NEUTRINO MASS SPECTRUM FROM GRAVITATIONAL WAVES GENERATEDBY DOUBLE NEUTRINO SPIN-FLIP IN SUPERNOVAE

Herman J. Mosquera Cuesta1and Gaetano Lambiase

2

Received 2008 May 28; accepted 2008 August 4

ABSTRACT

The supernova (SN) neutronization phase produces mainly electron (�e) neutrinos, the oscillations of which musttake place within a few mean free paths of their resonance surface located nearby their neutrinosphere. The latestresearch on the SN dynamics suggests that a significant part of these �e can convert into right-handed neutrinos byvirtue of the interaction of the electrons and the protons flowing with the SN outgoing plasma, whenever the Diracneutrinomagnetic moment is of strength �� < 10�11�B, with�B being the Bohr magneton. In the SN envelope, someof these neutrinos can flip back to the left-handed flavors due to the interaction of the neutrino magnetic moment withthe magnetic field in the SN expanding plasma (see the work by Kuznetsov & Mikheev; Kuznetsov, Mikheev, &Okrugin; Akhmedov & Khlopov; Itoh & Tsuneto; and Itoh et al.), a region where the field strength is currentlyaccepted to be Bk1013 G. This type of � oscillation was shown to generate powerful gravitational wave (GW)bursts (see the work byMosquera Cuesta; Mosquera Cuesta & Fiuza; and Loveridge). If such a double spin-flip mech-anism does run into action inside the SN core, then the release of both the oscillation-produced �� and �� particles andthe GW pulse generated by the coherent � spin-flips provides a unique emission offset �T emi

GW$� ¼ 0 for measuringthe � travel time to Earth. As massive � particles get noticeably delayed on their journey to Earth with respect to theEinstein GW they generated during the reconversion transient, then the accurate measurement of this time-of-flightdelay by SNEWS + LIGO, VIRGO, BBO, DECIGO, etc., might readily assess the absolute � mass spectrum.

Subject headinggs: elementary particles — gravitational waves — methods: data analysis — neutrinos —stars: magnetic fields — supernovae: general

Online material: color figure

1. INTRODUCTION

The determination of the absolute values of neutrinomasses iscertainly one of the most difficult problems from the experimentalpoint of view (Bilenky et al. 2003). One of the main difficulties ofthe issue of determining the � masses from solar or atmospheric �experiments concerns the ability of � detectors to be sensitive tothe species mass square difference instead of being sensitive tothe �mass itself. In this paper we introduce a model-independentnovel nonpareil method to achieve this goal.We argue that a highlyaccurate and largely improved assessment of the � mass scale canbe directly achieved bymeasurements of the delay in time of flightbetween the � particles themselves and the gravitational wave(GW) burst generated by the asymmetric flux of neutrinos under-going coherent (Pantaleone 1992) helicity (spin-flip) transitionsduring either the neutronization phase or the relaxation (diffusion)phase in the core of a Type II supernova (SN) explosion. Becausespecial relativistic effects do preclude massive particles from trav-eling at the speed of light, while massless particles are not (thegraviton in this case), the measurement of this � time lag leads toa direct accounting of its mass. We posit from the start that twobursts of GWs can be generated during the protoYneutron star(PNS) neutronization phase through spin-flip oscillations: (1) onesignal from the early conversion of active � particles into right-handed partners, at density � � few ; 1012 g cm�3, via the inter-action of the Dirac neutrino magnetic moment [of strength �� <(0:7Y1:5) ; 10�12�B, with �B being the Bohr magneton] with the

electrons and the protons in the SN outflowing plasma. Specifi-cally, the neutrino chirality flip is caused by the scattering via theintermediate photon (plasmon) off the plasma electromagnetic cur-rent presented by electrons,�Le

��!�Re�; protons, �Lp

þ�!�Rpþ;

etc. (2) A second signal exists by virtue of the reconversion pro-cess of these sterile � particles back into actives some time later,at lower density, via the interaction of the neutrino magnetic mo-ment with the magnetic field in the SN envelope (SNE). The GWcharacteristic amplitude, which depends directly on the luminosityand the mass square difference of the � species partaking in thecoherent transition (Pantaleone 1992), and the GW frequency ofeach of the bursts are computed. Finally, the time-of-flight delay� $ GW that can bemeasured upon the arrival of both signals toEarth observatories is then estimated, and the prospective of ob-taining the � mass spectrum from suchmeasurements is discussed.

