#  Finding diamonds in the rough: Targeted Sub-threshold Search for Strongly-
lensed Gravitational-wave Events

Alvin K. Y. Li<EMAIL_ADDRESS>Department of Physics, The Chinese
University of Hong Kong, Shatin, New Territories, Hong Kong  LIGO Laboratory,
California Institute of Technology, Pasadena, CA 91125, US  Rico K. L. Lo
<EMAIL_ADDRESS>LIGO Laboratory, California Institute of Technology,
Pasadena, CA 91125, US  Surabhi Sachdev  LIGO Laboratory, California Institute
of Technology, Pasadena, CA 91125, US  Department of Physics, The Pennsylvania
State University, University Park, PA 16802, USA  Institute for Gravitation
and the Cosmos, The Pennsylvania State University, University Park, PA 16802,
USA  C. L. Chan  Department of Physics, The Chinese University of Hong Kong,
Shatin, New Territories, Hong Kong  E.T. Lin  Institute of Astronomy, National
Tsing Hua University, Hsinchu $30013$, Taiwan  Tjonnie G. F. Li  Department of
Physics, The Chinese University of Hong Kong, Shatin, New Territories, Hong
Kong  Alan J. Weinstein  LIGO Laboratory, California Institute of Technology,
Pasadena, CA 91125, US

(August 27, 2024)

###### Abstract

Strong gravitational lensing of gravitational waves can produce duplicate
signals that are separated in time with different amplitudes. We consider the
case in which strong lensing produces identifiable gravitational-wave events
together with weaker sub-threshold signals that are hidden in the noise
background. We present a search method for the sub-threshold signals using
reduced template banks targeting specific confirmed gravitational-wave events.
We apply the method to an event from Advanced LIGO’s first observing run O1,
GW150914. We show that the method is effective in reducing the noise
background and hence raising the significance of (near-) sub-threshold
triggers. In the case of GW150914, we are able to improve the sensitive
distance by $2.0\%-14.8\%$. Finally, we present a list of possible lensed
candidates for O1/O2 events with significant sky location overlap with the
original events.

##  I Introduction

The LIGO-Virgo Collaboration Aasi  _et al._ (2015); Abbott  _et al._ (2018a);
Harry (2010); Acernese  _et al._ (2015) has made 11 confident detections of
gravitational waves from compact binary mergers Abbott  _et al._ (2018b) in
the first two observing runs O1 and O2. Gravitational waves has been applied
to many aspects of physics, including multi-messenger astronomy Abbott  _et
al._ (2017a, b); Burns  _et al._ (2018); Abbott  _et al._ (2016a); Aasi  _et
al._ (2014a); Abbott  _et al._ (2019), cosmology Abbott  _et al._ (2017c,
2018c); Aasi  _et al._ (2014b), testing general relativity Abbott  _et al._
(2016b, 2018d), and particle physics Albert  _et al._ (2017); Adrian-Martinez
_et al._ (2016).

General relativity predicts that emitted from a source can be refracted and
deflected before reaching the observer, an effect known as _gravitational
lensing_ which has been extensively studied for electromagnetic waves Rahvar
and Moffat (2019); Cunha and Herdeiro (2018); Turyshev and Toth (2017);
Dokuchaev and Nazarova (2017); Bartelmann and Maturi (2016); Narayan and
Bartelmann (1996); Metcalf  _et al._ (2018). Recent works have started
considering gravitational-wave lensing, including lensing rates Ohanian
(1974); Takahashi and Nakamura (2003); Takahashi (2004); Hannuksela  _et al._
(2019); Oguri (2018); Haris  _et al._ (2018); Ng  _et al._ (2018); Smith  _et
al._ (2018), strong-lensing Smith  _et al._ (2017); Cremonese and Mörtsell
(2018); Piórkowska  _et al._ (2013); Takahashi (2017); Sereno  _et al._
(2010); Li (2015); Lai  _et al._ (2018); Moylan  _et al._ (2007) and weak-
lensing effects imprinted onto gravitational waveforms Wang  _et al._ (1996);
Christian  _et al._ (2018). Claims or disclaims of detected pairs of
gravitational-wave events for being lensed images have also been made
Broadhurst  _et al._ (2019); Singer  _et al._ (2019). A recent study has
shown, however, that there was no strong evidence for the observation of
gravitational lensing during O1/O2 Hannuksela  _et al._ (2019). Nevertheless,
with the expected rise in future detection rate of gravitational waves from
binary black hole mergers Dent and Pankow (2018), chances are we will observe
lensed gravitational waves eventually.

Under strong lensing, multiple signals from the same source can arrive with
relative time delay and magnification. Refs. Haris  _et al._ (2018); Oguri
(2018) show that with a singular isothermal ellipsoidal model, most lensed
gravitational waves will have magnifications $\mu\rangle 1$, with relative
time delays $\Delta t$ ranging from approximately $8$s to $30$ days.
Therefore, it is possible to have gravitational waves with sufficiently strong
amplitudes that are identified as detections and corresponding weaker (sub-
threshold) signals that are burried in the noise background within the
observed data in O1/O2 (with $48.6$ and $268.3$ days of coincident detector
data respectively).

In this paper, we present a search method for sub-threshold lensed signals by
using information from confirmed gravitational-wave evets to limit the
parameter space for matched-filtering analysis.

##  II Methods

Suppose we want to search for strongly-lensed counterparts of a gravitational-
wave event, potentially being sub-threshold due to the _relative de-
magnification_ , meaning counterparts will appear to be further away from the
observing point and hence weaker. Here we assumed geometrical optics, i.e. the
magnification is frequency-independent and the gravitational lens is
achromatic, therefore will not distort waveform apart from an overall scaling,
with the assumption that the gravitational-wave wavelength $\lambda_{GW}$ is
much shorter than the Schwarzschild radius of the lens. For longer
wavelengths, diffraction effect will take a dominant role and the geometric
optics limit will become invalid. As an input, posterior samples of parameters
obtained from parameter estimation (e.g. from LALInference Veitch  _et al._
(2015)) of the targeted event are used to generate an injection set 111An
injection refers to a simulated lensed signal that we test a search pipeline
against to measure the effectiveness of the pipeline. For each sample, we
generate one injection with its original strength, and nine additional weaker
injections to mimic its lensed counterparts (See Fig. 1).

Having obtained an injection set, we run a matched-filtering based search
pipeline to see what templates in the full template bank can recover these
injections with _sufficiently high_ significance. In this work, we applied the
GstLAL pipeline GstLALMethodsPaper; Sachdev  _et al._ (2019). In principle,
the search will work equally well for other matched-filtering based pipelines
such as PyCBC Usman  _et al._ (2016). These templates will be used to form the
_reduced template bank_ to search for sub-threshold lensed signals.

The same search pipeline is rerun, with identical procedures as the usual
analyses but with the reduced template bank, to look for possible lensed
counterparts from all available data.

Compared to a search using the full bank, weaker lensed signals of the
targeted event will have higher statistical significance in a search using the
reduced bank because of the trials factor reduction.

The search outputs a list of possible lensed candidates of the targeted event.
These triggers will be followed-up using a Bayesian model selection analysis
being described in another upcoming paper Lo and Ignacio Magaña (2020).

Figure 1: Workflow of the targeted sub-threshold search method.

###  II.1 Reduced template bank

We start with the O2 template bank, with the parameter space of masses of the
binary components spanning from $2M_{\odot}$ to $400M_{\odot}$, covering
binary neutron stars (BNS), neutron-star-black-hole (NSBH) and binary black
hole (BBH) systems, with a total of $677000$ templates Sachdev  _et al._
(2019); Mukherjee  _et al._ (2018). For templates dedicated for BBH systems,
they carry (anti-)aligned dimensionless spin magnitudes $\leq 0.999$ as chosen
to be as close to the theoretical limit of $1$ as possible with current
waveform approximants222In our work, we are using the waveform approximants
“TaylorF2” and “SEOBNRv4_ROM”..

