Strong violations of the conserved-vector-current hypothesis in pion radiative decay?

Abstract

We study the question whether large isospin-breaking effects due to the quark mass matrix may be expected in the decay $\ensuremath{\pi}\ensuremath{\rightarrow}e{\ensuremath{\nu}}_{e}\ensuremath{\gamma}$. This has been predicted previously on the basis of quantum-chromodynamic perturbation theory. However, perturbation theory may be rather misleading in this context because it does not take into account the correct ${\ensuremath{\gamma}}_{5}$ structure of the vacuum. We show quite generally that isospin-breaking effects vanish in the chiral limit, independently of $\ensuremath{\xi}=\frac{{m}_{d}}{{m}_{u}}$. An illustration and quantitative estimate is given in the framework of the $\ensuremath{\sigma}$ model.