J-PARC E40 Collaboration

The interactions between octet baryons, that is, baryon-baryon ($BB$) interactions including hyperon-nucleon ($YN$) interactions are fundamental information for describing nuclear systems including hyperons such as hypernuclei and neutron stars. Historically, experimental data attributed to a pure two-body $YN$ system is quite limited due to various difficulties involved in conducting hyperon-proton scattering experiments [1, 2, 3, 4, 5, 6, 7, 8, 9]. However, there has been recent progress in obtaining the two-body $YN$ interaction from a two-body system.

We (J-PARC E40 collaboration) reported accurate measurements of the differential cross sections of the $\Sigma^{-}p$ elastic scattering in the momentum range from 470 to 850 MeV/$c$ [10]. This measurement first provided accurate differential information, which is essential for determining the $P$ and higher-wave interactions. The CLAS collaboration also reported the updated total cross sections of the $\Lambda p$ elastic scattering for the $\Lambda$ momentum between 0.9 and 2.0 GeV/$c$ [11]. The ALICE [12, 13, 14, 15, 16, 17, 18], and STAR [19, 20] collaborations measured particle correlations not only for the hyperon-nucleon pairs but also for the hyperon-hyperon pairs. These measurements, which are sensitive to small values of relative momentum, constitute new experimental methods for determining the $S$-wave interaction [21, 22, 23].

The Nijmegen group [24, 25, 26] and the Jülich group [27] developed theories behind $BB$ interactions using a boson-exchange model and considering a broken flavor SU(3) symmetry. The quark cluster model (QCM) was proposed to explain the origin of the short-range repulsive core in the nucleon-nucleon ($NN$) interactions by considering the effects of the Pauli principle for the quarks and the color magnetic interaction between them[28]. The Kyoto-Niigata group constructed a realistic description by incorporating an effective meson exchange potential into QCM to represent the middle- and long-range interactions [29].

$BB$ interactions have also been intensively studied using modern theoretical frameworks, such as the lattice QCD simulations [30, 31, 32] and the chiral effective field theory ($\chi$EFT). Lattice QCD potentials were used to analyze the particle correlations [33, 34]. $\chi$EFT is widely used for deriving the $NN$ force because it has an underlying chiral symmetry in QCD and a power counting feature to improve the calculation systematically by moving to a higher order [35]. $\chi$EFT has been extended to the hyperon sector too [36, 37, 38].

Realistic $YN$ interaction models, which should be constructed by gathering the theoretical and experimental efforts, will create a new trend in hypernuclear physics. For example, the no-core shell model calculations based on the $\chi$EFT extended to the $YN$ sector were recently performed to describe the $p$-shell hypernuclei [39, 40, 41, 42]. A realistic $YN$ interaction is also essential for constructing the equation of state of neutron stars with microscopic approaches using bare two-body $YN$ interactions [43].

In this letter, we present new results on the differential cross sections of the $\Sigma^{-}p\to\Lambda n$ reaction in the $\Sigma^{-}$ momentum range 470–650 MeV/$c$ measured in the J-PARC E40 experiment[10, 44]. The $\Sigma p$ scatterings (the $\Sigma^{-}p$ and $\Sigma^{+}p$ elastic scatterings and the $\Sigma^{-}p\to\Lambda n$ reaction) were systematically measured in the experiment.

The $\Sigma^{-}p$ channel is closely related with the $\Lambda N$ system because of the $\Lambda N$-$\Sigma N$ coupling [45]. The strength of the $\Lambda N$-$\Sigma N$ coupling has been intensively discussed in relation to the so-called hyperon puzzle in neutron stars [46]. In the nuclear (neutron) matter, the $\Lambda N$-$\Sigma N$ coupling, which is a dominant source of the attraction in some $\Lambda N$ interactions [46], can be suppressed as a result of Pauli blocking for the intermediate nucleon state. For interactions with a sizable $\Lambda N$-$\Sigma N$ coupling potential such as the one in $\chi$EFT NLO13 interaction, the $\Lambda N$ interaction becomes more repulsive at higher baryon densities compared to that in $\chi$EFT NLO19 interaction with a moderate coupling potential [37]. Such a scenario in which the $\Lambda N$ interaction becomes repulsive, together with an additional repulsive $\Lambda NN$ three-body force [47, 48], is hypothesized to prevent the $\Lambda$ particles from appearing in the neutron stars and to explain neutron stars with two-solar masses [49]. To constrain the strength of the two-body $\Lambda N$-$\Sigma N$ coupling, reactions involving the conversion such as $\Sigma^{-}p\to\Lambda n$ are potentially important.

