The Belle Collaboration

#
Measurement of the mass and width of the $\Lambda_{c}(2625)^{+}$ and the
branching ratios of $\Lambda_{c}(2625)^{+}\to\Sigma_{c}^{0}\pi^{+}$ and
$\Lambda_{c}(2625)^{+}\to\Sigma_{c}^{++}\pi^{-}$

D. Wang  0000-0003-1485-2143 J. Yelton  0000-0001-8840-3346 I. Adachi
0000-0003-2287-0173 J. K. Ahn  0000-0002-5795-2243 H. Aihara
0000-0002-1907-5964 D. M. Asner  0000-0002-1586-5790 H. Atmacan
0000-0003-2435-501X R. Ayad  0000-0003-3466-9290 V. Babu  0000-0003-0419-6912
Sw. Banerjee  0000-0001-8852-2409 M. Bauer  0000-0002-0953-7387 P. Behera
0000-0002-1527-2266 K. Belous  0000-0003-0014-2589 J. Bennett
0000-0002-5440-2668 M. Bessner  0000-0003-1776-0439 T. Bilka
0000-0003-1449-6986 D. Biswas  0000-0002-7543-3471 D. Bodrov
0000-0001-5279-4787 J. Borah  0000-0003-2990-1913 A. Bozek
0000-0002-5915-1319 M. Bračko  0000-0002-2495-0524 P. Branchini
0000-0002-2270-9673 T. E. Browder  0000-0001-7357-9007 A. Budano
0000-0002-0856-1131 M. Campajola  0000-0003-2518-7134 D. Červenkov
0000-0002-1865-741X M.-C. Chang  0000-0002-8650-6058 A. Chen
0000-0002-8544-9274 B. G. Cheon  0000-0002-8803-4429 H. E. Cho
0000-0002-7008-3759 K. Cho  0000-0003-1705-7399 S.-J. Cho  0000-0002-1673-5664
S.-K. Choi  0000-0003-2747-8277 Y. Choi  0000-0003-3499-7948 S. Choudhury
0000-0001-9841-0216 D. Cinabro  0000-0001-7347-6585 S. Das
0000-0001-6857-966X G. De Nardo  0000-0002-2047-9675 G. De Pietro
0000-0001-8442-107X R. Dhamija  0000-0001-7052-3163 F. Di Capua
0000-0001-9076-5936 Z. Doležal  0000-0002-5662-3675 T. V. Dong
0000-0003-3043-1939 D. Dossett  0000-0002-5670-5582 D. Epifanov
0000-0001-8656-2693 T. Ferber  0000-0002-6849-0427 D. Ferlewicz
0000-0002-4374-1234 B. G. Fulsom  0000-0002-5862-9739 R. Garg
0000-0002-7406-4707 V. Gaur  0000-0002-8880-6134 A. Giri  0000-0002-8895-0128
P. Goldenzweig  0000-0001-8785-847X E. Graziani  0000-0001-8602-5652 T. Gu
0000-0002-1470-6536 K. Gudkova  0000-0002-5858-3187 C. Hadjivasiliou
0000-0002-2234-0001 X. Han  0000-0003-1656-9413 K. Hayasaka
0000-0002-6347-433X H. Hayashii  0000-0002-5138-5903 M. T. Hedges
0000-0001-6504-1872 C.-L. Hsu  0000-0002-1641-430X K. Inami
0000-0003-2765-7072 N. Ipsita  0000-0002-2927-3366 A. Ishikawa
0000-0002-3561-5633 R. Itoh  0000-0003-1590-0266 M. Iwasaki
0000-0002-9402-7559 W. W. Jacobs  0000-0002-9996-6336 E.-J. Jang
0000-0002-1935-9887 S. Jia  0000-0001-8176-8545 Y. Jin  0000-0002-7323-0830 A.
B. Kaliyar  0000-0002-2211-619X K. H. Kang  0000-0002-6816-0751 T. Kawasaki
0000-0002-4089-5238 C. Kiesling  0000-0002-2209-535X C. H. Kim
0000-0002-5743-7698 D. Y. Kim  0000-0001-8125-9070 K.-H. Kim
0000-0002-4659-1112 Y.-K. Kim  0000-0002-9695-8103 K. Kinoshita
0000-0001-7175-4182 P. Kodyš  0000-0002-8644-2349 A. Korobov
0000-0001-5959-8172 S. Korpar  0000-0003-0971-0968 E. Kovalenko
0000-0001-8084-1931 P. Križan  0000-0002-4967-7675 P. Krokovny
0000-0002-1236-4667 M. Kumar  0000-0002-6627-9708 R. Kumar
0000-0002-6277-2626 K. Kumara  0000-0003-1572-5365 Y.-J. Kwon
0000-0001-9448-5691 T. Lam  0000-0001-9128-6806 J. S. Lange
0000-0003-0234-0474 S. C. Lee  0000-0002-9835-1006 P. Lewis
0000-0002-5991-622X L. K. Li  0000-0002-7366-1307 Y. Li  0000-0002-4413-6247
Y. B. Li  0000-0002-9909-2851 L. Li Gioi  0000-0003-2024-5649 J. Libby
0000-0002-1219-3247 K. Lieret  0000-0003-2792-7511 Y.-R. Lin
0000-0003-0864-6693 D. Liventsev  0000-0003-3416-0056 T. Matsuda
0000-0003-4673-570X D. Matvienko  0000-0002-2698-5448 F. Meier
0000-0002-6088-0412 M. Merola  0000-0002-7082-8108 F. Metzner
0000-0002-0128-264X K. Miyabayashi  0000-0003-4352-734X R. Mizuk
0000-0002-2209-6969 G. B. Mohanty  0000-0001-6850-7666 R. Mussa
0000-0002-0294-9071 I. Nakamura  0000-0002-7640-5456 T. Nakano
0000-0003-3157-5328 M. Nakao  0000-0001-8424-7075 Z. Natkaniec
0000-0003-0486-9291 A. Natochii  0000-0002-1076-814X L. Nayak
0000-0002-7739-914X M. Nayak  0000-0002-2572-4692 N. K. Nisar
0000-0001-9562-1253 S. Nishida  0000-0001-6373-2346 S. Ogawa
0000-0002-7310-5079 H. Ono  0000-0003-4486-0064 P. Oskin  0000-0002-7524-0936
G. Pakhlova  0000-0001-7518-3022 S. Pardi  0000-0001-7994-0537 H. Park
0000-0001-6087-2052 J. Park  0000-0001-6520-0028 S. Patra  0000-0002-4114-1091
S. Paul  0000-0002-8813-0437 T. K. Pedlar  0000-0001-9839-7373 R. Pestotnik
0000-0003-1804-9470 L. E. Piilonen  0000-0001-6836-0748 T. Podobnik
0000-0002-6131-819X E. Prencipe  0000-0002-9465-2493 M. T. Prim
0000-0002-1407-7450 N. Rout  0000-0002-4310-3638 G. Russo  0000-0001-5823-4393
S. Sandilya  0000-0002-4199-4369 A. Sangal  0000-0001-5853-349X L. Santelj
0000-0003-3904-2956 V. Savinov  0000-0002-9184-2830 G. Schnell
0000-0002-7336-3246 J. Schueler  0000-0002-2722-6953 C. Schwanda
0000-0003-4844-5028 Y. Seino  0000-0002-8378-4255 K. Senyo
0000-0002-1615-9118 M. E. Sevior  0000-0002-4824-101X W. Shan
0000-0003-2811-2218 M. Shapkin  0000-0002-4098-9592 C. Sharma
0000-0002-1312-0429 C. P. Shen  0000-0002-9012-4618 J.-G. Shiu
0000-0002-8478-5639 A. Sokolov  0000-0002-9420-0091 E. Solovieva
0000-0002-5735-4059 M. Starič  0000-0001-8751-5944 M. Sumihama
0000-0002-8954-0585 T. Sumiyoshi  0000-0002-0486-3896 W. Sutcliffe
0000-0002-9795-3582 M. Takizawa  0000-0001-8225-3973 U. Tamponi
0000-0001-6651-0706 K. Tanida  0000-0002-8255-3746 F. Tenchini
0000-0003-3469-9377 M. Uchida  0000-0003-4904-6168 S. Uno  0000-0002-3401-0480
R. van Tonder  0000-0002-7448-4816 G. Varner  0000-0002-0302-8151 K. E.
Varvell  0000-0003-1017-1295 A. Vinokurova  0000-0003-4220-8056 M.-Z. Wang
0000-0002-0979-8341 X. L. Wang  0000-0001-5805-1255 M. Watanabe
0000-0001-6917-6694 S. Watanuki  0000-0002-5241-6628 O. Werbycka
0000-0002-0614-8773 E. Won  0000-0002-4245-7442 X. Xu  0000-0001-5096-1182 B.
D. Yabsley  0000-0002-2680-0474 W. Yan  0000-0003-0713-0871 S. B. Yang
0000-0002-9543-7971 J. H. Yin  0000-0002-1479-9349 C. Z. Yuan
0000-0002-1652-6686 L. Yuan  0000-0002-6719-5397 Z. P. Zhang
0000-0001-6140-2044 V. Zhilich  0000-0002-0907-5565 V. Zhukova
0000-0002-8253-641X

