Coincidence Measurement of the Nonmesonic Weak Decay of ^{12}_{Lambda}C

We have measured the angular correlation of the pair nucleons np and nn emitted from the nonmesonic weak decay (NMWD) of ^{12}_{Lambda}C produced via the (pi^+,K^+) reaction in coincidence measurement. The Lambda p ->np and Lambda n ->nn modes were clearly identified by measuring the back-to-back correlation of the emitted nucleon pairs which is the characteristic of two-body kinematics. From the measured nucleon pair numbers N_{nn} and N_{np}, the ratio Gamma_n/Gamma_p of the partial decay widths Gamma_n(Lambda n ->nn) and Gamma_p(Lambda p ->np) of ^{12}_{Lambda}C was extracted to be 0.51+-0.13(stat) +- 0.05(syst); this result is almost free from the ambiguity due to the nuclear final state interaction and 3-body decay process, which were inherent in the previous results. The obtained Gamma_n/Gamma_p ratio of ^{12}_{Lambda}C (p-shell) is close to that of ^5_{Lambda}He (s-shell). The results are consistent with those of recent theoretical calculations.


Introduction
A Λ bound in a nucleus decays via either mesonic weak decay, Λ → Nπ, or nonmesonic weak decay (NMWD), ΛN → nN . Mesonic decay is essentially similar to free Λ decay and has been studied in detail. In a nucleus a Λ can decay via an interaction with a neighbor nucleon, either a proton (Λp → np; Γ p , proton-induced NMWD) or a neutron (Λn → nn; Γ n , neutroninduced NMWD); this is referred to as a one-nucleon-induced (1N) NMWD process. The two-nucleon induced (2N) process (ΛNN → nNN; Γ 2N ), which is another NMWD mode, has been predicted theoretically; however, it has not yet been verified experimentally. The NMWD process has attracted considerable attention since it provides the only practical means of studying the strangeness-changing baryonic weak interaction at present [1].
Since the discovery of hypernuclei, one of the primary concerns in the study of NMWD of Λ hypernuclei has been the relative strength of the two channels of 1N NMWD-the ratio of the decay widths (Γ n /Γ p ). Experimental ratios over the broad mass range of Λ hypernuclei are consistently greater than unity, thereby indicating the dominance of the neutron channel; on the other hand, the theoretical ratios are only one tenth of unity. The predominance of the proton channel in the theoretical prediction is due to the fact that the tensor term of one-pion exchange (OPE) contributing only to the proton channel is very high, and therefore, the contributions of other meson exchange terms become minor corrections. The OPE contribution is included in all model calculations for the long range contribution; hence, the Γ n /Γ p ratios have remained at around 0.1. This intriguing problem is referred to as the Γ n /Γ p puzzle.
In order to reduce the predominance of the proton-induced NMWD of the OPE model, many theoretical models for short range contribution, such as the heavy meson exchange (HME) models and the direct quark (DQ) model have been studied. All these models have been unsuccessful in increasing the Γ n /Γ p ratio significantly until recently. However, in recent important development, the incorrect sign in kaon exchange amplitudes was identified; the correction significantly increased the Γ n /Γ p values of these models [2]. Thus, the current theoretical values of the Γ n /Γ p ratio have increased to 0.34-0.70 for 5 Λ He and 0.29-0.53 for 12 Λ C [2,3,4].
Until recently experimental Γ n /Γ p values were extracted by fitting intranuclear cascade (INC) calculation yields to the experimental proton yields spectra. Almost all the experimental Γ n /Γ p ratios have been close to or greater than unity [1]. However, recent high-quality neutron yield spectra obtained by our group have made it possible to derive the Γ n /Γ p ratio directly by comparing the neutron yields to the corresponding proton yields; the Γ n /Γ p ratios obtained by this method are ∼0.5 for 12 Λ C [5,6] and ∼0.6 for 5 Λ He [6] agreeing well with the recent theoretical values. These ratios are significantly smaller than unity. They are the first experimental indications of the proton channel dominance (Λp → np) in the NMWD.
