# Compositeness of S-wave weakly-bound states
from next-to-leading order Weinberg’s relations

M. Albaladejo<EMAIL_ADDRESS>J. Nieves<EMAIL_ADDRESS>Instituto de Física Corpuscular (centro mixto CSIC-UV), Institutos de
Investigación de Paterna, C/Catedrático José Beltrán 2, E-46980 Paterna,
Valencia, Spain

###### Abstract

We discuss a model-independent estimator of the likelihood of the
compositeness of a shallow S-wave bound or virtual state. The approach is
based on an extension of Weinberg’s relations in Phys. Rev. 137, B672 (1965)
Weinberg (1965) and it relies only on the proximity of the energy of the state
to the two-hadron threshold to which it significantly couples. The scheme only
makes use of the experimental scattering length and the effective range low
energy parameters, and it is shown to be fully consistent for predominantly
molecular hadrons. As explicit applications, we analyse the case of the
deuteron, the ${}^{1}{\rm S}_{0}$ nucleon-nucleon virtual state and the exotic
$D^{\ast}_{s0}(2317)^{\pm}$, and find strong support to the molecular
interpretation in all cases. Results are less conclusive for the
$D^{\ast}_{s0}(2317)^{\pm}$, since the binding energy of this state is
significantly higher than that of the deuteron, and the approach employed here
is at the limit of its applicability. We also qualitatively address the case
of the recently discovered $T_{cc}^{+}$ state, within the isospin limit to
avoid the complexity of the very close thresholds $D^{0}D^{\ast+}$ and
$D^{+}D^{\ast 0}$, which could mask the ingredients of the approach proposed
in this work.

##  I Introduction

Quantum Chromodynamics (QCD), the theory of strong interactions, generates a
rich spectrum of hadrons, most of which can be classified according to simple
constituent quark models Gell-Mann (1964); Zweig (1964a, b); Godfrey and Isgur
(1985); Capstick and Isgur (1986) as $q\bar{q}$ (mesons) and $qqq$ (baryons)
states. Despite this fact, the last two decades have witnessed the discovery
of many states that defy this simple classification Zyla  _et al._ (2020).
Different worldwide experiments (BaBar, Belle, BES, LHCb, $\ldots$) have
reported the observation of a plethora of unstable states and peaks in mass-
distributions located surprisingly close to different two bottomed/charmed-
hadron thresholds, such as the $XYZ$ mesons, the $P_{c}$ pentaquarks Choi  _et
al._ (2003); Bondar  _et al._ (2012); Ablikim  _et al._ (2013a); Liu  _et al._
(2013a); Ablikim  _et al._ (2014, 2013b); Aaij  _et al._ (2015a, 2019);
Ablikim  _et al._ (2021), or the doubly charmed $T_{cc}^{+}$ state Aaij  _et
al._ (2022a, b). There exist also clear examples of exotic candidates in the
open charm and bottom sectors, e.g. $D^{\ast}_{s0}(2317)$,
$D^{\ast}_{0}(2300)$, $\Lambda_{c}(2595)$ Aubert  _et al._ (2003); Abe  _et
al._ (2004); Edwards  _et al._ (1995), $\ldots$, $B_{1}(5721)$,
$B_{2}^{*}(5747)$, $\Xi_{b}(6227)$ Abazov  _et al._ (2007); Aaltonen  _et al._
(2014); Aaij  _et al._ (2015b, 2018, 2021), etc. These states are often
interpreted as hadron molecules Barnes  _et al._ (2003); van Beveren and Rupp
(2003); Kolomeitsev and Lutz (2004); Szczepaniak (2003); Voloshin (2004);
Close and Page (2004); Swanson (2004); Tornqvist (2004); Guo  _et al._ (2006);
Gamermann  _et al._ (2007); Flynn and Nieves (2007); Faessler  _et al._
(2007); Lutz and Soyeur (2008); Dong  _et al._ (2009); Gamermann and Oset
(2009); Wu  _et al._ (2010); Bondar  _et al._ (2011); Wang  _et al._ (2011);
Yang  _et al._ (2012); Garcia-Recio  _et al._ (2009); Romanets  _et al._
(2012); Garcia-Recio  _et al._ (2013a); Wu  _et al._ (2012); Liu  _et al._
(2013b); Garcia-Recio  _et al._ (2013b); Wang  _et al._ (2013a); Guo  _et al._
(2013); Aceti  _et al._ (2014); Albaladejo  _et al._ (2016a, b, 2017); Nieves
_et al._ (2018); Du  _et al._ (2018); Nieves and Pavao (2020); Wang  _et al._
(2018); Karliner and Rosner (2015); Xiao  _et al._ (2013); Nieves  _et al._
(2020); Liu  _et al._ (2019a); Du  _et al._ (2020); Liu  _et al._ (2019b); Lu
_et al._ (2015); Dong  _et al._ (2021a); Du  _et al._ (2021, 2022); Dong  _et
al._ (2021b); Guo  _et al._ (2018); Guo and Oller (2021); Feijoo  _et al._
(2021); Albaladejo (2022), or compact tetraquarks/pentaquarks Esposito  _et
al._ (2017); Maiani  _et al._ (2005, 2014); Braaten (2013); Dias  _et al._
(2013); Maiani  _et al._ (2014); Qiao and Tang (2014); Deng  _et al._ (2014);
Ali  _et al._ (2012); Cheng and Hou (2003); Terasaki (2003); Dmitrasinovic
(2005); Bracco  _et al._ (2005); Wang and Wan (2006a); Chen and Li (2004); Kim
_et al._ (2022); Maiani  _et al._ (2015); Lebed (2015); Li  _et al._ (2015);
Ghosh  _et al._ (2017); Wang (2016); Zhu and Qiao (2016); Richard  _et al._
(2017); Hiyama  _et al._ (2018); Barnes and Godfrey (2004), and when allowed
by their quantum numbers and flavour content, there are also attempts to
describe these states as predominant $q\bar{q}$ or $qqq$ structures Colangelo
and De Fazio (2003); Cahn and Jackson (2003); Browder  _et al._ (2004);
Godfrey (2003); Narison (2005); Bardeen  _et al._ (2003); Nowak  _et al._
(2004); Browder  _et al._ (2004); Dai  _et al._ (2003); Lee  _et al._ (2007);
Lakhina and Swanson (2007); Segovia  _et al._ (2013); Yoshida  _et al._
(2015); Ortega  _et al._ (2016). Other possibilities (hybrids, virtual poles,
etc.) for the nature of these exotics Close and Godfrey (2003); Li (2005);
Wang and Wan (2006b); Dubynskiy and Voloshin (2008); Voloshin (2013);
Fernández-Ramírez  _et al._ (2019); Pilloni  _et al._ (2017) or the role of
kinematic (non-dynamical) effects (chiefly triangle singularities) in the
interpretation of the observed peaks have also been stressed Wang  _et al._
(2013b); Chen  _et al._ (2013); Swanson (2015); Guo  _et al._ (2015a, b); Liu
_et al._ (2016); Albaladejo  _et al._ (2016a); Szczepaniak (2015); Ikeda  _et
al._ (2016); Pilloni  _et al._ (2017); Bayar  _et al._ (2016); Guo  _et al._
(2020). Further discussions and references can be found in Refs. Lebed  _et
al._ (2017); Chen  _et al._ (2017); Ali  _et al._ (2017); Karliner  _et al._
(2018); Olsen  _et al._ (2018); Brambilla  _et al._ (2020). In addition to
knowing how many of these states exist and their masses and widths, it is
clearly a fundamental task in hadron physics to study the dynamical details of
their structure. Such analysis will be invaluable in improving our
understanding of strong interactions.

It is a direct consequence of unitarity that the inverse single-channel two-
particle scattering amplitude $f$ is given in terms of the phase shift
$\delta$ by $f(E)^{-1}=k\cot\delta(k)-ik$, with $k=\sqrt{2\mu E}$, for non-
relativistic kinematics, $\mu$ the reduced mass of the scattering particles,
and $E$ the energy of the system relative to the threshold $(m_{1}+m_{2})$.
The real part of the inverse scattering amplitude is a polynomial in even
powers of $k$ and, in the case of S-wave, it leads to the effective range
expansion (ERE):

$k\cot\delta(k)=\frac{1}{a}+\frac{1}{2}rk^{2}+{\cal O}\left(k^{4}\right)~{},$
(1)

where the parameters $a$ and $r$ are called the scattering length and
effective range, respectively. Weinberg’s compositeness rules Weinberg (1965)
connect these low energy observables with the probability $Z$ that an S-wave
shallow bound state is found in a bare elementary-particle state, which might
allow to interpret $X=(1-Z)$ as a kind of molecular probability:

$\displaystyle a$
$\displaystyle=\underbrace{-\frac{2}{\gamma_{b}}\left(\frac{1-Z}{2-Z}\right)}_{a_{\text{LO}}}\
+\ \mathcal{O}(\beta^{-1})~{},$ (2a) $\displaystyle r$
$\displaystyle=\underbrace{-\frac{1}{\gamma_{b}}\left(\frac{Z}{1-Z}\right)}_{r_{\text{LO}}}\
+\ \mathcal{O}(\beta^{-1})~{},$ (2b)

where $\gamma_{b}=\sqrt{2\mu|E_{b}|}$ (with $E_{b}<0$, the binding energy),
$\beta$ denotes the next momentum scale that is not treated explicitly in the
ERE, with $1/\beta$ providing an estimate for the interaction range
corrections. The work of Ref. Weinberg (1965) showed that the experimental
values for $a$ and $r$ from $pn$ scattering give strong model-independent
evidence that the deuteron is composite, i.e., the probability $Z$ of ending
the deuteron in a bare elementary particle state is very small. However, as
commonly acknowledged (see e.g. Refs. Matuschek  _et al._ (2021); Esposito
_et al._ (2022); Li  _et al._ (2022); Song  _et al._ (2022)), this does not
follow from the naive evaluation of $X(a,r)=1/\sqrt{1+2r/a}$, easily derived
from Eqs. (2), which gives the meaningless result of $X=1.68>1$ for a
probability. Indeed, the above formula and this numerical value for $X$ are
not given in Ref. Weinberg (1965). As explicitly pointed out by Weinberg, the
key token for the deuteron compositeness is the fact that $r$ is small and
positive of the order of the range $\sim m_{\pi}^{-1}$ of the $pn$
interaction, rather than large and negative. This discussion clearly shows the
ambiguities affecting any conclusion about the nature of an exotic state based
uniquely on a blind numerical computation of $X$, neglecting
$\mathcal{O}(\beta^{-1})$ corrections. Different applications, re-derivations,
re-interpretation and extensions of Weinberg’s compositeness relations have
been proposed Baru  _et al._ (2004); Gamermann  _et al._ (2010); Baru  _et
al._ (2010); Hanhart  _et al._ (2011); Aceti and Oset (2012); Hyodo  _et al._
(2012); Sekihara  _et al._ (2015); Kamiya and Hyodo (2016); Garcia-Recio  _et
al._ (2015); Guo and Oller (2016); Kamiya and Hyodo (2017); Sekihara (2017);
Oller (2018); Matuschek  _et al._ (2021); Esposito  _et al._ (2022); Li  _et
al._ (2022); Kinugawa and Hyodo (2022a); Song  _et al._ (2022); Sazdjian
(2022); Bruns (2022); Kinugawa and Hyodo (2022b), discussing or trying to
improve upon various aspects of the derivation of the criterion in Ref.
Weinberg (1965), in order to increase the cases to which it can be applied,
i.e., by considering also coupled channels.

