Novel Weak Decays in Doubly Strange Systems

The strangeness-changing ($\Delta S = 1$) weak baryon-baryon interaction is studied through the nonmesonic weak decay of double-$\Lambda$ hypernuclei. Besides the usual nucleon-induced decay $\Lambda N \to N N$ we discuss novel hyperon-induced decay modes $\Lambda \Lambda \to \Lambda N$ and $\Lambda \Lambda \to \Sigma N$. These reactions provide unique access to the exotic $\Lambda \Lambda$K and $\Lambda \Sigma$K vertices which place new constraints on Chiral Pertubation Theory ($\chi$PT) in the weak SU(3) sector. Within a meson-exchange framework, we use the pseudoscalar $\pi,\eta,K$ octet for the long-range part while parametrizing the short-range part through the vector mesons $\rho, \omega, K^*$. Realistic baryon-baryon forces for the $S=0,-1$ and -2 sectors account for the strong interaction in the initial and final states. For $^6_{\Lambda \Lambda}$He the new hyperon-induced decay modes account for up to 4% of the total nonmesonic decay rate. Predictions are made for all possible nonmesonic decay modes.


I. INTRODUCTION
The production and weak decay properties of strangeness-rich systems is of fundamental importance for our understanding of relativistic heavy-ion collisions and certain astrophysical phenomena, such as neutron stars. The simplest systems with strangeness, hypernuclei with one or two bound Λ's, have been used to study both the strong and the weak baryon-baryon (BB) interaction in the SU(3) sector. Until now, hypernuclear weak decay represents the only source of information on the | ∆S |= 1 four fermion interaction where, in contrast to the weak ∆S = 0 NN case, both the weak parity-conserving (PC) and parity-violating (PV) amplitudes can be studied.
In the absence of exact solutions to low-energy QCD, effective field-theory techniques based on chiral expansions have been fairly successful in the description of hadronic observables in the (non-strange) SU (2) sector. The stability of the chiral expansion is less clear for the SU(3) sector, due to the significant amount of SU(3) symmetry breaking. A wellknown failure of SU(3) chiral perturbation theory has been the prediction [1] of the four PC P -wave amplitudes in the weak nonleptonic decays of octet baryons, Y → Nπ, with Y = Λ, Σ or Ξ. Since large cancellations among tree-level amplitudes are held responsible for the problem with the weak P -wave octet amplitudes, it is imperative to assess whether this situation is universal within SU (3) χPT or limited to a few exceptional cases. However, other weak octet baryon-baryon-meson (BBM) vertices can only be accessed through reactions that allow for the virtual exchange of mesons, such as the reactions ΛN → NN and ΛΛ → Λ(Σ)N. The process ΛN → NN has been extensively studied in an approach where the long-range part of this interaction is based on the exchange of the SU(3) pseudoscalar meson octet (π, K, and η). The pseudoscalar baryon-baryon-meson vertices are considered fixed by experiment in the case of the pion, and by SU(3) chiral algebra for the K and the η.
Since the large momentum transfer in the reaction (typically 400 MeV/c) leads to a mechanism where short-range effects must be included, they have been modelled either through the exchange of the vector meson octet [2][3][4] (ρ,ω and K * ) or quark exchange [5]. The vector baryon-baryon-meson vertices are constrained by much weaker SU (6) considerations. The ΛN → NN process is then embedded in nuclear many-body matrix elements using either correlated Faddeev amplitudes in the case of few-body systems, hypernuclear shell model wave functions or nuclear matter solutions within the Local Density Approximation, depending on the mass number of the hypernucleus under investigation. While this description of hypernuclear weak decay is not as rigorous as effective field theory would require, it nevertheless has been reasonably successful in describing the available experimental data.
