Asymmetries in the Non-Mesonic Weak Decay of Polarized Lambda-Hypernuclei

The non-mesonic weak decay of polarized Lambda-hypernuclei is studied for the first time by taking into account, with a Monte Carlo intranuclear cascade code, the nucleon final state interactions. A one-meson-exchange model is employed to describe the Lambda N->n N processes in a finite nucleus framework. The relationship between the intrinsic Lambda asymmetry parameter a_\Lambda and the asymmetry a^M_\Lambda accessible in experiments is discussed. A strong dependence of a^M_\Lambda on nucleon final state interactions and detection threshold is obtained. Our results for a^M_\Lambda are consistent with ^{11}_\Lambda B and ^{12}_\Lambda C data but disagree with observations in ^5_\Lambda He.

The non-mesonic weak decay of polarized Λ-hypernuclei is studied for the first time by taking into account, with a Monte Carlo intranuclear cascade code, the nucleon final state interactions. A one-meson-exchange model is employed to describe the ΛN → nN processes in a finite nucleus framework. The relationship between the intrinsic Λ asymmetry parameter aΛ and the asymmetry a M Λ accessible in experiments is discussed. A strong dependence of a M Λ on nucleon final state interactions and detection threshold is obtained. Our results for a M Λ are consistent with 11 Λ B and 12 Λ C data but disagree with observations in 5 Λ He.
Despite this recent progress, the reaction mechanism for the hypernuclear NMWD is not fully understood. Indeed, an intriguing problem, of more recent origin, is open: it concerns the asymmetry of the angular emission of NMWD protons from polarized hypernuclei. This asymmetry is due to the interference between parityviolating and parity-conserving Λp → np transition amplitudes [14]. The study of the asymmetric emission of protons from polarized hypernuclei is supposed to provide information on the spin-parity structure of the ΛN → nN process and hence new constraints on the dynamics of the non-mesonic decay.
The intensity of protons emitted in Λp → np decays along a direction forming an angle θ with the polarization axis is given by (for details see Ref. [15]): where P y is the hypernuclear polarization and A y the hypernuclear asymmetry parameter. Moreover, I 0 is the (isotropic) intensity for an unpolarized hypernucleus, which we normalize as the total number of primary protons produced per NMWD, I 0 = 1/(1 + Γ n /Γ p ). In the shell model weak-coupling scheme, angular momentum algebra expresses the polarization of the Λ spin, p Λ , in terms of P y : p Λ = P y if J = J C + 1/2 and p Λ = −P y J/(J + 1) if J = J C − 1/2, J (J C ) being the hypernucleus (nuclear core) total spin. By introducing the intrinsic Λ asymmetry parameter, a Λ = A y if J = J C + 1/2 and a Λ = −A y (J + 1)/J if J = J C − 1/2, one obtains: A(θ) = p Λ a Λ cos θ. In the hypothesis that the weak-coupling scheme provides a realistic description of the hypernuclear structure, a Λ can be interpreted as the intrinsic Λ asymmetry parameter for the elementary process Λp → np taking place inside the hypernucleus. This scheme is known to be a good approximation for describing the ground state of Λ-hypernuclei and previous calculations [3,15] have proved that, thanks to the large momentum transfer, the non-mesonic decay is not much sensitive to nuclear structure details. Nucleon final state interactions (FSI), subsequent to the NMWD, are expected to modify the weak decay intensity of Eq. (1). Experimentally, one has access to a proton intensity I M (θ) which is generally assumed to have the same θ-dependence as I(θ): Then, the observable asymmetry a M Λ is determined as: Concerning the determination, from data, of the intrinsic Λ asymmetry parameter a Λ , it is important to stress the following two questions originated by nucleon FSI: i) one should demonstrate (experimentally and/or theoretically) that the angular dependence of I M (θ) employed in experimental analyses is realistic; ii) if this is verified, one should investigate the relationship between a Λ and a M Λ , since a M Λ is expected to depend on experimental conditions such as the proton detection threshold and the considered hypernucleus.
