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# A user-friendly tool to convert photon counting data to the open-source
Photon-HDF5 file format

Donald Ferschweiler  Mathematics, Sciences, and Engineering Division, Rio
Hondo College, Whittier, CA, USA  Department of Physics and Astronomy,
California State University, Long Beach, CA, USA  Maya Segal  Department of
Chemistry and Biochemistry, University of California, Los Angeles, CA, USA
Shimon Weiss  Department of Chemistry and Biochemistry, University of
California, Los Angeles, CA, USA  Xavier Michalet  Department of Chemistry and
Biochemistry, University of California, Los Angeles, CA, USA

###### Abstract

Photon-HDF5 is an open-source and open file format for storing photon-counting
data from single molecule microscopy experiments, introduced to simplify data
exchange and increase the reproducibility of data analysis. Part of the
Photon-HDF5 ecosystem, is phconvert, an extensible python library that allows
converting proprietary formats into Photon-HDF5 files. However, its use
requires some proficiency with command line instructions, the python
programming language, and the YAML markup format. This creates a significant
barrier for potential users without that expertise, but who want to benefit
from the advantages of releasing their files in an open format. In this work,
we present a GUI that lowers this barrier, thus simplifying the use of Photon-
HDF5. This tool uses the phconvert python library to convert data files
originally saved in proprietary data formats to Photon-HDF5 files, without
users having to write a single line of code. Because reproducible analyses
depend on essential experimental information, such as laser power or sample
description, the GUI also includes (currently limited) functionality to
associate valid metadata with the converted file, without having to write any
YAML. Finally, the GUI includes several productivity-enhancing features such
as whole-directory batch conversion and the ability to re-run a failed batch,
only converting the files that could not be converted in the previous run.

###### keywords:

photon-counting, single-molecule, fluorescence, open-source, Photon-HDF5,
phconvert, GUI, Python

##  1 INTRODUCTION

###  1.1 Photon-HDF5 File Format

The single-molecule fluorescence spectroscopy (SMFS) community has recently
promoted some level of standardization in measurement practices and analysis
pipelines [1]. In order to enable reproducibility of data analysis and thus
strengthen trust in scientific results, it is important for data to be freely
and easily available, for example by depositing it in online repositories.
Moreover, data should be readable and well-documented, which is best done
using standardized file formats and abundant, human-readable metadata.

Photon-counting SMFS methods implement widely varying setup geometries and
hardware. Common photon-counting techniques include single-molecule Förster
Resonance Energy Transfer (smFRET), fluorescence correlation and cross-
correlation spectroscopy (FCS and FCCS), and fluorescence lifetime
spectroscopy. In the case of smFRET, setups may have a variety of different
features, for example, multi-color detection schemes, nano or microsecond
alternative laser excitation (ns- or $\mu$s-ALEX), or polarization anisotropy.
In each of these cases, parameters change between measurements such as the
number and wavelength of light sources, the laser period in setups using
alternation, number of detectors, or the presence of polarizers, etc.

In addition to the wide variety of hardware, there is a wide variety of
acquisition software and file formats. Data is often recorded using custom-
built binary file formats, either supplied by an equipment manufacturer or
designed in-house by the laboratory in which the experiment was performed.
Oftentimes, the former are preferable to the latter, as manufacturers have a
vetted interest in long-term support of their customers, while graduate
students and postdocs follow each other in a lab, with variable degree of
overlap. However, manufacturers understandably develop their own formats
adapted to their hardware and market niche, with little concern for
compatibility with other products. The result is a slew of file formats, some
well-documented, some not at all, which leads to loss of effective access to
the data, even when the files remain available. Unifying common data storage
formats helps to alleviate the issues associated with proprietary file
formats, and as the number and variety of techniques continues to grow, it
becomes increasingly important to standardize an open and accessible format
for the burgeoning SMFS community.

A few years ago, we introduced a open and standardized photon-counting file
format, Photon-HDF5, based on the HDF5 open source file format [2]. The HDF5
organization is a vibrant community supporting multiple operating systems, and
provides a free HDFView software for these platforms, enabling users to open,
read, and copy data from these files without knowing any programming, making
it an ideal vehicle for universal data storage [3, 4]. Based on these
foundations, the Photon-HDF5 file format formalizes the organization of SMFS
data obtained with photon-counting hardware, by logically storing raw data
along with metadata for the sample and measurement. Storing data and metadata
together is extremely important for long-term reproducibility, see Photon-
HDF5.org or http://photon-hdf5.github.io/ for a complete documentation.

By virtue of the HDF5 format’s built-in features, Photon-HDF5 incorporates
efficient data compression, and can be easily read within computing
environment programming languages such as Python or MATLAB. Furthermore,
accompanying the format’s detailed description, code snippets for MATLAB and
LabVIEW are provided as examples, as well as a python based file converter
called phconvert (http://photon-hdf5.github.io/phconvert/) [5]. The format is
supported by an increasing number of SMFS data processing packages, such as
FRETBursts (python) [6] and ALiX (LabVIEW) [7].

###  1.2 phconvert

phconvert is a helper library which was released together with Photon-HDF5,
designed to convert some common proprietary file formats to the Photon-HDF5
format. Currently supported raw data files include PicoQuant’s
.HT3/.PT3/.PTU/.T3R, Becker & Hickl’s .SPC/.SET, and the .SM format used by
our lab and others for $\mu$s-ALEX smFRET. As mentioned earlier, one advantage
of such a conversion is that files can be viewed in human readable forms using
HDFView [4], and most importantly, when done properly, the data is stored with
relevant metadata, helping to make sense of the experimental and measurement
parameters.

Using phconvert requires basic familiarity with Python, something which is
becoming increasingly common in the community, but remains a potential
obstacle to the adoption of this simple action: converting ones’ data files
into Photon-HDF5 formatted files for deposition into a public online
repository. Additionally, some understanding of the different metadata fields
of Photon-HDF5 (and more generally, an understanding of the general logic of
the format organization) is required to generate fully documented files.
Finally, the metadata to be provided to phconvert needs to be prepared in a
YAML file, which requires some knowledge of this simple yet specific data
serialization language (YAML Ain’t Markup Language™) [8].

In order to lower the barrier to adoption of Photon-HDF5 for sharing photon-
counting data generated in SMFS experiments, we have developed a graphical
user interface (GUI) for the phconvert library that significantly reduces the
need for the specialized technical skills currently required for file
conversion. In this paper, we will describe the current state of the GUI,
detailing some of the ways it can currently be used, its limitations, and
future improvements that could be made.

##  2 Software design

###  2.1 Features

The goals set for this tool were to:

Simplify installation by packaging all dependencies

To obtain phconvert, users have several options, all of which involve
technical knowledge. They can download the package from GitHub, taking care to
properly install it so that it plays well with the system’s Python
installation, or they can use a package manager such as pip or conda. We
wanted to improve on this experience for users without technical experience.
Most users have likely not heard of these tools, let alone feel comfortable
using them. So installation of the GUI should be as straight-forward as
possible, and not require mucking about with language package managers or
stuff like the PYTHON_PATH variable.

Provide a user-friendly conversion tool

Minimal prior knowledge should be presumed of the user. Especially no
technical knowledge regarding programming, the command line, YAML, etc. Common
operations should be obvious, so that the user is not left hunting for what
they want. The correct thing to do should be made as easy as possible, and
there should be no hidden suprises that sabotage users.

Support batch conversion of multiple files

Users may have more than one file that they wish to convert, and it can be
convenient to deal with a large directory all at once. To streamline the
process of converting many files at a time, there should be batch conversion
mode.

Provide helpful feedback during and after conversion

As a minimum requirement, there should be a way for the users to know whether
the conversion process was successful. Users should also be able to find out
what when wrong in the case of a failure, so that they can fix the situation
and try again.

####  2.1.1 Additional nice-to-have features

YAML-free metadata specification

Many users won’t know what YAML is, nor the syntax for it, or even how to
create a YAML file in the first place. A method for users to provide the
metadata for the data file(s) being converted to Photon-HDF5 without having to
use YAML directly would definitely be useful.

Support for multiple platforms

For widest adoption, the GUI should be available on all the common platforms
so that most researchers who want to use it can.

Seamless upgrades to future phconvert versions

The user should be insulated from worrying about differences in the underlying
library version, and the GUI should be developed in such a way that the
maintainer(s) have as smooth a time as possible when the time comes to update.

