Nuclear structure in nonmesonic weak decay of hypernuclei

Abstract

A general shell model formalism for the nonmesonic weak decay of the hypernuclei has been developed. It involves a partial wave expansion of the emitted nucleon waves, preserves naturally the antisymmetrization between the escaping particles and the residual core, and contains as a particular case the weak lambda-core coupling formalism. The hypernuclei are grouped having in view their A - 1 cores, that is in those with even-even, even-odd and odd-oddcores. It is shown that in all three cases the nuclear structure manifests itself basically through Pauli Principle, and very simple expressions are derived for the neutron and proton induced decays rates, gamman and gammap, which does not involve the spectroscopic factors. For the strangeness-changing weak lambda N ->N N transition potential we use the One-Meson-Exchange Model (OMEM), which comprises the exchange of the complete pseudoscalar and vector meson octets (pi, <FONT FACE=Symbol>h,</FONT>K<FONT FACE=Symbol>, r</FONT>, omega, K*).We evaluate ³H, 4H, 4lambdaHe, 5lambdaHe, 11B, 12lambdaC, 16lambdaO, 17lambdaO, and 28Si hypernuclei, with commonly used parametrization for the OMEM, and compare the results with the available experimental information. The calculated rates gammaNM = gamman + gammap are consistent with the data, but the measurements of gamman/p = gamman/gammap are not well accounted for by the theory. It is suggested that, unless additional degrees of freedom are incorporated, the OMEM parameters should be radically modified.

Nuclear structure in nonmesonic weak decay of hypernuclei

F. Krmpotic^I; D. Tadic^II

^IDepartamento de Física, Universidad Nacional de La Plata, C. C. 67, 1900 La Plata, Argentina

^IIPhysics Department, University of Zagreb, 10 000 Zagreb, Croatia

ABSTRACT

A general shell model formalism for the nonmesonic weak decay of the hypernuclei has been developed. It involves a partial wave expansion of the emitted nucleon waves, preserves naturally the antisymmetrization between the escaping particles and the residual core, and contains as a particular case the weak L-core coupling formalism. The hypernuclei are grouped having in view their A – 1 cores, that is in those with even-even, even-odd and odd-oddcores. It is shown that in all three cases the nuclear structure manifests itself basically through Pauli Principle, and very simple expressions are derived for the neutron and proton induced decays rates, G_n and G_p, which does not involve the spectroscopic factors. For the strangeness-changing weak L N ® N N transition potential we use the One-Meson-Exchange Model (OMEM), which comprises the exchange of the complete pseudoscalar and vector meson octets (p, h, K, r, w, K*).We evaluate ³H, ⁴H, ⁴_LHe, ⁵_LHe, ¹¹B, ¹²_LC, ¹⁶_LO, ¹⁷_LO, and 28Si hypernuclei, with commonly used parametrization for the OMEM, and compare the results with the available experimental information. The calculated rates G_NM = G_n + G_p are consistent with the data, but the measurements of G_n/p = G_n/G_p are not well accounted for by the theory. It is suggested that, unless additional degrees of freedom are incorporated, the OMEM parameters should be radically modified.

I Introduction

Hypernuclear physics adds another flavor (strangeness) to the traditional nuclear physics, and its goal is to study the behavior of hyperons (L, S , X, W) in the nuclear environments, which are now bound system of neutrons, protons and one or more hyperons. Interesting strange nuclei with strangeness S = – 1 are the L hypernuclei, in which a L hyperon, having a mass of 1116 MeV and zero charge and isospin, replaces one of the nucleons. Same as the free L hyperon, they are mostly produced via the strong interactions, i.e., in the reaction processes p⁺n ® L K⁺, K^–n ® p^–L and K^–p ® p⁰L, by making use of the pion (p) and kaon (K) beams. They also basically decay through the weak interactions, as the free L does. Yet, as it is well known and explained below, there are some very important differences in the corresponding decaying modes.

First, it should be remembered that the free L hyperon decays nearly 100 % of the time by the L ® N p weak-mesonic mode (Fig. 1):

with the total transition rate = G⁰ = 2.50·10^–6 eV (which corresponds to the lifetime ⁰ = 2.63·10^–10 sec). For the decay at rest the energy-momentum conservation implies

Therefore the energy released is

Q₀ = M_L- M_N- m_p @ 37 MeV,

and the kinetic energies and momenta in the final state are:

It is clear that the isospin is changed by DT = 1/2 and 3/2 and its projection by DM_T = –1/2 in the free L ® N p decay. More, as

one sees that the above experimental data can be accounted for fairly well by neglecting the |1/2,1;3/2,–1/2ñ components in these relations (DT = 1/2 rule). In fact, one gets = 2, while the experimental result is 64.1/35.7 = 1.80.

