treated by a perturbational; properties of the other channels will not change
much if this $\chi_{c1}\pi^{0}$ channel is introduced. Moreover, since its
threshold is lower than the $X$(3872) by 230 MeV, it will be necessary to
consider the pion radiation from the $D^{(*)}\bar{D}{}^{(*)}$ components to
obtain the $\chi_{c1}\pi^{0}$ fraction. Here the model contains only the
relative $S$-wave hadron systems which have thresholds close to each other.

For the discussions of decay properties here, it is sufficient to consider the
formation of a loosely bound $D\bar{D}^{*}$ states, which couple to the
$c\bar{c}$ and to the $J\psi\rho$ and $J\psi\omega$ with finite decay widths
for $\rho$ and $\omega$. We assume effective couplings between $c\bar{c}$ and
$D\bar{{D}}{}^{*}$, which gives the attraction as we discussed in the previous
section, and between $D\bar{{D}}{}^{*}$ and $J\\!/\\!\psi\omega(\rho^{0})$,
which expresses the rearrangements. In this section we do not introduce OPEP;
the system is restricted only to the $S$-waves, and the attraction from the
OPEP is effectively taken into account by introducing the central attraction
between the $D^{(*)}$ and $\bar{{D}}{}^{(*)}$. The widths of the $\omega$ and
$\rho$ mesons are taken into account as an imaginary part in the
$J/\psi\omega$ and $J/\psi\rho$ propagators. In this way, we consider that the
model can simulate essential features of the decay properties of $X(3872)$.

From the quark model point of view, the $D\bar{{D}}{}^{*}$ states of total
charge 0 are the $c\bar{{c}}u\bar{{u}}$ or $c\bar{{c}}d\bar{{d}}$ states,
which contain also the $J\\!/\\!\psi\omega$ or $J\\!/\\!\psi\rho^{0}$ state
with the appropriate color configuration. The observed final
$J\\!/\\!\psi\pi^{3}$ and $J\\!/\\!\psi\pi^{2}$ decay modes are considered to
come from these components. The rearrangement between $D\bar{{D}}{}^{*}$ and
$\omega J\\!/\\!\psi$ or $\rho^{0}J\\!/\\!\psi$ occurs at the short distance,
where all four quarks exist in the hadron size region. The coupling between
the $c\bar{c}$ and $D\bar{D}{}^{*}$, however, is not known. Therefore, it is
treated as a phenomenological one, as shown below. Note that there is no
direct channel coupling between the $c\bar{{c}}$ channel and the $\omega
J\\!/\\!\psi$ or $\rho^{0}J\\!/\\!\psi$ channels in the present model setup.
They break the OZI rule, and the latter breaks the isospin symmetry.

The model Hamiltonian for the $c\bar{{c}}$, $D^{0}\bar{{D}}{}^{*0}$,
$D^{+}D^{-*}$, $J\\!/\\!\psi\omega$ and $J\\!/\\!\psi\rho$ channels is taken
as [38]:

$\displaystyle H$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccccccc}m_{c\bar{c}}&U^{(mc)}{}^{\dagger}\\\
U^{(mc)}&H^{(mm)}\\\ \end{array}\right),$ (197) $\displaystyle H^{(mm)}$
$\displaystyle=$ $\displaystyle H^{(mm)}_{0}+V^{(mm)},$ (198) $\displaystyle
H^{(mm)}_{0}$ $\displaystyle=$ $\displaystyle{\rm
diag}(K_{D{\bar{D}}0},K_{D{\bar{D}}\pm},K_{J/\psi\omega},K_{J/\psi\rho}),$
(199) $\displaystyle K_{D{\bar{D}}0,\pm}$ $\displaystyle=$ $\displaystyle
m_{D^{0,-}}+m_{D^{*0,+}}+{p^{2}\over 2\mu_{D{\bar{D}}0,\pm}},$ (200)
$\displaystyle K_{J\\!/\\!\psi\omega,\rho}$ $\displaystyle=$ $\displaystyle
m_{J\\!/\\!\psi}+m_{\omega,\rho}+{p^{2}\over 2\mu_{J/\psi\omega,\rho}},$ (201)
$\displaystyle U^{(mc)}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccccccc}-u\\\ u\\\ 0\\\ 0\\\
\end{array}\right),~{}~{}~{}V^{(mm)}=\left(\begin{array}[]{ccccccc}\tilde{v}&0&-v&-v\\\
0&\tilde{v}&v&-v\\\ -v&v&0&0\\\ -v&-v&0&0\\\ \end{array}\right),$ (210)

$\displaystyle u(q)$ $\displaystyle=$
$\displaystyle{g_{c\bar{c}}\over\sqrt{\Lambda_{q}}}L_{\Lambda_{q}}(q),~{}~{}~{}~{}L_{\Lambda_{q}}(q)={\Lambda_{q}^{2}\over\Lambda_{q}^{2}+q^{2}},$
(211) $\displaystyle v(q,q^{\prime})$ $\displaystyle=$
$\displaystyle{v_{0}\over\Lambda_{q}^{2}}L_{\Lambda_{q}}(q)L_{\Lambda_{q}}(q^{\prime}),~{}~{}~{}\tilde{v}(q,q^{\prime})={\tilde{v}_{0}\over\Lambda_{q}^{2}}L_{\Lambda_{q}}(q)L_{\Lambda_{q}}(q^{\prime}),$
(212)