###  2.2 Implementation

To make things truly point-and-click, a thin wrapper around phconvert was
needed that provided a single entry point for conversion. To make the
interface as simple and discoverable as possible, a minimum amount of
graphical elements were added, and a focus was placed on trying to make the
things that should be easy, as easy as possible. The button to convert a file
or batch is the biggest, most visible button. There is a status bar for each
conversion task that lets the user know if conversion succeeded, and the
status bar for batches tries to be informative regarding both the current
attempt and the overall status of the batch.

Implementing the resume feature for batch conversions required keeping track
of the files that had been converted already, and to not abort the whole
conversion process when an error is encountered. To deal with both of these, a
basic logging system was implemented. When a directory is selected for batch
conversion, a log file is created inside it. As the software goes through the
directory trying to convert (sensible) files, it writes to the log whether
each file was successful or not, and the reason for failure. If not all files
are successful, this log can be used by the GUI to reconstruct the list of
files it already converted successfully, saving time for the user by only
converting the files in the batch that failed, and which were presumably
fixed. It is also intended to be useful to users in finding out why a
conversion failed, and logs printed representations of all errors encountered.

###  2.3 Tools used

####  2.3.1 GUI design

The GUI is built using Python’s Tkinter toolkit [9]. The primary reason this
was chosen was because of all the graphical user interface toolkits available
to Python, Tkinter appeared to support the most interactivity during
development. A high degree of interactivity can improve development
productivity by reducing the time between writing code and seeing the results,
and in the ideal case allows updates on the fly so that bugs can be fixed
literally seconds after identification. Interactivity becomes even more
valuable when there is significant uncertainty in the development process,
such as when developing a novel graphical user interface for a workflow
previously based on the command line and Python programming. To this end, a
prototype of the GUI was developed with the Common Lisp language and the
Common Lisp Omnificent GUI (CLOG) toolkit [10]. when the code was changing
quickly, CLOG was a fast and convenient way to develop the GUI. After the
initial development steps, the GUI was ported to the target language and
toolkit, Python and Tkinter.

####  2.3.2 GUI distribution

Once the GUI was ready to be distributed to others for testing, a new set of
tools for deployment were required. Since we are writing this GUI for users
who are not expected to manually install a Python package and dependencies
with a package manager like pip or conda, the GUI should be distributed as a
self-contained executable that the user need only download and double click to
get started. To achieve this, we used pyinstaller to package up the GUI code
along with all of its dependencies, including the Python interpreter, with the
exception of things such as glibc which are provided by the operating system.
At this early stage, the GUI is packaged as an executable within a zipped up
directory structure that contains all the mentioned dependencies. This permits
easier debugging. In later stages, a single self-contained executable, rather
than the executable plus dependencies folder used for development, will be
created and released on Github, all that is needed being a command flag in the
build process.

####  2.3.3 Build automation

In order to provide multi-platform releases (at least on the most popular ones
in the SMSF community, namely Linux, Mac, and Windows) without the burden of
supporting dedicated machines for the builds, the AppVeyor online platform was
used [11]. As part of getting software into the hands of testers, a continuous
integration (CI) pipeline connected to the GUI’s Github repository was
developed using AppVeyor, starting from the configuration used by phconvert.
Each time a commit is pushed, the pipeline builds the executable bundles on
both Linux and Windows, then uploads them to the Github releases page.
Initially, we intended to release an executable running on macOS, but
unresolved issues related to Apple’s security policies have led to us to
postpone it. Contributions towards resolving these problems are welcome.

While developing the CI pipeline, the requirement that the code snippets in
the configuration file be runnable on every platform being targeted, except if
flagged otherwise, initially caused some issues. The crux of the matter was
that each platform had a different shell. Windows had batch/cmd.com, Linux had
Bash, and Mac had Zsh. At the face of it, this meant that large chunks of code
would need to be copied between the different platforms’ build processes, each
slightly tweaked for the specific platform. This is bad from a maintainability
standpoint, since it multiplies the amount of work necessary to make a single
change and the repetitiveness of the code provides a place for bugs to hide
and the opportunity for things to be forgotten to be changed. What was wanted
was a way to write the build process code in the AppVeyor configuration file
such that it would be valid on all platforms, with as much reuse as possible
so that only the things which absolutely needed to be different (such as
installation of software on Linux and Mac that was already present on Windows)
were different. Luckily, one common denominator across all the AppVeyor images
used is the installation of the Powershell or Powershell Core (pwsh) shell.
Additionally, AppVeyor provides a means in the configuration file to mark code
as pwsh snippets/blocks to be run. This provided the necessary common language
between platforms. As of now, all code in the AppVeyor configuration file with
the exception of one special case is written in pwsh, for better
maintainability.

##  3 Software demonstration

In the following subsections, we will demonstrate the GUI’s basic features by
illustrating how files can be converted to Photon-HDF5.

###  3.1 Installation

The current installation process for the GUI is as follows:

  1. 1.

Go to https://github.com/ramenbytes/phui, which is the working repository for
the GUI right now. PHUI stands for Phconvert User Interface.

  2. 2.

Click on Releases.

  3. 3.

Download the zip file of your choice corresponding to your platform and chosen
Python version. For Linux, choose an Ubuntu-labeled release, and for Windows,
choose a Visual Studio one.

  4. 4.

Extract the downloaded zip file.

  5. 5.

Inside the zip folder, descend into the dist/gui folder.

  6. 6.

Double-click the gui executable to run the GUI.

The only system requirements are that the user be on a platform at least as
up-to-date as the Appveyor image used to build the executable. Currently, that
is Ubuntu 18.04.4 LTS (Bionic Beaver) and Windows 10. They do not need to be
on the _same_ system, just one whose system libraries (such as glibc on Linux)
are contemporary to the build image’s. The rest of the dependencies are taken
care of by the build process and are stashed in the distributed folder
containing the executable which the user downloads.

###  3.2 Single-File Conversion

Figure 1: Initial window shown to users upon starting the GUI.

The simplest use case for the GUI is the conversion of a single file. Upon
starting the GUI, the user is greeted by the window in Figure 1. Starting in
the top left, the initially empty choose a target field displays the filename
of the data file the user wants to convert. To select a file for conversion,
the user clicks on the Select a file button to the immediate right of the
field. The user is presented with a file dialog which they can use to navigate
to and select the desired file. After a file is chosen, metadata needs to be
associated with it. There are currently several methods of doing so, and more
planned.

  * •

First, the user can provide a description in the white textbox near the bottom
of the window. The Photon-HDF5 format requires a description, so the
conversion process will fail if one isn’t provided.

  * •

Second, phconvert provides default values for certain metadata fields. Absent
user input, these values will be used. To override these default values or to
provide values for other fields, users currently need to use the third method,

  * •

Third, clicking the
bfSelect metadata file button to choose a YAML file containing the relevant
info. This is intended as a stopgap solution until a graphical metadata editor
is developed, freeing the users from worrying about YAML files. However, in
its current state, the conversion process is not entirely YAML-free.

After a data file has been chosen, and metadata associated with it, users will
need to press the Convert button. If the conversion is successful (no errors
are thrown during the process), then the status bar near the bottom of the
window (grey, with the label Unconverted in Fig. 1) will change to green, as
shown in Fig. 2.

Figure 2: A successful conversion

If an error occurs during the conversion process, the status bar changes to
red with a message noting the failure, and a dialog window with a printed
description of the error is presented to the user as seen in Fig. 3.

Figure 3: An unsuccessful conversion

The conversion can be retried by closing the error dialog, and pressing
Convert again. If the user fixes the error before trying again, the conversion
should succeed this time around.

###  3.3 Batch Conversion

Directly to the right of the Select a file button is the Select a directory
button, as seen in Fig. 1. This button allows a user to select an entire
directory as a candidate for conversion, as opposed to a single file. When the
user attempts to convert the directory, a subroutine will gather a list of all
the files in the provided directory that can be converted, based on the
filename extensions. Everything else is excluded from the conversion attempt.
Then, one by one, the program tries to convert each of these files, recording
in a log whether a file conversion was successful, and if not, what the error
was.

A user can try re-running the batch by clicking Convert again; this time, only
the files marked as failures in the log will be tried again. This is designed
such that long batch conversions are not repeated needlessly. At the end of
each attempt, the status bar shows the fraction of unconverted files that were
successfully converted on this attempt, how many were converted in previous
attempts, and where to find the log file. See Fig. 4 for an example.

Figure 4: Status from a partial batch conversion. At the start of this
attempt, one file was unconverted. Nine were logged as previously converted.
This attempt failed to convert the single unconverted file. The progress log
at the provided path lists the error which resulted in the failure.

