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We develop an extension of chiral perturbation theory to the$\Delta$(1232)-resonance energy region and apply it to investigate the pionelectroproduction off the nucleon ($e^- N \to e^- N\pi$). We present a completecalculation of this process, in the $\Delta$-resonance region, up tonext-to-leading order in the {\it $\delta$-expansion}. At this order, the onlyfree parameters are the three low-energy constants corresponding to themagnetic (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 amplitudesof pion electroproduction. These results compare favorably with thephenomenological multipole solutions and recent experimental results fromMIT-Bates and MAMI. Our prediction for the pion-mass dependence of the $\gammaN\Delta$ form factors offers an explanation for the discrepancy between therecent lattice-QCD results and the experimental value for the "C2/M1 ratio" atlow $Q^2$.Comment: 38 pages, 14 figures; version to be published in Phys. Rev.

physical review. d. particles, fields, gravitation, and cosmology/physical review. d, particles, fields, gravitation, and cosmology

Chiral effective-field theory in the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>Δ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>1232</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math>region: Pion electroproduction on the nucleon

Chiral effective-field theory in theΔ(1232)region: Pion electroproduction on the nucleon