Single and double coincidence nucleon spectra in the weak decay of Λ hypernuclei

Transcription

1 Single and double coincidence nucleon sectra in the weak decay of hyernuclei E. Bauer 1, G. Garbarino 2, A. Parreño 3 and A. Ramos 3 1 Deartamento de Física, Universidad Nacional de La Plata, C. C. 67 e Instituto de Física La Plata, CONICET, 1900 La Plata, Argentina 2 Diartimento di Fisica Teorica, Università di Torino and INFN, Sezione di Torino, I Torino, Italy 3 Deartament d Estructura i Constituents de la Matèria, Universitat de Barcelona, E Barcelona, Sain Abstract Single and double coincidence nucleon sectra in the weak decay of hyernuclei are studied within a nuclear matter formalism extended to finite nuclei via the local density aroximation. With resect to revious calculations, the resent work adots a unified microscoic aroach for both the one and two body induced mechanisms, including the channels nn nnn and n in addition to the mode n nn already considered in earlier henomenological studies. The roagation of the final nucleons in the residual nucleus is simulated by an intranuclear cascade code. Through the comarison of our redictions with 12 C KEK nucleon coincidence data obtained with a nucleon kinetic energy threshold of 30 MeV and an oening angle region cos θ NN 0.7 we determine Γ n /Γ = 0.66 ± We find that the value of Γ n /Γ extracted from single nucleon distributions is strongly affected by final state interaction effects. Some discreancies between measured and calculated roton sectra are also ointed out. Key words: Hyernuclei, Non Mesonic Weak Decay, Γ n /Γ ratio, Two Nucleon Induced Decay PACS: a, Pw Prerint submitted to Nuclear Physics A 20 January 2010

2 1 Introduction Diversified efforts have been devoted to the study of hyernuclear weak decay in the latest years. Theoretical reviews on the subject can be found in [1 3] and recent, related KEK, BNL and FINUDA exeriments in [4 17]. New exeriments are lanned at J PARC [18] and HyHI [19]. Strong evidences were obtained for a solution of the long standing roblem on the ratio Γ n /Γ between the widths for the weak non mesonic rocesses n nn and n. They originated from theoretical analysis [20, 21] of KEK two nucleon coincidence data [8 10]. The roblem was mostly due to a non trivial interretation of exerimental data, which required a careful analysis of nuclear medium effects on the weak decay nucleons, rather than to a oor understanding of the weak decay mechanisms themselves. The lately extracted Γ n /Γ values [20,21] turned out to be in agreement with the revious, ure theoretical estimates of [22 28] obtained by using one meson exchange otentials to describe the one nucleon induced, N nn weak transitions. Nevertheless, further theoretical and exerimental work is desirable in order to confirm the reviously mentioned evidence in favor of a solution of the Γ n /Γ uzzle and to have a more accurate determination of the ratio. Indeed, on the one hand such an evidence relies on articular theoretical descritions of both the weak decay mechanism and the subsequent roagation of the roduced nucleons within the residual nucleus. In this direction, the use of alternative weak decay and/or final state interaction models is of interest. On the other hand, one has to consider another roblem of the field: it concerns the asymmetry of the rotons emitted in the non mesonic decay of olarized hyernuclei, measured to be not far from zero in recent exeriments [29] while a large negative number is redicted by theoretical models [23, 30 32]. In the latest years, a strong effect of nucleon final state interactions (FSI) was ointed out [30], without bringing new hints for a ossible solution of the asymmetry uzzle. Recently, however, the dominating effect on the asymmetry of the scalar isoscalar channel in the weak decay mechanism has been emhasized [33 36], and the two ion exchange chiral model develoed in [22] gives romising results for the rates [22,37] and the asymmetries [37] of both 5 He and 12 C hyernuclei. Establishing the connections existing among the weak decay magnitudes: Γ n /Γ, Γ 2 (the width for two nucleon induced decays, NN nnn), Γ NM = Γ n + Γ + Γ 2 and the asymmetry arameters, and the question concerning the validity of the I = 1/2 isosin rule in the non mesonic decay are other imortant issues which deserve future investigations in the rosect of a better understanding of baryon baryon weak 2

3 interactions. On the exerimental side, very recent measurements of single [4 7, 13, 15, 17] and double coincidence [8 10] nucleon sectra from the non mesonic hyernuclear decay were reorted with imroved recision with resect to the ones at disosal in revious exeriments [38 42] and in forms that suggest suitable comarisons with theory. Some of these exeriments have somehow managed to derive values of Γ n /Γ. They are discussed in the following aragrahs. The Γ n /Γ ratio was determined by KEK E307 [4,5] for C, Si and Fe hyernuclei from single roton kinetic energy sectra measurements and by making use of the intranuclear cascade code of [43] (based on the olarization roagator formalism of [44]) to simulate nucleon re scattering in the residual nucleus. For 12 C, Γ n /Γ = 0.87 ± 0.23 was obtained in [5] by neglecting the two nucleon induced decay mechanism and Γ n /Γ = by assuming Γ 2 /(Γ n + Γ ) = Γ(n nn)/(γ n + Γ ) = As the very same authors of [4,5] noted in [6], these determinations of the ratio may be affected by the fact that in the exeriment the neutron induced decay width was estimated indirectly, from the roton measurement, using the relation Γ n = Γ T Γ Γ π Γ π 0( Γ 2 ). This method also required the measurement of the total decay width Γ T as well as the decay rates for the mesonic channels π (Γ π ) and π 0 n (Γ π 0, for which revious data from [39] were used in the analysis of [5]). Moreover, the severe energy losses suffered by rotons inside the (thick) target and detector materials and the consequently high kinetic energy threshold (about 40 MeV) for roton detection in KEK E307 did not ermit an easy reconstruction of the low energy art of the roton sectrum, which is essential for the indirect evaluation of Γ n. As a consequence, in [6] the hyothesis was advanced that Γ n (Γ ) might be overestimated (underestimated) in the analysis of [5] because of an underestimation in the number of emitted rotons. A controversial determination of Γ n /Γ from KEK E369 data, based on unroven, delicate hyotheses and theoretical inut (again from [43]), was reorted in [6]. In this exeriment, direct measurements of single neutron kinetic energy sectra were erformed (with a 10 MeV threshold) for 12 C and 89 Y; once analyzed together with the single roton sectra of [4,5], a ratio Γ n /Γ = 0.51 ± 0.15 for 12 C was derived by neglecting the two nucleon induced decay channel. To overcome the difficulties of the discussed exeriments, both single neutron and single roton energy sectra were measured simultaneously by KEK 3

