Neutrino quantum kinetics in compact objects

Neutrinos play a critical role of transporting energy and changing the lepton density within corecollapse supernovae and neutron star mergers. The quantum kinetic equations (QKEs) combine the effects of neutrino-matter interactions treated in classical Boltzmann transport with the neutrino flavor-changing effects treated in neutrino oscillation calculations. We present a method for extending existing neutrino interaction rates to full QKE source terms for use in numerical calculations. We demonstrate the effects of absorption and emission by nucleons and nuclei, electron scattering, electron-positron pair annihilation, nucleon-nucleon bremsstrahlung, neutrino-neutrino scattering. For the first time, we include all these collision terms self-consistently in a simulation of the full isotropic QKEs in conditions relevant to core-collapse supernovae and neutron star mergers. For our choice of parameters, the long-term evolution of the neutrino distribution function proceeds similarly with and without the oscillation term, though with measurable differences. We demonstrate that electron scattering, nucleon-nucleon bremsstrahlung processes, and four-neutrino processes dominate flavor decoherence in the protoneutron star (PNS), absorption dominates near the shock, and all of the considered processes except elastic nucleon scattering and neutrino-neutrino processes are relevant in the decoupling region. Finally, we propose an effective decoherence opacity that at most energies predicts decoherence rates to within a factor of 10 in our model PNS and within 20% outside of the PNS.

Figure 1

IsotropicSQA method summary. Neutrino oscillations are evolved explicitly via the evolution matrix $S$ (bottom horizontal lines) separately from collisions (top horizontal lines), and the effects are combined at time intervals of $d{t}_{\text{block}}$. This allows us to take very large time steps for the expensive and slowly acting interactions, and very short steps for the inexpensive and rapidly varying oscillations.

Figure 2

Two-point diagram for electron neutrino absorption and emission processes. The equivalent diagram for electron antineutrinos is the same diagram in reverse. The lack of internal neutrino lines makes this a particularly straightforward process to transform into QKE collision terms.

Figure 3

Absorption.—Evolution of the flavor–off-diagonal components of a maximally mixed Fermi-Dirac neutrino distribution [Eq. (17)] due to absorption and emission by nucleons and nuclei. On-diagonal components are not shown since they simply remain at their initial values. Interaction rates are based on a background described by $\rho ={10}^{12}\mathrm{g}{\mathrm{cm}}^{-3}$, $T=10\mathrm{MeV}$, and $Y_e=0.3$. We will show in Sec. 5c that absorption dominates decoherence of higher-energy neutrinos outside of the PNS.

Figure 4

Two-point diagrams for neutrino-electron scattering processes. The charged-current processes lead to flavor decoherence and all processes allow redistribution in energy.

Figure 5

Flavor structure of the isotropic part ${\mathrm{\Phi }}_0^{\pm }$ of electron scattering collision kernel $R^{\pm }$ [Eq. (34)] as used in Eq. (32). The reaction rates are calculated at an electron chemical potential of ${\mu }_e=3.16\mathrm{MeV}$ and a temperature of $kT=2.74\mathrm{MeV}$ as well as using ${\theta }_W=0.2223$. The top plot is for neutrinos and the bottom for antineutrinos. Red regions indicate that scattering events mostly preserve the sign and magnitude of the scattered neutrino’s off-diagonal elements, white regions collapse them to zero, and blue regions flip their sign.

