Probing late neutrino mass properties with supernova neutrinos
Joseph Baker,1 Haim Goldberg,2 Gilad Perez,3,4 and Ina Sarcevic1
1Department of Physics, University of Arizona, Tucson Arizona 85721, USA2Department of Physics, Northeastern University, Boston, Massachusetts 02115, USA
3Theoretical Physics Group, Ernest Orlando Lawrence Berkeley National Laboratory, University of California,Berkeley, California 94720, USA
4C.N. Yang Institute for Theoretical Physics State, University of New York, Stony Brook, New York 11794-3840, USA(Received 28 February 2007; published 12 September 2007)
Models of late-time neutrino mass generation contain new interactions of the cosmic backgroundneutrinos with supernova relic neutrinos (SRNs). Exchange of an on-shell light scalar may lead tosignificant modification of the differential SRN flux observed at earth. We consider an Abelian U(1) modelfor generating neutrino masses at low scales, and show that there are cases for which the changes inducedin the flux allow one to distinguish the Majorana or Dirac nature of neutrinos, as well as the type ofneutrino mass hierarchy (normal or inverted or quasidegenerate). In some region of parameter space thedetermination of the absolute values of the neutrino masses is also conceivable. Measurements of thepresence of these effects may be possible at the next-generation water Cerenkov detectors enriched withGadolinium, or a 100 kton liquid argon detector.
Neutrino flavor conversion has been observed in thesolar (SuperK, SNO) [1], atmospheric (SuperK) [2], andterrestrial (KamLand, K2K) [3] neutrino data, providingevidence for nonvanishing, subeV neutrino masses. Therenow remains the long-standing theoretical question of howthe neutrinos acquire their masses. The most elegant solu-tion to this puzzle is the seesaw mechanism [4]: oneassumes that lepton number is violated at some high scale�L in the form of right-handed neutrino, N, Majoranamasses, MN ��L. This induces, at a lower scale, aneffective operator of the form O�1� � �LH�2=�L, whereL denotes a lepton doublet and H the Higgs field. Theoscillation data then imply that �L � 1014 GeV. However,it is difficult to devise an experimental test of this mecha-nism (see however [5]). Therefore, it is important to ex-plore alternate natural mechanisms for neutrino massgeneration, especially those that may be tested in experi-ments at low energies.
A class of such models that have astrophysical andcosmological tests are the models of late-time neutrinomass generation [6–8]. In these models, neutrino massesare protected by some flavor symmetry different from theone related to the charged fermion masses, for example,some global U�1�N symmetry. The small neutrino massesare generated when the new symmetry is broken at lowscales. The effective Lagrangian for these models can beschematically written for either Dirac or Majorana parti-cles where the neutrino fields are neutrino mass eigenstatesas
L D� � Lkin � y���N � V���;
LM� � Lkin � y����� V���;
where Lkin is the kinetic piece of the Lagrangian, y� is a
dimensionless coupling, � is a standard model neutrinofield,N is an extra field introduced for the case of neutrinosbeing Dirac particles, � is the scalar field, and V��� is theassociated scalar potential. After spontaneous symmetrybreaking the neutrinos acquire masses given by
m� � y�f; (1)
where m� is the mass of a particular neutrino mass eigen-state and f is the symmetry breaking scale (f � h�i, whereh�i is the vacuum expectation value (VEV) of �). Withjust one scalar the couplings are diagonal in the mass basis.In addition, since the scalar couples to neutrinos only, theconstraints on the symmetry breaking scale f are weak [7].
In addition to generating neutrino mass through theirVEVs, the new light scalars provide another neutrino-neutrino interaction process aside from the standard modelZ0 exchange. The effects of the neutrinos coupling to thesescalars on the cosmic microwave background has beenpreviously studied [7]. Constraints have been placed onthe symmetry breaking scale, f, the scalar mass, MG, andthe scalar-neutrino couplings, y�, in these models fromcosmological considerations [7–9], as well as from de-manding that supernova cooling and the flux of the1987a neutrinos would not be significantly modified inthe presence of the additional fields [10,11] (for constraintsrelated to generating the observed baryon asymmetry ofthe Universe, see [12]).
In this paper we show that the presence of this newphysics significantly modifies the spectrum of supernovarelic neutrinos (SRNs) at earth. This modification occursbecause SRNs can interact with cosmic background neu-trinos through exchange of the new light scalar. In [11] thiseffect was studied assuming a single flavor, Majorana,case. Here we extend our study and include various inter-esting aspects related to the nature of the neutrino flavor
sector, for example, the breaking of lepton number, theeffect of multiple generation, etc. For simplicity we confineour study here to late neutrino mass models with a singleU(1) symmetry. We show that the energy spectrum of theSRN flux is sensitive to the type of neutrino mass hierarchyand whether neutrinos are Majorana or Dirac particles. Wediscuss how in some specific cases one can get additionalinformation about the neutrino masses as well. In addition,detection of this signal would also be direct evidence of thepresence of the cosmic background neutrinos. Hundreds ofevents per year from the flux of the SRN antineutrinoscould be seen at the next-generation large megaton waterCerenkov detectors [13,14] such as UNO, Hyper-Kamiokande, or MEMPHYS if they are enriched withGadolinium [15], or from the flux of the SRN neutrinosin a large 100 kton liquid argon neutrino detector [16].
In Sec. II we show how the SRN flux, including cosmo-logical evolution, is modified through the new interactions.In Sec. III we consider the normal and inverted neutrinomass hierarchy cases as well as quasidegenerate neutrinomasses. We also consider the possibility of neutrinos beingeither Majorana or Dirac particles. In addition we showthat there is a particularly interesting signal that leads to thedetermination of ratios of neutrino masses. The conditionsfor establishing a statistically significant signal abovebackground are discussed in subsection G of Sec. III.Finally, in Sec. IV, we conclude.
II. THE SUPERNOVA RELIC NEUTRINO FLUX
The resonance interaction of the SRN with cosmic back-ground neutrinos through the exchange of a new lightscalar was previously discussed in [11] for a single neu-trino mass eigenstate with m� � 0:05 eV.
In this paper we consider the case of the scalar interact-ing with three neutrino mass eigenstates and show the veryinteresting effect on the observed SRN spectrum for thenormal mass hierarchy, inverted mass hierarchy, and forquasidegenerate neutrino masses. We briefly discuss thecase of adding one sterile neutrino.
We start with the SRN flux without the new interactions.The diffuse SRN flux is a remnant of neutrinos emittedfrom all the supernova that have occurred in the Universe[17]. This flux is given by
F�E�� �Z zmax
0RSN�z�
dN��1� z�E��dE�
�1� z���������c dtdz
��������dz;(2)
where RSN�z� is the comoving rate of supernova formation,dN��1� z�E��=dE� is the neutrino energy spectrum emit-ted by supernova, dt=dz is for the cosmological expansion,c is the speed of light, z is the redshift, and E� is theneutrino energy.
