---
abstract: '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 $15\,M_\odot$ 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 Mikheyev-Smirnov-Wolfenstein effect. In this case one may distinguish the neutrino mass hierarchy in a SN neutrino signal if the mixing angle $\theta_{13}$ is as large as suggested by recent experiments.'
author:
- Srdjan Sarikas
- 'Georg G. Raffelt'
- Lorenz Hüdepohl
- 'Hans-Thomas Janka'
date: 15 September 2011
title: 'Suppression of Self-Induced Flavor Conversion in the Supernova Accretion Phase'
---

[*Introduction.*]{}—The huge neutrino fluxes emitted by core-collapse supernovae (SNe) are key to the explosion dynamics and nucleosynthesis [@Janka:2006fh] and detecting a high-statistics “neutrino light curve” from the next nearby SN is a major goal for neutrino astronomy [@Autiero:2007zj]. Besides probing the core-collapse phenomenon in unprecedented detail, one may detect signatures of flavor oscillations and extract information on neutrino mixing parameters [@Dighe:1999bi; @Chakraborty:2011ir].

The refractive effect caused by matter [@Wolfenstein:1977ue] suppresses flavor oscillations until neutrinos pass through the Mikheyev-Smirnov-Wolfenstein (MSW) region in the collapsing star’s envelope [@Mikheev:1986gs; @Kuo:1989qe]. However, neutrino-neutrino interactions, through a flavor off-diagonal refractive index [@Pantaleone:1992eq; @Sigl:1992fn], can trigger self-induced flavor conversions [@Samuel:1993uw; @Sawyer:2005jk; @Sawyer:2008zs]. This collective effect usually occurs between the neutrino sphere and the MSW region and can strongly modify neutrino spectra [@Duan:2006an; @Dasgupta:2009mg; @Duan:2010bg], although this would never seem to help explode the star [@Dasgupta:2011jf]. Actually, in low-mass SNe (not studied here) the density falls off so fast that MSW can occur first, leading to novel effects on the prompt $\nu_e$ burst [@Duan:2007sh].

Collective oscillations at first seemed unaffected by matter because its influence does not depend on neutrino energies [@Duan:2006an]. However, depending on emission angle, neutrinos accrue different matter-induced flavor-dependent phases until they reach a given radius. This “multi-angle matter effect” can suppress self-induced flavor conversion [@EstebanPretel:2008ni]. Based on schematic flux spectra, this was numerically confirmed for accretion-phase SN models where the density near the core is large [@Chakraborty:2011gd]. This epoch, before the delayed explosion finally takes off, is when the neutrino luminosity and the difference between the $\bar\nu_e$ and $\bar\nu_{\mu,\tau}$ fluxes are largest. If self-induced flavor conversion did not occur and the mixing angle $\theta_{13}$ was not very small [@Fogli:2011qn], the accretion phase would provide a plausible opportunity to determine the mass hierarchy [@Dighe:1999bi; @Chakraborty:2011gd].

Numerical multi-angle simulations of collective oscillations are very demanding [@Duan:2008eb], but it is much easier to study if such oscillations are suppressed for given density profile and neutrino distributions. Self-induced conversion requires that part of the spectrum is prepared in one flavor, the rest in another. The collective mode consists of pendulum-like flavor exchange between these parts without changing the overall flavor content [@Samuel:1993uw; @Hannestad:2006nj]. The inevitable starting point is a flavor instability of the neutrino distribution caused by neutrino-neutrino refraction. An exponentially growing mode can be detected with a linearized analysis of the evolution equations [@Sawyer:2008zs; @Banerjee:2011fj]. We here apply this method to a numerical accretion-phase SN model, for the first time using both realistic neutrino energy spectra and angular distributions.

