Comment on `Determination of light quark masses from $\eta \to 3\pi^0$'

TL;DR

This paper critically examines a previous claim of a model-independent determination of light quark mass differences from eta decay, showing that the original constraints are approximations and the numerical results are unreliable due to errors.

Contribution

It clarifies the limitations of the previous analysis and demonstrates that the claimed determination of $B_0(m_d-m_u)$ is not valid without additional information.

Findings

The two constraints are approximations of a single relation.

The original numerical results are due to computational errors.

Additional data is necessary for a reliable determination.

Abstract

In the commented paper [PRD 78, 034032 (2008); arXiv:0803.2956] Deandrea et al. claim that they provide a model-independent determination of the quantity $B_0(m_d-m_u)$ based on the analysis of the decay of eta to three neutral pions that takes the strong-interaction correction as an unknown parameter. In the following we show that the two constraints presented there as independent equations are in fact two approximations of a single constraint connecting together $B_0(m_d-m_u)$ with the strong-interaction correction to the $\eta\to3\pi^0$ decay amplitude. Thus, without some additional information on the latter (such as its correct value at least at one kinematic point) it does not lead to any determination of $B_0(m_d-m_u)$ and the numerical values presented there should not be taken seriously. We have also found that the numerical result for the studied quantity obtained in the commented paper is just an accidental number stemming from some numerical errors that had occurred in the computations presented there.