Ground state normalization in the nonmesonic weak decay of 12ΛC hypernucleus within a nuclear matter formalism

Nuclear Physics A 818 (2009) 174–187 www.elsevier.com/locate/nuclphysa

Ground state normalization in the nonmesonic weak decay of 12 Λ C hypernucleus within a nuclear matter formalism E. Bauer a,b,∗ a Departamento de Física, Universidad Nacional de La Plata, C.C. 67, 1900 La Plata, Argentina b Instituto de Física La Plata, CONICET, 1900 La Plata, Argentina

Received 13 August 2008; received in revised form 17 December 2008; accepted 19 December 2008 Available online 25 December 2008

Abstract The nonmesonic weak decay width of 12 Λ C hypernucleus has been evaluated within a nuclear matter formalism, using the local density approximation. In addition to the one-body induced decay (ΛN → nN ), it has been also considered the two-body induced decay (ΛN N → nN N ). This second decay is originated from ground state correlations, where a renormalization procedure to ensure a ground state normalized to one has been implemented. Our results show that the plain addition of the two-body induced decay implies a lost in the ground state-norm, which adds ∼ 38% of spurious strength to the nonmesonic weak decay width. Within our scheme, it is possible to reproduce the most recent data for the nonmesonic weak decay width of 12 Λ C. © 2009 Elsevier B.V. All rights reserved. PACS: 21.80.+a; 25.80.Pw Keywords: Λ-hypernuclei; Nonmesonic decay of hypernuclei; Γn /Γp ratio

1. Introduction A Λ-hypernucleus decays via the weak interaction mainly by two decay mechanisms: the socalled mesonic decay (Λ → πN) and the nonmesonic one (NM), where no meson is present in * Address for correspondence: Departamento de Física, Universidad Nacional de La Plata, C.C. 67, 1900 La Plata, Argentina. E-mail address: [email protected].

0375-9474/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2008.12.008

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the final state (for review articles see [1,2]). The NM decay width is denoted as ΓNM , which is defined as the sum of Γ1 ≡ Γ (ΛN → nN) plus Γ2 ≡ Γ (ΛN N → nN N ). The Γ1 decay width itself is the sum of Γn ≡ Γ (Λn → nn) plus Γp ≡ Γ (Λp → np). Experimental values are given for ΓNM , the Γn/p ≡ Γn /Γp ratio and the asymmetry of the protons emitted in the NM decay of polarized hypernuclei. In the present contribution, we focus on ΓNM , evaluated in nuclear matter together with the local density approximation which allows us to analyze the 12 Λ C hypernucleus. In the past, it has been an usual statement to assert that while the theory accounts for the experimental values of ΓNM , the same is not true for the ratio Γn/p . In fact, the disagreement between the theoretical and the experimental value for this ratio, has been named as “the Γn/p puzzle”. This situation has changed in recent years: new theoretical analysis together with more experimental information, have led us to a solution of the so-called puzzle. A typical theoretical value for the ratio for 12 0.3, while data analyzed by means of the intranuclear cascade Λ C is Γn/p ∼ exp code (INC) [3–6], gives a result Γn/p ∼ 0.4 ± 0.1.1 However, it should be noted that there still exist discrepancies with some nucleon spectra. For instance, the experimental single coincidence proton spectra for 12 Λ C is not well reproduced yet. In nuclear matter (using the local density approximation), some reported calculations for Γ1 for 12 Λ C, have values in the range between 0.5 [7] up to 1.45 [8] (given in units of the Λ free decay width, Γ 0 ). While typically Γ2 /Γ1 ∼ 0.3. The most recent experimental determination of ΓNM exp has been done by Outa et al. [9], who have reported a value ΓNM = 0.940 ± 0.035. Some previous exp exp experimental determinations are ΓNM = 1.14 ± 0.20 [10], ΓNM = 0.89 ± 0.15 ± 0.03 [11] and exp ΓNM = 0.828 ± 0.056 ± 0.066 [12]. There is some incompatibility between the result in [12], due to Sato et al. and both Noumi et al. [11] and Outa et al. We have relied on the Outa result, not only because it is the most recent one, but also due to its compatibility with both [10] and [11]-values. exp Beyond this controversy, the more precise determination of ΓNM offers us the opportunity to revise the theoretical determination of ΓNM . In two previous works, a model for the evaluation of Γ1 and Γ2 has been developed (see [13] and [14], respectively). Our scheme employs the same microscopic formalism and interactions for both Γ1 and Γ2 . Within this model, the reproduction exp of ΓNM does not seen possible: the predicted values are always too big. The main concern of this work is to understand and solve this problem. To solve this problem, here we revise the way in which Γ2 is added to Γ1 to build up ΓNM . The Γ2 contribution is originated from ground state correlations and the simple addition of Γ2 plus Γ1 , would add some spurious strength because the ground state is not normalized to one. This point turns out to be relevant not only in the determination of ΓNM , but also for the Γ2 /Γ1 -ratio, exp which is used as an input in the determination of Γn/p . As a further comment on this point, the lack in the normalization is not restricted to our particular model, but to any calculation where Γ2 is considered. A second point refers to the implementation of short range correlations (SRC) in nuclear matter. A frequently employed model for SRC is discussed, where we care about some numerical approximations in its implementation. Beyond these two points and in order to make our Γ2 model more complete, some additional Γ2 diagrams (not evaluated in [14]) have been considered. Our microscopic model for Γ1 and Γ2 , together with the norm-correction, gives a ΓNM value in good agreement with data. This work is organized as follows. In Section 2, our model for the renormalization of the ground state is presented, showing a scheme to incorporate Γ2 into ΓNM . In Section 3, a model 1 For this result, it has been considered the cos(θ) < −0.80 region (where this angle is the one between the two outgoing nucleons) and a kinetic detection threshold for nucleons TNth = 30 MeV.

