MO analysis of the high statistics Belle results on $\gamma\gamma\to \pi^+\pi^-,\pi^0\pi^0$ with chiral constraints

TL;DR

This paper applies Muskhelishvili-Omnès dispersive analysis to high-statistics Belle data on photon-photon to pion-pion scattering, incorporating chiral constraints and resonance contributions to improve understanding of low-energy QCD interactions.

Contribution

It introduces a modified MO formalism that simplifies polynomial ambiguities and includes tensor resonances up to 1.3 GeV, extending applicability beyond 1 GeV with inelastic effects.

Findings

Resonance contributions up to spin two are significant.

New constraints on pion polarizabilities are obtained.

Data helps distinguish isospin breaking scenarios in the $f_0(980)$ region.

Abstract

We reconsider Muskhelishvili-Omn\`es (MO) dispersive representations of photon-photon scattering to two pions, motivated by the very high statistics results recently released by the Belle collaboration for charged as well as neutral pion pairs and also by recent progress in the determination of the low-energy $\pi\pi$ scattering amplitude. Applicability of this formalism is extended beyond 1 GeV by taking into account inelasticity due to $K\bar{K}$ . A modified MO representation is derived which has the advantage that all polynomial ambiguities are collected into the subtraction constants and have simple relations to pion polarizabilities. It is obtained by treating differently the exactly known QED Born term and the other components of the left-hand cut. These components are approximated by a sum over resonances. All resonances up to spin two and masses up to $\simeq1.3$ GeV are included. The tensor contributions to the left-hand cut are found to be numerically important. We perform fits to the data imposing chiral constraints, in particular, using a model independent sum rule result on the $p^6$ chiral coupling $c_{34}$. Such theoretical constraints are necessary because the experimental errors are dominantly systematic. Results on further $p^6$ couplings and pion dipole and quadrupole polarizabilities are then derived from the fit. The relevance of the new data for distinguishing between two possible scenarios of isospin breaking in the $f_0(980)$ region is discussed.