Fast-pairwise Collective Neutrino Oscillations Associated with Asymmetric Neutrino Emissions in Core-collapse Supernovae

We present a linear stability analysis of the fast-pairwise neutrino flavor conversion based on a result of our latest axisymmetric core-collapse supernova (CCSN) simulation with full Boltzmann neutrino transport. In the CCSN simulation, coherent asymmetric neutrino emissions of electron-type neutrinos (νe) and their antiparticles ( ), in which the asymmetries of νe and are anticorrelated with each other, occur at almost the same time as the onset of aspherical shock expansion. We find that the asymmetric neutrino emissions play a crucial role on occurrences of fast flavor conversions. The linear analysis shows that unstable modes appear in both pre- and post-shock flows; for the latter, they appear only in the hemisphere of higher emissions (the same hemisphere with stronger shock expansion). We analyze the characteristics of electron–lepton number (ELN) crossing in depth by closely inspecting the angular distributions of neutrinos in momentum space. The ELN crossing happens in various ways, and the property depends on the radius: in the vicinity of neutron star, (νe) dominates over νe ( ) in the forward (backward) direction; at the larger radius, the ELN crossing occurs in the opposite way. We also find that the non-radial ELN crossing occurs at the boundary between no ELN crossing and the radial one, which is an effect of genuine multi-dimensional transport. Our findings indicate that the collective neutrino oscillation may occur more commonly in CCSNe and suggest that the CCSN community needs to accommodate these oscillations self-consistently in the modeling of CCSNe.


Introduction
More than three decades have passed since neutrinos emitted from SN1987A, a core-collapse supernova (CCSN) in the Large Magellanic Cloud, were directly detected by Kamiokande (Hirata et al. 1987) and the Irvine-Michigan-Brookhaven detector (Bionta et al. 1987). Those neutrinos were produced deep inside the stellar core during the development of explosion. A proto-neutron star (PNS) is supposed to be formed and most of its internal energy (∼10 53 erg) was radiated by 10-30 MeV neutrinos. This agrees qualitatively with the CCSN theory, although the sparse data sample was insufficient to unveil the explosion mechanism.
Significant progress has been made in the observational instruments, which will enable us to detect neutrinos from CCSNe with much higher statistics or at farther distances than those of SN1987A (see, e.g., Suwa et al. 2019 and references therein). Super-Kamiokande, one of the operating neutrino detectors, is capable of detecting ∼10 4 neutrinos for a Galactic CCSN and a few neutrinos for an event in Andromeda (M31) at the distance of ∼770 kpc (see, e.g., Raffelt 2011). Hyper-Kamiokande, one of the next generation detector, will improve the sensitivity by an order of magnitude (Abe et al. 2011;Hyper-Kamiokande Proto-Collaboration et al. 2018). The neutrino detections from multiple CCSNe or from a single CCSN event but with very high statistics will provide us vital information for comprehensive understanding of the CCSN mechanism.
On the theoretical side, tremendous progress has also been made very recently; for instance, three-dimensional (3D) CCSN simulations with spectral neutrino transport (Takiwaki et al. 2012(Takiwaki et al. , 2014Lentz et al. 2015;Melson et al. 2015;Kuroda et al. 2016;Roberts et al. 2016;Takiwaki et al. 2016;Müller et al. 2017Müller et al. , 2019O'Connor & Couch 2018;Ott et al. 2018;Summa et al. 2018;Burrows et al. 2019;Nagakura et al. 2019a;Nakamura et al. 2019;Vartanyan et al. 2019b) and those in axis symmetry but with multi-angle neutrino transport (Ott et al. 2008;Brandt et al. 2011;Nagakura et al. 2018Nagakura et al. , 2019cHarada et al. 2019) are nowadays available. Although the microphysics inputs, including neutrino-matter interactions and a nuclear equation-of-state (EOS), were implemented in these simulations at different levels of refinement and accuracy, some of these simulations successfully reproduced explosions without artifices. In the upcoming exa-scale era, a number of 3D CCSN simulations with approximate neutrino transport will be significantly increased, and 3D CCSN simulations with general relativistic full Boltzmann neutrino transport with further improved input physics will also become available . Both approaches will be complementary to each other in our efforts to make CCSN modeling more realistic.
