arX

iv:1

506.

0680

5v2

[he

p-ph

] 2

9 O

ct 2

015

Breaking the symmetries in self-induced flavor conversions

of neutrino beams from a ring

Alessandro Mirizzi1, 2

1Dipartimento Interateneo di Fisica “Michelangelo Merlin”, Via Amendola 173, 70126 Bari, Italy2Istituto Nazionale di Fisica Nucleare - Sezione di Bari, Via Amendola 173, 70126 Bari, Italy

Self-induced flavor conversions of supernova (SN) neutrinos have been characterized in the spher-ically symmetric “bulb” model, reducing the neutrino evolution to a one dimensional problem alonga radial direction. We lift this assumption, presenting a two-dimensional toy-model where neutrinobeams are launched in many different directions from a ring. We find that self-interacting neutrinosspontaneously break the spatial symmetries of this model. As a result the flavor content and thelepton number of the neutrino gas would acquire seizable direction-dependent variations, breakingthe coherent behavior found in the symmetric case. This finding would suggest that the previousresults of the self-induced flavor evolution obtained in one-dimensional models should be criticallyre-examined.

PACS numbers: 14.60.Pq, 97.60.Bw

I. INTRODUCTION

Dense neutrino gases in early universe or emitted fromcore-collapse supernovae (SNe) represent unique cases toprobe the effect of the neutrino-neutrino interactions onthe flavor conversions. Indeed, in these environmentsthe neutrino-neutrino interactions would generate a largeneutrino potential µ ∼

√2GFnν that in some cases can

exceed the ordinary matter term λ =√2GFne and the

neutrino vacuum oscillation frequency ω = ∆m2/2E.When this situation is encountered the neutrino-neutrinopotential would dominate the flavor evolution producinglarge self-induced flavor conversions (see [1] for a review).A vivid activity on these effects in the context of SN neu-trinos has flourished since a decade [2–5]. Indeed, it hasbeen realized that in the deepest SN regions self-inducedeffects can produce collective neutrino oscillations, lead-ing to peculiar spectral features in the oscillated neutrinospectra, dubbed as spectral swaps and splits [6–10].

The development of the self-induced flavor conversionsis associated with instabilities in the flavor space that aretriggered by the interacting neutrinos. The first one tobe noticed was the bimodal instability present even in anhomogeneous and isotropic neutrino gas [4]. In particu-lar, it was shown that an ensemble initially composed ofequal densities of νe and νe in the presence of a dominantneutrino-neutrino interaction term would exhibit in in-verted mass hierarchy (∆m2 < 0) large pair-conversionsof the type νeνe ↔ νxνx even with a small mixing angle.This behavior has been explained in terms of an unstablependulum in flavor space, where the instability is associ-ated with the tiny mixing angle [4, 11]. Furthermore, ithas been shown that if one introduces an anisotropy inthe neutrino gas, this can dramatically change the pre-vious solution. Indeed, in a non-isotropic neutrino en-semble, the neutrino-neutrino interaction term containsmulti-angle effects since the current-current nature of thelow-energy weak interactions introduces an angle depen-dent term (1−vp ·vq) between two interacting neutrino

modes [3, 12]. In the case of a gas completely symmet-ric in flavor content of ν and ν, even a small deviationfrom a perfect isotropy is enough to produce a multi-angle dechoerence leading to a flavor equilibrium amongthe different neutrino species in both the mass hierar-chies [13]. Multi-angle effects have been extensively stud-ied in the context of flavor evolution of SN neutrinos [14],whose emission is far from isotropic. It has been realizedthat in some cases they can destroy the collective behav-ior of the flavor evolution observed in an isotropic envi-ronment [13, 15, 16]. Multi-angle effects can also leadto a trajectory-dependent matter term, which if strongenough suppresses the self-induced conversions [17–20].In the context of SN neutrinos it has been often assumedan axially symmetric neutrino emission in oder to inte-grate out the azimuthal angle in the multi-angle kernel.However, it has been found that lifting this assumption,neutrino-neutrino interactions can break axial symme-try and lead to azimuthal-angle dependent flavor conver-sions [21–27].The lesson that has been gained from these situations

is that self-interacting neutrinos can spontaneously breakthe symmetries of the initial conditions, since small de-viations from them can be dramatically amplified duringthe further flavor evolution. This insight has recentlystimulated doubts about the validity of the solution ofthe SN neutrino equations of motion worked out in theso-called “bulb model” [3, 5, 16]. In this framework it isassumed the spherical symmetry about the center of theSN and the axial symmetry about any radial direction.These two symmetries allow one to reduce the problemto a one-dimensional evolution along a radial direction.Remarkably, removing the assumption of spherical sym-metry it would necessary to solve a challenging multi-dimensional problem to characterize the neutrino flavorevolution.In this context, in order to show how deviations from

the spatial symmetries of a system would affect the fla-vor evolution a simple two-dimensional model has beenrecently proposed in [28]. Namely, monochromatic neu-

