Double beta decay and neutrino masses

REVJ~;TA ;IEXlCA:<A
DE ¡.iSICA 48 SLl'LE~n~,,.o
2. 87-92
:<OVIE;lBRE
2002
Double beta decay and neutrino masses
Jorge G. Hirsch
InstituiD de Ciencias Nucleares, Universidad Nacional AUl6noma de Aléxico.
Apartado Postal 70-543, 04510 Mhico D.F, Mtxico
e-mail: [email protected]
Recibido el 19 de febrero de 2002; aceptado el 19 de abril de 2002
The flux oC snlar and atmosphcric nculrinos has becn measurcd wiLh increasing prcdsion, and olTers now direc! evidence oC (he prcscncc oC
nculrino oscillations hctwccn dilTcrent neutrino llavors. Thc nculrinolcss douhlc heta dccay, ir ddcClcd, would providc thc complementary
infonnation nceded 10 determine ncutrino masscs. and would a1so alTerdcfinitivc evidence Iha! the ncutrino is a Majorana particlc. A review
is presented on the ongoing and rulure doublc-beta-decay expcriments, the challenges still prcsent in the calculation of double-beta-dccay
nuclear matrix clemcnts, and of rccent improvernents based on the pscudo-SU(3) shell model.
Keywords: Double beta dcc<lY;neutrino masses
El flujo de neutrinos solares y atmosféricos se h<lmedido con gran precisión, ofreciendo actualmente evidencia directa de la existencia de
oscilaciones de neutrinos entre sus diferentes sabores. El decaimiento beta doble sin emisión de neutrinos, de ser detectado, proveerla la
información complementaria necesaria para determinar las masas de los neutrinos, al tiempo que mostraría al ncutrino como una partícula
de Majorana. Se presenta una revisión de los experimentos presentes y futuros asociados al decaimiento beta doble, de los desafíos presentes
en el cálculo de los elementos de matriz nucleares del decaimiento beta doble, y de los nuevos desarrollos basados en el modelo de capas
pseudo SU(3).
lJescriplores: Decaimiento beta doble; masa" de nculrinos
PACS: 21.60.Fw, 23.40.lIc,
27.70.+q
1. Introduction
Almospheric and solar neutrino experiments have provcnthat
neutrinos are malisive. ancllhc different flavors are largely mixed [1, 2]. The exislence of neulrino oscillalions allow !he delermination of!he difference belween lhe square of!he neutrino masses coming from differentfamilies [3]. However, the
Ilcutrinomass absolute scale must be obtaincd [rom direct kinemalical measuremenlS, or from !he double beta decay.
The neutrinoless double beta decay ({3{3o"),if delected,
would nO! only provide the complementary information ne-
eded to delennine neutrino malises. bUl also offer definitive
cvidcnce thatthe neulrino is a Majorana particle, i.e. iLis it'i
own antiparticle [4,5].
Theoretical nuclear malrix elements are needed to convert
experimental half-life limils, which are available for many
{3{3-unstableisolopes [6,7], inlo conslrains for !he effective
Majorana ma" of!he neulrino and lhe conlribution of righlhanded CUfrents to !he weak interactions. Thus, these matrix
elements are essentiallo undersland lhe underlying physics.
The IwO neulrino mode of the double beJa decay ({3132")
is allowed as a second order proccss in the standard modeL
II has been delected in len nuclei [8,7) and has served as a
lesl of a variety of nuclear models. The calculation of !he
f3/32v and /3/3011 matrix elcrnent'i rcquircs different thcoreti-
cal methods. Thcrcfore a successful prcdiction of me former
cannot be considcred a rigorous test of me laltcr. However,
in mosl cases it is the best available proof we can imposc
lo a nuclear model used lo predicl!he {3f30" matrix c1ements.
When il is possible,!he same models are used to calculale lhe
energy spcctrum, eleclromagnetic and bela decay lransitions
in !he initial, intermediate and final nuclei.