As parameters of a gravitational wave are not precisely known, posterior
samples of $N_{\text{ps}}=\mathcal{O}(1000)$ highest likelihood from parameter
estimation are used to represent the event. For a matched-filtering based CBC
search, templates are characterized by intrinsic parameters (e.g. component
masses and spins) since they govern the time evolution of a waveform. The
dependence on other extrinsic parameters, including luminosity distance
$D_{\rm L}$ and inclination $\iota$, are absorbed into the effective distance
$D_{\rm eff}$ defined in Allen  _et al._ (2012). The optimal signal-to-noise
(SNR) ratio $\rho_{\text{opt}}=\sqrt{\langle h|h\rangle}$ of a template
$h_{\text{template}}$ is then given by

$\rho_{\text{opt}}^{2}=\langle
h|h\rangle=4\int_{0}^{\infty}df\frac{|h_{\text{template}}|^{2}}{S_{n}(f)}\propto\left(\frac{1}{D_{\text{eff}}}\right)^{2},$
(1)

where $S_{n}(f)$ is the one-sided power spectral density of the noise in a
detector Allen  _et al._ (2012). Each template in a template bank is
_normalized by the optimal SNR squared_ such that the matched-filtering SNR is
independent of the strength of the templates. The SNR ratio
$\rho_{\text{opt}}$ thus scales inversely with effective distance.

For each sample, we make one injection with the original optimal SNR, and nine
additional injections with smaller optimal SNRs by varying their _effective
distances_ to mimic the weaker lensed images of the targeted event with
respect to Eq. 1. After generating a set of $10N_{\text{ps}}$ injections, we
inject these simulated signals into the data, and use a matched-filtering
based CBC search pipeline GstLAL to try recovering them. An injection is
considered “found” if the corresponding trigger 333That is the time of trigger
is within a certain window of the injected time. has _a False-Alarm-Rate (FAR)
of less than 1 per 30 days_. The templates that recovered the injections will
be put into the reduced template bank for the sub-threshold search. Performing
the injection campaign described above makes sure templates in the reduced
bank can identify sub-threshold lensed signals, if they truly exist.

###  II.2 Statistical significance of triggers

To measure its significance, each gravitational-wave candidate in the GstLAL
search is assigned a likelihood-ratio statistic $\mathcal{L}$ Cannon  _et al._
(2013, 2015); GstLALMethodsPaper; Sachdev  _et al._ (2019) based on the SNR, a
signal consistency check (GstLAL uses the normalized power in the difference
of the SNR time series and the autocorrelation time series), the sensitivity
of each detector at the time of the candidate, and the time and phase delays
between triggers from different detectors. Then for each event, a False-Alarm-
Probability (FAP), the probability that nosie would produce a trigger with a
ranking statistic $\ln\mathcal{L}$ greater or equal to the ranking statistic
$\ln\mathcal{L^{*}}$ of the trigger under consideration, is calculated. For a
single event, it is just the complementary cumulative distribution function of
the likelihood-ratio,

$P(\ln\mathcal{L}\geq\ln\mathcal{L^{*}}\mid\text{noise})=\int_{\ln\mathcal{L^{*}}}^{\infty}P(\ln\mathcal{L}\mid\text{noise})\mathrm{d}\ln\mathcal{L}.$
(2)

Alternatively, we can express an event’s significance in terms of the False-
alarm Rate (FAR), which refers to how often noise would produce a trigger with
ranking statistic $\ln\mathcal{L}$ greater or equal to the ranking statistic
$\ln\mathcal{L^{*}}$ of the trigger under consideration GstLALMethodsPaper,
given by $\text{FAP}/T$ where $T$ is the time length of the data under
analysis.

We discussed in the introduction that we expect to increase the significance
of sub-threshold triggers by reducing the template bank size. By targeting a
subset of the parameter space, we can increase the significance of events
present in the reduced subspace by reducing the trials factor. Since we are
only interested in signals belonging to the reduced subspace, the foreground
is kept constant while the background is lowered. This method can potentially
reveal sub-threshold signals belonging to a restricted parameter space,
allowing us to look for weaker lensed counterparts of the gravitational waves
detected by LIGO-Virgo.

When we use the reduced bank for the targeted search of lensed companions of
the targeted event, the GstLAL pipeline associates with each found trigger a
FAR (See Table LABEL:Table:_GW150914_results as an example). Readers are
reminded that the FARs evaluated here are _N_ OT indicators of how likely the
triggers are lensed counterparts of the targeted event, since the FAR
evaluated here only indicates how likely the identified trigger is a _g_
ravitational wave. Instead, the search output should only be interpreted as a
priority list for follow-up analysis that computes the likelihood that two
triggers are lensed gravitational waves from the same source. The Bayesian
model selection between lensed and un-lensed hypotheses will be explored in a
follow-up paper Lo and Ignacio Magaña (2020).

###  II.3 Skymap overlap

As a first-level validation for the possible lensed candidates found, we
perform a skymap-overlap consistency test. For each candidate found, we
localize the source by generating a skymap using BAYESTAR Singer and Price
(2016) and compute the percentage overlap Chan  _et al._ (2020) of the $90\%$
credible regions with the skymap of the targeted event. The deviation in sky
location from where the lensed signals are is insignificant for gravitational
waves when compared to the uncertainty in estimating the sky location of the
gravitational-wave source itself, and therefore lensed gravitational waves
from the same source should be observed at approximately the same sky
location. With this constraint, candidates with $0\%$ skymap overlap can be
immediately ruled out for being lensed counterparts of the targeted event.

##  III Example Search: GW150914

###  III.1 Constructing the Reduced Template Bank

We apply the method to the all O1/O2 reported gravitational waves. Here we
present the details for GW150914Abbott  _et al._ (2018b, 2016c) to study the
performance of the search method. GW150914 is the first reported gravitational
wave and has the highest SNR among all the identified O1 events. Table 1
summarises the information about GW150914.

Properties | Value
---|---
GPS time | $1126259462.43$
Primary mass (source frame) $m_{1}^{\text{source}}$ | $35.6^{+4.7}_{-3.1}\;M_{\odot}$
Secondary mass (source frame) $m_{2}^{\text{source}}$ | $30.6^{+3.0}_{-4.4}\;M_{\odot}$
Chirp mass (source frame) $\mathcal{M}_{c}^{\text{source}}$ | $28.6^{+1.7}_{-1.5}\;M_{\odot}$
Luminosity distance $D_{\text{L}}$ | $440^{+150}_{-170}\;\text{Mpc}$
Source redshift $z$ | $0.09^{+0.03}_{-0.03}$
Network SNR $\rho$ | 24.4
FAR | $<1.0\times 10^{-7}\;\text{yr}^{-1}$
Table 1: Information of the gravitational wave GW150914 Abbott  _et al._
(2016c, 2018b).

We generate the reduced template bank by first selecting posterior samples
with aligned spins released by the LIGO-Virgo collaboration O1_ . We use these
samples to simulate $7015$ lensed signals of GW150914 with varying strength.
Fig. 2 shows the distribution of injections in the chirp mass (detector frame)
$\mathcal{M}_{c}^{\text{detector}}$, - effective spin $\chi_{\text{eff}}$,
space. The orange cross indicates the median inferred value for
$\mathcal{M}_{c}^{\text{detector}}$ and $\chi_{\text{eff}}$ reported in Ref.
Abbott  _et al._ (2016c). We confirm that the injections primarily populate
around the median inferred value.

Figure 2: Injections used in the injection campaign for the GW150914 targeted
search in the chirp mass (detector frame) $\mathcal{M}_{c}^{\text{detector}}$
\- effective spin $\chi_{\text{eff}}$ space. The orange cross indicates the
median inferred value for $\mathcal{M}_{c}^{\text{detector}}$ and
$\chi_{\text{eff}}$ for GW150914 reported in Abbott  _et al._ (2016c).

