Hyperfine Interactions 78 (1993) 85-89 85

Induced pseudoscalar interaction in weak nucleon current and muon capture in hyperfine states of

muonic deuterium

M. Mor i ta

Josai lnternational University, Gumyo, Togane, Chiba 283, Japan

R. Mor i ta

Women's Junior College, Josai University, Sakado, Saitama 350-02, Japan

M. Doi, T. Sato, H. Ohtsubo

Department of Physics, Osaka University, Toyonaka, Osaka 560, Japan

and

K. Koshigiri

Department of Physics, Osaka Kyoiku University, Osaka 543, Japan

Muon capture rate in muonic deuterium can be calculated with less ambiguities in nuclear wave functions, since we know the bound and scattering states of the two nucleons with higher accuracy theoretically. It is, however, dependent on the pion-nucleon-delta coupling constant through the exchange current effect. This dependence is considerably reduced for the ratio of the capture rates from two hypert'me states of muonic deuterium. This ratio is, therefore, useful to study the strength of the induced pseudoscalar term. Several other physical quantities in light nuclei are also introduced here for the same purpose.

I . I n t r o d u c t i o n

In the V--A theory of nuclear weak processes, there are six coupling constants for weak charged currents. Except for the induced pseudoscalar term, they can be studied by using the experimental data on nuclear beta decays, and the latest status is given, e. g., in ref. [1]. The pseudoscalar term has a very small observable effect in the nuclear beta decays, but it has a sizable effect in muon capture reactions.

Since the magni tude of gp is not known with the same accuracy as that for the other five coupling constants, it has been one of the central problems in nuclear

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86 M. Morita et al. / Interaction in hyperfine states o f muonic deuterium

weak processes to find gp with an increasing accuracy, and to compare it with the prediction,

gPIgA = 2MM.I(M + g'), (1)

given by the PCAC theory. Here M, M~ and M~ are the masses for nucleon, muon and pion, respectively, q is the four momentum transfer. The matrix element of the axial vector current is expressed by

(pf l A~ l pi) = i~(pf)[gATa + i(gp/Mrt)qa + (gT/2M)aal~q13175"r(-)u(pi) , (2)

with qa = (Pi - Pf)a, oral3 --- (1/2i)[%, 7~], as usual. The induced tensor coupling constant gT is vanishingly small, as studied theoreti-

cally by our group [1,2] with experimental data on the angular distribution of the beta rays in the A = 12 system given by the Osaka, Leuvain, and ETH groups in 1977-92. This will be not described here in detail. We shall confine ourselves to the study ofgp in light nuclei.

In order to derive the strength of gp by analyzing experimental data, there are the following physical observables available for us:

(1) muon capture rates in complex nuclei; (2) muon capture rate in hydrogen; (3) physical quantities in radiative muon capture reaction; (4) nuclear polarizations and alignments in muon capture reaction; (5) ratio ofmuon capture rates from two hyperfine states of the muonic atom. First of all, partial muon capture rates in complex nuclei are, generally, strongly

dependent on the assumed nuclear model. And the dependence on gv is masked by the ambiguities in nuclear parameters, even in the case of the A = 12 system, where the wave functions are comparatively well established from investigation on the nuclear structure. The total muon capture rate seems to be less dependent on the nuclear parameters, since they are averaged over the final states. This makes also the gv dependence to be unclear. Secondly the muon capture rate in hydrogen is free from the nuclear physics, and the average value of the experimental data given by different groups in different years is consistent with the PCAC value. However, the individual data are scattered in a considerably wide range. Thirdly, the radiative muon capture rates are about 105 times smaller than nonradiative ones. The experi- mental data are, however, being accumulated on the ratio of the radiative to nonra- diative muon capture rates, gamma-ray spectrum, and gamma-ray angular distribution [3-5].

2. Muon capture in light nuclei

We have studied the nonradiative muon captures in light nuclei of A = 11-14.

M. Morita et al. / Interaction in hyperfine states o f muonic deuterium 87

In the case of liB, the ratio of the total muon capture rates from hyperfine states is rather insensitive to the nuclear model and the pseudoscalar coupling constant. A similar ratio of the partial muon capture rates leading to the bound 1/2- state is strongly dependent on gP/gA [6].

