Oscillations of High-Energy Cosmic Neutrinos in the Copious MeV Neutrino Background

The core-collapse of massive stars and merger of neutron star binaries are among the most promising candidate sites for the production of high-energy cosmic neutrinos. We demonstrate that the high-energy neutrinos produced in such extreme environments can experience efficient flavor conversions on scales much shorter than those expected in vacuum, due to their coherent forward scatterings with the bath of decohered low-energy neutrinos emitted from the central engine. These low-energy neutrinos, which exist as mass eigenstates, provide a very special and peculiar dominant background for the propagation of the high-energy ones. We point out that the high-energy neutrino flavor ratio is modified to a value independent of neutrinos energies, which is distinct from the conventional prediction with the matter effect. We also suggest that the signals can be used as a novel probe of new neutrino interactions beyond the Standard Model. This is yet another context where neutrino-neutrino interactions can play a crucial role in their flavor evolution.


I. INTRODUCTION
Core-collapse supernovae (CCSNe) and neutron star mergers (NSMs) commonly lead to a burst of thermal neutrinos in the MeV range, with a very active literature on the physics of their oscillations.These enviroenments are considered as the central engine of not only supernovae but also gamma-ray bursts (GRBs), and other energetic or transrelativistic supernovae driven by outflows such as jets and winds (i.e., engine-driven supernovae).It has been suggested that GeV-TeV neutrinos can be produced in such environments if neutron-loaded outflows are launched from a black hole with an accretion disk and/or a newborn magnetar [1][2][3][4][5].Even TeV-PeV neutrinos can be generated inside the outflows through shock acceleration or magnetic reconnections [6][7][8][9][10].
The IceCube discovery of high-energy neutrinos (HEν's) has opened a new avenue to investigate the physics of neutrino oscillations and related neutrino physics (see Refs. [11][12][13] and references therein).In this article, we investigate a novel effect caused by the interplay between the HEν's produced in outflows and lowenergy neutrinos (LEν's) directly from the central engine (see Fig. 1 for the schematic picture).Indeed, the decohered LEν's, which are in mass eigenstates, can provide a dominant unusual background for the propagation of the HE ones.In particular, we show that the resulting neutrino self-interactions (νSI) leads to a very intriguing phenomenon, in which the HEν's experience shortscale flavor oscillations in such a way that on average, they end up in the mass eigenstates.This phenomenon is noncollective in spirit and differs remarkably from the well-known phenomenon of collective oscillations of MeV neutrinos occurring in dense neutrino environments such as CCSNe and NSM remnants [14][15][16].HEν production occurs at R diss ≫ Reng, which may be beamed with the opening angle ∼ 1/Γ, with R diss , Reng, and Γ being the dissipation radius, the engine radius, and the outflow Lorentz factor.Note that the opening angle of the LEν beams is exaggerated for illustration purposes.

II. HE NEUTRINO INTERACTIONS IN JETS OR WINDS
Various scenarios for HEν production in GRBs, CC-SNe and NSMs have been suggested.In this work, we are interested in the fate of HEν's so we assume that they are produced at the dissipation radius R diss ∼ 10 8 − 10 10 cm, which is much larger than the engine radius R eng ∼ 10 6 cm.Then an interesting interplay is particularly plausible when the HEν production occurs within the duration of LEν's and outflow breakout time (that are longer than the light crossing time).Note that as seen below the effect on neutrino oscillation is largely model independent as long as R diss is so small that LEν's govern the neutrino potential.
The number density of LEν's at R diss (in the engine frame) is which can be much larger than the expected number density of HEν's therein, n HEν ≲ 10 24 cm −3 at 10 9 cm.Here L νe and ⟨E ν ⟩ are the electron neutrino luminosity and average energy, respectively.In addition, the electron number density in the outflow is Unlike the flavor evolution of the LEν's which is dominated by the mass Hamiltonian at such neutrino number densities, the evolution of HEν's can be dominated by their coherent scattering with the bath of the LEν's.This simply comes from the fact that for the HEν's, the strength of νSI (see Eq. ( 5)), can be much larger than their vacuum wavelength, ω ≈ ∆m 2 atm c 3 /(2ℏE ν ) ≃ 6×10 −10 cm −1 100 GeV/E ν , with G F being the Fermi constant.In the above equation ξ = 1 − cos Θ, where Θ is the opening angle of the neutrino beams, which is determined here mainly by the opening angle of HEν's.Note that as soon as the parameter µ is known, ξ and ν do not provide any more relevant information.For relativistic flows with Γ ∼ 2 − 100, one has ξ ≈ Θ 2 /2 ∼ 1/(2Γ 2 ).Note that the optical depth to incoherent neutrino scatterings is so small that the electron-positron pair production is negligible.Moreover, given the fact that the number density of LEν's is much larger than that of HEν's, one can assume that n ν is here exclusively determined by the LEν's.
Although the number density of the LEν's within the zones of interest is expected to be too small to allow for the νSI Hamiltonian to compete with or dominate their vacuum Hamiltonian, the evolution of HEν's is almost completely governed by the interaction term for appropriate LEν number densities (ω HEν ≪ µ ≲ ω LEν ).

