Chiral Effective-Field Theory in the Delta(1232) Region: Pion Electroproduction on the Nucleon

We develop an extension of chiral perturbation theory to the $\Delta$(1232)-resonance energy region and apply it to investigate the pion electroproduction off the nucleon ($e^- N \to e^- N\pi$). We present a complete calculation of this process, in the $\Delta$-resonance region, up to next-to-leading order in the {\it $\delta$-expansion}. At this order, the only free parameters are the three low-energy constants corresponding to the magnetic (M1), electric (E2), and Coulomb (C2) $\gamma N \to \Delta $ transition strength. After fitting these parameters to a few well-known data, our calculation provides a prediction for observables and multipole amplitudes of pion electroproduction. These results compare favorably with the phenomenological multipole solutions and recent experimental results from MIT-Bates and MAMI. Our prediction for the pion-mass dependence of the $\gamma N\Delta$ form factors offers an explanation for the discrepancy between the recent lattice-QCD results and the experimental value for the "C2/M1 ratio" at low $Q^2$.