Neutrino mass hierarchy and three-flavor spectral splits of supernova neutrinos

Basudeb Dasgupta,1 Alessandro Mirizzi,2 Irene Tamborra,1,3,4 and Ricard Tomas2

1Max-Planck-Institut fur Physik (Werner-Heisenberg-Institut), Fohringer Ring 6, 80805 Munchen, Germany2II Institut fur Theoretische Physik, Universitat Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany

3Dipartimento Interateneo di Fisica ‘‘Michelangelo Merlin,’’ Via Amendola 173, 70126 Bari, Italy4INFN, Sezione di Bari, Via Orabona 4, 70126 Bari, Italy

(Received 1 March 2010; published 28 May 2010)

It was recently realized that three-flavor effects could peculiarly modify the development of spectral

splits induced by collective oscillations, for supernova neutrinos emitted during the cooling phase of a

protoneutron star. We systematically explore this case, explaining how the impact of these three-flavor

effects depends on the ordering of the neutrino masses. In inverted mass hierarchy, the solar mass splitting

gives rise to instabilities in regions of the (anti)neutrino energy spectra that were otherwise stable under

the leading two-flavor evolution governed by the atmospheric mass splitting and by the 1–3 mixing angle.

As a consequence, the high-energy spectral splits found in the electron (anti)neutrino spectra disappear,

and are transferred to other flavors. Imperfect adiabaticity leads to smearing of spectral swap features. In

normal mass hierarchy, the three-flavor and the two-flavor instabilities act in the same region of the

neutrino energy spectrum, leading to only minor departures from the two-flavor treatment.

DOI: 10.1103/PhysRevD.81.093008 PACS numbers: 14.60.Pq, 97.60.Bw

I. INTRODUCTION

The neutrino flux from a core-collapse supernova (SN)is a powerful tool to probe fundamental neutrino propertiesas well as the dynamics of the explosion [1,2]. The diag-nostic role played by neutrinos during a stellar collapse islargely associated to the signatures imprinted on the ob-servable SN neutrino burst by flavor conversions occurringdeep inside the star.

It has been understood that the paradigm of neutrinoflavor transformation in supernovae [3], based primarily onthe Mikheyev-Smirnov-Wolfenstein (MSW) effect withthe ordinary matter [4], was incomplete. New surprisingand unexpected effects have been found to be important inthe region close to the neutrinosphere (see [5,6] for recentreviews). Here the neutrino density is so high that theneutrino-neutrino interactions dominate the flavor evolu-tion, producing collective oscillations. The most importantobservational consequence of �� � interactions is a swapof the �e and ��e spectra with the nonelectron �x and ��x

spectra in certain energy ranges [7,8].The development of the spectral swaps is strongly de-

pendent on the original SN neutrino fluxes. In the accretionphase, i.e. t & 0:5 ms after the core bounce, neutrino num-ber fluxes are expected to be ordered as�0

�e>�0

��e� �0

��x

[9–11]. In such a scenario, one finds that for normalneutrino mass hierarchy (NH, �m2

atm ¼ m23 �m2

1;2 > 0)

collective oscillations do not play a significant role. Forinverted hierarchy (IH, �m2

atm < 0), the end of collectiveoscillations is marked by a complete exchange of the e andx flavors for almost all antineutrinos. For neutrinos, theexchange happens only above a characteristic energy fixedby the lepton number conservation, giving rise to a spectralsplit in their energy distributions [12–15].

The neutrino number fluxes may be significantly differ-ent at later times, i.e. during the cooling phase. In Garchingsimulations [11], one finds a crossover among the different� spectra. As a consequence, the original fluxes exhibit adifferent ordering�0

�x* �0

�e* �0

��e. A study of this latter

case was performed [16], finding the occurrence of unex-pected multiple spectral splits for both neutrinos and anti-neutrinos, in normal and inverted mass hierarchies. Therich phenomenology of the spectral splits, and its depen-dence on the original neutrino energy spectra was furtherexplored in the extensive study performed in [17].Most of the collective flavor dynamics of SN neutrinos

can be explained in an effective two-flavor (2�) framework[18]. Collective oscillations are triggered by an instabilityin the two-flavor ‘‘H sector’’ associated with the atmos-pheric mass-squared difference �m2

atm and the mixingangle �13. Three-flavor (3�) effects are due to the‘‘L sector,’’ governed by the smaller solar mass splitting�m2

sol ¼ m22 �m2

1 > 0 and the mixing angle �12. Theseeffects have been studied for neutrino fluxes typical of theaccretion phase [18–21]. It was recently shown [22] thatthey are able to trigger collective flavor conversions, evenif the mixing angle �13 is exactly zero. However, apart fromthis initial kick, no new sizable effect was found in thesubsequent neutrino flavor evolution.Recently, the 3� case was studied for a scenario relevant

to the cooling phase [23]. It was found that in the invertedmass hierarchy, the presence of the solar sector can‘‘erase’’ the high-energy spectral splits that would haveoccurred in the �e and ��e spectra for a 2� flavor evolutiongoverned by the �m2

atm and �13. Moreover, the final elec-tron antineutrino energy spectrum exhibits a ‘‘mixed’’nature, i.e. the spectral swap is not complete. The phe-nomenological importance of this result is underlined by

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the fact that the high-energy spectral features are expectedto be significantly easier to observe at neutrino detectors.In [23] these three-flavor effects are associated with aninstability in the L sector and to the subsequent nonadia-batic flavor evolution driven by �m2

sol. Building on this

insight, we find it worthwhile to take a closer look at thethree-flavor effects during the cooling phase, and to under-stand the origin and nature of the 3� instability.

