IFUM-1090-FT

# Binary black hole system at equilibrium

Marco Astorino<EMAIL_ADDRESS>Istituto Nazionale di Fisica Nucleare
(INFN), Sezione di Milano
Via Celoria 16, I-20133 Milano, Italy  Adriano Viganò<EMAIL_ADDRESS>Università degli Studi di Milano
Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Milano
Via Celoria 16, I-20133 Milano, Italy

###### Abstract

An exact and analytical solution of four dimensional vacuum General Relativity
representing a system of two static black holes at equilibrium is presented.
The metric is completely regular outside the event horizons, both from
curvature and conical singularities. The balance between the two Schwarzschild
sources is granted by an external gravitational field, without the need of
extra matter fields besides gravity, nor strings or struts. The geometry of
the solution is analysed. The Smarr law, the first and the second law of black
hole thermodynamics are discussed.

##  I Introduction

Black holes are fundamental objects of our Universe. Nevertheless there are
scarce exact and analytical solutions that model black holes within the
context of General Relativity, which represents the standard framework of
gravitational interactions. The requirement of asymptotic flatness further
restricts the number of available solutions, basically due to the existence of
uniqueness theorems Chrusciel  _et al._ (2012).

The recent remarkable observational results from gravitational waves detection
reveal that the major source of gravitational waves is provided by the
interaction between massive black holes Abbott  _et al._ (2016).
Unfortunately, there are no exact analytical solutions describing well-founded
multi-source black holes in pure General Relativity. The few attempts built so
far need singular matter to sustain the gravitational attraction between the
sources, as can be seen from double Kerr attempts, see Kramer and Neugebauer
(1980); Dietz and Hoenselaers (1985). This matter is often interpreted as
struts or cosmological strings, however these objects not only have stability
issues and no phenomenological plausibility at the moment; they also rise some
fundamental theoretical issues related to the singular behaviour of the
spacetime, which therefore can hardly be thought as a proper manifold. The
$\delta$-like stress-energy tensor associated to the strut, located between
the black holes, violates all the reasonable energy conditions because, in
order to prevent the gravitational sources from the collapse, it has to mimic
a repulsive gravitational force generated by anti-gravitational matter.
Alternatively, the black holes can be pulled by axial strings of proper
matter, i.e. of positive energy density, which however are divergent
quantities that extend to infinity. Both scenarios are clearly out of current
empirical experience.

On the other hand, in the presence of electromagnetic fields, it is possible
to gain the equilibrium between two black holes thanks to the electromagnetic
repulsive force that balances the gravitational attraction. In fact some
solutions of these kind are known, basically the Majumdar–Papapetrou black
holes Majumdar (1947); Papapetrou (1945); Hartle and Hawking (1972) and the
Emparan–Teo dihole Emparan and Teo (2001). While these solutions are regular
outside the event horizons of the sources, they need to be composed of charged
matter, which is improbable from the astrophysical point of view. Furthermore
their physical parameters are very fine tuned to maintain the equilibrium,
therefore they are likely to be unstable under perturbations.

The purpose of this Letter is to explore the possibility to build analytical
metrics modelling two black hole sources at equilibrium, aiming at the most
phenomenologically realistic scenario as possible. For this reason, we will
discard any source of singularities outside the event horizons, not only Dirac
or cosmic strings, but also Misner strings. In fact, the NUT parameter is
known to bring in physical issues, e.g. closed timelike curves.

The strategy of our approach is based on an Ernst’s insight, who managed to
regularise the nodal singularity of the accelerating Schwarzschild black hole
in Ernst (1978). Similarly we examine the possibility of balancing the
gravitational attraction of the two (or more) sources by an external
gravitational field, which serves as a setting for embedding our multi-source
solution. Thus the spacetime asymptotic will not be restricted to the flat
one; instead we prefer to consider an axisymmetric multi-polar expansion
Tomimatsu (1984a); Breton  _et al._ (1998) able to model a generic
gravitational background 111An interior multi-polar deformation of the sources
can be achieved as well Erez and Rosen (1959), but it seems that these poles
bring curvature singularities not covered by an event horizon.. The multipolar
expansion is the key ingredient that allows to circumvent the no-go theorem
about the non-existence of static Beig and Schoen (2009) configurations of
many-body systems, with a suitable separation condition.

These kind of solutions are know in the literature, in the single source case,
as distorted black holes: they have been introduced by Doroshkevich,
Zel’dovich and Novikov Doroshkevich  _et al._ (1965), and lately Chandrasekhar
Chandrasekhar (1985) studied their equilibrium conditions. Geroch and Hartle
Geroch and Hartle (1982) enlightened then some of their general properties.
Distorted black holes are especially valuable when the black hole is not
isolated but embedded in a strong field regime, such as in the center of
galaxies, where approximate, perturbative or numerical methods loose accuracy.
Distorted metrics are generally considered local in the sense that are mostly
meaningful in a finite region surrounding the black holes. While the number of
gravitational multipoles of the background can in principle be left arbitrary
Astorino and Viganò (2021), in this Letter we focus only on the dipole and
quadrupole terms since it represents the simplest setting in which the black
holes distinctive parameters remain unconstrained. We stress that our
regularisation method relies only on the presence of (at least) two multipole
momenta, therefore it can be achieved for any couple on non-null poles.

In practice we superimpose a double black hole solution to the external field
background by means of the Belinski–Zakharov “inverse scattering method”
Belinsky and Zakharov (1978); Belinsky and Sakharov (1979); Belinski and
Verdaguer (2005). The details of the inverse scattering construction in the
context of external gravitational fields are left for a future work Astorino
and Viganò (2021).

##  II Metric and regularisation

Consider the static and axisymmetric metric, solution of the Einstein
equations in vacuum $R_{\mu\nu}=0$, in cylindrical Weyl coordinates
$(t,\rho,z,\phi)$

$\begin{split}{ds}^{2}&=-V(\rho,z){dt}^{2}+\rho^{2}V^{-1}(\rho,z){d\phi}^{2}\\\
&\quad+f(\rho,z)\bigl{(}{d\rho}^{2}+{dz}^{2}\bigr{)},\end{split}$ (1)

where

$\displaystyle V$
$\displaystyle=\frac{\mu_{1}\mu_{3}}{\mu_{2}\mu_{4}}\exp\biggl{[}2b_{1}z+2b_{2}\biggl{(}z^{2}-\frac{\rho^{2}}{2}\biggr{)}\biggr{]},$
(2)
$\displaystyle\begin{split}f&=16C_{f}\frac{\mu_{1}^{3}\mu_{2}^{5}\mu_{3}^{3}\mu_{4}^{5}}{W_{11}W_{22}W_{33}W_{44}W_{13}^{2}W_{24}^{2}Y_{12}Y_{14}Y_{23}Y_{34}}\exp\biggl{[}-b_{1}^{2}\rho^{2}+\frac{b_{2}^{2}}{2}\bigl{(}\rho^{2}-8z^{2}\bigr{)}\rho^{2}-4b_{1}b_{2}z\rho^{2}\\\
&\quad+2b_{1}(-z+\mu_{1}-\mu_{2}+\mu_{3}-\mu_{4})+b_{2}\bigl{(}-2z^{2}+\rho^{2}+4z(\mu_{1}-\mu_{2})+\mu_{1}^{2}-\mu_{2}^{2}+(\mu_{3}-\mu_{4})(4z+\mu_{3}+\mu_{4})\bigr{)}\biggr{]},\end{split}$
(3)

and $W_{ij}=\rho^{2}+\mu_{i}\mu_{j}$, $Y_{ij}=(\mu_{i}-\mu_{j})^{2}$. The
functions $\mu_{i}=\sqrt{\rho^{2}+(z-w_{i})^{2}}-(z-w_{i})$ contain the
constants $w_{i}$, that are chosen with ordering $w_{1}<w_{2}<w_{3}<w_{4}$ and
with values

$\displaystyle w_{1}$ $\displaystyle=z_{1}-m_{1},\qquad w_{2}=z_{1}+m_{1},$
(4a) $\displaystyle w_{3}$ $\displaystyle=z_{2}-m_{2},\qquad
w_{4}=z_{2}+m_{2}.$ (4b)

The parameters $m_{i}$ and $z_{i}$ refer to the mass and to the position of
the $i$-th black hole, respectively, while $b_{1}$ and $b_{2}$ are the dipole
and quadrupole momenta of the external gravitational field polar expansion.
The constant $C_{f}$ is a gauge parameter.

The metric (1) represents two Schwarzschild black holes immersed in an
external back-reacting gravitational field. The two event horizons extend in
the regions $w_{1}<z<w_{2}$ and $w_{3}<z<w_{4}$ when $\rho=0$, while the
curvature singularities are covered by the horizons. This can be understood
thanks to a criterion given in Harmark (2004): the kernel of the ($t,\phi$)
part of the metric can be 2-dimensional only inside the horizon. By removing
the external field ($b_{1}=b_{2}=0$), one straightforwardly recovers the
double-Schwarzschild solution (also known as Bach–Weyl metric) Bach and Weyl
(1922); Israel and Khan (1964); Costa and Perry (2000).

A physical solution should be regular on the $z$-axis, i.e. it must not be
affected by conical singularities. These angular defects are present when the
ratio between the length and the radius of small circles around the $z$-axis
is different from $2\pi$. A small circle around the $z$-axis has radius
$R=\sqrt{g_{zz}}\rho$ and length $L=2\pi\sqrt{g_{\phi\phi}}$ (cf. Alekseev and
Belinski (2019)), then the regularity condition corresponds to $L/(2\pi R)\to
1$ as $\rho\to 0$. It is easy to prove that, for the diagonal metric (1), the
above condition is equivalent to $fV\to 1$ as $\rho\to 0$. The latter must be
satisfied everywhere, outside the black holes, to avoid conical singularities.
We thus require

$fV=1,$ (5)

over the intervals $-\infty<z<w_{1}$, $w_{2}<z<w_{3}$ and $w_{4}<z<+\infty$ as
$\rho\to 0$. Assuming a $2\pi$-periodic azimuthal angle $\phi$, we have to fix
three out of seven parameters in order to satisfy (5).

Figure 1: Embedding diagram of the two black hole event horizons for the
parametric values $z_{1}=5$, $z_{2}=15$, $m_{1}=4$, $m_{2}=4$. This picture
shows the deformation of the horizons due to both the external gravitational
field and the mutual interaction between the sources. The horizon surface is
smooth because of the absence of any conical singularities.

We choose the gauge parameter $C_{f}$ and the external field parameters
$b_{1}$, $b_{2}$ as follows:

$\displaystyle C_{f}$
$\displaystyle=256m_{1}^{2}m_{2}^{2}(m_{1}+m_{2}+z_{1}-z_{2})^{2}(m_{1}+m_{2}-z_{1}+z_{2})^{2},$
(6) $\displaystyle b_{1}$
$\displaystyle=-\frac{(m_{1}z_{1}+m_{2}z_{2})}{4m_{1}m_{2}(z_{1}-z_{2})}\log\biggl{[}\frac{(m_{1}-m_{2}+z_{1}-z_{2})(m_{1}-m_{2}-z_{1}+z_{2})}{(m_{1}+m_{2}+z_{1}-z_{2})(m_{1}+m_{2}-z_{1}+z_{2})}\biggr{]},$
(7) $\displaystyle b_{2}$
$\displaystyle=\frac{(m_{1}+m_{2})}{8m_{1}m_{2}(z_{1}-z_{2})}\log\biggl{[}\frac{(m_{1}-m_{2}+z_{1}-z_{2})(m_{1}-m_{2}-z_{1}+z_{2})}{(m_{1}+m_{2}+z_{1}-z_{2})(m_{1}+m_{2}-z_{1}+z_{2})}\biggr{]}.$
(8)

It is understood that $C_{f}$, $b_{1}$, $b_{2}$ will assume the above values,
from now on. Note that the physical parameters characterising the black holes
were left unconstrained: this means that the physical properties of the binary
black hole system are completely generic. This fact guarantees a wide
flexibility in a possible phenomenological scenario 222In fact the presented
solution is not the minimal one which can be regularised. It is sufficient to
consider only one multipole term for the external gravitational field to
remove all the singularities from the double black hole configuration, at the
price of fixing the position or the mass of a black hole.. Of course one could
alternatively keep the $b_{i}$ generic, to model an arbitrary external
gravitational background: in that case the intrinsic black holes parameters
would adjust to fit the given background.

One can infer the order of magnitude of the multipole momenta (7), (8) for an
experimental setting. We make an explicit example by considering the data for
the black holes masses and separation provided by the GW150914 event Abbott
_et al._ (2016): we take $m_{1}=29M_{\odot}$, $m_{2}=36M_{\odot}$,
$|z_{1}-z_{2}|=520M_{\odot}$, and we find $b_{1}\sim 10^{-4}/M_{\odot}$ and
$b_{2}\sim-10^{-7}/M_{\odot}^{2}$. These are the values that ensure the
equilibrium, therefore in the case of a merging process they represent only an
upper bound. The relevance of the contributions of $b_{1}$ and $b_{2}$ in the
context of the Geroch–Hansen multipoles Hansen (1974) will be discussed in
Astorino and Viganò (2021).

We notice that it is possible, within our scheme, to build an array of
collinear black holes immersed in an external gravitational field with
complete multipolar expansion Astorino and Viganò (2021), thus generalising
the Israel–Kahn solution Israel and Khan (1964). In that case, the
regularisation would require the complete multipole expansion 333Actually an
arbitrary multipole expansion can be useful to model any generic stationary
and axisymmetric external gravitational field. of the external field to fix
the conical singularities, without constraining the physical parameters of the
black holes, such as their masses or positions on the $z$-axis.

