# Pole position of $\Lambda(1405)$ measured in $d(K^{-},n)\pi\Sigma$ reactions

J-PARC E31 Collaboration  S. Aikawa  S. Ajimura  T. Akaishi  H. Asano  G. Beer
C. Berucci111current affiliation: Leonardo SpA, Rome, Italy 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<EMAIL_ADDRESS>Y. Ishiguro
S. Ishimoto  T. Ishikawa  K. Itahashi  M. Iwai  M. Iwasaki  K. Kanno  K. Kato
Y. Kato  S. Kawasaki<EMAIL_ADDRESS>P. Kienle222deceased Y.
Komatsu  H. Kou  Y. Ma  J. Marton  Y. Matsuda  Y. Mizoi  O. Morra  R. Murayama
T. Nagae  H. Noumi<EMAIL_ADDRESS>H. Ohnishi  S. Okada333current
affiliation: Chubu Univ., Kasugai, Japan Z. Omar  H. Outa  K. Piscicchia  Y.
Sada  A. Sakaguchi  F. Sakuma  M. Sato  A. Scordo  M. Sekimoto  H. Shi  K.
Shirotori  D. Sirghi  F. Sirghi  K. Suzuki  S. Suzuki  T. Suzuki  K. Tanida
H. Tatsuno  A. O. Tokiyasu  M. Tokuda  D. Tomono  A. Toyoda  K.
Tsukada444current affiliation: ICR, Kyoto Univ., Uji, Japan O. Vazquez-Doce
E. Widmann  T. Yamaga  T. Yamazaki  H. Yim  Q. Zhang  J. Zmeskal  Department
of Physics, Tokyo Institute of Technology, Tokyo, 152-0551, Japan  Research
Center for Nuclear Physics (RCNP), Osaka University, Ibaraki, 567-0047, Japan
Department of Physics, Osaka University, Toyonaka, 560-0043, Japan  RIKEN
Cluster for Pioneering Research (CPR), RIKEN, Wako, 351-0198, Japan
Department of Physics and Astronomy, University of Victoria, Victoria BC V8W
3P6, Canada  Stefan-Meyer-Institut für subatomare Physik, A-1030 Vienna,
Austria  National Institute of Physics and Nuclear Engineering - IFINHH,
Romania  INFN Sezione di Torino, Torino, Italy  Dipartimento di Fisica
Generale, Universita’di Torino, Torino, Italy  Department of Physics, Seoul
National University, Seoul, 151-742, South Korea  Laboratori Nazionali di
Frascati dell’INFN, I-00044 Frascati, Italy  Accelerator Laboratory, High
Energy Accelerator Research Organization (KEK), Tsukuba, 305-0801, Japan
Department of Physics, The University of Tokyo, Tokyo, 113-0033, Japan
Laboratory of Physics, Osaka Electro-Communication University, Neyagawa,
572-8530, Japan  RIKEN Nishina Center for Accelerator-Based Science, RIKEN,
Wako, 351-0198, Japan  ASRC, Japan Atomic Energy Agency (JAEA), Ibaraki
319-1195, Japan  RIKEN SPring-8 Center, RIKEN, Hyogo, 679-5148, Japan
Cryogenics Science Center, High Energy Accelerator Research Organization
(KEK), Tsukuba, 305-0801, Japan  Department of Physics, Kyoto University,
Kyoto, 606-8502, Japan  Institute of Particle and Nuclear Studies, High Energy
Accelerator Research Organization (KEK), Tsukuba, 305-0801, Japan  Mechanical
Engineering Center, High Energy Accelerator Research Organization (KEK),
Tsukuba, 305-0801, Japan  Technische Universität München, D-85748, Garching,
Germany  Graduate School of Arts and Sciences, The Univeristy of Tokyo, Tokyo,
153-8902, Japan  Research Center for Electron Photon Science (ELPH), Tohoku
University, Sendai, 982-0826, Japan  Department of Physics, Al-Farabi Kazakh
National University, Almaty, 050040, Kazakhstan  Museo Storico della Fisica e
Centro Studi e Ricerche “Enrico Fermi”, Piazza del Viminale 1, 00184 Rome,
Italy  INFN, Laboratori Nazionali di Frascati, Via Enrico Fermi 40, 00044
Frascati, Italy  Department of Chemical Physics, Lund University, Lund, 221
00, Sweden  Excellence Cluster University, Technische Universität München,
D-85748, Garching, Germany  Korea Unstitute of Radiological and Medical
Sciences (KIRAMS), Seoul, 139-706, South Korea

###### Abstract

We measured a set of $\pi^{\pm}\Sigma^{\mp}$, $\pi^{0}\Sigma^{0}$, and
$\pi^{-}\Sigma^{0}$ invariant mass spectra below and above the $\bar{K}N$ mass
threshold in $K^{-}$-induced reactions on deuteron. We deduced the $S$-wave
$\bar{K}N\rightarrow\pi\Sigma$ and $\bar{K}N\rightarrow\bar{K}N$ scattering
amplitudes in the isospin 0 channel in the framework of a $\bar{K}N$ and
$\pi\Sigma$ coupled channel. We find that a resonance pole corresponding to
$\Lambda(1405)$ is located at 1417.7${}^{+6.0}_{-7.4}$(fitting
errors)${}^{+1.1}_{-1.0}$(systematic errors) + $[-26.1^{+6.0}_{-7.9}$(fitting
errors)${}^{+1.7}_{-2.0}$(systematic errors)]$i$ MeV/$c^{2}$, closer to the
$\bar{K}N$ mass threshold than the value determined by the Particle Data
Group.

