Neutrino-nucleon scattering in supernova matter from the virial expansion

We extend our virial approach to study the neutral-current neutrino response of nuclear matter at low densities. In the long-wavelength limit, the virial expansion makes model-independent predictions for neutrino-nucleon scattering rates and the density $S_V$ and spin $S_A$ responses. We find that $S_A$ is significantly reduced from one even at low densities. We provide a simple fit $S_A^f(n,T,Y_p)$ of the axial response as a function of density $n$, temperature $T$, and proton fraction $Y_p$, which can be incorporated into supernova simulations in a straightforward manner. This fit reproduces our virial results at low densities and the Burrows and Sawyer random-phase approximation (RPA) model calculations at high densities. Preliminary one-dimensional supernova simulations suggest that the virial reduction in the axial response may enhance neutrino heating rates in the gain region during the accretion phase of a core-collapse supernovae.

Figure 1

Vector response $S_V$ versus density $n$ for proton fractions of $Y_p=0$ (dashed lines) and 0.3 (solid lines). The curves are for temperatures of (from left to right with increasing thickness) 2.5, 5, 10, and 15 MeV. The solid circles show where $z_n=0.5$. The virial expansion is most valid to the left of these points.

Figure 2

Axial response $S_A$ versus density $n$ for temperatures of (a) $T=2.5$ MeV, (b) 5 MeV, (c) 10 MeV, (d) and 15 MeV. The red dashed lines are our virial expansion results, Eq. (28), for the indicated proton fractions. The solid red dots indicate where $z_n=0.5$. The virial expansion is most valid to the left of these points. The green dotted lines show the original Burrows and Sawyer RPA results [7]. Finally, the solid black lines show the interpolating fit $S_A^f$, Eq. (36).

Figure 3

Heating rate in the gain region for one-dimensional simulations of the accretion phase of core-collapse supernovae for various assumptions on the neutral-current neutrino-nucleon scattering cross section. The solid lines represent simulations that ignore the contribution to the rates from strange quarks, whereas dashed lines denote simulations that include strange-quark contributions, assuming $g_a^s=-0.2$ for comparison with Ref. [4]. The black lines show the heating rate for simulations that assume $S_A=1$, while orange lines show the larger heating rates with $S_A^f$ from Eq. (36).