We explain how the 3� effects are crucially dependenton the neutrino mass hierarchy. For normal mass hierarchy,there is no fundamental difference between H and L sec-tors, since in this case both mass splittings are positive andthe effective in-medium mixing angles are both small.Thus, even in a 2� setup one would expect spectral splitsdriven by the L-sector parameters, and expect them to beexactly where the splits appear in the case of the H sectorin normal hierarchy. However we find that it is not the case.In fact with typical SN parameters, collective oscillationsfail to produce any significant conversions for the L sector.This is so, because the L sector has a lower natural fre-quency (!L ¼ �m2

sol=2E, where E is a typical SN � en-

ergy) than the H sector and the collective interactionstrength drops at a rate much faster than it. This does notleave enough time for the instability to grow and makes thecollective evolution nonadiabatic. Therefore, spectralsplits fail to develop. Nevertheless, the system is in anunstable situation. Indeed, as we will show, a small pertur-bation in the initial conditions is enough to develop thecollective flavor conversions, producing spectral swapsalso in this two-flavor case.

In a realistic situation, this initial perturbation for theL sector is provided by 3� effects that couple this sector tothe H sector. Oscillations in the H sector are communi-cated to the L sector, allowing the instability to grow muchfaster. The spectral swapping still remains less adiabaticthan in the H sector. As a result, for normal mass hier-archy—where both H and L sectors trigger the sameinstability and ‘‘compete’’ to convert the high-energy �and �� spectra—the H sector wins. Three-flavor effects donot cause a significant change. The interesting bit happensfor inverted mass hierarchy. In this case, the H sector andthe L sector have instabilities in different parts of thespectrum and therefore do not compete with each other.Instead, they ‘‘cooperate’’ and act on complementary partsof the energy spectra. The L-sector instability catalyzed bythe H sector, operates in the high-energy region withouthindrance, and causes an additional swap, that erases thespectral split found in the 2� flavor evolution. The lowadiabaticity in the L sector is responsible for somewhatsmeared splits, and the effect is particularly important forsplits at higher energies, especially for the antineutrinos. Inthe remainder of this paper, we illustrate these aspectsusing simple examples and provide a semianalyticaltreatment.

The plan of our work is as follows. In Sec. II, we presentour formalism for the SN neutrino flavor evolution and set

up our numerical calculations, i.e. state our inputs forneutrino masses, mixing parameters, original supernovaneutrino energy spectra, and luminosities at late times. InSec. III, we present the 2� results in theH and L sectors, inparticular, showing the lack of adiabaticity in the L sectorand the role of the small perturbations in the initial con-ditions to circumvent that. In Sec. IV, we present a com-plete 3� calculation, showing that the H sector catalyzesthe L sector and then instabilities in both sectors develop—in cooperation for inverted mass hierarchy, and in compe-tition for normal mass hierarchy. We provide an estimate ofthe relative adiabaticity of the low-energy and high-energysplits—explaining the mixed spectra observed in the anti-neutrino sector. Finally in Sec. V, we comment on ourresults and conclude.

II. EQUATIONS OF MOTION AND NUMERICALFRAMEWORK

Mixed neutrinos are described by matrices of density �p

and ��p for each (anti)neutrino mode. The diagonal entries

are the usual occupation numbers whereas the off-diagonalterms encode phase information. The equations of motion(EoMs) are [8,24]

i@t�p ¼ ½Hp; �p�; (1)

where the Hamiltonian is

H p ¼ �p þ V þ ffiffiffi2

pGF

Z d3q

ð2�Þ3 ð�q � ��qÞð1� vq � vqÞ;(2)

with vp being the velocity. The matrix of the vacuum

oscillation frequencies for neutrinos is �p ¼ðm2

1; m22; m

23Þ=2jpj in the mass basis. For antineutrinos

�p ! ��p. The matter effect due to the background

electron density ne is represented, in the weak-interaction

basis, by V ¼ ffiffiffi2

pGFnediagð1; 0; 0Þ.

In spherical symmetry the EoMs can be expressed as aclosed set of differential equations along the radial direc-tion [25,26]. The factor (1� vq � vq) in the Hamiltonian,

implies ‘‘multiangle’’ effects for neutrinos moving ondifferent trajectories [7]. However, for realistic supernovaconditions the modifications are small, allowing for asingle-angle approximation [20,25]. We implement thisapproximation by launching all neutrinos with 45� relativeto the radial directions [19,25].For the numerical illustrations, we take the neutrino

mass-squared differences in vacuum to be �m2atm ¼ 2�

10�3 eV2 and �m2sol ¼ 8� 10�5 eV2, close to their cur-

rent best-fit values [27]. The values of the mixing parame-ters relevant for SN neutrino flavor conversions aresin2�12 ’ 0:31 and sin2�13 & 0:04 [27]. The matter effectin the region of collective oscillations (up to a few 100 km)can be accounted for by choosing small (matter sup-

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pressed) mixing angles [28], which we take to be ~�13 ¼~�12 ¼ 10�3, and considering as effective mass-square dif-ferences in matter �~m2

atm ¼ �m2atm cos�13 ’ �m2

atm and�~m2

sol ¼ �m2sol cos2�12 ’ 0:4�m2

sol [28,29]. We ignore

possible subleading CP violating effects [21] by setting�CP ¼ 0. MSW conversions typically occur after collec-tive effects have ceased [20,30]. Their effects then factor-ize and can be included separately. Therefore, we neglectthem in the following.

