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Extensions of the AZ-Algorithm and the Package MultiIntegrate

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Anti-Differentiation and the Calculation of Feynman Amplitudes

Part of the book series: Texts & Monographs in Symbolic Computation ((TEXTSMONOGR))

Abstract

We extend the (continuous) multivariate Almkvist-Zeilberger algorithm in order to apply it for instance to special Feynman integrals emerging in renormalizable Quantum field Theories. We will consider multidimensional integrals over hyperexponential integrals and try to find closed form representations in terms of nested sums and products or iterated integrals. In addition, if we fail to compute a closed form solution in full generality, we may succeed in computing the first coefficients of the Laurent series expansions of such integrals in terms of indefinite nested sums and products or iterated integrals. In this article we present the corresponding methods and algorithms. Our Mathematica package MultiIntegrate, can be considered as an enhanced implementation of the (continuous) multivariate Almkvist Zeilberger algorithm to compute recurrences or differential equations for hyperexponential integrands and integrals. Together with the summation package Sigma and the package HarmonicSums our package provides methods to compute closed form representations (or coefficients of the Laurent series expansions) of multidimensional integrals over hyperexponential integrands in terms of nested sums or iterated integrals.

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Notes

  1. 1.

    The Mathematica packages MultiIntegrate, HarmonicSums, Sigma and EvaluateMultiSums can be downloaded at https://risc.jku.at/software.

  2. 2.

    There exist upper bounds for a particular input. But usually, these bounds are too high and one tries smaller values.

  3. 3.

    There exist upper bounds for a particular input. But usually, these bounds are too high and one tries smaller values.

References

  1. J. Ablinger, Computer Algebra Algorithms for Special Functions in Particle Physics. PhD thesis, J. Kepler University Linz, 2012

    Google Scholar 

  2. J. Ablinger, The package HarmonicSums: Computer algebra and analytic aspects of nested sums, in PoS LL2014, 019 (2014)

    Google Scholar 

  3. J. Ablinger, Inverse mellin transform of holonomic sequences, in PoS LL2016, 067 (2016)

    Google Scholar 

  4. J. Ablinger, Computing the inverse mellin transform of holonomic sequences using Kovacic’s algorithm, in PoS RADCOR2017, 069 (2018)

    Google Scholar 

  5. J. Ablinger, Discovering and proving infinite pochhammer sum identities. Experimental Mathematics (2019)

    Google Scholar 

  6. J. Ablinger, J. Blümlein, C. Schneider, Harmonic sums and polylogarithms generated by cyclotomic polynomials. J. Math. Phys. 52(10), 102301 (2011)

    Google Scholar 

  7. J. Ablinger, J. Blümlein, M. Round, C. Schneider, Advanced computer algebra algorithms for the expansion of Feynman integrals, in PoS LL2012, 050 (2012)

    Google Scholar 

  8. J. Ablinger, J. Blümlein, C. Schneider, Analytic and algorithmic aspects of generalized harmonic sums and polylogarithms. J. Math. Phys. 54(8), 082301 (2013)

    Google Scholar 

  9. J. Ablinger, J. Blümlein, A. De Freitas, A. Hasselhuhn, A. von Manteuffel, M. Round, C. Schneider, F. Wißbrock, The transition matrix element A gq(N) of the variable flavor number scheme at \(O(\alpha _s^3)\). Nucl. Phys. B 882, 263–288 (2014)

    Google Scholar 

  10. J. Ablinger, J. Blümlein, C.G. Raab, C. Schneider, Iterated binomial sums and their associated iterated integrals. J. Math. Phys. 55(11), 112301 (2014)

    Google Scholar 

  11. J. Ablinger, A. Behring, J. Blümlein, A. De Freitas, A. von Manteuffel, C. Schneider, Calculating three loop ladder and V-topologies for massive operator matrix elements by computer algebra. Comput. Phys. Commun. 202, 33–112 (2016)