Similar works have been performed on the partial muon capture in 13C and 14N [7] in comparison with that in 12C [8]. Generally, the ratio of the muon capture rates from two hyperfine states and various nuclear orientations of the recoil nucleus are insensitive to the adopted nuclear model, and they are suitably dependent on gv/gA. Among these the most suitable quantities are, the average polarization PAV of 12B, average alignment of 13B, and the ratio of muon capture rates from two hyperfine states of 14N.

Extensive investigations have been made for the A = 12 system in connection with the study of the beta decay [1,8]. The ratio, R = PAv/PL, of average and long- itudinal polarizations of 12B is relatively free from systematic errors in experi- ment. And it is useful for the analysis with our formalism. There is an ambiguity in the branching ratio of the muon capture rates leading to the 1- excited state and the ground state of 12B in experiments. Except for this ambiguity, the magnitude of the induced pseudoscalar coupling constant derived from R is

gP/gA : 8.5 -4- 1.9, (3)

by taking into account the effects of the core polarization up to the second order, and those of the exchange currents, in addition to the general lp-sheU wave func- tion [8]. This study has a relation to the parameter y = - 2 M i f ysr/f~r, which is the ratio of the time component to the space component of the main axial vector term. In the formula for R, a combination (gP/gA + Y) appears and its value is found from the experimental data. Therefore, estimation o fy in theory is impor- tant. Since the higher-order core polarizations cancel the lower-order effects, we have to take into account the full effects. If we adopt a model M3Y with AE = 4.5 MeV (notation ofref. [1]), we have y = 4.6 and f o- = 0.94 in good agreement with experimental data, y = 4.76 -4- 0.44, the best fit value to the beta-ray angular distribution [9], and f a = 0.97 + 0.03 which is given by the average value of theft values and gA/gV = --1.26. Adopting the above theoretical value ofy and a simple formula for R, we have gP/gA = 7.4 in a good agreement with 7.2 of the PCAC value. Minor fluctuations ofgp/gA may take place by a full calculation of the for- mula for R, different assumption for nuclear parameters, and also by adopting dif- ferent experimental data on the branching ratios for the partial muon capture rates leading to the excited states of 12B. It is, however, interesting to note that the renormalization for gA and gp can be understood well as the effects of the higher- order many-particle wave functions and those of the meson exchange currents, at least, in the case of the A = 12 system. Here the muon capture takes place at the peripheral region of a light nuclus.

88 M. Morita et al. / Interaction in hyperfine states o f muonic deuterium

3. Muon capture in hypeffine states ofmuonic deuterium

The muon capture reaction in the deuterium is one of the basic processes for studying the weak nuclear exchange current because of its simple nuclear structure with less ambiguities. The V-A theory predicts the capture rate from the doublet state to be much larger than that from the quartet state. The total capture rate by the deuterium has a long history of theoretical and experimental investigations. The main subject of these has been the effect of the meson exchange current, since the observed capture rate is larger than the theoretical one if one assumes only one- body weak current.

We have investigated the effect of meson exchange current on the energy spec- trum of the emitted neutrons and the asymmetric angular distribution of the neu- trino [10], since there has been published an energy spectrum of the emitted neutrons [11]. New experimental data on the capture rate were also published,

W1/2 = 409 + 40 s -1 and 470 q- 29 s -1 , (4)

in refs. [ 12] and [ 13], respectively. Our theory gives 402 s-1 in agreement with a simi- lar work [14] which gives 415 s -1, if we adopt eq. (1) for gP/gA, where q2 is vari- able. We may limit the magnitude, for example, by adopting the data in ref. [13]. The 10% ambiguity of the capture rate causes about 50% ambiguity in g~/gA. Furthermore, the muon capture rate varies with the pion-nucleon-delta coupling constants through the meson exchange currents. Although these coupling con- stants can be determined by studying the pion-nucleon scattering, there remains still a certain ambiguity in these.

Finally we found the ratio of the muon capture rates from the doublet and quar- tet states which is insensitive to the pion-nucleon-delta coupling constants, and strongly dependent on g a / g A [15]. If the ratio, W1/2/W3/2, is measured with 10% uncertainty, we can limit gP/gA up to 10% accuracy in the vicinity of the PCAC value. Therefore, this is a very useful quantity to test PCAC.

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