Computer Algebra for Multi-Loop Feynman Integrals
Project Lead
Project Duration
01/09/2021 - 31/08/2025Project URL
Go to Website
Partners
The Austrian Science Fund (FWF)
Publications
2025
[de Freitas]
The three-loop single-mass heavy-flavor corrections to the structure functions $F_2(x, Q^2)$ and $g_1(x, Q^2)$
J. Ablinger, A. Behring, J. Blümlein, A. De Freitas, A. von Manteuffel, C. Schneider, K. Schönwald
Technical report no. 25-08 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). September 2025. Licensed under CC BY 4.0 International. [doi] [pdf]@techreport{RISC7178,
author = {J. Ablinger and A. Behring and J. Blümlein and A. De Freitas and A. von Manteuffel and C. Schneider and K. Schönwald},
title = {{The three-loop single-mass heavy-flavor corrections to the structure functions $F_2(x,Q^2)$ and $g_1(x,Q^2)$}},
language = {english},
abstract = {We report quantitative results on the single-mass heavy-flavor contributions up to three-loop order to the unpolarized structure function $F_2(x,Q^2)$ and the polarized structure function $g_1(x,Q^2)$ for the first time. These results are relevant for precision QCD analyses of the World deep-inelastic data and the data taken at future colliders, such as the Electron--Ion Collider, in order to measure the strong coupling constant $alpha_s(M_Z^2)$, and the twist-2 parton distribution functions consistently at next-to-next-to-leading order.},
number = {25-08},
year = {2025},
month = {September},
keywords = {single-mass heavy-flavor contributions, QCD, Feynman diagrams, computer algebra},
length = {6},
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)}
}
author = {J. Ablinger and A. Behring and J. Blümlein and A. De Freitas and A. von Manteuffel and C. Schneider and K. Schönwald},
title = {{The three-loop single-mass heavy-flavor corrections to the structure functions $F_2(x,Q^2)$ and $g_1(x,Q^2)$}},
language = {english},
abstract = {We report quantitative results on the single-mass heavy-flavor contributions up to three-loop order to the unpolarized structure function $F_2(x,Q^2)$ and the polarized structure function $g_1(x,Q^2)$ for the first time. These results are relevant for precision QCD analyses of the World deep-inelastic data and the data taken at future colliders, such as the Electron--Ion Collider, in order to measure the strong coupling constant $alpha_s(M_Z^2)$, and the twist-2 parton distribution functions consistently at next-to-next-to-leading order.},
number = {25-08},
year = {2025},
month = {September},
keywords = {single-mass heavy-flavor contributions, QCD, Feynman diagrams, computer algebra},
length = {6},
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]
The Single-Mass Variable Flavor Number Scheme at Three-Loop Order
J. Ablinger, A. Behring, J. Blümlein, d, A. De Freitas, A. von Manteuffel, C. Schneider, and K. Schönwald
Technical report no. 25-04 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). October 2025. arXiv:2510.02175 [hep-ph]. Licensed under CC BY 4.0 International. [doi] [pdf]@techreport{RISC7181,
author = {J. Ablinger and A. Behring and J. Blümlein and d and A. De Freitas and A. von Manteuffel and C. Schneider and and K. Schönwald},
title = {{The Single-Mass Variable Flavor Number Scheme at Three-Loop Order}},
language = {english},
abstract = {The matching relations in the unpolarized and polarized variable flavor number scheme at three-loop order are presented in the single-mass case. They describe the process of massive quarks becoming light at large virtualities $Q^2$. In this framework, heavy-quark parton distributions can be defined. Numerical results are presented on the matching relations in the case of the single-mass variable flavor number scheme for the light parton, charm and bottom quark distributions. These relations are process independent. In the polarized case we generally work in the Larin scheme. To two-loop order we present the polarized massive OMEs also in the $overline{rm MS}$ scheme. Fast numerical codes for the single-mass massive operator matrix elements are provided. },
number = {25-04},
year = {2025},
month = {October},
note = {arXiv:2510.02175 [hep-ph]},
keywords = {QCD, Feynman diagrams, computer algebra},
length = {27},
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)}
}
author = {J. Ablinger and A. Behring and J. Blümlein and d and A. De Freitas and A. von Manteuffel and C. Schneider and and K. Schönwald},
title = {{The Single-Mass Variable Flavor Number Scheme at Three-Loop Order}},
language = {english},
abstract = {The matching relations in the unpolarized and polarized variable flavor number scheme at three-loop order are presented in the single-mass case. They describe the process of massive quarks becoming light at large virtualities $Q^2$. In this framework, heavy-quark parton distributions can be defined. Numerical results are presented on the matching relations in the case of the single-mass variable flavor number scheme for the light parton, charm and bottom quark distributions. These relations are process independent. In the polarized case we generally work in the Larin scheme. To two-loop order we present the polarized massive OMEs also in the $overline{rm MS}$ scheme. Fast numerical codes for the single-mass massive operator matrix elements are provided. },