2. DOUBLE RESONANT CONVERSIONOF NEUTRINOS IN SUPERNOVAE

2.1. Interaction of �L Dirac Magnetic Momentwith SN Virtual Plasmon

The neutrino chirality conversion process �L $ �R in a SN hasbeen investigated inmany papers (see, for instance, Voloshin 1988;Peltoniemi 1992; Akhmedov et al. 1993;Dighe&Smirnov 2000).Next, we follow the reanalysis of the double � spin-flip in SNerecently revisited byKuznetsov&Mikheev (2007) andKuznetsovet al. (2008), who obtained a more stringent limit on the neutrinomagnetic moment, �� , after demanding compatibility with theSN 1987A � luminosity. The process becomes feasible in virtueof the interaction of the Dirac � magnetic moment with a virtualplasmon, which can be produced, �L�!�R þ �?, and absorbed,�L þ �?�!�R, inside a SN. Our main goal here is to estimate the�R luminosity after the first resonant conversion inside the SN.

A

1 Instituto de Cosmologia, Relatividade e Astrof ısica ( ICRA-BR), CentroBrasileiro de Pesquisas Fısicas (CBPF), Rua Dr. Xavier Sigaud 150, 22290-180,Rio de Janeiro, Brazil; and ICRANet Coordinating Centre, Piazzalle dellaRepubblica 10, 065100, Pescara, Italy.

2 Dipartimento di Fisica ‘‘E. R. Caianiello,’’ Universita di Salerno, 84081Baronissi (Sa), Italy; and INFN, Sezione di Napoli, Italy.

371

The Astrophysical Journal, 689:371Y376, 2008 December 10

# 2008. The American Astronomical Society. All rights reserved. Printed in U.S.A.

This quantity is one of the important parameters for estimatingthe GWamplitude of the signal generated at the transition (see x 3below). The calculation of the spin-flip rate of creation of the �Rin the SN core is given by (Kuznetsov & Mikheev 2007)

L�R �dE�R

dt¼ V

Z 1

0

dn�RdE 0

E 0 dE 0

¼ V

2�2

Z 1

0

E 03� E 0ð ÞdE 0; ð1Þ

where dn�R /dE0 defines the number of right-handed � particles

emitted in the 1 MeVenergy band of the � energy spectrum, andper unit time,�(E 0) defines the spectral density of the right-handed� luminosity, and V is the plasma volume. Thus, by using the SNcore conditions that are currently admitted (see, for instance, Jankaet al. 2007), plasma volume V ’ 4 ; 1018 cm3, temperaturerange T ¼ 30Y60 MeV, electron chemical potential range �e ¼280Y307MeV, neutrino chemical potential �� ¼ 160MeV,3 oneobtains

L�R ’��

�B

� �2

(0:4Y2) ; 1077 erg s�1; ð2Þ

which for a �� ¼ 3 ; 10�12�B compatible with SN 1987A neu-trino observations and preserving causalitywith respect to the left-handed diffusion � luminosity L�R < L�L P1053 erg s�1, rendersL�R ¼ 4 ; 1053 erg s�1. This constraint is on the order of the lu-minosities estimated in our earlier papers (MosqueraCuesta 2000,2002; Mosquera Cuesta & Fiuza 2004) to compute the GWam-plitude from � flavor conversions, which were different from theone estimated by (Loveridge 2004). More remarkable, this anal-ysis means that only�1%Y2% of the total number of �L particlesmay resonantly convert into �R particles.

