# SAGEX – Scattering Amplitudes: from Geometry to Experiment

### Project Lead

### Project Duration

01/09/2018 - 31/08/2022### Project URL

Go to Website## Partners

### EU

## Publications

### 2023

[Schneider]

### Hypergeometric Structures in Feynman Integrals

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

Annals of Mathematics and Artificial Intelligence, Special issue on " Symbolic Computation in Software Science" to appear, pp. ?-?. 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, Special issue on " Symbolic Computation in Software Science"},

volume = {to appear},

pages = {?--?},

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 = {55},

url = {https://doi.org/10.48550/arXiv.2111.15501}

}

**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, Special issue on " Symbolic Computation in Software Science"},

volume = {to appear},

pages = {?--?},

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 = {55},

url = {https://doi.org/10.48550/arXiv.2111.15501}

}

### 2022

[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 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 SAGEX Review on Scattering Amplitudes

#### G. Travaglini, A. Brandhuber, P. Dorey, T. McLoughlin, S. Abreu, Z. Bern, N. E. J. Bjerrum-Bohr, J. Bluemlein, R. Britto, J. J. M. Carrasco, D. Chicherin, M. Chiodaroli, P. H. Damgaard, V. Del Duca, L. J. Dixon, D. Dorigoni, C. Duhr, Y. Geyer, M. B. Green, E. Herrmann, P. Heslop, H. Johansson, G. P. Korchemsky, D. A. Kosower, L. Mason, R. Monteiro, D. O'Connell, G. Papathanasiou, L. Plante, J. Plefka, A. Puhm, A.-M. Raclariu, R. Roiban, C. Schneider, J. Trnka, P. Vanhove, C. Wen, C. D. White

J. Phys. A: Math. Theor. 55(44), pp. 1-12. 2022. arxiv.2203.13011 [hep-th], ISSN: 1751-8121. [doi]@

author = {G. Travaglini and A. Brandhuber and P. Dorey and T. McLoughlin and S. Abreu and Z. Bern and N. E. J. Bjerrum-Bohr and J. Bluemlein and R. Britto and J. J. M. Carrasco and D. Chicherin and M. Chiodaroli and P. H. Damgaard and V. Del Duca and L. J. Dixon and D. Dorigoni and C. Duhr and Y. Geyer and M. B. Green and E. Herrmann and P. Heslop and H. Johansson and G. P. Korchemsky and D. A. Kosower and L. Mason and R. Monteiro and D. O'Connell and G. Papathanasiou and L. Plante and J. Plefka and A. Puhm and A.-M. Raclariu and R. Roiban and C. Schneider and J. Trnka and P. Vanhove and C. Wen and C. D. White},

title = {{The SAGEX Review on Scattering Amplitudes}},

language = {english},

journal = {J. Phys. A: Math. Theor.},

volume = {55},

number = {44},

pages = {1--12},

isbn_issn = {ISSN: 1751-8121},

year = {2022},

refereed = {yes},

institution = {arxiv.2203.13011 [hep-th]},

keywords = {High Energy Physics - Theory (hep-th), General Relativity and Quantum Cosmology (gr-qc), High Energy Physics - Experiment (hep-ex), High Energy Physics - Phenomenology (hep-ph), FOS: Physical sciences},

length = {12},

url = {https://doi.org/10.1088/1751-8121/ac8380}

}

**article**{RISC6529,author = {G. Travaglini and A. Brandhuber and P. Dorey and T. McLoughlin and S. Abreu and Z. Bern and N. E. J. Bjerrum-Bohr and J. Bluemlein and R. Britto and J. J. M. Carrasco and D. Chicherin and M. Chiodaroli and P. H. Damgaard and V. Del Duca and L. J. Dixon and D. Dorigoni and C. Duhr and Y. Geyer and M. B. Green and E. Herrmann and P. Heslop and H. Johansson and G. P. Korchemsky and D. A. Kosower and L. Mason and R. Monteiro and D. O'Connell and G. Papathanasiou and L. Plante and J. Plefka and A. Puhm and A.-M. Raclariu and R. Roiban and C. Schneider and J. Trnka and P. Vanhove and C. Wen and C. D. White},

title = {{The SAGEX Review on Scattering Amplitudes}},

language = {english},

journal = {J. Phys. A: Math. Theor.},

volume = {55},

number = {44},

pages = {1--12},

isbn_issn = {ISSN: 1751-8121},

year = {2022},

refereed = {yes},

institution = {arxiv.2203.13011 [hep-th]},

keywords = {High Energy Physics - Theory (hep-th), General Relativity and Quantum Cosmology (gr-qc), High Energy Physics - Experiment (hep-ex), High Energy Physics - Phenomenology (hep-ph), FOS: Physical sciences},

length = {12},

url = {https://doi.org/10.1088/1751-8121/ac8380}

}

### 2021

[Schneider]

