2017/06/05 by Jakob Ablinger, J. Ablinger, J. Blümlein +13 · 115 citations
Computer Science · Mathematics · Physics and Astronomy · #Basic hypergeometric series #Black Holes and Theoretical Physics #Eisenstein series #Elliptic integral #Generalized hypergeometric function #Hypergeometric function #Hypergeometric function of a matrix argument #Jacobi elliptic functions #Mathematical analysis #Mathematics #Modular form #Particle physics theoretical and experimental studies #Pure mathematics #Quantum Chromodynamics and Particle Interactions #Theta function #cs.SC #hep-ph #hep-th #math-ph #math.AG #math.MP
paper · pdf · doi:10.1063/1.4986417
published in Journal of Mathematical Physics 59(6) (American Institute of Physics) · 68 pages LATEX, 10 Figures
arxiv created 2017/06/05 · openalex created_date 2017/06/15 · openalex publication_date 2018/06/01 · arxiv updated 2018/08/01 · openalex updated_date 2026/08/06
We calculate 3-loop master integrals for heavy quark correlators and the 3-loop quantum chromodynamics corrections to the ρ-parameter. They obey non-factorizing differential equations of second order with more than three singularities, which cannot be factorized in Mellin-N space either. The solution of the homogeneous equations is possible in terms of 2F1 Gauß hypergeometric functions at rational argument. In some cases, integrals of this type can be mapped to complete elliptic integrals at rational argument. This class of functions appears to be the next one arising in the calculation of more complicated Feynman integrals following the harmonic polylogarithms, generalized polylogarithms, cyclotomic harmonic polylogarithms, square-root valued iterated integrals, and combinations thereof, which appear in simpler cases. The inhomogeneous solution of the corresponding differential equations can be given in terms of iterative integrals, where the new innermost letter itself is not an iterative integral. A new class of iterative integrals is introduced containing letters in which (multiple) definite integrals appear as factors. For the elliptic case, we also derive the solution in terms of integrals over modular functions and also modular forms, using q-product and series representations implied by Jacobi’s ϑi functions and Dedekind’s η-function. The corresponding representations can be traced back to polynomials out of Lambert–Eisenstein series, having representations also as elliptic polylogarithms, a q-factorial 1/ηk(τ), logarithms, and polylogarithms of q and their q-integrals. Due to the specific form of the physical variable x(q) for different processes, different representations do usually appear. Numerical results are also presented.