Table Of ContentWeak nonmesonic decay spectra of hypernuclei
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1
Eduardo Bauera, Alfredo P. Galea˜ob, Mahir S. Husseinc, and Franjo Krmpoti´cd∗
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2
aFacultad de Ciencias Exactas, Departamento de F´ısica, Universidad Nacional de La
n
a Plata, 1900 La Plata, Argentina
J
7 bInstituto de F´ısica Te´orica, UNESP - Universidade Estadual Paulista,
Caixa Postal 70532-2, 01156-970 S˜ao Paulo, SP, Brazil
]
h
t cDepartamento de F´ısica Matem´atica, Instituto de F´ısica, Universidade de S˜ao Paulo,
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l
c Caixa Postal 66318, 05315-970 S˜ao Paulo, SP, Brazil
u
n dInstituto de F´ısica La Plata, CONICET, Universidad Nacional de La Plata,
[
1900 La Plata, Argentina
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v
4
We compute one- and two-nucleon kinetic-energy spectra and opening-angle distribu-
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1 tions for the nonmesonic weak decay of several hypernuclei, and compare our results
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with some recent data. The decay dynamics is described by transition potentials of
.
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the one-meson-exchange type, and the nuclear structure aspects by two versions of the
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0 independent-particle shell model (IPSM). In version IPSM-a, the hole states are treated
1 as stationary, while in version IPSM-b the deep-hole ones are considered to be quasi-
:
v stationary and are described by Breit-Wigner distributions.
i
X
r The mesonic mode, Λ → Nπ, with a rather small Q-value, QM = MΛ−MN −mπ ≈ 37
a
MeV, is heavily inhibited for Λ-hypernuclei, except for the very lightest, due to Pauli
blocking. With increasing mass number, A, a new mode quickly becomes dominant,
namely the nonmesonic weak decay (NMWD), ΛN → nN, whose Q-value, Q = M −
NM Λ
M + ε + ε ≈ 120 – 135 MeV, is sufficiently large to render this Pauli blocking less
N Λ N
and less effective. NMWD can be seen as one of the most radical transmutations of an
elementary particle inside the nuclear medium: the strangeness is changed by ∆S = −1
and the mass by ∆M = M − M = 176 Mev. From a practical point of view, the
Λ N
main interest in NMWD is that it is, at present, the only way available to probe the
strangeness-changing interaction between baryons.
Lately, the quality of experimental data on NMWD has improved considerably, and
today one has available, not only one-nucleon kinetic energy spectra, but also two-
nucleon coincidence spectra and opening-angle distributions obtained in several labo-
ratories around the world, such as, KEK, FINUDA, and BNL. Here we briefly discuss a
simple but fully quantum-mechanical formalism for the theoretical investigation of these
observables. For more details see Refs. [1] and [2].
∗E-mail address: krmpotic@fisica.unlp.edu.ar
2 E. Bauer, A. P. Galea˜o, M. S. Hussein and F. Krmpoti´c
We start from Fermi’s golden rule for the NMWD transition rate, Γ ≡ Γ(ΛN → nN),
N
with the states of the emitted nucleons approximated by plane waves and initial and
final short-range correlations implemented at a simple Jastrow-like level. The initial
hypernuclear state is taken as a Λ in single-particle state j = 1s weakly coupled to
Λ 1/2
an (A−1) nuclear core of spin J , i.e., |J i = |(J j )J i. For the description of nuclear
C I C Λ I
states we adopt the independent particle shell model (IPSM).
Let usconsider first thesimplest version ofthis model, IPSM-a, inwhich all therelevant
particle and hole states are assumed to be stationary. Thus, if the nucleon inducing the
decay isinstatej , thenthepossible statesoftheresidual nucleus are|J i = |(J j−1)J i
N F C N F
and the liberated energy is ∆ = M −M +ε +ε , where the ε’s are single-particle
jN Λ jΛ jN
energies. Within this scheme, we get
dp dp
Γ = 2π |hp p S;J |V|J i|2δ(E +E +E −∆ ) n N (1)
N Z Z n N F I n N r jN (2π)3(2π)3
X
SjNJF
4M3(A−2)
= dE dE F (p,P), (2)
π XZ N Z n jN
jN
where E = p2 /(2M) are the kinetic energies of the emitted nucleons, E = |p +
n,N n,N r n
p |2/[2(A−2)M]accountsfortherecoil, andV isthetransitionpotential. Itisunderstood
N
that all integrations on kinematical variables run over the allowed phase space. In Eq. (2),
jN+1/2
F (p,P) = FJ |hplPLλSJ|V|j j Ji|2, (3)
jN X jN X Λ N
J=|jN−1/2| SlLλJ
where FJ are spectroscopic factors, p and P are the relative and total momenta of
jN
the emitted nucleons, l and L are the corresponding orbital angular momenta, and the
couplings l + L = λ, λ + S = J and j + j = J are performed. The one-nucleon
Λ N
transition probability density S (E ) is obtained by taking the derivative of Γ , i. e.,
N N N
dΓ 4M3(A−2)
N
S (E ) = = dE F (p,P). (4)
N N dE π Z n jN
N XjN
The proton and neutron spectra are then given by ∆N (E) ∝ S (E) and ∆N (E) ∝
p p n
S (E) + 2S (E). Similar developments, with the appropriate choice of kinematical in-
p n
tegration variables, yield the coincidence energy spectra ∆N (E) ∝ S (E) and the
nN nN
opening-angle distributions ∆N (cosθ) ∝ S (cosθ), with N = n,p. More details, in-
nN nN
cluding the way to fix the normalization of these spectra and distributions, are given in
Ref. [2].