2.2. Conversion of �R�!�L in the SN Magnetic Field

Kuznetsov et al. (2008) have shown that by taking into accountthe additional energyCL, which the left-handed electron-type neu-trino �e acquires in the medium, the equation of the helicity evo-lution can be written in the form (Voloshin & Vysotsky 1986;Voloshin et al. 1986a, 1986b; Okun 1986, 1988)

i@

@t

�R

�L

� �¼ E0 þ

0 �� B?

�� B? CL

� �� ��R

�L

� �;

*CL ¼3GFffiffiffi2p �

mN

Ye þ4

3Y�e �

1

3

� �; ð3Þ

where the ratio �/mN ¼ nB is the nucleon density, Ye ¼ ne /nB ¼np /nB, Y�e ¼ n�e /nB, and ne;p; �e are the densities of the electrons,protons, and neutrinos, respectively, B? is the transverse compo-nent of themagnetic field with respect to the � propagation direc-tion, and the term E0 is proportional to the unit matrix, however,it is not crucial for the analysis below.

As pointed out by Kuznetsov et al. (2008), the additional en-ergy CL of left-handed � particles deserves a special analysis. Itis remarkable that the possibility exists for this value to be zerojust in the region of the SNE we are interested in. And, in turn,this is the condition of the resonant transition �R ! �L.When the� density in the SNE is low enough, one can neglect the value Y�ein the termCL, which gives the condition for the resonance in the

form Ye ¼ 1/3. (Typical values of Ye in SNE are Ye � 0:4Y0:5,which are rather similar to those of the collapsing matter). How-ever, the shock wave causes the nuclei dissociation and makesthe SNE material more transparent to � particles. This leads tothe proliferation of matter deleptonization in this region and, con-sequently, to the so-called short � outburst. According to the latestresearch on SNe, a typical gap appears along the radial distribu-tion of the parameter Ye where it can achieve values as low asYe � 0:1 (seeMezzacappa et al. 2001 and also Fig. 2 inKuznetsovet al. 2008, and references therein). Thus, a transition region un-avoidably exists where Ye takes the value of 1/3. It is remarkablethat only one such point appears where the Ye radial gradient ispositive, i.e., dYe /dr > 0. Nonetheless, the condition Ye ¼ 1/3 isthe necessary but yet not the sufficient one for the resonant con-version �R ! �L to occur. It is also required to satisfy the so-calledadiabatic condition. This means that the diagonal element CL inequation (3), at least, should not exceed the nondiagonal element�� B?, when the shift is made from the resonance point at thedistance of the order of the oscillation length. This leads to thecondition (Voloshin 1988)

��B?kdCL

dr

� �1=2

’ 3GFffiffiffi2p �

mN

dYe

dr

� �1=2

: ð4Þ

And values of these typical parameters inside the considered re-gion are dYe /dr � 10�8 cm�1 and � � 1010 g cm�3. Therefore,the magnetic field strength that realizes the resonance conditionreads as

B?k 2:6 ; 1014 G10�12�B

��

� �

;�

1010 g cm�3

� �1=2dYe

dr108 cm

� �1=2

: ð5Þ

Thus, one can conclude that the analysis performed above showsthat the Dar scenario of the double conversion of the neutrinohelicity (Dar 1987), �L ! �R ! �L, can be realized wheneverthe neutrinomagnetic moment is in the interval 10�13�B < �� <10�12�B and when the strength of the magnetic field reachesk1014 G (Kusenko 2004) in a region R between the neutrino-sphere R� and the shock wave stagnation radius Rs, where R� <R < Rs.4 Thus, the �L luminosity during this stagnation time,�Ts ’ 0:2Y0:4 s, is L�L ’ 3 ; 1053 erg s�1, as the conservationlaw allows one to expect�� < 10�12�B.Once onehas all these pa-rameters in hand, one can then proceed to compute the correspond-ing GW signal from each of the � resonant spin-flip transitions.