4 E462 for 5 12 He and KEK E508 for C [7]. From these measurements, the authors concluded that Γ n /Γ (N n /N 1)/2 0.5 for both 5 He and 12 C, N n (N ) being the total number of neutrons (rotons) with kinetic energies T N above 60 MeV. However, one has to note that the revious aroximate relation between Γ n /Γ and N n /N is only valid when FSI and two nucleon induced decay effects can be neglected. The redictions of [21] and the results resented in Sections 3.1 and 3.2 rove that FSI are not negligible even when a high detection threshold such as TN th = 60 MeV is used. In the exeriments KEK E462 ( 5 He) and KEK E508 ( 12 C), nucleon nucleon coincidence sectra were also measured [8 10]. Quite clean angular and energy correlations between neutron neutron and neutron roton emitted airs (i.e., back to back kinematics and T N1 + T N2 155 MeV) were observed, thus reresenting the first direct exerimental evidence of the existence of the one body induced decays n nn and n. The ratio, N nn /N n, between the numbers of emitted neutron neutron and neutron roton airs was measured to be around 0.5 for both 5 He and 12 C after alying the angular and energy restrictions: cosθ NN 0.8 and T N 30 MeV. The authors of [8,9] concluded that, under these constraints, Γ n /Γ N nn /N n 0.5 on the basis of a suosed cancellation of FSI and two nucleon stimulated decays effects in the ratio N nn /N n. In a more recent work [10], the result Γ n /Γ = 0.51±0.13±0.05 was deduced by the KEK collaboration for 12 C after correcting for FSI effects by making use of the number of detected roton roton airs in addition to measurements of N nn and N n. Two nucleon stimulated decays were not taken into account. A method similar to the one used in [6] was alied to determine Γ n /Γ. We shall widely discuss the non negligible effect of FSI in the extraction of Γ n /Γ from measurements of N nn /N n in Section 3.5. Desite this recent exerimental rogress, imroved and/or indeendent measurements are awaited for a really comlete understanding of the N nn reaction in nuclei. On this resect, the observation of the weak decay of neutron and roton rich hyernuclei [14,19] would also be source of new information. Beyond these considerations, from the above aragrahs it is clear that the extraction of the Γ n /Γ ratio from data is a comlex task, once FSI and the two body induced mechanism lay non negligible roles. A theoretical aroach for the extraction of Γ n /Γ from exerimental sectra (which, contrary to the decay rates, are the actual observable quantities) is required. Values of Γ n /Γ determined in such a way are thus model deendent; only when different aroaches lead to the same Γ n /Γ it is fair to call this ratio the exerimental value of Γ n /Γ. 4

5 On the theoretical side, an extensive study of single and double coincidence nucleon sectra for the non mesonic decay of 5 He and 12 C hyernuclei was resented in [20, 21]. A one meson exchange (OME) otential was used for the N nn transition in a finite nucleus framework. The model adoted for the two nucleon induced decay channel is described in [44]. In simle terms, this model results from the roduct of an aroximation to the 32h hase sace times a constant which embodies, henomenologically, the strong interaction dynamics. The rocess was assumed to roceed entirely from the n nn reaction. The calculation was erformed in nuclear matter together with the local density aroximation. The intranuclear cascade code of [43] was used to simulate the nucleon roagation inside the residual nucleus. Comarison with KEK E462 and KEK E508 two nucleon coincidence data [8, 9] lead to the determination of Γ n /Γ values around for both 5 12 He and C [21]. It was shown that FSI and two nucleon induced decays affect the extraction of Γ n /Γ from two nucleon coincidence data even when favorable energy and angular correlation restrictions are imosed on the observed nucleon airs. It should be noted that an asymmetric treatment was given to one and two body induced decay mechanisms in [20, 21]: while the one body stimulated decay is evaluated in a finite nucleus framework, an aroximated model in nuclear matter is used in the calculation of the two body induced decay. The rather simle two body induced decay model emloyed in those works, together with the non negligible role layed by the two body stimulated rocess in the interretation of data, have motivated us to exlore a unified aroach for both the one and the two body stimulated decay mechanisms. Our scheme for the decay rates, develoed in [25 28], uts on the same ground the microscoic evaluation of one and two body stimulated decay mechanisms and includes the channels nn nnn and n besides the standard mode n nn. This scheme uses the same OME weak transition otential (containing π, ρ, K, K, ω and η exchange) of [23]. The huge number of two body induced configurations makes the use of nuclear matter more convenient, the redictions being extended to finite nuclei via the local density aroximation. Most of the results shown in the resent work are for the intermediate mass hyernucleus 12 C, but we also resent some results for a heavier hyernucleus, 89 Y. FSI are evaluated by means of the Monte Carlo intranuclear cascade code of [43]. Our aim is twofold. First, we want to discuss redictions for the nucleon emission sectra using the unified microscoic aroach. Afterwards, by fitting exerimental sectra we address the question of the determination of the exerimental value of Γ n /Γ. Our main achievement is an agreement of the resent results with the ones of [20,21] concerning the ratio N nn /N n. This suorts the idea that it is ossible to determine from data a a basically model 5