We discuss now two subtle points about the interpretation of $Z$.111Our
discussion is similar to that of Ref. Guo  _et al._ (2018). In Eq. (18) of
Ref. Weinberg (1965), $Z$ is defined as a probability between the states of
the continuum and a bare, compact state, with the patent difficulty that the
latter is not, in general, a physical, asymptotic state. Therefore, the
effects produced by the interacting hadron cloud must be taken into account.
From an Effective Field Theory point of view, the effects of a bare coupling
(of a bare, compact state to a two-hadron continuum state) plus other
perturbative hadron–hadron interactions are translated into the physical
coupling of the state, $g^{2}$. The latter, as we will see below [cf. Eq.
(3)], enters into the LO term contribution to $Z$, and thus the interpretation
of $Z$ in terms of compositeness is unchanged Weinberg (1963a, b) (see also
Refs. Guo  _et al._ (2018); Albaladejo  _et al._ (2021)). From a different
point of view, $Z$ is also a renormalization field factor, and in this sense
it is in general a scheme-dependent quantity. However, it is crucial that in
the weak binding limit $(\gamma_{b}\ll\beta)$, the LO contribution to $Z$ is
determined by the pole position and the residue of the two-hadron scattering
amplitude at the physical pole Weinberg (1965); Matuschek  _et al._ (2021).
This term is non-analytical in $E$ and it is thus scheme-independent. Indeed,
the LO contribution is written in terms of two measurable quantities (position
and coupling of the bound state), and this model-independent contribution to
$Z$ becomes a measure of the compositeness. The scheme-dependent terms of $Z$,
for instance those analytic in $E$, need to be fixed by some renormalization
condition, but importantly they are suppressed by a factor of the order
$\mathcal{O}(\gamma_{b}/\beta)$ Weinberg (1965); Matuschek  _et al._ (2021).
Indeed, in the weak binding limit one has:

$\left(1-Z\right)=\frac{\mu g^{2}}{\gamma_{b}}+{\cal
O}\left(\gamma_{b}/\beta\right),\qquad g^{2}\equiv\lim_{E-E_{b}}(E_{b}-E)f(E)$
(3)

The above equation illustrates how the coupling $g^{2}$ and the binding
momentum $\gamma_{b}$ provide the leading contribution to the probability
$X=1-Z$, and that they do not fully determine the sub-leading
$\mathcal{O}(\gamma_{b}/\beta)$ contributions to $Z$. In addition, we note
that Eq. (3) is consistent with Eqs. (2) and the approximate relation $\mu
g^{2}\approx\gamma_{b}/(1-r\gamma_{b})$, valid for loosely bound states.

In this work, we will delve in the above discussion, and we will present an
estimator of the compositeness of a state relying only on the proximity of its
mass to a two-hadron threshold to which it significantly couples. Our
conclusions will apply only to weakly bound or virtual states since model-
independent statements are possible only if $\gamma_{b}\ll\beta$. The internal
structure of a non-shallow bound state cannot be studied without having a
knowledge of its wave-function more detailed than what can be inferred from
the experimental value of just a few scattering parameters. Hence, one cannot
extract model-independent conclusions on the compositeness of a particle
located far from the relevant threshold, since one would necessarily have to
rely on certain interaction, re-summation and renormalization models. In this
respect, the calculation of hadron form factors, which can be accessed by LQCD
simulations, can play an important role in determining the internal structure
of hadrons Flynn and Nieves (2007); Briceño  _et al._ (2021); Albaladejo and
Oller (2012).

For simplicity, we will ignore coupled-channels dynamics, though we will
comment in Sec. VI on this issue. We will also assume that the particle is
stable, otherwise $Z$ is complex. However, it might be an adequate
approximation to ignore the decay modes of a very narrow resonance.

The paper is organized as follows. After this introduction, we present in Sec.
II the model-independent estimator of the likelihood of the compositeness of a
shallow S-wave bound/virtual state proposed in this work. Next we show
explicit applications and analyse the case of the deuteron (Sec. III), the
${}^{1}{\rm S}_{0}$ nucleon-nucleon virtual state (Sec. V) and the exotic
$D^{\ast}_{s0}(2317)^{\pm}$ and $T_{cc}^{+}$ states in Secs. IV and VI,
respectively. We collect the most important conclusions of our work in Sec.
VII.

##  II Likelihood of the compositeness of a weakly bound state

The parameters of the ERE [cf. Eq. (1)] depend in turn on $\gamma_{b}$, which
is determined by the binding energy of the state. The large values for both
$a$ and $r$ when $Z$ is not zero appear because of the $1/\gamma_{b}$
contributions in Eqs. (2). They may suggest that the next-to-leading order
(NLO) approximation to the ERE, consisting in neglecting ${\cal
O}\left(k^{4}\right)$ terms, may itself break down when the particle is
elementary. This, however, does not happen since only the first two terms in
the expansion of $k\cot\delta(k)$ in powers of $k^{2}$ become of order
$\gamma_{b}$ for $Z\neq 0$ and $k\approx\gamma_{b}$. The third and higher
terms are smaller by powers of $\gamma_{b}/\beta$ Weinberg (1965). As a
consequence, the approximate relation

$\gamma_{b}\approx-\frac{1}{a}+\frac{1}{2}r\gamma_{b}^{2}$ (4)

is expected to be fulfilled with great accuracy for weakly bound states. This
is indeed the case for the deuteron. The above relation does not tell anything
about the elementarity of the particle, since it follows from the requirement
$\cot\delta(i\gamma_{b})=+i$. In fact, it is exactly satisfied by $a_{\rm LO}$
and $r_{\rm LO}$ for all $Z$ [cf. Eqs. (2)]. However, the deviations of the
actual scattering length and effective range from their $\gamma_{b}-$expansion
leading-order (LO) values $a_{\rm LO}$ and $r_{\rm LO}$ encode some valuable
information on the compositeness of the state. Moreover, from the discussion
above, the third and higher terms in the ERE of Eq. (1) could provide at most
corrections222The point that Weinberg makes in Ref. Weinberg (1965) is that
the coefficient of the $k^{4}$ term in the ERE does not diverge as
$1/\gamma_{b}^{3}$ in the weak-binding limit, which will affect to the
approximate relation of Eq. (4) at order $\gamma_{b}$. However, one cannot
discard that this coefficient could scale like $Z/\gamma_{b}^{2}$. The factor
of $Z$ in front is because this $1/\gamma_{b}^{2}$ behaviour will not appear
in the absence of compact bare states ($Z=0$). The reasoning runs in parallel
for any other higher order term of the ERE. Hence, we simply discard any new
contribution of order ${\cal O}(\gamma_{b})$ to Eq. (4) coming from the ERE
terms beyond the scattering length and effective-range. of order ${\cal
O}(Z\gamma_{b}^{2}/\beta)$ to the difference
$\gamma_{b}-\left(-1/a+r\gamma_{b}^{2}/2\right)$. With these ideas in mind, we
introduce a phenomenological term $\delta r$ to estimate the NLO contribution
to the effective range $r$, within its expansion in powers of the binding
momentum $\gamma_{b}$,

$r=\underbrace{-\frac{1}{\gamma_{b}}\left(\frac{Z}{1-Z}\right)+\delta
r}_{r_{\text{NLO}}}+\mathcal{O}\left(\frac{\gamma_{b}}{\beta^{2}}\right),\quad
r_{\text{NLO}}=r_{\text{LO}}+\delta r~{},$ (5a) This correction, $\delta r$,
is expected to be of the order of the range of the interaction [${\cal
O}(1/\beta)$]. The scattering length will have a similar NLO contribution,
$\delta a$, on top of the LO term in Eq. (2a). We fix this analogous NLO
contribution to the scattering length such that the difference
$\gamma_{b}-\left(-1/a_{\rm NLO}+r_{\rm NLO}\gamma_{b}^{2}/2\right)$ [cf. Eq.
(4)] deviates from zero in terms of the order ${\cal
O}\left(\gamma_{b}^{3}/\beta^{2}\right)$. This is to say, we require that the
$\gamma_{b}^{2}$ term in the Taylor expansion (in powers of $\gamma_{b}$) of
the quantity $\left[\gamma_{b}-\left(-1/a_{\rm NLO}+r_{\rm
NLO}\gamma_{b}^{2}/2\right)\right]$ vanishes. In this way, we obtain:
$a=\underbrace{-\frac{2}{\gamma_{b}}\left(\frac{1-Z}{2-Z}\right)-\frac{\delta
r}{2}\left(\frac{1-Z}{1-\frac{Z}{2}}\right)^{2}}_{a_{\text{NLO}}}+\mathcal{O}\left(\frac{\gamma_{b}}{\beta^{2}}\right),\,\quad
a_{\text{NLO}}=a_{\text{LO}}-\frac{\delta
r}{2}\left(\frac{1-Z}{1-\frac{Z}{2}}\right)^{2}.$ (5b)

The relations given in Eqs. (5) constitute the main result of this work. The
key point is that the same parameter $\delta r$ appear in both
$a_{\text{NLO}}$ and $r_{\text{NLO}}$. Therefore, they provide a model-
independent scheme to correlate the NLO corrections to $a$ and $r$, which
turns out to be consistent as long as $Z\simeq 0$, of the order of
$\mathcal{O}\left(\gamma_{b}/\beta\ll 1\right)$ or smaller. The fact that $Z$
is required to be at most of order $\mathcal{O}\left(\gamma_{b}/\beta\right)$
is because the ${\cal O}(k^{4})$ terms in the ERE expansion could lead to
corrections of order $\mathcal{O}\left(Z\gamma_{b}^{2}/\beta\right)$ in Eq.
(4), as mentioned before. These corrections should be, at most, comparable to
those of ${\cal O}\left(\gamma_{b}^{3}/\beta^{2}\right)$ neglected in the
present scheme to correlate $r_{\text{NLO}}$ and $a_{\text{NLO}}$. When $Z$
takes these small values, it would mean that the model-independent
contribution from the coupling $g^{2}$ in Eq. (3) should be large, giving a
strong support to the molecular nature of the weakly bound state. In other
words, a shallow bound state which couples to a two-hadron system, for which
the scattering length and effective-range could be accurately described by
$a_{\text{NLO}}$ and $r_{\text{NLO}}$ with values of $Z$ of the order of
$\left(\gamma_{b}/\beta\right)$, small but not necessarily zero, can be
reasonably seen as a hadron molecule. This is to say, its low energy
properties, including the binding, can be naturally accommodated as result of
the two-hadron interaction. To study systems for which $Z$ could take larger
values,333That is, larger than $\mathcal{O}(\gamma_{b}/\beta)$ but not
necessarily $Z\to 1$. For this latter case, corresponding to a purely compact
state, the scattering amplitude that arises by considering a bare state
propagator is exactly the ERE truncated at $\mathcal{O}(k^{2})$. Therefore, in
the $Z\to 1$ case, no additional terms in the ERE beyond $a$ and $r$. the
${\cal O}(k^{4})$ or higher terms in the ERE expansion will be required, which
points to the need in these cases to include additional details of the short
distance dynamics.

