The unknown neutrino parameters may leave detectable signatures in the supernova (SN) neutrino flux. However, even the contribution from the MSW flavor transition alone could cause ambiguity in the interpretation to the neutrino signals because of the uncertain local density profile of the SN matter and the model-dependent SN neutrino spectral parameters. A specific parametrization to the unknown local density profile is proposed in this work, and the contribution from the standard MSW effect is investigated through a multi-detector analysis of the SN neutrinos. In establishing the model-independent scheme, results based on the existing spectral models are included. The limitation of the analysis is also discussed.
We consider the problem of time-to-space conversion in quantum field theory of flavor mixing using a generalization of the wave-packet method in quantum mechanics. We work entirely within the canonical formalism of creation and annihilation operators that allows us, unlike the usual wave-packet formulation, to include the nontrivial effect due to flavor condensation in the vacuum.
In the dense supernova environment, neutrinos can undergo fast flavor conversions which depend on the large neutrino-neutrino interaction strength. It has been recently shown that both their presence and outcome can be affected when passing from the commonly used three neutrino species approach to the more general one with six species. Here, we build up on a previous work performed on this topic and perform a numerical simulation of flavor evolution in both space and time, assuming six neutrino species. We find that the results presented in our previous work remain qualitatively the same even …
We discuss the {\em intermediate} wave-packet formalism for analytically quantifying the energy dependence of the two-flavor conversion formula that is usually considered for analyzing neutrino oscillations and adjusting the focusing horn, target position and/or detector location of some flavor conversion experiments. Following a sequence of analytical approximations where we consider the second-order corrections in a power series expansion of the energy, we point out a {\em residual} time-dependent phase which, in addition to some well known wave-packet effects, can subtly modify the oscillat…
We consider active-sterile neutrino conversions in a supernova in the presence of random magnetic field domains. For large enough fields the magnetization of the medium may enhance the active to sterile neutrino conversion rates. Neglecting neutrino transition magnetic moments we show that for KeV neutrino mass squared differences these limits may overcome those that would apply in the isotropic case.
The coherent muon-electron conversion rates in various nuclei are calculated for general lepton flavor violating interactions. We solve the Dirac equations numerically for the initial state muon and the final state electron in the Coulomb force, and perform the overlap integrals between the wave functions and the nucleon densities. The results indicate that the conversion branching ratio increases for a light nucleus up to the atomic number Z ~ 30, is largest for Z = 30 -- 60, and becomes smaller for a heavy nucleus with Z > (~ 60). We also discuss the uncertainty from the input proton and neu…
The active-sterile neutrino conversion is studied for neutrino propagating in the axial potential generated by magnetised electron plasma in the supernova medium. We consider the effect of random magnetic field $B_{rms}$ on the average neutrino conversion probability. We obtained the constraint on $Δm^2$ and $\sin^2 2θ$ for different strength of the random magnetic fields, by considering the positive definiteness of the average neutrino conversion probability inside the supernova core. Our calculation shows that, $B_{rms}\le 0.63\times 10^{14}$ Gauss is preferable so that small values of $Δm^2…
Can we observe the solar eclipses in the neutrino light? In principle, this is possible by identifying the lunar matter effects on the flavor conversions of solar neutrinos when they traverse the Moon before reaching the detectors at the Earth. Unfortunately, we show that the lunar matter effects on the survival probability of solar $^8{\rm B}$ neutrinos are suppressed by an additional factor of $1.2\%$, compared to the day-night asymmetry. However, we point out that the matter effects on the flavor conversions of high-energy astrophysical neutrinos, when they propagate through the Sun, can be…
{Large enough random magnetic fields may affect in an important way neutrino conversion rates, even in the case where neutrinos have zero transition magnetic moments. We consider their effect in the case of active to sterile \neu conversions in a supernova and show that for KeV neutrino masses these limits may overcome those derived for the case of zero magnetic field.}
We perform a covariant wave-packet analysis of neutrino oscillations taking into account the lifetime of the neutrino production process . We find that flavor oscillations in space are washed out when the neutrinos are produced from long lived resonances - and what may be observed in appearance/disappearance experiments is a uniform conversion probability independent of distance. The lifetime of the resonance which produces the neutrinos acts as the the effective baseline of the experiment. For this reason the LSND experiment where neutrinos are produced from muon decay has two orders of magni…
Mass Varying neutrino mechanisms were proposed to link the neutrino mass scale with dark energy, addressing the coincidence problem. In some scenarios this mass can present a dependence on the baryonic density felt by neutrinos, creating an effective neutrino mass that depends both on the neutrino and baryonic densities. In this article we investigate the possibility that a neutrino effective mass is the only flavour conversion mechanism acting in neutrino oscillation experiments. We present a parameterization on the environmental effects on neutrino mass that produces the right flavour conver…
Using moment equations we analyze collective flavor transformation of supernova neutrinos. We study the convergence of moment equations and find that numerical results using a few moment converge quite fast. We study effects of emission angle distribution of neutrinos on neutrino sphere. We study scaling law of the amplitude of neutrino self-interaction Hamiltonian and find that it depends on model of emission angle distribution of neutrinos. Dependence of neutrino oscillation on different models of emission angle distribution is studied.