6 indeendent value for Γ n /Γ, which then truly qualifies as the exerimental value of the ratio. The aer is organized in the following way. The weak decay model emloyed in the calculation is outlined in Section 2.1. In Section 2.2 we give a very brief summary of the intranuclear cascade simulation. Numerical results for single and double coincidence nucleon distributions are resented and comared with data in Section 3. The contribution of the two nucleon induced decay channels is analyzed with secial regard in this Section. Finally, our conclusions are drawn in Section 4. 2 Model 2.1 Weak decay Let us consider the one and two nucleon induced non mesonic weak decay widths for a hyeron with four momentum k = (k 0, k) inside infinite nuclear matter with Fermi momentum k F. In a schematic way, one can write: Γ 1(2) (k, k F ) = f f V N NN 0 kf 2 δ(e f E 0 ), (1) where V N NN is the weak transition otential, 0 kf denotes the initial hyernuclear ground state with energy E 0 and f stands for a ossible 21h or 32h final state with energy E f. The 21h and 32h final states contribute to Γ 1 and Γ 2, resectively. The decay rates for a finite hyernucleus are obtained via the local density aroximation [45] Γ 1(2) = dk ψ (k) 2 dr ψ (r) 2 Γ 1(2) (k, k F (r)), (2) i.e., by averaging the artial width over the nuclear volume, assuming a local Fermi momentum k F (r) = {3π 2 ρ(r)/2} 1/3, where ρ(r) is the density rofile of the nuclear core. This average is weighted by the robability er unit volume of finding the at a given osition r, ψ (r) 2. In a hyernucleus, one needs to further average over the extended momentum distribution of the hyeron, ψ (k), the Fourier transform of ψ (r), for which we adot a 1s 1/2 harmonic oscillator wave function with frequency ω = 10.8 MeV adjusted to 6

7 the exerimental energy searation between the s and levels in 12 C. The hyeron energy is given by k 0 = m + k 2 /(2m ) + V, i.e., it also contains an exerimental binding term V = 10.8 MeV. Since V N NN is a two body oerator, the emission of two nucleons may originate either from the Hartree Fock vacuum or from ground state correlations induced by the nucleon nucleon interaction. At variance, the emission of three nucleons can only be achieved when V N NN acts over a ground state correlation. It is therefore convenient to introduce the following model for the hyernuclear ground state wave function [27]: 0 kf = N(k F ) 2h 2 3 h 3 V NN 2 2 h 2 3 h 3 ε 2 ε h2 + ε 3 ε h 2 3 h 3,(3) h3 where is the uncorrelated ground state wave function, i.e., the Hartree Fock vacuum, while the second term in the rhs reresents 22h correlations introduced by the nuclear residual interaction V NN. The articular labeling 2, h 2, 3 and h 3 (each symbol denoting article and hole four momenta and sin and isosin rojections) is exlained in Subsec Besides, is the normalized state of the, the article and hole energies are denoted by ε i and: N(k F ) = h 2 3 h 3 2 h 2 3 h 3 V NN 2 1/2 ε 2 ε h2 + ε 3 ε h3 (4) is the ground state normalization factor. The exlicit exression for N(k F ) is given in [28] One nucleon induced decay By inserting Eq. (3) into Eq. (1), one obtains for Γ 1 : Γ 1 (k, k F ) = N 2 (k F ) f f V N NN 2 δ(e f E 0 ), (5) the final states f being restricted to 21h states. Note that in this exression we have ket only the dominant Hartree Fock vacuum contribution. The corresonding Goldstone diagrams for the direct and exchange self energies are shown in Fig. 1. Exlicit exressions for Γ 1 are found in [25]. 7

8 q h 2 h 2 2 q 1 q Fig. 1. Direct and exchange self energy diagrams corresonding to the one nucleon induced decay channel in nuclear matter Two nucleon induced decay We consider now the two nucleon induced decay mode. From Eqs. (1) and (3), one gets for Γ 2 : Γ 2 (k, k F )=N 2 (k F ) f V N NN 2 h 2 3 h 3 ; f 2 h 2 3 h 3 2h 2 3 h 3 ; V NN 2 δ(e ε 2 ε h2 + ε 3 ε h3 f E 0 ), (6) where f stands now for the ossible 32h states. At variance with Γ 1, several Goldstone diagrams contribute to Γ 2. A microscoic evaluation of the twonucleon induced decay is a quite involved task which should be ursued on a ste by ste basis. The comlexity of the roblem makes the henomenological aroaches [44, 46] very valuable because they were able to establish a first estimation of the size of the two body induced decay mechanism, as comared to the dominant one nucleon induced one, as well as its effect on the determination of the ratio Γ n /Γ. In fact the first model for evaluating Γ 2 [46] was roosed in the hoe of understanding the large fraction of neutrons, and hence the large value of Γ n /Γ, observed in the exeriments. The work was insired by the revious knowledge on ion absortion. The idea was that the near on shell ion roduced at the weak vertex would be referentially absorbed by a neutron roton air roducing twice as many neutrons than rotons from this rocess. Thereafter, a hybrid model was emloyed in [44], where the strength of the mechanism was extracted from the henomenology of the two nucleon absortion of real ions, and the extension to other kinematical regions was done by taking an aroximation for the 22h hase sace. It was exlicitly found that the consideration of the two-nucleon induced channel has a 8