### Iterated integrals over letters induced by quadratic forms

#### J. Ablinger, J. Blümlein, C. Schneider

Physical Review D 103(9), pp. 096025-096035. 2021. ISSN 2470-0029. arXiv:2103.08330 [hep-th]. [doi]@

author = {J. Ablinger and J. Blümlein and C. Schneider},

title = {{Iterated integrals over letters induced by quadratic forms}},

language = {english},

journal = {Physical Review D },

volume = {103},

number = {9},

pages = {096025--096035},

isbn_issn = {ISSN 2470-0029},

year = {2021},

note = {arXiv:2103.08330 [hep-th]},

refereed = {yes},

length = {11},

url = {https://www.doi.org/10.1103/PhysRevD.103.096025}

}

**article**{RISC6294,author = {J. Ablinger and J. Blümlein and C. Schneider},

title = {{Iterated integrals over letters induced by quadratic forms}},

language = {english},

journal = {Physical Review D },

volume = {103},

number = {9},

pages = {096025--096035},

isbn_issn = {ISSN 2470-0029},

year = {2021},

note = {arXiv:2103.08330 [hep-th]},

refereed = {yes},

length = {11},

url = {https://www.doi.org/10.1103/PhysRevD.103.096025}

}

[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

Technical report no. 21-14 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). July 2021. Licensed under CC BY 4.0 International. [doi] [pdf]@

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.},

number = {21-14},

year = {2021},

month = {July},

keywords = {deep inelastic scattering, 3-loop Feynman integrals, symbolic summation, large moment method, special functions},

length = {13},

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**{RISC6350,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.},

number = {21-14},

year = {2021},

month = {July},

keywords = {deep inelastic scattering, 3-loop Feynman integrals, symbolic summation, large moment method, special functions},

length = {13},

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]

### The Logarithmic Contributions to the Polarized $O(alpha_s^3)$ Asymptotic Massive Wilson Coefficients and Operator Matrix Elements in Deeply Inelastic Scattering

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

Physical Review D 104(3), pp. 1-73. 2021. ISSN 2470-0029. arXiv:2105.09572 [hep-ph]. [doi]@

author = {J. Blümlein and A. De Freitas and M. Saragnese and K. Schönwald and C. Schneider},

title = {{The Logarithmic Contributions to the Polarized $O(alpha_s^3)$ Asymptotic Massive Wilson Coefficients and Operator Matrix Elements in Deeply Inelastic Scattering}},

language = {english},

abstract = {We compute the logarithmic contributions to the polarized massive Wilson coefficients fordeep-inelastic scattering in the asymptotic region $Q^2gg m^2$ to 3-loop order in the fixed-flavor number scheme and present the corresponding expressions for the polarized massiveoperator matrix elements needed in the variable flavor number scheme. The calculationis performed in the Larin scheme. For the massive operator matrix elements $A_{qq,Q}^{(3),PS}$ and $A_{qg,Q}^{(3),S}$the complete results are presented. The expressions are given in Mellin-$N$ space andin momentum fraction $z$-space.},

journal = {Physical Review D},

volume = {104},

number = {3},

pages = {1--73},

isbn_issn = {ISSN 2470-0029},

year = {2021},

note = {arXiv:2105.09572 [hep-ph]},

refereed = {yes},

keywords = {logarithmic contributions to the polarized massive Wilson coefficients, symbolic summation, harmonic sums, harmonic polylogarithm},

length = {73},

url = {https://doi.org/10.1103/PhysRevD.104.034030 }

}

**article**{RISC6333,author = {J. Blümlein and A. De Freitas and M. Saragnese and K. Schönwald and C. Schneider},

title = {{The Logarithmic Contributions to the Polarized $O(alpha_s^3)$ Asymptotic Massive Wilson Coefficients and Operator Matrix Elements in Deeply Inelastic Scattering}},

language = {english},

abstract = {We compute the logarithmic contributions to the polarized massive Wilson coefficients fordeep-inelastic scattering in the asymptotic region $Q^2gg m^2$ to 3-loop order in the fixed-flavor number scheme and present the corresponding expressions for the polarized massiveoperator matrix elements needed in the variable flavor number scheme. The calculationis performed in the Larin scheme. For the massive operator matrix elements $A_{qq,Q}^{(3),PS}$ and $A_{qg,Q}^{(3),S}$the complete results are presented. The expressions are given in Mellin-$N$ space andin momentum fraction $z$-space.},

journal = {Physical Review D},

volume = {104},

number = {3},

pages = {1--73},

isbn_issn = {ISSN 2470-0029},

year = {2021},

note = {arXiv:2105.09572 [hep-ph]},

refereed = {yes},

keywords = {logarithmic contributions to the polarized massive Wilson coefficients, symbolic summation, harmonic sums, harmonic polylogarithm},

length = {73},

url = {https://doi.org/10.1103/PhysRevD.104.034030 }

}

[Schneider]