To ensure phenomenological significance of the solution (1), it is essential
that the separation of the two sources is finite. The proper distance between
the two black hole horizons for $\rho=0$ and fixed $\phi$ is given by

$\begin{split}\ell&=\int_{w_{2}}^{w_{3}}dz\,\sqrt{g_{zz}}\big{|}_{\rho=0}\propto\sqrt{\frac{(w_{4}-z)(z-w_{1})}{(w_{3}-z)(z-w_{2})}}\\\
&\quad\times
e^{2b_{1}(2w_{3}-2w_{4}-z)+2b_{2}(2w_{3}^{2}-2w_{4}^{2}-z^{2})}.\end{split}$
(9)

It can be shown that the integral converges, and hence the equilibrium can be
achieved for finite proper distance.

##  III Near-horizon limit

We consider the near-horizon limit of the metric (1), in order to show that it
contains two distorted Schwarzschild black holes. We zoom in to the first
black hole horizon by performing the change of coordinates

$\rho=\sqrt{r(r-2m_{1})}\sin\theta,\quad z=z_{1}+(r-m_{1})\cos\theta,$ (10)

and by taking the limit $r\to 2m_{1}$, by which the metric (1) boils down to

$\begin{split}{ds}^{2}&\simeq
h(\theta)\biggl{[}-\biggl{(}1-\frac{2m_{1}}{r}\biggr{)}e^{F_{1}(\theta)}{dt}^{2}+\frac{D^{2}e^{F_{2}(\theta)}}{1-2m_{1}/r}{dr}^{2}\biggr{]}\\\
&\quad+(2m_{1})^{2}\biggl{[}D^{2}h(\theta)e^{F_{2}(\theta)}{d\theta}^{2}+\frac{\sin^{2}\theta}{h(\theta)}e^{-F_{1}(\theta)}{d\phi}^{2}\biggr{]},\end{split}$
(11)

where

$\displaystyle h(\theta)$
$\displaystyle=\frac{m_{1}\cos\theta+m_{2}+z_{1}-z_{2}}{m_{1}\cos\theta-
m_{2}+z_{1}-z_{2}},$ (12) $\displaystyle F_{1}(\theta)$
$\displaystyle=2[b_{1}+b_{2}(z_{1}+m_{1}\cos\theta)](z_{1}+m_{1}\cos\theta),$
(13)
$\displaystyle\begin{split}F_{2}(\theta)&=2b_{1}(m_{1}\cos\theta-2m_{1}-4m_{2}-z_{1})\\\
&\quad+2b_{2}(m_{1}^{2}\cos^{2}\theta+2m_{1}z_{1}\cos\theta-2m_{1}^{2}\\\
&\quad-z_{1}^{2}-4m_{1}z_{1}-8m_{2}z_{2}),\end{split}$ (14)

and

$D=\frac{m_{1}+m_{2}-z_{1}+z_{2}}{m_{1}-m_{2}-z_{1}+z_{2}}.$ (15)

One clearly recognises in (11) the structure of a distorted Schwarzschild
black hole Geroch and Hartle (1982). Actually the first black hole horizon is
deformed by the presence of both the external field and the second black hole.
Indeed, when the external field and the second black hole vanish, one recovers
the standard Schwarzschild metric. Obviously, a similar description holds for
the second black hole as well. A pictorial representation of the deformation
that the two horizons undergo is given in the embedding diagram of Fig. 1.

Another element about the geometry of the black hole horizon can be clarified
by computing the length of the equatorial and polar circles in the near-
horizon geometry (11). The equator ($\theta=\pi/2$) length is given by

$\begin{split}L_{\text{equator}}&=2m_{1}\int_{0}^{2\pi}\frac{e^{-F_{1}(\pi/2)/2}}{\sqrt{h(\pi/2)}}d\phi\\\
&=4\pi
m_{1}\sqrt{\frac{z_{1}-z_{2}-m_{2}}{z_{1}-z_{2}+m_{2}}}e^{-2(b_{1}+b_{2}z_{1})z_{1}},\end{split}$
(16)

while the polar length is

$L_{\text{polar}}=4m_{1}D\int_{0}^{\pi}\sqrt{h(\theta)}e^{F_{2}(\theta)/2}d\theta.$
(17)

It is not possible to analytically perform the latter integral, nevertheless
we can consider a numerical comparison between (16), (17) and
$L_{\text{Schwarzschild}}=4\pi m_{1}$. The result is

$\displaystyle L_{\text{equator}}$ $\displaystyle>L_{\text{Schwarzschild}},$
(18a) $\displaystyle L_{\text{polar}}$
$\displaystyle>L_{\text{Schwarzschild}}.$ (18b)

Thus, not only a deformation along the $z$-axis occurs, but there is also an
enlargement of both the equatorial and the polar circle with respect to the
Schwarzschild one. This is consistent with the behaviour of the black hole
temperature, as we will see below.

##  IV Smarr law and thermodynamics

Going back to the exact metric (1), we now compute some physical quantities
for that spacetime, in order to discuss the thermodynamics of the system.

The mass of the two black holes is found by means of the Komar–Tomimatsu
integral Komar (1959); Tomimatsu (1984b). The formula for the conserved mass,
in the case of the static spacetime (1), takes the form

$M=\alpha\int_{w_{2i-1}}^{w_{2i}}dz\,\rho\,g_{tt}^{-1}\,\partial_{\rho}g_{tt}\big{|}_{\rho=0},$
(19)

where $\alpha$ is a constant that takes into account the normalisation of the
timelike Killing vector $\xi\equiv\alpha\partial_{t}$, which generates the
stationary symmetry. It is well known that, in the absence of asymptotic
flatness, $\alpha$ is not necessarily equal to one, as happens for black holes
in AdS Henneaux and Teitelboim (1985) or Melvin Astorino  _et al._ (2016)
backgrounds. The integration is performed over the intervals $w_{1}<z<w_{2}$
and $w_{3}<z<w_{4}$, which correspond to the black holes, and the result is

$M_{1}=\alpha m_{1},\qquad M_{2}=\alpha m_{2}.$ (20)

The horizons areas are found integrating over the horizon surfaces, i.e.

$\mathcal{A}=\int_{0}^{2\pi}d\phi\int_{w_{2i-1}}^{w_{2i}}dz\,\sqrt{g_{zz}g_{\phi\phi}}\Big{|}_{\rho=0},$
(21)

thus

$\displaystyle\begin{split}\mathcal{A}_{1}&=16\pi
m_{1}^{2}\frac{m_{1}+m_{2}-z_{1}+z_{2}}{m_{1}-m_{2}-z_{1}+z_{2}}\\\
&\quad\times
e^{-2b_{1}(2m_{2}+m_{1}+z_{1})-2b_{2}((m_{1}+z_{1})^{2}+4m_{2}z_{2})},\end{split}$
(22) $\displaystyle\begin{split}\mathcal{A}_{2}&=16\pi
m_{2}^{2}\frac{m_{1}+m_{2}-z_{1}+z_{2}}{m_{2}-m_{1}-z_{1}+z_{2}}\\\
&\quad\times e^{-2b_{1}(m_{2}+z_{2})-2b_{2}(m_{2}+z_{2})^{2}}.\end{split}$
(23)

The temperature is obtained from the surface gravity as $T=\kappa/(2\pi)$.
Recalling that $\kappa^{2}=-\frac{1}{2}(\nabla_{\mu}\xi_{\nu})^{2}$, the
metric (1) gives rise to

$\kappa^{2}=-\frac{\alpha^{2}}{4}\frac{(\partial_{z}V)^{2}+(\partial_{\rho}V)^{2}}{fV}\bigg{|}_{\rho=0}.$
(24)

The temperatures are then

$\displaystyle\begin{split}T_{1}&=\frac{\alpha}{8\pi
m_{1}}\frac{m_{1}-m_{2}-z_{1}+z_{2}}{m_{1}+m_{2}-z_{1}+z_{2}}\\\ &\quad\times
e^{2b_{1}(2m_{2}+m_{1}+z_{1})+2b_{2}((m_{1}+z_{1})^{2}+4m_{2}z_{2})},\end{split}$
(25) $\displaystyle\begin{split}T_{2}&=\frac{\alpha}{8\pi
m_{2}}\frac{m_{2}-m_{1}-z_{1}+z_{2}}{m_{1}+m_{2}-z_{1}+z_{2}}\\\ &\quad\times
e^{2b_{1}(m_{2}+z_{2})+2b_{2}(m_{2}+z_{2})^{2}}.\end{split}$ (26)

One can verify that the same results are found via the Euclidean method Hartle
and Hawking (1976). We notice that the presence of the external field lowers
the black holes temperature, with respect to the Schwarzschild one, and hence
the surface gravity. A lower surface gravity means a lower gravitational
“pressure” on the horizon, which then can swell up. This feature is in
agreement with (18) and with the related observations. Moreover, it explains
how the external gravitational field acts, providing an external pressure in
the region of the holes, to sustain the mutual gravitational collapse of the
binary system.

Defining the entropy as $S=\mathcal{A}/4$, the above quantities satisfy the
Smarr law Smarr (1973) both for the individual black holes $M_{i}=2T_{i}S_{i}$
($i=1,2$) and for the whole spacetime

$M_{1}+M_{2}=2T_{1}S_{1}+2T_{2}S_{2}.$ (27)

This result holds regardless of the value of the constant $\alpha$.
Nevertheless, a choice for $\alpha$ must be done in order to study the
thermodynamics of the system.

We consider the system at thermal equilibrium from now on, i.e.
$T_{1}=T_{2}\equiv\bar{T}$. This condition is satisfied by imposing
$m_{1}=m_{2}$. We furthermore choose

$\begin{split}\alpha&=\sqrt{\frac{z_{1}-z_{2}-2m_{1}}{z_{1}-z_{2}}}\\\
&\quad\times e^{-(b_{1}+b_{2}(m_{1}+z_{2}))(m_{1}+z_{2})},\end{split}$ (28)

in order to fulfill the Christodoulou–Ruffini mass formula Christodoulou and
Ruffini (1971), as it happens for regular metrics in which the asymptotic
symmetry is different from the flat one Astorino  _et al._ (2016); Caldarelli
_et al._ (2000); Astorino (2017). It is worth noting that $m_{1}=m_{2}$ is not
the only possibility for the thermal equilibrium, but it is clearly the
simplest one. Moreover, this choice guarantees the integrability of the masses
(20).

Defining the total mass $\bar{M}\equiv M_{1}+M_{2}$ and the total entropy
$\bar{S}\equiv S_{1}+S_{2}$, the first law of thermodynamics

$\delta\bar{M}=\bar{T}\delta\bar{S},$ (29)

is verified. We notice that the variation in (29) is taken with respect to the
free parameters $m_{1}$, $z_{1}$, $z_{2}$, in which $\alpha$ depends: thus the
presence of $\alpha$ is crucial to the first law.

We now turn to the verification of the second law of thermodynamics. At this
scope we consider a process in which the initial state is described by two
black holes at finite distance with total mass $\bar{M}$ and entropy
$\bar{S}$, while the final state is modeled by a single black hole of mass

$M_{0}=m_{0}e^{-(b_{1}+(m_{0}+z_{0})b_{2})(m_{0}+z_{0})},$ (30)

and entropy

$S_{0}=4\pi m_{0}^{2}e^{2(b_{1}+(m_{0}+z_{0})b_{2})(m_{0}+z_{0})}.$ (31)

Here $m_{0}$ and $z_{0}$ are the mass parameter and the position of the single
black hole, respectively. $M_{0}$ and $S_{0}$ are computed from metric (1)
with a single black hole.

Note that, contrary to Tomimatsu (1984a); Breton  _et al._ (1998), we do not
assume $z_{0}=0$ _a priori_. The presence of the external gravitational field
breaks the translation invariance along the $z$-axis, since the field acts
differently on different points of the axis.

Figure 2: The blue surface represents the entropy variation $\Delta S$, while
the yellow surface is the 0 plane. The parameters vary over the values
$m_{1}\in[0,150]$ and $z_{1}\in[0,100]$. Within the whole parametric range the
single black hole state results more entropic than the disjoint pair.

This fact is relevant, because it allows to regularise the single black hole
metric with $b_{1}\neq 0$. In the single black hole case, the regularisation
condition is $b_{1}=-2b_{2}z_{0}$.

In order to have a meaningful comparison we need to compare the two states at
equal energy 444We are assuming that there is no energy loss due to emitted
radiation. and also their background must coincide. Being $b_{2}$
unconstrained in the single black hole configuration, we just have to equate
the values for $b_{1}$, which fixes the position of the single source at

$z_{0}=\frac{z_{1}+z_{2}}{2}.$ (32)

Then we require $M_{0}=\bar{M}$, that fixes $m_{0}$ and $z_{2}$ as

$\displaystyle m_{0}$
$\displaystyle=2m_{1}\sqrt{\frac{2m_{1}-z_{1}+z_{2}}{z_{2}-z_{1}}},$ (33)
$\displaystyle z_{2}$ $\displaystyle=z_{1}+(\sqrt{17}-1)m_{1}.$ (34)

The entropy variation is given by $\Delta S=S_{0}-\bar{S}$, where the value of
the parameters found above must be substituted into the expression. The
result, which is a function of $m_{1}$ and $z_{1}$, is quite involved, but it
can be plotted as in Fig. 2. It is clear from the plot that the function is
always positive or null, i.e.

$\Delta S\geq 0,$ (35)

which verifies the second law of thermodynamics.

##  V Conclusions

In this Letter we built the first analytical solution describing two black
holes at equilibrium in General Relativity. Its feasibility relies on two
novel fundamental features for this kind of model: $(i)$ the spacetime is
completely regular outside the black hole horizons and ($ii$) the metric,
thanks to its multipolar expansion, can be naturally embedded in
phenomenological settings, such as the center of galaxies. Thus the presented
solution can be considered of some realistic astrophysical interest, or even
as a reference for Numerical Relativity. In fact it is possible to transform
perturbatively our binary model to a rotating frame (relative to the center of
mass of the sources) Ni and Zimmermann (1978). Obviously, it would be
necessary a solution generating technique that deals with a metric depending
on three coordinates to exactly model the gravitational coalescence of two
sources, which is an intrinsically dynamical phenomenon; unfortunately such a
method is not available at the moment. However the system described in this
work could be at least interpreted, in the above co-rotating frame and at
first approximation, as a stationary phase of a binary black hole interaction
during a merging process.