Many experimenlal groups have reported mcasurements
of {3{3 processes [8,9J. In direct-counting experiments the
analysis of the sum-energy spectrum of the emilled elcclrons allowed!he idenlificalion ofthe different {3{3-decaymodes [1O].The {3132"-halflives have bcen mcasured in len nuc1ei, ranging betwecn 10'8 - 1022 yrs. Limils on the {3{3o"half tives go up to 1025 yrs which, based on !he !heoretical
estimates to the nuclear malrix elemcnls, put limits lO!he Majorana ma" of !he neutrino close lo 0.3 eV.
The struclUre of!he present arlicle is as follows. [n Sec. 2
the neUlrino masses and mixing are discussed, in Sec. 3 lhe
double beta decay fonnalism is inlroduced, and lhe {3{3experimenLs are reviewed in Sec. 4. Section 5 contains!he lheorelical calculations of the nuclear matrix elements in deformed
nuclei. The pseudo SU(3) formalism is describe<! in Sec. 6,
while in Sec. 7 old and new resulls for lhe {3{3half-Iives are
presenled. Conc1usions are drawn in Sec. 8.
2. Neutrino masses
The descriplion of the present resulLs of !he atmospheric and
solar neutrino cxpcriments requircs the neutrinos emiued in
weak decay proccsses like beta dccay (i.e. Ve, VIJl Vr) lo be a
88
JORGE G. HIRSCH
TABLE 1. Most probable valucs oC nculrino paramctcrs
LOW-QYO
LMA
ó.m~2 (eV2)
Óm!2 [cV2]
IU"I'
3.3 x 10-'
9.6
0.26
combinatian of several mass cigenstatcs
values mi [11,12],
vi
xlO-8
with mal)S eigcn-
3
Ve = ¿::UeiV¡.
(1)
i=1
NcuLrino mixing matrix
elerncnLI) are
denoted by Ud It is cus-
tomarily assumed lhal m, < m2 < m3. The effeelive electron neutrino mass m~ measured from lhe tritium {3 decay
cxpcriment"i can be written
m~ =
a<i
(IU,1I2m¡ + IUd2~
n'/2.
+ IU'312m
(2)
Presenl experimentallimi!s are m~ < 2.2 eV. It can be redueed lO0.3 eY along lhe presenl deeade.
The neutrinoless double beta deeay ({3{3ov), if delected,
would offer definitive evidenee lhal lhe neutrino is a Majorana particle [4,5]. This process violates lbe leplon number
eonservalion. being forbidden in lhe standard model of elementary particles.
The Majorana mass of the neulrino (mv) appears in lbe
f3f3ov-deeay in lbe form
(mv)
= ¡U,1I2m, + ei.'IUd2m,
+ ei.'IUd2m3'
S; 2.2eV,
i = 1,2,3,
Imi - mjl
< O.OBeV.
ó.m~3[eV2J
IU'31'
0.37
3.1 xlO-3
0.005
slandard model and would be useful in order to selecl Grand
Unifieation Theories [4]. Thcoretieal nuclear matrix elements
are needed to eonvert experimental haif-life limi!s, whieh are
available for many f3f3-unstable iSOlOpeS[B], into eonstrain!s
for particle physies pararneters sueh as lbe effective Majorana
mass of lbe neutrino and lbe eontribution of right-handed curren!s to lbe weak interaetion. The two neutrino mode of lhe
double beta decay ({3{3)is allowed as a seeond order process
in lbe standard mode!' It has been detecled in ten nuclei [B, 7]
and has served as a tesl of a variely of nuclear models. It is
the best available proof we can impose lo a nuclear model
used 10 prediet lbe f3f30v matrix elemen!s.
Many experimental groups have reported measurements
of {3f3 processes [B,7J. Nearly for all lbe cases lbe ground
state (g.S.) lO ground state (0+ --+ 0+) deeay was investigaledo In direct-eounting experimen!s lbe analysis of lhe sumenergy speetrum of lbe emitted electrons allowed lbe identifiealion of lhe different f3f3-deeay modes.