2

trinos streaming in a stationary way in two directions(“left” L and “right” R, respectively) from an infiniteboundary plane at z = 0 with periodic conditions onx and translation invariance along the y direction. Re-markably, there is a correspondence between the symme-tries of the bulb model and the ones of this planar case.Indeed, the translational symmetry in the x direction inthe planar model corresponds to the spherical symmetryof the bulb-model and the L-R symmetry is equivalentto the axial symmetry in the spherical case. By meansof a stability analysis of the linearized equations of mo-tion, it has been shown in the planar model that if oneperturbs the initial symmetries of the flavor content inboth the two emission modes and along the boundary inthe x direction, then self-induced oscillations can sponta-neously break both these spatial symmetries [28]. In [29]we have recently performed a numerical study of the fla-vor evolution for this case. We found that the initialsmall perturbations are amplified by neutrino interac-tions, leading to non-trivial two-dimensional structuresin the flavor content and lepton number of the neutrinoenseble, that would exhibit large space fluctuations.The purpose of this paper is to develop a two-

dimensional model to capture more closely some of thefeatures of the SN environment. In particular, with re-spect to the planar model considered in [28, 29] we makethe following improvements: (i) ν emission from a ringmimicking the neutrino-sphere, (ii) parameters inspiredby the SN neutrino emissivity, (iii) declining neutrinodensity from the boundary, (iv) multi-angle effects. Wealso assume that self-induced flavor conversions woulddevelop without any hindrance caused by a large matterterm. Perturbing the neutrino emission in the transla-tional symmetry on the ring and in the emission direc-tions, we find the spontaneous breaking of these symme-tries in both normal and inverted mass hierarchies. Asa consequence the flavor content and the lepton num-ber of the neutrino ensemble acquires seizable variationsalong different lines of sight. These findings are pre-sented as follows. In Sec. II we describe the features ofour two-dimensional model. We discuss the equations ofmotion to characterize the two-dimensional flavor evolu-tion. We show how it is possible to solve this problem byFourier transforming these equations, obtaining a towerof ordinary differential equations for the different Fouriermodes. In Sec. III we present the numerical results of ourstudy. We show how the breaking of the spatial symme-tries produce direction-dependent variations in the flavorcontent of the ensemble. Finally in Sec. IV we discussabout future developments and we conclude.

II. TWO-DIMENSIONAL MODEL

A. Equations of motion

Characterizing the SN neutrino flavor dynamicsamounts to follow the spatial evolution of the neutrino

fluxes. For a stationary neutrino emission, the Equa-tions of Motion (EoMs) of the ν space-dependent occu-pation numbers (r,p) with momentum p at position r

are [31, 32]

v · ∇r = −i[Ω, ] , (1)

where we indicate with sans-serif vectors in flavor space,while for the ones in real space we use the bold-face. Atthe left-hand-side of Eq. (1) there is the Liouville opera-tor representing the drift term proportional to the neu-trino velocity v, due to particle free streaming. Note thatwe are neglecting external forces and an explicit time de-pendence of the occupation numbers. On the right-hand-side of Eq. (1) the matrix Ω is the full Hamiltonian thatreads

Ω =M2

2E+√2GF

[

Nl +

∫ +∞

−∞

dE′E′2∫

dv′

(2π)3′(1− v · v′)

]

,

(2)where M2 is the matrix of the mass-squared, responsibleof the vacuum oscillations. The ordinary matter effectson neutrino flavor conversions is accounted by the matrixof charged lepton densities Nl. Finally, the neutrino-neutrino interaction potential is represented by the lastterm of the right-hand-side, where the integral in dv′ ison the unit sphere and the occupation numbers ′ dependon r, E′,v′. Note that we use negative E to denote anti-neutrinos.In order to show the effect of spontaneous breaking

of spatial symmetries we consider them to be emitted ina plane from a ring with radius r = R. We have thena two-dimensional model for which it is natural to usea system of polar coordinates to describe the neutrinoposition vector r = (r, φ) where r is the radius, φ ∈ [0; 2π]is the polar angle, as shown in Fig. 1.The neutrino velocity can be decomposed in the ra-

