Double beta decay in the generalized seniority scheme

Nuclear Physics A507 (1990) 239c-244c North-Holland

239c

D O U B L E B E T A D E C A Y IN T H E G E N E R A L I Z E D S E N I O R I T Y S C H E M E

Stuart PITTEL ~ Bartol Research Insntute, Unt'~ersltv of Delaware, Newark, DE 19716

J o n a t h a n E N G E L ' and Petr V O G E L ' N o r m a n Bridge Laboratory of Ph?mes, California Institute of Technology, Pasadena, CA 91125

Xmngdong J I W K 91125

Kellogg Radiation Laborator~

California Institute of Technologv, Pasadena, C-K

-k generahzed-semorltv truncation scheme is used ,n shell-model calculatmns of double beta decay matrix d e m e n t s Calculations are carried out for 76Ge. 828e and 128'la°Te Matrix elements calculated for the two-neutrino decay mode are small compared to v~eak-couphng shell-model calculations and support the suppression mechanism first observed in the quas> particle random phase approximation Matrix elements for the neutrinoless mode are similar to those of the weak-couphng shell model, suggesting that these matrix elements can be pinned down fairly accurately

1

INTRODUCTION In recent ?ears. there has been renewed interest in the use of neutrlnoless (0v) double beta

(33) decay to extract lntormatton on the mass of the Majorana neutrino

In order to relate the

neutrino mass to decay rates, it is necessary to haxe rehable calculations for the reletant nuclear matrix elements Several techniques have been used to calculate these matrix elements and their two neutrino (2v) ~ounterparts ~lth results ot uncertain accuracr l - a

\ n issue of par-

ticular Importance IS the apparent strong suppression ot the la°Te 2v decay relative to that of S2Se \ mechanism for this strong suppression was recently suggested in the Framework of the quaslpartlcle Random Phase a~pproximatlon (QRPA) 1

The QRPa~ has several shortcomings,

howexer Its equations do not conserve particle n u m b e r and, more Importantly, suffer an mstabtllt 5 for nuclear forces not much stronger than those expected from microscopic constderatmns Such problems would not arise in a ~hell-model treatment, which instead is limited by the need for truncation

Typically, truncations have been carried out In a weak-couphng framework 2

§ ~ o r k supported In part by NSF G r a n t # PHY86-17198 * Work supported in part b~ DOE Contract # DE-AS03-81ER40002 - Work supported in part by NqF G r a n t # s PHY85-05682 and PHY86-04197

0375-9474 / 90 / $3 50 © Elsevmr Science Publishers B V (North-Holland)

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S Ptttel et al I Double beta decay tn the generahzed semortty scheme

~hlch of necessity leaves out some of the orbltals t h a t the Q R P A suggests pla? an important role m these decays

In this work, we consLder an alternatwe shell-model truncation scheme

which p e r m n s us to incorporate the essential physics of the QRP-k without suffering from its drawbacks 4 The basis of our truncation ts the generalized s e m o n t y scheme 5

2

THE G E N E R A L I Z E D S E N I O R I T Y SHELL MODEL To better motivate our work, we first review briefly the physics of QRPA-mduced suppres-

stun The Q R P A method begins with BCS calculations for the m m a l and final nuclei Without any further modlficatmn, the BCS p m r m g serves to open up the 2u deea), which in zeroth order is P a u h blocked The amplitude is then reduced b~ RPA correlations, which effectively mix 4,8, etc q u a s l p a m c l e states into the quaslpartlcle vacua In the generahzed senlorlt) approach, we start b) replacing the BCS vacuum by a generallzed s e m o n t y zero (w = 0 )

state m~ol~lng a condensate of J = 0 + S pairs for neutrons and pro-

tons VlZ

I~', = where (p =

~'. = o > = ( s ~ ) " " (s~) ~ I o >

(1)

;,r U) _ * =

~

~ + i

+

J

and the c~f are variational coefficients to ~hlch we return shortl:~ The quaslpartlcle correlations generated by the QRPA correspond to proton-neutron pairs, In lowest order the R P ~ equations mix two of these pairs Into the ground states of the initial and final nuclei

In the generalized seniority framework, excitations In, oiling two neutron-

proton pairs can be expressed in the general form

(3) klrnn ,L

The state (3) {after projecting out w-----0 pieces} has generahzed seniority 2 for both neutrons and protons

-k similar procedure can be used to build states of higher generahzed ~enlont~,

which are analogous to QRP~k e,ccltatlons with greater numbers of quaslpartlcles nuclei

involved in the

double beta

deca?

processes under

Since the

consideration are not strongly

deformed, a truncation to IV.r,u __~ 2 should provide the dominant configurations for a descrlptmn of their ground states

Thus m the calculations reported here, we restrict our shell model barns

to all configuratmns of the form {1-3) We are currently investigating the posslblht:, of including (at least approximately) the effect of w = 4 configurations To fully speclf~ our generalized ~enlorlty barns we must first, determine the appropriate o 7 and aft structure coefficients that define the d o m i n a n t S pairs