Generalisation to an infinite array of sources can be achieved in a
straightforward manner: to this end, the complete multipolar expansion of the
external gravitational field will be necessary to obtain a regular solution.
Also the inclusion of electromagnetic charge, NUT and angular momentum
represents a natural generalisation of the spacetime studied here Astorino and
Viganò (2021).

Aknowledgments. The authors are grateful to Roberto Cotesta and Matteo
Macchini for useful discussions. They also thank Matteo Broccoli and Sergio
Cacciatori for critical readings of the draft of this paper. This work was
supported in part by Conicyt–Beca Chile, in part by MIUR-PRIN contract
2017CC72MK003 and also by INFN.

## References

  * Chrusciel  _et al._ (2012) P.T. Chrusciel, J. Lopes Costa, and M. Heusler, “Stationary Black Holes: Uniqueness and Beyond,” Living Rev. Rel. 15, 7 (2012), arXiv:1205.6112 [gr-qc] .
  * Abbott  _et al._ (2016) B.P. Abbott _et al._ (LIGO Scientific, Virgo), “Observation of Gravitational Waves from a Binary Black Hole Merger,” Phys. Rev. Lett. 116, 061102 (2016), arXiv:1602.03837 [gr-qc] .
  * Kramer and Neugebauer (1980) D. Kramer and G. Neugebauer, “The superposition of two kerr solutions,” Physics Letters A 75, 259 – 261 (1980).
  * Dietz and Hoenselaers (1985) W. Dietz and C. Hoenselaers, “Two mass solutions of einstein’s vacuum equations: The double kerr solution,” Annals of Physics 165, 319 – 383 (1985).
  * Majumdar (1947) S.D. Majumdar, “A class of exact solutions of Einstein’s field equations,” Phys. Rev. 72, 390–398 (1947).
  * Papapetrou (1945) A. Papapetrou, “A static solution of the equations of the gravitational field for an arbitary charge-distribution,” Proceedings of the Royal Irish Academy. Section A: Mathematical and Physical Sciences 51, 191–204 (1945).
  * Hartle and Hawking (1972) J.B. Hartle and S.W. Hawking, “Solutions of the Einstein-Maxwell equations with many black holes,” Commun. Math. Phys. 26, 87–101 (1972).
  * Emparan and Teo (2001) R. Emparan and E. Teo, “Macroscopic and microscopic description of black diholes,” Nucl. Phys. B 610, 190–214 (2001), arXiv:hep-th/0104206 .
  * Ernst (1978) F.J. Ernst, “Generalized c‐metric,” Journal of Mathematical Physics 19, 1986–1987 (1978).
  * Tomimatsu (1984a) A. Tomimatsu, “Distorted rotating black holes,” Physics Letters A 103, 374–376 (1984a).
  * Breton  _et al._ (1998) N. Breton, A.A. Garcia, V.S. Manko, and T.E. Denisova, “Arbitrarily deformed Kerr–Newman black hole in an external gravitational field,” Phys. Rev. D 57, 3382–3388 (1998).
  * Note (1) An interior multi-polar deformation of the sources can be achieved as well Erez and Rosen (1959), but it seems that these poles bring curvature singularities not covered by an event horizon.
  * Beig and Schoen (2009) R. Beig and R.M. Schoen, “On static n-body configurations in relativity,” Classical and Quantum Gravity 26, 075014 (2009).
  * Doroshkevich  _et al._ (1965) A.G. Doroshkevich, Y.B. Zel’dovich, and I.D. Novikov, “Gravitational Collapse of Non-Symmetric and Rotating Bodies,” Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki 49, 170 (1965).
  * Chandrasekhar (1985) S. Chandrasekhar,  _The mathematical theory of black holes_ (Clarendon Press, Oxford, 1985).
  * Geroch and Hartle (1982) R.P. Geroch and J.B. Hartle, “Distorted black holes,” J. Math. Phys. 23, 680 (1982).
  * Astorino and Viganò (2021) M. Astorino and A. Viganò, “Charged and rotating multi-black holes in external gravitational field,” IFUM-1091-FT (in preparation) (2021).
  * Belinsky and Zakharov (1978) V.A. Belinsky and V.E. Zakharov, “Integration of the Einstein Equations by the Inverse Scattering Problem Technique and the Calculation of the Exact Soliton Solutions,” Sov. Phys. JETP 48, 985–994 (1978).
  * Belinsky and Sakharov (1979) V.A. Belinsky and V.E. Sakharov, “Stationary Gravitational Solitons with Axial Symmetry,” Sov. Phys. JETP 50, 1–9 (1979).
  * Belinski and Verdaguer (2005) V. Belinski and E. Verdaguer,  _Gravitational solitons_, Cambridge Monographs on Mathematical Physics (Cambridge University Press, 2005).
  * Harmark (2004) T. Harmark, “Stationary and axisymmetric solutions of higher-dimensional general relativity,” Phys. Rev. D 70, 124002 (2004), arXiv:hep-th/0408141 .
  * Bach and Weyl (1922) R. Bach and H. Weyl, “Neue lösungen der einsteinschen gravitationsgleichungen,” Mathematische Zeitschrift 13, 134–145 (1922).
  * Israel and Khan (1964) W. Israel and K.A. Khan, “Collinear particles and bondi dipoles in general relativity,” Il Nuovo Cimento 33, 331–344 (1964).
  * Costa and Perry (2000) M.S. Costa and M.J. Perry, “Interacting black holes,” Nucl. Phys. B 591, 469–487 (2000), arXiv:hep-th/0008106 .
  * Alekseev and Belinski (2019) G.A. Alekseev and V.A. Belinski, “Superposition of fields of two rotating charged masses in general relativity and existence of equilibrium configurations,” Gen. Rel. Grav. 51, 68 (2019), arXiv:1905.05317 [gr-qc] .
  * Note (2) In fact the presented solution is not the minimal one which can be regularised. It is sufficient to consider only one multipole term for the external gravitational field to remove all the singularities from the double black hole configuration, at the price of fixing the position or the mass of a black hole.
  * Hansen (1974) R.O. Hansen, “Multipole moments of stationary space-times,” J. Math. Phys. 15, 46–52 (1974).
  * Note (3) Actually an arbitrary multipole expansion can be useful to model any generic stationary and axisymmetric external gravitational field.
  * Komar (1959) A. Komar, “Covariant conservation laws in general relativity,” Phys. Rev. 113, 934–936 (1959).
  * Tomimatsu (1984b) A. Tomimatsu, “Equilibrium of Two Rotating Charged Black Holes and the Dirac String,” Prog. Theor. Phys. 72, 73 (1984b).
  * Henneaux and Teitelboim (1985) M. Henneaux and C. Teitelboim, “Asymptotically anti-De Sitter Spaces,” Commun. Math. Phys. 98, 391–424 (1985).
  * Astorino  _et al._ (2016) M. Astorino, G. Compère, R. Oliveri, and N. Vandevoorde, “Mass of Kerr-Newman black holes in an external magnetic field,” Phys. Rev. D 94, 024019 (2016), arXiv:1602.08110 [gr-qc] .
  * Hartle and Hawking (1976) J.B. Hartle and S.W. Hawking, “Path Integral Derivation of Black Hole Radiance,” Phys. Rev. D 13, 2188–2203 (1976).
  * Smarr (1973) L. Smarr, “Mass formula for Kerr black holes,” Phys. Rev. Lett. 30, 71–73 (1973), [Erratum: Phys.Rev.Lett. 30, 521–521 (1973)].
  * Christodoulou and Ruffini (1971) D. Christodoulou and R. Ruffini, “Reversible transformations of a charged black hole,” Phys. Rev. D 4, 3552–3555 (1971).
  * Caldarelli  _et al._ (2000) M.M. Caldarelli, G. Cognola, and D. Klemm, “Thermodynamics of Kerr-Newman-AdS black holes and conformal field theories,” Class. Quant. Grav. 17, 399–420 (2000), arXiv:hep-th/9908022 .
  * Astorino (2017) M. Astorino, “Thermodynamics of Regular Accelerating Black Holes,” Phys. Rev. D 95, 064007 (2017), arXiv:1612.04387 [gr-qc] .
  * Note (4) We are assuming that there is no energy loss due to emitted radiation.
  * Ni and Zimmermann (1978) W.T. Ni and M. Zimmermann, “Inertial and gravitational effects in the proper reference frame of an accelerated, rotating observer,” Phys. Rev. D 17, 1473–1476 (1978).
  * Erez and Rosen (1959) G. Erez and N. Rosen, “The gravitational field of a particle possessing a multipole moment,” Bull. Research Council Israel Sect. F.8 (1959).


# Is two-poles’ $\Lambda(1405)$ one state or two?

Zhong-Yu Wang  School of Physics and Electronics, Hunan Key Laboratory of
Nanophotonics and Devices, Central South University, Changsha 410083, China
Hiwa A. Ahmed  School of Physics and Electronics, Hunan Key Laboratory of
Nanophotonics and Devices, Central South University, Changsha 410083, China
Physics Department, College of Science, University of Sulaimani, Kurdistan
Region 46001, Iraq  C. W. Xiao<EMAIL_ADDRESS>School of Physics and
Electronics, Hunan Key Laboratory of Nanophotonics and Devices, Central South
University, Changsha 410083, China

###### Abstract

To understand the nature of two poles for the $\Lambda(1405)$ state, we
revisit the interactions of $\bar{K}N$ and $\pi\Sigma$ with their coupled
channels, where two-poles structure is found in the second Riemann sheet. We
also dynamically generate two poles in the single channel interaction of
$\bar{K}N$ and $\pi\Sigma$, respectively. Moreover, we make a further study of
two poles’ properties by evaluating the couplings, the compositeness, the wave
functions, and the radii for the interactions of four coupled channels, two
coupled channels and the single channel. Our results show that the nature of
two poles is unique. The higher-mass pole is a pure $\bar{K}N$ molecule, and
the lower-mass one is a compositeness of mainly $\pi\Sigma$ with tiny
component $\bar{K}N$. From our results, one can conclude that the
$\Lambda(1405)$ state would be overlapped with two different states of the
same quantum number.

##  I Introduction

Quantum chromodynamics (QCD) has become the fundamental theory for describing
the strong interaction between quarks and gluons, and perturbative QCD has
also achieved great success in explaining the strong interactions in the high
energy region. However, in the low energy region where many resonances appear,
the non-perturbative QCD should be applied to understand the properties of the
hadronic states under the confinement. There are many non-perturbative QCD
approaches proposed, seen more details in the reviews Chen:2016qju ;
Hosaka:2016pey ; Guo:2017jvc  and references therein. In recent years, due to
the development of the high energy experiments, many exotic states were found,
such as tetraquark-like or pentaquark-like states, which were a chance and
challenge to understand the features of the non-perturbative QCD and caught
much attentions both in theories and experiments, see the reviews Chen:2016qju
; Hosaka:2016pey ; Guo:2017jvc ; Chen:2016spr ; Lebed:2016hpi ;
Esposito:2016noz ; Ali:2017jda ; Briceno:2017max ; Olsen:2017bmm ;
Karliner:2017qhf ; Yuan:2018inv ; Liu:2019zoy ; Brambilla:2019esw .
Understanding the structures and the decay properties of the exotic states is
an important and long term issue in the particle physics. In fact, as
discussed in Ref. Guo:2017jvc , most of the exotic states are located nearby
the thresholds of certain channels and show up clearly molecular nature. A
further study was given in Ref. Dong:2020hxe  to reveal this mysterious
feature using a nonrelativistic effective field theory with open channels, and
more investigations can be found in Ref. Dong:2021juy  where the spectrum of
hadronic molecules in the charm sector were systematically researched with the
single channel Bethe-Salpeter equation.

Even though more and more new resonances in recent experiments need to
understand their nature, there are also some states found in early stage, such
as $f_{0}(980)$, $a_{0}(980)$, $\Lambda(1405)$, and so on, to be still under
debate for their structures and other properties. As we already known, the
$\Lambda(1405)$ resonance was discovered earlier in 1961 in the invariant mass
spectrum of $\pi\Sigma$ below the $K^{-}p$ threshold Alston:1961zzd , which
was theoretically predicted in Refs. Dalitz:1959dn ; Dalitz:1960du  as a
quasi-bound molecular state existed in the $\bar{K}N$ interaction. The
$\Lambda(1405)$ state has been studied extensively in theory after its
discovery by the experiments. At the beginning, considered it as the normal
three-quark baryon, the constituent quark models failed to reproduce its mass
and describe its nature Isgur:1978xj ; Capstick:1986bm . Treated as the five-
quark baryon, a $\Lambda^{*}$ state with the mass about 1400 MeV was obtained
using the MIT bag model Strottman:1979qu , but its assignment to the
$\Lambda(1405)$ resonance was still questionable due to the wave function
having other important components except for the part of $\bar{K}N$ and the
small ratio of the coupling constants for the channels of $\bar{K}N$ and
$\pi\Sigma$. Using Jaffe and Wilczek’s diquark model, it was possible to
interpret the $\Lambda(1405)$ state as $J^{P}=\frac{1}{2}^{-}$ pentaquark
state Zhang:2004xt . In the Skyrme model, the $\Lambda(1405)$ state was
explained as the kaon bound state Callan:1985hy . Constructed an effective
potential from the chiral meson-baryon Lagrangian and using the coupled
channel Lippmann-Schwinger equation, the $\Lambda(1405)$ resonance was
reproduced as quasi-bound state of $\bar{K}N$ and the experimental data was
well described in Refs. Kaiser:1995eg ; Kaiser:1996js . Furthermore, also with
the coupled channel Lippmann-Schwinger equation (later denoted as the coupled
channel Bethe-Salpeter equation under the on-shell description Oller:2000ma ),
but derived the interaction potential only from the lowest order chiral
Lagrangian, Ref. Oset:1997it  also dynamically generated the $\Lambda(1405)$
resonance with one cutoff for the loop functions, and also described the
experimental data well.

