Observation of a K ¯ NN bound state in the 3 He( K − ,Λp)n reaction
T. Yamaga, S. Ajimura, H. Asano, G. Beer, H. Bhang, M. Bragadireanu, P. Buehler, L. Busso, M. Cargnelli, S. Choi, C. Curceanu, S. Enomoto, H. Fujioka, Y. Fujiwara, T. Fukuda, C. Guaraldo, T. Hashimoto, R. S. Hayano, T. Hiraiwa, M. Iio, M. Iliescu, K. Inoue, Y. Ishiguro, T. Ishikawa, S. Ishimoto, K. Itahashi, M. Iwai, M. Iwasaki, K. Kanno, K. Kato, Y. Kato, S. Kawasaki, P. Kienle, H. Kou, Y. Ma, J. Marton, Y. Matsuda, Y. Mizoi, O. Morra, T. Nagae, H. Noumi, H. Ohnishi, S. Okada, H. Outa, K. Piscicchia, Y. Sada, A. Sakaguchi, F. Sakuma, M. Sato, A. Scordo, M. Sekimoto, H. Shi, K. Shirotori, D. Sirghi, F. Sirghi, S. Suzuki, T. Suzuki, K. Tanida, H. Tatsuno, M. Tokuda, D. Tomono, A. Toyoda, K. Tsukada, O. Vazquez Doce, E. Widmann, T. Yamazaki, H. Yim, Q. Zhang, J. Zmeskal
aa r X i v : . [ nu c l - e x ] J un Observation of a ¯
KNN bound state in the He( K − , Λ p ) n reaction T. Yamaga , ∗ S. Ajimura , H. Asano , G. Beer , H. Bhang , M. Bragadireanu , P. Buehler , L. Busso , ,M. Cargnelli , S. Choi , C. Curceanu , S. Enomoto , H. Fujioka , Y. Fujiwara , T. Fukuda ,C. Guaraldo , T. Hashimoto , R. S. Hayano , T. Hiraiwa , M. Iio , M. Iliescu , K. Inoue , Y. Ishiguro ,T. Ishikawa , S. Ishimoto , K. Itahashi , M. Iwai , M. Iwasaki , † K. Kanno , K. Kato , Y. Kato ,S. Kawasaki , P. Kienle , ‡ H. Kou , Y. Ma , J. Marton , Y. Matsuda , Y. Mizoi , O. Morra ,T. Nagae , H. Noumi , , H. Ohnishi , S. Okada , H. Outa , K. Piscicchia , , Y. Sada , A. Sakaguchi ,F. Sakuma , M. Sato , A. Scordo , M. Sekimoto , H. Shi , K. Shirotori , D. Sirghi , , F. Sirghi , ,S. Suzuki , T. Suzuki , K. Tanida , H. Tatsuno , M. Tokuda , D. Tomono , A. Toyoda ,K. Tsukada , O. Vazquez Doce , , E. Widmann , T. Yamazaki , , H. Yim , Q. Zhang , and J. Zmeskal RIKEN Cluster for Pioneering Research, RIKEN, Wako, 351-0198, Japan Research Center for Nuclear Physics (RCNP), Osaka University, Osaka, 567-0047, Japan Department of Physics and Astronomy, University of Victoria, Victoria BC V8W 3P6, Canada Department of Physics, Seoul National University, Seoul, 151-742, South Korea National Institute of Physics and Nuclear Engineering - IFIN HH, Romania Stefan-Meyer-Institut f¨ur subatomare Physik, A-1090 Vienna, Austria Istituto Nazionale di Fisica Nucleare (INFN) Sezione di Torino, Torino, Italy Dipartimento di Fisica Generale, Universita’ di Torino, Torino, Italy Laboratori Nazionali di Frascati dell’ INFN, I-00044 Frascati, Italy Department of Physics, Osaka University, Osaka, 560-0043, Japan Department of Physics, Kyoto University, Kyoto, 606-8502, Japan Department of Physics, The University of Tokyo, Tokyo, 113-0033, Japan Laboratory of Physics, Osaka Electro-Communication University, Osaka, 572-8530, Japan High Energy Accelerator Research Organization (KEK), Tsukuba, 305-0801, Japan Department of Physics, Tokyo Institute of Technology, Tokyo, 152-8551, Japan Technische Universit¨at M¨unchen, D-85748, Garching, Germany Graduate School of Arts and Sciences, The University of Tokyo, Tokyo, 153-8902, Japan Department of Physics, Tohoku University, Sendai, 980-8578, Japan Excellence Cluster Universe, Technische Universit¨at M¨unchen, D-85748, Garching, Germany Korea Institute of Radiological and Medical Sciences (KIRAMS), Seoul, 139-706, South Korea ASRC, Japan Atomic Energy Agency, Ibaraki 319-1195, Japan Department of Chemical Physics, Lund University, Lund, 221 00, Sweden Research Center for Electron Photon Science (ELPH), Tohoku University, Sendai, 982-0826, Japan Engineering Science Laboratory, Chubu University, Aichi, 487-8501, Japan and Centro Fermi-Museo Storico della Fisica e Centro studi e ricerche ”Enrico Fermi”, 000184 Rome, Italy (J-PARC E15 Collaboration) (Dated: June 25, 2020)We have performed an exclusive measurement of the K − + He → Λ pn reaction at an inci-dent kaon momentum of 1 GeV /c . In the Λ p invariant mass spectrum, a clear peak was ob-served below the mass threshold of ¯ K + N + N , as a signal of the kaonic nuclear bound state,¯ KNN . The binding energy, decay width, and S -wave Gaussian reaction form-factor of this statewere observed to be B K = 42 ± . ) +3 − (syst . ) MeV, Γ K = 100 ± . ) +19 − (syst . ) MeV, and Q K = 383 ± . ) +4 − (syst . ) MeV /c , respectively. The total production cross-section of ¯ KNN ,determined by its Λ p decay mode, was σ totK · BR Λ p = 9 . ± . . ) +1 . − . (syst . ) µ b. We estimatedthe branching ratio of the ¯ KNN state to the Λ p and Σ p decay modes as BR Λ p /BR Σ p ∼ .
7, byassuming that the physical processes leading to the Σ NN final states are analogous to those of Λ pn . I. INTRODUCTION
The bound system of an anti-kaon ( ¯ K ) and a nucleon( N ) has been studied ever since the Λ(1405) was sug-gested as a ¯ KN molecular state [1, 2]. Based on numer- ∗ [email protected] † [email protected] ‡ deceased ous theoretical calculations of the chiral SU(3) dynam-ics and lattice-QCD, the interpretation that the Λ(1405)has an internal structure as a ¯ KN molecular-state ratherthan a three-quark baryon has gained stronger theoreti-cal support [3–5].The possibility of a more general system containing a¯ K , called a kaonic nucleus, has also been discussed. Muchtheoretical work on these kaonic nuclei, especially in the¯ KNN bound state, has been undertaken with various ¯ KN interaction models and calculation methods [6–21]. The¯ KNN bound state has charge +1 and isospin I = 1 / K − pp for the I z = +1 / J P = 0 − .The existence of the ¯ KNN bound state is generally sup-ported by all the calculations mentioned above; however,the estimated binding energies and widths of the stateare widely spread.To search for the ¯
KNN bound state, we conducted theexperiment J-PARC E15 using the in-flight K − beamat J-PARC. In the first measurement of the experi-ment, we demonstrated a significant yield excess well be-low the ¯ KNN mass threshold ( M ¯ KNN = m ¯ K + 2 m N ∼ .
