Novel method for determining the light quark mass ratio using $\eta'\to\eta \pi\pi$ decays

We propose a novel approach for extracting symmetry breaking effects from symmetry conserving three-body decays. The method is based on mapping the Dalitz plot to a unit disk, and the difference of the disk distributions of two related decays isolates purely symmetry breaking effects. We demonstrate this method by extracting the fundamental parameter $Q$, an isospin breaking ratio of light quark masses defined as $Q^2\equiv (m_s^2-\hat m^2)/(m_d^2-m_u^2)$ with $\hat m$ the average of up and down quark masses, from the decays $\eta'\to\eta\pi^+\pi^-$ and $\eta'\to\eta\pi^0\pi^0$. With the Dalitz plot distributions for these two decays reported by BESIII, we illustrate the method and obtain $Q=22.5\pm1.0$, which is consistent with previous determinations and has a comparable uncertainty. With the full BESIII data set, which is eight times larger than the one used here, a more precise determination of $Q$ should become possible. This promising and novel method can be generalized to other three-body decays to extract symmetry breaking effects.