An analytical solution to a parametric resonance effect for antineutrinos in a 3+1 flavor (active+sterile) scheme using multiple non-adiabatic density shifts is presented. We derive the conditions for a full flavor conversion for antineutrino oscillations $\barν_α\to \barν_s$ $(α=e,μ,τ)$ under the assumption that LSND best-fits for the mixing parameters are valid in a short-baseline accelerator experiment. We show that the parametric resonance effect can be exploited to increase the effective antineutrino oscillation length by a factor of 10-40, thus sustaining a high oscillation probability f…
In general, attempts to extend the standard model will include extra gauge structure. We parameterize string and technicolor models for a Z' boson with primitive lepton flavor violating interactions. Calculations for its muon conversion rate (mu N -> e N) on Titanium are made and used to show the potential of forthcoming experiments MECO at Brookhaven National Laboratory and PRIME at the Japan Hadron Facility (to be renamed). For reasonable choices of parameters, such U(1) bosons with masses the order 10 TeV and mu-e Z' charge as low as ~10^-5 are demonstrated to be accessible for MECO and PRI…
The supernova neutrino flavor evolution in the presence of the non-trivial neutrino magnetic moment and strong magnetic field is numerically derived using the two-flavor and single-angle approximation. The novel properties of collective neutrino oscillations are studied and distinct patterns of flavor and spin-flavor spectral splits are presented. Finally we also discuss how the neutrino magnetic moment affects the observable supernova neutrino energy spectra.
Fast neutrino flavor conversion can occur in core-collapse supernovae or compact binary merger remnants when non-forward collisions are also at play, and neutrinos are not fully decoupled from matter. This work aims to shed light on the conditions under which fast flavor conversion is enhanced or suppressed by collisions. By relying on a neutrino toy model with three angular bins in the absence of spatial inhomogeneities, we consider two angular configurations: The first one with angular distributions of $ν_e$ and $\barν_e$ that are almost isotropic as expected before complete neutrino decoupl…
We calculate fast conversions of two flavor neutrinos by considering Boltzmann collisions of neutrino scatterings. In an idealized angular distribution of neutrinos with electron-lepton number crossing, we find that the collision terms of the neutrino scattering enhance the transition probability of fast flavor conversions as in the previous study. We analyze the dynamics of fast flavor conversions with collisions in detail based on the motion of polarization vectors in cylindrical coordinate analogous to a pendulum motion. The phase of the all polarization vector synchronizes in the linear ev…
Self induced neutrino flavor conversions in the dense regions of stellar core collapse are almost exclusively studied in the standard two flavor scenario. Linear stability analysis has been successfully used to understand these flavor conversions. This is the first linearized study of $\textit{three flavor}$ fast instabilities. The `fast' conversions are fascinating distinctions of the dense neutrino systems. In the fast modes the collective oscillation dynamics are independent of the neutrino mass, growing at the scale of the large neutrino-neutrino interaction strength ($10^5$ km$^{-1}$) of …
During the accretion phase of a core-collapse supernova (SN), the deleptonization flux has recently been found to develop a global dipole pattern (LESA---Lepton Emission Self-sustained Asymmetry). The $ν_e$ minus $\barν_e$ flux essentially vanishes in one direction, potentially facilitating self-induced flavor conversion. On the other hand, below the stalled shock wave, self-induced flavor conversion is typically suppressed by multi-angle matter effects, preventing any impact of flavor conversion on SN explosion dynamics. In a schematic model of SN neutrino fluxes, we study the impact of modif…
In the early Universe, or near a supernova core, neutrino flavor evolution may be affected by coherent neutrino-neutrino scattering. We develop a microscopic picture of this phenomenon. We show that coherent scattering does not lead to the formation of entangled states in the neutrino ensemble and therefore the evolution of the system can always be described by a set of one-particle equations. We also show that the previously accepted formalism overcounts the neutrino interaction energy; the correct one-particle evolution equations for both active-active and active-sterile oscillations contain…