###### keywords:

Hyperon resonance, Meson-baryon bound state, Kaon-nucleon interation,
Scattering amplitude

††journal: Physics Letters B

##  1 Introduction

$\Lambda(1405)$ is a well-known hyperon resonance with strangeness $-1$, spin-
parity 1/2-, and isospin 0 ($I$ = 0). It is classified as the first orbital
excited state in the constituent quark model. However, the properties of
$\Lambda(1405)$ are not easily explained, such as the fact that it has the
lightest mass among the negative parity baryons even though it contains a
heavier strange quark, and the large mass difference it exhibits compared to
that for the so-called spin–orbit partner state of $\Lambda(1520)$. It has
been argued that $\Lambda(1405)$ is a bound state of an anti-kaon ($\bar{K}$)
and a nucleon ($N$) since it is located just below the $\bar{K}N$ mass
threshold, Dalitz and Tuan first predicted a possible quasi-bound state of
$\bar{K}N$ with $I$ = 0 in 1959, based on low-energy $K^{-}$-proton scattering
experiments [1, 2]. The first observation of a hyperon resonance sitting just
below the $\bar{K}N$ mass threshold in $\pi^{-}\Sigma^{+}/\pi^{+}\Sigma^{-}$
invariant mass spectra was reported in 1961 [3]. Since then, several sets of
experimental data on $\Lambda(1405)$ have been reported [4, 5, 6, 7, 8, 9, 10,
11, 12, 13, 14, 15, 16]. Dalitz and Deloff deduced a resonance energy and
width of 1406.5 $\pm$ 4.0 MeV and 50 $\pm$ 2 MeV by analyzing the measured
$\pi^{-}\Sigma^{+}$ mass spectrum [7] based on $\bar{K}N$ scattering theory
[17]. The latest edition of the Review of Particle Physics [18] gives average
values of 1405.1${}^{+1.3}_{-0.9}$ MeV and 50.5 $\pm$ 2.0 MeV, including two
later works [19, 20] which demonstrate that a so-called phenomenological
approach giving the $\Lambda(1405)$ mass at $\sim$1405 MeV [21, 22] is
favorable for fitting the $(\pi\Sigma)^{0}$ invariant mass spectra from
$K^{-}$ stopped on 4He [23] and proton–proton collisions (HADES) [12]. A
recent review on $\Lambda(1405)$ is available in Ref. [24].

Over the last two decades, there have been intensive discussions about the so-
called chiral unitary approach, which is a coupled-channel meson–baryon
scattering theory employing chiral Lagrangians. Several calculations indicate
that there are two resonance poles between the $\pi\Sigma$ and $\bar{K}N$ mass
thresholds [25, 26, 27, 28, 29], where the higher pole, coupled to $\bar{K}N$,
is located at around 1420 MeV or greater. The chiral unitary approach is in
contradiction with the phenomenological approach. There is a discussion of
differences in different theoretical treatments of chiral unitary approaches
and the phenomenological approach [30].

The experimental situation is also controversial. Recent measurements of
$(\pi\Sigma)^{0}$ mass spectra have been reported in photo-induced reactions
on protons [10, 13, 14, 15, 16] and proton–proton collisions [11, 12]. The
CLAS collaboration reported precise $\pi^{-}\Sigma^{+}$, $\pi^{+}\Sigma^{-}$,
and $\pi^{0}\Sigma^{0}$ spectra for a wide range of incident photon energies
[13, 14]. Theoretical analyses have been made on these data and reproduced the
spectral shapes fairly well, even though they involved many parameters [31]
and/or reaction diagrams [32]. The HADES collaboration reported invariant mass
spectra of $\pi^{-}\Sigma^{+}$, $\pi^{+}\Sigma^{-}$, and their sum [12]. Their
spectral shapes were different from those for photo-production. In particular,
they observed peaks even below 1400 MeV. Theoretical analyses of these spectra
have also been made [33, 20]. However, the locations of $\Lambda(1405)$
determined by a chiral unitary model [33] and a phenomenological model [20]
are not compatible with each other. Therefore, experimental data to directly
determine the $\bar{K}N$ scattering amplitude coupled to $\Lambda(1405)$ are
required.

##  2 Experiment

We carried out an experimental study of kaon-induced $\pi\Sigma$ production
via $d(K^{-},n)\pi\Sigma$ reactions [34]. Our expectation was to measure a
reaction sequence consisting of a 1-GeV/$c$ incident negative kaon knocking
out a neutron at a very forward angle (less than 6 degrees in the laboratory
frame) from a deuteron, with the $\bar{K}$ recoiled backward reacting with the
residual nucleon ($N_{2}$) to produce $\pi$ and $\Sigma$, as shown in the
reaction diagram in the inset of Fig. 1. In the second step of the reaction
sequence, $\bar{K}N_{2}\rightarrow\pi\Sigma$ scattering takes place even below
the $\bar{K}N$ mass threshold. Since the typical momentum of a recoiled
$\bar{K}$ is as low as $\sim$250 MeV/$c$ for a $\pi\Sigma$ invariant mass of
around 1405 MeV/$c^{2}$, $S$-wave scattering is expected to be dominant. We
measured the $\pi\Sigma$ invariant mass spectra, from which we deduced the
$\bar{K}N$ scattering amplitude in the $I$ = 0 channel.