Since the late 1960's, the production and decay of single-Λ hypernuclei has been studied experimentally in great detail, but only very few events involving doubly-strange objects have been reported [6][7][8][9][10]. Double Λ hypernuclei are produced via the (K − , K + ) reaction at KEK (Japan) and BNL (USA), where a ΛΛ hypernuclear fragment can be formed by Ξ − capture on a nucleus. The FINUDA experiment at DAΦNE (Frascati, Italy) can produce double-Λ hypernuclei by stopping slow K − (coming from the Φ decay) into thin targets to obtain data with higher energy resolution. Studying the weak decay of those objects opens the door to a number of new exotic Λ-induced decay modes: ΛΛ → Λn and ΛΛ → ΣN. Both of these decays would involve hyperons in the final state and should be distinguishable from the ordinary ΛN → NN mode. The ΛΛ → Λn channel is especially intriguing since the dominant pion exchange is forbidden, thus forcing this reaction to occur mostly through kaon exchange. One would therefore gain access to the weak ΛΛK and ΛΣK vertices.
In this paper, we extend previous weak decay calculations of single-Λ hypernuclei into the S = −2 sector, thus exploring the power of the ΛΛ → Λ(Σ)N process to shed light on the novel weak vertices. In order to take into account the effects of the strong interaction between the baryons, correlated wave functions are obtained from a G-matrix calculation for the initial ΛN and ΛΛ states, while a T -matrix equation is solved for the final NN and YN states using the Nijmegen interaction models [11], in particular the NSC97f one.

II. DECAY RATE
In the weak nonmesonic decay of double-Λ hypernuclei, new hyperon-induced mechanisms, the ΛΛ → ΛN and the ΛΛ → ΣN transitions (denoted as ΛΛ → Y N throughout the text) become possible in addition to the dominant ΛN → NN decay mode. Assuming the initial hypernucleus to be at rest, the NMD rate is given by: where the quantities M H , E R , E 1 and E 2 are the mass of the hypernucleus, the energy of the residual (A − 2)-particle system, and the total asymptotic energies of the emitted baryons, respectively. The integration variables k 1 and k 2 stand for the momenta of the two baryons in the final state. In the expression above, the momentum-conserving delta function has been used to integrate over the momentum of the residual nucleus. The sum, together with the factor 1/(2J + 1), indicates an average over the initial hypernucleus spin projections, M I , and a sum over all quantum numbers of the residual (A − 2)-particle system, {R}, as well as the spin and isospin projections of the emitted particles, {1} and {2}. The total nonmesonic decay rate can be written as: where the rate corresponding to ΛN → NN transitions is divided into a neutron-induced rate, Γ n : Λn → nn and a proton-induced one, Γ p : Λp → np, while the Λ-induced transitions ΛΛ → Y N give rise to Λn, Σ 0 n and Σ − p final states, hence: Note that while the individual rates have been written above as exclusive observables, in principle, charge-exchange final state interactions (FSI) with the residual nucleus obscurs a clean experimental discrimination of these channels on the basis of the charge of the emitted particles. Monte Carlo intranuclear cascade models are necessary [12] to extract the partial weak decay rates of hypernuclei from experiment [13]. The impossibility of measuring the partial rates directly from the charge of the emitted particles was also shown in the case of the weak decay of the hypertriton [14], a system where the strong interaction effects can be treated exactly using Fadeev equations with a realistic baryon-baryon potential.
In order to draw conclusions regarding the weak dynamics, one has to write the hypernuclear transition amplitude, M i→f , in terms of the elementary two-body transitions, B 1 B 2 → B 3 B 4 , and nuclear structure details have to be treated with as few approximations and ambiguities as possible. We work in a shell-model framework, hence the Λ hyperons and the nucleons are described by single-particle orbitals. In addition, we assume a weak coupling scheme by virtue of which the Λ hyperons couple only to the ground state of the nuclear core. Therefore, in the case of the 6 ΛΛ He hypernucleus studied here, with quantum numbers J I = M I = 0, T I = M T I = 0, the state will be given by where antisymmetry forces the two Λ hyperons to be in a 1 S 0 state, since they are assumed to be in the lowest s-shell before the decay occurs. This is so because, in general, hypernuclei with Λ's in excited orbitals will rapidly decay into the ground state through electromagnetic or strong de-excitation processes, which are orders of magnitude faster than those mediated by the weak interaction. The single-particle orbitals for nucleons and Λ's are taken as harmonic oscillator states with parameters b N = 1.4 fm and b Λ = 1.6 fm, respectively. The