dial (vr) and transverse (vt) component defined as v =(vr, v±) = (cos θr,± sin θr), where θr ∈ [0, π/2] is theangle between the radial direction and the one of theneutrino propagation (see, e.g., [33]), and the ± signindicate a transverse velocity in the clock-wise (v+) oranti-clockwise (v−) direction with respect to the radialdirection, respectively. We mention that in the recentmulti-angle study [34], where neutrinos emitted from aplane were considered, the range in the emission angleswas θ ∈ [0, π]. With this choice it is not necessary todistinguish clock-wise or anti-clockwise modes. However,in our work we preferred to use have the θr ∈ [0, π/2] inorder to start with a situation symmetric in the two emis-sion directions ± and show the effect of breaking of thisdiscrete symmetry. Note that the local angle θr woulddepend on the radius r. In order to avoid this effect, inthe literature it is preferred to label the neutrino modesin terms of their emission angle ϑR ∈ [0, π/2] along theboundary at r = R. The two angles θr and θR are relatedby [16]

R sin θR = r sin θr . (3)

3

FIG. 1: Two-dimensional model for the neutrino beams emit-ted from a ring with radius r = R.

Furthermore, we introduce the angular variable u =sin θR, u ∈ [0, 1]. With this choice the components ofthe neutrino velocity are [21]

vr = cos θr =

√

1− R2

r2u2 ,

vt ≡ v± = ± sin θr = ±R

ru . (4)

We assume that the neutrino distributions at r = R inthe energy E and in the angular variables u and in theR,L directions can be factorized as

F±(E, u) = Fν(E)× Fν(u)× F± . (5)

We assume the neutrino angular distributions to be flatin u and equal for all the flavors, i.e. Fν(u) = 1.Concerning the distributions in the clock-wise (+) or

anti-clockwise (−) directions, we assume that these aregiven by

F± =(1 + β±)

2 + β+ + β−

, (6)

where the quantities β± ≪ 1 are introduced to slightlyperturb the ± of a given u mode at the boundary.The neutrino number flux Fν(E) at the ring is given

by

Fν(E) =1

4πR2

Lν

〈Eν〉fν(E) , (7)

where we have normalized the neutrino emission on asphere with radius R.

B. Two-flavor case

In the following we will consider only a two-flavor sys-tem (νe, νx) where x = µ, τ and we will describe the

neutrino energy modes in terms of the two neutrino fre-quency ω = ∆m2/2E0, where ∆m2 = m2

2 − m21 is the

mass-squared difference. We have assumed a monochro-matic neutrino emission with E = E0. In the two-flavorcase the density matrices are projected over the Paulimatrices σ obtaining the polarization vectors in the usualway [16], where we normalize the (anti)neutrino polariza-tion vectors to the difference of the anti-neutrino fluxesat the boundary.The Liouville operator on the left-hand-side of the

EoMs [Eq. (1)] assumes the form

v · ∇r = vrd

dr+

v±r

d

dφ, (8)

so that the EoMs read (see also [21])

d

drP±,u = − v±

vrr

d

dφP±,u

+

[

ω

vrB+Ωνν

±

]

× P±,u , (9)

where we indicated with sans-serif the vectors in flavorspace. In particular, the unit vector B = (B1,B2,B3)points in the mass eigenstate direction in flavor space,such that B · e3 = − cosϑ, where ϑ is the vacuum mix-ing angle. For simplicity we neglect a possible mattereffect, assuming that its only role would be to reducethe effective in-medium mixing angle, ϑ ≪ 1 [4]. Theneutrino-neutrino interaction terms has a multi-anglekernel (1 − vp · vq) which takes the form

1

vr

∫

dθ′r[1− vrv′

r − vtv′

t]D′

=R

r

∫

dϑ′

R cosϑ′

R

[

1

vrv′r− 1− vtv

′t

vrv′r

]

D′ (10)

where D is the difference between the neutrino and anti-neutrino polarization vector of a given mode, and we usedfrom Eq. (3)

dθr =R

r

cosϑR

cos θrdϑR =

R

r

cosϑR

vrdϑR =

R

r

du

vr. (11)

In the large-distance limit (r ≫ R) one can expandEq. (10) obtaining

1

2

R

r

∫

du′[vt − v′t]2D

′ . (12)

We note that for the case we are studying the self-interaction term declines as r−3, while in the SN caseit declines as r−4. Considering the contribution of theclockwise (+) and anti-clockwise (−) modes in the pre-vious equation one gets

vt2D

′ =

(

R

r

)2

u2(D+,u′ + D−,u′)

vtv′

tD′ = ∓

(

R

r

)2

uu′(D+,u′ − D−,u′) . (13)

4

Then, the neutrino self-interaction term in the large dis-tance limit r ≫ R assumes the form

Ωνν± = µr

∫ 1

0

du′[(u2 + u′2)(D+,u′ + D−,u′)