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for the SRC in nuclear matter is discussed in detail. Numerical results are shown in Section 4, together with an analysis of the implications of the corrections. Finally, in Section 5, some conclusions are given. 2. Ground state correlations (GSC) To start with, we write down the partial decay width ΓNM (kF ) in a schematic way as   f |V ΛN |0k 2 δ (4) (pf − p0 ), ΓNM (kF ) = F

(1)

f

where |0kF and |f  are the ground state and the final state, respectively; V ΛN is the two-body transition potential and pi represents an energy–momentum four-vector. The Fermi momentum is denoted as kF . By performing the integration over kF using the local density approximation (see [8]), the total decay width ΓNM is obtained. Now, the ground state can be written as    p1 p2 h1 h2 |V N N | kF 1 |p1 p2 h1 h2  , (2) |0kF = N (kF ) | kF − 4 p1 + p2 − h1 − h2 p1 ,p2 ,h1 ,h2

where the second term in the right-hand side of the equation represents 2p2h-correlations. In this equation | kF is the Hartree–Fock vacuum. In the denominator, i are the single particle energies. The nuclear residual interaction is represented by V N N and N (kF ) is the normalization as a function of kF :      p1 p2 h1 h2 |V N N | k 2 −1/2 1 F   N (kF ) = 1 + . (3)  +  −  −   16 p1 p2 h1 h2 p1 ,p2 ,h1 ,h2

We will show soon that the inclusion of N (kF ) has an important effect over ΓNM . The importance of a proper treatment of the ground state normalization has been already pointed out by Van Neck et al. [15]. When Eq. (2) is inserted into the expression of ΓNM (kF ) given by Eq. (1) with the arbitrary selection of N (kF ) = 1, the usual expressions for Γ1 and Γ2 are obtained. The first one comes from the first term in Eq. (2), while Γ2 is originated from the second term in the same equation. The use of N (kF ) = 1 means that the ground state is not properly normalized and therefore some spurious strength is added to ΓNM . 3. Short range correlations (SRC) In momentum space one way to take care of SRC is by the use of a modified transition potential obtained as (see [16])   dp  ξ˜ |p + q| V (p), (4) VSRC (q) = V (q) − 3 (2π) where ξ˜ (p) =

2π 2 δ(p − qc ), qc2

(5)

with qc = 780 MeV/c, is a particular correlation function in momentum space. We have limited our discussion of SRC to this model and its implementation in the evaluation of ΓNM deserves

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some care. We show this with an example. Let us show the result of Eq. (4) with the central part of the parity conserving one pion exchange potential, which we write in a simplified manner as VπC (q) = Cπ

q2 σ 1 · σ 2τ 1 · τ 2 q 2 + m2π

(6)

¯ where M¯ is the average between the nucleon and Λ with Cπ = −GF m2π (gN N π /2M)(Bπ /2M), masses. Using this potential in Eq. (4) we obtain VπSRC,C (q) = VπC (q)

 2 

 qc + m2π + q 2 − 2qc |q|  m2π 1   σ 1 · σ 2τ 1 · τ 2. − Cπ 2 + ln 2 2qc |q|  qc2 + m2π + q 2 + 2qc |q| 

(7)

By calling κ = 2qc |q|/(qc2 + m2π + q 2 ) and making the approximation ln(1 + κ) ≈ κ,

(8)

we finally obtain VπSRC,C (q) = VπC (q) − Cπ

qc2 + q 2 σ 1 · σ 2τ 1 · τ 2, qc2 + m2π + q 2

(9)

which is equivalent to the following general prescription to build up the modified potential due to the action of the SRC:   (10) V SRC,C (q) = V C (q) − V C q 2 → qc2 + q 2 . To the best of our knowledge, this way of taking care of SRC in nuclear matter is the most frequently used one. However, we should call attention on the non-equivalence between the approximation in Eq. (10) and the one in Eq. (4) for the kinematical conditions of the nonmesonic Λ decay. This is because the approximation given by Eq. (8) is a bad approximation for the momentum transfer in the ΛN → N N decay channel, where q ≈ 400 MeV/c (the ΛN N → N N N decay channel is analyzed soon). A simple numerical test shows that Eq. (10) fairly accounts for the expression given by Eq. (4) only for q  50 MeV/c. For the full V ΛN -transition potential (which includes q-dependent form factors), the integral in Eq. (4) can be also performed analytically. The numerical result shows that ΓNM evaluated with the inclusion of SRC given by the model in Eqs. (4) and (5), is ∼ 35% smaller than the same quantity with the prescription in Eq. (10) (employing the same qc -value). In the present contribution we present results only for the model in Eqs. (4) and (5), as this model gives us some confidence about its applicability within the kinematical conditions of the nonmesonic Λ decay. Alternatively, the use of Eq. (10) can be seen as a model in itself. Due to the frequent use of the model in Eq. (10) and to the fact that under certain conditions these two models give the same result, we think it is of interest to give here some details on the kinematical conditions of the nonmesonic Λ decay. Let us write down explicit expressions for both Γ1 and Γ2 . We do this in a very schematic way,   ΛN 2  (q) Im Π1p1h (q0 , q) (11) Γ1 (k, kF ) = C dq θ (q0 )θ |k − q| − kF VSRC and

