Self-induced spectral splits in supernova neutrino fluxes
In the dense-neutrino region above the neutrino sphere of a supernova ( km), neutrino-neutrino refraction causes collective flavor transformations. They can lead to “spectral splits” where an energy splits the transformed spectrum sharply into parts of almost pure but different flavors. Unless there is an ordinary MSW resonance in the dense-neutrino region, is determined by flavor-lepton number conservation alone. Spectral splits are created by an adiabatic transition between regions of large and small neutrino density. We solve the equations of motion in the adiabatic limit explicitly and provide analytic expressions for a generic example.
At large densities, neutrino-neutrino refraction causes nonlinear flavor oscillation phenomena with sometimes perplexing results Pantaleone:1992eq (); Samuel:1993uw (); Samuel:1996ri (); Qian:1995ua (); Fuller:2005ae (); Pastor:2001iu (); Pastor:2002we (); Duan:2005cp (); Duan:2006an (); Duan:2006jv (); Hannestad:2006nj (); Duan:2007mv (); Mirizzi2007 (); Raffelt:2007yz (); EstebanPretel:2007ec (). In the region between the neutrino sphere and a radius of about 400 km in core-collapse supernovae (SNe), the neutrino flavor content evolves dramatically Pastor:2002we (); Duan:2005cp (); Duan:2006an (); Duan:2006jv (); Hannestad:2006nj (); Duan:2007mv (); Mirizzi2007 (). The global features of this self-induced transformation are equivalent to the motion of a gyroscopic pendulum in flavor space Hannestad:2006nj (); Duan:2007mv (). However, this picture does not explain the “spectral splits” that have been numerically observed in the transformed fluxes Duan:2006an (); Duan:2006jv (); Mirizzi2007 (). In a typical case, the primary flux below a split energy emerges from the dense-neutrino region in its original flavor, whereas above , it is completely transformed to (some mixture of and ), the step at being very sharp. (To be specific we explore the – system with the atmospheric and the small 13-mixing angle.)
It has been suggested that an adiabatic transition from high to low neutrino density is the primary cause for the split Duan:2006an (); Duan:2007mv (). Dense neutrinos perform synchronized oscillations: all modes oscillate with a common frequency , even though their individual frequencies vary as . Flavor oscillations can be visualized as the precession of polarization vectors in a “flavor field.” The “stick together” by the ––interaction, thus forming a collective object that precesses around . The collectivity is lost when the neutrino density decreases. However, if the decrease is slow, all align themselves with or against in the process of decoupling from each other. Eventually they all precess with their individual around , but without visible consequences because of their (anti-)alignment with .
We extend this interpretation of the split phenomenon in several ways. We (i) show that flavor-lepton number conservation determines , (ii) solve the equations of motion explicitly in the adiabatic limit, and (iii) provide an analytic result for a generic case.
Ii Equations of motion
We represent the flavor content of an isotropic – gas by flavor polarization vectors and , where overbarred quantities correspond to . We define their global counterparts as and and introduce , representing the net lepton number. The equations of motion (EOMs) are Hannestad:2006nj (); Sigl:1992fn ()
and the same for with . Here represents the usual matter potential and the – interaction strength, where and are the electron and neutrino densities. We work in the mass basis where corresponds to the normal and to the inverted mass hierarchies. The interaction direction is a unit vector such that with being the vacuum mixing angle. Unless there is an MSW resonance in the dense-neutrino region, one can eliminate from Eq. (1) by going into a rotating frame, at the expense of a small effective mixing angle Duan:2005cp (); Hannestad:2006nj (). The only difference for antineutrinos is that in vacuum they oscillate “the other way round.” Therefore, instead of using we may extend to negative frequencies such that () and use only with . In these terms, , where .
After elimination of , the EOM for can be obtained by integrating Eq. (1) with :
It shows that so that is conserved Hannestad:2006nj (). The in-medium mixing angle above a SN core is small and therefore the mass and interaction basis almost coincide. Collective effects then only induce pair transformations of the form , whereas the excess flux from deleptonization is conserved.