Core collapse supernovae are a huge source of all flavor neutrinos. The flavor composition, energy spectrum and time structure of the neutrino burst from a galactic supernova can provide information about the explosion mechanism and the mechanisms of proto neutron star cooling. Such data can also give information about the intrinsic properties of the neutrino such as flavor oscillations. One important question is to understand to which extend can the supernova and the neutrino physics be decoupled in the observation of a single supernova. In this paper we discuss the possibility to probe the n…
We have studied lepton flavor violation processes $μ\to eγ$ and $μ-e$ conversion in nuclei induced by unparticle. Both ${\rm Br}(μ\to eγ)$ and $μ-e$ conversion rate ${\rm CR}(μ-e,{\rm Nuclei})$ strongly depend on the scale dimension $d_{\cal U}$ and the unparticle coupling $λ^{ff'}_{\rm K}$(K=V, A, S, P). Present experimental upper bounds on ${\rm Br}(μ\to eγ)$, ${\rm CR}(μ-e,{\rm Ti})$ and ${\rm CR}(μ-e,{\rm Au})$ put stringent constraints on the parameters of unaprticle physics. The scale dimensions $d_{\cal U}$ around 2 are favored for the unparticle scale $Λ_{\cal U}$ of ${\cal O}(10 {\rm …
There is no guarantee that the violation of lepton number, assuming it exists, will primarily manifest itself in neutrinoless double beta decay ($0νββ$). Lepton-number violation and lepton-flavor violation may be related, and much-needed information regarding the physics that violates lepton number can be learned by exploring observables that violate lepton flavors other than electron flavor. One of the most promising observables is $μ^- \rightarrow e^+$ conversion, which can be explored by experiments that are specifically designed to search for $μ^- \rightarrow e^-$ conversion. We survey lep…
Several extensions of the standard electroweak model allow new four-fermion interactions (nu_a nu_b * ff) with strength eps_ab*G_F, where (a,b) are flavor indices. We revisit their effects on flavor oscillations of massive (anti)neutrinos in supernovae, in order to achieve, in the region above the protoneutron star, an analytical treatment valid for generic values of the neutrino mixing angles (omega,phi,psi)=(theta_12,theta_13,theta_23). Assuming that eps_ab<<1, we find that the leading effects on the flavor transitions occurring at high (H) and low (L) density along the supernova matter prof…
Neutrino flavor evolution in core-collapse supernovae, neutron-star mergers, or the early universe is dominated by neutrino-neutrino refraction, often spawning "self-induced flavor conversion", i.e., shuffling of flavor among momentum modes. This effect is driven by collective run-away modes of the coupled "flavor oscillators" and can spontaneously break the initial symmetries such as axial symmetry, homogeneity, isotropy, and even stationarity. Moreover, the growth rates of unstable modes can be of the order of the neutrino-neutrino interaction energy instead of the much smaller vacuum oscill…
Supernova neutrinos can exhibit a rich variety of flavor conversion mechanisms. In particular, they can experience "fast" self-induced flavor conversions almost immediately above the core. Very recently, a novel method has been proposed to investigate these phenomena, in terms of the dispersion relation for the complex frequency and wave number ($ω$,$k$) of disturbances in the mean field of the $ν_eν_x$ flavor coherence. We discuss a systematic approach to such instabilities, originally developed in the context of plasma physics, and based of the time-asymptotic behavior of the Green's functio…
The modeling of fast flavor evolution of neutrinos in dense environments has been traditionally carried out by relying on a two flavor approximation for simplicity. In addition, vacuum mixing has been deemed negligible. For the first time, we highlight that the fast flavor evolution in three flavors is intrinsically different from the one obtained in the two flavor approximation. This is due to the exponential growth of flavor mixing in the $e$--$μ$ and $e$--$τ$ sectors generated by the vacuum term in the Hamiltonian. As a result, substantially larger flavor mixing is found in three flavors. O…
Neutrino-neutrino refraction in dense media can cause self-induced flavor conversion triggered by collective run-away modes of the interacting flavor oscillators. The growth rates were usually found to be of order a typical vacuum oscillation frequency $Δm^2/2E$. However, even in the simple case of a $ν_e$ beam interacting with an opposite-moving $\barν_e$ beam, and allowing for spatial inhomogeneities, the growth rate of the fastest-growing Fourier mode is of order $μ=\sqrt{2} G_{\rm F} n_ν$, a typical $ν$--$ν$ interaction energy. This growth rate is much larger than the vacuum oscillation fr…