 Γ2 (k, kF ) = C

 ΛN 2  dq θ (q0 )θ |k − q| − kF VSRC (q) Im Π2p2h (q0 , q)

(12)

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Fig. 1. Imaginary part of the polarization functions Π1p1h and Π2p2h as a function of the momentum transfer q. The polarization functions and q, are in units of 10−3 MeV−1 fm−3 and MeV/c, respectively.

where C = −6(GF m2π )2 π/(2π)3 and q0 = k0 − E(k − q) − VN , k being the energy–momentum of the Λ. Final values for Γ1 and Γ2 are obtained after integrating over k and kF . The functions Π1p1h and Π2p2h are the 1p1h and 2p2h-polarizations functions, respectively. We do not go through the derivation of these expressions (details can be found in [18], for instance). In Fig. 1, we have plotted Im Π1p1h and Im Π2p2h as a function of the momentum transfer q. For simplicity, this is done for a Λ at rest: k = 0 and for kF = 210 MeV/c. The function Im Π1p1h is peaked around q ≈ 400 MeV/c. From the same figure, the situation for Im Π2p2h is different as it is spread over a wider q-region, with a peak around q ≈ 320 MeV/c. Note that from the second step function in Eqs. (11) and (12), and for the particular conditions in Fig. 1 (the Λ at rest: k = 0 and kF = 210 MeV/c), the q-integral starts in q = kF (this is depicted in the figure by means of a vertical dashed line). The actual integration includes non-zero k-values and several Fermi momenta. In any case, from Fig. 1 it is clear that the q-integration interval is far from the region q  50 MeV/c, which explains the non-equivalence between the two models for the physical problem under consideration. As a final point for this section, note that this discussion is not restricted to our particular Γ2 -evaluation. There are two former models for the calculation of Γ2 . The starting point of all

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these determinations is Eq. (12), but they differ between each other in the model for Π2p2h . The first work which has called attention on Γ2 is the one due to Alberico et al. [17], where a so-called semi-phenomenological Γ2 has been adopted, which results from a microscopic evaluation of the polarization propagator Π2p2h in nuclear matter, originally performed for electron scattering in [19]. Using this electron scattering calculation, a constant Im Π2p2h is proposed, which is appropriate for pion absorption. Thereafter, Ramos et al. [18] has used also a semiphenomenological Γ2 , where an approximate value for the function Im Π2p2h is obtained as the product of the phase space corresponding to the ΛN N → N N N -reaction times a constant taken from real pion absorption. It should be noted that in the ΛN N → N N N -reaction, all mesons are strictly off the mass shell and the use of the pion absorption results is an approximation to take care of the dynamics involved in Γ2 . Beyond the same starting point of Eq. (12) and the difference in the calculation of the function Im Π2p2h these two works also differ from the one in [14], by the way in which the isospin is taken into consideration and some minor points. In any case, the just quoted consideration in the inclusion of the SRC is valid for all these Γ2 -models. 4. Results and discussion We give now the numerical results. The transition potential V ΛN is represented by the exchanges of the π , η, K, ρ, ω and K ∗ mesons, whose formulation has been taken from [20], and the values of the different coupling constants and cutoff parameters appearing in the transition potential have been taken from [21] and [22], named as Nijimegen and NSC97f, respectively. For the nuclear residual interaction V N N (which is employed in both Γ2 and N (kF )), we have used the Bonn potential [23] in the framework of the parametrization presented in [24], which contains the exchange of π , ρ, σ and ω mesons and neglects the η and δ mesons. In addition, for V N N we have also considered a g + Vπ+ρ potential, where the different parameters entering into Vπ+ρ , are the ones in the Bonn potential. The Landau–Migdal g -parameter is adjusted so to reproduce the same function N (kF ) obtained from the Bonn potential, a value which turns out to be g = 0.63 (in pionic units). When implementing the LDA, the hyperon is assumed to be in the 1s1/2 orbit of a harmonic oscillator well with frequency h¯ ω = 10.8 MeV (for details see [25]). The partial decay widths Γ1 (kF ) and Γ2 (kF ) have been evaluated using the scheme developed in [13] and [14], respectively; but with the implementation of the SRC described above, together with some new diagrams which are described soon. The Γ2 -contribution has three isospin terms: Γnn ≡ Γ (Λnn → nnn), Γnp ≡ Γ (Λnp → nnp) and Γpp ≡ Γ (Λpp → npp), where Γ2 = Γnn + Γnp + Γpp . The dominant term is Γnp , where the relative magnitude of each contribution follows approximately the relation, Γnp : Γpp : Γnn ≈ 0.81 : 0.15 : 0.04.2 There are several Goldstone diagrams which contribute to each of these terms. In [14], it has been considered the diagrams where the transition potential V ΛN is attached to the same bubble (these diagrams are depicted in Fig. 1 in that work). In addition to these diagrams, in the present contribution we have added the ones where the transition potential is attached to different bubbles. These diagrams are shown in Fig. 2. In this figure, the transition potential is attached either to two different particles (graph pp ), to two different holes (graph hh ) or to a particle and a hole of different bubbles (graph ph ). The analytical expressions for the decay widths are displayed in Appendix A. 2 This relation is valid for the whole set of diagrams considered in this contribution. For the diagrams evaluated in [14], this relation is Γnp : Γpp : Γnn ≈ 0.78 : 0.17 : 0.05.