    Article  MathSciNet  Google Scholar 

  12. S. Abramov, M. Petkovšek, D’Alembertian solutions of linear differential and difference equations, in Proceedings of ISSAC’94, ed. by J. von zur Gathen, pp. 169–174 (1994)

    Google Scholar 

  13. S. Abramov, E. Zima, D’Alembertian solutions of inhomogeneous linear equations (differential, difference, and some other), in Proceedings of ISSAC’96, ed. by Y.N. Lakshman, pp. 232–240 (1996)

    Google Scholar 

  14. M. Apagodu, D. Zeilberger. Multi-variable Zeilberger and Almkvist-Zeilberger algorithms and the sharpening of Wilf-Zeilberger theory. Adv. Appl. Math. 37(2), 139–152 (2006)

    Article  MathSciNet  Google Scholar 

  15. I. Bierenbaum, J. Blümlein, S. Klein, Mellin moments of the \(o(\alpha ^3_s)\) heavy flavor contributions to unpolarized deep-inelastic scattering at q 2 ≫ m 2 and anomalous dimensions. Nucl. Phys. B 820(1), 417–482 (2009)

    Google Scholar 

  16. J. Blümlein, S. Kurth, Harmonic sums and Mellin transforms up to two-loop order. Phys. Rev. D 60(1), 014018 (1999)

    Google Scholar 

  17. J. Blümlein, M. Kauers, S. Klein, C. Schneider, Determining the closed forms of the \(O(a_s^3)\) anomalous dimensions and Wilson coefficients from Mellin moments by means of computer algebra. Comput. Phys. Commun. 180(11), 2143–2165 (2009)

    Google Scholar 

  18. J. Blümlein, S. Klein, C. Schneider, F. Stan, A symbolic summation approach to Feynman integral calculus. J. Symb. Comput. 47(10), 1267–1289 (2011)

    Article  MathSciNet  Google Scholar 

  19. J. Blümlein, M. Round, C. Schneider, Refined holonomic summation algorithms in particle physics. Adv. Comput. Algebra WWCA 2016 226, 51–91 (2018)

    Article  MathSciNet  Google Scholar 

  20. C. Bogner, S. Weinzierl, Feynman graph polynomials. Int. J. Modern Phys. A 25, 2585–2618 (2010)

    Article  MathSciNet  Google Scholar 

  21. D. Broadhurst, W. Zudilin, A magnetic double integral. J. Aust. Math. Soc. 1, 9–25 (2019)

    Article  MathSciNet  Google Scholar 

  22. M. Bronstein, Linear ordinary differential equations: breaking through the order 2 barrier, in Proceedings of ISSAC’92, pp. 42–48 (1992)

    Google Scholar 

  23. S. Chen, M. Kauers, Trading order for degree in creative telescoping. J. Symb. Comput. 47(8), 968–995 (2012)

    Article  MathSciNet  Google Scholar 

  24. S. Chen, M. Kauers, Order-degree curves for hypergeometric creative telescoping, in Proceedings of ISSAC’12, pp. 122–129 (2012)

    Google Scholar 

  25. S. Chen, M. Kauers, C. Koutschan, A generalized apagodu-zeilberger algorithm, in Proceedings of ISSAC’14, pp. 107–114 (2014)

    Google Scholar 

  26. Y. Frishman, Operator products at almost light like distances. Ann. Phys. 66, 373–389 (1971)

    Article  MathSciNet  Google Scholar 

  27. M.Y. Kalmykov, O. Veretin, Single scale diagrams and multiple binomial sums. Phys. Lett. B 483, 315–323 (2000)

    Article  MathSciNet  Google Scholar 

  28. C. Koutschan, A fast approach to creative telescoping. Math. Comput. Sci. 4(2–3), 259–266 (2010)

    Article  MathSciNet  Google Scholar 

  29. J.J. Kovacic, An algorithm for solving second order linear homogeneous differential equations. J. Symb. Comput. 2, 3–43 (1986)