2

∓ uu′(D+,u′ − D−,u′)] (14)

where the ∓ refers to the ± modes respectively, and

µr = [Fνe(R)− Fνx(R)]R3

2r3

= 3.5× 105 km−1

(

R

r

)3 (Lνe

〈Eνe〉− Lνx

〈Eνx〉

)

× 15 MeV

1051 MeV/s

(

10 km

R

)2

. (15)

An equation analogous to Eq. (10) can be written for theanti-neutrinos.One can define a conserved “lepton current” Lµ =

(L0,L) whose components are (see also [35])

L0 =

∫ 1

0

du′1

2(D+,u′ + D−,u′) · B , (16)

Lr =

∫ 1

0

du′vr1

2(D+,u′ + D−,u′) · B , (17)

Lt =

∫ 1

0

du′|vt|1

2(D+,u′ − D−,u′) · B , (18)

where L is a two-dimensional vector (Lr,Lθ), andDR(L),u · B ≃ D

3R(L),u. From Eq. (1) one realizes that

the lepton current satisfies a continuity equation

∂0L0 +∇r · L = ∇r · L = 0 , (19)

where first equality follows since ∂0L0 = 0 having weassumed a stationary solution. Eq. (19) generalizes thelepton-number conservation law of the one dimensionalcase [4].

C. Equations of motion in Fourier space

The differential operators in Eq. (9) implies that theflavor evolution is characterized by a partial differentialequation problem in r and φ. In [29, 36] (see also [28]) ithas been shown how it is possible to solve such a prob-lem by Fourier transforming the equations of motion withrespect to the coordinate along which a perturbation isintroduced. We assume a perturbation of the polariza-tion vectors at r = R with period 2π in φ so that

P±,u(R, φ) = P0± + 2ezδ cosφ , (20)

where P0± is the unperturbed value of the polarization

vector, and δ ≪ 1 is the amplitude of the perturbation.Up to the small difference in the emission modes ± [seeEq. (6)] the initial values of the polarization vectors are

P0±(ν) ≃ (1 + α)ez , (21)

P0±(ν) ≃ ez , (22)

where the initial flavor asymmetry is given by

α =Fνe − Fνe

Fνe − Fνx

. (23)

The functions P±,u(r, φ) are periodic in φ with period2π. Their Fourier transform is then

P±,u,n(r) =1

2π

∫ 2π

0

P±,u(r, φ)e−inφdφ , (24)

so that

P±,u(r, φ) =+∞∑

n=−∞

P±,u,n(r)e+inφ . (25)

The EoMs for the Fourier modes at large r ≫ R as-sume the form

d

drP±,u,n(r) = ∓inu

R

r2P±,u,n

+ω

vrB× P±,u,n

+ µr

+∞∑

j=−∞

∫ 1

0

du′[(u2 + u′2)(D+,u′,n−j + D−,u′,n−j)

2

∓uu′(D+,u′,n−j − D−,u′,n−j)]× P±,u,j .

We stress that it is enough to follow the evolution for pos-itive modes n ≥ 0, since the P±,u(r, φ) are real functionsand therefore

P∗

±,u,n = P±,u,−n . (26)

Once the evolution of the harmonic modes is obtainedfrom Eq. (26), the polarization vector in configura-tion space can be obtained by inverse Fourier transform[Eq. (25)].

III. NUMERICAL EXAMPLES

We present the results of the flavor evolution in thetwo-dimensional model described above. To calculate theν-ν interaction strength in Eq. (15) and the flavor asym-metry parameter in Eq. (23) we use benchmark valuesoften used in previous studies of self-induced neutrinooscillations (see, e.g. [14]), i.e. we take as average ener-gies

(〈Eνe〉, 〈Eνe 〉, 〈Eνx〉) = (12, 15, 18) MeV , (27)

while for the neutrino luminosities (in units of 1051 erg/s)we assume

Lνe = 2.40 , Lνe = 2.0 , Lνx = 1.50 . (28)

These values are typical of the early time SN accre-tion phase, and corresponds to an asymmetry parameterα = 1.34. As specified before we work in a single-energy

5

50 100

150 200

250 1

2 3

4 5

6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

P3

r(km)φ(rad)

P3

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

50 100 150 200 250 300 1

2 3

4 5

6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

P3

r(km)φ(rad)

P3

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

FIG. 2: Two-dimensional evolution of the 3-rd component P3

of the ν polarization vector in the r-φ plane, and its map onthe bottom plane breaking only the ± symmetry. Upper plotrefers to NH, while lower panel is for IH.

