Flavor conversions with energy-dependent neutrino emission and absorption

Decades of theoretical research on core-collapse supernovae (CCSNe) and binary neutron star mergers (BNSMs) have shown that neutrinos play important roles in the dynamics of these explosive transient events. The influence of neutrino physics in these phenomena has been much discussed. One of the quantum kinetic features receiving increased attention throughout the community is collective neutrino oscillations, which are neutrino flavor conversions induced by neutrino self-interactions [1, 2, 3, 4]. Fast neutrino flavor conversions (FFCs) [4] and collisional flavor instabilities (CFIs) [5] potentially affect the dynamics. The sensitivity of CCSN and BNSM dynamics to these flavor conversions have been recently discussed [6, 7, 8, 9, 10], although there is very little known about their nonlinear properties.

FFC can occur independently of vacuum oscillations and its dynamics is essentially energy-independent. It has been found that electron neutrino lepton number crossings (ELN crossings) in angular distributions are a necessary and sufficient condition for FFCs [11, 12]. The associated criterion, namely ELN-XLN crossing, also offers to determine asymptotic states of FFCs in nonlinear regime [13, 14], where XLN denotes a heavy-leptonic neutrino lepton number. Recent studies have found that the ELN crossings are ubiquitous in CCSNe [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31] and BNSMs [32, 33, 34, 35, 6, 36, 7, 37]. This suggests that we need to accommodate FFCs in the theoretical modeling of CCSNe and BNSMs.

Nonlinear dynamics of FFCs have been simulated by solving the neutrino quantum kinetic equation (QKE) [38, 39, 7, 37, 40, 41, 42, 43, 44, 45, 46, 47]. Under the assumption of homogeneous neutrinos, the dynamics can be described in analogy with pendulum motions [48, 49]. Although the pendulum analogy offers deeper insights of FFCs, this toy model can not be applied in the case with neutrino-matter collisions [50, 51]; indeed, the FFC dynamics becomes qualitatively different [52, 53, 54, 55, 50, 56, 57, 58]. Recent studies also performed large-scale simulations of FFCs with collisions at various levels of approximations [59, 60, 13, 61, 8, 9]. We refer readers to [62, 63, 64] for recent reviews.

As mentioned above, matter collisions potentially have a significant impact on the dynamics of flavor conversions. CFI, which is a flavor conversion induced by matter collisions, has recently attracted much attention [5]. One of the necessary conditions for CFIs is the difference of reaction rates between neutrinos and antineutrinos [65]. A notable feature of CFIs is that flavor conversions occur even for isotropic neutrino distributions [5]. Some recent studies suggested that CFIs can ubiquitously occur in CCSN [8] and BNSM [65] environments. It is also noteworthy that the growth rate of CFIs has a resonant structure, if the number density of electron neutrinos ($\nu_{e}$) and their antipartners ($\bar{\nu}_{e}$) is nearly equal to each other [65, 66]. If FFCs are parasitically induced by CFIs, the flavor conversions may be further accelerated [67].

One thing we do notice is that the nonlinear interplay between flavor conversions and matter collisions remains an open question, even in the case with homogeneous neutrino assumption. As for FFCs, the coupling with isoenergetic scattering such as nucleon scattering has been studied [55, 56, 57, 50], whereas the impact of emission and absorption on flavor conversions has not yet been investigated in detail. It should also be mentioned that, although the interplay between CFIs and emission/absorption has been already investigated [5, 67, 65, 68, 66], there are little studies on the cases with anisotropic distributions or non-monochromatic neutrinos [67, 65]. The better understanding of roles of emission/absorption on these flavor conversions can contribute valuable information to interpret results obtained from more self-consistent and global simulations of FFCs and CFIs [9, 8], and it may also offer new quantum kinetic features of neutrinos.