The inverse half !ife of lbe two neutrino mode of the {3{3deeay can be expressed in the form [13]
(5)
(3)
The relalive Majorana CP-violating phases are denoted 4>i.
Notiee lhat, even if the mass eigenslales are as large as their
upper limi!s ('" 2 eV), lhe CP-phases could conspire to generate a strong eaneellation, eausing (mv) to be very small,
evenlually zero.
From the solar and almospherie neutrino experiments lbe
squarcd mass differences D.mrj = m~- mr have beco delermined. In lbe most probable scenarios, ~m¡2' IU,tI' =
e",,' 80 and ¡Ud' = sin280 are obtained from solar neutrino experiments, ~~3 from atrnospheric neutrino experimen!s and IU'3I' is restrieted from experimen!s looking for
eleetron disappearanee like CHOOZ. The phases 4>i and lbe
mass ofthe lighlest neutrino musl be oblained from the f3f30vdeeayexperimen!s.
The most probable values of the neutrino parameters,
obtained from global analysis of known experimental results [11], are lisled in Table 1.
Combined wilb lbe tritium f3-decay data, lbey establish
the following !im;!s lo lhe neutrino masses [11]:
mi
IU"I'
where G,v is a kinematieal factor, whieh depends strongly
on total energy released in lbe decay.
In the case thc Ov decay exists. the virtual ncutrino must
be emitted in one vertex, and absorbed in lbe other. Sinee
in lbe standard theory lbe emilted particle is a righl-handed
antineutrino and lbe absorbed one a left-handed neutrino the
process requires lbal: (a) lbe exehanged neutrino is a Majorana particle and (b) both neutrinos have a eommon helicity eomponent. The helieily matehing can be satisfied in two
ways: (i) lbe neutrinos have a nonvanishing mass and therefore a "wrong" helieity eomponenl proportional to mv/ Ev.
The deeay rate will be proportionallo (mv)'. Or (ii) lbe heI¡eity restrietion eould be satisfied if lbere is a right handed
current interaction. In this ca<;ca nonvanishing mass allowing
mixing of neutrino lypes is also required [13, 14].
For ma'isive Majorana ncutrinos one can peñonn the integration over lbe four-momentum of lbe exehanged particle
and obtain a "neutrino potential" whieh for a light neutrino
(mv < 10 MeY) has !he form
(4)
H(r, E) = 2R
1rr
3. Double beta decay
The deteetion of the neutrinoless double beta decay (f3f30v)
would imply indisputable evidenee of physics beyond lbe
roo dq sin(q,:! ,
Jo
q
+E
(6)
where E is lbe average exeilalion energy of lbe intermediale
odd-odd nueleus and lbe nuclear radius R has becn added to
Rev. Mex. Fls. 4852 (2002) 87-92
OOUBLE BETA OECAY ANO NEUTRINO MASSES
make lhe neutrino polential dimensionless. The final fonnula. rcst.rictcd lO the [errn proJX>rlional 10 [he ncutrino maCiS,
is 113, 15J
(1/2)_1
TOv
_ ((rnv))2C
-
m,
M2
01.'
Ov'
(7)
where COv is lhe phase spaee integral a"ocialed with the
cmission of the lWQ electrons. Thc nuclear matrix elcrncnts
Mov are {l31
(8)
wíth
(9)
where lhe kets IOn and loj) denole lhe emresponding initial
and final nuclear slales, the quantities gv and 9 A are the dimcnsionlcss coupling constanL'i oC the vector and axial vector
nuclear currcnlS, and
m,"
m,n
rn,"
m,n
The (j being the Pauli matriees related wilh the spin operalor and t- lhe isospin lowering operalOr, whieh satisfies
t-In) = Ip). The superindex GT denoles the Gamow-Teller
spin-isospin transfer channcl, while F indicales lhe Fenni
isospin one.
