Neutrino pair bremsstrahlung due to electromagnetic collisions in neutron star cores revisited

TL;DR

The paper re-evaluates neutrino pair bremsstrahlung from electromagnetic collisions in npeμ neutron star cores, considering both normal and proton-superconducting matter. Using a Fermi’s Golden Rule framework with an in-medium photon propagator decomposed into longitudinal and transverse parts, it shows that the transverse channel governs the emissivity, exhibiting a nonstandard $Q^{\mathrm{em}}_{\mathrm{Br}} \propto T^{23/3}$ scaling in normal matter due to dynamic screening, while remaining numerically small. When protons are superconducting, nucleon contributions are suppressed and lepton-lepton bremsstrahlung acts as a residual source, with static screening restoring the conventional $Q^{\mathrm{em}}_{\mathrm{Br}} \propto T^{8}$; the authors provide simple analytical expressions for both limiting regimes. The results indicate that electromagnetic bremsstrahlung is generally subdominant to nucleon processes unless nucleons are strongly paired, and they offer practical formulas to incorporate this channel in NS cooling calculations.

Abstract

We reconsider the problem of neutrino pair bremsstrahlung emission originating from the electromagnetic collisions of charged particles in nucleonic ($npeμ$) neutron star cores. Two limiting cases are considered: (i) protons are in the normal state and (ii) protons are in the superconducting state. In both cases, the dominant contribution to the bremsstrahlung emissivity $Q^{\mathrm{em}}_{\mathrm{Br}}$ comes from the transverse part of in-medium electromagnetic interactions. For non-superconducting matter, we obtain an unusual $Q^{\mathrm{em}}_{\mathrm{Br}}\propto T^{23/3}$ temperature dependence due to the dynamical character of plasma screening in the transverse channel, but considerably smaller values of $Q^{\mathrm{em}}_{\mathrm{Br}}$ than in previous studies, rendering the considered process unimportant in practice. In contrast, for superconducting and superfluid matter, the neutrino emission processes involving nucleons are suppressed and $Q^{\mathrm{em}}_{\mathrm{Br}}$ due to lepton collisions provides the residual contribution to the neutrino emissivity of neutron star core matter. In the superconducting case, the plasma screening becomes static and the standard $Q^{\mathrm{em}}_{\mathrm{Br}}\propto T^{8}$ temperature scaling is restored. Simple analytical expressions for $Q^{\mathrm{em}}_{\mathrm{Br}}$ in both limiting cases are provided.

Figures (6)

• Figure S1: Feynman diagrams describing the direct part of the bremsstrahlung process amplitude. Thick solid lines refer to colliding charged fermions with four-momenta $P_{1}$, $P_2$ before and $P_3$, $P_4$ after the collision, respectively. Rectangle blocks correspond to weak vertices (in general, renormalized in medium), and thin solid lines refer to the emitted neutrino pair (with four-momenta $Q_1$ and $Q_2$). Dashed lines with bubbles correspond to the in-medium photon exchange with $\Pi(K)$ being the polarization function and $K=P_2-P_4$ being the 4-momentum transfer in a collision. An exchange contribution in the case of like particles collisions is given by an additional four diagrams obtained from the plotted ones by interchanging indices $3\longleftrightarrow 4$ (not shown in the figure).
• Figure S2: Partial contributions to bremsstrahlung emissivity due to electromagnetic collisions in normal NS cores with BSk 24 EoS as functions of density for $T=10^8$ K. Each panel marked $ij$ with $i,j=e,p,\mu$ corresponds to neutrino pair emission by particle species $i$ scattered off particle $j$ as indicated in the panels. In each panel, different line types correspond to different contributions to emissivity as shown in legend and discussed in the text, while total contributions from each collision are shown with black solid lines. Thin lines of corresponding line types show the results of calculations in the lowest-order weak screening approximation.
• Figure S3: Different approximations to the total bremsstrahlung emissivity due to electromagnetic interactions in non-superfluid BSk24 NS cores as a function of density for $T=10^8$ K. The results obtained via Eq. \ref{['eq:Q_12noscr']} with all contributions included are shown with the thick black line. The red line shows the contribution solely from the axial-vector channel. The blue line corresponds to retaining only the transverse channel of the electromagnetic interaction, while the green line shows the contribution solely from the transverse electromagnetic channel to the axial-vector current emissivity. Thin lines of corresponding types represent the lowest-order weak screening approximation.
• Figure S4: Bermsstrahlung emissivity (scaled by $T_8^{-8}$) due to electromagnetic collisions as a function of density in normal NS cores with BSk24 EoS. Solid red, blue, and magenta lines shows the results of the present study for $T=10^7$ K, $10^8$ K, and $10^9$ K, respectively. Dash-dotted lines show the results of Ref. Jaikumar2005PhRvD for $ee$ bremsstrahlung for the same three values of temperature, and by dotted lines the results of Ref. Kaminker1999 for $ee$ bremsstrahlung are shown (in this case, combination $Q^{ee}_{\mathrm{Br}}T_8^{-8}$ does not depend on temperature). The black dashed line show the $ep$ bremmstrahlung emissivity according to Ref. Kaminker1999.
• Figure S5: Partial contributions to bremsstrahlung emissivity due to electromagnetic collisions in superconducting NS cores with the BSk24 EoS, as functions of density for $T=10^8$ K and $T_{Cp}=10^{10}$ K. Each panel marked $ij$ with $i,j=e,\mu$ corresponds to neutrino pair emission by particle species $i$ scattered off particle $j$ as indicated in the panels. In each panel, different line types correspond to different contributions to emissivity as shown in the legend and discussed in the text, while total contributions from each collision are shown with black solid lines (cf. Fig. \ref{['fig:Qnorm_comp']}).