Analytical dispersive construction of $\eta\to3\pi$ amplitude: First order in isospin breaking

TL;DR

This paper develops an analytic dispersive approach to determine quark mass ratios from η→3π decays, addressing discrepancies between theory and experiment by combining chiral perturbation theory and experimental data.

Contribution

It introduces two dispersive methods that integrate NNLO chiral results and experimental measurements to accurately extract the quark mass ratio R.

Findings

Final quark mass ratio R=37.7±2.2

Determined Q=23.1±0.7 using lattice inputs

Provided updated values for current quark masses

Abstract

Because of their small electromagnetic corrections, the isospin-breaking decays $\eta\to3\pi$ seem to be good candidates for extracting isospin-breaking parameters $ (m_d-m_u)$. This task is unfortunately complicated by large chiral corrections and the discrepancy between the experimentally measured values of the Dalitz parameters describing the energy dependence of the amplitudes of these decays and those predicted from chiral perturbation theory. We present two methods based on an analytic dispersive representation that use the information from the NNLO chiral result and the one from the measurement of the charged $\eta\to3\pi$ decay by KLOE together in a harmonized way in order to determine the value of the quark mass ratio $R$. Our final result is $R=37.7\pm 2.2$. This value supplemented by values of $m_s/\hat{m}$ or even $\hat{m}$ and $m_s$ from other methods (as sum-rules or lattice) enables us to obtain further quark mass characteristics. For instance the recent lattice value for $m_s/\hat{m}~27.5$ leads to $Q= 23.1\pm0.7$. We also quote the corresponding values of the current masses $m_u$ and $m_d$.