3. � OSCILLATIONYDRIVEN GWDURING SN NEUTRONIZATION

The characteristic GWamplitude of the signal produced by the� outflow can be estimated by using the general relativistic quad-rupole formula (Burrows & Hayes 1996)

hTTij (t) ¼4G

c4D

Z t

�1� t 0ð ÞL� t 0ð Þdt 0 ei � ej

�!h ’ 4G

c4D��L��T�fL!�fR

; ð6Þ

3 These conditions could exist in the time interval before the first second afterthe core bounce.

4 These kinds of magnetic field strengths have been extensively said to bereached after the SN core collapse forms just-born pulsars (magnetars), in the centralengines of gamma-ray burst outflows, and during the quantum-magnetic collapseof newborn neutron stars, etc.

MOSQUERA CUESTA & LAMBIASE372 Vol. 689

where D is the source distance, L�(t) is the total � luminosity,ei � ej is the GW polarization tensor, the superscript TT standsfor the transverse-traceless part, and finally, �(t) is the instan-taneous quadrupole anisotropy. Above, we estimated the �R lu-minosity; next, we estimate the degree of asymmetry of the PNSthrough the anisotropic parameter � and the timescale �T�fL!�fRfor the resonant transition to take place, as discussed above.

To estimate the star asymmetry, let us recall that the resonancecondition for the transition �eL ! ��R is given by (at the reso-nance r )

V�e ( r )þ B( r ) = p� 2�c2 ¼ 0: ð7Þ

Thus, the PNSmagnetic field vectorB in equation (7) distorts thesurface of resonance due to the relative orientation of p with re-spect to B (see vector B in Fig. 1). The deformed surface of res-onance can be parameterized as r() ¼ r þ % cos , where % (<r)is the radial deformation and cos ¼ B = p. The deformation en-forces a nonsymmetrical outgoing neutrino flux, i.e., the net fluxof neutrinos emitted from the upper hemisphere is different fromthe one emitted from the lower hemisphere (see Fig. 1). There-fore, a geometrical definition of the quadrupole anisotropy canbe � ¼ (Sþ � S�)/(Sþ þ S�), where S� is the area of the upper/lower hemisphere, whence one obtains � ’ %/ r.5 The anisotropyof the outgoing neutrinos is also related to the energy flux Fs

emitted by the PNS and, in turn, to the fractional momentumasymmetry �jpj/jpj (Kusenko & Segre 1996; Barkovich et al.2002; Lambiase 2005a, 2005b; Mosquera Cuesta & Fiuza 2004).To compute Fs, one has to take into account the structure of theflux at the resonant surface, which acts as an effective emis-sion surface, and the � distribution in the diffusive approxima-

tion (Barkovich et al. 2002). As a result, one gets �jpj/jpj ¼16(R �0Fs = u dS )/(

R �0Fs = n dS ) ’ 2%/9r (n is a unit vector nor-

mal to the resonance surface and u ¼ B/jBj).6 An anisotropy of�1% would suffice to account for the observed pulsar kicks(Kusenko & Segre 1996; Loveridge 2004; Mosquera Cuesta2000, 2002); hence,� ’ 0:045 � O(0:01)YO(0:1), which is con-sistent with numerical results (Burrows & Hayes 1996; Muller& Janka 1997). Finally, the conversion probability is P�eL!��R ¼1/2� 1/2 cos 2i cos 2f (Okun 1986, 1988), where is definedas

tan 2(r) ¼ 2�� B?=(B = pþ V�e � 2�c2): ð8Þ

The quantities i ¼ (ri) and f ¼ (rf ) are the values of the mix-ing angle at the initial point ri and the final point rf of the neutrinopath.7

Meanwhile, the average timescale of this first � spin-flip con-version is (Dar 1987; Voloshin 1988)

