Some dynamical properties of the pion inπ+→e+νγ andπ+→e+e+e−ν decays

We investigate some of the dynamical properties of the pion by explicitly calculating the γ and ξ parameters in ${\pi }^{+}$→$e^{+}$νγ and ${\pi }^{+}$→$e^{+}$ $e^{+}$ $e^{\mathrm{-}}$ν decays within the relativistic quark model, nucleon-loop approximation, and the σ model. One of the purposes in this study is to clarify a sign ambiguity raised recently in the prediction of γ within the quark model. The second purpose is to give some precise predictions for ξ, which has not yet been discussed extensively. The last purpose in this analysis involves a possible connection between ξ and chiral-symmetry breaking. Within the σ model, this parameter is found to depend logarithmically upon $m_{\pi }$, the explicit-breaking parameter of the symmetry.