The quantity RSN is the comoving rate of supernovaformation, which can be parametrized as [17]
RSN�z� ��0:013
M�
�_���z�; (3)
where _���z� is the star formation rate given by
_� ��z� � �1 2� � 102M� yr1 Mpc3 � �1� z��:
(4)
We take RSN�0� � 2� 104 yr1 Mpc3, � � 2 (for 0<z< 1), and � � 0 (for z > 1) [17]. These are ‘‘median’’values for the parameters which have uncertainties in themcoming from the uncertainty in the present knowledge ofthe cosmic star formation rate [17,18].1 The factor dt=dz isgiven by
dtdz�
�100
km
s Mpch�1� z�
����������������������������������������M�1� z�
3 ���
q �1;
(5)
with �M � 0:3, �� � 0:7, and h � 0:7.The energy spectrum of the neutrinos emitted by a
supernova has been modeled by several groups [19–21].One of the models leads to perfect equipartition of theenergy radiated into each neutrino flavor [19]. A secondmodel makes a detailed one dimensional calculation of allrelevant neutrino processes in the collapsing star and uses avariety of supernova progenitor masses [21]. Anothermodel proposed by Keil, Raffelt, and Janka [20] (KRJmodel) performs their calculations using a MC simulation.These models give spectra which have a width narrowerthan that of a thermal spectrum (so-called ‘‘pinched’’spectrum [22]). It also predicts an average energy for themuon and tau flavor neutrinos very close to the averageenergy of the electron antineutrinos. As an illustrativeexample we take the neutrino spectrum given by the KRJmodel with the additional assumption that the total energycarried by each neutrino flavor is L�e � L ��e � L�x � 5�1052 ergs, where x stands for the muon and tau neutrinosand antineutrinos. The KRJ energy spectrum of neutrinoflavor eigenstates produced by a supernova is given by
dN���E���
dE����1� ����
1����L���1� ����E
2v�
�E��E��
����
� exp��1� ����
E��E��
�; (6)
with the Ev� representing the average neutrino energiesand ��� characterizing the amount of spectral pinching.Here for simplicity we set the values of the parameters to[20] (for recent numerical studies see e.g. [23])
1Future SN observatories will have the power to significantlyreduce the related uncertainties [14].
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�e: ��e � 3:4; E�e � 13:0 MeV;
��e: � ��e � 4:2; E ��e � 15:4 MeV;
�x: ��x � 2:5; E�x � 15:7 MeV;
(7)
and also neglect effects such as shock wave and turbulence[24].
Because of matter oscillation effects, neutrinos emergefrom a supernova as coherent fluxes of mass eigenstateswhich we label as F�i , where i � 1, 2, or 3 represents theparticular neutrino mass eigenstate [25].
If neutrino flavor evolution inside of the collapsing staris either fully adiabatic or fully nonadiabatic (the flavorevolution is adiabatic if the mixing angle sin2�13 * 103
and nonadiabatic if sin2�13 & 105) then the energy spec-trum of each neutrino mass eigenstate that leaves thesurface of the star corresponds to the original energyspectrum of some particular neutrino flavor eigenstate atemission from the neutrinosphere, i.e., there is a one-to-one correspondence between each dN���E���=dE�� andsome dN�i�E�i�=dE�i .
2 The original produced flux ofsome neutrino flavor at the neutrinosphere will be labeledas F0
�� . Translated back into the flavor basis, the expres-sions for the �e and ��e fluxes emerging from a supernovacan be written as
F�e � PHjUe2j2F0
�e � �1 PHjUe2j2�F0
�x ;
F ��e � jUe1j2F0
��e � jUe2j2F0
�x ;(8)
for the normal mass hierarchy and
F�e � jUe2j2F0
�e � jUe1j2F0
�x ;
F ��e ��PHjUe1j
2F0��e � �1
�PHjUe1j2�F0
�x ;(9)
for the inverted mass hierarchy, where PH�and �PH� � 0for the adiabatic case and PH�and �PH� � 1 for the non-adiabatic case [25]. In the equations above, jUe1j
2 �cos2�12, jUe2j
2 � sin2�12, �12 � �� (�� is the solar mix-ing angle [27]), where sin22�12 � 0:86� 0:3, andjUe3j
2 0 [28]. When supernova neutrino flavor evolutionis nonadiabatic then the �e and ��e flux for the normal andinverted hierarchies are identical.
We show that the addition of a new light scalar opens thepossibility of determining the neutrino mass hierarchyindependent of the neutrino fluxes, and also independentof the adiabatic or nonadiabatic nature of supernova neu-trino flavor evolution.
A. Modifications due to new physics
We consider the modifications to the SRN flux due to theresonance interaction of an SRN with a neutrino in thecosmic neutrino background. In this process a supernovaneutrino with energy ESN
� will go through the resonance
when the kinematic condition
ESN� �
M2G
2mi� ERes
i (10)
is satisfied. More specifically, a neutrino observed withenergy EObs
� will have gone through resonance if its energylies in the region
EResi
1� z< EObs
� < EResi ; (11)
where z is the redshift. There will be large depletion of theSRN flux in the energy domain given in Eq. (11) as long asthe neutrino-scalar coupling satisfies [11]
y > 4:6� 108 MG
1 keV: (12)
This condition comes from requiring that the mean freepath for absorption is much smaller than the Hubble scale.It is important to note that in the narrow width approxima-tion for the resonance, the condition equation (12) is asufficient condition to guarantee the absorption of all threeneutrino flavors. After the neutrinos that have energies inthe region given by Eq. (11) go through the resonance theyare redistributed to lower energies when the producedscalar decays back to neutrino mass eigenstates [29]. Inparticular, neutrinos after interaction will be redistributedwith a flat energy distribution from zero energy up to theoriginal energy of the incident supernova neutrino.
To find the effect of these interactions on the flux of theSRN we note that the neutrinos leaving a supernova atredshift z emerge as the mass eigenstates. However, thesemass eigenstate fluxes are now modified through interac-tion with the cosmic background neutrinos as they propa-gate to the Earth. We consider for simplicity an AbelianU(1) late neutrino mass model. This implies that theYukawa interaction between the scalars (in particular theGoldstone) and the neutrinos are diagonal in the mass basis(this is not the case in a model with non-Abelian symme-tries). Supernova neutrinos are in their mass eigenstatesand each mass eigenstate interacts only with the same masseigenstate background neutrino via Goldstone exchange.To illustrate how the interactions modify the neutrino masseigenstate flux we consider as an example the flux of the �1
mass eigenstate, F�1.