Figure 1: Schematic illustration of the experimental setup. CDS: cylindrical
detector system, CDH: cylindrical scintillator hodoscope, CDC: cylindrical
drift chamber, D2 TGT: liquid deuterium target, T0: time zero counter, BPD:
backward proton detector, BPC: backward proton drift chamber, DEF: beam
defining counter, FDC: forward drift chamber, NC: neutron counter array, CVC:
charged particle veto counter, and PC: proton counter array. The reaction
diagram expected for $d(K^{-},N)\pi\Sigma$ is shown in the inset.

The experiment was performed at the K1.8BR beam line [35] of the Japan Proton
Accelerator Research Complex (J-PARC). Negatively charged kaons delivered from
K1.8BR were incident on a liquid deuterium (D2) target of 125 mm thickness.
The momentum of the incident kaons was analyzed by the K1.8BR-D5 magnetic
spectrometer. A schematic layout of the experimental setup [36, 37] is
illustrated in Fig. 1. A time-zero counter (T0), placed 1100 mm upstream from
the D2 target, defined a time origin triggered by the incident kaon for time-
of-flight measurements of scattered particles. A drift chamber (BPC) and
scintillator hodoscopes (BPD) were placed 143.2 mm and 482.5 mm upstream from
the D2 target center, respectively. Kaon beam tracks were measured by the BPC,
which were used to determine the reaction vertex. The BPD and the BPC were
used to detect backward emitted protons from $\pi^{0}\Sigma^{0}$ productions,
as mentioned later. The kaon beam was finally defined by a beam defining
counter (DEF) placed just in front of the D2 target. The integrated luminosity
of the kaon beam used in the present analysis was ($5927\pm 158$) [($8078\pm
248$)] $\mu$b-1, which was the product of the beam intensity 5.56$\times$1010,
the number of target deuterons 4.82$\times$1023 [6.03$\times$1023],
efficiencies of the beam line detectors (28.1$\pm$1.7)% [(31.5$\pm$1.9)%)],
trigger system (95.2$\pm$2.6)%, and the data acquisition system
(76.5$\pm$5.9)%. Here, the numbers in square brackets are for the
$\pi^{0}\Sigma^{0}$ mode. Charged particles from the D2 target were measured
by a cylindrical detector system (CDS), consisting of a cylindrical drift
chamber (CDC) and scintillator hodoscopes (CDHs) surrounding the D2 target. An
efficiency of the CDC for one-charged particle tracking was estimated to be
(97.7$\pm$0.4)%. The CDS was operated in a solenoid magnet with a magnetic
field of 0.714 Tesla. Scattered neutrons were detected by neutron counters
(NCs), consisting of an array of 112 plastic scintillator slabs (200 mm width,
1500 mm height, and 50 mm thickness each), placed approximately 15 m from the
D2 target. Since the solid angle of the NCs seen from the target is
(21.5$\pm$0.2)% msr, the angular coverage for the emitted neutrons is less
than 6 degrees. The detection efficiency of the NC was estimated to be
(31.7$\pm$1.6)%. It was measured by finding a neutron at the NC to the
predicted neutron emission direction in the $p(K^{-},\bar{K}^{0})n$ reaction,
where a backward-recoiled $\bar{K}^{0}$ was reconstructed by the CDS. In
reality, a factor of (91.9$\pm$0.7)% due to a charged-particle veto counter
(CVC) placed in front of the NC to veto charged particles, was multiplied as
an effective efficiency of the NC. Charged particles emitted in a forward
angle, including the incident beams, were swept out by a dipole magnet placed
behind the solenoid magnet. Protons knocked out from deuterons by the incident
kaon beam were bent by the dipole magnet in the opposite direction of the
beam. The time-of-flight of the knocked-out proton was measured by proton
counters (PC), consisting of hodoscopes of 27 scintillator slabs (100 mm
width, 1500 mm height, and 50 mm thickness each), which were placed beside the
CVC. The solid angle of the PC is slightly momenum dependent and is typically
22.6 msr. The trajectory of each scattered proton was determined using the
position information from the reaction vertex at the target, a drift chamber
(FDC) placed at the entrance of the dipole magnet, and the hit slab of the PC.
A tracking efficiency for the proton is estimated to be (81.9$\pm$4.2)%. We
measured the $d(K^{-},p)\pi^{-}\Sigma^{0}$ reaction, where the $p$ was
detected by the PC and the two $\pi^{-}$s were detected by the CDS.

