Fast neutrino flavor conversion: roles of dense matter and spectrum crossing

The flavor conversion of a neutrino usually occurs at densities $\lesssim G_F^{-1} \omega$, whether in the ordinary matter or the neutrino medium, and on time/distance scales of order $\omega^{-1}$, where $G_F$ is the Fermi weak coupling constant and $\omega$ is the vacuum oscillation frequency of the neutrino. In 2005 Sawyer and more recently both he and other groups have shown that neutrino flavor conversions can occur on scales much shorter than $\omega^{-1}$ in a very dense, anisotropic neutrino gas such as that in a core-collapse supernova or a binary neutron star merger. It has also been suggested that the fast neutrino flavor conversion require a crossing in the electron lepton number (ELN) angular distribution or spectrum. We study a few simple analytical models with which we elucidate the roles of the dense matter and the ELN spectrum crossing in neutrino fast flavor conversions. We show that a large matter density can induce fast neutrino flavor conversions in certain astrophysical scenarios such as at the early epoch of a core-collapse supernova.


I. INTRODUCTION
Neutrinos are copiously produced in core-collapse supernovae and binary neutron star mergers and play important roles in these astrophysical events. Because neutrinos can experience flavor transformation or oscillations during propagation (see, e.g., Ref. [1] for a review), it is of great interest to know whether neutrino oscillations can have any signficant impact on the chemical evolution or even the dynamics of these environments.
In vacuum the flavor conversion of a neutrino occurs on a length scale determined by its vacuum oscillation frequency ω = δm 2 /2E, where δm 2 and E are the (effective) mass-squared difference and the energy of the neutrino, respectively. For a 10 MeV neutrino and the atmospheric mass splitting the flavor conversion occurs on the scale of 1 km. A new length scale λ −1 = ( √ 2G F n e ) −1 comes into the play when the neutrino propagates through a dense matter such as inside the sun, where G F is the Fermi weak coupling constant, and n e is the net electron number density. However, the neutrino flavor conversion is suppressed in dense matter until λ ω where the Mikheyev-Smirnov-Wolfenstein (MSW) mechanism can operate [2]. As a result, the length scale of the neutrino flavor conversion through the MSW mechanism is also of order ω −1 .
As early as in 2005 Sawyer has envisioned that flavor conversions can occur on the scale of ( √ 2G F n ν ) −1 inside a core-collapse supernova, where n ν is the num- * abbar@apc.in2p3.fr; Permenant address: Astro-Particule et Cosmologie (APC), Universit Denis Diderot, France † duan@unm.edu ber density of the neutrinos [3,4]. It is expected that this fast flavor conversion or fast (neutrino) oscillations can take place over the distance of 0.1 − 1 cm. However, such fast oscillations were not observed in more sophisticated numerical simulations (see, e.g., Refs. [5][6][7][8][9][10]; see also Ref. [11] for a review). It turns out that all these calculations except those in Refs. [9,10] adopted the neutrino bulb model where the neutrinos of different flavors are emitted isotropically from the same neutrino sphere. Recently Sawyer reported fast flavor conversions in a multi-bulb model where different neutrino flavors have distinct neutrino spheres [12]. This finding seems to have been confirmed by other groups [13,14].
Although a few works have been put out with the aims to understand fast neutrino flavor conversions from the perspective of the dispersion relation [15,16] and in the nonlinear regime [17], the origin of fast oscillations remains largely elusive. In investigating a multi-bulb model with uniform neutrino angular distributions (on the surface of the neutrino spheres), Dasgupta et al observed that "a necessary condition to have fast instabilities appear to be a crossing between the angular spectra of ν e andν e " [14]. However, this finding was not conclusive because no closed-form analytical solution was found. They have also concluded that "background matter suppresses instabilities in space (but not in time)". The main goals of this paper are to understand the origin of fast oscillations and to re-examine the above-mentioned two conclusions about fast oscillations of Ref. [14].