Given the neutrino emissions from the core either by numerical simulations or by some simplified models, one may be able to calculate the expected signals on terrestrial detectors by taking into account neutrino oscillations (see, e.g., Dighe & Smirnov 2000). Neutrinos should experience the ordinary vacuum oscillation in the intervening space and will also go through the so-called Mikheyev-Smirnov-Wofenstein resonance in the stellar envelope. Since neutrino signals from CCSNe sensitively depend on the neutrino mass hierarchy (but they may be insensitive to the neutrino oscillation itself at later times in the neutron-star (NS) cooling; Suwa et al. 2019), the future detections of supernova neutrinos may reveal the ordering of neutrino masses. As such, the connection between theory and observation will be more tight in the next decades toward the comprehensive understanding of neutrino physics (see, e.g., Nakamura et al. 2016;Seadrow et al. 2018 for more details).
Unfortunately, however, there remains a crucial concern in establishing realistic templates of neutrino signals and theoretical modelings of CCSNe-that is, collective neutrino oscillations or the oscillations induced by neutrinos themselves. Even the most up-to-date simulations neglect these effects despite the fact that they may have an impact on both the explosion mechanism and the neutrino signals. There are mainly two reasons for the defect: (1) it is still uncertain whether the collective neutrino oscillations really occur in CCSNe or not; (2) if they do indeed, the treatment in CCSN simulations is not easy because of the disparity in scales and the nonlinearity of the phenomenon. Nevertheless, considerable efforts have been made in the CCSN community to address these issues by using various approaches (see recent reviews, e.g., Chakraborty et al. 2016a;Mirizzi et al. 2016;Horiuchi & Kneller 2018, and references therein). The first issue has been studied by the linear stability analysis (see, e.g., Izaguirre et al. 2017;Delfan Azari et al. 2019;Yi et al. 2019) or by searching for the so-called electron-lepton number (ELN) crossing , in which the energy-integrated angular distributions of ν e and n ē in momentum space intersect each other. Note that the ELN crossing is supposed to be a necessary condition for occurrences of the fast flavor conversion, which is one of the collective oscillations modes. The second issue has been, on the other hand, addressed by solving nonlinear quantum kinetic equations under many simplifications (see, e.g., Richers et al. 2019).
In this paper, we tackle the former issue, focusing on fast flavor conversions (Sawyer 2005(Sawyer , 2016Chakraborty et al. 2016b;Capozzi et al. 2017;Dasgupta et al. 2017Izaguirre et al. 2017;Abbar & Duan 2018;Airen et al. 2018;Delfan Azari et al. 2019). Note that our CCSN simulations are capable of assessing occurrences of fast flavor conversion, since it feeds on the difference in the angular distributions among different species of neutrinos in momentum space, which is accessible only to the multi-angle neutrino transport like ours (Nagakura et al. 2018(Nagakura et al. , 2019cHarada et al. 2019). The methodology in this study is essentially the same as that in Delfan Azari et al. (2019). We carry out the linear stability analysis as postprocessing for the results of the CCSN simulation, but in this paper, we employ one of the latest CCSN models, in which stronger asymmetric neutrino emissions (~10%) were observed to be associated with PNS kick. More interestingly, the asymmetries of ν e and n ē emissions are anticorrelated with each other in this model-that is, the higher ν e emissions occur the opposite direction to the higher n ē emissions. 5 We expect that such an anticorrelation will give an impact on fast flavor conversions and, indeed, find its positive sign, unlike in the previous paper (Delfan Azari et al. 2019).
This paper is organized as follows. In Section 2, we briefly summarize our numerical modeling of the CCSN. The basic equations of the linear stability analysis are given in Section 3, and then we present our main results in Section 4. Finally, we conclude the paper with a summary in Section 5. Unless otherwise stated, we use the unit with c=G=ÿ=1, in which c, G, and ÿ are the light speed, the gravitational constant, and the reduced Planck constant, respectively. We use the metric signature of −++ +. Greek and Latin indices run over 03 and 13, respectively.

CCSN Model
Neutrino distribution functions, f ν , as solutions of classical Boltzmann equations are fundamental quantities for the linear stability analysis of the fast flavor conversions (see Section 3 for more details). In our CCSN simulations, we solve the Boltzmann equations for neutrino transport self-consistently but neglect possible neutrino oscillations entirely. We use the neutrino data as the background (a fixed point) for the linear stability analysis (see Section 3 for more details). In this section, we give an overview of our latest CCSN simulation.
