2008/01/31 by David Broadhurst, Broadhurst, David · 2 citations
Mathematics · Physics and Astronomy · #Algebraic and Geometric Analysis #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Mathematical functions and polynomials #Particle physics theoretical and experimental studies
paper · pdf · doi:10.48550/arxiv.0801.4813
openalex publication_date 2008/01/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A proof is found for the elliptic integral evaluation of the Bessel moment M:=∫0^∞ t I02(t)K02(t)K0(2t) \rm dt =1/12 \bf K(sin(π/12))\bf K(cos(π/12)) =\fracΓ6(\frac13)64π222/3 resulting from an angular average of a 2-loop 4-point massive Feynman diagram, with one internal mass doubled. This evaluation follows from contour integration of the Green function for a hexagonal lattice, thereby relating M to a linear combination of two more tractable moments, one given by the Green function for a diamond lattice and both evaluated by using W.N. Bailey's reduction of an Appell double series to a product of elliptic integrals. Cubic and sesquiplicate modular transformations of an elliptic integral from the equal-mass Dalitz plot are proven and used extensively. Derivations are given of the sum rules ∫0^∞(I0(a t)K0(a t)-\frac2π K0(4a t) K0(t))K0(t) \rm dt=0 with a>0, proven by analytic continuation of an identity from Bailey's work, and ∫0^∞ t I0(a t)(I03(a t)K0(8t)- (1)/(4π2) I0(t)K03(t)) \rm dt=0 with 2≥ a≥0, proven by showing that a Feynman diagram in two spacetime dimensions generates the enumeration of staircase polygons in four dimensions.