Furthermore, using the chiral unitary approach (ChUA) Oset:1997it ;
Oller:2000ma , where more details about the ChUA can be referred to the recent
reviews Oller:2019opk ; MartinezTorres:2020hus ; Oller:2020guq ; Guo:2020hli ,
more further investigations had been done in Refs. Oller:2000fj ; Oset:2001cn
; Jido:2002yz ; GarciaRecio:2002td ; Jido:2003cb ; GarciaRecio:2003ks ;
Hyodo:2007jq ; Hyodo:2008xr  for the interactions of $\bar{K}N$ and its
coupled channels to reveal the nature of the $\Lambda(1405)$ state.
Especially, two-poles structure was found in different Riemann sheets for the
$\Lambda(1405)$ state in Ref Oller:2000fj , which was confirmed in the later
work of Jido:2002yz ; GarciaRecio:2002td  and investigated with more details
about the nature of two poles in Ref. Jido:2003cb . In Ref. GarciaRecio:2003ks
the quark mass dependence for these poles was discussed. The pole structure of
the $\Lambda(1405)$ state was looked inside with the interactions of coupled
channels and single channel in Ref. Hyodo:2007jq . In fact, the two-poles
structure of the $\Lambda(1405)$ resonance was predicted for the first time in
Ref. Fink:1989uk  using the coupled channel Lippmann-Schwinger equation where
the cloudy quark bag model was taken for the potentials. The progress on the
ChUA for the $\Lambda(1405)$ resonance can be found in the reviews
Hyodo:2011ur .

Note that, Refs. Kaiser:1995eg ; Kaiser:1996js  had taken into account the
next leading order of chiral Lagrangian to extract the potential. And the
direct and crossed Born terms were considered to the interaction potentials in
Ref. Oller:2000fj . Moreover, with the ChUA, Ref. Borasoy:2005ie  considering
both the leading order contact (Weinberg-Tomozawa) term, the direct and
crossed Born terms (the $s$\- and $u$-channel contributions Xiao:2016ogq ),
and the next leading order (NLO) contact terms to derived the interaction
potentials, where the contributions to the fitting results were discussed in
details and they found that the ones for the Born terms were small, for more
investigations see Refs. Oller:2005ig ; Oller:2006jw ; Borasoy:2006sr ;
Ikeda:2011pi ; Ikeda:2012au ; Guo:2012vv ; Ramos:2016odk . Following Ref.
Bruns:2010sv , a Bethe-Salpeter framework with full off-shell dependence was
adopted in Ref. Mai:2012dt  for the interactions of $\bar{K}N$ and its coupled
channels, where the two-pole structure was confirmed and the off-shell effect
was impacted slightly the description of experimental data. A further
investigation was given in Ref. Mai:2014xna  with more constraints from the
new experimental data. Based on the results of lattice QCD and the ChUA in the
finite volume, Ref. Meissner:2020khl  argued that the two-pole structure was a
fact of QCD and suggested that the two-pole structure should be put in the
main listings of Particle Data Group (PDG) Zyla:2020zbs . To view the origin
of these poles, a comparative analysis of different theoretical models was
performed in Ref. Cieply:2016jby . The wave function and the compositeness for
the $\Lambda(1405)$ resonance were discussed in details in Refs.
Sekihara:2014kya ; Kamiya:2016jqc  to address the critical issues of its
molecular picture. Using the ChUA and a full analysis of low-energy
experimental data including the differential cross section data for the first
time, Ref. Sadasivan:2018jig  determined the two poles of $\Lambda(1405)$ by a
simultaneous analysis of $s$\- and $p$-waves of the scattering amplitude. With
more constraints from the vector-baryon interactions in the ChUA, the two
poles was well reproduced in Ref. Khemchandani:2018amu , where one new state
of isospin $I=1$ was claimed. In a chiral quark model, it was also supported
that the $\Lambda(1405)$ state was a $\bar{K}N$ bound state in Ref.
Qin:2020gxr  and had the two-pole structure. Whereas, in the Skyrme model, the
$\Lambda(1405)$ state was assumed to be a narrow $\bar{K}N$ Feshbach resonance
in Ref. Ezoe:2020piq . Using a model-independent of the Uniformized Mittag-
Leffler expansion, Ref. Yamada:2021cjo  fitted the experimental data well and
supported the single-pole picture of the $\Lambda(1405)$. Recently, also with
the full off-shell dependence for the effective potential, the two-pole
structure was confirmed once again in Ref. Ren:2021yxc  by applying
subtractive renormalization for the coupled channel scattering amplitudes.
Besides, there were also some experimental results for the two-pole structure
of the $\Lambda(1405)$ state Zychor:2007gf ; Lu:2013nza . More about the
status of the $\Lambda(1405)$ resonance can be found in the review of PDG
Zyla:2020zbs  by Meißner and Hyodo, and in the recent reviews of Mai:2020ltx ;
Hyodo:2020czb .

In the present work, to understand more about the properties of
$\Lambda(1405)$, we revisit the coupled channel interactions of $\bar{K}N$ and
its coupled channels with the ChUA following the method of Ref. Ahmed:2020kmp
, where the pole structures of the $\sigma$ or called $f_{0}(500)$,
$f_{0}(980)$ and $a_{0}(980)$ were investigated by their corresponding
compositeness, wave functions and radii in the coupled and single channel
interactions. Our work is organized as follows. In the next section, we
briefly introduce the formalism of coupled channel interaction within the
ChUA, where Bethe-Salpeter equation and the meson-baryon interaction
potentials are provided. Furthermore, we discuss the evaluations of the
couplings, the compositeness, the wave functions and the radii for two poles
of $\Lambda(1405)$. In the following section, we show our results of the
coupled and single channel interactions. At the end, it is our conclusions.

##  II Formalism

In this section, we first revisit the formalism of the ChUA for the meson-
baryon interactions, where the interaction potentials are derived from the
lowest order chiral Lagrangian, and the scattering amplitudes are evaluated
with the coupled channel Bethe-Salpeter equation. Next, we introduce the
calculations of the couplings, the wave functions, the compositeness and the
radii of the resonances by extending the work of Ahmed:2020kmp  to the
baryonic sector.

###  II.1 S-wave scattering amplitude in coupled channels

The lowest order baryonic Lagrangian that describes the low-energy interaction
of pseudoscalar mesons with baryon fields was discussed in details in Refs.
Meissner:1993ah ; Pich:1995bw ; Ecker:1994gg ; Bernard:1995dp , where the
interaction Lagrangian at the lowest order momentum comes from the
$\Gamma_{\mu}$ term in the covariant derivative, given by

$L_{1}^{(B)}=\left\langle\bar{B}i\gamma^{\mu}\frac{1}{4f^{2}}\left[\left(\Phi\partial_{\mu}\Phi-\partial_{\mu}\Phi\Phi\right)B-B\left(\Phi\partial_{\mu}\Phi-\partial_{\mu}\Phi\Phi\right)\right]\right\rangle.$
(1)

where the symbol $\langle\rangle$ stands for the trace of $SU(3)$ matrices,
and $SU(3)$ matrices for the pseudoscalar mesons and the lowest-lying
$\frac{1}{2}^{+}$ baryon octet are written as

$\Phi=\left(\begin{array}[]{ccc}\frac{1}{\sqrt{2}}\pi^{0}+\frac{1}{\sqrt{6}}\eta&\pi^{+}&K^{+}\\\
\pi^{-}&-\frac{1}{\sqrt{2}}\pi^{0}+\frac{1}{\sqrt{6}}\eta&K^{0}\\\
K^{-}&\bar{K}^{0}&-\frac{2}{\sqrt{6}}\eta\end{array}\right),$ (2)

$B=\left(\begin{array}[]{ccc}\frac{1}{\sqrt{2}}\Sigma^{0}+\frac{1}{\sqrt{6}}\Lambda&\Sigma^{+}&p\\\
\Sigma^{-}&-\frac{1}{\sqrt{2}}\Sigma^{0}+\frac{1}{\sqrt{6}}\Lambda&n\\\
\Xi^{-}&\Xi^{0}&-\frac{2}{\sqrt{6}}\Lambda\end{array}\right).$ (3)

Then, from the lowest order of chiral Lagrangian in Eq. (1), we obtain the
$s$-wave interaction potentials for the coupled channels of $\bar{K}N$. There
are four coupled channels in isospin $I=0$ and strangness $S=-1$ sector,
$\bar{K}N$, $\pi\Sigma$, $\eta\Lambda$ and $K\Xi$, denoted as channel 1, 2, 3
and 4, respectively. Finally, we get the interaction potentials as follow
Jido:2003cb ,

$V_{ij}=-C_{ij}\frac{1}{4f^{2}}\left(2\sqrt{s}-M_{i}-M_{j}\right)\left(\frac{M_{i}+E}{2M_{i}}\right)^{1/2}\left(\frac{M_{j}+E^{\prime}}{2M_{j}}\right)^{1/2},$
(4)

where $E$, $E^{\prime}$ are the energies of the initial and final mesons,
$M_{i}$, $M_{j}$ the masses of the initial and final baryons, and the
coefficient matrix $C_{ij}$ is symmetric, given in Ref. Oset:1997it . For the
meson decay constant, we take $f=1.123f_{\pi}$ Jido:2003cb , where the one
$f_{\pi}$ is the weak pion decay constant, having $f_{\pi}=93$ MeV Oset:1997it
.

The scattering amplitudes of the coupled channels ($T$) can be solved by
Bethe-Salpeter equations with the on-shell description Oset:1997it ,

$T=[1-VG]^{-1}V,$ (5)

where the matrix $V$ is constructed by Eq. (4). The element of diagonal matrix
$G$ the loop function of intermediate mesons and baryons in the $l$-th channel
is given by

$G_{l}=i\int\frac{d^{4}q}{(2\pi)^{4}}\frac{2M_{l}}{(P-q)^{2}-M_{l}^{2}+i\epsilon}\frac{1}{q^{2}-m_{l}^{2}+i\epsilon}\text{
,}$ (6)

where $P=p_{1}+k_{1}$ is the total momentum of the meson-baryon system. Note
that the $G$ function is logarithmically divergent, which need to be
regularized. Using the three-momentum cutoff (CO) method, one can obtain
Oset:1997it ,

$G_{l}(s)=\int_{0}^{q_{max}}\frac{q^{2}dq}{2\pi^{2}}\frac{1}{2\omega_{l}(q)}\frac{M_{l}}{E_{l}(q)}\frac{1}{p^{0}+k^{0}-\omega_{l}(q)-E_{l}(q)+i\epsilon},$
(7)

where $\omega_{l}(q)=\sqrt{q^{2}+m_{l}^{2}}$,
$E_{l}(q)=\sqrt{q^{2}+M_{l}^{2}}$, and $p^{0}+k^{0}=\sqrt{s}$, $q_{max}$ is
the only free parameter. On the other hand, with the dimensional
regularization (DR) method, we have the one from Ref. Oller:2000fj ,

$\displaystyle G_{l}(s)=$
$\displaystyle\frac{2M_{l}}{16\pi^{2}}\left\\{a_{\mu}+\ln\frac{M_{l}^{2}}{\mu^{2}}+\frac{m_{l}^{2}-M_{l}^{2}+s}{2s}\ln\frac{m_{l}^{2}}{M_{l}^{2}}\right.$
(8)
$\displaystyle+\frac{q_{cml}(s)}{\sqrt{s}}\left[\ln\left(s-\left(M_{l}^{2}-m_{l}^{2}\right)+2q_{cml}(s)\sqrt{s}\right)\right.$
$\displaystyle+\ln\left(s+\left(M_{l}^{2}-m_{l}^{2}\right)+2q_{cml}(s)\sqrt{s}\right)$
$\displaystyle-\ln\left(-s-\left(M_{l}^{2}-m_{l}^{2}\right)+2q_{cml}(s)\sqrt{s}\right)$
$\displaystyle\left.\left.-\ln\left(-s+\left(M_{l}^{2}-m_{l}^{2}\right)+2q_{cml}(s)\sqrt{s}\right)\right]\right\\},$

where $\mu$ is the regularization scale, $a_{\mu}$ the subtraction constant,
and $q_{cml}(s)$ the three momentum of the particles in the center-of-mass
(CM) frame, given by

$q_{cml}(s)=\frac{\lambda^{1/2}\left(s,M_{l}^{2},m_{l}^{2}\right)}{2\sqrt{s}},$
(9)

with the usual Källen triangle function
$\lambda(a,b,c)=a^{2}+b^{2}+c^{2}-2(ab+ac+bc)$.

In the ChUA, the mass ($M_{r}$) and decay width ($\Gamma_{r}$) of the
resonance for the coupled channels interactions can be determined from the
pole ($s_{p}$) in the general second Riemann sheets, having
$\sqrt{s_{p}}=M_{r}-i\Gamma_{r}/2$ 111Note that for the case of a bound state
without open (coupled) channel to decay, the pole will be located in the first
Riemann and the real axes, and thus, the pole has no width, written
$\sqrt{s_{p}}=M_{r}$.. Thus, to search for the pole in the second Riemann
sheet, the scattering amplitude should be extrapolated to the complex plane,
which can be fulfilled by the $G_{l}(s)$ function defined to the second
Riemann sheet,

$G_{l}^{(II)}(s)=G_{l}^{(I)}(s)-2i\operatorname{Im}G_{l}^{(I)}(s)=G_{l}^{(I)}(s)+\frac{i}{2\pi}\frac{M_{l}q_{cml}(s)}{\sqrt{s}},$
(10)

where $G_{l}^{(I)}$ corresponds to the propagator function in the first
Riemann sheet, see Eq. (7) or Eq. (8).