In this paper, we perform single and multi-energy dynamical simulations of flavor evolution with neutrino emission and absorption. Neutrinos are assumed to be homogeneous space and axial-symmetric (but anisotropic) in momentum space. It is also assumed that the system consists of two-neutrino flavor. In this study, we pay particular attention to the energy- and angular-dependent features of flavor conversions. One of the intriguing features found in this study is that the so-called angular swap, that corresponds to the full exchange between $\nu_{e}$ and heavy-leptonic neutrinos ($\nu_{x}$) at a certain angular region, is facilitated by emission and absorption. We will discuss the possible generation mechanism.

This paper is organized as follows. We start with the numerical setup for dynamical simulations of flavor conversions in Section II. We then move on to investigate effects of neutrino emission and absorption on flavor conversions with the energy-independent reaction rates from various perspectives in Section III. In Section IV, we extend the study to the case of energy-dependent emission and absorption. Finally, we summarize our key findings and conclusions in Section V.

Time evolution of neutrinos in phase space follows the quantum kinetic equations (QKEs) [2, 69],

In these expressions, $\rho$ is the density matrix for neutrinos; $\mathcal{H}$ is the Hamiltonian potential; $C$ is the collision term. The bar description indicates the quantities for antineutrinos hereafter. We consider two flavor system comprised of electron and heavy-leptonic neutrinos. Accordingly, $\rho$ and $\bar{\rho}$ have four components,

It is also assumed that the neutrino distribution is spacial homogeneous and axial asymmetry in momentum space. In this study, we include neutrino emission and absorption via charged-current interactions by surrounding matters. Since heavy-leptons are unlikely to appear in CCSNe and BNSMs (but see [70]), we ignore the charged-current reactions for $\nu_{x}$ and their anti-partners ($\bar{\nu}_{x}$).

Under these assumptions, the QKEs can be rewritten as

with the azimuthal-angle integrated density matrix $\rho_{a}\equiv\rho_{a}(E_{\nu},\cos{\theta_{\nu}},t)=\int d\phi_{\nu}\rho(E_{\nu},\cos{\theta_{\nu}},\phi_{\nu},t)$, the emission and absorption rates for electron neutrinos, $R_{e}$ and $R_{a}$, and neutrino self-interaction potentials,

We refer to results of SN simulation by Sumiyoshi and Yamada [71] to the numerical setup. In detail, we focus on the situation of $\sim$50 km from the stellar center at 100 ms after the core bounce. The detailed parameters are summarized in Table 1. In this situation, the chemical potentials of $\nu_{e}$ and their anti-partners ($\bar{\nu}_{e}$) are almost zero and the necessary condition for FFCs is in place. In SN matter, the matter density decreases towards the stellar surface and neutrinos are decoupled from matters at a certain radius (neutrino sphere). The position of neutrino sphere is generally $r_{\nu_{e}}>r_{\bar{\nu}_{e}}>r_{\nu_{x}}$, $r_{\bar{\nu}_{x}}$, although it depends on a neutrino energy. Here we focus on the region between the neutrino spheres of $\bar{\nu}_{e}$ and $\nu_{x}$ around average neutrino energy¹¹1Specifically, we consider the transport sphere for $\nu_{x}$ and $\bar{\nu}_{x}$.. On the basis of this fact, we employ the initial energy spectrum of neutrinos in Figure 1. Here the quantity on the vertical axis is defined as $I\equiv E_{\nu}^{2}/(2\pi)^{3}\int\rho_{a}d\cos{\theta_{\nu}}$. It is assumed that $\nu_{e}$ and $\bar{\nu}_{e}$ are in the thermal equilibrium at all energies and at $E_{\nu}\gtrsim 20$ MeV, respectively. The energy spectrum for $\nu_{e}$ and the higher energy part of $\bar{\nu}_{e}$ match with the FD of $T_{\nu}=4.5$ MeV and $\mu_{\nu}=0$ MeV (dotted line). On the other hand, $\nu_{x}$ and $\bar{\nu}_{x}$ are already decoupled from matters at all neutrino energies in more inner region of the star. Therefore, $\nu_{x}$ and $\bar{\nu}_{x}$ spectra are consistent with the FD of $T_{\nu}=6$ MeV and $\mu_{\nu}=-8.91$ MeV (dot-dashed line) at high energies. It should be noted that the spectrum of $\nu_{e}$ and $\nu_{x}$ are crossed with each other at $E_{\nu}\sim 25$ MeV.