The metadata for batch conversions is obtained in the same way as for single
file conversions, however it will be applied to _all_ files in the batch. That
means that currently, batch conversions are only suitable for groups of files
that have identical metadata. For example, data from several runs of the same
experiment. Offering more flexibility in different scenarios is a top priority
for future development of the GUI.

###  3.4 Multiple Targets

The last two buttons of interest are Remove target in the top right, and Add a
conversion target on the bottom of Fig. 1. These allow the addition of
additional individual conversion “tasks” for the GUI to perform. Adding
targets allows the user to add files or batches of files, each of which is
handled individually with its own associated metadata, status bar, and button
for conversion. See Fig. 5 for an example.

Figure 5: Multiple targets for conversion. One successful file (top), one
partially successful batch (second from top), one failed file (second from
bottom), and one target that hasn’t been attempted yet and does not have any
data or metadata (bottom).

Using the Remove target button, users can remove files or batches of files
from the GUI’s display, for whatever the reason may be.

##  4 CONCLUSION

We have developed a GUI which lowers, but does not yet eliminate, the barriers
faced by users in the adoption of Photon-HDF5. Absolutely no Python or command
line experience is necessary to use the GUI, which is certainly an
improvement. The primary remaining barriers are in metadata storage and
discoverability.

###  4.1 Metadata

The GUI allows users to associate metadata with the data sets being converted,
but in a very limited manner. Anything different than the defaults provided by
phconvert must be provided via the fallback approach, i.e., manually entering
the metadata into a YAML file using a text editor. Unfortunately, because of
the variety of different experimental setups in the photon-counting community,
most data sets will not perfectly match the defaults provided and the YAML
file option will need to be used to correct the default values.

At this stage, it is also not immediately apparent to the user that there are
default values for fields, or even which fields exist. As of now, nothing is
displayed in the GUI that indicates the presence of these fields and that the
defaults from phconvert are being used. This means that if a user is not
familiar with the GUI, it is very easy for them to be fooled into thinking
that they have successfully converted a file to Photon-HDF5, when in fact that
is only partially true. While their experimental data may have successfully
been converted, it is now associated with the default values provided by
phconvert, which are not guaranteed to make sense in the user’s specific case.

However, this temporary limitation can easily be avoided since most users will
typically generate similar files with the hardware at their disposal.
Therefore, proper use of the GUI will be guaranteed if the users use the same
YAML file, adapted to their setup, simply modifying the few fields identifying
each experiment before running the GUI.

Since the GUI allows entering a new data description for each new converted
file, we believe that this functionality of the software will help a number of
laboratories to convert their files, once they are in possession of an
appropriate template YAML file. We are planning to provide a few examples of
such YAML files with the first release, and encourage users to publicize
theirs, together with a detailed description of their setups, so that the
community can benefit from best practices from the leading labs.

Future developments of the GUI will explore the possibility of implementing a
basic metadata editor in the GUI, which presents all the fields to the user
and allows them to enter values into them.

A related issue is that for batch conversions, there is currently no way of
specifying metadata on a per-file basis. This restricts the set of data files
which can be usefully and correctly batch converted to those which are
differing minimally or not at all. If even one metadata field is different
between two files, they will have to be processed in separate batches or
individually. Ideally, users would be able to have a fine-grained control over
the metadata for each file in a batch, perhaps being able to set defaults for
the entire batch and overriding or supplementing them on a per-file basis.

Finally, even if users can easily associate any metadata they want with a file
and/or batch, they still need to know what fields are required in the first
place. The Photon-HDF5 specification is quite rich, and learning what metadata
needs to be associated with the data from which experimental setup is
challenging for a first time user. Beyond the suggestion made above of using a
standardized YAML file for a given setup or set of experiments, one possible
(partial) solution would be to add a workflow in the GUI that walks the user
through the process, guiding them towards the fields they need to include.

###  4.2 Documentation

A definite possibility for future improvements would be documentation for the
GUI, complete with clear instructions displayed in the repository README file
detailing how to install the GUI on each of the supported platforms and the
system requirements for each.

###  4.3 Releases and Contributions

This project is hosted on Github at https://github.com/ramenbytes/phui, and
the executables and source code can be found there. During this stage of
development, all contributions such as pull requests or issues should be
opened against this repository. However, it is hoped that in the future PHUI
will be merged with the phconvert repository, in which case a notice will be
posted to the above repository directing users to the new location.

##  5 ACKNOWLEDGEMENTS

This work was funded in part by NIH grant R01 GM130942, NSF awards MCB 1818147
and EAGER 1842951, and by HFSP grant RGP0061/2019.

DF is grateful for SPIE’s and MKS Instruments’ Student Conference Support
Program grant.

DF also thanks Rio Hondo College, the UCLA-RHC Scientific Exchange, and
Christian Vaca for initiating this collaboration, and for providing support
and mentorship throughout.

## References

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# A possible interpretation of $\Lambda$ baryon spectrum with pentaquark
components

Kai Xu1, Attaphon Kaewsnod1, Zheng Zhao1, Thiri Yadanar Htun1,2, Ayut
Limphirat1, Yupeng Yan1 1 School of Physics and Center of Excellence in High
Energy Physics and Astrophysics, Suranaree University of Technology, Nakhon
Ratchasima 30000, Thailand  2 Department of Physics, University of Mandalay,
05032 Mandalay, Myanmar<EMAIL_ADDRESS><EMAIL_ADDRESS>and
<EMAIL_ADDRESS>
###### Abstract

The $\Lambda$ baryon spectrum is studied within the SU(3) flavor symmetry in a
constituent quark model. We found that it is rather difficult to accommodate
some negative-parity $\Lambda$ resonances as single $q^{2}s$ ($q=u,\,d$
quarks) states in the conventional three-quark picture. The ground
$q^{3}s\bar{q}$ pentaquark mass spectrum is evaluated and a possible
interpretation is proposed in the work: the observed $\Lambda(1405)1/2^{-}$,
$\Lambda(1670)1/2^{-}$ and $\Lambda(1800)1/2^{-}$ are three-state mixtures of
two $p$-wave $q^{2}s$ states and one ground $q^{3}s\bar{q}$ pentaquark state,
so are the $\Lambda(1520)3/2^{-}$, $\Lambda(1690)3/2^{-}$ and
$\Lambda(2050)3/2^{-}$ resonances.

Keywords: $\Lambda$ baryon mass spectrum, pentaquark mass spectra, three-body
mixture

  * October 2022

##  1 Introduction

In recent decades, new experimental approaches have been applied to study the
$\Lambda$ resonances. Experimental data are mainly from $K^{-}p$ invariant
mass spectra of hyperons below 1.7 GeV [1, 2, 3], partial wave analyses of
$K^{-}p$ reactions with energies from 1480 to 2100 MeV [4] and photoproduction
of $\Lambda(1405)$ from JLab [5, 6]. The $\Lambda$ resonances are listed in
table 1 [7].

Recent data analyses have confirmed most 3* and 4* $\Lambda$ states, but
derived rather different information of some little known states. Recent
couple channel analyses from the Kent group (KSU) and JLab have extracted more
hyperons [8, 11, 9, 10, 12, 13, 14, 15]. The Kent group found some new states
like $\Lambda$(1710)$1/2^{+}$ and $\Lambda$(2050)$3/2^{-}$, which were listed
in Particle Data Group (PDG) [7] with one star, by fitting partial-wave
amplitudes using multichannel parametrization [8]. A similar couple-channel
fit has been done by JPAC group [11], where $\Lambda$(2050) is confirmed by
this analysis but $\Lambda$(1800) is not seen. A dynamical coupled-channel
model applied by the Osaka group [9, 10] results in no clear existence of
$\Lambda$(1710), $\Lambda$(1800), $\Lambda$(2000) and $\Lambda$(2050). Bonn-
Gatchina (BnGa) group has recently reported $\Lambda$(1710)$1/2^{+}$,
$\Lambda$(2070)$3/2^{+}$ and $\Lambda$(2080)$5/2^{-}$, suggesting all the
states together with $\Lambda(1405)$$1/2^{-}$ as SU(3) singlet states [12, 13,
14, 15]. The experimental data from $K^{-}p$ induced reactions have made a
great contribution to our knowledge of the $\Lambda$ baryon spectrum, but it
is still insufficient to reveal the nature of all $\Lambda$ resonances.

Table 1: $\Lambda$ resonances (Data from PDG).

