INFN - Laboratori Nazionali di Frascati, May 1995
From: The Second DAFNE Physics Handbook
Chapter 12: Nuclear Physics with Kaons
Eds. L. Maiani, G. Pancheri, N. Paver
Supported by the INFN, by the EC under the HCM contract number CHRX-CT920026 and by the authors home institutions.
I=1/2 Rule
Dipartimento di Fisica Sperimentale, Università di Torino (Italy)
and
Istituto Nazionale di Fisica Nucleare, Sezione di Torino (Italy)
cm
The decay of 
-hypernuclei (
X) is driven
by the weak interaction and the two main mechanisms are:
,
which is just the free decay of the 
but now occurring in the nuclear
medium
which is only possible in a nucleus and is related to
parity non-conserving weak interactions
A quite recent and complete review on the subject is due to Cohen [1].
The branching ratios (BR) for the two branches of 
decay in
free space, 
(64 %) and
(36 %) are, within 5 %, in
agreement with the isospin rule 
I = 1/2, as it can be
readily checked by an elementary Clebsch-Gordan analysis. Such a
phenomenological rule has never been explained on sound theoretical grounds.
However the weak decay of the free 
is strongly hindered when
the 
is embedded in nuclear matter, in the ground state of an
hypernucleus (I remind that, due to different lifetimes for 
weak decay
and strong or e.m. decay of excited states of hypernuclei, the
decay of hyperfragments occurs practically always from the ground state).
The binding energy B
of the 
-hyperon in the ground state
increases linearly with A, reaching the saturation value of 25 MeV for heavy
hypernuclei, and the phase space for the mesonic decay is greatly reduced.
The outgoing nucleon from the mesonic decay has a very low momentum
(< 100 MeV/c), quite less than the Fermi momentum of a nucleon in a nucleus
(
270 MeV/c), and the process is substantially Pauli-blocked.
In all but the lightest hypernuclei the primary decay channel
would then be the NM one, for which the energy release is approximately
176 MeV, corresponding to a momentum of each final nucleon of 
417 MeV/c.
The NM mode has a much larger phase space and the final nucleons are not
Pauli-blocked.
A measurement of the relative BRs for the two channels:
could provide information about the structure of the weak Hamiltonian,
in particular on the relative importance of the 
I=1/2 and
I=3/2 amplitudes that can contribute to the process.
Notwithstanding the considerable interest of this subject, experimental
data are scarce and not precise, due to the difficulty of producing
abundantly 
-hypernuclei in their ground states and detecting their
decay, in particular the NM one (2). However some recent data and analyses
showed interesting new features: Schumacher [2], by analyzing
the data on NM decay for the light hypernuclei 
H,
He and 
He suggested that a violation of the
I=1/2 rule could be present. At the 1.6 
level
the data were not consistent with pure 
I=1/2
hypothesis and suggested
that the 
I=3/2 amplitude could be comparable or even larger
that the 
I=1/2 piece.
In the near future an answer to this challenging question could be given
by the FINUDA spectrometer [3], one of the two experiments
approved for running at DA
NE, the Frascati 
-Factory that will be
commissioned next year. The production reaction of hypernuclei will be
the (
,
) one and the ``battle horse'' of the apparatus
will be just the study of NM decays, that will profit at best from the
cleanliness of the 
source [4] and the transparent
structure of the detector [5], whose solid angle is
larger than 2
sr.
I remind the most important features of FINUDA [6]:
about 75 hypernuclear states/hour for a production rate of 10
, about 6 NM decays (1)/hour and about 1 NM decay (2)/hour
at the initial luminosity of DA
NE of 10
cm
s
.
These numbers are between one and two orders of magnitude larger than those
of current or planned experiments at existing machines, and a further
order of magnitude will be gained at the design luminosity of DA
NE
of 10
cm
s
.