nucleon parameter is chosen to account for the 4 He form factor. The parameter for the Λ wave function reproduces the Hartree-Fock (HF) probability of finding the two Λ particles at relative distance r, obtained in Ref. [15] by adjusting their model parameters to the binding energies of the three observed double Λ-hypernuclei, 6 ΛΛ He, 10 ΛΛ B and 13 ΛΛ B [6][7][8][9][10]. To obtain the rate corresponding to the ΛN → NN transition, with the initial N being n(t N = −1/2) or p(t N = 1/2), we have to write the non-strange nuclear core as one nucleon coupled to a 3-particle system (with quantum numbers J 3 , M 3 , T 3 and M T 3 ) and decouple one of the two Λ's in the cluster. Therefore, the initial ΛN pair will convert into a final NN pair with quantum numbers k 1 s 1 t 1 , k 2 s 2 t 2 , leaving a residual 4-particle system with quantum numbers J R , M R , T R , M T R . Performing all the necessary decoupling and recoupling operations, we finally arrive at: where We have assumed the Λ at the weak vertex to behave as a | 1 2 − 1 2 isospurion in order to naturally incorporate the experimentally observed ∆I = 1/2 rule. The momentum states k 1 and k 2 have been transformed to total ( K = k 1 + k 2 ) and relative ( k = ( k 1 − k 2 )/2) momenta and the final two-nucleon state, k, SM S , T M T |, must be properly antisymmetrized, which means that the factor (1 − (−1) L+S+T )/ √ 2 will appear when a decomposition in partial waves is performed for the outgoing plane wave.
For the ΛΛ → Y N transition we don't need to decouple one of the hyperons from the cluster, neither a nucleon from the core. The residual 4-particle system, which coincides with the 4 Λ He core in Eq. (5), contains no strangeness, while the final two-particle state contains one hyperon that can be either a Λ (|Y t Y = |00 ), a Σ − (|Y t Y = |1 − 1 ) or a Σ 0 (|Y t Y = |10 ). The ΛΛ → Y N hypernuclear amplitude is given by: where

III. THE MESON EXCHANGE POTENTIAL
In a meson-exchange description, the ΛN → NN and ΛΛ → Y N transitions are assumed to proceed via the exchange of virtual mesons belonging to the ground-state pseudoscalar and vector meson octets. The amplitude, which is schematically represented by the Feynman diagram depicted in Fig. [1], reads where Ψ p (x) = e −ipx u(p) is the free baryon field of positive energy, Γ i the Dirac operator characteristic of the baryon-baryon-meson (BBM) vertex and ∆ M (x − y) the meson propagator.
In Table I we show the strong and weak hamiltonians for pseudoscalar (PS) and vector (V) mesons. The constants A, B, α, β and ǫ correspond to the weak couplings, while g (g V , g T ) represents the strong (vector, tensor) one.
The nonrelativistic reduction of this amplitude gives us the potential in momentum space [3], which for pseudoscalar mesons reads: where G F m π 2 = 2.21 × 10 −7 is the Fermi weak constant, q is the momentum carried by the meson directed towards the strong vertex, µ the meson mass and M 2 (M 1 ) is the average of the baryon masses at the strong (weak) vertex. For vector mesons the potential reads: In Eqs. (9), (10) the operatorsÂ,B,α,β andε contain, apart from the weak coupling constants, the specific isospin dependence of the potential, which is τ 1 τ 2 for the isovector π and ρ mesons,1 for the isoscalar η and ω mesons and a combination of both operators for the isodoublet K and K * . Ref. [3] introduces a compact way to write the transition potential in r-space: where the index i runs over the different mesons exchanged (i = 1, . . . , 6 represents π, η,K,ρ, ω,K * ) and α over the different spin operators: central spin independent (C), central spin dependent (SS), tensor (T) and parity violating (PV) In order to account for the finite size and structure of the particles involved in the process, form factors (FF) are included. In our previous calculations of the decay of single-Λ hypernuclei [3] we used a monopole FF, , at both the weak and strong vertices, with different cut-offs depending on the exchanged meson, Λ i . Those cut-offs were taken from the Jülich YN interaction model [16]. The latest version of the Nijmegen potentials [11], which is used here, also gives meson-dependent cut-offs but uses an exponential FF at each vertex of the type F ( q 2 ) = exp(− q 2 /2Λ i 2 ). Therefore, the results presented here are also obtained with the exponential-type FF, although, for technical reasons, it is matched to a function of the type Λ 2 i /( Λ 2 i + q 2 ) at | q |≃ 400 MeV/c, the most relevant momentum transfer in the weak decay transition. By construction, this type of monopole FF is equivalent to the exponential one at q = 0 and we have verified that, up to a momentum transfer of 600 MeV/c above which the transition amplitude is negligible, the differences between both functions are less than 2%. The modified cut-offs, Λ i , to be used in the monopole-type FF are listed in Table III.
