Experimental determination of the isospin of $\Lambda_c(2765)^+/\Sigma_c(2765)^+$

TL;DR

This paper experimentally determines the isospin of a charmed baryon, establishing it as a $ ext{Lambda}_c$ with isospin zero, using extensive data from the Belle detector at KEKB.

Contribution

The study provides the first experimental determination of the isospin of $ ext{Lambda}_c(2765)^+/ ext{Sigma}_c(2765)^+$, clarifying its classification as a $ ext{Lambda}_c$ baryon.

Findings

Isospin of the particle is zero.

No evidence of isospin partners in the searched channels.

Confirmed as a $ ext{Lambda}_c$ baryon.

Abstract

We report an experimental determination of the isospin of $\Lambda_c(2765)^+/\Sigma_c(2765)^+$ using 980 fb$^{-1}$ data in the $e^+e^-$ annihilation around $\sqrt{s} = 10.6$ GeV collected by the Belle detector located at the KEKB collider. The isospin partners are searched for in the $\Sigma_c(2455)^{++/0} \pi^{0}$ channels, and no evidence was obtained. Thus the isospin is determined to be zero, and the particle is established to be a $\Lambda_c$.

Tables

Table 1. Table 1: Summary of the systematic uncertainties for R I subscript 𝑅 𝐼 R_{I} .

Source	($\%$)
${\pi }^0$ reconstruction efficiency	2.1
${\pi }^{+/-}$ reconstruction efficiency	0.9
${\pi }^{+/-}$ Track reconstruction efficiency	0.4
Fit bias	0.3
Yield for $I_3=0$ state	25.3
Width for $I_3=0$ state	17.1
Mass of $I_3=0$ state	1.7
Total	30.7

Equations

$$R_I=\frac{{\sigma }_{+1}{\mathcal{B}}_{+1}+{\sigma }_{-1}{\mathcal{B}}_{-1}}{2{\sigma }_0{\mathcal{B}}_0},$$

$$R_I=\frac{{\sigma }_{\pm 1}}{{\sigma }_0}.$$

Full text

††preprint:

BELLE-CONF-1905

I Introduction

Charmed baryons are an excellent system to study quark dynamics, especially possible diquark correlations in baryons, as they consist of one heavy charm quark and two light quarks. There are a few dozens of states known so far, but in most cases their quantum numbers have not been determined experimentally PDG . For ground states and low-lying states with excitation energies up to about 400 MeV, classification by quark models still works well and plausible quantum numbers can be given. However, for higher excitation energies, unique identification is not possible because quark models predict several states within their typical mass uncertainties of around 50 MeV/$c^{2}$ th_capstick ; th-Roberts ; th-yoshida .

$\Lambda_{c}(2765)^{+}/\Sigma_{c}(2765)^{+}$ (for the notation, see note footnote-notation in the reference section) is the lightest charmed baryon for which there is no assumed quark-model identification. It was first observed by CLEO Lc27CLEO in the $\Lambda_{c}^{+}\pi^{+}\pi^{-}$ final state and later reported by Belle Lc28Belle in the $\Sigma_{c}^{++/0}\pi^{\mp}$ final state. The experimental status is very poor: the spin, parity and isospin are not determined yet and CLEO did not determine the uncertainty on its width.

In a quark model calculation by Capstick and Isgur th_capstick , there are six states predicted in this mass region, one state each for $I(J^{P})=0(1/2^{+}),~{}0(1/2^{-})$, and two states each for $I(J^{P})=1(1/2^{-}),~{}(3/2^{-})$. On the other hand, in the most recent quark model calculation by Yoshida et al. th-yoshida , the $I=0$ states are found in the higher mass region, while the $I=1$ states are still in the range of $2765\pm 50$ MeV/$c^{2}$. In addition, there are many theoretical calculations in various models (for example th-oh ; th-Zhong ; th-Ebert ; th-Roberts ; th-chen ; th-chen2 ; th-kumakawa ; th-cheng ; th-Lu ), and almost any combination of $I=0$ or 1, $J=1/2$ or 3/2, and $P=+$ or $-$ seems to be possible. Therefore, the nature of $\Lambda_{c}(2765)^{+}/\Sigma_{c}(2765)^{+}$ is quite controversial and experimental determination of its quantum numbers is awaited.

In this paper, we present a determination of the isospin of $\Lambda_{c}(2765)^{+}/\Sigma_{c}(2765)^{+}$ by searching for the possible isospin partners of the state in the decay modes to $\Sigma_{c}(2455)^{++/0}\pi^{0}\to\Lambda_{c}^{+}\pi^{\pm}\pi^{0}$.

