Revüta Mexicana de Física 41, Suplemento 1 (1995) 81-89

Double beta decay in heavy deformed nuclei: what havewe learned?JORGE G. HIRSCH

Departamento de Física, Centro de Investigación de Estudios Avanzados del IPNApartado postal 14-740, 07000 México, D.F., México

ABSTRACT. The pseudo SU(3) approach is used to describe low-Iying states and BE(2) intensitiesof rare earth and actinide nuclei which are {3{3decay candidates. The {3{3half lives of sorne of theseparent nuclei to the ground and excited states of the daughter ones are evaluated for the two andzero neutrino emitting modes. The existen ce of selection rules which strongly restricts the decaysis discussed. These restrictions represent a possible test of the mode!. Up to now the predictionsare in good agreement with the available experimental data.

RESUMEN. El enfoque pseudo SU(3) se usa para describir los estados de menor energía y susintensidades BE(2) de núcleos en la región de las tierras raras y los actínidos que son candidatosa decaimientos {3{3.Se evalúan las vidas medias de esta desintegración para algunos de estosnúcleos padres yendo al estado fundamental y a estados excitados de los núcleos hijos, en losmodos con emisión de dos y cero neutrinos. Se discute la existencia de reglas de selección querestringen fuertemente los decaimientos y representan una posible prueba para el modelo. Hastaahora la concordancia entre las predicciones y los datos experimentales disponibles es bastantebuena.

PAes: 23.40.Hc; 21.60.Fw

1. INTRODUCTION

The detection of the neutrinoless double beta decay (/3/3ov) would imply indisputableevidence of physics beyond the standard model and would be useful in order to selectGrand Unification Theories [1]. Theoretical nuclear matrix elements are needed to convertexperimental half-life limits, which are available for many /3/3-unstable isotopes [2], intoconstraints for particle physics parameters such as the effective Majorana mass of theneutrino and the contribution of right-handed currenb to the weak interaction. The twoneutrino mode of the double beta decay (/3/32v) is allowed as a second order process inthe standard mode!. It has been detected in nine nuclei [2] and has served as a test of avariety of nuclear models. It is the best available proof we can impose to a nuclear modelused to predict the /3/3ov matrix elements.

Many experimental groups have reported meaBurements of /3/32v processes [2]. Nearlyfor all the cases the ground state (g.s.) to ground state (0+ -> 0+) decay was investigated.In direct-counting experiments the analysis of the sum-energy spectrum of the emittedelectrons allowed the identification of the different /3/3-decay modes. The experimentalresearch on the double beta decay of heavy deformed nuclei enjoys an increasing activity.Thus, the /3/32v half-life of 238U [3] and 150Nd [2,4,5] have been measured aud search is inprogress for 244pU [6].

82 JORGEG. HIRSCH

In previous papers [7,8] we used the pseudo SU(3) shell model to evaluate the two neu-trino double beta half lifes of eleven heavy deformed potential double beta emitters. Wefound good agreement with the available experimental information. The radiochemicallymeasured 238Udecay [3) raised sorne expectations, given the experimental arrangementcannot discriminate between the different double beta decay modes, and sorne theoreti-cal estimates [9) predicted similar decay ratios for both the Ov and the 2v modes. Ourcalculations for the two neutrino mode were consistent with the experimental resulto

We have also calculated the zero neutrino matrix elements for six heavy deformed doublebeta emitters [10). Using the upper limit for the neutrino mass we also estimated theirf3f3ov half-lives. In the case of 238Uwe found the zero neutrino half-life at least three ordersof magnitude greater than the two neutrino one, giving strong support of the identificationof the observed half-life as being the two neutrino double beta decay. The f3f32v of 150Ndto the ground and excited states of 150Smhas also been studied [11). In the present paperwe review the aboye mentioned calculations as well as the pseudo SU(3) formalism usedin those works.

2. THE PSEUDOSU(3) FORMALISM

In the pseudo SU(3) shell model coupling scheme [12) normal parity orbitals (1],I,j) areidentified with orbitals of a harmonic oscillator of one quanta less i¡ = 1]- 1. This set oforbitals with J = j = i+ S, pseudo spin s = 1/2 and pseudo orbital angular momentum¡define the so called pseudo space. The orbitals with j = ¡:1:1/2 are nearly degenerate.For configurations of identical partic1es occupying a single j orbital of abnormal parity aconvenient characterization of states is made by means of the seniority coupling scheme.