9 non negligible effect on the extraction of the ratio Γ n /Γ from the number of emitted rotons and neutrons. A following work [43] determined the energy distribution of nucleons, in order see whether the two decay mechanisms, one and two nucleon induced, could be disentangled for a roer evaluation of Γ n /Γ. These revious studies, based on a henomenological evaluation of Γ 2, established the basis for the analyses erformed in [20,21] of exerimental two nucleon coincidence nucleon sectra [8,9], which contributed to the solution of the so called Γ n /Γ uzzle. It is therefore imortant to check whether the value of the ratio is stable against the use of different models of Γ 2, in other words, to try to establish a model indeendent value for Γ n /Γ. This is the main urose of this work. In articular, we will emloy a model which is based on the same microscoic aroach as that adoted for the calculation of the one nucleon induced mechanisms. Our unified scheme considers a full meson exchange model in the weak transition otential and 22h intermediate states excited by a realistic strong NN interaction. This roduces a quite different energy distribution of the three nucleons emitted in the two nucleon absortion rocesses, as we will see in Sect In the resent work we will kee only the dominating terms of the two nucleon stimulated contributions, namely those obtained by attaching the weak transition otential V N NN to the same 11h bubble, an examle of which is seen in Fig. 2. Actually, a more comlete set of diagrams, including terms with the two weak transition otentials connected to different 11h bubbles as well as Pauli exchange contributions, were discussed in [27,28]. The imlementation of the full set of diagrams in the evaluation of the sectra is quite involve and goes beyond the scoe of the resent contribution. Moreover, by keeing the dominating terms it is exected that these missing terms would not modify the value extracted for Γ n /Γ in a significant way. Therefore the conclusions of the resent work should not be altered. Note that there are three different contributions to the dominating terms, which we denote by Γ 2, Γ h 2 and Γ hh 2 (Γ 2 = Γ 2 +Γ h 2 +Γ hh 2 ). In the first one, the two V N nn are attached to the same article (see Fig. 2, from which the notation emloyed in Eqs. (3) and (4) becomes clear). In the h contribution one V N nn is connected to a article and the other one to a hole. Finally, the two otentials are attached to the same hole for the hh art. The corresonding h and hh Goldstone diagrams can be found in [26] (Fig. 1) together with their analytical exressions. In addition, the isosin summation in Eq. (6) allows three isosin decay channels, Γ 2 = Γ nn + Γ n + Γ, with: Γ nn Γ(nn nnn), Γ n Γ(n nn) and Γ Γ( n). The results obtained in [26] imly Γ n : Γ : Γ nn 0.78 : 0.17 : 0.05, which oints to the need of incororating also the effect of the isosin contributions Γ and Γ nn, neglected in 9

10 q q 2 h 2 3 h 3 q 0 2 q 0 Fig. 2. art of the two article two hole contribution to the self energy in nuclear matter. the henomenological models both for the evaluation of the nucleon sectra and for the extraction of the Γ n /Γ ratio from data. Finally, we oint out that for the nuclear residual interaction V NN (which enters both Γ 2 and N(k F )) we have used the Bonn otential [47] in the framework of the arametrization resented in [48], which contains the exchange of π, ρ, σ and ω mesons. 2.2 Intranuclear cascade After the interacts with one (two) nucleons, the two (three) roduced nucleons will interact with other nucleons in their way out of the nucleus. This rocess, which will generate secondary nucleons, is accounted for by the intranuclear cascade model described in [43]. This model considers a semiclassical roagation of rimary (i.e., weak decay) and secondary nucleons. Nucleons move along classical, straight trajectories between collision oints and under the influence of a local (i.e., r deendent) mean otential, V (r) = k F (r) 2 /2m, where k F (r) = [3π 2 ρ(r)/2] 1/3 is the local nucleon Fermi momentum corresonding to the nucleon density ρ(r). Proagating nucleons collide with the nucleons of the medium according to free sace nucleon nucleon cross sections roerly corrected to take into account the Pauli blocking and Fermi motion effects. At small energies, for which the nucleon wavelength is comarable in size to the average distance between nucleons, a semiclassical nucleon roagation is not aroriate. For this reason we do not consider our sectra below 30 MeV kinetic energy to be realistic. A quantum mechanical simulation of FSI would be referred but it is not yet available. We finally note that the adoted code was tested successfully in a variety of other reactions [49]. For more details of the code we refer to [43]. 10

11 3 Results We start our discussion by observing that the models of [23,25,26] on which the resent work is based have been recently imroved. On one hand, a more realistic wave function, obtained in terms of the exerimental hyeron binding energy, has been considered in the local density aroximation (LDA) calculation of [25,26], leading to a reduction of about 25% on the one nucleon induced (hereafter referred to as 1N induced) decay rates, Γ n and Γ, and of 30-40% on the two nucleon induced (2N induced) ones, Γ nn, Γ n and Γ, with resect to the original results. In addition, the normalization of the hyernuclear ground state (see Eqs. (3) and (4)) roduces a further reduction of both Γ 1 and Γ 2 of about 30%. The finite nucleus aroach of [23] has been imroved by using numerically more accurate distorted final state wave functions and the rates increase by about 15%. In both cases the ratio Γ n /Γ is essentially left unchanged. Fortunately, being normalized er non mesonic weak decay, the sectra discussed in [20,21] are not affected by the corrections of [23]. The udated redictions, used in the resent work, are given in Table 1 together with exerimental results for Γ n, Γ, Γ NM, Γ n /Γ and Γ 2 /Γ NM. The two sets of results for the finite nucleus calculation, denoted by OMEa and OMEf, are obtained by using, resectively, the Nijmegen otentials NSC97a and NSC97b [50] for the strong vertices and final state NN interactions. The weak transition otential V N NN is modeled, in both finite nucleus and LDA models, by the exchange of the π, η, K, ρ, ω and K mesons, whose formulation has been taken from [23]. The finite nucleus results for Γ NM underestimate data, as exected, since they lack the contribution of the 2N stimulated decay mechanism. However, the results obtained for Γ n and Γ within the model OMEf are in good agreement with the data of [12]. The LDA rediction for Γ NM slightly underestimates the most recent data of [8,12] but agrees with the revious determinations of [5,41]. Concerning Γ n /Γ, with the excetion of the data from [12], we have only quoted in Table 1 exerimental determinations obtained from single nucleon measurements. Most of these determinations aear to be underestimated by any theoretical rediction found in the literature, which leads to the well known Γ n /Γ uzzle. We note that large exerimental errors affect all but the most recent data, as mentioned in the Introduction. On the contrary, the analyses of recent two nucleon coincidence data erformed in [21] enabled to determine central values for Γ n /Γ around , in agreement with ure theoretical estimates. 11