We start by defining the modified flux of the �1 eigen-states as ~F�1. For each redshift z the �1 eigenstates thatsatisfy the condition given by Eq. (11) will have resonanceinteraction with cosmic background neutrinos, producingthe intermediate scalar. A neutrino mass eigenstate will gothrough the resonance when the coupling satisfies Eq. (12).The cross section (averaged over the width of the reso-nance) for this to occur is approximately given by �Res ’�=M2
G. This will lead to a mean free path much smallerthan the typical distance a supernova neutrino will travel toarrive at the Earth, for the values of MG that we consider.2See, for example, Table 1, Fogli, et. al. [26].
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We label the absorbed flux asFRes�1
. Naively, the modifiedflux would be given by
~F �1� F�1
FRes�1: (13)
However, this expression does not take into account thatthe scalar decays back into neutrino mass eigenstates. Weneed to add this contribution to Eq. (13). The scalar candecay to any of the neutrino mass eigenstates. The proba-bility that the scalar decays to a particular neutrino masseigenstate is proportional to the square of the Yukawacoupling of that particular neutrino mass eigenstate to thescalar. From Eq. (1) we note that the relative probabilitiesare proportional to the ratios of squares of the neutrinomasses,
Pj m2jP3
i�1 m2i
: (14)
Then the probability that a scalar decays to the neutrinomass eigenstate �1 is P1. These decays result in redistrib-ution of the neutrino energies from zero energy up to theenergy of the incident SRN with a flat energy distribution.We define P1 � FRes
1!10 as the fraction of the flux of �1 thatinitiates a resonance, producing a scalar which then decaysback into a �1 eigenstate with degraded energy (indicatedby the notation 10). Then, Eq. (13) is modified to
~F �1� F�1
FRes�1� P1 � F
Res1!10 : (15)
We still need to take into account the contributions fromthe decays of scalars produced by other neutrino masseigenstates. Therefore, there should be a sum over all ofthe initial states, and Eq. (15) becomes
~F �1� F�1
FRes�1� P1
Xi�1;2;3;�1;�2;�3
FResi!10 : (16)
In more general notation, for the jth neutrino mass eigen-state,
~Fj � Fj Fresj � Pj �
Xi�1;2;3;�1;�2;�3
FResi!j0 : (17)
The contributions over a range of redshift must be taken todetermine the total flux at Earth. If neutrinos are Diracparticles then there is factor of 1=2 multiplying the lastterm (see discussion in Sec. III E).
The modified flux of electron neutrinos and electronantineutrinos can then be written as
~F �e � cos2�12~F�1� sin2�12
~F�2; (18)
and
~F ��e � cos2�12~F ��1� sin2�12
~F ��2: (19)
Finally, we note that each neutrino mass eigenstate goesthrough resonance at different energies given by Eq. (10)when there is just a single scalar of mass MG. Dependingon the details of the neutrino mass hierarchy, these reso-
nance energies can either be very close to one another, orwidely spaced apart.
III. SIGNALS OF MODELS OF LATE-TIMENEUTRINO MASS GENERATION
In this section we discuss the signals for the neutrinomass hierarchy in the observed SRN flux. We consider thecase of the normal neutrino mass hierarchy, the invertedneutrino mass hierarchy, and the possibility that the neu-trino masses are quasidegenerate. We also show the effectsof the neutrinos being Dirac or Majorana particles on theSRN flux signal. For these cases, unless otherwise noted,we choose the value of ERes
i , defined in Eq. (10), at z � 0 tobe equal to 15 MeV for one of the neutrino mass eigen-states. This choice is made to illustrate the effects of theresonance process and to determine a region of the pa-rameter space of the late-time neutrino mass generationmodels where the effect of the SRN modification would beseen. This is so, since for water Cerenkov detectors nearreactors the background becomes negligible above about13 MeV [13,14]. As will be discussed in the conclusions,the energy resolution is about 2 MeV. In subsections Athrough E we consider the case where neutrino flavorevolution in the SN is adiabatic, and in subsection F weshow that the same features are obtained for a case wherethe flavor evolution is nonadiabatic. In subsection G wediscuss the detection of the new interactions.
In the following we focus on the flux of electron anti-neutrinos that arrive at Earth, since the proposed waterCerenkov experiments for detection are sensitive to thisneutrino flavor through the interaction of electron antineu-trinos with protons with a cross section given by [30]
where Ee� and pe� are the energy and momentum of thedetected positron. Note that detection of the electron neu-trino component of the SRN flux at a large liquid argondetector would provide complementary information [31].
A. Normal neutrino mass hierarchy
As an example to illustrate how resonance interactionsbetween the SRN and the cosmic background neutrinos canaffect the SRN flux, we first consider a normal masshierarchy of neutrino mass eigenstates. As a particularexample of this hierarchy we choose the masses of themass eigenstates to be
m1 � 0:002 eV; m2 � 0:009 eV;
m3 � 0:05 eV:(21)
This conforms to the best value of the atmospheric masssplitting jm2
3 m21;2j ’ 2:4� 103 eV2 [32]. The value of
the lightest mass, m1, was chosen to be 0.002 eV for the
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purpose of our numerical study, however there is no lowerlimit on the value of the mass of the lightest neutrino ineither the normal or inverted hierarchies. If the mass of thelightest neutrino is lowered below the neutrino backgroundtemperature, TC�B, then the resonance for this lightest stateis governed by the corresponding thermal energy of thebackground neutrinos (see footnote 3 below). The particu-lar choice for the neutrino masses in Eq. (21) results in thefollowing features:
(1) When a scalar is produced, it decays predominantlyto the m3 mass eigenstate. Equation (14) gives P3 0:967, P2 0:031, and P1 0:002.
(2) Because there is one scalar with mass MG, Eq. (10)implies that the ratios of the neutrino masses governthe resonance energy positions, so that ERes
2 �2=9� ERes
1 and ERes3 � 1=25� ERes
1 .3
If we choose the lightest neutrino mass eigenstate to havethe resonance at 15 MeV, to illustrate the effect, then thecorresponding scalar mass is MG 245 eV, and point 2above implies that the other two resonances are ERes
2 3:8 MeV and ERes
3 0:60 MeV. These two resonance en-ergies are both well below experimental detection thresh-olds. Additionally, because the modified electronantineutrino flux is composed only of F ~��1
and F ~��2the
overall effect on F ~��e is a depletion since decays of thescalar primarily contribute to F ~��3
. The resulting flux ofelectron antineutrinos can be seen in Fig. 1. Folding theflux with the cross section for electron antineutrinos onprotons, Eq. (20), gives the spectrum in Fig. 2.