37 GeV /c ) in the inclusive analysis of the He( K − , n )reaction [22], which suggests the strongly attractive na-ture of the ¯ KN interaction. We therefore extended theanalysis focusing on the simplest exclusive channel, theΛ pn final state, which consists of three baryons includingthe lightest hyperon [23]. Because s -quark conservationis secured in nuclear reactions governed by the strong in-teraction, we can trace the s -quark flow. Thus, the inter-action between a recoiled ¯ K and two spectator nucleons,¯ K – NN , can be studied by YN -pair analysis, which willtell us the reaction dynamics and formation signature of¯ KNN , if it exists. As described in Ref. [23], a kinemat-ical anomaly, a concentration of events around M ¯ KNN ,was observed only in the Λ p invariant mass spectrum.To study this anomaly, we performed a second measure-ment and found a peak structure in the Λ p invariant massspectrum located below M ¯ KNN , which we interpreted asa signal of the ¯
KNN bound state [24].In Ref. [24], the Λ pn final state was selected by detect-ing Λ p and by the kinematical consistency of the reactionincluding a missing neutron. However, we cannot entirelyexclude the two final states Σ pn and Σ − pp by the selec-tion. We treated the effect of the contamination of theΣ NN final state (the Σ N decay channel of ¯ KNN ) as asource of systematic error for simplicity. In this article,we evaluated the effect of the Σ NN final state contami-nation and estimated the ¯ KNN decay branch to the Σ p channel in a self-consistent way. II. J-PARC E15 EXPERIMENT
We measured the K − + He → Λ pn reaction to searchfor the ¯ KNN bound state by its Λ p decay mode. Theincident momentum of the K − beam is chosen to be p K =1 GeV /c to maximize the cross-section of the elementary K − N → ¯ KN reaction, corresponding to √ s = 1 . p invariant mass of the Λ pn final state, we analyzedthe process as two successive reactions, i.e. , K − + He → X + n,X → Λ p. (1)The former two-body reaction can be characterized bytwo parameters, the invariant mass of X ( m X ) and mo-mentum transfer to X ( q X ). We interpret the X forma-tion reaction in a more microscopic way, described in the framework of the cascade reactions K − + N → ¯ K + n, ¯ K + N N → X, (2)in which a virtual kaon ¯ K is produced in the primaryreaction between a K − and a nucleon followed by a for-mation reaction of the X resonance together with twospectator nucleons. In the microscopic view, m X cor-responds to the invariant mass of the ¯ K + NN system,and q X is the 3-momentum of the intermediate virtual ¯ K that can be measured by the momentum of Λ + p in thefinal state in the laboratory frame. At p K = 1 GeV /c ,the minimum q X is as small as ∼
200 MeV /c when theneutron is formed in the forward direction, so we can ex-pect a large ¯ K sticking probability to the two residualnucleons.The experiment was performed at the hadron exper-imental facility of J-PARC. A high-intensity secondary K − beam, produced by bombarding a primary gold tar-get with a 30-GeV proton beam, is transported alongthe K1.8BR beam line. Other secondary particles in thebeam are removed by an electrostatic separator.A beam-line detector system and a cylindrical detectorsystem (CDS) are used to measure incident K − and scat-tered charged particles, respectively. A detailed descrip-tion of the experimental setup is given in Refs. [25–27];however, we summarize the basics as follows.The beam-line detector system measures the time offlight and momentum of the K − beam. At the on-linelevel, K − is identified by an aerogel Cherenkov detector.The position and direction of the beam are measured bya drift chamber located just in front of the experimentaltarget of liquid He. The liquid He target is locatedat the final focus point of the beam line. The targetcell of He has a cylindrical shape with a diameter of 68mm and a length along the beam direction of 137 mm,and has a density of ∼
80 mg / cm . We accumulated He-filled data as the experimental run, and the emptytarget data as a background study. The CDS surroundingthe He target is composed of a cylindrical drift chamberand a cylindrical hodoscope. The detectors are installedinside a solenoid magnet to measure the momenta of thescattered charged particles.
III. ANALYSIS
Particle identification and momentum reconstructionof the K − beam and scattered charged-particles wereperformed. Then, the K − + He → Λ pn final state wasselected, where Λ and p were detected by CDS and themissing- n was identified kinematically. For the selectedΛ pn events, we measured a 2D distribution of the invari-ant mass of the Λ p and the momentum transfer to the Λ p .To investigate the production of the ¯ KNN bound state,we conducted a spectral fitting to the 2D distribution.
A. Beam and scattered particle analysis
For the K − beam, we applied time-of-flight-based PIDselection to achieve a high purity of kaon identification.Contamination from the in-flight kaon decay was elim-inated by checking the track inconsistency as a parti-cle trajectory recorded by drift chambers. The beammomentum was determined with a second-order transfermatrix of the final beam-line dipole spectrometer magnetcalculated using the TRANSPORT code [28]. A typicalmomentum resolution was estimated to be 0.2%.The trajectories of the charged particles from the K − + He reaction were measured by the CDS. We de-signed the magnet to have sufficient magnetic uniformityin the effective region of the CDS to apply a simple helicalfit to each trajectory to analyze its momentum. The ab-solute magnetic field strength was 0.715 T, calibrated us-ing monochromatic invariant-mass peaks of K s → π + π − and Λ → pπ − decays. The PID was conducted by a con-ventional method based on the 2D event distribution overthe mass-square and momentum. In the present analysis,a ± . σ region from the intrinsic mass was selected foreach particle. Any overlap of two different PID regionswas rejected to reduce miss-identification [23]. The inef-ficiency due to the overlap rejection was corrected in theanalysis efficiency. After the particle identification, anenergy-loss correction was applied by considering all thematerials on the trajectory of the particle to obtain itsinitial momentum. B. Event selection of Λ pn final state To select the K − + He → Λ pn reaction, three chargedparticles, ppπ − , were required. From the ppπ − , we ex-amined two possible pπ − -pairs as for Λ candidates (Λ ′ ).A candidate trajectory is tentatively defined by the pπ − vertex (the nearest point of the two trajectories) and syn-thetic momentum vector of the two. Then, we checkedif the event kinematics is consistent with the Λ pn finalstate, by a kinematical fitting. In the kinematical fit-ting, the pπ − -pair invariant mass ( m pπ − ) and the ppπ − missing mass ( m R in the He( K − , ppπ − ) R reaction)are used to derive the χ (degrees of freedom = 2, in thepresent case) as an indicator of the kinematical consis-tency to be the Λ pn final state. The “ KinFitter ” packagebased on the
Root classes [29] was used to search for theminimum χ .To include geometrical consistency of the event topol-ogy in the consistency test, a log-likelihood l ( x ) is intro-duced as l ( x ) = − ln Y i =1 p i ( x i ) , (3)where p i is the probability density function of the i -thvariable estimated by a Monte Carlo simulation, and themaximum value is renormalized to be one, so as to make l ( x ) = 0 at the most probable density point of the pa-rameter set. x stands for x = (cid:0) χ , D K − p , D K − Λ ′ , D Λ ′ p , D pπ − (cid:1) , (4)where the five variables are the χ given by the kinemati-cal fitting, the distances of closest approach for incoming K − with p ( D K − p ) and with Λ ′ ( D K − Λ ′ ), the distance ofclosest approach of Λ ′ and p ( D Λ ′ p ), and the minimumapproach of the pπ − -pair at the Λ ′ decay point ( D pπ − ).Finally, both the K − Λ ′ and K − p vertices were requiredto be in the fiducial volume of the target, to reduce thebackground from the target cell. In this examination,more than 99.5% of the pπ − were paired correctly in thesimulation.The event distribution of m R and l ( x ) is shown as a2D plot in Fig. 1-(a). A strong event concentration is seenat the bottom of the figure, which corresponds to the non-mesonic Λ pn final state. As shown in the m R spectrum,Fig. 1-(b), Λ pn events make a clear peak at m n , and theevents are clearly separated from the mesonic ( YNN + π )final states located at m R > m N + m π . To improve the ) c (GeV/ R m ) x ( l C oun t s (a) C oun t s allselected (b) n m ) π +m n m ( Λ m FIG. 1. (color online) (a) 2D plot of m R in the He( K − , ppπ − ) R reaction, and l ( x ). (b) Projected spectrumon the m R axis by selecting l ( x ) <
30. The vertical blackdashed lines are the masses of n ( m n ), N + π ( m N + m π ), andΛ ( m Λ ). Events from the Λ pn final state make a strong eventconcentration at the bottom of the 2D plot, where m R ∼ m n .The Λ pn event was selected below the red line in the 2D plot.The projection of selected events is shown by the red his-togram in (b). ) c (GeV/ - π p m ) x ( l C oun t s (a) × C oun t s (b) Λ m -selection pn Λ Events in the window are plotted.
