Calculation of the parity-forbidden decay width of 6Li*(Jpi =0+, T=1; E*=3.56 MeV)→α+d

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1984 J. Phys. G: Nucl. Phys. 10 L21

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J. Phys. G: Nucl. Phys. 10 (1984) L21-L24. Printed in Great Britain

LETTER TO THE EDITOR

Calculation of the parity-forbidden decay width of 6Li*(J" = 0' , T= 1; E* = 3.56 MeV)+ a + d

V V Burovt, V M Dubovik?, S G KadmenskyS, Yu M Tchuvil'sky 0 and L A Tosunyan 5 t Joint Institute for Nuclear Research, Dubna, USSR $ Voronezh State University, Voronezh, USSR 0 Nuclear Physics Institute, Moscow State University, Moscow, USSR

Received 11 October 1983

Abstract. The decay of 6Li*(O+, 1; 3.56 M e V ) + a + d occurs due to the admixture of continuous-spectrum 0 - states with the 0' state considered. To calculate the width, r, of this decay, we have elaborated a method of fractional parentage coefficients for a six-nucleon system in the continuous spectrum. Finally, after numerical calculations, we obtained an extremely small value for r.

Previously, there have been no precise experimental measurements of any parity-non- conserving (PNC) nuclear transition with T= 1. Such transitions arise due to the generation of a virtual intermediate Z boson in the 7rNN vertex (smaller contributions give PNC constants at the p(w)NN vertices). The PNC constant at the MNN vertices, recently calculated by Dubovik and Zenkin (1983a, b) on the basis of the gauge theory SU(2)L x U(l) x SU(3) without any fitting parameters, turns out to be smaller than the widely used semi-phenomenological so called 'best value' = 4.6 x given by Desplanques et a1 (1980). Thus theoretical estimates, which are used by experimentalists studying PNC decays, may be overestimated by a factor of 10 (see, e.g., Robertson 1982).

In this Letter we give for the first time the scheme and results of a realistic calculation of the width r of the experimentally studied decay (Barrette et a1 1975, Bellotti et a1 1975, Robertson et a1 1980) 6Li*(O+, T= 1)+ad and the inverse process ad+6Li*. Specific features of the nuclear structure of 6Li lead to essential differences between these processes compared with others: first, as the spectrum of the nucleus does not contain 0- levels (except the distant 0 - , E* = 21.5 MeV level, which corresponds to T= 1, and therefore decays in the a d channel with a very small width), the decay of Li*(3.56 MeV) is a result of the admixture of a 0- continuous-spectrum state with the wavefunction of the 0' level through the operator V,,, . Second, the main contribution to the given decay comes from the exchange amplitudes, i.e. the amplitudes that contain the permutation of nucleon coordinates between the a particle and the deuteron. A rough estimate of the decay probability taking the PNC in the deuteron wavefunction only into account has been made by Robertson and Riska (the results are cited by Robertson 1982): r=5 x lo-' for the Weinberg-Salam model (and r = 5 x lo-'' for the Cabibbo model). Our calculation of the effect taking the many-particle properties of the nuclear wavefunction of 6Li and of the continuous a d spectrum into account gives a result very different from this estimate.

The 6Li nuclear level scheme in the level region studied is rather simple (see, e.g.,

0305-46 16/84/0 1002 1 + 04 $02.25 0 1984 The Institute of Physics L2 1

L2 2 Letter to the Editor

Ajzenberg-Selove 1979). It is essential that there are no other O+ levels in the distant region of the spectrum. Therefore, neglecting the weak coupling with the closed channel apn and the admixture of the distant 0- level (E* = 2 1.5 MeV), we come to the scattering problem between one level and one channel (Mahaux and Weidenmuller 1969) where V,,, realises their coupling. In this case we have

r = 2n1 ( @6Li I VPNC lA@a @d qad ( P I ) I ’* (1)

Here q a d ( p ) is the wavefunction of the continuous spectrum with angular momentum 1= 1 and d is the antisymmetriser. This wavefunction is a solution of the Schrodinger equation where the deep attractive potential (Kurdiumov et a1 1972) fitted to the scattering phases is used as the nuclear part of the a d interaction potential. The states of a , d and 6Li are described by functions of the translational-invariant shell model of Kurdiumov et a1 (1970).

