Neutrino and antineutrino charge-exchange reactions on ${}^{12}\mathrm{C}$

We extend the formalism of weak interaction processes, obtaining new expressions for the transition rates, which greatly facilitate numerical calculations, for both neutrino-nucleus reactions and muon capture. Explicit violation of the conserved vector current hypothesis by the Coulomb field, as well as development of a sum-rule approach for inclusive cross sections, has been worked out. We have done a thorough study of exclusive (ground-state) properties of ${}^{12}\mathrm{B}$ and ${}^{12}\mathrm{N}$ within the projected quasiparticle random phase approximation (PQRPA). Good agreement with experimental data achieved in this way put into evidence the limitations of the standard RPA and QRPA models, which come from the inability of the RPA to open the $p_{3/2}$ shell and from the nonconservation of the number of particles in the QRPA. The inclusive neutrino/antineutrino ($\nu /\mathrm{\nu ̃}$) reactions ${}^{12}\mathrm{C}$($\nu ,e^{-}){}^{12}$N and ${}^{12}\mathrm{C}$($\mathrm{\nu ̃},e^{+}){}^{12}$B are calculated within both the PQRPA and the relativistic QRPA. It is found that (i) the magnitudes of the resulting cross sections are close to the sum-rule limit at low energy, but significantly smaller than this limit at high energies, for both $\nu$ and $\mathrm{\nu ̃}$; (ii) they increase steadily when the size of the configuration space is augmented, particularly for $\nu /\mathrm{\nu ̃}$ energies $>$200 MeV; and (iii) they converge for sufficiently large configuration space and final-state spin. The quasi-elastic ${}^{12}\mathrm{C}$($\nu ,{\mu }^{-}){}^{12}$N cross section recently measured in the MiniBooNE experiment is briefly discussed. We study the decomposition of the inclusive cross section based on the degree of forbiddenness of different multipoles. A few words are dedicated to the $\nu /\mathrm{\nu ̃}$-${}^{12}\mathrm{C}$ charge-exchange reactions related to astrophysical applications.

Figure 1

${}^{12}\mathrm{B}$ and ${}^{12}\mathrm{N}$ ground-state energies (in units of MeV) and GT $B$ values within the PQRPA for different s.p. spaces, as a function of the $\mathit{pp}$-coupling $t$. Experimental values are $E({}^{12}$B$)=13.37$ MeV and $E({}^{12}$N$)=17.33$ MeV [95], and $B({}^{12}$B$)=0.466$, and $B({}^{12}$N$)=0.526$ [96].

Figure 2

Exclusive ${}^{12}\mathrm{C}$($\nu ,e^{-}){}^{12}$N cross section ${\sigma }_e(E_{\nu },1_1^{+})$ (in units of ${10}^{-42}$ cm${}^2$), plotted as a function of the incident neutrino energy $E_{\nu }$. Results for several single-particle spaces $S_N$, and $t=0$, within three energy intervals, are shown. Experimental data in the DAR region are from Ref. [25].

Figure 3

Same as Fig. 2, but here $t=0$ for $S_2$ and $S_3$, $t=0.2$ for $S_4$, and $t=0.3$ for $S_6$. SM and EPT calculations are, respectively, from Refs. [98] and [16]. Experimental data in the DAR region are from Ref. [25].

Figure 4

Calculated ${}^{12}\mathrm{C}$($\mathrm{\nu ̃},e^{+}){}^{12}$B cross section ${\sigma }_{e^{+}}(E_{\mathrm{\nu ̃}},1_1^{+})$ (in units of ${10}^{-42}$ cm${}^2$), plotted as a function of the incident antineutrino energy $E_{\mathrm{\nu ̃}}$. As in Fig. 3, the value of $t$ is $0$ for s.p. spaces $S_2$, and $S_3$, $0.2$ for $S_4$, and $0.3$ for $S_6$. The EPT calculation from Ref. [16] is also shown.

Figure 5

Muon-capture transition rate to the ${}^{12}\mathrm{B}$ ground state (in units of ${10}^2$ ${\mathrm{s}}^{-1}$, and electron and muon folded ECSs to the ${}^{12}\mathrm{N}$ ground state in units of ${10}^{-42}$ cm${}^2$ and ${10}^{-41}$ cm${}^2$, respectively. Experimental values, in these units, are $\Lambda ({}^{12}$B$)=6.2\pm 0.3$ [45], ${\stackrel{̲}{\sigma }}_e({}^{12}$N$)=9.1\pm 0.4\pm 0.9$ [25], and ${\stackrel{̲}{\sigma }}_e({}^{12}$N$)=8.9\pm 0.3\pm 0.9$ [26] and ${\stackrel{̲}{\sigma }}_{\mu }({}^{12}$N$)=6.6\pm 1.0\pm 1.0$ [28], and ${\stackrel{̲}{\sigma }}_{\mu }({}^{12}$N$)=5.6\pm 0.8\pm 1.0$ [29], respectively.

