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Dispersive construction of two-loop P->3π(P=K,η) amplitudes

The branching ratio of the η->3πdecay is an important source of information on the value of the quark mass ratio 1/R=(m_d-m_u)/(m_s-\hat m). Furthermore, isospin breaking effects in the decays K->3πprovide information on the pion scattering lengths. The cusp effect in the K->3πdecays is presently being analyzed by the NA48 and KTeV experiments. From the theoretical point of view, these processes have been studied by different methods. We propose a unified and relativistic treatment relying on very general principles, unitarity, analyticity and crossing symmetry, combined with chiral counting, in order to construct model-independent representations of the corresponding amplitudes that are valid at two loops. A general description of the procedure is given and is illustrated in the case of the ηdecay amplitude in the leading order in the isospin breaking.