To calculate (1) the two-particle fractional parentage coefficients (FPC) of the right and left brackets are needed. To find the FPC of the six-nucleon a d functions of the continuous spectra, the expansion is carried out in a divergent series over the oscillator wavefunctions with po =are, where ro is the one-particle oscillator parameter in the wavefunction @ 6 ~ i ,

@, and @C/d (Filippov and Ohrimenko 1980):

It is essential that in the amplitude (1) the term with N = 1 goes to zero due to the antisymmetrisation (A). This eliminates the contribution of non-exchange amplitudes to the given process in the final result. As a result, the spin and isospin parts of the amplitude (1) are expressed through the standard spin-isospin FPC and the moment recoupling symbols (Kurdiumov et a1 1970). The problem of calculating the matrix elements of the space permutation operators contained in d remains. These matrix elements are reduced to the coefficients of the transition from some Jacobi coordinates to others. In this case a standard recipe is the application of a number of successive Talmi transformations. However, the calculations in this way are rather cumbersome. It is more convenient to use the transformation Jacobi brackets (TJB), introduced by Efros (1976), which solve such problems in one step. However, Efros considered TJB for definite projections of 1 only. Here we obtain the TJB expressions required in the SU(3) scheme in the case when the transformations of the Jacobi coordinates correspond to the permutation of two nucleons between a and d:

In our case the TJB (the matrix element of the permutation) has a very simple form:

((N0)(00)(00):(N0)L = 1 IP,, I(NO)(N, O)(N, O):(NO)L = 1)

N , !n!N, ! (4)

For JBT with given 1, L , and L 2 , equation (4) has to be multiplied by the respective trivial Clebsch-Gordan coefficients of the SU(3) group ( (Ni O)Li(NiO)Lj J(N’0)L’) (see, e.g., Ichimura et a1 1973). The problem under investigation is essentially simplified by the fact

Letter to the Editor L2 3

that, according to the selection rules of the matrix elements, both the trivial and four- particle permutations d o not contribute to the amplitude. Moreover, the selection rules with respect to the principal quantum number lead to the condition N , + n = 1; hence the decrease in matrix elements with N is provided by a factor (i\/t)"-'. Thus the calculational method for the FPC of the continuous spectrum function of composite particles is demonstrated for this concrete process.

The numerical computation of the PNC effect we are interested in has resulted in an extremely small value of the width =0.5 x lo-'' eVt. Such a small value, which is due to the intricate interference of many matrix elements, requires a careful check of the stability to the approximations made. Let us discuss them.

(1) The choice of the a d potential has a continuous and discrete ambiguity. Our problem is stable with respect to the continuous ambiguity. At the same time the result of the action of V,,, on I,96Li is the wavefunction of relative motion of a and d, ~ ( p ) , the node of which is near the node of qad(p). Consequently, the change of the potential Vad within the discrete ambiguity may significantly change the overlap integral of the functions. Nevertheless, about one half of the value of the integral is accumulated in the region of the Coulomb barrier, where the form of the potential is known exactly. This contribution is a measure of the stability with respect to the discrete ambiguity of Vad.

(2) Estimates of the non-exchange term, which arises due to the admixture to @ 6 ~ j of the state with E=Eo +2hw, [ f ]=[411] , S = 1, T= 1 and with a small and unknown weight p, show that this term is in any case smaller than the sum of the exchange terms.

(3) The problem is sensitive to the choice of the N N correlation function at short distances, f ( p ) (results of calculations of the width taking the correlations into account and the methodical calculation without the latter differ from each other by about 30 times, although the scale of the change of separate two-nucleon matrix elements is as usual). In view of this, the value of r obtained for another reasonable choice of Vad, f ( p ) and p is also changed by a few times.

Let us proceed to our brief conclusions. (1) The scale of r does not permit us to hope that it will be measured in the near

future. (2) The measurement even of linear PNC effects by using the 0' level of the nucleus 6Li

may prove to be difficult. (3) The reason for (2) is the absence in the spectrum of the nucleus 6Li of suitable 0-

resonances close to 0' with a sufficiently large width r, whereas the direct connection with the continuous spectrum is rather small. The effect of coupling with the continuous spectrum, which defines our value of r, can be estimated from a resonance width of approximately 1 MeV at a distance of 20 MeV.

(4) The calculation shows that the contribution of vector-meson amplitudes to the given decay is rather great (the interference term is approximately equal in magnitude to the pion amplitude squared), i.e. the given process is extremely sensitive to the values of the PNC constants of the weak N N interactions.

eV measured by Fifield et a1 (1983) for the process with T = l , 20Ne ( J " = l ' , T = l ; E = 1 1 . 2 6 M e V ) - + ' 6 0 + a , is anomalously large and requires serious theoretical consideration and experimental verification.

The authors are indebted to Drs R A Eramzhyan, F A Gareev, V Z Goldberg, G Ia Korenman, V I Kukulin and V M Lobashev for useful discussions. ? W e use the set of PNC constants h:=1.3 x lo-', hd=0.39 x lo-' and h h = - 2 . 2 ~ lo-' calculated by Dubovik and Zenkin (1983a, b).

( 5 ) The value of r = 4 . 2 x

L24 Letter to the Editor

References

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Robertson R G H 1982 International Science Series, Neutrino Physics and Astrophysics ed. E Fiorini (New

Robertson R G H et a1 1980 Chalk River Progress Report PR-P-124: 2.3

Holland) p 46

York: Plenum) pp 2 1 9 4 0