4. Double beta decay experiments
Afler lhe first direCl deleClion of the {3{32v-decayof B2Se in
1987 using a lime projection chamber, lhis rare decay has
beco obscrvcd in len nuclci using a varicty of experimental
lcehniques 18J. Geochemical deleclion was the firsl method
employed, allowing lhe delennination of the {3{32v-halflife
ofB2Se, 96Zr, 128,130Te.The {3{32v-half!ife Of238U was measored using radiochemistry, Time projeclion chambers were
used with "Ca, 82Se, looMo and IsoNd, counting spcctromelers with 82Se, 96Zr, looMo and 116Cd, and ultrapure semiconductor gennanium dctcctors with 76GC.
Geochemical and radiochcmical methods counl the number of daughler nuclei in an old sample, being unable lO distinguish bctwccn the modes with and wi1houl ncutrino cmissioo. Direct rnca'iuremenl'i dctcct the simultancous crnission
of the lwo elcetrons, and the shape of lhe energy speclrum
provides a clear signalure for lhe different {3{3-decaymodes.
Thesc experiments musl be perfonned in a very low background enviroument, usually provided in underground laboratories, and make use of ultrapure material s with radioae.
(ive eontaminanL<;prescnt in less than onc part in 1012. Thc
{3{32v decay of lOoMo lO lhe first exciled sUUe in looRu has
also bccn detceled.
89
In most of the direcl counler experiments lower bounds
were estahlished for the {3f30v decay half lives. The mosl
stringcnl Iimit for thc Majorana mass oC the ncutrino was
oblained by lhe Heidelberg-Moscow experiment using 76Ge.
They repon
> 1.9 X 1025 yr, which, depending
on the nuclear malrix elements employed, imply (mv) <
0.35 eY [16]. Evidence for the firsl positive detection of lhe
{3f30v dceay in 76Ge was rceently reponed [17]. BUl immediately lhe whole remaining communily of {3{3experimenlalisls
sellt a commellt were they showed "lhallhere is no ba,is for
lhe claim presenled in the paper"¡18J.
There are many running experimenls [8] aimed lo improve the present limils for the Majorana mass of the neutrino,
10 measure !he {3[3,v decay in other polenlial emillers, a, well
as lo detcel other forms of this dceay, like lhe double positron emission and lhe double electron capture. ITEP ha,
commissioned a large TPC chamber wilh 7.5 kg of 136Xe,
and plans lO study also IsoNd. NEMO-3 has started measuring with 8.5 kg of looMo and smaller samples of olher isotopes. Boiometric delectors buiit wilh Te02 crystals are measuring the 130Te {3{3-halflife. AH of lhem, logelher Wilh lhe
Heidelberg-Moscow experiment, are probing Majorana ma,ses in the range of 0.2--D.5 eV.
Experiment, are heing planned to detcel Majorana ma,ses as low a, 0.01 eY 181.The GENIUS projecl plans lo use
1 Ion of 76Ge with a furlher background reduclion, and could
be upgraded up lO 10 lons, The projecl CUORE is being developed lo use 1 Ion of 130Te in lhe fonn of 1000 Te02 bolomelric deteclors. A !iquid 136Xe chamber wilh a few hundred
kg would have nearly ,"ero backgrouud, The MOON experiment would use a Ion of 1OOMo.AIl these experiments have
a real pOlential for delecting the {3{3ovdceay, or lo exclude
many of the LMA and LOW scenarios [12].
T¿£2
5. Theoretical calculations for heavy deformed
nuclei
The calculalion of lwo neutrino double beta decay matrix clements ha, proveo to be extremely sensilive lo lhe delails of
lhe wave funclions of the inilial and final nuclei 15J. While
.QRPA calculations are ea,y lo perfonn, lhe uncenainties in
the residual particle-particle proton-neutron interaction strongly limillheir predictive power [5]. Shell model sludies in lhe
full fp shell provide rcliable matrix clements fm 48Ca [19J
and other !ight nuclei 120J. Matrix clement, for 76Ge, 82Se
and 136Xc were obtained using very large shell model spaces, which are however strongIy truncated [21J, For heavier
nuclei staudard shell model calculations are impracticable.
