Collective neutrino oscillations in two spatial dimensions
Shashank Shalgar
University of New MexicoWork done in collaboration with Sajad Abbar and Huaiyu Duan
July 17, 2015
Shashank Shalgar (University of New Mexico Work done in collaboration with Sajad Abbar and Huaiyu Duan )Collective neutrino oscillations in two spatial dimensions July 17, 2015 1 / 19
Motivation
I Core-collapse supernovae are one of the most intense sourcesneutrinos
I 1053 ergs (or 1058 neutrinos) are released in an interval of 10 seconds
I Core-collapse supernovae are one of the most favored sites forR-process nucleosynthesis
I Question: Do we understand neutrino flavor oscillations in the interiorof core-collapse supernovae?
Shashank Shalgar (University of New Mexico Work done in collaboration with Sajad Abbar and Huaiyu Duan )Collective neutrino oscillations in two spatial dimensions July 17, 2015 2 / 19
Neutrino Oscillations
ρ =
(〈ψ∗νeψνe 〉 〈ψ
∗νeψνµ〉
〈ψ∗νµψνe 〉 〈ψ∗νµψνµ〉
)ρ =
(〈ψ∗νe ψνe 〉 〈ψ
∗νe ψνµ〉
〈ψ∗νµψνe 〉 〈ψ∗νµψνµ〉
)
ρ(L) = e−iHLρ(0)e iHL
H =1
2
(−ω cos(2θv) ω sin(2θv)ω sin(2θv) ω cos(2θv)
)ω =
m22 −m2
1
2E
P(νe → νµ) = sin2(2θv)sin2(ω
2L)
Shashank Shalgar (University of New Mexico Work done in collaboration with Sajad Abbar and Huaiyu Duan )Collective neutrino oscillations in two spatial dimensions July 17, 2015 3 / 19
Hamiltonian
H = Hvac + Hmat + Hself
Hvac =1
2
(−ω cos(2θv) ω sin(2θv)ω sin(2θv) ω cos(2θv)
)Hmat =
(√2GFne 0
0 0
)Hself =
√2GF
∫d3p′(1− v · v ′)(ρp′ − ρp′)
Shashank Shalgar (University of New Mexico Work done in collaboration with Sajad Abbar and Huaiyu Duan )Collective neutrino oscillations in two spatial dimensions July 17, 2015 4 / 19
Neutrino oscillations in a medium
I Neutrino flavor oscillations can be modified in presence of matterwhen at least one flavor experiences a potential that is different fromthe potential experienced by other flavors
I The potential may be a result of matter or due to presence ofneutrino gas
I The modification of neutrino flavor oscillation due to neutrino gas isdifferent from the effect of matter in two crucial ways:
1 Neutrino-neutrino interaction leads to non-linear effect, while thematter effect is linear
2 Neutrino-neutrino potential is dependent on the relative angle betweenthe neutrinos
Shashank Shalgar (University of New Mexico Work done in collaboration with Sajad Abbar and Huaiyu Duan )Collective neutrino oscillations in two spatial dimensions July 17, 2015 5 / 19
Neutrino bulb model
H. Duan, G. M. Fuller and Y. Z. Qian, Ann. Rev. Nucl. Part. Sci. 60, 569 (2010) [arXiv:1001.2799 [hep-ph]].
Shashank Shalgar (University of New Mexico Work done in collaboration with Sajad Abbar and Huaiyu Duan )Collective neutrino oscillations in two spatial dimensions July 17, 2015 6 / 19
Neutrino oscillations in the bulb model
H. Duan, G. M. Fuller, J. Carlson and Y. Z. Qian, Phys. Rev. Lett. 97, 241101 (2006) [astro-ph/0608050].
Shashank Shalgar (University of New Mexico Work done in collaboration with Sajad Abbar and Huaiyu Duan )Collective neutrino oscillations in two spatial dimensions July 17, 2015 7 / 19
Assumptions of the neutrino bulb model revisited
I. Tamborra, F. Hanke, H. T. Janka, B. Mller, G. G. Raffelt and A. Marek, Astrophys. J. 792, 96 (2014) [arXiv:1402.5418
[astro-ph.SR]].
Shashank Shalgar (University of New Mexico Work done in collaboration with Sajad Abbar and Huaiyu Duan )Collective neutrino oscillations in two spatial dimensions July 17, 2015 8 / 19
Scrutiny of spherical symmetry
I How is the neutrino flavor oscillation modified if the assumption ofspherical symmetry is removed?
I A model of neutrino flavor oscillations in three spatial dimensions isdifficult to formulate
I We consider a two dimensional toy model to investigate the collectiveneutrino oscillations in multi dimensional space
Shashank Shalgar (University of New Mexico Work done in collaboration with Sajad Abbar and Huaiyu Duan )Collective neutrino oscillations in two spatial dimensions July 17, 2015 9 / 19
Neutrino Line model
x
θ′θ
z
θ,θ′ ∈ [−θmax,θmax]
I At each point (x , z) the neutrino flavor structure in the direction θ isgiven by a 2× 2 density matrix for neutrinos and anti-neutrino.
I We assume mono-energetic neutrinos of electron-type emanating fromeach point x .