###### Abstract

Using the entire data sample of $980\,\textrm{fb}^{-1}$ collected with the
Belle detector operating at the KEKB asymmetric-energy collider, we report the
measurement of the mass, width, and the relative branching ratios of the
$\Lambda_{c}(2625)^{+}$ charmed baryon. The mass difference between
$\Lambda_{c}(2625)^{+}$ and $\Lambda_{c}^{+}$ is measured to be
$M(\Lambda_{c}(2625)^{+})-M(\Lambda_{c}^{+})=341.518\pm 0.006\pm 0.049\
\textrm{MeV}/\textit{c}^{2}$. The upper limit on the width is measured to be
$\Gamma(\Lambda_{c}(2625)^{+})<0.52\,\textrm{MeV}/\textit{c}^{2}$ at 90%
credibility level. Based on a full Dalitz plot fit, branching ratios with
respect to the mode $\Lambda_{c}(2625)^{+}\to\Lambda_{c}^{+}\pi^{+}\pi^{-}$
are measured to be
$\frac{\mathcal{B}(\Lambda_{c}(2625)^{+}\to\Sigma_{c}^{0}\pi^{+})}{\mathcal{B}(\Lambda_{c}(2625)^{+}\to\Lambda_{c}^{+}\pi^{+}\pi^{-})}=(5.19\pm
0.23\pm 0.40)\%$ and
$\frac{\mathcal{B}(\Lambda_{c}(2625)^{+}\to\Sigma_{c}^{++}\pi^{-})}{\mathcal{B}(\Lambda_{c}(2625)^{+}\to\Lambda_{c}^{+}\pi^{+}\pi^{-})}=(5.13\pm
0.26\pm 0.32)\%$. These measurements can be used to further constrain the
parameters of the underlying theoretical models.

##  I Introduction

The $\Lambda_{c}^{+}$ charmed baryons consist of a heavy charm quark and two
light ($ud$) quarks with the ground state having quantum numbers
$J^{P}=\frac{1}{2}^{+}$. The $\Lambda_{c}(2595)^{+}$ and
$\Lambda_{c}(2625)^{+}$ are the two lowest-lying excited states observed, and
are generally believed to have $J^{P}=\frac{1}{2}^{-}$ and
$J^{P}=\frac{3}{2}^{-}$, respectively. The $\Lambda_{c}(2595)^{+}$
predominantly decays to the $J^{P}=\frac{1}{2}^{+}$ $\Sigma_{c}(2455)^{++/0}$
states via an $s$-wave decay. The analogous decay for the
$\Lambda_{c}(2625)^{+}$ to the $J^{P}=\frac{3}{2}^{-}$
$\Sigma_{c}(2520)^{++/0}$ states is kinematically suppressed as it can only
happen through the low-mass tail of the $\Sigma_{c}(2520)^{++/0}$. The
$d$-wave decay to the $J^{P}=\frac{1}{2}^{+}$ $\Sigma_{c}(2455)^{++/0}$ states
is allowed, but its contribution is known to be small. Thus, the
$\Lambda_{c}(2625)^{+}$ decay is thought to proceed primarily via the direct
three-body, $p$-wave decay
$\Lambda_{c}(2625)^{+}\to\Lambda_{c}^{+}\pi^{+}\pi^{-}$.

The $\Lambda_{c}(2625)^{+}$ was first observed in 1993 [1]. The CDF
collaboration reported the most recent measurements of $\Lambda_{c}(2625)^{+}$
properties in 2011 using a data sample of 6.2k events [2]. Their measurement
for the $\Lambda_{c}(2625)^{+}$ mass with respect to the $\Lambda_{c}^{+}$
mass is much more precise compared with previous measurements, and an upper
limit on the $\Lambda_{c}(2625)^{+}$ width was reported. The limited decay
phase space of $\Lambda_{c}(2625)^{+}\to\Lambda_{c}^{+}\pi^{+}\pi^{-}$ makes
it difficult to extract the $\Sigma_{c}(2455)^{++/0}$ yields by fitting the
$\Lambda_{c}^{+}\pi^{\pm}$ invariant mass due to the presence of reflection
peaks formed by the combination of the $\Lambda_{c}^{+}$ and the other final
state pion. The large data sample collected by Belle, together with the use of
an amplitude model [3] to describe the decay, allows us to use a full Dalitz
fit that naturally includes the reflections.