###  II.2 Definition of the couplings and wave functions

To probe more structure properties of a resonance, we make a further
investigation on its couplings to the other coupled channels, the wave
functions, and so on. For the case of the meson-baryon interaction, the
couplings can be defined by the residue of the pole,

$g_{i}^{2}=\lim_{\sqrt{s}\rightarrow\sqrt{s_{p}}}\left(\sqrt{s}-\sqrt{s_{p}}\right)T_{ii}.$
(11)

Furthermore, we can generalize Weinberg’s rule Weinberg:1965zz  for the bound
state or resonance to the formalism of the ChUA with the couplings obtained
Aceti:2012dd ,

$-\sum_{i}g_{i}^{2}\left[\frac{dG_{l}}{d\sqrt{s}}\right]_{\sqrt{s}=\sqrt{s_{p}}}=1,$
(12)

which is valid only for the case of a pure molecular state. However, in some
cases, a bound state or resonance contains not only the molecular parts, but
also some other non-molecular components. Thus, for the case of the composite
state, this sum rule can be generalized as,

$-\sum_{i}g_{i}^{2}\left[\frac{dG_{l}}{d\sqrt{s}}\right]_{\sqrt{s}=\sqrt{s_{p}}}=1-Z,$
(13)

where $Z$ represents the probability of the other non-molecular components
contained in the bound state or resonance. Note that for a specified channel,
the $G_{l}$ function should be extrapolated to the right Riemann sheet for a
corresponding pole. More details about the sum rule can be referred to Refs.
Baru:2003qq ; Sekihara:2012xp ; Xiao:2012vv ; Hyodo:2013iga ; Aceti:2014ala ;
Guo:2015daa

In order to make further study on the components and sources of the states, we
construct the wave function of resonances in a small distances. As done in
Ref. YamagataSekihara:2010pj , the wave function of a resonance in coordinate
space was given by

$\phi(\vec{r})=\int_{q_{\max}}\frac{d^{3}\vec{p}}{(2\pi)^{3/2}}e^{i\vec{p}\cdot\vec{r}}\langle\vec{p}\mid\Psi\rangle.$
(14)

Performing the angle integration of the momentum, one can have
Ozpineci:2013qza

$\displaystyle\phi(\vec{r})=\frac{1}{(2\pi)^{3/2}}\frac{4\pi}{r}\frac{1}{C}\int_{q_{\max}}pdp\sin(pr)\times\frac{\Theta\left(q_{\max}-|\vec{p}|\right)}{E-\omega_{1}(\vec{p})-\omega_{2}(\vec{p})}.$
(15)

where $C$ is the normalization constant, and $E\equiv\sqrt{s_{p}}$. From the
wave function obtained, we can calculate the form factor,

$\displaystyle F(\vec{q})$ $\displaystyle=\int
d^{3}\vec{r}\phi(\vec{r})\phi^{*}(\vec{r})e^{-i\vec{q}^{\prime}\cdot\vec{r}}$
(16) $\displaystyle=\int
d^{3}\vec{p}\times\frac{\theta(\Lambda-p)\theta(\Lambda-|\vec{p}-\vec{q}|)}{\left[E-\omega_{1}(p)-\omega_{2}(p)\right]\left[E-\omega_{1}(\vec{p}-\vec{q})-\omega_{2}(\vec{p}-\vec{q})\right]},$

with a normalization to keep $F(q=0)\equiv 1$ and $\theta(\Lambda-p)$ is the
step function. Finally, we can get the radius of the resonance with the form
factor at origin,

$\left\langle
r^{2}\right\rangle=-6\left[\frac{\mathrm{d}F(q)}{\mathrm{d}q^{2}}\right]_{q^{2}=0},$
(17)

which is consistent with the one obtained from the tail of the wave functions
Sekihara:2012xp  for the state not close to the threshold, see more
discussions in Ref. Ahmed:2020kmp .

##  III Results

We first revisit the $\bar{K}N$ interactions with coupled channels of
$\bar{K}N$, $\pi\Sigma$, $\eta\Lambda$ and $K\Xi$ in the isospin basis as done
in Ref. Oset:1997it . Since the two relevant poles found in the second Riemann
sheet (shown below) couple strongly to the $\bar{K}N$ and $\pi\Sigma$
channels, we pay more attention to these two channels. Then we research the
interactions with only two coupled channels of $\bar{K}N$ and $\pi\Sigma$, and
the interactions of these two singled channels in the isospin $I=0$ and
strangeness $S=-1$ sector. The $\Lambda(1405)$ resonance is dynamically
generated in the coupled channels interactions, where two-pole structure is
found. In order to make further understanding of the components and properties
for this resonance, and reveal the nature of two poles, we investigate the
couplings, the compositeness, the wave functions and the radii of two poles
for this state in the coupled channel and single channel interactions.

###  III.1 Coupled channels approach

Figure 1: $\pi\Sigma$ invariant mass spectra of
$\pi\Sigma\rightarrow\pi\Sigma$ with a normalization coefficient $C=113$ for
four coupled channels and $C=110$ for two coupled channels. Data were taken
from Ref. Hemingway:1984pz .

First, we show our results for the coupled channel interactions with four
coupled channels $\bar{K}N$, $\pi\Sigma$, $\eta\Lambda$, $K\Xi$, and two
coupled channels $\bar{K}N$, $\pi\Sigma$. As discussed in the last section,
the free parameters of the ChUA are $q_{max}$ for the cutoff method, or the
regularization scale $\mu$ and subtraction constant $a_{\mu}$ for the
dimensional regularization method, which are used to regularize the propagator
functions. The values of these parameters for the case of four coupled
channels are $\mu=q_{max}=630$ MeV Oset:1997it  and $a_{\bar{K}N}=-1.84$,
$a_{\pi\Sigma}=-2.00$, $a_{\eta\Lambda}=-2.25$, $a_{K\Xi}=-2.67$, which were
taken from Ref. Jido:2003cb . For the case of two coupled channels, we
obtained $\mu=q_{max}=623$ MeV and $a_{\bar{K}N}=-1.86$, $a_{\pi\Sigma}=-2.09$
by fitting the $\pi\Sigma$ invariant mass distributions, see our results of
Fig. 1, where the results of both cases described the experimental data well.
In Fig. 1, we show the results of the $\pi\Sigma$ invariant mass
distributions, which are given by

$\frac{d\sigma}{dM_{inv}}=C\left|T_{\pi\Sigma\rightarrow\pi\Sigma}\right|^{2}k_{\pi},$
(18)

where $C$ is a normalized constant, $T_{\pi\Sigma\rightarrow\pi\Sigma}$ the
scattering amplitude of $\pi\Sigma$ channel obtained from Eq. (5) for $I=0$
sector, and $k_{\pi}$ the $\pi$ momentum in the CM frame of $\pi\Sigma$
system. For the case of four coupled channel, the results of Fig. 1 are
consistent with the ones in Ref. Oset:1997it . The results for the case of two
coupled channel in Fig. 1 look like a bit better. There are also some other
experimental data as shown in Fig. 2. The results of Fig. 2 are the low energy
total cross sections of $K^{-}p$ scattering to various final states 222Since
we focus on the origin of two poles and their different properties in the
present work, we did not include these data for the fit mentioned above in the
two coupled channels case. Due to large uncertainties from the data, for the
simplicity we do not show the results of the combined fit with all these data,
of which the total chi square is not much different., which are given by

$\sigma_{ij}=\frac{1}{4\pi}\frac{MM^{\prime}}{s}\frac{k^{\prime}}{k}\left|T_{ij}\right|^{2},$
(19)

where $M$($M^{\prime}$) is the initial(final) baryon mass and
$k$($k^{\prime}$) the initial(final) three momentum in the CM frame. It is
worth to mention that, since some of the physical decay channels has mixing
with isospin $I=0$ and $I=1$ components, we consider the mixing in our
calculations, where the details of the amplitudes for $I=1$ are not shown and
can be referred to Ref. Oset:1997it  for more details. As one can see in Fig.
2, the results obtained for four and two coupled channels are consistent with
the experimental data within the uncertainties.

Figure 2: Cross sections for $K^{-}p$ $\to$ $K^{-}p$, $\bar{K^{0}}n$,
$\pi^{+}\Sigma^{-}$, $\pi^{-}\Sigma^{+}$ , and $\pi^{0}\Sigma^{0}$ in the lab
frame. Solid (Red) line: results of four coupled channels. Dashed (blue) line:
results of two coupled channels. See Refs. Humphrey:1962zz ; Sakitt:1965kh ;
Kim:1967zze ; Kittel:1966zz ; Ciborowski:1982et ; Evans:1983hz ; Kim1966 ;
Martin:1980qe  for the details of the experimental data.

In the presence of bound state or resonance, a peak will appear in modulus
square of the scattering amplitude. Thus, to look for the resonance of the
interaction, we plot the modulus squares of the scattering amplitudes of
$\bar{K}N$ and $\pi\Sigma$ for the case of the coupled channel interactions,
which are shown in Fig. 3. As one can see from Fig. 3, there are clear peak
structures above the $\pi\Sigma$ threshold in all the scattering amplitudes,
and the line shape of the curves for two and four coupled channels are
consistent with each other, where only the maximum values and the peak
positions are a little different in the $\bar{K}N\rightarrow\bar{K}N$ and
$\bar{K}N\rightarrow\pi\Sigma$ channels. These results also show that the
other two coupled channels $\eta\Lambda$ and $K\Xi$ have small contribution to
these peaks. But, the properties of these peaks are different. The one
appearing in the $\bar{K}N\rightarrow\bar{K}N$ interaction has a mass close to
the $\bar{K}N$ threshold with a narrower width, which is similar to the one of
the $\bar{K}N\rightarrow\pi\Sigma$ transition. However, the one from the
$\pi\Sigma\rightarrow\pi\Sigma$ interaction has a lower mass with a wider
width. These two poles’s coherence behaviour are similar the ones as shown in
Ref. Sombillo:2021rxv . Indeed, this is the two-pole structures, as found in
Refs. Oller:2000fj ; Jido:2002yz ; GarciaRecio:2002td ; Jido:2003cb , as shown
clearly in Fig. 4, where we plot the modulus of the scattering amplitude
$\left|T_{\bar{K}N}\right|$ in the second Riemann sheet for the two cases of
coupled channel interactions.

Figure 3: Modulus square of $\bar{K}N\rightarrow\bar{K}N$,
$\bar{K}N\rightarrow\pi\Sigma$ and $\pi\Sigma\rightarrow\pi\Sigma$ amplitudes.

Figure 4: Modulus of the scattering amplitude $\left|T_{\bar{K}N}\right|$ in
the second Riemann sheet of the complex energy plane $(\sqrt{s}=x+iy)$ for the
interactions of four (left) and two (right) coupled channels.

In order to get the masses and widths of these peaks, we look for the pole in
the unphysical Riemann sheet, which correspond to the singularity in the
scattering amplitude in complex energy plane. As discussed in the last
section, in the ChUA the Riemann sheets can be defined through the loop
function $G_{l}(s)$. By doing that, we find the corresponding poles of these
peaks in Fig. 3 in the second Riemann sheet, as summarized in Table 1, where
two regularization methods were used for the loop function $G_{l}(s)$, the CO
method and DR method. In the case of four coupled channels, using DR method
two poles are found in the second Riemann sheet, which correspond to the
$\Lambda(1405)$ state 333In the latest PDG Zyla:2020zbs , the lower-mass pole
was assigned as a $\Lambda(1380)$ resonance and the higher one considered for
the $\Lambda(1405)$ state.. Both poles are below the $\bar{K}N$ threshold and
above the $\pi\Sigma$ threshold, and thus the $\pi\Sigma$ channel is the only
open channel in $I=0$. Another pole is found, which corresponds to the
$\Lambda(1670)$ state. These results are consistent with the ones obtained in
Ref. Jido:2003cb . However, using CO method two similar poles are found,
whereas the third pole is far above the region where the $\Lambda(1670)$ state
appeared. The inconsistency for the third pole indicates that using one CO can
not reproduced well for both $\Lambda(1405)$ and $\Lambda(1670)$ state, and
different CO should be used for the $K\Xi$ interaction to dynamically generate
the $\Lambda(1670)$ state. Since we mainly concern about the $\Lambda(1405)$
state, we will use the DR method for the following calculations. Thus, to
investigate more details of these two poles for the $\Lambda(1405)$ state, we
omit the other two channels $\eta\Lambda$ and $K\Xi$, and study the coupled
channel interactions of two channels $\bar{K}N$ and $\pi\Sigma$. Two poles
$\sqrt{s_{p}}=(1384.54+60.98i)$ MeV and $\sqrt{s_{p}}=(1438.95+12.44i)$ MeV
are present in the second Riemann sheet, which are consistent with the ones
obtained by the CO method. Compared with the results of four coupled channels,
the mass and width of the lower-mass pole is slightly smaller. On the
contrary, for the higher-mass pole the mass is become larger and the width is
smaller. From these results, one can see that the position of the poles do not
change remarkably by reducing the coupled channels from four channels to two
channels. Hence, these two poles are coming from the strong interactions of
the $\bar{K}N$ and $\pi\Sigma$ channels, which can also be confirmed by the
results of the couplings and the single channel interaction below. Note that
the mass of higher-mass pole in two coupled channels case is slightly above
the threshold of $\bar{K}N$ channel.