The $\Sigma p$ scattering experiment (J-PARC E40) was performed at the K1.8 beam line in the J-PARC Hadron Experimental Facility. A 1.33 GeV/$c$ $\pi^{-}$ beam of $2.0\times 10^{7}$/spill was produced from a 30 GeV proton beam with a cycle of 5.2 seconds and a beam duration of 2 seconds. The experimental concept and the experimental setup are shown in Fig. 1 in [10]. $\Sigma^{-}$ particles were produced by the $\pi^{-}p\to K^{+}\Sigma^{-}$ reaction in a liquid hydrogen (LH₂) target, and the produced $\Sigma^{-}$ moving in the LH₂ target interacted with protons. The momentum of each $\Sigma^{-}$ was measured with an accuracy of approximately 5 MeV/$c$ as the missing momentum calculated from the momenta of the $\pi^{-}$ beam and the outgoing $K^{+}$ analyzed by the K1.8 beam line spectrometer [50] and the forward magnetic spectrometer (KURAMA), respectively. In total, 1.62 $\times 10^{7}$ $\Sigma^{-}$ particles were used to search for the $\Sigma^{-}$-induced secondary reactions. Secondary reactions such as the $\Sigma^{-}p\to\Lambda n$ reaction were identified kinematically from the data of the charged particles in the final state using the CATCH system surrounding the LH₂ target, which comprises a cylindrical scintillation fiber tracker (CFT), a bismuth germanate calorimeter (BGO), and a scintillator hodoscope (PiID) coaxially arranged outwards from the center [51, 52]. The tracks of the charged particles were reconstructed using CFT, and their kinetic energies were measured using BGO. A detailed analysis of the $\Sigma^{-}$ identification and the secondary reaction with CATCH is found in [10]. In this letter, we focus on the analysis to derive the differential cross sections of the $\Sigma^{-}p\to\Lambda n$ reaction.

For the analysis of the $\Sigma^{-}p\to\Lambda n$ reaction in the LH₂ target, both a proton and a $\pi^{-}$ were required to be detected by CATCH in coincidence with the $\Sigma^{-}$ production within a time interval of $\pm$10 ns. Particle identification between $\pi^{-}$ and proton was performed by the $dE$-$E$ method between the partial energy deposit ($dE$) in CFT and the total energy deposit ($E$) in BGO. One can refer to Fig. 8 in [10]. The kinetic energy and direction of the proton were measured using CATCH. However, the thickness of BGO was not sufficient for $\pi^{-}$ to be stopped, and only the direction of the $\pi^{-}$ was obtained by the tracking using CFT. Therefore, a certain kinematic assumption is necessary to estimate the magnitude of the $\pi^{-}$ momentum. In the analysis of the $\Sigma^{-}p\to\Lambda n$ reaction, the $\pi^{-}$ momentum was determined such that the invariant mass of the $\pi^{-}$ and proton became the $\Lambda$ mass using the momentum of the proton and the opening angle between the two tracks. A vertex point defined as the closest point between the two tracks was required to be within 40 mm from the target center in the $xy$ plane, which is perpendicular to the beam axis. We assume that this vertex point is the decay point ($vtx_{decay}$) of the scattered $\Lambda$. The closest distance at $vtx_{decay}$ was also required to be less than 5 mm.