The content of a given neutrino species �� with energyEat a position r is given by

���ðE; rÞ ¼F��

ðE; rÞFðE; rÞ ; (3)

where F��is the energy distribution of �� and F is the sum

of the energy distributions of all flavors, for neutrinos andantineutrinos, respectively. For the three relevant SN �energy distributions at the neutrinosphere F0

��, we take

F0��ðEÞ ¼ �0

��’��

ðEÞ; (4)

where �0�� ¼ L��=hE��

i is the number flux, defined in

terms of the neutrino luminosity L��and the neutrino

average energy hE��i. ’��

ðEÞ is the normalized neutrino

spectrum (RdE’��

¼ 1), parametrized as [11]

’��ðEÞ ¼��

�ð�ÞE��1

hE��i� e

��E=hE�� i; (5)

where � is a spectral parameter, and �ð�Þ is the Eulergamma function. The values of the parameters are modeldependent [9,10]. For our numerical illustrations, wechoose

ðhE�ei; hE ��ei; hE ��x

iÞ ¼ ð12; 15; 18Þ MeV; (6)

and � ¼ 4, from the admissible parameter ranges [11]. Wetake ratios of the fluxes at late times to be [10]

�0�e:�0

��e:�0

�x¼ 0:85:0:75:1:0; (7)

where we have assumed equal � and neutrino andantineutrino initial fluxes.

The strength of the neutrino-neutrino interactions can beparametrized as [25]

�0 ¼ffiffiffi2

pGFjF0

��e� F0

��xj; (8)

where the fluxes are taken at the neutrinosphere radius R ¼10 km. Numerically, we assume �0 ¼ 7� 105 km�1.When formally deriving the single-angle approximation,one explicitly obtains that the radial dependence of theneutrino-neutrino interaction strength can be written as[26]

�ðrÞ ¼ �0

R2

r2Cr; (9)

where the r�2 scaling comes from the geometrical fluxdilution, and the collinearity factor

Cr ¼ 4

�1� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1� ðR=rÞ2pðR=rÞ2

�2 � 1 ¼ 1

2

�R

r

�2

for r ! 1;

(10)

arises from the (1� cos�) structure of the neutrino-neutrino interaction. The asymptotic behavior for large ragrees with what one obtains by considering that all neu-trinos are launched at 45� to the radial direction [19]. Theknown decline of the neutrino-neutrino interactionstrength, �ðrÞ � r�4 for r � R, is evident.

III. MULTIPLE SPECTRAL SPLITS IN ATWO-FLAVOR SCENARIO

A. 2� H system

We start our investigation with the flavor evolution in theH sector characterized by (�m2

atm, �13). As explained in[16], spectral swaps can develop around energies E ¼ Ec

of the original neutrino spectra (except at E ¼ 0), wherespectra of different flavors cross each other, i.e. at thecrossing points

F�eðEcÞ � F�x

ðEcÞ ¼ 0; F ��eðEcÞ � F ��xðEcÞ ¼ 0;

(11)

for neutrinos and antineutrinos, respectively. A givencrossing point is unstable if

dðF�e� F�xÞ=dE < 0 for IH;

dðF�e� F�x

Þ=dE > 0 for NH;(12)

and analogously for antineutrinos.In Fig. 1, we show the results of the flavor evolution for

neutrinos (left panel) and antineutrinos (right panel) ininverted mass hierarchy. In the upper panels we show theinitial and final �e and �x energy spectra, while in the lowerpanels we show the electron neutrino survival probabilityPee. Following the instability conditions stated above, onefinds that in IH the spectral swap would develop forneutrinos around Ec ’ 13 MeV and for antineutrinosaround Ec ’ 9 MeV. The development of a spectral swapin the middle of the energy spectra produces two splits inthe final � spectra. In particular, in the �e final spectrum theswapped region is between 5 and 23 MeV, while for ��e isbetween 3 and 17 MeV.In Fig. 2, the corresponding results for normal hierarchy

are shown. In this case the only unstable crossing point is atEc ! 1 in the tail of the energy spectra, therefore theresulting � and �� spectra exhibit only a high-energyswap and a single split each. In this case, the split is at E ’23 MeV for �e, and at E ’ 17 MeV for ��e.From this example, we realize that in the two mass

hierarchies the instabilities occur around different and

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well-separated crossing points. This leads to spectral swapsthat occur in different energy ranges for the two masshierarchies. In fact, these ranges are nonoverlapping andalmost complementary, i.e. the high-energy ends of theswaps in IH are the low-energy ends of the swaps in NH.This implies that if we take the � spectra swapped by theconversions in IH, as an input for conversions in NH, theselatter would swap also the high-energy spectrum of theelectron (anti)neutrinos, giving the impression that thehigh-energy split has been erased. This observation willplay an important role in our understanding of the fullthree-flavor evolution.