Table 1: Poles positions 555Note that the poles are always a pair of
conjugated solutions in the complex Riemann sheet. But, the pole corresponding
to a resonance should be the one having negative imaginary part,
$\sqrt{s_{p}}=M_{r}-i\Gamma_{r}/2$ as defined above. With the definition of
Eq. (10), one can find the positive imaginary one without strictly specified
the signs of the three momenta for the definition of the Riemann sheets, which
is sometimes assigned as the pole of the resonance. compared with other works
and experimental results (Unit: MeV).

|  $1^{st}$ pole |  $2^{st}$ pole |  $3^{st}$ pole
---|---|---|---
Four coupled channels (DR) | $1390.61+65.91i$ | $1426.77+16.32i$ | $1680.38+20.18i$
Four coupled channels (CO) | $1377.35+58.46i$ | $1427.96+18.10i$ | $1762.8+27.51i$
Two coupled channels (DR) | $1384.54+60.98i$ | $1438.95+12.44i$ |
Two coupled channels (CO) | $1375.56+62.63i$ | $1438.04+11.44i$ |
Ref. Ikeda:2012au , NLO | $1381_{-6}^{+18}-81_{-8}^{+19}i$ | $1424_{-23}^{+7}-26_{-14}^{+3}i$ |
Ref. Guo:2012vv  solution II | $1388_{-9}^{+9}-114_{-25}^{+24}i$ | $1421_{-2}^{+3}-19_{-5}^{+8}i$ |
Ref. Mai:2014xna  solution II | $1330_{-5}^{+4}-56_{-11}^{+17}i$ | $1434_{-2}^{+2}-10_{-1}^{+2}i$ |
Ref. Mai:2014xna  solution IV | $1325_{-15}^{+15}-90_{-18}^{+12}i$ | $1429_{-7}^{+8}-12_{-3}^{+2}i$ |
Ref. Anisovich:2020lec  one-pole solution |  | $(1421\pm 3)-(23\pm 3)i$ |
Ref. Anisovich:2020lec  two-pole solution |  $1380-90i$ (fixed) | $(1423\pm 3)-(20\pm 3)i$ |

From the former results, we can estimate that each pole mainly contributed
from which channels. To show how the poles couple to the channels
quantitatively, we calculated the couplings. The coupling strength of the
resonance to each coupled channel is extracted from the residues of the
scattering amplitude around the pole position as discussed in last section.
The corresponding couplings for four coupled channels and two coupled channels
are present in Table 2. For the case of four coupled channel interactions, the
couplings of the two channels $\bar{K}N$ and $\pi\Sigma$ for the lower-mass
pole $\sqrt{s_{p}}=1390.61+65.91i$ are about three times larger than those of
the other two coupled channels, $\eta\Lambda$ and $K\Xi$. Among them, the one
of the $\pi\Sigma$ channel is bigger than that of the $\bar{K}N$ channel,
which means that the lower-mass pole is dominant by the $\pi\Sigma$ channel.
The coupling of the $\bar{K}N$ channel for the higher-mass pole
$\sqrt{s_{p}}=1426.77+16.32i$ is about 2.71, which is two times bigger than
the ones for the channels $\pi\Sigma$ and $\eta\Lambda$, whereas the one of
the last $K\Xi$ channel is quit small. Thus, the higher-mass pole is mainly
contributed by the $\bar{K}N$ channel. Besides, the third pole, which
corresponds to the $\Lambda(1670)$ state, is strongly coupled to the $K\Xi$
channel. These results are compatible with the results of Refs. Jido:2003cb ;
Sekihara:2010uz  and crossed check by two coupled channels results.

Using the sum rule of Eq. (13), the compositeness of the two poles was
computed with the couplings obtained above, where one can check whether these
poles are a pure molecular state or have something else. Our results are shown
in Table 3. As one can see in Table 3, the first pole is highly dominated by
the $\pi\Sigma$ components with the amount of 0.63, whereas the one of
$\bar{K}N$ is about three times smaller. The components of second pole are
mainly constructed by the part of $\bar{K}N$, which is up to 98%, and have a
tiny part of $\pi\Sigma$. Note that the contributions from the other two
channels, $\eta\Lambda$ and $K\Xi$, are quite small for these two poles, owing
to their thresholds far away above two poles. Indeed, this conclusion can be
confirmed by the results of the two coupled channels. Besides, in the results
of four coupled channels, there is the third pole, corresponding to the
$\Lambda(1670)$ state, which is significantly contributed by the $K\Xi$
channel. Note that, as discussed in Ref. Matuschek:2020gqe , the compositeness
condition of Eq. (13) is valid for the case of the weak-binding limit and the
pole with narrow width. Thus, the compositeness of the third pole is
questionable, since it is the bound state of the $K\Xi$ with large binding
energy of about 134 MeV. Indeed, this is similar to the case of the
$f_{0}(500)$ (or called $\sigma$) resonance as found in Ref. Ahmed:2020kmp ,
which has a large width of about 400 MeV. This is why we investigate the
couplings and the compositeness together. Thus, our final conclusions are
based on both results.

Table 2: Couplings for each coupled channel compared with the other works.

$\sqrt{s_{p}}$ | $g_{\bar{K}N}$ | $\left|g_{\bar{K}N}\right|$ | $g_{\pi\Sigma}$ | $\left|g_{\pi\Sigma}\right|$ | $g_{\eta\Lambda}$ | $\left|g_{\eta\Lambda}\right|$ | $g_{K\Xi}$ | $\left|g_{K\Xi}\right|$
---|---|---|---|---|---|---|---|---
Four coupled channels |  |  |  |  |  |  |  |
$1390.61+65.91i$ | $1.20+1.74i$ | $2.12$ | $2.46+1.51i$ | $2.88$ | $0.01+0.76i$ | $0.76$ | $0.45+0.41i$ | $0.61$
$1426.77+16.32i$ | $2.55-0.94i$ | $2.71$ | $0.42-1.45i$ | $1.51$ | $1.40-0.20i$ | $1.41$ | $0.11-0.33i$ | $0.35$
$1680.38+20.18i$ | $0.30+0.72i$ | $0.78$ | $0.01+0.27i$ | $0.27$ | $1.05+0.11i$ | $1.06$ | $3.45+0.12i$ | $3.46$
Two coupled channels |  |  |  |  |  |  |  |
$1384.54+60.98i$ | $1.46+1.45i$ | $2.06$ | $2.38+1.29i$ | $2.71$ | $-$ | $-$ | $-$ | $-$
$1438.95+12.44i$ | $2.00-1.34i$ | $2.41$ | $0.07+1.42i$ | $1.42$ | $-$ | $-$ | $-$ | $-$
Ref. Jido:2003cb  |  |  |  |  |  |  |  |
$1390+66i$ | $1.2+1.7i$ | $2.1$ | $-2.5-1.5i$ | $2.9$ | $0.010+0.77i$ | $0.77$ | $-0.45-0.41i$ | $0.61$
$1426+16i$ | $-2.5+0.94i$ | $2.7$ | $0.42-1.4i$ | $1.5$ | $-1.4+0.21i$ | $1.4$ | $0.11-0.33i$ | $0.35$
$1680+20i$ | $0.30+0.71i$ | $0.77$ | $-0.003-0.27i$ | $0.27$ | $-1.1-0.12i$ | $1.1$ | $3.4+0.14i$ | $3.5$
Ref. Sekihara:2010uz  |  |  |  |  |  |  |  |
$1391-66i$ | $-0.86+1.26i$ | $-$ | $-1.42+0.88i$ | $-$ | $-0.01+0.79i$ | $-$ | $-0.33+0.30i$ | $-$
$1426-17i$ | $1.84+0.67i$ | $-$ | $0.26+0.85i$ | $-$ | $1.44+0.21i$ | $-$ | $0.09+0.24i$ | $-$

Table 3: Compositeness of the poles in coupled channels approach.

$\sqrt{s_{p}}$ | $(1-Z)_{\bar{K}N}$ | $\left|(1-Z)_{\bar{K}N}\right|$ | $(1-Z)_{\pi\Sigma}$ | $\left|(1-Z)_{\pi\Sigma}\right|$ | $(1-Z)_{\eta\Lambda}$ | $\left|(1-Z)_{\eta\Lambda}\right|$ | $(1-Z)_{K\Xi}$ | $\left|(1-Z)_{K\Xi}\right|$
---|---|---|---|---|---|---|---|---
Four coupled channels |  |  |  |  |  |  |  |
$1390.61+65.91i$ | $-0.20+0.12i$ | $0.24$ | $0.36-0.52i$ | $0.63$ | $-0.01-0.002i$ | $0.01$ | $-0.0003+0.01i$ | $0.01$
$1426.77+16.32i$ | $0.98-0.04i$ | $0.98$ | $-0.04+0.15i$ | $0.15$ | $0.05-0.01i$ | $0.05$ | $-0.002-0.001i$ | $0.002$
$1680.38+20.18i$ | $0.02+0.002i$ | $0.02$ | $0.002+0.002i$ | $0.003$ | $-0.07-0.16i$ | $0.17$ | $0.52+0.09i$ | $0.52$
Two coupled channels |  |  |  |  |  |  |  |
$1384.54+60.98i$ | $-0.12+0.18i$ | $0.22$ | $0.27-0.51i$ | $0.58$ | $-$ | $-$ | $-$ | $-$
$1438.95+12.44i$ | $0.96-0.09i$ | $0.97$ | $0.05+0.12i$ | $0.13$ | $-$ | $-$ | $-$ | $-$

The CLAS collaboration reported the observation of two pole structure of
$\Lambda(1405)$ from the electroproduction process, and found that this state
was not a simple Breit-Wigner resonance Lu:2013nza . One can see from the
results of Fig. 3 that the scattering amplitudes of
$T_{\pi\Sigma\to\pi\Sigma}$ and $T_{\bar{K}N\to\bar{K}N}$
($T_{\bar{K}N\to\pi\Sigma}$) are dominated by the lower and higher-mass poles,
respectively. As discussed in Ref. Hyodo:2011ur , as a consequence of the
interference of these two poles in the amplitudes, one could observe only a
single resonance peak experimentally in one channel. Therefore, if the
$\Lambda(1405)$ is not a single resonance, it should be a superposition of two
independent states with the same quantum numbers. To show this interference
effect on the final observation, as done in Ref. Jido:2003cb , we take

$T_{\bar{K}N\rightarrow\pi\Sigma}=g_{\bar{K}N}^{p_{1}}\frac{1}{\sqrt{s}-M_{p_{1}}+i\Gamma_{p_{1}}/2}g_{\pi\Sigma}^{p_{1}}+g_{\bar{K}N}^{p_{2}}\frac{1}{\sqrt{s}-M_{p_{2}}+i\Gamma_{p_{2}}/2}g_{\pi\Sigma}^{p_{2}},$
(20)
$T_{\pi\Sigma\rightarrow\pi\Sigma}=g_{\pi\Sigma}^{p_{1}}\frac{1}{\sqrt{s}-M_{p_{1}}+i\Gamma_{p_{1}}/2}g_{\pi\Sigma}^{p_{1}}+g_{\pi\Sigma}^{p_{2}}\frac{1}{\sqrt{s}-M_{p_{2}}+i\Gamma_{p_{2}}/2}g_{\pi\Sigma}^{p_{2}},$
(21)

where the upper indexes $p_{1}$ and $p_{2}$ denote the first pole and second
pole found in the second Riemann sheet, respectively, and $g_{i}^{p}$ are the
couplings of $i$-channel for a certain pole ($p$), which were given in Table
2. The results are shown in Fig. 5, where one can see that the results of Eq.
(21) for four coupled channels (the top-right one) are well consistent with
the one of $\pi\Sigma$ invariant mass distributions in Fig. 1. For two coupled
channels, the higher mass pole looks like to have stronger strengths. Indeed,
the results of Eq. (20) are mainly contributed by the higher-mass pole, which
are consistent with the ones of Fig. 3 for both four channels and two channels
cases. Furthermore, to show the inference effect more clearly, we take for the
conjugated pole,

$T_{\bar{K}N\rightarrow\pi\Sigma}=g_{\bar{K}N}^{p_{1}}\frac{1}{\sqrt{s}-M_{p_{1}}-i\Gamma_{p_{1}}/2}g_{\pi\Sigma}^{p_{1}}+g_{\bar{K}N}^{p_{2}}\frac{1}{\sqrt{s}-M_{p_{2}}-i\Gamma_{p_{2}}/2}g_{\pi\Sigma}^{p_{2}},$
(22)
$T_{\pi\Sigma\rightarrow\pi\Sigma}=g_{\pi\Sigma}^{p_{1}}\frac{1}{\sqrt{s}-M_{p_{1}}-i\Gamma_{p_{1}}/2}g_{\pi\Sigma}^{p_{1}}+g_{\pi\Sigma}^{p_{2}}\frac{1}{\sqrt{s}-M_{p_{2}}-i\Gamma_{p_{2}}/2}g_{\pi\Sigma}^{p_{2}},$
(23)

where the results are shown in Fig. 6. From Figs. 5 and 6, one can see that
the peak position of the $\Lambda(1405)$ is mainly contributed by the two
poles, due to their interference consequence, which is originated from the
strong coupled effect between the channels of $\bar{K}N$ and $\pi\Sigma$.
Thus, the mass and width of the peak $\Lambda(1405)$ are between two poles’
masses and widths, respectively. For more discussions about the interference
effects of the Breit-Wigner parameterization can be referred to Ref.
Bai:2019jrb .

Figure 5: The $\pi\Sigma$ spectra using Breit-Wigner parameterization with
Eqs. (20) and (21) for four coupled channels (upper) and two coupled channels
(lower).

Figure 6: The same as Fig. 5 but with Eqs. (22) and (23) for four coupled
channels (upper) and two coupled channels (lower).

The wave function also contains the information of the structure of a state,
since the wave function in coordinate space indicates the spatial composition
of the components. The wave functions of the two poles for the cases of four
and two coupled channels are shown in Figs. 7 and 8 for the components of
$\bar{K}N$ and $\pi\Sigma$, respectively, which are consistent with the ones
obtained in Ref. YamagataSekihara:2010pj  qualitatively (with different
normalization). From Figs. 7 and 8, one can see that the real part of wave
functions are almost become zero after a certain size of about 4 fm, and the
imaginary parts are similar, which means that the wave functions would be
confined within a few fm. Note that the results of Figs. 7 and 8 are just the
scattering wave functions. In principle, for the case of the open channel, the
$\pi\Sigma$ channel in our case, the wave functions in Fig. 8 would have the
parts of the outgoing plane wave function, which were out of our concerns and
ignored. More details can be found in Refs. YamagataSekihara:2010pj ;
Aceti:2012dd .