It appears that the discrepancy between the experimental and theoretical values of the Γ n /Γ p ratios of NMWD has been removed. However, the Γ n /Γ p ratios of 5 Λ He and 12 Λ C derived by comparing the nucleon yields still contain the ambiguity due to the possible contributions of 2N NMWD and the final state interaction (FSI) effects on the emitted nucleons. The current FSI model calculation which takes into account the 2N NMWD and FSI can not reproduce the yield spectra [6,15]. Therefore, the uncertainty of the ratios due to the ambiguity could not be estimated in such single nucleon (singles) measurements. In order to remove such ambiguity experimentally, the decay channel has to be explicitly identified for each event. In order to determine each decay channel of 1N-induced NMWD exclusively, we have performed a coincidence measurement of both n+p (np pair) and n+n (nn pair) pair nucleons from the NMWD of the hypernuclei, 5 Λ He and 12 Λ C. Since ΛN → NN is a two-body process, the outgoing nucleon pairs will exhibit a clear back-to-back (bb) correlation with regards to their opening angle, and their energy sum distribution would exhibit a peak at around the decay Q value, ∼155 MeV, although they are broadened due to nuclear medium effects. On the other hand, in 2N NMWD, three nucleons are produced in the final state and a pair of nucleons among them would exhibit neither the bb feature in the opening angle nor the energy sum peak at the Q value. The nucleon from 1N NMWD that suffered intense FSI on the way out of the residual nucleus would be deflected from the original momentum direction and lose kinetic energy, thereby degrading the bb kinematics characteristic of 1N NMWD. Therefore, by applying the event selection criteria of bb kinematics, we can exclusively select 1N NMWD events suppressing possible contributions from the 2N process and the events that suffered serious FSI.
Recently we have reported the Γ n /Γ p value of s-shell 5 Λ He that was determined in the two-nucleon coincidence measurement of the NMWD [7]. By applying the bb kinematical condition for the opening angle, we obtained the pair number ratio of nn to np pairs of 1N NMWD, N nn /N np , to be 0.45±0.11±0.03. This ratio becomes essentially the Γ n /Γ p ratio when FSI is weak as in 5 Λ He. The s-shell hypernucleus, 5 Λ He, was selected for the first exclusive coincidence measurement, since the FSI effect for such a light hypernucleus must be small. However, the unambiguous determination of Γ n /Γ p for a p-shell hypernucleus also is essential in the study of baryonic weak interaction since only the p-shell hypernuclei can provide the p-wave initial state for the interaction while the sshell 5 Λ He does only that of s-wave. The importance of the p-wave contribution in the NMWD in the finite nucleus has been discussed in the reference [8]. In such respect, 12 Λ C has been extensively studied together with 5 Λ He. Exclusive determination is even more important for the Γ n /Γ p ratio of heavier p-shell hypernuclei such as 12 Λ C since the nucleons propagating there will suffer more intense FSI. Although 12 Λ C is the most extensively studied p-shell Λ hypernucleus, no unambiguous Γ n /Γ p ratio for it is available yet.
The values of the asymmetry parameter α nm of protons from Λp → np NMWD of the polarized hypernuclei were measured and reported for both 5 Λ He and 12 Λ C formed via the (π + ,K + ) reaction. Although small positive values of 0.24±0.22 and 0.11±0.08±0.04 [9,10] were observed for 5 Λ He, a large negative value (α nm ≃ −1.0 ± 0.4) was reported for the p-shell hypernuclei [11]. Authors indicated a possible explanation that the interaction mechanism of NMWD for short and long range could be different and therefore the initial p-state contribution in 12 Λ C made the difference. The theoretical predictions for the asymmetry parameter for 5 Λ He are about -0.7 [4,12] from both the recent heavy meson exchange (HME) and the direct quark (DQ) models. They significantly overestimate the magnitude of the parameter. The recent HME model prediction for α nm of 12 Λ C is 0.36 that is again far off from the experimental value. The current difficulty in understanding the decay asymmetries of s-and p-shell hypernuclei implies that an accurate and unambiguous determination of Γ n /Γ p ratios for both s-and p-shell hypernuclei is essential for NMWD studies.