Templates which found our injected signals are added to the reduced template
bank. Fig. 3 shows the original template bank (dark blue) used by the GstLAL
search in O1 and O2, and the reduced template bank (orange) generated for the
sub-threshold search of GW150914 projected onto the $m_{1}-m_{2}$ plane. The
reduced bank is sparser than the original bank. In fact, there are only $139$
templates in the reduced bank, compared to the $677000$ templates in the full
bank.

Figure 3: (Color online) The templates in the original and reduced template
bank, plotted in dark blue and orange respectively on the $m_{1}-m_{2}$ plane.
The best-match template for GW150914 is indicated by a red star. The reduced
bank is sparser than the original bank. However, because templates in the
reduced bank can recover the lensed signal injections in an injection
campaign, these templates can identify the sub-threshold lensed signals, if
they truly exist.

###  III.2 Search Sensitivity

To assess the search sensitivity of our search, we analyse the same injection
set as above but now in all the data from O1 using the original and reduced
bank. Figure 4 compares the distribution of ranking statistic (log likelihood
$\ln\mathcal{L}$).

Figure 4: (Colour online) Ranking statistic threshold curves (on the log
likelihood $\ln\mathcal{L}$ \- Number of events $N\geq\ln\mathcal{L}$ plane)
for GW150914. The results combined all O1 data. The blue and red curve
corresponds to result using the original full and reduced bank respectively.
The red curve is below the blue one, indicating that the noise background has
been reduced through reducing the template bank size. The shaded regions
correspond to the $1$-, $2$-, and $3$-sigma regions respectively (in
decreasing order of transparency).

Note that the red curve (with reduced bank) is below the blue curve (with
original bank), indicating that the noise background is reduced when we use
the reduced bank.

From our set of injections with known distances, we can assess the distance
out to which we may identify gravitational waves (observable range). Fig. 5
shows the observable range - FAR curves for GW150914-like signals using the
original and reduced bank.

Figure 5: (Colour online) The observable range - FAR plots for GW150914 using
the original (blue) and reduced bank (orange). The curve with the reduced bank
is above the one with the original full template bank. In general the
improvement in observable range is at least $2.0\%$. Particularly, around the
usual threshold for FAR (i.e. $\text{FAR}\approx 10^{-7}\text{Hz}$), the
observable range is improved by approximately $14.8\%$.

Note that the red curve (with reduced bank) is shifted upward compared to the
blue curve (using the original bank), indicating an increase in observable
range. In fact, the observable range is improved by at least $2.0\%$. In
particular, around the usual FAR threshold of $10^{-}7$ Hz, the observable
range is improved by approximately $14.8\%$, suggesting that with the use of a
reduced bank, we can see gravitational waves to further distances at a given
FAR threshold. From Eq. 1 we can conclude that by being able to see further,
equivalently we can search for lensed gravitational waves which seem to be
further away.

##  IV Strong-lensing Candidates for O1/O2 events

Finally, we present the list of top $5$ candidates (excluding known
gravitational-wave events) of strongly-lensed images of confirmed events in
O1/O2 identified by applying this method over all O1/O2 data in Table 2.
Restraining to the usual FAR threshold of $1$ in $30$ days, the expectation
number of candidates found in our search is slightly larger than $1$.

Rank | Targeted Event | GPS Time | SNR | FAR${}_{\text{reduced bank}}$ (Hz) | Skymap overlap(%) | Remark
---|---|---|---|---|---|---
1 | GW170818 | 1169069154.58 | 10.2 | $2.37\times 10^{-9}$ | 18.1 | Reported in Venumadhav  _et al._ (2019). Also found by GW170104.
2 | GW170814 | 1170079035.73 | 8.8 | $1.54\times 10^{-8}$ | 53.7 | Reported in Venumadhav  _et al._ (2019). Also found by GW170818.
3 | GW170818 | 1128755912.91 | 8.9 | $6.90\times 10^{-8}$ | 100.0 | -
4 | GW170608 | 1134291607.28 | 9.5 | $7.12\times 10^{-8}$ | 23.4 | -
5 | GW170608 | 1174511607.75 | 8.6 | $7.32\times 10^{-8}$ | 47.9 | -
Table 2: Top 5 candidates of strongly-lensed images for O1/O2 events found by
this search over all O1/O2 data. Note that confirmed gravitational-wave events
are excluded in this table.

Note that our reduced banks retrieved the corresponding targeted event. In the
full search results, we retrieved in total $124$ candidates. The sky-location
consistency test preliminarily reduces the number of candidates to $64$. In
table 2 we report the top 10 candidates (in ascending order of FAR).
Nevertheless, readers should be reminded that the FARs shown in Table 2 do not
indicate how likely these triggers are lensed counterparts of the targeted
gravitational-wave event, but instead a priority ranking for follow-up
analysis that computes the likelihood that two triggers share the same set of
source parameters. These candidates are passed to a follow-up analysis that
computes the likelihood that two triggers are lensed counterparts of each
other.

##  V Concluding Remarks

Gravitational-wave lensing has been the focus of increased attention in the
recent literature Smith  _et al._ (2017); Oguri (2018); Haris  _et al._
(2018); Cremonese and Mörtsell (2018); Ng  _et al._ (2018); Piórkowska  _et
al._ (2013); Takahashi (2017); Sereno  _et al._ (2010); Smith  _et al._
(2018); Wang  _et al._ (1996); Christian  _et al._ (2018); Li (2015); Lai  _et
al._ (2018); Moylan  _et al._ (2007). A recent study has found no compelling
evidence for lensing magnification, repeated signals from a single strongly
lensed source, and wave optics effects due to solar-mass compact objects
Hannuksela  _et al._ (2019) in the GWTC-1 catalog Abbott  _et al._ (2018b).
Nevertheless, possibility remains that strong lensing produces strong signals
that are identified as a gravitational-wave event and weaker signals that are
buried inside the noise background. The latter are referred to as sub-
threshold and may not be labelled as a gravitational-wave detection without
further analysis.

We proposed a search for these sub-threshold signals by means of matched-
filtering using a reduced template bank produced to target specific
gravitational-wave events. We construct this targeted bank by selecting high-
likelihood samples from the parameter estimation results, simulating a set of
signals by varying distances of these samples and selecting templates that
found them. We show that the resulting bank reduces the noise background and
increases the search sensitivity of the search. Applied to GW150914 as an
example, we found that the sensitive distance increases by at most $14.8\%$.

We also present the list of top $5$ triggers which pass the usual FAR
threshold of $\text{FAR}\leq 1/30$ days from the targeted search of all O1/O2
gravitational-wave events. Nevertheless, the FAR associated with these
triggers should not be interpreted as a measure of the likelihood that these
triggers are indeed duplicated signals of the same source. Instead, the output
of this search should be interpreted as a priority list for follow-up analysis
that computes the likelihood that two triggers share the same set of source
parameters Haris  _et al._ (2018); Lo and Ignacio Magaña (2020).

The performance of our search can be further optimized. For example, selection
procedure for templates that are included in the reduced bank may require
tuning to find the optimal balance between coverage and sensitivity. Moreover,
it has been shown that the subdivision of templates within a bank can also
affect the performance. Furthermore, since we are looking for lensed
counterparts of targeted events, from the sky location of the target, we
should be able to set a consistent range for the difference in arrival time
between the detectors for the lensed counterparts.