Relative to the electron antineutrino flux without inter-actions, the case with interactions has large depletionbecause of the dominant decay into them3 mass eigenstate.If we choose a heavier scalar so that them2 mass eigenstategoes through the resonance at 15 MeV instead of the m1
mass eigenstate (i.e.,MG 490 eV), then we get the samefeature, but the depletion is smaller. This is because F ~��1
ismultiplied by cos2�12 0:70, while F ~��2
is multiplied bysin2�12 0:30, so F ~��2
is the smaller component of thefinal electron antineutrino flux. If the scalar were evenheavier (i.e., MG 1:2 keV) so that the m3 mass eigen-state goes through the resonance at 15 MeV, then there isno depletion in the electron antineutrino flux since both them1 and m2 eigenstates would have resonances at very highenergies. As a result, neither of the corresponding fluxeswould be visibly modified.
The mass of the lightest neutrino can also be loweredwithout bound (although as mentioned above we do notconsider the cases where the mass of the lightest neutrino is
below the mass of the cosmic background neutrino tem-perature). For a fixed value of MG, as the mass of thelightest state is lowered, the position of the resonancemoves to higher energies until it is in a region where theflux of the SRN is too small for the feature to be experi-mentally observable.
To illustrate the effect on the electron neutrino spectrum,relevant to argon detectors such as Icarus and a future100 kton argon detector [16], we show the event rates inFig. 3 for the normal hierarchy case. A next-generationliquid argon detector with a size of 100 ktons could mea-
5 10 15 20 25 30 35 40Eν( MeV)
0.5
1
1.5
2
F( cm–2 s –1 MeV–1 )
FIG. 1 (color online). The SRN electron antineutrino fluxwithout interactions (solid, red curve) and with interactions(dashed, blue curve) when neutrinos are Majorana particlesand for the normal mass hierarchy.
FIG. 2 (color online). The event rates for Hyper-Kamiokandewithout interactions (red, solid curve) and with interactions(blue, dashed curve) when neutrinos are Majorana particlesand for the normal mass hierarchy.
FIG. 3 (color online). The SRN electron neutrino flux for anormal hierarchy folded with the cross section for electronneutrinos to interact with argon in a 100 kton detector runningfor 5 years. The solid (red) curve is for no interactions and thedashed (blue) curve is with interactions.
3The value of ERes1 for the lightest neutrino could be in the
range�������������������2E�TC�Bp
& ERes1 �
���������������2E�m1
p, where the lower limit cor-
responds to the transition to the relativistic case and TC�B is thebackground neutrinos temperature. Since TC�B � 2� 104 eVwhich is not far from m1 (given that the effect goes like thesquare root of the mass in that range), our results will only beslightly modified when this is taken into account.
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sure a significant number of electron neutrinos over just5 years [16]. At neutrino energies lower than about 19 MeVthe solar neutrino flux dominates the SRN flux, and atenergies greater than about 40 MeV the atmospheric neu-trino flux begins to dominate. In Fig. 3, to obtain eventrates, we have folded the SRN flux with the cross sectionfor electron neutrinos to interact with argon [16]. We haveused a resonance energy of 25 MeV for the lightest neu-trino mass state (taken to be 0.001 eV so that MG 225 eV). In this case we find significant reduction in theintegrated event rate over the region in which the SRN fluxis dominant.
B. Inverted neutrino mass hierarchy
We now consider an example of the inverted masshierarchy, characterized by m1 ’ m2 � m3, with neutrinomasses chosen to be
m1 � 0:05 eV; m2 0:05 eV; m3 � 0:008 eV:
(22)
This reflects the best value of 7:92� 105 eV2 [32] for�m2
21, the �1-�2 mass splitting. Whenever a scalar isproduced it dominantly decays into these two heavy eigen-states with equal probabilities. Because F ~��1
and F ~��2are the
contributing components to the electron antineutrino flux,in this scenario there are both regions of depletion but alsoregions of overall enhancement of the flux due to rescatter-ing. We choose the case where the m1 and m2 masseigenstates go through resonance at 15 MeV, giving MG 1:2 keV, and show the results in Fig. 4 (and Fig. 5 forweighting with cross section).
We find that the enhancement is large because all initialneutrino mass eigenstate fluxes produce scalars which addto the low energy m1 and m2 eigenstate fluxes. The m1 andm2 eigenstate flux is depleted and is redistributed to lowerenergies. Once we fold the flux with the cross section, wesee in Fig. 5 that in contrast to the case of the normal masshierarchy, the peak at low energies is more pronounced.This is a result of the electron antineutrino flux beingcomposed only of the m1 andm2 neutrino mass eigenstates
(since jUe3j2 0), both of which get depleted by the
resonance at the same energy since the two eigenstatesare nearly mass degenerate. Therefore the electron anti-neutrino flux is almost completely depleted near the reso-nance cutoff.
We apply a similar analysis to the electron neutrino flux,of relevance to a liquid argon detector. In Fig. 6 we showevent rates for 100 kton liquid argon detector. We use aresonance energy of 25 MeV for the two heavier masseigenstates with mass 0.05 eV, which corresponds toMG 1580 eV. The integrated event rate for energies above thesolar background are reduced relative to the no interactioncase.
C. Quasidegenerate neutrino masses
There still remains the possibility that the neutrino masseigenstates are quasidegenerate. For example, a mass hier-archy structure with
m1 0:06 eV; m2 0:06 eV; m3 0:08 eV;
(23)
satisfies the requirements for the two independent masssplittings as well as cosmological constraints on the sum ofthe neutrino masses [33]. We consider the case where thetwo eigenstates with mass 0.06 eV have the same reso-nance energy as before, which would correspond to MG �1340 eV. The third mass eigenstate then has a resonance
FIG. 4 (color online). The SRN electron antineutrino fluxwithout interactions (solid, red curve) and with interactions(dashed, blue curve) when neutrinos are Majorana particlesand for the inverted mass hierarchy.
FIG. 5 (color online). The event rates for Hyper-Kamiokandewithout interactions (solid, red curve) and with interactions(dashed, blue curve) when neutrinos are Majorana particlesand for the inverted mass hierarchy.
FIG. 6 (color online). The SRN electron neutrino flux for aninverted hierarchy folded with the cross section for electronneutrinos to interact with argon in a 100 kton detector runningfor 5 years. The solid (red) curve is for no interactions and thedashed (blue) curve is with interactions.
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energy at approximately 11 MeV, however there is nocorresponding dip since only the m1 and m2 mass eigen-states contribute to the final ��e flux. The result for the fluxcan be seen in Fig. 7 and for the event rate in Fig. 8.
