2017/12/12 by Martijn Hidding, Hidding, Martijn, Francesco Moriello +1 · 11 citations
Computer Science · Mathematics · Physics and Astronomy · #Algebraic Geometry and Number Theory #Algebraic and Geometric Analysis #Algorithm #Computation #Cryptography and Residue Arithmetic #Differential equation #Elliptic integral #FOS: Physical sciences #Feynman diagram #Feynman integral #Geometry #High Energy Physics - Phenomenology (hep-ph) #Integral equation #Mathematical analysis #Mathematical physics #Mathematics #Order (exchange) #Parametric equation #Pure mathematics #Volume integral #hep-ph
paper · pdf · doi:10.48550/arxiv.1712.04441
published in arXiv (Cornell University) (Cornell University) · The differential equations method is applied to linearly reducible elliptic Feynman integrals, the solutions are in terms of elliptic polylogarithms, JHEP version, 50 pages
openalex publication_date 2017/12/12 · arxiv created 2019/01/15 · arxiv updated 2019/01/17 · openalex created_date 2022/10/04 · openalex updated_date 2026/08/06
We define linearly reducible elliptic Feynman integrals, and we show that\nthey can be algorithmically solved up to arbitrary order of the dimensional\nregulator in terms of a 1-dimensional integral over a polylogarithmic\nintegrand, which we call the inner polylogarithmic part (IPP). The solution is\nobtained by direct integration of the Feynman parametric representation. When\nthe IPP depends on one elliptic curve (and no other algebraic functions), this\nclass of Feynman integrals can be algorithmically solved in terms of elliptic\nmultiple polylogarithms (eMPLs) by using integration by parts identities. We\nthen elaborate on the differential equations method. Specifically, we show that\nthe IPP can be mapped to a generalized integral topology satisfying a set of\ndifferential equations in \ε-form. In the examples we consider the\ncanonical differential equations can be directly solved in terms of eMPLs up to\narbitrary order of the dimensional regulator. The remaining 1-dimensional\nintegral may be performed to express such integrals completely in terms of\neMPLs. We apply these methods to solve two- and three-points integrals in terms\nof eMPLs. We analytically continue these integrals to the physical region by\nusing their 1-dimensional integral representation.\n