The rest of the paper is organized as follows. After establishing the formalism in Sec. II, we will revisit the bipolar model in Sec. III from the perspective of spectrum crossing. In Sec. IV we will explain the origin of fast oscillations in a four-beam neutrino model through arXiv:1712.07013v2 [hep-ph] 17 Aug 2018 its connection to the bipolar model. We will also give an explicit proof of the necessity of a crossed neutrino angular distribution for fast oscillations to occur in an outward flowing, axially symmetric neutrino flux. In Sec. V we will re-examine several important results about fast neutrino flavor conversions in the literature that are relevant to supernova physics. Finally, we will discuss the implications of our work and conclude in Sec. VI.

II. EQUATION OF MOTION
We consider the mixing between two neutrinos flavors, e and x, where x stands for an appropriate linear combination of the µ and τ flavors. We adopt the neutrinoflavor-isospin convention [18] in which the neutrino and antineutrino of energy E are labeled by their oscillation frequencies respectively, where the plus (minus) sign applies to the neutrino (antineutrino), and a positive (negative) neutrino mass-squared difference δm 2 indicates the normal (inverted) neutrino mass hierarchy. We assume that the physical conditions including the densities of all neutrino species are essentially constant and homogeneous on the time and distance scales of interest. Following Refs. [15,19] we define the electron lepton number (ELN) distribution or spectrum as where v and p are the velocity and momentum of the neutrino, respectively, and f να/να (α = e, x) are the occupation numbers of the corresponding neutrino species.
Here we have assumed that neutrinos are relativistic so that |v| = 1 and |p| = E. There exist multiple conventions in the literature for the normalization of the ELN spectrum. Here we choose to normalize it by the ν e density such that where n νe (n νx ) is the total number density of ν e (ν x ), and NH and IH stand for the normal and inverted neutrino mass hierarchies, respectively. Note that, according to Eq. (1), the neutrino has a positive vacuum oscillation frequency ω if δm 2 > 0 and negative ω if δm 2 < 0. In the rest of the paper we will sometimes simply use the words "spectrum" and "distribution" to refer to the ELN energy or angular distributions.
We will examine the situation where neutrinos are almost in pure weak interaction states so that the (reduced, traceless) neutrino density matrix (in the weak interaction basis) is of the form where | | 1. We will look for the criteria for the onset of collective neutrino oscillations by performing the linear flavor-stability analysis in which the terms of order 2 or higher are ignored [20].
In the absence of collisions neutrinos obey the equation of motion (EoM) [21] where H 0 is the conventional neutrino oscillation Hamiltonian in the absence of ambient neutrinos, and H νν is the potential due to neutrino-neutrino forward scattering. We assume a small neutrino mixing angle so that where λ = √ 2G F n e is (a measure of the strength of) the matter potential, and σ 3 = diag[1, −1] is the third Pauli matrix. For the situation that we are interested in, where µ = √ 2G F n νe is (a measure of the strength of) the neutrino potential. For simplicity we sometimes suppress the arguments or subscripts of certain physical quantities, and the primed symbols such as g = g(ω , v ) are understood as the corresponding quantities that depend on the integration variables ω and v . As usual we have ignored the trace terms which do not affect neutrino oscillations.

III. FLAVOR INSTABILITIES IN HOMOGENEOUS AND ISOTROPIC NEUTRINO GASES
It will prove helpful to first consider the relation between spectrum crossing and the flavor instabilities in a homogeneous and isotropic neutrino gas. The importance of spectrum crossing was first pointed out by Dasgupta et al in a report of multiple spectral splits resulted from collective neutrino oscillations [19]. An explicit proof of this relation was later given by Banerjee et al in an analysis on the flavor instabilities of dense neutrino gases [20]. As a useful example we will inspect the flavor instabilities in the bipolar model from the perspective of spectrum crossing. For completeness we will also include a brief proof of this relation for a neutrino gas with a continuous energy spectrum as that in Ref. [20] which will be used later in the paper.

