Suppression of the neutrinoless ββ decay?

Volume 199, number 4

PHYSICS LETTERS B

31 December 1987

SUPPRESSION OF THE NEUTRINOLESS I][~ DECAY?~ T. T O M O D A J and A m a n d FAESSLER Institut ~r Theoretische Physik, Universitdt Tiibingen, D-7400 Tiibingen, Fed. Rep. Germany Received 2 July 1987: revised manuscript received 13 October 1987

The neutrinoless !313decay rates of 76Ge, 828e, 128. 130Teare calculated in the quasi-particle random phase approximation using a realistic effective NN interaction. The reduction of the 0v1313decay nuclear matrix elements due to ground-state correlations is much weaker than that of the 2v[313decay matrix elements, and we can deduce stringent limits on the Majorana neutrino mass and the right-handed leptonic currents from experimental data on 0v13!3decay.

The property of the neutrinos has been one o f the most important issues in the recent development of particle physics. Since the neutrinoless double beta (0v~13) decay gives practically the only possibility o f distinguishing between Majorana and Dirac neutrinos much effort has been devoted to the problem of 0v13~ decay (for reviews see refs. [ 1-4]). We have shown in ref. [ 5] that a recoil term in the nuclear vector current gives the dominant contribution to the 0viii3 decay caused by a right-handed leptonic current, and deduced a very stringent limit on the relevant coupling strength from an experimental lower bound for the 0v~ [3 decay half-life of 76Ge. Since such an analysis involves evaluation o f nuclear matrix elements, it is important to use a reliable nuclear model. There have been discussions about tests of nuclear wave functions by calculating the experimentally known rates of 1313 decay with emission o f two neutrinos (2v[3~ decay) [ 1-4]. The difficulty encountered in such attempts was that the calculated 2v~[3 decay rates turned out to be too large compared with experimental data by one to two orders of magnitude. Trying to overcome this difficulty, Klapdor and Grotz [6] as well as Vogel and Zirnbauer [7] have shown that the nuclear ground state correlations reduce the 2v~ [3 decay rates considerably. However, their calculated 2v~[~ decay rates for the case o f the heavier nuclei ~2s, 13OTeare still by an order of magnitude larger [6,7] than the experimental data. In a recent work [ 8] we have calculated 2v[313 decay rates in the p r o t o n - n e u t r o n quasi-particle r a n d o m phase approximation ( Q R P A ) [9,7] using a realistic effective interaction, and shown that they are strongly suppressed, in agreement also with the experimental data on ~2s,~3OTe' when a reasonable amount of particle -particle interaction is taken into account. In the present paper we want to discuss the interesting and important problem: Will the 0 v ~ nuclear matrix elements be equally reduced by the mechanism responsible for the suppression of the 2v[$~ matrix elements? If the answer were "Yes", we would obtain limits on the Majorana neutrino mass and the right-handed leptonic currents much less stringent than those deduced in ref. [5], where the ground-state Correlations due to prot o n - n e u t r o n modes are not taken into account. We will calculate the 0vl313 decay rates for the 0+-~0 + transitions of 76Ge~76Se, 82Se~S2Kr, J28Te~ ~28Xe and J3°Te--, 13°Xe, and show that the answer to the question above is fortunately " N o " . We assume the following effective weak interaction hamiltonian density [2,3,5]: Hw = ( G cos O , . / x ~ ) (J'Lt,Y~ ~ + ~Cjl~¢,J~*+ qjR~,Y~* + 2jR~,J~*) + h.c.,

(1)

¢~Supported by the Bundesministerium for Forschung und Technologie under contract number 06 Tfi 778/9. Present address: Swiss Institute for Nuclear Research, CH-5234 Villigen, Switzerland. 0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

475

Volume 199, number 4

PHYSICS LETTERS B

31 December 1987

with G = 1.16637 × 10- 5 G e V - 2, cos 0c = 0.9737, and with the left- and right-handed leptonic currents j~ =0~';'(1 --y,)V~L, j~ =e~'~'(1 +~5)PeR,

