Medium modification of the charged-current neutrino opacity and its implications

Previous work on neutrino emission from proto-neutron stars which employed full solutions of the Boltzmann equation showed that the average energies of emitted electron neutrinos and antineutrinos are closer to one another than predicted by older, more approximate work. This in turn implied that the neutrino driven wind is proton rich during its entire life, precluding $r$-process nucleosynthesis and the synthesis of Sr, Y, and Zr. This work relied on charged-current neutrino interaction rates that are appropriate for a free nucleon gas. Here, it is shown in detail that the inclusion of the nucleon potential energies and collisional broadening of the response significantly alters this conclusion. Isovector interactions, which give rise to the nuclear symmetry energy, produce a difference between the neutron and proton single-particle energies $\Delta U=U_n-U_p$ and alter the kinematics of the charged-current reactions. In neutron-rich matter, and for a given neutrino/antineutrino energy, the rate for ${\nu }_e+n\to e^{-}+p$ is enhanced while ${\overline{\nu }}_e+p\to n+e^{+}$ is suppressed because the $Q$ value for these reactions is altered by $\pm \Delta U$, respectively. In the neutrino decoupling region, collisional broadening acts to enhance both ${\nu }_e$ and ${\overline{\nu }}_e$ cross sections, and random-phase approximation (RPA) corrections decrease the ${\nu }_e$ cross section and increase the ${\overline{\nu }}_e$ cross section, but mean field shifts have a larger effect. Therefore, electron neutrinos decouple at lower temperature than when the nucleons are assumed to be free and have lower average energies. The change is large enough to allow for a reasonable period of time when the neutrino driven wind is predicted to be neutron rich. It is also shown that the electron fraction in the wind is influenced by the nuclear symmetry energy.

Figure 1

(a) The electron chemical potential (dashed lines) and $\Delta U=U_n-U_p$ (solid lines) are shown as a function of density for the two equation of state models (IUFSU: red curves; GM3: black curves) in beta equilibrium for $Y_{\nu }=0$ and $T=8$ MeV. The grey band shows an approximate range of values for inverse spin relaxation time calculated in Ref. [9] and is discussed in connection with collisional broadening. (b) The equilibrium electron fraction as a function of density for the two equations of state shown in the top panel.

Figure 2

Angle integrated differential cross sections for a 12 MeV neutrino. The solid lines correspond to the reaction ${\nu }_e+n\to e^{-}+p$ and the dashed lines correspond to ${\overline{\nu }}_e+p\to e^{+}+n$. The black lines are calculations in which mean field effects have been included, while the red lines are calculations in which the mean field effects have been ignored. The green dotted line corresponds to the available electron phase space, arbitrarily scaled. The assumed background conditions are $T$ $=$ 8 MeV and $n_B=0.02{\mathrm{fm}}^{-3}$. The electron fraction is 0.027, which corresponds to beta equilibrium for the given temperature, density, and the assumed nuclear interactions. The nucleon potential difference is $U_n-U_p=\Delta U=9\mathrm{MeV}$. All cross sections are for the same baryon density and electron fraction (i.e., all assume the same $\tilde{\mu }$ for the neutrons and protons).

Figure 3

(a) The total absorption inverse mean free path as a function of incoming neutrino energy for electron neutrinos (solid lines) and electron antineutrinos (dashed lines). The dot-dashed line shows the effective bremsstrahlung inverse mean free path. In both panels the black lines include mean field effects and the red lines assume a free gas response function. (b) The ratio of the total electron neutrino capture rate to the total electron antineutrino capture rate. Beta equilibrium has been assumed and the temperature has been fixed at 8 MeV.

Figure 4

The axial portion of the ${\nu }_e$ (main panel) and ${\overline{\nu }}_e$ (inset) absorption cross sections including collisional broadening. This shifts a significant fraction of the response to larger $E_e$ where there is larger lepton phase space available. The ambient conditions and neutrino energy are the same as those in Fig. 2. The dashed lines show the RPA response, including both mean fields and collisional broadening. The dotted line in the inset panel shows the free gas response.

Figure 5

(a) First energy moment of the outgoing electron neutrino and antineutrino as a function of time in three PNS cooling simulations. The solid lines are the average energies of the electron neutrinos and the dashed lines are for electron antineutrinos. The black lines correspond to a model which employed the GM3 equation of state, the red lines to a model which employed the IU-FSU equation of state, and the green lines to a model which ignored mean field effects on the neutrino opacities (but used the GM3 equation of state). (b) Predicted neutrino driven wind electron fraction as a function of time for the three models shown in the top panel (solid lines), as well as two models with the bremsstrahlung rate reduced by a factor of 4 (dot-dashed lines). The colors are the same as in the top panel.

Figure 6

The thermodynamic conditions and nucleon potential difference characterizing the region where electron neutrinos decouple as a function of neutrino energy. These values are taken from the PNS model which employed the IU-FSU equation of state at 3.3 seconds after core bounce. At this point the average electron neutrino energy is 8.3 MeV.