A. Bipolar model
For a homogeneous and isotropic neutrino medium one has and The bipolar model describes a homogeneous and isotropic neutrino gas initially consisting of mono-energetic ν e and ν e (see, e.g., Refs [22][23][24]). It has a discrete energy spectrum where g 1 = 1 and ω 1 = δm 2 /2E 1 for the neutrino with energy E 1 , and g 2 = −nν e /n νe and ω 2 = −ω 1 for the antineutrino. Keeping only the linear terms in the EoM one has In the linear regime collective neutrino oscillations are represented by the normal modes where Q i (i = 1, 2) are two complex constants, and Ω is the collective oscillation frequency which can be solved from the characteristic equation The bipolar model becomes flavor unstable if there exists a solution with From Eq. (13) it is straightforward to show that a flavor instability can exist only if or, equivalently, A few remarks are in order.
• Because inequalities (15) do not depend on λ, the presence of a uniform matter density does not affect flavor instability. In fact, it is obvious from Eq. (11) that the presence of the ambient matter only shifts the collective oscillation frequency Ω by −λ [18].
• Eq. (11) implies that, if the oscillation frequencies of both neutrino modes are shifted by a common value ∆Ω, then the collective oscillation frequency Ω is also shifted by ∆Ω which again does not affect the flavor instability [18].
• Eq. (15a) implies that a neutrino flavor instability exists only if the ELNs of the two neutrino modes, g 1 and g 2 , have opposite signs, i.e., g 1 g 2 < 0. (See Fig. 1.) • Eq. (15b) gives another necessary condition of flavor instability, (ω 1 − ω 2 )(g 1 − g 2 ) < 0. For the bipolar system that we considers, this is possible only if ω 1 < 0 which requires δm 2 < 0, i.e., the neutrino mass hierarchy is inverted. If the neutrino has the inverted mass hierarchy, it is straightforward to show from Eq. (15) that the bipolar model is unstable for µ ∈ (µ − , µ + ), where the lower and upper boundaries of the instability region are respectively. Unless |g 1 | and |g 2 | are almost the same, µ + and µ − are of the same magnitude as the spread of the 1 In Ref. [19] and some other references, ω is defined to be always positive for the neutrino. In that convention, an energy spectrum with a negative (positive) crossing can be unstable if the neutrino has a normal (inverted) neutrino mass hierarchy. In our convention, however, g(ω) → g(−ω) as the neutrino mass hierarchy is flipped, and a negative crossing in the energy spectrum becomes positive and vice versa. The flavor stabilities will change accordingly. Either way, a neutrino system with a single crossed energy spectrum can be unstable with one neutrino mass hierarchy but stable with the other. oscillation frequencies ∆ω = |ω 1 − ω 2 |. By the dimension argument one expects the maximum value of κ to be of the same magnitude as ∆ω, which in turn is usually of the same order as |ω 1 |. Therefore, like the MSW mechanism, the flavor conversion in the bipolar model, and also in homogeneous and isotropic neutrino media in general, usually occurs at neutrino densities G −1 F ω and on time/distance scales of order ω −1 with ω being a characteristic vacuum oscillation frequency of the neutrino gas.
B. Neutrino gas of a continuous spectrum 2 The linearized EoM of a homogeneous and isotropic neutrino gas with a continuous energy spectrum is Assuming solution ω (t) = Q ω e iΩt to Eq. (17) one obtains which implies Again one sees that,λ, a combined potential due to the matter and neutrino background, has the effect of shifting the collective oscillation frequency Ω but does not affect the flavor instability. For the purpose of analyzing flavor instabilities we will assumeλ = 0 in this section.
where γ = Re(Ω). Suppose that there exists a flavor instability for µ within the range of (µ − , µ + ). In the limit µ → µ − + 0 + , κ → 0 + , and Eq. (21) becomes 2 The proof for the necessity of a crossed spectrum for neutrino flavor instability was first given by Banerjee et al in Ref. [20]. The content of this section is included for completeness.
where P signifies the Cauchy principal value of the integral, and ω c = Ω| µ=µ− . 3 Because µ is real, Eq. (22) implies that the spectrum g(ω) must cross 0 at ω = ω c . In addition, unless there are other crossing points nearby, the integral in Eq. (22) has the same sign as the crossing of g(ω) at ω c . This in turn implies that the spectrum crossing at ω c must be negative because we have defined µ to be positive. This result can be considered as a generalization of that in the bipolar model. (See Fig. 1.)