To incorporate the effects of the strong baryon-baryon (BB) interaction we solve a Tmatrix scattering equation in momentum space for the outgoing two-particle system (NN or YN). For the initial two-particle system (ΛN or ΛΛ) we take into account medium effects, thus intermediate states only propagate into states allowed by the Pauli operator (G-matrix). For the initial interacting ΛN pair we previously found [17] that multiplying the independent two-particle wave function by a spin-independent correlation function of the type: with a = 0.5 fm, b = 0.25 fm −2 , c = 1.28 fm, n = 2, produced a correlated wave function which averaged the ones obtained from a microscopic finite-nucleus G-matrix calculation [18] using the soft-core and hard-core Nijmegen models [19]. In the case of the interacting ΛΛ system, we again assume a correlation function of the type of Eq. (12). The parameters, a = 0.80 fm, b = 0.12 fm −2 , c = 1.28 fm and n = 1, are determined to reproduce the ratio between the correlated and uncorrelated 1 S 0 ΛΛ wave functions in nuclear matter at saturation density, Ψ1 S 0 (k, r)/j 0 (kr), taking k = 100 MeV/c as a representative momentum of the two Λ's in a finite nucleus. The nuclear-matter correlated wave function, Ψ1 S 0 (k, r), has been obtained from a coupled-channel G-matrix calculation using the NSC97f model of the new Nijmegen potentials. In Fig. 2, we show the probability per unit length of finding two Λ particles at a distance r, given by 4πr 2 |Ψ rel ΛΛ | 2 . The solid line corresponds to the uncorrelated two-Λ wave function, which in this work is given by the 1s harmonic oscillator wave function with the parameter b rel = √ 2b Λ . The dashed line incorporates the calculated correlation function, Ψ1 S 0 (k, r)/j 0 (kr), a result that is well parametrized by a phenomenological correlation function of the form of Eq. (12), as shown by the dotted line.
As mentioned before, the strong BBM couplings are taken from the NSC97f [11] potential. In the weak sector only the couplings corresponding to decays involving pions are experimentally known. The weak couplings for heavy mesons are obtained following Refs. [2,20]. The PV amplitudes for the nonleptonic decays B → B ′ + M involving pseudoscalar mesons are derived using the soft-meson reduction theorem and SU (3) symmetry. This allows us to relate the physical amplitudes of the nonleptonic hyperon decays into a pion plus a nucleon, B → B ′ + π, with the unphysical amplitudes involving other members of the meson and baryon octets. On the other hand, SU(6) W permits relating the amplitudes involving pseudoscalar mesons with those for vector mesons. The weak PC coupling constants are obtained using a pole model, where the pole terms due to the ( 1 2 ) + ground state (singular in the SU(3) soft meson limit) are assumed to be the dominant contribution. No meson pole terms are included. The values of these couplings for the weak vertices are listed in Table  IV.

IV. RESULTS
The results for the decay rates of 6 ΛΛ He into the different channels are summarized in Tables V,VI and VII, where the rate is presented in units of the free Λ decay, Γ Λ = 3.8 × 10 8 s −1 . Apart from giving the various partial rates, the ratio between the neutron-induced and proton-induced decay rates, Γ n /Γ p , is also quoted. Only ∆I = 1/2 transitions have been considered.
We begin by discussing the results of Table V, which list calculations absent of either strong correlations (initial and final) and FF. Although the results are clearly unrealistic, they serve to illustrate, in comparison with the following table, the effect of the strong interaction on the weak decay process. The ΛN → NN transitions receive contributions mainly from the π, ρ, K and K * mesons; however, once short-range strong correlations are included (see Table VI), the role of the heavier mesons is strongly reduced. The results in Table V illustrate that isospin symmetry forbids isovector meson contributions, (π, ρ), for the uncorrelated ΛΛ → Λn decay rate and isoscalar meson contributions, (η, ω), for the uncorrelated ΛΛ → ΣN rates. Finally, we note that the nucleon-induced rate is about 20 times larger than the total hyperon-induced contribution.