The details of the code development of our Boltzmann solver are described in a series of papers Nagakura et al. 2014Nagakura et al. , 2017Nagakura et al. , 2019d and its reliability has been well established by a detailed comparison to another Monte-Carlo neutrino transport code (Richers et al. 2017). Some results of axisymmetric CCSN simulations by using our code can be seen in Nagakura et al. (2018Nagakura et al. ( , 2019c and Harada et al. (2019).
In the present study, we employ the result of one of our latest axisymmetric CCSN models in Nagakura et al. (2019c). In the simulation, the initial condition of the matter profile is taken from a 11.2 M ☉ progenitor model in Woosley et al. (2002), and the most up-to-date version of our code was run. The Boltzmann solver for neutrino transport is the same as that used in Nagakura et al. (2018), while we recently improved input physics (Nagakura et al. 2019b) under a multi-nuclear variational method (VM) EOS (Furusawa et al. 2017b). The homogeneous nuclear matter is treated with the VM (Togashi & Takano 2013;Togashi et al. 2017), in which Argonne v18  and UIX (Carlson et al. 1983;Pudliner et al. 1995) potentials are adopted for the two-and three-body potentials, respectively. Inhomogeneous matter composed of various nuclei and dripped nucleons in nuclear statistical equilibrium are handled with various finite-density and thermal effects (see Furusawa et al. 2017a for more details). Based on the nuclear abundances provided by this EOS, we constructed new weak interaction tables that includes electron captures by heavy and light nuclei and positron captures by light nuclei.
Below, we briefly summarize some characteristics in this CCSN model that deserve mentioning. As shown in Figure 1, the shock wave expands strongly in the northern hemisphere (see panel (a)) and the PNS receives a linear momentum in the opposite direction (see panel (b)). Note that our code is capable of treating the PNS proper motion directly and self-consistently in real time. We also find that strong asymmetric emissions of electron-type neutrino (ν e ) and its antiparticle (n ē ) occur at almost the same time as the onset of the aspherical shock expansion, which seems to be associated with the PNS kick (Nagakura et al. 2019c). The emissions of n ē are higher in the hemisphere of stronger shock expansion (i.e., the northern hemisphere) whereas the ν e emissions have the opposite trend, i.e., higher in the hemisphere, into which the PNS is kicked (see also the top panel of Figure 3 in Nagakura et al. 2019c).
The asymmetric ν e and n ē emissions are mainly caused by the nonspherical distributions of Y e around 10r25 km (see panel (b)), which are sustained by coherent lateral motions of matter. Interestingly, the linear momentum carried by neutrinos including the heavy-leptonic neutrino (ν x ) contributions is comparable to that of the PNS proper motion up to 300 ms after the bounce, which indicates that the asymmetric neutrino emissions play an important role in the acceleration of the PNS. We refer the reader to Nagakura et al. (2019c) for more details. As shown below, these asymmetric neutrino emissions are also important for the fast flavor conversion; indeed, higher ν e (n ē ) emissions stabilize (trigger) the fast flavor conversion in the post-shock flows, the details of which will be discussed in Section 4.

Linear Stability Analysis of Fast Flavor Conversion
We conduct a linear stability analysis by employing the dispersion relation (DR) approach (Izaguirre et al. 2017), which is probably one of the most convenient methods for the stability analysis in the literature. Below, we derive the DR of the fast flavor conversion. We refer readers to Izaguirre et al. We start with the equation of motion (EOM) for neutrino, where v μ , ρ ν , and H denote the neutrino four velocity = , the density matrix of neutrinos, and the Hamiltonian matrix, respectively. Equation (1) is expressed on the flavor basis, i.e., the diagonal components of the density matrix correspond to the distribution functions of flavor eigenstates, whereas the flavor coherence is expressed in the form of off-diagonal elements. The EOM of the anti-neutrinos can be included in Equation (1) by using the flavor isospin convention, in which the density matrix of anti-neutrinos has an opposite sign of that of neutrinos and corresponds to negative frequencies (energies).
Noted that several simplifications have been done in Equation (1): we take the ultra-relativistic limit, the spacetime is flat, and we ignore the collision term of classical Boltzmann equations and the spin coherence (Tian et al. 2017). We further impose two-flavor approximation, in which we consider two flavors alone: ν e and ν x . Note that our numerical setup of CCSN simulations is compatible with the two-flavor approximation, in which we assumed that the neutrino distribution functions were identical among all heavy-leptonic neutrinos (ν μ and ν τ ). Further studies are required to assess the impact of these simplifications but are beyond the scope of this paper. Notwithstanding these uncertainties, Equation (1) seems to contain the primary terms for the fast flavor conversion in CCSNe.