Given the experimental values for the mass, or equivalently the binding
momentum $\gamma_{b}^{\text{exp}}$, the associated scattering length ($a_{\rm
exp}$), and the effective range ($r_{\rm exp}$) parameters of a two-particle
weakly bound state, we propose to study the following two dimensional (2D)
distribution:

$\displaystyle\mathcal{L}(Z,\delta r)=$
$\displaystyle\frac{1}{3}\left[\left(\frac{a_{\rm exp}-a_{\rm NLO}}{\Delta
a_{\rm exp}}\right)^{2}+\left(\frac{r_{\rm exp}-r_{\rm NLO}}{\Delta r_{\rm
exp}}\right)^{2}+\left(\frac{\gamma^{\rm exp}_{b}-\gamma^{\rm
NLO}_{b}}{\Delta\gamma_{b}^{\rm exp}}\right)^{2}\right],$ (6)

to estimate the likelihood of the compositeness of the state. Above,
$\gamma^{\rm NLO}_{b}$ is given by

$\gamma^{\text{NLO}}_{b}=\frac{1-\sqrt{1+2r_{\text{NLO}}/a_{\text{NLO}}}}{r_{\text{NLO}}}~{},$
(7)

which exactly satisfies $\left[\gamma^{\rm NLO}_{b}-\left(-1/a_{\rm
NLO}+r_{\rm NLO}(\gamma^{\rm NLO}_{b})^{2}/2\right)\right]=0$. Consistent with
the order at which we are working, we evaluate $a_{\rm NLO}$ and $r_{\rm NLO}$
using $\gamma^{\rm exp}_{b}$. It is important to notice that
$\Delta\gamma_{b}^{\rm exp},\Delta a_{\rm exp}$, and $\Delta r_{\rm exp}$
should be fixed taking into account not only the uncertainties on the
determinations of these observables, but also the expected accuracy of their
NLO approximation, i.e., an $\mathcal{O}(\gamma_{b}^{2}/\beta^{2})$ relative
error. If the actual experimental errors are smaller than this expected
accuracy, then a relative error $(\gamma_{b}/\beta)^{2}$ should be taken
instead.

Figure 1: Left: Distribution of Eq. (6) for different values of the
probability $Z$ of ending the deuteron in a bare elementary particle

, as a function of the NLO contribution $\delta r$ [Eq. (5a)] to the effective
range of the $pn$ interaction in the isoscalar ${}^{3}{\rm S}_{1}$ wave.
Middle and right: $Z$ dependence of $a$ (middle) and $r$ (right) at LO (red,
dashed-dotted lines) and NLO (blue, solid lines), compared with the
experimental values (green bands).

##  III The deuteron

We first apply the compositeness distribution of Eq. (6) to the paradigmatic
case of the deuteron, whose properties are known very precisely: $E_{b}^{\rm
exp}=-2.224575(9)$ MeV [or equivalently $\gamma_{b}^{\rm
exp}=0.2316068(5)\,{\rm fm}^{-1}$] Van Der Leun and Alderliesten (1982), and
$a_{\rm exp}=-5.42(1)$ fm and $r_{\rm exp}=1.75(1)$ fm from the Granada-group
analysis of the $pn$ isoscalar ${}^{3}{\rm S}_{1}$ wave Navarro Pérez  _et
al._ (2016) (see also Ref. Ruiz Arriola  _et al._ (2020)). The analysis of the
latter work includes statistical errors, stemming from the data uncertainties
for a fixed form of the potential, and systematic errors arising from the
different most-likely forms of the potentials. Assuming they are independent,
the total uncertainty corresponds to adding both errors in quadrature. Despite
including systematic uncertainties, the errors of $a$ and $r$ turn out to be
much smaller than the accuracy that can be expected from the NLO
approximation, $(\gamma_{b}^{2}/m_{\pi}^{2})\sim 10\%$, taking $\beta\simeq
m_{\pi}$. Therefore, instead of taking the small errors quoted above, we fix
$\Delta\gamma^{\rm exp}_{b}$, $\Delta a_{\rm exp}$, and $\Delta r_{\rm exp}$
in Eq. (6) assuming a relative error of 10%. With all these inputs, we show in
Fig. 1 the 2D distribution ${\cal L}(Z,\delta r)$ for the case of the
deuteron. It strongly supports molecular probabilities $(1-Z)$ quite close to
one, in conjunction with values of the NLO $\delta r$ contribution of the
order of $1/m_{\pi}\sim 1.4\,\text{fm}$, as expected. We also see in Fig. 1
(middle and right plots) that for a value $\delta r=1.75\,\text{fm}$,
$a_{\text{NLO}}$ and $r_{\text{NLO}}$ are closer to the experimental values
than the corresponding LO predictions [cf. Eq. (2)]. This model-independent
analysis provides a consistent picture, where values of $Z$ greater than
$\gamma_{b}^{\rm exp}/m_{\pi}\sim 0.3$ are very implausible, which confirms
the dominant molecular structure of the deuteron Weinberg (1965).

##  IV The $D^{\ast}_{s0}(2317)^{\pm}$

This scalar narrow resonance ($\Gamma<3.8$ MeV), which lies $45\,\text{MeV}$
below the $DK$ threshold and has valence-quark content $c\bar{s}$, was
discovered in 2003 by the BaBar Collaboration Aubert  _et al._ (2003). Its
abnormally light mass cannot be easily accommodated within constituent quark
models Fayyazuddin and Riazuddin (2004); Sadzikowski (2004); Lakhina and
Swanson (2007); Segovia  _et al._ (2013); Ortega  _et al._ (2016), and as a
consequence, it is common to describe this exotic resonance as the result of
the S-wave $DK$ interaction Guo  _et al._ (2006); Flynn and Nieves (2007); Guo
_et al._ (2008); Liu  _et al._ (2013b); Mohler  _et al._ (2013); Lang  _et
al._ (2014); Martínez Torres  _et al._ (2015); Bali  _et al._ (2017); Guo  _et
al._ (2018); Albaladejo  _et al._ (2018). Unitarized heavy-meson chiral
approaches predict that the $D^{\ast}_{s0}(2317)^{\pm}$ would belong to a
light-flavor SU(3) anti-triplet, completed by an isospin doublet associated to
the scalar $D^{*}_{0}(2300)$ resonance Albaladejo  _et al._ (2017); Du  _et
al._ (2018). Here, we will discuss, within the scheme outlined above, the $DK$
molecular probability of the $D^{\ast}_{s0}(2317)^{\pm}$, neglecting the
isospin-violating $D_{s}\pi$ decay channel. Given the lack of $DK$ scattering
data, we take the values of the isoscalar S-wave $DK$ scattering length and
effective range obtained in Ref. Martínez Torres  _et al._ (2015) from the
finite volume QCD levels reported in Refs. Mohler  _et al._ (2013); Lang  _et
al._ (2014). Namely, we use $a_{\rm exp}=-1.3(5)\,\text{fm}$ and $r_{\rm
exp}=-0.1(3)\,\text{fm}$. In addition, we take $E_{b}^{\rm
exp}=-45(4)\,\text{MeV}$, estimated from the experimental masses compiled in
the PDG Zyla  _et al._ (2020), which leads to $\gamma_{b}^{\rm
exp}=0.95(4)\,\text{fm}^{-1}$ when isospin averaged masses are used for the
kaon and $D$ mesons. Due to the large uncertainties affecting both $a_{\rm
exp}$ and $r_{\rm exp}$, it is not necessary to take into account the
subtleties associated with the $D^{0}K^{+}$–$D^{+}K^{0}$ isospin breaking
effects. The scale $\beta$ is in this case of the order of $300\,\text{MeV}$,
estimated from the expected effects induced by the nearest $D_{s}\eta$
threshold Matuschek  _et al._ (2021) and/or by the two-pion exchange
interaction, none of which are explicitly treated in the ERE. Hence, we expect
the accuracy of the NLO $\gamma_{b}-$expansion to be of the order of
$(\gamma_{b}^{2}/\beta^{2})\sim 40\%$, which we adopt for $\Delta\gamma^{\rm
exp}_{b}$, while for $\Delta a_{\rm exp}$ and $\Delta r_{\rm exp}$, we use the
errors quoted above.

The compositeness 2D distribution of Eq. (6) for the
$D^{\ast}_{s0}(2317)^{\pm}$ is shown in Fig. 2. The results favor $DK$
molecular probabilities of at least 50%, which is in agreement with previous
calculations Liu  _et al._ (2013b); Martínez Torres  _et al._ (2015);
Albaladejo  _et al._ (2016c); Guo  _et al._ (2018); Albaladejo  _et al._
(2018); Matuschek  _et al._ (2021); Song  _et al._ (2022). We cannot be as
predictive in this case as for the deuteron, not only because of the bigger
uncertainties of the input, but also because of the larger size of the power-
counting parameter $\gamma_{b}/\beta\sim 0.6$. Nevertheless, the approach is
still consistent since values of $Z$ of order ${\cal
O}\left(\gamma_{b}/\beta\right)$, or smaller, are favored by the compositeness
distribution for this state. We however should note that the binding energy of
the $D^{\ast}_{s0}(2317)^{\pm}$ is significantly higher than that of deuteron,
and the NLO approach employed here is at the limit of its applicability. To be
quantitatively more precise, it would be necessary to know more details about
the dynamics at short distances of this exotic state than those encoded in the
first two parameters of the ERE.

