Multipole Theory of Neutrino-Nuclear Reactions: Application toC12

The cross section for neutrino-induced nuclear reactions is presented in the form of a multipole expansion, appropriate for the excitation of nuclear levels of a definite spin and parity. The theory is then applied to the $T=1\mathrm{}$ levels of ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$, knowledge of their excitation cross sections being required for both low-energy $E_{\nu }\le 53$ MeV, with neutrinos from stopped muons) and high-energy ($E_{\nu }\sim \mathrm{GeV}$) neutrino experiments that use counters containing carbon. We take a phenomenological approach, determining the needed transition densities from fits to the measured form factors of these levels for the closely related electroexcitation process. While the low-energy cross section is dominated by the excitation of the $1^{+}$ ground state of ${}_{}{}^{12}{}_{}{}^{}\mathrm{N}$, the high-energy cross sections receive their largest contributions from both positive-parity spin-flip states and especially from the $1^{-}$ giant-resonance levels, with additional strength from $3^{+}$ and $2^{-}$, $4^{-}$ levels.