Gauge symmetric Δ(1232) couplings and the radiative muon capture in hydrogen

Abstract

By use of the difference between the gauge symmetric and standard $\ensuremath{\pi}N\ensuremath{\Delta}$ couplings, a contact $\ensuremath{\pi}\ensuremath{\pi}\mathit{NN}$ term, quadratic in the $\ensuremath{\pi}N\ensuremath{\Delta}$ coupling, is explicitly constructed. In addition, contribution from the \ensuremath{\Delta} excitation mechanism to the photon spectrum for the radiative muon capture in hydrogen is derived from the gauge symmetric $\ensuremath{\pi}N\ensuremath{\Delta}$ and $\ensuremath{\gamma}N\ensuremath{\Delta}$ couplings. It is shown for the photon spectrum, studied experimentally recently for photon momentum $k>60\phantom{\rule{0.3em}{0ex}}\text{MeV}$, that this contribution is smaller by 4--10% than the one obtained from the standardly used couplings for the on-shell \ensuremath{\Delta}'s.