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

• Received 7 September 2004

DOI:https://doi.org/10.1103/PhysRevC.72.015501