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Suppression of the two-neutrino double β decay
Volume 194, number 1
PHYSICS LETTERS B
30 July 1987
S U P P R E S S I O N OF THE T W O - N E U T R I N O DOUBLE [i DECAY "~ O. CIVITARESE ~,2, Amand FAESSLER and T. TOMODA Institut J~r Theoretische Physik, Universiti~t Tiibingen, D-7400 Tiibingen, Fed. Rep. Germany
Received 12 March 1987; revised manuscript received 24 May 1987
Two-neutrino 1513decay rates of 76Ge, S~Se,~28.t3OTeare calculated in the quasi-particle random phase approximation using a realistic effectiveNN interaction. The decaysare stronglysuppressed, in agreementwith the experimentaldata, when a reasonable amount of particle-particle interaction is taken into account.
The nuclear double beta (1313) decay has been attracting considerable attention in recent years [1-4]. The principal interest in the process originates from the possibility of distinguishing between Majorana and Dirac neutrinos by observing a neutrinoless (0v) 1313decay. In order to relate the neutrino mass and the coupling strengths of right-handed leptonic currents to 0v1313 decay rates [5], a reliable calculation of nuclear matrix elements is necessary. A comparison of a 0v1313 decay process with a pion double charge exchange reaction may be helpful in reducing ambiguities in the nuclear models [6,7], although there are at present only few experimental data with a 13~ decaying nucleus as a target. Tests of nuclear wave functions by 1313 decay with emission of two neutrinos (2v1313 decay) have been discussed more intensively elsewhere [ 1-4]. The difficulty encountered in such attempts was that the calculated 2v~13 decay rates turned out to be too large compared with experimental data by one to two orders of magnitude [ 2,4,8,9 ]. Theoretical estimates of nuclear maxtrix elements for 2v1313 decay have been reported recently [ 10, l l ]. Klapdor and Grotz [ 10] performed a particle number projected BCS calculation with a schematic spin-isospin as well as a quadrupole-quadrupole ~"Supported by the Bundesministerium far Forschung und Technologie under contract number 06 TiJ 778/9. Permanent address: Department of Physics, University of La Plata, 1900 La Plata, Argentina. 2 Fellowof the CONICET,Argentina.
residual interaction. They have found that ground state correlations due to quadrupole modes in the initial and final nuclei reduce the 2v1313 decay rates by a factor 1.2 to 7. A different mechanism of suppression of 2v1313 decay has been proposed by Vogel and Zirnbauer [ 11 ]. They have used the proton-neutron quasi-particle random phase approximation (QRPA) [12] to calculate the matrix elements connecting both initial and final 0 ÷ ground states with each 1 ÷ state of intermediate odd-odd nuclei [13-15]. In contrast to previous treatments of 1313decay using the QRPA formalism they also took into account a particle-particle interaction. They found a reduction of 2v1313 decay rates by a factor 3 to 25 using a zero-range spin-isospin interaction where the strength of the particle-particle matrix elements were taken to be ~ 2/5 of the corresponding particle-hole ones. In the present paper we want to discuss whether the suppression mechanism found in ref. [ 11 ] persists when a more realistic effective interaction is used. We also briefly consider the relationship between the two mechanisms proposed in refs. [10,11]. The half-life z~/2 for 2v13~ decay can be given in a factorized form involving leptonic and nuclear parts
[3]
as
(rl/2)--I = F I M G ' r I ~ ,
(l)
where F is a lepton phase space integral and
0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
11
Volume 194, n u m b e r 1
PHYSICS LETTERS B
,~ 0"81
MGT
0
_ ~ ( 1 , + I1 + >(1 +llv+~rllO~ + > jk me c2 + ½Q~ + Ej-E~ (2) Here Q ~ is the Q-value for the 1313decay. The matrix elements of z +o" in the QRPA can be written [ 13 ] as
. .
6
.
.