###### Acknowledgements.

The authors acknowledge the generous support from the National Science
Foundation in the United States. The authors would also like to acknowledge
Jonah Kanner and Bruce Allen for their useful suggestions. RKLL and TGFL would
also like to gratefully acknowledge the support from the Croucher Foundation
in Hong Kong. The work described in this paper was partially supported by a
grant from the Research Grants Council of the Hong Kong (Project No. CUHK
14306218) and the Direct Grant for Research from the Research Committee of the
Chinese University of Hong Kong. SS was supported in part by the LIGO
Laboratory and in part by the Eberly Research Funds of Penn State, The
Pennsylvania State University, University Park, PA 16802, USA. The authors are
also grateful for computational resources provided by the LIGO Laboratory and
supported by National Science Foundation Grants PHY-0757058 and PHY-0823459.
This research has made use of data, software and/or web tools obtained from
the Gravitational Wave Open Science Center (https://www.gw-openscience.org)
Vallisneri  _et al._ (2015), a service of LIGO Laboratory, the LIGO Scientific
Collaboration and the Virgo Collaboration. LIGO was constructed by the
California Institute of Technology and Massachusetts Institute of Technology
with funding from the National Science Foundation and operates under
cooperative agreement PHY-0757058. Virgo is funded by the French Centre
National de Recherche Scientifique (CNRS), the Italian Istituto Nazionale
della Fisica Nucleare (INFN) and the Dutch Nikhef, with contributions by
Polish and Hungarian institutes. This paper carries LIGO Document Number
LIGO-P1900059.

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# Several conjectures from the hadron physics: The transient worlds beside(s)
ours

Hua-Xing Chen<EMAIL_ADDRESS>School of Physics, Southeast University,
Nanjing 210094, China

###### Abstract

The past decades witnessed the golden era of hadron physics, which gives us a
good opportunity to study the physics happening in a transient period of time.
The development on the singly heavy baryons indicates that there exists the
fine structure of hadron spectrum caused by the direct strong interaction, and
the development on the exotic hadrons indicates that the residue strong
interaction is capable of forming the hadronic molecules. Similar to the
electromagnetic interaction, the strong interaction may be capable of forming
some imaginable hadronic worlds. Moreover, there can be various worlds formed
by various fundamental physical laws. Some hadrons have so transient lifetimes
that they may not even be formed. We discuss whether these non-existent
particles are capable of affecting our realistic world. We discuss what kinds
of particles exist, and so can be observed, in our realistic world. We
conjecture that the ratios $R\equiv M/\Gamma\gg 1/2$ ($M\tau\gg\hbar/2$) and
$R\equiv M/\Gamma<1/2$ ($M\tau<\hbar/2$) can be used to describe the particles
existing and not-existing in our realistic world, respectively. Here $M$,
$\Gamma$, and $\tau$ are the mass, width, and lifetime, respectively. We
propose to use the ratio $R\equiv M/\Gamma\sim 1/2$ ($M\tau\sim\hbar/2$) to
describe the particles quasi-existing in our realistic world, whose studies
may allow us to go beyond the quantum physics and arrive at another type of
boundary of our realistic world, that is from the existent to the non-
existent. We propose to investigate the quasi-existent particles by studying
the singly heavy baryons through various imperfect symmetries among them. We
obtain an incidental conjecture that the lifetime and width may be quantized
for the particles existing in our realistic world as

$M\tau=n\hbar/2~{}~{}~{}{\rm{and}}~{}~{}~{}M/\Gamma=n/2\,,~{}~{}~{}n=1,2,3\cdots\,.$

hadron physics, singly heavy baryon, exotic hadron, uncertainty principle,
quantization condition

##  I What do we learn from the hadron physics

The book of Chuang Tzu is an ancient and important Chinese spiritual text,
where the author wrote:“Take a foot long stick and remove half every day. In
ten thousands years it will not run out.” This relates to the old
philosophical question:“Is matter infinitely divisible?” Although this
question has not yet been fully answered, we have gained much knowledge about
our microscopic world, e.g., the divisibility stops at quarks and leptons
according to the Standard Model, while some scientists believe that these
elementary particles also have a structure pdg . A similar question is
that:“Can the lifetime of an unstable particle be infinitely short?” This is a
philosophical question that has not yet been well investigated scientifically.
The past decades witnessed the golden era of hadron physics, which gives us a
good opportunity to investigate this question by studying the physics that
happens in a transient period of time. Two of the major developments are about
the singly heavy baryons and the exotic hadrons. Let us briefly introduce them
as follows, and we refer to the recent review Chen:2022asf  for their detailed
discussions.

The electromagnetic interaction holds the atomic nucleus and electrons
together inside an atom, and it leads to the fine/hyperfine structure of the
light spectrum. The strong interaction among quarks and gluons is similar to
the electromagnetic interaction in some aspects, and it can also lead to the
fine structure of the hadron spectrum. An ideal platform to study this is the
singly heavy baryon containing one valence heavy quark ($charm/bottom$) and
two valence light quarks ($up/down/strange$), where the light quarks and
gluons circle around the nearly static heavy quark, so the whole system
behaves as the QCD analogue of the atom bound by the electromagnetic
interaction Copley:1979wj ; Korner:1994nh ; Manohar:2000dt ; Klempt:2009pi ;
Chen:2016spr .

Important progresses have been made in this field in the past five years. In
2017 the LHCb collaboration discovered as many as five excited
$\Omega_{c}^{0}$ baryons in the $\Xi_{c}^{+}K^{-}$ mass spectrum LHCb:2017uwr
, as shown in Fig. 1(a):

$\Omega_{c}(3000)^{0},\,\Omega_{c}(3050)^{0},\,\Omega_{c}(3066)^{0},\,\Omega_{c}(3090)^{0},\,\Omega_{c}(3119)^{0}.$

In 2020 they discovered three excited $\Xi_{c}^{0}$ baryons in the
$\Lambda^{+}_{c}K^{-}$ mass spectrum LHCb:2020iby , as shown in Fig. 1(b):

$\Xi_{c}(2923)^{0}\,,\,\Xi_{c}(2939)^{0}\,,\,\Xi_{c}(2965)^{0}\,.$

Also in 2020 they discovered four excited $\Omega_{b}^{-}$ baryons in the
$\Xi_{b}^{0}K^{-}$ mass spectrum LHCb:2020tqd , as shown in Fig. 1(c):

$\Omega_{b}(6316)^{-}\,,\,\Omega_{b}(6330)^{-}\,,\,\Omega_{b}(6340)^{-}\,,\,\Omega_{b}(6350)^{-}\,.$

These excited $\Omega_{c}/\Xi_{c}/\Omega_{b}$ baryons are good candidates for
the $1P$-wave singly heavy baryons. Their fine structures indicate the rich
internal structures of hadrons, which are caused directly by the strong
interaction.

Figure 1: (a) The $\Xi^{+}_{c}K^{-}$ invariant mass spectrum measured by LHCb
in 2017 and taken from Ref. LHCb:2017uwr . (b) Distribution of the invariant
mass difference $\Delta
m=m(\Lambda^{+}_{c}K^{-})-m(\Lambda^{+}_{c})-m(K^{-})$, measured by LHCb in
2020 and taken from Ref. LHCb:2020iby . (c) Distribution of the invariant mass
difference $\Delta m^{\prime}=m(\Xi_{b}^{0}K^{-})-m(\Xi_{b}^{0})$, measured by
LHCb in 2020 and taken from Ref. LHCb:2020tqd . These fine structures indicate
the rich internal structures of the excited $\Omega_{c}/\Xi_{c}/\Omega_{b}$
baryons.

An atomic nucleus is made of protons and neutrons, which are themselves
composed of quarks and gluons. In the past century a huge number of hadrons
were observed in particle experiments, which are similar to the proton and
neutron. One naturally conjecture that these hadrons are capable of forming
some subatomic particles that are similar to the atomic nucleus
Weinberg:1965zz ; Voloshin:1976ap . Nowadays we call these subatomic particles
“hadronic molecules”. Since the discovery of the $X(3872)/\chi_{c1}(3872)$ by
Belle in 2003 Belle:2003nnu , there have been significant progresses in this
field and many candidates for the hadronic molecules have been observed in
experiments. Note that some of these candidates can also be explained as the
compact multiquark states that are still hadrons but not hadronic molecules
Jaffe:1976ig ; Maiani:2004vq .