�T�fL!�fR¼ �B

��

� �2m2

e

�� 2fsc(1þ hZi)Ye

� �mp

�

� �; ð9Þ

where hZi � O(1Y30) is the average electric charge of the nucleiand �fsc is the fine-structure constant. Using the current boundson the neutrino magnetic moment �� P3 ; 10�12�B, Ye ’ 1/3,hZi � 10, � � 2 ; 1012 g cm�3, and � � 0:04, it follows that�T�f L!�f R ’ (1� 10) ; 10�2 s (parameters have been chosen fromSN simulations evolving the PNS on timescales of�3ms aroundcore bounce;Mayle et al. 1987;Walker&Schramm1987;Burrows&Hayes 1996; Mezzacappa et al. 2001; van Putten 2002; Arnaudet al. 2002; Beacom et al. 2001). In such a case, the above time-scale suggests that the GW burst would be as long as the expectedduration of the pure neutronization phase itself, i.e., �Tneut �10Y100 ms, according to most SN analyses and models (Mayleet al. 1987; Walker & Schramm 1987; Burrows & Hayes 1996;Mezzacappa et al. 2001; van Putten 2002; Arnaud et al. 2002;Beacom et al. 2001), with the maximum GW emission takingplace around�T max

neut � 3ms (van Putten 2002;Arnaud et al. 2002;Mosquera Cuesta 2000, 2002; Mosquera Cuesta & Fiuza 2004).Hence, the outcoming GW signal will be the evolute ( linearsuperposition) of all the coherent �eL ! ��;�R oscillations takingplace over the neutronization transient, in analogy with the GWsignal from the collective motion of neutronmatter in a just-bornpulsar. This implies a GW frequency of fGW �1/�Tneut�100 Hz,for the overall GWemission, and fGW � 1/�T max

neut � 330Hz at itspeak. Meanwhile, according to our probability discussion above,about 1%Y2% of the total � particles released during the SN neu-tronization phase may oscillate (Voloshin 1988; Peltoniemi 1992;Akhmedov et al. 1993; Dighe & Smirnov 2000), carrying awayan effective power L� ¼ 3 ; 1054Y1053 erg s�1, i.e., 0:01 ; 3 ;1053 erg, emitted during �Tneut � 10Y100 ms (this is similar tothe upper limit computed in Peltoniemi [1992], L� ¼ (2/10) ;1053½��e /(10

�12�B)� erg s�1). Moreover, as is evident from equa-tion (6), the GWamplitude is a function of the helicity-changing� luminosity, i.e.,h ¼ h(L

�eL!��;�Rmax ). The � luminosity itself depends

Fig. 1.—Illustration of the combined effect of the � spin coupling to the starmagnetic field and rotation. This figure was taken fromMosquera Cuesta& Fiuza(2004).

5 A detailed analysis of the asymmetry parameter � requires one to study itstime evolution during the SN collapse. Such a task goes beyond the aim of thispaper.Working in the stationary regime, wemay assume� constant (see Burrows& Hayes 1996; Burrows et al. 1995; Zwerger & Muller 1997; van Putten 2002).

6 To compute�jpj/jpj one uses the standard resonance condition V� ¼ 2�c2(see Barkovich et al. 2002 for details). According to Mezzacappa et al. (2001),during the first 10Y200 ms, Ye may assume values’1/3 so that V�e � (3Ye � 1)is suppressed by several orders of magnitude. At �10 ms, � � 1012 g cm�3, r �50 km, and jpj � 10MeV, the resonance condition leads to a range for�m2 cos 2consistent with solar (or atmospheric) neutrino data.

7 By using the typical values Bk1010 G, �� P9 ; 10�11�B, and the profile� ’ �core(rc /r)

3 for rk rc (rc � 10 km is the core radius and �core � 1014 g cm�3),one can easily verify that the adiabatic parameter � � 2(��B?)