[*Numerical SN model.*]{}—Our spherically symmetric simulation of an accretion-phase SN model was performed with the [Prometheus-Vertex]{} code as in Ref. [@Huedepohl:2009wh], now with a $15\,M_\odot$ progenitor [@Woosley:1995ip]. The transport module computes the energy and angle distributions of $\nu$ and $\bar\nu$ for all flavors by a tangent-ray discretization of the Boltzmann transport equation [@Rampp:2002bq]. We used 21 nearly geometrically spaced energy bins up to 380 MeV and 672 tangent rays. We do not artificially trigger an explosion, but otherwise our model is comparable to Ref. [@Chakraborty:2011gd]. We use several snapshots and illustrate our findings with one at 280 ms post bounce. The flux spectra (Fig. \[fig:fluxes\]) show a $\nu_e$ excess from deleptonization and a $\bar\nu_e$ flux almost twice that of $\nu_x$ (representing any of $\nu_{\mu,\tau}$ or $\bar\nu_{\mu,\tau}$) and average $\nu_e$, $\bar\nu_e$ and $\nu_x$ energies of 15.3, 18.1, and 16.9 MeV.

We study neutrino propagation in the free-streaming limit, so we can describe the angular distribution by the angle $\vartheta_R$ relative to the radial direction at a chosen inner-boundary radius $R$. Actually it is more convenient to use $u=\sin^2\vartheta_R=(1-\cos^2\vartheta_r)\,r^2/R^2$, which is uniformly distributed on $0\leq u\leq1$ if emission is isotropic at a “neutrino sphere” with radius $R$ [@EstebanPretel:2008ni; @Banerjee:2011fj]. We choose $R=44.7$ km and show the corresponding $u$ distribution in Fig. \[fig:fluxes\]. Isotropic emission from a neutrino sphere is not a good description because neutrinos emerge from a thick layer. The $\bar\nu_e$ and $\nu_x$ intensities are similar in the radial direction: the excess $\bar\nu_e$ flux largely arises from its broader angular distribution (larger emission region). Flavor oscillations depend on the difference of the $e$ and $x$ distributions, which is small in the radial direction (Fig. \[fig:fluxes\]). The angular distributions do not cross, although in principle there could have been a forward $\nu_x$ excess.

![Flux spectra for our 280 ms SN model. The angle variable $0\leq u\leq1$ is based on $R=44.7$ km. \[fig:fluxes\]](distrib_energy_280_ms-eps-converted-to.pdf "fig:"){width="0.75\columnwidth"} ![Flux spectra for our 280 ms SN model. The angle variable $0\leq u\leq1$ is based on $R=44.7$ km. \[fig:fluxes\]](distrib_u_280_ms-eps-converted-to.pdf "fig:"){width="0.75\columnwidth"}

In the context of neutrino oscillations, $\omega=\Delta m^2/2E$ is a preferred energy variable, where $\Delta m^2=(50~{\rm meV})^2$ is the “atmospheric” neutrino mass-squared difference relevant for 1–3 oscillations studied here. Moreover, treating anti-neutrinos formally as negative-energy neutrinos with negative occupation numbers vastly simplifies the formalism. Flavor oscillations can exchange $\nu_e$ with $\nu_x$, leaving the overall neutrino flux unchanged, so only $F_{\nu_e}{-}F_{\nu_x}$ matters. Our sign convention means that for anti-neutrinos we then use $F_{\bar\nu_x}-F_{\bar\nu_e}$, corresponding to the flavor isospin convention [@Duan:2006an]. The neutrino flux difference distribution $g(\omega,u)$ thus defined is shown in Fig. \[fig:g\_distribution\]. It is negative for anti-neutrinos ($\omega<0$) because $F_{\bar\nu_e}>F_{\bar\nu_x}$. For $\omega\sim0.2~{\rm km}^{-1}$ there is a spectral crossing as a function of $u$, i.e. for large $E$ the $\nu_x$ flux does exceed the $\nu_e$ flux in the forward direction.