IV. FAST FLAVOR CONVERSIONS IN ANISOTROPIC NEUTRINO MEDIA
We will now turn to the relation between fast flavor conversions and the crossings in the neutrino angular distribution. The first, albeit inconclusive, observation of this relation was made by Dasgupta et al in a study of fast neutrino oscillations in a multi-bulb supernova model [14]. Here we will consider stationary, anisotropic neutrino media which consist of neutrino fluxes emitted constantly from planar surfaces. We assume that the neutrino emission is uniform on the emission plane and that the matter distribution is homogeneous in the whole space. We will first use the four-beam neutrino model to demonstrate the origin of fast flavor conversions by relating it to the bipolar model. We will then give an explicit proof of the necessity of a crossed neutrino angular distribution for fast oscillations to occur in an outward flowing, axially symmetric neutrino flux. The four-beam neutrino model has the ELN distribution where v i and ω i (i = 1, 2, 3, 4) are the velocities and oscillation frequencies of the four neutrino modes of the system. (See Fig. 2). We will assume the reflection symmetry between neutrino modes (1, 2) and (4,3) in the four-beam model except for small differences among i , i.e., ( 1 , 2 ) = ( 4 , 3 ). We define where the z direction is perpendicular to the neutrino emission plane. Because of the reflection symmetry in the EoM, the symmetric modes ( i+ ) and the anti-symmetric modes ( i− ) evolve independently in the linear regime.

Symmetric mode
The symmetric mode of the four-beam model obeys the EoM in the linear regime, wherẽ with v z being the z component of v, and µ ij = (1 − v i · v j )µ. Here we have rewritten the EoM of the symmetric mode in a form similar to that of the bipolar model, Eq. (11), so that the results of the bipolar model can be applied here. Eq. (26) shows that the presence of the ordinary matter can contribute to the spread of effective oscillation frequencies ∆ω = |ω 1+ −ω 2+ |, which is generally true for a system with an anisotropic neutrino emission. This multi-angle matter effect is responsible for the suppression of neutrino oscillations in the accretion phase of a core-collapse supernova if all neutrino species are emitted from the same neutrino sphere and with the same angular distribution [25].
Here we want to emphasize that the same multi-angle matter effect can also induce fast flavor conversions. To see this we take the limit δm 2 → 0 so that The discussion in Sec. III A implies that there can exist flavor instability even in this limit. Assuming v 1z < v 2z , the requirement of a negatively crossed effective spectrum g + (ω + ) means that g 1 > 0 > g 2 , e.g., beams (1,4) are emitted in the ν e state and beams (2,3) in theν e state. In other words, the existence of a flavor instability in the symmetric mode requires that there exist a negative crossing in the angular distribution When this is the case the neutrino medium becomes unstable against the flavor conversion when µ is comparable to λ. [See Eqs. (16) and (27).] The exponential growth rate κ of the flavor instability is also of the same order of λ which can be much larger than vacuum oscillation frequencies (taken to be zero in this limit).
As an example we consider the case where beams (1, 4) are emitted in the ν e state with v 1z = 1/2 and g 1 = 1 and beams (2,3) in theν e state with various values of v 2z and g 2 . In the two top panels of Fig. 3 we plot the exponential growth rate κ of the flavor instability as a function of v 2z and g 2 with fixed ratios λ/µ = 1 and 3, respectively. We only plotted the results with g 2 < 0 where there exits a crossing in the effective spectrumg + (ω + ). From the figure one sees that the symmetric fast flavor conversions are possible in the four-beam model only if theν e beams are emitted in a more forward direction than the ν e beams so thatg + (ω + ) has a negative crossing.

Anti-symmetric mode
The EoM for the anti-symmetric mode is the same as Eq. (25) except that the "plus" quantities are replaced by their "minus" counterparts: Without losing generality we will assume the configuration with v 1 ·v 2 > v 1 ·v 3 so thatμ − > 0. The comparison between Eq. (29) and Eq. (26) reveals two important features of the anti-symmetric collective mode in the fourbeam model: • Both the ordinary matter and the background neutrino medium contribute to the spread of effective oscillation frequencies ∆ω − = |ω 1− −ω 2− |. This multi-angle effect due to the neutrino medium itself can suppress collective neutrino oscillations in the supernova model in which all neutrino species have the same angular distribution [8].