12 Table 1 Non mesonic weak decay rates (in units of the free decay width) redicted for 12 C by the udated finite nucleus aroach (OMEa and OMEf calculations) of [23] and LDA model of [25 27]. Γ n Γ Γ nn Γ n Γ Γ n /Γ Γ NM FNa FNf LDA Γ n Γ Γ 2 /Γ NM Γ n /Γ Γ NM FINUDA [17] 0.24 ± 0.10 KEK E508 [12] 0.23 ± ± ± ± ± 0.04 KEK E508 [8] ± KEK E369 [6] 0.51 ± 0.15 KEK E307 [5] 0.87 ± 0.09 ± ± ± KEK [41] 1.87 ± ± 0.15 ± 0.03 BNL [40] ± 0.20 BNL [38] 0.58 ± 0.18 Concerning the 2N stimulated decay widths, the dominant contribution (about 75%) of the microscoic calculation originates from the n induced decay, n nn. Nevertheless, the other 2N induced channels cannot be neglected. The LDA rediction Γ 2 /Γ NM = 0.21 is in agreement with the henomenological estimate of [20,21], Γ 2 /Γ NM = 0.20, as well as with the very recent exerimental determinations of the FINUDA collaboration [17] and of KEK E508 [12] reorted in Table Single nucleon sectra In Figure 3 we comare results of the resent LDA calculation with those of the revious finite nucleus evaluation of [20, 21]. The number of rimary rotons emitted in the 1N induced non mesonic weak decay of 12 C is given as a function of the roton kinetic energy. In order to make the comarison model indeendent, the sectra are normalized er 1N induced decay assuming Γ n /Γ = (Γ n /Γ ) LDA = 0.285, which is equivalent to normalize er the corresonding roton induced decay rate. We observe that the Full Width at Half Maximum (FWHM) of the LDA distribution is MeV larger than the one of the finite nucleus sectrum, which should be considered more realistic. This is essentially due to the larger disersion of Fermi motion in the nuclear matter calculation. In Figure 4 we show the single roton kinetic energy sectra for the non mesonic weak decay of 12 C once 2N induced decays and FSI effects are in- 12

13 Finite Nucleus LDA N T (MeV) Fig. 3. Kinetic energy sectra of rimary rotons from the 1N induced non mesonic weak decay of 12 C. The continuous line refers to the resent LDA calculation, the dashed one to the finite nucleus evaluation of [20, 21]. Both curves are normalized er 1N induced decay assuming Γ n /Γ = (Γ n /Γ ) LDA = LDA Finite Nucleus old LDA with Γ n /Γ =0.285 old LDA with Γ n /Γ =1 KEK - E508 BNL-CERN N T (MeV) Fig. 4. Single roton kinetic energy sectra for the non mesonic weak decay of 12 C after 2N induced decays and FSI effects are included. Data are from KEK E508 [7] and Montwill et al. [38]. All results are normalized er non mesonic weak decay. See detailed exlanation in the text. cluded. All sectra are normalized er non mesonic weak decay. The continuous line refers to the resent LDA result. The dashed line has been obtained with the LDA of [43], where the weak transition otential was described by a correlated ion exchange. The old LDA sectrum has been obtained assuming same Γ n /Γ of the resent model, Γ n /Γ = (Γ n /Γ ) LDA = Good agreement is found between the two LDA calculations desite the different weak decay models emloyed. The dot dashed line has been taken from the finite nucleus evaluation of [20,21] fixing Γ n /Γ = (Γ n /Γ ) LDA = Again, the 13

14 differences between LDA and finite nucleus estimates are mainly due to the different hase saces in the two cases. As it is aarent from Figure 4, all the theoretical sectra are in strong disagreement with KEK E508 data [7]. The LDA results are closer to the data by Montwill et al. [38] 1, esecially if we a large value of Γ n /Γ such as 1 is enforced in the old LDA (dotted curve). We recall that such LDA model [43] was also able to reroduce the KEK E307 single roton data of [5] with values of Γ n /Γ larger than 0.6, as reorted in the Introduction. Note, however, that the roton sectrum obtained using the more realistic hase sace of the finite nucleus model (dot dashed line) is in disagreement with all exerimental single roton sectra. Aarently, given the smooth deendence of the normalized roton yield on Γ n /Γ, the only way to reconcile the theoretical redictions of models having Γ n /Γ 1 with the exerimental data would be the assumtion of substantial modifications in the FSI aroach emloyed, but this would then soil the success found by the same aroach in other nuclear reactions [49]. Note also that there exist in the literature two other exeriments giving the distribution of rotons following the weak decay of 12 C, that of Ref. [40] and the recent one from the FINUDA collaboration [15, 17]. Since these sectra are unnormalized they cannot be dislayed directly in Fig. 4. However, there is a noticeable qualitative difference: the shae of the recent FINUDA sectrum shows a shallow eak at T 80 MeV (the back to back kinematics) that no theoretical calculation obtains and which is also lacking in any of the other exerimental data sets. This is an interesting oint that deserves a more careful study. Given the resent disersion of the different exerimental results (not only in shae but also in size), we cannot yet use this observable to constrain the various non mesonic weak decay models. In Fig. 5 the results of the resent LDA calculation for the the single neutron sectrum are shown, both for the rimary neutrons coming from the 1N induced mechanism (dashed line) and after including 2N induced decays and FSI effects (solid line). The full result is in good agreement with KEK E369 data but overestimates somewhat the more recise KEK E508 sectrum. Note also that a more realistic finite nucleus calculation, based on the same one meson exchange model for the weak transition, also obtains agreement with the KEK E369 data (see Fig. 6 of Ref. [21]), but going through the higher 1 Actually, Montwill s emulsion data refer to a a mixture of boron ( 9 B, 10 B and 11 B), carbon (11 C, 12 C and 13 C) and nitrogen (13 N, 14 N and 15 N) hyernuclei. As exected, the roton sectra we redict for all these hyernuclei are very similar to each other. It is thus justified to comare Montwill s sectrum with other data and our results for 12 C, as done in Figure 4. 14