It is clear by comparing Figs. 4 and 7 that the case of theinverted mass hierarchy is nearly indistinguishable fromthe case of quasidegenerate neutrino masses. However thequasidegenerate case is distinguishable from the specificcase of the normal mass hierarchy when one neutrino massis much lighter than the other two masses.
D. Multiple depletion dips
There is the possibility within the normal hierarchy thatthe m1 and m2 mass eigenstates are nearly, but not exactly,degenerate, and also still much lighter than the m3 masseigenstate. One example of the possible values for themasses in such a scenario is
m1 0:01 eV; m2 0:013 eV; m3 0:05 eV:
(24)
If we choose the resonance of the m2 eigenstate to be at15 MeV, then the m1 mass eigenstate goes through reso-nance at ERes
1 20 MeV. This corresponds to MG 630 eV.
As can be seen in Figs. 9 and 10, this leads to twodepletion dips in the final electron antineutrino spectrum
and three peaks. There is always a signal corresponding toeach neutrino mass eigenstate interacting with theGoldstone. The presence of two distinct depletion dips inan experimentally interesting region, however, is sensitiveto the ratio of the masses of two of the neutrino masseigenstates, in this case m1 and m2. For example, inSec. III A there is only one depletion dip because the ratioof the masses, and therefore the ratio of the resonanceenergies, for the m1 and m2 states is 4.5, so that with m1
resonance at 15 MeV the m2 resonance is at 3.8 MeV,outside of the observable region.
Experimental observation of the energy position of thesedips could determine the ratio of the m1 and m2 masses,which together with the measured value of �m2
21 and �m232
allows one to determine the neutrino masses. This is aremarkable possibility since it is extremely hard to experi-mentally determine the exact values of the neutrinomasses, especially the mass of the lightest state.
E. Dirac vs Majorana neutrinos
If neutrinos are Majorana particles, then each scalardecay produces a �L�L or �R�R for each mass eigenstate.If the neutrinos are Dirac particles then the scalar candecay to � �N or toN ��, where �N andN are the extra neutrino
FIG. 8 (color online). The event rates for Hyper-Kamiokandewithout interactions (solid, red curve) and with interactions(dashed, blue curve) when neutrinos are Majorana particlesand for quasidegenerate neutrino masses.
FIG. 10 (color online). The event rates for Hyper-Kamiokandewithout interactions (solid, red curve) and with interactions(dashed, blue curve) when neutrinos are Majorana particles,for the normal mass hierarchy and where two neutrinos havedistinct resonance features in the experimentally observableregion.
FIG. 9 (color online). The SRN electron antineutrino fluxwithout interactions (solid, red curve) and with interactions(dashed, blue curve) when neutrinos are Majorana particles,for the normal mass hierarchy and where two neutrinos havedistinct resonance features in the experimentally observableregion.
FIG. 7 (color online). The SRN electron antineutrino fluxwithout interactions (solid, red curve) and with interactions(dashed, blue curve) when neutrinos are Majorana particlesand for quasidegenerate neutrino masses.
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fields added for the case of neutrinos being Dirac particlesin the late-time neutrino mass generation models. Thenonly half of the decays of the scalar produce an antineu-trino that will be seen in the detector. Therefore for the caseof neutrinos being Dirac particles there is an overall factorof 1=2 multiplying the last term of Eq. (17) relative to thecase of neutrinos being Majorana particles.
If the neutrinos are arranged in a normal mass hierarchyas in Sec. III A then the ability to distinguish between theneutrinos being Dirac or Majorana particles is confoundedby the small amount of scalar decays into the m1 and m2
eigenstates. However, in the case of the inverted masshierarchy of Sec. III B there can be a visible difference inthe electron antineutrino flux if the neutrinos are Dirac orMajorana particles as seen by comparing the Majoranaparticle case of Fig. 4 with the Dirac particle case ofFig. 11.
The resonance energy of them1 andm2 mass eigenstateshave been set to 15 MeV in this case, exactly the same asfor the Majorana particle case considered in Sec. III B.Because of the extra factor of 1=2 the overall scale of theenhancement is much smaller than in the case of neutrinosbeing Majorana particles.
F. Nonadiabatic case (sin2�13 & 105)
If supernova neutrino flavor evolution is nonadiabatic,then the flux of electron antineutrinos that leaves a super-nova is independent of the neutrino mass hierarchy (seeEqs. (8) and (9)). If the supernova neutrinos interact withthe cosmic background neutrinos via new light scalars thenthe flux observed at Earth will be different for the normalmass hierarchy and the inverted mass hierarchy. However,this difference is not detectable at the present or near-futureneutrino experiments because it relies on the ability toobserve neutrinos in the SRN flux at low energies wherethere is a large reactor background.
The difference between the two neutrino mass hierar-chies is present, even in the case of small sin2�13, becausefor a normal mass hierarchy, the heavy neutrino masseigenstate is m3 which does not contribute to the � �e flux.
All of the scalars produced through the neutrino-neutrinointeractions will dominantly decay into this heaviest neu-trino mass eigenstate, and the final flux will have an overalldepletion relative to the SRN flux without interactions.However, for the inverted hierarchy, the m1 and m2 masseigenstates are the heavy states, while the m3 mass eigen-state is the light state. The scalars produced through theneutrino-neutrino interactions will dominantly decay intothe two heavy states, leading to a low energy enhancementas well as the higher energy dip.
We show in Fig. 12 the SRN flux for the normal masshierarchy (dotted, red curve) with m1 � 0:001 eV, m2 �0:008 eV, m3 � 0:05 eV, and MG � 173 eV, the invertedmass hierarchy (dashed, blue curve) with m1 � 0:05 eV,m2 � 0:05 eV, m3 � 0:008 eV, and MG � 1225 eV, andthe SRN flux without interactions (solid, black curve).
G. Signal detection
Here we show an example of an inverted mass hierarchy.The resonant energy for the m1 and m2 neutrino masseigenstates is taken to be 16 MeV, which for m1 �m2 �0:05 eV gives MG � 1265 eV. We present both the ex-pected SRN flux as well as the SRN flux folded with thecross section given in Eq. (20) in Figs. 13 and 14.
FIG. 11 (color online). The SRN electron antineutrino fluxwithout interactions (solid, red curve) and with interactions(dashed, blue curve) when neutrinos are Dirac particles andfor the inverted mass hierarchy.
FIG. 12 (color online). The SRN electron antineutrino fluxwithout interactions (solid, black curve), with interactions andnormal mass hierarchy (dotted, red curve), and with interactionsand inverted mass hierarchy (dashed, blue curve), when neutri-nos are Majorana particles and for sin2�13 & 105.
