Electromagnetic Meson Decays from an Effective Lagrangian with Chiral Symmetry

Abstract

A nonet-symmetrical and PCAC-violating term is introduced into a Lagrangian with $S{U}_{3}\ensuremath{\bigotimes}S{U}_{3}$ chiral symmetry and PCAC (hypothesis of partially conserved axial-vector current). Breaking of the nonet symmetry of this term is achieved through the renormalization of the fields. This renormalization is necessary because of the requirement that the kinetic-energy terms of the Lagrangian be correctly normalized after the removal of the tadpole terms. Using as input the decay rate for ${\ensuremath{\pi}}^{0}\ensuremath{\rightarrow}2\ensuremath{\gamma}$, we predict $\ensuremath{\Gamma}(\ensuremath{\eta}\ensuremath{\rightarrow}2\ensuremath{\gamma})=0.41\ifmmode\pm\else\textpm\fi{}0.08$ keV and $\ensuremath{\Gamma}({X}^{0}\ensuremath{\rightarrow}2\ensuremath{\gamma})=6\ifmmode\pm\else\textpm\fi{}1$ keV. The result for $\ensuremath{\Gamma}(\ensuremath{\eta}\ensuremath{\rightarrow}2\ensuremath{\gamma})$ agrees with experiment within 2 standard deviations. It is argued that errors of about 20% should be expected in our calculations, mainly because of the neglect of the effects due to the finite width of the various particles.