Strangeness-changing rates and hyperonic bulk viscosity in neutron star mergers

In this paper, we present a computation of the rates of strangeness-changing processes and the resultant bulk viscosity in matter at the densities and temperatures typical of neutron star mergers. To deal with the high temperature in this environment, we go beyond the Fermi surface approximation in our rate calculations and numerically evaluate the full phase space integral. We include processes where quarks move between baryons via meson exchange: These have generally been omitted in previous analyses but provide the dominant contribution to the rates of strangeness-changing processes and the bulk viscosity. The calculation of these rates is an essential step toward any calculation of dissipation mechanisms in hyperonic matter in mergers. As one application, we calculate the dissipation times for density oscillations at the frequencies seen in merger simulations. We find that hyperon bulk viscosity for temperatures in the MeV regime can probably be neglected in this context but becomes highly relevant for keV-range temperatures.

Figure 1

Logarithm of the ratios of all baryonic particle densities over the total baryon density at $T=2$ MeV (solid lines) and $T=50$ MeV (dashed lines) plotted as a function of total baryon density in units of the saturation density. In the used parametrization $\mathrm{PK}1+\mathrm{H}$, saturation density is given by $n_0=0.148{\mathrm{fm}}^{-3}$. Although the $\mathrm{\Lambda }$ hyperon is less massive than the ${\mathrm{\Sigma }}^{-}$hyperon, the order of their onset is reversed because of charge neutrality.

Figure 2

Panels (a) and (b) show Feynman and quark-flow diagrams for the OME contribution to process I. The flavor-changing weak-interaction vertex $F_{n\mathrm{\Sigma }}^W$ connecting the incoming neutron $n$ with a pion and the ${\mathrm{\Sigma }}^{-}$-hyperon represents a combination of a flavor-changing $W\text{−}$boson exchange within the baryon and a quark exchange (modeled via one meson exchange) with the spectator baryon. The strong-interaction vertex ${\mathrm{F}}_{np}^S$ connects the nucleons $n$ and $p$ with a pion. For the matrix element in Eq. (7), we have to subtract a second Feynman diagram with the two initial neutrons exchanged. Panel (c) shows the Feynman diagram for the contact interaction contribution, where the two nucleons exchange a charged $W$ boson that is integrated out. This is the basis for the matrix element in Eq. (12). All coupling constants can be found in Appendix pp2. The remaining diagrams are shown in Appendix pp1.

Figure 3

Individual rates of all four strangeness-changing processes defined in Eqs. (5) at $T=5$ MeV as a function of baryon density normalized to saturation density $n_0$ in the OME channel. Below the hyperon onset at $n_B\approx 1.85n_0$, the rates drop quickly to zero as the thermal population of hyperons becomes highly suppressed. The individual rates grow roughly as $T^3$. Extrapolation using this scaling is accurate to within $40\%$ up to $T=50\mathrm{MeV}$.

Figure 4

Net strangeness changing rate (processes I $+$ II $+$ III $+$ IV) at $T=5$ MeV (solid blue line) in comparison to the FS approximation (black, dashed line labeled “OME FS approx;” see Appendix pp3). We also show the contribution to the rate from the contact interaction (black, dotted line), estimated using the FS approximation and the nonrelativistic limit [Eqs. (12) and (13)] . Rates in the FS approximation are only defined above the hyperon density threshold.

Figure 5

Comparison of the rate for process III for the $\mathrm{PK}1+\mathrm{H}$ EOS, which we use in in this work, and the GM1'B EOS (see Sec. 2) at $T=5$ MeV. Above the hyperon onset density, the rates level out to similar values, and below the onset they show the same exponential suppression.

Figure 6

Susceptibility $B$ for the $\mathrm{PK}1+\mathrm{H}$ EOS [defined in Eq. (31)] as a function of baryon density at temperatures $T=3,4,5,10,25,50$ MeV, where the highest (blue solid) curve at low densities corresponds to $T=3$ MeV and smaller susceptibilities correspond to higher temperatures. As the density drops below the hyperon onset at $n_B\approx 1.85n_0$ (marked by the thin, black, dashed, vertical line), the susceptibility shows an exponential increase, arising from the exponential suppression of the hyperon fraction in this regime.