We decompose the Hamiltonian in Equation (1) into three contributions: From top to bottom, they represent the vacuum, matter, and neutrino-self-interaction contributions, respectively; M 2 , σ 3 , and G F denote the mass-squared matrix, the third Pauli matrix, and the Fermi constant, respectively. Λ μ represents the matter potential in a covariant form, which can be written as where -+ n n , e e and u μ are the electron and positron number densities and their four velocities, respectively. Note that we have already subtracted the trace part of the matter potential, which does not affect the flavor conversion. In Equation (4), muon and tau contributions are neglected, which may be good approximations in the supernova core (but see Bollig et al. 2017). Following the common practice, we divide the integral in momentum space 6 into the energy part (E 2 dE/(2π 2 )) and the angular one (dΓ) in the expression of H ν , where E denotes the neutrino energy and dΓ corresponds to the measure for the solid angle normalized by 4π Hereafter, we ignore the vacuum contribution (H V ) since we focus only on the fast mode in the neutrino flavor conversions. 7 Then, Equation (1) becomes energy-independent and one can integrate out the energy dependence. The energy-integrated form of the EOM can be written as and H ν defined in Equation (3) can be also rewritten in terms of It is well known that the matter potential, which dominates the vacuum contribution in the supernova core, suppresses the neutrino flavor conversion as long as the neutrino contribution is neglected (Wolfenstein 1979). It is hence reasonable to use the neutrino distribution functions obtained in our CCSN simulation, which neglected the neutrino oscillations, as unperturbed states in the linear stability analysis. They are indeed fixed points in Equation (5).
For latter convenience, we decompose the energy-integrated density matrix into the trace and traceless parts: The coefficients, n f , are related with the unperturbed distribution function, f νi , as in this study. Hence, the unperturbed density matrix is expressed as We linearize Equation (5), assuming that the off-diagonal component is small (S 1 v  ), to obtain the following equation for S v : Note that the diagonal component remains conserved in the linear order (see also Airen et al. 2018).
To obtain solutions of Equation (11), we take a plane-wave ansatz, which can be written in the form Then, the EOM can be rewritten as ). We can further rewrite the equation as which is called the polarization vector. Inserting Equation (16) into the right-hand side of Equation (17), we obtain the following relation: In this equation, h = -mn diag 1, 1, 1, 1 ( ) is the Minkowski metric and Π μν is called the polarization tensor. The nontrivial solutions can be obtained only when which gives a relation between ω and k or the DR.
We numerically search the solutions of Equation (20) that give instability. As is well known, however, we need to take care when numerically finding these solutions, since the socalled spurious modes (Sarikas et al. 2012) are artificially generated if we conduct integrations numerically by discretization. More recently, two of the authors of this paper developed a novel method to avoid this unpleasant issue, in which the integrations are done analytically with some basis functions (Morinaga & Yamada 2018). The validity of the method was confirmed in our previous paper (Delfan Azari et al. 2019). This method is a bit computationally costly, however, since high-order polynomials are required to compute the DR for strongly forward-peaked angular distributions accurately and is not suitable for a survey of wide spatial regions in many snapshots. We hence use a simpler formula for the maximum growth rate of an unstable solution (see also Equation (8) in 6 The integral domain for the neutrino energy is from negative to positive infinity in Equation (3), since we take the flavor isospin convection. 7 Note that the vacuum contribution may play an important role as a seed perturbator to trigger the flavor conversion. We also refer the reader to Airen et al. (2018) for the case where slow and fast modes mix. Morinaga et al. 2019 This approximate expression is partially motivated by the fact that unstable solutions appear when the ELN crossing occurs. Its validity will be checked at some selected points (see Section 4.1 for more details).

Results
In this section, we present main results of this paper. We first summarize the stability analysis of the fast flavor conversion in Section 4.1. Then, we discuss the role of asymmetric neutrino emissions, paying particular attention to the ELN crossings in Section 4.2.

Stability of Fast Flavor Conversion
Figure 2 displays spatial maps of the growth rate of the fast flavor conversion for some selected snapshots, in which colors other than white represent the unstable region. Note that the growth rate is computed based on Equation (3). One of the common properties among all these snapshots is that unstable modes exist in the wide areas at large radii (see, e.g., the preshock region with r200 km in the top panels of Figure 2). The occurrence of ELN crossing is also confirmed in the region: ν e is dominant over n ē in the outgoing (cos θ ν =1) direction whereas the trend is opposite in the backward (cos θ ν =−1) direction. This implies that the crossing occurs somewhere in between. Below, we describe the essence of the mechanism (but see Morinaga et al. 2019 for more details).