The mass of the $\Lambda_{c}(2625)^{+}$, relative to the $\Lambda_{c}^{+}$
mass, is already relatively well known, but the large Belle data sample allows
for a more precise measurement. No intrinsic width of the
$\Lambda_{c}(2625)^{+}$ has yet been measured, and the current upper limit
$\Gamma<0.97\ \textrm{MeV}/\textit{c}^{2}$ at 90% credibility level by the
Particle Data Group (PDG) [4] is based on the CDF measurement.

Theoretical predictions for the width vary for this narrow state [3, 5, 6, 7].
An improved limit on the width of the $\Lambda_{c}(2625)^{+}$ will help to
constrain these predictions, and provide insights into other charmed baryons
since their widths are related through common coupling constants [8].

##  II Detector and dataset

The measurement presented here is based on the entire dataset collected by the
Belle detector [9, 10] operating at the KEKB asymmetric-energy $e^{+}e^{-}$
collider [11, 12]. The total integrated luminosity of the dataset is $980\
\textrm{fb}^{-1}$, which is mostly collected at or near the $\Upsilon(4S)$
resonance.

The Belle detector is a large-solid-angle magnetic spectrometer that consists
of a silicon vertex detector (SVD), a 50-layer central drift chamber (CDC), an
array of aerogel threshold Cherenkov counters (ACC), a barrel-like arrangement
of time-of-flight scintillation counters (TOF), and an electromagnetic
calorimeter comprised of CsI(Tl) crystals (ECL) located inside a super-
conducting solenoid coil that provides a 1.5 T magnetic field. An iron flux-
return located outside of the coil is instrumented to detect $K_{L}^{0}$
mesons and to identify muons (KLM). The detector is described in detail
elsewhere [9]. Two inner detector configurations were used. The first
consisted of a 2.0 cm radius beampipe and a 3-layer silicon vertex detector,
while the second used a 1.5 cm radius beampipe, a 4-layer silicon detector and
a small-cell inner drift chamber.

Monte Carlo (MC) events are generated using EVTGEN [13] to optimize selection
criteria and to be used in the Dalitz plot fit. The
$\Lambda_{c}(2625)^{+}\to\Lambda_{c}^{+}\pi^{+}\pi^{-}$ and
$\Lambda_{c}^{+}\to pK^{-}\pi^{+}$ samples are generated using a phase space
model. A $D^{*+}\to D^{0}\pi^{+}$, with $D^{0}\to K^{-}\pi^{+}$, sample is
also generated to compare the mass-resolution function in the MC sample and
the experimental data, and thus to estimate the systematic uncertainties on
the measurements. The detector response is simulated with GEANT3 [14] and the
event reconstruction is performed using data converted with the Belle-to-
Belle-II (B2BII) software package [15] and then analyzed using Belle II
software [16, 17].

##  III Analysis

The candidate $\Lambda_{c}(2625)^{+}$ baryons are reconstructed from the decay
chain $\Lambda_{c}(2625)^{+}\to\Lambda_{c}^{+}\pi^{+}\pi^{-}$,
$\Lambda_{c}^{+}\to pK^{-}\pi^{+}$ [18]. The final-state charged particles,
$\pi^{\pm}$, $K^{-}$ and $p$, are selected based on the likelihood information
from the tracking (SVD, CDC) and particle identification (CDC, ACC, TOF)
systems into a combined likelihood,
$\mathcal{L}(h_{1}{:}h_{2})=\mathcal{L}(h_{1})/(\mathcal{L}(h_{1})+\mathcal{L}(h_{2}))$,
where $h_{1}$ and $h_{2}$ are $p$, $K$ or $\pi$ [19]. We require the proton
candidates to have $\mathcal{L}(p{:}K)>0.6$ and $\mathcal{L}(p{:}\pi)>0.6$,
kaon candidates to have $\mathcal{L}(K{:}p)>0.6$ and
$\mathcal{L}(K{:}\pi)>0.6$, and pion candidates to have
$\mathcal{L}(\pi{:}K)>0.6$ and $\mathcal{L}(\pi{:}p)>0.6$. Electrons are
suppressed by requiring
$\mathcal{L}(e^{-})/(\mathcal{L}(e^{-})+\mathcal{L}(\mathrm{hadrons}))<0.1$
for all candidates; the likelihoods $\mathcal{L}(e^{-})$ and
$\mathcal{L}(\mathrm{hadrons})$ include information from the ECL in addition
to the tracking and particle identification systems [19]. The particle
identification efficiency is approximately 87% for proton, 85% for kaons and
96% for pions. Charged tracks are also required to have a point of closest
approach with respect to the interaction point less than 3 cm in the $e^{+}$
beam direction and less than 1 cm in the plane perpendicular to it.

A vertex fit is applied to the daughter particles of the $\Lambda_{c}^{+}$
candidates and the resultant $\chi^{2}$ probability of the fit is required to
be greater than 0.001. Candidates within $\pm 7.0\
\textrm{MeV}/\textit{c}^{2}\ (\approx 1.6\sigma)$ are selected and mass-
constrained to the $\Lambda_{c}^{+}$ PDG mass of
$2286.46\,\textrm{MeV}/\textit{c}^{2}$ [4]. Two pions of opposite charge are
then combined with the constrained $\Lambda_{c}^{+}$ candidate to form a
$\Lambda_{c}(2625)^{+}$ candidate. The $\Lambda_{c}(2625)^{+}$ daughters are
then kinematically fitted to come from a common vertex, with a constraint that
the vertex has to be within the beamspot since the $\Lambda_{c}(2625)^{+}$ is
short-lived. The $\chi^{2}$ probability of this fit is required to be greater
than 0.001 to ensure the quality of the fit. As excited charmed baryons
including the $\Lambda_{c}(2625)^{+}$ typically have a hard momentum
distribution, we only keep $\Lambda_{c}(2625)^{+}$ candidates with
$x_{p}>0.7$, where
$x_{p}=p^{*}/\sqrt{E_{\mathrm{beam}}^{2}/\mathit{c}^{2}-M^{2}\mathit{c}^{2}}$
and $p^{*}$ is momentum of the $\Lambda_{c}(2625)^{+}$ in the $e^{+}e^{-}$
center of mass frame. As the mass of the $\Lambda_{c}^{+}$ is constrained to
its PDG value, the reconstructed mass $M(\Lambda_{c}^{+}\pi^{+}\pi^{-})$ has
the resolution of the mass difference
$M(\Lambda_{c}(2625)^{+})-M(\Lambda_{c}^{+})$.