When FF and strong correlations in the initial and final states are included, the picture changes considerably, as can be seen from the results shown in Table VI. Most of the ΛN → NN rate comes from the exchange of the π and K mesons. Note that the small Γ n /Γ p ratio for the one-pion exchange mechanism substantially increases when the K-meson is incorporated due to a destructive (constructive) interference in the p-induced (n-induced) rate. This aspect is discussed recently in Ref. [4], which updates the meson-exchange model of Ref. [3] and finds agreements with results obtained by groups combining the π and K exchange with either a direct quark mechanism [5] or 2π exchange [21]. Since there are now two Λ particles contributing to the ΛN → NN decay, one would expect the result to be roughly twice the rate in 5 Λ He, which for this potential turns out to be 0.32Γ Λ , as shown in Ref. [4]. We obtain instead a significantly larger value of Γ NN = 0.96Γ Λ . This finding can be traced to the Λ which is more strongly bound in 6 ΛΛ He than it is in 5 Λ He by about 80% (i.e., the Λ single-particle separation energy changes from 3.12 MeV to around 5 MeV). This increased binding is reflected in the Λ single-particle wave function, with a harmonic oscillator parameter b Λ = 1.6 fm instead of the value b Λ = 1.87 fm used in the decay of single-Λ hypernuclei [3,4]. As mentioned in Sec. II, the parameter b Λ = 1.6 fm simulates the uncorrelated ΛΛ probability of Ref. [15], which reproduces the binding energy of the three observed double-Λ hypernuclei. There is also some influence from the slightly different kinematics involved in the ΛN → NN decay of 6 ΛΛ He with respect to that in 5 Λ He. From the results presented in Table VI, we observe that the ΛΛ → Λn rate receives a tiny contribution from the isovector π and ρ mesons. This is due to the combined transition ΛΛ weak → ΣN strong → ΛN, which is possible through the coupling between ΛN and ΣN states induced by the strong interaction. Analogously, the strong ΛN − ΣN coupling also induces very small contributions from the isoscalar mesons to the ΣN rate, Γ ΣN .
We find the rate Γ Λn to be dominated by K-exchange. We note that the π + K contribution is very similar to the value for K-exchange alone; adding the η meson decreases the rate by 35%. However, there is strong evidence that we are overestimating the role of the η meson by using strong ηNN and ηΛΛ couplings that rely on SU(3) symmetry. All indications from a careful analysis of reactions like η photoproduction [22] point towards a ηNN coupling much smaller than the one provided by SU(3) symmetry. In this case, the weak decay of double-Λ hypernuclei going to final Λn states would be almost solely determined by K exchange and the measurement of this decay channel would provide valuable information on the weak ΛΛK couplings. In particular, since this partial rate is dominated by PC transitions, as can be inferred from the values of the ΛΛK 0 weak couplings listed in Table  IV, one would gain access to the PC ΛΛK 0 amplitude. The rate to final ΣN states, Γ ΣN , on the other hand is very small and, as expected, dominated by the π-exchange mechanism.
The ratios between the rates that include correlations are: Correlations have a stronger influence on the YN channels, which are a factor 30 − 300 smaller compared to the NN rate, even though the ΣN transition allows a final state with a lower relative momentum and, therefore, is less sensitive to the strongly repulsive core of the YN interaction. However, the NN rate is dominated by tensor transitions that are absent in the YN channels because the interacting ΛΛ pair is in a 1 S 0 state. With the tensor strength distributed towards larger distances compared to those for central and parity-violating transitions, the NN transition becomes comparatively less affected by strong correlations than the YN ones. Note that correlations tend to increase the ΛN rate while decreasing the ΣN rate. The reason is that the uncorrelated ΣN rate is dominated by π and K * exchanges, the K * contribution being about twice that of the π meson, whereas the ΛN rate comes mainly from strange meson exchange, with the K contribution twice as large as that of the K * . Therefore, the supression of short-range physics induced by correlations affects the ΣN rate more than the ΛN one. In addition, it is known [23,24] that the ΣN interaction for the new Nijmegen potentials is quite repulsive in the I = 1/2 channel, affecting the ΣN wave functions significantly, which are pushed out from the weak potential interaction range.