Accreted matter in pre-shock region is mostly composed by heavy nuclei (see, e.g., the right panel of Figure 11 in Nagakura et al. 2019b). Some neutrinos emitted from PNS experiences scatterings by these nuclei and then turn their directions. Since the outgoing neutrinos are several orders of magnitude more abundant than those in the inward direction, the scattered neutrinos govern the neutrino population in the inward direction (see also Figure 2(b) in Morinaga et al. 2019). Note also that, since the average energy of n ē is higher than that of n n , e ē experiences more scatterings with nuclei than ν e , which makes n ē be more abundant than ν e in the inward direction. Thus, the ELN sign is negative in the inward direction, which is opposite to that in outgoing direction (see, e.g., Tamborra et al. 2017), i.e., the ELN crossing appears.
Noted that the ELN crossing in the pre-shock region has been overlooked so far even in most recent papers (see, e.g., are more abundant than in the pre-shock region. In addition to this, although the ELN property in the inward direction at pre-shock region is crucial ingredient for the fast flavor conversion, it has received little attention. This is because neutrinos have strongly forwardpeaked angular distributions at large radii, which have eluded recognition of the ELN crossing. As pointed out in Morinaga et al. (2019), the ELN crossing is tiny but the growth rate is large enough to induce the flavor conversion, which may give an impact on terrestrial observations of CCSN neutrinos. Note that the neutrino signals in early post-bounce phase will not be affected by the fast flavor conversion, since n ē emissions are much smaller than ν e at this phase. Indeed, we observe the ELN crossings from T b 50 ms in this model. In the post-shock regions, on the other hand, most heavy nuclei are broken up into lighter nuclei or nucleons; thus, the above mechanism does not operate. In fact, n e is dominant over n ē for all flight directions up to T b ∼150 ms, and there is no positive sign of fast flavor conversion (see first and second panels from left on the bottom row in Figure 2). This is qualitatively consistent with our previous paper (Delfan Azari et al. 2019). As shown in other plots on the bottom row in Figure 2, however, unstable modes appear in the northern hemisphere (the same hemisphere with stronger shock expansion) from T b 200 ms and persist throughout the late phase. The role of the asymmetric neutrino emissions in the fast flavor conversion will be discussed in detail in Section 4.2.
In the top row of Figure 3, we show the DR for wave number vectors, which are chosen to be radial, at two unstable locations in the post-shock region. Equation (20) is solved with spherical harmonics up to ℓ=9. As we have already mentioned, the analytic integration with base functions is the key to avoid spurious modes. In the figure, we find some peaks in the DR, which may be a good indicator for the existence of unstable models (see Delfan Azari et al. 2019 for more details). In the bottom panels, we show the growth rates as a function of the wave number. There exist unstable modes, indeed, as indicated by the DR as well as by the approximate prescription of Equation (3).

Role of Asymmetric ν Emissions
Next, we turn our attention to the role of the asymmetric neutrino emission in the occurrence of the fast flavor conversion. Figures 4 and 5 portray the asymmetry in neutrino emissions: the former displays the radial profile of number density of ν e (N νe ) and n ē ( n N ē ) along two selected radial rays (θ=45 • and 135 • ) for two snapshots at T b =100 and 250ms, and the latter displays their ELN asymmetry, which is defined by the ratio of the ELN along each radial ray to the angle average. At T b =100 ms, the radial distributions of N νe and n N ē are roughly spherically symmetric except for the region of 15r30 km, in which violent matter motions produced by convections in the PNS disturb the neutrino distributions. Occasionally, N νe and n N ē become close each other (see, e.g., the black and brown solid lines in Figure 4). However, N νe is roughly one order of magnitude larger than n N ē , and the ELN crossing hardly occurs. 8 The dominance of ν e over n ē can be understood through chemical potential of ν e (μ νe ). 9 At T b =100 ms, μ νe is 8 MeV in the region of 15r30 km (see the black lines in the bottom panel of Figure 6), and ν e is more abundant than n ē . Although the difference of N νe and n N ē becomes smaller with an increasing radius, ν e still dominates over n ē in the above region. In such environments, the ELN crossing does not occur in the post-shock flows, which is qualitatively same results as those found in spherically symmetric CCSN simulations (see, e.g., Tamborra et al. 2017).