Correctly calibrating the momentum scale for low momentum pions is critical
for this analysis. We calibrate the momentum scale using copious
$K^{0}_{S}\to\pi^{+}\pi^{-}$ events in the experimental data. Low-momentum
tracks are iteratively calibrated as a function of the curvature, polar angle,
and momentum of each track in the laboratory frame by comparing the
reconstructed and world-average mass of the $K^{0}_{S}$ meson as a function of
the $K^{0}_{S}$ momentum. This correction has been used in a previous
$\Sigma_{c}^{++/0}$ study using Belle data [20]. Since the mass-resolution
function is crucial for the precise measurement of the $\Lambda_{c}(2625)^{+}$
mass and width, the MC tracks are smeared using the analysis software during
reconstruction, as otherwise the MC mass resolution is known to be better than
that of the experimental data. This track smearing affects the width of the
mass-resolution function but not its central value. The mass-resolution
function of the $\Lambda_{c}(2625)^{+}$ mass is parameterized as a sum of two
Gaussian functions with parameters fixed according to a signal MC sample with
both corrections as detailed above.

The consistency between the MC sample and the experimental data is checked by
comparing the mass resolution of $D^{*+}$ events, which have similar
kinematics to the events under study. The low-momentum track correction
ensures that the measured $D^{*+}$ mass in data and MC are independent of the
momentum of the soft pion [20]. The resolution of the $D^{*+}$ mass relative
to the $D^{0}$ mass in the experimental data is found by fitting the
$M(D^{0}\pi^{+})-M(D^{0})$ mass distribution in the experimental data with a
Breit-Wigner distribution convoluted with a double-Gaussian mass-resolution
function, where the width of the Breit-Wigner is fixed to the PDG value of
$83.4\ \textrm{keV}/\textit{c}^{2}$ [4]. In this study, without track
smearing, the mass resolution in the experimental data is measured to be 114%
of the value obtained from the MC sample. However, with track smearing, the
mass resolution in the experimental data is measured to be 86% of the value
obtained from the MC sample. In all other narrow signals studied, for instance
the $\Lambda_{c}^{+}$, the track smearing ensures that the MC and data agrees
reasonably. The track smearing has negligible effect on the mass measurement.
The results of these consistency checks are used in the estimation of the
systematic uncertainties described below.

The reconstructed $M(\Lambda_{c}^{+}\pi^{+}\pi^{-})$ mass distribution in the
experimental is fitted using RooFit [21]. Figure 1 shows the
$M(\Lambda_{c}^{+}\pi^{+}\pi^{-})$ mass distribution in the experimental data
overlaid with the fit result. The signal function is a Breit-Wigner
distribution convolved with a double-Gaussian mass-resolution function, and
the background function is a second-order Chebychev polynomial. The resolution
function for the invariant mass distribution is obtained from the MC sample,
without track smearing, and scaled by 114% in accordance with the $D^{*+}$
study. The solid line shows the overall fit and the dashed lines show the
individual signal and background components of the fit. The fitted mass is
$2628.025\pm 0.006\,\textrm{MeV}/\textit{c}^{2}$, independent of which version
of the mass-resolution function we use. If we use the track-smearing
correction without any rescaling, the fitted width is found to be zero, so we
have no definitive evidence of a non-zero width and will present only an upper
limit for the measurement of the intrinsic width of the
$\Lambda_{c}^{+}(2625)$. If we use the mass resolution without track smearing,
scaled by 114%, the fitted width is $0.490\pm
0.025\,\textrm{MeV}/\textit{c}^{2}$. With the mass-resolution function with
track smearing scaled by 86%, the fitted width is $0.293\pm
0.026\,\textrm{MeV}/\textit{c}^{2}$. These finite values for the fitted width
after scaling the mass resolution are only used to find the limit on the
intrinsic width including systematic uncertainties.

The fitted mass of $\Lambda_{c}(2625)^{+}$ in the signal MC sample is slightly
different from the generated value. Applying a bias correction, determined by
the mass shift observed in the signal MC sample, the mass of the
$\Lambda_{c}(2625)^{+}$ is measured to be $2627.978\pm
0.007\,\textrm{MeV}/\textit{c}^{2}$, where the uncertainty is statistical.

Two upper limits on the width are calculated based on the two fit methods
described in the previous section and the larger upper limit is reported as
the final answer. Using the mass-resolution function determined from MC scaled
by 114% without track smearing, the upper limit is determined to be

$\Gamma(\Lambda_{c}(2625)^{+})<0.52\,\textrm{MeV}/\textit{c}^{2}$ (1)

at 90% credibility level by integrating the likelihood function to find the
value for which the integral contains 90% of the total area. Using the mass-
resolution function scaled by 86% with track smearing would yield a tighter
upper limit. Therefore, we conservatively report the former as the upper limit
on the width of $\Lambda_{c}(2625)^{+}$.

Figure 1: Distribution of the invariant mass
$M(\Lambda_{c}^{+}\pi^{+}\pi^{-})$ where the $\Lambda_{c}^{+}$ mass is fixed
to the PDG value. The solid line shows the overall distribution and the dashed
lines show the individual signal and background components.

##  IV Dalitz plot fit

A Dalitz plot fit is made in order to determine the relative branching ratios
of $\Lambda_{c}(2625)^{+}$ with respect to the mode
$\Lambda_{c}(2625)^{+}\to\Lambda_{c}^{+}\pi^{+}\pi^{-}$. For the Dalitz plot
fit, only $\Lambda_{c}(2625)^{+}$ candidates within $\pm
2\,\textrm{MeV}/\textit{c}^{2}$ of the $\Lambda_{c}(2625)^{+}$ PDG mass are
accepted [4]. The invariant mass of the $\Lambda_{c}(2625)^{+}$ candidates is
then constrained to the $\Lambda_{c}(2625)^{+}$ PDG mass of
$2628.11\,\textrm{MeV}/\textit{c}^{2}$, and the four-vectors of the daughter
particles are updated accordingly. A fit is made to the Dalitz plot using an
amplitude model as presented by Arifi et al. [3] using the AmpTools software
package [22]. The $\Lambda_{c}(2625)^{+}$ signal distribution is calculated
from the squared amplitude with spin sum of final states and spin average of
the initial states