  * Particle | Status | $J^{P}$ |  $M_{BW}^{exp}$ ${\rm MeV}$ |  $\Gamma_{BW}^{exp}$ ${\rm MeV}$
---|---|---|---|---
$\Lambda(1116)$ | **** | $\frac{1}{2}^{+}$ | 1116 | -
$\Lambda(1600)$ | **** | $\frac{1}{2}^{+}$ | 1570-1630 | 150-250
$\Lambda(1710)$ | * | $\frac{1}{2}^{+}$ | 1700-1726 | 138-222
$\Lambda(1810)$ | *** | $\frac{1}{2}^{+}$ | 1740-1840 | 50-170
$\Lambda(1820)$ | **** | $\frac{5}{2}^{+}$ | 1815-1825 | 70-90
$\Lambda(1890)$ | **** | $\frac{3}{2}^{+}$ | 1870-1910 | 80-160
$\Lambda(2070)$ | * | $\frac{3}{2}^{+}$ | 2046-2094 | 320-420
$\Lambda(2085)$ | ** | $\frac{7}{2}^{+}$ | 2000-2040 | 130-190
$\Lambda(2110)$ | *** | $\frac{5}{2}^{+}$ | 2050-2130 | 200-300
$\Lambda(2350)$ | *** | $\frac{9}{2}^{+}$ | 2340-2370 | 100-250
$\Lambda(1380)$ | ** | $\frac{1}{2}^{-}$ | 1310-1340 | -
$\Lambda(1405)$ | **** | $\frac{1}{2}^{-}$ | 1404-1406 | 48.5-52.5
$\Lambda(1520)$ | **** | $\frac{3}{2}^{-}$ | 1518-1520 | 15-17
$\Lambda(1670)$ | **** | $\frac{1}{2}^{-}$ | 1670-1678 | 25-35
$\Lambda(1690)$ | **** | $\frac{3}{2}^{-}$ | 1685-1695 | 60-80
$\Lambda(1800)$ | *** | $\frac{1}{2}^{-}$ | 1750-1850 | 150-250
$\Lambda(1830)$ | **** | $\frac{5}{2}^{-}$ | 1820-1830 | 60-120
$\Lambda(2000)$ | * | $\frac{1}{2}^{-}$ | 2000-2060 | 100-150
$\Lambda(2050)$ | * | $\frac{3}{2}^{-}$ | 2034-2078 | 432-554
$\Lambda(2080)$ | * | $\frac{5}{2}^{-}$ | 2069-2095 | 152-210
$\Lambda(2100)$ | **** | $\frac{7}{2}^{-}$ | 2090-2110 | 100-250
$\Lambda(2325)$ | * | $\frac{3}{2}^{-}$ | 2307-2347 | 120-200

The $\Lambda$ spectra have been studied in various quark models theoretically,
for example, the Isgur-Karl model [18, 19, 20] which includes the one-gluon
exchange quark-quark interaction, models taking into account the Goldstone
boson exchange interaction [21], models considering the relativistic
instanton-induced quark-quark interaction [22], and relativistic quark models
with quark-diquark interactions [23, 24].

The traditional constituent quark models always have difficulties in
describing the low mass of $\Lambda(1405)$ [25] in the $q^{3}$ picture since a
heavier strange quark is contained. The $\Lambda$(1405) resonance, since
discovered in the $\pi\Sigma$ invariant mass spectrum [26] in 1960s, has been
an interesting state in the hyperon spectrum, the nature of the resonance is
still under debate. $\Lambda$(1405) was first generated dynamically as a
$N\bar{K}$ quasi-bound molecular state [27, 28], then followed a large number
of studies in chiral unitary approach (ChUA) [29, 30, 31, 32, 33] and other
chiral dynamics on this resonance [34, 35, 36, 37, 43, 42, 44, 45, 38, 39, 40,
41, 46]. By solving the coupled channel Lippmann-Schwinger equation with a
chiral meson-baryon Lagrangian at the next-to-leading order, the
$\Lambda$(1405) resonance as a quasi-bound state of $N\bar{K}$ described the
low-energy $K^{-}p$ scattering data well [34]. Limited to the lowest-order
chiral Lagrangian with coupled channel method, $\Lambda$(1405) was identified
as quasi-bound states of $N\bar{K}$ using the N/D method and dispersion
relations [35]. The two-poles structure of $\Lambda(1405)1/2^{-}$ was firstly
proposed in the complex energy plane for the different Riemann sheets using
the ChUA in 2001 [29]. Then a variety of investigations on the two-poles
structure of the $\Lambda$(1405) resonance were carried out. The contributions
of the two-pole structure to the lineshape of $\Lambda$(1405) were shown in
Refs. [30, 36]. After the SIDDHARTA kaonic hydrogen result [16, 17] was
reported, different works [37, 38, 39, 40, 41, 42] revealed the real and
imaginary parts of the two poles of $\Lambda(1405)1/2^{-}$. New fits including
the CLAS photoproduction data were carried out in the works [31, 32, 33, 43]
and the origin of two poles of $\Lambda(1405)1/2^{-}$ were discussed as well.
In recent years, the first high-mass pole of $\Lambda(1405)1/2^{-}$ was shown
to be mainly a $\bar{K}N$ molecular, the second pole was interpreted as a
composite state of mainly $\pi\Sigma$ with tiny component $\bar{K}N$ in Refs.
[44, 45, 46], and the influence of SU(3) flavor symmetry was analyzed in the
last work [46]. Exotic configurations for $\Lambda(1405)$ have also been
proposed, such as a mixed pentaquark and three-quark state [47], a hybrid
baryon [48], a mixed strange hybrid/three-quark baryon [49] which all applied
the method of the QCD sum rule. A mixed three-quark and meson-baryon molecular
states for the $\Lambda(1405)$ resonance are studied in Refs. [50, 51] in
quark models. The electromagnetic form factors of the $\Lambda(1405)$
resonance were calculated in Lattice QCD for the first time to reveal the
inner structure of $\Lambda(1405)$ [52, 53].

In the present work we extend the non-relativistic quark model employed in the
works [54, 55] to study the $\Lambda$ mass spectra including light
$q^{3}s\bar{q}$ pentaquark components. The lowest two pairs of negative-parity
$\Lambda$ resonances are reproduced by mixing up the three-quark
$\Lambda_{1}^{*}$ (flavor-singlet), $\Lambda_{8}$ (flavor-octet) states with
the ground $q^{3}s\bar{q}$ pentaquark states. The paper is organized as
follows. In section 2 we briefly introduce the Hamiltonian for multi-quark
systems including the spin-orbit couplings. The masses of low-lying $q^{2}s$
states are also calculated in section 2. In section 3, we derive the mass
spectra of ground $q^{3}s\bar{q}$ pentaquark states and propose an
interpretation of the lowest two pairs of negative-parity $\Lambda$ resonances
by introducing the ground $q^{3}s\bar{q}$ pentaquark components in the three-
state mixtures. A discussion and a short summary are given in sec. 4.

##  2  $\Lambda$ resonances as $q^{2}s$ states

We calculate the mass of low-lying $q^{2}s$ states in the Hamiltonian, which
is an extend of the model in Ref. [54] by including the spin-orbit
interactions,

$\displaystyle H=H_{0}+H_{hyp}^{OGE}+H^{SO},$ $\displaystyle
H_{0}=\sum_{k=1}^{N}(m_{k}+\frac{p_{k}^{2}}{2m_{k}})+\sum_{i<j}^{N}(-\frac{3}{8}\lambda^{C}_{i}\cdot\lambda^{C}_{j})(A_{ij}r_{ij}-\frac{B_{ij}}{r_{ij}}),$
$\displaystyle
H_{hyp}^{OGE}=-C_{OGE}\sum_{i<j}\frac{\lambda^{C}_{i}\cdot\lambda^{C}_{j}}{m_{i}m_{j}}\,\vec{\sigma}_{i}\cdot\vec{\sigma}_{j},$
$\displaystyle H^{SO}=2k_{i}\,d_{j}\;\vec{L}\cdot\vec{S}.$ (1)

where $A_{ij}$ and $B_{ij}$ are mass-dependent coupling parameters, taking the
form

$\displaystyle
A_{ij}=a\sqrt{\frac{m_{ij}}{m_{u}}},\;\;B_{ij}=b\sqrt{\frac{m_{u}}{m_{ij}}}.$
(2)

with $m_{ij}$ being the reduced mass of $i$th and $j$th quarks, defined as
$m_{ij}=\frac{m_{i}m_{j}}{m_{i}+m_{j}}$. The hyperfine interaction,
$H_{hyp}^{OGE}$ includes only one-gluon exchange contribution (OGE), where
$C_{OGE}=C_{m}\,m_{u}^{2}$, with $m_{u}$ being the constituent $u$ quark mass
and $C_{m}$ a constant. $\lambda^{C}_{i}$ in the above equations are the
generators of color SU(3) group. The three model coupling constants and two
constituent quark masses are taken from the previous work [55],

