Damping of neutrino flavor Damping of neutrino flavor conversion in the wake of conversion in the wake of the supernova shock wavethe supernova shock wave

by G.L. Fogli, E. Lisi, D. Montanino, A. Mirizziby G.L. Fogli, E. Lisi, D. Montanino, A. Mirizzi

Based on hep-ph/0603033:Damping of supernova neutrino transitions in

stochastic shock-wave density profiles

Based on hep-ph/0603033:Damping of supernova neutrino transitions in

stochastic shock-wave density profiles

Core collapse SN’s is one of the most energetic event in nature. It corresponds to the terminal phase of a massive star which becomes instable at the end of its life. It collapses and ejects its outer mantle in a shock wave driven explosion.

ENERGY SCALES: 99% of the released energy available in the core collapse (~1053 erg) is emitted by (anti)neutrinos of all flavors with energies of order of a tenth of MeVs.

TIME SCALE: The duration of the burst lasts ~10s

EXPECTED RATE: 1-3 SN/century in our galaxy (dO(10)

kpc).

cooling

“Hot” PN star

Shock wave

Stellar core

Schirato & Fullerastro-ph/0205390

T. Totani et al., Astrophys. J. 496, 216 (1998).

Rc104km RPN102km

transitions The flavor evolution in matter is described by the MSW equation:

where:

not relevant for SNenot known but from solar & KamLAND

12

3

m122

Normal Hierarchy (NH)

m132

kH>0

12

3

Inverted Hierarchy (IH)

m122

kH<0

from solar & KamLAND

from atmospheric & K2K

ii

Neutrino potential in matter

Behind this phenomenology and neglecting Earth matter crossing, the (relevant) survival probability PeeP(ee) can be decomposed into two effective “high” (H) and “low” (L) 2 subsystems, up to small terms of the order of O(m12

2/m132, sin213):

[see, e.g., G.L. Fogli et al., PRD68 (2003) 033005, Dighe and Smirnov, PRD62, (2000) 033007] where the only dependence on the matter effect is encoded in terms of P2,H

ee.

Clearly, a time dependent potential V=V(t) induced by the passage of the shock modulates the survival probability P2,H

ee and thus leaves an “imprint” on the time spectrum of the neutrino burst.

Livermore group (Schirato & Fuller), astro-ph/0205390

Forward shock

Models of shock

Forward shock

Reverse shock

Garching group (Tomàs et al.), astro-ph/0205390

Forward shockNo reverse shock

Tokio group (Kawagoe et al.), unpublished

Our parameterization of the shock profile

Kifonidis (PhD thesis)

0 5 10 15X 109cm

t=4s

t=10s

t=20s

Stochastic scale density fluctuation of various magnitudes and correlation lengths may reasonably arise in the wake of a shock front (i.e., for r<rshock). A SN neutrino “beam” traveling to the Earth might thus experience stochastic matter effects while traversing the stellar envelope.

We shall assume L0=10km. With this hypothesis the density fluctuations can be considered “-correlated”, i.e.:

We will consider only “small” scale fluctuations, i.e., fluctuations whose correlation length is smaller than the typical oscillation wavelength in matter at resonance:

with

represents the “r.m.s” of the amplitude of the stochastic fluctuations and, in principle, =(r). Unfortunately, there is not an ab initio theory of small scale fluctuation, so we make the simplifying assumption that fluctuation arise only after the passage of the shock wave:

We conservatively assume that 4%.

Evolution in fluctuating potentialsSuppose that a system is described by an

Hamiltonian which is composed by one “deterministic” and one “stochastic” component:

where (t) is random fluctuating -correlated function:

In this case Schrödinger equation is no longer adequate to describe the evolution of the system:

[see, e.g., Balantekin et al., Burgess et al.]

“damping” term (perturbation for up to ~10%)

In our case Q=|ee|. The previous modified “Liouville” equation can be written as a “Bloch” equation in term of the polarization vector

Application to SN ’s

standard term (leading)

with , and the probability of observing an electron neutrino at distance r can be calculated as

We suppose that the fluctuations are sufficiently small to affect only the “High” subsector.

After some calculations, the probability P2,Hee in

presence of random noise can be recast as

with

where is the effective “13” mixing angle in matter.

The effect of noise is thus to suppress the MSW effect into the stellar medium. In the limit of large fluctuations, one gets P2,H

ee1/2, which correspond to a sort of complete “flavor depolarization” for the effective states in the H subsystem.

Conclusions The observation of a modulation in the survival

probability caused by the passage of the shock wave inside the exploding supernova can give us valuable information on the unknown oscillation parameters (mass hierarchy, 13) as well as on the internal structure of the exploding star

But small-scale fluctuations can partially hide this effect and cause a dangerous confusion scenario (no prompt shock? “wrong” hierarchy? too small 13?)

For this reason, a better theoretical understanding of stochastic density fluctuations behind the shock front would be of great benefit for future interpretation of SN neutrino events

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Bonus slides

() in inverse (direct) hierarchy

Analytical vs

Runge Kutta

0 1 2 33 2 1 X 108cm

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are needed to see this picture.

Kifonidis et al., Astrophys. J. Lett., 531, L123 (2000)

Evolution of entropy in SN explosion