B. 2� L system

We now consider the L system. Since �m2sol > 0, we

expect a behavior similar to that of theH system in normalhierarchy. However, we find that in this case no flavorconversion occurs. This has to be attributed to two rea-sons—insufficient growth of the instability, and lack ofadiabaticity. The strength of the instability is given bythe off-diagonal components in the density matrix. For asimple pendular system with energy E, the time period forgrowth of off-diagonal components is a few times thependular time period pend [8,16]

pend �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2E

�~m2sol�

s; (13)

which scales logarithmically with the small in-medium mixing angle. This time period is roughlyffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�~m2

atm=�~m2sol

q� 8 times larger for the L sector, causing

the instability to develop relatively slowly. The slowgrowth is further exacerbated by a relatively fast decreasein collective neutrino interaction strength �.During the spectral swapping phenomenon, the spec-

trum near the crossing acts like an inverted pendulum[16]. The swap sweeps through the spectrum on eachside of the crossing, and the modes at the edge of theswap precess at an average oscillation frequency [16].Adiabaticity requires [23]��������d ln�

dr

��������<: (14)

Since the collective neutrino interaction strength� goes asr�4, the rate at which neutrino refractive effects are de-creasing is jd ln�=drj ’ 1=50 km�1 at r ’ 200 km, ap-proximately where the swapping takes place. One finds � �~m2

sol=2E ’ 1=400 km�1 for a typical energy E ’32 MeV in the region of the swap, so the adiabaticitycondition is not met.

neutrinos

Fν

(a.u

.)

1

2

3

4

5

67

e, initx, inite, finx, fin

antineutrinos

0 10 20 30 40 50 600.0

0.2

0.4

0.6

0.8

1.0

Pee

E (MeV)

10 20 30 40 50 60

E (MeV)

FIG. 2 (color online). Flavor evolution in the 2� H sector forneutrinos (left panels) and antineutrinos (right panels) in normalmass hierarchy. Upper panels: the �e (red) and �x (blue) energyspectra, before (dashed curves) and after (continuous curves) thecollective oscillations. Lower panels: survival probability Pee forelectron (anti)neutrinos. The gray regions represent the range inwhich the spectral swap occurs.

neutrinosF

ν (a

.u.)

1

2

3

4

5

67

e, initx, inite, finx, fin

antineutrinos

0 10 20 30 40 50 600.0

0.2

0.4

0.6

0.8

1.0

Pee

E (MeV)

10 20 30 40 50 60

E (MeV)

FIG. 1 (color online). Flavor evolution in the 2� H sector forneutrinos (left panels) and antineutrinos (right panels) in invertedmass hierarchy. Upper panels: the �e (red) and �x (blue) energyspectra, before (dashed curves) and after (continuous curves) thecollective oscillations. Lower panels: survival probability Pee forelectron (anti)neutrinos. The gray regions represent the range inwhich the spectral swap occurs.

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Nevertheless, the L system has the same instability asthe H system in NH, around the crossing point Ec ! 1 ofthe (anti)neutrino energy spectra. Indeed, if we slightlyperturb the initial conditions of our � ensemble, e.g. putexplicit off-diagonal terms in the initial neutrino densitymatrix, the instability grows easily and leads to significantflavor conversions.1

In Fig. 3, we show the effects of these artificially trig-gered flavor conversions on the electron neutrinos (leftpanels) and antineutrinos (right panels). Upper panelsshow the energy spectrum, while the lower panels displaythe neutrino survival probability. We find that it is suffi-cient to take off-diagonal seeds in the density matrix

j�0exj ¼ �� ð�0

ee þ �0xxÞ; (15)

with � * 10�5, to produce significant flavor conversions.However, relative to the H sector in the normal hierarchy,the swap is less sharp. This is naturally expected becausethe adiabaticity does not change significantly by taking aninitial perturbation.

In Fig. 4, we show the j�exj component of the � densitymatrix for E ¼ 32 MeV for three cases: (i) the H systemwith NH, (ii) the L system with � ¼ 10�6, and (iii) theL system seeded with � ¼ 10�4. We realize that the growthof the off-diagonal component is almost absent for theL system with � ¼ 10�6. Even when there are adequatelylarge off-diagonal elements, as in the case of � ¼ 10�4, thegrowth is relatively slow compared to theH system, so thatsignificant flavor changes can start only at large radius (r *300 km). Consequently, due to the violation of adiabatic-ity, collective flavor conversions do not have enough timeto develop complete splits before the effects of theneutrino-neutrino interactions become negligible.

IV. MULTIPLE SPECTRAL SPLITS IN ATHREE-FLAVOR SCENARIO

A. Speedup of the �m2sol instability

Equipped with the insights of the previous section, weare ready to analyze the behavior of the flavor conversionsin the complete three-flavor scenario. We work in therotated basis ð�e; �x; �yÞ ¼ RTð�23Þð�e; ��; �Þ [18]. Thisis equivalent to take �23 ¼ 0 in the neutrino mixing matrix(which makes no difference to �e and ��e evolution, if �and flavors are treated identically). The vacuumHamiltonian is written as [18]

�ðEÞ ¼�~m2atm

2E

s213 0 c13s13

0 0 0

c13s13 0 c213

0BB@

1CCA

þ�~m2sol

2E

c213s212 c12c13s12 �c13s

212s13

c12c13s12 c212 �c12s12s13

�c13s212s13 �c12s12s13 s212s

213

0BB@

1CCA;

(16)

where cij ¼ cos~�ij and sij ¼ sin~�ij. In the limit of

neutrinosF

ν (a

.u.)

1

2

3

4

5

67

e, init

ε≤10-6x, init

ε=10-5

ε=3×10-5

ε=10-4

antineutrinos

0 10 20 30 40 50 600.0

0.2

0.4

0.6

0.8

1.0

Pee

E (MeV)10 20 30 40 50 60

E (MeV)

FIG. 3 (color online). Flavor evolution in the 2� L sector forneutrinos (left panels) and antineutrinos (right panels). Upperpanels: initial �e (red) and �x (blue) energy spectra and the �e

spectrum after the collective oscillations, for different values ofan initial off-diagonal component of the density matrix j�0

exj ¼�� ð�0

ee þ �0xxÞ (see the text for details). Lower panels: survival

probability Pee for electron (anti)neutrinos.