$\displaystyle m_{u}=m_{d}=327\ {\rm MeV}\,,\quad m_{s}=498\ {\rm MeV}\,,$
$\displaystyle C_{m}=18.3\ {\rm MeV},\quad a=49500\ {\rm MeV^{2}},\quad
b=0.75$

$H^{SO}$ in equation (2) is the spin-orbit interaction, taking the simplified
form as in Refs. [18, 19, 20], where $L$ and $S$ are the total orbital angular
and spin operators of baryon states. $d_{j}$ are $SU(3)_{F}$ flavor multiplet-
dependent constants, taking the values,

$\displaystyle d_{1}=2/3\,,\quad d_{8}=1\,,\quad d_{10}=1/3$ (4)

$k_{i}$ are $SU(6)_{SF}$ spin-flavor supermultiplet-dependent constants, which
are determined by fitting the theoretical results to the mass splitting of
spin-orbit pairs in the $\Lambda$, $\Sigma$ and $\Sigma^{*}$ spectra in the
pure $q^{3}$ picture. The mass difference between the two
$\Lambda(70,{}^{4}8,1,1^{-})$ states $\frac{1}{2}^{-}$ $\Lambda$(1800) and
$\frac{5}{2}^{-}$ $\Lambda$(1830) leads to $k_{1}\approx 4$ MeV while the gap
between the $\frac{3}{2}^{-}$ $\Sigma$(1580) and $\frac{1}{2}^{-}$
$\Sigma$(1620) states results in $k_{1}\approx 3$ MeV. The $(70,1^{-})$ and
$(20,1^{+})$ supermultiplets have a direct relation in the lowest harmonic
oscillation approximation, which results in $k_{2}\approx 1.5\;k_{1}\approx
6\;{\rm MeV}$. $k_{3}$ is determined about $-14$ MeV by the
$\Lambda(56,{}^{2}8,2,2^{+})$ states $\Lambda(1820)$ and $\Lambda(1890)$, and
about $-5$ MeV by the $\Sigma(56,{}^{2}8,2,2^{+})$ states $\Sigma(1915)$ and
$\Sigma(1940)$. The mass gap between the $\Lambda(70,{}^{4}8,2,2^{+})$ states
$\Lambda(2110)$ and $\Lambda(2085)$ leads to $k_{4}\approx-14$ MeV, and the
one between the $\Sigma(70,{}^{4}8,2,2^{+})$ states $\Sigma(2070)$ and
$\Sigma(2030)$ leads to $k_{4}\approx-6$ MeV. In this work, we use averaged
values,

$\displaystyle k_{1}([70,1^{-}])=4\ {\rm MeV}\,,k_{2}([20,1^{+}])=6\ {\rm
MeV}\,,$ $\displaystyle k_{3}([56,2^{+}])=-10\ {\rm
MeV}\,,k_{4}([70,2^{+}]))=-10\ {\rm MeV}$ (5)

The positive-parity and negative-parity $\Lambda$ resonances of the first and
second excitation bands are studied in the Hamiltonian in equation (2) in the
three-quark picture. The color-orbital-spin-flavor wave functions of $q^{2}s$
baryon states in the $SU(6)_{SF}$ representations, which are applied in the
study, are listed in Appendix A. The theoretical masses and tentative
matchings between data and the theoretical results are shown in table 2.

Table 2: Theoretical $\Lambda$ resonances matched with data up to the second
excitation band. $\Lambda^{*}$ stands for the flavor-singlet states. $\Gamma$
and $D$ stands respectively for the $SU(6)_{SF}$ and $SU(3)_{F}$ multiplets,
$N$, $S$, $L^{P}$ and $J^{P}$ are the principle quantum number, total spin,
orbital and angular momentum with $P$ being the parity.

  * $(\Gamma,{}^{2S+1}D,N,L^{P})$ | Status | $J^{P}$ | $M^{exp}{\rm(MeV)}$ | $M^{cal}{\rm(MeV)}$
---|---|---|---|---
$\Lambda(56,{}^{2}8,0,0^{+})$ | **** | $\frac{1}{2}^{+}$ | $\Lambda(1116)$ | 1112
$\Lambda(56,{}^{2}8,2,0^{+})$ | **** | $\frac{1}{2}^{+}$ | $\Lambda(1600)$ | 1689
$\Lambda(56,{}^{2}8,2,2^{+})$ | **** | $\frac{5}{2}^{+}$ | $\Lambda(1820)$ | 1825
$\Lambda(56,{}^{2}8,2,2^{+})$ | **** | $\frac{3}{2}^{+}$ | $\Lambda(1890)$ | 1875
$\Lambda(20,{}^{2}8,2,1^{+})$ | - | $\frac{1}{2}^{+}$ | missing | 1927
$\Lambda(20,{}^{4}8,2,1^{+})$ | - | $\frac{3}{2}^{+}$ | missing | 1945
$\Lambda^{*}(20,{}^{4}1,2,1^{+})$ | - | $\frac{1}{2}^{+}$ | missing | 2162
$\Lambda^{*}(20,{}^{4}1,2,1^{+})$ | - | $\frac{3}{2}^{+}$ | missing | 2174
$\Lambda^{*}(20,{}^{4}1,2,1^{+})$ | - | $\frac{5}{2}^{+}$ | missing | 2194
$\Lambda(70,{}^{2}8,2,0^{+})$ | *** | $\frac{1}{2}^{+}$ | $\Lambda(1810)$ | 1821
$\Lambda(70,{}^{4}8,2,0^{+})$ | * | $\frac{3}{2}^{+}$ | $\Lambda(2070)$ | 2081
$\Lambda^{*}(70,{}^{2}1,2,0^{+})$ | - | $\frac{1}{2}^{+}$ | $\Lambda(1710)$ | 1838
$\Lambda(70,{}^{2}8,2,2^{+})$ | **** | $\frac{3}{2}^{+}$ | $?\Lambda(1890)$ | 1922
$\Lambda(70,{}^{2}8,2,2^{+})$ | **** | $\frac{5}{2}^{+}$ | $?\Lambda(1820)$ | 1872
$\Lambda(70,{}^{4}8,2,2^{+})$ | - | $\frac{1}{2}^{+}$ | missing | 2242
$\Lambda(70,{}^{4}8,2,2^{+})$ | - | $\frac{3}{2}^{+}$ | missing | 2212
$\Lambda(70,{}^{4}8,2,2^{+})$ | *** | $\frac{5}{2}^{+}$ | $\Lambda(2110)$ | 2142
$\Lambda(70,{}^{4}8,2,2^{+})$ | ** | $\frac{7}{2}^{+}$ | $\Lambda(2085)$ | 2092
$\Lambda^{*}(70,{}^{2}1,2,2^{+})$ | - | $\frac{3}{2}^{+}$ | missing | 1929
$\Lambda^{*}(70,{}^{4}1,2,2^{+})$ | - | $\frac{5}{2}^{+}$ | missing | 1896
$\Lambda(70,{}^{2}8,1,1^{-})$ |  | $\frac{1}{2}^{-}$ |  | 1579
$\Lambda(70,{}^{2}8,1,1^{-})$ |  | $\frac{3}{2}^{-}$ |  | 1591
$\Lambda(70,{}^{4}8,1,1^{-})$ | *** | $\frac{1}{2}^{-}$ |  $\Lambda$(1800) | 1810
$\Lambda(70,{}^{4}8,1,1^{-})$ | **** | $\frac{5}{2}^{-}$ |  $\Lambda$(1830) | 1842
$\Lambda(70,{}^{4}8,1,1^{-})$ | - | $\frac{3}{2}^{-}$ | missing | 1822
$\Lambda^{*}(70,{}^{2}1,1,1^{-})$ |  | $\frac{1}{2}^{-}$ |  | 1582
$\Lambda^{*}(70,{}^{2}1,1,1^{-})$ |  | $\frac{3}{2}^{-}$ |  | 1590

In the second excitation band shown in table 2, the positive-parity $\Lambda$
resonances belonging to the 20-supermultiplet for both the singlet and octet
flavors are missing. $\Lambda$(1820) and $\Lambda$(1890) are matched with the
spin $\frac{5}{2}$ and $\frac{3}{2}$ states in the 56-multiplets respectively,
but it can not be ruled out that the two resonances are matched with the
70-multiplet $\frac{5}{2}$ and $\frac{3}{2}$ states. The 3-star
$\Lambda$(1810) state is assigned as a $\Lambda(70,{}^{2}8,2,0^{+})$ state in
this work. This resonance $\Lambda$(1710) reported by KSU [8] in 2013 is
interpreted as the $\Lambda^{*}(70,{}^{2}1,2,0^{+})$ singlet state in Ref.
[15], but it can’t be described by the relativistic interacting quark-diquark
models [23, 24]. Though a higher mass is predicted for the flavor-singlet
state $\Lambda^{*}(70,{}^{2}1,2,0^{+})$ in the work, we may tentatively assign
the 1-star $\Lambda$(1710) resonance as the $\Lambda^{*}(70,{}^{2}1,2,0^{+})$
state, considering the huge width of the $\Lambda$(1710) resonance. The
resonances $\Lambda(2070)$$\frac{3}{2}^{+}$ reported by BnGa group in 2019
[13] is assigned as the $\Lambda^{*}(70,{}^{2}1,2,2^{+})$ state in Ref. [15],
but this work prefers it being the $\Lambda(70,{}^{4}8,2,0^{+})$ state. And
the resonance $\Lambda(2110)$$\frac{5}{2}^{+}$ may be assigned as the
$\Lambda(70,{}^{4}8,2,2^{+})$ state.