![Distribution $g(\omega,u)$ describing the neutrino fluxes.\[fig:g\_distribution\]](distrib_all_280_ms-png-converted-to.pdf){width="0.72\columnwidth"}

Self-induced oscillations exchange the positive and negative parts of $g(\omega,u)$, leaving fixed the overall flavor content $\varepsilon=(F_{\nu_e}{-}F_{\nu_x})/(F_{\bar\nu_e}{-}F_{\bar\nu_x})-1=\int
d\omega d u g(\omega,u)$. Our $g(\omega,u)$ is mostly negative for $\bar\nu$ and mostly positive for $\nu$, so collective oscillations largely correspond to pair conversions $\nu_e\bar\nu_e\leftrightarrow\nu_x\bar\nu_x$. Accretion-phase distributions are “single crossed” in this sense, i.e.$g(\omega,u)$ changes sign essentially only on the line $\omega=0$, because of the large excess of the $\nu_e$ and $\bar\nu_e$ fluxes. Significant multiple crossings are typical for the cooling phase [@Dasgupta:2009mg].

[*Equations of motion (EoM).*]{}—We describe three-flavor neutrino propagation by energy- and angle-dependent $3{\times}3$ matrices ${\bf\Phi}_{E,u}(r)$. Boldface characters denote matrices in flavor space. The diagonal ${\bf\Phi}_{E,u}$ elements are the ordinary number fluxes $F_{E,u}^\alpha$ (flavor $\alpha$) integrated over a sphere of radius $r$, with negative $E$ and negative number fluxes for anti-neutrinos. The off-diagonal elements, which are initially zero, represent phase information caused by flavor oscillations. The flavor evolution is then provided by the “Schrödinger equation” ${\rm i}\partial_r{\bf\Phi}_{E,u}=[{\bf H}_{E,u},{\bf\Phi}_{E,u}]$ with the Hamiltonian [@EstebanPretel:2008ni] $$\begin{aligned}
\label{eq:hamiltonian}
{\bf H}_{E,u}&=&\frac{1}{v_{u}}\,\left(\frac{{\bf M}^2}{2E}+
\sqrt{2}\,G_{\rm F}{\bf N}_\ell\right)\\
&+&\frac{\sqrt{2}\,G_{\rm F}}{4\pi r^2}\int_{-\infty}^{+\infty}dE'
\int_0^1 du'
\frac{1-v_{u}v_{u'}}{v_{u}v_{u'}}\,{\bf\Phi}_{E',u'}\,.
\nonumber\end{aligned}$$ The matrix ${\bf M}^2$ of neutrino mass-squares causes vacuum flavor oscillations and that of net charged lepton densities ${\bf
N_\ell}={\rm diag}(n_e{-}n_{\bar
e},n_\mu{-}n_{\bar\mu},n_\tau{-}n_{\bar\tau})$ adds the Wolfenstein matter effect. The third term provides neutrino-neutrino refraction and is analogous to matter except for Pantaleone’s off-diagonal elements and except that in the SN context neutrinos are not isotropic. A neutrino radial velocity at radius $r$ is $v_u=(1-u\,R^2/r^2)^{1/2}$. The factor $1-v_u v_{u'}$ arises from the current-current nature of the weak interaction and causes multi-angle effects. Moreover, $v_u$ appears in the denominator because we follow the flavor evolution projected on the radial direction, causing the multi-angle matter effect [@EstebanPretel:2008ni].

Up to the MSW region, the matter effect is so large that ${\bf\Phi}_{E,u}$ is very nearly diagonal in the weak-interaction basis, the usual approximation made in SN neutrino transport. Neutrinos remain stuck in flavor eigenstates unless the off-diagonal ${\bf\Phi}_{E,u}$ elements start growing by the self-induced instability. To find the latter we linearize the EoM in the small off-diagonal amplitudes.

[*Stability condition.*]{}—We study the instability driven by the atmospheric $\Delta m^2$ and the mixing angle $\theta_{13}\ll1$, we work in the two-flavor limit, and switch to the $\omega=\Delta
m^2/2E$ variable. We write the flux matrices in the form $${\bf\Phi}_{\omega,u}=\frac{{\rm Tr}\,{\bf\Phi}_{\omega,u}}{2}
+\frac{F_{\omega,u}^e-F_{\omega,u}^x}{2}\,
\begin{pmatrix}
s_{\omega,u}&S_{\omega,u}\\
S_{\omega,u}^*&-s_{\omega,u}
\end{pmatrix}\,,$$ where $F_{\omega,u}^{e,x}$ are the flavor fluxes at the inner boundary radius $R$. The flux summed over all flavors, ${\rm
Tr}\,{\bf\Phi}_{\omega,u}$, is conserved in our free-streaming limit. The $\nu_e$ survival probability is $\frac{1}{2}[1+s_{\omega,u}(r)]$ in terms of the “swap factor” $-1\leq s_{\omega,u}(r)\leq1$. The off-diagonal element $S_{\omega,u}$ is complex and $s^2_{\omega,u}+|S_{\omega,u}|^2=1$.