III. TWO-BEAM MODEL
In order to demonstrate how the flavor content of HEν's is impacted by their propagation in the bath of the LEν's, we study neutrino flavor conversions in a one-dimensional two-beam model, which consists of two energy bins, and a three-flavor neutrino gas with two angular beams.The neutrino energies are taken to be E ν = 10 MeV and 100 GeV for the bins representing the LEν's and the HEν's, respectively, unless otherwise stated.Thus in brief, our model consists of two angle beams each including neutrinos and antineutrinos with two energies representing high-and low-energy neutrinos.We also assume that the neutrino density is constant within the bath of LEν's.
In order to study the flavor evolution of neutrinos in our model, we solve the Liouville-von Neumann equation for the neutrino density matrix, ϱ (c = ℏ = 1) [19] with being the neutrino potential stemming from the neutrinoneutrino forward scattering [20][21][22].Here p is the neutrino momentum, E ν = |p|, v = p/E ν , and M 2 are the energy, velocity, and mass-square matrix of the neutrino, respectively, and U is the Pontecorvo-Maki-Nakagawa-Sakata matrix.Moreover, H m is the contribution from the matter term which is proportional to matter (electron) density [23,24], which is ignored in our calculations due to the relatively small matter density inside the outflow.Hence, there are only two nonzero terms in H (vacuum and νSI) which are both diagonal in the mass basis and constant (see below), but with different eigenvalues.
As mentioned above, H νν is almost exclusively determined by LEν's here due to their much larger number densities.In this study, we assume that LEν's are in mass eigenstates because they are expected to be already decohered within the zones of interests, which are very far from their emission region (with a typical coherence length of a few 10 6 cm for the atmospheric mass difference) [25][26][27].In addition, the HEν's do not significantly disturb the flavor state of the LEν bath due to their much smaller number densities.This implies that the LEν's do not evolve since their evolution is dominated by the mass term and they are already in the mass eigenstates.Consequently, H νν remains approximately constant.(We will later discuss how the neutrino flavor evolution changes once the oscillations of LEν's are taken into account.)On the other hand, the HEν's find themselves in the bath of the LEν's as soon as they are produced, and start flavor conversions on relatively short scales due to their interactions with the LEν's.It should be kept in mind that the coherence length of HEν's are expected to be much longer than those of the LE ones [28].