Assuming the DT = 1/2 rule, the phenomelogical weak Hamiltonian for the process depicted in Fig. 1 can be expressed as:

where = 2.21×10^–7 is the weak coupling constant. The empirical constants A_p = 1.05 and B_p = –7.15, adjusted to the observables of the free L decay, determine the strengths of parity violating and parity conserving amplitudes, respectively. The nucleon, L and pion fields are given by y_N and y_L and f_p, respectively, while the isospin spurion is included in order to enforce the empirical DT = 1/2 rule. (Note that: _Nf_p · y_L = (L).

The free L hyperon weak decay is radically modified in the nuclear environment because the nucleon and the hyperon now move, respectively, in the mean fields U_N and U_L, which come from the NN and NL interactions. U_N and U_L are characterized by the single particle energies (s.p.e.) e_N and e_L and we have to differentiate between:

1. Mesonic Decay (MD): The basic process is again represented by the graph shown in Fig. 1 and described by the hamiltonian (1). Yet, the energy-momentum conservation is different:

where M_A = AM_N and p_A are, respectively, the mass and the momentum of the whole nucleus; A is the mass number, and are the s.p.e. of the loosely bound states above the Fermi energy . They are of the order of a few MeV, while e_L is the energy of the 0s_1/2 state and goes from –11.7 MeV for to –26.5 MeV for [1]. Thus, the corresponding Q-values

Q_M = M_L – M_N – m_p + e_L – ,

are significantly smaller than Q₀, particularly for medium and heavy nuclei. The experimental decay rates G_p^– + = G_M º G_M (L ® N p) are of the order of G⁰ only for nuclei with A < 4, and they rapidly fall as a function of nuclear mass. For instance, in : /G⁰ = 0.217 ± 0.084 and G_p–/G⁰ = . This hindrance effect, as is illustrated in Fig. 2 for the hypernucleus , is due to Pauli principle. In fact, the population of states that are below the Fermi level, with energies < , is totally blocked by the Pauli principle, while transitions to states that lie above are strongly hindered due to the selection rule DN = 0, N being the harmonic oscillator quantum numbers. That is, only a few second forbidden transitions (DN = 2) can occur in the case of , and it is clear that the degree of forbiddiness increases with A.

2. Nonmesonic Decay: New nonmesonic decay (NMD) channels L N ® N N become open inside the nucleus, where there are no pions in the final state. The corresponding transition rates can be stimulated either by protons, G_pº G(Lp ® np), or by neutrons, G_n º G(Ln ® nn). The energy-momentum conservation and the Q-value are, respectively:

and

Q_M = M_L – M_N + e_L – ,

where p₁ and p₂ are the momenta of the two outgoing nucleons. As the mean value of is ~ 30 MeV one gets that Q_NM ~ 120-135 MeV, which is basically the kinetic energy of the two particles that are ejected from the hypernucleus. This means that the nonnesonic decay process possesses a large phase space in the continuum, as is outlined in Fig. 3 for the case of the hypernucleus . The theoretical models reproduce fairly well the experimental values of the total width G_NM = G_n + G_p but the ratio G_n/p º G_n/G_p (0.5 < < 2) remains a puzzle.

Very often it is assumed that the hypernuclear NMD LN ® N N is triggered via the exchange of a virtual meson, and the obvious candidate is the one-pion-exchange (OPE) mechanism, where the strong Hamiltonian

(with g_NNp = 13.4) accompanies the weak Hamiltonian (1). Following the pioneering investigations of Adams [12] several calculations have been done within this coupling scheme yielding: @ G⁰ and @ 0.1-0.2 [14-39]. The importance of the r meson in the weak decay mechanism was first discussed by McKellar and Gibson [13], and the present-day consensus is, however, that the effect of the r-meson on both G_NM and G_n/p is small [21, 23, 24, 26, 38]. The full one meson-exchange model (OMEM), which encompasses all pseudoscalar mesons (p, h, K) and all vector mesons (r, w, K^*), has been also considered by several authors [23, 24, 33, 34, 35, 36, 38]. From these works we have learned that, although the K meson contribution significantly increases the ratio G_n/p, the OMEM is unable to account for the corresponding experimental values.