Figure 2: The $J/\psi p$ invariant mass spectrum of the $\Lambda_{b}^{0}\to
J/\psi K^{-}p$ decay, measured by LHCb (a) in 2015 LHCb:2015yax  and (b) in
2019 LHCb:2019kea . The purple and blue solid curves in the left panel
describe the $P_{c}(4380)^{+}$ and $P_{c}(4450)^{+}$, respectively. The latter
$P_{c}(4450)^{+}$ is further separated into two substructures,
$P_{c}(4440)^{+}$ and $P_{c}(4457)^{+}$, in the right panel. Taken from Refs.
LHCb:2015yax ; LHCb:2019kea .

As shown in Fig. 2, in the LHCb experiments performed in 2015 LHCb:2015yax
and 2019 LHCb:2019kea , the famous hidden-charm pentaquark states,

$P_{c}(4312)^{+}\,,\,P_{c}(4380)^{+}\,,\,P_{c}(4440)^{+}\,,\,P_{c}(4457)^{+}\,,$

were discovered; besides, in the LHCb experiments performed in 2020
LHCb:2020jpq  and 2022 Collaboration:2022boa , the hidden-charm pentaquark
states with strangeness,

$P_{cs}(4338)^{0}\,,\,P_{cs}(4459)^{0}\,,$

were discovered. These $P_{c}/P_{cs}$ structures are just below their
corresponding meson-baryon thresholds, and their discoveries strongly support
the existence of hadronic molecules Wu:2010jy ; Yang:2011wz ; Wang:2011rga ;
Karliner:2015ina ; Chen:2016qju . Moreover, it was proposed in Ref.
Chen:2021xlu  that the residue strong interaction is capable of forming the
covalent hadronic molecules through the shared light quarks, similar to the
chemical molecules formed by the residue electromagnetic interaction through
the shared electrons.

##  II Imaginable worlds

Based on the recent developments of the hadron physics, let us imagine a world
without the weak interaction. To start with, we assume that there is only a
small net number of the $strange$, $charm$, and $bottom$ quarks. In our
realistic world most of the hadrons are not stable, while in this imaginal
world some of them become stable besides the proton and neutron, such as

$\Lambda^{0}\,,\,\Lambda_{c}^{+}\,,\,\Lambda_{b}^{+}\,\cdots$

As depicted in Fig. 3, protons and neutrons compose various atomic nuclei, and
then compose various atoms and chemical molecules together with electrons,
which further compose our realistic world. After turning off the weak
interaction, there exist more stable hadrons in the imaginal world, which are
capable of forming more hadronic molecules besides the atomic nuclei.
Accordingly, the imaginal world can be more complicated.

Figure 3: A picture of the world from the viewpoint of hadron physics.

Let us further assume that the net number of the $strange$ quarks is larger
than that of the $up$ and $down$ quarks, while the net number of the $charm$
and $bottom$ quarks is still small. In this imaginal world the stable hadrons
are

$\Lambda^{0}\,,\,\Xi^{0,-}\,,\,\Xi_{c}^{+,0}\,,\,\Xi_{b}^{0,-}\,\cdots$

while the lower-lying proton and neutron are limited by the reactions:

$p+\Xi^{-}\to 2\Lambda^{0}~{}~{}~{}{\rm and}~{}~{}~{}n+\Xi^{0}\to
2\Lambda^{0}\,.$ (1)

Hence, there can be various worlds formed by various fundamental physical laws
with certain fundamental parameters, and the world can become much different
as long as its fundamental physical laws or its fundamental parameters change
a little. Especially, our realistic world is so ingenious that the free
neutrons decay through the weak interaction, but the neutrons bound inside
many atomic nuclei do not.

To get a feeling of the above imaginable worlds, we show several time scales
as follows:

  * •

The lifetime of our universe is $1.38\times 10^{10}$ years $\sim 10^{+17}$ s.

  * •

The lifetime of the neutron is $877.75$ s $\sim 10^{+3}$ s.

  * •

The lifetimes of the $\Lambda^{0}$ and $\Xi^{0,-}$ hyperons are all at the
order of $10^{-10}$ s.

  * •

The time needed to form a hadron is less than $1$ GeV-1 $\sim 10^{-24}$ s.

We compare the lifetime of the $\Lambda^{0}$ hyperon with its forming time,
and the ratio is about $10^{+14}$. We compare the lifetime of our universe
with that of the neutron as its fundamental element, and the ratio is also
about $10^{+14}$. Hence, the imaginable worlds, with a small number of the
$p/n/\Lambda^{0}$ or $\Lambda^{0}/\Xi^{0}/\Xi^{-}$ baryons, may exist in a
very short time, e.g., $10^{-15}$ s. Besides these imaginable worlds made of
hadrons, there might be the imaginable worlds directly made of quarks and
gluons, which might exist in a time scale much shorter than that needed to
form a hadron.

##  III Existing or not-existing

In our realistic world most of the hadrons and hadronic molecules are not
stable partly due to the weak interaction. Some of them have so transient
lifetimes that they may not even be formed. Then are these non-existent
particles still capable of affecting our realistic world? Since we can not
directly observe them in experiments, we build a simple toy model towards an
answer to this question:

  * •

$A$ is a stable particle.

  * •

$B_{1}$ and $B_{2}$ both decay to $A$. The width of $B_{1}$ to $A$ is limited,
while the width of $B_{2}$ to $A$ is so large that $B_{2}$ does not exist.

  * •

$C_{1}$ and $C_{2}$ both decay to $B_{1}$. The width of $C_{1}$ to $B_{1}$ is
limited, while the width of $C_{2}$ to $B_{1}$ is so large that $C_{2}$ does
not exist.

  * •

$C_{3}$ and $C_{4}$ both decay to $B_{2}$. The width of $C_{3}$ to $B_{2}$ is
limited, while the width of $C_{4}$ to $B_{2}$ is so large that $C_{4}$ does
not exist.

  * •

The widths of $C_{1\cdots 4}$ to $A$ are all negligible.

Let us investigate two possibilities as follows.

Figure 4: A simple toy model based on the assumptions: (a) the non-existent
particles do not affect our realistic world, and (b) the non-existent
particles can still affect our realistic world.

Firstly, we assume that the non-existent particles do not affect our realistic
world. As depicted in Fig. 4(a), we derive:

  * •

$A$ is a stable particle, so it exists. The widths of $B_{1}$ and $C_{1}$ are
both limited, so they also exist. The widths of $B_{2}$ and $C_{2}$ are both
too large, so they do not exist.

  * •

Under the assumption that the non-existent particle $B_{2}$ does not affect
our realistic world, the particles $C_{3}$ and $C_{4}$ are both stable, so
they exist.

The last item is not so reasonable that the present assumption is problematic.

Secondly, we assume that the non-existent particles can still affect our
realistic world. As depicted in Fig. 4(b), we derive:

  * •

$A$ is a stable particle, so it exists. The widths of $B_{1}$ and $C_{1}$ are
both limited, so they also exist. The widths of $B_{2}$ and $C_{2}$ are both
too large, so they do not exist.

  * •

Under the assumption that the non-existent particle $B_{2}$ can still affect
our realistic world, the width of $C_{4}$ to $B_{2}$ is too large, so $C_{4}$
does not exist.

  * •

Under the assumption that the non-existent particle $B_{2}$ can still affect
our realistic world, $C_{3}$ may exist since the width of $C_{3}$ to $B_{2}$
is limited, while it does not exist if the width of the decay process
$C_{3}\to B_{2}\to A$ turns out to be too large.

These analyses are more reasonable than the previous analyses, so the present
assumption is more reasonable than the previous one. We may further derive
from this assumption that the higher-lying particles are generally more
unstable.

Based on the above analyses, we conjecture that there are hadrons and hadronic
molecules that are too unstable to exist, but they can still affect our
realistic world. We may further conjecture that the imaginal worlds, made of
these non-existent particles, do not exist but still weakly affect our
realistic world. Note that some new physics particles may behave similarly to
the intermediate particle $B_{2}$, given that their widths are too large for
them to be observed in our realistic world.

