Nucleon-Nucleon Coincidence Spectra in the Non-Mesonic Weak Decay of Lambda-Hypernuclei and the n/p Puzzle

The main open problem in the physics of Lambda-hypernuclei is the lack of a sound theoretical interpretation of the large experimental values for the ratio G_n/G_p. To approach the problem, we have incorporated a Lambda N->nN one-meson-exchange model in an intranuclear cascade code for a finite nucleus calculation of the nucleon-nucleon (angular and energy) coincidence distributions in the non-mesonic weak decay of 5_Lambda-He and 12_Lambda-C hypernuclei. The two-nucleon induced decay mechanism, Lambda np->nnp, has been taken into account by means of the polarization propagator formalism in local density approximation. With respect to single nucleon spectra studies, the treatment of two-nucleon correlations allow a cleaner and more direct determination, from data, of G_n/G_p. Our results agree with the preliminary coincidence data of KEK-E462 for 5_Lambda-He. This allows us to conclude that G_n/G_p for 5_Lambda-He should be close to 0.46, the value predicted by our one-meson-exchange model for this hypernucleus. Unfortunately, to disentangle one- and two-body induced decay channels from experimental non-mesonic decay events, three-nucleon coincidences should be measured.

Since many years, a sound theoretical explanation of the large experimental values of the ratio, Γ n /Γ p , between the neutron-and proton-induced non-mesonic decay widths, Γ(Λn → nn) and Γ(Λp → np), of Λhypernuclei is missing [1,2]. The calculations underestimate the central data for all considered hypernuclei, although the large experimental error bars do not allow one to reach any definite conclusion. Moreover, in the experiments performed until now it has not been possible to distinguish between nucleons produced by the one-body induced and the (non-negligible) two-body induced decay mechanism, ΛN N → nN N . Because of its strong tensor component, the one-pion-exchange (OPE) model with the ∆I = 1/2 isospin rule supplies very small ratios, typically in the interval 0.05 ÷ 0.20. On the contrary, the OPE description can reproduce the total non-mesonic decay rates observed for light and medium hypernuclei. Other interaction mechanisms beyond OPE might then be responsible for the overestimation of Γ p and the underestimation of Γ n . Those which have been studied extensively in the literature are the following ones: 1) the inclusion in the ΛN → nN transition potential of mesons heavier than the pion (also including the exchange of correlated or uncorrelated two-pions) [3][4][5][6][7]; 2) the inclusion of interaction terms that explicitly violate the ∆I = 1/2 rule [1,8,9]; 3) the inclusion of the two-body induced decay mechanism [10][11][12] and 4) the description of the short range ΛN → nN transition in terms of quark degrees of freedom [13,14], which automatically introduces ∆I = 3/2 contributions.
Recent progress has been made on the subject.
(1) On the one hand, a few calculations [5][6][7]13] with ΛN → nN transition potentials including heavy-meson-exchange and/or direct quark contributions obtained ratios more in agreement with data, without providing, nevertheless, a satisfactory explanation of the puzzle. In particular, these calculations found a reduction of the proton-induced decay width due to the opposite sign of the tensor component of K-exchange with respect to the one for π-exchange. Moreover, the parity violating ΛN ( 3 S 1 ) → nN ( 3 P 1 ) transition, which contributes to both the n-and p-induced processes, is considerably enhanced by K-exchange and direct quark mechanisms and tends to increase Γ n /Γ p [6,13].
(2) On the other hand, an error in the computer program employed in Ref. [15] to evaluate the single nucleon energy spectra from non-mesonic decay has been detected [16]. It consisted in the underestimation, by a factor ten, of the nucleon-nucleon collision probabilities. The correction of such an error leads to quite different spectral shapes and allows one to extract smaller values of Γ n /Γ p (which is a free parameter in the polarization propagator model of Refs. [15,16]) when a comparison with old experimental data for 12 Λ C [17] is done. In the light of these recent developments and of new experiments [18][19][20], it is important to develop different theoretical approaches and strategies for the determination of the Γ n /Γ p ratio. In this Letter we present a finite nucleus calculation of the nucleon-nucleon coincidence distributions in the non-mesonic weak decay of 5 Λ He and 12 Λ C hypernuclei. The work is motivated by the fact that, unlike the single nucleon spectra, correlation observables permit a cleaner and more direct extraction of Γ n /Γ p from data [1]. An experiment performed very recently at KEK [18] has actually measured the angular and energy correlations that we discuss in this paper. Some prelimi-nary results of the experiment can already be compared with our calculations. The one-meson-exchange (OME) weak transition potential we employ to describe the one-nucleon stimulated decays contains the exchange of ρ, K, K * , ω and η mesons in addition to the pion [6]. The final state interactions acting between the two primary nucleons are taken into account by using a scattering N N wave function from the Lippmann-Schwinger (T -matrix) equation obtained with the NSC97f potential [21]. The OME decay rates predicted by this model are the following ones [6]: Γ 1 ≡ Γ n + Γ p = 0.32, Γ n /Γ p = 0.46 for 5 Λ He and Γ 1 = 0.55, Γ n /Γ p = 0.34 for 12 Λ C. The distributions of primary nucleons emitted in twonucleon induced decays and the ratio Γ 2 /Γ 1 has been determined by the polarization propagator method in local density approximation of Ref. [11]: Γ 2 /Γ 1 = 0.20 for 5 Λ He and Γ 2 /Γ 1 = 0.25 for 12 Λ C [22]. In their way out of the nucleus, the primary nucleons, due to collisions with other nucleons, continuously change energy, direction and charge. As a consequence, secondary nucleons are also emitted. We simulate the nucleon propagation inside the residual nucleus with the Monte Carlo code of Ref. [16].