Figure 2: Compositeness distribution for the $D^{\ast}_{s0}(2317)^{\pm}$.

##  V The ${}^{1}{\rm S}_{0}$ nucleon-nucleon virtual state

The $pn$ scattering length and effective range determined in Ref. Ruiz Arriola
_et al._ (2020) for this isovector partial wave are $a_{\rm exp}=23.735(16)\
\text{fm}$ and $r_{\rm exp}=2.68(3)\ \text{fm}$, respectively, with the total
uncertainties obtained by adding statistical and systematic errors in
quadrature. This partial wave has a shallow virtual state (pole on the real
energy-axis below the threshold on the unphysical sheet), the position of
which is determined by the condition $\cot\delta(-i\gamma_{v})=+i$, with
$\gamma_{v}=\sqrt{2\mu|E_{v}|}$ and $E_{v}<0$, the binding energy of the
virtual state. From the ERE, it follows

$\gamma_{v}\approx\frac{1}{a}-\frac{1}{2}r\gamma_{v}^{2}+{\cal
O}(\gamma^{4}_{v})~{}.$ (8)

Neglecting ${\cal O}(\gamma^{4}_{v})$ corrections and using the experimental
$a_{\rm exp}$ and $r_{\rm exp}$ values, the above equation leads to

$\gamma_{v}^{\rm exp}\approx\frac{-1+\sqrt{1+2r_{\rm exp}/a_{\rm exp}}}{r_{\rm
exp}}=0.03999(5)\,{\rm fm}^{-1}~{},$ (9)

where the error comes from the uncertainties of the experimental ERE
parameters. This error turns out to be around a factor of ten greater than the
corrections induced by the ${\cal O}(\gamma^{4}_{v})$ term in Eq. (8). This
virtual binding momentum corresponds to $E_{v}=-0.0663(4)$ MeV.

It is not trivial to extend the notion of compositeness to states other than
bound states, since wave functions derived from poles on the unphysical sheet
are not normalizable and the probabilistic interpretation is lost. They are
not QCD asymptotic states, and thus it seems difficult to argue about wave-
function components. However, one could think of some variation of QCD
parameters, e.g. quark masses, such that these virtual states could become
physical, bound states. From this perspective, it would make sense to
generalize the notion of compositeness. On the other hand, formally relying on
the definition of the field renormalization $Z$ in the nonrelativistic theory
Matuschek  _et al._ (2021), relations between $a$, $r$ and $Z$ can be derived
also for a virtual state with a pole at $k=-i\gamma_{v}$, and they are similar
to those of a bound state. One should simply replace $\gamma_{b}$ with
$-\gamma_{v}$ in $a_{\rm NLO}$ and $r_{\rm NLO}$ given in Eqs. (2) and (5). In
addition, $\gamma_{v}^{\rm NLO}$ should be evaluated using Eq. (9), but with
the NLO ERE parameters. Thus, we can use the definition of Eq. (6) for the
compositeness distribution of a virtual state. The accuracy of the NLO
$\gamma-$expansion $(\gamma_{v}^{2}/m_{\pi}^{2})\sim 0.3\%$ is larger than the
experimental errors on $a_{\rm exp}$ and $\gamma_{v}^{\rm exp}$, and we take
it to set $\Delta\gamma_{v}^{\rm exp}$ and $\Delta a_{\rm exp}$, while we use
the experimental error to fix $\Delta r_{\rm exp}$. Due to the high precision
of the input parameters, very small variations of $Z$ and $\delta r$ produce
quite large changes of ${\cal L}(Z,\delta r)$. In Fig. 3, we show the neperian
logarithm of the compositeness distribution for this case. We see that for
$Z>0.2$, the 2D function ${\cal L}(Z,\delta r)$ takes values larger than 2,
while its minimum values are found for $Z<0.1$. This is consistent with the
general understanding that virtual states are the result of two-particle
interactions, and in this sense can be considered of molecular type Matuschek
_et al._ (2021).

Figure 3: Compositeness distribution for the ${}^{1}{\rm S}_{0}$ nucleon-
nucleon virtual state. Note that the $\ln(1+{\cal L})$ function is displayed
for ${\cal L}<36$.

Figure 4: Left: Distribution of Eq. (6) for different values of the
probability $Z$ of the $T_{cc}^{+}$ state ending in a bare elementary particle
(compact tetraquark), as a function of the NLO contribution $\delta r$ [Eq.
(5a)] to the effective range of the isoscalar $D\bar{D}^{\ast}$ interaction.
Middle and right: $Z$ dependence of $a$ (middle) and $r$ (right) at LO (red,
dashed-dotted lines) and NLO (blue, solid lines), compared with the
phenomenological values (green bands) discussed in Sec. VI.

##  VI An estimation of $T_{cc}^{+}$ compositeness

The $T_{cc}^{+}$ state has been recently discovered by the LHCb collaboration
Aaij  _et al._ (2022a, b) as a prominent peak in the $D^{0}D^{0}\pi^{+}$
spectrum. It is very close to the $D^{\ast+}D^{0}$ threshold, since the
experimental analysis throw $\Delta
M_{T_{cc}^{+}}=M_{T_{cc}^{+}}-m_{D^{\ast+}}-m_{D^{0}}=-273(61)\,\text{keV}$
Aaij  _et al._ (2022a) or $\Delta M_{T_{cc}^{+}}=-360(40)\,\text{keV}$ Aaij
_et al._ (2022b).444Here we do not discuss the width, the importance of which
has been highlighted in the literature, e.g. in Refs. Yan and Valderrama
(2022); Meng  _et al._ (2021); Fleming  _et al._ (2021); Ling  _et al._
(2022); Lin  _et al._ (2022). However, the $D^{\ast 0}D^{+}$ threshold is only
$1.4\,\text{MeV}$ above, and therefore one must consider coupled channels in
order to have a fully accurate description of the state. In this situation one
cannot straightforwardly apply Weinberg’s compositeness criteria, which
requires the coupling of the bound state to a single channel.

The approach should be modified in the presence of close coupled channels that
play an important role on the long-distance dynamics of the state, and as a
consequence, they significantly modify the effective range parameter Matuschek
_et al._ (2021). In Ref. Baru  _et al._ (2022), it is shown for the
$\chi_{c1}(3872)$ and $T_{cc}^{+}$ exotic states that the appearance of a
large and negative effective range, which in the one-channel case would
indicate the dominance of a compact component Weinberg (1965), can be
naturally generated by the coupled-channel dynamics. Hence, in presence of
coupled channels, and before analyzing the estimator of the compositeness
proposed in this work, it would be necessary to correct the effective range by
a term Matuschek  _et al._ (2021); Baru  _et al._ (2022) that stems from
coupled-channel effects, and which clearly needs to be attributed to the
molecular component of the state. Other efforts beyond the one just mentioned
have been devoted to extend the compositeness condition to coupled channels
(see references in Sec. I).

However, in this work we adopt a different perspective and for illustrative
purposes, we have qualitatively applied our generalization of Weinberg
compositeness condition to the $T_{cc}^{+}$ state within a simplified
scenario. We have reduced the coupled-channel problem to a single-channel one
by considering the model of Ref. Albaladejo (2022) in the exact isospin limit,
which we briefly outline below. The latter work performs an S-wave
$D^{\ast+}D^{0}$, $D^{\ast 0}D^{+}$ coupled-channel analysis in terms of two
coupling constants $C_{0,1}$ and an ultraviolet cutoff $\Lambda$. If one takes
common masses for the $D^{(\ast)}$ mesons,
$m_{D^{(\ast)}}=(m_{D^{(\ast)+}}+m_{D^{(\ast)0}})/2$, then the two-channel
problem diagonalizes and one ends up with two independent amplitudes for each
of the definite-isospin ($I=0$ and $I=1$) sectors. Taking this limit, and
using the values for the constant $C_{0}$ fitted in Ref. Albaladejo (2022), we
obtain $a_{\text{ph}}=-5.38(30)\,\text{fm}$ and
$r_{\text{ph}}=0.95(32)\,\text{fm}$ for the isoscalar555We assume that the
$T_{cc}^{+}$ state is mostly an isoscalar state, as suggested by additional
experimental information besides the $D^{0}D^{0}\pi^{+}$ spectrum, see Refs.
Aaij  _et al._ (2022a, b); Albaladejo (2022). scattering length and effective
range, respectively. The central values account for the averages of Eqs.
(3.9a) and (3.9b) in Ref. Albaladejo (2022), computed for two different values
of the cutoff $\Lambda=0.5\,\text{GeV}$ and $1.0\,\text{GeV}$. The quoted
errors represent the addition in quadrature of the statistical error and half
of the dispersion between both determinations. In this isospin limit, the
$T_{cc}^{+}$ binding energy (respect to the average threshold) increases, and
we find $\Delta M_{T_{cc}^{+}}^{\text{ph}}=-833(67)\,\text{keV}$ and
$-856(53)\,\text{keV}$ for $\Lambda=0.5\,\text{GeV}$ and
$\Lambda=1.0\,\text{GeV}$, respectively, which lead to a binding momentum
$\gamma_{b}^{\text{ph}}=40.2(1.7)\,\text{MeV}$. Taking the scale $\beta\simeq
m_{\pi}$, one obtains $(\gamma_{b}^{\text{ph}}/\beta)^{2}\simeq 8\%$, which is
of the order of the uncertainty in the scattering length and smaller than that
of the effective range.

For the $T_{cc}^{+}$ and contrary to the case of the deuteron, we are not
determining the binding momentum, the scattering length and the effective
range from an experimental analysis, but instead from a phenomenological model
Albaladejo (2022). For this reason, we have denoted these quantities as
$\gamma_{b}^{\text{ph}},a_{\text{ph}}$ and $r_{\text{ph}}$, respectively. It
is worth pointing out that two recent LQCD simulations Padmanath and Prelovsek
(2022); Chen  _et al._ (2022) have also computed the scattering length and
effective range for $S$-wave $I=0$ $D\bar{D}^{\ast}$ scattering, and have
found a value $r_{0}\simeq 1\,\text{fm}$, similar to the one obtained in Ref.
Albaladejo (2022), to be used in our work. We also note that, while here
$r_{\text{ph}}$ is small and positive, the experimental value for the
effective range obtained in Ref. Aaij  _et al._ (2022b) is large and negative.
However, in the latter work, the negative value is a built-in property of the
model used in the analysis. More importantly, it is obtained in a coupled
channel scheme, whereas $r_{\text{ph}}$ obtained here refers to the single
channel case in the exact isospin limit. Therefore, the comparison between the
two values is not meaningful. Further discussions on this subject can be found
in Refs. Matuschek  _et al._ (2021); Du  _et al._ (2022); Baru  _et al._
(2022).

