Evaluating ‘elliptic’ master integrals at special kinematic values: using differential equations and their solutions via expansions near singular points

Journal of High Energy Physics, Jul 2018

Abstract This is a sequel of our previous paper where we described an algorithm to find a solution of differential equations for master integrals in the form of an ϵ-expansion series with numerical coefficients. The algorithm is based on using generalized power series expansions near singular points of the differential system, solving difference equations for the corresponding coefficients in these expansions and using matching to connect series expansions at two neighboring points. Here we use our algorithm and the corresponding code for our example of four-loop generalized sunset diagrams with three massive and tw massless propagators, in order to obtain new analytical results. We analytically evaluate the master integrals at threshold, p2 = 9m2, in an expansion in ϵ up to ϵ1. With the help of our code, we obtain numerical results for the threshold master integrals in an ϵ-expansion with the accuracy of 6000 digits and then use the PSLQ algorithm to arrive at analytical values. Our basis of constants is build from bases of multiple polylogarithm values at sixth roots of unity.

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Evaluating ‘elliptic’ master integrals at special kinematic values: using differential equations and their solutions via expansions near singular points

Journal of High Energy Physics July 2018, 2018:102 | Cite as Evaluating ‘elliptic’ master integrals at special kinematic values: using differential equations and their solutions via expansions near singular points AuthorsAuthors and affiliations Roman N. LeeAlexander V. SmirnovVladimir A. Smirnov Open Access Regular Article - Theoretical Physics First Online: 16 July 2018 Received: 07 June 2018 Revised: 07 July 2018 Accepted: 10 July 2018 Abstract This is a sequel of our previous paper where we described an algorithm to find a solution of differential equations for master integrals in the form of an ϵ-expansion series with numerical coefficients. The algorithm is based on using generalized power series expansions near singular points of the differential system, solving difference equations for the corresponding coefficients in these expansions and using matching to connect series expansions at two neighboring points. Here we use our algorithm and the corresponding code for our example of four-loop generalized sunset diagrams with three massive and tw massless propagators, in order to obtain new analytical results. We analytically evaluate the master integrals at threshold, p2 = 9m2, in an expansion in ϵ up to ϵ1. With the help of our code, we obtain numerical results for the threshold master integrals in an ϵ-expansion with the accuracy of 6000 digits and then use the PSLQ algorithm to arrive at analytical values. Our basis of constants is build from bases of multiple polylogarithm values at sixth roots of unity. Keywords Perturbative QCD Scattering Amplitudes  ArXiv ePrint: 1805.00227 Download to read the full article text Notes Open Access This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. References [1] E. 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Copyright information © The Author(s) 2018 Authors and Affiliations Roman N. Lee1Alexander V. Smirnov24Vladimir A. Smirnov3Email author1.Budker Institute of Nuclear PhysicsNovosibirskRussia2.Research Computing CenterMoscow State UniversityMoscowRussia3.Skobeltsyn Institute of Nuclear Physics of Moscow State UniversityMoscowRussia4.Institut für Theoretische Teilchenphysik, KITKarlsruheGermany


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Roman N. Lee, Alexander V. Smirnov, Vladimir A. Smirnov. Evaluating ‘elliptic’ master integrals at special kinematic values: using differential equations and their solutions via expansions near singular points, Journal of High Energy Physics, 2018, 102, DOI: 10.1007/JHEP07(2018)102