We wish to restate that the OMEM transition potential is purely phenomenological and that it is not derived from a fundamental underlying form, as happens for instance, in the case of electro-magnetic transitions or the semileptonic weak decays. As in the case of the OPE, in the OMEM, a weak baryon-baryon-meson (BBM) coupling is always combined with a strong BBM coupling. The strong one is determined experimentally with some help from the SU(3) symmetry, and the involving uncertainties have been copiously discussed in the literature [40-43]. It is the weak BBM couplings which could become the largest source of errors. In fact, only the weak NLp amplitude can be taken from the experiment, at the expense of neglecting the off-mass-shell corrections. All other weak BBM couplings are derived theoretically by using SU(3) and SU(6)_w symmetries, octet dominance, current algebra, PCAC, pole dominance, etc. [18,24,25,44-51]. Assortments of such methods have been developed and employed for a long time in weak interaction physics to explain the hyperon nonleptonic decays. One should also keep in mind that both the strong and weak BBM couplings, as well the meson masses, can become significantly renormalized by the nuclear environment [51].

The nuclear structure frameworks utilized in the literature for the formal derivations of the NMD rates are: i) the nuclear matter, and ii) the nuclear shell model. There are relatively few works where the second method was employed, and up to quite recently all they were involved the technique of coefficients of fractional parentage, with the spectroscopic factors (SF) explicitly appearing in the expressions for the transition rates [19, 24, 29, 37]. At variance, Barbero et al. [38] have developed a fully general shell model formalism and have specified it for hypernuclei with odd-mass core, such as and , were the cores are: ³H, ³He, ¹¹C and ²⁷Si. Here we also discuss the hypernuclei with even-even and odd-odd number of protons and neutrons, namely and , and , and .

II Shell Model Formalism

The shell model framework for the NMD rates has been developed in detail in Ref. [38] and here we will just sketch the main steps, employing the same notation. One starts from Fermi's Golden Rule for the decay rate,

where V is the weak hypernuclear potential, and the wave functions for the kets |p₁p₂SM_S, J_FM_F; TM_Tñ and |J_IM_Iñ are assumed to be antisymmetrized and normalized. After performing: 1) the transformation to the relative and center of mass (c.m.) momenta, p and P, and angular momenta l and L, and 2) the angular momentum couplings: l+L = , + S = J one obtains:

where , t_L = – 1/2, t_p = 1/2, t_n = –1/2, and the label a goes over all final states with the same spin and parity. Thus, that the NMD rates, in principle, depend on both: i) the nuclear structure effects through the two-particle N L parentage coefficients , and on the transition potential via the elementary transition amplitudes

Here (¼|V|¼) is the direct matrix element and the factor in front takes care of the antisymmetrization. To evaluate the nuclear matrix element one has to carry out the jj – LS recoupling and the Moshinsky transformation [52] on the ket |_{j L j N}J):

where (¼|¼) are the Moshinsky brackets [52], and l and L stand for the quantum numbers of the relative and c.m. orbital angular momenta in the LN system. The explicit expressions for the transition potentials are given in the Ref. [38].

When the hyperon is assumed to be weakly coupled to the A-1 core, which implies that the interaction of L with core nucleons is disregarded, one has that , where J_C is the spin of the core, and gets

To evaluate the one-particle spectroscopic amplitudes we will use the BCS approximation and, it will be assumed that the even-even, odd-even and odd-odd cores are described, respectively, as zero, one and two quasiparticle states. Correspondingly, the state |J_Cñ goes into |BCSñ, |BCSñ and |BCSñ, where |BCSñ is the BCS vacuum and is the quasiparticle creation operator [38]. In all three cases the NMD rate can be cast in the form:

where

and

The geometrical factors F_J(j_Nt_N) are:

1. Even-Even core: |J_Cñ ® |BCSñ,

2. Odd-Even core: |J_Cñ ® |BCSñ,

3. Odd-Odd core: |J_Cñ ® |BCSñ,

It is clear that in (13) t₂ = – t₁.

The nuclear structure manifests itself basically through the factors F_J(j_Nt_N), which are engendered by the Pauli principle. Their values for a few cases are given in Table 1. We would like to stress that the quantum numbers j₁t₁ and j₂t₂ stand for the hyperon partners in the initial state, and that j_N runs over all proton and neutron occupied states in the initial nucleus. It is amazing to notice that the the last three equations are valid for any hypernucleus, which could be so light as or so heavy as . One should also add that the Eq. (8) contains the same physics as the Eq. (5) in Ref. [24] or the Eq. (30) in Ref. [37], with the advantage that we do not have to deal with spectroscopic factors. Of course, neither the initial and final wave functions are needed. From the results displayed in Table 1 it can be seen that in all three cases the coefficients F_J(j_Nt_N) are of the same order of magnitude, which indicates that the nuclear structure effects in the NMD are of minor importance.