scheme, where we take as representative vacuum oscilla-tion frequency the one corresponding to the average ofthe ν ensemble with the emissivity parameters chosenabove (see [14]). Namely we take ω = 0.68 km−1. Con-cerning the neutrino oscillation parameters we choose asmall mixing angle ϑ = 10−2. Moreover, we assumeNu = 100 modes for the angular variable u ∈ [0; 1] inorder to have numerical convergence of the results andto avoid spurious instabilities due to few angular modes.It is known that forcing the ± symmetry (taking

β+ = β− = 0 in Eq. (6)) and the translational sym-metry on the ring (taking δ = 0 in Eq. (20)) the ensem-ble is stable in normal mass hierarchy (NH, ∆m2 > 0)while in inverted mass hierarchy (IH, ∆m2 < 0) it ex-hibits large bimodal flavor changes in the form of pairconversions νeνe → νxνx [4]. If we perturb the ± sym-metry taking small seeds β+ = −β− in the distributionsof Eq. (6) the system now exhibits the analogous of theso-called multi-azimuthal-angle (MAA) instability of thebulb model [21, 24, 25]. The result of the flavor evolutionis shown in Fig. 2 where it is represented the behaviorof the 3-rd component of the (integrated over u) anti-neutrino polarization vector P3 = 1/2(P+ + P−) in the(r, φ) plane for NH (upper panel) and IH (lower panel).In these numerical examples we have chosen β+ = 10−2.The most striking effect of the MAA instability is that

50 100 150 200 250 300 1

2 3

4 5

6

-0.6-0.4-0.2

0 0.2 0.4 0.6 0.8

1 1.2

P3

r(km)φ(rad)

P3

-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1 1.2

50 100

150 200

250 1

2 3

4 5

6

-0.4-0.2

0 0.2 0.4 0.6 0.8

1 1.2

P3

r(km)φ(rad)

P3

-0.4-0.2 0 0.2 0.4 0.6 0.8 1 1.2

FIG. 3: Two-dimensional evolution of the 3-rd component P3

of the ν polarization vector in the r-φ plane, and its map onthe bottom plane breaking the azimuthal invariance for the± modes and the translational symmetry on the ring. Upperplot refers to NH, while lower panel is for IH.

now also NH exhibits flavor conversions at r >∼ 60 km.The choice of the initial seed β± determines the onsetradius of the flavor conversions: the largest the seed,the earliest flavor conversions start. In IH (lower panel)flavor conversions start as in the ± symmetric case atr >∼ 50 km and MAA has a minor impact on the flavorevolution. From the Figure we realize that the behav-ior of the flavor conversions is uniform in the φ variablesince the translational symmetry has remained unbroken.Indeed we have solved only the EoMs [Eq. (26)] for then = 0 Fourier mode in this case.

The next step is to perturb also the translational sym-metry on the ring assuming a seed δ in the longitutudinaldistribution of the polarization vectors on the boundary[see Eq. (20)]. In this case we consider the evoltion of thefirst N=100 Fourier modes in Eq. (26). In this way weare sensitive to variations occuring at an angular scale∆φ >∼ 3. Results are shown in Fig. 3 with the same for-mat of the previous Figure. We used as seed to break theφ-symmetry δ = 3×10−3. In both NH (upper panel) andIH case (lower panel) flavor conversions start as in thetranslational invariant case, i.e. the planes of commonoscillation phase are flat in φ direction. However this be-havior is not stable. In the NH case around r ≃ 100 kmsomething occurs: The P3 component is no longer flat in

6

x(km)-300

-200

-100

0

100

200

300

y(km)

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

1.55

x(km)-300

-200

-100

0

100

200

300

y(km)

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

1.55

FIG. 4: Component Lr of the vector lepton number L in carte-sian coordinates in NH (upper panel) and IH (lower panel),respectively.

φ, while it starts to acquire notable variations (∼ 20 %)at different longitude. In IH after flavor conversions de-velop, also the translational symmetry is perturbed atr >∼ 100 km, with variations at different φ with values upto 30 %.In Fig. 4 we represent the component Lr of the vector

lepton number L [Eq. (18)] in cartesian coordinates

x = r cosφ ,

y = r sinφ . (29)

We realize that when the spherical symmetry is broken,the lepton number acquires significant variations in dif-ferent directions at a given r with respect to the initialuniform value Lr = α = 1.34. In particular one finds∼ 20 % variations.In order to clarify better this flavor dynamics, in Fig. 5

we show a contour plot representing the growth of thedifferent Fourier modes |Pn| (in logarithmic scale) in theplane n-r for NH (upper panel) and IH (lower panel). Weconsider the evolution of the first N = 100 modes. Werealize that the breaking of the translational symmetryat r ≃ 60 km in NH corresponds to the rapid excitationof the n > 0 harmonics that reach values |Pn| ∼< 10−2. In-

stead in IH for r >∼ 50 km the modes start to get excited

log10|Pn|

0 10 20 30 40 50 60 70 80 90 100n

0

50

100

150

200

250

300

r(km)