The small-amplitude limit means $|S_{\omega,u}|\ll1$ and to linear order $s_{\omega,u}=1$. Assuming in addition a large distance from the source so that $1-v_u\ll 1$, the evolution equation linearized in $S_{\omega,u}$ and in $u$ is [@Banerjee:2011fj] $$\begin{aligned}
\label{eq:smallEoM}
{\rm i}\partial_r S_{\omega,u}&=&
\left[\omega+u(\lambda+\varepsilon\mu)\right]S_{\omega,u}
\nonumber\\
& &-\mu \int du'\,d\omega'\,(u+u')\,g_{\omega'u'}\,S_{\omega',u'}\,.
\label{stability-eom}\end{aligned}$$ Weak-interaction effects are encoded in the $r$-dependent parameters with dimension energy $$\begin{aligned}
\lambda&=&\sqrt{2}\,G_{\rm F}\,[n_e(r)-n_{\bar e}(r)]\,\frac{R^2}{2r^2}\,,
\nonumber\\
\mu&=&\frac{\sqrt{2}\,G_{\rm F}\,[F_{\bar\nu_e}(R)-F_{\bar\nu_x}(R)]}{4\pi r^2}
\,\frac{R^2}{2r^2}\,.\end{aligned}$$ The factor $R^2/2r^2$ signifies that only the multi-angle impact of the $\nu$-$\nu$ and matter effects are relevant for the stability analysis, not the densities themselves. Both $\lambda$ and $\mu$ depend on $R$, but so does the occupied $u$-range: physical results do not depend on the choice of $R$. We choose $R=44.7$ km such that the occupied $u$-range is 0–1. We normalize the $\nu$-$\nu$ interaction strength $\mu$ to the $\bar\nu_e$-$\bar\nu_x$ flux difference at $R$, i.e. $\int_{-\infty}^0 d\omega\int_0^1 du\,g_{\omega,u}=-1$, but the only physically relevant quantity is $\mu(r)\,g_{\omega,u}$. Our SN model provides $\mu(R)=1.73 \times 10^{4}~{\rm km}^{-1}$ and an “asymmetry” $\varepsilon=\int d\omega\,
du\,g_{\omega,u}=0.35$.

Writing solutions of the linear differential equation, Eq. (\[eq:smallEoM\]), in the form $S_{\omega,u}=Q_{\omega,u}\,e^{-{\rm i}\Omega r}$ with complex frequency $\Omega=\gamma+{\rm i}\kappa$ and eigenvector $Q_{\omega,u}$ leads to the eigenvalue equation [@Banerjee:2011fj], $$\label{fourier-eom}
(\omega + u \bar\lambda - \Omega)\, Q_{\omega,u}=
\mu \int du'\,d\omega'\,(u+u')\,g_{\omega'u'}\,Q_{\omega',u'}\,,$$ where $\bar\lambda \equiv \lambda + \varepsilon \mu$. The solution has to be of the form $Q_{\omega,u}=(A+Bu)/(\omega+u\bar\lambda-\Omega)$. Solutions exist if $\mu^{-1}=I_1\pm\sqrt{I_0I_2}$, where $I_n=\int
d\omega\,du\,g_{\omega,u}\,u^n/(\omega+u\bar\lambda-\Omega)$. The system is stable if all $\Omega$ are purely real. A possible imaginary part, $\kappa$, is the exponential growth rate.