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Fig. 2. Goldstone diagrams for the 2p2h-contributions to the Λ decay width. The meaning of pp , ph and hh is explained in the text. The dashed and wavy lines stand for V ΛN and V N N , respectively. An up (down) going arrow represents a particle (hole), while an up going arrow with a wide line represents the Λ. Table 1 Contributions from the diagrams in Fig. 2, to the weak decay width of 12 Λ C. All Γ ’s are in units of free Λ decay rate, Γ 0 = 2.52 × 10−6 eV. The notation Γ2 indicates that this sum is a particular sub-set of the diagrams contributing to Γ2 . Although there is no null value, the ones smaller than 10−3 are represented as ∼ 0. Diagram

Γnn

Γnp

Γpp

Γ2

pp ph hh

0.001 ∼ 0. ∼ 0.

0.036 0.003 0.001

∼ 0. ∼ 0. ∼ 0.

0.037 0.003 0.001

0.001

0.040

∼ 0.

0.041

Sum

In Table 1, we show the results from the new diagrams in Fig. 2. In this table, the transition potential is the one named as Nijimegen [21], while the nuclear residual interaction is the Bonn potential. The values displayed in this table already contain the correction due to the normalization of the ground state and they should be compared with the sum of all the diagrams where the transition potential is attached to the same bubble. For the interactions just described, this value is 0.209, meaning that the whole Γ2 represents ∼ 16% of the total Γ2 . In Table 2, we present our values for Γ1, 2 and ΓNM for the two sets of nuclear residual interactions and transition potential parameters, with and without the action of N (kF ). In first place, it is clear that the effect of the ground state renormalization is important: the spurious part in ΓNM (i.e. 100 × |ΓNM (without renorm.) − ΓNM (with renorm.)|/ΓNM (with renorm.)), is ∼ 38%. It should be noted that once the GSC are considered, the partial decay widths Γ1 (kF ) and Γ2 (kF ) are multiplied by the function N (kF ) and then, the kF -integration gives the final ΓNM . That is, the action of the ground state renormalization is not the plain multiplication of Γ1,2 by a constant. It is also important to note that the two nuclear residual interactions have been chosen to produce the same N (kF )-value. Therefore, the Γ1 result (with renormalization) is the same for both nuclear residual interactions. Also from Table 2, it is clear that the action of the ground state renormalization is important for the ΓNM -value, but it acts uniformly over Γ1 and Γ2 , leaving the Γ2 /Γ1 -ratio unchanged. Within the present model, this ratio ranges from 0.24 for the Nijimegen transition potential and the Vg +π+ρ -residual interaction, up to 0.54 for NSC97f and the Bonn potential. It should be noted that our ΓNM -result for the interaction NSC97f (Nijimegen) and the Bonn potential (Vg +π+ρ residual interaction), is in close agreement with the data from [9]. And so does the result for the interaction NSC97f with the data from [12] (using the Vg +π+ρ -residual interaction). As a final comment on Table 2, our result for ΓNM with renormalization shows a small decrease (increase) with respect to Γ1 without renormalization, for the interaction Nijimegen

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Table 2 N N and V ΛN -potentials, The nonmesonic weak decay width of 12 Λ C. The first and second columns represent the V respectively. The third one refers to the inclusion or not of the normalization factor in the ground state. Units are the same as in Table 1. V NN

V ΛN

Renorm.

Γ1

Γ2

ΓNM

Bonn [23,24]

Nijimegen [21] NSC97f [22] Nijimegen [21] NSC97f [22] Nijimegen [21] NSC97f [22] Nijimegen [21] NSC97f [22]

no no yes yes no no yes yes

1.031 0.814 0.747 0.590 1.031 0.814 0.747 0.590

0.344 0.437 0.250 0.317 0.255 0.367 0.180 0.261

1.375 1.251 0.997 0.907 1.286 1.181 0.927 0.851

Vg +π +ρ

Experiment [9] Experiment [12]