The many partic1e states of no nuc1eonsin a given shell1]o, Ct = v or ", can be defined bythe totally antisymmetric irreducible representations {1n;;'} and {1n:} of unitary groups.The dimensions of the normal (N) parity space is n{'[ = (i¡o + 1)(i¡0 + 2) and that of theunique (A) space is n~= 21]0+ 4 with the constraint no = n~ + n{'[. Proton and neutronstates are coupled to angular momentum JN and JA in both the normal and uniqueparity sectors, respectively. The wave function of the many-partic1e state with angularmomentum J and projection M is expressed as a direct product of the normal and uniqueparity ones, as

IJM) = L [IJN)Q9IJA)(JN JA

(1)

We are interested in describing the low-Iying energy states. In the normal subspace onlypseudo spin zero configurations are taken into account. Additionally in the abnormal parityspace only seniority zero configurations are taken into account. This simplification impliesthat J:; = J~ = O.This is a very strong assumption qnite useful in order to simplify thecalculations. Its effects upon the present calcnlation are discussed below.

The double beta decay, when described in the pseudo SU(3) scheme, is strongly depen-dent on the occupation numbers for protons and neutrons in the normal and abnormalparity states nr:, n{;', n:, n;) [81. These numbers are determined filling the Nilsson levels

DOUBLE BETA DECAY IN IIEAVY DEFORMED NUCLEI:... 83

from below, as discussed in [S}. In particular the /3f3zv decay is allowed only if they fulfilthe following relationships

(2)

If these relation are not satisfied the /3/32v decay becomes forbidden. This is the firstselection rule the pseudo SU(3) formalism impose on the double beta decay. As an example,for 150Nd we have obtained the occupation numbers n: = 4, nt; = 6, n~ = 2, n{:= 6.We have selected the standard version of the pseudo SU(3) Hamiltonian [13]. It is

constructed by a spherical Nilsson hamiltonian which describes the single-particle motionof neutrons or protons, a quadrupole-quadrupole interaction and a residual force. Thelatter allows the fine tuning of low lying spectral features like K band splitting and theeffective moments of inertia.With the occupation numbers and the hamiltonian discussed aboye, the wave function

of the deformed ground state of 150Nd can be written [SI

¡150Nd, 0+) = l(hll/2);, J: = M: = O; (i13/2)~, J~= M~ = O) A

1{l6h {23}. (12, O).; {l6}v {23}v (1S, O)v; 1 (30, O) K = 1J = M = O) N'

(3)

3. TIIE DOUBLE llETA DECAY

The inverse half life of the two neutrinu mude of the /3/3-decay can be expressed in theform [14)

(4)

where G2v(J:) are kinematical factors. They depend on EJ:q = !IQ¡¡¡¡ - E(Jq)1 + mec2which is half of the total energy rcleased. The nuclear matrix element is evaluated usingthe pseudo SU(3) formalismo Fur the 150Nd --+ 150Sm case it can be written as

Mi:7(J:) = a(J) b(n:) [;~J+1)

L ((O~)li, (Ol¡)IiII(.\ol'o)KoJ) L((30, 0)10, (.\Ol'o)KoJII(.\I')qlJ)(Ao~o)Ko I P P

[

(12, O) (O, O) (12, O) 1]'" (1S,O) (.\01'0) (12,2) p' ((12 2)III[ii _ ii _ 1](Ao~o)lll(lS O») (5)L.; (30,O) (.\01'0) (.\/t). p , Ory).4 Ory)., 'p'P 1 1 1

84 JORGE G. HIRSCH

In the aboye fprmula (... , ... 11 ... ) denotes the SU(3) Clebsch.Gordan coefficients [15]'the symbol [... ) represents a 9 - A/l recoupling coefficient [16], (... 111 ... /11 ... ) is the triplereduced matrix elements [17] and the following notation was introduced:

atO) _ 4'1- (2'1 + l),¡z,¡=¡'

a(2) = _2_ [5'1('1- 1))2'1- 3)] 1/2

2'1+ 1 3(2'1 + 1)

[ ]1/2

b(n:) = (n: + 2)(1) + 1 - n: /2) , (6)

with L\c = ~.,~~[2Z + 1- 0.76((Z + 1)4/3 - Z4/3)] MeV.