We assume that $\nu_{e}$ are isotropic initially, while we use four anisotropic distributions for $\bar{\nu}_{e}$, $\nu_{x}$ and $\bar{\nu}_{x}$, following the previous papers [55, 49]. Hereafter we call the angular distribution models from Shalgar and Tamborra [55] and Padilla-Gay et al. [49] as ST and PTR models, respectively. We define energy-integrated angular distributions as $g\equiv\int\rho_{a}E_{\nu}^{2}dE_{\nu}/(2\pi)^{3}$,

$n_{\nu_{e}}$ is the total number of electron neutrinos; $a$ and $b$ are angular-shape parameters in Table 2. These initial angular distributions are shown in the top panel of Figure 2. The bottom panel shows the angular distributions of ELN-XLN, or $L_{e}-L_{x}\equiv g_{ee}-\bar{g}_{ee}-g_{xx}+\bar{g}_{xx}$. All models have the angles with both of the positive and negative $L_{e}-L_{x}$ and hence FFCs are expected to be induced in all models. It should be noted that isotropic or non-isotropic in each flavor is consistent with whether or not the flavor is in thermal equilibrium.

The emission and absorption rates ($R_{e}$ and $R_{a}$) are also adopted by reference to the same situation in the realistic SN simulation. For the reaction rate of $\nu_{e}$, we emulate the electron capture by free protons, which is the dominant process in this situation. Specifically, we take $R_{a}(E_{\nu})=2\times 10^{-5}\ {\rm cm^{-1}}(E_{\nu}/30{\rm MeV})^{2}$ and $R_{e}(E_{\nu})$ is determined through the detailed balance relation, or $R_{e}(1-\rho_{ee,{\rm eq}})=R_{a}\rho_{ee,{\rm eq}}$. Here $\rho_{ee,{\rm eq}}$ is the Fermi-Dirac distribution (FD) for $\nu_{e}$ with $T_{\nu}$ and $\mu_{\nu}$ in Table 1. For $\bar{\nu}_{e}$, the absorption rate is assumed to be $\bar{R}_{a}(E_{\nu})=0.1R_{a}(E_{\nu})$, emulating the positron capture by free neutrons, and $\bar{R}_{e}(E_{\nu})$ is determined through the detailed balance relation in the same manner as that for $\nu_{e}$. We neglect the emission and absorption for $\nu_{x}$ and $\bar{\nu}_{x}$ except in Section III.3.

We solve the QKEs with Monte Carlo (MC) method [72]. This code can treat neutrino transport, matter collisions and neutrino flavor conversions self-consistently. We ask readers to refer to [72] for more details.

To extract the essence of emission and absorption effects, we first perform dynamical simulations of flavor conversions with a single neutrino energy. In section III.1, we investigate these effects using the PTR-B model, compared to the results in the absence of emission/absorption (PTR-Bwo). We also discuss how initial angular distributions and the emission/absorption of $\nu_{x}$ and $\bar{\nu}_{x}$ affect the results in Sections III.3 and III.2, respectively. Through this section, it is assumed that all neutrinos have $E_{\nu}=$ 13 MeV, which is the typical average energy of $\nu_{e}$ in CCSNe. The total numbers of neutrinos for each flavor are shown in Table 1 and we adjust the energy bin width so that the initial $\nu_{e}$ distribution function matches the thermal equilibrium value. The emission and absorption rates for $\nu_{e}$ and $\bar{\nu}_{e}$ are set to be the values at $E_{\nu}=$ 13 MeV and they satisfy the detailed-balance relation at $T_{\nu}=4.5$ MeV and $\mu_{\nu}=0$ MeV. We consider 128 propagation directions of neutrinos as a reference resolution, following the angular distributions in Eq. 9.