$\sum|\mathcal{T}_{1}+\mathcal{T}_{2}+\mathcal{T}_{3}+\mathcal{T}_{4}+\mathcal{T}_{5}|^{2}$
(2)

where $\mathcal{T}_{1}$ through $\mathcal{T}_{5}$ are the decay amplitudes
through the intermediate states $\Sigma_{c}^{0}$, $\Sigma_{c}(2520)^{0}$,
$\Sigma_{c}^{++}$, $\Sigma_{c}(2520)^{++}$, and the direct three-body decay,
respectively. Each amplitude is modeled as a Breit-Wigner function multiplied
by a form factor specific to each decay channel. A constant amplitude is used
to model the background $\Lambda_{c}^{+}\pi^{+}\pi^{-}$ combinations, which
are not decay products of $\Lambda_{c}(2625)^{+}$. The yield of each decay
channel is calculated using AmpTools by an integration of the individual
component over the Dalitz plot. The contribution of the three-body decay in
the signal model is different from the background phasespace decay in that the
former is not flat across the Dalitz plot. During the fit, the masses and
widths of these intermediate particles are constrained to their respective PDG
values to facilitate the convergence. The small variations of the detector
acceptance across the Dalitz plot are taken into account by using the output
of a phasespace MC sample passed through the GEANT3 detector simulation as
input to the AmpTools fitting package.

Figure 2 shows the Dalitz plots for candidates in the signal region. On the
left subplot, the contributions from $\Sigma_{c}^{++}$ and the reflection from
$\Sigma_{c}^{0}$ constitute the two horizontal stripes. The upper and lower
parts of the Dalitz plot show slight excesses due to $\Sigma_{c}(2520)^{++/0}$
decays. There is also a clear excess on the left side of the Dalitz plot
compared to the right in agreement with the three-body decay taking into
account the spin, as predicted in the amplitude model [3]. On the right
subplot, the horizontal and vertical stripes indicate the $\Sigma_{c}^{++}$
and $\Sigma_{c}^{0}$ decays respectively. It is straightforward to see the
origin of the reflection peaks on the $M(\Lambda_{c}^{+}\pi^{+})$ mass
projection from this 2D Dalitz plot.

Figure 3 shows the projections of the fitted results with each component
labeled on the plot. The $\Sigma_{c}^{++}$ peak and the reflection peak from
$\Sigma_{c}^{0}$ are evident on the $M(\Lambda_{c}^{+}\pi^{+})$ mass
projection. The shoulders on the left and right side of the mass region are
mostly formed by the decays from the off-shell $\Sigma_{c}(2520)^{++/0}$. The
three-body $p$-wave decay in the signal model shows up in the
$M(\pi^{+}\pi^{-})$ mass projection as an assymetric distribution, in contrast
to the symmetric distribution from the background phasespace decay. The
$\Lambda_{c}(2625)^{+}$ yield in the signal region is
$N_{\mathrm{sig}}(\Lambda_{c}(2625)^{+}))=30319\pm 371$. The $\Sigma_{c}^{0}$
yield is $N_{\mathrm{sig}}(\Sigma_{c}^{0})=1964\pm 66$ and the
$\Sigma_{c}^{++}$ yield is $N_{\mathrm{sig}}(\Sigma_{c}^{++})=2022\pm 76$.

Figure 2: Dalitz plot for $\Lambda_{c}(2625)^{+}$ candidates in the signal
region. Explanations of the patterns in the text.

Figure 3: Dalitz plot fit result plotted as projections. Solid lines shows the
overall fitted distribution and its individual components as indicated in the
legend. More explanations in the text.

To account for $\Sigma_{c}(2455)^{++/0}$ candidates that are not decay
products of the $\Lambda_{c}(2625)^{+}$, the $\Sigma_{c}(2455)^{++/0}$ yields
from the $M(\Lambda_{c}^{+}\pi^{+}\pi^{-})$ sidebands are subtracted from the
$\Sigma_{c}(2455)^{++/0}$ yields found from the amplitude fit. The sidebands
are six $4\,\textrm{MeV}/\textit{c}^{2}$ regions near the
$\Lambda_{c}(2625)^{+}$ signal region, as shown in Fig. 4. Each sideband
region is fitted as an incoherent sum of the contributions from the
$\Sigma_{c}(2455)^{0}$, the $\Sigma_{c}(2455)^{++}$, and the three-body
phasespace decay. Figures 5 and 6 show the projections of the fit results for
each sideband region with each component labeled on the plot. The
$\Sigma_{c}(2455)^{++/0}$ yields in the signal region are determined by
extrapolating the yields from the sidebands according to a linear fit, as
shown in Fig. 7 and tabulated in Table 1. The background yields to be
subtracted are $N_{\mathrm{bkg}}(\Sigma_{c}^{0})=391\pm 11$ and
$N_{\mathrm{bkg}}(\Sigma_{c}^{++})=467\pm 12$. The branching ratio of
$\Lambda_{c}(2625)^{+}\to\Sigma_{c}^{0}\pi^{+}$ relative to the reference mode
$\Lambda_{c}(2625)^{+}\to\Lambda_{c}^{+}\pi^{+}\pi^{-}$ is calculated using

$\begin{split}\frac{\mathcal{B}(\Lambda_{c}(2625)^{+}\to\Sigma_{c}^{0}\pi^{+})}{\mathcal{B}(\Lambda_{c}(2625)^{+}\to\Lambda_{c}^{+}\pi^{+}\pi^{-})}=\frac{N_{\mathrm{sig}}(\Sigma_{c}^{0})-N_{\mathrm{bkg}}(\Sigma_{c}^{0})}{N_{\mathrm{sig}}(\Lambda_{c}(2625)^{+})}\end{split}$
(3)

and similarly for the $\Sigma_{c}^{++}\pi^{-}$ mode.

Figure 4: Signal region and the six sideband regions on either side of the
signal region used for sideband subtraction.

Figure 5: Projections of the Dalitz plot fits of the 3 sidebands on the left
side of the signal region. Overall fitted distribution and the individual
fitted components are shown alongside the experimental data.

Figure 6: Projections of the Dalitz plot fits of the 3 sidebands on the right
side of the signal region. Overall fitted distribution and the individual
fitted components are shown alongside the experimental data.

Figure 7: $\Sigma_{c}^{0}$ and $\Sigma_{c}^{++}$ yields from the sideband
Dalitz plot fits, overlaid with linear and quadratic extrapolations.