FIG. 2. (color online) (a) 2D plot of m pπ − and l ( x ). (b)Projected spectrum on the m pπ − axis. Events in the Λ pn -selection window (shown in Fig. 1-(a)) are plotted. The ver-tical black dashed line is the Λ mass. Λ pn -selection, we selected Λ pn events on the 2D plane of m R and l ( x ), as indicated by the red line in Fig. 1-(a).The 2D plot of m pπ − and l ( x ), applying the Λ pn -selection window, is shown in Fig. 2-(a), and the pro-jection onto m pπ − is shown in Fig. 2-(b). As shown inthe figure, Λ is clearly selected. The tail of the Λ-peakis quite small; however, we should note that it does notsecure the purity of the Λ pn final state, in that the tailis removed by the kinematical fitting procedure throughthe χ evaluation. In the present Λ pn -selection, the otherfinal states may come in, as is indicated in Fig. 1-(a).To evaluate the contamination yields of the other fi-nal states, we conducted a detailed simulation as shownin Fig. 3. In this simulation, we generated non-mesonic YNN final states (Λ pn , Σ pn , and Σ − pp ) according tothe fit result (described in Sec. IV A) to make the simu-lation realistic. For simplicity, the event distribution ofmesonic final states, which make smaller contributions tothe Λ pn -selection window, are generated proportional tothe phase space.As shown in Fig. 3-(a), it is difficult to eliminate theΣ pn and Σ − pp final state events in the Λ pn -selectionwindow, since the m R spectra of contaminations of thetwo components are very similar. In particular, the Λ pn and Σ − pp final states have the same m R distribution.This is because R = n , γ + n , and n for the Λ pn , Σ pn , and Σ − pp final states, respectively. Thus, we plotted the m R − spectrum of He( K − , pp ) R − , as shown in Fig. 3-(b), to give R − = π − + n , π − + γ + n , and Σ − for theΛ pn , Σ pn , and Σ − pp final states, respectively. As shownin the figure, the relative yields can be evaluated easily,since the Σ − pp final state makes a peak at the Σ − intrin-sic mass, while the Λ pn final state becomes even broaderin the m R − distribution. Figure 3-(c) is the projectionof the events onto l ( x ), where the Λ pn final state hassmaller l ( x ) than the other final states.The relative yields of the signal and contaminationsin the present Λ pn -selection window were estimated bythe simultaneous fitting of these three spectra. The re-sult is summarized in Tab. I. The fit result improvedsubstantially by applying realistic Λ pn distribution, to-gether with Σ pn and Σ − pp contributions to the spec-tra. However, the fit result, chi-square 917 over degreesof freedom 506 of Fig. 3, might not be very sufficientby number. This is because we accepted events havingrelatively large l ( x ) to evaluate the contamination fromthe mesonic final states, whose distribution is simply as-sumed to be proportional to the phase space. Thus, thesystematic uncertainties of the table were evaluated bylimiting the fitting data region of Fig. 3 to l ( x ) <
10 toreduce the contamination effect from mesonic final states.Contaminations from the mesonic final state and fromthe K − reaction at the target cell are negligible. Thus, wefocused on the non-mesonic Σ pn and Σ − pp final states(Σ NN ) in the following analysis (Sec. III E). TABLE I. Relative yields of signal and contaminations in thepresent Λ pn -selection. The first and second errors are statis-tical and systematic, respectively.Source Relative yield ( R j ) (%)Λ pn (signal) 76 . ± . ± . pn . ± . ± . − pp . ± . ± . . ± . ± . K − reaction at the targetcell 3 . ± . ± . C. m X and q X distributions For the Λ pn -selected events, we measured the invariantmass of the Λ p system ( m X ) and the momentum transferto the Λ p system ( q X ). As shown in Eq. 1, q X can begiven by the momenta of Λ ( p Λ ) and p ( p p ) as q X = | p Λ + p p | . (5)Figure 4 shows the 2D event distribution on the m X and q X plane. As shown in the figure, there are verystrong event-concentrating regions. To show these event-concentrations unbiased manner, an acceptance correc-tion was applied to the data, to make the results indepen-dent of both the experimental setup and analysis code. ) c (GeV/ R m C oun t s (a) pp)R - π , - He(K of R m ) c (GeV/ - R m (b) - , pp)R - He(K of - R m ) x ( l data fit totalpn Λ pn Σ pp - Σ π YNN+ (c) ) x l( FIG. 3. (color online) Distributions of (a) m R of He( K − , π − pp ) R (the same as Fig. 1-(b)), (b) m R − of He( K − , pp ) R − ,and (c) l ( x ). For the m R and m R − spectra, l ( x ) <
30 was selected. These three distributions were simultaneously fittedby simulated spectra shown by colored lines. The fitting chi-square and number of degrees of freedom were 917 and 506,respectively. For mesonic (
YNN + π ), the final states all of possible charged states and combinations were summed. ) c (GeV/ X m ) c ( G e V / X q ) X dq X / ( d m σ d )) c / ( nb / ( M e V FIG. 4. (color online) 2D plot on the m X and q X plane afteracceptance correction. The black dotted line shows the kine-matical limit of the reaction. The vertical gray dotted lineand blue dotted curve are M ¯ KNN and M F ( q ), respectively.The gray hatched regions indicate where the experimental ef-ficiency is < . The events density, represented by a color code, is givenin units of the double differential cross-section: d σdm X dq X = N ( m X , q X ) ε ( m X , q X ) 1∆ m X q X L , (6)where N ( m X , q X ) is the obtained event number in∆ m X = 10 MeV /c and ∆ q X = 20 MeV /c (bin widthsof m X and q X , respectively). L is the integrated lumi-nosity, evaluated to be 2 . ± .