In CCSNe and BNSMs, neutrinos have non-monochromatic energy spectrum and the reaction rates of neutrino-matter interactions are energy dependent. Hence it is natural to consider the evolution of neutrino flavors with matter collisions in the multi-energetic treatment. In this section, we employ the neutrino energy spectrum and the energy-dependent reaction rates into numerical simulations. The initial energy spectrum are described in Figure 1 and they are consistent with the realistic CCSN simulation. We adopt 10 energy meshes in the range of $0-100$ MeV. As the single-energy case, the energy-integrated angular distributions follow Eqs. 9 of four models (see also Figure 2) and the same number of neutrinos are employed in Table 1. We distribute 1,280 MC samples in a neutrino phase space.

Figure 13 shows the time evolution of $n_{\nu_{e}}$ (top) and $n_{\nu_{x}}$ (bottom). Colors distinguish the models, and the results of single-energy cases are also plotted in darker dotted lines. We find that these evolution has two phases, FFC and collision driven phases, as same as the single-energy case. First, we discuss effects of multi-energetic treatment using the PTR-B model as a reference (green line). At $t\lesssim 4\times 10^{-6}$ s, FFCs drive the evolution as well as the single-energy case. Table 3 shows the results of linear stability analysis. We find that the growth rate is almost the same as that in the single-energy case. This fact is attributed to the property that the dynamics of pure FFCs is energy-independent and depends on the number densities of neutrinos (see Eq. 4). On the other hand, the overall flavor evolution is delayed by effects of energy-dependent collisions. For $E_{\nu}\lesssim 13$ MeV, the reaction rates are smaller than that in the single-energy case, and the decoherence by the absorption of off-diagonal components is weak. As see in Figure 1, the fraction of neutrinos with $E_{\nu}\lesssim 13$ MeV to the total number of neutrinos is $\sim 84\%$ initially and these neutrinos contribute to the self-interaction Hamiltonian $H_{\nu\nu}$. This dominancy of low energy neutrinos leads to the slower attenuation of FFCs, and hence the lifetime of FFCs is extended in the multi-energy treatment. The extension of FFC lifetime is also confirmed by the evolution of $L_{e}-L_{x}$ evolution in the left bottom panel of Figure 14. The multi- and single-energy results are shown in solid and dashed lines, respectively. At $t=4\times 10^{-6}$ s, $L_{e}-L_{x}$’s are positive at all angles in the single-energy case, whereas the negative $L_{e}-L_{x}$ appears at $\cos{\theta_{\nu}}\lesssim-0.2$ in the multi-energy case. This indicates that FFCs in the multi-energy case still survive until this time. The crossings in this case disappear at $t=8\times 10^{-6}$ s.

In our previous paper [56], we have proposed the $\chi$ diagnostics to quantify the impact of multi-energy effects on FFCs with a $\chi$ parameter, or $\chi=\mid(\langle E_{\nu}\rangle-\langle RE_{\nu}\rangle)/(\langle E_{\nu}\rangle+\langle RE_{\nu}\rangle)\mid$. $\langle E_{\nu}\rangle$ and $\langle RE_{\nu}\rangle$ are the average energy of neutrinos and reaction-weighted one, respectively. In the monochromatic neutrino case, $\chi$ exactly becomes 0. If $\chi\sim 0$ in the multi-energy case, neutrinos with a particular energy dominantly contribute to a flavor evolution and results are close to single-energy ones. As $\chi$ increases, the deviation from single-energy results becomes larger, and the multi-energy treatment is essential. We derive $\chi=0.281$ for the PTR-B model, which indicates that the multi-energy treatment is essential in this model. This is consistent with numerical results.