FIG. 13 (color online). The SRN electron antineutrino fluxwithout interactions (solid, red curve), and with interactionsand inverted mass hierarchy (dashed, blue curve) when neutrinosare Majorana particles.
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Comparing the dashed (blue) curve to the solid (red)curve in Fig. 14 we see that there is a depletion of the SRNevent rate above approximately 8 MeV if the SRNs interactvia the light scalar at resonance with the cosmic back-ground neutrinos (dashed, blue curve). This depletion ispresent up to the location of the dip, which in this case is16 MeV. The main source of background, assuming theaddition of Gd to the water Cerenkov detector, are nuclearreactor electron antineutrinos, but this background is smallabove neutrino energies of about 12 MeV [13,14]. Thisbackground is dependent on the location of the experimentand could be nearly absent. Clearly the detector which isnot near nuclear reactors would have a better chance ofseeing the signal for lower values of the resonance energy[13,14].
To demonstrate the significance of our signal we look atan energy of 15 MeV (where the effect is most significantand the reactor background is negligible). At this energywe expect approximately 11 events per year at HyperKfrom the SRN flux without new interactions (solid, redcurve). However, if the neutrinos interact via the lightscalar at resonance, then we instead predict approximately2 events per year (dashed, blue curve) at this energy bin(assuming a 2 MeV bin). The average fluctuation in thenumber of events expected in 5 years with no interactionscan be estimated as ��
������55p
� 7:5 events. If we take asour signal the number of events expected without interac-tions minus the number of events expected with interac-tions (i.e., the event deficit) over the 5 yr period (this is55 10 � 45 events), then in 5 years one expects approxi-mately a 6��45=7:5� effect. A similar analysis can beperformed for the previous cases discussed in this paper,but since the depletion in these cases is at lower energiesone must pay careful attention to the reactor background.
This analysis also requires a side band study, in order todetermine the SRN flux in a region where interactions areineffective (in our present example, this would be above16 MeV). This would provide the overall normalizationnecessary for establishing the background. Clearly, oncethe shape of the signal without resonance and the non-SRNbackground are known, a more sophisticated analysis (in-
cluding a bin-by-bin fit to the shape of the curves) can beachieved, and may even provide enhanced significance.This, however, is beyond the scope of this work, whoseaim we regard to be a discussion of the qualitative aspect ofour new physics signal.
For a liquid argon detector, if the resonance energy isabove the cutoff for the solar neutrino flux at about19 MeV, the integrated number of SRN electron neutrinoevents would be visibly reduced in the presence of newinteractions. This is shown for a normal mass hierarchy andan inverted mass hierarchy in Figs. 3 and 6 respectively.While our total number of events is a conservative estimate(we use a z evolution that is flat above z � 1, instead ofstronger dependence [16]), the depletion of events up to theresonance cutoff is a robust feature. We find that there isapproximately a 25% reduction in the number of electronneutrino absorption events with new interactions comparedto without new interactions for both the normal and in-verted mass hierarchies for the energy window from19 MeV to 40 MeV.
IV. CONCLUSION
The late-time neutrino mass generation models could betested by detecting unique features of the SRN flux in bothits electron antineutrino and neutrino components (forother tests of new physics that can be done with the SRNflux, see [26]). To illustrate this new effect we have con-sidered an Abelian U(1) model that generates neutrinomasses at low scales. However, it is clear that the mainfeatures would still hold for a more complicated, non-Abelian model, although these models are already moreconstrained by big bang nucleosynthesis (BBN) consider-ations. For example, one could still have observable dips inthe SRN spectrum if the resonances are in a desirableenergy window, but the couplings are no longer propor-tional to the neutrino masses, and so some predictive poweris lost. However, one could hope to correlate the observa-tions of the dip locations in the SRN spectrum with signalsproposed to be present in the cosmic microwave back-ground [7] for this case. We expect that the future genera-tion water Cerenkov detectors enriched with gadoliniumsuch as UNO, Hyper-Kamiokande, or MEMPHYS wouldbe able to detect a substantial number of SRN antineutrinoevents in a year [15]. Note that the threshold for this is onthe order of 10 MeV and depends on the location of thedetector, especially due to reactor backgrounds [13,14].The effects of smearing due to the energy resolutionof the water Cerenkov-type detector needs to also betaken into account in a detailed analysis. For a Gaussianenergy resolution function with width � (�=MeV�
0:6�����������������E=MeV
p), the smearing is only at most a few MeV
[13] in the energy domain we considered. This smearing isthen always smaller than the width of the depletion featuresconsidered. The neutrino component of the SRN flux couldbe detected by a large 100 kton liquid argon detector [16].
FIG. 14 (color online). The event rates for Hyper-Kamiokandewithout interactions (solid, red curve) and with interactions(dashed, blue curve) when neutrinos are Majorana particles,for the inverted mass hierarchy.
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If there are neutrino-neutrino interactions through the lightscalars present in these models, there is a possibility todistinguish between normal and inverted mass hierarchiesand Dirac versus Majorana neutrinos, as well as to deter-mine the absolute values of the neutrino masses. Theability to distinguish the neutrino mass hierarchy is inde-pendent of whether supernova neutrino flavor evolution isadiabatic or nonadiabatic.
The qualitative features of the signal of new interactionsvia light scalar, such as the depletion, enhancement atlower energies, and the possibility to distinguish betweenneutrino mass hierarchy, as well as the nature of theneutrino are independent of the theoretical model for thesupernova neutrino energy spectrum, which predictslightly different shape and wider range of average ener-gies for different neutrino (antineutrino) flavors than theKRJ model [19–21]. This is not surprising because theproduced neutrino spectrum does not depend on the de-tailed shape and normalization of the initial supernovaneutrino fluxes but rather on the coupling of the scalar tothe final neutrino mass eigenstates. As shown in Sec. A (3)of the first paper [11], the presence of a deep dip isuniversal, and its position depends only on the masses ofthe scalar and the target neutrino, rather than any feature ofthe neutrino spectrum. For a normal neutrino mass hier-archy there is an overall depletion of the SRN flux, whilefor an inverted neutrino mass hierarchy there is an en-hancement of the SRN flux at low energies and a regionof depletion at higher energies. If a sterile neutrino with amuch larger mass than the active neutrinos were also tocouple to the new scalar, then independent of the details ofthe masses of the active neutrinos the effect would bealmost complete depletion of the spectrum in some energywindow since the scalar would decay predominantly to themassive sterile neutrino.