2/ ��j�0/�jð Þ > 1 atthe resonance r.

NEUTRINO MASS SPECTRUM FROM GRAVITATIONAL WAVES 373No. 1, 2008

on the probability of conversion (Peltoniemi 1992; MosqueraCuesta 2000, 2002; Mosquera Cuesta & Fiuza 2004; Loveridge2004), i.e., L

�eL!��;�Rmax ¼ (P�eL!��;�R )L

�tot.

The characteristic GW strain [per (Hz)1=2] from the outgoing

flux of spin-flipping (first transition) � particles is

h�f L!�f 0R � h ’ 1:1 ; 10�23 Hz�1=2� � P�f L!�f 0R

0:01

;Ltot�

3 ; 1054 erg s�12:2 Mpc

D

�T

10�1 s

�

0:1; ð10Þ

for a SN exploding at a fiducial distance of 2.2 Mpc, e.g., at theAndromeda galaxy (see Table 18). The GW strain in this mech-anism (see Fig. 2) is several orders of magnitude larger than in theSN � diffusive escape (Burrows & Hayes 1996; Muller & Janka1997; Arnaud et al. 2002; Loveridge 2004) because of the huge �luminosity the � oscillations provide by virtue of being a highlycoherent process (Pantaleone 1992;Mosquera Cuesta 2000, 2002;Mosquera Cuesta & Fiuza 2004). This makes it detectable fromvery far distances. These GW signals are right in the bandwidthof the highest sensitivity (10Y300 Hz) of most ground-basedinterferometers.

Spin flavor oscillations �eL ! ��R, which according to the latestresearch on SN dynamics do take place during the neutronizationphase of core-collapse SNe (Mayle et al. 1987;Walker&Schramm1987; Voloshin 1988; Dighe & Smirnov 2000; Kuznetsov &Mikheev 2007), allow powerful GW bursts to be released fromone side (according to eq. [6]) and a stream of ��R particles to begenerated from the other side, over a timescale given by equation (9).The latter would in principle escape from the PNS were it not forthe appearance of several resonances that catch up with them(Voloshin 1988; Peltoniemi 1992;Akhmedov et al. 1993). If therewere no such resonance, the �fL ! �f 0R oscillation process wouldleak away all the binding energy of the star, leaving no energy atall for the left-handed �L particles that are said to drive the actualSN explosion and for us to have observed themduring SN 1987A.A new resonance may occur at rk100 km from the center, whichconverts�90%Y99%of the spin-flipYproduced �R particles backinto �L ones (Voloshin 1988; Akhmedov 1988; Akhmedov &Khlopov 1988a, 1988b; Itoh & Tsuneto 1972; Itoh et al. 1996;Peltoniemi 1992; Akhmedov et al. 1993; Athar et al. 1995). As dis-cussed in these papers, in fact, in the outer layer of the SN core

the amplitude of the coherent weak interaction of �L with the PNSmatter (V�e ) can cross smoothly enough to ensure adiabatic res-onant conversion of �f R into �f L.

9 Following Mezzacappa et al.(2001), the region where V�e ¼ 0 as Ye ¼ 1/3 corresponds to apostbounce timescale �100 ms and radius �150 km at whichthe � luminosity is L� � 3 ; 1052 erg s�1, and the matter densityis � � 1010 g cm�3. There, the adiabaticity condition demandsB?k 1010 G for the �� quoted above (such a field is characteristicof young pulsars). This reverse transition (rt) should resonantlyproduce an important set of ordinary (muon and tau) �L particles,which could be found far from their ownneutrinosphere and, hence,can stream away from the PNS. Whence a second GW burst withthe characteristics h ’ 1 ; 10�23 Hz�1/2 for D ¼ 2:2 Mpc and�Trt ’ 1:4 s is released in this region. Notice that this h issimilar to the one for the first transition despite the � luminositybeing lower. This featuremakes it similar to theGWmemory prop-erty of the �-driven signal, i.e., time-dependent strain amplitudewith average value nearly constant (Burrows & Hayes 1996).To obtain this result, equations (9) and (10) were used. Where-fore, the GW frequency fGW � 1/�Trt � 0:7 Hz falls in the low-frequency band and could be detected by the planned BBO andDECIGOGW interferometric observatories. Notice also that thetime lag for the event at LIGO, VIRGO, etc., and the one at BBOand DECIGO is then about 100 ms. It is this transition whichdefines the offset to measure the time-of-flight delay, since both��;� and GW free-stream away from the PNS at this point.