At T b =250 ms, the asymmetric neutrino emissions are noticeable: ν e at θ=135°is more abundant than at θ=45°( see the red lines in Figure 4), whereas n ē has an opposite trend (see the blue lines in Figure 4). The characteristics of asymmetric neutrino emissions can also be seen in Figure 5: the ELN distribution along the ray with θ=45°(135°) is  50 % lower (higher) than that of angle average in the region between the PNS and shock radii. 10 This indicates that the number densities of ν e and n ē become close to each other along the radial ray with θ=45°, whereas they are markedly different in the ray with θ=135°. As pointed out by , the ELN crossings potentially occur if the n ē -to-ν e ratio becomes close to unity, i.e., the radial ray with θ=45°is preferable for the ELN crossing.
The ELN crossing occurs at r50 km on this ray, indeed. 11 Figure 7 shows the angular distributions of ν e and n ē along the  . Radial profile of ELN asymmetry, which is defined as the ratio of the ELN along each radial ray (θ=45°and 135°) to their spherical average. According to the definition, the deviation from unity (the thin blue line) corresponds to the degree of asymmetries of ELN distributions. Figure 6. Radial profiles of the electron fraction (top), entropy per baryon (middle), and chemical potential of electron-type neutrinos (bottom). The line type distinguishes the different radial ray with θ=45°(solid) and θ=135°( dashed), respectively. The black and red lines denote T b =100 ms and T b =250 ms. radial ray with θ=45°but at different radii. In these plots, the f ν dependence is integrated out. Note also that we normalize the vertical axis by f max , which is defined as the maximum value of f ν -integrated distribution functions for both ν e and n ē at the same spatial point. As shown in these plots, there are ELN crossings at r=50, 100, and 150 km (but see below for the case with r = 75 km). At r=50 km, n ē is dominant over ν e for the forward direction (cos θ ν >0.9), whereas the trend is opposite in other directions. For the other two radii (r = 100 and 150 km), the ELN crossing occurs in the opposite way to that of r=50 km. In Figure 8, we also display the same quantities but on a different radial ray (θ=135°) for comparison. It clearly shows that ν e always dominates over n ē and, indeed, the neutrino distributions are stable to the fast flavor conversion.
As mentioned above, how the ELN crossing occurs along the radial ray with θ=45°depends on the radius, which indicates that different mechanisms are responsible for the ELN crossings. At r∼50 km, the angular distribution of n ē is more forward-peaked than that of ν e , since n ē decouples from matter at a smaller radius than ν e . Although N νe is larger than n N ē (see the red and blue solid lines in Figure 4), its difference is much smaller than that along the ray with θ=135°. This is mainly due to the fact that ν e is more efficiently absorbed by neutrons and the positron capture is also facilitated in the lower Y e environment along the radial ray with θ=45°(see the top panel of Figure 6). As a result, n ē dominates over ν e in the forward direction alone and then the ELN crossing occurs.
At larger radii (r80 km), N νe and n N ē are gradually deviated from each other with an increasing radius, since n ē is more frequently absorbed or scattered by matter due to its higher average energy than that of ν e . As a consequence, ν e dominates over n ē in the forward direction again. We find, however, that n ē dominates over ν e in the inward directions along the radial ray with θ=45°at r100 km (see the green and purple lines in Figure 7). This is due to the fact that the  positron capture by neutrons is more frequent than the electron capture by protons, which can be seen in Figure 9, and the low-Y e environment is responsible (see the top panel in Figure 6). The negative μ νe (see the solid red line in the bottom panel of Figure 6) is consistent with this interpretation.
The low-Y e matter environment along a radial ray with θ=45°at r100 km is as a consequence of ejections of neutron-rich matter and higher n ē emissions in the same hemisphere. We find that neutron-rich matter at r100 km are dredged up by the neutrino-driven convections and, more interestingly, some of them are ejected, which is supposed to be associated with stronger shock expansion in the same hemisphere. Note that the higher n ē emission than in the opposite hemisphere also provides a preferable condition to create low-Y e ejecta (see also Fujimoto & Nagakura 2019 for more details).
As shown in Figure 7, the occurrence of ELN crossing is determined by the delicate balance of angular distributions between ν e and n ē . In such circumstances, asymmetric neutrino emissions affect the ELN angular distributions not only directly but also indirectly through the change of matter state. It should be noted, however, that they do not always play a positive role for the ELN crossing. Indeed, they tend to make the number densities of ν e and n ē more different in the opposite hemisphere (the same hemisphere with a PNS proper motion), which is negative for the ELN crossing. Figure 8 vindicates this: ν e dominates over n ē in all directions. This can be also seen in the first two panels from right on the top row in Figure 2, in which the stable region is widely spread in the same hemisphere.