We measured the $\pi^{\pm}\Sigma^{\mp}$ production associated with a knocked-
out neutron detected by the NC, where $\pi^{+}$ and $\pi^{-}$ were detected by
the CDS and the missing neutron was identified separately in a
$d(K^{-},n\pi^{+}\pi^{-})$ missing mass spectrum, as shown in Fig. 2. In these
modes, three background processes are relevant as they all give the same final
state of $n\pi^{+}\pi^{-}n_{\rm miss}$, where $n_{\rm miss}$ represents the
neutron identified in the $d(K^{-},n\pi^{+}\pi^{-})$ missing mass spectrum:
(1) $K^{-}d\rightarrow n\bar{K}^{0}n_{\rm miss}$, (2)
$K^{-}d\rightarrow\pi^{-}\Sigma^{+}n_{\rm miss}$, and (3)
$K^{-}d\rightarrow\pi^{+}\Sigma^{-}n_{\rm miss}$. In (2) and (3),
$\pi^{\mp}\Sigma^{\pm}$ are produced with an incident $K^{-}$ interacting with
a bound proton in a deuteron. A neutron from the $\Sigma$ decay is emitted at
a forward angle and detected by the NC. The $n_{\rm miss}$ is a spectator
neutron in the processes. They are the so-called one-step
$\pi^{\mp}\Sigma^{\pm}$ production processes, which occurs in a quite
differrent kinematical region compared with that for the two-step process that
we concern in the present article. Above three processes can be excluded since
we can identify the $\bar{K}^{0}$, $\Sigma^{+}$, and $\Sigma^{-}$ peaks in the
invariant mass spectra of $\pi^{+}\pi^{-}$, $n\pi^{+}$, and $n\pi^{-}$, as
shown in Fig. 2(b), (c), and (d), respectively. We obtained the
$\pi^{\pm}\Sigma^{\mp}$ missing mass spectra in the
$d(K^{-},n){\pi^{\pm}\Sigma^{\mp}}$ reactions separately, as we will show
later. The production ratio of $\pi^{-}\Sigma^{+}$ to $\pi^{+}\Sigma^{-}$ was
obtained to reproduce the $\Sigma^{\pm}$ peak and its kinematic reflection
(continuum-like distribution). The decomposed $d(K^{-},n\pi^{\pm})$ missing
mass spectra are shown in Fig. 3(a) and (b), respectively.

Figure 2: (a) Missing mass spectrum of $d(K^{-},n\pi^{+}\pi^{-})$ togather
with an illustration of a typical signal event topology. The missing neutron
($n_{\rm miss}$) is identified separately. The dashed lines indicate a
selected neutron mass region. (b), (c), and (d) Invariant mass spectra of
$\pi^{+}\pi^{-}$, $n\pi^{+}$, and $n\pi^{-}$, respectively, in the
$d(K^{-},n\pi^{+}\pi^{-})n_{\rm miss}$ reactions. The peak regions shown with
the dashed lines were excluded as $K^{0}$ and one-step $\pi^{\mp}\Sigma^{\pm}$
production processes as typical event topologies are illustrated,
respectively. Figure 3: (a) and (b) Decomposed $\Sigma^{\pm}$ peaks in the
$d(K^{-},n\pi^{\mp})$ missing mass spectra, respectively. (c) Invariant mass
spectrum of $\pi^{-}$ and $p$ measured by CDS and BPD/BPC, respectively. A
$\Lambda$ peak was selected for the $\pi^{0}\Sigma^{0}$ mode (vertical lines)
as the $\Sigma^{0}$ immediately decays into $\Lambda+\gamma$. A typical event
topology for the $\pi^{0}\Sigma^{0}$ mode is illustrated in the figure. (d)
Missing mass spectrum of $d(K^{-},n\Lambda)$. The expected $\pi^{0}$,
$\pi^{0}\gamma$, and background components (BG) are overlaid as histograms.
See text for BG. A $\pi^{0}\gamma$ region (0.18–0.3 GeV/$c^{2}$) was gated for
the $\pi^{0}\Sigma^{0}$ mode. (e) Scatter plot of two possible
$d(K^{-},p\pi^{-})$ missing masses for the $\pi^{-}\Sigma^{0}$ mode. A typical
event topology for the $\pi^{-}\Sigma^{0}$ mode is illustrated in the figure.
A $\Sigma^{0}$ mass region was selected, as indicated by the blue lines. (f)
Missing mass spectrum of $d(K^{-},p\pi^{-}\pi^{-})$ (crosses) and selected
$\Sigma^{0}$ region in (e) (histogram). A $p\gamma$ peak was selected to
identify the $\pi^{-}\Sigma^{0}$ mode (vertical lines).

In the $\pi^{0}\Sigma^{0}$ production, $\Sigma^{0}$ immediately decays to
$\Lambda\gamma$. The $\Lambda$ hyperon decays to $\pi^{-}$ and a proton. The
$\pi^{-}$ is emitted in a wide angular region and could be detected by the
CDS. While, the proton is generally emitted backward because most of the
momentum of the $\pi^{0}\Sigma^{0}$ that recoiled backward in the $d(K^{-},n)$
reaction is carried by the heavier particle. We measured the time-of-flight of
the backward proton, detected by the BPC and the BPD. We identified the
decaying $\Lambda$ in the invariant mass spectrum reconstructed from the
measured momenta of the $\pi^{-}$ and the proton [Fig. 3(c)]. Then, the
missing mass spectrum of $d(K^{-},n\Lambda)$ was obtained as shown in Fig.
3(d). The missing $\pi^{0}$, $\pi^{0}\gamma$, and background components (BG)
contributions were decomposed based on a Monte Carlo simulation, as indicated
in the figure, which were 12%, 70%, and 18%, respectively. Here, hyperon
($Y$)-production processes that associate with a backward proton
($K^{-}d\rightarrow p(Y\pi)^{-}$) and those induced by quasi-free backward
kaons that react with an another deuteron ($d^{\prime}$) in the deuterium
target ($K^{-}d\rightarrow\bar{K}X,\bar{K}d^{\prime}\rightarrow YX^{\prime}$)
are taken into accout as the BG components. By gating the mass window for 0.18
to 0.3 GeV/$c^{2}$ in the spectrum, we obtained the $\pi^{0}\Sigma^{0}$ mode
with only a small amount of contamination from the $\pi^{0}\Lambda$ mode and
background components, which were reduced to be 1.0% and 3.9%, respectively.
The contribution of the contamination is subtracted in the present
$\pi^{0}\Sigma^{0}$ missing mass spectrum.