Nevertheless, we should mention that the results of Ref. Albaladejo (2022) are
consistent with those found in the state of the art work of Ref. Du  _et al._
(2022), where i) the $D^{\ast+}D^{0}$, $D^{\ast 0}D^{+}$ coupled-channel
dynamics, ii) long-range interactions provided by the one pion exchange
mechanism and iii) effects from the three-body $DD\pi$ thresholds, which lie
very close to and below the two-body $D^{\ast}D$ ones, are accurately taken
into account. Indeed, the parameters of the effective range expansion from the
low-energy scattering amplitude are also reliably extracted in Ref. Du  _et
al._ (2022), and it turns out that both scattering length and effective range
are compatible, within uncertainties, with $a_{\text{ph}}$ and $r_{\text{ph}}$
used here, and obtained from the analysis of Ref. Albaladejo (2022). This is
so once the the largest contribution to the effective range, originated from
isospin breaking related to the $D^{(*)}$-meson mass differences, that is,
from the coupling of $D^{\ast+}D^{0}$ to the slightly higher $D^{\ast 0}D^{+}$
channel is discounted666Such correction is so large that it leads to a
positive residual finite range ($\approx r_{\text{ph}}\sim 1$ fm), which is
the quantity entering in the Weinberg’s relations. (see Table IV and Eq. (40)
of Ref. Du  _et al._ (2022)), as mentioned in the beginning of the section.
Therefore, though the results shown below for the compositeness of the
$T_{cc}^{+}$, obtained within the model of Ref. [76] in the isospin limit,
should be considered as only qualitative ones, they might be sufficiently
realistic to illustrate the performance of the estimator of the compositeness
proposed in this work.

Bearing all these caveats in mind, we have computed the distribution of Eq.
(6), using the numerical values discussed above for $a_{\text{ph}}$,
$r_{\text{ph}}$ and $\gamma_{b}^{\text{ph}}$. The results are shown in Fig. 4,
similarly as done for the deuteron in Fig. 1. As can be seen in the leftmost
panel, $\mathcal{L}(Z,\delta r)<0.5$ for $Z<0.2$ with
$\gamma_{b}^{\text{ph}}/\beta\sim 0.3$, and the minimum of
$\mathcal{L}(Z,\delta r)$ is quite compatible with $Z=0$. Indeed in the middle
panel, for $\delta r=0.95\,\text{fm}(=r_{\text{ph}})$ such that
$r_{\text{NLO}}(Z=0,\delta r=0.95\,\text{fm})=r_{\text{ph}}$, it can be seen
that the scattering length $a_{\text{NLO}}$ coincides with $a_{\text{ph}}$ at
$Z=0$. This analysis supports a molecular picture for the $T_{cc}^{+}$ state,
as previous works have also concluded Feijoo  _et al._ (2021); Du  _et al._
(2022), and in particular the model of Ref. Albaladejo (2022), whose
predictions for the binding momentum, scattering length and effective range
have been used here.

##  VII Summary and discussion

We have discussed a model-independent estimator of the likelihood of the
compositeness of a shallow S-wave bound or virtual state. It relies only on
the proximity of the energy of the state to the two-hadron threshold to which
it significantly couples and on the experimental scattering length and
effective range low energy parameters. The approach is based on NLO Weinberg’s
relations and it is self-consistent as long as the obtained $Z$ is small of
the order of ${\cal O}(\gamma_{b}/\beta)$. To systematically study systems
where $Z$ could take larger values, the order ${\cal O}(k^{4})$ or higher
terms in the ERE would be required. This is because in those cases, it would
be necessary to include additional details to further constrain the short-
range structure of the wave-function. We have analysed the case of the
deuteron (Sec. III), the exotic $D^{\ast}_{s0}(2317)^{\pm}$ resonance (Sec.
IV), and the ${}^{1}{\rm S}_{0}$ nucleon-nucleon virtual state (Sec. V), and
found strong support to the molecular interpretation in all cases.
Nevertheless, results are less conclusive for the $D^{\ast}_{s0}(2317)^{\pm}$
due to the large size of the power-counting parameter $\gamma_{b}/\beta\sim
0.6$ and therefore the NLO approach employed here is at the limit of its
applicability.

As discussed in Sec. VI, the Weinberg compositeness criteria as well as the
extension presented here are only valid for the case of a single channel, and
the approach needs to be modified when coupled channels are present Matuschek
_et al._ (2021); Baru  _et al._ (2022). To avoid this problem, in Sec. VI we
have discussed the case of the $T_{cc}^{+}$, by reducing the coupled-channel
($D^{\ast+}D^{0}$-$D^{\ast 0}D^{+}$) problem to a single-channel one in the
isospin limit using the phenomenological model of Ref. Albaladejo (2022).
Applying the generalization of Weinberg’s compositeness condition proposed in
the present work, we have shown that the parameters are compatible with a
molecular interpretation of the $T_{cc}^{+}$ state. These results are,
however, only qualitative and for illustrating purposes, since we have not
considered the complexity of the very close thresholds $D^{0}D^{\ast+}$ and
$D^{+}D^{\ast 0}$, which could make more difficult to appreciate the main
ingredients of the approach (expansion) derived in this paper.

## Acknowledgements

We warmly thank E. Ruiz-Arriola for useful discussions. This research has been
supported by the Spanish Ministerio de Ciencia e Innovación (MICINN) and the
European Regional Development Fund (ERDF) under contract PID2020-112777GB-I00,
the EU STRONG-2020 project under the program H2020-INFRAIA-2018-1, grant
agreement no. 824093 and by Generalitat Valenciana under contract
PROMETEO/2020/023. M. A. is supported by Generalitat Valenciana under Grant
No. CIDEGENT/2020/002.