IV. RESULTS
In the upper panel of Fig. 2, we show the survival probabilities of HEν e 's propagating in vacuum (dashed curve), and in the bath of the LEν's (solid curve).As can be clearly seen, the oscillation scales of HEν's can change by orders of magnitude when coherent scatterings with LEν's are taken into account.As a matter of fact, the oscillation scale of HEν's in a bath of the LE ones is determined by the number density of the LEν's, namely l osc ∼ |H νν | −1 ∼ µ −1 (∼ 10 5 cm for this simulation).This scaling behavior can be immediately deduced from Eq. ( 4) given the fact that for the HEν's, the dominant contribution to the Hamiltonian comes from coherent scatterings with LEν's as long as ω HEν ≪ µ (we indeed observe this behaviour for 10 ω HEν ≲ µ).Note that here the only relevant physical parameter is µ/ω HEν , therefore one can play with µ in Fig. 2 as long as this ratio is constant (provided that r is also appropriately rescaled for the upper panel).
The fact that HEν's oscillate on scales ∼ µ −1 might remind an astute reader of the phenomenon of fast flavor conversions occurring for MeV neutrinos in dense neutrino media [29,30].This similarity becomes more obvious once one notes that the oscillations of HEν's can even occur when ω HEν = 0. However and in spite of this resemblance, it should be kept in mind that these two phenomena completely differ in spirit and have nothing to do with each other.Although for the occurrence of fast conversions certain criteria need to be fulfilled [31], the short-scale conversions of HEν's in a bath of the LE ones is a generic phenomenon provided that there are two populations of neutrinos of which one is dominant.
Flavor conversions of HEν's induced by the static bath of the LEν's is also distinct from the phenomenon of ordinary collective oscillations in dense neutrino media.While the latter is a nonlinear phenomenon with a high level of coupling, the former is a linear phenomenon where H νν solely provides a constant background for the flavor evolution of HEν's.This implies that such flavor conversions of HEν's is a noncollective phenomenon.Also note that the relevant n ν 's can be many orders of magnitude smaller than the values for which collective oscillations of MeV neutrinos are expected, due to the much smaller ω HEν .
In order to see how the short-scale flavor conversions of HEν's changes the expected ν e : ν µ : ν τ ratio on earth, one can average the survival probabilities over a few oscillations.As indicated in our upcoming work [32], such an averaging process in our two-beam model corresponds to averaging over the neutrino angular distribution in a more-realistic, multiangle neutrino gas.The average survival probabilities then reach a steady state which does not depend on the details of the simulation (apart from the neutrino mixing parameters as discussed in the following), shown in the lower panel of Fig. 2.This behaviour can be understood analytically as follows.In the mass basis, the νSI Hamiltonian is diagonal with its kk- ρ αα are the initial (anti)neutrino occupation numbers in flavor α.This comes from the fact that H νν is nearly determined only by the LEν's which are in the mass eigenstates.Then the HEν density matrix in the mass basis, ρ, evolves as, implying that the averaged flavor ratio, ν e : ν µ : ν τ , can be written as, where there is a summation over α and k, and f e : f µ : f τ is the initial flavor ratio at the production region.The black lines in the lower panel of Fig. 2 indicate the analytical flavor ratios in Eq. ( 7), which show a perfect agreement with the numerical results.Note that the average density matrix in the flavor basis is equal to the one expected after the neutrino decoherence, and is also independent of the neutrino energy.This behaviour indeed results from the fact that the HEν's oscillate very quickly about H νν and consequently, they end up in the mass eigenstates (on average).Hence, in summary, short-scale conversions of HEν's induced by the ambient gas of the LEν's lead to their decoherence on scales which can be shorter than their natural decoherence length [28] by many orders of magnitude.
Although such short-scale oscillations and the resulting decoherence of HEν's is an interesting phenomenon by itself and could in principle impact the physics of their propagation by modifying their flavor ratio at the source, we here discuss a few important cases in which the induced conversions of HEν's can be observable on Earth.
Once HEν's leave the LEν bath, they should propagate in the dense ejecta, where n e ≫ n ν [33][34][35][36][37].In particular, for engine-driven transients, the HEν production region is surrounded with the stellar or merger ejecta, whose density is much larger than that in the production region.This means that in solving Eq. ( 4) for the neutrino propagation in this region, we ignore the H νν term.In order to account for the decoherence experienced by neutrinos in the LEν bath, we start with an initial density matrix which is a time average of the one in Eq. ( 6).In addition, for the H m term we consider a blue supergiant matter profile from Ref. [38], as an example (30 M ⊙ BSG in Ref. [37]).Needless to say, the short-scale conversions of HEν's will impact the outcome of the matter effect and correspondingly, their flavor ratio on Earth.This is illustrated clearly in Fig. 3 for a FIG. 3. The expected νe : νµ : ντ ratio on Earth in the absence and presence of νSI for different neutrino energies.Note that the matter effect is included in both cases, assuming the density profile of a blue supergiant, and the ratio with νSI is very close to the total flavor equipartition.In addition, the green star indicates the total flavor equipartition expected from the propagation of HEν's in a bath of oscillating LEν's, as discussed in the text.Note that apart from the matter only case for the 1 TeV neutrinos, the other ones are almost on top of each other.It is also illuminating to keep in mind that the final flavor states are specific to the initial flavor composition of 1:2:0 and can vary under different circumstances.
case with the initial flavor ratio ν e : ν µ : ν τ = 1 : 2 : 0 in the normal mass ordering.This is particularly interesting considering the energy-independent nature of the LEν-induced short-scale oscillations of HEν's.Although the pure matter effect shows a clear sign of energy dependence here [37], it is almost independent of the neutrino energy in the presence of νSI.This is even the case at high energies where the muon damping is expected to occur in such dense environments [39,40].This can provide one with a new observable indication of neutrino flavor mixing caused by νSI.
So far we have assumed that the bath of the LEν's is completely decohered.However, the situation could be different.On the one hand, since R diss can be as low as 10 8 cm, the solar-mass channel LEν's could be still in phase since l coh,⊙ ∼ σ x E2 ν /∆m 2 sol ≳ 10 8 cm.Moreover, due to the possibility for the existence of µ ≳ 10 −4 cm −1 , the decoherence of LEν's might be suppressed [41] in the atmospheric channel as well and LEν's can experience a sort of (partial) collective oscillations in the production region of HE ones.Such erratic conversions of LEν's will lead to the total flavor equipartition of HEν's regardless of their initial flavor content, as indicated by the green star in Fig. 3.This phenomenon will be discussed in more details in our upcoming work [32] 1 .
In some of the beyond Standard Model (SM) theories of particle physics, neutrinos can experience neutrino nonstandard self-interactions (νNSSI) [42,43].Such νNSSI modify Eq. ( 5) to [44][45][46] where Ĝ contains information about νNSSI ( Ĝ = 1 in SM).For example, in the vector mediator scenario, we may have and its νNSSI components are related to the vector mediator mass m V and the coupling strength g by | Ĝαβ | ∝ g 2 /m 2 V .The current constraints on νNSSI are model dependent and strong for the mediator mass below MeV energies.For heavier mediators, the constraints from the early universe are rather weak, e.g., | Ĝαβ | ≲ 10 7 [47], although laboratory constraints can be stronger in the limited parameter space [48].It has been suggested that spectral modulations and time delays of HEν's enable us to study the unexplored parameter space of νNSSI [49][50][51][52][53][54].We point out that coherent νSI-induced oscillations of HEν's can be used as a novel probe of νNSSI.This is illustrated in Fig. 4 where the red region shows the impact of νNSSI.The flavor content is expected to have observable sensitivity to νNSSI, i.e., a ∼ 10% change of flavor ratio is caused by | Ĝαβ | ∼ 0.1.This means that one could probe such weak couplings with this effect.

