Neutrino and antineutrino cross sections in $^{12}$C

We extend the formalism of weak interaction processes, obtaining new expressions for the transition rates, which greatly facilitate numerical calculations, both for neutrino-nucleus reactions and muon capture. We have done a thorough study of exclusive (ground state) properties of $^{12}$B and $^{12}$N within the projected quasiparticle random phase approximation (PQRPA). Good agreement with experimental data is achieved in this way. The inclusive neutrino/antineutrino ($ν/\tildeν$) reactions $^{12}$C($ν,e^-)^{12}$N and $^{12}$C($\tildeν,e^+)^{12}$B are calculated within both the PQRPA, and the relativistic QRPA (RQRPA). It is found that the magnitudes of the resulting cross-sections: i) are close to the sum-rule limit at low energy, but significantly smaller than this limit at high energies both for $ν$ and $\tildeν$, ii) they steadily increase when the size of the configuration space is augmented, and particulary for $ν/\tildeν$ energies $> 200$ MeV, and iii) converge for sufficiently large configuration space and final state spin.

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