We measure relative branching ratios to be

$\displaystyle\frac{\mathcal{B}(\Lambda_{c}(2625)^{+}\to\Sigma_{c}^{0}\pi^{+})}{\mathcal{B}(\Lambda_{c}(2625)^{+}\to\Lambda_{c}^{+}\pi^{+}\pi^{-})}=(5.19\pm
0.23)\%$ (4)
$\displaystyle\frac{\mathcal{B}(\Lambda_{c}(2625)^{+}\to\Sigma_{c}^{++}\pi^{-})}{\mathcal{B}(\Lambda_{c}(2625)^{+}\to\Lambda_{c}^{+}\pi^{+}\pi^{-})}=(5.13\pm
0.26)\%$

where the errors are statistical only.

##  V Systematic uncertainties

The primary source of systematic uncertainty on the $\Lambda_{c}(2625)^{+}$
width is the inconsistency of the mass-resolution function between the MC
sample and the experimental data. We use the $D^{*+}\to D^{0}\pi^{+}$ decay as
a control sample to determine the under- or over-estimation of the mass
resolution in the MC sample relative to the experimental data. The mass
resolution in the experimental data is found to be 86% of that in the MC
sample with track smearing, 114% without track smearing. Both mass-resolution
functions are used to determine the upper limit on the $\Lambda_{c}(2625)^{+}$
width in the experimental data. When applied to the experimental data, the
mass resolution without track smearing scaled by 114% results in a more
conservative upper limit on the $\Lambda_{c}(2625)^{+}$ width, thus reported
as the final result.

The systematic uncertainty on the $\Lambda_{c}(2625)^{+}$ mass is not greatly
affected by the uncertainty on the $M(\Lambda_{c}^{+}\pi^{+}\pi^{-})$ mass
resolution, but is limited by the precision with which the Belle detector can
measure the mass in this range. Studies with $D^{*+}\to D^{0}\pi^{+}$ decays
show that the measured $D^{*+}$ mass difference with respect to the world-
average value is $0.004\,\textrm{MeV}/\textit{c}^{2}$. Any imperfection in the
soft pion momentum calibration changes the measured mass of the
$\Lambda_{c}(2625)^{+}$ more than that of the $D^{*+}$. We determine the scale
factor required to correct the soft pion momentum such that the $D^{*+}$ mass
matches its PDG value, then apply the same scale factor to the daughter pions
from $\Lambda_{c}(2625)^{+}$ candidates. The $\Lambda_{c}(2625)^{+}$ mass
changed by $0.042\,\textrm{MeV}/\textit{c}^{2}$, which we assign as the
systematic uncertainty due to the mass scale. The track smearing correction
applied to tracks in the MC sample has a negligible effect on the mass
measurement. The systematic uncertainty due to the low momentum correction is
$0.025\,\textrm{MeV}/\textit{c}^{2}$, which is found by comparing the measured
$\Lambda_{c}(2625)^{+}$ mass with and without the low momentum correction.
Summing the contributions from the mass scale and low momentum track
correction in quadrature, the total systematic uncertainty on the mass
measurement is $0.049\,\textrm{MeV}/\textit{c}^{2}$.

The systematic uncertainties on the relative branching ratios, which are
calculated from Eq. (3), are derived from the uncertainty of the
$\Lambda_{c}(2625)^{+}$ yield in the signal region, the $\Sigma_{c}^{++/0}$
yields fitted in the signal Dalitz plot fit, and the $\Sigma_{c}^{++/0}$
subtracted yields extrapolated from the sideband fits. The systematic
uncertainty associated with each is tabulated in Table 2. The
$\Lambda_{c}(2625)^{+}$ signal yield is most affected by the mass-resolution
function. We fit the experimental data with a mass-resolution function
determined with and without track smearing. The difference in the yields is
taken as the systematic uncertainty on the $\Lambda_{c}(2625)^{+}$ signal
yield. The $\Sigma_{c}^{++/0}$ signal yields are determined from the Dalitz
plot fit, with their masses and widths fixed to PDG values. The masses,
widths, and mass resolutions are systematically varied within the PDG
uncertainties, and the maximum change in the fitted $\Sigma_{c}^{++/0}$ yield
is taken as the systematic uncertainty. In order to determine the sideband
subtracted yield, the six sidebands are fitted individually to determine the
$\Sigma_{c}^{++/0}$ yields, with the yield and uncertainty in each sideband
region shown in Fig. 7. The extrapolated yield at the nominal
$\Lambda_{c}(2625)^{+}$ mass is a weighted average of the yields of the six
sidebands. We take the difference between the linear and quadratic
extrapolation as shown in Table 1 as the systematic uncertainty due to the
extrapolation. To account for the statistical fluctuation due the finite MC
sample sizes in the Dalitz plot fits, we compare the fitted results using two
statistically independent MC samples of the same size. The difference is taken
as the systematic uncertainty due to the MC sample size. With systematic
uncertainties on the yields in Eq. (3) listed in Table 2, the total systematic
uncertainties on the ratio of branching fractions are calculated from the
propagation of error and are listed in Table 3. the total systematic
uncertainty on the ratio of branching fractions is found to be 0.40% for the
$\Sigma_{c}^{0}\pi^{+}$ channel, and 0.32% for the $\Sigma_{c}^{++}\pi^{-}$
channel.

Table 1: Subtracted yields for $\Sigma_{c}^{++/0}$ Method |  $\Sigma_{c}^{0}$ yield |  $\Sigma_{c}^{++}$ yield
---|---|---
Linear |  $391\pm 11$ | $467\pm 12$
Quadratic |  $348\pm 26$ | $436\pm 28$
Difference | 11.00% | 6.64%
Table 2: The percentage systematic uncertainties of the signal yields used for the branching ratio calculation. Source |  $\Sigma_{c}^{0}$ signal |  $\Sigma_{c}^{++}$ signal |  $\Sigma_{c}^{0}$ sideband |  $\Sigma_{c}^{++}$ sideband |  $\Lambda_{c}(2625)^{+}$ signal
---|---|---|---|---|---
Resolution | 1.97% | 1.42% | 2.74% | 1.08% | 3.64%
$\Sigma_{c}^{0/++}$ width | 4.00% | 2.26% | 2.52% | 2.38% | -
$\Sigma_{c}^{0/++}$ mass | 1.25% | 1.11% | 0.08% | 0.08% | -
Extrapolation | - | - | 11.00% | 6.64% | -
MC sample size | 1.91% | 2.09% | 0.71% | 0.22% | -
Total | 5.01% | 3.57% | 11.63% | 7.14% | 3.64%
Table 3: Systematic uncertainties on the ratio of branching fractions. Source |  $\frac{\mathcal{B}(\Lambda_{c}(2625)^{+}\to\Sigma_{c}^{0}\pi^{+})}{\mathcal{B}(\Lambda_{c}(2625)^{+}\to\Lambda_{c}^{+}\pi^{+}\pi^{-})}$ | $\frac{\mathcal{B}(\Lambda_{c}(2625)^{+}\to\Sigma_{c}^{++}\pi^{-})}{\mathcal{B}(\Lambda_{c}(2625)^{+}\to\Lambda_{c}^{+}\pi^{+}\pi^{-})}$
---|---|---
$\Sigma_{c}^{0/++}$ resolution | 0.13% | 0.10%
$\Sigma_{c}^{0/++}$ width | 0.26% | 0.16%
$\Sigma_{c}^{0/++}$ mass | 0.08% | 0.07%
Extrapolation | 0.14% | 0.10%
MC sample size | 0.12% | 0.14%
$\Lambda_{c}^{+}(2625)$ resolution | 0.19% | 0.19%
Total | 0.40% | 0.32%