##  IV Beyond the quantum physics

In this section we further discuss what kinds of particles exist, and so can
be observed, in our realistic world. Note that the concept of “exist” has been
widely discussed in philosophy, while in this study we simply treat it the
same as “observable” or “capable of being observed”.

Undoubtedly, the answer closely relates to the lifetimes of particles. All the
stable hadron (the proton) and hadronic molecules (some of the atomic nuclei)
have been well established. Some of the unstable hadrons and hadronic
molecules can be observed in experiments, given that their widths are not too
large. On the contrary, some of the unstable hadrons and hadronic molecules
have so transient lifetimes that they may not even be formed, not to mention
being observed in experiments.

However, the answer not only relates to the lifetimes, for examples:

  * •

The widths of the $W$ and $Z$ bosons are $2.085\pm 0.042$ GeV and $2.4952\pm
0.0023$ GeV, respectively. The width of the $top$ quark is
$1.42^{+0.19}_{-0.15}$ GeV, that is also larger than 1 GeV. Although these
three particles are all very broad, they have been well established in
experiments. For comparisons, their masses are $80.379\pm 0.012$ GeV,
$91.1876\pm 0.0021$ GeV, and $172.76\pm 0.30$ GeV, respectively.

  * •

However, all the hadrons well observed (or at least not badly observed) in
experiments have the widths smaller than 1 GeV. As two of the broadest hadrons
observed so far, the $\pi_{2}(2100)$ meson and the $N(2600)$ baryon have the
widths $625\pm 50$ MeV and $500-800$ MeV, respectively. For comparisons, their
masses are $2090\pm 29$ MeV and $2550-2750$ MeV, respectively.

  * •

The pole parameter of the $f_{0}(500)$ meson is estimated in the PDG2020 pdg
to be $(400-550)-i(200-350)$ MeV, so its mass and width are

$\displaystyle M_{f_{0}(500)}$ $\displaystyle=$ $\displaystyle 400-550{~{}\rm
MeV}\,,$ (2) $\displaystyle\Gamma_{f_{0}(500)}$ $\displaystyle=$
$\displaystyle 400-700{~{}\rm MeV}\,.$

Besides, the pole parameter of the $K_{0}^{*}(700)$ meson is estimated to be
$(630-730)-i(260-340)$ MeV pdg , so its mass and width are

$\displaystyle M_{K_{0}^{*}(700)}$ $\displaystyle=$ $\displaystyle
630-730{~{}\rm MeV}\,,$ (3) $\displaystyle\Gamma_{K_{0}^{*}(700)}$
$\displaystyle=$ $\displaystyle 520-680{~{}\rm MeV}\,.$

The establishments of these two mesons are probably the most difficult among
all the hadrons Pelaez:2015qba .

Therefore, the answer seems to relate to not only the lifetimes but also the
masses of particles.

Generally speaking, the particle with a larger ratio

$R\equiv M/\Gamma\sim M\tau/\hbar\,,$ (4)

can be more easily observed, while that with a smaller ratio becomes more
difficult to be observed. Here $M$, $\Gamma$, and $\tau$ are the mass, width,
and lifetime of the particle, respectively.

Especially, the $f_{0}(500)$ and $K_{0}^{*}(700)$ mesons both satisfy $R\equiv
M/\Gamma\sim 1$. We use $\Delta E$ and $\Delta t$ to denote the uncertainties
of $M$ and $\tau$, respectively. Given that the mass $M$ and the lifetime
$\tau$ are two well-defined positive-definite parameters, their uncertainties
should be smaller than themselves:

$\displaystyle\Delta E$ $\displaystyle\lesssim$ $\displaystyle M\,,$ (5)
$\displaystyle\Delta t$ $\displaystyle\lesssim$ $\displaystyle\tau\,,$ (6)

so the $f_{0}(500)$ and $K_{0}^{*}(700)$ mesons further satisfy

$\Delta E\Delta t\lesssim M\tau\sim M\hbar/\Gamma=R\hbar\sim\hbar\,.$ (7)

This indicates that the uncertainty principle $\Delta E\Delta
t\geqslant\hbar/2$ may play an important role in determining “what kinds of
particles exist, and so can be observed, in our realistic world”.

Let us consider an unstable particle with $R\equiv M/\Gamma<1/2$
($M\tau<\hbar/2$). This particle does not exist in our realistic world
probably, since it does not obey the uncertainty principle:

$\Delta E\Delta t\lesssim M\tau\sim M\hbar/\Gamma=R\hbar<\hbar/2\,.$ (8)

This follows the argument that the particles existing in our world should
satisfy some of its fundamental physical laws such as the quantum physics,
while those not-existing do not need to. We shall further discuss this at the
end of this paper.

Another example is the sea quarks confined inside hadrons, which do not
satisfy the uncertainty principle $\Delta p\Delta x\geqslant\hbar/2$, and
correspondingly, the free quarks do not exist in our realistic world. This
triggers the question on the definition of “free”: does the free quark still
not exist in the time scale much shorter than that needed to form a hadron,
and correspondingly, does the free electric charge still exist in the time
scale larger than the lifetime of our universe?

Figure 5: Another type of boundary of our realistic world, that is from the
existent to the non-existent. The ratio $R$ is defined as $R\equiv
M/\Gamma\sim M\tau/\hbar$, where $M$, $\Gamma$, and $\tau$ represent the mass,
width, and lifetime of the unstable particle, respectively.

Based on the above analyses, we conjecture that the ratios $R\equiv
M/\Gamma\gg 1/2$ ($M\tau\gg\hbar/2$) and $R\equiv M/\Gamma<1/2$
($M\tau<\hbar/2$) can be used to describe the particles existing and not-
existing in our realistic world, respectively. Besides, the ratio $R\equiv
M/\Gamma\sim 1/2$ ($M\tau\sim\hbar/2$) can be used to describe the particles
quasi-existing in our realistic world. These quasi-existent particles are of
particular interest, since they may not well obey the quantum physics but they
may still be observed in experiments. As depicted in Fig. 5, studying these
quasi-existent particles may allow us to go beyond the quantum physics and
arrive at another type of boundary of our realistic world, that is from the
existent to the non-existent.

##  V The singly heavy baryon system as a possible platform

As discussed in the previous section, the quasi-existent particles satisfying
$R\equiv M/\Gamma\sim 1/2$ ($M\tau\sim\hbar/2$) are of particular interest,
since they may not well obey the quantum physics but they may still be
observed in experiments. Especially, the $f_{0}(500)$ and $K_{0}^{*}(700)$
mesons both satisfy $R\equiv M/\Gamma\sim 1$, so they are worthy of further
investigations. However, as isolated cases, their studies are probably not
enough. Instead, we propose to study the singly heavy baryon system, which may
serve as an even better platform due to various imperfect symmetries among the
many singly heavy baryons.

As discussed in Sec. I, the singly heavy baryon contains one valence heavy
quark ($charm/bottom$) and two valence light quarks ($up/down/strange$), where
the light quarks and gluons circle around the nearly static heavy quark. This
system behaves as the QCD analogue of the atom bound by the electromagnetic
interaction, but its internal structure is much more complicated.

Figure 6: Jacobi coordinates $\vec{\lambda}$ and $\vec{\rho}$ for the singly
heavy baryon system. Figure 7: Categorization of the $1P$-wave singly charmed
baryons into eight multiplets within the framework of heavy quark effective
theory.

As depicted in Fig. 6, we focus on the $Qq_{1}q_{2}$ system, with $Q$ the
heavy quark and $q_{1,2}$ the two light quarks:

  * •

Its total orbital angular momentum is

$L=l_{\rho}\otimes l_{\lambda}\,,$ (9)

where $l_{\rho}$ is the orbital angular momentum between the two light quarks,
and $l_{\lambda}$ is the orbital angular momentum between the heavy quark and
the two-light-quark system.