The SU(3) tensorial components (AO, /10) of the normal part of the double Gamow-Teller operator must be able to couple the proton and total irreps (18, O) and (30, O)associated with the ground state of 150Nd to the corresponding irreps (12,2) and (A/l)awhich characterize the gronnd and excited rotational bands in 150Sm. Ir these irreps cannotbe connected by (AO, /lo) the {3{32v decays to the 0+ and 2+ states are forbidden. This is asecond selection rule imposed by the model to the (3{32v decay.

In the case the Ov decay exists, the virtual nentrino mnst be emitted in one vertex, andabsorbed in the other. Since in the standard theory the emitted partiele is a right-handedantineutrino and the absorbed one a left.handed nentrino the process requires that: a) theexchanged nentrino is a Majorana partiele and b) both nentrinos have a common helicitycomponent. The helicity matching can he satisfied in two ways: a) the neutrinos have anonvanishing mass and therefore a "wrong" helicity component proportional to mv/ Ev.The decay rate will he proportional to (mv}2 Or b) the helicity restriction could besatisfied if there is a right handed cnrrent interaction. In this caBe a nonvanishing massallowing mixing of neutrino types is also reqnired [14,181.

The (3/30v matrix elements were evalnated using the pseudo SU(3) formalism [ll]. Thefinal expressions are similar to those related with the two nentrino mode, given in Eq. (5),but restricted to the decay to the gronnd state of the final nnelei. They explicitly ineludethe "neutrino potential" originated by the interchange of a Majorana neutrino between thetwo emitting neutrons. The 0/1 mode allows transitions with L\l t O, whose contribntionto the {3{3ov matrix elements were fonnd an order of magnitude less than those which donot change the orbital angnlar momentnm.

The five nnelei which were fonnd lo ha,.e the (3{32v mode striclly forbidden withinthe psendo SU(3) model are allowed lo have {3/30v decays becanse the nentrino potentialallows parity mixing transitions. \Ve wonld expect that these Ov matrix elements, relatedwith L\/ = 1,3,5 transitions were of the same order of magnitude of the L\/ = 2,4,6transitions and about a factor ten lesser than the L\l = O matrix elements. This hindranceis comparable with the modifications that the symplectic model would introdnce, allowingmore than one shell active for eaeh kind of nueleons. These questions are under currentinvestigation.

DOUBLEBETADECAYIN HEAVYDEFORMEDNUCLEI:... 85

TABLE1. Theoretical estimates for the {3{3-decayhalf-Hfe in the 2v and Ov mode for several heavydeformed nuelei are given alld compared with the available experimental data.

r;!' [yrl r~!' [yrlTransition

146Nd ~ '46Sm

148Nd --+ 148Sm

,soNd ~ ISOSm186W --+ 18608

19208 --+ 192pt

238U-.238pu

theo

2.1 x 1031

6.0 X 10'0

6.0 X 10'8

6.1 X 1024

9.0 X 1025

1.4 X 10'1

exp

9-17 x 1018 [2,4,5J

2 X 1021 [31

theo1.18 x 10'8

6.75 X 1024

1.05 X 1024

5.13 x 1025

3.28 x 10'6

1.03 X 1024

exp

> 2.1 X 1021 [4J

> 2.3 x 10'° [261

> 2.0 X 10'1 [3J

4. RESULTS AND DISCUSSION

We predict, according to the assumptions of the model, that the two neutrino double betatransitions of the following five nuelei:

I54Sm -> 154Gd160Gd -> I60DyI76Yb -> I76Hf232Th -> 232U244Pu -> 244Cm

are jorbidden. In particular we predict a null result for the present search in 244pu [6]'a potentially strong test of our model. Inelusion of pairing within the same truncationscheme wiJI allow mixing of different pseudo SU(3) irreps, and is in progress [191. Thelack of a spin-isospin channel in our Hamiltonian is perhaps an additional limitation,but it is ineluded, at least partially, in the quadrupole-quadrupole interaction, when it isrecoupled.In Table I are indicated the results for the six f3f3 emitters in which we get a f3f32v