01 nb − . ε ( m X , q X ) is the experimental efficiency, which is quite smooth, as shownin Fig. 5-(a), around all the events-concentrating regionsof Fig. 4.After the acceptance correction, if no intermediatestate, such as X , exists in the K − + He → Λ pn reaction,then the event distribution will simply follow the Λ pn phase space ρ ( m X , q X ) without having a specific form-factor as given in Fig. 5-(b). In contrast to the data inFig. 4, ρ ( m X , q X ) is smooth for the entire kinematicallyallowed region.To account for the observed event distribution, threephysical processes were introduced as in Ref. [24]. Detailsof the physical processes, the formulation of each fittingfunction, and the fitting procedures are described in thefollowing sections. D. 2D model fitting functions
We considered the following three processes: K ) the¯ KNN bound state, F ) the non-mesonic quasi-free kaonabsorption (QF ¯ K - abs ) process, and B ) a broad distribu-tion covering the whole kinematically allowed region ofthe Λ pn final state. To decompose those processes, weconducted 2D fitting for the event distribution.The production yields of these three processes( F i ( m X , q X ) for i = K, F, B ) observed in the Λ pn fi-nal state should be proportional to the Λ pn phase space ρ ( m X , q X ). Thus, F i ( m X , q X ) can be described as theproduct of ρ ( m X , q X ) and specific spectral terms for the i -th process of a component f i ( m X , q X ), as F i ( m X , q X ) = ρ ( m X , q X ) f i ( m X , q X ) . (7)Figure 6 shows typical 2D distributions of f i ( m X , q X )for the three processes. All the parameters of the fittingfunctions described below are fixed to the final fittingvalues. ) c (GeV/ X m ) c ( G e V / X q (a) ) X q , X m ( ε ) c (GeV/ X m - × . - × . - × . - × . - × . - × . (b) ) X q , X m ( ρ FIG. 5. (color online) (a) Simulated spectra of experimental efficiency ε ( m X , q X ) for Λ pn final states. ε ( m X , q X ) includesgeometrical acceptance of CDS and analysis efficiency (decay branching ratio of Λ is also taken into account). The efficiencyis calculated bin by bin. The hatched regions are insensitive in the present setup, where ε < . pn phase space ρ ( m X , q X ) taking into account the kaon beam momentum bite. The ratio is normalized by one generated event.The roughness of the contours in both (a) and (b) is due to the limited statistics of the simulation. The vertical gray dottedlines and blue dotted curves are the same as in Fig. 4. ) c (GeV/ X m ) c ( G e V / X q (a) bound state z-scales are logarithmic ) c (GeV/ X m (b) quasi-free ) c (GeV/ X m (c) broad FIG. 6. (color online) 2D spectral functions for (a) the ¯
KNN bound state f K ( m X , q X ) (see Eq. 8), (b) the QF ¯ K - abs process f F ( m X , q X ) (see Eq. 10), and (c) a broad distribution f B ( m X , q X ) (see Eq. 12). For all the figures, the function strength isgiven in a logarithmic scale, where the contours are in the steps of 10% (red–orange), 1% (orange–cyan), and 0.1% (cyan–blue)compared to the maximum density of each function. The vertical gray dotted lines and blue dotted curves are the same as inFig. 4. To make f i automatically fulfill time-reversal symme-try, we limited ourselves to using q X -even terms to for-mulate the fitting functions described below, with oneexception. The details and the reason for the exceptionare described below. ¯ KNN production ( i = K ) As described in Ref. [24], we formulated the formationcross-section of the ¯
KNN bound state according to thereaction in Eq. 1 with a plane-wave impulse approxima-tion (PWIA) with a harmonic oscillator wave function.In this way, we simplified the microscopic reaction mech- anism in Eq. 2. We assumed that the spatial size of thebound state is much smaller than that of He, so the sizeterm of He was ignored in the formula. The time inte-gral gives a Breit–Wigner formula in the m X -direction,and the spatial-integral gives a Gaussian-form factor as f K ( m X , q X ) = (Γ K / ( m X − M K ) + (Γ K / × A K exp (cid:18) − q X Q K (cid:19) , (8)where M K , Γ K , and Q K are the mass, decay width,and S -wave reaction form-factor (involving microscopicreaction dynamics) parameter of the bound state, respec-tively.
2. Non-mesonic QF ¯ K - abs process ( i = F ) When the invariant mass m X of the secondary reactionin Eq. 2 is larger than the threshold M ¯ KNN , the recoil-kaon can behave as an approximately free particle; i.e. , X can be any channel, such as ¯ K + N + N , Y + N , or othermesonic channels. Among these, we denote the Y + N channel as the non-mesonic QF ¯ K - abs process. Specifically, Y and N are Λ and p in the Λ pn final state. In the non-mesonic QF ¯ K - abs process, a recoiled ¯ K is almost on-shelland absorbed by the two spectator nucleons. In QF ¯ K - abs , q X is predominantly defined by the neutron emission an-gle, because the residual nucleons are spectators (almostat-rest). Thus, the m X distribution-centroid is given as M F ( q X ) = r m N + m K + 4 m N q m K + q X , (9)where m N and m ¯ K are the intrinsic mass of N and ¯ K ,respectively. We plotted the M F ( q X )-curve in Fig. 4 as ablue dotted line. In the figure, two event concentrationson M F ( q X ) are clearly seen around q X ∼ . /c and ∼ . /c . These event concentrations correspond tothe backward and forward scattered ¯ K in the elementary K − N → ¯ Kn reaction. The QF ¯ K - abs should distributearound M F ( q X ) in the m X direction due to the Fermi-motion of the two nucleons. To describe the distribution,a Gaussian function is utilized, as f F ( m X , q X ) = exp " − ( m X − M F ( q X )) σ ( q X ) × (cid:20) A F exp (cid:18) − q X Q F (cid:19) + A F + A F exp (cid:18) m X m + q X q (cid:19)(cid:21) . (10)In the formula, we allowed the m X distribution width tohave a q X dependence as σ ( q X ) = σ + σ q X . (11)The second angle bracket in Eq. 10 represents the q X dependence of the production yield of the QF ¯ K - abs pro-cess, while the middle term is for flat distribution, andthe first and third terms correspond to backward andforward scattered ¯ K events, respectively.The forward ¯ K part of the QF ¯ K - abs process is locatedfar from the region of interest (distributed around theprojectile K − momentum ∼ c ), as shown in Fig. 4and Fig. 6-(b), so we phenomenologically formulated ourmodel fitting function as an exponential for simplicity, asgiven in Eq. 10.
