In the framework of Chiral Perturbation Theory including photons, we found that the contribution of the photon exchange between two intermediate charged Kaons to the slope parameter of the decay (etarightarrow 3pi^{0}) amounts to (-0.0221pm 0.0034).
When compared with the experimental value, (alpha =-0.0317pm 0.0016), on the one hand, and with the contribution of the up and down quark mass difference, (+0.013pm 0.032), on the other hand, our result leads to the direct conclusion: textit{The} (etarightarrow 3pi^{0}) textit{decay uline{cannot} be used to determine} (m_{d}-m_{u}).
Sutherlands theorem dictates that the contribution of the electromagnetic interaction to the decay process (etarightarrow 3pi^{0}) is neglected with respect to the one coming from the difference between the up and down quark masses. In the framework
of chiral perturbation theory including virtual photons, we calculated the main diagram concerning the exchange of a virtual photon between two intermediate charged pions. The correction induced by this diagram on the slope parameter amounts to (17%) of the correction induced by the pure strong interaction at one-loop level. If this result is maintained when considering all the diagrams at the chiral order we are working, we can say without any doubt that Sutherlands theorem is strongly violated. As a direct consequence, any determination of light quark masses from the present decay textit{should} take into account the electromagnetic interaction.
We provide a model-independent determination of the quantity B_0(m_d-m_u). Our approach rests only on chiral symmetry and data from the decay of the eta into three neutral pions. Since the low-energy prediction at next-to-leading order fails to repro
duce the experimental results, we keep the strong interaction correction as an unknown parameter. As a first step, we relate this parameter to the quark mass difference using data from the Dalitz plot. A similar relation is obtained using data from the decay width. Combining both relations we obtain B_0(m_d-m_u)=(4495+/-440) MeV^2. The preceding value, combined with lattice determinations, leads to the values m_u(2 GeV)=(2.9+/-0.8) MeV and m_d(2 GeV)=(4.7+/-0.8) MeV.