All of these signals, and especially their observation,depend on the parameters of the model. In Fig. 15 we showconstraints on the parameter space for which the SRNeffects can be obtained in the y�-MG plane. The signalsproposed here are present in the SRN flux only if thecouplings of the neutrino mass eigenstates to the scalarare larger than the condition given in Eq. (12) for a givenvalue of MG. This condition comes from requiring that themean free path for absorption of a SRN neutrino on acosmic background neutrino is much smaller than theHubble scale [11]. It is a sufficient condition to guaranteethe absorption of all three neutrino flavors. This lowerbound on the coupling is represented by the diagonal solid(blue) line. If the resonance energy is below 12 MeV thenthere is a large background from nuclear reactor antineu-trinos [13,14]. To have a significant signal we take MG to
be above���������������������2m�ERes
�;min
q, where ERes
�;min is approximately
15 MeV. These threshold values are represented by thethree vertical dashed, red lines which are calculated forvalues of m� � 0:001 eV, 0.008 eV, and 0.05 eV. If the
mass of the scalar is larger than these values, then thesignal would be above the reactor background. Similarly,the signal would not be observable if the mass of the scalaris large so that the heaviest neutrino mass eigenstate has aresonance energy in the region where the SRN flux issmall. We also show the constraint imposed by BBN con-siderations, which is similar to the bound obtained from SNcooling and to the bound from the observation of unde-graded SN1987A neutrino flux [11]. The SRN flux is alsosensitive to the nonresonant process, for example 2�!�! 2G! 4�, but only in a very small region of theparameter space, above the horizonal black dashed lineand below the horizontal red solid line [11]. The area abovethe diagonal dashed, green line corresponds to the BBNconstraint for a non-Abelian Majorana case. We note thatthere is still a large range of parameter space where thecouplings are large enough to give SRN flux modificationin an energy window that large neutrino detectors coulddirectly probe.
We have shown that the cosmic background neutrinosinteracting with supernova relic neutrinos through ex-change of the light scalar lead to significant modificationof the SRN flux observed at Earth. These signals would bedetectable for a large region of parameter space, in somecases at a significance of more than 5�, and measurementsof the presence of these effects are well within the reach of
FIG. 15 (color online). The cosmological bounds and theregions for the supernova neutrino spectrum distortion due tothe resonance and nonresonance processes for a single Majorana(Dirac) neutrino for an Abelian (non-Abelian) model are shownin the �y�;MG� plane. The region above the solid horizontal (red)line is excluded by the BBN constraint (for the Dirac case), SNcooling (for Majorana case), and due to the observation of(undegraded) SN1987A neutrinos. In the region below the solidslanting (blue) line the mean free path is too long for theresonance to occur. The region above the dashed slanting (green)line, which is relevant only for the non-Abelian Majorana case,is the region excluded by the BBN constraint. The region abovethe dashed horizontal (black) line is the region of the futureexperimental sensitivity to the observation/nonobservation of theSRN neutrinos due to nonresonant processes. The verticaldashed lines correspond to the minimum values of MG thatlead to a depletion signal that is above the nuclear reactorantineutrino background for the neutrino masses of 0.001 eV,0.008 eV, and 0.05 eV from left to right, respectively.
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the next-generation water Cerenkov detectors enrichedwith gadolinium, or a large 100 kton liquid argon detector.Specifically we have shown that the changes induced in theflux by the exchange of the light scalars might allow one todistinguish between neutrinos being Majorana or Diracparticles, the type of neutrino mass hierarchy (normal orinverted or quasidegenerate), and could also possibly de-termine the absolute values of the neutrino masses. Aninteresting feature is that the ability to distinguish neutrinomass hierarchy does not depend on the dynamics of theflavor evolution of neutrinos leaving the supernova(whether it is adiabatic or nonadiabatic), or on the specificshape and normalization of the initial supernova neutrinoflux. Note that the hierarchy determination can be made bysolely looking at the spectrum of supernova relic electronantineutrinos, without the need to do the measurement ofthe flux of supernova relic electron neutrinos. In addition,
the modification of the SRN flux in any of the proposedscenarios is a clear indication of the presence of the cosmicbackground neutrinos left over from the era of big bangnucleosynthesis.
ACKNOWLEDGMENTS
We thank Tom Weiler for many discussions and for hisvaluable comments and suggestions. This research wassupported in part by the DOE under contracts No. DE-FG02-04ER41319 (J. B. and I. S.), No. DE-FG02-04ER41298 (J. B. and I. S.), No. DE-AC03-76SF00098(G. P.), and the NSF under Grant No. PHY-0244507(H. G.). J. B. and I. S. would like to thank the LBLTheory Group for their hospitality while this work wasbeing completed.
[1] B. T. Cleveland et al., Astrophys. J. 496, 505 (1998); Y.Fukuda et al., Phys. Rev. Lett. 81, 1158 (1998); J. N.Abdurashitov et al., J. Exp. Theor. Phys. 95, 181 (2002);Q. R. Ahmad et al., Phys. Rev. Lett. 89, 011301 (2002).
[2] Y. Fukuda et al., Phys. Rev. Lett. 81, 1562 (1998); Y.Ashie et al., ibid. 93, 101801 (2004).
[3] K. Eguchi et al., Phys. Rev. Lett. 90, 021802 (2003); M. H.Ahn et al., ibid. 90, 041801 (2003); T. Araki et al., ibid.94, 081801 (2005); E. Aliu et al., ibid. 94, 081802 (2005).
[4] P. Ramond, M. Gell-Mann, and R. Slansky, Supergravity(North-Holland, Amsterdam, 1979); T. Yanagida, inProceedings of the Workshop on the Unified Theory andthe Baryon Number in the Universe, Tsukuba, 1979 (un-published); R. N. Mohapatra and G. Senjanovic, Phys.Rev. Lett. 44, 912 (1980).
[5] M. R. Buckley and H. Murayama, Phys. Rev. Lett. 97,231801 (2006).
[6] N. Arkani-Hamed and Y. Grossman, Phys. Lett. B 459,179 (1999).
[7] Z. Chacko, L. J. Hall, T. Okui, and S. J. Oliver, Phys. Rev.D 70, 085008 (2004); L. J. Hall and S. J. Oliver, Nucl.Phys. B, Proc. Suppl. 137, 269 (2004); Z. Chacko, L. J.Hall, S. J. Oliver, and M. Perelstein, Phys. Rev. Lett. 94,111801 (2005); T. Okui, J. High Energy Phys. 09 (2005)017.
[8] H. Davoudiasl, R. Kitano, G. D. Kribs, and H. Murayama,Phys. Rev. D 71, 113004 (2005).
[9] S. Hannestad, Annu. Rev. Nucl. Part. Sci. 56, 137 (2006);M. Cirelli and A. Strumia, J. Cosmol. Astropart. Phys. 12(2006) 013.