(2)

and the current neutrinos 2n P e L = E Ue~Nin, i=1

2n P'eR~--VeiUiR" i=1

E

(3)

Here Ni is the eigenstate of the Majorana neutrino mass matrix with the eigenvalue m~, and n the number of generations. The left- and right-handed nuclear currents are assumed to be [5]

(J°kt.R(X),J[,R(X))=

A3

n=l

"~+n~-~exp(--AIx--rnl) (gv~-gACn,-T-gA~n-t-gvDn),

(4)

with the recoil terms

C. = (p. +p'.) .~rJ2M, D,, = [p. +p;,-ict~e. X (p~ -p'.) ]/2M,

(5)

where p. and p; are the initial and final nucleon momenta, M the nucleon mass and gv = 1, gA = 1.254,/~ = 4.7. The exponential function in eq. (4) is a dipole form factor in coordinate space representing the finite size of the nucleons, and the cutoff m o m e n t u m A = 850 MeV/c is used. The term proportional to K in eq. (1) will be neglected in the following (see ref. [ 5 ]). The 0 + - , 0 + 0v[~[~ decay due to a finite Majorana neutrino mass involves the two nuclear matrix elements defined by [ 5 ]

Mg%~ with r = Ir~ -r21, H(r) = 0 (.4r)/r, A= -E~ + mec 2+ ½Q~. Here E~ and Ej are the energies of the initial and the intermediate nuclear states, Q ~ the Q-value of the ~[~ decay. The form of the function 0 is given in ref. [ 5 ]. On the other hand, the 0 +--,0+0vIiI3 decay caused by the right-handed leptonic current coupled to the left-handed nuclear current [i.e., by the term proportional to t/in eq. (1)] is dominantly determined by the recoil matrix element [ 5 ]

MR(°v)=/4Rltf~ffl'@2( r(°)(r)-~g(l\3meM

~

i(r) ) ) ,

(7)

where A3

V(°)(r) = 6-~n e x p ( - A r ) [1 +Ar+ -~(Ar)2],

(8a)

1( A 2 ~4 V~|)(r) = ~1 J(dkexp(ik.r) ~:\A 2+k2J ,

(8b)

and V~°~(r)~ ( r ) , V" )(r) --, l/r 2 in the limit A--,oo. We neglect the small correction terms eqs. (2.22b)-(2.22d) and (2.25c) of ref. [5]. In eqs. (6) and (7), the nuclear matrix element of a two-body operator O~2 is defined as

476

Volume 199, number 4

PHYSICS LETTERS B

31 December 1987

(O~2 > = 12pc,,,,, ~ ( 0 + II[c~,gn,](P'"')J[IJ~- > (J~-IJ~ > (JTII. [c*~g~](P")JIIOi~ > J,jlcJ'

X(_)n+p.+j+j,(2j,.jt_l) {pn' p't7 J'J }

(PP'J'r~r+O121nn'J')'

(9)

where ~j,,,= ( - )J+'"cj .... If both sets of the intermediate states { ]J~ > } and { IJ~ ) } are complete, eq. (9) re+ O,m ]0 + ). We calculate the matrix elements of the one-body transition denduces to (O12) = (0~- I ~1 ~ , , , ~ + ~,,, sities in the QRPA formalism [9] and obtain (J7 [I[ c*pg~]
[c~,Cn,]°""')J]lJ'~) - - ~

( upv~x~,j + vvu,yJ~,j),

(10a)

1 (OplJnXf~,1,.+UpV-,,fJp~j,.),

(10b)

where u and v are the quasi-particle transformation coefficients, xp,,,j J" and yp~,j J" the forward and backward QRPA amplitudes of the jth J~ state of the intermediate odd-odd nucleus; the barred quantities indicate that quasiparticles and phonons are defined with respect to the final ground state 0~-. The overlaps between two J~ states belonging to two different sets are given by

(j~lj7)

~ tI. -Y Jp,n . k - ~J~ ~ pn.j

(11)

- - Y pJ'~ n . k . V pJ'~ n . j "~ l .

pn

The half-life rl/2 0~ for the 0vl313 decay can be expressed as [5] (r%)-'

~o~ ( ( m,,)lme) = C ......