In Fig. 1 we show the kinetic energy correlation of np coincidence pairs emitted in the non-mesonic decay of 5 Λ He. The spectra are normalized per non-mesonic weak decay. To facilitate a comparison with experiments, whose kinetic energy threshold for proton (neutron) detection is typically of about 30 (10) MeV, and to avoid a possible non-realistic behaviour of the intranuclear cascade simulation for low nucleon energies, in all the figures of the paper we required T n , T p ≥ 30 MeV. A narrow peak is predicted close to the Q-value (153 MeV) expected for the proton-induced three-body process 5 Λ He → 3 H + n + p: it is mainly originated by the back-to-back kinematics (cos θ np < −0.8). A broad peak, predominantly due to Λp → np or Λn → nn weak transitions followed by the emission of secondary (less energetic) nucleons, has been found around 140 MeV for cos θ np > −0.8. The kinetic energy correlation for 5 Λ He nn pairs (Fig. 2) shows essentially the same structure of the np distribution just discussed.
In Fig. 3, which corresponds to the energy correlation of 12 Λ C np pairs, the narrow peak appearing at T n + T p ≃ 155 MeV again gets the dominant contribution from back-to-back coincidences. The relevance of the nucleon final state interactions (FSI) in 12 Λ C relative to 5 Λ He can be seen from the second, broader peak appearing in the region around 110 MeV for 12 Λ C and 140 MeV for 5 Λ He. This peak is in fact more pronounced for the heavier hypernucleus. Another consequence of the different FSI effects in 5 Λ He and 12 Λ C is the different magnitude of the tail of the back-to-back distribution at low energies.
Figs. 4 and 5 show the opening angle correlations of nn, np and pp pairs emitted in the decay of 5 Λ He and 12 Λ C, respectively. Comparing both figures for nn and np coincidences, one sees that the back-to-back peaks are more pronounced for 5 Λ He (less sensitive to FSI than 12 Λ C) than for 12 Λ C, while the (almost uniform) tail of these distributions (feeded by FSI) is more significant in 12 Λ C than in 5 Λ He. Since at least one proton of any pp coincidence is a secondary particle, the pp spectra are quite uniform: actually, due to the relevance of the back-to-back kinematics in the weak decay, these distributions slowly decrease as cos θ pp increases. Again as a consequence of FSI, the number of pp pairs is considerably larger in 12 Λ C than in 5 Λ He. The ratio Γ n /Γ p is defined as the ratio between the number of primary weak decay nn and np pairs, N wd nn and N wd np . However, due to nucleon FSI, one expects the inequality: to be valid in a situation, such as the experimental one, in which particular intervals of variability of pair opening angle, ∆θ 12 , and sum energy, ∆(T 1 +T 2 ), are employed in the determination of N nn and N np . Actually, as one can deduce from Figs. 1-5, not only N nn and N np but also the ratio N nn /N np depends on ∆θ 12 and ∆(T 1 + T 2 ). The numbers of nucleon pairs N nn , N np and N pp discussed up to now are related to the corresponding quantities for the one-nucleon (N 1B N N ) and two-nucleon (N 2B N N ) induced processes [the former (latter) being normalized per onebody (two-body) stimulated non-mesonic weak decay] via the following equation: (2) Table I shows the dependence of N nn , N np and N nn /N np on ∆θ 12 and ∆(T 1 + T 2 ) for 12 Λ C. For comparison, the same quantities for the primary weak decay nucleons are listed as well. Without any restriction on θ N N and the nucleon energies, one notes a great increase (by about one order of magnitude) of both the nn and np numbers when the effect of the FSI is taken into account. Again because of FSI, the use of an energy threshold T th N of 30 MeV supplies N nn and N np values considerably reduced with respect to the ones obtained at T th N = 0 MeV. On the contrary, the ratio between the number of nn and np pairs is much less sensitive to FSI effects.  In Table II (III), the ratio N nn /N np for 5 Λ He ( 12 Λ C) is given for different combinations of opening angle interval and nucleon energy threshold. In parentheses we also report the predictions obtained when the two-nucleon induced decay channel is neglected. The results of the figures and tables presented in this work are in a form that permits a direct comparison with KEK-E462 data [18], which are now under analysis. The first preliminary results of KEK-E462 are quoted in Table II. We note that while the central values of the data are more in agreement with the calculation that neglects the effect of the two-body induced decay mechanism, the complete results are compatible with the upper limits of the same data. This could indicate an almost negligible effect of the Λnp → nnp process and/or a Γ n /Γ p ratio slightly lower than the one (0.46) predicted by our OME model for 5 Λ He. To clarify this point, on the one hand one has to wait for the final results of KEK-E462. On the other hand, our calculations show that three-nucleon coincidences are required to disentangle the effects of one-and two-body stimulated decay channels from observed decay events. The simplistic picture that the back-to-back kinematics is able to select one-nucleon induced processes is in fact far from being realistic.
In conclusion, our OME weak interaction model supplemented by FSI through an intranuclear cascade simulation provides two-nucleon coincidence observables which reproduce the preliminary KEK-E462 results for 5 Λ He. This allows us to conclude that Γ n /Γ p for 5 Λ He should be close to 0.46. Although further (theoretical and experimental) confirmation is needed, in this paper we think we have proved how the study of nucleon coincidence observables can offer a promising possibility to solve the longstanding puzzle on the Γ n /Γ p ratio.
In a forthcoming (long) paper [23] we shall discuss the nucleon correlation observables for the (one-and twonucleon stimulated) non-mesonic decay of 5 Λ He and 12 Λ C in a sistematic way. Single nucleon spectra will be further subject of this work. In addition, one should treat the case of 4 Λ H, which is of extreme importance in order to test the validity of the ∆I = 1/2 isospin rule in the ΛN → nN weak transition [1,9,20]