15 N without FSI 1N+2N+FSI KEK - E508 KEK - E369 N n T n (MeV) Fig. 5. Single-neutron kinetic energy sectra for the non-mesonic decay of 12 C, obtained from our LDA model which redicts Γ n /Γ = The dashed line corresonds to the distribution of rimary neutrons, while the continuous line is obtained once 2N induced decays and FSI effects are included. Data are from KEK E369 [6] and KEK E508 [7]. The sectrum of rimary neutrons (exerimental data and the full theoretical result) is (are) normalized er 1N induced (total) non mesonic weak decay. side of the error bars. The evolution of the neutron sectrum with the nuclear size can be easily exlored in a LDA tye aroach, whereas it would become a rohibitive task in a finite nucleus model, esecially in the higher mass region. In Fig. 6 we comare our LDA single neutron sectrum with that measured by the exeriment KEK E369 [6] in the case of 89 Y. The decent descrition of data rovided by the resent LDA model for a hyernucleus as heavy as 89 Y is already an indicator of the reliability of the intranuclear cascade code used to simulate the nucleon FSI in hyernuclear decay. 3.2 Ratio N n /N We would like to comare now our results for the ratio between the number of neutrons and the number of rotons roduced in the decay of 12 C with exerimental observations. Since our definitive aim is to determine Γ n /Γ by such a comarison, it is convenient [21] to start by introducing the number of nucleons of tye N (N = n or ) roduced, for a given energy cut, in n induced (NN 1Bn ), induced (N1B N ), nn induced (NN 2Bnn ), n induced (N 2Bn N ) and induced (N 2B N ) decays. By normalizing these quantities er n,, 15

16 N+2N KEK E369 N n T n (MeV) Fig. 6. Single neutron kinetic energy sectra for the non mesonic weak decay of 89 Y. Data are from KEK E369 [6]. All results are normalized er non mesonic weak decay. nn, n and induced decay, resectively, the total number of nucleons of the tye N normalized er non mesonic weak decay is given by: N N = N1Bn N Γ n + N 1B N Γ + NN 2Bnn Γ nn + N 2Bn N Γ n + N 2B N Γ. (7) Γ n + Γ + Γ nn + Γ n + Γ By definition, the nucleon numbers NN 1Bn, N1B N, etc, are indeendent of the model emloyed to describe the weak decay. In the absence of FSI effects, one would have N 1Bn n 0,N 2Bn n = N 2B = 2, N 1B n = 2, N 2Bn = N 1B = N 2B n = 1, N 1Bn = 0, Nn 2Bnn = 3,N 2Bnn = = 1. Their actual values deend on the strong interaction art of the roblem, related to nucleon FSI, and on the framework (finite nucleus or nuclear matter) used for treating the hyernuclear structure effects, which roduces different hase sace factors. The deendence on the weak decay model enters Eq. (7) via the various artial decay widths. In Table 2 we reort our results for the weak decay model indeendent nucleon numbers for two kinetic energy thresholds for nucleon detection, TN th = 40 and 60 MeV. From Eq. (7), our results of Table 1, and taking the values of Table 2 for TN th = 60 MeV, we then determine: N n N = (8) We have to note that this result is close to the ones obtained in the finite nucleus calculation of [21], namely N n /N = 1.38 and 1.42, using the OMEa 16

17 Table 2 Predictions for the weak interaction model indeendent quantities NN 1Bn, N1B N, NN 2Bnn, N 2Bn N, N 2B N and for NN 2B of Eqs. (7) and (10) (12) for 12 C by alying nucleon kinetic energy thresholds of TN th = 40 and 60 MeV. T th N (MeV) N1Bn n N 1B n N 2Bnn n N 2Bn n N 2B n N 2B n T th N (MeV) N1Bn N 1B N 2Bnn N 2Bn N 2B N 2B and OMEf model, resectively. If the 2N stimulated decay mode is neglected, the resent calculation redicts ( Nn ) 1N = (9) N These results underestimate the value N n /N = 2.00±0.17 obtained by KEK E508 [7] for TN th = 60 MeV. Such an occurrence is related to the disagreement already observed for the single roton sectra of Fig. 4. Our results for TN th = 40 MeV, namely N n /N = 1.26 and (N n /N ) 1N = 1.23, do not comare well either with the revious exerimental determination, N n /N = 1.73±0.22, obtained from KEK E369 and KEK E307 data [6] by alying the same detection threshold. This discreancy will obviously be reflected in the value of the Γ n /Γ ratio that can be extracted from the one nucleon observables. From Eq. (7) one obtains: N n N = Nn 1Bn Γ n + Nn 1B + 1 ( Γ Γ 1 ( Γ N 1Bn n + N 1B + 1 Γ Γ Γ n Γ 1 + Γ n Γ ) (N 2Bnn n ) (N 2Bnn ) Γ nn + Nn 2Bn Γ n + Nn 2B Γ ),(10) Γ nn + N 2Bn Γ n + N 2B Γ from which one derives the ratio Γ n /Γ : Γ n Γ = N 1B n ( N 1Bn + Nn 2B Γ 2 + N 2B Γ 1 Γ 2 Γ 1 ( N 1B + N 2B ) Nn Nn 1Bn N ) Γ 2 Nn Γ 1 N Γ 2 N 2B n Γ 1, (11) 17

