Fast Flavor Depolarization of Supernova Neutrinos

Flavor-dependent neutrino emission is critical to the evolution of a supernova and its neutrino signal. In the dense anisotropic interior of the star, neutrino-neutrino forward-scattering can lead to fast collective neutrino oscillations, which has striking consequences. We present a theory of fast flavor depolarization, explaining how diffusion to smaller angular scales causes the neutrino flavor differences to become smaller. We show that transverse relaxation determines the epoch of this irreversible depolarization. We give a method to compute the depolarized fluxes, leading to a simple formula, which can be a crucial input for supernova theory and neutrino phenomenology.

Metronomes sway in lockstep, crickets chirp in a chorus, and neurons fire in sync -all examples of coordinated action by seemingly unregulated agents [1]. Neutrinos emitted by collapsing stars also exhibit collective behavior in their quantum mechanical flavor oscillations due to neutrino-neutrino forward-scattering [2 -21]. Astonishingly, this collective evolution can be much faster than that of its constituents [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36]. It's as if a marching band could move faster than Usain Bolt. Such fast evolution has been alluded to erase the differences between neutrino fluxes within picoseconds and over distances smaller than a pinhead. In this Letter, we propose a theory for "fast flavor depolarization", which has major consequences for supernova (SN) explosions and their signals at neutrino telescopes.
Fast oscillation is a peculiar avatar of neutrino oscillation. Like ordinary neutrino oscillations it allows interconversions between neutrino flavors. However, unlike them, it proceeds at a rate √ 2G F n ν ∼ 10 cm −1 , proportional to the local neutrino density ∼ (10 35 -10 30 ) cm −3 at radii r ∼ (10 -100) km in a SN [37], which greatly exceeds the oscillation rate in vacuum ω = |∆m 2 |/(2E) ∼ km −1 . (We use = c = 1, expressing everything in units of length or time.) As such, fast oscillation is quite insensitive to the size or sign of the neutrino-mass-square difference ∆m 2 , stemming from an instability that can be triggered by any nonzero ω [25].
Neutrino distributions along different directions, If the ν µ,τ andν µ,τ flavors are almost identical (hereafter denoted as ν x ), as motivated by the much lower µ ± and τ ± densities than those of e ± , the criterion for instability is met if the ν e andν e distributions are equal along some direction(s) [25][26][27][28][29][30][31][32][33][34][35]. Fig. 1 shows a sketch of the decoupling region in the SN. The different neutrino flavors have hierarchical interaction rates, and they kinetically decouple at R νe > Rν e > R νx . In the decoupling region, this can produce relative forward excesses in the fluxes of ν x overν e , andν e over ν e [38][39][40][41][42], as shown in the schematic polar plots. This allows the ν e andν e distributions to develop a crossing, as believed to be required for the fast instability. The SN neutrino decoupling region with illustrative polar plots of angle-dependent neutrino distributions, initially (thick dashed ellipses) with a forward-excess ofνe (green) over νe (red), producing a crossing, and of νx (blue) overνe, and finally (thin ellipses) their differences reduced due to depolarization.
Although the triggering and initial growth of fast oscillations are understood [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36], owing to its complicated nonlinear evolution [29,33,36], the final impact is not yet known. Stellar explosion and the neutrino signal are sensitive to the processed flavor-dependent fluxes, and the required neutrino theory prediction is lacking. In this work, we address this crucial theoretical and phenomenological obstacle and pave a clear path forward. We present a theory that explains how, when, and to what extent do the flavor differences change.
For two flavors, say e and µ, the final distributions after depolarization can be written as where the depolarization factor f D p , which is the same for ν andν, is equal to ½ for perfect equality of distributions and 0 for no change. Values between ½ and 1 indicate effective flavor conversion. We will present an explicit formula for f D v [in Eq. (7)], assuming an azimuth-symmetric F . This result for f D v is previewed in Fig. 2. As predicted analytically, the extent of depolarization depends on the radial velocity v = cos θ and lepton asymmetry A ∝ (n νe − nν e ). In the following, we set up the problem, present our theory that leads to this result, and conclude by discussing the relevance of our results to SN physics and neutrino phenomenology.