The $\pi^{-}\Sigma^{0}$ mode was identified by selecting the $\Sigma^{0}$ and
$p\gamma$ mass regions in a scatter plot of the two possible
$d(K^{-},p\pi^{-})$ missing masses and the $d(K^{-},p\pi^{-}\pi^{-})$ missing
mass spectrum, as shown in Fig. 3(e) and (f), respectively. The missing
$p\gamma$ mass distribution is isolated since $\Sigma^{0}$ is moving slowly.

##  3 $\pi\Sigma$ mass spectra

The mass spectra of $\pi^{\pm}\Sigma^{\mp}$, $\pi^{0}\Sigma^{0}$, and
$\pi^{-}\Sigma^{0}$ were obtained, as shown in Fig. 4. Errors in the vertical
axes include statistical errors, scaling factor errors, and systematic
uncertainties. The scaling factor errors arise from uncertainties in
corrections of the target thickness and beam intensity, the efficiencies of
the data acquisition system, detectors, and event selections in the analysis
codes, and the geometrical acceptances and efficienceies of the relevant
particle detectors. Geometrical acceptances of the detector setup for
$\pi^{\pm}$, $\pi^{-}p$, and $2\pi^{-}$ from $\pi^{\pm}\Sigma^{\mp}$,
$\pi^{0}\Sigma^{0}$, and $\pi^{-}\Sigma^{0}$, respectively, are shown in Fig.
4(d), which were evaluated by Monte Carlo simulations in conditions that
knocked-out neutron and proton are detected at the NC and the PC,
respectively. The scaling factor errors relative to the obtained cross
sections for $\pi^{\pm}\Sigma^{\mp}$, $\pi^{0}\Sigma^{0}$, and
$\pi^{-}\Sigma^{0}$ are estimated to be 5.8%, 6.2%, and 3.5%, respectively.
For the $\pi^{\pm}\Sigma^{\mp}$ spectra, the fitting errors to separate the
two modes as described in Fig. 3(a) and (b) are also taken into account, which
are dominant sources of systematic uncertainties in estimations of the cross
sections. The fitting errors relative to the cross sections are typically 7%
and 6% at the $K^{-}p$/$K^{0}n$ mass thresholds for the
$\pi^{\pm}\Sigma^{\mp}$ modes, respectively. In the case of the
$\pi^{0}\Sigma^{0}$ mode, the fitting error to decompose the $\pi^{0}\Lambda$
mode and the other background mentioned in the previous section is dependent
on the missing mass. It is typically 1.5% at the $\bar{K}N$ mass threshold. On
the other hand it is 13% at around 1475 MeV/$c^{2}$, where contamination of
the BG conponents is maximum. The statistical and total errors are shown
separately as inner and outer bars in Fig. 4(a) and (b), while only the total
errors are shown in Fig. 4(c).

Figure 4: Measured spectra of (a) $\pi^{\pm}\Sigma^{\mp}$, (b)
$\pi^{0}\Sigma^{0}$ and $\pi^{-}\Sigma^{0}$, and (c) $\pi^{0}\Sigma^{0}$ and
$(\pi^{+}\Sigma^{-}+\pi^{-}\Sigma^{+}-\pi^{-}\Sigma^{0})/2$. (d) Acceptances
of the detectors for $\pi^{\pm}$, $\pi^{-}p$, and $2\pi^{-}$ from
$\pi^{\pm}\Sigma^{\mp}$, $\pi^{0}\Sigma^{0}$, and $\pi^{-}\Sigma^{0}$,
respectively. Statistical and total errors are shown separately as inner and
outer bars in (a) and (b), while only total errors are shown in (c). The
vertical thin lines shows the $K^{-}p$ and $K^{0}n$ mass thresholds.

We observed different line shapes in the $\pi^{\pm}\Sigma^{\mp}$ modes [Fig.
4(a)]. Since both the $I$ = 0 and 1 amplitudes contribute to the modes, the
difference is due to interference between the two amplitudes. In the
$\pi^{-}\Sigma^{+}$ mode, we find a bump around 1450 MeV/$c^{2}$ with a small
shoulder below the $K^{-}p$ mass threshold. On the other hand, the
$\pi^{+}\Sigma^{-}$ spectrum shows a broad distribution with a maximum
strength just below the $K^{-}p$ mass threshold. The $\pi^{0}\Sigma^{0}$ and
$\pi^{-}\Sigma^{0}$ modes [Fig. 4(b)] contain only the $I$ = 0 and 1
amplitudes, respectively. The strength of the $\pi^{-}\Sigma^{0}$ spectrum is
smaller than that of the $\pi^{0}\Sigma^{0}$ spectrum. We find that the $I$ =
0 amplitude is dominant, particularly below the $\bar{K}N$ mass threshold. We
find no structure at around 1385 MeV/$c^{2}$ in the $\pi^{-}\Sigma^{0}$, where
we might expect a structure of the $\Sigma^{*}(1385)$ resonance. This fact
suggests dominance of $S$-wave $\pi\Sigma$ production in the present
reactions, since $\Sigma^{*}(1385)$ decays into a $P$-wave $\pi\Sigma$ state.