To accomphsh this, we use a

number-conserving variational procedure, whereby the pairs are chosen so as to minimize the expectation value of the Hamlhonlan for a w . = w~ = 0 trial wave functmn (1,2)

S Ptttel et al I Double beta decay m the generahzed semonty scheme

241c

To evaluate the shell-model Hamlltonlan matrix m our [ow-generahzed-seniorlty basis, we use formulae 6 derived earlier in the context of the Interacting Boson Model Large single-particle spaces can be accommodated in these calculatmns, so that potentially important spin-orbit partners need not be discarded

3

CALCULATIONS }~¢e have apphed our method to the 33 decay of 78Ge, 82Se, t28Te and 13°Te In the two

4 ~

80 nuclei, we use a single-particle space comprised of the full lf-2p shell plus the lg9. 2 and

lg7 2 orbltals, for both neutrons and protons

In the Te calculations, we include the 3s-2d-lg

shell and the l h l h 2 and lh 9 2 orbltals Ideally, we would like to choose our single-particle energies from experiment (at the nearest d o u b b magic core) and our effeeu~e interaction from an appropriate realistic force Regrettably. this is not feasible ~hen

using so large a single-particle space

The problem is that

the

monopole-monopole component of the n-p force strongly modifies the effective spherical smgleparuele energies as we proceed through a shell, thus It is i m p o r t a n t that the input singleparticle energies and the monopole-monopole interaction be chosen consistently Unfortunatel), monopole n-p interactions obtained from reallsUe G matrices do not properly reproduce singleparticle systematlcs [n the absence of a more consistent mtcros(ople input procedure, we have adopted the following prescription for our effective Hamtltonlan Our single-particle energies are extracted from a Woods Saxon potential 7, parametrlzed for the nuclei of interest Our effective two-body interaction is obtained from a p a r a m e t n z e d fit to a Reid-soft-core G matrix s

How-

e~er, to avoid double eounung, we turn off the monopole-monopole contribution to the n-p interaction

4

RESULTS AND DISCUSSION Our results for the 2~, matrix elements, calculated in closure approximation, are s u m m a r -

Ized m Table 1 The first point to note is t h a t our 2v calculations produce smaller matrix elements than m the earher weak-coupling shell-model calculations

Furthermore our calculations

suggest a re[auve suppresston of the 13°Te deca) compared to that of g2Se, although not as

2~ Table 1 Calculated Matr,x Elements I MaT(cl) I for the 2u Mode

76Ge

82Se

128Te

130Te

Present ~ o r k

1 14

0 90

0 87

0 68

Weak-couphng shell model 2

2 6

19

2 9

30

S Ptttel et al I Double beta decay m the generahzed semortty scheme

242c

much as is requ,red by experimental data Our calculatmns s u p p o r t the cancellatmn mechanism first observed in Q R P A calculations 1 In 13°Te, for example the to-~-0 ~

w~-0

pairing contribution is -2 82 t,h e w = 2

contrtbutmn is -073, and the to = 0

and t 0 = 2

~

to=2

generalized ,emorlty-changmg cont~ributmn is

+2 57 The t ~ o ,ets of shell-model results s h o ~ n in Table 1 are both based on fairly realistic calculations

The fact that they differ so dramatically suggests that, at present, t,he 2v closure

matrix elements cannot be reliably calculated Next, ~e consider the 0v mode, for which our results are summarized in Table 2 The same cancellation mechanism as described abo',e for the 2u mode is operative here too, but is much less effectixe

-ks a consequence, our 0v matrix elements are fairly large and in good agreement

~lth the weak-couphng results

This suggests that the matrix elements rele'~ant to the neutri-

noless mode can be determined much more reliably than those for the 211 mode

Table 2 Calculated Matrix Elements I ~[g} - 3[,~" [ for the 0u Mode

76Ge

82q e

128Te

t30Te

Present work

50

3 7

6 3

57

Weak-couphng shell model 2

5 5

4 i

67

68

Finally, if" we include the results of the QIRPA calculations in our discussion, the uncertainties in both sets of matrix elements increase

Nevertheless the same basic conclusion ~tlll holds

that the 0u matrix elements can be pinned down much more accurately than the 2v matrl'~ elements Clearly, some improvements in our anabs~s are still needed

Perhaps the two most Impor-

tant are a) an improved input H a m d t o m a n and b) the incorporation of u' = 4 admixtures The limitations of our calculations notwithstanding ~e believe that the generalized seniorlr~ shell model proxldes an appropriate framework for calculations of double beta decay matrix elements

Double beta deca~ remains the most sensitive probe of' the Majorana neutrino mass

and a reliable determination of the relexant matrix elements merits continuing effort

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S Plttel et al I Double beta decay tn the generahzed semonty scheme

243c

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