number = {25-04},
year = {2025},
month = {October},
note = {arXiv:2510.02175 [hep-ph]},
keywords = {QCD, Feynman diagrams, computer algebra},
length = {27},
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]
The two-mass contributions to the three-loop massive operator matrix elements $tilde{A}_{Qg}^{(3)}$ and $Delta tilde{A}_{Qg}^{(3)}$
J. Ablinger, J. Bluemlein, A. De Freitas, A. von Manteuffel, C. Schneider, Kay Schoenwald
To appear in Journal of High Energy Physics, pp. ?-?. 2025. ISSN 1029-8479. arXiv:2510.09403 [hep-ph]. [doi]@article{RISC7198,
author = {J. Ablinger and J. Bluemlein and A. De Freitas and A. von Manteuffel and C. Schneider and Kay Schoenwald},
title = {{The two-mass contributions to the three-loop massive operator matrix elements $tilde{A}_{Qg}^{(3)}$ and $Delta tilde{A}_{Qg}^{(3)}$}},
language = {english},
abstract = {We calculate the two-mass three-loop contributions to the unpolarized and polarized massive operator matrix elements $tilde{A}_{Qg}^{(3)}$ and $Delta tilde{A}_{Qg}^{(3)}$ in $x$-space for a general mass ratio by using a semi-analytic approach. We also compute Mellin moments up to $N = 2000 (3000)$ by an independent method, to which we compare the results in $x$-space. In the polarized case, we work in the Larin scheme. We present numerical results. The two-mass contributions amount to about $50 %$ of the full textcolor{blue}{$O(T_F^2)$} and textcolor{blue}{$O(T_F^3)$} terms contributing to the operator matrix elements. The present result completes the calculation of all unpolarized and polarized massive three-loop operator matrix elements.},
journal = {To appear in Journal of High Energy Physics},
volume = {?},
pages = {?--?},
isbn_issn = {ISSN 1029-8479},
year = {2025},
note = {arXiv:2510.09403 [hep-ph]},
refereed = {yes},
length = {50},
url = {https://doi.org/10.35011/risc.25-07}
}
author = {J. Ablinger and J. Bluemlein and A. De Freitas and A. von Manteuffel and C. Schneider and Kay Schoenwald},
title = {{The two-mass contributions to the three-loop massive operator matrix elements $tilde{A}_{Qg}^{(3)}$ and $Delta tilde{A}_{Qg}^{(3)}$}},
language = {english},
abstract = {We calculate the two-mass three-loop contributions to the unpolarized and polarized massive operator matrix elements $tilde{A}_{Qg}^{(3)}$ and $Delta tilde{A}_{Qg}^{(3)}$ in $x$-space for a general mass ratio by using a semi-analytic approach. We also compute Mellin moments up to $N = 2000 (3000)$ by an independent method, to which we compare the results in $x$-space. In the polarized case, we work in the Larin scheme. We present numerical results. The two-mass contributions amount to about $50 %$ of the full textcolor{blue}{$O(T_F^2)$} and textcolor{blue}{$O(T_F^3)$} terms contributing to the operator matrix elements. The present result completes the calculation of all unpolarized and polarized massive three-loop operator matrix elements.},
journal = {To appear in Journal of High Energy Physics},
volume = {?},
pages = {?--?},
isbn_issn = {ISSN 1029-8479},
year = {2025},
note = {arXiv:2510.09403 [hep-ph]},
refereed = {yes},
length = {50},
url = {https://doi.org/10.35011/risc.25-07}
}
[Fadeev]
Computer algebra for special functions
Nikolai Fadeev
RISC, Johannes Kepler University Linz. PhD Thesis. May 2025.@phdthesis{RISC7201,
author = {Nikolai Fadeev},
title = {{Computer algebra for special functions}},
language = {english},
abstract = {Calculations done in different mathematical areas — such as computer algebra,combinatorics, number theory, differential equations — and physical areas — suchas particle physics — give rise to a plethora of problems involving special functionsthat need to be dealt with efficiently. In this PhD, we concentrated on two suchparticular problems.In the first part of this PhD thesis, we explored the relation between iteratedbinomial sums, an extension of general harmonic sums, and their integral representations, in order to compute their asymptotic expansions. To do that in a fullyautomatic way, we created a dedicated package, RICA. Using Mellin representations,we first formalised and extended a classical recursive method to compute Mellininverses of such sums, and together with it implemented several methods to compute asymptotic expansions of such integrals. In the process, we introduced andexplored a new class of functions related to Mellin convolutions. Those allowed usto automatically compute asymptotic expansions for more general classes of sumsin a new and efficient way, while providing a way to get symbolic representationsfor the constants appearing in the calculation of the Mellin inversions.In the second part of this PhD