Figure 7

Re-equilibration rate $\gamma$ defined in Eq. (30) for the $\mathrm{PK}1+\mathrm{H}$ EOS as a function of baryon density. The color coding is identical to Fig. 6. All rates are obtained by evaluating the full phase space integral for the OME matrix elements. The black, dotted, horizontal line marks the optimal equilibration rate for maximal bulk viscosity where it would match the external oscillation, $\gamma =\omega$. The exponential behavior of the rates from Fig. 4 balances that of the susceptibility $B$ from Fig. 6, leading to nearly density-independent re-equilibration rates $\gamma$. Therefore, the rates are, even at lower temperatures, too fast to match the external oscillation. Only at temperatures in the keV regime, $\gamma$ matches $\omega$; see the gray, dashed line where we show $\gamma$ for a temperature of $4.5\mathrm{keV}$ computed from the EOS at $T=0$ using the FS approximation for the rates.

Figure 8

Solid lines show the bulk viscosity as function of baryon density at temperatures of $T=2,5,10,25,50$ MeV for the finite temperature $\mathrm{PK}1+\mathrm{H}$ EOS computed from the full phase space integral. Above the hyperon onset, the bulk viscosity decreases with temperature, since even for $T=2$ MeV the re-equilibration rate, which generally further increases with temperature, is too fast to match the external frequency (see Fig. 7). Below the hyperon onset (marked by a black dashed vertical line), the bulk viscosity drops more drastically for low temperatures due to the faster decrease of the hyperon fraction. For temperatures of $2\mathrm{MeV}$ or below, the rates and susceptibilities cannot be computed reliably below the hyperon onset because of the small hyperon fraction. At temperatures of over 20 MeV, bulk viscosity is completely smooth due to the higher thermal hyperon population. The dashed lines show the bulk viscosity for $T=0.5\mathrm{MeV}$ and $T=4.5$ keV. At temperatures of a few keV, the re-equilibration time and the external oscillation match, leading to a maximal bulk viscosity. The kink at $n_B\approx 2.2n_0$ is a result of the $\mathrm{\Lambda }$ onset.

Figure 9

Density and temperature dependence of the dissipation time for density oscillations, using the $\mathrm{PK}1+\mathrm{H}$ EOS. Within the 1-ms contour, the shortest dissipation time occurs at a temperature of $T=4\mathrm{keV}$, where the re-equilibration time $\gamma$ matches the external frequency $\omega$, leading to maximal bulk viscosity. For lower temperatures, the re-equilibration rate is slower than the external oscillation, whereas for higher temperatures the rates are too fast.

Figure 10

Feynman- and quark-flow diagram for process II, $n+p\to p+\mathrm{\Lambda }$, in the OME channel [panels (a) and (b)] and the contact interaction channel [panel (c)].

Figure 11

Feynman- and quark-flow diagram for process II, $n+p\to p+\mathrm{\Lambda }$, in the OME channel [panels (a) and (b)] and the contact interaction channel [panel (c)], both with the initial nucleons exchanged.

Figure 12

Feynman- and quark-flow diagram for process III, $n+n\to n+\mathrm{\Lambda }$ in the OME channel. The corresponding contact interaction channel would be mediated by neutral $Z\text{−}$boson exchange and is therefore suppressed by the GIM mechanism. For the calculation of the OME matrix element, a diagram with the two incoming neutrons exchanged has to be subtracted from the depicted one.

Figure 13

Feynman- and quark-flow diagram for process IV, $\mathrm{\Lambda }+\mathrm{\Lambda }\to \mathrm{\Lambda }+n$ in the OME channel. The corresponding contact interaction channel is suppressed due to the GIM mechanism. For the calculation of the OME matrix element, a diagram with the two incoming hyperons exchanged has to be subtracted from the depicted one.