  Neutrino-induced productions (neutrinoproduction) of photons and pions from
nucleons and nuclei are important for the interpretation of
neutrino-oscillation experiments, as they are potential backgrounds in the
MiniBooNE experiment [A. A. Aquilar-Arevalo et al. (MiniBooNE Collaboration),
Phys. Rev. Lett. {\bf 100}, 032301 (2008)]. These processes are studied at
intermediate energies, where the \Delta (1232) resonance becomes important. The
Lorentz-covariant effective field theory, which is the framework used in this
series of study, contains nucleons, pions, \Delta s, isoscalar scalar (\sigma)
and vector (\omega) fields, and isovector vector (\rho) fields. The lagrangian
exhibits a nonlinear realization of (approximate) $SU(2)_L \otimes SU(2)_R$
chiral symmetry and incorporates vector meson dominance. In this paper, we
focus on setting up the framework. Power counting for vertices and Feynman
diagrams is explained. Because of the built-in symmetries, the vector current
is automatically conserved (CVC), and the axial-vector current is partially
conserved (PCAC). To calibrate the axial-vector transition current (N
$\leftrightarrow$ \Delta), 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 small.