~°), ( ( m ~ ) / m e ) + 2 C ....

~ " J I - C ( A ~ ) ( 2 ) 2 " ~ - C ~ q~o) (

rl) 2 + 2C}°2( ( mv)/m~) ( 2 )

( r / ) +2C~ °) (2 ) ( r / ) ,

(12)

where

( m ~ ) = ~ ' m ~ U 2 , ., i

(2)=2~'U~,Vo,, i

(~l)=q~'U~,V¢,,

(13)

i

and r~ m~(°) (see eqs. (3.1) and (A.2) of ref. m ~ etc. consist of the products of electron phase space integrals L-~o) i jk [5]) and the nuclear matrix elements M ~ ), Zv, 26T, V, )~(I'T,F,T,P,R. Here

Zv = ( g v / g A ) 2 M b ° ~ / M ~ ~, Z~ = ( g v / g A ) M f ~ ° ~ ' / M ~ ~,

(14)

and see eqs. (2.13) of ref. [5] for the definition of the other nuclear matrix elements (relative to M ~ ~) 2~v,v and X~r,v,r,p. In the present work we calculate all these nuclear matrix elements except Z+, the contribution of the latter being negligibly small [ 5 ]. The electron phase space integrals F~°~ are calculated using the solutions of the Dirac equation with the finite size of the nucleus taken into account (see the case " W F I " described in ref. [ 5]) and listed in table 1 together with the experimental values of Q~. We perform a QRPA calculation using the nuclear matter G-matrix derived from the Bonn potential [12]. We introduce the parametrization gp,~ " G (o~=p, n), gphG and gppG, for the pairing, particle-hole and particle-particle interactions, respectively. The same model space and values of parameters as in ref. [ 8 ] are used in the present calculation. The readers are referred to ref. [8] for the method of determination of gpair, p g~i~, and gph. These parameters are kept fixed in the following and only the particle-particle interaction strength gpp is considered as a variable. In contrast to ref. [ 8], the excited states of the intermediate odd-odd nucleus are not restricted to those with J ' = 1 +. We take all possible J " which can be formed with proton-neutron two quasi-particle states and use the same gp~ and the same gpo for all these multipolarities. The short-range repulsive NN correlations are treated in the same way as in ref. [ 5 ]. Fig. 1 shows the calculated nuclear matrix elements M ~ ) , v and M~ t°~) for the OvlB[3 decay of 78Ge as functions of gr,p. The solid lines are the results of summation over all multipolarities in the intermediate nuclear 477

Volume 199, number 4

PHYSICS LETTERS B

31 December 1987

Table 1 Phase space integrals U~" calculated for the 0vfllB decays of 76Ge, 82Se, L28Te and ~3°Te using the experimental values of QI~ given in the last line.

jk F}2 > [10 t3y i f m2 ]

11 33 44 55 66 13 16 14 34 15 56

6.697 11.11 3.363 6241 59.90 - 1.838 -9.843 2.581 -4.927 97.61 -611,3

Qf~l~(exp) [MeV] ") Ref.[10].

82Se

76Ge

2,041 ~>

t28Te

30.63 106.5 22.78 30940 262.1 -10.37 -35.13 12.95 -42.43 365.9 -2847 2.995 a)

2.701 0.8353 0.6946 4550 27.26 -0.3568 -5.725 0.7838 -0.5163 73.28 -352.2 0.868 b)

t3°Te 66.40 176.9 42.46 104300 573.4 -20.75 -81.15 27.29 - 72.78 1062 -7734 2.533 b)

h) Ref. [ l l ].

states, whereas the dashed lines represent the contributions from the indicated multipolarities. Since the neutrino emitted from (or absorbed by) a nucleon is a virtual particle in the case of 0v]3~ decay, the typical momentum transfer from the nucleon can be as large as the Fermi momentum of the nucleon. It follows that the angular momentum transfer can be also large and there are considerable contributions of the virtual single [3 transitions to and from the intermediate nuclear states with J ~ # l + ( J ~ # 0 +) for the matrix elements M~°~ ) and M~°V)(M~°V>).