In this paper we report the results of the two-nucleon observables measured in the coincidence experiment of the decay of 12 Λ C especially focusing on its Γ n /Γ p ratio. The experiment was carried out at the K6 beam line of 12-GeV proton synchrotron (PS) in High Energy Accelerator Research Organization (KEK). Λ hypernuclei 12 Λ C were produced via the (π + ,K + ) reaction at the beam momentum of 1.05 GeV/c on the 12 C target. In order to improve the vertex resolution, we used thin segmented plastic scintillators for an active 12 C target. Fig. 1 shows the schematic view of coincidence counters for detecting the decay product particles from the weak decay of 12 Λ C. It comprises three sets of coincidence counter-two located at the top and bottom of the target, optimized for bb coincidence events and one at the side to allow the detection of non-back-to-back events. Each of top and bottom counter sets comprise a fast timing counter (T2), a drift chamber (PDC), veto or stop timing counter (T3) and neutron counter arrays (T4). The side set is similar; however, the PDC is absent.
The inclusive 12 Λ C excitation energy spectrum derived from the momenta of incoming pion and the outgoing kaon is shown in Fig. 2(a). The two single particle Λ states (1s and 1p) are clearly identified. The 1p Λ state is located just above proton emission threshold; it decays into 11 Λ B by emitting a proton. Fig. 2(b) and 2(c) show the excitation energy spectra in coincidence with the emitted pair nucleons np and nn, respectively. The vertical lines show the applied gates of the decay measurement for ground state of 12 Λ C. The enhancement of the yield in the quasi-free Λ region of the nn coincidence excitation energy spectrum as compared with that of the np pairs occurs due to the emission of two neutron via the absorption of π − from the quasi-free Λ mesonic decay. Neutral particles, γ and neutrons, were measured using six layers of 5 cm thick plastic scintillators (T4) vetoed with T3. The particle identification (PID) of the neutrons and γ was achieved by using the flight time from the decay vertex to the fired T4 counter. The neutron and γ were well separated, and the accidental background within the neutron gate was negligible. The charged particles were identified by combining dE/dx, TOF 23 and E T . dE/dx denotes the energy loss per unit length measured by T 2 ; TOF 23 , flight time between T 2 and T 3 ; and E T , the total energy deposited in the sequentially fired counters, T 2 , T 3 , and T 4 . Protons are well separated from pions and deutrons. The kinetic energies of neutrons and protons were calculated from the flight times and the ranges, respectively. Proton energy was corrected for the energy loss inside the target material thanks to the good vertex resolution. Further experimental details are provided in the previous report [6]. Fig. 3(a) and 3(b) show the raw distribution of np and nn pair yields Y np (cosθ) and Y nn (cosθ) in the opening angle θ, respectively. Only the pair events of which each nucleon has an energy greater than 30 MeV are counted. The dominance of the bb peaks shown in the raw spectra is enhanced by the maximized acceptance for the coincidence back-to-back kinematic events. A total of 116 and 43 events were observed in the bb angular region of cosθ < −0.7 for the np and nn pair yields, respectively. The shoulder bump observed in the bb region of the nn pair can be understood as a combined result of a statistical fluctuation due to the limited statistics of nn pair events (total 43 counts only in the bb region) and the binning effect of neutron z-position analysis (refer to Fig. 1). It was revealed that the background nucleon pairs produced through the absorption of π − from the mesonic decay of 12 Λ C in the materials around the target were insignificant unlike the situation in the case of 5 Λ He [7]. The angular resolutions for the np and nn pairs were estimated to be σ cos(θnp) = 0.018 and σ cos(θnn) = 0.029 at cosθ = −0.9, respectively.   