matrix element different from zero [8]. The value for the integrated kinematical factorswere obtained following the procedure indicated by Doi et al. [201 with gA/gV = 1.0. Inthe second and third columns the theoretical and experimental f3f32v.half lives are given.The agreement with the available data for 150Nd and 232U is good. Improvements in theexperimental information about the decay of these nuelei together with 148Nd is possible,but the other three are almost exeluded of detection beca use their lepton kinematicalfactors G2v are very small. This is particularly trne for I46Nd, which has the smallest Qvalue for the f3f3 decay and thns becomes strongly inhibited.We also evaluated the f3f30v matrix elements between the the same six nuelei [lOJ. In the

last two columns of Table 1 the theoretical predictions and experimental lower limits ofthe f3f30v.half lives are given. In order to calculate the zero-neutrino half-life we assumed(mv) = 1 eV, which is the larger neutrino mass parameter compatible with the 76Geexperiment [21]. In the case of 238U the predicted Ov half life is three orders of magnitude

86 JORGE G. HIRSCH

TABLEn. The dimensionless matrix elements and predicted half-lives for the f3ih" decay of lSoNdto the ground stale, the !irst 2+ and the !irst and second excited 0+ states of ISOSm.

0+ ~ 0+ (g.s.)0+ ~ 0+ (1)0+ ~ 0+ (2)0+ ~ 2+

MGT(J+)'" ..0549

O

.004995.38 x 10-5

r~!'(O+ ~ J:) [yr]6.73 x lO,s

00

4.31 X lO"

7.21 x 10"

greater than the predicted 2" half life, which essentially agrees with the experimental one,confirming that the observed (3(3decay of 238U has to be the two neutrino mode. Thetransition is strongly dominated by a pair of neutrons in the normal parity orbital ir'/2decaying into two protons in the unique orbital i~3/2'

In the case of 150Nd, the pseudo SU(3) O" matrix element reported here is about a factorfour less than the QRPA estimations [9]. This is a very relevant resulto First, it exhibits thestability of the neutrinoless double beta decay matrix elements evaluated in quite differentnuclear models, in the case of deformed nuclei. Second, this factor of four, which is littlecompared with the order of magnitude variations in the 2" theoretical estimations, is stil!important in order to extract the parameter (me). In this case the transition is dominatedby a pair of neutrons in the normal parity orbital h9/2 decaying into two protons in theunique orbital h~I/2' again resembling the two neutrino case.

As can be seen in the last two columns of Table 1, the r¿£2 predicted for (m") = 1 eYare at least three orders of magnitude greater than the experimental limits. These re-sults reflect the fact that at, the present stage of the experimental (3(3research, the limits(mv) ~ 1.1 eY obtained by the Heidelberg-Moscow collaboration 1211using significativevolumes of ultrapure 76Ge are the most sensitive. l3ut, if the (3(3ov decay is observedin 76Ge, at least a second observation wil! be essential, ami 150Nd is a likely candidateto do this job [4,18]. In the next few years the limit for (mv) extracted from (3(3ov ex-periments is expected to be improved up lo 0.1 eY and 150Nd is one of the selectedisotopes [221.

Finally we review the two neutrino mode of the douole beta decay (3(32v of 150Ndinto the ground state, the first excited 2+ and the first and second excited 0+ statesof 150Sm150Sm [11].

In Table TI the matrix elements and predicted half-lives for the (3(32v decay of 150Ndto the ground sta te, the first 2+ and the first and second exdted 0+ sta tes of 150Sm arepresented. The matrix elements are given in units of (mec2)-(J+l). It must be mentionedthat the phase space factors differ by about 10% from those presented in Table I [81wherea different renormalization procedure was llsed.

The f3f32" decay to the first excited 0+ state is jorbidden. In this model this is imposedby the fulfilment of an exact selection rule. The pair of annihilation operators ii(04)!' whenexpanded in their SU(3) components, cannot couple the 150Nd g.s. irrep (30, O) to the irrep(20,4) which we assodated with the 6rst exdted 0+ state. Thus the transition betweenmembers of these particular irreps are forbidden.