The results for various approaches to the treatment of FSI are compared in Table VII. Those labelled with "FSI eff." have been obtained by multiplying the free NN, ΛN and ΣN final states by the momentum and channel independent function 1 − j 0 (q c r), with q c = 3.93 fm −1 . As recently discussed in Ref. [4], treating FSI in this simplified phenomenological fashion overestimates the rate Γ NN calculated rigorously with a T -matrix by about a factor of two. The rate Γ Λn obtained with the phenomenological FSI treatment is almost one order of magnitude smaller compared with the result obtained within a T -matrix approach, while on the other hand, the rate Γ ΣN is overpredicted by almost an order of magnitude. As in Ref. [4], we again find that in order to be able to extract information on the weak decay amplitudes a careful treatment of FSI is essential in the study of the weak decay of double-Λ hypernuclei.
We finally compare our results with the ones presented recently by Itonaga et al. [25], where the weak decay mechanism contains π, ρ and correlated two-pion exchange with scalar (σ) and vector (ρ) quantum numbers. The Λ wave function in 6 ΛΛ He was taken the same as in their 5 Λ He calculation [26], thus neglecting the possible effect of a stronger binding of the Λ in a double-Λ hypernucleus. Hence, the nucleon-induced rate obtained in [25] is Γ NN = 0.753Γ Λ , equal to twice their decay rate in 5 Λ He. This rate is about 20% smaller than the one obtained here. The rates for Λn and ΣN final states, Γ Λn = 0.025Γ Λ and Γ ΣN = 0.0012Γ Λ , are about a factor two smaller than the ones obtained in this work. We note that the ΛΛ correlation function used in Ref. [25] is in fact the 1 S 0 ΛN one used in their study of the decay of 5 Λ He. We also note that the K-exchange mechanism, dominant in the ΛΛ → Λn transition amplitude, is absent in the work of Ref. [25].
In Ref. [3], we explored SU(3) symmetry breaking effects in the decay of single-Λ hypernuclei by using weak NNK couplings derived in the framework of heavy baryon chiral perturbation theory [27], where one-loop SU(3) corrections to leading order were evaluated. Here we explore similar effects not only for the NNK but also for the ΛΛK and ΛΣK couplings, which were calculated in Refs. [28,29]. In Table VIII we list the values for the weak PV (S-wave) and PC (P -wave) couplings involving the kaon. Compared to the ones quoted in Ref. [3], the numbers in Table VIII differ since they were obtained using a common mass splitting of 200 MeV between the octet and decuplet baryons [28,29].
Comparing the values listed in Table VIII with those in Table IV we observe that the one-loop corrected NNK P -wave constants are approximately a factor two smaller than the tree level ones, while the S-wave are very similar, except for the pnK + coupling, which, again, is half the tree-level value. This explains the reduced K-exchange contribution to the nucleon-induced rates with respect to the tree-level result. Note that the magnitude of the (constructive) interference between π and K in the nn channel has not changed much, while it has become less destructive in the np channel, thereby producing a smaller Γ n /Γ p ratio. Once all the mesons are included, the NN rate obtained with the one-loop corrected NNK couplings is a factor 1.4 larger than that obtained with the tree-level values, while the ratio Γ n /Γ p is smaller by a factor of two.
Although the one-loop corrected ΛΛK couplings of Table VIII are not much different from the corresponding tree-level values, they lead to a reduction of the K-meson exchange contribution to the Λn rate by about a factor two. Hence, the contribution of the η meson and the vector mesons becomes more relevant, making the extraction of the ΛΛK coupling from this ΛΛ → Λn decay mode more difficult.
The decay rate into final ΣN states is the only one that increases when the one-loop corrected ΛΣK couplings are used, despite the magnitude of the P -wave amplitudes being ten times smaller than the corresponding tree-level values, whereas the S-wave amplitudes are only moderately larger. In fact, Table VI reveals that the K-exchange contribution to the ΣN rate is two times smaller than that obtained with the tree-level ΛΣK couplings, as expected from the reduced magnitude of the couplings. However, due to the opposite sign of the loop-corrected P -wave coupling with respect to the corresponding tree level value, the interference between the π and K contributions is now constructive instead of destructive, which explains the final increased loop-corrected rate.