cosθ). N N N (cosθ) can be expressed as N nn(p) (cosθ)= Y nn(p) (cosθ)/(Y nm · ǫ nn(p) ), where Y nm and ǫ nn(p) are the total number of NMWD observed and the overall efficiency for detecting two nucleons from the nn and np pairs in coincidence including the detector acceptance. Y nm is obtained by multiplying the number of hypernuclei produced in the ground state of 12 Λ C with the NMWD branching ratio b nm , which is defined as b nm = 1 − b π − − b π 0 , where b π − and b π 0 denote the π − and π 0 branching ratios, respectively. The accuracy of the b nm value of 0.768±0.012±0.005 has been improved drastically due to the accurate measurement of b π 0 in the present experiment [13] in comparison with the previous one [14]. This accurate b nm value made it possible to normalize the pair yields per NMWD without introducing significant systematic errors. The measured N N N angular correlation of nn and np pairs could now be directly compared with those obtained from the FSI model calculation [15]. ǫ N N (cosθ) was calculated with energy dependent event-by-event simulations, and it was tested to be in good agreement with the data [16]. The neutron detection efficiency was calculated by the Monte-Carlo simulation code-the modified DEMONS code-which is applicable to a multi-element neutron detector and has been tested for producing various experimental data well [5]. The angular correlation of np pair clearly exhibits a peak at cosθ ≃ −1, thereby indicating the bb emission of two nucleons, which is the signature of two-body decay final state. The bb kinematic regions defined as cosθ < −0.7 essentially include all the events in the bb kinematics peak. However, although dominance of bb kinematics events is degraded to a certain extent in the nn pairs, the bb events continue to be a major contribution. The dominance of the bb kinematic events observed in both np and nn pair confirms that 1N NMWD, Λp → np and Λn → nn, is the main NMWD mode of 12 Λ C. In non-bb region, we observe pair events more or less uniformly distributed over all angle in cosθ whose total number reaches ∼40 % of the total pair numbers as shown in Table 1. Table 1 shows the pair numbers in each opening angle region. We consider the uniform distribution event extended even in bb region. Therefore, we will subtract the corresponding component of the uniform distribution events in bb region from the pair numbers N N N for the later discussion of 1N NMWD.

Results and Discussion
In Fig. 3(e) and 3(f), the spectra of the np and nn pair yields in the NMWD of 12 Λ C in the bb region are shown in the energy sum of the two emitted nucleons. The rms energy-sum resolutions, σ Esum , for nn and np pairs were estimated to be 11 and 9 MeV for typical cases, such as 75 MeV nucleons, respectively. The energy-sum spectra show that the pair yields are distributed over a broad energy region from 70∼80 MeV to the Q value at around 155 MeV; on the other hand, a few pair yields exist above the Q values. Those present above the Q values are considered to be due to the rapid deterioration in the neutronenergy resolution with an increase in the neutron energy. Since the energy sum spectrum of the two nucleons emitted in the two-body decay process, Table 1 The yields of coincidence events Y np , Y nn and Y pp and the normalized pair numbers per NMWD N np ,N nn and N pp are shown in the opening angle regions bb (back-toback,cosθ < −0.7) and non-bb (cosθ ≥ −0.7). N N N values listed are the numbers simply integrated over the corresponding angular regions. The pair numbers after the subtraction of the uniform background in the bb region are also listed. Errors are statistical. ΛN → NN , would exhibit a peak at around the Q value, ∼155 MeV, the broad energy distribution indicates that a large number of the bb kinematics nucleons suffered an energy loss on the way out of the residual nucleus with a mass of ∼ 10. On the other hand, a dominant narrow peak around Q value was clearly visible in the np pair energy sum spectrum of 5 Λ He [7]. Therefore even if we consider the pair numbers confined in the bb region only, they include a considerable number of events that suffered FSI. We observe a similar or even more energy degraded energy-sum spectrum in nn pairs; however, the statistics are considerably lower.  