For the $L=1$ first excitation band, we may interpret the
$\Lambda$(1800)$\frac{1}{2}^{-}$ and $\Lambda$(1830)$\frac{5}{2}^{-}$
resonances as three-quark states in the $\Lambda(70,{}^{4}8,1,1^{-})$
multiplet, as shown in table 2. The spin $\frac{3}{2}^{-}$ state, the spin-
orbit coupling partner of the $\Lambda$(1800)$\frac{1}{2}^{-}$ and
$\Lambda$(1830)$\frac{5}{2}^{-}$ resonances has not been reported. The two
predicted flavor-singlet and two flavor-octet $\Lambda$ states of spin
$\frac{1}{2}^{-}$ and $\frac{3}{2}^{-}$ lay rather far from the observed
lowest negative-parity $\Lambda$ resonance states, thus we may not make any
matching for these four negative-parity $\Lambda$ resonances in the pure 3q
picture.

Table 3: $q^{3}s\bar{q}$ ground state pentaquark masses in ${\rm MeV}$.
$J^{P}$ is the total angular momentum with $P$ being parity, and $S$ is the
total spin of ground state pentaquarks, from the coupling of the spins of
$q^{3}s$ and $\bar{q}$.

  * $J^{P}$ |  $q^{3}s\bar{q}$ configurations | $(S^{q^{3}s},S^{\bar{q}},S)$ | $M(q^{3}s\bar{q})$
---|---|---|---
$\frac{5}{2}^{-}$ | $\Psi^{csf}_{[211]_{C}[31]_{FS}[31]_{F}[4]_{S}}(q^{3}s\bar{q})$ | (2,1/2,5/2) | 2408
$\frac{3}{2}^{-}$ | $\Psi^{csf}_{[211]_{C}[31]_{FS}[4]_{F}[31]_{S}}(q^{3}s\bar{q})$ | (1,1/2,3/2) | 2392
| $\left\\{\begin{array}[]{c}\Psi^{csf}_{[211]_{C}[31]_{FS}[31]_{F}[4]_{S}}(q^{3}s\bar{q})\\\ \Psi^{csf}_{[211]_{C}[31]_{FS}[31]_{F}[31]_{S}}(q^{3}s\bar{q})\end{array}\right\\}$ |  (2,1/2,3/2) (1,1/2,3/2) | $\left(\begin{array}[]{c}1966\\\ 2407\end{array}\right)$
| $\Psi^{csf}_{[211]_{C}[31]_{FS}[211]_{F}[31]_{S}}(q^{3}s\bar{q})$ | (1,1/2,3/2) | 2116
| $\Psi^{csf}_{[211]_{C}[31]_{FS}[22]_{F}[31]_{S}}(q^{3}s\bar{q})$ | (1,1/2,3/2) | 2229
$\frac{1}{2}^{-}$ | $\Psi^{csf}_{[211]_{C}[31]_{FS}[4]_{F}[31]_{S}}(q^{3}s\bar{q})$ | (1,1/2,1/2) | 2659
| $\left\\{\begin{array}[]{c}\Psi^{csf}_{[211]_{C}[31]_{FS}[31]_{F}[31]_{S}}(q^{3}s\bar{q})\\\ \Psi^{csf}_{[211]_{C}[31]_{FS}[31]_{F}[22]_{S}}(q^{3}s\bar{q})\end{array}\right\\}$ |  (1,1/2,1/2) (0,1/2,1/2) | $\left(\begin{array}[]{c}2162\\\ 2314\end{array}\right)$
| $\left\\{\begin{array}[]{c}\Psi^{csf}_{[211]_{C}[31]_{FS}[211]_{F}[31]_{S}}(q^{3}s\bar{q})\\\ \Psi^{csf}_{[211]_{C}[31]_{FS}[211]_{F}[22]_{S}}(q^{3}s\bar{q})\end{array}\right\\}$ |  (1,1/2,1/2) (0,1/2,1/2) | $\left(\begin{array}[]{c}1742\\\ 2052\end{array}\right)$
| $\Psi^{csf}_{[211]_{C}[31]_{FS}[22]_{F}[31]_{S}}(q^{3}s\bar{q})$ | (1,1/2,1/2) | 1894

##  3 Pentaquark components in negative-parity $\Lambda$ resonances

As shown in the previous section, most $\Lambda$ resonances are reasonably
described in the three-quark picture, except for the two pairs of the lowest
negative-parity $\Lambda$ resonances. The flavor-singlet and flavor-octet
$\Lambda$ states may be mixed to give the mass of $\Lambda$(1405) and other
hyperon resonances, as shown in Refs. [15, 49, 56]. In the present model,
however, the masses of the $\Lambda_{1}$ singlet-flavor state and
$\Lambda_{8}$ octet-flavor state are so close that one may make the mixture
unreasonable for deriving the mass of the lowest negative-parity $\Lambda$
resonances. In this work, we assume that the $l=1$ negative-parity $\Lambda$
states may mix with the $q^{3}s\bar{q}$ ground pentaquark states which are
also of negative-parity.

The $q^{3}s\bar{q}$ pentaquarks are studied in the Hamiltonian in equation
(2), where we have considered the couplings among different configurations due
to the one-gloun-exchange (OGE) hyperfine interaction. Listed in table 3 are
the theoretical masses of $q^{3}s\bar{q}$ ground state pentaquarks. There are
one $5/2$, five $3/2$ and six $1/2$ states in the spectra. As shown in Ref.
[54], the color and total spin-flavor parts of pentaquark states are always in
the $[211]_{C}$ and $[31]_{SF}$ configurations. The $q^{3}s$ $[31]$ flavor
configurations can be [4], [31] and [211], where the two possible Weyl
tableaux $\young(uud,s)$ and $\young(uus,d)$ for the $q^{3}s$ $[31]$ flavor
configuration correspond respectively to isospin $I=3/2$ and $I=1/2$ $q^{3}s$
components while the flavor configuration [4] leads to isospin $I=1$ and 2
pentaquark states. We focus only on isospin $I=0$ pentaquark states which may
mix with negative-parity $q^{2}s$ states.