In the fast oscillation limit, we neglect the vacuum and matter term in Eq.(2), compared to the neutrino selfinteraction term. The self-term then enters the Hamiltonian only through the difference of distributions integrated over "energy" [25], defined by the electron lepton number (ELN) distribution G v = +∞ −∞ dω g ω, v , and the equation for S ω, v becomes essentially ω-independent. For locally azimuth-symmetric ELNs, Eq.(2) becomes where v is the radial velocity and µ 0 is the collective potential. Initial conditions are S ω, v | ini = +ê 3 and Eq. (3) is the same for all ω, so ν andν have identical solutions. In our algebra, hereafter, t = µ 0 t and z = µ 0 z, which are dimensionless. For concreteness, ELNs are taken to be piecewise-constant with one crossing at v = 0, and the lepton asymmetry A = +1 −1 dv G v takes values in {0.0, 0.2, 0.4, 0.5, 0.6, 0.9}. For our numerical examples, we solve Eq.(3) with µ 0 = 33 cm −1 , corresponding to n ν ≈ 5 × 10 33 cm −1 . Modes are seeded at z = 0 with periodic boundary conditions on z ∈ (−3, +3) cm. The numerical method and error estimates are as in [36].
where T n = n+1 2n+1 M n+1 + n 2n+1 M n−1 that approximates to M n + ∂ n M n /(2n + 1) + ∂ 2 n M n /2 in the continuum limit of the discrete variable n [7]. After dotting Eq.(5) with M n , approximating |M n × M 1 | M n · M 1 , using 2n + 1 ≈ 2n, and averaging over z, one finds for large n: where M n denotes spatial average of M n = |M n |, which we call "power" in the multipole. Eq.(6) is a diffusionadvection equation where n plays the role of space and M 1 of the diffusion constant. G v and initial conditions for S v are smooth in v, so that M n are initially small for n 1. As time passes, the power in lower multipoles diffuses to higher values of n.
One can obtain an analytical solution to the above partial differential equation if M 1 is approximately constant. First we note that Eq.(6) remains invariant under the scaling n → an and t → a 2 t with a > 0. Therefore, the solution for M n in Eq.(6) can depend on n and t only through the scaling variable ξ = n 2 /t. Using ξ as the independent variable, Eq. to M n fin at large times. The finite behavior at large t is crucial to be able to truncate the multipole expansion. The solution shows that kinematic decoherence has a strong dependence on M 1 , which is 1 − A/2 initially. Thus, for small lepton asymmetry the effective diffusion coefficient M 1 is larger. Further, shrinking of M 1 results in less kinematic decoherence at later times, and as time progresses the system reaches an almost steady state with no further diffusion in multipole space. On the other hand for larger lepton asymmetry, i.e., smaller M 1 ini , there is less diffusion and depolarization throughout.
To verify the above analytical solution, we numerically solve Eq.(3) for our suite of ELNs. In Fig. 3, we show an illustrative result for S v , M 1 for all the ELNs, and various M n for A = 0.2. The top left panel shows how the flavor composition, even for a single v mode, is scrambled within picoseconds and sub-mm distances. In the right panel, we see M 1 is approximately constant at early and late epochs, but decreases at t ≈ 4 ps. We will explain the decrease in just a moment, but using the approximately constant M 1 in our analytical solutions for M n , we find qualitative agreement with the numerical results shown in the bottom panel. The sharp change in M 1 ini at t ≈ 4 ps prevents a perfect agreement. One can also see that the higher multipoles (fainter curves) peak one-by-one, each rising exponentially and falling asymptotically, diffusing as per our theory.
Transverse Relaxation.-For the lower-n multipoles, e.g., M 1 , the preceding discussion does not apply and there is a different way to understand their depolarization. Kinematic decoherence is controlled by the Hamil- tonian, which is H v ≈ − 1 3 M 0 + vM 1 for our ELNs in a frame corotating with the M 0 -M 1 plane [36]. Naively, the spatial average of Eq.
which can be visualized as a spin S v precessing around the magnetic field H v , implying that S v remains constant. However, the above conclusion is incorrect, and the length of S v in fact becomes smaller. Initially, S v is alongê 3 , and it starts tilting away due to the action of H vac ω . So, H ⊥ v starts growing as well. The motion has been shown to be similar to an inverted pendulum [6,29,35,36] and gives spectral-swap like features [8,36]. If H ⊥ v becomes of the same order as H v , then S v makes a large precession angle and reaches the transverse plane. Here, the averaging procedure does not factorize over the cross product; S v at different spatial locations relatively dephase and S ⊥ v shrinks [36]. As S v swings past the plane, S v , and thus M 1 , has shrunk. This mechanism is called T2 relaxation in the context of magnetic resonance imaging [43]. Comparing the top and bottom panels in Fig. 4, one sees that S v shrinks almost exactly when H ⊥ v ≈ H v . Fig. 4 shows that the v < 0 modes, for which H ⊥ v overshoots H v , are depolarized completely and S v<0 → 0. For v > 0, the relaxation is less prominent, especially when A is large. To zeroth order in v, one has S v>0 | fin ≈ A, where we use S v<0 | fin → 0 and enforce conservation of lepton asymmetry. For our chosen form of G v , it further implies that M 1 | fin ≈ A/2, as opposed to its initial value 1 − A/2. The above was for A > 0 and a forward excess. For ELNs with a backward excess and/or A < 0, similar arguments apply.