1.0

I

I

I

I

50

I

I

I

£ "-x

0.8

&

i: t

4o~

°~,,°~O.6

30

0.4

20

0.2 R

10

(1+cont.)

..... ,

......................

0.0

I

~

,,

i....... (">",I....

V (O+co~t)

........ ":( -

-0.2

-0.4

i

0.0 0 2

i

0.4

i

t

0.6 0.8

-20

i

1.0

g~ 478

-10

1.2

Fig. I. The calculated nuclear matrix elements M~;T, or)~. and M~ (°') for the 0v13!3 decay of ¢6Ge as functions ofgpp.

Volume 199, number 4

31 December 1987

PHYSICS LETTERS B

Table 2 The calculated nuclear matrix elements with gpp= 1 for the 0v[313decays of 76Ge, 82Se, 128Teand ~3°Te.

M ~ ' [fro-~ ]

Xv )CGv 2r X('~v X~ X~ X~

76Ge

82Se

0.330 -0.290 0.951 -0.262 1.049 -0.318 - 0.485 70.3

0.293 -0.286 0.952 -0.258 1.048 -0.314 - 0.525 71.2

~-~STe

~3OTe

0.246 -0.296 0.951 -0.266 1.049 -0.326 - 0.491 78.5

0.2i2 -0.299 0.948 -0.268 1.052 -0.331 - 0.496 79.6

The contribution to M~,°~) from intermediate 1 + states decreases as gpp increases, and vanishes at gpp~ 1 as in the case o f the 2v13j3 decay matrix elements (see ref. [8]). The ground-state correlations induced by the p r o t o n - n e u t r o n 1 + mode are relatively weak at gpo = 0, but they become stronger as we increase gpp, and finally the contribution of the second term in eq. (10b) cancels that of the first term completely. On the other hand, the contributions to M~-°~) from higher multipolarities such as 2 - , 3 +, 4 - do not decrease so rapidly as that of 1 + states, reflecting the slow growth of ground-state correlations induced by these higher multipolarity modes. Since the total matrix element M~°~ ~ is obtained by summing over all these contributions, its dependence on gpp becomes much weaker than that of the 2vl313 decay matrix elements. The present value o f M~°~ ~ at gpp=0 is larger than M~°~ ) = 0 . 5 6 3 fm -j obtained in ref. [5] mainly because a larger model space is employed in the present calculation. It becomes smaller than the latter when the particle-particle interaction is turned on. Although the particle-particle interaction was taken into account in ref. [ 5], the ground-state correlations due to p r o t o n - n e u t r o n modes were absent in the variational trial wave functions. The matrix element M~ (°v~ is dominated by the contribution from V (°) (r), the range o f which is much shorter than that o f H ( r ) involved in M~-~ ~. In this case the fraction of the contribution from the intermediate states with higher multipolarities becomes larger. And even for the contribution of intermediate 1 + states, the cancellation between the two terms in eq. (10b) becomes incomplete because the virtual single 13 transitions which change the orbital q u a n t u m numbers become more important. Consequently the matrix element M~ (°v) decreases much more slowly than M~°~ ~ as a function of gpp. The matrix element M~ °v~ also shows a moderate decrease in magnitude after summing over all possible natural-parity intermediate states. Since it has been shown in ref. [8 ] that the experimental half-lives for the 2uflfl decays of the four nuclei under consideration are all consistent with the calculation with gpp= 1, we use this value in the following. Table 2 shows the calculated 0vl313 decay nuclear matrix elements with gpp= 1. Using these nuclear matrix elements and the phase space integrals of table 1, we calculate the coefficients C},°,~,, etc. o f eq. (12) using eq. (2.17) of ref. [ 5 ]. Comparing eq. (12) with experimental lower limits for T°y2, we obtain upper bounds for , <2> and . Such an analysis is summarized in table 3. While the 76Ge [ 13] and 82Se [ 14] data are taken from counter experiments, the geochemical decay rates [ 15 ], which are the sum o f 0v and 2v1313 decay rates, are used as upper limits for the 0v1313 decay rates for ~28.~3OTe"Fig. 2 shows the allowed regions deduced from the experimental data for several fixed values o f <2>. Comparing the limits on < my), <2> and < ~/) listed in table 3, we obtain the most stringent limits [ I < 1.9 eV and I I < 1.8× 10 -8 from the 128Te data and 1(2)1 < 3 . 6 × 10 -6 from the 78Ge data. In summary, the reduction of the 0v1313 decay nuclear matrix elements due to ground state correlations is much weaker than that of the 2v1313 decay matrix elements, and we can deduce stringent limits on the Majorana neutrino mass and the right-handed leptonic currents from the experimental data on 0v1313 decay. 479