  * •

Its total spin angular momentum is

$S={s}_{q_{1}}\otimes{s}_{q_{2}}\otimes{s}_{Q}=s_{l}\otimes{s}_{Q}\,,$ (10)

where $s_{q_{1,2}}$ and $s_{Q}$ are the light and heavy quark spin angular
momenta, and $s_{l}={s}_{q_{1}}\otimes{s}_{q_{2}}$ is the total spin angular
momentum of the two light quarks.

  * •

In the heavy quark limit $m_{Q}\to\infty$, its properties are mainly
determined by the light degrees of freedom. Accordingly, its total angular
momentum

${J}=\left(l_{\rho}\otimes
l_{\lambda}\right)_{L}\otimes\left({s}_{q_{1}}\otimes{s}_{q_{2}}\otimes{s}_{Q}\right)_{S}\,,$
(11)

can be written as

${J}=\left(\left(l_{\rho}\otimes
l_{\lambda}\right)_{L}\otimes\left({s}_{q_{1}}\otimes{s}_{q_{2}}\right)_{s_{l}}\right)_{j_{l}}\otimes{s}_{Q}\,,$
(12)

where $j_{l}=L\otimes s_{l}=l_{\rho}\otimes
l_{\lambda}\otimes{s}_{q_{1}}\otimes{s}_{q_{2}}$ is the total angular momentum
of the light component.

To well understand the singly heavy baryon system, we need to carefully
examine the four internal symmetries between the two light quarks:

  * •

The color structure of the two light quarks is antisymmetric
($\mathbf{\bar{3}}_{C}$).

  * •

The flavor structure of the two light quarks is either antisymmetric or
symmetric. Within the flavor $SU(3)$ symmetry, the antisymmetric structure is
the $\mathbf{\bar{3}}_{F}$ representation composed of the
$\Lambda_{Q}/\Xi_{Q}$ baryons, and the symmetric structure is the
$\mathbf{6}_{F}$ representation composed of the
$\Sigma_{Q}/\Xi^{\prime}_{Q}/\Omega_{Q}$ baryons.

  * •

The spin structure of the two light quarks is either antisymmetric ($s_{l}=0$)
or symmetric ($s_{l}=1$).

  * •

The orbital structure of the two light quarks is either antisymmetric
($l_{\rho}=1/3/\cdots$) or symmetric ($l_{\rho}=0/2/\cdots$).

According to the Pauli principle, the total wave function of the two light
quarks is antisymmetric, based on which we can categorize the singly heavy
baryons into several multiplets. According to the heavy quark effective theory
Grinstein:1990mj ; Eichten:1989zv ; Falk:1990yz , two states can have the same
quantum numbers $(j_{l},s_{l},l_{\rho},l_{\lambda})$ when $j_{l}\neq 0$, and
they form a degenerate doublet with $J=j_{l}\otimes s_{Q}=j_{l}\pm 1/2$; there
is only one state having the quantum numbers
$(j_{l},s_{l},l_{\rho},l_{\lambda})$ when $j_{l}=0$, and it forms a singlet
itself with $J=j_{l}\otimes s_{Q}=1/2$.

As shown in Fig. 7, we can categorize the $1P$-wave singly charmed baryons
into eight multiplets, denoted as $[Flavor,j_{l},s_{l},\rho/\lambda]$. There
can be as many as seven $\Lambda_{c}$ and seven $\Xi_{c}$ baryons, which form
four multiplets of the $\mathbf{\bar{3}}_{F}$ representation; there can also
be seven $\Sigma_{c}$, seven $\Xi^{\prime}_{c}$, and seven $\Omega_{c}$
baryons, which form four multiplets of the $\mathbf{6}_{F}$ representation. As
discussed in Sec. I, their rich internal structure causes the fine structure
of the strong interaction, which has been observed in recent LHCb experiments
LHCb:2017uwr ; LHCb:2020iby ; LHCb:2020tqd .

In the past years important experimental progresses have been made in the
field of singly heavy baryon system, and many candidates for the excited
singly heavy baryons have been observed in experiments pdg . These
observations have attracted much attention among the theorists, and various
theoretical methods have been applied to study the singly heavy baryons
Chen:2022asf . Especially, both the constitute quark model and the QCD sum
rule method have been applied to systematically study their mass spectra and
decay properties. The results obtained using these two methods are roughly
consistent with each other, although not exactly the same, e.g., see the
constitute quark model calculations of Ref. Liang:2020hbo  and the QCD sum
rule calculations of Refs. Yang:2020zrh ; Yang:2021lce .

Based on these two theoretical methods, some singly heavy baryons are
calculated to have rather large widths, for examples:

  * •

The width of the $\Omega_{b0}(1/2^{-})$ baryon belonging to the
$[\mathbf{6}_{F},0,1,\lambda]$ singlet is calculated to be about 1.0 GeV in
Ref. Liang:2020hbo  through the constitute quark model. This width is
calculated to be about 2.7 GeV in Ref. Yang:2020zrh  through the QCD sum rule
method. Note that the uncertainties of these two calculations are rather
large, so these two results are still roughly consistent with each other. For
comparison, the mass of this baryon is calculated to be about 6.3 GeV.

  * •

The width of the $\Omega_{c0}(1/2^{-})$ baryon belonging to the
$[\mathbf{6}_{F},0,1,\lambda]$ singlet is calculated to be about 1.0 GeV in
Ref. Yang:2021lce  through the QCD sum rule method. For comparison, its mass
is calculated to be about 3.0 GeV.

  * •

Especially, the widths of the $\Lambda_{c1}(1/2^{-})$ and $\Xi_{c1}(1/2^{-})$
baryons belonging to the $[\mathbf{\bar{3}}_{F},1,0,\lambda]$ doublet are
calculated in Ref. Yang:2022oog  through the QCD sum rule method to be about
2.4 GeV and 6.6 GeV, respectively. For comparisons, their masses are
calculated to be about 2.7 and 2.9 GeV, respectively.

Besides, there can exist the $1D$\- and $1F$-wave singly heavy baryons, etc.
As higher-lying particles, their widths are generally larger than the
$1P$-wave ones.

Therefore, there might exist some singly heavy baryons satisfying $R\equiv
M/\Gamma\sim 1/2$. Moreover, there exist various imperfect symmetries in the
singly heavy baryon system:

  * •

The fine structure of the singly heavy baryons within the same multiplet. In
the heavy quark limit the properties of a singly heavy baryon are mainly
determined by its light degrees of freedom, so the baryons within the same
multiplet are similar to some extent. For example, the $\Lambda_{c1}(1/2^{-})$
and $\Lambda_{c1}(3/2^{-})$ baryons belonging to the
$[\mathbf{\bar{3}}_{F},1,0,\lambda]$ doublet are similar to each other to some
extent.

  * •

The heavy quark flavor symmetry between the $charm$ and $bottom$ quarks. This
symmetry indicates that the singly charmed baryons
$\Lambda_{c}/\Xi_{c}/\Sigma_{c}/\Xi^{\prime}_{c}/\Omega_{c}$ and the singly
bottom baryons $\Lambda_{b}/\Xi_{b}/\Sigma_{b}/\Xi^{\prime}_{b}/\Omega_{b}$
are respectively similar to each other. However, the heavy quark effective
theory works better for the singly bottom baryons than for the singly charmed
baryons, since the mass of the $bottom$ quark is significantly larger than
that of the $charm$ quark.

  * •

The isospin $SU(2)$ symmetry between the $up$ and $down$ quarks as well as the
flavor $SU(3)$ symmetry among the $up$, $down$, and $strange$ quarks. The
former symmetry works quite well, while the latter still works but not so
well. For examples, the former symmetry indicates that the singly bottom
baryons $\Xi_{b}^{0}$ and $\Xi_{b}^{-}$ are quite similar to each other, and
the latter indicates that the singly charmed baryons $\Sigma_{c}$,
$\Xi^{\prime}_{c}$, and $\Omega_{c}$ are similar to some extent.