[*Application to our SN model.*]{}—Ignoring the effect of matter ($\lambda=0$), we show $\kappa(\mu)$ for our 280 ms SN model in Fig. \[fig:kappa\]. The system is essentially stable above $\mu$ of a few 100 km$^{-1}$. It is noteworthy that $\kappa$ is of the same order as a typical $\omega$ of the $g_{\omega,u}$ distribution, in our case a few km$^{-1}$. We also show $\kappa(\mu)$ for $\lambda=10^2$ and $10^3$ km$^{-1}$ and observe a shift to larger $\mu$-values [@Banerjee:2011fj].

![Growth rate $\kappa$ for our SN model as a function of $\mu$ for various $\lambda$ values as indicated.\[fig:kappa\]](kappa-eps-converted-to.pdf){width="0.78\columnwidth"}

![Contours for the growth rate $\kappa$ in km$^{-1}$. Also shown is the profile for our SN model. The vertical axis essentially denotes the density, the horizontal axis the radius ($\mu\propto r^{-4}$). \[fig:trajectory\]](vision_280-png-converted-to.pdf){width="0.78\columnwidth"}

-12pt

In Fig. \[fig:trajectory\] we show contours of $\kappa$ in the $(\mu,\lambda)$ plane. For large $\mu$ and $\lambda$ values, the system is unstable for $\lambda\sim\mu$ [@Banerjee:2011fj]. In other words, if the local neutrino number density is much smaller or much larger than the local electron density, the system is stable.

We also show the locus of $[\mu(r),\lambda(r)]$ along the radial direction. Since $\mu(r)\propto r^{-4}$, the red solid line in Fig. \[fig:trajectory\] is essentially the SN density profile. The step-like feature is the shock wave where the matter density drops by an order of magnitude. Without matter ($\lambda=0$), neutrinos would enter the instability strip at $\mu\sim100$, corresponding to $r\sim
150$ km. We find similar results for other snapshots at times 150 and 400 ms postbounce, i.e., neutrinos do not encounter a self-induced instability.

[*Comparison with earlier work.*]{}—A similar accretion-phase model of the Basel group was used to study flavor stability by numerically solving the EoMs [@Chakraborty:2011gd]. A mono-energetic neutrino distribution and isotropic emission at a neutrino sphere were assumed. For some snapshots, an instability occurred at a large radius. Applying our method to the same matter profile and schematic neutrino distribution we find perfect agreement with Ref. [@Chakraborty:2011gd] and even reproduce their onset radius for those cases where partial flavor conversion occurs [@Sarikas:2011jc]. It would be interesting to repeat our study with realistic Basel distributions to see if partial flavor conversion is an artifact of their schematic energy and angle distributions.

[*Conclusions.*]{}—We have performed a linearized flavor stability analysis of accretion-phase SN models and neutrino fluxes with realistic energy and angle distributions. For these models, self-induced flavor conversions do not occur. One should apply this method to a broader class of models to see if this conclusion is generic. It also remains to extend a linearized analysis to cases without cylindrical symmetry of the angular distribution in view of Sawyer’s concerns about a significant multi-angle instability [@Sawyer:2008zs]. In realistic 3D models, the neutrino distribution is not cylindrically symmetric and even if this were the case, in principle even a small fluctuation could trigger a novel instability if it were to exist.

Recent evidence suggests that the neutrino mixing angle $\theta_{13}$ is not very small [@Fogli:2011qn], implying that the MSW conversion in the SN is adiabatic. One can then distinguish the neutrino mass hierarchy by Earth matter effects [@Dighe:1999bi] or the early rise time [@Chakraborty:2011ir], but only if collective oscillations do not swap flavors before the MSW region. (In the presence of the collective flavor swap, the hierarchy could be distinguished for an extremely small $\theta_{13}$ where the MSW conversion is not adiabatic [@Dasgupta:2008my].) The suppression of self-induced flavor conversion during the accretion phase, if generic, is good news for the possibility of determining the mass hierarchy with SN neutrinos if a “large” value for $\theta_{13}$ is experimentally confirmed.

We thank T. Hahn for helping to implement a numerical library [@Hahn:2004fe]. We acknowledge partial support by DFG Grants No. TR 7, TR 27, EXC 153 and computer time at the HLRS in Stuttgart and NIC in Jülich.

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