First Experimental Determination of the
One-Proton Induced Non-Mesonic Weak Decay Width
for p-shell
Λ-hypernuclei
The FINUDA Collaboration: M. Agnello1,2, L. Benussi3, M. Bertani3, H.C. Bhang4, G. Bonomi5,6, E. Botta7,2, T. Bressani7,2, S. Bufalino2, D. Calvo2, P. Camerini8,9, B. Dalena10,11,a, F. De Mori7,2, G. D’Erasmo10,11, A. Feliciello2,∗, A. Filippi2, H. Fujioka12, P. Gianotti3, N. Grion9, V. Lucherini3, S. Marcello7,2, T. Nagae12,
H. Outa13, V. Paticchio10, S. Piano9, R. Rui8,9, G. Simonetti10,11and A. Zenoni5,6 1DISAT, Politecnico di Torino, Corso Duca degli Abruzzi 24, I-10129 Torino, Italy 2INFN, Sezione di Torino, Via P. Giuria 1, I-10125 Torino, Italy
3Laboratori Nazionali di Frascati dell’INFN, Via E. Fermi 40, I-00044 Frascati, Italy 4Department of Physics, Seoul National University, 151-742 Seoul, South Korea
5Dipartimento di Ingegneria Meccanica ed Industriale, Universit`a di Brescia, Via Branze 38,
I-25123 Brescia, Italy
6INFN, Sezione di Pavia, Via A. Bassi 6, I-27100 Pavia, Italy
7Dipartimento di Fisica, Universit`a di Torino, Via P. Giuria 1, I-10125 Torino, Italy 8Dipartimento di Fisica, Universit`a di Trieste, Via Valerio 2, I-34127 Trieste, Italy 9INFN, Sezione di Trieste, Via Valerio 2, I-34127 Trieste, Italy
10INFN, Sezione di Bari, Via E. Orabona 4, I-70125 Bari, Italy
11Dipartimento di Fisica, Universit`a di Bari, Via G. Amendola 173, I-70126 Bari, Italy 12Department of Physics, Kyoto University, Kitashirakawa, Sakyo-ku, Kyoto 606-8502, Japan 13RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan
anow at CEA/SACLAY, DSM/Irfu/SACM, F-91191 Gif-sur-Ivette, France
∗corresponding author
E-mail: Alessandro.Feliciello@to.infn.it
(Received August 22, 2014)
A further study of the FINUDA data about non-mesonic weak decay of eightΛ-hypernuclei in the A range (5 – 16) was carried out. New values of the ratio between the two-nucleon and the one-proton induced decay widths,Γ2N/Γp, were determined from single proton spectra,Γ2N/Γp= 0.50 ± 0.24,
and from neutron and proton coincidence spectra,Γ2N/Γp= 0.36 ± 0.14stat+0.05−0.04syssys, in full agreement with the previously published ones. Starting from these values, a method was developed to extract the one-proton induced decay width in units of the freeΛ decay width, Γp/ΓΛ, without resorting to
Intra-Nuclear Cascade models but by exploiting measured quantities only and by assuming a linear dependence on A of the effects due to Final State Interaction. The present work is the first systematic Γp/ΓΛdetermination ever done and it agrees with recent theoretical predictions within the errors.
KEYWORDS: DAΦNE, FINUDA experiment, p-shell Λ-hypernuclei, non-mesonic weak decay
1. Introduction
The weak interaction mediates theΛ-hypernucleus decay to ordinary non-strange nuclear systems through two channels: the mesonic (MWD) and the non-mesonic (NMWD) one. The MWD is further
1
split into two branches, corresponding to the decay modes of theΛ hyperon in free space: A
ΛZ→ A(Z+ 1) + π− (Γπ−), (1)
A
ΛZ → AZ+ π0 (Γπ0), (2)
whereAΛZ indicates the parentΛ-hypernucleus with mass number A and atomic number Z,A(Z+ 1) andAZ stand for residual nuclear systems (usually the daughter nucleus in its ground state) and finally
Γ denotes the partial decay width.