The n(π + , K + )Λ reaction is able to produce Λ hypernuclear states with a sizeable amount of spin-polarization [16] preferentially aligned along the axis normal to the reaction plane. Until now, four KEK experiments measured the proton asymmetric emission from polarized Λ-hypernuclei. The 1992 KEK-E160 experiment [17], which studied p-shell hypernuclei, suffered from large uncertainties: only poor statistics and energy resolution could be used; moreover, the values of the Λ polarization p Λ needed to determine the asymmetry a M Λ , had to be evaluated theoretically. More recently, a M Λ was measured by KEK-E278 [18] for the decay of 5 Λ He. The values of p Λ used to obtain a M Λ were determined by observing the asymmetry, A π − = p Λ a π − Λ , in the emission of negative pions in the 5 Λ He mesonic decay, after assuming [19] a π − Λ to be equal to the value for the free Λ → π − p decay, a π − Λ = −0.642 ± 0.013. A similar measurement of p Λ is very difficult, instead, for p-shell hypernuclei due to their small branching ratio and expected asymmetry A π − for the mesonic decay; even the recent and more accurate experiment KEK-E508 [20] had to resort to theoretical estimates [21] for the Λ polarization in 12 Λ C and 11 Λ B. Recently, a M Λ was measured again for 5 Λ He, by KEK-E462 [20], but with improved statistics.
In Table I we report the results for a M Λ obtained by the above mentioned experiments, together with recent theoretical estimates for a Λ . While theoretical models predict negative a Λ values [22], with a moderate dependence on the hypernucleus, the experiments seem to favor negative values for a M Λ ( 12 Λ C) but positive values for a M Λ ( 5 Λ He). Concerning the above comparison between theory and experiment, it is important to stress that, while one predicts a Λ ( 5 Λ He) ≃ a Λ ( 12 Λ C), there is no known reason to expect this approximate equality to be valid for a M Λ . Indeed, the relationship between I(θ) of Eq. (1) and I M (θ) of Eq. (2) can be strongly affected by FSI of the emitted protons: this fact prevents establishing a direct relation between a Λ and a M Λ and to make a direct comparison among results for these quantities. In order to overcome this problem, in the present work we evaluate the effects of the nucleon FSI on the NMWD of 5 Λ He, 11 Λ B and 12 Λ C and we perform the first theoretical estimate of a M Λ . The ΛN → nN weak transition is described with the one-meson-exchange potential of Ref. [3], which accounts for the exchange of π, ρ, K, K * , ω and η mesons and well reproduces the new Γ n /Γ p ratios extracted from KEK data [7] via the weak-interaction-model independent analysis of Refs. [5,11]. The strong final state interactions acting between the weak decay nucleons are taken into account through a scattering nN wave function from the Lippmann-Schwinger equation obtained with the Nijmegen Soft-Core NSC97 (versions "a" and "f") potentials [25]. The two-nucleon stimulated process ΛN N → nN N [10,26] is safely neglected in our analysis. The fraction of protons from two-nucleon induced decays which escapes from the nucleus with an energy above the typical detection threshold is predicted [5] to be small with respect to the fraction originating from ΛN → nN . The propagation of primary (i.e., weak decay) and secondary nucleons (due to FSI) inside the residual nucleus is simulated with the Monte Carlo code of Ref. [27].