thesis, we studied first order inhomogeneous systems of differential equations involving an extra parameter epsilon coming from particlephysics computations. Since usually those systems could only be solved up to someorder in epsilon, we aimed at developing a method to optimise the solving task of suchsystems. We studied an approach centered on the minimisation of the epsilon-order in theexpansion of the inhomogeneous part. In particular, we proposed a method basedon separating the system in smaller subsystems called triangularization, beforeanalysing each of those individually using dffierent uncoupling schemes, selectedpriorization of equations and through comparisons of epsilon-orders. This method hasbeen implemented in a package called SystemAnalysis.},
year = {2025},
month = {May},
translation = {0},
school = {RISC, Johannes Kepler University Linz},
length = {292}
}
author = {Nikolai Fadeev},
title = {{Computer algebra for special functions}},
language = {english},
abstract = {Calculations done in different mathematical areas — such as computer algebra,combinatorics, number theory, differential equations — and physical areas — suchas particle physics — give rise to a plethora of problems involving special functionsthat need to be dealt with efficiently. In this PhD, we concentrated on two suchparticular problems.In the first part of this PhD thesis, we explored the relation between iteratedbinomial sums, an extension of general harmonic sums, and their integral representations, in order to compute their asymptotic expansions. To do that in a fullyautomatic way, we created a dedicated package, RICA. Using Mellin representations,we first formalised and extended a classical recursive method to compute Mellininverses of such sums, and together with it implemented several methods to compute asymptotic expansions of such integrals. In the process, we introduced andexplored a new class of functions related to Mellin convolutions. Those allowed usto automatically compute asymptotic expansions for more general classes of sumsin a new and efficient way, while providing a way to get symbolic representationsfor the constants appearing in the calculation of the Mellin inversions.In the second part of this PhD thesis, we studied first order inhomogeneous systems of differential equations involving an extra parameter epsilon coming from particlephysics computations. Since usually those systems could only be solved up to someorder in epsilon, we aimed at developing a method to optimise the solving task of suchsystems. We studied an approach centered on the minimisation of the epsilon-order in theexpansion of the inhomogeneous part. In particular, we proposed a method basedon separating the system in smaller subsystems called triangularization, beforeanalysing each of those individually using dffierent uncoupling schemes, selectedpriorization of equations and through comparisons of epsilon-orders. This method hasbeen implemented in a package called SystemAnalysis.},
year = {2025},
month = {May},
translation = {0},
school = {RISC, Johannes Kepler University Linz},
length = {292}
}
[Schneider]
Creative Telescoping for Hypergeometric Double Sums
P. Paule, C. Schneider
J. Symb. Comput. 128(102394), pp. 1-30. 2025. ISSN: 0747-7171. Symbolic Computation and Combinatorics: A special issue in memory and honor of Marko Petkovšek, edited by Shaoshi Chen, Sergei Abramov, Manuel Kauers, Eugene Zima. [doi]@article{RISC7068,
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.},
journal = {J. Symb. Comput.},
volume = {128},
number = {102394},
pages = {1--30},
isbn_issn = {ISSN: 0747-7171},
year = {2025},
note = {Symbolic Computation and Combinatorics: A special issue in memory and honor of Marko Petkovšek, edited by Shaoshi Chen, Sergei Abramov, Manuel Kauers, Eugene Zima},
refereed = {yes},
keywords = {creative telescoping; symbolic summation, hypergeometric multi-sums, contiguous relations, parameterized recurrences, rational solutions},
length = {30},
url = {https://doi.org/10.1016/j.jsc.2024.102394}
}
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.},
journal = {J. Symb. Comput.},
volume = {128},
number = {102394},
pages = {1--30},
isbn_issn = {ISSN: 0747-7171},
year = {2025},
note = {Symbolic Computation and Combinatorics: A special issue in memory and honor of Marko Petkovšek, edited by Shaoshi Chen, Sergei Abramov, Manuel Kauers, Eugene Zima},
refereed = {yes},
keywords = {creative telescoping; symbolic summation, hypergeometric multi-sums, contiguous relations, parameterized recurrences, rational solutions},
length = {30},
url = {https://doi.org/10.1016/j.jsc.2024.102394}
}
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]@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}
}
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]@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}
}
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]
Challenges for analytic calculations of the massive three-loop form factors
