Neutrino-pair bremsstrahlung from nucleon-$\alpha$ versus nucleon-nucleon scattering

We study the impact of the nucleon-$\alpha P$-wave resonances on neutrino-pair bremsstrahlung. Because of the noncentral spin-orbit interaction, these resonances lead to an enhanced contribution to the nucleon spin structure factor for temperatures $T\lesssim 4$ MeV. If the $\alpha$-particle fraction is significant and the temperature is in this range, this contribution is competitive with neutron-neutron bremsstrahlung. This may be relevant for neutrino production in core-collapse supernovae or other dense astrophysical environments. Similar enhancements are expected for resonant noncentral nucleon-nucleus interactions.

Figure 1

Diagrams contributing to the $n\alpha ⟷n\alpha \nu \overline{\nu }$ process. The solid line denotes neutrons $n$, the dashed lines $\alpha$ particles, the dotted line the $\nu \overline{\nu }$ pair, and the wiggly line is a $Z^0$ boson exchange. For $nn$ bremsstrahlung, the $\nu \overline{\nu }$ pair can also be radiated from the second neutron.

Figure 2

Phase shifts $\delta$ as a function of relative momentum $p$ for $n\alpha$ scattering. The different partial waves are labelled by the standard notation ${}^{2S+1}{\ell }_j$. Phase shifts are shown following Ref. [20]: for low momenta from the fits to data of Ref. [21] and for higher momenta (lines with points) from optical model calculations [22]. We do not show the ${}^2{S}_{1/2}$ phase shift, because the corresponding $\mathcal{T}$ matrix commutes with the spin operator in Eq. (1) and therefore, this channel does not contribute to $n\alpha$ bremsstrahlung.

Figure 3

$\Xi \left(\omega \right)$ as a function of $\omega /T$ for various temperatures. The solid lines are for $n\alpha$, while the dashed lines are for $nn$. For fixed $\omega /T,\Xi \left(\omega \right)$ for $n\alpha$ bremsstrahlung decreases with increasing $T$ (except for low $\omega /T$) and increases for the $nn$ case.

Figure 4

Energy-loss rates $Q_{\nu \overline{\nu }}$ as a function of temperature $T$. The solid (dashed) line is for $n\alpha \left(nn\right)$ bremsstrahlung. The $Q_{\nu \overline{\nu }}$ shown is divided by $T^5$ (in MeV), by the neutron mass fraction $f_n$, and by the mass density over the mass number ${\rho }_{X,12}/A_X$ of $X=n$ or $\alpha$ particles (in ${10}^{12}{\mathrm{g}\mathrm{cm}}^{-3}$), with $A_X=1$ or 4, respectively. The darker (red) curves are numerical results using Eq. (9) and the lighter (cyan) curves are the fits, Eqs. (14) and (15).