The pseudo SU(3) shell modcl 122,23] is a microscopic modcl which allows !he descriplion of heavy defonned
nuclei in the laboralory frame through the use of a fennionie many-parlicle basis with good angular momenlum. The
microscopic Hamiitonian employed includes single-parlicle
energies as well as pairing and quadrupole-quadrupole interaclions. MOSl of lhe Hamillonian paramelers are fixed by
Rev. Me.x. Fís. 4852 (2002) 87-92
90
JORGEG.
known systematics. Three "rotor tenns" allow a fine tuning
of lhe energy speetra and are fiued for eaeh nuelei. The pseudo SU(3) model has been used lo describe many low-Iying
rotalional hands, as well as B(E2) and B(M 1) inlensilies in
rare earth and aelinide nuclei, bolh with even-even and oddmass numbers [24-29]. lt has been exhibited as a powerful
amI predielive 1001 in the deseriplion of heavy deforIned nuclei.
Tbe {3{3half lives of sorne of lhese parent nuclei 10 the
ground and exeited StaleS of the daughter ones were evalualed for lhe IwO and zero neutrino emitling modes [3G-34]
llsing lhe pseudo SU(3) seheme. The predielions were in good agreemem with the available experimenlal data for lI;oNd
and 23'U. EXlending the previous researeh [30-34] Ihe {3{3
half lives of six nuclei, whose deeays were previously reporled as Iheoretieally forbidden, have been evalualed using the
pseudo SU(3) mode!' While lhe 2v mode is forbidden when
lhe mosl probable oceupalions are eonsidered, StaleS with differenl oceupalion numbers can be mixed through the pairing
imeraelion. In this way Ihe possibilily of observing the {3{3
decay in l,oGd and 2•• Po was diseussed for both the 2v and
Ov modes [35-37].
6. The pseudo SU(3) formalism
In lhe pseudo SU(3) shell-model eoupling seheme [22], normal parily orbitals (1/,I,j) are identified Wilh orbitals of a
harmonie oseillalor of one quanla less ij
1/ - 1. The sel of
orbilals with ]
j
i+ 8, pseudo spin 8 1/2, and pseudo orbilal angular momenlum i, define the so-called pseudo
spaee. Tbe orbitals wilh j
i:l: 1/2 are nearly degenerate.
For configuralions of identieal parlicles oceupying a single j
orbilal of abnormal parity, a eonvenient eharaeterizalion of
Slates is made by means of lhe seniority eoupling seheme.
The many-particle slates of no nucleons in a given
shell 1/0' a = v or ", can be defined by the tolally anlisymmetrie irredueible representations {1n;:'} and {1n:} of
unilary groups. The dimensions of the nOrInal (N) parily spaee is
= (ijo + l)(ijo + 2) and that of the unique (A) spaee
is n~ 2(1/0 + 2) with the eonstraint no
+
Proton and neutron states are eoupled to angular momentum JN
and J A in both the nOrInal and unique parity sectors, respecti"ely. The wave funclion of the many-parlicIe state with angular momentum J and projection M is expressed as a direct
product of the normal and unique parity ones, as:
=
= =
=
=
n;;
= n~
=
IJM)
=
¿
[IJN)
0
iJA)L.
n;;.
(12)
JN JA
Sincc wc are inlereslcd in describing low-Iying energy statcs.
only pseudo spin zero configurations are taken ioto accounl
in the nOrInal parity space and only seniority lOro configurations in the abnorInal parity spaee. Tbis simplifiealion implies
that
=
= O. This is a strong assumption, bUl ane that
is physieally molivated and very useful for simplifying the
ea!culations. Recent researeh show thal lhe quasi SU(3) for-
J: J:
HIRSCH
TABLE JI. CaJculated double beta half-Iives for the two- neutrino
and the zero-neutrino modcs.
Transition
T~t2Iyr]
T¿(2
Iyr]
1..••
6Nd-t 146Sm
2.1 x 10
1.18 x 10"
148Nd-t148Sm
6.0 x 10'°
6.75 x 10"
1150Nd-+1.~OSm
31
6.0 x
1016
1.05 x 10"
186W-+1860s
6.1 x 1O:il4.