Figure 6

Inclusive ${}^{12}\mathrm{C}$($\nu ,e^{-}){}^{12}$N cross section ${\sigma }_{e^{-}}(E_{\nu })$ (in units of ${10}^{-39}$ cm${}^2$) plotted as a function of the incident neutrino energy $E_{\nu }$. PQRPA results within s.p. spaces $S_2$, $S_3$, and $S_6$, and with the same values of $s=t$ as in Fig. 3, are compared with two sum-rule limits (as explained in the text), ${\mathrm{SR}}_{\mathrm{GT}}$ and ${\mathrm{SR}}_{1\mathrm{F}}$, obtained with an average excitation energy $\stackrel{̲}{{\omega }_{{\mathsf{J}}_n^{\pi }}}$ of $17.34$ and $42$ MeV, respectively. Several previous RPA-like calculations, namely, the RPA [43], CRPA [102], and RQRPA within $S_{20}$ for $E_{2\mathit{qp}}=100$ MeV [51], as well as the SM [43] and the TDA [34], are also shown.

Figure 7

Inclusive ${}^{12}\mathrm{C}$($\mathrm{\nu ̃},e^{+}){}^{12}$B cross section ${\sigma }_{e^{+}}(E_{\nu })$ (in units of ${10}^{-39}$ cm${}^2$) plotted as a function of the incident neutrino energy $E_{\mathrm{\nu ̃}}$. All results were obtained in the same way as in the neutrino case in Fig. 6.

Figure 8

Inclusive ${}^{12}\mathrm{C}$($\nu ,e^{-}){}^{12}$N cross-section ${\sigma }_{e^{-}}(E_{\nu })$(in units of ${10}^{-39}$ cm${}^2$) plotted as a function of the incident neutrino energy $E_{\nu }$, evaluated in RQRPA with different configuration spaces. These cross sections are plotted as functions of the incident neutrino energy with different cut-off of the $E_{2\mathit{qp}}$ quasiparticle energy as it is explained in the text. The left and right panels show the cross section evaluated with $S_{20}$, and $S_{30}$ s.p. spaces. The last cross section shows that the convergence of the calculation is achieved up to 600 MeV of incident neutrino energy.

Figure 9

Same as Fig. 8 but for the ${}^{12}\mathrm{C}$($\mathrm{\nu ̃},e^{+}){}^{12}$B cross section ${\sigma }_{e^{+}}(E_{\mathrm{\nu ̃}})$.

Figure 10

The calculated RQRPA (within $S_{30}$ and $E_{2\mathit{qp}}=500$ MeV) quasielastic (${\nu }_e,{}^{12}$C) cross section per neutron (solid line) is compared with that for the (${\nu }_{\mu },{}^{12}$C) scattering data measured at MiniBooNE [13]; the dotted line shows the same calculation but renormalized by a factor of $1.5$. Also displayed are the calculations done by Martini et al. [73, 103] within the RFGM for pure (1p-1h) excitations (dashed line) and with the inclusion of the np-nh channels (dot-dashed line).

Figure 11

Left and right panels show, respectively, the cross sections ${\sigma }_{e^{-}}(E_{\nu })$, and ${\sigma }_{e^{+}}(E_{\mathrm{\nu ̃}})$ (in units of ${10}^{-39}$ cm${}^2$) evaluated in RQRPA for $S_{20}$, and $S_{30}$ s.p. spaces with the cutoff $E_{2\mathit{qp}}=500$ MeV, and different maximal values of ${\mathsf{J}}^{\pm }$, with $\mathsf{J}$ going from $1$ up to $14$ for neutrinos, and from $1$ up to $11$ for antineutrinos.

Figure 12

Left: Allowed ($J^{\pi }=0^{+},1^{+}$), first-forbidden ($J^{\pi }=0^{-},1^{-},2^{-}$), second-forbidden ($J^{\pi }=2^{+},3^{+}$), and third-forbidden ($J^{\pi }=3^{-},4^{-}$) inclusive ${}^{12}\mathrm{C}$($\mathrm{\nu ̃},e^{+}){}^{12}$B cross section ${\sigma }_{e^{+}}(E_{\mathrm{\nu ̃}})$ (in units of ${10}^{-42}$ cm${}^2$), plotted as a function of the incident neutrino energy $E_{\mathrm{\nu ̃}}$. Right: Same as left, but for $d{\sigma }_{e^{+}}(E_{\mathrm{\nu ̃}})/{\mathit{dE}}_{\mathrm{\nu ̃}}$ (in units of ${10}^{-42}$ cm${}^2$ MeV${}^{-1})$.

Figure 13

Flux-averaged neutrino and antineutrino cross sections ${\stackrel{̲}{\sigma }}_{e^{\pm }}$ in ${}^{12}\mathrm{C}$ with typical supernova fluxes.