Finally, we have investigated how our results would change if we allowed the one-loop corrected BBK couplings to vary within their estimated error bands. The corresponding range of variation on the rates is quoted in Table IX, where the results obtained with the tree-level BBK couplings have also been included to facilitate the comparison between both calculations. The results summarized in Table IX show clearly that the inclusion of weak BBK couplings obtained with one-loop corrections to the leading order in χPT produces a ΛN → NN rate 40 % larger than the tree-level value, a ratio Γ n /Γ p twice smaller, a ΛΛ → Λn rate roughly twice smaller, and a ΛΛ → ΣN rate almost twice as large.

V. CONCLUSIONS
In this study we investigated the nonmesonic weak decay modes of double-Λ hypernuclei within a one-meson-exchange framework. Our results are given for the specific double-Λ hypernucleus 6 ΛΛ He, but can easily be extended to heavier systems. The standard nucleon-induced nonmesonic ΛN → NN rate is found to be more than twice as large as in 5 Λ He due to the increased binding of the second Λ hyperon. Two new hyperon-induced modes become possible, ΛΛ → Λn and ΛΛ → ΣN, the latter coming in two charge states. We find the total hyperon-induced rate to be as large as 4% of the total nonmesonic rate. This new rate is dominated by the ΛΛ → Λn mode. In fact, this transition turns out to be the more interesting one (rather than the ΛΛ → ΣN) since it allows direct access to exotic vertices like ΛΛK, unencumbered by the usually dominant pion exchange. Indeed, one-loop log corrected χPT results modify the ΛΛ → Λn by 50% while changing the ΛN → NN only at the 15% level, demonstrating the power of this weak mechanism to test χPT in the weak SU(3) sector. With a free Λ in the final state this new mode should be distinguishable from the usual nucleon-induced decay channels. Given the potential benefits to hadronic physics, the experimental program of investigating the production and decay of double-Λ hypernuclei should be intensified. the authors, A.P., would like to thank the Institute for Nuclear Theory and the University of Washington (Seattle, USA) where an important part of this work was developped. This work has been partially supported by the U.S. Dept. of Energy under Grant No. DE-FG03-00-ER41132, by the DGICYT (Spain) under contract PB98-1247 and by the Generalitat de Catalunya project SGR2000-24.

A. Weak PC couplings
In this subsection we present the expression for the PC weak coupling constants derived by using a pole model [2] with only baryon poles.
• η-exchange • K-exchange A(p, pK 0 ) = g(p, To compute the former amplitudes, we use the following values for the weak transitions at the baryon lines A BB ′ : If we desire only the ∆I = 1 2 part of the above expressions, as we do in the present work, the following replacements have to be made [30]: • For the ΛNρ couplings: a V → 3a V and a T → 3a T .

C. Strong coupling constants
In this subsection we present the convention used for the strong couplings.
For the vector ρ, ω and K * mesons, the expressions are equivalent to the ones quoted above by making the replacements of Eq. (23).  φΨ   TABLE II. Possible 2S+1 L J channels present in the weak decay of 6 ΛΛ He. The notation used is the following: Lr stands for the relative orbital momentum for the initial ΛΛ or ΛN pair, L ′ for the relative angular momentum of the two-particle state right after the weak transition occurs, and L for the relative angular momentum of the final two-particle state after including the effects of the strong interaction (FSI). The symbols C, T and PV stand for Central, Tensor and Parity Violating.

TABLES
, which matches the exponential-type FF used in Ref. [11] .   6 ΛΛ He in absence of either strong correlations and FF. The nonmesonic decay rate is in units of Γ Λ = 3.8 × 10 9 s −1 . The NSC97f [11] strong coupling constants have been used.  6 ΛΛ He. Strong correlations (initial and final) and FF are included. The nonmesonic decay rate is in units of Γ Λ = 3.8 × 10 9 s −1 . The NSC97f [11] strong interaction model has been used. The values between parenthesis have been obtained using the weak kaon couplings from Refs. [28,29] .