5.13
1920S-+
9.0 x 10"
3.28 X 102•
1.4 x 1021
1.03 x 10"
Hl2Pt
238U-t238pu
X
10215
malism ean be used lO improve the deseriplion of the intruder
sector [38]. Its appliealion to the {3{3deeay are under developmen!.
Double beta decay, when deseribed in the pseudo SU(3)
seheme, is strongly dependent on lhe oceupation numbees for
protons and neutrons in the nOrInal and abnormal parity Slates:
1
I
[30]. These numbers are detennined by
filling lhe Nilsson levels from below, as diseussed in Ref. 30.
The effeet of allowing ehanges in these numbers is diseussed
in Refs. 36 and 37 and brieny mentioned in the next section.
In lhe first series of papees [30-34] we evalualed the
{3{3malrix elements by taking into aeeounl only the leading SU(3) eoupled proton-neutron irredueible representation. which in rceent calculaLion was shown lO represent
around 60% of the wave funetion in even-even Dy and Er
isotopes [29].
n~ n':,n: n~
7. Predicted,8-,8-
half-Iivcs
In all of the ea!culations only one active shell was allowed foc
protons, and likewise, only one for neutrons. Tbis is a very
slrong truneation. For the {3{3decay this implies thal only one
uneorrelated Garnow- Teller lransilion is allowed: that whieh rcmovcs a neutron froro a normal parity state wilh maxi~
mum angular momenlum and ereates a pro Ion in the .intruder
shell (h~/2 ~ h~I/2 in rafe earth nuclei, i~1/2 -+ i~3/2 in
aetinides). This unique Garnow-Teller transition eontrols the
{3{3decay. Under these assumplions, if the oceupation of the
Nilsson levels is sueh that the number of protons in the abnormal parity states docs not ehange for the initial and final
stale eonfigurations, the deeay is forbidden.
The published resulLs for the six allowed two neutrino
{3- {3- emillers are given in Table 11.
The agreement belween the theoretieal two neutrino half
lives [30] with lhe available data for 150Nd (T~t2 = 9(17) x
1018 yr) and 23'u (T~t2 = 2 X 1021 yr) is good. For the
theorelieal {3{3ovhalflives [31] we assumed (mv) = leV.
As menlioned above, the number of nucIeons in nOrInal
and unique parity orbital s is delerInined by the filling of lhe
deforIned Nilsson orbitals. In this way lhe theory predieLs the
complete suppression of the {3{3deeay for the following five
nucIei: 15'Sm, 1.oGd, 17'Yb, 232Th and 24'Po [30). 11was
Rev. Mex. Frs. 4882 (2002) 87-92
91
DOlJBLE BETA DECAY AND NElJTRINO ~ASSES
Prcdíctcd half-livcs for Ihe 2v and Ov double beta dc~
cay. for the nuclci wcrc Ihe {3(32IJ dccay was reported as forhiddcn.
Tr\AI.E
1[1.
Nuclei
154Sm
T~!2lyr]
r¿!'(m.)'
23
1.04 x 10
[yreV'¡
9.38 x lO"
1021
I60Gd
6:02 x
170Er
LID x 10"
2.92 X lO'.
176Yb
2.77 x 10'3
6.43 x 10"
232Th
244
Pu
5.30 x
2.09 x 10"
1022
5.44 x
6.43 x 10'0
la"
1.30 x lO"
expeeled lhal lhese forbidden deeays would have, in !he besl
case, matrix clernents lha1 w()uld be no grcatcr than 20% of
lhe allowcd oncs, rcsulting
in al
Ica'it
Qne ordcr of magnitudc
reduelion in !he predieled half-life [34]. ExpcrimentallimiLs
for lhe 313 deeay of ¡60Ud have been reported [39,40J. Reeemly;l was argued lhat lhe strong eaneellation of!he 2v mode in lhe 1313 deeay of 160Ud would suppress the baekground
for lhe deleclion of lhe 0/1 mode 41].