Although the singly heavy baryons with $R\equiv M/\Gamma\sim 1/2$ are rather
difficult to be observed in experiments, their partner states due to the above
imperfect symmetries may have a significantly larger ratio $R\equiv M/\Gamma$,
and so become much easier to be observed. For example, there are five excited
$\Omega_{c}^{0}$ baryons but only three excited $\Xi_{c}^{0}$ baryons observed
in the LHCb experiments LHCb:2017uwr ; LHCb:2020iby , as shown in Fig. 1(a,b).
Correspondingly, there can be two excited $\Xi_{c}^{0}$ baryons still missing,
and they are expected to have significantly larger widths Yang:2021lce . The
above imperfect symmetries can be applied to overcome some of the difficulties
in studying these missing states.

There might exist some other symmetries and more fine/hyperfine structures in
the singly heavy baryon system. These possibly-existing symmetries and the
above imperfect symmetries make the singly heavy baryon system serve as a
possible platform to study the quasi-existent particles with $R\equiv
M/\Gamma\sim 1/2$ ($M\tau\sim\hbar/2$).

##  VI Summary and Perspective

In this paper we discuss what do we learn from the hadron physics after its
quick developments in the past decades. The development on the singly heavy
baryons indicates that there exists the fine structure of hadron spectrum
caused by the direct strong interaction, and the development on the exotic
hadrons indicates that the residue strong interaction is capable of forming
the hadronic molecules. As depicted in Fig. 3, the direct and residue
electromagnetic interactions form atoms and chemical molecules respectively,
which further compose our realistic world. Similarly, the strong interaction
may also be capable of forming some imaginable hadronic worlds, although these
worlds might only exist in a very short time.

In our realistic world most of the hadrons and hadronic molecules are not
stable. Some of them have so transient lifetimes that they may not even be
formed. We discuss whether these non-existent particles are still capable of
affecting our realistic world. Based on a simple toy model, we argue its
answer to be possibly positive. Accordingly, we conjecture that there are
hadrons and hadronic molecules that are too unstable to exist, but they can
still affect our realistic world; there might be the imaginal worlds made of
these non-existent particles, and these worlds do not exist neither but still
weakly affect our realistic world.

We further discuss what kinds of particles exist, and so can be observed, in
our realistic world. We argue that its answer possibly relates to not only the
lifetimes but also the masses of particles, and moreover, the uncertainty
principle may play an important role in answering this question. We conjecture
that the ratios $R\equiv M/\Gamma\gg 1/2$ ($M\tau\gg\hbar/2$) and $R\equiv
M/\Gamma<1/2$ ($M\tau<\hbar/2$) can be used to describe the particles existing
and not-existing in our realistic world, respectively. Here $M$, $\Gamma$, and
$\tau$ are the masses, widths, and lifetimes of particles, respectively.

Actually, we can use a rather simple argument to show that the particle with
$R\equiv M/\Gamma<1/2$ ($M\tau<\hbar/2$) does not obey the uncertainty
principle. Let us imagine an unstable particle with the mass $M$ and the
lifetime $\tau\rightarrow 0$. The uncertainty principle demands

$\Delta E\Delta t\geqslant{\hbar/2}\,,$ (13)

where $\Delta E$ and $\Delta t$ are the uncertainties of $M$ and $\tau$,
respectively. Given that the mass $M$ and the lifetime $\tau$ are two well-
defined positive-definite parameters, their uncertainties should be smaller
than themselves:

$\displaystyle\Delta E$ $\displaystyle\lesssim$ $\displaystyle M\,,$ (14)
$\displaystyle\Delta t$ $\displaystyle\lesssim$ $\displaystyle\tau\,.$ (15)

Therefore, the uncertainty principle further demands

$M\tau\gtrsim\hbar/2\,,$ (16)

and the particle with $R\equiv M/\Gamma<1/2$ ($M\tau<\hbar/2$) does not obey
the uncertainty principle.

The above analyses indicate that the quantum physics, which rules our
microscopic world, may set a lower limit on the lifetime of the particle
existing in our realistic world. To better understand this, we relate the
Bohr’s quantization condition for the angular momentum of the electron around
the proton

$L=mvr=n\hbar\,,~{}~{}~{}n=1,2,3\cdots\,,$ (17)

to the condition of the circular standing wave

$2\pi r=n\lambda\,,$ (18)

through the de Broglie relation $\lambda=h/p=h/(mv)$. Comparing with the two
uncertainty principles $\Delta p\Delta x\geqslant\hbar/2$ and $\Delta E\Delta
t\geqslant{\hbar/2}$ as well as the correspondences

$(p,\Delta p)\rightarrow(M,\Delta E)\,,~{}(r,\Delta x)\rightarrow(\tau,\Delta
t)\,,$ (19)

we conjecture that the lifetime may also be quantized:

$M\tau=n\hbar\,,~{}~{}~{}n=1,2,3\cdots\,.$ (20)

Considering that the angular momentum quantum number can be either an integer
or a half-integer actually, this quantization condition is finalized to be

$M\tau=n\hbar/2{~{}~{}~{}\rm
and~{}~{}~{}}M/\Gamma=n/2\,,~{}~{}~{}n=1,2,3\cdots\,,$ (21)

for the particles existing in our realistic world as the “standing waves” in
the time axis.

Besides the existent and non-existent particles, we propose to use the ratio
$R\equiv M/\Gamma\sim 1/2$ ($M\tau\sim\hbar/2$) to describe the particles
quasi-existing in our realistic world. These quasi-existent particles are of
particular interest, since they may not well obey the quantum physics but they
may still be observed in experiments. Their studies may allow us to go beyond
the quantum physics and arrive at another type of boundary of our realistic
world, that is from the existent to the non-existent. Especially, the
$f_{0}(500)$ and $K_{0}^{*}(700)$ mesons both satisfy $R\equiv M/\Gamma\sim
1$, so they are worthy of further investigations. Moreover, we propose to
study the singly heavy baryon system, which serves as a possible platform to
study the quasi-existent particles due to various imperfect symmetries among
the many singly heavy baryons.

Based on the above analyses, we further conjecture that some fundamental
physical laws, such as the quantum physics, rule our realistic world simply
because they are the boundaries stabilizing our realistic world (or blocking
our realistic world from some other worlds); the particles existing in our
realistic world should satisfy these fundamental physical laws, while those
not-existing do not need to. This argument has already been stated in Sec. IV.
Therefore, we may meet some new physics when investigating the particles not-
existing in our realistic world, and we might also meet some other new physics
when investigating the particles quasi-existing in our realistic world, e.g.,
the non-existent particle $B_{2}$ depicted in Fig. 4(b) may be investigated
through the decay process $C_{3}\to B_{2}\to A$, and the possibly-existing
quasi-existent singly heavy baryons may be investigated through various
imperfect symmetries among the many singly heavy baryons.

Besides our realistic world, there can be various worlds formed by various
fundamental physical laws with certain fundamental parameters, and the world
can become much different as long as its fundamental physical laws or its
fundamental parameters change a little. Especially, there may exist the
imaginal hadronic worlds formed by the strong interaction, whose time and
space scales are much smaller than those of our realistic world. If there
exist some civilizations in these mini-worlds, there might be a tiny chance
that these civilizations are willing and able to send out some of their
information, and at the same time we are able to receive these information.

To end this paper, let us imagine that someday in the future our human beings
finally exhaust all the resources of our universe, but still we can not break
through its boundary. What we can do at that time might be exchanging our
belongings to some external world for the necessary resources, and the most
valuable belonging of ours is probably the knowledge of our universe. On the
contrary, we might also serve as the external world, and exchange our
resources to the civilizations from some mini-worlds for the knowledge of
their universes.

## Acknowledgments

This project is supported by the National Natural Science Foundation of China
under Grant No. 12075019, the Jiangsu Provincial Double-Innovation Program
under Grant No. JSSCRC2021488, and the Fundamental Research Funds for the
Central Universities.

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