For p-shell and heavier hypernuclei, some of the |j−1i are deep-hole states, having
N
considerable spreading widths, γ , as revealed for instance in quasifree (p,2p) reactions.
jN
It is, therefore, unreasonable to treat such cases as stationary, zero-width, states. Rather,
they are better approximated as Breit-Wigner distributions in the liberated energy ε,
P (ε) = 2γjN 1 . This leads to a slightly more sophisticated version of the
jN π γj2N+4(ε−∆jN)2
IPSM, which we call IPSM-b. It turns out that the final expressions we need can be
obtained from those of IPSM-a through the replacements: ∆ 7→ ε, and ··· 7→
jN jN
+∞dεP (ε)···. This is explained in more detail, for the case of S (E),Pin Ref. [1].
jN −∞ jN nN
P R
Weak nonmesonic decay spectra of hypernuclei 3
The full one-meson exchange potential (OMEP), that comprises the (π,η,K,ρ,ω,K∗)
mesons, has been employed for V in numerical evaluations. In Fig. 1 we confront the two
IPSM approaches for the spectra S (E). One sees that, except for the ground states,
np
the narrow peaks engendered by the recoil effect within the IPSM-a become pretty wide
bumps within the IPSM-b.
On the other hand, preliminary calculations indicate that the two IPSM versions yield
similar results for S (E) and S (cosθ). In Fig. 2 are compared the experimental [3]
N nN
and theoretical kinetic energy spectra ∆N (E) for 12C, using the just mentioned OMEP.
p Λ
The theoretical spectrum is peaked around 85 MeV, and reproduces quite well the data
for energies larger than 50 MeV. Yet, it differs quite a lot at smaller energies, where the
effect of final state interactions (FSI) is likely to be quite important.
1122
ΛΛCC:: pprroottoonnss
-1-1ΓΓ (MeV) (MeV)pp 0000 0000....1212....23235555 IIPPSSMM--aa 225454111188HHHH6622SSOOCCeeeeii 11 00880000 FFiinnaall SSttaattee IInntteerrEEaaxxppcceeTTrrtthhiimmiieeooeeoonnrrnnyyttss!!
S(E)/S(E)/npnp 00 00..00..1155 6600
00 E)E)
0.3 IPSM-b N(N(pp
0.25 ∆∆
4400
0.2
0.15
0.1
2200
0.05
0
100 110 120 130 140 150 160 170
Kinetic Energy Sum (MeV)
00
00 2200 4400 6600 8800 110000 112200 114400
KKiinneettiicc EEnneerrggyy ((MMeeVV))
Figure 1. Normalized energy spectra Figure 2. Comparison between the exper-
S (E)/Γ for 4He, 5He, 12C, 16O, and 28Si imental [3] and theoretical kinetic energy
np p Λ Λ Λ Λ Λ
hypernuclei, taken from Ref. [1]. spectra for protons from 12C decay.
Λ
TheIPSMreproduceswell [2]theBNLexperiment for4He[4], butitdoesnotreproduce
Λ
well the FINUDA experiment for the S (E) spectra in 5He, 7Li, and 12C [3]. Once
N Λ Λ Λ
normalized to the transition rate, all the spectra are tailored basically by the kinematics
of the corresponding phase space, depending very weakly on the dynamics governing the
ΛN → nN transition proper. The IPSM is the appropriate lowest-order approximation
for the theoretical description of the NMWD of hypernuclei. It is in comparison to this
picture that one should appraise the effects of the FSI and of the two-nucleon-induced
decay mode.
REFERENCES
1. C. Barbero, et al., Phys. Rev. C 78 (2008) 044312. See also ibid. 059901(E).
2. E. Bauer, et al., Phys. Lett. B 674 (2009) 103.
3. M. Agnello, et al., Nucl. Phys. A 804 (2008) 151.
4. J.D. Parker, et al., Phys. Rev. C 76 (2007) 035501.