Since the nucleon from MWD carries a momentum p≈ 100 MeV (QMWD-value≈ 37 MeV) which is
much lower than the Fermi momentum, such a process is strongly suppressed by the Pauli exclusion principle in all but the lightestΛ-hypernuclei.
NMWD involves the constituentΛ and one core nucleon. The relevance of this process was pointed out soon after the firstΛ-hypernucleus discovery [1] and it was explained by assuming that the pion emitted in the weak vertexΛ → πN is virtual and it is then absorbed by the nuclear medium, resulting in one of the following processes, usually referred to as one-nucleon induced decays:
A
ΛZ→ (A−2)(Z− 1) + n + p (Γp), (3)
A
ΛZ→ (A−2)Z+ n + n (Γn). (4)
More recently it was realized that sometimes the virtual pion from theΛ decay process can be ab-sorbed by a pair of core nucleons (np, pp or nn), correlated by the strong interaction [2], according to the following reactions:
A ΛZ → (A−3)(Z− 1) + n + n + p (Γnp), (5) A ΛZ→ (A−3)(Z− 2) + n + p + p (Γpp), (6) A ΛZ → (A−3)Z+ n + n + n (Γnn). (7)
These three decay channels have different widths and, according to a microscopic calculation [3], the so called two-nucleon induced decay is largely dominated by process (5). Therefore, very often the
np-induced channel widthΓnp is assumed to be equivalent to the total two-nucleon induced process widthΓ2N and then
ΓT= ΓMWD+ ΓNMWD= Γπ−+ Γπ0+ Γ1N + Γ2N ≈ Γπ−+ Γπ0 + Γp+ Γn+ Γnp. (8)
The NMWD mode can occur only in nuclei. In this case the Q-value of the elementary weak reaction underlying processes (3)-(7) is high enough (QNMWD-value≈ 175 MeV) to avoid any Pauli blocking
effect; then the final state nucleons have a sizeable probability to escape from the parent nucleus. Actually it is expected that NMWD dominates over MWD for all but the s-shellΛ-hypernuclei and only for very light systems the two modes are predicted to be competitive.
Hypernuclear NMWD was scarcely studied up to few years ago essentially because its exper-imental observation represents a challenging task. First, it is necessary to produce and to identify Λ-hypernuclei in their ground state by means of a performing magnetic spectrometer. Then, all the final state nucleons in (3), (4) and (5) should be detected in coincidence and their energy should be precisely measured. Moreover, there is an inherent limitation in extracting the physical observables of interest due to the not negligible distortion introduced by Final State Interaction (FSI) on the final state nucleon kinetic energy spectra. Actually, the information on the particle initial bare momenta and flight direction may be completely lost; also, their contribution can be mixed and then possible additional quantum-mechanical interference effects can arise [4].
First generation experiments were carried out at BNL and KEK [5, 6]. Afterwards, the SKS Collaboration measured at KEK proton and neutron kinetic energy distributions from 5ΛHe and12ΛC NMWD [7]. Very recently, the FINUDA Collaboration measured at DAΦNE the proton and the neu-tron kinetic energy spectra from5ΛHe and from seven p-shellΛ-hypernuclei NMWD.
2. Refined determination ofΓ2N/ΓNMWD
In an early paper [8],5
ΛHe,7ΛLi and12ΛC NMWD proton kinetic energy distributions measured by
the FINUDA Collaboration were presented and discussed. Afterwards, proton spectra from NMWD of 9ΛBe, 11ΛB,13ΛC,15ΛN and16ΛO were obtained and analyzed [9]. Fig. 1 shows all the experimental distributions: up to now, they represent a unique database for p-shell Λ-hypernuclei in the A range (5 – 16) and they allowed to perform very interesting investigations and to drawn several important conclusions. Ref. [9] describes the method developed to isolate in the spectra of Fig. 1 the contribution from process (5), without resorting to any Intra-Nuclear Cascade (INC) calculation, as it was done instead in Refs. [10, 11]. This way, it was possible to get a first evaluation ofΓ2N/ΓNMWDratio.
