Charged-current reactions in the supernova neutrino-sphere

We calculate neutrino absorption rates due to charged-current reactions ${\nu }_e+n\to e^{-}+p$ and ${\overline{\nu }}_e+p\to e^{+}+n$ in the outer regions of a newly born neutron star called the neutrino-sphere. To improve on recent work which has shown that nuclear mean fields enhance the ${\nu }_e$ cross section and suppress the ${\overline{\nu }}_e$ cross section, we employ realistic nucleon-nucleon interactions that fit measured scattering phase shifts. Using these interactions we calculate the momentum-, density-, and temperature-dependent nucleon self-energies in the Hartree-Fock approximation. A potential derived from chiral effective field theory and a pseudopotential constructed to reproduce nucleon-nucleon phase shifts at the mean-field level are used to study the equilibrium proton fraction and charged-current rates. We compare our results to earlier calculations obtained using phenomenological mean-field models and to those obtained in the virial expansion valid at low density and high temperature. In the virial regime our results are consistent with previous calculations, and at higher densities relevant for the neutrino-sphere, $\rho \gtrsim {10}^{12}$ ${\mathrm{g}/\mathrm{cm}}^3$, we find the difference between the ${\nu }_e$ and ${\overline{\nu }}_e$ absorption rates to be larger than predicted earlier. Our results may have implications for heavy-element nucleosynthesis in supernovae, and for supernova neutrino detection.

Figure 1

Energy and momentum constraints on the charged-current reactions for conditions discussed in the text. Reactions are possible when the allowed region for lepton kinematics, shown by the shaded region enclosed by the black lines, overlaps the allowed region for nucleon kinematics, shown by the regions enclosed by blue and red lines corresponding to the ${\nu }_e$ and ${\overline{\nu }}_e$ reactions, respectively. The region enclosed by the solid blue lines includes the nuclear self-energy difference for the transition $n\to p$ associated with the ${\nu }_e$ reaction, and regions enclosed by the dashed lines are for noninteracting nucleons. The $p\to n$ transition associated with the ${\overline{\nu }}_e$ reaction is kinematically forbidden as there is no overlap when nucleon self-energy corrections are included.

Figure 2

Feynman diagrams depicting the self-consistent HF self-energy. The double lines are the dressed nucleon propagators and wavy lines represent the NN interaction.

Figure 3

Change in the energy per particle of neutron matter from NN interactions in the Hartree-Fock (HF) approximation. Results for both the chiral NN potential and the pseudopotential are shown and compared to the model-independent virial equation of state [40]. The shaded area denotes the density region in which the fugacity $z<0.5$.

Figure 4

The proton fraction $Y_p$ as a function of density for matter in beta equilibrium at temperature $T=8\mathrm{MeV}$. Results for the chiral NN potential and the pseudopotential in the Hartree-Fock (HF) approximation are shown. The shaded $Y_p$ band is enclosed by solid and dashed lines resulting from the pseudopotential and modified pseudopotential calculations, respectively. The region between the HF chiral and HF pseudopotential bands should be considered as a conservative uncertainty range. In addition, we compare to the model-independent virial equation of state [40] as well as the predictions from relativistic mean-field (RMF) theory [23]. The shaded area denotes the density region in which the fugacity $z<0.5$.

Figure 5

Momentum dependence of the neutron and proton single-particle energies in hot ($T=8\mathrm{MeV}$) and dense ($n_B=0.02$ ${\mathrm{fm}}^{-3}$) beta-equilibrated nuclear matter calculated in the HF approximation from the pseudopotential. The solid and dashed lines are parametrized fits, with the form given in Eq. (11), of the nonrelativistic dispersion relations for protons and neutrons respectively.

Figure 6

Difference in the momentum-dependent self-energies of neutrons and protons in the HF approximation for beta-equilibrated matter at $n_B=0.02{\mathrm{fm}}^{-3}$ and temperature $T=8\mathrm{MeV}$. Results for the chiral NN potential and pseudopotential are shown.

Figure 7

Difference in the HF proton and neutron energy shifts $U$, defined in Eq. (11), as a function of the density. The results from the pseudopotential and chiral NN interaction are compared to those from RMF models [23].

Figure 8

Proton and neutron effective masses $M^{*}/M$ in the HF approximation as a function of the density for the pseudopotential and chiral NN interaction.

Figure 9

Tree-level scattering amplitude for the process ${\nu }_e+n\to e^{-}+p$.

Figure 10

Effect of the in-medium neutron (proton) dispersion relation on the differential cross section for (anti)neutrino absorption as a function of the outgoing lepton energy $E_e$. We consider an incoming neutrino energy $E_{\nu }=24$ MeV and matter in beta equilibrium at a density $n_B=0.02$ ${\mathrm{fm}}^{-3}$ and temperature $T=8$ MeV, including weak magnetism and leading $|\vec{p}|/M$ corrections. The chiral NN interaction and pseudopotential are both used in the HF approximation. This provides a range for the theoretical uncertainty due to the many-body treatment, which can be improved by performing higher-order calculations.

Figure 11

Effect of the in-medium neutron (proton) dispersion relation on the (anti)neutrino absorption mean free path in beta-equilibrated matter at density $n_B=0.02$ ${\mathrm{fm}}^{-3}$ and temperature $T=8$ MeV. The chiral NN potential and pseudopotential are both used in HF approximation. This provides a conservative range for the theoretical uncertainty due to the many-body treatment, which can be improved by performing higher-order calculations. Also shown is the mean free path for the neutrino-pair absorption process.

Figure 12

The relative importance of the scattering continuum contribution to the second virial coefficient (normalized with respect to the sum of bound- and scattering-state contributions). Results assuming a free-space deuteron binding energy and a medium-reduced binding energy of $B_d=1.0\mathrm{MeV}$ are shown.

Figure 13

${}^3{S}_1$ phase shift as a function of laboratory energy $T_{\mathrm{lab}}$ from the Nijmegen partial-wave analysis (PWA) [38] as well as those used in the modified pseudopotential.