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Weak Pion and Photon Production off Nucleons in a Chiral Effective Field Theory

Neutrino-induced pion and photon production from nucleons and nuclei are important for the interpretation of neutrino-oscillation experiments, and these processes are potential backgrounds in the MiniBooNE experiment [A. A. Aquilar-Arevalo \textit{et al.} (MiniBooNE Collaboration), Phys.\ Rev.\ Lett.\ {\bf 100}, 032301 (2008)]. Pion and photon production are investigated at intermediate energies, where the $Δ$ resonance becomes important. The Lorentz-covariant effective field theory contains nucleons, pions, Deltas, isoscalar scalar ($σ$) and vector ($ω$) fields, and isovector vector ($ρ$) fields. The lagrangian exhibits a nonlinear realization of (approximate) $SU(2)_L \otimes SU(2)_R$ chiral symmetry and incorporates vector meson dominance. Power counting for vertices and Feynman diagrams involving the $Δ$ is explained. Because of the built-in symmetries, the vector currents are automatically conserved, and the axial-vector currents satisfy PCAC. The irrelevance of so-called off-shell $Δ$ couplings and the structure of the dressed $Δ$ propagator, which has a pole only in the spin-3/2 channel, are discussed. To calibrate the axial-vector transition current $(N\! \leftrightarrow Δ)$, pion production from the nucleon is used as a benchmark and compared to bubble-chamber data from Argonne and Brookhaven National Laboratories. At low energies, the convergence of our power-counting scheme is investigated, and next-to-leading-order tree-level corrections are found to be very small.