Hyperspherical approach to quantum electrodynamics: sixth-order magnetic moment

Abstract

We present the details of a powerful new technique used in an exact calculation of the contribution of 6 sixth-order vertex graphs to the anomalous magnetic moment of the electron. By introducing four-dimensional spherical coordinates, the momentum-space integrals can be evaluated directly. The angular integrations areperformed using some simple properties of the Gegenbauer polynomials. For these graphs, the remaining convergent radial integrands are simple rational functions which are readily integrated. The divergent integrals are treated separately to extract all terms which do not vanish in the infrared and ultraviolet limits. The actual processing of the many terms involved was done using the algebraic

Hadronic light-by-light corrections to the muong−2:The pion-pole contribution

“…Let us mention that in the language of Refs. [45,35] the loop integrals we have to deal with are planar. We also note that these integrals are of the two-loop self-energy type.…”

“…Since the external momentum P flows only through the massive fermion propagators, we do not need to deform the integration contour for the radial integrals over Q 2 1 and Q 2 2 (see the discussion in Refs. [45,35]). Note that I 5 and I 6 depend only on one variable, Q 2 1 and Q 2 2 , respectively, whereas the expression for I 2 factorizes into a product of two single-variable functions.…”

“…(3.4) using the technique of Gegenbauer polynomials (hyperspherical approach); see Refs. [44,45,35]. In order to do so, we perform a Wick rotation of the momenta q µ 1 , q µ 2 , and p µ , denoting by capital letters the rotated Euclidean momenta, with Q 2 i = −q 2 i , P 2 = −m 2 , etc.…”

The Anomalous Magnetic Moment of the Muon: A Theoretical Introduction

“…With a representation of the form (5.31), the angular integrations can be performed, using for instance standard Gegenbauer polynomial techniques [hyperspherical approach], see Refs. [145,146,76]. This leads to a two dimensional integral representation:…”

High-precision ϵ-expansions of massive four-loop vacuum bubbles

“…The analytical expressions of V 1 and S 1 are strictly related. In fact, introducing hyperspherical variables [25,26] and performing the angular integrations one finds that the finite parts of V 1 and S 1 can be expressed as a sum of many double elliptic integrals with the same three square roots in the denominator. The simplest of these integrals is that with unity as numerator…”