Volume 199, number 4

PHYSICS LETTERS B

31 December 1987

Table 3 The coefficients C~°), the experimental 0v[313decay half-lives flY2, and the upper limits on the Majorana neutrino mass (my), and the right-handed current coupling strengths (2) and ( t/).

C~,°,I, [ y - ' ] C},~! [y-l] C~O,] [y t] C~ ~ [y ~] C}~°) [y ~1 CI °) [y '] rlY2(exp) [y] I Km,DI [eW] Ifk)l IKr/>l ~1 Ref.[13].

76Ge

82Se

128Te

] 3OTe

1.2IX10 -~3 _3.02X 10-14 1.04X10-[, 174xl0 -13 3.69Xl0 -9 -5.72X10 -~4 >4.7 X I 0 ?3a) <2.5

4.36X 10 -t3 _ 1.34x 10-,3 2.98X10-~1 1.31XI0 -t2 1.33x10 -8 -4.73X10 -t3 >1.1 X102abl <8.2 <8.5 XI0 6 <9.0 XI0 -8

2.73X 10-t4 _3.24X 10-~5 3.79X10-~ 7.27X10 -15 1.18x10 -9 -1.66X10 -'5 >5 X10 ~4~1 < 1.9

5.05X 10-~3 _ 1.44X 10-~3 4.09x10-~ 1.16x10 -12 1.92x10 8 -3.85X10 ]3 >1.5 X1021c) <21 <2.4 X10 -~ <2.1 XI0 7

<3.6

X I 0 -6

<2.8 X10 -8

<5.5 X I 0 -6

<1.8 XI0 -8

b>Ref.[14]. °Ref.[15].

-3

/

.,"

~

-r,=p> 4 . T x ' ~ y

-...

- 2

bc,X>=lxlO-6

t- °'°°

I,~,,~& ( . . , ~i. 2 1X% " 2 ~I

-1 \ \

'~\\ \ I

-3

-2

-1

0

1

2

<,','v> ~

""

-8

--6

-4

-2

I

3 lOa
-2

0

",,,,

'~,\,\

\,

\.,.~

',,

\ \\\

i

-1

\ i\\\ 1

0

25

LX<.X>=2x10''6

-r~>LSx~y

20

!'

.. /

f

o

',

i <~=20x10 ~

,}

,

2

4

\, ',,',~ 6

8

i 2

t-~-e _, t~Xe -

" , , ",,N, • " " \ N,,NN -4 ... "..

I

-10

I

,,

.,r.~>l.lxlO~y

-8

/

\

a~_.~ _. ,n~,.

- 10

i

-%.

',,,'.,, \

'X,X

r=~>~
\ ,

I

10

10e
-25-20

I

\ \\\

5

10

%

t;" t,.

",a

-15

-10

-5

0

\S

t

15

i

I

20 25 lOe~rp

Fig. 2. The allowed regions (inside the ellipses) deduced from the experimental bounds [ 13-15 ] on the 0v1313decay half-lives of 76Ge, 8-'Se, '28Teand ]3°Te, for several fixed values of (2). 480

