Simple analytical expression for vector hypernuclear asymmetry in nonmesonic decay of $^5_Lambda$He and $^{12}_\Lambda$C

Abstract

We present general explicit expressions for a shell-model calculationof the vector hypernuclear parameter in nonmesonic weak decay. We use a widely accepted effective coupling Hamiltonian involving theexchange of the complete pseudoscalar and vector meson octets(<FONT FACE=Symbol>p, h</FONT>, K, <FONT FACE=Symbol>r, w</FONT>, K*). In contrast to the approximated formula widely used in the literature,we correctly treat the contribution of transitions originated fromsingle-proton states beyond the s-shell. Exact and simple analytical expressions are obtained for the particularcases of $^5_Lambda$He and $^{12}_\Lambda$C, within theone-pion-exchange model. Numerical computations of the asymmetry parameter, $a_\Lambda$, are presented. Our results show a qualitative agreementwith other theoretical estimates but also acontradiction with recent experimental determinations. Our simple analytical formulas provide a guide in searchingthe origin of such discrepancies, andthey will be useful for helping to solve the hypernuclearweak decay puzzle.

Hypernuclear decay; Asymmetry parameter; One-meson-exchange model

Simple analytical expression for vector hypernuclear asymmetry in nonmesonic decay of $^5_Lambda$He and $^{12}_\Lambda$C

César Barbero^I; Francisco Krmpotic^II; Alfredo P. Galeão^III

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

^IIInstituto de Física, Universidade de São Paulo, C. P. 66318, 05315-970 São Paulo, SP, Brazil, Instituto de Física La Plata, CONICET, 1900 La Plata, Argentina

^IIIInstituto de Física Teórica, Universidade Estadual Paulista, Rua Pamplona 145, 01405-900 São Paulo, SP, Brazil

ABSTRACT

We present general explicit expressions for a shell-model calculationof the vector hypernuclear parameter in nonmesonic weak decay. We use a widely accepted effective coupling Hamiltonian involving theexchange of the complete pseudoscalar and vector meson octets(p, h, K, r, w, K^*). In contrast to the approximated formula widely used in the literature,we correctly treat the contribution of transitions originated fromsingle-proton states beyond the s-shell. Exact and simple analytical expressions are obtained for the particularcases of $^5_Lambda$He and $^{12}_\Lambda$C, within theone-pion-exchange model. Numerical computations of the asymmetry parameter, $a_\Lambda$, are presented. Our results show a qualitative agreementwith other theoretical estimates but also acontradiction with recent experimental determinations. Our simple analytical formulas provide a guide in searchingthe origin of such discrepancies, andthey will be useful for helping to solve the hypernuclearweak decay puzzle.

Keywords: Hypernuclear decay; Asymmetry parameter; One-meson-exchange model

The free decay of a L hyperon occurs almost exclusivelythrough the mesonic mode, L ® pN, emergingthe nucleon with a momentum of about 100 MeV/c. Inside thenuclear medium (p_F ~ 270 MeV/c) this mode is Pauli blockedand, for all but the lightest L hypernuclei (A > 5),the weak decay is dominated by the nonmesonic channel, LN ® NN, with enough kinetic energy to put the twoemitted nucleons above the Fermi surface. The nonmesonichypernuclear weak decay (NMHWD) offers us a very uniqueopportunity to investigate strangeness-changing weak interactionbetween hadrons. Assuming that the NMHWD is saturated by theL N ® NN mode, the transitions receivecontributions either from neutrons (Ln ® nn) andprotons (Lp ® np), with rates G_n and G_p, respectively (G_NM = G_n + G_p, G_n/p = G_n/G_p). Over the last three decades amount oftheoretical and experimental effort has been invested in solvingan interesting puzzle: the theoretical models cannot reproduce simultaneouslythe experimental values of G_n/p and a_L.