Λ He(open circle) [7] and 12 Λ C (filled circle). We observe dominant bb peaks in both 5 Λ He and 12 Λ C; this indicates that the 1N NMWD is the major mechanism of NMWD. By comparing the angular correlations of 5 Λ He and 12 Λ C, we observe two clear features-broadening of the bb peak in angular correlations and quenching of the bb pair numbers of 12 Λ C over those of 5 Λ He in both np and nn pairs. The total bb pair numbers of np and nn in 12 Λ C are reduced in comparison with those of 5 Λ He by ∼40% and ∼20%, respectively. The features can be understood in terms of the same physics responsible for the energy degradation in the E sum ( 12 Λ C) spectra (Fig. 3(e)) in comparison with those of 5 Λ He. By considering that 2N theoretical branching ratio does not vary considerably with the mass number, and therefore, the relative importance of 2N with respect to 1N would be more or less the same for 5 Λ He and 12 Λ C [15], we infer that FSI is the major cause of broadening and quenching in the angular correlation and the degradation in the energy-sum spectrum. This implies that the effect of FSI on the pair numbers, even in the bb region, would not be negligible, and we must take it into account in order to extract Γ n /Γ p ratio from the pair number ratio. A significant portion (∼40%) of the pair number is observed in the non-back-to-back region as shown in Table 1. The possible contributions in the non-back-to-back region are such as 2N(or multinucleon)-induced NMWD events, those that suffer strong FSI, π − absorption contamination and random coincidence events etc. However, in 12 Λ C, the contributions from the last two cases are found to be almost negligible, and the first two are considered to be the major contributions in non-back-to-back region. These non-bb events, whose strength reaches ∼40 percent of NMWD, are the sources of the ambiguity in the previous results of Γ n /Γ p since they can not be distinguished from 1N NMWD in the singles measurement. However, we can reject them by considering only the coincidence pair numbers in the bb region in the derivation of Γ n /Γ p , thereby resulting the ratio which is almost free from the ambiguity inherent in the previous results.

Angular region
Next, we would like to derive the Γ n /Γ p ratio from the pair number ratio, N nn /N np , of the bb region. If neither Γ 2N contribution nor FSI existed, the ratio of N nn /N np would become Γ n /Γ p . However, as discussed, the effects of FSI still remain in the pair numbers of 12 Λ C. The most harmful effect of FSI is the channel crossover such as that from nn to np or from np to nn. It directly contaminates the pair number ratio. We observed 8 pp pair coincidence events in the bb region, which are possible only via the FSI process. A neutron in the np pair emitted at the vertex is converted into a proton via the nuclear interaction on the way out of the nucleus, thereby registering as a pp coincidence event. Therefore, they represent the channel crossover information of FSI on the nucleons emitted. Table 1 shows N pp yields-the normalized pp pair numbers per NMWD. We will utilize N pp for the correction of the FSI crossover effect on Γ n /Γ p ratio. The pair numbers in the bb region N nn (bb), N np (bb) and N pp (bb) can be expressed as N nn (bb) = r n f 2 n p +r p f n g n q N np (bb) = r p f n f p p +2r n f n g p q (1) N pp (bb) = r p f p g p q +r n g 2 p q ′ where r n (= Γ n /Γ nm ) and r p (= Γ p /Γ nm ) are the fractions of neutron and proton induced channels out of the NMWD. f n(p) is the loss factor of neutrons (protons) from the energy region of interest due to FSI and g n(p) is the resulting crossover influx factor of neutrons (protons) from protons (neutrons) channel. The first terms of N nn and N np represent the pair numbers emitted out of the nucleus that maintain the channel starting at the initial weak vertex point. The second terms represent the pair numbers that cross the channel and end in a channel different from the starting one. p, q, and q ′ are the loss factors from the bb angular region of the original and channel crossing pairs; however, their actual values are not required in the following derivation. Instead, they are replaced with the observed pp pair numbers. If there were no FSI effects, f n(p) and g n(p) would be one and zero, respectively. We put f n = f p = f and g n = g p = g by considering the charge symmetry of a strong interaction. A similar shape of proton and neutron spectrum obtained from the present experiment [6] supports this approximation. A simple simulation reveals that the g 2 term in Eq. 1 is less than 1 percent of the noncrossing terms and can therefore be neglected. Then, the second terms of N nn (bb) and N np (bb) of Eq. 1 are replaced by N pp (bb) and the r n /r p (= x =Γ n /Γ p ) ratio can be expressed in terms of pair numbers only as follows: The N N N (bb) numbers are the integrated pair numbers in the bb region after the subtraction of the background, which is assumed to be uniformly distributed in all cosθ as we pointed out in the discussion of Fig. 3(c) and 3(d).