Figure 6

Diagram demonstrating how the first term in Eq. (32) (electron scattering collision integral) affects the phase of 2-flavor mixed-state neutrinos after scattering when we follow the neutrino from its initial to its final energy with each scatter. We show a maximal-length single particle flavor isospin vector $\overrightarrow{\rho }$ for an initial neutrino flavor state described by the density matrix ${\rho }_{ab}$ (unrelated to the fluid density $\rho$) that is mostly electron neutrino (Initial 1) and one that is mostly muon neutrino (Initial 2) for illustration purposes (${\rho }_{(x)}=\sqrt{7}/4$, ${\rho }_{(y)}=0$ and ${\rho }_{(z)}=\pm 3/4$). We set the oscillation Hamiltonian to zero and only consider electron scattering. Red regions in Fig. 5 drive the distribution function closer to the flavor axis with each scatter, as shown by the series of red arrows. Blue regions in Fig. 5 follow the same pattern, except that the neutrino phase is negated on each scatter, as shown by the series of red arrows. In both cases, the density matrix approaches the flavor states more quickly as ${\mathrm{\Phi }}_{0e\mu }^{\pm }/⟨\mathrm{\Phi }{⟩}_{0e\mu }^{\pm }$ approaches 0. To see this, compare the Initial 1 case to the Initial 2 case, where the magnitude of the off-diagonal component of the scattering kernel is larger for the latter. The dashed circle represents the sphere along which oscillations can rotate the vector, described by ${\rho }_{(x)}^2+{\rho }_{(y)}^2+{\rho }_{(z)}^2=1$.

Figure 7

Flavor structure of the linearly anisotropic part ${\mathrm{\Phi }}_1^{\pm }$ of electron scattering collision kernel $R^{\pm }$ as used in Eq. (32). The reaction rates are calculated at an electron chemical potential of ${\mu }_e=3.16\mathrm{MeV}$ and a temperature of $kT=2.74\mathrm{MeV}$ using ${\theta }_W=0.2223$. The top plot is for neutrinos and the bottom for antineutrinos. The anisotropic collision term for off-diagonal elements (middle panels) reflect both the anisotropy of the flavor-diagonal parts (left and right panels) and the flavor structure of Fig. 5.

Figure 8

Inelastic electron scattering.—Evolution of maximally mixed Fermi-Dirac neutrino distribution [Eq. (17)] due to inelastic electron scattering ($\nu +e^{-}\leftrightarrow \nu +e^{-}$) interactions. The top panel contains the flavor-diagonal neutrino distribution components, the middle panel contains the flavor-diagonal antineutrino distribution components, and the bottom panel contains the flavor–off-diagonal neutrino and antineutrino components. Interaction rates are based on a background described by $\rho ={10}^{12}\mathrm{g}{\mathrm{cm}}^{-3}$, $T=10\mathrm{MeV}$, and $Y_e=0.3$. The charged-current parts of the electron scattering interaction drive the off-diagonal elements to zero, and the diagonal elements redistribute in energy to non-Fermi-Dirac values as long as the off-diagonal elements are nonzero.

Figure 9

Elastic electron scattering.—Evolution of maximally mixed Fermi-Dirac neutrino distribution [Eq. (17)] due to elastic electron scattering ($\nu +e^{-}\leftrightarrow \nu +e^{-}$) interactions. The top panel contains the flavor–off-diagonal neutrino and antineutrino components. The bottom panel shows the difference between the elastic ($f$) and inelastic ($f_{\text{inelastic}}$ from Fig. 8) results. On-diagonal components are not shown since they simply remain at their initial values. Interaction rates are based on a background described by $\rho ={10}^{12}{\mathrm{g}\mathrm{cm}}^{-3}$, $T=10\mathrm{MeV}$, and $Y_e=0.3$. Though these approximate results are qualitatively similar to the bottom panel of Fig. 8, there are significant quantitative differences.

Figure 10

Two-point diagram for neutrino-nucleon scattering. This is very similar to Fig. 4, but the lack of charged-current processes makes the flavor structure less complex.

Figure 11

Flavor structure of the isotropic part ${\mathrm{\Phi }}_0^{\pm }$ of the pair annihilation kernel $R^{\pm }$ as used in Eq. (39). The reaction rates are calculated at an electron chemical potential of ${\mu }_e=17.8\mathrm{MeV}$ and a temperature of $kT=2.74\mathrm{MeV}$ using ${\theta }_W=0.2223$. The top plot is for neutrinos and the bottom for antineutrinos. Red and dark blue regions show where electron degeneracy has a significant impact on the flavor structure of the scattering kernel.