-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

log10|Pn|

0 10 20 30 40 50 60 70 80 90 100n

0

50

100

150

200

250

300

r(km)

-5

-4

-3

-2

-1

0

1

FIG. 5: Contour plots of the first 100 Fourier modes |Pn| (inlogarithmic scale) in the plane n-r in NH (upper panel) andIH (lower panel) respectively.

and can grow to |Pn| >∼ 10−1.5 at r >∼ 150 km. This ex-plains why the effect of the breaking of the translationalsymmetry is more pronounced in IH rather than in NH.

IV. CONCLUSIONS

We have considered a simple two-dimensional toy-model, namely neutrino beams emitted from a ring, topoint-out the effect of spontaneous breaking of axial andspherical symmetries in the self-induced flavor conver-sions of SN neutrinos.We found that if the slightly perturb the space symme-

tries on the boundary, these perturbation seeds are dra-matically amplified altering the flavor conversions foundin a symmetric model. Therefore, the flavor content ofthe self-interacting SN neutrinos would acquire signifi-cant direction-dependent variations. These results arequalitatively similar to what we found in the planarmodel we studied in [14]. Our findings suggest that thecharacterization on the flavor conversions obtained beforeshould be critically reconsidered, including these sponta-neous symmetry breaking effects. In order to have a re-alistic characterization of the possible SN neutrino spec-tra our simple toy model should be improved on differ-ent aspects. In particular, one should extend this model

7

to a realistic three-dimensional spherical case. In thiscase one would have the possibility to break the sphericalsymmetry in both longitudinal and latitudinal directions.Moreover, in order to get our numerical solution we haveconsidered N = 100 Fourier modes. In this cases we havenot found the presence of flavor converions at lower radiithan in the spherically symmetric case. However, in [28]it has been shown that harmonics with sufficiently highn could become unstable also at low-radii. Increasing Nto 500 we have not found any sizeble change in the onsetof the flavor changes and in the subsequent flavor evolu-tion in the non-linear regime. However, it remains to beseen if with a much higher number of harmonics low-radiieffects could occur. In this regard, it would be useful astability analysis performed along the lines of [37].Continuous energy spectra should also be taken into

account to understand how the spectral splitting fea-tures found in the bulb model would be modified inthis case. The role of matter effects that would sup-press self-induced flavor conversions during the accre-tion phase should also be investigated. The final goalwould be to study of the self-induced neutrino flavor con-

versions in realistic multi-dimensional supernova modelsaccounting for largely aspherical neutrino emission andmatter profiles. This objective is particularly timely nowsince in the last recent years, SN model simulations haveexperienced several breakthroughs. After 1D [38] and2D [33] models, the forefront has reached 3D SN simula-tions [39, 40]. Therefore, it seems the perfect juncture toconnect realistic SN simulations with nonlinear neutrinooscillations. This open issue makes compulsory the needfor further dedicated studies to fully clarify the fascinat-ing behavior of the interacting neutrino field.

Acknowledgements

The author warmly thanks Pasquale Serpico for use-ful comments on this manuscript. This work is sup-ported by the Italian Ministero dell’Istruzione, Univer-sita e Ricerca (MIUR) and Istituto Nazionale di FisicaNucleare (INFN) through the “Theoretical AstroparticlePhysics” projects.

[1] H. Duan, G. M. Fuller and Y. Z. Qian, “Collective Neu-trino Oscillations,” Ann. Rev. Nucl. Part. Sci. 60, 569(2010) [arXiv:1001.2799 [hep-ph]].

[2] H. Duan, G. M. Fuller and Y. Z. Qian, “Collective neu-trino flavor transformation in supernovae,” Phys. Rev. D74, 123004 (2006) [astro-ph/0511275].

[3] H. Duan, G. M. Fuller, J. Carlson and Y. -Z. Qian, “Sim-ulation of Coherent Non-Linear Neutrino Flavor Trans-formation in the Supernova Environment. 1. CorrelatedNeutrino Trajectories,” Phys. Rev. D 74, 105014 (2006)[astro-ph/0606616].

[4] S. Hannestad, G. G. Raffelt, G. Sigl and Y. Y. Y. Wong,“Self-induced conversion in dense neutrino gases: Pendu-lum in flavour space,” Phys. Rev. D 74, 105010 (2006)[Erratum-ibid. D 76, 029901 (2007)] [astro-ph/0608695].