3. Broad distribution ( i = B ) The two reaction processes described above have spe-cific regions where events concentrate. However, there isa broad distribution, which cannot be explained easily,over the entire kinematically allowed region in ( m X , q X ).In contrast to other processes, Λ, p , and n share thekinetic energy rather randomly, resulting in a relativelyweak m X and q X dependence, similar to a point-like in-teraction whose cross-section should be proportional to ρ ( m X , q X ), and thus f i ( m X , q X ) ∼ constant. A natu-ral interpretation of this component is the three-nucleonabsorption (3NA) reaction of an incident K − . On theother hand, there is a weak but yet clear m X and q X de-pendence over the whole kinematical region. The eventdensity at higher m X and lower q X is much weaker thanthat at the opposite side. On the other hand, there isno clear event density correlation between m X and q X ,which indicates that the distribution could be describedby the Cartesian product of centroid concentrating func-tions in both m X and q X . The most natural formula canbe written as an extension of Eq. 8 as f B ( m X , q X ) = (Γ B / ( m X − M B ) + (Γ B / × (cid:18) A B + A B q X Q B (cid:19) exp (cid:18) − q X Q B (cid:19) . (12) m X spectra of Λ pn final state To demonstrate the applicability of the model fittingfunctions conceptually, we present the m X spectrum ofthe data in the Λ pn -selection window and compare itwith the m X spectral shapes, restricting ourselves to theΛ pn final state, for K ) ¯ KNN , F ) QF ¯ K - abs , and B ) thebroad distribution, as shown in Fig. 7. For comparison,the acceptance was corrected for the data Fig. 7-(a) bydividing the data by ε ( m X , q X ) bin by bin (except for ε ( m X , q X ) < . ρ ( m X , q X ). Both figures wereintegrated over the whole q X region. All the parametersof the fitting functions of Fig. 7-(b) were fixed to the finalfitting value. For the figure, the 2D experimental resolu-tion (depending on both m X and q X ) was considered inthe Monte Carlo simulation. The magenta band is thesum of all the reaction components and the band widthindicates the fit error.As shown in the figure, the global structure of the m X spectrum is qualitatively described only with the Λ pn fi-nal state, even before considering the Σ NN contribution,as expected. The quantitative fitting was performed byconsidering Σ NN effects, as described in the followingsection. ) c (GeV/ X m )) c ( nb / ( M e V / X d m / σ d (a) data
22 2.2 2.4 2.6 2.8 ) c (GeV/ X m total bound state quasi-free broad (b) pn Λ FIG. 7. (color online) m X distribution (integrated by q X over the whole kinematically allowed region) of (a) data and(b) model functions. The model function is limited to theΛ pn final state; i.e. , the Σ NN contribution is excluded. Thecolored lines are spectra of three processes. The magentathick curve is the sum of all the processes with an error bandof the 95% confidence level. The vertical gray dotted lines are M ¯ KNN . E. Effect of Σ NN contamination As we described in Sec. III B, the selected Λ pn eventsare not free from contamination from the Σ NN (Σ pn and Σ − pp ) final states. The effect of these contami-nations should be taken into account in generating thefinal spectral fitting. It is clear that an ideal methodto evaluate the contaminations is to observe the Σ NN final state separately. Unfortunately, this is not possi-ble with the present experimental setup. In the presentanalysis, we assumed the Σ NN channels are producedin analogue reaction processes with that of Λ pn , i.e., K )¯ KNN , F ) QF ¯ K - abs , and B ) the broad distribution, andthus the same functions f i as the Λ pn final state canbe applied to represent the ( m X , q X ) event distributionof the Σ NN final states. When f i and its parametersare given as a common function, the YNN final statesand their contributions to the spectra through the Λ pn -selection window can be reliably evaluated by expanding F i to F ji so the formula is also applicable to Σ NN , where j = (Λ pn, Σ pn, Σ − pp ) and F ji = ρ j f i . ρ and ε can alsobe expanded to account for each final state in the samemanner.For the Σ pn final state, X is produced in the sameway as the Λ pn final state, but X goes to Σ p instead ofΛ p . Because Σ decays to γ Λ (100%), part of the Σ pn final state leaks in the Λ pn -selection window. As shownin Fig. 8-(a), the simulated acceptance over ( m Σ p , q Σ p )is smaller but similar to Fig. 5-(b). The expected m X and q X for the contaminating events are also simulated,and the resulting m X spectrum is shown in Fig. 9-(a).As shown in the figure, the structure in the spectrum issimilar but shifted to the lower side compared to Fig. 7-(b), due to the missing energy of the γ -ray.In contrast, the situation is very different for the Σ − pp final state. We simulated this channel in a similar mannerto that used for the Σ pn final state by replacing a Σ p -pair with a Σ − p -pair. The Σ − decays to nπ − ( ∼ π − and one of the pro-tons in this final state happen to be close to the Λ intrin-sic mass, the event may enter the Λ pn -selection window.This makes the simulated acceptance over ( m Σ − p , q Σ − p )given in Fig. 8-(b) very different from the other two.We simulated m X and q X of the contaminated eventsfor the incorrect Λ p -pair (pseudo-Λ p -pair), which wouldbe analyzed as the Λ pn final state in the analysis code.The resulting m X spectrum is given in Fig. 9-(b). Asshown in the figure, the structure in the spectrum is alsototally different from the other m X spectra. It should benoted that we generated the I z = − / KNN ( ¯ K nn )bound state instead of I z = +1 / − pp simula-tion at the same relative yield with the other two finalstates. This assumption might not be valid, because theisospin combination in the formation channel is different.However, it does not affect the fitting, because eventsfrom ¯ KNN concentrate at the lower q X -side, as shown inFig. 6-(a), where our detector system does not have sen-sitivity for the Σ − pp final state, as shown by the hatchedregion in Fig. 8-(b). For the same reason, the contribu-tion from the QF ¯ K - abs process to this final state is muchsmaller than those in the other final states. F. Iterative fitting procedure