The $\pi\Sigma$ production cross sections can be described with $T_{1}^{I}$
and $T_{2}^{I^{\prime}}$ as follows;

$\displaystyle\frac{d\sigma}{d\Omega}(\pi^{\pm}\Sigma^{\mp})\propto\left|C_{1}^{0}T_{2}^{I^{\prime}=0}\mp{C_{1}^{1}T_{2}^{I^{\prime}=1}}\right|^{2},$
(1)
$\displaystyle\frac{d\sigma}{d\Omega}(\pi^{-}\Sigma^{0})\propto\left|C_{1}^{1}T_{2}^{I^{\prime}=1}\right|^{2},$
(2)
$\displaystyle\frac{d\sigma}{d\Omega}(\pi^{0}\Sigma^{0})\propto\left|C_{1}^{0}T_{2}^{I^{\prime}=0}\right|^{2},$
(3) $\displaystyle C_{1}^{0}=\frac{3T_{1}^{I=0}-T_{1}^{I=1}}{4\sqrt{3}},\ \
C_{1}^{1}=\frac{T_{1}^{I=1}+T_{1}^{I=1}}{4}.$ (4)

Here, $T_{1}^{I}$ and $T_{2}^{I^{\prime}}$ represent the scattering amplitude
of the first-step and second-step two-body $K^{-}N_{1}\rightarrow\bar{K}N$ and
$\bar{K}N_{2}\rightarrow\pi\Sigma$ reactions with isospin $I$ and
$I^{\prime}$, respectively. The coefficients are determined by the sums of the
products of the Clebsch–Gordan coefficients in terms of the isospin in the
possible processes in the two-step reaction, as described as follows:

$\displaystyle\sum_{m_{X},I,m,I^{\prime},m^{\prime}}{\langle\frac{1}{2}m_{N_{1}}\frac{1}{2}m_{N_{2}}|00\rangle\langle\frac{1}{2}m_{\bar{K}}\frac{1}{2}m_{N}|Im\rangle}$
$\displaystyle\times\langle\frac{1}{2}m_{K^{-}}\frac{1}{2}m_{N_{1}}|Im\rangle{T_{1}^{I}}$
$\displaystyle\times\langle\frac{1}{2}m_{\pi}\frac{1}{2}m_{\Sigma}|I^{\prime}{m^{\prime}}\rangle\langle\frac{1}{2}m_{\bar{K}}\frac{1}{2}m_{N_{2}}|I^{\prime}{m^{\prime}}\rangle{T_{2}^{I^{\prime}}},$
(5)

where $m_{X=N_{1},N_{2},\bar{K},N,K^{-},\pi,\Sigma}$ is a $z$-component of the
isospin of a relevant particle $X$. Then, one finds a relation among the four
reaction cross sections as

$\displaystyle\frac{1}{2}\frac{d\sigma}{d\Omega}(\pi^{+}\Sigma^{-}+\pi^{-}\Sigma^{+}-\pi^{-}\Sigma^{0})=\frac{d\sigma}{d\Omega}(\pi^{0}\Sigma^{0}).$
(6)

We confirmed the relationship, as demonstrated in Fig. 4(c).

##  4 Discussion

Several authors have discussed $\pi\Sigma$ production associated with nucleon
emission in kaon induced reactions on deuterons [38, 39, 40, 41, 42], and
hence we describe the $\pi\Sigma$ spectral shape assuming that the two-step
reaction is dominant when the knocked-out nucleon is emitted at a very forward
angle. We neglect the direct production of $\pi\Sigma$ by collisions of
incident $K^{-}$ with nucleons in deuteron as its contribution is negligibly
small at the very forward angle of knocked-out neutron. Then, the $\pi\Sigma$
production cross section can be described as

$\displaystyle\frac{d^{2}\sigma}{dM_{\pi\Sigma}d\Omega_{n}}\sim\left|\langle{n\pi\Sigma}|T_{2}G_{0}(\bar{K},N_{2})T_{1}|K^{-}\Phi_{d}\rangle\right|^{2},$
(7) $\displaystyle T_{2}=T_{2}^{I^{\prime}}(\bar{K}N_{2},\pi\Sigma),$ (8)
$\displaystyle T_{1}=T_{1}^{I}(K^{-}N_{1},\bar{K}N),$ (9)

where $|K^{-}\Phi_{d}\rangle$ and $|n\pi\Sigma\rangle$ denote the initial
$K^{-}$ and deuteron and final $n\pi\Sigma$ wave functions, respectively.
$G_{0}(\bar{K},N_{2})$ is the Green’s function which describes the
intermediate $\bar{K}$ propagation between the two vertices. More detailed
expressions can be found in Refs. [40, 38, 42]. The cross section can be
simplified by a factorization approximation, as follows:

$\displaystyle\frac{d^{2}\sigma}{dM_{\pi\Sigma}d\Omega_{n}}\approx\left|T_{2}^{I^{\prime}}\right|^{2}F_{\rm
res}(M_{\pi\Sigma}),$ (10) $\displaystyle F_{\rm
res}(M_{\pi\Sigma})=\left|\int{G_{0}T_{1}^{I}\Phi_{d}(q_{N_{2}})d^{3}q_{N_{2}}}\right|^{2}.$
(11)