Thc pairing imcraction allows lhe mixing of slatcs with
diffcrcnt occupancics in lhe nonnal ami intrudcr scctors. In
this way il is possible 10 ealeulate a finile /3132" half-life for
lhe nuele; where lhis deeay was previously reported as forbidden. In Refs. 35 ami 36 il was assumed lhallhe final nuelei
r
wave function has
(\VO
componenl,..; wilh diffcrcnt occupan-
eies: one 10 whieh lhe /3/32" is forbidden, and lhe olher to
whieh il is allowed. The /3/30" has finile eontributions for the
two componen!s wiLh diffcrcnt occupation numbers in lhe final slale. There are two lerms in !he /3130" deeay: one to lhe
oasis state whieh has allowed /3/32"deeay, and one lo the stale
1. S. Fukuda el. al., Pllys. Rev. Lell. 86 (2001) 5651.
2. Q.R. Ahmad el. ,,1..1'lrys.
Re\'. l.el/. ll7 (~OOI) 071301.
3. J.N. Bahcall. P.1. Kr.lstt.:v, and A. Yu. Smirnov. J. I/¡gll ElIergy
Phys. 05 (20DI) 015; John N. Baheall. M.e. Gonzalez-Garcia,
and Carlos Pcna.C1aray, J. I¡¡gll ErJergy Pllys. 08 (2001) 014.
4. J.D. Vcrgados, Plly!>. Rep. 111 (1986) 1; A. Facssler, Prog. Parlo
Nuc!. Phys. 21 (19X8) 183; T. Tomoda. Rep. Prog. Pllys. 54
(1991)53.
5. J. Suhoncn amI O. Civit.arese, Phys. Rcp. 300 (1998) 123.
6. M.K. Moc. 1m. Jour. Mod. Phys. E 2 (1993) 507; S.R. Elliot,
M.K. Moc. M.A. Nclson, and M.A. Vicnt, Nud. Phys. B I'me.
Suppl. 31 (1993) 6X.
7. A.S. Barabash, Czech. .faum. "lIy.\ .. (2001) in prcss.
8. I.v. Kirpichniko\', Phys. Al. Nucl. 63 (2(x)O) 1417.
9. Angel Morales, Nud. Phys. B (Proc. Suppl.) 77 (1999) 335.
wilh forbidden /3/h." decay. The resulLs, laken fmm Rer. 36,
are presenled in Table 1lI. A detailcd sludy oflhe /3/3decay of
16oUd, ineluding toe mixing of different oeeupaneies in the
parent and daughler nuelei, ean be found in Ref. 37.
As a eonsequence of the explieit inelusion of deformation
in !he presenl model, !he /3/30" half-lives are larger than those reported in Rer. 42. In 160Ud !he 13130" decay half-life is
at least lhree orders of magnitude larger than lhe /3/h." deeay
half-life. lt implies lhal the background suppression due lO a
large /3/32" half-life would be effeetive, allhough nol as notieeably as was optimistieally envisioned in Rer. 41. In any
case, !he resulls presenled strongly suggesl !hat !he planned
expcriments using USO eryslals [41] would be able to delecl
lhe /3/h." decay of 160Ud, and lO establish eompctitive limils
lO !he /3/30"decay.
8. Conclusions
We have reviewed the eompclling evidenee exhibiting lhe
neutrinos as massive partieles. The observation of lhe neutrinoless double bela decay would eomplement !hese dala by
fixing !he mass seale. Running and planned expcrimenLs will
push lhe upper limils of!he Majorana mass ofthe neutrino lo
0.01 eV.
The pseudo SU(3) predietions for !he /3/h." and /3/30"
half-Iives have bccn eXlended by !he inelusion of lhe pairing
mixing belween states wi!h differcnt oceupaneies in !he normal and intruder sCClors. Thc nuclei 1600d and 244pU emerge
as viable eandidates where these deeays can be delecled.
Acknowledgment
Work supported in parl by CONACyT.
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