Volume 199, number 4

PHYSICS LETTERS B

31 December 1987

Note added. A f t e r c o m p l e t i n g this work, we r e c e i v e d a p r e p r i n t by Engel et al. [ ! 6 ] w h i c h treats a s i m i l a r p r o b l e m b u t w i t h q u i t e a d i f f e r e n t c o n c l u s i o n . U s i n g a m o r e p h e n o m e n o l o g i c a l z e r o - r a n g e N N i n t e r a c t i o n , they p e r f o r m e d a Q R P A c a l c u l a t i o n a n d o b t a i n e d M~.q~)v.T h e y f o u n d a drastic r e d u c t i o n o f these m a t r i x e l e m e n t s w h e n b o t h the p a r t i c l e - p a r t i c l e i n t e r a c t i o n a n d s h o r t - r a n g e N N c o r r e l a t i o n s ( S R C s ) w e r e t a k e n into a c c o u n t , w h i l e the r e d u c t i o n was m o d e r a t e if S R C s were neglected. In the p r e s e n t w o r k S R C s h a v e b e e n t a k e n into acc o u n t as was m e n t i o n e d before, a n d we d o n o t find such a drastic effect o f SRCs. A n u m e r i c a l c a l c u l a t i o n shows that the typical t w o - b o d y 0v[313 t r a n s i t i o n m a t r i x e l e m e n t ((;g0fs/2) 2 J = 01T + T+ O"1 -0"2H(r) ] (pOg9/2)2J=O) has the v a l u e 0.162 f m - ~ (0.199 f m - ~) w i t h ( w i t h o u t ) SRCs. A l t h o u g h the S R C s r e d u c e i n d i v i d u a l t w o - b o d y m a t r i x e l e m e n t s differently, t h e y r e d u c e d the total m a t r i x e l e m e n t s l a~(ov) 20% in the p r e s e n t v z G T , F also by a b o u t c a l c u l a t i o n as well as in the c a l c u l a t i o n s o f refs. [5,17].

References [ 1] H. Primakoffand S.P. Rosen, Annu. Rev. Nucl. Part. Sci. 31 (1981) 145. [2] W.C. Haxton and G.J. Stephenson Jr., Progr. Part. Nucl. Phys. 12 (1984) 409. [3] M. Doi, T. Kotani and E. Takasugi, Progr. Theor. Phys. Suppl. 83 (1985) 1. [4] J.D. Vergados, Phys. Rep. 133 (1986) 1. [5] T. Tomoda, A. Faessler, K.W. Schmid and F. Griimmer, Nucl. Phys. A 452 (1986) 591. [6] H.V. Klapdor and K. Grotz, Phys. Lett. B 142 (1984) 323; K. Grotz and H.V. Klapdor, Nucl. Phys. A 460 (1986) 395. [7] P. Vogel and M.R. Zirnbauer, Phys. Rev. Lett. 57 (1986) 3148. [ 8 ] O. Civitarese, A. Faessler and T. Tomoda, Phys. Lett. B 194 (1987) 11. [9] J.A. Halbleib and R.A. Sorensen, Nucl. Phys. A 98 (1967) 542. [ 10] R.J. Ellis, B.J. Hall, G.R. Dyck, C.A. Lander, K.S. Sharma, R.C, Barber and H.E. Duckworth, Phys. Len. B 136 (1984) 146. [ 111 A.H. Wapstra and G. Audi, Nucl. Phys. A 432 (1985) 55. [12] K, Holinde, Phys, Rep. 68 (1981) 121. [ 13] D.O. Caldwell, Proc. Intern, Conf. on High energy physics (Berkeley, July 1986). [14] S.R. Ellion, A.A. Hahn and M.K. Moe, Proc. Intern. Syrup. on Weak and electromagnetic interactions in nuclei (July 1986), ed. H.V. Klapdor (Springer, Berlin, 1986) p. 692. [ 151 T. Kirsten, E. Heusser, D. Kaether, J. Oehm, E. Pernicka and H. Richter, Proc. Intern. Syrup. on Nuclear beta decays and neutrino (Osaka, Japan, June 1986), eds. T. Kotani, H. Ejiri and E. Takasugi (World Scientific, Singapore, 1986) p. 81. [ 16] J. Engel, P. Vogel and M.R. Zirnbauer, California Institute of Technology preprint MAP-95 (June 1987). [ 17] W.C. Haxton, G.J. Stephenson Jr. and D. Strottman, Phys. Rev. D 25 (1982) 2360.

481