=Y~ (pllalln)(UpV,~Xp.j+VpU.yp.j), pn
pn
0.2
(3a)
""'-)).\"~ "%\,,~, - -----
76Ge-~ 76Se 82Se-'* 82Kr
\~,\ l/
. . . . . . . . 130 Te --'130Xe
/
I O~6
(3b)
where Xp,,j and yp,j are forward and backward going QRPA amplitudes o f t h e j t h 1 ÷ state of the odd-odd nuclei; the barred quantities indicate that quasi-particles and phonons are defined with respect to the final ground state 07-. The overlaps between two 1 ÷ states belonging to two different sets, ( 1~ I I f ), are given by
( 1; I1 ,+ ) = E (xo,,*xp,J-.fp,,,~Yp,J) •
.
04. . . . . . . . . .
-0.2
( 1 E I I r tr[10?- )
= Z (Pllcrlln)(Vpa,.¢p,.k +aoVr, yp,,,D ,
.
~
o.o
<1~+ IIr+~llO~+ >
(4)
pn
We have neglected in eq. (4) a factor ( 0 ~- I0~ ) [ 14 ] which would further reduce the overlaps by several percent. We calculate the 2vl313 decay half-lives for the following 0 + --*07" transitions: 76Ge---~76Se, 825e---,82Kr, J28Te--. ~28Xe and 13°Te---.13°Xe. We take the full 3ho9 and 4ho9 major oscillator shells for the description of the nuclei with A = 76, 82 and the model space consisting of 1P3/2,i/2, the full 4ho9 shell, Oh ~~/2.9/2and 1f7/2.5/2 subshells for A = 128, 130. The single-particle energies are calculated with a Coulomb-corrected Woods-Saxon potential. As a realistic two-body interaction we use the nuclear matter G-matrix calculated from the Bonn one-boson-exchange potential [16]. In order to take account of a renormalization for finite nuclei a relatively small (in absolute value) starting energy of - 2 5 MeV is used in solving the Bethe-Goldstone equation and in addition the parametrization, gp"a~rG ( a = p , n), gphG and gppG, is introduced for the pairing, parti12
30 July 1987
- 0.8
0.0
I
i
i
I
0.2
0.4.
0.6
0.8
i
1.0 1.2 gpp
Fig. 1. The calculated 2v[313decay matrix elements MOTas functions of the strength gppof the panicle-particle interaction. cle-hole and particle-particle interactions, respectively. First, the BCS equation is solved and the strengths gPair and g~,air of the proton and neutron pairing interactions are determined so as to reproduce the experimental odd-even mass difference for each nucleus. They vary for the different nuclei in the n ~<1.20. Then range 0.93 ~
Volume 194, number 1
PHYSICS LETTERS B
increase gpp, they become stronger and finally the contribution of the second term in the parentheses on the right-hand side of eq. (3b), which involves a backward-going Q R P A amplitude, becomes as large as that of the first term and cancels this completely. In contrast to the results of ref. [ 11 ], where MOT vanishes at gpp~ ~ ~gph, l a complete cancellation in the present calculation takes place at gpp~ 1, i.e. at gr,o.~gr,, within 30% (see above for the values ofgph), which means that the particle-particle and particle-hole interactions are related to each other approximately by a simple recoupling of the angular momenta. This is a nice feature of the present calculation using a realistic interaction in view of the experimentally known suppression o f 2vl3~ decay rates. Although in principle particle-particle and particle-hole interactions undergo different renormalization in the nucleus, they are not expected to differ very much from each other. It is noted that the values of MOT at gpp = 0 in ref. [ 11 ] are smaller than the present results by factors 2 to 3. One of the reasons is that the MOT in ref. [ 11 ] has been calculated by replacing the final state Of~ by 0~, or 0i+ by 07-, in eq. (2) and taking the average of these two matrix elements. This simplified treatment yields a smaller value for MOT. Second, a zerorange spin-isospin interaction which gives the same excitation energy for the G T G R as the realistic interaction used in the present calculation, yields weaker 13- and 13+ strengths for low-lying 1 ÷ states. This also causes MOT to be smaller. Fig. 2 shows the 2vl313 decay half-lives rl/2, eq. (1), as functions o f gpo. We used the free nucleon value gA= 1.254 o f the axial vector coupling constant for the calculation of the phase space integral F. Experimental half-lives [ 1 8 - 2 2 ] are indicated either by hatched regions or, in the case of lower limits, by lines with arrows. It is seen that all the experimental data are consistent with the calculated half-lives with gpp ~ 1. From another point o f view, it is rather difficult to predict the half-lives since they are very sensitive to the value of gpp in the neighborhood of gop = 1. A possible solution might be to compare the calculated 13÷ strength functions with experimental data on the (n,p) reaction using the 1313decay daughter nuclei as targets, or data on single 13- decay from the 1 ÷ states o f the intermediate o d d - o d d nuclei to the ground states of the 1313decay daughters. Unfor-
30 July 1987
1023
1022
ucI
1o
\\\\II\
\ \ \ \
\\
1020
76Ge.76Se
82Se~82Kr lg
10 O0
02
Or,
06
08 •
10
12
10 00
02
0/,
06
08
1.0 gpp
12
02
04
05
08
10 gpp
12
gPP
1025
•
UM
102 ~
1023
1022
1°~00 0'2 0'~ 46 Q'8 1'0 1.2 gpp
lO
O0
Fig. 2. The calculated 2vl313decay half-lives ~,/2 as functions of gpp.Experimental half-livesare indicated either by hatched regions or lines with arrows (for lower limits), and taken from the following sources: UCSB-LBL,ref. [ 18]; USC-PNL, ref. [ 19]; UCI, ref. [20]; UM, ref. [21 ]; MPIH, ref. [22]. tunately there are only few such data at present. In order to estimate the amount of the groun d state correlations (g.s.c.'s), which are essential for the suppression o f MeT, one can alternatively start from a Q R P A equation for the quadrupole mode and first determine the g.s.c.'s due to the quadrupole mode. Out of this correlated ground state, one obtains that portion of g.s.c.'s which is relevant to the pn 1 + mode by angular m o m e n t u m recoupling, and then one can consider modifications of these g.s.c.'s due to the pn 13
Volume 194, number 1
PHYSICS LETTERS B
i n t e r a c t i o n in the 1 + c h a n n e l . Such a p r o c e d u r e was a d o p t e d in ref. [ 1 0 ] . H o w e v e r , i f we start d i r e c t l y f r o m the Q R P A for the p n l + m o d e as in ref. [ 1 1 ] or in the p r e s e n t work, all the g.s.c.'s p e r t a i n i n g to this m o d e are a u t o m a t i c a l l y t a k e n i n t o a c c o u n t a n d we d o not h a v e to w o r r y a b o u t the effect o f q u a d rupole, o c t u p o l e , or w h a t e v e r m o d e o n the g.s.c.'s r e l e v a n t to the pn 1 + m o d e . T h e f o r m a l i s m as well as n u m e r i c a l c a l c u l a t i o n s b e c o m e m u c h simpler, a n d we can get a clearer insight i n t o the m e c h a n i s m o f the s u p p r e s s i o n o f Mew. T h e i m p o r t a n t p o i n t is t h a t we s h o u l d not neglec t the p a r t i c l e - p a r t i c l e interaction. O t h e r w i s e we w o u l d u n d e r e s t i m a t e the g.s.c.'s c o n s i d e r a b l y a n d c o n s e q u e n t l y o b t a i n t o o large values for MeT. In s u m m a r y we h a v e c a l c u l a t e d 2vl313 d e c a y rates in the q u a s i - p a r t i c l e R P A f o r m a l i s m using a realistic i n t e r a c t i o n a n d f o u n d that t h e y are strongly suppressed w h e n a r e a s o n a b l e a m o u n t o f partic l e - p a r t i c l e i n t e r a c t i o n is t a k e n into a c c o u n t . We are grateful to P r o f e s s o r T.W. D o n n e l l y for v a l u a b l e d i s c u s s i o n s a n d for his critical r e a d i n g o f the m a n u s c r i p t . T h a n k s are also d u e to P r o f e s s o r R. M a d e y for p r o v i d i n g us w i t h e x p e r i m e n t a l d a t a before publication.