Figure 12

Flavor structure of the linearly anisotropic part ${\mathrm{\Phi }}_1^{\pm }$ of the pair annihilation kernel $R^{\pm }$ as used in Eq. (39). The reaction rates are calculated at an electron chemical potential of ${\mu }_e=17.8\mathrm{MeV}$ and a temperature of $kT=2.74\mathrm{MeV}$ using ${\theta }_W=0.2223$. The top plot is for neutrinos and the bottom for antineutrinos. The anisotropic annihilation term for off-diagonal elements (middle panels) reflects both the anisotropy of the flavor-diagonal parts (left and right panels) and the flavor structure of Fig. 11.

Figure 13

Pair annihilation.—Evolution of maximally mixed Fermi-Dirac neutrino distribution [Eq. (17)] due to electron-positron pair ($e^{+}+e^{-}\leftrightarrow \nu +\overline{\nu }$) interactions. The top panel contains the flavor-diagonal neutrino distribution components, the middle panel contains the flavor-diagonal antineutrino distribution components, and the bottom panel contains the flavor–off-diagonal neutrino and antineutrino components. Interaction rates are based on a background described by $\rho ={10}^{12}{\mathrm{g}\mathrm{cm}}^{-3}$, $T=10\mathrm{MeV}$, and $Y_e=0.3$. The nonlinear terms cause the flavor-diagonal elements to redistribute away from Fermi-Dirac values as long as the off-diagonal elements are nonzero. The long evolution timescale makes this process a subdominant driver of flavor decoherence.

Figure 14

Effective absorption pair annihilation.—Evolution of maximally mixed Fermi-Dirac neutrino distribution [Eq. (17)] due to electron-positron pair ($e^{+}+e^{-}\leftrightarrow \nu +\overline{\nu }$) interactions treated as an effective absorption/emission process. The top panel contains the flavor–off-diagonal neutrino and antineutrino components. The bottom panel shows the difference between the elastic ($f$) and inelastic ($f_{\text{inelastic}}$ from Fig. 8) results. On-diagonal components are not shown since they simply remain at their initial values. Interaction rates are based on a background described by $\rho ={10}^{12}{\mathrm{g}\mathrm{cm}}^{-3}$, $T=10\mathrm{MeV}$, and $Y_e=0.3$. There are significant differences between how an effective absorption treatment leads to flavor decoherence compared to the full treatment.

Figure 15

Two-point diagram for nucleon-nucleon bremsstrahlung radiation. This diagram represents many diagrams with various permutations of $Z$ bosons connecting the lines. A single internal neutrino line makes the structure similar to electron-positron pair annihilation (Fig. 4).

Figure 16

Effective absorption bremsstrahlung.—Evolution of the flavor–off-diagonal components of a maximally mixed Fermi-Dirac neutrino distribution [Eq. (17)] due to nucleon-nucleon bremsstrahlung ($N+N\leftrightarrow N+N+\nu +\overline{\nu }$) interactions. On-diagonal components are not shown since they simply remain at their initial values. Interaction rates are based on a background described by $\rho ={10}^{12}{\mathrm{g}\mathrm{cm}}^{-3}$, $T=10\mathrm{MeV}$, and $Y_e=0.3$. Bremsstrahlung processes decohere low-energy neutrinos more rapidly than high-energy neutrinos and become a dominant driver of decoherence at nuclear densities. The significant errors associated with the effective absorption treatment of pair processes (Fig. 14) calls for a more realistic treatment of bremsstrahlung processes.

Figure 17

Two-point diagrams for neutrino-neutrino scattering and pair annihilation. The multiple internal neutrino lines make this a difficult process to treat in the QKEs. The lack of charged-current interactions precludes phase decoherence.