To determine the spectroscopic parameters, we con-ducted 2D fitting for the 2D event distribution, as de-scribed in Ref. [24]. As shown in Fig. 5-(a) by the grayhatching, the present setup has insensitive regions dueto the geometrical coverage of the CDS. To avoid spu-rious bias caused by the acceptance correction, we di-rectly compared the data and the fitting function in thecount base by computing the expected event-numbers λ ( m X , q X ) to be observed in a ( m X , q X )-bin by λ ( m X , q X ) = X i,j R j ε j ( m X , q X ) F ji ( m X , q X ) ∆ m X ∆ q X = X i,j R j ε j ( m X , q X ) ρ j ( m X , q X ) f i ( m X , q X ) ∆ m X ∆ q X , (13)where ∆ m X and ∆ q X are the bin widths. Then, we eval-uated the probability of observing data in the ( m X , q X )-bin as P ( Z = N ( m X , q X )), where P is the Poisson dis-tribution function, N ( m X , q X ) is the data counts at the( m X , q X )-bin, and Z is a random Poisson variable forthe expectation value of λ ( m X , q X ). The log-likelihoodfor the 2D fitting ln.L can be defined as an ensemble ofprobabilities as ln.L = − X m X ,q X ln( P ( Z = N ( m X , q X ))) , (14)and the maximum ln.L was obtained to fit the data byoptimizing the spectroscopic parameters. There are atotal of 17 parameters in this fitting, consisting of four ) c (GeV/ p Σ m ) c ( G e V / p Σ q pn Σ (a) ) c (GeV/ p - Σ m ) c ( G e V / p - Σ q pp - Σ (b) FIG. 8. (color online) Experimental acceptance for each Σ NN contamination: (a) Σ pn final state and (b) Σ − pp final state.The vertical and horizontal axes for Σ pn (Σ − pp ) are the momentum transfer and invariant mass of the Σ p (Σ − p ) system.The hatched regions are insensitive in the present setup, where ε < . ) c (GeV/ X m )) c ( nb / ( M e V / X d m / σ d (a) pn Σ : partial invariant X m ( -pair) p Σ mass of
22 2.2 2.4 2.6 2.8 ) c (GeV/ X m total bound state quasi-free broad (b) pp - Σ : invariant mass X m ( -pair) p Λ of pseudo- FIG. 9. (color online) Expected spectral shapes of m X for the(a) Σ pn final state and (b) Σ − pp final state. The horizontalaxis of (a) is the Λ p invariant mass, which is a partial invariantmass of Σ p where γ of Σ → γ Λ is missing. The axis of (b)is the Λ p invariant mass after miss-identification as the Λ pn final state; i.e. , X is a pair of pseudo-Λ and p . The verticalgray dotted lines are M ¯ KNN . Note that the full scale of thedifferential cross-section is different from that of Fig. 7. parameters for the ¯
KNN bound state, eight parametersfor the non-mesonic QF ¯ K - abs process, and five parame-ters for the broad component. For the summation for ln.L , we omitted the ( m X , q X )-bin having no statisticalsignificance where ε j ( m X , q X ) < . pn -selection win-dow after the fitting procedure converged by dividing thespectra by ε Λ pn ( m X , q X ) bin by bin for both the data andfit results, except for Figs. 1-3.Due to the asymmetrical kinematical limits (seeFig. 4), the spectral function largely depends on the q X -region. We performed a first fitting for the whole regionas the global fit, then performed a second fitting for onlythe q X region from 0.3 to 0.6 GeV/ c to focus on ¯ KNN . The second fitting was conducted to deduce the param-eters of ¯
KNN under a better S/N region, so the otherparameters are fixed in the second fitting. After an iter-ation of a spectral fitting for the data shown in Fig. 4,we looped back to evaluate the ratio of the final stateyields of Λ pn : Σ pn : Σ − pp : other in the Λ pn -selectionwindow by the fitting procedure described in Sec. III B(see Fig. 3 and Tab. I). To obtain self-consistent results,we looped back over the two procedures iteratively untilboth the ratio parameters and spectroscopic parametersconverge. IV. RESULTS AND DISCUSSIONA. 2D fitted spectra
To demonstrate the accuracy of the fit result in 2D,we plotted the fit result for the m X -spectra in the q X -slice (as shown in Fig. 10) and for the q X -spectra in the m X -slice (as shown in Fig. 11), i.e., projections of 2Ddata onto the m X -axis and q X -axis at the same time.In other words, Figs. 10 and 11 show the compilationof event projections of the two-dimensional four-by-four m X - and q X -regions of Fig. 4 onto each axis. In eachspectrum, data are compared with the fit result as shownin the magenta band (95% confidence level), and decom-posed as colored lines. All the regions are well repro-duced for both the m X and q X spectra. The maximumlog-likelihood and total number of degrees of freedom ofthe fitting were 2425 and 2234, respectively. We plot-ted the signal of ¯ KNN formation and its Λ p decay as ared line, and ¯ KNN → Σ p in the Λ pn -selection windowas a red dashed-line. To simplify the plot, we summedthe QF ¯ K - abs and broad contributions from the Λ pn finalstate and from contaminations of the Σ NN final states,because the spectra for each reaction process are rela-0 )) c ( nb / ( M e V / X d m / σ d (a) ≤ X q (b) ≤ X q < data fit total p Λ→ KNN p Σ→ KNN quasi-free broad ) c (GeV/ X m )) c ( nb / ( M e V / X d m / σ d (c) ≤ X q < ) c (GeV/ X m (d) > X q FIG. 10. (color online) m X spectra for various intervals of q X : (a) q X ≤ . /c , (b) 0 . < q X ≤ . /c , (c)0 . < q X ≤ . /c , and (d) 0 . /c < q X . The dottedlines correspond to the m X -slice regions given in Fig. 11. tively similar (see Figs. 7-(b) and 9). As expected, the¯ KNN formation signal is clearly seen in Fig. 10-(b) in the m X spectrum, and in Fig. 11-(b) in the q X spectrum.At the lowest q X region of the m X spectrum in Fig. 10-(a), the spectrum is confined in a medium mass regiondue to the kinematical boundary (see Fig. 4 and Fig. 5).In this region, the backward ¯ K part of the QF ¯ K - abs pro-cess K − + N → ¯ K + n becomes dominant. In Fig. 10-(b),the ¯ KNN formation signal is dominant and contributionsfrom other processes, in particular the QF ¯ K - abs process,are relatively suppressed. In the relatively large q X re-gion in Fig. 10-(c), the broad component becomes dom-inant, while the ¯ KNN formation signal becomes weaker.At an even larger q X region in Fig. 10-(d), the forward¯ K part of the QF ¯ K - abs process becomes large, which dis-tributes to the large m X side. This events concentrationmay partially arise from direct K − absorption on twoprotons in He (2NA), but the width is too great to beexplained by the Fermi motion. Therefore, it is difficultto interpret 2NA as the dominant process of this eventsconcentration. In this q X region, there is also a largecontribution from the broad component.Figure 11 shows the q X spectra sliced on m X . Fig-ure 11-(a) shows the region below the ¯ KNN formationsignal where the broad distribution is dominant, havingsmall leakage from the signal. As shown in the spec-trum, the broad distribution has no clear structure andhas a larger yield at a higher q X region than at a lower q X region. Figure 11-(b) shows the ¯ KNN formation sig-nal region, in which the events clearly concentrate at the )) c ( nb / ( M e V / X dq / σ d (a) ≤ X m (b) ≤ X m < data fit total p Λ→ KNN p Σ→ KNN quasi-free broad ) c (GeV/ X q )) c ( nb / ( M e V / X dq / σ d (c) ≤ X m < ) c (GeV/ X q (d) > X m FIG. 11. (color online) q X spectra for various intervals of m X :(a) m X ≤ .