The $\Lambda$ momentum ($p_{\Lambda}^{(\Lambda\to p\pi^{-})}$) reconstructed with the assumption of the $\Lambda\to p\pi^{-}$ decay is checked to determine whether $p_{\Lambda}^{(\Lambda\to p\pi^{-})}$ is consistent with the momentum ($p_{\Lambda}^{(\Sigma^{-}p\to\Lambda n)}$) calculated based on the $\Sigma^{-}p\to\Lambda n$ kinematics from the initial $\Sigma^{-}$ momentum and the $\Sigma^{-}p\to\Lambda n$ scattering angle. We define $\Delta p_{\Lambda}(\Sigma^{-}p\to\Lambda n)$ as the difference between $p_{\Lambda}^{(\Lambda\to p\pi^{-})}$ and $p_{\Lambda}^{(\Sigma^{-}p\to\Lambda n)}$; that is, $\Delta p_{\Lambda}(\Sigma^{-}p\to\Lambda n)=p_{\Lambda}^{(p\pi^{-})}-p_{\Lambda}^{(\Sigma^{-}p\to\Lambda n)}$. Data points in Fig. 1 (d) show the $\Delta p_{\Lambda}(\Sigma^{-}p\to\Lambda n)$ distribution for the scattering angle of $0.3\leq\cos\theta\leq 0.4$ in the c.m. system for the $\Sigma^{-}$ momentum between 470 and 550 MeV/$c$. Here, the scattering angle $\theta$ is defined as the angle between the $\Sigma^{-}$ beam and the scattered $\Lambda$. The peak structure around $\Delta p_{\Lambda}(\Sigma^{-}p\to\Lambda n)=0$ represents the $\Sigma^{-}p\to\Lambda n$ events.

The broad structure in the $\Delta p_{\Lambda}(\Sigma^{-}p\to\Lambda n)$ distribution on the left side of the peak is attributed to other secondary reactions. As the source of the other secondary reactions, the $\Sigma^{-}p$ elastic scattering and the $\Sigma^{-}p\to\Sigma^{0}n$ reactions are considered. The scatterings between a target proton and decay products of the $\Sigma^{-}\to n\pi^{-}$ decay, that is, $np$ and $\pi^{-}p$ scatterings, are also taken into account. To identify the source of the background reaction, the measured proton energy was compared with the calculated energies based on the background kinematics. For example, in the $np$ scattering case, the energy of the recoil proton was calculated from the initial neutron momentum and the scattering angle between the initial neutron and recoil proton. In this calculation, the momentum of the initial neutron was obtained by assuming that a $\pi^{-}$ is emitted from the $\Sigma^{-}\to n\pi^{-}$ decay. Fig. 1 (a) shows the $\Delta E_{p}(np)$ distribution, which is the difference between the measured and calculated kinetic energies of the proton for the $np$ scattering kinematics. The peak around $\Delta E_{p}(np)=0$ corresponds to the $np$ scattering event. We also define the $\Delta p_{p}(\pi^{-}p)$ (and $\Delta E_{p}(\Sigma^{-}p)$ ) values, representing the difference between the measured momentum (and the measured kinetic energy) of the proton and the calculated one assuming the $\pi^{-}p$ ( and $\Sigma^{-}p$) scatterings, respectively, as shown in Figs. 1 (b) and (c). The effect of misidentification of the initial $\Sigma^{-}$ particle owing to the contamination of $\pi^{+}$ and protons in the $K^{+}$ selection is also shown as green-hatched histograms in Fig. 1, obtained by selecting the side-band region of $K^{+}$ in the mass-squared distribution detected by the KURAMA spectrometer [10].

To estimate the contribution from each secondary reaction, we fit the four $\Delta E$ and $\Delta p$ spectra simultaneously with the simulated spectra of five possible reactions, as shown by the colored spectra in Fig. 1. Realistic resolutions of the detectors and efficiencies were taken into consideration in the simulation. Refer to [10] for a detailed description. The sum of these spectra reproduces the $\Delta E$ and $\Delta p$ spectra well. Fortunately, the background reactions are kinematically separated from the $\Sigma^{-}p\to\Lambda n$ reaction, except for the $\Sigma^{-}p$ elastic scattering, as shown by the histogram with a black line in Fig. 1 (d).

The differential cross section is defined as

where $\rho$, $N_{A}$, and $L$ represent the target density, Avogadro’s number, and the total flight length of the $\Sigma^{-}$ hyperons in the LH₂ target, respectively. $i_{vtz}$ represents the index of the $z$ vertex position from $-150$ mm to 150 mm with an interval of 30 mm. For a scattering angle $\theta$ in the c.m. frame and a $z$ vertex position of $i_{vtz}$, $N_{scat}(i_{vtz},\cos\theta)$ and $\epsilon(i_{vtz},\cos\theta)$ represent the number of $\Sigma^{-}p\to\Lambda n$ reaction events and the detection efficiency of the CATCH system, respectively. The numerator is the efficiency-corrected number of scattering events. $\Delta\Omega$ represents the solid angle for each scattering angle. Regarding the total flight length $L$, the same value as in the $\Sigma^{-}p$ elastic scattering analysis was used [10].