• The effective spectrumg − (ω − ) has a sign opposite to that of the actual ELN distribution. This implies that a flavor instability may exist when there is a positive crossing in the angular distribution G(v z ).
As in the case of the symmetric mode, there can exist a flavor instability for the anti-symmetric mode in the limit δm 2 → 0. A crossing in the effective ELN spectrum g − (ω − ) (i.e.g 1−g2− < 0) also implies a crossing in the actual angular distribution G(v z ) (g 1 g 2 < 0). Unlike the symmetric mode, the anti-symmetric mode can have fast flavor conversions even when λ = 0. In addition, the multi-angle neutrino effect can change the the order of ω i− , i.e. whetherω 1− <ω 2− orω 1− >ω 2− . As a result, the regimes where the effective spectrumg − (ω − ) has a negative crossing are not solely determined by v 2z as in the case of the symmetric mode. In the lower panels of Fig. 3 we plot the exponential growth rate κ of this instability as a function of v 2z and g 2 with fixed ratios λ/µ = 0 and 3, respectively.

B. Outward flowing, axially symmetric neutrino flux
We next consider an outward flowing, nearly axially symmetric neutrino flux with and small perturbations to the initial states of the neutrinos which can depend the azimuthal angle ϕ of v. We will again consider flavor instabilities in the limit δm 2 → 0. The linearized EoM of this model is where Following Ref. [20] we assume a solution of the form to Eq. (31) and obtain whereλ = λ+Φ 0 . Comparing Eqs. (19) and (34) one sees that Φ z shifts the collective oscillation frequency Ω while −λ/v z plays the role of vacuum oscillation frequency ω.
Eq. (34) implies where and where Eq. (37) has a solution only if which corresponds to the collective modes that preserve and break the axial symmetry, respectively. 4 4 The derivation of the characteristic equation (39) for this model is similar to that in Refs. [13,14] and is included here for completeness.
Let us consider the symmetry-breaking solution first. We rewrite Eq. (39b) as and (ω min ,ω max ) = (−λ, ∞) ifλ < 0 and (−∞, −λ) if λ > 0. Using the result of the homogeneous, isotropic medium (in Sec. III B), we determine that fast flavor conversions that break the axial symmetry can occur in this medium only if there exists a positive crossing in the angular distribution G(v z ) within [0, 1]. One can also perform a similar analysis to Eq. (39a) and show that fast flavor conversions that preserve the axial symmetry can take place if there exists a crossing in G(v z ), although the sign of the crossing is not constrained. (See Appendix A for details.)

V. MULTI-BULB SUPERNOVA MODEL
Now we consider the simple supernova model proposed by Sawyer in Ref. [12] in which ν e ,ν e , ν x are emitted constantly and half-isotropically from three spherically symmetric neutrino spheres or "bulbs" of radii R νe , Rν e and R νx , respectively, andν x has the identical emission characteristics as ν x . This multi-bulb model is a generalization of the neutrino (single-)bulb model in Ref. [5] where R νe = Rν e = R νx . Because the typical length scale of the fast flavor conversion is much smaller than R ν (ν = ν e ,ν e , ν x ), we can ignore the spherical nature of the model and apply the results of Sec. IV B.