H

L (ε=10-4)

L (ε=10-6)

102 10310-4

10-3

10-2

10-1

1

|ρex

|

r (km)

FIG. 4 (color online). Growth of the off-diagonal componentj�exj of the neutrino density matrix for the 2� H system innormal hierarchy and for the L system taking two differentvalues of j�0

exj ¼ �� ð�0ee þ �0

xxÞ. The neutrino energy is takento be E ¼ 32 MeV.

1The speedup of flavor instabilities under the effects of verysmall seeds in the initial conditions was already pointed out in[31].

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�13 ¼ 0, the �e � �y (H) and the �e � �x (L) sectors are

completely decoupled.When the neutrino-neutrino interaction is strong, all the

density matrix mode �p stay pinned to each other, exhib-

iting collective flavor conversions. Now we review thefactorization of the H sector from other subleading con-tributions, as shown in [18]. For each energy mode, werewrite our density matrix and Hamiltonian as

� ¼ �ð0Þ þ �ð1Þ; (17)

H ¼ Hð0Þ þ Hð1Þ; (18)

where the superscript (0) refers to off components in theL sector and all diagonal components, while (1) to allothers, namely,

�ð0Þ ¼�ee �ex 0�ex �xx 00 0 �yy

0B@

1CA; (19)

and

�ð1Þ ¼0 0 �ey

0 0 �xy

�ey �

xy 0

0B@

1CA: (20)

Analogous expressions hold for Hð0Þ and Hð1Þ.Note from Eq. (16) that Hð0Þ, which contains the e� x

block, causes oscillations due to �m2sol, while Hð1Þ, con-

taining the e� y off-diagonal terms, gives �m2atm-driven

oscillations. Putting this decomposition into the equationsof motion, one finds [18]

i _�ð0Þ ¼ ½Hð0Þ; �ð0Þ� þ ½Hð1Þ; �ð1Þ�; (21)

i _�ð1Þ ¼ ½Hð1Þ; �ð0Þ� þ ½Hð0Þ; �ð1Þ�: (22)

Analogous equations hold for the antineutrinos. Note theinteresting structure of the EoMs that is an outcome of the

commutation relations— _�ð0Þ depends only on commuta-

tors ½Hð0Þ; �ð0Þ� and ½Hð1Þ; �ð1Þ�, while _�ð1Þ depends only onthe cross terms [18].

In a pure 2� L-system evolution, Hð1Þ and �ð1Þ are zero.Thus Eq. (21) has only the first term on the right-hand side,and Eq. (22) is irrelevant. Once off-diagonal components

of �ð0Þ, driven byHð0Þ, develop asymmetrically in neutrinosand antineutrinos, the instability grows under the action of

Hð0Þ (proportional to �m2sol), i.e. the pendular time scale is

pend � 1=ffiffiffiffiffiffiffiffiffiffiffi!L�

p. The addition of off-diagonal elements to

�ð0Þ kick starts the process.

In a complete 3� system, Hð1Þ is nonzero and thus

produces the off-diagonal components in �ð0Þ and �ð1Þmore quickly. Not only do initial off-diagonal terms getgenerated in the L sector, but also those terms grow at amuch faster rate. The growth is speeded up by the second

term in the equation of motion (21), i.e. by Hð1Þ which

induces oscillations �m2atm dependent at the leading order.

Therefore, it leads to a growth of the instability at almostthe same rate as for the H system, and much faster than anisolated L system.In Fig. 5, we show the diagonal components of the

neutrino density matrix �ee, �xx, �yy (upper panel) and

the off-diagonal �ex, �ey (lower panel) for a given energy

mode with E ¼ 32 MeV as a function of r in inverted masshierarchy. The initial behavior is qualitatively similar to theone in normal hierarchy (not shown). The pendular oscil-lations of �ee begin to develop at r ’ 35 km for the e� ysector, and proceed as pure 2� transitions till r ’ 50 km.Up to this point, �ex has not evolved significantly. All off-diagonal components increase rapidly, but the �ex starts todevelop only after �ey, and saturates at r ’ 60 km, when

e� x conversions start. Note that, �ex grows faster than a2� L system (shown in Fig. 4), as predicted. The�13 coupling between the L and H sectors induces three-flavor effects in the neutrino conversions: The initial kick,associated with �m2

atm, is necessary to trigger the instabil-ity in the L system.

B. Inverted mass hierarchy

In Fig. 6, we show the complete development of the �ee,�xx, and �yy components of the density matrix in inverted

mass hierarchy for neutrinos (left panels) and antineutrinos(right panels) for four representative energy modes. Weobserve that once the conversions have been started, the

ρee

ρyy

ρxx

ρ αα

0.1

0.2

0.3

0.4

0.5

|ρey|

|ρex|

20 30 40 50 60 70 8010-7

10-6

10-5

10-4

10-3

10-2

10-11

|ρeα

|

r (km)

FIG. 5 (color online). Initial radial evolution of the diagonal�ee, �xx, �yy (upper panel) and off-diagonal �ex, �ey (lower

panel) components of the neutrino density matrix for invertedmass hierarchy, shown for a neutrino energy E ¼ 32 MeV.