0.940 ± 0.035 0.828±0.056±0.066

(NSC97f). In fact, while for Nijimegen the value for Γ1 is greater than the one for NSC97f; just the opposite occurs for Γ2 . This is a consequence of the different weight of each spin–isospin component in each interaction, together with the different structure in the spin–isospin sums between Γ1 and Γ2 . Clearly the choice of different transition potentials produces different results. Our analysis emphasized the relevance of making a careful spin and isospin summation, as the same transition potential has a different effect when these summations are not the same. Beyond these considerations, the renormalization procedure refers to the lack of the ground state norm, and should be implemented regardless the selection of the transition potential. It should be mentioned that the transition potential employed in [17] limits itself to a one pion exchange while in [18] the ρ-meson exchange is also considered. In both works the transition potential includes polarization effects by means of the ring approximation, whose effect is a reduction in the ΓNM -result. In this sense, the RPA (or equivalently the ring approximation) also implies some kind of ground state renormalization. This produces a reduction in the polarization insertion, whose imaginary part is related to the decay width. In Table 3, a similar analysis to the one in Table 2, is done for Γn,p and the ratio Γn/p , where the theoretical values are obtained with the scheme in [13], (but using the oscillator frequency h¯ ω = 10.8 MeV, just mentioned). The decay widths Γn,p,nn,... refers to primary decays. This exp means that to extract the ratio Γn/p , from the experimental spectra, a model for the analysis of data is required, where the Γ2 /Γ1 -ratio plays an important role. In the present table, two experimental values are shown: the one from Outa et al. [9], whom have used the approximation Nnn /Nnp Γn/p , where Nij represents the total number of ij -pairs emitted in the Λ-weak decay (this result is denoted as preliminary by the author). In this table it is also reported the value by Sato et al. [12], extracted under the assumption of Γ2 /Γ1 = 0.35 and obtained from single-proton exp energy spectra. From the same work, it is reported Γn/p = 0.87 ± 0.09 ± 0.21 for Γ2 /Γ1 = 0, exp which shows the importance of the Γ2 /Γ1 -ratio for the determination of Γn/p . In this table, it is also observed that the ratio Γn/p is unaffected by the renormalization procedure. exp exp Note that ΓNM and Γn/p are deduced from two independent measurements and it is expected that any theoretical model should care about both magnitudes. The renormalization of the correlated ground state is important for the ΓNM -evaluation and leaves the Γn/p - and Γ2 /Γ1 -ratio unaffected. Our scheme evaluates both Γ1 and Γ2 and only by taking care of the ground state normalization, it is possible to account for the nonmesonic weak decay. As discussed, the Γ2 /Γ1 -

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Table 3 The same as Table 2, but for Γn , Γp and the ratio Γn/p . Model int.

Renorm.

Γn

Γp

Γn /Γp

Nijimegen [21] NSC97f [22] Nijimegen [21] NSC97f [22]

no no yes yes

0.213 0.155 0.154 0.112

0.819 0.660 0.593 0.478

0.260 0.235 0.260 0.234 0.56 ± 0.12 ± 0.04 +0.11+0.23 0.60−0.09−0.21

Experiment [9] Experiment [12]

exp

ratio is relevant in the determination of Γn/p . By using distinct transition potentials and nuclear residual interactions we have obtained different values for the Γ2 /Γ1 ratio and values of ΓNM in agreement with data. From these information, it is not possible to disentangle the best set of transition potential and nuclear interaction. This could be achieved by the comparison with data of the nucleon emission spectra. However, this analysis goes beyond the present contribution. Our main concern here is to account for ΓNM , using the same microscopic model, transition potential and residual interaction for both Γ1 and Γ2 . 5. Conclusions The evaluation of Γn and Γp , where Γ1 = Γn + Γp , has attracted an important amount of theoretical attention. In spite of it sizable magnitude, the same has not happened with Γ2 . In the present contribution, it has been called attention on the lost in the ground state norm due to the inclusion of ground state correlations. In addition, three Γ2 -diagrams not evaluated before have been considered. These diagrams are the ones where the transition potential is attached to different bubbles of the Π2p2h -polarization propagator. In a recent work, Bhang et al. [26], by analyzing the experimental information, conclude that a not-null Γ2 is required to reproduce the spectrum-data. This goal is not achieved with only Γ1 and in fact, he has suggested a value for the ratio Γ2 /ΓNM ∼ = 0.4. The immediate consequence of this information is that ground state correlations are important. Once the ground state correlations comes into play, an adequate treatment of the ground state norm is required. The lack of the ground state norm adds ∼ 38% of spurious strength to the nonmesonic weak decay (using the Bonn potential for the residual nuclear interaction). And the new diagrams increase the former Γ2 value by ∼ 16%. Using the same nuclear matter model, transition potential and nuclear residual interaction, we have been able to reproduce the total nonmesonic weak decay width for 12 Λ C. More specifically, using the transition potential named as Nijimegen and the Vg +π+ρ residual interaction, we have obtained ΓNM = 0.927 and Γ2 /Γ1 = 0.24. The corresponding values for the NSC97f and the Bonn potential are ΓNM = 0.907 and Γ2 /Γ1 = 0.54. These two values for ΓNM are in agreement with the experimental measurement in [9]. Finally, our results for the NSC97f potential and the Vg +π+ρ residual interaction, are ΓNM = 0.851 and Γ2 /Γ1 = 0.44, where the nonmesonic weak decay rate is in agreement with the data in [12]. The next step is the evaluation of the nucleon emission spectra. The study of the spectra is a goal in itself and a ΓNM -value in agreement with data is a good starting point. In addition, the shape of the spectra depend on the balance between Γ1 and Γ2 and therefore, due to their different Γ2 /Γ1 values, it is expected that this analysis would help us to choose among the different transition potentials and residual interactions. This work is in progress.