Figure 18

Four-neutrino processes.—Evolution of maximally mixed Fermi-Dirac neutrino distribution [Eq. (17)] due to neutrino-neutrino scattering ($\nu +\nu \leftrightarrow \nu +\nu$) and pair ($\nu +\overline{\nu }\leftrightarrow \nu +\overline{\nu }$) interactions. The top panel contains the flavor-diagonal neutrino distribution components, the middle panel contains the flavor-diagonal antineutrino distribution components, and the bottom panel contains the flavor–off-diagonal neutrino and antineutrino components. Interaction rates are based on a background described by $\rho ={10}^{12}{\mathrm{g}\mathrm{cm}}^{-3}$, $T=10\mathrm{MeV}$, and $Y_e=0.3$. The neutrinos never decay to flavor-diagonal Fermi-Dirac distributions, but instead redistribute such that the distribution flavor vector at all neutrino energies are aligned, and the neutrino vectors are antialigned with the antineutrino vectors.

Figure 19

All processes.—Evolution of maximally mixed Fermi-Dirac neutrino distribution [Eq. (17)] due to absorption/emission, inelastic electron scattering, electron-positron pair annihilation, nucleon-nucleon bremsstrahlung, neutrino-neutrino scattering, and neutrino-antineutrino pair interactions. The top panel contains the flavor-diagonal neutrino distribution components, the middle panel constrains the flavor-diagonal antineutrino distribution components, and the bottom panel contains the flavor–off-diagonal neutrino and antineutrino components. Interaction rates are based on a background described by $\rho ={10}^{12}{\mathrm{g}\mathrm{cm}}^{-3}$, $T=10\mathrm{MeV}$, and $Y_e=0.3$. This quantitatively provides an estimate of decoherence timescales on the order of a millisecond for these parameters depending on energy.

Figure 20

Oscillation terms.—Evolution of all components of a two-flavor neutrino distribution function due to the action of the oscillation Hamiltonian in Eq. (3) in a background fluid described by $\rho ={10}^{12}{\mathrm{g}\mathrm{cm}}^{-3}$, $T=10\mathrm{MeV}$, and $Y_e=0.3$. There is a short precession timescale associated with the number density of electrons visible in the flavor–off-diagonal components (bottom two panels) and a longer nutation timescale associated with the total neutrino density most obvious in the flavor-diagonal components (top two panels). Both are much shorter than the collision timescale.

Figure 21

Top panels: Evolution of the $z$ component of the neutrino distribution flavor vectors relative to the equilibrium vector length at several neutrino energies. Dark (light) shades show (anti)neutrino values. The equivalent values from the nonoscillating calculations are plotted in green for neutrinos (solid lines) and antineutrinos (dashed lines). The spread is due to violent oscillations on much shorter timescales, and one can see collisions decreasing the amplitude of the oscillations (width of the spread) and driving the distributions toward the flavor axis. Center panels: Component of the distribution flavor vector perpendicular to the $z$ direction [Eq. (44)] relative to the equilibrium vector length using the same color/line conventions as above. The spread is due to the same oscillations that cause the spread in the top panels. The results roughly agree with those of the nonoscillating calculations, plotted as solid green for neutrinos and dashed green for antineutrinos. Bottom panels: Length of the neutrino distribution flavor vector relative to the equilibrium length using the same color and line conventions as above. There is not spread because the distribution flavor vector length is not directly sensitive to oscillations.

Figure 22

Background fluid snapshot from a one-dimensional neutrino radiation hydrodynamics core-collapse supernova simulation at 100 ms after core bounce [79]. We perform separate isotropic, homogeneous quantum kinetics calculations without the oscillation term at each radial point using the matter density $\rho$, electron fraction $Y_e$, and temperature $T$ at that radius as input. The bottom panel also shows for reference the chemical potentials of electrons ${\mu }_e$ and electron neutrinos ${\mu }_{{\nu }_e}$ as determined by the HShen equation of state [70]. The location of the shock is at the right edge of the plot at 168 km, and the approximate location of the outer edge of the protoneutron star is shown at 10 km with a vertical dashed green line.