Figure 7: Wave functions of two poles for the $\bar{K}N$ channel.

Figure 8: Wave functions of two poles for the $\pi\Sigma$ channel.

To quantify the spatial size of the $\Lambda(1405)$, we calculated the radii
of states using Eq. (17) with the form factors obtained. Since the form
factors are complex with the poles of the resonance having widths, the radii
for two poles become complex, as shown in Table 4. The modulus of radius of
the higher-mass pole is $1.57$ fm and $1.58$ fm for four and two coupled
channels, respectively. However, the one for the lower-mass pole is $0.67$ fm
and $0.66$ fm, respectively, which are within the typical hadronic scale $\leq
0.8$ fm Sekihara:2012xp . These results for the higher-mass pole are
consistent with the ones obtained in Ref. Sekihara:2012xp , about 1.69 fm with
binding energy 10 MeV. Consequently, the radius of the higher-mass pole is
about two times larger than the typical size of the hadrons, which has quite
different size nature compared with the one for the low-mass pole.

Table 4: Radii of the poles evaluated with Eq. (17) in coupled channels approach. $\sqrt{s_{p}}$ | $\sqrt{\left\langle r^{2}\right\rangle_{\bar{K}N}}$ | $\mid\sqrt{\left\langle r^{2}\right\rangle_{\bar{K}N}}\mid$ | $\sqrt{\left\langle r^{2}\right\rangle_{\pi\Sigma}}$ | $\mid\sqrt{\left\langle r^{2}\right\rangle_{\pi\Sigma}}\mid$
---|---|---|---|---
Four coupled channels |  |  |  |
$1390.61+65.91i$ | $-$ | $-$ |  $(0.67+0.06i)$ fm |  $0.67$ fm
$1426.77+16.32i$ |  $(1.46+0.57i)$ fm |  $1.57$ fm | $-$ | $-$
Two coupled channels |  |  |  |
$1384.54+60.98i$ | $-$ | $-$ |  $(0.65+0.10i)$ fm |  $0.66$ fm
$1438.95+12.44i$ |  $(1.16+1.07i)$ fm |  $1.58$ fm | $-$ | $-$

Moreover, we can study the scattering length of the elastic channels at the
threshold Oset:1997it , written as

$a_{ii}=\frac{-1}{4\pi}\frac{M}{\sqrt{s}}T_{ii},$ (24)

which is an important parameter in analyzing low energy interactions. The
scattering lengths of $K^{-}p$ channel are summarized in Table 5. The result
of four coupled channels is well consistent with the experimental values and
the one of next to leading order calculations. Its value is slightly changed
for two coupled channels case.

Table 5: Scattering length of $K^{-}p$ channel. Approaches |  $a_{K^{-}p}$ (fm)
---|---
Four coupled channels | $-0.73+1.31i$
Two coupled channels | $-0.15+1.48i$
Ref. Ikeda:2012au , NLO | $-0.70+0.89i$
Ref. Guo:2012vv  (Fit I) | $-0.67^{+0.13}_{-0.12}+0.92^{+0.07}_{-0.08}i$
Ref. Guo:2012vv  (Fit II) | $-0.61^{+0.13}_{-0.15}+0.89^{+0.07}_{-0.06}i$
Exp. Iwasaki:1997wf  | $(-0.78\pm 0.15\pm 0.03)+(0.49\pm 0.25\pm 0.12)i$
Exp. Martin:1980qe  | $-0.67+0.64i$
Exp Ohnishi:2015iaq  | $(-0.65\pm 0.10)+(0.81\pm 0.15)i$

###  III.2 Single channel approach

For the sake of a comprehensive understanding of the pole structure of the
$\Lambda(1405)$, we studied the single channel interaction, where the
interactions for the channels $\bar{K}N$ and $\pi\Sigma$ were done
individually without the coupled channel effect. The results of modulus
squares for the scattering amplitudes are shown in Fig. 9. From Fig. 9, we can
see a sharp peak with zero width in the $\bar{K}N$ channel and a wide bump
structure in the $\pi\Sigma$ channel. Since we have found two poles which
coupled to the $\bar{K}N$ and $\pi\Sigma$ channels in the coupled channel
interactions, these structures were the corresponding ones to these channels
in the singled channel interaction. In this case, there are only two Riemann
sheets for each channel. As shown in Table 6, we find one pole of
$\sqrt{s_{p}}=1429.71$ located at the first Riemann sheet and below the
threshold in the case of $\bar{K}N$ interaction, which is a bound state of
$\bar{K}N$ with zero width for no channel to decay in the single channel
interaction. This indicates that the interaction of $\bar{K}N$ is strong
enough to bound a state. Whereas, the one of $\sqrt{s_{p}}=1380.75+80.95i$ is
appeared in the second Riemann sheet and above the threshold of $\pi\Sigma$
channel, which is a broad resonance for the weak potential of the $\pi\Sigma$
channel and analogous to the case of $f_{0}(500)$ in the $\pi\pi$ interaction,
see the results of Ref. Ahmed:2020kmp . Hence, the pole of $\pi\Sigma$ has
large decay width as in the case of the coupled channels, because it is above
the $\pi\Sigma$ threshold and thus it still can decay to $\pi\Sigma$ channel.
From these results, one can see clearly that two poles found in the coupled
channel interactions above can be generated directly in the single channel
interaction even without coupled channel effects, which is different from the
case of $a_{0}(980)$ in the case of $\bar{K}K$ interactions in isospin $I=1$
sector as found in Ref. Ahmed:2020kmp .

Figure 9: Modulus squares of $T_{\bar{K}N\rightarrow\bar{K}N}$ (left) and
$T_{\bar{K}N\rightarrow\pi\Sigma}$ (right) amplitudes in single channel
interaction.

Even though the coupling lose the physical meaning since there is no coupled
channel interaction, we calculated the corresponding couplings, that are given
in Table 6 for the purpose of evaluating the compositeness. The results of the
compositeness are given in Table 7, which are consistent with the ones
obtained in the case of coupled channel interactions above, see Table 3. Note
that the value of compositeness for the higher-mass pole is real for the real
pole of a bound state.

Table 6: Pole positions and couplings in single channel interaction. $\sqrt{s_{p}}$ | $g_{\bar{K}N}$ | $\left|g_{\bar{K}N}\right|$ | $g_{\pi\Sigma}$ | $\left|g_{\pi\Sigma}\right|$
---|---|---|---|---
$1380.75+80.95i$ | $-$ | $-$ | $1.97+1.45i$ | $2.44$
$1429.71$ | $1.81$ | $1.81$ | $-$ | $-$
Table 7: Compositeness of the poles in single channel interaction. $\sqrt{s_{p}}$ | $(1-Z)_{\bar{K}N}$ | $\left|(1-Z)_{\bar{K}N}\right|$ | $(1-Z)_{\pi\Sigma}$ | $\left|(1-Z)_{\pi\Sigma}\right|$
---|---|---|---|---
$1380.75+80.95i$ | $-$ | $-$ | $0.30-0.35i$ | $0.45$
$1429.71$ | $0.89$ | $0.89$ | $-$ | $-$

To check whether the spatial distributions of the poles found in singled
channel affected by the coupled channel effect or not, we also evaluated the
corresponding wave functions, which are shown in Fig. 10. Now, due to the
higher-mass pole is a bound state of no width, it has a real wave function.
But, the behaviour of the function is still the same as the one in the coupled
channel case. With the form factors obtained from the wave functions, the
radii evaluated with Eq. (17) for two poles are given in Table 8. Analogously,
the radius for the higher-mass pole of $\bar{K}N$ is real now, where its value
is $2.79$ fm and much greater than the ones $1.57$ fm or $1.58$ fm obtained in
the case of coupled channel interactions, see Table 4. But, the one for the
lower-mass pole of $\pi\Sigma$ is $0.78$ fm, which is just slightly greater
than the ones $0.67$ fm or $0.66$ fm in Table 4 and still kept in the the
typical hadronic scale.

Figure 10: Wave functions of the poles of $\bar{K}N$ (upper) and $\pi\Sigma$ (lower) in single channel approach. Table 8: Radii of the poles evaluated with Eq. (17) in single channel approach. $\sqrt{s_{p}}$ | $\sqrt{\left\langle r^{2}\right\rangle_{\bar{K}N}}$ | $\mid\sqrt{\left\langle r^{2}\right\rangle_{\bar{K}N}}\mid$ | $\sqrt{\left\langle r^{2}\right\rangle_{\pi\Sigma}}$ | $\mid\sqrt{\left\langle r^{2}\right\rangle_{\pi\Sigma}}\mid$
---|---|---|---|---
$1380.75+80.95i$ | $-$ | $-$ |  $(0.78+0.08i)$ fm |  $0.78$ fm
$1429.71$ |  $2.79$ fm |  $2.79$ fm | $-$ | $-$

##  IV Conclusions

In the present work, to examine the properties of the $\Lambda(1405)$ state,
we revisit the meson-baryon interactions with the chiral unitary approach,
where we investigate the interactions of coupled channels (four coupled
channels and two coupled channels) and singled channel (two singled channel)
in isospin $I=0$ and $S=-1$ sector. Both in the cases of four coupled channels
and two coupled channels, two poles corresponding to the $\Lambda(1405)$ state
are found in the second Riemann sheet. We determine the free parameters of
$a_{\mu}$ in the loop functions with the constraint of some experimental data.
With the Breit-Wigner parameterization, we show that how the $\Lambda(1405)$
peak found in the experimental invariant mass distribution of $\pi\Sigma$ can
be demonstrated through the interference effects of two poles. In the case of
the singled channel calculation, we find a pole from each channel interaction
of $\bar{K}N$ and $\pi\Sigma$, which indicates that two poles can be
dynamically generated from the single channel interactions even without the
coupled channel effects. Based on these results, one can see that the nature
of two poles is different.

Furthermore, we studied the couplings, the compositeness, the wave functions,
and the mean-squared distance of these dynamically generated poles both in
coupled channels and singled channel interactions. From the results of the
couplings and compositeness, we find that the higher-mass pole is strongly
coupled to $\bar{K}N$ channel and contributed mostly by the $\bar{K}N$
components, which is nearly up to 100%. However, the lower-mass pole is
tightly coupled to the $\pi\Sigma$ channel with some contributions from the
$\bar{K}N$ interaction, where about 60% of the pole component is $\pi\Sigma$
and about 20% is $\bar{K}N$. With the wave functions obtained, we calculate
the radii of two poles. From our results, the radius of the higher-mass pole
is about 1.57 fm for the case of coupled channel interactions and 2.79 fm in
the single channel interaction, which are larger than the typical hadronic
scale. Whereas, the one for the lower-mass pole is about 0.66 fm and 0.78 fm
for the coupled channel and single channel interactions, respectively, which
are within the typical hadronic scale. Thus, the properties of the components
and the sizes for the two poles are unique.

Eventually, based on these results obtained, one can conclude that the nature
of two poles corresponding to the $\Lambda(1405)$ state is quite different.
The higher-mass pole is a pure $\bar{K}N$ molecule, and the lower-mass one is
a compositeness of main components of $\pi\Sigma$ and small part of
$\bar{K}N$. Therefore, we achieve that the $\Lambda(1405)$ state is the
superposition of two states with the same quantum number rather than only one
ordinary particle.

## Acknowledgments

We thank Profs. Christoph Hanhart and Eulogio Oset for useful discussions, and
acknowledge Prof. Eulogio Oset for careful reading the manuscript and valuable
comments.

