A new possibility of the fast neutrino-flavor conversion in the pre-shock region of core-collapse supernova

We make a strong case that the fast neutrino-flavor conversion, one of the collective flavor oscillation modes, commonly occurs in core-collapse supernovae (CCSNe). It is confirmed in the numerical data obtained in realistic simulations of CCSNe but the argument is much more generic and applicable universally: the coherent neutrino-nucleus scattering makes the electron lepton number (ELN) change signs at some inward direction and trigger the flavor conversion in the outward direction in the pre-shock region. Although the ELN crossing is tiny and that is why it has eluded recognition so far, it is still large enough to induce the flavor conversion. Our findings will have an important observational consequences for CCSNe neutrinos.

Introduction.-Neutrinos will give us vital clues not only to the explosion mechanism of core-collapse supernovae (CCSNe) but also to their flavor structures.In fact, prediction of the luminosities and energy spectra for all neutrino species requires taking into account neutrino oscillations appropriately.This is more difficult than previously thought, however, if collective neutrino oscillations occur , since they are nonlinear phenomena described with integro-partial differential equations.No consensus has been reached thus far on whether, when and how the collective oscillation occurs in CCSNe.In this Letter we make a strong case that the fast neutrinoflavor conversion, one of the collective neutrino oscillation modes, should commonly occurs in the post-bounce phase of CCSNe.
The fast flavor conversion has been extensively studied in the literature [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36].A convenient criterion for its occurrence is supposed to be the ELN crossing, i.e., ν e is dominant over νe in some propagation directions whereas νe overwhelms ν e in the other directions.Tamborra et al. [18] searched for such ELN crossings in the numerical data of CCSNe simulations under the assumption of spherical symmetry.Paying attention mainly to outward-going neutrinos, they reported negative results.More recently, such investigations are extended to the results of multi-dimensional simulations [30,34].Abbar et al. [30] found the ELN crossing in some extended domains in the post-shock region.On the other hand, Delfan Azari et al. [34] reported no detection of ELN crossing based on a 2D CCSN model in Nagakura et al. [37].We stress that these results depend strongly on multi-dimensional effects and may change from model to model.
In this Letter, we discuss a new possibility of the fast flavor conversion, based on a more robust argument.We focus on the pre-shock region.This is the region ahead of the shock wave, in which cold matter mainly composed of heavy nuclei is imploding toward the shock.We argue that the ELN crossing is produced rather commonly by the coherent scattering of neutrinos on these heavy nuclei, with νe being scattered more often than ν e , which sets the stage for the fast flavor conversion.
Capozzi et al. [31] pointed out recently that collisional processes are important to generate the fast flavor conversion.What they have in mind in their paper, however, is completely different from what we consider in this Letter.They studied scatterings that occur in the vicinity of the neutrinosphere whereas we investigate the region at much larger radii; the scattering processes are also different.Cherry et al. [8,12], Cirigliano et al. [19] and Zaizen et al. [21] also explored the possible effect of scattering of neutrinos on nucleons in the post-shock region, the socalled neutrino halo.Time-independence and spherical symmetry they imposed, however, obscured the role of the fast flavor conversion unfortunately.
As we shall see below, our argument is quite simple and robust: the existence of ELN crossing is demonstrated analytically; it is then vindicated by more realistic CCSN simulations.Note that our findings have been overlooked so far probably because the ELN crossing is tiny.However, such a tiny crossing is actually large enough for the fast flavor conversion to grow substantially.It is also intriguing that the flavor conversion always propagates outward, which will hence have an impact on the terrestrial observation of supernova neutrinos.Backward scattering on heavy nuclei.-Nowthe main claim of this paper: coherent scatterings of neutrinos on heavy nuclei produce the ELN crossing in the pre-shock region which is tiny but still sufficient to induce the fast flavor conversion.Interestingly, the conversion propagates outward as convective instability [26,33,38,39].We will substantiate this contention shortly.