References [ 1] H. Primakoffand S.P. Rosen, Ann. Rev. Nucl. Part. Sci. 31 (1981) 145. [2] W.C. Haxton and G.J. Stephenson Jr., Prog. Pail. Nucl. Phys. 12 (1984) 409. [3] M. Doi, T. Kotani and E. Takasugi, Prog. Theor. Phys. Suppl. 83 (1985) 1. [4] J.D. Vergados, Phys. Rep. 133 (1986) 1.
14
30 July 1987
[5] T. Tomoda, A. Faessler, K.W. Schmid and F. Grfimmer, Nucl. Phys. A 452 (1986) 591. [ 6 ] A. Fazely and L.C. Liu, Phys. Rev. Lett. 57 (1986) 968. [ 7 ] T. Tomoda, University of Tiibingen preprint (1986), submitted to Phys. Rev. Lett. [8] L. Zamick and N. Auerbach, Phys. Rev. C 26 (1982) 2185. [9] T. Tsuboi, K. Muto and H. Horie, Phys. Lett. B 143 (1984) 293. [ 10] H.V. Klapdor and K. Grotz, Phys. Lett. B 142 (1984) 323; K. Grotz and H.V. Klapdor, Nucl. Phys. A 460 (1986) 395. [11] P. Vogel and M.R. Zirnbauer, Phys. Rev. Lett. 57 (1986) 3148. [ 12] J.A. Halbleib and R.A. Sorensen, Nucl. Phys. A 98 (1967) 542. [ 13] A.H. Huffman, Phys. Rev. C 2 (1970) 742. [ 14] K. Grotz and H.V. Klapdor, Phys. Len. B 157 (1985) 242. [ 15] P. Vogel and P. Fisher, Phys. Rev. C 32 (1985) 1362. [ 16 ] K. Holinde, Phys. Rep. 68 (1981 ) 121. [ 17] R. Madey, B.D. Anderson, B.S. Flanders and J.W. Watson, Proc. Intern. Symp. on Weak and electromagnetic interactions in nuclei (July 1986), ed. H.V. Klapdor (Springer, Berlin, 1986) p. 280; R. Madey, private communication. [ 18 ] D.O. Caldwell, R.M. Eisberg, D.M. Grumm, D.L. Hale, M.S. Witherell, F.S. Goulding, D.A. Landis, N.W. Madden, D.F. Malone, R.H. Pehl and A.R. Smith, Phys. Rev. D 33 (1986) 2737. [19] F.T. Avignone 11I, R.L. Brodzinski, J.C. Evans Jr., W.K. Hensley, H.S. Miley and J.H. Reeves, Phys. Rev. C 34 (1986) 666. [20] S.R. Elliott, A.A. Hahn and M.K. Moe, Phys. Rev. Lett. 56 (1986) 2582. [21 ] O.K. Manuel, Proc. Intern. Symp. on Nuclear beta decays and neutrinos (Osaka, Japan, June 1986), eds. T. Kotani, H. Ej iri and E. Takasugi (World Scientific, Singapore, 1986 ) p. 71. [22] T. Kirsten, E. Heusser, D. Kaether, J. Oehm, E. Pernicka and H. Richter, Proc. Intern. Symp. on Nuclear beta decays and neutrinos (Osaka, Japan, June 1986), eds. T. Kotani, H. Ejiri and E. Takasugi (World Scientific, Singapore, 1986) p. 81.
PHYSICS LETTERS B
30 July 1987
S U P P R E S S I O N OF THE T W O - N E U T R I N O DOUBLE [i DECAY "~ O. CIVITARESE ~,2, Amand FAESSLER and T. TOMODA Institut J~r Theoretische Physik, Universiti~t Tiibingen, D-7400 Tiibingen, Fed. Rep. Germany
Received 12 March 1987; revised manuscript received 24 May 1987
Two-neutrino 1513decay rates of 76Ge, S~Se,~28.t3OTeare calculated in the quasi-particle random phase approximation using a realistic effectiveNN interaction. The decaysare stronglysuppressed, in agreementwith the experimentaldata, when a reasonable amount of particle-particle interaction is taken into account.