4. TIME-OF-FLIGHT DELAY � $ GW

The measurement of the � $ GW time delay from � oscil-lations in SNe promises to be an innovative procedure to obtainthe � mass spectrum. Provided that Einstein’s GWs do propagate

TABLE 1

Time Delay between GW and (jpj ¼ 10 MeV) � Bursts

from a SN Neutronization, as a Function of � Mass and Distance

�T arrGW$�

(s)

Source

Distance

(kpc) �1 �2 �3

GC......................... 10 5:15 ; 10�9 5:15 ; 10�3 0.32

LMC...................... 55 2:83 ; 10�8 2:83 ; 10�2 1.7

M31....................... 2:2 ; 103 1:13 ; 10�6 1.13 68.8

Source.................... 1:1 ; 104 5:66 ; 10�6 5.66 344.0

Note.—The � masses in eVare 10�3, 1.0, and 2.5, for flavors �1, �2, and �3,respectively.

Fig. 2.—Characteristics (h(�fL!�f 0R), fGW) of the GW burst generated via the �spin-flip oscillation mechanism vs. detector noise spectral density. For sources ateither the GC or LMC, the pulses will be detectable by LIGO-I and VIRGO. Todistances�10Mpc (farther out than the Andromeda galaxy), such radiation wouldbe detectable byAdvancedLIGO andVIRGO.ResonantGWantennas, tuned at thefrequency interval indicated, could also detect such events. Highlighted is the GWsignal of a SN neutronization phase at Andromeda, which would have a frequencyfGW � 100 Hz. [See the electronic edition of the Journal for a color version of thisfigure.]

8 The mass eigenstates listed are masses supposed to be estimated throughoutthe � detection in a future SN event, not the mass constraints already establishedfrom solar and atmospheric neutrinos, the expected time delay of which is com-putable straightaway. If a nonstandard mass eigenstate is detected, then one canuse the seesaw mechanism to infer the remaining part of the spectrum.

9 The cross level condition once again involves the termsB = p. Nevertheless,at that point the deformation of the resonance surface may be neglected, whenceno relevant GW burst is expected (yet � is quite low).

MOSQUERA CUESTA & LAMBIASE374 Vol. 689

at the speed of light, the GW burst produced by spin-flip oscilla-tions during the neutronization phase will arrive to GWobserva-tories earlier than its source (the massive � particles from thesecond conversion) will get to � telescopes.

As pointed out above, the mechanism to generate GWs at theinstant at which the second transition �f 0R ! �fL takes place canby itself define a unique emission offset, �T emi

GW$� ¼ 0, whichmakes possible a cleaner and highly accurate determination of the� mass spectrum by ‘‘following’’ the GW and neutrino propaga-tion to Earth observatories. The time lag in arrival is (Beacomet al. 2001)

�T arrGW$� ’ 0:12 s

D

2:2 Mpc

� �m�

0:2 eV

10 MeV

jpj

� �2

: ð11Þ

5. DISCUSSION

Inmost SNmodels (Burrows&Hayes 1996;Mezzacappa et al.2001; Beacom et al. 2001; van Putten 2002), the neutroniza-tion burst is a well-characterized process of intrinsic duration�T ’ 10 ms, with its maximum occurring within 3:5� 0:5 msafter core collapse (Mayle et al. 1987;Walker & Schramm 1987;van Putten 2002; Burrows&Hayes 1996). This timescale relatesto the detectors’ approximate sensitivity to � masses beyond themass limit

m� > 6:7 ; 10�2 eV2:2 Mpc

D

�T

10 ms

� �1=2 jpj10 MeV

� �: ð12Þ

This threshold is in agreement with the current bounds on �masses(Fukuda et al. 1998).