In the multi-bulb model the number flux of the neutrino species at radius r takes the form where ϑ is the polar angle of the flux with respect to the radial direction, and n ν,0 is the number density of the neutrino species ν on the corresponding neutrino sphere. The total number luminosity of neutrino species ν iṡ For R νe > Rν e > R νx andṄ νx :Ṅν e :Ṅ νe = 0.62 : 0.77 : 1 Sawyer found that there could exist a flavor instability even on the ν e sphere if Rν e R νe > 0.44 + 0.55 For R νe = Rν e /0.93 = R νx /0.8 = 15 km and nν e ,0 = 10 32 cm −3 he calculated the instability growth rate to be κ = 0.32 m −1 on the surface of the ν e sphere. However, since j νx = jν x in the multi-bulb model and, therefore, the contribution of ν x andν x to the angular distribution G(v z ) cancel each other, the criterion for "pure" fast neutrino flavor conversions (with δm 2 → 0) should not depend on R νx or n νx,0 , which is contrary to the condition set forth in Eq. (44). (For this reason the multi-bulb model was called the two-bulb model by Chakraborty et al in Ref. [13].) Further, it was observed by Dasgupta et al [14] that, in order for fast flavor conversions to take place in the multi-bulb model, there should exist at least one crossing in the angular distribution. This requirement has been explicitly shown in Sec. IV B, and implies that (n νe,0 − nν e,0 )(R νe − Rν e ) < 0.
The example given above by Sawyer in Ref. [12] does not satisfy this criterion. He also did not include the matter effect which can significantly affect the fast neutrino flavor conversion. We have performed a linear flavor stability analysis to the multi-bulb model, and we did not find any flavor instability with the parameters given in Ref. [12]. In Ref. [13] Chakraborty et al applied the linear flavor instability analysis to the multi-bulb model but only in the far regime where r R ν . In this regime where u = sin 2 ϑ| r=R , and n νe and nν e are parameterized as with the upper and lower signs applying to ν e andν e , respectively. In the above parameterization, the asymmetry parameter a and the width parameter b are both within the range of (−1, 1) and determineṄ νe /Ṅν e and R νe /Rν e , respectively, and R is the radius at which the above parameterization is valid. In this parameterization, the angular distribution has a crossing only if Indeed, Figs. 5 and 6 of Ref. [13] show that the flavor instabilities exist only in the parameter space where the above criterion is satisfied. Those figures also demonstrate that a large matter density (i.e. λ µ) can dramatically affect the neutrino flavor instabilities in the multi-bulb model.
As another example of the multi-bulb model, we calculated the neutrino flavor instability region in a onedimensional, electron-capture supernova model with an 8.8M progenitor which was computed by the Garching group [26]. At t ≈ 200 ms after the core bounce this model hasṄ νe = 9.80 × 10 56 s −1 ,Ṅν e = 8.36 × 10 56 s −1 , N νx = 6.92×10 56 s −1 , R νe = 40.2 km, Rν e = 36.6 km and R νx = 36.1 km, respectively. It is easy to check that the FIG. 4. The flavor instability region of the neutrino gas (in the light shadow) with various matter densities (proportional to the matter potential λ) in a one-dimensional, electron-capture supernova model at t ≈ 200 ms after the core bounce [26]. The thick solid curve is the actual matter profile as a function of radius r. The hatched region is where the "pure" fast flavor conversions (with the neutrino mass-squared splitting δm 2 → 0) become possible, and the dark shadowed region is where collective neutrino oscillations can occur if the neutrino spheres of νx andνe coincide with that of νe.
neutrino gas in this supernova model has a crossed angular distribution. Using the results in Sec. IV B we plot the flavor unstable region in the supernova as the light shadowed area in Fig. 4 for various matter densities. To simplify the calculation, we have assumed the two-flavor mixing with the inverted neutrino mass hierarchy and a single vacuum oscillation frequency ω 0 = 0.68 km −1 for all neutrino species. This simplification should not have a dramatic impact on the flavor unstable region in Fig. 4 which is a log-log plot. We depict the flavor unstable region of the pure fast modes with δm 2 → 0 as the hatched area in the same figure. As comparison we also plot the flavor unstable region as dark shadowed region for the case where the neutrino spheres of all other neutrino species coincide with that of ν e . This corresponds to the single-bulb model where only "slow" neutrino oscillations can exist since there is no crossing in the angular distribution. Finally we plot the actual matter profile as the thick solid curve in the figure. Fig. 4 clearly demonstrates the important influence of the ambient matter on neutrino oscillations in a supernova. In the absence of a dense matter background, collective neutrino oscillations are self-suppressed near the neutrino spheres where the neutrino density is large [8].