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Acknowledgements I would like to thank G. Garbarino for very helpful discussions and for a careful and critical reading of the manuscript. This work has been partially supported by the CONICET, under contract PIP 6159. Appendix A In this appendix we show explicit expressions for the decay widths corresponding to graphs pp , ph and hh in Fig. 2. In first place, we write down the nuclear strong interaction as follows: VτNNN (t) =

fπ2  VC,τN (t) + Vσ,τN (t)σ 1 · σ 2 + VL,τN (t)σ 1 · tˆ σ 2 · tˆ , 2 mπ

(A.1)

where the functions VC,τN (q), Vσ,τN (q) and VL,τN (q) are adjusted to reproduce any effective nuclear residual interaction. The τ -index refers to the isospin. The corresponding expression for the transition potential is   (q) = GF m2π SτΛ (q)σ 1 · qˆ + Sτ Λ (q)σ 2 · qˆ + PC,τΛ (q) VτΛN Λ ˆ 2 · qˆ + Pσ,τΛ (q)σ 1 · σ 2 + PL,τΛ (q)σ 1 · qσ + iSV ,τΛ (q)(σ 1 × σ 2 ) · qˆ ,

(A.2)

Sτ Λ (q),

where the quantities SτΛ (q), PC,τΛ (q), Pσ,τΛ (q), PL,τΛ (q), and SV ,τΛ (q) are also adjusted to reproduce any transition potential. To arrive at the expressions for the decay widths, we start by showing the following partial decay widths,

pp Γ˜τΛ τ Λ ,τN τ N (k, kF )       2 (GF m2π )2 fπ2 2 1 = dq dt dh dh θ (q0 ) 4π m4π (2π)2 (2π)5         × θ |k − q| − kF θ |h − t| − kF θ |h − t + q| − kF θ kF − |h|       × θ |h + t| − kF θ |h + t − q| − kF θ kF − |h |    × δ q0 − EN (h − t + q) − EN (h) + EN (h + t) − EN (h ) pp

×

WτΛ τ Λ ,τN τ N (q, t, t )

q0 − EN (h − t + q) + EN (h − t) 1 × , q0 − EN (t + h ) + EN (t + h − q)

ph Γ˜τΛ τ Λ ,τN τ N (k, kF )       2 (GF m2π )2 fπ2 2 1 = dq dt dh dh θ (q0 ) 4π m4π (2π)2 (2π)5         × θ |k − q| − kF θ |h − t| − kF θ |h − t + q| − kF θ kF − |h|       × θ |h + t| − kF θ kF − |h + q| θ kF − |h |

(A.3)

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E. Bauer / Nuclear Physics A 818 (2009) 174–187

   × δ q0 − EN (h − t + q) − EN (h) + EN (h + t) − EN (h ) ph

×

WτΛ τ Λ ,τN τ N (q, t, t )

q0 − EN (h − t + q) + EN (h − t) 1 × , q0 − EN (h + q) + EN (h )

(A.4)

and Γ˜τhh (k, kF ) Λ τ Λ ,τN τ N       2 (GF m2π )2 fπ2 2 1 dq dt dh dh θ (q0 ) = 4π m4π (2π)2 (2π)5         × θ |k − q| − kF θ kF − |h + q| θ |h − t + q| − kF θ kF − |h|       × θ |h + t| − kF θ kF − |h + q| θ kF − |h |    × δ q0 − EN (h − t + q) − EN (h) + EN (h + t) − EN (h )

×

Wτhh (q, t, t ) Λ τ Λ ,τN τ N

[q0 − EN (h + q) + EN (h)][q0 − EN (h + q) + EN (h )]

,

(A.5)

where, q0 = k0 − EN (k − q) − VN , k0 being the energy of the Λ and VN is the nucleon binding pp ,ph ,hh energy. In these expressions the function WτΛ τ Λ ,τN τ N (q, t, t ) is defined as pp

WτΛ τ Λ ,τN τ N (q, t, t ) = PC,τΛ (q)PC,τΛ (q)VCΛ,τN τ N (t, t ) + Pσ,τΛ (q)Pσ,τΛ (q)Vσ Λ,τN τ N (t, t ) + PL,τΛ (q)PL,τΛ (q)VLΛ,τN τ N (t, t ) + Pσ,τΛ (q)PL,τΛ (q)Vσ LΛ,τN τ N (t, t ) + SτΛ (q)SτΛ (q)VSΛ,τN τ N (t, t ) + Sτ Λ (q)Sτ (q)VS Λ,τN τ N (t, t ) Λ

+ SV ,τΛ (q)SV ,τΛ (q)VSV Λ,τN τ N (t, t ),

(A.6)

where the functions ViΛ,τN τ N (t, t ) are VCΛ,τN τ N (t, t ) = VC,τ (t )VC,τN (t) + 3Vσ,τ (t )Vσ,τN (t) N

N

+ 8(tˆ · tˆ )2 VL,τN (t )VL,τN (t) + Vσ,τN (t )VL,τN (t) + VL,τN (t )Vσ,τN (t), Vσ Λ,τN τ N (t, t ) = −s6Vσ,τ (t )Vσ,τN (t) − s(tˆ × tˆ )2 VL,τ (t )VL,τN (t) N N

  − s2 Vσ,τN (t )VL,τN (t) + VL,τN (t )Vσ,τN (t) + VC,τN (t )VL,τN (t)   + VL,τN (t )VC,τN (t) + 3 Vσ,τN (t )VC,τN (t) + VC,τN (t )Vσ,τN (t) ,



2

VLΛ,τN τ N (t, t ) = −s2Vσ,τ (t )Vσ,τN (t) − s qˆ · (tˆ × tˆ ) VL,τ (t )VL,τN (t) N