II The Belle experiment and the data set

We use a dataset of 980 fb*-1* of integrated luminosity around $\sqrt{s}=10.6$ GeV collected by the Belle detector at the KEKB asymmetric-energy $e^{+}e^{-}$ collider KEKB . 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 superconducting 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. The detector is described in detail elsewhere Belle . Two inner detector configurations were used. A 2.0 cm radius beam-pipe and a 3-layer silicon vertex detector were used for the first sample of 156 fb*-1*, while a 1.5 cm radius beam pipe, a 4-layer silicon detector and a small-cell inner drift chamber were used to record the remaining 824 fb*-1* svd2 . We use a Monte Carlo (MC) simulation to model the detector response and its acceptance to obtain the reconstruction efficiency, the mass resolution for signals and feed-down signals. The MC events are generated with PYTHIA pythia , decays of unstable particles are simulated with EVTGEN evtgen , and the detector response is simulated by using GEANT geant .

III Reconstruction of $\Lambda_{c}(2765)^{+}$ in $\Lambda_{c}^{+}\pi^{+}\pi^{-}$

We reconstruct $\Lambda_{c}(2765)^{+}$ candidates in the $\Lambda_{c}^{+}\pi^{+}\pi^{-}$ final state. $\Lambda_{c}^{+}$ baryons are reconstructed using the $\Lambda_{c}^{+}\to pK^{-}\pi^{+}$ decay mode (the inclusion of the charge-conjugate modes is implied throughout this paper). For the primary selection of charged particles, we use the same track quality and particle identification as the $\Lambda_{c}(2880)^{+}$ study by Belle Lc28Belle . In addition, in order to reject duplicated tracks, when there are multiple tracks of the same particle species (pion, kaon, or proton), we check the invariant mass of each track pair. If the invariant mass is smaller than 0.28 GeV/$c^{2}$, 0.989 GeV/$c^{2}$, and 1.878 GeV/$c^{2}$ for pion, kaon, and proton pairs, respectively, one of the tracks is rejected by the track quality. This selection reduces the number of $\Lambda_{c}^{+}\pi^{+}\pi^{-}$ candidates by 0.1%. The invariant mass of the $pK^{-}\pi^{+}$ combination is required to be within $\pm 8$ MeV/$c^{2}$ (3.7$\sigma$) of the mass of $\Lambda_{c}^{+}$ ci_lc_w_babar . We perform a $\Lambda_{c}^{+}$-mass-constrained fit to the $pK^{-}\pi^{+}$ vertex to improve the $\Lambda_{c}^{+}$ momentum resolution. We combine the $\Lambda_{c}^{+}$ candidate with the remaining $\pi^{+}\pi^{-}$ candidates. The scaled momentum $x_{p}=p^{*}/p_{\rm max}$, where $p^{*}$ is the momentum of the $\Lambda_{c}(2765)^{+}$ candidate in the beam center-of-mass frame, $p_{\rm max}$ is given by $\sqrt{E_{\rm beam}^{2}-M^{2}c^{4}}$ with beam energy $E_{\rm beam}$ and mass of $\Lambda_{c}(2765)^{+}$ candidate $M$, is required to be greater than 0.8 to reduce combinatorial backgrounds. To improve the invariant mass resolution, we perform a common vertex fit to each $\Lambda_{c}^{+}\pi^{+}\pi^{-}$ candidate. The resulting $\Lambda_{c}^{+}\pi^{+}\pi^{-}$ invariant mass [$M(\Lambda_{c}^{+}\pi^{+}\pi^{-})$] spectrum is shown in Fig. 1(a), which presents clear peaks of $\Lambda_{c}(2765)^{+}$ and $\Lambda_{c}(2880)^{+}$.

We can also see intermediate states, $\Sigma_{c}(2455/2520)$, in the $M(\Lambda_{c}^{+}\pi)$ distribution as shown in Fig. 2. In the following analysis, we define the signal region for $\Sigma_{c}(2455)$ as $\left|M(\Lambda_{c}^{+}\pi)-m_{\Sigma_{c}(2455)}\right|<5$ MeV/$c^{2}$. Figure 1(b) shows the $M(\Lambda_{c}^{+}\pi^{+}\pi^{-})$ spectrum with $\Sigma_{c}(2455)$ selection.