Here, $q_{N_{2}}$ is the momentum of the residual nucleon. In this way, the
$\pi\Sigma$ spectrum can be decomposed into $T_{2}^{I^{\prime}}$ and the
response function $F_{\rm res}$. Using the $K^{-}N\rightarrow\bar{K}N$
scattering amplitudes based on a partial wave analysis [43] and the deuteron
wave function $\Phi_{d}$ [44], we evaluate $F_{\rm res}$ as a function of the
$\pi\Sigma$ mass $M_{\pi\Sigma}$, as shown by the dashed line in Fig. 5(b).
Here, we took 3 degrees as a typical scattering angle of the knocked-out
nucleon in the laboratory frame. The line shapes of the $\pi\Sigma$ mass
spectra above the $\bar{K}N$ mass threshold are characterized by $F_{\rm
res}$, the distribution of which reflects the Fermi motion of a nucleon in the
dueteron. For $S$-wave $T_{2}^{I^{\prime}}$, we consider the
$\bar{K}N$-$\pi\Sigma$ coupled channel $T$ matrix. The diagonal and off-
diagonal matrix elements can be parametrized similarly to the case in Ref.
[48] as

$\displaystyle
T_{2}^{I^{\prime}}(\bar{K}N,\bar{K}N)=\frac{A^{I^{\prime}}}{1-iA^{I^{\prime}}k_{2}+\frac{1}{2}A^{I^{\prime}}R^{I^{\prime}}k_{2}^{2}},$
(12) $\displaystyle
T_{2}^{I^{\prime}}(\bar{K}N,\pi\Sigma)=\frac{e^{i\delta^{I^{\prime}}}}{\sqrt{k_{1}}}\frac{\sqrt{{\rm
Im}A^{I^{\prime}}-\frac{1}{2}|A^{I^{\prime}}|^{2}{\rm
Im}R^{I^{\prime}}k_{2}^{2}}}{1-iA^{I^{\prime}}k_{2}+\frac{1}{2}A^{I^{\prime}}R^{I^{\prime}}k_{2}^{2}},$
(13)

where $A^{I^{\prime}}$, $R^{I^{\prime}}$, and $\delta^{I^{\prime}}$ are the
complex scattering length, complex effective range, and real phase,
respectively. $k_{1}$ and $k_{2}$ are respectively the momenta of $\pi$ and
$\bar{K}$ in the center of mass frame. Here, $k_{2}$ becomes a pure imaginary
number below the $\bar{K}N$ mass threshold, to satisfy analytic continuity.

Figure 5: (a) Experimental resolution as a function of the $\pi\Sigma$ mass.
(b) Calculated $\pi\Sigma$ spectrum to fit the measured spectra in the $I$ = 0
channel. The solid thick and thin lines are the spectrum with and without the
resolution function convoluted, respectively. The response function $F_{\rm
res}$ is shown as a dashed line in arbitrary units. (c) Deduced scattering
amplitude of $\bar{K}N\rightarrow\bar{K}N$ in the $I$ = 0 channel. The real
and imaginary parts are shown as solid and dashed lines, respectively. The
vertical thin lines show the $K^{-}p$ and $K^{0}n$ mass thresholds.

