expansion coefficients can be determined using either the package {tt Sigma} in the case of linear difference equations or by applying heuristic methods in the case of partial linear difference equations. In the present context a new type of sums occurs, the Hurwitz harmonic sums, and generalized versions of them. The code {tt HypSeries} transforming classes of differential equations into analytic series expansions is described. Also partial difference equations having rational solutions and rational function solutions of Pochhammer symbols are considered, for which the code {tt solvePartialLDE} is designed. Generalized hypergeometric functions, Appell-,~Kamp'e de F'eriet-, Horn-, Lauricella-Saran-, Srivasta-, and Exton--type functions are considered. We illustrate the algorithms by examples.},

journal = {Annals of Mathematics and Artificial Intelligence},

volume = { 91},

number = {5},

pages = {591--649},

isbn_issn = {ISSN 1573-7470},

year = {2023},

note = {arXiv:2111.15501 [math-ph]},

refereed = {yes},

keywords = {hypergeometric functions, symbolic summation, expansion, partial linear difference equations, partial linear differential equations},

length = {59},

url = {https://doi.org/10.1007/s10472-023-09831-8}

}

[Schneider]

### Refined telescoping algorithms in $RPiSigma$-extensions to reduce the degrees of the denominators

#### C. Schneider

In: ISSAC '23: Proceedings of the 2023 International Symposium on Symbolic and Algebraic Computation, Gabriela Jeronimo (ed.), pp. 498-507. July 2023. ACM, ISBN 9798400700392. arXiv:2302.03563 [cs.SC]. [doi]@

author = {C. Schneider},

title = {{Refined telescoping algorithms in $RPiSigma$-extensions to reduce the degrees of the denominators}},

booktitle = {{ISSAC '23: Proceedings of the 2023 International Symposium on Symbolic and Algebraic Computation}},

language = {english},

abstract = {We present a general framework in the setting of difference ring extensions that enables one to find improved representations of indefinite nested sums such that the arising denominators within the summands have reduced degrees. The underlying (parameterized) telescoping algorithms can be executed in $RPiSigma$-ring extensions that are built over general $PiSigma$-fields. An important application of this toolbox is the simplification of d'Alembertian and Liouvillian solutions coming from recurrence relations where the denominators of the arising sums do not factor nicely.},

pages = {498--507},

publisher = {ACM},

isbn_issn = {ISBN 9798400700392},

year = {2023},

month = {July},

note = {arXiv:2302.03563 [cs.SC]},

editor = {Gabriela Jeronimo},

refereed = {yes},

keywords = {telescoping, difference rings, reduced denominators, nested sums},

length = {10},

url = {https://doi.org/10.1145/3597066.3597073}

}

**inproceedings**{RISC6699,author = {C. Schneider},

title = {{Refined telescoping algorithms in $RPiSigma$-extensions to reduce the degrees of the denominators}},

booktitle = {{ISSAC '23: Proceedings of the 2023 International Symposium on Symbolic and Algebraic Computation}},

language = {english},

abstract = {We present a general framework in the setting of difference ring extensions that enables one to find improved representations of indefinite nested sums such that the arising denominators within the summands have reduced degrees. The underlying (parameterized) telescoping algorithms can be executed in $RPiSigma$-ring extensions that are built over general $PiSigma$-fields. An important application of this toolbox is the simplification of d'Alembertian and Liouvillian solutions coming from recurrence relations where the denominators of the arising sums do not factor nicely.},

pages = {498--507},

publisher = {ACM},

isbn_issn = {ISBN 9798400700392},

year = {2023},

month = {July},

note = {arXiv:2302.03563 [cs.SC]},

editor = {Gabriela Jeronimo},

refereed = {yes},

keywords = {telescoping, difference rings, reduced denominators, nested sums},

length = {10},

url = {https://doi.org/10.1145/3597066.3597073}

}

[Schneider]

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

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

Rocky Mt J Math to appear, pp. ?-?. 2023. ISSN: 357596. arXiv:2209.07887 [math.NT]. [doi]@

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**{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}

}

[Schneider]

### Computing Mellin representations and asymptotics of nested binomial sums in a symbolic way: the RICA package

#### Johannes Bluemlein, Nikolai Fadeev, Carsten Schneider

ACM Communications in Computer Algebra 57(2), pp. 31-34. June 2023. ISSN:1932-2240. arXiv:2308.06042 [hep-ph]. [doi]@

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**{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}

}

### 2022

[de Freitas]

### The Unpolarized and Polarized Single-Mass Three-Loop Heavy Flavor Operator Matrix Elements $A_{gg, Q}$ and $Delta A_{gg, Q}$

