Dispersion relation for hadronic light-by-light scattering: theoretical foundations

TL;DR

This paper advances a dispersive, model-independent framework for analyzing the hadronic light-by-light scattering tensor, clarifying its structure and contributions to the muon g-2, with explicit relations and unambiguous definitions.

Contribution

It provides a Lorentz decomposition of the HLbL tensor using a generalized approach, enabling transparent gauge invariance, crossing symmetry, and a simplified dispersive representation for the muon g-2.

Findings

Dispersive approach unambiguously defines pion-pole and pion-loop contributions.

Tensor decomposition clarifies gauge invariance and crossing symmetry.

Pion loop matches scalar QED amplitude with pion form factors.

Abstract

In this paper we make a further step towards a dispersive description of the hadronic light-by-light (HLbL) tensor, which should ultimately lead to a data-driven evaluation of its contribution to $(g-2)_\mu$. We first provide a Lorentz decomposition of the HLbL tensor performed according to the general recipe by Bardeen, Tung, and Tarrach, generalizing and extending our previous approach, which was constructed in terms of a basis of helicity amplitudes. Such a tensor decomposition has several advantages: the role of gauge invariance and crossing symmetry becomes fully transparent; the scalar coefficient functions are free of kinematic singularities and zeros, and thus fulfill a Mandelstam double-dispersive representation; and the explicit relation for the HLbL contribution to $(g-2)_\mu$ in terms of the coefficient functions simplifies substantially. We demonstrate explicitly that the dispersive approach defines both the pion-pole and the pion-loop contribution unambiguously and in a model-independent way. The pion loop, dispersively defined as pion-box topology, is proven to coincide exactly with the one-loop scalar QED amplitude, multiplied by the appropriate pion vector form factors.