The decay of 
--hypernuclei (
) to normal nuclei (S=0)
occurs through the weak interaction which can change the strangeness.
In contrast to a 
--hypernucleus a strong decay channel
(
) is open.
The study of the weak decay of a 
--hypernucleus aims to the
understanding of the stability of the system on the one side and
of how the many--body nuclear system affects the products of
the 
decay on the other (clearly the two issues are related).
Moreover the decay process also reflects the influence of
the medium on the 
itself and on the weak interactions it experiences.
Basically two are the mechanisms for the decay of an hypernucleus:
i) the mesonic decay (for a review see ref. [11]), namely
which is just the free decay of the 
but now occurring in the medium;
ii) the non--mesonic (NM) decay, commonly viewed as being driven by the process
which is only possible in a nucleus.
They are displayed in Fig.8.
Figure 8: Weak--decay diagrams for mesonic decays (left--hand side), and for
non--mesonic decays (right--hand side) of a 
(from ref.[6]).
For the mesonic decay the following two branches are open in free space
The above branching ratios are in approximate accord with the isospin rule
as it can be easily checked with a Clebsch--Gordan analysis.
It is disturbing that such a rule, discovered at the empirical level, has
never been convincingly understood on a sound theoretical ground.
Now, if the process (24), whose Q--value is about 40 MeV (see (2)), occurs
with the 
at rest, then most
of the energy is carried away by the pion and it is easily verified that the
outgoing proton remains with only about 5 MeV of energy. Accordingly its
momentum turns out to be
which is much less than the Fermi momentum 
(in the above
is the nucleon mass). Thus the mesonic decay in nuclei is
substantially hindered by the Pauli principle.
However some high momentum components in the proton wave function are
supplied by the interaction both with the outgoing pion and with the medium:
thus the restriction on the mesonic 
decay of a hypernucleus is less
severe than implied by (27), even in heavy nuclei, although
the suppression remains impressive.
In this respect the situation is well illustrated in Fig.9.
Figure 9: The calculated mesonic decay rates of the free 
decay rate
(from ref.[6]).
The NM decay, on the other hand, is not affected by the Pauli blocking. Indeed, assuming the available energy to be equally splitted between the two nucleons, one gets
hence
Owing to the large energy and momentum transfer characterizing the
NM decay channel it is clear that the short range component of
the 
N interaction is going to play an important role in the
process.
Indeed Alberico et al. [12] showed that in nuclear matter the stronger
the 
-N short--range repulsion is,
the larger the reduction of the
non--mesonic decay is going to be. This result was achieved by parametrizing
the short range interaction with the Landau--Migdal parameter 
and
allowing for sizable variations of the latter.
Central questions to be asked concerning the NM decay of hypernuclei are:
i) how does the NM decay compare with the free 
life--time?
ii) which of the two mechanisms
and
is more relevant for the NM decay?
iii) how the NM decay is realized in a many--body framework?
Fig.10 provides an answer to the first question: indeed it is clearly seen that
for 
the NM decay mode not only is the dominant one, but also brings
the total width for the decay of a hypernucleus 
back to the free value
(if not to values larger than 
).
Figure 10: Ratio 
of non--mesonic to 
mesonic decay rates as a
function of A.
Concerning the second question, which of course is not disjoint from the third one, a simple evaluation of Clebsch--Gordan coefficients leads to the geometrical estimate
(of course (32) ignores a 
contribution in the denominator).
The experimental indication, on the other hand, is more consistent with a value for the ratio (32) close to one (although the data are rather scarce).
A variety of calculations have been performed to improve upon the
geometrical estimate. These approaches range from models accounting for
all possible mesons exchanges between 
and NN propagation lines
with one strong and one weak vertex to models accounting for the quarks
degrees of freedom. Not surprisingly the results thus obtained show a marked
model dependence.
This brings us to address the third question, which is more easily answered
in a nuclear matter context. In fact in this framework is transparently
seen that the occurence
of the elementary processes (30) and (31) might be prevented.
Indeed it is easily realized that if the pion emitted from the 
,
an allowed process now since
it is taking place in a weak vertex, is close to the mass--shell ( in the
considered process this correspond to a pion carrying relatively large energy
and little momentum) then it cannot be absorbed by a single nucleon
because, as shown in Fig.11, this would induce a particle--hole excitation
in a region forbidden in nuclear matter. It can do so only when it is
highly virtual (Fig.11).
Figure 11: The intersection in the 
plane of the parabola
(solid line) representing the energy--momentum relation for the 
emitted in
the decay of a 
at rest with the excitation spectrum of nuclear matter.
Dashed region: particle--hole excitation. Dashed line: the collective pionic
branch. Also shown (dotted line) the free pion branch
(from ref.[12]).
Therefore a close to the mass--shell pion can only be absorbed by a
pair of nucleons, leading to two particle -- two hole excitations. More
specifically the absorption can occur in the pionic branch, a collective
nuclear state embodying a coherent superposition of pionic and
--hole elementary excitations, which lies at an energy lower than
the physical pion. Its decay mode in the kinematical region of concern for
the decay of a hypernucleus is by two nucleons emission. It thus appears
that the 
decay, particularly in heavy nuclei, is occurring by
a substantial fraction not via the (25), but rather via the process
The above outlined situation is somewhat reminiscent of the absorption of a real photon which, even when of relatively high energy, carries nevertheless little momentum and accordingly can only be absorbed on a correlated pair of nucleons (hence the quasi--deuteron model) in order to conserve both energy and momentum. In ref.(11) it is shown that these excitations provide an important contribution to the width for the NM decay.
The identification of the process, because is associated with the emission of three nucleons, poses however a serious experimental challenge.
