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

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) µ . 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 doubledispersive representation; and the explicit relation for the HLbL contribution to (g − 2) µ 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.

Dispersion relation for hadronic light-by-light scattering: two-pion contributions

“…The first step is the decomposition of the HLbL tensor into Lorentz structures and scalar functions that are free of kinematic singularities and zeros. We have solved this problem in [31] and recapitulate the results in section 2.1. This representation, referred to as BTT tensor decomposition [51,52] in figure 2, allows us to write the HLbL contribution to (g−2) µ in full generality as a master formula that involves only three integrals.…”

“…In [31], we have used the Mandelstam representation for the scalar functions to study the pion-box contribution. In section 2.2, we extend the dispersive treatment and derive from the Mandelstam representation single-variable dispersion relations for general two-pion contributions.…”

“…We slightly modify and improve the master formula presented in [30,31] in such a way that crossing symmetry between all three off-shell photons remains manifest. The dynamical input in the master formula is encoded in only six different scalar functions and their crossed versions.…”

“…In order to improve the determination of the HLbL contribution, we proposed a dispersive framework [28], based on the fundamental principles of analyticity, unitarity, gauge invariance, and crossing symmetry, which opens up a path towards a data-driven evaluation [29]. As the next step [30,31], we presented a comprehensive solution to the task of constructing a basis for the HLbL tensor devoid of kinematic singularities, defining scalar functions that are amenable to a dispersive treatment. In particular, we derived a Lorentz decomposition of the HLbL tensor that manifestly implements crossing symmetry and gauge invariance, with scalar coefficient functions free of kinematic singularities and zeros that fulfill the Mandelstam double-spectral representation.…”

“…Based on a recipe by Bardeen, Tung [51], and Tarrach [52] (BTT), we have derived in [30,31] a decomposition of the HLbL tensor…”