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different energy modes for the three neutrino species os-cillate in phase, confirming the collective behavior of theflavor conversions. However, the final fate of �ee, �xx, and�yy depends on their energy. As a result of the three-flavor

effects, the �e mixes with both �x and �y. Therefore, the �e

flavor conversions can be described in terms of the com-bined effects of the L andH two-neutrino systems. We findthat the effects of the H sector on the �e � �y conversions

saturate before the ones of the L sector, as expected by thehierarchy between the two mass splittings. As we havediscussed in Sec. III A, �m2

atm < 0 and �m2sol > 0 are

expected to process complementary parts of the neutrinoenergy spectra. Therefore, their effects do not interfere inthe same energy range. In particular, in the part of theneutrino energy spectra unstable under the effects of�m2

atm, the �ee and �yy would swap, while �xx, which

has been perturbed from its stable equilibrium configura-tion, would relax to it again. Conversely, in the part of theneutrino energy spectra unstable under the effects of�m2

sol,

the �ee and �xx swap, while �yy comes back to its initial

value.In the three-flavor space, the neutrino ensemble behaves

like a pendulum. Once it is perturbed from its initialconfiguration, it would evolve toward its stable equilibriumposition which may be different for different energymodes. In inverted mass hierarchy, the highest energymodes relax to the x state, while the intermediate ones tothe y state, while lowest energy modes remain in the eflavor.

For the modes represented in Fig. 6, we see that for E ¼2:5 MeV none of the three neutrino species is affected bysignificant flavor changes. For E ¼ 7:5 MeV the flavorconversions produce a swap between �ee and �yy, while

�xx comes back to its original value. At 23 MeV, we are inthe transition region between e� y and e� x conversions,therefore �yy and �xx are both partially swapped into �ee.

Finally, at E ¼ 40 MeV, �ee and �xx exchange their initialvalues, while �yy returns to its original value. Therefore,

the behavior of the diagonal components of the densitymatrix at different energies would produce a single split inthe �e energy spectrum, since all the �e modes at suffi-ciently high energy would swap with either �y or �x.

Therefore, the high-energy spectral split at E ’ 23 MeVobserved in inverted mass hierarchy in the 2� evolution ofSec. III A is washed out by the three-flavor effects. In theantineutrinos (right panels), we find an analogous behavior.At E ¼ 2:5 MeV �ee, �xx, and �yy are essentially un-

changed. At E ¼ 13 MeV the flavor conversions producea swap between �ee and �yy, while �xx comes back to its

original value. For E ¼ 23 MeV, �yy returns to its original

value, while �ee and �xx tend to exchange their initialvalues. However, as we will discuss later, due to the lessadiabatic behavior around the splitting region, the ��e � ��x

swap is not complete. Finally, for E ¼ 40 MeV �ee and�xx completely convert into each other and �yy is stable.

In the upper panels of Fig. 7, we show the neutrino (left)and antineutrino (right) spectra before and after the com-plete 3� flavor conversions in inverted hierarchy. In thelower panel we show the corresponding Pee ¼ Pð�e !�eÞ, Pex ¼ Pð�e ! �xÞ, and Pey ¼ Pð�e ! �yÞ probabil-ities, which can be defined approximately as the off-diagonal elements of the density matrix are small at theend of the collective evolution. Namely, these three prob-abilities read

Pð�e ! ��Þ ¼ �f�� � �0

xx

�0ee � �0

xx

; (23)

where 0 and f indicate the initial and final values of �,respectively. We first consider the neutrino flavor evolu-tion. As we have already discussed, the effects of the L andH systems process complementary parts of the neutrinoenergy spectra. Indeed, as we can see from the conversionprobabilities, in the energy range where Pex ¼ 1, we havePey ¼ 0 and vice versa. In particular, the solar mass split-

ting�m2sol induces an electron survival probability Pee ¼ 0

at high energies, erasing the H sector high-energy split atE ’ 23 MeV, as was pointed out in [23]. Thus, the finalelectron neutrino spectrum reads

Ff�e

¼ PeeF0�eþ ð1� PeeÞF0

�x’�F0�e

for E & 5 MeV;F0�x

for E * 5 MeV:

(24)

Conversely, the behavior of the y and x spectra is the same

ρ eeneutrinos

0.0

0.2

0.4

0.6

antineutrinos

0.1

0.2

0.3

0.4

0.5ρ yy

0.0

0.2

0.4

0.6

0.1

0.2

0.3

0.4

0.5

40 MeV

2.5 MeV

7.5 MeV23 MeVρ xx

0.0

0.2

0.4

0.6

102 103

r (km)

40 MeV

2.5 MeV23 MeV

13 MeV

102 103

r (km)

0.1

0.2

0.3

0.4

0.5

FIG. 6 (color online). Three-flavor evolution in inverted masshierarchy. Radial evolution of the diagonal components of thedensity matrix � for neutrinos (left panels) and antineutrinos(right panels) for different energy modes.

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as the that of the nonelectron species in 2� flavor evolutionin IH and NH, respectively, i.e.