[5] G. L. Fogli, E. Lisi, A. Marrone and A. Mirizzi, “Col-lective neutrino flavor transitions in supernovae and therole of trajectory averaging,” JCAP 0712, 010 (2007)[arXiv:0707.1998 [hep-ph]].

[6] G. G. Raffelt and A. Y. Smirnov, “Self-induced spec-tral splits in supernova neutrino fluxes,” Phys. Rev. D76, 081301 (2007) [Phys. Rev. D 77, 029903 (2008)][arXiv:0705.1830 [hep-ph]].

[7] H. Duan, G. M. Fuller, J. Carlson and Y. Z. Qian, “Neu-trino Mass Hierarchy and Stepwise Spectral Swappingof Supernova Neutrino Flavors,” Phys. Rev. Lett. 99,241802 (2007) [arXiv:0707.0290 [astro-ph]].

[8] B. Dasgupta, A. Dighe, G. G. Raffelt and A. Y. .Smirnov,“Multiple Spectral Splits of Supernova Neutrinos,” Phys.Rev. Lett. 103, 051105 (2009) [arXiv:0904.3542 [hep-ph]].

[9] A. Friedland, “Self-refraction of supernova neutrinos:mixed spectra and three-flavor instabilities,” Phys. Rev.Lett. 104, 191102 (2010) [arXiv:1001.0996 [hep-ph]].

[10] B. Dasgupta, A. Mirizzi, I. Tamborra and R. Tomas,

“Neutrino mass hierarchy and three-flavor spectral splitsof supernova neutrinos,” Phys. Rev. D 81, 093008 (2010)[arXiv:1002.2943 [hep-ph]].

[11] H. Duan, G. M. Fuller and Y. Z. Qian, “A Simple Pic-ture for Neutrino Flavor Transformation in Supernovae,”Phys. Rev. D 76, 085013 (2007) [arXiv:0706.4293 [astro-ph]].

[12] Y. Z. Qian and G. M. Fuller, “Neutrino-neutrino scat-tering and matter enhanced neutrino flavor transfor-mation in Supernovae,” Phys. Rev. D 51, 1479 (1995)[astro-ph/9406073].

[13] G. G. Raffelt and G. Sigl, “Self-induced decoherence indense neutrino gases,” Phys. Rev. D 75, 083002 (2007)[hep-ph/0701182].

[14] A. Mirizzi and R. Tomas, “Multi-angle effects in self-induced oscillations for different supernova neutrinofluxes,” Phys. Rev. D 84, 033013 (2011) [arXiv:1012.1339[hep-ph]].

[15] R. F. Sawyer, “The multi-angle instability in denseneutrino systems,” Phys. Rev. D 79, 105003 (2009)[arXiv:0803.4319 [astro-ph]].

[16] A. Esteban-Pretel, S. Pastor, R. Tomas, G. G. Raffeltand G. Sigl, “Decoherence in supernova neutrino trans-formations suppressed by deleptonization,” Phys. Rev. D76, 125018 (2007) [arXiv:0706.2498 [astro-ph]].

[17] S. Chakraborty, T. Fischer, A. Mirizzi, N. Saviano andR. Tomas, “No collective neutrino flavor conversions dur-ing the supernova accretion phase,” Phys. Rev. Lett. 107,151101 (2011) [arXiv:1104.4031 [hep-ph]].

[18] S. Chakraborty, T. Fischer, A. Mirizzi, N. Saviano andR. Tomas, “Analysis of matter suppression in collec-tive neutrino oscillations during the supernova accretionphase,” Phys. Rev. D 84, 025002 (2011) [arXiv:1105.1130[hep-ph]].

[19] N. Saviano, S. Chakraborty, T. Fischer and A. Mirizzi,

8

“Stability analysis of collective neutrino oscillations inthe supernova accretion phase with realistic energy andangle distributions,” Phys. Rev. D 85, 113002 (2012)[arXiv:1203.1484 [hep-ph]].

[20] S. Sarikas, G. G. Raffelt, L. Hudepohl and H. T. Janka,“Suppression of Self-Induced Flavor Conversion in theSupernova Accretion Phase,” Phys. Rev. Lett. 108,061101 (2012) [arXiv:1109.3601 [astro-ph.SR]].

[21] G. Raffelt, S. Sarikas and D. de Sousa Seixas, “AxialSymmetry Breaking in Self-Induced Flavor Conversionof Supernova Neutrino Fluxes,” Phys. Rev. Lett. 111,no. 9, 091101 (2013) [Erratum-ibid. 113, no. 23, 239903(2014)] [arXiv:1305.7140 [hep-ph]].