Depolarization.-To quantify the effect of relaxation we define the depolarization factor as the relative reduction in the flavor purity of each Bloch vector, For flavor-pure initial conditions, S v ini = 1. As noted, f D v is 0 (½) when there is no (perfect) depolarization, and lies between ½ and 1 if there is effective conversion to the other flavor.
The extent of depolarization can be readily found. For positive lepton asymmetry, A > 0, the negative velocity modes are almost completely depolarized, so clearly f D v ≈ ½. For positive velocity modes the functional behavior of f D v can be obtained by using the multipole expansion: , dropping the higher multipoles. As we found, M 0 = A is a constant in time but M 1 flips from 1−A/2 to A/2. This brings us to the promised formula for the depolarization factor that was shown in Fig. 2: dropping the higher multipoles. For ELNs with a backward excess and/or A < 0 the analogous formula for f D v is easy to obtain using the mirror symmetries +v ↔ −v and +G and a rescaling of µ 0 [36]. Summary & Outlook.-We have presented an analytical theory of fast neutrino flavor conversions in the nonlinear regime. We showed how, as time passes, flavor differences over large ranges of velocity diffuse into variations over smaller velocity ranges, or equivalent ranges of emission angles, causing depolarization. Coarse-graining, by averaging over a small spatial volume and over small ranges of v, introduces loss of information that leads to an apparent arrow of time out of the time-reversible Eq.
(2). f D v used in Eq.(1) must be understood in a spatially averaged sense. Without coarse-graining there cannot be irreversibility. These features, including both v > 0 and v < 0 modes, are carefully verified using our state-ofthe-art numerics [36]. We then showed that the epoch of T2 relaxation determines when depolarization occurs, and the initial lepton asymmetry A determines the rate of flavor depolarization. Finally, we gave a strategy and a formula for computing the extent of depolarization, which is the ultimate outcome for fast collective oscillations pointed out by Sawyer [22][23][24]. Unlike the Landau-Zener conversion [44][45][46][47] for Mikheyev-Smirnov-Wolfenstein effect in ordinary matter [48,49], this encodes decoherence and not flavor conversion. While there are many ways of adding more layers of complication, we hope to have clarified the core underlying physics, and given a method that is quite universal and generic.
The neutrino flux after suffering fast conversions can be determined using the depolarization factor f D v . In a SN, these fluxes are responsible for heating and cooling processes [50]. The net heating rateQ that is responsible for shock revival depends on the product of cross section σ α ∝ E 2 α and luminosity L α ∝ v E α F α , with theν e and ν e dominating owing to their larger cross sections [51]. It is clear that depolarization can changeQ, becauseν e and ν e energies move closer to that of ν x , and the increase proportional to E 3 νx /E 3 νe can be quite large [52]. Including the effects of subsequent slow collective oscillations [5,12], MSW conversions, propagation and earth effects [53], allows one to determine the final neutrino signal from a SN explosion. These can be measured at current and upcoming neutrino telescopes and may provide a remarkable way to directly test neutrino-neutrino interactions [54][55][56]. These Standard Model interactions have never been directly tested in a laboratory. Of course, a variety of other particle physics and astrophysics information may be gleaned from such a signal [57][58][59][60][61][62][63][64]. In many such analyses, knowing f D v is important. For the first time, our work provides this crucial input.
What lies ahead? For the more exciting, one could use this set-up as a test of possible secret neutrino-neutrino interactions [65], that have been proposed as a solution to the Hubble tension [66,67]. Collective flavor conversions may also occur in the disk of merging neutron stars [11,68,69]. These possibilities are not yet fully explored. Sticking to basics, however, several improvements, extensions, and applications are possible. Threeflavor effects were ignored here [10,32,70,71]. It will be interesting to see if our approach can be extended to include higher order terms in v and A, break the azimuthal symmetry, and include more complicated ELNs. These are important, but won't qualitatively change the picture we painted. As regards experiments, the diffuse SN neutrino background may soon become detectable [72,73], and hopefully the next galactic SN is not too far in the future [74][75][76]. These effects may also have observable impact on the neutron star merger events at LIGO [77]. It is therefore of paramount importance that predictions for neutrinos are put on a firm footing and the experiments are well-prepared [78,79], so that we can reliably extract all the physics out of these once-in-a-lifetime events. Acknowledgements