Recently, it was noticed [12, 13] that the description of the FINUDA experimental data, provided by means of a Gaussian fit to the high energy region of the spectrum, could be possibly improved by shifting down the lower edge of the fitting interval. As a preliminary step of the present analysis, this hypothesis was checked and it was found that indeed an overall better result can be achieved by starting the fit from 70 MeV, instead of 80 MeV. The main consequence of this shift was the change of the blue areas in Fig. 1 with respect to the previous analysis. Since their values are the main input to evaluate the two-nucleon induced NMWD partial width, the same procedure described in Ref. [9] led to a new determination for Γ2N/Γp= 0.50 ± 0.24 and for Γ2N/ΓNMWD= 0.25 ± 0.12 which turned
out to be fully consistent with the previous one.
In a second approach [14], the Γ2N/Γp andΓ2N/ΓNMWD ratios were evaluated by considering
both protons and neutrons emitted in coincidence with the π−from the Λ-hypernucleus formation reaction. Also in this case, new estimations forΓ2N/Γp= 0.36 ± 0.14stat+0.05−0.04sys
sys and forΓ2N/ΓNMWD
= 0.20 ± 0.08stat+0.04−0.03syssys were done. Again, the new outcome is compatible with the previous one and
moreover it is in agreement with the experimental result from KEK [10] and with recent theoretical predictions [15].
3. First determination ofΓp/ΓΛfor eightΛ-hypernuclei (A = 5 – 16)
Previous studies [9, 14] demonstrated that the high energy portion of the eight spectra of Fig. 1 are essentially populated by protons from decay process (3) even though their shapes are severely distorted by FSI. However, in quantifying Γ2N/Γp it was possible to parameterize the sizeable FSI contribution through measured quantities. On the contrary, in order to evaluate the Γp/ΓΛabsolute values from the measured proton spectra, it is unavoidable to carefully take into account the effective impact of the FSI effect. More in detail one should keep in mind that:
a) the actual number of primary protons due to decays (3) and (5) is reduced by FSI;
b) in any portion of the spectrum, the number of protons is increased not only owing to FSI suffered
by higher energy protons, but by higher energy neutrons from (4) as well;
c) quantum-mechanical interference effect may occur among protons of the same energy from
different sources (primary ones from (3) and (5), secondary ones from FSI).
The original approach of the present analysis was to try to evaluate the FSI effect on measured spectra without performing complex INC calculations but just by exploiting experimental data under basic assumptions. On the basis of the conclusions drawn in Ref. [4], the consequences of the effect
b) on the blue areas of Fig. 1 can be safely neglected and the contribution of the decay channel (5)
above 70 MeV can be considered less than 5% ofΓNMWD; then, by taking into account the FINUDA
result for Γ2N/Γpreported in Section 2, the total amount of primary protons from (5) would not be larger than 2% of those from (3). As a consequence, also the interference effect c) may be neglected.