N

− s2(qˆ · tˆ )2 Vσ,τN (t )VL,τN (t) + (qˆ · tˆ )2 VL,τN (t )Vσ,τN (t)

+ (qˆ · tˆ )2 VC,τN (t )VL,τN (t) + (qˆ · tˆ )2 VL,τN (t )VC,τN (t) + Vσ,τN (t )VC,τN (t) + VC,τN (t )Vσ,τN (t),

E. Bauer / Nuclear Physics A 818 (2009) 174–187



185

2

Vσ LΛ,τN τ N (t, t ) = −s4Vσ,τ (t )Vσ,τN (t) − s2 qˆ · (tˆ × tˆ ) VL,τ (t )VL,τN (t)

  + s −1 + (qˆ · tˆ )2 Vσ,τN (t )VL,τN (t)   + s −1 + (qˆ · tˆ )2 VL,τN (t )Vσ,τN (t) N

N

+ (qˆ · tˆ )2 VC,τN (t )VL,τN (t) + (qˆ · tˆ )2 VL,τN (t )VC,τN (t)   + 2 Vσ,τN (t )VC,τN (t) + VC,τN (t )Vσ,τN (t) ,

2



VSΛ,τN τ N (t, t ) = −s2Vσ,τ (t )Vσ,τN (t) − s qˆ · (tˆ × tˆ ) VL,τ (t )VL,τN (t) N N

  + s −1 + (qˆ · tˆ )2 Vσ,τN (t )VL,τN (t)   + s −1 + (qˆ · tˆ )2 VL,τN (t )Vσ,τN (t)

+ (qˆ · tˆ )2 VC,τN (t )VL,τN (t) + (qˆ · tˆ )2 VL,τN (t )VC,τN (t) + Vσ,τN (t )VC,τN (t) + VC,τN (t )Vσ,τN (t),

VS Λ,τN τ N (t, t ) = VC,τ (t )VC,τN (t) + 3Vσ,τ (t )Vσ,τN (t) N

N

+ 8(tˆ · tˆ )2 VL,τN (t )VL,τN (t) + Vσ,τN (t )VL,τN (t) + VL,τN (t )Vσ,τN (t),



2 VSV Λ,τN τ N (t, t ) = −4Vσ,τ (t )Vσ,τN (t) − s (tˆ × tˆ )2 − qˆ · (tˆ × tˆ ) VL,τ (t )VL,τN (t) N

N

    + s 1 + (qˆ · tˆ )2 Vσ,τN (t )VL,τN (t) + s 1 + (qˆ · tˆ )2 VL,τN (t )Vσ,τN (t)     + 1 − (qˆ · tˆ )2 VC,τN (t )VL,τN (t) + 1 − (qˆ · tˆ )2 VL,τN (t )VC,τN (t)   (A.7) + 2 Vσ,τN (t )VC,τN (t) + VC,τN (t )Vσ,τN (t) .

= In these expressions the constant s = 1. For the hh and ph , we have Wτhh Λ τ Λ ,τN τ N pp

ph

pp

WτΛ τ Λ ,τN τ N and WτΛ τ Λ ,τN τ N = WτΛ τ Λ ,τN τ N (with s = −1). The summation over isospin leads to pp

pp

pp

pp

pp

pp

pp

pp

pp

Γnn = Γ11,11 + Γ00,00 + Γ11,00 + Γ00,11 + Γ11,10 + Γ11,01 + Γ10,11 + Γ01,11 pp

pp

pp

pp

pp

pp

pp

pp

+ Γ10,10 + Γ01,01 + Γ00,01 + Γ00,10 + Γ01,00 + Γ10,00 + Γ10,01 + Γ01,10 , pp

pp

pp

pp

pp

pp

pp

pp

pp

pp

pp

Γnp = −26Γ11,11 + 2Γ00,00 − 2Γ11,00 + 10Γ00,11 10Γ11,10 + 10Γ11,01 − 2Γ00,01 − 2Γ00,10 , pp

pp

pp

pp

pp

pp

pp

Γpp = Γ11,11 + Γ00,00 + Γ11,00 + Γ00,11 + Γ11,10 + Γ11,01 − Γ10,11 − Γ01,11 pp

pp

pp

pp

pp

pp

pp

pp

− Γ10,10 − Γ01,01 + Γ00,01 + Γ00,10 − Γ01,00 − Γ10,00 − Γ10,01 − Γ01,10 , ph

ph

ph

ph

ph

ph

ph

ph

ph

Γnn = Γ11,11 + Γ00,00 + Γ11,00 + Γ00,11 + Γ11,10 + Γ11,01 + Γ10,11 + Γ01,11 ph

ph

ph

ph

ph

ph

ph

ph

+ Γ10,10 + Γ01,01 + Γ00,01 + Γ00,10 + Γ01,00 + Γ10,00 + Γ10,01 + Γ01,10 , ph

ph

ph

ph

ph

ph

ph

Γnp = 14Γ11,11 + 2Γ00,00 − 2Γ11,00 + 10Γ00,11 2Γ11,10 + 10Γ11,01 ph

ph

− 2Γ00,01 − 2Γ00,10 ,

(A.8)