Similar as scatterings, the energy-dependent emission/absorption generates rich energy-dependent features in FFCs. Figure 15 shows the time evolution of $I_{ee}$ (blue) and $I_{xx}$ (red). We normalize $I$’s by the initial values $I_{0}$’s in each neutrino energy. Colors get darker with neutrino energies. It is found that $I_{ee}/I_{ee,0}$’s for $E_{\nu}\lesssim 25$ MeV increase with time, whereas they decrease for $E_{\nu}\gtrsim 25$ MeV. This is attributed to the initial spectral crossing between $I_{ee}$ and $I_{xx}$ at $E_{\nu}\sim 25$ MeV (see Figure 1). Since FFCs attempt to eliminate the difference between $\nu_{e}$ and $\nu_{x}$, the direction of conversions differs between high and low energies. Related to this, we find that flavor conversions occur strongly at the higher energy because of the larger difference between $I_{ee}$ and $I_{xx}$.

In the late phase ($t\gtrsim 4\times 10^{-6}$ s), after FFCs are sufficiently attenuated, matter collisions and CFIs drive the evolution. In top panel of Figure 13, we find that $n_{\nu_{e}}$ increases by emission more slowly than the single-energy case. This indicates the energy-averaged emission rate is smaller in the multi-energy case, which is consistent with the result in the FFC driven phase. On the other hand, since reactions for $\nu_{x}$ and $\bar{\nu}_{x}$ are neglected, the constant evolution is observed as same as the single-energy case. As shown by the linear stability analysis, the unstable mode is observed at $t=8\times 10^{-6}$ s, but the growth rate $\gamma$ is longer than the timescale of interest (see Table 3). Considering that the crossings in the $L_{e}-L_{x}$ distribution disappears at $t=8\times 10^{-6}$ s, this unstable mode is presumed to be CFIs. Moreover, it should be noted that the asymptotic values of $n_{\nu_{x}}$ are nearly identical between the single- and multi-energy cases. This is understood by the evolution of angular distributions in Figure 16. We find the angular swaps between $\nu_{e}$ and $\nu_{x}$ at wider angular range as same as the single-energy case. As mentioned in Section III.1, FFCs at each angle are determined by the total lepton number and the quantity $P$. Since adopting the same angular distributions for both cases, the asymptotic states are almost the same.

We also investigate the dependence of initial angular distributions using the ST and PTR models. In all the models, the initial growth rates $\gamma$ are almost the same as the single-energy case in Table 3, which again describes the property of FFCs. The results in the nonlinear phase are shown in Figures 13 and 14. In all models, the overall evolution becomes slower than the single-energy cases as same as the PTR-B model, while the features of each model are similar to those of the single-energy case. Below are some comments on individual models. The effects of multi-energy treatment in the PTR-C model are similar to that in the PTR-B model. Specifically, the asymptotic states in the PTR-C model ($t\gtrsim 4\times 10^{-6}$ s) are almost identical between the single- and multi-energy cases due to the fact that the maximum amount of FFCs is limited. After the $L_{e}-L_{x}$ crossing disappears at $t\sim 8\times 10^{-6}$ s, CFIs drive the weak flavor conversions. In the PTR-D model, the $L_{e}-L_{x}$ crossings are still observed even at $t=8\times 10^{-6}$ s in Figure 14 and FFCs survive to the end in the timescale of interest. As same as the single case of ST model, high angular resolutions are necessary in the multi-energy case. In the high-resolution results ($N_{\theta_{\nu}}=512$), the $L_{e}-L_{x}$ crossings are found even at $t=8\times 10^{-6}$ s and hence the unstable mode at this time may be the weak FFCs.

We perform dynamical simulations of neutrino flavor conversions with neutrino emission and absorption under the assumption of homogeneous neutrinos. We adopt physically motivated setups in this study, which is set based on the results of realistic SN simulations [71]. We start with discussing the case with energy-independent reaction rates (Section III), which presents some key features of the interplay between flavor conversion and emission/absorption. The dynamics is qualitatively different from FFCs without collisions; indeed, the periodic (or pendulum-like) features in flavor conversions disappear. The result is essentially in line with the case with isoenergetic neutrino-matter scattering [55, 50, 56].