Excluding the spin $\frac{5}{2}^{-}$ state, we have only four spin
$\frac{3}{2}^{-}$ and five $\frac{1}{2}^{-}$ $q^{3}s\bar{q}$ ground pentaquark
states which may mix up with the two pairs of the low $p$-wave $q^{2}s$ states
shown in the lower block of table 2. We mix the two $q^{2}s$ three-quark
states and one $q^{3}s\bar{q}$ pentaquark state,

$\displaystyle\left(\begin{array}[]{c}\psi_{1}\\\ \psi_{2}\\\
\psi_{3}\end{array}\right)=U\left(\begin{array}[]{c}\Lambda_{1}\\\
\Lambda_{8}\\\ q^{3}s\bar{q}\end{array}\right)$ (12)

with U takes the form as,

$\displaystyle\left(\begin{array}[]{ccc}\cos(\theta)\cos(\psi)\cos(\phi)+\sin(\psi)\sin(\phi)&\cos(\psi)\sin(\phi)-\cos(\theta)\sin(\psi)\cos(\phi)&\sin(\theta)\cos(\phi)\\\
\sin(\psi)\cos(\phi)-\cos(\theta)\cos(\psi)\sin(\phi)&\cos(\theta)\sin(\psi)\sin(\phi)+\cos(\psi)\cos(\phi)&-\sin(\theta)\sin(\phi)\\\
-\sin(\theta)\cos(\psi)&\sin(\theta)\sin(\psi)&\cos(\theta)\\\
\end{array}\right)$

Table 4: The three-state mixtures of $q^{2}s$ singlet $\Lambda_{1}^{*}$, octet $\Lambda_{8}$ and ground $q^{3}s\,\bar{q}$ pentaquark states. The $\Lambda_{1}^{*}$ states, $\Lambda_{8}$ states for spin $1/2^{-}$ and $3/2^{-}$ take the theoretical values from table 2. All the possible $1/2^{-}$ and $3/2^{-}$ pentaquark states and masses (in MeV) are from table 3. $\psi_{1}$ State |  $\psi_{2}$ State |  $\psi_{3}$ State | $J^{P}$ |  $\Lambda_{1}$ State |  $\Lambda_{8}$ State | $\theta$ | $\phi$ | $\psi$ |  $q^{3}s\bar{q}$ states
---|---|---|---|---|---|---|---|---|---
1405 |  1670 ($\Lambda(1670)$) | 2248 | $\frac{1}{2}^{-}$ | 1582 | 1579 | $157.3^{\circ}$ | $135.0^{\circ}$ | $90.4^{\circ}$ | 2162
1405 |  1670 ($\Lambda(1670)$) | 2400 | $\frac{1}{2}^{-}$ | 1582 | 1579 | $159.9^{\circ}$ | $135.0^{\circ}$ | $90.3^{\circ}$ | 2314
1405 |  1670 ($\Lambda(1670)$) |  1828 ($\Lambda(1800)$) | $\frac{1}{2}^{-}$ | 1582 | 1579 | $132.8^{\circ}$ | $133.0^{\circ}$ | $94.4^{\circ}$ | 1742
1405 |  1670 ($\Lambda(1670)$) | 2138 | $\frac{1}{2}^{-}$ | 1582 | 1579 | $154.6^{\circ}$ | $134.9^{\circ}$ | $90.6^{\circ}$ | 2052
1405 |  1670 ($\Lambda(1670)$) | 1980 | $\frac{1}{2}^{-}$ | 1582 | 1579 | $148.2^{\circ}$ | $134.9^{\circ}$ | $90.6^{\circ}$ | 1894
1510 |  1660 ($\Lambda(1690)$) |  1977 ($\Lambda(2050)$) | $\frac{3}{2}^{-}$ | 1590 | 1591 | $169.3^{\circ}$ | $136.6^{\circ}$ | $88.0^{\circ}$ | 1966
1510 |  1660 ($\Lambda(1690)$) | 2418 | $\frac{3}{2}^{-}$ | 1590 | 1591 | $173.1^{\circ}$ | $137.0^{\circ}$ | $87.6^{\circ}$ | 2407
1510 |  1660 ($\Lambda(1690)$) | 2127 | $\frac{3}{2}^{-}$ | 1590 | 1591 | $171.2^{\circ}$ | $137.3^{\circ}$ | $87.3^{\circ}$ | 2116
1510 |  1660 ($\Lambda(1690)$) | 2240 | $\frac{3}{2}^{-}$ | 1590 | 1591 | $172.1^{\circ}$ | $137.6^{\circ}$ | $87.0^{\circ}$ | 2229

where $\psi_{1},\psi_{2},\psi_{3}$ are the resulting three negative-parity
physical states from low to high energies corresponding to the observed
negative-parity $\Lambda$ resonances, and $\Lambda_{1}$, $\Lambda_{8}$ and
$q^{3}s\bar{q}$ stand for the flavor-singlet and flavor-octet $q^{2}s$ states
in table 2 and the ground $q^{3}s\bar{q}$ pentaquark states in table 3
respectively. The $3\times 3$ unitary matrix $U$ is parameterized as in Ref.
[57], which is made of three rotation matrices of the angles $\theta$, $\psi$
and $\phi$. The mixing angles are in the domain $0\leq\theta\leq\pi$,
$-\pi<\phi\leq\pi$ and $0\leq\psi<\pi$. Through the unitary transformation of
the mass matrices, one get the mass relation,

$\displaystyle
M_{\Lambda_{1}}+M_{\Lambda_{8}}+M_{q^{3}s\bar{q}}=M_{\psi_{1}}+M_{\psi_{2}}+M_{\psi_{3}}$
(15)

where $M_{\Lambda_{1}}$, $M_{\Lambda_{8}}$ and $M_{q^{3}s\bar{q}}$ are the
masses of the $\Lambda_{1}$ flavor-singlet state, the $\Lambda_{8}$ flavor-
octet state and the ground $q^{3}s\bar{q}$ pentaquark state. $M_{\psi_{3}}$
and the mixing angles are determined by adjusting the $J=1/2$ lower states
$\psi_{1}$ and $\psi_{2}$ respectively to $\Lambda(1405)$ and $\Lambda(1670)$,
and the $J=3/2$ lower states $\psi_{1}$ and $\psi_{2}$ to $\Lambda(1520)$ and
$\Lambda(1690)$. Listed in table 4 are all possible three-state mixtures of
the ground $q^{3}s\bar{q}$ pentaquark states with $J^{P}=\frac{1}{2}^{-}$ and
$\frac{3}{2}^{-}$ in table 3 with the $q^{2}s$ singlet $\Lambda_{1}$ and octet
$\Lambda_{8}$ pairs. And the mixing matrices in equation (25) show the
composition of the physical states in terms of two $q^{2}s$ states and the
lowest pentaquark state. Since it is impossible to derive $M_{\psi_{1}}=1520$
MeV and $M_{\psi_{2}}=1690$ MeV in the present model, we let 1510 MeV and 1660
MeV for $\Lambda(1520)$ and $\Lambda(1690)$ respectively.

##  4 Discussion and Summary

In this work, the masses of low-lying $q^{2}s$ states and ground
$q^{3}s\bar{q}$ states are evaluated. Most positive-parity $\Lambda$
resonances and some negative-parity resonances are well reproduced in the
three-quark picture. A tentative interpretation of the low-lying negative-
parity $\Lambda$ resonances $\Lambda(1405)$, $\Lambda(1520)$, $\Lambda(1670)$
and $\Lambda(1690)$ is proposed, that is, the resonances might be the mixture
of the $p$-wave $q^{2}s$ states and ground $q^{3}s\bar{q}$ pentaquark states.
The theoretical results in equation (25) have shown that
$\Lambda(1405)1/2^{-}$ and $\Lambda(1520)3/2^{-}$ are mainly $q^{2}s$ states,
but $\Lambda(1670)1/2^{-}$ and $\Lambda(1690)3/2^{-}$ contain considerable
$q^{3}s\bar{q}$ pentaquark contributions. Except for the
$\Lambda(1380)1/2^{-}$ resonance which can not be interpreted either as a
three-quark or the mixture of three-quark and pentaquark states, we have
reproduced the whole $\Lambda$ baryon spectrum for $N\leq 2$ bands.

As shown in table 4, the lowest $J=1/2$ and $J=3/2$ $M_{\psi_{3}}$ states are
close to the $\Lambda(1800)$$1/2^{-}$ and $\Lambda(2050)\frac{3}{2}^{-}$
resonance, respectively. The 3-star $\Lambda(1800)$$1/2^{-}$ resonance of
$M=1800\pm 50$ MeV and $\Gamma=200\pm 50$ MeV is tentatively assigned in the
$q^{2}s$ $\Lambda(70,{}^{4}8,1,1^{-})$ multiplet in Section 2. However, the
much larger BW width of the $\Lambda(1800)$$1/2^{-}$ resonance than the spin-
orbit pair state $\Lambda(1830)\frac{5}{2}^{-}$ and the fact that Osaka group
[9, 10] and JPAC [11] analysis found no evidence of this state make it
debatable that the $\Lambda(1800)$$1/2^{-}$ resonance is a pure three-quark
state. One may make a bold guess that the $\Lambda(1800)\frac{1}{2}^{-}$ may
be composed of two resonances, one is the $q^{2}s$
$\Lambda(70,{}^{4}8,1,1^{-})$ state and another the highest state of the
three-state mixture of two lowest negative-parity $q^{2}s$ states and one
ground $q^{3}s\bar{q}$ pentaquark state.