Kinetic energy (MeV) 0 20 40 60 80 100 120 140 160 180 200 counts/(10 MeV) 0 20 40 60 80 HeΛ 5
Kinetic Energy (MeV)
0 20 40 60 80 100 120 140 160 180 200 counts/(10 MeV) 40 30 20 10 0 50 60 LiΛ 7
Kinetic Energy (MeV)
0 20 40 60 80 100 120 140 160 180 200 counts/(10 MeV) 0 20 40 60 80 100 Be Λ 9
KInetic Energy (MeV)
0 20 40 60 80 100 120 140 160 180 200 counts/(10 MeV) 0 20 40 60 80 100 120 140 160 B Λ 11
Kinetic Energy (MeV)
0 20 40 60 80 100 120 140 160 180 200 counts/(10 MeV) 0 20 40 60 80 100 120 140 160 180 C Λ 12
Kinetic Energy (MeV)
0 20 40 60 80 100 120 140 160 180 200 counts/(10 MeV) 0 20 40 60 80 100 120 C Λ 13
Kinetic energy (MeV)
0 20 40 60 80 100 120 140 160 180 200 counts/(10 MeV) 0 50 100 150 200 250 300 350 400 N Λ 15
Kinetic Energy (MeV)
0 20 40 60 80 100 120 140 160 180 200 counts/(10 MeV) 0 10 20 30 40 50 60 O Λ 16 Fig . 1. Proton kinetic ener gy spectra from the NMWD of (from right to left, top to bottom): 5 He,Λ 7 Li,Λ 9 Be,Λ 11 Λ B, 12 Λ C, 13 Λ C, 15 Λ N and 16 Λ O. The curv es represent the ne w analysis Gaussian fits to the experimental data: the solid line part indicates the actual fit range, while the dashed one indicates the estimated one-proton induced NMWD contrib ution to the lo wer ener gy spectrum re gion. The blue hatched area is the higher ener gy half Gaussian area, where the tw o-nucleon induced NMWD is ne gligible.
Table I. Γp/ΓΛvalues for the eightΛ-hypernuclei under study obtained with (9) (fourth column). Present
results are compared with previous experimental data, when available, (fifth column) and with the outcome of a recent theoretical work [20].
ΓT/ΓΛ α(A) Γp/ΓΛ Γp/ΓΛ Γp/ΓΛ
(input to (9)) (input to (9)) [present work] [previous works] [20]
5 ΛHe 0.96± 0.03 [5,16] 1.08± 0.16 0.22± 0.05 0.21± 0.07 [5] 0.237 7 ΛLi 1.12± 0.12 1.51± 0.22 0.28± 0.07 0.297 9 ΛBe 1.15± 0.13 1.94± 0.28 0.30± 0.07 0.401 11 ΛB 1.28± 0.10 [6] 2.37± 0.34 0.47± 0.11 0.30± 0.07 [6] 0.444 12 ΛC 1.242± 0.042 [16, 17] 2.58± 0.37 0.65± 0.19 0.31± 0.07 [6] 0.535 0.45± 0.10 [18] 13 ΛC 1.21± 0.16 2.80± 0.40 0.60± 0.14 0.495 15 ΛN 1.26± 0.18 3.23± 0.47 0.49± 0.11 0.555 16 ΛO 1.28± 0.19 3.44± 0.50 0.44± 0.12 0.586
Finally the effect a) was parameterized by means of the following relationship: Γp ΓΛ = ΓT ΓΛ · BR(p) = ΓT ΓΛ · 2(Np− N2N)+ α(Np− N2N) Nhyp (9) where BR(p) is the branching ratio for (3), Npis the number of protons in each of the blue areas of Fig. 1, N2N the number of protons from (5) (. 2%), Nhyp the number of producedΛ-hypernuclei, the factor 2 takes into account the total area of the fitting Gaussian andα is the unknown coefficient which accounts for the number of protons moved from the blue area in the lower energy region of the distribution due to FSI. More precisely,α/(2 + α) is the fraction of protons undergoing FSI.
In order to calculate α for all the Λ-hypernuclei under study, published Γp/ΓΛ values for 5ΛHe and12ΛC were used and a linear scaling law with A was assumed for the FSI contribution and, con-sequently, forα. The ratio ΓT/ΓΛwas instead evaluated by substituting in theΓT expression (8) the
experimental values existing in literature. This way, it was possible to getα5(5ΛHe)= 1.15 ± 0.26 for 5
ΛHe (based on5ΛHe experimental data) and α12(12ΛC) = 2.48 ± 0.46 for12ΛC (based on12ΛC
experi-mental data). Under the assumption of a linear dependence ofα on A, the crossed evaluations were obtained:α5(12ΛC)= 1.04 ± 0.19 and α12(5ΛHe)= 2.77 ± 0.63. Finally, it was possible to write down
the following general expression forα(A): α(A) = α5
5 · A = α12
12 · A = (0.215 ± 0.031) · A (10)
which allowed to infer that the fraction of primary protons from NMWD (3) lost (i.e. degraded by FSI from the blue area to lower energy) is 35% in the case of5ΛHe and 63% in the case of16ΛO.