18 where N 2B N N2Bnn N Γ nn + N 2Bn N Γ n + N 2B N Γ. (12) Γ nn + Γ n + Γ Therefore, the ratio Γ n /Γ can be determined in terms of the observed ratio N n /N, the weak model indeendent nucleon numbers NN 1Bn, N 1B N, etc (which however deend on FSI effects), and a theoretical estimation for the two nucleon stimulated widths Γ nn /Γ 1, Γ n /Γ 1 and Γ /Γ 1. Using our redictions of Tables 1 and 2 together with the datum of KEK E508 [7] for TN th = 60 MeV, N n /N = 2.00 ± 0.17, we obtain: Γ n Γ = 0.95 ± (13) Neglecting the 2N stimulated channel the result is: ( Γn ) 1N = 0.88 ± 0.16, (14) Γ while enhancing arbitrarily the 2N induced rates by a factor of two we obtain: ( Γn ) Γ2 2Γ 2 = 1.02 ± (15) Γ The sensitivity of the ratio Γ n /Γ to the values of the 2N induced decay widths turns out to be moderate, i.e., well within the error bars. This is due not only to the minor role of the 2N stimulated rocesses in Eq. (11) (N 1B n() and Nn() 1Bn are larger than N2B n() Γ 2/Γ 1 since a quite high energy threshold is emloyed), but also to the articular value of N n /N used in the analysis, which causes a certain cancellation among the 2N stimulated contributions in both the numerator and denominator of Eq. (11): N 2B n N2B N n/n. These occurrences can be better visualized in Fig. 7, which shows the relation between the observable ratio N n /N and Γ n /Γ for different choices of Γ 2 /Γ 1 and assuming a detection threshold of T N = 60 MeV. The dot dashed line refers to the calculation in which Γ 2 is set to 0, the continuous line to the Γ 2 /Γ 1 ratio redicted by the resent LDA model and the dashed line to the case in which the size of Γ 2 is arbitrarily doubled. The dotted line corresonds to the case in which 2N induced decays and FSI are neglected. 18

19 T N > 60 MeV 2.4 N n / N KEK E508 Γ 2 = Γ 1 Γ 2 = Γ 1 Γ 2 = 0 Γ 2 = 0, no FSI Γ n /Γ Fig. 7. Deendence of the observable ratio N n /N on Γ n /Γ and Γ 2 /Γ 1 for 12 C and a nucleon energy threshold of 60 MeV. The horizontal lines show KEK E508 data [7]. See text for further details. Note that, even after alying a high kinetic thresholds such as 60 MeV, FSI effects are not negligible. This is why the reviously extracted Γ n /Γ values are in disagreement with the value 0.5 given in [7] and exected on the basis of the relation Γ n /Γ = (N n /N 1)/2, which holds if 2N induced decays and FSI effects are ignored. On the contrary, it is well manifest from Fig. 7 that the 2N induced decay mechanism lays a relatively small role in the whole range of reasonable Γ n /Γ values. We end this subsection by remarking that the value of for Γ n /Γ redicted by the weak decay model emloyed in the resent aer, as well as those of the other available weak decay models, strongly underestimate the values of Eqs. (13) (15). Again, this is just a consequence of the discreancy found with the measured single roton sectra, which in turn show an unfortunate disersion among themselves. This, together with the strong effect of FSI deicted in Fig. 7 makes it advisable to resort to better suited quantities for the determination of Γ n /Γ, such as the two nucleon coincidence observables, in articular the ratio of nn to n airs discussed in Sect Double coincidence nucleon sectra Now we discuss the NN coincidence sectra obtained from our model for the decay of 12 C. Figs. 8 and 9 show, resectively, the distribution of nn and n airs, as a function of the cosine of the oening angle, where FSI effects have 19

20 from 1N induced from 2N induced Total T N > 30 MeV N nn cos θ nn Fig. 8. Oening angle distribution of nn airs normalized er non mesonic weak decay for 12 C from 1N induced from 2N induced Total T N > 30 MeV N n cos θ n Fig. 9. Oening angle distribution of n airs normalized er non mesonic weak decay for 12 C. been incororated and a kinetic energy cut of 30 MeV has been alied. We observe that the distribution of airs from the 1N induced rocesses (dash dotted line) is more back to back dominated than the one coming from the 2N induced rocesses (dashed line). In any case, the 1N induced channel still rovides the larger contribution of airs in the whole range of oening angles. For cos θ NN < 0.4 the results of Figs. 8 and 9 reasonably reroduce the ones 20

21 of the revious finite nucleus calculation reorted in Fig. 5 of [20] and Fig. 9 of [21]. On the contrary, for cos θ NN > 0.4 a discreancy is evident, the finite nucleus distributions being nearly flat in this region and the nuclear matter ones going monotonically to almost vanishing values with increasing values of cosθ NN. This is just a reflection of the recoil effects resent in a finite nucleus calculation, which make the NN airs to be less back to back correlated. Our distributions of Figs. 8 and 9 can also be comared with those obtained by KEK E508 and shown in Figure 3 of the aer by Kim et al. [10]. Due to the limited statistics of data, we concentrate on the angular region with cos θ NN < 0.7. In this region, the values N nn = ± and N n = ± have been determined exerimentally, whereas our corresonding results are N nn = and N n = While there is decent agreement in the case of N nn, we overestimate N n by a factor of about 2. This discreancy is a consequence of the overestimation of the KEK E508 single roton sectra of Fig. 4. Another effect of the difference between the redicted and the measured number of rotons can be seen when comaring our result for the number of roton roton airs for TN th = 30 MeV and cos θ NN < 0.7, N = 0.050, with the exerimental value N = ± The nn and n air distributions as functions of the air kinetic energy are shown in Figs. 10 and 11, resectively, where the energy of each nucleon is larger than a threshold kinetic energy of 30 MeV. The uer anels show the distributions obtained without any cut in the oening angle, while in the bottom anels only the back to back events are ket by alying the restriction cos θ NN < 0.7. We observe that the 2N induced events (dashed lines) in the uer anels scarcely contribute at the osition of the the rimary eak of the 1N induced contribution (dot dashed lines). Instead, they enhance the total distribution at a air energy of around 100 MeV, making the secondary eak (associated to the re scattering events) there to become even higher (see the case of nn airs) than the rimary one at the Q value of about 155 MeV. When the angular cut is alied, many of the events in the low energy region are removed and the so called back to back eak at 155 MeV stands out more clearly, although there is still an imortant fraction of events that lie outside this eak. This is in quantitative agreement with the finite nucleus results of our revious works (see Fig. 3 of [20] and Fig. 10 of [21]), the only qualitative difference being the width of the back to back eak, which aears more smeared out in the resent work due to Fermi motion. Note also that the distributions from 1N induced decays of Figs. 10 and 11 extend above the largest ossible Q values, Q 152 MeV and Q 157 MeV, corresonding to the three body non mesonic decays 12 C 10 C + nn and 12 C 10 B + n, 21