The nuclear double beta (1313) decay has been attracting considerable attention in recent years [1-4]. The principal interest in the process originates from the possibility of distinguishing between Majorana and Dirac neutrinos by observing a neutrinoless (0v) 1313decay. In order to relate the neutrino mass and the coupling strengths of right-handed leptonic currents to 0v1313 decay rates [5], a reliable calculation of nuclear matrix elements is necessary. A comparison of a 0v1313 decay process with a pion double charge exchange reaction may be helpful in reducing ambiguities in the nuclear models [6,7], although there are at present only few experimental data with a 13~ decaying nucleus as a target. Tests of nuclear wave functions by 1313 decay with emission of two neutrinos (2v1313 decay) have been discussed more intensively elsewhere [ 1-4]. The difficulty encountered in such attempts was that the calculated 2v~13 decay rates turned out to be too large compared with experimental data by one to two orders of magnitude [ 2,4,8,9 ]. Theoretical estimates of nuclear maxtrix elements for 2v1313 decay have been reported recently [ 10, l l ]. Klapdor and Grotz [ 10] performed a particle number projected BCS calculation with a schematic spin-isospin as well as a quadrupole-quadrupole ~"Supported by the Bundesministerium far Forschung und Technologie under contract number 06 TiJ 778/9. Permanent address: Department of Physics, University of La Plata, 1900 La Plata, Argentina. 2 Fellowof the CONICET,Argentina.
residual interaction. They have found that ground state correlations due to quadrupole modes in the initial and final nuclei reduce the 2v1313 decay rates by a factor 1.2 to 7. A different mechanism of suppression of 2v1313 decay has been proposed by Vogel and Zirnbauer [ 11 ]. They have used the proton-neutron quasi-particle random phase approximation (QRPA) [12] to calculate the matrix elements connecting both initial and final 0 ÷ ground states with each 1 ÷ state of intermediate odd-odd nuclei [13-15]. In contrast to previous treatments of 1313decay using the QRPA formalism they also took into account a particle-particle interaction. They found a reduction of 2v1313 decay rates by a factor 3 to 25 using a zero-range spin-isospin interaction where the strength of the particle-particle matrix elements were taken to be ~ 2/5 of the corresponding particle-hole ones. In the present paper we want to discuss whether the suppression mechanism found in ref. [ 11 ] persists when a more realistic effective interaction is used. We also briefly consider the relationship between the two mechanisms proposed in refs. [10,11]. The half-life z~/2 for 2v13~ decay can be given in a factorized form involving leptonic and nuclear parts
[3]
as
(rl/2)--I = F I M G ' r I ~ ,
(l)
where F is a lepton phase space integral and
0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
11
Volume 194, n u m b e r 1
PHYSICS LETTERS B
,~ 0"81
MGT
0
_ ~
. .
6
.
.
=Y~ (pllalln)(UpV,~Xp.j+VpU.yp.j), pn
pn
0.2
(3a)
""'-)).\"~ "%\,,~, - -----
76Ge-~ 76Se 82Se-'* 82Kr
\~,\ l/
. . . . . . . . 130 Te --'130Xe
/
I O~6
(3b)
where Xp,,j and yp,j are forward and backward going QRPA amplitudes o f t h e j t h 1 ÷ state of the odd-odd nuclei; the barred quantities indicate that quasi-particles and phonons are defined with respect to the final ground state 07-. The overlaps between two 1 ÷ states belonging to two different sets, ( 1~ I I f ), are given by
( 1; I1 ,+ ) = E (xo,,*xp,J-.fp,,,~Yp,J) •
.
04. . . . . . . . . .
