Unified approach to structure factors and neutrino processes in nucleon matter

We present a unified approach to neutrino processes in nucleon matter based on Landau's theory of Fermi liquids that includes one and two quasiparticle-quasihole pair states as well as mean-field effects. We show how rates of neutrino processes involving two nucleons may be calculated in terms of the collision integral in the Landau transport equation for quasiparticles. Using a relaxation time approximation, we solve the transport equation for density and spin-density fluctuations and derive a general form for the response functions. We apply our approach to neutral-current processes in neutron matter, where the spin response function is crucial to the calculation of neutrino elastic and inelastic scattering and neutrino-pair bremsstrahlung and absorption from strongly interacting nucleons. We calculate the relaxation rates using modern nuclear interactions and including many-body contributions, and find that rates of neutrino processes are reduced compared with estimates based on the one-pion exchange interaction, which is used in current simulations of core-collapse supernovae.

Figure 1

Spin relaxation rate given by $C_{\sigma }$ of Eq. (34) as a function of Fermi momentum $k_F$ obtained from the OPE interaction, from low-momentum interactions $V_{\mathrm{low}\hspace{0.5em}k}$, and including second-order many-body contributions. In addition, we show that the result obtained from $V_{\mathrm{low}\hspace{0.5em}k}$ plus second-order contributions is dominated by tensor interactions (dotted vs solid line).

Figure 2

Relaxation rate for decay of an excess of quasiparticles in a particular momentum state given by $C$ of Eq. (29) as a function of Fermi momentum $k_F$ obtained from the OPE interaction, from low-momentum interactions $V_{\mathrm{low}\hspace{0.5em}k}$, and including second-order many-body contributions. In addition, we show that the result obtained from $V_{\mathrm{low}\hspace{0.5em}k}$ plus second-order contributions is dominated by central interactions (dotted vs solid line).

Figure 3

Ratio of the spin relaxation rate to the relaxation rate for an excess of quasiparticles in a single momentum state $(1/{\tau }_{\sigma })/(1/\tau )$ as a function of Fermi momentum $k_F$ for purely tensor scattering amplitudes (in which case the value is 2), for the OPE interaction (which gives the value $4/3$), from low-momentum interactions $V_{\mathrm{low}\hspace{0.5em}k}$, and including second-order many-body contributions.

Figure 4

Imaginary part of the spin response function $\mathrm{Im}{\chi }_{\sigma }/N(0)$ of Eq. (21) in units of the density of states vs $\omega /(v_{F}q)$. Results are shown for the noninteracting system, without and with mean-field effects, $G_0=0$ and $G_0=0.8$, respectively, and for different values of the spin relaxation rate $1/{\tau }_{\sigma }=0,v_{F}q{\tau }_{\sigma }=2$ and $v_{F}q{\tau }_{\sigma }=5$.