DOUBLE BETA DECAY IN HEAVY DEFORMED NUCLEI:... 87

The decay to the second excited 0+ state is allowed but strongly cancelled. The predictedhalf-life is four orders of magnitude larger than that of the decay to the g.S. The /3/32vdecay to the 2+ state is inhibited by the JLt dependence of the matrix element as it isdiscussed in Section 3. The matrix element of the /3/32v decay to the lirst excited 2+ stateM2v(2ts,) is three orders of magnitude less than the matrix element of the decay to theg.s. M2v(Ots.l.The present results contradict those previously published [23,24]were it was speculated

that the /3f32v decay of IsoNd to the lirst excited 0+ state of IS0Sm could have a similarintensity of that to the g.s. \Ve found that in the present formalism this decay is forbidden.Ir we select different occupation numbers for both lsoNd and IS0Sm,taken the deformationof the latter nucleus instead of that of the former, we found very similar results for thedecay to the g.s. and the lirst 2+ state, but the matrix elements of the decay to the lirstand second 0+ sta tes becomes interchanged with essentially the same values. Consideringthe difference in the phase space integrals we predict a half live of the order 1021years forthe decay to the lirst excited 0+ state and the decay to the second excited one becomesforbidden.The aboye discussed reduction of the matrix element of the /3/32v decay to the excited

0+ state as compared with the decay to the g.S. is not a general result of the pseudo SU(3)scheme. A recent analysis of the case of lOoMo [251shows that both matrix elementsare very similar and that they are in agreement with the experimental information. Inconclusion, the appearance of selection rules which can produce the suppression of thematrix elements governing a /3/32v transition is a consequence of the details of the irrepsinvolved.

5. CONCLUSIONS

The pseudo SU(3) model is a very powerful machinery to describe the collective behavior ofheavy deformed nucleL It has been used to reproduce very accurately the rotational spectraof heavy deformed nuclei, inclnding the K-band splitting, the amplitudes for transitionsof the E2, MI and M3 type.\Ve have used this model to evaluate the /3f32v half lives of 11 heavy deformed nuclei,

using the pseudo SU(3) approach together with a summation method, which avoids theclosure approximation, obtaining good agreement with the available data and makingtestable predictions, quite different from the QRPA ones in sorne cases.\Ve have also evaluated the /3/3ov half lives of six of these heavy deformed nuclei. \Ve

obtained an order of magnitude agreement with the QRPA estimations, and a factor fourof difference. \Ve exhibited predictions for the neutrinoless double beta half-lives assuming(mv) = 1 eV, and discussed the relevance of our results. In the case of 238U this resultcomplements those obtained for the two neutrino /3/3decay, and it conlirms the observedhalf-life as two neutrino in originoAt last we have studied the f3f32v decay mode of IsoNd to the ground and excited states

of lS0Sm. The /3/32v decay to the lirst excited 0+ state was found forbidden in the modeland the decay to the second excited 0+ state has a half-life four orders of magnitude

88 JORGE G. HIRSCH

greater than that to the g.S. The decay to the 2+ state is strongly inhibited due to theenergy dependence of the matrix elements M2v(2+), two powers greater than that of thematrix eIement M2v(0+).

In all the calculations only one active shell was allowed for protons, and one for neutrons.This is a very strong truncation, of the same type as used in shell model calculations. Animportant consequence of this truncation is the fact that only one uncorrelated Gamow-Teller transition is allowed: that which removes a neutro n from a normal parity state withmaximum angular momentum, and creates a proton in the intruder shell (h9/2 ---> hil/2 inrare earth nuclei, ill/2 ---> ií3/2 in actinides). This unique Gamow-Teller transition controlsthe double beta decay. If the occupation of the Nilsson levels is such that the number ofprotons in the abnormal states does not change going from the initial to the final stateconfigurations, the decay becomes forbidden.

The pseudo SU(3) model uses a quite restrictive Hilbert space. The mode! could beimproved by incorporating mixing between different irreps, via pairing by example [19].Also other active shells can be taken into account in the symplectic extension [271. Inboth cases the selection rules that. impose such strong restrictions to the {3{32v decays ofsorne nuclei can be superseded. However, if the main part of the wave function is wellrepresented by the pseudo SU(3) model, those forbidden decays will have, in the bettercase, matrix elements that will be no greater than 20% of the allowed ones, resulting inat least one order of magnitude cancellation in the half-Jife.

ACKNOWLEDGEMENTS

This work was supported in part by CONACyT under contract 3513-E9310. The articlesreviewed here were done in collaboration with Octavio Castaños, Peter O. Hess and O.Civitarese.

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