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

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

Nucl. Phys. B 971, pp. 1-44. 2021. ISSN 0550-3213. arXiv:2107.06267 [hep-ph]. [doi]@

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

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

language = {english},

abstract = {We calculate the unpolarized and polarized three--loop anomalous dimensions and splitting functions $P_{rm NS}^+, P_{rm NS}^-$ and $P_{rm NS}^{rm s}$ in QCD in the $overline{sf MS}$ scheme by using the traditional method of space--like off shell massless operator matrix elements. This is a gauge--dependent framework. For the first time we also calculate the three--loop anomalous dimensions $P_{rm NS}^{rm pm, tr}$ for transversity directly. We compare our results to the literature. },

journal = {Nucl. Phys. B},

volume = {971},

pages = {1--44},

isbn_issn = {ISSN 0550-3213},

year = {2021},

note = {arXiv:2107.06267 [hep-ph]},

refereed = {yes},

length = {44},

url = {https://doi.org/10.1016/j.nuclphysb.2021.115542}

}

**article**{RISC6362,author = {J. Blümlein and P. Marquard and C. Schneider and K. Schönwald},

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

language = {english},

abstract = {We calculate the unpolarized and polarized three--loop anomalous dimensions and splitting functions $P_{rm NS}^+, P_{rm NS}^-$ and $P_{rm NS}^{rm s}$ in QCD in the $overline{sf MS}$ scheme by using the traditional method of space--like off shell massless operator matrix elements. This is a gauge--dependent framework. For the first time we also calculate the three--loop anomalous dimensions $P_{rm NS}^{rm pm, tr}$ for transversity directly. We compare our results to the literature. },

journal = {Nucl. Phys. B},

volume = {971},

pages = {1--44},

isbn_issn = {ISSN 0550-3213},

year = {2021},

note = {arXiv:2107.06267 [hep-ph]},

refereed = {yes},

length = {44},

url = {https://doi.org/10.1016/j.nuclphysb.2021.115542}

}

### 2020

[Fadeev]

### First-order factorizable systems of differential equations in one variable

#### N. Fadeev

Technical report no. 20-20 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). 2020. [pdf]@

author = {N. Fadeev},

title = {{First-order factorizable systems of differential equations in one variable}},

language = {english},

number = {20-20},

year = {2020},

length = {29},

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**{RISC6222,author = {N. Fadeev},

title = {{First-order factorizable systems of differential equations in one variable}},

language = {english},

number = {20-20},

year = {2020},

length = {29},

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]

### Three loop QCD corrections to heavy quark form factors

#### J. Ablinger, J. Blümlein, P. Marquard, N. Rana, C. Schneider

In: Proc. ACAT 2019, , J. Phys.: Conf. Ser. 1525012018, pp. 1-10. 2020. ISSN 1742-6596. arXiv:1905.03728 [hep-ph]. [doi]@

author = {J. Ablinger and J. Blümlein and P. Marquard and N. Rana and C. Schneider},

title = {{Three loop QCD corrections to heavy quark form factors}},

booktitle = {{Proc. ACAT 2019}},

language = {english},

series = {J. Phys.: Conf. Ser.},

volume = {1525},

number = {012018},

pages = {1--10},

isbn_issn = {ISSN 1742-6596},

year = {2020},

note = {arXiv:1905.03728 [hep-ph]},

editor = {?},

refereed = {yes},

length = {10},

url = {https://www.doi.org/10.1088/1742-6596/1525/1/012018}

}

**inproceedings**{RISC4826,author = {J. Ablinger and J. Blümlein and P. Marquard and N. Rana and C. Schneider},

title = {{Three loop QCD corrections to heavy quark form factors}},

booktitle = {{Proc. ACAT 2019}},

language = {english},

series = {J. Phys.: Conf. Ser.},

volume = {1525},

number = {012018},

pages = {1--10},

isbn_issn = {ISSN 1742-6596},

year = {2020},

note = {arXiv:1905.03728 [hep-ph]},

editor = {?},

refereed = {yes},

length = {10},

url = {https://www.doi.org/10.1088/1742-6596/1525/1/012018}

}