The shock wave generated at core bounce is stalled in the core and becomes an accretion shock at r ∼ 200 km.Matter outside this stagnant shock is cold and hence mainly composed of heavy nuclei and is falling almost freely onto the shock front.Neutrinos emitted from the neutrinosphere located much deeper inside (r 50 km) are moving outward almost freely outside the shock, since the matter density is low there.A small fraction of these neutrinos are back-scattered by nuclei, however, and produce the inward-going population.Since νe has higher energies than ν e on average while the luminosities are similar between them, the inward-going population is dominated by νe .
This can be demonstrated more quantitatively with the so-called bulb model, in which neutrinos are emitted from the neutrino surface half-isotropically.For concreteness, we assume that the energy spectra of neutrinos are expressed as f ν (E) ∝ E αν e −(3+αν )E/ Ēν [9,[40][41][42].Hereafter the index ν represents ν e or νe .Without interactions with matter, all neutrinos are going outward, being confined in a cone.Their angular distributions are given as [42] where R ν is the radius of the neutrinosphere, L ν and Ēν are the luminosity and average energy of neutrino, respectively, µ is cosine of the zenith angle measured from the local radial direction and Θ is the step function.The ELN angular distribution is given by G νe − G νe .It is normally found that the ELN is positive and its intensity is of the order of 10 −1 cm −1 in the outward direction (µ ∼ 1).
The population of inward-going neutrinos (µ ∼ −1) can be estimated from this outward-going population and the matter distribution as follows.The density profile outside the shock front is approximately expressed as ρ(r) ∝ r −β as a function of the radial position r.The rate of coherent scattering is estimated with the formula given in Bruenn [43] together with the assumption A const., Z−N A 0 for the average mass (A), proton (Z) and neutron (N ) numbers of nuclei.Then the angular distribution of ν is derived by line integrations as [42] up to the lowest order of (µ + 1) and (R ν /r), where R sh is the shock radius and ρ sh is the matter density just outside the shock front.The leading angular dependence reflects the fact that the coherent scattering is strongly forward-peaked, ∝ (1 + cos θ), where θ is the scattering angle [43].In the limit of r → ∞, the outward-going neutrinos become all radially-going actually and there is no neutrino going radially-inward.At finite radii, however, there remains a small finite contribution, giving the second term in the last factor.Note that the difference in R ν between ν e and νe is included only in this term.As a result, the ELN (G scat νe − G scat νe ) becomes negative as long as L ν Ēν is larger for νe than for ν e at angles that satisfy 1 (1 + µ) (R ν /r) 2 /4.The absolute value of ELN is estimated typically to be 10 −6 cm −1 , which will be also vindicated later by realistic simulations.
The different signs of ELN for the outward and inward directions imply that there occurs an ELN crossing in between.The growth rate of the fast flavor conversion is roughly given by the geometric mean of the ELN intensities at their positive and negative parts (see below) [44].It is estimated to be 10 −4 cm −1 = 1/(100 m), which is large enough for the fast flavor conversion to develope sufficiently in the time scale of CCSNe.Growth rates of flavor conversion.-Beforemoving to the realistic numerical models, we give here some mathematical formulae that will be employed there for quantitative analyses.The initial phase of the collective neutrino flavor conversion can be studied by the linear stability analysis [5,25,29].Flavor evolutions are described by the kinetic equations for the density matrices of neutrinos f: where x ≡ (t, x) denotes the position in spacetime, Γ ≡ (E, v) the energy (E > 0 for neutrino and E < 0 for antineutrino) and flight direction and is the masssquared matrix and Λ is the 4-current of leptons defined as In the region of our current concern, H vac is smaller than H int and is dropped in the following analysis.This implies that only the fast flavor conversion is considered.The vacuum-mass term, H vac , plays the role of an instigator of the flavor conversion in this context, generating initial perturbations.If the maximum wave number of vacuum oscillation, [45], becomes comparable to the growth rate σ (see below) of the fast flavor conversion, however, H vac should be reinstated and the slow mode needs to be also considered [29].The collision term C[f ] is also neglected, since it is important not in the flavor conversion itself but in setting the background for it [31].
We work in the framework of 2-flavor mixing.Then a small perturbation around the flavor eigenstate is expressed as where f c (Γ) ≡ f νe (Γ) − f νx (Γ) and the small off-diagonal component is denoted by S.