Neutrino flavor conversions may dramatically affect the inner working of compact astrophysical objects as well as the synthesis of the heavier elements. We present the first sophisticated numerical solution of the neutrino flavor conversion within a (2+1+1) dimensional setup: we include the advective term in the neutrino equations of motion and track the flavor evolution in two spatial dimensions, one angular variable, and time. Notably, the advective term hinders the development of neutrino pairwise conversions, if the conditions favoring such conversions (i.e., crossings between the angular …
A simple covariant formalism to describe flavor and CP violation in the left-handed quark sector in a model independent way is provided. The introduction of a covariant basis, which makes the standard model approximate symmetry structure manifest, leads to a physical and transparent picture of flavor conversion processes. Our method is particularly useful to derive robust bounds on models with arbitrary mechanisms of alignment. Known constraints on flavor violation in the K and D systems are reproduced in a straightforward manner. Assumptions-free limits, based on top flavor violation at the L…
Loop quantum gravity theory incorporates a new scale length ${\cal L}$ which induces a Lorentz invariance breakdown. This scale can be either an universal constant or can be fixed by the momentum of particles (${\cal L}\sim p^{-1}$). Effects of the scale parameter ${\cal L}$ and helicity terms occurring in the dispersion relation of fermions are reviewed in the framework of spin-flip conversion of neutrino flavors.
Expected flavor composition of supernova neutrinos
Gran Sasso Science Institute (GSSI)（格兰萨索科学研究所）；INFN - Laboratori Nazionali del Gran Sasso (LNGS)（意大利国家核物理研究所-格兰萨索国家实验室）；Technische Universität München（慕尼黑工业大学）；RIKEN iTHEMS（理化学研究所 iTHEMS）；Department of Physics, University of California, Berkeley（加州大学伯克利分校物理系）
The lepton flavor violation $μ\to eγ$, $τ\to eγ$, $τ\toμγ$, $μ\to3e$, $τ\to3e$, $τ\to3μ$ and $μ-e$ conversion in Al and Ti are studied in both the Altarelli-Feruglio $A_4$ model and the $S_4$ model of Ding. The rates of these lepton flavor violation process for the inverted hierarchy neutrino mass spectrum are enhanced considerably by the next to leading order contributions. For both models, we find that the $μ-e$ conversion in Ti is within the precision of next generation experiments in all the parameter space considered, the $μ-e$ conversion in Al should be observable at least in a very sign…
Self-induced flavor conversions of supernova (SN) neutrinos can strongly modify the flavor dependent fluxes. We perform a linearized flavor stability analysis with accretion-phase matter profiles of a 15 M_sun spherically symmetric model and corresponding neutrino fluxes. We use realistic energy and angle distributions, the latter deviating strongly from quasi-isotropic emission, thus accounting for both multi-angle and multi-energy effects. For our matter and neutrino density profile we always find stable conditions: flavor conversions are limited to the usual MSW effect. In this case one may…
In environments with high dense neutrino gases, such as in core-collapse supernovae, the neutrinos can experience collective neutrino oscillation due to their self-interactions. In particular, fast flavor conversion driven by the crossings in the neutrino angular distribution can affect explosion mechanism, nucleosynthesis, and neutrino observation. We perform the numerical computation of nonlinear flavor evolution on the neutrino angular distribution with tiny crossings expected to be generated in the preshock region. We demonstrate that the fast instability is triggered and a cascade develop…
A dense gas of neutrinos and antineutrinos can undergo fast pairwise conversions near the decoupling regions of core-collapse supernovae and in compact binary neutron star mergers. The flavor dependent neutrino heating can play a role in sustaining convection inside the hot and dense matter. In this paper, we study the unexplored effect of the bulk velocity of matter on fast pairwise conversions and demonstrate that depending on the direction and the magnitude of the bulk velocity, neutrino flavor conversions could be significantly enhanced or suppressed. The bulk velocity of matter, which is …
Fast flavor conversions of supernova neutrinos, possible near the neutrinosphere, depends on an interesting interplay of collisions and neutrino oscillations. Contrary to naive expectations, the rate of self-induced neutrino oscillations, due to neutrino-neutrino forward scattering, comfortably exceeds the rate of collisions even deep inside the supernova core. Consistently accounting for collisions and oscillations, we present the first calculations to show that collisions can create the conditions for fast flavor conversions of neutrinos, but oscillations can continue without significant dam…