Ff�y

¼ PeyF0�eþ ð1� PeyÞF0

�x

’8><>:F0�x

for E & 5 MeV;F0�e

for 5 MeV & E & 23 MeV;F0�x

for E * 23 MeV;(25)

and

Ff�x ¼ PexF

0�eþ ð1� PexÞF0

�x

¼�F0�x

for E & 23 MeV;F0�e

for E * 23 MeV:(26)

Indeed, the �x and �y spectra will be affected, respectively,

only by the L or the H sector, since x� y conversions arestrongly suppressed as evident from the vacuumHamiltonian in Eq. (16). One should note that while thehigh-energy spectral split is no longer present in the �e

final energy spectrum, �y and �x final spectra still present

this high-energy feature. Therefore, MSWeffects in super-nova, vacuummixing, and Earth effects that occur later can

make the high-energy split reappear in the observableelectron neutrino spectra at Earth.Moving on to the antineutrinos, we realize that the e� x

swap is not sharp, due to the imperfect adiabaticity in the Lsector. This implies that the swap of ��e and ��x spectra atintermediate energies (14 MeV & E & 40 MeV) is notcomplete and this explains the mixed ��e spectrum, ob-served also in the numerical simulations in [23]. This effecthas to be attributed to the low adiabaticity of the�m2

sol-induced conversions, which is particularly visible

for antineutrinos. The adiabaticity of a spectral split de-pends on the condition in Eq. (14). The ��e and ��x spectraare closer to each other than the corresponding neutrinospectra. This implies that in the high-energy region, wherethe swap takes place,

jF ��eðEÞ � F ��x

ðEÞj jF�eðEÞ � F�x

ðEÞj: (27)

In the final phases of the swapping dynamics, the neutrinoand antineutrino spectra evolve quite independently, andthe precession frequencies of the two blocks are not gov-erned by a common �, but by individual �’s proportionalto the flux differences in Eq. (27). They behave essentiallyas two uncoupled oscillators because the neutrino-neutrinointeraction � is now smaller than the frequency differenceof the two blocks. One can see this clearly in the numericalsimulations of Ref. [16], shown at [32]. The frequency�1pend � ffiffiffiffiffiffiffiffi

!�p

is lower for antineutrinos [16], and thus

adiabaticity tends to be broken more severely forantineutrinos.

C. Normal mass hierarchy

In Fig. 8, we show the complete development of the �ee,�xx, and �yy components of the density matrix in normal

hierarchy for neutrinos (left panels) and for antineutrinos(right panels) for different energy modes. For both neutri-nos and antineutrinos, the swapping dynamics occursmainly between the �ee and the �yy, while �xx experiences

only nutations around its initial equilibrium value, beforerelaxing completely to it when the flavor conversions aresaturated. Therefore, the flavor evolution is close to the 2�H case in normal hierarchy discussed in Sec. III A. Indeed,both �m2

atm and �m2sol are positive. Therefore, the L sector

would behave as a replica of the H sector, but with asmaller mass splitting. In this condition, both H and Lsectors process the same regions of the electron neutrinoenergy spectra. However, the hierarchy between the twomass splittings produces the dominance of the �e � �y

swaps, while conversion effects in the �e � �x sectorremain inhibited by adiabaticity violation. The only regionwhere the L instability can compete withH is very close tothe split. In particular, for low-energy neutrino modes (E ¼2:5, 18 MeV in Fig. 8) the �ee comes back to its initialvalue, while at higher energies (E ¼ 23, 40 MeV)exchanges its initial value with �yy. For the antineutrinos,

neutrinosF

ν (a

.u.)

1

2

3

4

5

67

e, initx,y init

y, finx, fin

e, fin

antineutrinos

PeyPex

Pee

0 10 20 30 40 50 600.0

0.2

0.4

0.6

0.8

1.0

Peα

E (MeV)

10 20 30 40 50 60

E (MeV)

FIG. 7 (color online). Three-flavor evolution in inverted masshierarchy for neutrinos (left panels) and antineutrinos (rightpanels). Upper panels: Energy spectra initially (dashed curves)and after collective oscillations (solid curves) for �e (red), �x

(black), and �y (blue). Lower panels: probabilities Pee (solid red

curve), Pey (dashed blue curve), and Pey (dotted black curve).

The gray bands represent the region where the spectral swapsoccur. For antineutrinos the light gray band indicates the regionwhere the spectral swap is partial (see text for details).

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�ee remains at its initial value for the modes at E ¼2:5 MeV. At E ¼ 18 MeV, we are around the splittingregion where the adiabaticity in the H system is moreseverely violated, and the e� y conversions are not com-plete. Under these conditions, the L instability can alsoplay a role, producing a weak swap in �xx. Finally, at E ¼40 MeV, conversions again occur only between e and ystates, producing a complete swap of the initial �ee and �yy

values.In Fig. 9, we show the neutrinos (left panels) and anti-

neutrino (right panels) spectra before and after the com-plete 3� flavor conversions in NH. In the lower panels werepresent the corresponding Pee, Pex, and Pey probabilities.

Once more, we realize that the flavor evolution can bemostly described in terms of two-flavor �e � �y transi-

tions, while the role of �e � �x conversions is subleading.The spectral splitting features are remarkably similar to theones found for the 2� H system in normal hierarchy(Sec. III A). In particular, the �e spectrum presents a singlesplit at E ’ 23 MeV and ��e spectrum around E ’ 17 MeV.Only close to the splitting regions, we find subleadingeffects associated with �e � �x conversions.

V. CONCLUSIONS

Collective neutrino flavor conversions in supernovae,associated with neutrino-neutrino interactions, have beenrecognized to induce peculiar spectral swaps among thedifferent neutrino species. The development of these fea-

tures is associated with instabilities in the flavor space. Inparticular, these instabilities would develop around thecrossing points of the original SN neutrino spectra. Then,the neutrino mass hierarchy determines if a crossing pointis unstable under the effects of the collective oscillations. Aparticularly intriguing case is the one in which the originalSN neutrino fluxes exhibit an ordering with �0

�x* �0

�e*

�0��e, possible during the cooling phase. In this case, the

two-flavor study realized by [16] found the occurrence ofmultiple spectral splits for both neutrinos and antineutri-nos, depending on the neutrino mass hierarchy. A recentnumerical exploration of this case performed in [23] hasfound that when three-flavor effects are taken into accountin inverted mass hierarchy, the high-energy spectral swapsobserved in the 2� evolution are erased by effects related to�m2

sol.