#### J. Ablinger, A. Behring, J. Bluemlein, A. De Freitas, A. Goedicke, A. von Manteuffel, C. Schneider, K. Schoenwald

Journal of High Energy Physics 2022(12, Article 134), pp. 1-55. 2022. ISSN 1029-8479. arXiv:2211.05462 [hep-ph]. [doi]@

author = {J. Ablinger and A. Behring and J. Bluemlein and A. De Freitas and A. Goedicke and A. von Manteuffel and C. Schneider and K. Schoenwald},

title = {{The Unpolarized and Polarized Single-Mass Three-Loop Heavy Flavor Operator Matrix Elements $A_{gg, Q}$ and $Delta A_{gg, Q}$}},

language = {english},

abstract = {We calculate the gluonic massive operator matrix elements in the unpolarized and polarized cases, $A_{gg,Q}(x,mu^2)$ and $Delta A_{gg,Q}(x,mu^2)$, at three--loop order for a single mass. These quantities contribute to the matching of the gluon distribution in the variable flavor number scheme. The polarized operator matrix element is calculated in the Larin scheme. These operator matrix elements contain finite binomial and inverse binomial sums in Mellin $N$--space and iterated integrals over square root--valued alphabets in momentum fraction $x$--space. We derive the necessary analytic relations for the analytic continuation of these quantities from the even or odd Mellin moments into the complex plane, present analytic expressions in momentum fraction $x$--space and derive numerical results. The present results complete the gluon transition matrix elements both of the single-- and double--mass variable flavor number scheme to three--loop order.},

journal = {Journal of High Energy Physics},

volume = {2022},

number = {12, Article 134},

pages = {1--55},

isbn_issn = { ISSN 1029-8479},

year = {2022},

note = {arXiv:2211.05462 [hep-ph]},

refereed = {yes},

keywords = {Feynman integrals, linear difference equations, linear differential equations, binomial sums, harmonic sums, iterative integrals, computer algebra},

length = {48},

url = {https://doi.org/10.1007/JHEP12(2022)134}

}

**article**{RISC6632,author = {J. Ablinger and A. Behring and J. Bluemlein and A. De Freitas and A. Goedicke and A. von Manteuffel and C. Schneider and K. Schoenwald},

title = {{The Unpolarized and Polarized Single-Mass Three-Loop Heavy Flavor Operator Matrix Elements $A_{gg, Q}$ and $Delta A_{gg, Q}$}},

language = {english},

abstract = {We calculate the gluonic massive operator matrix elements in the unpolarized and polarized cases, $A_{gg,Q}(x,mu^2)$ and $Delta A_{gg,Q}(x,mu^2)$, at three--loop order for a single mass. These quantities contribute to the matching of the gluon distribution in the variable flavor number scheme. The polarized operator matrix element is calculated in the Larin scheme. These operator matrix elements contain finite binomial and inverse binomial sums in Mellin $N$--space and iterated integrals over square root--valued alphabets in momentum fraction $x$--space. We derive the necessary analytic relations for the analytic continuation of these quantities from the even or odd Mellin moments into the complex plane, present analytic expressions in momentum fraction $x$--space and derive numerical results. The present results complete the gluon transition matrix elements both of the single-- and double--mass variable flavor number scheme to three--loop order.},

journal = {Journal of High Energy Physics},

volume = {2022},

number = {12, Article 134},

pages = {1--55},

isbn_issn = { ISSN 1029-8479},

year = {2022},

note = {arXiv:2211.05462 [hep-ph]},

refereed = {yes},

keywords = {Feynman integrals, linear difference equations, linear differential equations, binomial sums, harmonic sums, iterative integrals, computer algebra},

length = {48},

url = {https://doi.org/10.1007/JHEP12(2022)134}

}

[Schneider]

### The three-loop polarized singlet anomalous dimensions from off-shell operator matrix elements

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

Journal of High Energy Physics 2022(193), pp. 0-32. 2022. ISSN 1029-8479 . arXiv:2111.12401 [hep-ph]. [doi]@