Iii Adiabatic solution
We rewrite the EOMs in terms of an “effective Hamiltonian” for the individual modes as
In the adiabatic limit each moves slowly compared to the precession of so that the latter follows the former. We assume that initially all represent the same flavor and thus are aligned. If initially is large, every is practically aligned with . Therefore, in the adiabatic limit it stays aligned with for the entire evolution:
which solves the EOMs. Here and is a unit vector. Here and henceforth we assume an excess flux of neutrinos over antineutrinos, implying that initially and are collinear and .
According to Eq. (3) all lie in the plane spanned by and which we call the “co-rotating plane.” In the adiabatic limit all , and consequently , also stay in that plane. Therefore we can decompose
and rewrite the EOM of Eq. (2) as
Therefore and the co-rotating plane precess around with the common or “co-rotation frequency” .
We conclude that the system evolves simultaneously in two ways: a fast precession around determined by and a drift in the co-rotating plane caused by the explicit variation. To isolate the latter from the former, we go (following Ref. Duan:2005cp ()) into the co-rotating frame where the individual Hamiltonians become
We use the same notation because the relevant components , , , and remain invariant.
Initially () the oscillations are synchronized, , and all form a collective . As decreases, the zenith angles spread out while remaining in a single co-rotating plane. In the end () the co-rotation frequency is and Eqs. (4) and (7) imply that all final and therefore all with are aligned with , the others anti-aligned: a spectral split is inevitable with being the split frequency. The lengths are conserved and eventually all point in the directions. Therefore the conservation of flavor-lepton number gives us , for , by virtue of
In general, .
For individual modes the EOMs given by are completely solved if we find and , the component transverse to , since is conserved and given by the initial condition. From Eq. (4) we infer , from Eq. (7) and so that
Integration of the second equation over gives us
Projecting Eq. (5) on the ––plane we find or explicitly
For large when the oscillations are synchronized, this agrees with the usual expression for Pastor:2001iu (), but it changes when the spread out in the zenith direction. Inserting Eq. (10) into Eq. (12) we find
We have assumed that all are initially aligned. One can relax this restriction and allow some to have opposite orientation. If different species are emitted from a SN core with equal luminosities but different average energies, the spectra will cross over so that some range of modes is prepared, say, as and another as .
Iv Neutrinos only
We illustrate the power of our new results with a generic neutrino-only example (). The spectrum is taken box like with for and 0 otherwise. With being conserved we find from Eq. (8)
The case is special because remains fixed. For we have and no flavor evolution. We use to show the initial and final in Fig. 1 (left). The dotted line denotes the adiabatic final state where . The solid line is from a numerical solution of the EOMs with and , typical for a SN. We have checked numerically that the split indeed becomes sharper with increasing and thus increasing adiabaticity.
In Fig. 2 we show for 51 individual modes. They start with the common value . Later they spread and eventually split, some of them approaching and the others . Some modes first move down and then turn around as changes. A few modes do not reach because of imperfect adiabaticity.
where . For and the limits of agree with Eq. (14) from lepton-number conservation. For we obtain , representing the initial condition , and for we find .
With Eq. (9) these results provide analytic solutions for the adiabatic . We show examples in Fig. 2 (bottom left) for comparison with the numerical solution of the EOMs. The agreement is striking and confirms the picture of adiabatic evolution in the co-rotating plane. The agreement is poor for modes close to the split () at low neutrino densities () where the evolution becomes nonadiabatic.
V Adiabaticity condition
For our neutrino-only () box spectrum Eqs. (IV) give and . For we obtain and so that the last term in the numerator of Eq. (16) dominates: . With decreasing, increases and at when , adiabaticity violation begins. For the denominator of Eq. (16) gives the dependence , and therefore the closer to the stronger the adiabaticity violation.
Vi Including antineutrinos.
As a second generic case we now add antineutrinos. One important difference is that even a very small initial misalignment between and is enough to cause a strong effect. Consider a single energy mode for with and one for with that are initially aligned in the flavor direction, now taken very close to the mass direction, and assume an inverted hierarchy. From the dynamics of the flavor pendulum Hannestad:2006nj (); Duan:2007mv () we know that in the end is antialigned with , whereas retains a large transverse component because is conserved: The system prepares itself for a spectral split.