Γp/ΓΛ values for the eight Λ-hypernuclei under study, evaluated with (9), are listed in Table I along with measuredΓT/ΓΛused as input to (9) when available; for the remainingΛ-hypernuclei it
was used the parametrization ΓT/ΓΛ(A) = (0.990 ± 0.094) + (0.018 ± 0.010) · A from Ref. [19].
The upper panel of Fig. 2 shows the comparison among the presentΓp/ΓΛdeterminations, three sets of experimental data [5, 6, 18] and a theoretical prediction [20]. The general trend of the FINUDA experimental points is quite well reproduced by the theoretical calculation, even though the experi-mental errors are quite large, with the exception of9ΛBe, which is lower by 1.5σ and of16ΛO which is lower by 1.3σ. In the lower panel of Fig. 2 the FINUDA Γπ−/ΓΛdeterminations [19] are plotted
mass number [A] 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Λ Γ / p Γ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
a)
mass number [A]
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Λ Γ / -π Γ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
b)
Fig. 2. a)Γp/ΓΛ experimental values as a function of A for 5ΛHe,7ΛLi,9ΛBe,11ΛB, 12ΛC,13ΛC,15ΛN and16ΛO
from the present analysis (blue stars). Theoretical predictions forΓp/ΓΛ[20] (violet squares) are shown for
comparison. Measurements ofΓp/ΓΛfor5ΛHe [5] (brown full circle), for11ΛB and12ΛC [6] (green full circles)
and for12
ΛC [18] (orange full circle) are plotted as well.
b) Γπ−/ΓΛ experimental values as a function of A for5ΛHe, Λ7Li,9ΛBe, 11ΛB and15ΛN [19] (red stars) and for
12
ΛC [18] (orange cross). Theoretical calculations ofΓπ−/ΓΛfor5ΛHe [21] (gray up triangle), for7ΛLi,9ΛBe,11ΛB, 12
ΛC and15ΛN [20] (down violet triangles) and for5ΛHe,7ΛLi,9ΛBe,11ΛB and15ΛN [22] (cyan diamonds) are also reported.
together with another set of experimental data [18] and the outcome of three different theoretical cal-culations [20–22]. By looking at the picture as a whole, it is possible for the first time to verify the long time advocated complementary behavior of hypernuclear MWD and NMWD in the A range (5 – 16), at least as far as channels with charged particle in the final state are concerned.
4. Conclusions
The one-proton induced NMWD partial decay width was determined for eightΛ-hypernuclei in the A range (5 – 16). Such a measurement was essentially based on the analysis of the experimental proton kinetic energy spectra and it is the first systematic survey ever done for p-shellΛ-hypernuclei. The obtained values, even though affected by quite large errors, agree reasonably well with those predicted by a recent theoretical calculation [20].
Further investigations require much more precise experimental data on bothΓT/ΓΛand Np(Nhyp) (see Eq. (9)). In the current international scenario, the only facility with a dedicated physics program and equipped with the beams and detectors required for this kind of measurements is J-PARC. At the same time, an effort should be made in order to implement a simple INC calculation, without two-nucleon induced NMWD contribution, to check the validity of the hypothesis about the linear dependence on A of the FSI correction.
When all these requirements will be satisfied, it could be possible to try to face the problem of the experimental study of theΛN → NN weak interaction starting from hypernuclear data. Actually, a precise study of the hypernuclear NMWD is the only experimental way to get information on the four-baryon weak processΛN → NN and to realize the idea of using a nuclear system as laboratory where to study interactions among elementary particles not otherwise accessible in vacuo.
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