-0.2
( 1 E I I r tr[10?- )
= Z (Pllcrlln)(Vpa,.¢p,.k +aoVr, yp,,,D ,
.
~
o.o
<1~+ IIr+~llO~+ >
(4)
pn
We have neglected in eq. (4) a factor ( 0 ~- I0~ ) [ 14 ] which would further reduce the overlaps by several percent. We calculate the 2vl313 decay half-lives for the following 0 + --*07" transitions: 76Ge---~76Se, 825e---,82Kr, J28Te--. ~28Xe and 13°Te---.13°Xe. We take the full 3ho9 and 4ho9 major oscillator shells for the description of the nuclei with A = 76, 82 and the model space consisting of 1P3/2,i/2, the full 4ho9 shell, Oh ~~/2.9/2and 1f7/2.5/2 subshells for A = 128, 130. The single-particle energies are calculated with a Coulomb-corrected Woods-Saxon potential. As a realistic two-body interaction we use the nuclear matter G-matrix calculated from the Bonn one-boson-exchange potential [16]. In order to take account of a renormalization for finite nuclei a relatively small (in absolute value) starting energy of - 2 5 MeV is used in solving the Bethe-Goldstone equation and in addition the parametrization, gp"a~rG ( a = p , n), gphG and gppG, is introduced for the pairing, parti12
30 July 1987
- 0.8
0.0
I
i
i
I
0.2
0.4.
0.6
0.8
i
1.0 1.2 gpp
Fig. 1. The calculated 2v[313decay matrix elements MOTas functions of the strength gppof the panicle-particle interaction. cle-hole and particle-particle interactions, respectively. First, the BCS equation is solved and the strengths gPair and g~,air of the proton and neutron pairing interactions are determined so as to reproduce the experimental odd-even mass difference for each nucleus. They vary for the different nuclei in the n ~<1.20. Then range 0.93 ~
Volume 194, number 1
PHYSICS LETTERS B
increase gpp, they become stronger and finally the contribution of the second term in the parentheses on the right-hand side of eq. (3b), which involves a backward-going Q R P A amplitude, becomes as large as that of the first term and cancels this completely. In contrast to the results of ref. [ 11 ], where MOT vanishes at gpp~ ~ ~gph, l a complete cancellation in the present calculation takes place at gpp~ 1, i.e. at gr,o.~gr,, within 30% (see above for the values ofgph), which means that the particle-particle and particle-hole interactions are related to each other approximately by a simple recoupling of the angular momenta. This is a nice feature of the present calculation using a realistic interaction in view of the experimentally known suppression o f 2vl3~ decay rates. Although in principle particle-particle and particle-hole interactions undergo different renormalization in the nucleus, they are not expected to differ very much from each other. It is noted that the values of MOT at gpp = 0 in ref. [ 11 ] are smaller than the present results by factors 2 to 3. One of the reasons is that the MOT in ref. [ 11 ] has been calculated by replacing the final state Of~ by 0~, or 0i+ by 07-, in eq. (2) and taking the average of these two matrix elements. This simplified treatment yields a smaller value for MOT. Second, a zerorange spin-isospin interaction which gives the same excitation energy for the G T G R as the realistic interaction used in the present calculation, yields weaker 13- and 13+ strengths for low-lying 1 ÷ states. This also causes MOT to be smaller. Fig. 2 shows the 2vl313 decay half-lives rl/2, eq. (1), as functions o f gpo. We used the free nucleon value gA= 1.254 o f the axial vector coupling constant for the calculation of the phase space integral F. Experimental half-lives [ 1 8 - 2 2 ] are indicated either by hatched regions or, in the case of lower limits, by lines with arrows. It is seen that all the experimental data are consistent with the calculated half-lives with gpp ~ 1. From another point o f view, it is rather difficult to predict the half-lives since they are very sensitive to the value of gpp in the neighborhood of gop = 1. A possible solution might be to compare the calculated 13÷ strength functions with experimental data on the (n,p) reaction using the 1313decay daughter nuclei as targets, or data on single 13- decay from the 1 ÷ states o f the intermediate o d d - o d d nuclei to the ground states of the 1313decay daughters. Unfor-