Figure 23

Decoherence times of flavor–off-diagonal components of the neutrino distribution function in conditions relevant to CCSNe. We perform an isotropic, homogeneous nonoscillating quantum kinetic calculation using fluid values from each radial point in Fig. 22. Top panels show the decoherence times for neutrinos and bottom panels show those for antineutrinos. Neutrino energies from 1 MeV to 100 MeV are colored according to the color bar on the right, while energies from 101 to 200 MeV are all colored gold. Each panel shows the results using a different set of interactions as indicated by the text in the panel. The approximate outer edge of the protoneutron star is depicted with a dashed green line at 10 km. For reference, we also show the location where the radial coordinate is equal to the decoherence length scale with a dashed blue line. Electron scattering, bremsstrahlung processes, and four-neutrino processes dominate decoherence in the PNS, while absorption dominates decoherence just under the shock. All processes except neutrino-nucleon elastic scattering and neutrino-neutrino processes contribute in the decoupling region.

Figure 24

Ratio of the computed flavor decoherence length scale to the effective decoherence opacity [Eq. (45)] in the nonoscillating calculations including absorption, inelastic electron scattering, electron-positron pair annihilation, and effective-absorption nucleon-nucleon bremsstrahlung. This effective opacity is a much better predictor of flavor decoherence rates than the mean free path.

Figure 25

Vacuum oscillations test.—The top panel shows the evolution of the ${\nu }_e$ and ${\nu }_{\mu }$ distribution functions for the 10 MeV and 49 MeV energy bins, along with the analytic solution. The bottom panel shows the maximum error in the solution for simulations with three different time step sizes. The error increases with decreasing step size. See text for details.

Figure 26

MSW oscillations test.—The top panel shows the evolution of the ${\nu }_e$ and ${\nu }_{\mu }$ distribution functions for the 10 MeV and 49 MeV energy bins, along with the analytic solution. The bottom panel shows the maximum error in the solution for simulations with three different time step sizes. The error increases with decreasing step size. See text for details.

Figure 27

Bipolar oscillation test.—First panel: Evolution of a neutrino distribution function $f$ initially consisting of pure $50\mathrm{MeV}{\nu }_{\mu }$ and ${\overline{\nu }}_{\mu }$ in equal amounts, ignoring collision terms. Second panel: Maximum error between the fiducial calculation (integration accuracy of ${10}^{-13}$ and $d{t}_{\text{block}}={10}^{-3}\mathrm{s}$) and that with lower integration accuracy (red line) and longer $d{t}_{\text{block}}$ (blue line). Third panel: Relative change in the length of the distribution flavor vector, which should be 0 since collision terms are set to zero. Fourth panel: How non-Hermitian the distribution function becomes at each step before enforcing Hermitivity.

Figure 28

Fidelity tests.—Calculations evolving the QKEs including oscillations and collisions. Top panel: Neutrino distribution flavor vector length relative to the flavor-diagonal Fermi-Dirac length. Center panel: Antineutrino distribution flavor vector length. Bottom panel: Relative error compared to the fiducial calculation. Black curves show the results of a fiducial calculation using 25 energy bins and an energy grid spacing of 4 MeV, while red curves show the results from Sec. 5 with 50 energy bins and an energy spacing of 2 MeV. Green curves result from worsening the accuracy of the integrator to ${10}^{-11}$. Blue curves result from increasing the target impact to ${10}^{-3}$. Red curves result from doubling the neutrino energy resolution with the same energy domain. Gold curves result from increasing the upper bound of the neutrino energy domain and the number of energy bins by 20%, keeping the energy grid spacing constant. In the bottom panel, solid lines take the maximum over all energies and helicities, dashed lines show only errors for neutrinos at 28 MeV, and dotted lines show only errors for antineutrinos at 28 MeV.