Shashank Shalgar (University of New Mexico Work done in collaboration with Sajad Abbar and Huaiyu Duan )Collective neutrino oscillations in two spatial dimensions July 17, 2015 10 / 19
Neutrino Line model
If two points on the line start with almost identical flavor will they remainalmost identical? We can quantify this effect in two different ways,
ρθ(x , z) =
(1 εθ(x , z)
ε∗θ(x , z) 0
)or in terms of Fourier modes,
ρm,θ(z) = 1L
∫ L0 e
i2πmxL ρθ(x , z)dx =
(δm,0 εm,θ(z)
ε∗−m,θ(z) 0
)We have very similar expressions for anti-neutrinos.
Shashank Shalgar (University of New Mexico Work done in collaboration with Sajad Abbar and Huaiyu Duan )Collective neutrino oscillations in two spatial dimensions July 17, 2015 11 / 19
Fourier modes
= + +
= + +
z
m=0 m=1 m=2νe flux
Shashank Shalgar (University of New Mexico Work done in collaboration with Sajad Abbar and Huaiyu Duan )Collective neutrino oscillations in two spatial dimensions July 17, 2015 12 / 19
Fourier modes
I All Fourier modes m except m = 0 encode the spatial dependence offlavor in the x direction
I In the linear regime (off-diagonal element of ρm 1) the equation ofmotion for all modes is decoupled
i cos θ∂zεm,θ(z) =2πm
Lsin θεm,θ(z)− ωηεm,θ(z)
+ µ(1− α)
∫ θmax
−θmax
εm,θ(z)(1− cos(θ − θ′))dθ′
− µ
∫ θmax
−θmax
(εm,θ′(z)− αεm,θ′(z))(1− cos(θ − θ′))dθ′
Shashank Shalgar (University of New Mexico Work done in collaboration with Sajad Abbar and Huaiyu Duan )Collective neutrino oscillations in two spatial dimensions July 17, 2015 13 / 19
Linear stability analysis
i∂z
εm,θ1
εm,θ1
εm,θ2
εm,θ2
...εm,θNεm,θN
=
2N × 2N
εm,θ1
εm,θ1
εm,θ2
εm,θ2
...εm,θNεm,θN
Shashank Shalgar (University of New Mexico Work done in collaboration with Sajad Abbar and Huaiyu Duan )Collective neutrino oscillations in two spatial dimensions July 17, 2015 14 / 19
What does instability mean?
The solution of the off-diagonal term of the density matrix for each modeand angle εm,θ is of the exponential form,
εm,θ ∝ exp(−iΩmz).
Complex Ωm for m 6= 0 implies that the one-dimensional system isqualitatively and quantitatively different from a two-dimensional system inthe linear regime.
Shashank Shalgar (University of New Mexico Work done in collaboration with Sajad Abbar and Huaiyu Duan )Collective neutrino oscillations in two spatial dimensions July 17, 2015 15 / 19
Region of instability
0 1000 2000 3000 40000
2
4
6
8
10
/ω
# angle bins = 100
IH
0 1000 2000 3000 4000 5000
# angle bins = 500
IH
0 5000 10000 15000
µ/ω
0
2
4
6
8
10
/ω
NH
0 5000 10000 15000 20000
µ/ω
NH
Figure : Growth rate (Im(Ωm)) for m=5000 and number of angle bins, N = 100and 500 (first and second column respectively). The top row assumes invertedmass hierarchy while the bottom row assumes normal mass hierarchy. Hereµ =√
2GFnν/2θmaxShashank Shalgar (University of New Mexico Work done in collaboration with Sajad Abbar and Huaiyu Duan )Collective neutrino oscillations in two spatial dimensions July 17, 2015 16 / 19
Region of instability
0 3000 6000 9000 12000 15000
µ/ω
0
1000
2000
3000
4000
5000
6000
m
0
1
2
3
4
5
6
7
8
0 1000 2000 3000 4000 5000 6000
µ/ω
0
1000
2000
3000
4000
5000
6000
m
0
1
2
3
4
5
6
7
8
9
Figure : Region of instability for normal (left) and inverted hierarchy (right) withnν/nν = 0.8, L = 40π(ω−1) and θ in the range −π/6 to π/6.µ =√
2GFnν/2θmax
Shashank Shalgar (University of New Mexico Work done in collaboration with Sajad Abbar and Huaiyu Duan )Collective neutrino oscillations in two spatial dimensions July 17, 2015 17 / 19
The matter effect
0 1000 2000 3000 4000 5000 6000
µ/ω
0
1000
2000
3000
4000
5000
6000
m
0
1
2
3
4
5
6
7
8
9
0 1000 2000 3000 4000 5000 6000
µ/ω
0
1000
2000
3000
4000
5000
6000
m
0
1
2
3
4
5
6
7
8
Figure : Region of instability for inverted hierarchy without matter (left) and withmatter (right) with nν/nν = 0.8, L = 40π(ω−1) and θ in the range −π/6 to π/6.µ =√
2GFnν/2θmax
Shashank Shalgar (University of New Mexico Work done in collaboration with Sajad Abbar and Huaiyu Duan )Collective neutrino oscillations in two spatial dimensions July 17, 2015 18 / 19
Conclusion and Discussion
I In a multi-dimensional model of collective neutrino oscillations theinstability can occur and much larger effective neutrino numberdensity than in one dimensional model
I It there a region near the proto-neutron star where effect of bothcollisions and flavor instability can be seen simultaneously?
I At what length scale will we have to revisit the assumption ofcoherent forward scattering?
Shashank Shalgar (University of New Mexico Work done in collaboration with Sajad Abbar and Huaiyu Duan )Collective neutrino oscillations in two spatial dimensions July 17, 2015 19 / 19