30 July 1987
1023
1022
ucI
1o
\\\\II\
\ \ \ \
\\
1020
76Ge.76Se
82Se~82Kr lg
10 O0
02
Or,
06
08 •
10
12
10 00
02
0/,
06
08
1.0 gpp
12
02
04
05
08
10 gpp
12
gPP
1025
•
UM
102 ~
1023
1022
1°~00 0'2 0'~ 46 Q'8 1'0 1.2 gpp
lO
O0
Fig. 2. The calculated 2vl313decay half-lives ~,/2 as functions of gpp.Experimental half-livesare indicated either by hatched regions or lines with arrows (for lower limits), and taken from the following sources: UCSB-LBL,ref. [ 18]; USC-PNL, ref. [ 19]; UCI, ref. [20]; UM, ref. [21 ]; MPIH, ref. [22]. tunately there are only few such data at present. In order to estimate the amount of the groun d state correlations (g.s.c.'s), which are essential for the suppression o f MeT, one can alternatively start from a Q R P A equation for the quadrupole mode and first determine the g.s.c.'s due to the quadrupole mode. Out of this correlated ground state, one obtains that portion of g.s.c.'s which is relevant to the pn 1 + mode by angular m o m e n t u m recoupling, and then one can consider modifications of these g.s.c.'s due to the pn 13
Volume 194, number 1
PHYSICS LETTERS B
i n t e r a c t i o n in the 1 + c h a n n e l . Such a p r o c e d u r e was a d o p t e d in ref. [ 1 0 ] . H o w e v e r , i f we start d i r e c t l y f r o m the Q R P A for the p n l + m o d e as in ref. [ 1 1 ] or in the p r e s e n t work, all the g.s.c.'s p e r t a i n i n g to this m o d e are a u t o m a t i c a l l y t a k e n i n t o a c c o u n t a n d we d o not h a v e to w o r r y a b o u t the effect o f q u a d rupole, o c t u p o l e , or w h a t e v e r m o d e o n the g.s.c.'s r e l e v a n t to the pn 1 + m o d e . T h e f o r m a l i s m as well as n u m e r i c a l c a l c u l a t i o n s b e c o m e m u c h simpler, a n d we can get a clearer insight i n t o the m e c h a n i s m o f the s u p p r e s s i o n o f Mew. T h e i m p o r t a n t p o i n t is t h a t we s h o u l d not neglec t the p a r t i c l e - p a r t i c l e interaction. O t h e r w i s e we w o u l d u n d e r e s t i m a t e the g.s.c.'s c o n s i d e r a b l y a n d c o n s e q u e n t l y o b t a i n t o o large values for MeT. In s u m m a r y we h a v e c a l c u l a t e d 2vl313 d e c a y rates in the q u a s i - p a r t i c l e R P A f o r m a l i s m using a realistic i n t e r a c t i o n a n d f o u n d that t h e y are strongly suppressed w h e n a r e a s o n a b l e a m o u n t o f partic l e - p a r t i c l e i n t e r a c t i o n is t a k e n into a c c o u n t . We are grateful to P r o f e s s o r T.W. D o n n e l l y for v a l u a b l e d i s c u s s i o n s a n d for his critical r e a d i n g o f the m a n u s c r i p t . T h a n k s are also d u e to P r o f e s s o r R. M a d e y for p r o v i d i n g us w i t h e x p e r i m e n t a l d a t a before publication.
References [ 1] H. Primakoffand S.P. Rosen, Ann. Rev. Nucl. Part. Sci. 31 (1981) 145. [2] W.C. Haxton and G.J. Stephenson Jr., Prog. Pail. Nucl. Phys. 12 (1984) 409. [3] M. Doi, T. Kotani and E. Takasugi, Prog. Theor. Phys. Suppl. 83 (1985) 1. [4] J.D. Vergados, Phys. Rep. 133 (1986) 1.
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