## References

  * (1) H. X. Chen, W. Chen, X. Liu and S. L. Zhu, Phys. Rept. 639, 1 (2016) [arXiv:1601.02092 [hep-ph]].
  * (2) A. Hosaka, T. Iijima, K. Miyabayashi, Y. Sakai and S. Yasui, PTEP 2016, no. 6, 062C01 (2016) [arXiv:1603.09229 [hep-ph]].
  * (3) F. K. Guo, C. Hanhart, U.-G. Meißner, Q. Wang, Q. Zhao and B. S. Zou, Rev. Mod. Phys. 90, no. 1, 015004 (2018) [arXiv:1705.00141 [hep-ph]].
  * (4) H. X. Chen, W. Chen, X. Liu, Y. R. Liu and S. L. Zhu, Rept. Prog. Phys. 80, no. 7, 076201 (2017) [arXiv:1609.08928 [hep-ph]].
  * (5) R. F. Lebed, R. E. Mitchell and E. S. Swanson, Prog. Part. Nucl. Phys. 93, 143 (2017) [arXiv:1610.04528 [hep-ph]].
  * (6) A. Esposito, A. Pilloni and A. D. Polosa, Phys. Rept. 668, 1 (2016) [arXiv:1611.07920 [hep-ph]].
  * (7) A. Ali, J. S. Lange and S. Stone, Prog. Part. Nucl. Phys. 97, 123 (2017) [arXiv:1706.00610 [hep-ph]].
  * (8) R. A. Briceño, J. J. Dudek and R. D. Young, Rev. Mod. Phys. 90, no. 2, 025001 (2018) [arXiv:1706.06223 [hep-lat]].
  * (9) S. L. Olsen, T. Skwarnicki and D. Zieminska, Rev. Mod. Phys. 90, no. 1, 015003 (2018) [arXiv:1708.04012 [hep-ph]].
  * (10) M. Karliner, J. L. Rosner and T. Skwarnicki, Ann. Rev. Nucl. Part. Sci. 68, 17 (2018) [arXiv:1711.10626 [hep-ph]].
  * (11) C. Z. Yuan, Int. J. Mod. Phys. A 33, no. 21, 1830018 (2018) [arXiv:1808.01570 [hep-ex]].
  * (12) Y. R. Liu, H. X. Chen, W. Chen, X. Liu and S. L. Zhu, Prog. Part. Nucl. Phys. 107, 237 (2019) [arXiv:1903.11976 [hep-ph]].
  * (13) N. Brambilla, S. Eidelman, C. Hanhart, A. Nefediev, C. P. Shen, C. E. Thomas, A. Vairo and C. Z. Yuan, Phys. Rept. 873, 1 (2020) [arXiv:1907.07583 [hep-ex]].
  * (14) X. K. Dong, F. K. Guo and B. S. Zou, Phys. Rev. Lett. 126, no. 15, 152001 (2021) [arXiv:2011.14517 [hep-ph]].
  * (15) X. K. Dong, F. K. Guo and B. S. Zou, Progr. Phys. 41, 65 (2021) [arXiv:2101.01021 [hep-ph]].
  * (16) M. H. Alston, L. W. Alvarez, P. Eberhard, M. L. Good, W. Graziano, H. K. Ticho and S. G. Wojcicki, Phys. Rev. Lett. 6, 698 (1961).
  * (17) R. H. Dalitz and S. F. Tuan, Phys. Rev. Lett. 2, 425 (1959).
  * (18) R. H. Dalitz and S. F. Tuan, Annals Phys. 10, 307 (1960).
  * (19) N. Isgur and G. Karl, Phys. Rev. D 18, 4187 (1978).
  * (20) S. Capstick and N. Isgur, Phys. Rev. D 34, 2809 (1986) [AIP Conf. Proc. 132, 267 (1985)].
  * (21) D. Strottman, Phys. Rev. D 20, 748 (1979).
  * (22) A. Zhang, Y. R. Liu, P. Z. Huang, W. Z. Deng, X. L. Chen and S. L. Zhu, HEPNP 29, 250 (2005) [hep-ph/0403210].
  * (23) C. G. Callan, Jr. and I. R. Klebanov, Nucl. Phys. B 262, 365 (1985).
  * (24) N. Kaiser, P. B. Siegel and W. Weise, Nucl. Phys. A 594, 325-345 (1995) [arXiv:nucl-th/9505043 [nucl-th]].
  * (25) N. Kaiser, T. Waas and W. Weise, Nucl. Phys. A 612, 297-320 (1997) [arXiv:hep-ph/9607459 [hep-ph]].
  * (26) J. A. Oller, E. Oset and A. Ramos, Prog. Part. Nucl. Phys. 45, 157 (2000) [hep-ph/0002193].
  * (27) E. Oset and A. Ramos, Nucl. Phys. A 635, 99 (1998) [nucl-th/9711022].
  * (28) J. A. Oller, Prog. Part. Nucl. Phys. 110, 103728 (2020) [arXiv:1909.00370 [hep-ph]].
  * (29) A. Martinez Torres, K. P. Khemchandani, L. Roca and E. Oset, Few Body Syst. 61, no.4, 35 (2020) [arXiv:2005.14357 [nucl-th]].
  * (30) J. A. Oller, Symmetry 12, no.7, 1114 (2020) [arXiv:2005.14417 [hep-ph]].
  * (31) Z. H. Guo, arXiv:2006.16855 [hep-ph].
  * (32) J. A. Oller and U.-G. Meißner, Phys. Lett. B 500, 263 (2001) [hep-ph/0011146].
  * (33) E. Oset, A. Ramos and C. Bennhold, Phys. Lett. B 527, 99 (2002) Erratum: [Phys. Lett. B 530, 260 (2002)] [nucl-th/0109006].
  * (34) D. Jido, A. Hosaka, J. C. Nacher, E. Oset and A. Ramos, Phys. Rev. C 66, 025203 (2002) [hep-ph/0203248].
  * (35) C. Garcia-Recio, J. Nieves, E. Ruiz Arriola and M. J. Vicente Vacas, Phys. Rev. D 67, 076009 (2003) [arXiv:hep-ph/0210311 [hep-ph]].
  * (36) D. Jido, J. A. Oller, E. Oset, A. Ramos and U.-G. Meißner, Nucl. Phys. A 725, 181 (2003) [nucl-th/0303062].
  * (37) C. García-Recio, M. F. M. Lutz and J. Nieves, Phys. Lett. B 582, 49-54 (2004) [arXiv:nucl-th/0305100 [nucl-th]].
  * (38) T. Hyodo and W. Weise, Phys. Rev. C 77, 035204 (2008) [arXiv:0712.1613 [nucl-th]].
  * (39) T. Hyodo, D. Jido and A. Hosaka, Phys. Rev. C 78, 025203 (2008) [arXiv:0803.2550 [nucl-th]].
  * (40) P. J. Fink, Jr., G. He, R. H. Landau and J. W. Schnick, Phys. Rev. C 41, 2720 (1990).
  * (41) T. Hyodo and D. Jido, Prog. Part. Nucl. Phys. 67, 55 (2012) [arXiv:1104.4474 [nucl-th]].
  * (42) B. Borasoy, R. Nissler and W. Weise, Eur. Phys. J. A 25, 79-96 (2005) [arXiv:hep-ph/0505239 [hep-ph]].
  * (43) C. W. Xiao, Phys. Rev. D 95, no. 1, 014006 (2017) [arXiv:1609.02712 [hep-ph]].
  * (44) J. A. Oller, J. Prades and M. Verbeni, Phys. Rev. Lett. 95, 172502 (2005) [hep-ph/0508081].
  * (45) J. A. Oller, Eur. Phys. J. A 28, 63-82 (2006) [arXiv:hep-ph/0603134 [hep-ph]].
  * (46) B. Borasoy, U. G. Meissner and R. Nissler, Phys. Rev. C 74, 055201 (2006) [arXiv:hep-ph/0606108 [hep-ph]].
  * (47) Y. Ikeda, T. Hyodo and W. Weise, Phys. Lett. B 706, 63-67 (2011) [arXiv:1109.3005 [nucl-th]].
  * (48) Y. Ikeda, T. Hyodo and W. Weise, Nucl. Phys. A 881, 98-114 (2012) [arXiv:1201.6549 [nucl-th]].
  * (49) Z. H. Guo and J. A. Oller, Phys. Rev. C 87, no. 3, 035202 (2013) [arXiv:1210.3485 [hep-ph]].
  * (50) A. Ramos, A. Feijoo and V. K. Magas, Nucl. Phys. A 954, 58-74 (2016) [arXiv:1605.03767 [nucl-th]].
  * (51) P. C. Bruns, M. Mai and U.-G. Meißner, Phys. Lett. B 697, 254-259 (2011) [arXiv:1012.2233 [nucl-th]].
  * (52) M. Mai and U.-G. Meißner, Nucl. Phys. A 900, 51 - 64 (2013) [arXiv:1202.2030 [nucl-th]].
  * (53) M. Mai and U.-G. Meißner, Eur. Phys. J. A 51, no.3, 30 (2015) [arXiv:1411.7884 [hep-ph]].
  * (54) U.-G. Meißner, Symmetry 12, no.6, 981 (2020) [arXiv:2005.06909 [hep-ph]].
  * (55) P. A. Zyla et al. [Particle Data Group], PTEP 2020, no.8, 083C01 (2020).
  * (56) A. Cieplý, M. Mai, U.-G. Meißner and J. Smejkal, Nucl. Phys. A 954, 17-40 (2016) [arXiv:1603.02531 [hep-ph]].
  * (57) T. Sekihara, T. Hyodo and D. Jido, PTEP 2015, 063D04 (2015) [arXiv:1411.2308 [hep-ph]].
  * (58) Y. Kamiya, K. Miyahara, S. Ohnishi, Y. Ikeda, T. Hyodo, E. Oset and W. Weise, Nucl. Phys. A 954, 41-57 (2016) [arXiv:1602.08852 [hep-ph]].
  * (59) D. Sadasivan, M. Mai and M. Döring, Phys. Lett. B 789, 329-335 (2019) [arXiv:1805.04534 [nucl-th]].
  * (60) K. P. Khemchandani, A. Martínez Torres and J. A. Oller, Phys. Rev. C 100, no.1, 015208 (2019) [arXiv:1810.09990 [hep-ph]].
  * (61) L. Qin, Y. Tan, X. Hu and J. Ping, Phys. Rev. C 103, no.1, 015203 (2021) [arXiv:2008.03734 [hep-ph]].
  * (62) T. Ezoe and A. Hosaka, Phys. Rev. D 102, no.1, 014046 (2020) [arXiv:2006.03788 [hep-ph]].
  * (63) W. A. Yamada and O. Morimatsu, Phys. Rev. C 103, no.4, 045201 (2021) [arXiv:2101.10029 [hep-ph]].
  * (64) X. L. Ren, E. Epelbaum, J. Gegelia and U. G. Meißner, [arXiv:2102.00914 [hep-ph]].
  * (65) I. Zychor et al., Phys. Lett. B 660, 167 (2008) [arXiv:0705.1039 [nucl-ex]].
  * (66) H. Y. Lu et al. [CLAS Collaboration], Phys. Rev. C 88, 045202 (2013) [arXiv:1307.4411 [nucl-ex]].
  * (67) M. Mai, [arXiv:2010.00056 [nucl-th]].
  * (68) T. Hyodo and M. Niiyama, [arXiv:2010.07592 [hep-ph]].
  * (69) H. A. Ahmed and C. W. Xiao, Phys. Rev. D 101, no. 9, 094034 (2020) [arXiv:2001.08141 [hep-ph]].
  * (70) U.-G. Meißner, Rept. Prog. Phys. 56, 903 (1993) [hep-ph/9302247].
  * (71) A. Pich, Rept. Prog. Phys. 58, 563 (1995) [hep-ph/9502366].
  * (72) G. Ecker, Prog. Part. Nucl. Phys. 35, 1 (1995) [hep-ph/9501357].
  * (73) V. Bernard, N. Kaiser and U.-G. Meißner, Int. J. Mod. Phys. E 4, 193 (1995) [hep-ph/9501384].
  * (74) S. Weinberg, Phys. Rev. 137, B672 (1965).
  * (75) F. Aceti and E. Oset, Phys. Rev. D 86, 014012 (2012) [arXiv:1202.4607 [hep-ph]].
  * (76) V. Baru, J. Haidenbauer, C. Hanhart, Y. Kalashnikova and A. E. Kudryavtsev, Phys. Lett. B 586, 53-61 (2004) [arXiv:hep-ph/0308129 [hep-ph]].
  * (77) T. Sekihara and T. Hyodo, Phys. Rev. C 87, no. 4, 045202 (2013) [arXiv:1209.0577 [nucl-th]].
  * (78) C. W. Xiao, F. Aceti and M. Bayar, Eur. Phys. J. A 49, 22 (2013) [arXiv:1210.7176 [hep-ph]].
  * (79) T. Hyodo, Phys. Rev. Lett. 111, 132002 (2013) [arXiv:1305.1999 [hep-ph]].
  * (80) F. Aceti, L. R. Dai, L. S. Geng, E. Oset and Y. Zhang, Eur. Phys. J. A 50, 57 (2014) [arXiv:1301.2554 [hep-ph]].
  * (81) Z. H. Guo and J. A. Oller, Phys. Rev. D 93, no. 9, 096001 (2016) [arXiv:1508.06400 [hep-ph]].
  * (82) J. Yamagata-Sekihara, J. Nieves and E. Oset, Phys. Rev. D 83, 014003 (2011) [arXiv:1007.3923 [hep-ph]].
  * (83) A. Ozpineci, C. W. Xiao and E. Oset, Phys. Rev. D 88, 034018 (2013) [arXiv:1306.3154 [hep-ph]].
  * (84) R. J. Hemingway, Nucl. Phys. B 253, 742 (1985).
  * (85) W. E. Humphrey and R. R. Ross, Phys. Rev. 127, 1305 (1962).
  * (86) M. Sakitt, T. B. Day, R. G. Glasser, N. Seeman, J. H. Friedman, W. E. Humphrey and R. R. Ross, Phys. Rev. 139, B719 (1965).
  * (87) J. K. Kim, Phys. Rev. Lett. 19, 1074 (1967).
  * (88) W. Kittel, G. Otter and I. Wacek, Phys. Lett. 21, 349 (1966).
  * (89) J. Ciborowski et al., J. Phys. G 8, 13 (1982).
  * (90) D. Evans, J. V. Major, E. Rondio, J. A. Zakrzewski, J. E. Conboy, D. J. Miller and T. Tymieniecka, J. Phys. G 9, 885 (1983).
  * (91) J. K. Kim, Columbia University Report No. NEVIS-149, 1966.
  * (92) A. D. Martin, Nucl. Phys. B 179, 33 (1981).
  * (93) D. L. B. Sombillo, Y. Ikeda, T. Sato and A. Hosaka, [arXiv:2105.04898 [hep-ph]].
  * (94) A. V. Anisovich, A. V. Sarantsev, V. A. Nikonov, V. Burkert, R. A. Schumacher, U. Thoma and E. Klempt, Eur. Phys. J. A 56, no. 5, 139 (2020).
  * (95) T. Sekihara, T. Hyodo and D. Jido, Phys. Rev. C 83, 055202 (2011) [arXiv:1012.3232 [nucl-th]].
  * (96) I. Matuschek, V. Baru, F. K. Guo and C. Hanhart, Eur. Phys. J. A 57, no.3, 101 (2021) [arXiv:2007.05329 [hep-ph]].
  * (97) Y. Bai and D. Y. Chen, Phys. Rev. D 99, no.7, 072007 (2019) [arXiv:1901.01394 [hep-ph]].
  * (98) M. Iwasaki, R. S. Hayano, T. M. Ito, S. N. Nakamura, T. P. Terada, D. R. Gill, L. Lee, A. Olin, M. Salomon, S. Yen, K. Bartlette, G. A. Beer, G. Mason, G. Trayling, H. Outa, T. Taniguchi, Y. Yamashita and R. Seki, Phys. Rev. Lett. 78, 3067-3069 (1997).
  * (99) S. Ohnishi, Y. Ikeda, T. Hyodo and W. Weise, Phys. Rev. C 93, no.2, 025207 (2016) [arXiv:1512.00123 [nucl-th]].