Inclusive neutrino scattering off the deuteron at low energies in chiral effective field theory

Cross sections for inclusive neutrino scattering off the deuteron induced by neutral and charge-changing weak currents are calculated from threshold up to 150 MeV energies in a chiral effective field theory including high orders in the power counting. Contributions beyond leading order (LO) in the weak current are found to be small, and increase the cross sections obtained with the LO transition operators by a couple of percent over the whole energy range 0–150 MeV. The cutoff dependence is negligible, and the predicted cross sections are within $\sim 2\%$ of, albeit consistently larger than, corresponding predictions obtained in conventional meson-exchange frameworks.

Figure 1

Diagrams illustrating one- and two-body electromagnetic currents entering at $Q^{-2}$ (LO), $Q^{-1}$ (N1LO), $Q^0$ (N2LO), and $Q^1$ (N3LO). Nucleons, pions, and photons are denoted by solid, dashed, and wavy lines, respectively. The square in panel (d) represents the ${(Q/m)}^2$ relativistic correction to the LO one-body current ($m$ is the nucleon mass). The solid circle in panel (j) is associated with the $\gamma \pi N$ coupling involving the LECs $d_8, d_9$, and $2d_{21}-d_{22}$ in the $\pi N$ chiral Lagrangian ${\mathcal{L}}_{\pi N}^{\left(3\right)}$ [22]. The solid circle in panel (k) denotes two-body contact terms of minimal and nonminimal nature, the latter involving two unknown LECs (see text). Only one among all possible time orderings is shown for the N1LO and N3LO currents, so that all direct- and crossed-box contributions are accounted for.

Figure 2

Diagrams illustrating one- and two-body electromagnetic charge operators entering at $Q^{-3}$ (LO), $Q^{-1}$ (N2LO), $Q^0$ (N3LO), $Q^1$ (N4LO). The square in panel (b) represents the ${(Q/m)}^2$ relativistic correction to the LO one-body charge operator, whereas panel (c) represents the charge operator ${\rho }_{\gamma }^{\left(0\right)}\left(\mathrm{OPE}\right)$ given in Eq. (3.1). As in Fig. 1, only a single time ordering is shown for the N3LO and N4LO contributions.

Figure 3

Diagrams illustrating one- and two-body axial currents entering at $Q^{-3}$ (LO), $Q^{-1}$ (N2LO), $Q^0$ (N3LO), and $Q^1$ (N4LO). Nucleons, pions, and axial fields are denoted by solid, dashed, and wavy lines, respectively. The squares in panels (c) and (d) denote relativistic corrections to the one-body axial current, while the circles in panels (e) and (f) represent vertices implied by the ${\mathcal{L}}_{\pi N}^{\left(2\right)}$ chiral Lagrangian [22], involving the LECs $c_i$ (see Ref. [18] for additional explanations). As in Fig. 1, only a single time ordering is shown.

Figure 4

Diagrams illustrating one- and two-body axial charge operators entering at $Q^{-2}$ (LO), $Q^{-1}$ (N1LO), and $Q^1$ (N3LO). Nucleons, pions, and axial fields are denoted by solid, dashed, and wavy lines, respectively. The diamonds in panels (l) and (m) indicate higher-order $A\pi N$ vertices implied by the ${\mathcal{L}}_{\pi N}^{\left(3\right)}$ chiral Lagrangian [22], involving the LECs $d_i$ (see Ref. [18] for additional explanations). As in Fig. 1, only a single time ordering is shown.

Figure 5

Total cross sections in ${\mathrm{fm}}^2$ for the ${\nu }_e-\mathrm{CC}$ induced process on the deuteron. The solid line corresponds to the $\chi \mathrm{EFT}$ calculation with cutoff $\mathrm{\Lambda }=500\mathrm{MeV}$, based on the chiral potential of Ref. [13] and including electroweak contributions up to N3LO in the vector current and axial charge, and up to N4LO in the axial current and vector charge; see Figs. 1, 2, 3, 4. The dashed line is obtained within the conventional meson-exchange picture of Ref. [3]. The inset shows the ratio of conventional to $\chi \mathrm{EFT}$ predictions.

Figure 6

Same as in Fig. 5 but for the ${\overline{\nu }}_e-\mathrm{CC}$-induced process on the deuteron.

Figure 7

Same as in Fig. 5 but for the ${\nu }_e$-NC-induced process on the deuteron.

Figure 8

Same as in Fig. 5 but for the ${\overline{\nu }}_e$-NC-induced process on the deuteron.

Figure 9

The convergence pattern as function of increasing order in the chiral expansion of the weak current. Ratios of corrections at a given order relative to the preceding order are shown. Note that the $y$-axis scale in the right-hand-side panels is doubled relative to that in the left-hand-side panels.

Figure 10

Double-differential cross sections in ${\mathrm{fm}}^2/\left(\mathrm{MeV}\mathrm{sr}\right)$ for the ${\nu }_e-\mathrm{CC}$- and ${\overline{\nu }}_e-\mathrm{CC}$-induced process on the deuteron. The solid line corresponds to the $\chi \mathrm{EFT}$ calculation with cutoff $\mathrm{\Lambda }=500\mathrm{MeV}$, based on the chiral potential of Ref. [13] and including electroweak contributions up to N3LO in the vector current and axial charge, and up to N4LO in the axial current and vector charge; see Figs. 1, 2, 3, 4. The dashed line is obtained within the conventional meson-exchange picture of Ref. [3]. The inset shows the ratio of conventional to $\chi \mathrm{EFT}$ predictions.