We finally discuss the importance of non-axisymmetric properties of the angular distributions of neutrinos in momentum space. We note that there are no ELN crossings in the f ν -integrated angular distribution at r=75 km along a radial way with θ=45°(see the blue lines in Figure 7), although unstable modes exist, as shown in the right panels of Figure 3. The unexpected result comes from the fact that the crossing is simply smeared out by the f ν integration in Figure 7, but it actually exists in the original distribution. In Figure 10, we display the ELN angular distributions with retaining the f ν dependence for two spatial locations (r=50 and 75 km for the left and right panels, respectively) at the same zenith angle of θ=45°at T b =250 ms. The cyan and red colors imply the positive and negative ELN, respectively. As is clear in these plots, the red prolate spheroids in both panels are tilted with respect to the radial direction, which is a noticeable sign of non-axis symmetry. In particular, it is inclined more strongly at r=75 km than at r=50 km, i.e., the non-axisymmetric is more remarkable in the former, and then the non-radial ELN crossing occurs as a consequence.
Such non-radial ELN crossings are not rare, in fact. Figure 11 exhibits the type of ELN crossings by color on the 2D spatial map at T b =250 ms. The regions colored with red and blue denote the radial and non-radial ELN crossings, respectively. As shown in Figure 11, non-radial ELN crossings occur in wide spatial ranges of both pre-and post-shock regions in this model. We speculate that the difference in lateral fluxes between ν e and n ē is responsible for the non-radial ELN crossing. The lateral ν e flux tends to be negative sign, i.e., ν e advects from south to north, which is due to the fact that ν e emissions are stronger in the southern-hemisphere. On the other hand, n ē has an opposite trend. This generates a coherent lateral flux of ELN and then induces the non-radial ELN crossings. Noted, however, that our Boltzmann solver suffers from the numerical diffusion, in particular at larger radii as demonstrated in Richers et al. (2017), which may artificially enhance the region of non-radial ELN crossing. For more quantitative discussions, we need to carry out higher-resolution simulations, which are beyond the scope of this paper.

Summary
In this paper, we conducted the linear stability analysis of the fast flavor collective neutrino oscillations based on a result of our latest axisymmetric CCSN model obtained with the full Boltzmann neutrino transport. In this model, we found remarkable asymmetric neutrino emissions associated with a nonspherical shock expansion and a PNS proper motion (Nagakura et al. 2019c). We reckoned that such coherent asymmetric neutrino emissions have an impact on the ELN crossing, in turn affecting the fast flavor conversion.
We found that unstable modes exist in both pre-and postshock regions in this model. In the former, the ELN crossing is not triggered by multi-dimensional effects but rather by coherent scatterings of neutrinos off heavy nuclei. Thanks to its higher average energy, n ē experiences coherent scatterings of heavy nuclei more frequently than ν e and, as a result, n ē is more abundant than ν e in the inward flight directions. Since ν e is dominant over n ē in the forward directions, the ELN crossing occurs somewhere in between, which will trigger the fast flavor conversion. In the post-shock flows, on the other hand, we found that ν e dominates over n ē in all directions until the initiation of nonspherical shock expansion accompanied by asymmetric neutrino emissions. Thereafter (from T b ∼200 ms on), however, we did find the ELN crossings and hence unstable modes in the hemisphere of higher n ē emissions. The disparity in the number densities between ν e and n ē is reduced by the anticorrelation of their number fluxes in the same hemisphere, and a preferable condition for the ELN crossing is produced. Noted that asymmetric neutrino emissions were not observed in our previous CCSN model (Nagakura et al. 2018), Figure 9. Radial profiles of the ratio of positron capture on free protons to electron captures on free neutrons. We display them along two different radial rays (θ=45°and θ=135°) at two different snapshots (T b =100 ms and T b =250ms). The line type and colors distinguish the radial ray and time, respectively, which are the same as in Figure 6. which would be the main reason why Delfan Azari et al. (2019) found no ELN crossings in the post-shock region.