II Numerical Results and Discussion

The numerical values of the parameters necessary to specify the transition potential, were taken from Ref. [24], where, in turn, the strong couplings have been taken from Refs. [39, 40] and the weak ones from Ref. [23]. The energy difference D_{j N t N} in (10) is evaluated from the experimental single nucleon and hyperon energies. It is a general belief nowadays that, in any realistic evaluation of the hypernuclear NMD rates, the finite nucleon size (FNS) and the short range correlations (SRC) have to be included simultaneously. Therefore, in the present paper both the FNS and SRC renormalization effects are considered, in the way described in Ref. [38]. Under these circumstances, and because of the relative smallness of pion mass, the transition is dominated by the OPE [24, 38].

The numerical calculations were done in the extreme shell model, which implies that the pairing factors u_{j N} were taken to be equal to one (zero) for the occupied (empty) levels. Thus, from the nuclear structure point of view, the only free parameter is the harmonic oscillator length b. We evaluate it from the relation b = 1/, and the oscillator energy was estimated from the relation w = 45A^–1/3 – 25A^–2/3 MeV, which is frequently used for light nuclei.

The calculations for G_{N M} and G_{n / p} are confronted the experimental data in Tables 2 and in ³, respectively. Three different OMEM have been employed for the transition potential. Namely: (p) only the pion was taken into the account, (PS) all three pseudoscalar mesons (p + h + K) were included, and (PS + V) also the vector (r + w + K^*) mesons are considered. The same remarks are pertinent here as in the study [38] where only has been analyzed. That is: (1) the simple OPE model accounts for G_{N M}, but it fails badly regarding G_{n / p}, (2) when h and K mesons are included, the total transition rate is only slightly modified, while G_{n / p} change significantly, coming somewhat closer to the measured values, and (3) the results are not drastically modified when all vector mesons are built-in.

IV Summary and Conclusions

The shell model formalism for the nonmesonic weak decay of the hypernuclei involves a partial wave expansion of the emitted nucleon waves and preserves naturally the antisymmetrization between the escaping particles and the residual core. The general expression (4) is valid for any nuclear model and it shows that the nonmesonic transition rates should depend, in principle, on both: (i) the weak transition potential, through the elementary transition amplitudes (pPlL

SJT;j_L j_N,t_N), and (ii) the nuclear structure, through the two-particle NL parentage coefficients . The latter explicitly depend on the initial and final wave functions. Yet, as explained in Ref. [38], and because of: a) the inclusive nature of the nonmesonic decay, and b) the peculiar properties of the coefficients F_J(j_Nt_N), this dependence is washed out. In this way we have arrived at a very simple result for transition rates, given by the Eq. (8), which is valid, not only for the hypernuclei with odd-even core (as shown previously [38]), but also for those which have even-even and odd-odd cores, which has been demonstrated here.

We reproduce satisfactorily the data for the total transition rates with the OMEM parametrization from the literature [24], but the n / p-ratios are not well accounted for. Thus, after having acquired full control of the nuclear structure involved in the process, and after having convinced ourselves that the nuclear structure correlations can not play a crucial role, we firmly believe that the currently used OMEM should be radically changed. Either its parametrization has to be modified or additional degrees of freedom have to be incorporated, such as the factorizable terms [53], the axial-vector-meson exchanges [54], or the correlated the correlated 2p from Ref. [37].

The authors acknowledge the support of ANPCyT (Argentina) under grant BID 1201/OC-AR (PICT 03-04296), of CONICET under grant PIP 463, of Fundación Antorchas (Argentina) under grant 13740/01-111, and of Croatian Ministry of Science and Technology under grant # 119222. D.T. thanks the Abdus Salam ICTP Visiting Scholar programme for travel fares. F.K. is a Fellow of the CONICET Argentina. We would like to thank A.P. Galeão, C. Barbero and E. Bauer for very helpful and illuminating discussions.

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Received on 30 October, 2002

Publication Dates

• Publication in this collection
  25 Aug 2003
• Date of issue
  June 2003

History

• Accepted
  30 Oct 2002
• Received
  30 Oct 2002