    Article  MathSciNet  Google Scholar 

  30. S.-O. Moch, P. Uwer, S. Weinzierl, Nested sums, expansion of transcendental functions, and multiscale multiloop integrals. J. Math. Phys. 43(6), 3363–3386 (2002)

    Article  MathSciNet  Google Scholar 

  31. M. Mohammed, D. Zeilberger, Sharp upper bounds for the orders of the recurrences outputted by the Zeilberger and q-Zeilberger algorithms. J. Symb. Comput. 39(2), 201–207 (2005)

    Article  Google Scholar 

  32. M. Petkovšek, Hypergeometric solutions of linear recurrences with polynomial coefficients. J. Symb. Comput. 14, 243–264 (1992)

    Article  MathSciNet  Google Scholar 

  33. E. Remiddi, J.A.M. Vermaseren, Harmonic polylogarithms. Int. J. Modern Phys. A 15, 725–754 (2000)

    Article  MathSciNet  Google Scholar 

  34. C. Schneider, Symbolic summation in difference fields. PhD thesis, RISC, J. Kepler University Linz, May 2001

    Google Scholar 

  35. C. Schneider, A new Sigma approach to multi-summation. Adv. Appl. Math. 34, 740–767 (2005)

    Article  MathSciNet  Google Scholar 

  36. C. Schneider, Solving parameterized linear difference equations in terms of indefinite nested sums and products. J. Differ. Equ. Appl. 11(9), 799–821 (2006)

    Article  MathSciNet  Google Scholar 

  37. C. Schneider, Symbolic summation assists combinatorics. SÃminaire Lotharingien de Combinatoire 56, 1–36 (2007)

    MathSciNet  MATH  Google Scholar 

  38. C. Schneider, Simplifying multiple sums in difference fields, in Computer Algebra in Quantum Field Theory: Integration, Summation and Special Functions, ed. by C. Schneider, J. Blümlein. Texts & Monographs in Symbolic Computation (Springer, Wien, 2013), pp. 325–360

    Google Scholar 

  39. C. Schneider, Modern summation methods for loop integrals in quantum field theory: The packages sigma, EvaluateMultiSums and SumProduction. J. Phys. Conf. Ser. 523, 012037 (2014)

    Article  Google Scholar 

  40. J.A.M. Vermaseren, Harmonic sums, Mellin transforms and integrals. Int. J. Modern Phys. A 14(13), 2037–2076 (1999)

    Article  MathSciNet  Google Scholar 

  41. K. Wegschaider, Computer generated proofs of binomial multi-sum identities. Master’s thesis, RISC, J. Kepler University, May 1997

    Google Scholar 

  42. S. Weinzierl, Feynman graphs, in Computer Algebra in Quantum Field Theory: Integration, Summation and Special Functions, ed. by C. Schneider, J. Blümlein. Texts & Monographs in Symbolic Computation (Springer, Wien, 2013), pp. 381–406

    Google Scholar 

  43. H.S. Wilf, D. Zeilberger, An algorithmic proof theory for hypergeometric (ordinary and “q”) multisum/integral identities. Inventiones Mathematicae 108, 575–633 (1992)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This work was supported by the Austrian Science Fund (FWF) grant SFB F50 (F5009-N15) and by the bilateral project WTZ BG 03/2019 (KP-06-Austria/8/2019), funded by OeAD (Austria) and Bulgarian National Science Fund. The author would like to thank C. Schneider for useful discussions.

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Ablinger, J. (2021). Extensions of the AZ-Algorithm and the Package MultiIntegrate. In: Blümlein, J., Schneider, C. (eds) Anti-Differentiation and the Calculation of Feynman Amplitudes. Texts & Monographs in Symbolic Computation. Springer, Cham. https://doi.org/10.1007/978-3-030-80219-6_2

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  • DOI: https://doi.org/10.1007/978-3-030-80219-6_2

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