## References

  * Weinberg (1965) S. Weinberg, Phys. Rev. 137, B672 (1965).
  * Gell-Mann (1964) M. Gell-Mann, Phys. Lett. 8, 214 (1964).
  * Zweig (1964a) G. Zweig,  _An SU 3 model for strong interaction symmetry and its breaking; Version 1_, Tech. Rep. CERN-TH-401 (1964).
  * Zweig (1964b) G. Zweig,  _An SU 3 model for strong interaction symmetry and its breaking; Version 2_, Tech. Rep. CERN-TH-412 (1964).
  * Godfrey and Isgur (1985) S. Godfrey and N. Isgur, Phys. Rev. D 32, 189 (1985).
  * Capstick and Isgur (1986) S. Capstick and N. Isgur, Phys. Rev. D 34, 2809 (1986).
  * Zyla  _et al._ (2020) P. A. Zyla _et al._ (Particle Data Group), PTEP 2020, 083C01 (2020).
  * Choi  _et al._ (2003) S. K. Choi _et al._ (Belle), Phys. Rev. Lett. 91, 262001 (2003), arXiv:hep-ex/0309032 .
  * Bondar  _et al._ (2012) A. Bondar _et al._ (Belle), Phys. Rev. Lett. 108, 122001 (2012), arXiv:1110.2251 [hep-ex] .
  * Ablikim  _et al._ (2013a) M. Ablikim _et al._ (BESIII), Phys. Rev. Lett. 110, 252001 (2013a), arXiv:1303.5949 [hep-ex] .
  * Liu  _et al._ (2013a) Z. Q. Liu _et al._ (Belle), Phys. Rev. Lett. 110, 252002 (2013a), [Erratum: Phys.Rev.Lett. 111, 019901 (2013)], arXiv:1304.0121 [hep-ex] .
  * Ablikim  _et al._ (2014) M. Ablikim _et al._ (BESIII), Phys. Rev. Lett. 112, 132001 (2014), arXiv:1308.2760 [hep-ex] .
  * Ablikim  _et al._ (2013b) M. Ablikim _et al._ (BESIII), Phys. Rev. Lett. 111, 242001 (2013b), arXiv:1309.1896 [hep-ex] .
  * Aaij  _et al._ (2015a) R. Aaij _et al._ (LHCb), Phys. Rev. Lett. 115, 072001 (2015a), arXiv:1507.03414 [hep-ex] .
  * Aaij  _et al._ (2019) R. Aaij _et al._ (LHCb), Phys. Rev. Lett. 122, 222001 (2019), arXiv:1904.03947 [hep-ex] .
  * Ablikim  _et al._ (2021) M. Ablikim _et al._ (BESIII), Phys. Rev. Lett. 126, 102001 (2021), arXiv:2011.07855 [hep-ex] .
  * Aaij  _et al._ (2022a) R. Aaij _et al._ (LHCb), Nature Commun. 13, 3351 (2022a), arXiv:2109.01056 [hep-ex] .
  * Aaij  _et al._ (2022b) R. Aaij _et al._ (LHCb), Nature Phys. 18, 751 (2022b), arXiv:2109.01038 [hep-ex] .
  * Aubert  _et al._ (2003) B. Aubert _et al._ (BaBar), Phys. Rev. Lett. 90, 242001 (2003), arXiv:hep-ex/0304021 .
  * Abe  _et al._ (2004) K. Abe _et al._ (Belle), Phys. Rev. D 69, 112002 (2004), arXiv:hep-ex/0307021 .
  * Edwards  _et al._ (1995) K. W. Edwards _et al._ (CLEO), Phys. Rev. Lett. 74, 3331 (1995).
  * Abazov  _et al._ (2007) V. M. Abazov _et al._ (D0), Phys. Rev. Lett. 99, 172001 (2007), arXiv:0705.3229 [hep-ex] .
  * Aaltonen  _et al._ (2014) T. A. Aaltonen _et al._ (CDF), Phys. Rev. D 90, 012013 (2014), arXiv:1309.5961 [hep-ex] .
  * Aaij  _et al._ (2015b) R. Aaij _et al._ (LHCb), JHEP 04, 024 (2015b), arXiv:1502.02638 [hep-ex] .
  * Aaij  _et al._ (2018) R. Aaij _et al._ (LHCb), Phys. Rev. Lett. 121, 072002 (2018), arXiv:1805.09418 [hep-ex] .
  * Aaij  _et al._ (2021) R. Aaij _et al._ (LHCb), Phys. Rev. D 103, 012004 (2021), arXiv:2010.14485 [hep-ex] .
  * Barnes  _et al._ (2003) T. Barnes, F. E. Close, and H. J. Lipkin, Phys. Rev. D 68, 054006 (2003), arXiv:hep-ph/0305025 .
  * van Beveren and Rupp (2003) E. van Beveren and G. Rupp, Phys. Rev. Lett. 91, 012003 (2003), arXiv:hep-ph/0305035 .
  * Kolomeitsev and Lutz (2004) E. E. Kolomeitsev and M. F. M. Lutz, Phys. Lett. B 582, 39 (2004), arXiv:hep-ph/0307133 .
  * Szczepaniak (2003) A. P. Szczepaniak, Phys. Lett. B 567, 23 (2003), arXiv:hep-ph/0305060 .
  * Voloshin (2004) M. B. Voloshin, Phys. Lett. B 579, 316 (2004), arXiv:hep-ph/0309307 .
  * Close and Page (2004) F. E. Close and P. R. Page, Phys. Lett. B 578, 119 (2004), arXiv:hep-ph/0309253 .
  * Swanson (2004) E. S. Swanson, Phys. Lett. B 588, 189 (2004), arXiv:hep-ph/0311229 .
  * Tornqvist (2004) N. A. Tornqvist, Phys. Lett. B 590, 209 (2004), arXiv:hep-ph/0402237 .
  * Guo  _et al._ (2006) F.-K. Guo, P.-N. Shen, H.-C. Chiang, R.-G. Ping, and B.-S. Zou, Phys. Lett. B 641, 278 (2006), arXiv:hep-ph/0603072 .
  * Gamermann  _et al._ (2007) D. Gamermann, E. Oset, D. Strottman, and M. J. Vicente Vacas, Phys. Rev. D 76, 074016 (2007), arXiv:hep-ph/0612179 .
  * Flynn and Nieves (2007) J. M. Flynn and J. Nieves, Phys. Rev. D 75, 074024 (2007), arXiv:hep-ph/0703047 .
  * Faessler  _et al._ (2007) A. Faessler, T. Gutsche, V. E. Lyubovitskij, and Y.-L. Ma, Phys. Rev. D 76, 014005 (2007), arXiv:0705.0254 [hep-ph] .
  * Lutz and Soyeur (2008) M. F. M. Lutz and M. Soyeur, Nucl. Phys. A 813, 14 (2008), arXiv:0710.1545 [hep-ph] .
  * Dong  _et al._ (2009) Y. Dong, A. Faessler, T. Gutsche, S. Kovalenko, and V. E. Lyubovitskij, Phys. Rev. D 79, 094013 (2009), arXiv:0903.5416 [hep-ph] .
  * Gamermann and Oset (2009) D. Gamermann and E. Oset, Phys. Rev. D 80, 014003 (2009), arXiv:0905.0402 [hep-ph] .
  * Wu  _et al._ (2010) J.-J. Wu, R. Molina, E. Oset, and B. S. Zou, Phys. Rev. Lett. 105, 232001 (2010), arXiv:1007.0573 [nucl-th] .
  * Bondar  _et al._ (2011) A. E. Bondar, A. Garmash, A. I. Milstein, R. Mizuk, and M. B. Voloshin, Phys. Rev. D 84, 054010 (2011), arXiv:1105.4473 [hep-ph] .
  * Wang  _et al._ (2011) W. L. Wang, F. Huang, Z. Y. Zhang, and B. S. Zou, Phys. Rev. C 84, 015203 (2011), arXiv:1101.0453 [nucl-th] .
  * Yang  _et al._ (2012) Z.-C. Yang, Z.-F. Sun, J. He, X. Liu, and S.-L. Zhu, Chin. Phys. C 36, 6 (2012), arXiv:1105.2901 [hep-ph] .
  * Garcia-Recio  _et al._ (2009) C. Garcia-Recio, V. K. Magas, T. Mizutani, J. Nieves, A. Ramos, L. L. Salcedo, and L. Tolos, Phys. Rev. D 79, 054004 (2009), arXiv:0807.2969 [hep-ph] .
  * Romanets  _et al._ (2012) O. Romanets, L. Tolos, C. Garcia-Recio, J. Nieves, L. L. Salcedo, and R. G. E. Timmermans, Phys. Rev. D 85, 114032 (2012), arXiv:1202.2239 [hep-ph] .
  * Garcia-Recio  _et al._ (2013a) C. Garcia-Recio, J. Nieves, O. Romanets, L. L. Salcedo, and L. Tolos, Phys. Rev. D 87, 034032 (2013a), arXiv:1210.4755 [hep-ph] .
  * Wu  _et al._ (2012) J.-J. Wu, T. S. H. Lee, and B. S. Zou, Phys. Rev. C 85, 044002 (2012), arXiv:1202.1036 [nucl-th] .
  * Liu  _et al._ (2013b) L. Liu, K. Orginos, F.-K. Guo, C. Hanhart, and U.-G. Meissner, Phys. Rev. D 87, 014508 (2013b), arXiv:1208.4535 [hep-lat] .
  * Garcia-Recio  _et al._ (2013b) C. Garcia-Recio, J. Nieves, O. Romanets, L. L. Salcedo, and L. Tolos, Phys. Rev. D 87, 074034 (2013b), arXiv:1302.6938 [hep-ph] .
  * Wang  _et al._ (2013a) Q. Wang, C. Hanhart, and Q. Zhao, Phys. Rev. Lett. 111, 132003 (2013a), arXiv:1303.6355 [hep-ph] .
  * Guo  _et al._ (2013) F.-K. Guo, C. Hidalgo-Duque, J. Nieves, and M. P. Valderrama, Phys. Rev. D 88, 054007 (2013), arXiv:1303.6608 [hep-ph] .
  * Aceti  _et al._ (2014) F. Aceti, M. Bayar, E. Oset, A. Martinez Torres, K. P. Khemchandani, J. M. Dias, F. S. Navarra, and M. Nielsen, Phys. Rev. D 90, 016003 (2014), arXiv:1401.8216 [hep-ph] .
  * Albaladejo  _et al._ (2016a) M. Albaladejo, F.-K. Guo, C. Hidalgo-Duque, and J. Nieves, Phys. Lett. B 755, 337 (2016a), arXiv:1512.03638 [hep-ph] .
  * Albaladejo  _et al._ (2016b) M. Albaladejo, P. Fernandez-Soler, and J. Nieves, Eur. Phys. J. C 76, 573 (2016b), arXiv:1606.03008 [hep-ph] .
  * Albaladejo  _et al._ (2017) M. Albaladejo, P. Fernandez-Soler, F.-K. Guo, and J. Nieves, Phys. Lett. B 767, 465 (2017), arXiv:1610.06727 [hep-ph] .
  * Nieves  _et al._ (2018) J. Nieves, R. Pavao, and L. Tolos, Eur. Phys. J. C 78, 114 (2018), arXiv:1712.00327 [hep-ph] .
  * Du  _et al._ (2018) M.-L. Du, M. Albaladejo, P. Fernández-Soler, F.-K. Guo, C. Hanhart, U.-G. Meißner, J. Nieves, and D.-L. Yao, Phys. Rev. D 98, 094018 (2018), arXiv:1712.07957 [hep-ph] .
  * Nieves and Pavao (2020) J. Nieves and R. Pavao, Phys. Rev. D 101, 014018 (2020), arXiv:1907.05747 [hep-ph] .
  * Wang  _et al._ (2018) Q. Wang, V. Baru, A. A. Filin, C. Hanhart, A. V. Nefediev, and J. L. Wynen, Phys. Rev. D 98, 074023 (2018), arXiv:1805.07453 [hep-ph] .