27 GeV /c , (b) 2 . < m X ≤ .
37 GeV /c , (c)2 . < m X ≤ . /c , and (d) 2 . /c < m X , withthe fitting results shown as colored lines. The dotted linescorrespond to the q X -slice regions given in Fig. 10. lower q X side. In Fig. 11-(c), we can see the backward¯ K part of the QF ¯ K - abs process, together with the leakagefrom the signal and broad distribution. In contrast to¯ KNN , the QF ¯ K - abs process even more strongly concen-trates in the lower q X region (neutron is emitted to thevery forward direction). To compare the q X dependencewith that of the ¯ KNN formation process, we formulatedour model fitting function for the forward ¯ K QF ¯ K - abs process to have a Gaussian form (see Eq. 10). The q X spectrum at the highest m X region is given in Fig. 11-(d). The major components are the broad distributionand the forward ¯ K part of the QF ¯ K - abs process. Thecentroid of the event concentration locates at an incidentkaon momentum of 1 GeV/ c , but the width in q is againtoo great to interpret it as being due to the 2NA reaction.Thus, the 2NA process would be rather small in the caseof the Λ pn final state of the present reaction.To check the Σ pn contamination effect in the presentfitting, we divided Fig. 10-(b) into two regions for m R ≤ m n and m R > m n , as shown in Fig. 12. The figure showsthat the spectra are consistent with the Σ pn final statedistribution in Fig. 3-(a), i.e., that the ¯ KNN → Σ p con-tribution exists only on the m R > m n side. As shownin the figure, the m X spectrum of Fig. 12-(b) below themass threshold of M ¯ KNN is slightly wider and deeper thanthat of Fig. 12-(a) in both the data and total fitting func-tion, as expected, due to the presence of Σ pn contami-nation.1 ) c (GeV/ X m )) c ( nb / ( M e V / X d m / σ d (a) n m ≤ R m ≤ X data fit total p Λ→ KNN p Σ→ KNN quasi-freebroad
22 2.2 2.4 2.6 2.8 ) c (GeV/ X m (b) n > m R m ≤ X FIG. 12. (color online) m X spectra for (a) m R ≤ m n and(b) m R > m n , with the fitting result. The selected q X re-gion is the same as for Fig. 10-(b) (0 . < q X ≤ . /c ).The vertical dotted line is M ¯ KNN . Note that the bin widthis exceptional, ∆ m X = 20 MeV/ c , for these figures to havesufficient statistics. B. Fitted parameters
The converged 17 spectroscopic parameters are listedin Tab. II. We improved the fitting procedure to fullytake into account the Σ NN final states in the presentanalysis, as well as the ( m X , q X ) dependence of the de-tector resolution. As a result, the values of the spectro-scopic parameters were updated from our recent publica-tion [24], though the updated values are within the errorrange of the previous publication. TABLE II. Converged 17 spectroscopic parameters and theirerrors. ¯
KNN bound state Value ± (stat . ) + − (syst . ) A K (1 . ± . +0 . − . ) × M K . ± . +0 . − . GeV /c Γ K . ± . +0 . − . GeV Q K . ± . +0 . − . GeV /c Non-mesonic QF ¯ K - abs Value ± (stat . ) + − (syst . ) A F (4 . ± . +0 . − . ) × A F (1 . ± . +0 . − . ) × A F (2 . ± . +0 . − . ) × − σ . ± . +0 . − . GeV /c σ . ± . +0 . − . GeV − Q F . ± . +0 . − . GeV /cm . ± . +0 . − . MeV /c q . ± . +0 . − . MeV /c Broad distribution Value ± (stat . ) + − (syst . ) A B (0 . ± +3767 − . ) × − A B (2 . ± . +0 . − . ) × M B . ± . +0 . − . GeV /c Γ B . ± . +0 . − . GeV Q B . ± . +0 . − . GeV /c The mass position of the ¯
KNN bound state M K (orthe binding energy B K ≡ M ¯ KNN − M K ) and its decaywidth Γ K are M K = 2 . ± . . ) +0 . − . (syst . ) GeV /c ( B K = 42 ± . ) +3 − (syst . ) MeV) , Γ K = 100 ± . ) +19 − (syst . ) MeV , respectively. The S -wave Gaussian reaction form factorparameter of the ¯ KNN bound state Q K is Q K = 383 ± . ) +4 − (syst . ) MeV /c. The total production cross-section of the ¯
KNN boundstate going to the Λ p decay mode σ totK · BR Λ p was evalu-ated by integrating the spectrum to be σ totK · BR Λ p = 9 . ± . . ) +1 . − . (syst . ) µ b . In the present analysis, the strength of the ¯
KNN → Σ p decay mode is deduced based on the Σ pn contam-ination yield given by Fig. 3. By assuming that therelative yields of the three physical processes of Σ NN and those of Λ pn are equal, we estimated the differen-tial cross-section of ¯ KNN decaying into the Σ p mode σ totK · BR Σ p as σ totK · BR Σ p = 5 . ± . . ) +0 . − . (syst . ) µ b . Therefore, the branching ratio of the Λ p and Σ p decaymodes was estimated to be BR Λ p /BR Σ p ∼ .
7. The es-timated branching ratio is higher than the value of thetheoretical calculation based on the chiral unitary ap-proach, predicting a ratio of almost one [30].
C. Systematic errors
The systematic errors were evaluated by consideringthe uncertainties of the absolute magnetic field strengthof the solenoid, the binning effect of spectra, and system-atic errors of the branch of the final states (Tab. I). Forproduction cross-sections, we considered the luminosityuncertainty. To be conservative, the evaluated system-atic errors are added linearly.We succeeded in reproducing the data distribution byour model fitting functions. However, for the broad dis-tribution, we cannot simply specify the physical processof its formation. Thus, we also tried an independentmodel fitting functions, which are intentionally unphysi-cal but still able to reproduce the global data structure.A typical model fitting function fulfilling the require-ments can be obtained by replacing the q X -even polyno-mial term with a simple q X -proportional one in Eq. 12.The q X -proportional term is not physical by itself, andcan only be possible as a comprehensive interference ofan S -wave and a P -wave. As yet another extreme of the2model fitting function of the broad distribution, we alsoexamined a fit by replacing Loretnzian term of Eq. 12 tothe second order polynomials. Although these alternativemodel functions are unphysical, we treated the centroidshifts of the other parameters as a source of systematicerror for safety.The systematic uncertainties are much reduced fromRef. [24], due to the improved analysis procedure by con-sidering a precise and realistic evaluation of the Σ NN contamination in the Λ pn -selection window. D. Discussion
We introduced three physical processes to account forthe data, K ) ¯ KNN state production, F ) QF ¯ K - abs pro-cess, and B ) the broad distribution, and found that thepresence of K ) ¯ KNN is essential to explain the spectraself-consistently, which cannot be formed as an artifact.The presence of F ) is naturally expected from the anal-ysis on inclusive channel presented in Ref. [22], but therelative yield of the quasi-free component is substantiallyreduced because we focused on the non-mesonic Λ pn finalstate in the present paper. For process B ), we pointedout that the possibility that it could be due to point-like 3NA kaon absorption, because of the weakness of its( m X , q X ) dependence.For the ¯ KNN bound state, B K ∼
40 MeV agrees nicelywith phenomenological predictions[8, 12, 14, 17, 31].However, it should be noted that the obtained B K isthe spectral Breit–Wigner pole position, neglecting themicroscopic reaction dynamics given in Eq. 2. Thus, thepresent Breit–Wigner pole might be different from thephysical pole predicted by theoretical calculations.Γ K ∼
100 MeV is wide, as for a quasi-bound state,compared to the binding energy B K . It is also wider thanthe Λ(1405) → π Σ decay width of ∼
50 MeV (100%). IfΛ(1405) is the ¯ KN quasi-bound state, then it is naturallyexpected that the ¯ KNN → π Σ N decay will occur in thesame order as the YN decay channels.As shown in Fig. 11-(b), the production yield of the¯ KNN bound state is much larger in a smaller q X region.This trend is a common feature of nuclear bound-stateformation reactions in general. In the K − + He → X + n formation channel, we can achieve a minimum momen-tum transfer to X as small as ∼
200 MeV/ c , whichmakes this channel the ideal formation process. How-ever, σ totK · BR YN (=Λ p, or Σ p ) is still small compared tothe total cross-section of the elementary K − N → ¯ Kn reaction by the order of O (10 − ). Even if we take intoaccount a mesonic decay branch similar to YN decay,the total ¯ KNN formation branch would still be less than O (10 − ) of the elementary cross-section. In spite of thesmall formation yield and large decay width near thebinding threshold, we have succeeded in observing kaonicbound state formation. This is because the YNN finalstates, which strongly limit the number of possible com-plicated intermediate states such as mesonic processes, allow the s -quark flow in the reaction to be traced by Y , and moreover, the ¯ KNN signal and remaining non-mesonic QF ¯ K - abs processes can be effectively separatedby q X -slicing.Let us consider the physical meaning of Q K in Eq. 8. Q K is quite large, more than twice the Q F of the non-mesonic QF ¯ K - abs process. The value of Q F is natural inview of the size of the He radius, as well as the strong an-gular dependence of the elementary process K − N → ¯ Kn observed in Ref. [22] at p K − = 1 GeV/ c , which is theprimary reaction of Eq. 2. Instead, the value of Q K may carry information on the spatial size of the ¯ KNN state. We formulated the model fitting function basedon a simple PWIA calculation, assuming that the ¯
KNN wave function can be written in the ground state of a har-monic oscillator (HO). The spatial size of the HO wavefunction can be given as R K = ~ /Q K ∼ . m N + m ¯ K ) / m N , R K ∼ . B K ∼
40 MeV.Finally, we briefly discuss the broad component. Thepresent data show that the 2NA kaon absorption chan-nels are weak, in contrast to kaon absorption at-rest ex-periments [32], so we need to understand why 3NA stillexists while the 2NA channels are weak. The distribu-tion of this component f B , given in Fig. 6-(c), becomes abroad P -wave resonance-like structure characterized by M B between m Λ + m p and M ¯ KNN , A B << A B , as shownin Tab. II. This phenomenon might be simply due tothe nature of the formula of the fitting function, given inEq. 12, but it is worth studying in more detail to clarifythe physics of this component. To be conservative, wekeep our interpretation open for the physical process ofthis broad distribution, and treated that as a source ofthe systematic error.Open questions still remain, such as the spin-parity J P of the ¯ KNN state, and the relationship between thepresent ¯
KNN signal and Λ(1405) resonance. Also, in theanalysis, we have not taken into account the interferenceeffects between the three introduced physical processes.More comprehensive studies are required to clarify theseremaining questions.