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

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

language = {english},

abstract = {We calculate the polarized three--loop singlet anomalous dimensions and splitting functions in QCD in the M--scheme by using the traditional method of space--like off--shell massless operator matrix elements. This is a gauge--dependent framework. Here one obtains the anomalous dimensions without referring to gravitational currents. We also calculate the non--singlet splitting function $Delta P_{rm qq}^{(2), rm s, NS}$ and compare our results to the literature. },

journal = {Journal of High Energy Physics},

volume = {2022},

number = {193},

pages = {0--32},

isbn_issn = {ISSN 1029-8479 },

year = {2022},

note = {arXiv:2111.12401 [hep-ph]},

refereed = {yes},

keywords = {particle physics, solving recurrences, large moment method, harmonic sums},

length = {33},

url = {https://doi.org/10.1007/JHEP01(2022)193}

}

**article**{RISC6435,author = {J. Blümlein and P. Marquard and C. Schneider and K. Schönwald},

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

language = {english},

abstract = {We calculate the polarized three--loop singlet anomalous dimensions and splitting functions in QCD in the M--scheme by using the traditional method of space--like off--shell massless operator matrix elements. This is a gauge--dependent framework. Here one obtains the anomalous dimensions without referring to gravitational currents. We also calculate the non--singlet splitting function $Delta P_{rm qq}^{(2), rm s, NS}$ and compare our results to the literature. },

journal = {Journal of High Energy Physics},

volume = {2022},

number = {193},

pages = {0--32},

isbn_issn = {ISSN 1029-8479 },

year = {2022},

note = {arXiv:2111.12401 [hep-ph]},

refereed = {yes},

keywords = {particle physics, solving recurrences, large moment method, harmonic sums},

length = {33},

url = {https://doi.org/10.1007/JHEP01(2022)193}

}

[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]

### The massless three-loop Wilson coefficients for the deep-inelastic structure functions $F_2, F_L, xF_3$ and $g_1$

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

Journal of High Energy Physics(Paper No. 156), pp. 1-83. 2022. ISSN 1029-8479. arXiv:2208.14325 [hep-ph]. [doi]@

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

title = {{The massless three-loop Wilson coefficients for the deep-inelastic structure functions $F_2, F_L, xF_3$ and $g_1$}},

language = {english},

abstract = {We calculate the massless unpolarized Wilson coefficients for deeply inelastic scattering for thestructure functions $F_2(x,Q^2), F_L(x,Q^2), x F_3(x,Q^2)$ in the $overline{sf MS}$ scheme and the polarized Wilson coefficients of the structure function $g_1(x,Q^2)$ in the Larin scheme up to three--loop order in QCD in a fully automated way based on the method of arbitrary high Mellin moments. We workin the Larin scheme in the case of contributing axial--vector couplings or polarized nucleons. For the unpolarized structure functions we compare to results given in the literature. The polarized three--loop Wilson coefficients are calculated for the first time. As a by--product we also obtain the quarkonic three--loop anomalous dimensions from the $O(1/ep)$ terms of the unrenormalized forward Compton amplitude. Expansions for small and large values of the Bjorken variable $x$ are provided.},

journal = {Journal of High Energy Physics},

number = {Paper No. 156},

pages = {1--83},

isbn_issn = {ISSN 1029-8479},

year = {2022},

note = {arXiv:2208.14325 [hep-ph]},

refereed = {yes},

keywords = {massless unpolarized Wilson coefficients, large moment method, linear difference equations, computer algebra,coupled systems of linear differential equations},

length = {83},

url = {https://doi.org/10.1007/JHEP11(2022)156}

}

**article**{RISC6627,author = {J. Blümlein and P. Marquard and C. Schneider and K. Schönwald},

title = {{The massless three-loop Wilson coefficients for the deep-inelastic structure functions $F_2, F_L, xF_3$ and $g_1$}},

language = {english},

abstract = {We calculate the massless unpolarized Wilson coefficients for deeply inelastic scattering for thestructure functions $F_2(x,Q^2), F_L(x,Q^2), x F_3(x,Q^2)$ in the $overline{sf MS}$ scheme and the polarized Wilson coefficients of the structure function $g_1(x,Q^2)$ in the Larin scheme up to three--loop order in QCD in a fully automated way based on the method of arbitrary high Mellin moments. We workin the Larin scheme in the case of contributing axial--vector couplings or polarized nucleons. For the unpolarized structure functions we compare to results given in the literature. The polarized three--loop Wilson coefficients are calculated for the first time. As a by--product we also obtain the quarkonic three--loop anomalous dimensions from the $O(1/ep)$ terms of the unrenormalized forward Compton amplitude. Expansions for small and large values of the Bjorken variable $x$ are provided.},

journal = {Journal of High Energy Physics},

number = {Paper No. 156},

pages = {1--83},

isbn_issn = {ISSN 1029-8479},

year = {2022},

note = {arXiv:2208.14325 [hep-ph]},

refereed = {yes},

keywords = {massless unpolarized Wilson coefficients, large moment method, linear difference equations, computer algebra,coupled systems of linear differential equations},

length = {83},

url = {https://doi.org/10.1007/JHEP11(2022)156}

}