Defining further the energy
to the linear order of S .Note that the variation of Λ is neglected, since we consider a patch of space much smaller than the background scale height and a period of time much shorter than the typical hydrodynamical time scale.For the plane wave ansatz S (x, v) ≡ Q(v)e ik•x , a nontrivial solution of Eq. ( 5) exists iff is satisfied for the polarization tensor given as The fast flavor conversion, which is regarded here as instability of the flavor eigenstate, occurs when the solution of Eq. ( 6): k 0 = ω(k) has a positive imaginary part for some k ∈ R 3 .We normally need to solve Eq. ( 6) numerically, not an easy task [46].Realistic models.-Belowwe vindicate the above argument given for the bulb model by quantitatively analyzing the data obtained in our CCSN simulations with the full Boltzmann neutrino transport.Importantly, the ELN crossings in the pre-shock region are confirmed in many of our models [47,48] and also in those of Garching group, which are publicly available [49].Tamborra et al. [18] reported that there was no ELN crossing in the latter models, which is not true, however.In the following analysis, we employ a numerical data of a spherically symmetric 11.2M CCSN model [50] as a representative case.Figure 1 portrays the radial profiles of neutrino number densities and baryonic mass density as well as the approximate estimate of the growth rate of the fast flavor conversion given by the following formula: which is not bad indeed as confirmed later by linear analysis.
As shown in Fig. 1, the fast flavor conversion occurs at the pre-shock region and its growth rate is ∼ 10 −4 cm −1 .It should be stressed that the result is not an artifact by numerical diffusions in our CCSN simulations; indeed, the same simulation but with much higher resolutions yields essentially the same results (the gray dashed line in the same figure).On the other hand, the fast flavor conversion is suppressed in the post-shock region.It is attributed to the fact that almost all heavy nuclei are photo-dissociated in the post-shock flows, which substantially reduces scattering opacities.In addition, the isotropic emission of ν e via the electron capture by free protons is enhanced by shock heating and becomes the dominant weak-process for inward-going neutrinos behind the shock wave [51].As a result, the tiny ELN crossing that could be induced by the scattering is washed out and ν e dominates over νe in all directions.We turn our attention to the detailed characteristics of the neutrino distributions in momentum space.For outward-going neutrinos, the average energy, which roughly corresponds to the energy at the peak of the number spectrum, is higher for νe than ν e , whereas the height of the peak of the spectrum is higher for ν e than νe (see solid lines in Fig. 2(a)); as a result, the number density of ν e is slightly larger than that of νe , i.e., the ELN is positive (see also the solid lines at µ > 0 in Fig. 2(b)).For inward-going neutrinos, on the other hand, both the height of the peak of the spectrum and the average energy are higher for νe than ν e and hence νe is more abundant than ν e (see dashed lines in Fig. 2(a)), i.e., the ELN is negative.This indicates that the ELN crossing occurs, which is exactly what we predicted from our toy model.Indeed, it is confirmed that the neutrino angular distributions intersect at µ ∼ 0.2 as shown in Fig. 2

(b).
To see more clearly the role of the scattering by heavy nuclei, we perform an additional simulation, in which we turn it off.The angular distributions of neutrinos obtained in this simulation are displayed as dashed lines in Fig. 2(b).It is apparent that the outward-going neutrinos are almost intact whereas the inward-going neutrinos are strongly affected, in which neutrinos are much less abundant and, more importantly, the ELN crossing disappears.We can hence conclude that the coherent scattering by heavy nuclei plays a crucial role in generating the ELN crossing.Fig. 2(c) displays the dispersion relation (DR) at r = 241 km for k parallel to the radial direction, which gives the growth rate of the fast flavor conversion more precisely than Eq.(8).Note that the maximum growth rate derived from DR is ∼ 10 −4 cm −1 , which is roughly the same value estimated by Eq. (8) (2.94 × 10 −4 cm −1 ).More interestingly, the group velocity of these unstable modes (v g = d Re ω/dk) is ∼ 0.7c and always positive, which implies that the flavor conversion proceeds in the outward direction.8) (solid lines) and the ratio of nν e to nν e , α (dashed lines) as functions of radius at some different times: 10 ms (red), 30 ms (orange), 50 ms (lime), 100 ms (green), 200 ms (blue) and 400 ms (purple).The thin vertical lines indicate the shock positions at the same times.The lack of red and orange solid lines in this figure means no fast flavor conversion at the corresponding times.