$\displaystyle\left(\begin{array}[]{c}\Lambda(1405)\\\ \Lambda(1670)\\\
\Lambda(1800)\end{array}\right)=\left(\begin{array}[]{ccc}0.69&-0.72&0.06\\\
-0.52&-0.44&0.73\\\ -0.50&-0.54&-0.68\\\
\end{array}\right)\left(\begin{array}[]{c}\Lambda_{1}^{*}\\\ \Lambda_{8}\\\
q^{3}s\bar{q}\end{array}\right)$ (25)
$\displaystyle\left(\begin{array}[]{c}\Lambda(1520)\\\ \Lambda(1690)\\\
\Lambda(2050)\end{array}\right)=\left(\begin{array}[]{ccc}0.71&-0.70&-0.01\\\
-0.69&-0.70&0.19\\\ -0.14&-0.13&-0.98\\\
\end{array}\right)\left(\begin{array}[]{c}\Lambda_{1}^{*}\\\ \Lambda_{8}\\\
q^{3}s\bar{q}\end{array}\right)$ (35)

The $\Lambda(2050)\frac{3}{2}^{-}$ resonance of $M=2056\pm 22$ MeV and
$\Gamma=493\pm 61$ MeV was first reported by the KSU group [8], and the bump
was confirmed by JPAC analysis [11] in 2016, with a smaller width
$\Gamma=269\pm 35$ MeV. The fact that the $P_{c}$(4450) state was turned out
to be two much narrower $P_{c}$ states in the more precise LHCb experiment
[58] has inspired one to guess that there may be more resonance states in the
pole mass region of $\Lambda(2050)\frac{3}{2}^{-}$ since the width of this
resonance is much larger than all other $\Lambda$ resonances in PDG.

This research has received funding support from the NSRF via the Program
Management Unit for Human Resources and Institutional Development, Research
and Innovation (Grant No. B05F640055). Z.Z., A.K., T.Y. Htun, A.L., and Y.Y.
all acknowledge support from Suranaree University of Technology (SUT). A.K.
would like to thank for the support of the Thailand Science Research and
Innovation (TSRI) for their support through the Royal Golden Jubilee Ph.D.
(RGJ-PHD) Program (Grant No. PHD/0242/2558) and the SUT-OROG scholarship
(Contract No. 46/2558).

##  Appendix A Explicit $q^{2}s$ $\Lambda$ wave functions

In this Appendix the $q^{2}s$ color-orbital-spin-flavor wave functions with
the principle quantum number $N\leq 2$ are listed in table 5, where
$\chi_{i}$, $\Phi_{j}$, and $\phi^{N^{\prime}}_{L^{\prime}M^{\prime}y}$ are
the spin, flavor, and spatial wave functions, respectively. The $SU(3)_{F}$
decuplet states are excluded since $\Sigma$ and $\Sigma^{*}$ resonances are
not discussed in the present work.

Table 5: Explicit $q^{2}s$ color-orbital-spin-flavor wave functions for isospin I=0. All the color wave function is $\psi^{c}_{[111]}$ singlet. |  $SU(6)_{SF}$ | $l^{P}$ |  $SU(6)_{SF}$$\times O(3)$ wave functions
---|---|---|---
$\phantom{-}N$ |  Representations | O(3) |  $SU(3)_{F}$ singlet |  $SU(3)_{F}$ octet
$\phantom{-}0$ | $\phantom{-}56$ | $0^{+}$ |  | $J^{P}=\frac{1}{2}^{+}$
|  |  |  | $\frac{1}{\sqrt{2}}\phi^{0}_{00s}(\Phi_{\lambda}\chi_{\lambda}+\Phi_{\rho}\chi_{\rho})$
$\phantom{-}1$ | $\phantom{-}70$ | $1^{-}$ | $\;J^{P}=\frac{1}{2}^{-}\;,\;\frac{3}{2}^{-}$ | $J^{P}=\frac{1}{2}^{-}\;,\;\frac{3}{2}^{-}$
|  |  | $\frac{1}{\sqrt{2}}(\chi_{\rho}\phi^{1}_{1m\lambda}-\chi_{\lambda}\phi^{1}_{1m\rho})\Phi_{A}$ | $\frac{1}{2}[\phi^{1}_{1m\rho}(\Phi_{\lambda}\chi_{\rho}+\Phi_{\rho}\chi_{\lambda})+\phi^{1}_{1m\lambda}(\Phi_{\rho}\chi_{\rho}-\Phi_{\lambda}\chi_{\lambda})]$
|  |  |  | $J^{P}=\frac{1}{2}^{-}\;,\;\frac{3}{2}^{-},\;\frac{5}{2}^{-}$
|  |  |  | $\frac{1}{\sqrt{2}}\chi_{S}(\phi^{1}_{1m\lambda}\Phi_{\lambda}+\phi^{1}_{1m\rho}\Phi_{\rho})$
$\phantom{-}2$ | $\phantom{-}56$ | $0^{+}$ |  | $J^{P}=\frac{1}{2}^{+}$
|  |  |  | $\frac{1}{\sqrt{2}}\phi^{2}_{00s}(\Phi_{\lambda}\chi_{\lambda}+\Phi_{\rho}\chi_{\rho})$
| $\phantom{-}70$ | $0^{+}$ | $J^{P}=\frac{1}{2}^{+}$ | $J^{P}=\frac{1}{2}^{+}$
|  |  | $\frac{1}{\sqrt{2}}(\chi_{\rho}\phi^{2}_{00\lambda}-\chi_{\lambda}\phi^{2}_{00\rho})\Phi_{A}$ | $\frac{1}{\sqrt{2}}[\phi^{2}_{00\rho}(\Phi_{\lambda}\chi_{\rho}+\Phi_{\rho}\chi_{\lambda})+\phi^{2}_{00\lambda}(\Phi_{\rho}\chi_{\rho}-\Phi_{\lambda}\chi_{\lambda})]$
|  |  |  | $J^{P}=\frac{3}{2}^{+}$
|  |  |  | $\frac{1}{\sqrt{2}}\chi_{S}(\phi^{2}_{00\lambda}\Phi_{\lambda}+\phi^{2}_{00\rho}\Phi_{\rho})$
$\phantom{-}2$ | $\phantom{-}20$ | $1^{+}$ | $J^{P}=\frac{1}{2}^{+}\;,\;\frac{3}{2}^{+},\;\frac{5}{2}^{+}$ | $J^{P}=\frac{1}{2}^{+}\;,\;\frac{3}{2}^{+}$
|  |  | $\phi^{2}_{1mA}\chi_{S}\Phi_{A}$ | $\phi^{2}_{1mA}(\Phi_{\rho}\chi_{\rho}-\Phi_{\lambda}\chi_{\lambda})$
$\phantom{-}2$ | $\phantom{-}56$ | $2^{+}$ |  | $J^{P}=\frac{3}{2}^{+}\;,\;\frac{5}{2}^{+}$
|  |  |  | $\frac{1}{\sqrt{2}}\phi^{2}_{2mS}(\Phi_{\rho}\chi_{\rho}+\Phi_{\lambda}\chi_{\lambda})$
| $\phantom{-}70$ | $2^{+}$ | $J^{P}=\frac{3}{2}^{+}\;,\;\frac{5}{2}^{+}$ | $J^{P}=\frac{3}{2}^{+}\;,\;\frac{5}{2}^{+}$
|  |  | $\frac{1}{\sqrt{2}}(\chi_{\rho}\phi^{2}_{2m\lambda}-\chi_{\lambda}\phi^{2}_{2m\rho})\Phi_{A}$ | $\frac{1}{2}[\phi^{2}_{2m\rho}(\Phi_{\lambda}\chi_{\rho}+\Phi_{\rho}\chi_{\lambda})+\phi^{2}_{2m\lambda}(\Phi_{\rho}\chi_{\rho}-\Phi_{\lambda}\chi_{\lambda})]$
|  |  |  | $J^{P}=\frac{1}{2}^{+}\;,\;\frac{3}{2}^{+}\;,\;\frac{5}{2}^{+}\;,\;\frac{7}{2}^{+}$
|  |  |  | $\frac{1}{\sqrt{2}}\chi_{S}(\phi^{2}_{2m\lambda}\Phi_{\lambda}+\phi^{2}_{2m\rho}\Phi_{\rho})$

The explicit flavor wave functions of flavor-singlet $\Lambda_{1}$ and flavor-
octet $\Lambda_{8}$ are shown below:

$\displaystyle\Phi^{\Lambda_{1}}_{A}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{6}}(uds-dus+sud-usd+dsu-sdu),$
$\displaystyle\Phi^{\Lambda_{8}}_{\lambda}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(sud+usd-dsu-sdu),$
$\displaystyle\Phi^{\Lambda_{8}}_{\rho}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{12}}(2uds-2dus+sdu+usd-dsu-sud)$

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