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E. Bauer / Nuclear Physics A 818 (2009) 174–187 ph

ph

ph

ph

ph

ph

ph

ph

ph

ph

ph

ph

ph

Γpp = 9Γ11,11 + Γ00,00 + Γ11,00 + Γ00,11 + 9Γ11,10 + Γ11,01 − Γ10,11 − Γ01,11 ph

ph

ph

ph

− Γ10,10 − Γ01,01 + Γ00,01 + Γ00,10 − Γ01,00 − Γ10,00 − Γ10,01 − Γ01,10 ,

(A.9)

and

















hh hh hh hh hh hh hh hh hh Γnn = Γ11,11 + Γ00,00 + Γ11,00 + Γ00,11 + Γ11,10 + Γ11,01 + Γ10,11 + Γ01,11















hh hh hh hh hh hh hh hh + Γ10,10 + Γ01,01 + Γ00,01 + Γ00,10 + Γ01,00 + Γ10,00 + Γ10,01 + Γ01,10 ,

















hh hh hh hh hh hh hh = −10Γ11,11 + 2Γ00,00 − 2Γ11,00 + 10Γ00,11 2Γ11,10 + 2Γ11,01 Γnp hh hh − 2Γ00,01 − 2Γ00,10 ,

















hh hh hh hh hh hh hh hh hh = −15Γ11,11 + Γ00,00 + Γ11,00 + Γ00,11 + 9Γ11,10 + 9Γ11,01 − Γ10,11 − Γ01,11 Γpp















hh hh hh hh hh hh hh hh − Γ10,10 − Γ01,01 + Γ00,01 + Γ00,10 − Γ01,00 − Γ10,00 − Γ10,01 − Γ01,10 ,

(A.10)

where the (k, kF )-dependence has been omitted for simplicity. From these expressions and using the local density approximation, it is straightforward to obtain the final result for the decay widths (for details in the implementation of the local density approximation, see [8,25]). References [1] E. Oset, A. Ramos, Prog. Part. Nucl. Phys. 41 (1998) 191. [2] W.M. Alberico, G. Garbarino, Phys. Rep. 369 (2002) 1; W.M. Alberico, G. Garbarino, in: T. Bressani, A. Filippi, U. Wiedner (Eds.), Hadron Physics, Proceedings of the International School of Physics “Enrico Fermi”, Course CLVIII, Varenna (Italy), 22 June–2 July 2004, IOS Press, Amsterdam, 2005, p. 125. [3] A. Ramos, M.J. Vicente-Vacas, E. Oset, Phys. Rev. C 55 (1997) 735; A. Ramos, M.J. Vicente-Vacas, E. Oset, Phys. Rev. C 66 (2002) 039903, Erratum. [4] G. Garbarino, A. Parreño, A. Ramos, Phys. Rev. Lett. 91 (2003) 112501. [5] G. Garbarino, A. Parreño, A. Ramos, Phys. Rev. C 69 (2004) 054603. [6] E. Bauer, G. Garbarino, A. Parreño, A. Ramos, nucl-th/0602066. [7] J.F. Dubach, G.B. Feldman, B.R. Holstein, L. de la Torre, Ann. Phys. (N.Y.) 249 (1996) 146. [8] E. Oset, L.L. Salcedo, Nucl. Phys. A 443 (1985) 704. [9] H. Outa, et al., Nucl. Phys. A 754 (2005) 157c. [10] J.J. Szymanski, et al., Phys. Rev. C 43 (1991) 849. [11] H. Noumi, et al., Phys. Rev. C 52 (1995) 2936. [12] Y. Sato, et al., Phys. Rev. C 71 (2005) 025203. [13] E. Bauer, F. Krmpoti´c, Nucl. Phys. A 717 (2003) 217. [14] E. Bauer, F. Krmpoti´c, Nucl. Phys. A 739 (2004) 109. [15] D. Van Neck, M. Waroquier, V. Van der Sluys, J. Ryckebusch, Phys. Lett. B 274 (1992) 143. [16] E. Oset, H. Toki, W. Weise, Phys. Rep. 83 (1982) 281. [17] W.M. Alberico, A. De Pace, M. Ericson, A. Molinari, Phys. Lett. B 256 (1991) 134. [18] A. Ramos, E. Oset, L.L. Salcedo, Phys. Rev. C 50 (1994) 2314. [19] W.M. Alberico, M. Ericson, A. Molinari, Ann. Phys. 154 (1984) 356. [20] A. Parreño, A. Ramos, C. Bennhold, Phys. Rev. C 56 (1997) 339; A. Parreño, A. Ramos, Phys. Rev. C 65 (2002) 015204. [21] M.N. Nagels, T.A. Rijiken, J.J. de Swart, Phys. Rev. D 15 (1977) 2547; P.M.M. Maessen, T.A. Rijiken, J.J. de Swart, Phys. Rev. C 40 (1989) 2226.

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[22] V.G.J. Stoks, Th.A. Rijken, Phys. Rev. C 59 (1999) 3009; Th.A. Rijken, V.G.J. Stoks, Y. Yamamoto, Phys. Rev. C 59 (1999) 21. [23] R. Machleidt, K. Holinde, Ch. Elster, Phys. Rep. 149 (1987) 1. [24] M.B. Barbaro, A. De Pace, T.W. Donnelly, A. Molinari, Nucl. Phys. A 596 (1996) 553. [25] E. Bauer, Nucl. Phys. A 796 (2007) 11. [26] H. Bhang, et al., Eur. Phys. J. A 33 (2007) 259.

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