Growth Rate [cm
Fig. 3 shows the growth rates of the fast flavor conversion as a function of radius at different times.One can see that the conversion is suppressed in the early post-bounce phase (up to ∼ 30 ms after bounce in this CCSN model).This is simply because νe emissions are suppressed at early times.Once νe is produced substantially (at ∼ 50 ms), it is confirmed that the ELN crossing occurs in the pre-shock region and is sustained for the rest of the post-bounce phase.

Conclusion.-In this Letter we have presented a new possibility of the fast neutrino-flavor conversion in CCSNe.
We have argued that it should be ubiquitous in the preshock region in the post bounce phase except for the very early period ( 30 ms after bounce).The key ingredient is the coherent neutrino-nucleus scattering.We have demonstrated both analytically and numerically that the scattering induces the ELN crossing and then triggers the fast flavor conversion.We also found that the group velocities of unstable modes are always positive irrespective of their phase velocities, i.e., the fast flavor conversion should have an influence on the terrestrial observation of supernova neutrinos.

Neutrinosphere
Shock front Neutrinosphere Shock front O a e A c 1 7 i R 1 q W y Z E t u q 1 T q i D X j + B Z S / R N Y b q E L < / l a t e x i t > r < l a t e x i t s h a 1 _ b a s e 6 4 = " f 4 z y w < l a t e x i t s h a 1 _ b a s e 6 4 = " L 5 V 7 X q e K J e b q c E u 1 G t H v + d Y Y Z n M = " > A A A C x X i c h V F N L 8 R Q F D 1 T 3 5 8 z 2 A g b M S E 2 J q 9 I i J X E g i X G I D F M 2 n p m X q Z f a with the following scattering kernel1 : where n A (r), A(r), Z(r) and N (r) are the number density and average mass, proton and neutron numbers of nuclei, respectively, as a function of r; G F is the Fermi-coupling constant, θ W is the Weinberg angle (sin 2 θ W 0.231); θ v v is the angle between v and v; the form factor e −bE 2 (1−cos θ vv ) accounts for the coherency of the scattering and is determined by the ratio of the ν wavelength to the radius of nuclei.Since the distribution function of scattered neutrinos at a given radius r is a sum of all neutrinos scattered at larger radii, it is obtained by line-integration (orange line in Fig. S1 We assume further that the density profile satisfies a power-law: In addition, we employ the following approximations: n A (r) ρ(r) Am a , (S15)

1 ]
FIG. 1.The radial profiles of the baryonic mass density (orange) and the number density of νe (cyan) and νe (red) multiplied with √ 2GF ( c) 2 .The black solid-and gray dashed lines represent the growth rate of the fast flavor conversion for the standard-((NE, Nµ) = (20, 10)) and high-((NE, Nµ) = (30, 40)) resolution simulations, respectively (NE and Nµ denote numbers of the energy-and angular grid points, respectively).The time is 100 ms after bounce and the shock wave is located at ∼ 223 km.

k 1 ]FIG. 2 .
FIG. 2. (a): The energy spectra of νe (cyan) and νe (red) at r = 241 km.The solid and dashed lines are for µ = 0.97 (outgoing neutrinos) and for µ = −0.87(ingoing neutrinos), respectively.A factor of 10 5 is multiplied for the latter.(b): The angular distributions of neutrinos at the same radius with the same notation for colors.Dashed lines represent the results for the simulation without scatterings of heavy nuclei.(c): Complex ω as a function of real k for unstable modes at r = 241 km derived by solving Eq. (6).The solid and dashed lines represent Im ω and 0.05 × Re ω, respectively.The time is 100 ms after bounce.

FIG. 3 .
FIG.3.Growth rates of the fast flavor conversion estimated by Eq. (8) (solid lines) and the ratio of nν e to nν e , α (dashed lines) as functions of radius at some different times: 10 ms (red), 30 ms (orange), 50 ms (lime), 100 ms (green), 200 ms (blue) and 400 ms (purple).The thin vertical lines indicate the shock positions at the same times.The lack of red and orange solid lines in this figure means no fast flavor conversion at the corresponding times.
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d 9 M
FIG. S1.Schematic pictures of (a) the neutrino bulb model and (b) coherent scattering backwards.