##  VI Discussion

We report the most precise $\Lambda_{c}(2625)^{+}$ mass, width, and branching
ratio measurements to date. The measured mass is consistent with previous
results. The measured upper limit on $\Lambda_{c}(2625)^{+}$ width is
$\Gamma(\Lambda_{c}(2625)^{+})<0.52\,\textrm{MeV}/\textit{c}^{2}$ at the 90%
credibility level. Theoretical predictions for the $\Lambda_{c}(2625)^{+}$
width vary. Arifi et al. predict the width to be 0.570 MeV/$\textrm{c}^{2}$
based on chiral and heavy quark symmetry [3]. The width is revised to be
between 0.09 and 0.26 MeV/$\textrm{c}^{2}$ in a subsequent publication with
the inclusion of relativistic corrections [5]. Kawakami et al. predict a width
in the range of 0.11 - 0.73 MeV/$\textrm{c}^{2}$ based on chiral symmetry [6].
Guo et al. predict a much smaller width of $1.13\times 10^{-2}$
MeV/$\textrm{c}^{2}$, based on the ${}^{3}P_{0}$ model [7].

The relative branching ratios of
$\Lambda_{c}(2625)^{+}\to\Sigma_{c}^{0}\pi^{+}$ and
$\Lambda_{c}(2625)^{+}\to\Sigma_{c}^{++}\pi^{-}$ relative to the reference
mode $\Lambda_{c}(2625)^{+}\to\Lambda_{c}^{++}\pi^{+}\pi^{-}$ are extracted
from a full Dalitz plot fit. Backgrounds from non-$\Lambda_{c}(2625)^{+}$
decays are subtracted from the $\Sigma_{c}^{++/0}$ yields. Our measurements
align with the prediction by Arifi et al., who assume $\Lambda_{c}(2625)^{+}$
is a $\lambda$ mode excitation [3]. Kawakami et al. predicted a wide range [6]
and Guo et al.’s predicted the ratio
$\Gamma(\Sigma_{c}^{++}\pi^{-})/\Gamma_{\mathrm{total}}$ to be 29.9% [7],
which is already in contradiction with the previous measurement. Our
measurements of the properties of the $\Lambda_{c}(2625)^{+}$ charmed baryon
will be useful to further constrain the parameter space of the quark models
and can be applied to other heavy quark systems.

##  VII Conclusions

We measure the mass of the $\Lambda_{c}(2625)^{+}$ to be $2627.978\pm 0.006\pm
0.049\,\textrm{MeV}/\textit{c}^{2}$, where the uncertainty on the
$\Lambda_{c}^{+}$ mass is not included since it is constrained to the PDG
value during reconstruction. This is equivalent to

$M(\Lambda_{c}(2625)^{+})-M(\Lambda_{c}^{+})=341.518\pm 0.006\pm
0.049\,\textrm{MeV}/\textit{c}^{2}$ (5)

The mass measurement is consistent with the previous CDF measurement but with
approximately half the uncertainty [2].

An upper limit on the $\Lambda_{c}(2625)^{+}$ width is determined to be

$\Gamma(\Lambda_{c}(2625)^{+})<0.52\,\textrm{MeV}/\textit{c}^{2}\ $ (6)

at 90% credibility level which is around a factor of two more stringent than
the previous limit.

Based on a full Dalitz plot fit and with sideband subtraction of the
$\Sigma_{c}^{++/0}$ yields, the branching ratios relative to the mode
$\Lambda_{c}(2625)^{+}\to\Lambda_{c}^{+}\pi^{+}\pi^{-}$ are obtained:

$\displaystyle\frac{\mathcal{B}(\Lambda_{c}(2625)^{+}\to\Sigma_{c}^{0}\pi^{+})}{\mathcal{B}(\Lambda_{c}(2625)^{+}\to\Lambda_{c}^{+}\pi^{+}\pi^{-})}=(5.19\pm
0.23\pm 0.40)\%$ (7)
$\displaystyle\frac{\mathcal{B}(\Lambda_{c}(2625)^{+}\to\Sigma_{c}^{++}\pi^{-})}{\mathcal{B}(\Lambda_{c}(2625)^{+}\to\Lambda_{c}^{+}\pi^{+}\pi^{-})}=(5.13\pm
0.26\pm 0.32)\%$

This is the first measurement made of these branching fractions as previously
only limits have been presented. These measurements can be used as inputs to
theoretical models to generate predictions for other heavy quark baryons.