The detection efficiency for the $\Sigma^{-}p\to\Lambda n$ scattering events ($\epsilon(i_{vtz},\cos\theta)$) was studied using a realistic Monte Carlo simulation based on the Geant4 package [53], where the realistic angular resolution, the tracking efficiency of CFT, and the realistic energy resolution for BGO were taken into account [10]. The generated data of the secondary reactions were analyzed by the same analysis program. The detection ratio for the $\Sigma^{-}p\to\Lambda n$ reaction was obtained for each scattering angle as its detection efficiency. Figs. 2 (c) and (d) show the efficiencies averaged for the $z$ vertex region, which is denoted as $\bar{\epsilon}(\cos\theta)$, for the momentum regions 470–550 MeV/$c$ and 550–650 MeV/$c$, respectively. The branching ratio of the $\Lambda\to p\pi^{-}$ decay is included in the efficiency. The effect of the systematic uncertainty of the tracking efficiency of CFT, which was estimated from calibration measurements of the $pp$ scattering cross sections [10], is typically 0.5% - 3% as represented by the red box in Figs. 2 (c) and (d). In the backward angle around $\cos\theta=-1$, the kinetic energy of the proton from the $\Lambda$ decay is too small to be detected. Therefore, the efficiency decreases in the backward angles. The decrease in the efficiency at the forward angles is due to the decreased acceptance of CATCH.

The number of scattering events was estimated from the $\Delta p_{\Lambda}(\Sigma^{-}p\to\Lambda n)$ spectra for each scattering angle, as shown in Fig. 1 (d). The sum of the simulated background reactions was used as the background spectrum. The efficiency-corrected number of scattering events was estimated in several ways by changing the estimation of the scattering events and by changing the background estimation. In the $z$ vertex-dependent manner, $N_{scat}(i_{vtz},\cos\theta)$ was obtained by subtracting the simulated background spectrum from the $\Delta p_{\Lambda}(\Sigma^{-}p\to\Lambda n)$ spectrum in each $z$ vertex region. The efficiency-corrected scattering number was obtained in the form of the numerator of equation (1). Alternatively, we also estimated the efficiency-corrected scattering number with a modified form of $\sum_{i_{vtz}}N_{scat}(i_{vtz},\cos\theta)/\bar{\epsilon}(i_{vtz},\cos\theta)$. It implies that the scattering event number of all $z$ vertex bins was corrected by the averaged efficiency for the $z$ vertex position. In this method, the scattering event number was obtained from the reproduced spectrum for the $\Sigma^{-}p\to\Lambda n$ reaction (the spectrum with a brown line in Fig. 1 (d)) in the $\Delta p(\Sigma^{-}p\to\Lambda n)$ spectrum. Figs. 2 (a) and (b) show the scattering event numbers with the statistical errors for each scattering angle for the two $\Sigma^{-}$ momentum regions. The detected event numbers are approximately 1700 and 630 in total for the momentum regions 470–550 MeV/$c$ and 550–650 MeV/$c$, respectively. To estimate the effect of the background estimation, we also derived the efficiency-corrected number of scattering events based on a different background spectrum obtained by fitting the $\Delta p(\Sigma^{-}p\to\Lambda n)$ spectrum alone with the simulated spectra. The difference in the efficiency-corrected number of scattering events was treated as the systematic uncertainty. The sizes of the systematic uncertainties for each angle are indicated by the boxes in Figs. 2 (a) and (b).

Figs. 3 (a) and (b) show the measured differential cross sections for the $\Sigma^{-}p\to\Lambda n$ scattering. These results are obtained from approximately 50 times more scattering events than that in the past experiment [7]. The statistical and systematic errors are represented as the error bars and boxes, respectively. The sources of the systematic errors are (1) total $\Sigma^{-}$ track length; (2) estimation of the efficiency-corrected number of scattering events, shown by the boxes in Figs. 2 (a) and (b); and (3) CATCH efficiency, shown by the red boxes in Figs. 2 (c) and (d). The uncertainty in the total $\Sigma^{-}$ track length is less than 1% [10]. The main contribution to the systematic errors comes from the uncertainty of the efficiency-corrected number of scattering events including the systematic error in the background estimation. These sources were quadratically summed.

