Analytical expressions of 3 and 4-loop sunrise Feynman integrals and 4-dimensional lattice integrals

Abstract

In this paper we continue the work begun in 2002 on the identification of the analytical expressions of Feynman integrals which require the evaluation of multiple elliptic integrals. We rewrite and simplify the analytical expression of the 3-loop self-mass integral with three equal masses and on-shell external momentum. We collect and analyze a number of results on double and triple elliptic integrals. By using very high-precision numerical fits, for the first time we are able to identify a very compact analytical expression for the 4-loop on-shell self-mass integral with 4 equal masses, that is one of the master integrals of the 4-loop electron g-2. Moreover, we fit the analytical expressions of some integrals which appear in lattice perturbation theory, and in particular the 4-dimensional generalized Watson integral.

13 pages, 1 figure, LaTex; v2: some rephrasing of text; v3: reference added, minor modifications; v4: checks of lattice integrals up to 2400 digits; some modifications of text; version accepted for publication in IJMPA, needs the document class ws-ijmpa.cls

References in corpus (10)

• Harmonic Polylogarithms
• Harmonic sums, Mellin transforms and Integrals
• Harmonic Sums and Mellin Transforms up to two-loop Order
• Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops
• Analytic treatment of the two loop equal mass sunrise graph
• Elliptic integral evaluations of Bessel moments
• Lattice Perturbation Theory
• High-precision e-expansions of massive four-loop vacuum bubbles
• Calculation of master integrals by difference equations
• Lattice Green Function (at 0) for the 4d Hypercubic Lattice

Cited by in corpus (14)

• CODATA Recommended Values of the Fundamental Physical Constants: 2010
• Space-time dimensionality D as complex variable: calculating loop integrals using dimensional recurrence relation and analytical properties with respect to D
• High-precision calculation of the 4-loop contribution to the electron g-2 in QED
• Feynman graph polynomials
• Analytic Structure of all Loop Banana Amplitudes
• Embedding Feynman Integral (Calabi-Yau) Geometries in Weighted Projective Space
• Generalized Harmonic, Cyclotomic, and Binomial Sums, their Polylogarithms and Special Numbers
• High-precision calculation of the 4-loop QED contribution to the slope of the Dirac form factor
• Three-loop banana integrals with four unequal masses
• Some algebraic and arithmetic properties of Feynman diagrams
• On Laporta's 4-loop sunrise formulae
• High-precision four-loop mass and wave function renormalization in QED
• On Borwein's conjectures for planar uniform random walks
• $\mathbb{Q}$-linear dependence of certain Bessel moments