J Bluemlein, A. De Freitas, P. Marquard, C. Schneider
In: Proceedings of Loops and Legs in Quantum Field Theory, P. Marquard, M. Steinhauser (ed.)PoS(LL2024)03124-05, pp. 1-18. 2024. ISSN 1824-8039. arXiv:2408.07046 [hep-ph]. [doi]@inproceedings{RISC7066,
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}},
booktitle = {{Proceedings of Loops and Legs in Quantum Field Theory}},
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.},
volume = {PoS(LL2024)031},
number = {24-05},
pages = {1--18},
isbn_issn = {ISSN 1824-8039},
year = {2024},
note = { arXiv:2408.07046 [hep-ph]},
editor = {P. Marquard and M. Steinhauser},
refereed = {no},
keywords = {Form factor; computer algebra, coupled systems, differential equations, recurrences, analytic continuation, holonomic functions},
length = {18},
url = {https://doi.org/10.22323/1.467.0031 }
}
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}},
booktitle = {{Proceedings of Loops and Legs in Quantum Field Theory}},
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.},
volume = {PoS(LL2024)031},
number = {24-05},
pages = {1--18},
isbn_issn = {ISSN 1824-8039},
year = {2024},
note = { arXiv:2408.07046 [hep-ph]},
editor = {P. Marquard and M. Steinhauser},
refereed = {no},
keywords = {Form factor; computer algebra, coupled systems, differential equations, recurrences, analytic continuation, holonomic functions},
length = {18},
url = {https://doi.org/10.22323/1.467.0031 }
}
[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
In: Proceedings of Loops and Legs in Quantum Field Theory, P. Marquard, M. Steinhauser (ed.)PoS(LL2024)047 , pp. 1-12. 2024. SSN 1824-8039. arXiv:2407.02006 [hep-ph]. [doi]@inproceedings{RISC7067,
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}},
booktitle = {{Proceedings of Loops and Legs in Quantum Field Theory}},
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.},
volume = {PoS(LL2024)047 },
pages = {1--12},
isbn_issn = {SSN 1824-8039},
year = {2024},
note = {arXiv:2407.02006 [hep-ph]},
editor = {P. Marquard and M. Steinhauser},
refereed = {no},
keywords = {Feynman integrals, deep-inelastic scattering, numerical results},
length = {12},
url = {https://doi.org/10.22323/1.467.0047 }
}
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}},
booktitle = {{Proceedings of Loops and Legs in Quantum Field Theory}},
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.},
volume = {PoS(LL2024)047 },
pages = {1--12},
isbn_issn = {SSN 1824-8039},
year = {2024},
note = {arXiv:2407.02006 [hep-ph]},
editor = {P. Marquard and M. Steinhauser},
refereed = {no},
keywords = {Feynman integrals, deep-inelastic scattering, numerical results},
length = {12},
url = {https://doi.org/10.22323/1.467.0047 }
}
[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]@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}
}
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]
Error bounds for the asymptotic expansion of the partition function
Koustav Banerjee, Peter Paule, Cristian-Silviu Radu, Carsten Schneider
Rocky Mt J Math 54(6), pp. 1551-1592. 2024. ISSN: 357596. arXiv:2209.07887 [math.NT]. [doi] [pdf]@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 = {54},
number = {6},
pages = {1551--1592},
isbn_issn = {ISSN: 357596},
year = {2024},
note = {arXiv:2209.07887 [math.NT]},
refereed = {yes},
keywords = {partition function, asymptotic expansion, error bounds, symbolic summation},
length = {43},
url = {https://www.doi.org/10.1216/rmj.2024.54.1551}
}
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 = {54},
number = {6},
pages = {1551--1592},
isbn_issn = {ISSN: 357596},
year = {2024},
note = {arXiv:2209.07887 [math.NT]},
refereed = {yes},
keywords = {partition function, asymptotic expansion, error bounds, symbolic summation},
length = {43},
url = {https://www.doi.org/10.1216/rmj.2024.54.1551}
}
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]@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}
}
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]@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 }
}
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]@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}
}
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]@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}
}
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]@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}
}
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]
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]@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}
}
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]@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}
}
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]@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}
}
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]@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}
}
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}
}