##  VIII Acknowledgements

This work, based on data collected using the Belle detector, which was
operated until June 2010, was supported by the Ministry of Education, Culture,
Sports, Science, and Technology (MEXT) of Japan, the Japan Society for the
Promotion of Science (JSPS), and the Tau-Lepton Physics Research Center of
Nagoya University; the Australian Research Council including grants
DP180102629, DP170102389, DP170102204, DE220100462, DP150103061, FT130100303;
Austrian Federal Ministry of Education, Science and Research (FWF) and FWF
Austrian Science Fund No. P 31361-N36; the National Natural Science Foundation
of China under Contracts No. 11675166, No. 11705209; No. 11975076; No.
12135005; No. 12175041; No. 12161141008; Key Research Program of Frontier
Sciences, Chinese Academy of Sciences (CAS), Grant No. QYZDJ-SSW-SLH011;
Project ZR2022JQ02 supported by Shandong Provincial Natural Science
Foundation; the Ministry of Education, Youth and Sports of the Czech Republic
under Contract No. LTT17020; the Czech Science Foundation Grant No. 22-18469S;
Horizon 2020 ERC Advanced Grant No. 884719 and ERC Starting Grant No. 947006
“InterLeptons” (European Union); the Carl Zeiss Foundation, the Deutsche
Forschungsgemeinschaft, the Excellence Cluster Universe, and the
VolkswagenStiftung; the Department of Atomic Energy (Project Identification
No. RTI 4002) and the Department of Science and Technology of India; the
Istituto Nazionale di Fisica Nucleare of Italy; National Research Foundation
(NRF) of Korea Grant Nos. 2016R1D1A1B02012900, 2018R1A2B3003643,
2018R1A6A1A06024970, RS202200197659, 2019R1I1A3A01058933, 2021R1A6A1A03043957,
2021R1F1A1060423, 2021R1F1A1064008, 2022R1A2C1003993; Radiation Science
Research Institute, Foreign Large-size Research Facility Application
Supporting project, the Global Science Experimental Data Hub Center of the
Korea Institute of Science and Technology Information and KREONET/GLORIAD; the
Polish Ministry of Science and Higher Education and the National Science
Center; the Ministry of Science and Higher Education of the Russian
Federation, Agreement 14.W03.31.0026, and the HSE University Basic Research
Program, Moscow; University of Tabuk research grants S-1440-0321, S-0256-1438,
and S-0280-1439 (Saudi Arabia); the Slovenian Research Agency Grant Nos.
J1-9124 and P1-0135; Ikerbasque, Basque Foundation for Science, Spain; the
Swiss National Science Foundation; the Ministry of Education and the Ministry
of Science and Technology of Taiwan; and the United States Department of
Energy and the National Science Foundation. These acknowledgements are not to
be interpreted as an endorsement of any statement made by any of our
institutes, funding agencies, governments, or their representatives. We thank
the KEKB group for the excellent operation of the accelerator; the KEK
cryogenics group for the efficient operation of the solenoid; and the KEK
computer group and the Pacific Northwest National Laboratory (PNNL)
Environmental Molecular Sciences Laboratory (EMSL) computing group for strong
computing support; and the National Institute of Informatics, and Science
Information NETwork 6 (SINET6) for valuable network support.

## References

  * Albrecht  _et al._ [1993] H. Albrecht _et al._ (ARGUS), Observation of a new charmed baryon, Phys. Lett. B 317, 227 (1993).
  * Aaltonen  _et al._ [2011] T. Aaltonen _et al._ (CDF), Measurements of the properties of $\Lambda_{c}(2595)$, $\Lambda_{c}(2625)$, $\Sigma_{c}(2455)$, and $\Sigma_{c}(2520)$ baryons, Phys. Rev. D 84, 012003 (2011), arXiv:1105.5995 [hep-ex] .
  * Arifi  _et al._ [2018] A. J. Arifi, H. Nagahiro, and A. Hosaka, Three-body decay of $\Lambda_{c}^{*}(2595)$ and $\Lambda_{c}^{*}(2625)$ with the inclusion of a direct two-pion coupling, Phys. Rev. D 98, 114007 (2018), arXiv:1809.10290 [hep-ph] .
  * Zyla  _et al._ [2020] P. A. Zyla _et al._ (Particle Data Group), Review of Particle Physics, Prog. Theo. Exp. Phys. 2020, 083C01 (2020).
  * Arifi  _et al._ [2021] A. J. Arifi, D. Suenaga, and A. Hosaka, Relativistic corrections to decays of heavy baryons in the quark model, Phys. Rev. D 103, 094003 (2021), arXiv:2102.03754 [hep-ph] .
  * Kawakami and Harada [2018] Y. Kawakami and M. Harada, Analysis of $\Lambda_{c}(2595)$, $\Lambda_{c}(2625)$, $\Lambda_{b}(5912)$, $\Lambda_{b}(5920)$ based on a chiral partner structure, Phys. Rev. D 97, 114024 (2018), arXiv:1804.04872 [hep-ph] .
  * Guo  _et al._ [2019] J.-J. Guo, P. Yang, and A. Zhang, Strong decays of observed $\Lambda_{c}$ baryons in the ${}^{3}P_{0}$ model, Phys. Rev. D 100, 014001 (2019), arXiv:1902.07488 [hep-ph] .
  * Cheng [2015] H.-Y. Cheng, Charmed baryons circa 2015, Front. Phys. (Beijing) 10, 101406 (2015).
  * Abashian  _et al._ [2002] A. Abashian _et al._ (Belle), The Belle Detector, Nucl. Instrum. Meth. A 479, 117 (2002).
  * Brodzicka  _et al._ [2012] J. Brodzicka _et al._ (Belle), Physics Achievements from the Belle Experiment, Prog. Theo. Exp. Phys. 2012, 04D001 (2012), arXiv:1212.5342 [hep-ex] .
  * Kurokawa and Kikutani [2003] S. Kurokawa and E. Kikutani, Overview of the KEKB accelerators, Nucl. Instrum. Meth. A 499, 1 (2003).
  * Abe  _et al._ [2013] T. Abe _et al._ , Achievements of KEKB, Prog. Theo. Exp. Phys. 2013, 03A001 (2013).
  * Lange [2001] D. J. Lange, The EvtGen particle decay simulation package, Nucl. Instrum. Meth. A 462, 152 (2001).
  * Brun  _et al._ [1987] R. Brun, F. Bruyant, M. Maire, A. C. McPherson, and P. Zanarini, GEANT3 (1987).
  * Gelb  _et al._ [2018] M. Gelb _et al._ , B2BII: Data Conversion from Belle to Belle II, Comput. Softw. Big Sci. 2, 9 (2018), arXiv:1810.00019 [hep-ex] .
  * Kuhr  _et al._ [2019] T. Kuhr, C. Pulvermacher, M. Ritter, T. Hauth, and N. Braun (Belle-II Framework Software Group), The Belle II Core Software, Comput. Softw. Big Sci. 3, 1 (2019), arXiv:1809.04299 [physics.comp-ph] .
  * [17] Belle II Analysis Software Framework (basf2), https://doi.org/10.5281/zenodo.5574115.
  * [18] Reconstruction of charge conjugates is implied unless otherwise stated.
  * Nakano [2002] E. Nakano, Belle PID, Nucl. Instrum. Meth. A 494, 402 (2002).
  * Lee  _et al._ [2014] S. H. Lee _et al._ (Belle), Measurements of the masses and widths of the $\Sigma_{c}(2455)^{0/++}$ and $\Sigma_{c}(2520)^{0/++}$ baryons, Phys. Rev. D 89, 091102 (2014), arXiv:1404.5389 [hep-ex] .
  * Verkerke and Kirkby [2003] W. Verkerke and D. P. Kirkby, The RooFit toolkit for data modeling, eConf C0303241, MOLT007 (2003), arXiv:physics/0306116 .
  * [22] H. Matevosyan, R. Mitchell, and M. Shepherd, https://github.com/mashephe/AmpTools/wiki.