We demonstrate the fitting result for the $\pi\Sigma$ ($I$ = 0) channel, as
shown in Fig. 5(b). $A^{0}$ and $R^{0}$ are determined to fit the measured
$\pi^{0}\Sigma^{0}$ and
$(\pi^{+}\Sigma^{-}+\pi^{-}\Sigma^{+}-\pi^{-}\Sigma^{0})/2$ spectra,
simultaneously. We took the $\bar{K}N$ mass threshold at the average of
$K^{-}p$ and $K^{0}n$ since the differential cross sections of
$K^{-}n\rightarrow K^{-}n$ [45] and $K^{-}p\rightarrow K^{0}n$ [46] are almost
equal at a neutron forward angle at an incident kaon momentum of $\sim$1
GeV/$c$. However, we took into account the differences from the fitting
results for the cases of the $K^{-}p$ and $K^{0}n$ mass thresholds as
systematic errors. In the present fitting, $\delta^{I^{\prime}}$ could not be
determined since it deos not appear explicitly in the fitting function that
depends on $|T_{2}^{I^{\prime}}(\bar{K}N,\pi\Sigma)|^{2}$. In the fitting, the
experimental resolution function [Fig. 5(a)] was convoluted with the
calculated spectrum and the vertical scale is arbitrarily adjusted. We
obtained $A^{0}=[-1.12\pm 0.11(\mbox{fit})^{+0.10}_{-0.07}(\mbox{syst.})]$ \+
$[0.84\pm 0.12(\mbox{fit})^{+0.08}_{-0.07}(\mbox{syst.})]i$ fm,
$R^{0}=[-0.18\pm 0.31\mbox{(fit)}^{+0.08}_{-0.06}(\mbox{syst.})]$ \+
$[-0.40\pm 0.13\mbox{(fit)}\pm 0.09(\mbox{syst.})]i$ fm, where the fitting
errors are indicated as “(fit)”. As mentioned above, the differences of the
different $\bar{K}N$ mass threshold were taken into account as systematic
errors indicated as “(syst.)”. The reduced chi-square was 1.76 with 24 degrees
of freedom. The present scattering length is smaller than a recent theoretical
calculation, $-1.77+1.08i$, which is based on the lattice QCD [47]. The thick
and thin solid lines in Fig. 5(b) show the resolution-convoluted and no-
resolution-convoluted spectra, respectively, calculated with the best fit
values. The energy dependence of the deduced $T_{2}^{0}(\bar{K}N,\bar{K}N)$ is
shown in Fig. 5(c). We find a zero-crossing in the real part and a bump in the
imaginary part at the same place. This is a typical structure of a resonance.
We find a resonance pole at
$1417.7^{+6.0}_{-7.4}(\mbox{fit})^{+1.1}_{-1.0}(\mbox{syst.})$ \+
$[-26.1^{+6.0}_{-7.9}(\mbox{fit})^{+1.7}_{-2.0}(\mbox{syst.})]i$ MeV/$c^{2}$
in the $I$ = 0 channel of the $\bar{K}N\rightarrow\bar{K}N$ scattering. The
errors are estimated by fluctuations of the pole position due to the errors
for the best fit values of $A^{0}$ and $R^{0}$. The real part of the deduced
pole is closer to the $K^{-}p$ mass threshold than the so-called PDG value of
1405.1 MeV/$c^{2}$. It is worthy of evaluating the following quantity,
$|T_{2}^{0}(\bar{K}N,\bar{K}N)|^{2}/|T_{2}^{0}(\bar{K}N,\pi\Sigma)|^{2}\sim
2.2^{+1.0}_{-0.6}$(fit)$\pm 0.3$(syst.) at the pole energy, which corresponds
to the ratio of the two partial widths in the Flatté formula [49, 50]. This
suggests that the coupling of $\Lambda(1405)$ to $\bar{K}N$ is predominant,
which does not contradict a picture of $\Lambda(1405)$ as a $\bar{K}N$-bound
state. Meißner and Hyodo have reviewed and discussed the pole structure of the
$\Lambda(1405)$ region based on chiral unitary approaches with a constraint on
the scattering length obtained from kaonic hydrogen atom $X$-ray data by the
SIDDHARTA collaboration [51, 52][53]. They collected four sets of two poles
deduced by several authors in the relevant region. Poles 1 and 2 are the so-
called higher and lower poles, respectively, which are thought to be coupled
to $\bar{K}N$ and $\pi\Sigma$, respectively. The suggested higher poles are
located at the region of 1421–1434 MeV on the real axis and 10–26 MeV on the
imaginary axis in the complex energy plane. The pole position determined by
the present experiment is consistent to the higher poles though it is located
at slightly smaller and larger values for the real and imaginary parts,
respectively. A lattice QCD calculation has reported two poles and the so-
called higher pole is located at $1430-22i$ MeV/$c^{2}$ [54]. Our result is
smaller and similar in real and imaginary part, respectively. Recently,
Anisovich $et\ al.$ reported one single pole of $\Lambda(1405)$ contribution
to fit the data of $\gamma$ and $K^{-}$ induced reactions on proton and the
kaonic hydrogen atom, as $1422\pm 3-(21\pm 3)i$ MeV/$c^{2}$ [55]. The present
result is consistent with the reported pole position.

##  5 Conclusion

We measured $\pi^{\pm}\Sigma^{\mp}$, $\pi^{0}\Sigma^{0}$, and
$\pi^{-}\Sigma^{0}$ mass spectra below and above the $\bar{K}N$ mass threshold
in $d(K^{-},N)\pi\Sigma$ reactions at a forward angle, of $N$ knocked out by
an incident kaon momentum of 1 GeV/$c$. We obtained decomposed $\pi\Sigma$
spectra in terms of $I$ = 0 and 1, and confirmed a relation between the four
reactions with respect to the isospin states. We find that the $I$ = 0
amplitude is dominant. We demonstrated that the $\pi\Sigma$ spectral shape in
the $I$ = 0 channel is well reproduced by the two-step reaction of a neutron
knocked out at a forward angle by an incident negative kaon and a recoiled
$\bar{K}$ reacting with a residual nucleon in deuteron to produce $\pi\Sigma$
in the $I$ = 0 state. We deduced the two-body $\bar{K}N$ scattering amplitude
in the $I$ = 0 channel around the $\bar{K}N$ mass threshold, from which we
find a resonance pole at
$1417.7^{+6.0}_{-7.4}(\mbox{fit})^{+1.1}_{-1.0}(\mbox{syst.})$ \+
$[-26.1^{+6.0}_{-7.9}(\mbox{fit})^{+1.7}_{-2.0}(\mbox{syst.})]i$ MeV/$c^{2}$.
The present data provide fundamental information on the $\bar{K}N$ interaction
and kaonic nuclei [56, 57].

## Acknowledgements

The authors would like to express their thanks to the J-PARC PAC members and
the crews of the J-PARC accelerator and hadron facility group for their
encouragement, support, and stable delivery of beams for the E31 experiment.
We are grateful to Professor D. Jido, Dr. T. Sekihara, and Professor J.
Yamagata-Sekihara for their support since the planning stage of the E31
experiment. We are grateful to Professor K. Miyagawa and Dr. H. Kamano for
their contributions to the calculations of the $\pi\Sigma$ spectral shapes.
The present work was supported by MEXT Grants-in-Aid of Innovative Area No.
21105003, No. 18H05402, and a Grant-in-Aid of Scientific Research A No.
16H02188 and S No. 22H04940.

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