From the experimental side there is actually an intense activity,as can be seen in the light of the experiments under way and/orplanned at (KEK) [1], (FINUDA) [2] and Brookhaven NationalLaboratory (BNL) [3]. The preliminary results give a G_n/p ratiovalue very close to 0. 5 [4-6] and themeasurements of a_L favor a negative value for and a positive value for [5-9]. On the other hand, fromthe pioneering work of Block and Dalitz [10] there have beenmany theoretical attempts dedicated to solving the puzzle. Theearlier studies were based on the simplest model of the virtualpion exchange [11]. This model naturally explains the longrange part of the two body interaction, and it reproduces reasonablywell the total decay rate but fails badly in reproducing the otherobservables. In order to achieve a better description,some improvements have been introduced: (i) models that include the exchange of different combinations of otherheavier mesons, like h, K, r, w and K^* [12]-[19], to reproduce theshort range part of the interaction ; (ii) analysis of the two nucleon (2N)stimulated process LNN ® NNN [20, 21]; (iii) inclusion of interaction terms that violate the isospin DT = 1/2 rule [22]-[24]; (iv) descriptionof the short range baryon-baryon interaction in terms of quarkdegrees of freedom [25, 26]; (v) correlated (in the formof s and r mesons) and uncorrelated two-pion exchanges [27]-[31]. We emphasize that none of these modelsgive a fully satisfactory description of all the NMHWD observablessimultaneously, even though a consistent (though not sufficient)increase of G_n/p has been found.

All such models reproduce quite well G_nm, but seem to stronglyunderestimate the experimental n/p branching ratio, G_n/G_p. Besides, the measurements favor a negative value of a_L for and a positive one for , meanwhile all existing calculations based on strict one-meson-exchange (OME) models [14-16, 18, 19, 26, 32, 33] findvalues between -0. 73 and -0. 19 for [34, 35] and, when results are available in the samemodel, very similar values for .

Most calculations of the asymmetry parameter make use of an approximateformula (Eq. (18), below) which, however, is valid only for s-shellhypernuclei. Since an essential aspect in the asymmetry puzzle presentedabove concerns the comparison of its values for and , it would be of great interestto have an expression that is applicable to both cases. Besides, simple formulas containing kinematical factors and nuclearmatrix elements separately, willprovide a manageable and useful tool. They will be a guide in searchingthe origin of discrepancies between theoretical calculations andexperimental data.

Thus, we present in this paper a general formalism for the vector asymmetryparameter in nonmesonic decay and derive exact and simple analyticalexpressions, within the simplest one-pionexchange model (OPEM), for the particular cases of and hypernuclei.

Single-L hypernuclei produced in a (p⁺, K⁺) reactionend up with considerable vector polarization along the directionnormal to the reaction plane, = (p_p+ × p_K+) / |p_p+ × p_K+|. Theangular distribution of protons emitted in Lp ® np decay is given by [34]

where G_p is the full proton-induced decay rate, and

is the vector hypernuclear asymmetry, with

being

Here V is the OME potential involving thecomplete pseudoscalar and vector meson octets(p, h, K, r, w, K^*)which has been extensively discussed in Refs. [17, 18], p₁s₁ (p₂s₂) are the momenta and spinprojections of the emitted neutron (proton) and

with M and M_F being the nucleon and residual nucleusmasses and the liberated energy.

We assume the hyperon to stay in the j_L = 1s_1/2single-particle state, and the initial hypernuclear state is builtas |J_Iñ º |(j_LJ_C)J_I|, where |J_Cñ is the ^A-1Z ground state. Thus, the intrinsic asymmetry parameter isdefined as

Rewriting the transition amplitude in (4) in the totalspin (S, M_S) and isospin (T, M_T) basis we get

where p = (p₂ - p₁), P = p₁ + p₂ and

For the integration in (7) we take the z-axis asindicated in Fig. 1 and choose p₂, q_p1 and f_p1 as independent variables. Thus, expanding the finalstate in terms of relative and center-of-mass partial waves of theemitted nucleons [17], after integration on f_p1and recoupling of angular momentum, we get

where (plPLlSJTJ_F;J_I|V|J_I) = [1 - (-)^l+S+T] áplPLlSJTJ_F;J_I|V|J_Iñ.

From [17] we have

where

The two-particle spectroscopic amplitudes are cast as [17]

To continue, we will adopt the extreme shell model andrestrict our attention to cases where the single-proton states arecompletely filled in |J_Cñ. This is so, forthe cores of both (J_I = 1/2,J_C = 0), and (J_I = 1,J_C = 3/2) [36].

In such cases, the final nuclear states take the form |J_Fñ º |(J_C)J_Fñ. This leads to

Putting all this together and performing the summation on J_F, we obtain

where

The intrinsic asymmetry parameter can finally be written as [31]

where

with L = 0 for the 1s_1/2 state, and L = 0 and 1 for the 1p_3/2 state. It has been verified that w₀ = G_p, in such way thatthe new information is carried by w₁.