V. CONCLUSION
We have brought to light a novel phenomenon, in which a class of high-energy cosmic neutrino emission can ex-perience flavor conversions induced by the copious LEν background, on scales much shorter than their intrinsic vacuum oscillation wavelengths.Unlike the celebrated phenomenon of collective oscillations of MeV neutrinos in a dense neutrino medium, the unearthed flavor conversions of high-energy cosmic neutrinos is a noncollective phenomenon in spirit.This intriguing phenomenon can occur when HEν's from relativistic outflows launched at the core-collapse of massive stars or at the mergers propagate in the bath of the already-decohered lower energy neutrinos from the central engine.Despite the small number density of HEν's which can be insufficient to result in their own collective oscillations, their presence can lead to short-scale conversions of HEν's on scales determined by the density of LEν's.The background-induced conversions of HEν's change their flavor content in an energy-independent manner and takes the HEν gas to a state which is diagonal in the mass basis.This way they cause an induced decoherence of HEν's on scales which are many orders of magnitude shorter than their natural decoherence lengths.Such a modification of the HEν's at source can impact the physics of the phenomena occurring during their propagation, such as neutrino decay, scattering, etc.We also point out a few possibilities where such short-scale induced decoherence can directly impact the flavor ratio of HEν on Earth, including the matter effect of HEν's, the νNSSI, and the total flavor equipartition due to an oscillating ambient LEν gas.
Our study provides the first step toward understanding this intriguing phenomenon and further exploration is needed to better understand its implications.This is yet another context where neutrino-neutrino interactions can play a crucial role in their flavor evolution, and also motivates further investigations into multimessenger high-energy emission from GRBs, CCSNe and NSMs.

FIG. 1 .
FIG.1.Schematic picture of HEν (GeV-PeV) production and their interactions with LEν's (MeV-GeV) from the central engine such as a black hole with an accretion disk or a newborn magnetar.HEν production occurs at R diss ≫ Reng, which may be beamed with the opening angle ∼ 1/Γ, with R diss , Reng, and Γ being the dissipation radius, the engine radius, and the outflow Lorentz factor.Note that the opening angle of the LEν beams is exaggerated for illustration purposes.

FIG. 2 .
FIG.2.Upper panel: Survival probability of HEνe's propagating in vacuum (dashed red and in the bath of the LEν's (solid blue curve).HEν's can experience flavor conversions on scales much shorter than those expected in vacuum.Here, for illustrative purposes we have assumed µ = 10 −7 cm −1 , ELEν = 10 MeV and EHEν = 100 GeV.For the LEν bath, nν e /nν e = 1.3 is also fixed, although the survival probabilities are independent of nν e /nν e as long as it is not too close to 1. Lower panel: The survival probabilities as a function of the HEν energy, where the diamonds, points, and squares are the survival probabilities of νe, νµ, and ντ obtained from the simulations, respectively, and the black lines are the corresponding analytical solutions.We assume an initial flavor ratio of νe : νµ : ντ = 1 : 0 : 0. We here set θ12 = 33.6 • , θ23 = 47.2 • , θ13 = 8.5 • and δCP = 0. Antineutrinos behave exactly in the same manner.

FIG. 4 .
FIG. 4. The expected νe : νµ : ντ ratio after HEν's escape their production region, in the presence of νNSSI.The triangle, circle, and square indicate the ratio in SM while the red region shows how the ratio changes in the presence of νNSSI for the 1 : 2 : 0 case.Here the red region is created by choosing a large set of randomly populated Ĝαβ assuming that | Ĝαβ | < 1 (for α ̸ = β).Via coherent νSI, the final HEν flavor ratio is very sensitive to the νNSSI.