[22] G. Raffelt and D. d. S. Seixas, “Neutrino flavor pendu-lum in both mass hierarchies,” Phys. Rev. D 88, 045031(2013) [arXiv:1307.7625 [hep-ph]].

[23] H. Duan, “Flavor Oscillation Modes In Dense Neu-trino Media,” Phys. Rev. D 88, 125008 (2013)[arXiv:1309.7377 [hep-ph]].

[24] A. Mirizzi, “Multi-azimuthal-angle effects in self-inducedsupernova neutrino flavor conversions without axialsymmetry,” Phys. Rev. D 88, no. 7, 073004 (2013)[arXiv:1308.1402 [hep-ph]].

[25] S. Chakraborty and A. Mirizzi, “Multi-azimuthal-angleinstability for different supernova neutrino fluxes,” Phys.Rev. D 90, no. 3, 033004 (2014) [arXiv:1308.5255 [hep-ph]].

[26] S. Chakraborty, A. Mirizzi, N. Saviano andD. d. S. Seixas, “Suppression of the multi-azimuthal-angle instability in dense neutrino gas during supernovaaccretion phase,” Phys. Rev. D 89, no. 9, 093001 (2014)[arXiv:1402.1767 [hep-ph]].

[27] S. Chakraborty, G. Raffelt, H. T. Janka and B. Mueller,“Supernova deleptonization asymmetry: Impact on self-induced flavor conversion,” arXiv:1412.0670 [hep-ph].

[28] H. Duan and S. Shalgar, “Flavor instabilities in theneutrino line model,” Phys. Lett. B 747, 139 (2015)[arXiv:1412.7097 [hep-ph]].

[29] A. Mirizzi, G. Mangano and N. Saviano, “Self-inducedflavor instabilities of a dense neutrino stream in a two-dimensional model,” Phys. Rev. D 92, no. 2, 021702(2015) [arXiv:1503.03485 [hep-ph]].

[30] S. Sarikas, D. d. S. Seixas and G. Raffelt, “Spuri-

ous instabilities in multi-angle simulations of collectiveflavor conversion,” Phys. Rev. D 86, 125020 (2012)[arXiv:1210.4557 [hep-ph]].

[31] G. Sigl and G. Raffelt, “General kinetic description ofrelativistic mixed neutrinos,” Nucl. Phys. B 406, 423(1993).

[32] P. Strack and A. Burrows, “Generalized Boltzmann for-malism for oscillating neutrinos,” Phys. Rev. D 71,093004 (2005) [hep-ph/0504035].

[33] R. Buras, M. Rampp, H.-T. Janka and K. Kifoni-dis, “Two-dimensional hydrodynamic core-collapse su-pernova simulations with spectral neutrino transport. 1.Numerical method and results for a 15 solar mass star,”Astron. Astrophys. 447, 1049 (2006) [astro-ph/0507135].

[34] S. Abbar, H. Duan and S. Shalgar, “Flavor instabilitiesin the multiangle neutrino line model,” Phys. Rev. D 92,no. 6, 065019 (2015) [arXiv:1507.08992 [hep-ph]].

[35] H. Duan, G. M. Fuller and Y. Z. Qian, “Symmetries incollective neutrino oscillations,” J. Phys. G 36, 105003(2009) [arXiv:0808.2046 [astro-ph]].

[36] G. Mangano, A. Mirizzi and N. Saviano, “Dampingthe neutrino flavor pendulum by breaking homogeneity,”Phys. Rev. D 89, no. 7, 073017 (2014) [arXiv:1403.1892[hep-ph]].

[37] S. Chakraborty, R. S. Hansen, I. Izaguirre and G. Raffelt,“Self-induced flavor conversion of supernova neutrinos onsmall scales,” arXiv:1507.07569 [hep-ph].

[38] T. Fischer, S. C. Whitehouse, A. Mezzacappa, F.-K. Thielemann and M. Liebendorfer, “Protoneutron starevolution and the neutrino driven wind in general rela-tivistic neutrino radiation hydrodynamics simulations,”Astron. Astrophys. 517 (2010) A80 [arXiv:0908.1871[astro-ph.HE]].

[39] A. Wongwathanarat, E. Mueller and H.-T. Janka,“Three-Dimensional Simulations of Core-Collapse Super-novae: From Shock Revival to Shock Breakout,” Astron.Astrophys. 577, A48 (2015) [arXiv:1409.5431 [astro-ph.HE]].

[40] E. J. Lentz et al., “Three-dimensional Core-collapse Su-pernova Simulated Using a 15 M⊙ Progenitor,” Astro-phys. J. 807, 2, L31 (2015) [arXiv:1505.05824 [astro-ph.HE]]