The presence of the Clebsch-Gordan coefficient in Eq. (17), for k = 1, ensures that l and l¢ have oppositeparities. Since the initial state in the two matrix elements has a definiteparity, this implies that all contributions to w₁ come frominterference terms between the parity-conserving and the parity-violatingparts of the transition potential. Furthermore, the antisymmetrizationfactor in Eq. (11) shows that the two final states have T ¹ T¢. These are general properties of the asymmetry parameter.

We have used these general expressions for the numerical computation of a_L within several OME models. The values of w₀ and w₁ are shown in Tables I and ^II, for and Crespectively, in units of the free L decayconstant, = 2. 50 ×10^-6 eV. In the case of , we have also included, between parentheses,the values obtained with the approximateformula usually adopted in the literature [34], which is strictlyvalid only for s-shell hypernuclei:

where

We find values ranging from -0.62 to -0.29, in qualitative agreement with other theoretical estimates [14-16, 18, 19, 26, 32-35] but in contradiction with some recent experimental determinations [5-9].

We remark here that the approximate formula ignores the fact that the final state of nonmesonic decay is a three-body one and does not include the full contribution of the transitions coming from proton states beyond the s-shell, being therefore of only limited validity, and should not be used for p-shell hypernuclei such as , or, even worse, for heavier ones. This being said, comparison of the corresponding values for a_L in Table I shows that the formula works well within its range of validity.

Also, from ^{Table II} we observe that the p-shell contributions to w₀ and w₁ for are by no means negligible. However, they are also in approximately the same ratio, so that the effect on a_L is much smaller.

In order to obtain simple and manageable analytical expressions for the asymmetry parameter, we adopt now the simplest OME model, which takes into account only the contribution of one pion exchange. As can be seen form our numerical results, the main characteristics of the nonmesonic hypernuclear weak decay asymmetry are well described within this OPEM. Then, as a starting point, this model will be enough to obtain such analytical formula[37].

After a straightforward analytical evaluation of the summation terms indicated in (17), together with the calculation of the nuclear matrix elements indicated in Eqs. (11)-(13) within the strict OPEM, we can write

for , and

for . Here , and , and R = (bP)²/3 are, respectively, the nuclear moments and the ratio between the 1p_3/2 and 1s_1/2 contributions defined in Section IV C from Ref. [17]. These formulas clearly show that the s-part of results agrees with the ones. This has also been tested numerically, as can be seen from Tables I and ^II. There we observe that this conclusion is also valid within the complete OME model. Simultaneously, our numerical computation shows that the contribution of the p-shell of has approximately the same magnitude that the s-one, both for w₀ and w₁ (this has been extensively discussed in Ref. [17] concerning the decay rate).

On the other side, it is also well known that the main contribution comes from the lower quantum numbers and, as it has been discussed in Ref. [17], the tensorial term is the dominant one in the hypernuclear weak decay case. Thus, approximating the p-shell contribution of by the s-one, and retaining only the terms containing the tensorial moment in our formula, we arrive to a very simple expression for a_L:

This expression is valid both for and C hypernuclear decays, within the OPEM. Our results presented in Eqs. (20), (21) and (22) suggest that, in order to obtain a stronger theoretical dependence of a_L on the particular hypernucleus considered (as indicated by experimental data) future improvements of the model are required.

We believe that a different decay mechanism should be introduced for s-and p-shell hypernuclei, beyond the Born approximation. X

Summarizing, we have derived simple formulas for the evaluation of the asymmetry parameter, which exactly include the effects of three-body kinematics in the final states and correctly treat the contribution of transitions originated from proton states beyond the s-shell.

Besides, we have deduced exact analytical expressions for the particular cases of and written, within the OPEM, in terms of nuclear moments.

The numerical values of a_L, calculated within different OME models, variate from -0.62 to -0.29.

The negative value systematically obtained for a_L for the two hypernuclei indicates that it will be hard to get a positive or zero value for it in the case, at least within strict OME models. The puzzle posed by the experimental results for a_L in s- and p-shell hypernuclei remains unexplained.

However, our results suggest that additional changes in the nuclear model and/or the decay mechanism should be introduced for s- and p-shell hypernuclei, in such way that the improvements lead to a stronger theoretical dependence of a_L on the hypernucleus considered, which could reverse the asymmetry sign in the case of , as required to reproduce the more recent experimental data on this observable.

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Received on 18 March, 2006

Publication Dates

• Publication in this collection
  13 Apr 2007
• Date of issue
  Dec 2006

History

• Received
  18 Mar 2006