We then analyzed the ELN crossings in detail by closely inspecting the angular distributions of neutrinos in momentum space. It turns out that these ELN crossings have different origins. In the inner region (see, e.g., r = 50 km in Figure 7), n ē is dominant over ν e in the outward directions, whereas the trend is opposite in the inward directions. This happens because the enhanced emissions of n ē in one hemisphere in this model makes the number densities of ν e and n ē comparable to each other, while the angular distribution of n ē is, in general, more forward-peaked owing to its earlier decoupling with matter. In the outer region (see, e.g., r = 100 and 150 km in Figure 7), on the other hand, the ELN crossing occurs in the opposite way: ν e is dominant over n ē in the outward directions and vice versa in the inward directions. This occurs because the positron capture on free neutrons is more frequent than the electron capture by free protons in low-Y e environments, which results in the negative ELN in the inward direction and then induces the ELN crossing. The low-Y e matter environment is as a consequence of ejections of neutron-rich matter and asymmetric neutrino emissions. The former would be associated with the nonspherical shock expansion and also would be aided by the dredged up neutrino-driven convection at r100 km. On the other hand, the latter in the hemisphere with higher n ē emissions also provides a preferable condition to create low-Y e ejecta, as discussed in Fujimoto & Nagakura (2019). In contrast, the enhancement of higher ν e emissions in the opposite hemisphere suppresses the occurrence of ELN crossing, which implies that the asymmetric neutrino emissions can give rise to stabilize the fast flavor conversion there.
We also find that the non-radial ELN crossing occurs between the regions with no ELN crossing and radial ELN crossing. The non-radial ELN crossing never happens in spherical symmetry, since the axis symmetry is imposed in the neutrino angular distribution in momentum space; hence, the non-radial ELN crossing is a purely multi-dimensional effect. Indeed, the difference of lateral fluxes between ν e and n ē is a primal cause of the non-radial ELN crossing. Whether it really occurs in CCSN core is a subtle problem, however, and the further studies with higher angular resolutions are needed to ascertain it. We will address the issue in the forthcoming paper.
As discussed in Nagakura et al. (2019c), the asymmetric neutrino emissions observed in this model are correlated with the shock morphology and the NS kick. This implies that the occurrence of fast flavor conversion will be also correlated with them and may have strong impacts on observables, such as nucleosynthetic yields and neutrino signals. As for the former, Fujimoto & Nagakura (2019) recently discussed the possible consequences in the explosive nucleosynthesis by asymmetric neutrino emissions. Our findings in this study show a need for the further study of the impact of fast flavor conversions on their outcomes. As for the latter issue, the self-consistent CCSN simulations that take into account the fast flavor conversion somehow are required, which are one of the top priorities in our future project. We also note that the axisymmetric condition, which was imposed on our CCSN model, may artificially enhance the asymmetry of neutrino emissions. This issue will Figure 10. Angular distributions of the ELN in momentum space. They are displayed for two different spatial locations; r=50 km for the left and r=75 km for the right panels, respectively, with the same radial ray (θ=45°) and the same time snapshot (T b =250 ms). The cyan and red colors imply the positive and negative ELN, respectively. The arrow represents the local radial direction (along with z-direction in the panel) from the coordinate origin of momentum space. Figure 11. Color-coded 2D map (r−θ plane) to see the type of ELN crossing. Red and blue colors are radial and non-radial ELN crossing (see the main text for the definition of the radial and non-radial crossing), respectively. The shock radius is marked as a black solid line. The time is T b =250 ms. be addressed once 3D CCSN simulations with full Boltzmann neutrino transport are available.
Finally, we make a few remarks. Our findings in this paper indicate that the enhancement of n ē is a key to the occurrence of fast flavor conversion in the post-shock region, which was also pointed out by previous studies (see, e.g., . Importantly, such asymmetric neutrino emissions may be a common property in CCSNe; for instance, the LESA appears in many 3D CCSN simulations regardless of numerical methods (Tamborra et al. 2014;O'Connor & Couch 2018;Glas et al. 2019;Powell & Müller 2019;Vartanyan et al. 2019a). Moreover the fast flavor conversion is likely to occur commonly in the pre-shock region  unless the number density of ν e is much larger than that of n ē . Other recent works (see, e.g., Sasaki et al. 2019) also found the occurrence of collective neutrino oscillations in CCSNe. In order to treat all these phenomena more rigorously, the quantum kinetic treatment of neutrino transport should be explored further. The entire CCSN community will tackle its intricate problems,including the aspect of technical issues, more considerably in future and will address them toward unveiling the explosion mechanism of CCSNe, although it may be a long way to go.