  * Karliner and Rosner (2015) M. Karliner and J. L. Rosner, Phys. Rev. Lett. 115, 122001 (2015), arXiv:1506.06386 [hep-ph] .
  * Xiao  _et al._ (2013) C. W. Xiao, J. Nieves, and E. Oset, Phys. Rev. D 88, 056012 (2013), arXiv:1304.5368 [hep-ph] .
  * Nieves  _et al._ (2020) J. Nieves, R. Pavao, and L. Tolos, Eur. Phys. J. C 80, 22 (2020), arXiv:1911.06089 [hep-ph] .
  * Liu  _et al._ (2019a) M.-Z. Liu, Y.-W. Pan, F.-Z. Peng, M. Sánchez Sánchez, L.-S. Geng, A. Hosaka, and M. Pavon Valderrama, Phys. Rev. Lett. 122, 242001 (2019a), arXiv:1903.11560 [hep-ph] .
  * Du  _et al._ (2020) M.-L. Du, V. Baru, F.-K. Guo, C. Hanhart, U.-G. Meißner, J. A. Oller, and Q. Wang, Phys. Rev. Lett. 124, 072001 (2020), arXiv:1910.11846 [hep-ph] .
  * Liu  _et al._ (2019b) M.-Z. Liu, T.-W. Wu, M. Pavon Valderrama, J.-J. Xie, and L.-S. Geng, Phys. Rev. D 99, 094018 (2019b), arXiv:1902.03044 [hep-ph] .
  * Lu  _et al._ (2015) J.-X. Lu, Y. Zhou, H.-X. Chen, J.-J. Xie, and L.-S. Geng, Phys. Rev. D 92, 014036 (2015), arXiv:1409.3133 [hep-ph] .
  * Dong  _et al._ (2021a) X.-K. Dong, F.-K. Guo, and B.-S. Zou, Commun. Theor. Phys. 73, 125201 (2021a), arXiv:2108.02673 [hep-ph] .
  * Du  _et al._ (2021) M.-L. Du, V. Baru, F.-K. Guo, C. Hanhart, U.-G. Meißner, J. A. Oller, and Q. Wang, JHEP 08, 157 (2021), arXiv:2102.07159 [hep-ph] .
  * Du  _et al._ (2022) M.-L. Du, V. Baru, X.-K. Dong, A. Filin, F.-K. Guo, C. Hanhart, A. Nefediev, J. Nieves, and Q. Wang, Phys. Rev. D 105, 014024 (2022), arXiv:2110.13765 [hep-ph] .
  * Dong  _et al._ (2021b) X.-K. Dong, F.-K. Guo, and B.-S. Zou, Progr. Phys. 41, 65 (2021b), arXiv:2101.01021 [hep-ph] .
  * Guo  _et al._ (2018) F.-K. Guo, C. Hanhart, U.-G. Meißner, Q. Wang, Q. Zhao, and B.-S. Zou, Rev. Mod. Phys. 90, 015004 (2018), [Erratum: Rev.Mod.Phys. 94, 029901 (2022)], arXiv:1705.00141 [hep-ph] .
  * Guo and Oller (2021) Z.-H. Guo and J. A. Oller, Phys. Rev. D 103, 054021 (2021), arXiv:2012.11904 [hep-ph] .
  * Feijoo  _et al._ (2021) A. Feijoo, W. H. Liang, and E. Oset, Phys. Rev. D 104, 114015 (2021), arXiv:2108.02730 [hep-ph] .
  * Albaladejo (2022) M. Albaladejo, Phys. Lett. B 829, 137052 (2022), arXiv:2110.02944 [hep-ph] .
  * Esposito  _et al._ (2017) A. Esposito, A. Pilloni, and A. D. Polosa, Phys. Rept. 668, 1 (2017), arXiv:1611.07920 [hep-ph] .
  * Maiani  _et al._ (2005) L. Maiani, F. Piccinini, A. D. Polosa, and V. Riquer, Phys. Rev. D 71, 014028 (2005), arXiv:hep-ph/0412098 .
  * Maiani  _et al._ (2014) L. Maiani, F. Piccinini, A. D. Polosa, and V. Riquer, Phys. Rev. D 89, 114010 (2014), arXiv:1405.1551 [hep-ph] .
  * Braaten (2013) E. Braaten, Phys. Rev. Lett. 111, 162003 (2013), arXiv:1305.6905 [hep-ph] .
  * Dias  _et al._ (2013) J. M. Dias, F. S. Navarra, M. Nielsen, and C. M. Zanetti, Phys. Rev. D 88, 016004 (2013), arXiv:1304.6433 [hep-ph] .
  * Qiao and Tang (2014) C.-F. Qiao and L. Tang, Eur. Phys. J. C 74, 3122 (2014), arXiv:1307.6654 [hep-ph] .
  * Deng  _et al._ (2014) C. Deng, J. Ping, and F. Wang, Phys. Rev. D 90, 054009 (2014), arXiv:1402.0777 [hep-ph] .
  * Ali  _et al._ (2012) A. Ali, C. Hambrock, and W. Wang, Phys. Rev. D 85, 054011 (2012), arXiv:1110.1333 [hep-ph] .
  * Cheng and Hou (2003) H.-Y. Cheng and W.-S. Hou, Phys. Lett. B 566, 193 (2003), arXiv:hep-ph/0305038 .
  * Terasaki (2003) K. Terasaki, Phys. Rev. D 68, 011501 (2003), arXiv:hep-ph/0305213 .
  * Dmitrasinovic (2005) V. Dmitrasinovic, Phys. Rev. Lett. 94, 162002 (2005).
  * Bracco  _et al._ (2005) M. E. Bracco, A. Lozea, R. D. Matheus, F. S. Navarra, and M. Nielsen, Phys. Lett. B 624, 217 (2005), arXiv:hep-ph/0503137 .
  * Wang and Wan (2006a) Z.-G. Wang and S.-L. Wan, Nucl. Phys. A 778, 22 (2006a), arXiv:hep-ph/0602080 .
  * Chen and Li (2004) Y.-Q. Chen and X.-Q. Li, Phys. Rev. Lett. 93, 232001 (2004), arXiv:hep-ph/0407062 .
  * Kim  _et al._ (2022) Y. Kim, M. Oka, and K. Suzuki, Phys. Rev. D 105, 074021 (2022), arXiv:2202.06520 [hep-ph] .
  * Maiani  _et al._ (2015) L. Maiani, A. D. Polosa, and V. Riquer, Phys. Lett. B 749, 289 (2015), arXiv:1507.04980 [hep-ph] .
  * Lebed (2015) R. F. Lebed, Phys. Lett. B 749, 454 (2015), arXiv:1507.05867 [hep-ph] .
  * Li  _et al._ (2015) G.-N. Li, X.-G. He, and M. He, JHEP 12, 128 (2015), arXiv:1507.08252 [hep-ph] .
  * Ghosh  _et al._ (2017) R. Ghosh, A. Bhattacharya, and B. Chakrabarti, Phys. Part. Nucl. Lett. 14, 550 (2017), arXiv:1508.00356 [hep-ph] .
  * Wang (2016) Z.-G. Wang, Eur. Phys. J. C 76, 70 (2016), arXiv:1508.01468 [hep-ph] .
  * Zhu and Qiao (2016) R. Zhu and C.-F. Qiao, Phys. Lett. B 756, 259 (2016), arXiv:1510.08693 [hep-ph] .
  * Richard  _et al._ (2017) J. M. Richard, A. Valcarce, and J. Vijande, Phys. Lett. B 774, 710 (2017), arXiv:1710.08239 [hep-ph] .
  * Hiyama  _et al._ (2018) E. Hiyama, A. Hosaka, M. Oka, and J.-M. Richard, Phys. Rev. C 98, 045208 (2018), arXiv:1803.11369 [nucl-th] .
  * Barnes and Godfrey (2004) T. Barnes and S. Godfrey, Phys. Rev. D 69, 054008 (2004), arXiv:hep-ph/0311162 .
  * Colangelo and De Fazio (2003) P. Colangelo and F. De Fazio, Phys. Lett. B 570, 180 (2003), arXiv:hep-ph/0305140 .
  * Cahn and Jackson (2003) R. N. Cahn and J. D. Jackson, Phys. Rev. D 68, 037502 (2003), arXiv:hep-ph/0305012 .
  * Browder  _et al._ (2004) T. E. Browder, S. Pakvasa, and A. A. Petrov, Phys. Lett. B 578, 365 (2004), arXiv:hep-ph/0307054 .
  * Godfrey (2003) S. Godfrey, Phys. Lett. B 568, 254 (2003), arXiv:hep-ph/0305122 .
  * Narison (2005) S. Narison, Phys. Lett. B 605, 319 (2005), arXiv:hep-ph/0307248 .
  * Bardeen  _et al._ (2003) W. A. Bardeen, E. J. Eichten, and C. T. Hill, Phys. Rev. D 68, 054024 (2003), arXiv:hep-ph/0305049 .
  * Nowak  _et al._ (2004) M. A. Nowak, M. Rho, and I. Zahed, Acta Phys. Polon. B 35, 2377 (2004), arXiv:hep-ph/0307102 .
  * Dai  _et al._ (2003) Y.-B. Dai, C.-S. Huang, C. Liu, and S.-L. Zhu, Phys. Rev. D 68, 114011 (2003), arXiv:hep-ph/0306274 .
  * Lee  _et al._ (2007) I. W. Lee, T. Lee, D. P. Min, and B.-Y. Park, Eur. Phys. J. C 49, 737 (2007), arXiv:hep-ph/0412210 .
  * Lakhina and Swanson (2007) O. Lakhina and E. S. Swanson, Phys. Lett. B 650, 159 (2007), arXiv:hep-ph/0608011 .
  * Segovia  _et al._ (2013) J. Segovia, D. R. Entem, F. Fernandez, and E. Hernandez, Int. J. Mod. Phys. E 22, 1330026 (2013), arXiv:1309.6926 [hep-ph] .
  * Yoshida  _et al._ (2015) T. Yoshida, E. Hiyama, A. Hosaka, M. Oka, and K. Sadato, Phys. Rev. D 92, 114029 (2015), arXiv:1510.01067 [hep-ph] .
  * Ortega  _et al._ (2016) P. G. Ortega, J. Segovia, D. R. Entem, and F. Fernandez, Phys. Rev. D 94, 074037 (2016), arXiv:1603.07000 [hep-ph] .
  * Close and Godfrey (2003) F. E. Close and S. Godfrey, Phys. Lett. B 574, 210 (2003), arXiv:hep-ph/0305285 .
  * Li (2005) B. A. Li, Phys. Lett. B 605, 306 (2005), arXiv:hep-ph/0410264 .
  * Wang and Wan (2006b) Z. G. Wang and S. L. Wan, Phys. Rev. D 73, 094020 (2006b), arXiv:hep-ph/0603007 .
  * Dubynskiy and Voloshin (2008) S. Dubynskiy and M. B. Voloshin, Phys. Lett. B 666, 344 (2008), arXiv:0803.2224 [hep-ph] .
  * Voloshin (2013) M. B. Voloshin, Phys. Rev. D 87, 091501 (2013), arXiv:1304.0380 [hep-ph] .
  * Fernández-Ramírez  _et al._ (2019) C. Fernández-Ramírez, A. Pilloni, M. Albaladejo, A. Jackura, V. Mathieu, M. Mikhasenko, J. A. Silva-Castro, and A. P. Szczepaniak (JPAC), Phys. Rev. Lett. 123, 092001 (2019), arXiv:1904.10021 [hep-ph] .
  * Pilloni  _et al._ (2017) A. Pilloni, C. Fernandez-Ramirez, A. Jackura, V. Mathieu, M. Mikhasenko, J. Nys, and A. P. Szczepaniak (JPAC), Phys. Lett. B 772, 200 (2017), arXiv:1612.06490 [hep-ph] .
  * Wang  _et al._ (2013b) Q. Wang, C. Hanhart, and Q. Zhao, Phys. Lett. B 725, 106 (2013b), arXiv:1305.1997 [hep-ph] .
  * Chen  _et al._ (2013) D.-Y. Chen, X. Liu, and T. Matsuki, Phys. Rev. D 88, 036008 (2013), arXiv:1304.5845 [hep-ph] .
