What can we learn from new measurements of Dalitz plot parameters for K $\rightarrow 3 \pi$ decays ?

Abstract

We give a simple expression in linear and quadratic Dalitz--plot slopes which does not depend on the charge combination of the $3\pi$ state $(K^\pm \to \pi^{\pm}\pi^{+}\pi^{-}$ or $\pi^{\pm}\pi^{0}\pi^{0}$ and $K_L^{0} \to \pi^{+}\pi^{-}\pi^{0}$ or $\pi^{0}\pi^{0}\pi^{0})$, if all phases between final states are negligible. After investigating the influence of radiative corrections, it is shown how new measurements, especially of quadratic slopes in the $\pi^{\pm} \pi^{0} \pi^{0}$ channel, could help to test theoretical predictions more stringently. A FORTRAN code for the radiatiative corrections is available on request.