The value of the constant background was determined by χ 2 fitting. The ratio of the raw pair numbers simply integrated over the bb region 0.60±0.12 becomes 0.53±0.13 after the subtraction of the uniform background. The obtained Γ n /Γ p ratio 0.51 ± 0.13 ± 0.05 in the Eq. 2 is only 4 percent reduced from the pair number ratio N nn (bb)/N np (bb). This is surprising considering that we observed the strong FSI effect both in the energy-sum spectra and in the angular correlations. However, the small correction can be understood considering that N nn /N np ∼ 1/2 and hence the crossover pair numbers from nn to np and from np to nn are almost balanced, and therefore, the crossoverinflux and -outflux due to FSI cancel each other. The systematic errors mainly originate from the uncertainties of the neutron detection efficiency (6%) and the pp pair acceptance (4%). Many sources of uncertainty such as b nm , f n and detection efficiencies are canceled out by taking the ratio [7].
The present result determined with the pair nucleon number ratio N nn /N np is in good agreement with the recent theoretical results based on heavy meson exchange (HME) model calculations [3,4]. It finally resolved the longstanding Γ n /Γ p ratio puzzle without ambiguity by removing the effect of FSI selfconsistently and by rejecting non-bb events applying bb kinematic condition. It is quite accidental that the present result is close to that of the previous singles measurement which were determined only by assuming the nonexistence of 2N (or multi-nucleon induced) NMWD [5]. However, we have pointed out that there exist a significant amount (∼ 40% of the total) of non-bb events, which probably are the candidates of them. The present Γ n /Γ p ratio of 12 Λ C is in good agreement with that of the previous exclusive measurement of 5 Λ He (0.45 ± 0.11 ± 0.03), which indicates that the NMWD mechanism of s-shell ( 5 Λ He) and p-shell ( 12 Λ C) hypernuclei are rather similar. Though the Γ n /Γ p ratio determined from the ratio of pair numbers N nn /N np agrees well with those of the theoretical predictions [4,15], there remain discrepancies between the pair numbers N N N of our measurement and those of the theoretical calculation [15]: the pair numbers for bb events and non-bb events, and the significant quenching of pair numbers of 12 Λ C compared with those of 5 Λ He. For example, the observed pair number of bb events is about half of the prediction while that of non-bb events is only a fraction.

Conclusion
We have measured, for the first time, the angular correlation N N N (cosθ) of the pair nucleons np and nn from the NMWD of 12 Λ C in coincidence measurement. We have exclusively identified the dominant 1N NMWD process, Λp → np and Λn → nn, by requiring the bb angular correlation of the two emitted pair nucleons, which is the characteristic of two-body decay. Then we have obtained the Γ n /Γ p ratio from the ratio of pair numbers N nn /N np of 1N NMWD in which most of the non-bb events are excluded and FSI effects have been corrected using the simultaneously measured pp pair events. Therefore, the present Γ n /Γ p result is almost free from the ambiguities due to 2N(or multinucleon induced) NMWD contribution and the FSI effects which were inherent in the previous results obtained with singles spectra.
The obtained Γ n /Γ p ratio of 0.51 ± 0.13 ± 0.05 is in good agreement with the recent theoretical results calculated based on HME models. This has finally resolved the long standing Γ n /Γ p ratio puzzle unambiguously. This ratio is very close to that of the previous exclusive measurement of 5 Λ He, thereby indicating the decay mechanisms of s-shell ( 5 Λ He) and p-shell ( 12 Λ C) hypernuclei are rather similar. Moreover, we have obtained the pair number yields in the non-backto-back kinematics region, which would provide information on the possible 2N NMWD contribution.