In these momentum ranges, the differential cross sections of the $\Sigma^{-}p\to\Lambda n$ reaction show a slightly forward peak structure. In contrast to the $\Sigma^{-}p$ elastic scattering [10], sizable contributions also exist for the backward angular region. Figs. 3 also shows predictions by various theoretical models. In the momentum region between 470 and 550 MeV/$c$, theoretical calculations by the fss2 including QCM [29] and the extended $\chi$EFT model [36, 37], reproduced the measured data adequately. On the other hand, the Nijmegen models (ESC08c [25] and ESC16 [26]) clearly underestimate the forward angular region. In the higher momentum range between 550 and 650 MeV/$c$, the differential cross section becomes flatter in its angular dependence. Predictions from the fss2 and the $\chi$EFT seem to overestimate the differential cross section at forward angles.

The theoretical predictions by $\chi$EFT NLO13 and NLO19 are similar, as shown by the red and yellow lines, respectively, in Figs. 3 (a) and (b), even though the strength of the $\Lambda N$-$\Sigma N$ coupling potential is quite different for these two models [37]. Although the experimental accuracy of the present data is still comparable to the difference between the two models, the present data and $\Lambda p$ scattering data in future experiments proposed at J-PARC [54] will provide new insight into the $\Lambda N$-$\Sigma N$ coupling. Haidenbauer et al. also pointed out that a study of the $\Lambda p$ scattering near the $\Sigma N$ threshold would be quite helpful for constraining the $\Lambda N$-$\Sigma N$ coupling [55].

The integrated cross sections for $-0.7\leq\cos\theta\leq 1.0$ were obtained as $22.5\pm 0.68\rm{(stat.)}\pm 0.65\rm{(syst.)}$ mb and $15.8\pm 0.83\rm{(stat.)}\pm 0.52\rm{(syst.)}$ mb for the momentum regions 470–550 MeV/$c$ and 550–650 MeV/$c$, respectively. These values were compared with the past measurements [7], as shown in Fig. 3 (c). The measured integrated and differential cross sections are invaluable experimental inputs for improving the $BB$ interaction models. A systematic theoretical investigation of both the $\Sigma^{-}p$ elastic scattering and the $\Sigma^{-}p\to\Lambda n$ reaction can be performed based on these data.

Finally, we compare the present results with the $\Sigma^{-}p$ elastic scattering presented in [10]. Both results are rather consistent with the predictions by the fss2 and $\chi$EFT. On the other hand, the ESC models underestimate the differential cross sections at the forward angular regions for both channels. These models should also be compared with the $\Sigma^{+}p$ data, for which analysis is presently ongoing [56].

In summary, we successfully measured the differential cross sections of the $\Sigma^{-}p\to\Lambda n$ reaction for the momentum region 470–650 MeV/$c$ at J-PARC. These results are part of a series of systematic studies of $\Sigma N$ interactions from the two-body $\Sigma^{\pm}p$ scatterings. The differential cross sections were measured for the wide angular region of $-0.7\leq\cos\theta\leq 1.0$ by detecting approximately 100 scattering events for each angular bin of $\Delta\cos\theta=0.1$. The total number of the reaction events is approximately 50 times larger than the past experiment. The differential cross section of the $\Sigma^{-}p\to\Lambda n$ reaction shows a moderate forward-peaking angular distribution. The integrated cross sections for angular coverage were also obtained with a drastically improved accuracy. These accurate measurements will play an essential role in establishing realistic $BB$ interaction models.

We would like to thank the staff of the J-PARC accelerator and the Hadron Experimental Facility for their support in providing the beam during the beam time. We would also like to thank the staff of CYRIC and ELPH at Tohoku University for their support in providing beams for test experiments for our detectors. We would like to express our gratitude to Y. Fujiwara for the theoretical support from an early period of the experimental design and thank T. A. Rijken and J. Haidenbauer for their theoretical calculations. We also thank KEKCC and SINET4. This work was supported by the JSPS KAKENHI Grant Numbers 23684011, 15H00838, 15H05442, 15H02079, and 18H03693. This work was also supported by the Grants-in-Aid Number 24105003 and 18H05403 for Scientific Research from the Ministry of Education, Culture, Science, and Technology (MEXT), Japan.