The yield for $\Lambda_{c}(2765)^{+}\to\Sigma_{c}(2455)^{++/0}\pi^{\mp}(\to\Lambda_{c}^{+}\pi^{+}\pi^{-})$ is determined by a fit to Fig. 1(b) using Breit-Wigner functions for $\Lambda_{c}(2765)^{+}$, $\Lambda_{c}(2880)^{+}$, and $\Lambda_{c}(2940)^{+}$ and a second-order polynomial as background. In addition, there is a small contribution from $\Sigma_{c}(2520)$ feed-down which is due to an accidental coincidence of $\Sigma_{c}(2520)$ and a random pion that produces a rather narrow peak in the $M(\Lambda_{c}^{+}\pi^{+}\pi^{-})$ spectrum around 2.68 GeV/$c^{2}$. The details are to be reported elsewhere Lc2765full .

IV Search for isospin partners in

$\Lambda_{c}^{+}\pi^{\pm}\pi^{0}$ final state

In order to determine the isospin, we search for the isospin partners with $I_{3}=\pm 1$ in the $\Sigma_{c}(2765)^{++/0}\to\Sigma_{c}^{++/0}\pi^{0}\to\Lambda_{c}^{+}\pi^{\pm}\pi^{0}$ final states. The $\Sigma_{c}(2765)^{++/0}\to\Sigma_{c}^{+}\pi^{\pm}\to\Lambda_{c}^{+}\pi^{0}\pi^{\pm}$ decay modes are not used because the momentum of $\pi^{0}$ in these modes is low and thus the background level is very high. The ratio for the product of cross section and branching fractions between $I_{3}=0$ and $I_{3}=\pm 1$ states ($R_{I}$)

[TABLE]

where $\sigma_{i}$ is the production cross section for the $I_{3}=i$ states, ${\cal B}_{\pm 1}$ is the branching fraction for the $I_{3}=\pm 1$ states decaying into $\Sigma_{c}(2455)^{++/0}\pi^{0}$ decay modes, and ${\cal B}_{0}$ is the sum of the branching fractions of the $I_{3}=0$ state decaying into $\Sigma_{c}(2455)^{++}\pi^{-}$ and $\Sigma_{c}(2455)^{0}\pi^{+}$, is measured. If we assume the cross sections are the same for $I_{3}=+1$ and $I_{3}=-1$ and all the branching fractions are the same, this ratio simplifies to

[TABLE]

If resonances are observed with masses and widths close to those of the $I_{3}=0$ state, it is strong evidence that the isospin is one. If resonances are not observed and obtained upper limit for the $R_{I}$ is much smaller than one, it is strong evidence that the isospin is zero.

In this analysis, the signal region, defined as $2.65$ GeV/$c^{2}<M(\Lambda_{c}^{+}\pi^{\pm}\pi^{0})<2.9$ GeV/$c^{2}$ after selecting the $\Sigma_{c}^{++/0}(2455)$ region, is blinded until all the selection criteria and combinatorial background are determined. Except for the $\pi^{0}$, the event selection is the same as that applied for the $\Lambda_{c}^{+}\pi^{+}\pi^{-}$ analysis. A $\pi^{0}$ is reconstructed from two photons, energies of which are required to be greater than 50 MeV. The $\pi^{0}$ candidates are selected by requiring 120 MeV/$c^{2}<M(2\gamma)<150$ MeV/$c^{2}$. In order to suppress the combinatorial background further, the $\pi^{0}$ helicity angle, defined as the angle between the momentum of the $\Lambda_{c}(2765)$ in the center-of-mass frame and momentum of the $\pi^{0}$ in the $\Lambda_{c}(2765)^{+}$ rest frame is required to be greater than $-0.35$.

The mass distributions for combinatorial background events are checked from the $\Sigma_{c}(2455)^{++/0}$ sideband region defined as $\left|M(\Lambda_{c}^{+}\pi^{\pm})-m_{\Sigma_{c}(2455)^{++/0}}\pm 20\text{ MeV}/c^{2}\right|<5\text{ MeV}/c^{2}$. The $M(\Lambda_{c}^{+}\pi^{\pm}\pi^{0})$ distribution for the sideband region is fitted with a second order exponential curve $f(M)=\exp(aM+bM^{2})$, where $a$ and $b$ are the fit parameters. The values of the $\chi^{2}/d.o.f.$ are obtained as $162.0/184$ for $M(\Lambda_{c}^{+}\pi^{+}\pi^{0})$ and 212.2/184 for $M(\Lambda_{c}^{+}\pi^{-}\pi^{0})$, confirming that this background parameterization is appropriate. Therefore it is decided to use this function for the background PDF. The validity to use this function for the $\Sigma_{c}(2455)$ signal regions is checked by comparing mass distributions for sideband region and $\Sigma_{c}(2455)$ signal region outside of the signal box. They are found to be consistent with each other.