We study the fully nonlinear fast flavor evolution of neutrinos in 1+1 dimensions. Our numerical analysis shows that at late time the system reaches an approximately steady state. Using the steady state approximation we analytically show that the spatial variation of the polarization vectors is given by their precession around a common axis, which itself has a motion reminiscent of a gyroscopic pendulum. We then show that the steady state solution to the equations of motion cannot be separated in position and velocity, that is the motion is not collective in the usual sense. However, the fast …
The magnetic field in the solar convective zone has a random small-scale component with the r.m.s. value substantially exceeding the strength of a regular large-scale field. For two Majorana neutrino flavors $\times$ two helicities in the presence of a neutrino transition magnetic moment and nonzero neutrino mixing we analyze the displacement of the allowed ($Δm^2 - \sin^22θ$) - parameter region reconciled for the SuperKamiokande(SK) and radiochemical (GALLEX, SAGE, Homestake) experiments in dependence on the r.m.s. magnetic field value $b$. We find an effective production of electron antineut…
The two dominant processes of lepton flavor violation, mu to 3e and mu to e conversion in atomic nuclei caused by neutral KK gauge boson exchanges at tree level, are studied in the context of five dimensional gauge-Higgs unification scenario. The key point of the flavor violation is a fact that the bulk masses and the brane ones cannot be simultaneously diagonalized. We estimate the branching ratio of each processes and obtain the lower bound of compactification scale around weak scale from the current experimental data. We discuss the reasons why the final result is not so severe although the…
Neutrino flux streaming from a supernova can undergo rapid flavor conversions almost immediately above the core. Focusing on this region, we study these fast conversions using a linear stability analysis. We find that, for realistic angular distributions of neutrinos, fast conversions can occur within a few nanoseconds in regions just above the neutrinosphere. Our results also show that neutrinos travelling towards the core make fast conversions more rapid. These conversions, if they exist, can have significant implications for supernova explosion mechanism and nucleosynthesis.
It has been recently pointed out that neutrino fluxes from a supernova can show substantial flavor conversions almost immediately above the core. Using linear stability analyses and numerical solutions of the fully nonlinear equations of motion, we perform a detailed study of these fast conversions, focussing on the region just above the supernova core. We carefully specify the instabilities for evolution in space or time, andfind that neutrinos travelling towards the core make fast conversions more generic, i.e., possible for a wider range of flux ratios and angular asymmetries that produce a…
The dense neutrino flux streaming from a core-collapse supernova can undergo self-induced flavor conversion caused by neutrino-neutrino refraction. Numerical studies of these nonlinear effects are challenging because representing the neutrino radiation field by discrete energy and angle bins can easily lead to unphysical solutions. In particular, if the number of angle bins N_a is too small, flavor conversion begins too deep and produces completely spurious results. At the same time, N_a=1 (single-angle approximation) can be a good proxy for the N_a -> infinity limit. Based on a linearized sta…
Multiple scattering and induced parton splitting lead to a medium modification of the QCD evolution for jet fragmentation functions and the final hadron spectra. Medium-induced parton splittings not only lead to energy loss of leading partons and suppression of leading hadron spectra, but also modify the flavor composition of a jet due to induced flavor conversion via gluon emission, quark pair production and annihilation. Through a numerical study of the medium-modified QCD evolution, leading $K^-$ strange meson spectra are found to be particularly sensitive to the induced flavor conversion i…
Neutrinos in a core-collapse supernova can undergo fast flavor conversions with a possible impact on the explosion mechanism and nucleosynthesis. We perform the first non-linear simulations of fast conversions in the presence of three neutrino flavors. The recent supernova simulations with muon production call for such an analysis, as they relax the standard $ν_{μ,τ}=\barν_{μ,τ}$ (two-flavor) assumption. Our results show the significance of muon and tau lepton number angular distributions, together with the traditional electron lepton number ones. Indeed, our three-flavor results are potential…