Motivated by this intriguing result, in our paper we haveperformed a detailed study of the three-flavor effects in thecollective oscillations for supernova neutrino spectra typi-cal of the cooling phase. We have found that the effects of�m2

sol in the three-flavor evolution are important only in

inverted mass hierarchy. In this case, the presence of �m2sol

gives rise to instabilities in regions of the neutrino energy

ρ eeneutrinos

0.0

0.2

0.4

0.6

antineutrinos

0.1

0.2

0.3

0.4

0.5ρ yy

0.0

0.2

0.4

0.6

0.1

0.2

0.3

0.4

0.5

40 MeV

2.5 MeV

23 MeV

18 MeVρ xx

0.2

0.4

0.6

102 103

r (km)102 103

r (km)

0.2

0.3

0.4

0.5

FIG. 8 (color online). Three-flavor evolution in normal masshierarchy. Radial evolution of the diagonal components of thedensity matrix � for neutrinos (left panels) and antineutrinos(right panels) for different energy modes.

neutrinos

Fν

(a.u

.)

1

2

3

4

5

67

e, initx,y init

y, finx, fin

e, fin

antineutrinos

PeyPex

Pee

0 10 20 30 40 50 600.0

0.2

0.4

0.6

0.8

1.0

Peα

E (MeV)

10 20 30 40 50 60

E (MeV)

FIG. 9 (color online). Three-flavor evolution in normal masshierarchy for neutrinos (left panels) and antineutrinos (rightpanels). Upper panels: Energy spectra initially (dashed curves)and after collective oscillations (solid curves) for �e (red), �x

(black), and �y (blue). Lower panels: corresponding probabilities

Pee (solid red curve), Pey (dashed blue curve), and Pey (dotted

black curve). The gray bands represent the region where thespectral swaps occur (see text for details).

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spectra that were stable under the two-flavor evolutiongoverned by �m2

atm and �13. Therefore, the combinationsof these two different instabilities would produce a wash-out of the high-energy splitting features in the �e and ��e

spectra. Conversely, in normal mass hierarchy the three-flavor instabilities and the two-flavor one act in the sameregions of the neutrino energy spectrum, leading only tominor departures from the two-flavor evolution.Essentially, the system behaves like a pendulum in 3�flavor space. It can topple toward either the �y state or

the �x state. In inverted hierarchy, when the H and Linstabilities are in different regions of energy, the pendu-lum topples toward �y for the H instability, and toward �x

for the L instability. In normal hierarchy, when the insta-bilities are in the same region of energy, the pendulumtopples toward �y, as the L instability is relatively non-

adiabatic. As a consequence, in inverted mass hierarchy theelectron (anti)neutrino spectrum at the end of the collectiveoscillations would present only a very low-energy (E &5 MeV) splitting feature, being completely swapped to theoriginal nonelectron spectra at higher energies.

We wish to emphasize that the high-energy splittingfeatures may survive in the observable electron (anti)neu-trino spectrum at Earth, even in inverted hierarchy. Indeed,MSW matter effects in SN, vacuum mixing, and Eartheffects would further mix the �e and �x spectra. The non-electron spectrum at the end of the collective oscillationsstill contains high-energy splitting features, since for thenonelectron species the collective flavor conversions haveoccurred essentially as in the two-flavor case. Therefore,especially for neutrinos, which have sharper spectralswaps, the electron neutrino signal at Earth could stillpresent observable splitting features at high energy. Also,due to the lower neutrino luminosity at sufficiently late

times, the initial collective interaction strength �0 can besomewhat lower than is assumed in Ref. [23] and this work.We found that in certain regions of the spectral parameterspace, reducing�0 by a factor of 10 makes the three-flavoreffect disappear due to a stronger adiabaticity violation inthe L sector. In principle, this effect could produce inter-esting signatures in the time evolution of the SN neutrinosignal.In conclusion, the nonlinear equations that govern the

flavor evolution of neutrinos emitted during a stellar col-lapse are a continuous source of surprises and new effects.During this last year, dramatic changes have occurred inthe picture consolidated after the initial exploration ofcollective supernova neutrino oscillations. The discoveryof this new three-flavor effect is the most recent of thesechanges. After our study, it appears that its impact on thecollective neutrino flavor conversions is conceptually andquantitatively well under control.

ACKNOWLEDGMENTS

We thank Alois Kabelschacht for fruitful discussionsduring the development of our work and for a carefulreading of our manuscript. R. T. thanks Andreu Esteban-Pretel for useful discussions. B. D. thanks AlexanderFriedland for useful correspondence. B.D. and I. T. werepartly supported by the Deutsche Forschungsgemeinschaftunder Grant No. TR-27 ‘‘Neutrinos and Beyond’’ and theCluster of Excellence ‘‘Origin and Structure of theUniverse’’ (Munich and Garching). The work of I. T. hasbeen partly supported by the Italian MIUR and INFNthrough the ‘‘Astroparticle Physics’’ research project. Herstay in Munich has been partly supported by the ItalianSociety of Physics (Borsa SIF ‘‘Antonio Stanghellini’’).

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