V. SUMMARY
We have measured the Λ pn final state in the in-flight reaction on a He target at a kaon momentum of1 GeV /c . We observed the kaonic nuclear quasi-boundstate, I z = +1 / KNN , and obtained its parameters by2D fitting of the Λ p invariant mass and momentum trans-fer.The binding energy and the decay width of the statewere B K = 42 ± . ) +3 − (syst . ) MeV and Γ K =100 ± . ) +19 − (syst . ) MeV, respectively. The S -wave Gaussian reaction form-factor was Q K = 383 ± . ) +4 − (syst . ) MeV /c . The total production cross-sections of the ¯ KNN bound state decaying into non-3mesonic Λ p and Σ p modes were obtained to be σ totK · BR Λ p = 9 . ± . . ) +1 . − . (syst . ) µ b and σ totK · BR Σ p =5 . ± . . ) +0 . − . (syst . ) µ b, respectively. Thus, the ra-tio Λ p/ Σ p decay branch was approximately 1 . KNN from a simplePWIA-based model fitting function, the implied size isquite small compared to the mean nucleon distance innormal nuclei. However, the observed value of Q K =383 ± . ) +4 − (syst . ) MeV /c is unexpectedly large(about twice as large as an elementary process), whichmakes the theoretical microscopic study difficult. There-fore, a more realistic theoretical calculation including de-tailed reaction dynamics and a more detailed experimen-tal study are essential to understand the observed q X distribution. ACKNOWLEDGMENTS
The authors are grateful to the staff members of J-PARC/KEK for their extensive efforts, especially on thestable operation of the facility. We are also grateful to thecontributions of Professors Daisuke Jido, Takayasu Seki-hara, Dr. Rie Murayama, and Dr. Ken Suzuki. This workis partly supported by MEXT Grants-in-Aid 26800158,17K05481, 26287057, 24105003, 14102005, 17070007, and18H05402. Part of this work is supported by the Min-istero degli Affari Esteri e della Cooperazione Inter-nazionale, Direzione Generale per la Promorzione del Sis-tema Paese (MAECI), StrangeMatter project. [1] R. H. Dalitz and S. F. Tuan, Ann. Phys. , 100 (1959).[2] R. H. Dalitz, T. C. Wong, and G. Rajasekaran, Phys.Rev. , 1617 (1967).[3] K. Miyahara and T. Hyodo, Phys. Rev. C , 1 (2016).[4] Y. Kamiya and T. Hyodo, Phys. Rev. C , 1 (2016).[5] J. M. Hall, W. Kamleh, D. B. Leinweber, B. J. Menadue,B. J. Owen, A. W. Thomas, and R. D. Young, Phys. Rev.Lett. , 1 (2015).[6] T. Yamazaki and Y. Akaishi, Phys. Lett. B , 70(2002).[7] Y. Akaishi and T. Yamazaki, Phys. Rev. C , 044005(2002).[8] Y. Ikeda and T. Sato, Phys. Rev. C , 1 (2007).[9] N. V. Shevchenko, A. Gal, and J. Mareˇs, Phys. Rev.Lett. , 1 (2007).[10] N. V. Shevchenko, A. Gal, J. Mareˇs, and J. R´evai, Phys.Rev. C (2007), 10.1103/PhysRevC.76.044004.[11] A. Dot´e, T. Hyodo, and W. Weise, Nucl. Phys. A ,197 (2008).[12] S. Wycech and A. M. Green, Phys. Rev. C , 1 (2009).[13] A. Dot´e, T. Hyodo, and W. Weise, Phys. Rev. C , 1(2009).[14] Y. Ikeda and T. Sato, Phys. Rev.C (2009),10.1103/PhysRevC.79.035201.[15] N. Barnea, A. Gal, and E. Liverts, Phys. Lett. B ,132 (2012).[16] M. Bayar and E. Oset, Phys. Rev. C , 1 (2013).[17] J. R´evai and N. V. Shevchenko, Phys. Rev. C , 1(2014).[18] T. Sekihara, E. Oset, and A. Ramos, Prog. Theor. Exp. Phys. , 123D03 (2016).[19] A. Dot´e, T. Inoue, and T. Myo, Phys. Rev. C , 1(2017).[20] S. Ohnishi, W. Horiuchi, T. Hoshino, K. Miyahara,and T. Hyodo, Phys. Rev. C , 065202 (2017),arXiv:1701.07589.[21] A. Dot´e, T. Inoue, and T. Myo, Phys. Lett. B , 405(2018).[22] T. Hashimoto et al. , Prog. Theor. Exp. Phys. ,61D01 (2015).[23] Y. Sada et al. , Prog. Theor. Exp. Phys. , 051D01(2016).[24] S. Ajimura et al. , Phys. Lett. B , 620 (2019).[25] K. Agari et al. , Prog. Theor. Exp. Phys. , 02B009(2012).[26] K. Agari et al. , Prog. Theor. Exp. Phys. , 02B011(2012).[27] M. Iio et al. , Nucl. Instrum. Methods, A , 1 (2012).[28] K. L. Brown, F. Rothacker, D. C. Carey, and F. C. Iselin,CERN-80-04 (1980).[29] “A Kinematic Fit with Constraints [online],”http://github.com/goepfert/KinFitter/wiki/KinFitter(2011).[30] T. Sekihara, J. Yamagata-Sekihara, D. Jido, andY. Kanada-En’Yo, Phys. Rev. C 10.1103/Phys-RevC.86.065205.[31] T. Yamazaki and Y. Akaishi, Phys. Rev. C 10.1103/Phys-RevC.76.045201.[32] R. Del Grande et al. , Eur. Phys. J. C79