IV.1 Result

Fig. 3 shows $M(\Lambda_{c}^{+}\pi^{\pm}\pi^{0})$ distributions for the signal regions with fit lines ovarlaid. A simultaneous unbinned extended maximum likelihood fit is performed with the constraint that the signal yields for $I_{3}=\pm 1$ states are the same. The signal PDF is made based on the signal MC simulation assuming that the mass and width are the same as those of the $I_{3}=0$ state. The sum of the signal yields for $I_{3}=\pm 1$ is obtained to be $230\pm 250$. Note that, if we assume that the cross-section and branching fractions are the same, the expected signal yields would be 8764, estimated from the yields of $\Lambda_{c}(2765)^{+}\to\Sigma_{c}(2455)^{++/0}\pi^{\mp}\to\Lambda_{c}^{+}\pi^{+}\pi^{-}$ taking the difference of acceptance and efficiency into account.

The $\chi^{2}/d.o.f.$ of the fit is calculated after filling the data in bins as 211/232 (215/232) for $M(\Lambda_{c}^{+}\pi^{+}\pi^{0})$ [$M(\Lambda_{c}^{+}\pi^{-}\pi^{0})$] without including the signal component. The 95$\%$ upper limit on the $R_{I}$ is obtained based on Bayesian approach, after incorporating the following systematic uncertainties which are summarized in Table 1. The obtained 95$\%$ C.L upper limit for $R_{I}$ is 0.09, which is much smaller than unity, demonstrating that isospin of $\Lambda_{c}(2765)^{+}$ is indeed zero. This is consistent with the fact that the state was not seen in the $\Lambda_{c}^{+}\pi$ spectra by Belle Belle-Sc2800 , that would be an expected decay mode of an excited $\Sigma_{c}$ baryon.

V Summary and discussion

In summary, we determined the isospin of $\Lambda_{c}(2765)^{+}/\Sigma_{c}(2765)^{+}$ by searching for possible isospin partners in the decays to $\Sigma_{c}(2455)^{++/0}\pi^{0}\to\Lambda_{c}^{+}\pi^{\pm}\pi^{0}$. The isospin is determined to be $I=0$, therefore the name should indeed be $\Lambda_{c}(2765)^{+}$. The result is rather inconsistent with the most recent quark model calculation th-yoshida , but not with the older ones th_capstick ; th-Roberts . Also, some theoretical predictions are ruled out by the result. However, in order to discuss the nature of this state in more detail, the determination of its other quantum numbers (i.e., spin and parity) is highly desirable and we are planning a publication on this subject, using this data set, in the near future.

Acknowledgment

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 5 (SINET5) for valuable network support. We acknowledge support from the Ministry of Education, Culture, Sports, Science, and Technology (MEXT) of Japan, the Japan Society for the Promotion of Science (JSPS) under KAKENHI Grant No. JP19H05148, and the Tau-Lepton Physics Research Center of Nagoya University; the Australian Research Council including grants DP180102629, DP170102389, DP170102204, DP150103061, FT130100303; Austrian Science Fund (FWF); the National Natural Science Foundation of China under Contracts No. 11435013, No. 11475187, No. 11521505, No. 11575017, No. 11675166, No. 11705209; Key Research Program of Frontier Sciences, Chinese Academy of Sciences (CAS), Grant No. QYZDJ-SSW-SLH011; the CAS Center for Excellence in Particle Physics (CCEPP); the Shanghai Pujiang Program under Grant No. 18PJ1401000; the Ministry of Education, Youth and Sports of the Czech Republic under Contract No. LTT17020; the Carl Zeiss Foundation, the Deutsche Forschungsgemeinschaft, the Excellence Cluster Universe, and the VolkswagenStiftung; the Department of Science and Technology of India; the Istituto Nazionale di Fisica Nucleare of Italy; National Research Foundation (NRF) of Korea Grants No. 2015H1A2A1033649, No. 2016R1D1A1B01010135, No. 2016K1A3A7A09005603, No. 2016R1D1A1B02012900, No. 2018R1A2B3003643, No. 2018R1A6A1A06024970, No. 2018R1D1 A1B07047294; 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 Grant of the Russian Federation Government, Agreement No. 14.W03.31.0026; the Slovenian Research Agency; 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.