The energy of the hypernuclear final states produced in the reaction:
will be measured with a resolution of 0.6 MeV FWHM by means of the
fine spectroscopy (
p/p = 0.3 % FWHM) of the 
, whereas
the energy of the products from the NM decay of 
X:
will be measured with a precision of 1.3 MeV for the protons, about 10 MeV
for the neutrons. These resolution will not allow, in general, the
identification of the final nuclear states 
(X-1) and
X in (4) and (5).
In these inclusive measurements an averaging over the final states of the
weak decays 
will be inherently present, and a possible violation of the 
I
=1/2 rule could be inferred only by a systematic study of the ratio
of the BRs for (1) and (2) as a function of the mass number.
I remind that in the simplified hypothesis of final total isospin 1,
this ratio must be 1/2 for a pure 
I=1/2 interaction.
However, for a 
Li stopping target, the above resolutions may allow
an exclusive measurement, in which also the final state of the residual
nucleus is determined.
Let me consider the following
chains of production and decay processes for a 
Li target:
where the squares contain the hypernucleus undergoing the following decay.
In process (7) 
is not spectroscopized, even nor detected, and there are
two neutrons in the final state. However, if I consider that the weak decay of
an hypernucleus always occurs from the ground state, and that the slowing down
time (
10
s) of 
He
in the target following its production is much lower
than the 
lifetime (2.63
10
s), the detection of the
two neutrons in coincidence with a total missing mass resolution of
14 MeV allows to isolate the final state of 
He,
which has a high threshold for particle emission (19.8 MeV) and no low-lying
excited states.
The shell model picture of 
He is two
neutrons, two protons and the 
in the 1s
shell
and one neutron in the 1p
shell.
If the residual nucleus is 
He (two protons and two neutrons
in the 1s
shell) the NM exclusive decay
is the result of a weak interaction among a 
in an
1s
state and a neutron in a 1p
state.
To my knowledge, there are no theoretical calculations for exclusive
NM decays. A guess on the relative weight of the exclusive process over
the total NM decay rate can be inferred by analogy with some similar
nuclear processes.
(
, 2n) [7] or
(
, 2p) [8] reactions on nuclei are quite similar
concerning the energy released to a nucleon pair: 140 MeV plus the
kinetic energy compared to 176 MeV for NM decays.
For a 
Li target, the residual nucleus distribution in (
, 2
)
reactions is quite interesting: 
He in the ground state is produced
with a relative frequency of 
30 %, for the remaining 
70 %
being produced in the continuum. These fractions may be
taken into account for evaluating the relative BRs for the exclusive
process (7). (
, 2
) reactions
in 
Li producing 
He in the ground state involve the proton
and the neutron of 
Li in the 1p
shell, whereas processes
with 
He unbound involve a proton and a neutron of 1s
and 1p
shells, like in the case of NM decays (7)
with 
He in the ground state.
Information on the exclusive process (7) will be unique and could
lead to relevant constraints on the 
I = 1/2 rule, by a proper
handling of the nuclear wave functions involved. However, also the continuum
part of the missing mass spectrum of the two neutrons emitted in (7),
leading to 
He unbound states, is of great importance, since it is linked to
the weak interaction of a 
and a neutron both in the 1s
state. Similar information on the weak interaction of a 
and a proton
in the 1s
state could be obtained by a study of process
(6) in which 
Li is unstable for proton emission.
The total missing mass resolution for this channel could be better than
10 MeV, not enough for a clean separation of the exclusive channel with 
H in
the final state. However, if the collected statistics will be sufficient,
unfolding techniques and angular correlation constraints might allow a quite
precise determination also for this exclusive channel.
With the expected counting rates of NM decays in FINUDA, and considering that
for process (7) the coincidence with a 
is not required, I
expect about 3 NM decays (6) and (7)/hour. Brs
(6) and (7) could then be measured to a precision of
% in a reasonable time. At the same level of precision I expect the
knowledge of absolute efficiencies of detection and acceptances, as well as the
corrections for nuclear structure effects, quite known for the simple light
nuclei involved. Deviation of the ratio of BRs for (6) and
(7) from the value 1/2, expected by the pure 
I
= 1/2 rule, would be an indication of challenging new physics.