  * Swanson (2015) E. S. Swanson, Phys. Rev. D 91, 034009 (2015), arXiv:1409.3291 [hep-ph] .
  * Guo  _et al._ (2015a) F.-K. Guo, C. Hanhart, Q. Wang, and Q. Zhao, Phys. Rev. D 91, 051504 (2015a), arXiv:1411.5584 [hep-ph] .
  * Guo  _et al._ (2015b) F.-K. Guo, U.-G. Meißner, W. Wang, and Z. Yang, Phys. Rev. D 92, 071502 (2015b), arXiv:1507.04950 [hep-ph] .
  * Liu  _et al._ (2016) X.-H. Liu, Q. Wang, and Q. Zhao, Phys. Lett. B 757, 231 (2016), arXiv:1507.05359 [hep-ph] .
  * Szczepaniak (2015) A. P. Szczepaniak, Phys. Lett. B 747, 410 (2015), arXiv:1501.01691 [hep-ph] .
  * Ikeda  _et al._ (2016) Y. Ikeda, S. Aoki, T. Doi, S. Gongyo, T. Hatsuda, T. Inoue, T. Iritani, N. Ishii, K. Murano, and K. Sasaki (HAL QCD), Phys. Rev. Lett. 117, 242001 (2016), arXiv:1602.03465 [hep-lat] .
  * Bayar  _et al._ (2016) M. Bayar, F. Aceti, F.-K. Guo, and E. Oset, Phys. Rev. D 94, 074039 (2016), arXiv:1609.04133 [hep-ph] .
  * Guo  _et al._ (2020) F.-K. Guo, X.-H. Liu, and S. Sakai, Prog. Part. Nucl. Phys. 112, 103757 (2020), arXiv:1912.07030 [hep-ph] .
  * Lebed  _et al._ (2017) R. F. Lebed, R. E. Mitchell, and E. S. Swanson, Prog. Part. Nucl. Phys. 93, 143 (2017), arXiv:1610.04528 [hep-ph] .
  * Chen  _et al._ (2017) H.-X. Chen, W. Chen, X. Liu, Y.-R. Liu, and S.-L. Zhu, Rept. Prog. Phys. 80, 076201 (2017), arXiv:1609.08928 [hep-ph] .
  * Ali  _et al._ (2017) A. Ali, J. S. Lange, and S. Stone, Prog. Part. Nucl. Phys. 97, 123 (2017), arXiv:1706.00610 [hep-ph] .
  * Karliner  _et al._ (2018) M. Karliner, J. L. Rosner, and T. Skwarnicki, Ann. Rev. Nucl. Part. Sci. 68, 17 (2018), arXiv:1711.10626 [hep-ph] .
  * Olsen  _et al._ (2018) S. L. Olsen, T. Skwarnicki, and D. Zieminska, Rev. Mod. Phys. 90, 015003 (2018), arXiv:1708.04012 [hep-ph] .
  * Brambilla  _et al._ (2020) 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] .
  * Matuschek  _et al._ (2021) I. Matuschek, V. Baru, F.-K. Guo, and C. Hanhart, Eur. Phys. J. A 57, 101 (2021), arXiv:2007.05329 [hep-ph] .
  * Esposito  _et al._ (2022) A. Esposito, L. Maiani, A. Pilloni, A. D. Polosa, and V. Riquer, Phys. Rev. D 105, L031503 (2022), arXiv:2108.11413 [hep-ph] .
  * Li  _et al._ (2022) Y. Li, F.-K. Guo, J.-Y. Pang, and J.-J. Wu, Phys. Rev. D 105, L071502 (2022), arXiv:2110.02766 [hep-ph] .
  * Song  _et al._ (2022) J. Song, L. R. Dai, and E. Oset, (2022), arXiv:2201.04414 [hep-ph] .
  * Baru  _et al._ (2004) V. Baru, J. Haidenbauer, C. Hanhart, Y. Kalashnikova, and A. E. Kudryavtsev, Phys. Lett. B 586, 53 (2004), arXiv:hep-ph/0308129 .
  * Gamermann  _et al._ (2010) D. Gamermann, J. Nieves, E. Oset, and E. Ruiz Arriola, Phys. Rev. D 81, 014029 (2010), arXiv:0911.4407 [hep-ph] .
  * Baru  _et al._ (2010) V. Baru, C. Hanhart, Y. S. Kalashnikova, A. E. Kudryavtsev, and A. V. Nefediev, Eur. Phys. J. A 44, 93 (2010), arXiv:1001.0369 [hep-ph] .
  * Hanhart  _et al._ (2011) C. Hanhart, Y. S. Kalashnikova, and A. V. Nefediev, Eur. Phys. J. A 47, 101 (2011), arXiv:1106.1185 [hep-ph] .
  * Aceti and Oset (2012) F. Aceti and E. Oset, Phys. Rev. D 86, 014012 (2012), arXiv:1202.4607 [hep-ph] .
  * Hyodo  _et al._ (2012) T. Hyodo, D. Jido, and A. Hosaka, Phys. Rev. C 85, 015201 (2012), arXiv:1108.5524 [nucl-th] .
  * Sekihara  _et al._ (2015) T. Sekihara, T. Hyodo, and D. Jido, PTEP 2015, 063D04 (2015), arXiv:1411.2308 [hep-ph] .
  * Kamiya and Hyodo (2016) Y. Kamiya and T. Hyodo, Phys. Rev. C 93, 035203 (2016), arXiv:1509.00146 [hep-ph] .
  * Garcia-Recio  _et al._ (2015) C. Garcia-Recio, C. Hidalgo-Duque, J. Nieves, L. L. Salcedo, and L. Tolos, Phys. Rev. D 92, 034011 (2015), arXiv:1506.04235 [hep-ph] .
  * Guo and Oller (2016) Z.-H. Guo and J. A. Oller, Phys. Rev. D 93, 096001 (2016), arXiv:1508.06400 [hep-ph] .
  * Kamiya and Hyodo (2017) Y. Kamiya and T. Hyodo, PTEP 2017, 023D02 (2017), arXiv:1607.01899 [hep-ph] .
  * Sekihara (2017) T. Sekihara, Phys. Rev. C 95, 025206 (2017), arXiv:1609.09496 [quant-ph] .
  * Oller (2018) J. A. Oller, Annals Phys. 396, 429 (2018), arXiv:1710.00991 [hep-ph] .
  * Kinugawa and Hyodo (2022a) T. Kinugawa and T. Hyodo, EPJ Web Conf. 262, 01019 (2022a), arXiv:2112.00249 [hep-ph] .
  * Sazdjian (2022) H. Sazdjian, Symmetry 14, 515 (2022), arXiv:2202.01081 [hep-ph] .
  * Bruns (2022) P. C. Bruns, (2022), arXiv:2203.16909 [hep-ph] .
  * Kinugawa and Hyodo (2022b) T. Kinugawa and T. Hyodo, Phys. Rev. C 106, 015205 (2022b), arXiv:2205.08470 [hep-ph] .
  * Weinberg (1963a) S. Weinberg, Phys. Rev. 130, 776 (1963a).
  * Weinberg (1963b) S. Weinberg, Phys. Rev. 131, 440 (1963b).
  * Albaladejo  _et al._ (2021) M. Albaladejo, J. M. Nieves, and L. Tolos, Phys. Rev. C 104, 035203 (2021), arXiv:2102.08589 [hep-ph] .
  * Briceño  _et al._ (2021) R. A. Briceño, A. W. Jackura, F. G. Ortega-Gama, and K. H. Sherman, Phys. Rev. D 103, 114512 (2021), arXiv:2012.13338 [hep-lat] .
  * Albaladejo and Oller (2012) M. Albaladejo and J. A. Oller, Phys. Rev. D 86, 034003 (2012), arXiv:1205.6606 [hep-ph] .
  * Van Der Leun and Alderliesten (1982) C. Van Der Leun and C. Alderliesten, Nucl. Phys. A 380, 261 (1982).
  * Navarro Pérez  _et al._ (2016) R. Navarro Pérez, J. E. Amaro, and E. Ruiz Arriola, J. Phys. G 43, 114001 (2016), arXiv:1410.8097 [nucl-th] .
  * Ruiz Arriola  _et al._ (2020) E. Ruiz Arriola, J. E. Amaro, and R. Navarro Pérez, Front. in Phys. 8, 1 (2020), arXiv:1911.09637 [nucl-th] .
  * Fayyazuddin and Riazuddin (2004) Fayyazuddin and Riazuddin, Phys. Rev. D 69, 114008 (2004), arXiv:hep-ph/0309283 .
  * Sadzikowski (2004) M. Sadzikowski, Phys. Lett. B 579, 39 (2004), arXiv:hep-ph/0307084 .
  * Guo  _et al._ (2008) F.-K. Guo, C. Hanhart, S. Krewald, and U.-G. Meissner, Phys. Lett. B 666, 251 (2008), arXiv:0806.3374 [hep-ph] .
  * Mohler  _et al._ (2013) D. Mohler, C. B. Lang, L. Leskovec, S. Prelovsek, and R. M. Woloshyn, Phys. Rev. Lett. 111, 222001 (2013), arXiv:1308.3175 [hep-lat] .
  * Lang  _et al._ (2014) C. B. Lang, L. Leskovec, D. Mohler, S. Prelovsek, and R. M. Woloshyn, Phys. Rev. D 90, 034510 (2014), arXiv:1403.8103 [hep-lat] .
  * Martínez Torres  _et al._ (2015) A. Martínez Torres, E. Oset, S. Prelovsek, and A. Ramos, JHEP 05, 153 (2015), arXiv:1412.1706 [hep-lat] .
  * Bali  _et al._ (2017) G. S. Bali, S. Collins, A. Cox, and A. Schäfer, Phys. Rev. D 96, 074501 (2017), arXiv:1706.01247 [hep-lat] .
  * Albaladejo  _et al._ (2018) M. Albaladejo, P. Fernandez-Soler, J. Nieves, and P. G. Ortega, Eur. Phys. J. C 78, 722 (2018), arXiv:1805.07104 [hep-ph] .
  * Albaladejo  _et al._ (2016c) M. Albaladejo, D. Jido, J. Nieves, and E. Oset, Eur. Phys. J. C 76, 300 (2016c), arXiv:1604.01193 [hep-ph] .
  * Yan and Valderrama (2022) M.-J. Yan and M. P. Valderrama, Phys. Rev. D 105, 014007 (2022), arXiv:2108.04785 [hep-ph] .
  * Meng  _et al._ (2021) L. Meng, G.-J. Wang, B. Wang, and S.-L. Zhu, Phys. Rev. D 104, 051502 (2021), arXiv:2107.14784 [hep-ph] .
  * Fleming  _et al._ (2021) S. Fleming, R. Hodges, and T. Mehen, Phys. Rev. D 104, 116010 (2021), arXiv:2109.02188 [hep-ph] .
  * Ling  _et al._ (2022) X.-Z. Ling, M.-Z. Liu, L.-S. Geng, E. Wang, and J.-J. Xie, Phys. Lett. B 826, 136897 (2022), arXiv:2108.00947 [hep-ph] .
  * Lin  _et al._ (2022) Z.-Y. Lin, J.-B. Cheng, and S.-L. Zhu, (2022), arXiv:2205.14628 [hep-ph] .
  * Baru  _et al._ (2022) V. Baru, X.-K. Dong, M.-L. Du, A. Filin, F.-K. Guo, C. Hanhart, A. Nefediev, J. Nieves, and Q. Wang, Phys. Lett. B 833, 137290 (2022), arXiv:2110.07484 [hep-ph] .
  * Padmanath and Prelovsek (2022) M. Padmanath and S. Prelovsek, Phys. Rev. Lett. 129, 032002 (2022), arXiv:2202.10110 [hep-lat] .
  * Chen  _et al._ (2022) S. Chen, C. Shi, Y. Chen, M. Gong, Z. Liu, W. Sun, and R. Zhang, (2022), arXiv:2206.06185 [hep-lat] .