In the single-bulb model where all neutrino species have an identical angular emission, collective neutrino oscillations can take place but only in the parameter space where the matter potential is finely matched with the neutrino potential which is not the case in our example. In contrast, in the multi-bulb model which has a crossed neutrino angular distribution, fast flavor conversions can occur near the neutrino spheres for a broad range of matter densities spanning over two orders of magnitude. Since the actual matter profile falls into the regime of the flavor instability in this particular example, we expect that the fast flavor conversion can develop there.

VI. DISCUSSION AND CONCLUSIONS
We have used the four-beam neutrino model to explain the origin of fast neutrino oscillations in a dense, anistropic neutrino gas even in the limit that the neutrino mass splitting δm 2 → 0, although a finite flavor mixing is necessary to initiate such a flavor conversion. It is the multi-angle effect of the neutrinos as they travel through a dense matter and neutrino background that induce the fast flavor conversion. This flavor conversion can take place inside a very dense medium over very short distance scales of order (G F n e ) −1 or (G F n ν ) −1 .
We have shown that the fast flavor conversion is generally prohibited in a stationary, outward flowing, axially symmetric neutrino flux unless there is a crossing in the neutrino angular distribution. This provides an explicit proof for a similar conjecture by Dasgupta et al in Ref. [14] for the multi-bulb supernova model. However, this conclusion does not apply to a medium where both inward and outward flowing neutrino fluxes are present. This can be easily seen from the four-beam model when the directions of two of the neutrino beams are flipped, e.g., v 2z → −v 2z and v 3z → −v 3z . This would result a sign change in the effective spectrum [see Eqs. (26) and (29)]. Therefore, even if there is no crossing in the actual angular distribution G(v z ), there can still be a crossing in the effective spectrumg(ω) because of this sign change in the part of the effective spectrum for the inward neutrino fluxes. In other words, fast flavor conversions can exist in a stationary neutrino gas with both inward and outward neutrino fluxes, e.g. inside the ν e sphere of a core-collapse supernova, even if there is no crossing in the neutrino angular distribution. We note that the convective operator in the EoM (5) becomes for a collective oscillation wave of collective frequency Ω and wave vector K [15,27]. It is the different signs in v · K of the inward and outward neutrino fluxes that produce the crossing in the effective spectrumg(ω) in a stationary neutrino gas when there is no crossing in the actual neutrino angular distribution. Obviously, this argument does not apply to the temporal flavor instability (i.e. with Im(Ω) = 0) which still requires a crossed neutrino angular distribution [15].
We have applied the linear flavor stability analysis to a multi-bulb supernova model. As envisioned by Sawyer fast flavor conversions can take place in this model because different neutrino species have different neutrino spheres. However, it is somewhat surprising that it is because of the presence of a large matter density that this flavor conversion is possible. It has been pointed out that the matter effect can be (partially) "cancelled" in a non-stationary neutrino gas [28,29]. However, a significant cancellation of the matter effect occurs only for rapidly pulsating neutrino oscillation modes of frequency ∼ λ which is extremely high near or inside the neutrino sphere. The matter effect discussed in this work is still relevant to most other neutrino oscillation modes which are probably more important in the real supernova environment.
Of course, our result about supernova neutrinos is only suggestive because of the various assumptions of the multi-bulb model. A real supernova is dynamic and does not have clear-cut neutrino spheres. There also exist neutrinos propagating inward which can affect neutrino flavor conversions [15,30]. A more definite conclusion requires simulations of neutrino flavor oscillations in time-dependent, multi-dimensional supernova models with accurate neutrino angular distributions.
At this limit, the imaginary part of Eq. (39a) can be written as which is possible if G(v c ) = 0.
If G(v z ) has only one simple crossing point (at v c ) within the range of [0, 1], then function has no crossing within the same range. In this case, we can define as an average of f (v z ) within [0, 1], and the real part of Eq. (39a) implies that The two solutions in the above equation require opposite crossing directions of G(v z ) at v c . Therefore, a condition for the axially symmetric flavor oscillations to occur in an outward flowing neutrino flux is that its angular distribution G(v z ) has a crossing point within the range of [0, 1], but the sign of this crossing is not constrained.