Matter-enhanced antineutrino flavor transformation between $\barν_e$ and $\barν_{μ(τ)}$ can occur in supernovae if the vacuum masses for these species satisfy $m_{\barν_e}>m_{\barν_{μ(τ)}}$. For $δm^2> 1$ eV$^2$, such flavor transformation can affect the electron fraction $Y_e$ in the neutrino-heated supernova ejecta. We point out that such flavor transformation will not drive $Y_e>0.5$ at the $r$-process nucleosynthesis epoch in the best available supernova model for such nucleosynthesis. Consequently, there is no obvious conflict between matter-enhanced antineutrino flavor transformation $\b…
Neutrinos propagating in dense neutrino media such as core-collapse supernovae and neutron star merger remnants can experience the so-called fast flavor conversions on scales much shorter than those expected in vacuum. A very generic class of fast flavor instabilities is the ones which are produced by the backward scattering of neutrinos off the nuclei at relatively large distances from the supernova core. In this study we demonstrate that despite their ubiquity, such fast instabilities are unlikely to cause significant flavor conversions if the population of neutrinos in the backward directio…
Fast and collisional flavor instabilities possibly occur in the neutrino decoupling regions of core-collapse supernovae and neutron-star mergers. To gain a better understanding of the relevant flavor dynamics, we numerically solve for the collisionally unstable evolution in a homogeneous, anisotropic model. In these calculations collisional instability is precipitated by unequal neutrino and antineutrino scattering rates. We compare the solutions obtained using neutral-current and charged-current interactions. We then study the nonlinear development of fast instabilities subjected to asymmetri…
Neutrinos in dense environments undergo collective pair conversions $ν_e\barν_e \leftrightarrow ν_x\barν_x$, where $x$ is a non-electron flavor, due to forward scattering off each other that may be a crucial ingredient for supernova explosions. Depending on the flavor-dependent local angular distributions of the neutrino fluxes, the conversion rate can be "fast", i.e., of the order $μ=\sqrt{2}G_F n_ν$, which can far exceed the usual neutrino oscillation frequency $ω=Δm^2/(2E)$. Until now, this surprising nonlinear phenomenon has only been understood in the linear regime and explored further us…
Several recent numerical studies have examined the effects of neutrino neutral-current scattering on fast flavor conversion (FFC). Those studies are in apparent conflict, with some finding enhancement of flavor conversion and others finding suppression. The ones that report enhancement all use homogeneous models, and we discuss in detail the self-consistency issues that they face as a result. We reproduce scattering-enhanced FFC in our own homogeneous calculations, showing that it occurs for both neutral- and charged-current scattering and that it is due to isotropization of the angular distri…
If $ν_μ$ or $ν_τ$ mix with $ν_e$, neutrino oscillations and collisions in a supernova (SN) core allow these flavors effectively to participate in $β$ equilibrium and thus to obtain a large chemical potential. If a sterile species mixes with $ν_e$, these effects lead to an anomalous loss of energy and lepton number. We study flavor conversion in a SN core on the basis of a new kinetic equation which rigorously includes neutrino interference and degeneneracy effects. Our discussion serves as an example and illustration of the properties of this ``non-abelian Boltzmann equation''.
High energy cosmic neutrino fluxes can be produced inside relativistic jets under the envelopes of collapsing stars. In the energy range E ~ (0.3 - 1e5) GeV, flavor conversion of these neutrinos is modified by various matter effects inside the star and the Earth. We present a comprehensive (both analytic and numerical) description of the flavor conversion of these neutrinos which includes: (i) oscillations inside jets, (ii) flavor-to-mass state transitions in an envelope, (iii) loss of coherence on the way to observer, and (iv) oscillations of the mass states inside the Earth. We show that con…
We perform numerical simulations of fast collective neutrino flavor conversions in an one-dimensional box mimicking a system with the periodic boundary condition in one spatial direction and translation symmetry in the other two dimensions. We evolve the system over several thousands of the characteristic timescale (inverse of the interaction strength) with different initial $\barν_e$ to $ν_e$ number density ratios and different initial seed perturbations. We find that small scale structures are formed due to the interaction of the flavor waves. This results in nearly flavor depolarization in …