In Fig. 1 (2) we show the proton intensity obtained for the non-mesonic decay of 5 Λ He ( 12 Λ C) using the full one-meson-exchange model with the NSC97f potential (the NSC97a potential predicts very similar results). We note that the hypernuclear polarization has been taken to be P y = 1 in these figures, so that the hypernuclear asymmetry parameter A y can be directly extracted from the values of the weak decay intensity at θ = 0 • and θ = 180 • . The continuous histograms correspond to the intensity I(θ) of primary protons [Eq. (1)]. The inclusion of the nucleon FSI strongly modifies the spectra. With vanishing kinetic energy detection threshold, T th p , the intensities are strongly enhanced, especially for 12 Λ C. For T th p = 30 or 50 MeV, the spectra are closer to I(θ), although with a different slope, reflecting the fact that FSI are responsible for a substantial fraction of outgoing protons with energy below these thresholds. A further reduction of I M (θ) is observed for T th p = 70 MeV.  It is evident from Figs. 1 and 2 that the simulated intensities turn out to be well fitted by the linear law in cos θ of Eq. (2). We can thus estimate a M Λ by using Eq. (3) with p Λ = 1 for 5 Λ He, p Λ = −1/2 for 12 Λ C and p Λ = −5/7 for 11 Λ B. To do this, I M (0 • ) (I M (180 • )) is evaluated numerically as the proton intensity in the bin with cos θ ∈ [0.9, 1] (∈ [−1, −0.9]). In Table II (III) we show our predictions for I 0 , I M 0 , a Λ and a M Λ for 5 Λ He ( 11 Λ B and 12 Λ C). They refer to the one-pion-exchange (OPE) and the full one-meson-exchange (OME) models, both using the NSC97f potential. As a result of the nucleon FSI, |a Λ | > ∼ |a M Λ | for any value of the proton threshold:  [20], which correspond to a proton detection threshold varying (from event to event) between 30 and 50 MeV. For these conditions, we obtain OME asymmetries a M Λ rather independent of the hypernucleus and in the range −0.55 ÷ −0.37. The a M Λ values are smaller in size than the corresponding asymmetries before FSI effects, a Λ , by 25 to 50%. It is evident that our OME results are in agreement with the 12 Λ C datum, barely compatible with the 11 Λ B datum and inconsistent with the 5 Λ He datum. One also sees that the OPE asymmetries are systematically smaller, though less realistic from the theoretical point of view, than the OME ones.
In view of the above large discrepancy, we have proved, numerically, that positive a M Λ values -such as the ones measured at KEK for 5 Λ He-can be obtained only if positive values for the intrinsic asymmetry a Λ are enforced in the weak decay intensity I(θ) of Eq. (1): indeed, a Λ and a M Λ always have the same sign. However, unless there are large SU(3) violations in the coupling constants, it seems unlikely that the meson-exchange models give rise to a positive or vanishing value of the intrinsic Λ asymmetry. Indeed, we have analyzed the origin of the large and negative asymmetry parameter in the one-meson-exchange model of Ref. [3], by calculating the two-body ΛN ( 2S+1 L J ) → nN ( 2S ′ +1 L ′ J ) amplitudes a, b, c, d, e, f for 5 Λ He, and determining the intrinsic asymmetry through the following relation [28]: .
In a framework with real ΛN and nN wave functions, the OPE mechanism produces a large and negative a Λ value due, mainly, to an interference between a large and negative tensor amplitude d ( 3 S 1 → 3 D 1 ) and the parity violating amplitudes b ( 1 S 0 → 3 P 0 ) and f ( 3 S 1 → 3 P 1 ), which are both positive and of moderate size. The inclusion of kaon exchange modifies this picture drastically. Destructive interference with the pion in the tensor channel re-duces the d amplitude by a factor of 4, which would lead to a sensitive decrease in the size of a Λ . However, the negative a ( 1 S 0 → 1 S 0 ) and c ( 3 S 1 → 3 S 1 ) amplitudes become one order of magnitude larger in size. Their interference with the positive e ( 3 S 1 → 1 P 1 ) and f ( 3 S 1 → 3 P 1 ) amplitudes end up producing a final value for a Λ which is even 50% larger in size than for OPE alone. The inclusion of the heavier mesons does not change this qualitative behavior. Summarizing, we have seen how FSI are an important ingredient when studying the NMWD of polarized hypernuclei. The first relationship between the intrinsic asymmetry a Λ and the observable asymmetry a M Λ has been established. Unfortunately, not even an analysis including FSI can explain the present experimental data. From the theoretical point of view, we believe it unlikely that new reaction mechanisms are responsible for the present discrepancies. Only small and positive values of a Λ , not predicted by any existing model, could reduce a M Λ to small and positive values.