Nearby SNe will somehow be seen. Apart fromGWs and neu-trinos, �-rays, X-rays, visible, infrared, or radio signals will bedetected. Therefore, their position on the sky and distance (D)maybe determined quite accurately, including—if far from the MilkyWay—their host galaxy (Ando et al. 2005). Besides, the UniversalTime of arrival of the GW burst to three or more gravitationalradiation interferometric observatories or resonant detectors willbe precisely established (Schutz 1986; Arnaud et al. 2002). Theuncertainty in the GW timing depends on the signal-to-noise ratio(S/N) as�T (GWjD¼10 kpc) � 1:45� /(S/N) � 0:15ms, with � �1 ms being the rms width of the main GW peak (Arnaud et al.2002). Meanwhile, the type of � and its energy and UniversalTime of arrival to � telescopes of the SNEWS network will behighly accurately measured (Antonioli et al. 2004; Beacom &Vogel 1999). The � timing uncertainty is�T max

� ¼ �Cash(N�)�1=2,

with �Cash � (2:3� 0:3) ms and N� being the event statistics(proportional toD). This leads to the SN distance-dependent un-certainty in the �mass, �m2

� / �T max� /D � 0:5Y0:6 eV2 (Arnaud

et al. 2002), which impliesm� � 7 ; 10�1 eV, which is consistentwith our previous estimate from equation (12). Hence, those �particles and their spin-flip conversion signals must be detected.

Therefore, the left-hand side of equation (11), i.e., the time-of-flight delay�TGW$�, will bemeasuredwith a very high accuracy.With these quantities, a very precise and stringent assessment ofthe absolute �mass eigenstate spectrumwill be readily set out bymeans not explored earlier in astroparticle physics: an innovativetechnique involving not only particle but also GWastronomy. Forinstance, at a 10 kpc distance, e.g., to the Galactic center (GC inFig. 2), the resulting time delay should approximate�TGW !� ¼5:2 ; 10�3 s, for a flavor of mass m� � 1 eV and jpj � 10 MeV.A SN event from the GC or Large Magellanic Cloud (LMC)would provide enough statistics in SNO, SK, etc., �5000Y8000events, so as to allow for the definition of the � mass eigenstates(Beacom et al. 2001). Farther out, � events are less promisingin this perspective, but we stress that one � event collected bythe plannedmegaton � detector, from a large-distance source,mayprove sufficient (see further arguments in Ando et al. 2005).

6. SUMMARY

In this paper, it has been emphasized that knowing the � ab-solute mass scale with enough accuracy would turn out to be afundamental test of the physics beyond the standardmodel of fun-damental interactions. By virtue of the very important two-stepmechanism of � spin-flavor conversions in SNe, very recentlyrevisited byKuznetsov et al. (2008), we suggest that by combiningthe detection of the GW signals generated by those oscillations andthe � signals collected by SNEWS from the same SN event, onemight conclusively assess the � mass spectrum. In particular, sortingout the neutronization phase signal from both the � light curveand the second peak in the GW waveform (with its memorylikefeature; Burrows &Hayes 1996) might allow one to achieve thisgoal in a nonpareil fashion.

H. J. M. C. thanks FAPERJ, Brazil for financial support andICRANet Coordinating Centre, Pescara, Italy for hospitalityduring the early stages of this work. G. L. acknowledges supportto this work provided by MIUR through PRIN AstroparticlePhysics 2007 and by research funds of the Universita di Salerno.He also acknowledges ASI for financial support.

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