[10] Y. Farzan, Phys. Rev. D 67, 073015 (2003).[11] H. Goldberg, G. Perez, and I. Sarcevic, J. High Energy
Phys. 11 (2006) 023.[12] L. J. Hall, H. Murayama, and G. Perez, Phys. Rev. Lett. 95,
111301 (2005).
[13] J. F. Beacom and M. R. Vagins, Phys. Rev. Lett. 93,171101 (2004); G. L. Fogli, E. Lisi, A. Mirizzi, and D.Montanino, J. Cosmol. Astropart. Phys. 04 (2005) 002.
[14] L. J. Hall, H. Murayama, M. Papucci, and G. Perez,arXiv:hep-ph/0607109.
[15] C. K. Jung, AIP Conf. Proc. 533, 29 (2000); K. Nakamura,Int. J. Mod. Phys. A 18, 4053 (2003); J. E. Campagne, M.Maltoni, M. Mezzetto, and T. Schwetz, J. High EnergyPhys. 04 (2007) 003.
[16] A. Ereditato and A. Rubbia, Nucl. Phys. B, Proc. Suppl.154, 163 (2006).
[17] L. E. Strigari, M. Kaplinghat, G. Steigman, and T. P.Walker, J. Cosmol. Astropart. Phys. 03 (2004) 007; S.Ando et al., Astropart. Phys. 18, 307 (2003); M. Fukugitaand M. Kawasaki, Mon. Not. R. Astron. Soc. 340, L7(2003); S. Ando and K. Sato, Phys. Lett. B 559, 113(2003); New J. Phys. 6, 170 (2004); L. Strigari et al., J.Cosmol. Astropart. Phys. 04 (2005) 017; H. Yuksel, S.Ando, and J. F. Beacom, Phys. Rev. C 74, 015803 (2006).
[18] D. Schiminovich et al., Astrophys. J. 619, L47 (2005);P. G. Perez-Gonzalez et al., ibid. 630, 82 (2005); A. M.Hopkins and J. F. Beacom, Astrophys. J. 651, 142 (2006);M. A. Fardal, N. Katz, D. H. Weinberg, and R. Dav’e,arXiv:astro-ph/0604534; F. Mannucci, H. Buttery, R.Maiolino, A. Marconi, and L. Pozzetti, arXiv:astro-ph/0607143.
[19] T. Totani, K. Sato, H. E. Dalhed, and J. R. Wilson,Astrophys. J. 496, 216 (1998).
[20] M. Th. Keil, G. G. Raffelt, and H. Th. Janka, Astrophys. J.590, 971 (2003).
[21] T. A. Thompson, A. Burrows, and P. A. Pinto, Astrophys.J. 592, 434 (2003).
[22] A. Mirizzi and G. G. Raffelt, Phys. Rev. D 72, 063001(2005); C. Lunardini, Astropart. Phys. 26, 190 (2006);Phys. Rev. D 73, 083009 (2006).
PROBING LATE NEUTRINO MASS PROPERTIES WITH . . . PHYSICAL REVIEW D 76, 063004 (2007)
063004-11
[23] K. Kifonidis, T. Plewa, L. Scheck, H. Th. Janka, and E.Mueller, arXiv:astro-ph/0511369; C. L. Fryer, G.Rockefeller, and M. S. Warren, Astrophys. J. 643, 292(2006); A. Marek, H. Dimmelmeier, H. Th. Janka, E.Muller, and R. Buras, Astron. Astrophys. 445, 273(2006); A. Burrows, E. Livne, L. Dessart, C. Ott, and J.Murphy, Astrophys. J. 640, 878 (2006); C. L. Fryer and A.Kusenko, Astrophys. J. Suppl. Ser. 163, 335 (2006); L.Scheck, K. Kifonidis, H. Th. Janka, and E. Mueller,arXiv:astro-ph/0601302.
[24] M. Rampp, R. Buras, H. Th. Janka, and G. Raffelt,arXiv:astro-ph/0203493; R. C. Schirato and G. M. Fuller,arXiv:astro-ph/0205390; K. Takahashi, K. Sato, H. E.Dalhed, and J. R. Wilson, Astropart. Phys. 20, 189(2003); G. L. Fogli, E. Lisi, A. Mirizzi, and D.Montanino, Phys. Rev. D 68, 033005 (2003); R. Tomaset al., J. Cosmol. Astropart. Phys. 09 (2004) 015; G. L.Fogli, E. Lisi, A. Mirizzi, and D. Montanino, ibid. 06(2006) 012; A. Friedland and A. Gruzinov, arXiv:astro-ph/0607244.
[25] A. S. Dighe and A. Y. Smirnov, Phys. Rev. D 62, 033007(2000).
[26] S. Ando, Phys. Lett. B 570, 11 (2003); G. L. Fogli, E. Lisi,A. Mirizzi, and D. Montanino, Phys. Rev. D 70, 013001(2004). These papers consider the effects of neutrinodecay on the supernova relic neutrino flux observed at
Earth. The neutrinos are coupled to very light or masslessMajorons and the process considered is �Heavy ! �Light ��.
[27] M. C. Gonzalez-Garcia and Y. Nir, Rev. Mod. Phys. 75,345 (2003).
[28] S. Eidelman et al. (Particle Data Group), Phys. Lett. B592, 1 (2004).
[29] The decay is almost instantaneous compared to a Hubbletime: the decay rate in the lab frame of the scalar producedat resonance is easily calculated to be ��y2=16��m�,which is much larger than an inverse Hubble time if y�4� 1016�1 eV=m���
1=2. In the light of the muchstronger requirement Eq. (12), this is easily satisfied forall neutrino masses we consider.
[30] P. Vogel and J. F. Beacom, Phys. Rev. D 60, 053003(1999); A. Strumia and F. Vissani, Phys. Lett. B 564, 42(2003).
[31] A. G. Cocco, A. Ereditato, G. Fiorillo, G. Mangano, and V.Pettorino, J. Cosmol. Astropart. Phys. 12 (2004) 002; J. F.Beacom and L. E. Strigari, Phys. Rev. C 73, 035807(2006); C. Lunardini, Phys. Rev. D 73, 083009 (2006).
[32] G. L. Fogli, E. Lisi, A. Marrone, A. Palazzo, and A. M.Rotunno, arXiv:hep-ph/0506307.
[33] A. Goobar, S. Hannestad, E. Mortsell, and H. Tu, J.Cosmol. Astropart. Phys. 06 (2006) 019.
BAKER, GOLDBERG, PEREZ, AND SARCEVIC PHYSICAL REVIEW D 76, 063004 (2007)