22 0.02 T N > 30 MeV N nn N-induced 2N-induced Total T N > 30 MeV cos θ nn < 0.7 N nn T 1 + T 2 (MeV) Fig. 10. Distribution of the total kinetic energy of nn airs normalized er non mesonic weak decay for 12 C. The energy threshold is 30 MeV for each nucleon in the air. In the uer anel there is no angular restriction, while the condition cos θ nn < 0.7 has been imosed in the results of the lower anel. resectively. These unwanted effects are tyical of nuclear matter LDA tye calculations, which assume a continuum of states within the Fermi sea, and, therefore, cannot aroriately handle the discrete transition of the hyernuclear state to the ground state of the residual system, which should be signaled by a narrow eak in the distribution of nucleon airs at a total kinetic energy equal to the Q value of the reaction. This effect, however, does not affect the analysis nor the conclusions of the resent work, which are based on the use of integrated quantities N induced strength In this subsection we comare the results of our microscoic model for the 2N induced channel with those of the henomenological model of Ref. [44]. In Fig. 12 we comare the momentum distributions of each of the three rimary nucleons emitted in 2N induced rocesses (rior to FSI effects) for both 22

23 T N > 30 MeV N n N n N-induced 2N-induced Total T N > 30 MeV cos θ n < T 1 + T 2 (MeV) Fig. 11. Distribution of the total kinetic energy of n airs normalized er non mesonic weak decay for 12 C. The energy threshold is 30 MeV for each nucleon in the air. In the uer anel there is no angular restriction, while the condition cos θ n < 0.7 has been imosed in the results of the lower anel. hen. micr. N( i ) (arb. units) i (MeV/c) i (MeV/c) Fig. 12. Momentum distribution of each of the three rimary nucleons emitted in 2N induced decay rocesses for 12 C. models. The differences observed are a consequence of the articular dynamics embodied in each model. The henomenological model assumes a one ion exchange icture for the weak interaction and accounts for the hase sace 23

24 of 22h excitations in the two nucleon absortion rocess. Since the articular kinematical constraints of the 2N induced channel allow for ions to be close to its mass shell, this configuration will be esecially enhanced. Consequently, the nucleon emitted at the Nπ vertex, whose momentum is denoted by 1, has very little kinetic energy left, whereas the other two nucleons of the 2N induced channel are quite energetic and come out in a ractically back to back geometry. In contrast, the microscoic one meson exchange model of Ref. [26] used in the resent work weights each 22h excitation with the strong interaction resonsible for its couling to the ground state. Therefore, the momentum 2 of the nucleon emitted after absorbing the exchanged meson (see Fig. 2) can reach high values (close to 500 MeV/c) from combining the momentum q of the virtual meson with the momentum 2 of the correlated nucleon, which can be large due to the hard nature of the short range NN interaction. The momentum 1, carried by the nucleon emitted at the vertex, and the momentum 3, corresonding to the sectator nucleon, are quite similar, around 300 MeV/c. This value is characteristic of the range of the interactions generating these nucleons, which are, resectively, the weak and strong one meson exchange models used in [26] hen. micr no FSI FSI T N > 30 MeV N n cos θ n cos θ n Fig. 13. Oening angle distribution of n airs from the n induced channel for C, normalized er n induced decay. 12 We now comare the henomenological and microscoic models after the three rimary nucleons emitted in the weak decay rocess are allowed to undergo collisions with the other nucleons as they move out of the nucleus. We restrict here to the n induced decay mode, which is the most imortant one in the microscoic aroach and the only one considered by the henomenological model. The distribution of n airs, normalized er its corresonding n induced transition rate, is shown in Fig. 13 as a function of the oening angle, where the threshold kinetic energy is 30 MeV for each nucleon of the 24

25 air. The dotted (solid) line shows the results without (with) FSI effects. It is clear that, at variance to the microscoical aroach, the henomenological model roduces a distinct back to back distribution which is artly ket even when FSI effects are included. The distribution of airs as a function of the total kinetic energy and for a threshold TN th = 30 MeV is dislayed in Fig. 14. We observe that the amount of N n airs er n induced decay event in the absence of FSI (area under the dotted curves) is smaller in the henomenological model, since the slow nucleon is always eliminated by the kinetic energy cut of 30 MeV. This situation is comensated when the oening angle cut is also alied (dashed lines), since it removes more events in the microscoic distribution, which is not so back to back dominated. Therefore, even if the kinematics of the rimary nucleons look quite different, at the end, once the effect of FSI is considered and the energy and angular cuts are alied, both models roduce similar neutron roton angular and energy sectra er n induced decay event. This is a welcome feature since it will reduce the model deendence in the extraction of the Γ n /Γ ratio hen. T N > 30 MeV micr. no FSI FSI FSI, cos θ n < 0.8 N n T 1 + T 2 (MeV) T 1 + T 2 (MeV) Fig. 14. Total kinetic energy distribution of n airs from the n induced channel C, normalized er n induced decay. for Ratio N nn /N n We start this section by comaring our redictions for the ratio between the number of nn and n airs, N nn /N n, in 12 C with the exerimental value obtained by the KEK E508 exeriment [8 10]. As in revious aers [20, 21], we introduce the numbers of NN airs (NN = nn, n or ) coming from one nucleon induced (NNN 1Bn and N1B NN ) and two nucleon induced (N2Bnn NN, N2Bn NN and N 2B NN ) rocesses, each one of them being normalized er the rate of the corresonding rocess. In the absence of FSI and ignoring the two nucleon 25