We also find that the time evolution of flavor conversions with emission/absorption can be divided into two phases, (1)FFC driven phase and (2)collisional one. The multiple phase appearing in nonlinear regime is accounted for by the disparity of timescales between FFC and collisions. In the former phase, FFCs are initially more vigorous than the case without collisions due to breaking the symmetry of pendulum motion by collisions. On the other hand, FFCs are gradually attenuated by matter decoherence. After the competition of these collision effects, the larger number of $\nu_{e}$ is converted to $\nu_{x}$ by FFCs. In the latter phase, the flavor conversion is driven by CFIs, while the timescale is much longer than FFCs. $\nu_{e}$ and $\bar{\nu}_{e}$, which are less populated than the initial state due to strong FFCs in the earlier phase, are replenished by emission. The asymptotic states are characterized by matter collisions in this phase. Although the detailed features depend on the initial condition, we show that the overall trend is similar among all our models. It should also be worthy to note that $\nu_{x}$ emission/absorption do not change our conclusion, unless their reactions become comparable to $\nu_{e}$ and $\bar{\nu}_{e}$.

We also find that angular swap between $\nu_{e}$ ($\bar{\nu}_{e}$) and $\nu_{x}$ ($\bar{\nu}_{x}$) is facilitated by emission/absorption. In the pure FFC cases, the swap occurs very narrow angular region where ELN crossing occurs nearby. On the other hand, the crossing point can substantially deviate from the initial angular position in the case with emission/absorption, that expands the region where the angular swap occurs.

We extend our discussion in the case with energy-dependent reaction rates. The initial energy spectrum are set so as to be consistent with realistic SN simulations. In the low energy regime, the isotropization of neutrinos by emission/absorption is less remarkable, and therefore the lifetime of FFCs can be longer than the single-energy case. As a result, the overall trend becomes similar to the case with the lower reaction rate for monochromatic neutrinos. It should be mentioned that, although the flavor conversion is less vigorous than the single-energy case, the long-lived FFCs compensates for this. In fact, the degree of flavor conversion is almost identical to the signle-energy case, and the angular swap is also observed in a wide angular region. We also find that the energy-dependent emission/absorption generates rich energy-dependent features in FFCs similar as scatterings. In initial condition, the energy spectrum of $\nu_{e}$ and $\nu_{x}$ is comparable at $E_{\nu}\sim 25$ MeV, while $\nu_{e}$ and $\nu_{x}$ are dominant at the lower and higher energy regions, respectively. This exhibits that the increase or decrease of each flavor of neutrinos due to flavor conversion becomes qualitatively different between the low and high energy region.

Although the present study does not address all issues in the interplay between flavor conversions and collision term, it provides some valuable insights on quantum kinetic features of neutrinos in CCSN and BNSM environments. One of them is to determine asymptotic states of neutrinos. As shown in our models, FFCs promptly establish a quasi-steady state, which corresponds to, however, just a pseudo asymptotic state. After FFCs subside, matter collisions or CFIs take over to dictate secure evolution of neutrinos, asymptoting to another state. This indicates that the actual asymptotic state of neutrinos can not be determined solely by either neutrino flavor conversion or neutrino-matter interaction, but by self-consistent treatments of feedback between both of them.

This study also reveals a new possibility that flavor conversion offers a new path to absorb $\nu_{x}$ in the high energy region. Since the $\nu_{x}$ is initially larger than $\nu_{e}$, flavor conversions facilitate the $\nu_{x}$ conversion to $\nu_{e}$, and then it is absorbed through charged current reactions of $\nu_{e}$, exhibiting the increase of neutrino heating. It is an intriguing question how the heating can influence on CCSN and BNSM dynamics, although we leave the detailed investigations for future work.

In this work, we impose many simplifications and assumptions such as homogeneous neutrino gas, axial symmetry in momentum space, and two flavor system. Recent works have shown that the dynamics of flavor conversion is very sensitive to these conditions (e.g., [38, 40, 76, 42, 14, 13, 61, 77]). We also note that the homogeneous approximation suffers from a self-consistent problem in studying the impact of collisions on flavor conversion [78]. It should be stressed, however, that some intrinsic features of collision effects can be studied in our simplified approach. This offers new insights into roles of flavor conversions on CCSNe/BNSMs dynamics.