Assuming box spectra for both and , we show the initial and final in Fig. 1 (right), for the inverted hierarchy, , and . From Eq. (12) one infers . For this is . Therefore, all modes have negative frequencies in the co-rotating frame and tilt away from (see also the numerical in Fig. 2). The final split frequency is found from flavor lepton number conservation to be , using for . With we find in agreement with Fig. 1. For we have so that the final split always occurs among the neutrinos. According to Fig. 2 the split starts when the vector D develops a significant transverse component, and it proceeds efficiently in a region .
The “wiggles” in the curves in the right panels of Fig. 2 stem from the nutation of the flavor pendulum Hannestad:2006nj (); Duan:2007mv (). We have chosen a relatively fast evolution (), implying poor adiabaticity, to avoid too many nutation periods on the plot. For a very slow the nutations disappear and the co-rotating frame removes the full global evolution of the system.
We have studied the phenomenon of spectral splits that is caused by neutrino-neutrino refraction in the SN dense-neutrino region. We have carried previous explanations of this novel effect Duan:2006an (); Duan:2007mv () to the point of explicit solutions in the adiabatic limit.
A spectral split occurs when a neutrino ensemble is prepared such that the common direction of the flavor polarization vectors deviates from the mass direction. An adiabatic density decrease turns all modes below a split energy into the mass direction, and the others in the opposite direction. Remarkably, during this phase all modes remain in a single rotating plane, even after losing full synchronization. is determined by lepton number conservation in the mass basis.
The spectral split is a generic feature of the adiabatic evolution when the density changes from large to small values. It can appear even in the absense of neutrino-neutrino interactions. Indeed, in the usual MSW case the evolution to zero density transforms to and to for all energies. This corresponds to . The neutrino-neutrino interactions shift to non-zero values.
A spectral split is caused in the SN neutrino (but not antineutrino) flux by neutrino-neutrino interactions alone, especially during the accretion phase when ordinary MSW resonances occur far outside the dense-neutrino region. Later the matter profile may become so shallow that the H-resonance moves into this region Duan:2006an (); Duan:2006jv (); Duan:2007mv (). The simultaneous action of collective effects and an ordinary MSW resonance may then cause spectral splits for both neutrinos and antineutrinos, leading to a rich phenomenology, perhaps modifying r-process nucleosynthesis Duan:2006an (); Duan:2007mv (). Of course, the fluxes will be further processed by ordinary conversion in the SN envelope Dighe:1999bi (); Dighe:2004xy (), thus modifying observable signatures. Still, observing spectral splits would provide a smoking gun signature both for the relevant neutrino properties and, if it occurs among antineutrinos at late times, for the occurrence of a shallow density profile above the neutrino sphere.
The neutrino flux emitted by a SN is anisotropic so that neutrinos on different trajectories experience different neutrino-neutrino interaction histories Duan:2006an (); Duan:2007mv () that would be expected to cause kinematical flavor decoherence of different angular modes Raffelt:2007yz (). A numerical exploration reveals, however, that in a typical SN scenario the deleptonization flux suppresses decoherence and the evolution is almost identical to that of an isotropic ensemble EstebanPretel:2007ec (). Our treatment of the spectral evolution is apparently applicable in a realistic SN context.
Collective neutrino oscillation phenomena in a SN may well be important for the explosion mechanism, r-process nucleosynthesis and may provide detectable signatures in a high-statistics signal from the next galactic SN. Building on previous ideas, our formalism gives a simple, elegant and quantitative explanation of seemingly impenetrable numerical results. Our approach provides the basis for developing a quantitative understanding of realistic consequences of collective neutrino oscillations for SN physics and observational signatures.
Acknowledgements.We acknowledge support by the Deutsche Forschungsgemeinschaft (TR 27 “Neutrinos and beyond”), the European Union (ILIAS project, contract RII3-CT-2004-506222), the Alexander von Humboldt Foundation, and The Cluster of Excellence “Origin and Structure of the Universe” (Munich and Garching).
- preprint: MPP-2007-53
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