where $m_{c\bar{c}}$ is the $c\bar{{c}}$ bare mass when the coupling to
$D\bar{{D}}{}^{*}$ is switched off. The reduced masses,
$\mu_{D{\bar{D}}0,\pm}$ and $\mu_{J/\psi\omega,\rho}$, are for the
$D{}^{0}\bar{{D}}{}^{*0}$, $D^{\pm}D^{*\mp}$, $J\\!/\\!\psi\omega$ and
$J\\!/\\!\psi\rho$ systems, respectively. The coupling between the
$c\bar{{c}}$ state and the $D\bar{{D}}{}^{*}$ state is expressed by the
transfer potential, $U^{(mc)}$, which is chosen to be Lorentzian in the
momentum space with the strength $g_{c\bar{c}}$. The rearrangement between the
$D\bar{{D}}{}^{*}$ states and the $J\\!/\\!\psi\omega$ and $J\\!/\\!\psi\rho$
meson is expressed by a separable potential, $v$ in $V^{(mm)}$. The basis of
the matrix expression in (199) and in (210) are ($D^{(*)0}\bar{{D}}{}^{(*)0}$,
$D^{(*)\pm}D^{(*)\mp}$, $J\\!/\\!\psi\omega$, $J\\!/\\!\psi\rho^{0}$). The
strength of the interaction between the $D$ and $\bar{{D}}$ mesons,
$\tilde{v}_{0}$, is taken to be the maximal value which does not create a
bound state in the $B\bar{B}$ systems, where no bound states has been observed
yet. The strengths $g_{c\bar{c}}$ and $v_{0}$ are free parameters under the
condition that the mass of $X$(3872) can be reproduced. The value of
$\Lambda_{q}$ is the same as the one used in the previous section,
$\Lambda_{q}$ = 0.5 GeV. The parameters are summarized in table 10.

Table 10: Model parameters and the amounts of each component. Masses and widths are in MeV, and taken from [184] $m_{J/\psi}$ | $m_{\omega}$ | $\Gamma_{\omega}$ | $m_{\rho}$ | $\Gamma_{\rho}$ | $g_{c\bar{c}}$ | $v_{0}$ | $\tilde{v}_{0}$
---|---|---|---|---|---|---|---
3096.916 | 782.65 | 8.49 | 775.26 | 147.8 | 0.04136 | 0.1929 | $-0.1886$
$|c_{c\bar{c}}|^{2}$ | $|c_{0}(^{3}S)|^{2}$ | $|c_{\pm}(^{3}S)|^{2}$ | $|c_{J/\psi\omega}|^{2}$ | $|c_{J/\psi\rho}|^{2}$ |  |  |
0.036 | 0.913 | 0.034 | 0.010 | 0.006 |  |  |

The amount of each component in the $X$(3872) bound state is also listed in
table 10. The bulk feature is similar to the models in the previous section:
the dominant component is ${D^{0}\bar{{D}}{}^{*0}}$ while the ${D^{+}D^{*-}}$
component is considerably smaller because of the threshold difference. The
amount of the $c\bar{c}$ component is somewhat smaller but still sizable. The
$J/\psi\omega$ and $J/\psi\rho$ components are small comparing to the
$D\bar{{D}}{}^{*}$ components. The fact that the $J/\psi\rho$ and
$J/\psi\omega$ components of $X$(3872) are comparable in size is reproduced in
the present model.

Figure 13: The $B$ meson weak decay. See text.

As listed in table 6, the $X$(3872) is produced by various processes. As a
typical example, we discuss the $X$(3872) decay process in the $B$ meson weak
decay in the following. As illustrated in figure 13, the mass spectrum of
$X$(3872) from the $B$ meson weak decay is proportional to the sum of the
transfer strength from the $c\bar{{c}}$ to the two-meson states, $f$,
$W(c\bar{{c}}\rightarrow f;E)$, which can be expressed as

$\displaystyle\sum_{f}{\rmd W(c\bar{{c}}\rightarrow f;E)\over\rmd
E}=-{1\over\pi}{\rm Im}\langle
c\bar{{c}}|{G^{(c\bar{c})}}(E)|c\bar{{c}}\rangle.$ (213)

Here ${G^{(c\bar{c})}}$ is the full propagator of the $c\bar{{c}}$ state,
which can be written by using the self energy $\Sigma_{c\bar{c}}$ as:

$\displaystyle{G^{(c\bar{c})}}(E)$ $\displaystyle=$
$\displaystyle(E-m_{c\bar{c}}-\Sigma_{c\bar{c}}(E))^{-1},$ (214)
$\displaystyle\Sigma_{c\bar{c}}(E)$ $\displaystyle=$ $\displaystyle
U^{(mc)}{}^{\dagger}{G^{(mm)}}(E)U^{(mc)}.$ (215)

Here we define the free and the full propagators within the two-meson space,
${G^{(mm)}_{0}}(E)$ and ${G^{(mm)}}(E)$, respectively, with the decay widths
as

$\displaystyle{G^{(mm)}_{0}}(E)$ $\displaystyle=$
$\displaystyle\Big{(}E-H_{0}^{(mm)}+\rmi{\Gamma\over 2}\Big{)}^{-1},$ (216)
$\displaystyle{G^{(mm)}}(E)$ $\displaystyle=$
$\displaystyle\Big{(}E-H^{(mm)}+\rmi{\Gamma\over 2}\Big{)}^{-1},$ (217)
$\displaystyle\Gamma$ $\displaystyle=$ $\displaystyle{\rm
diag}(0,0,\Gamma_{\omega},\Gamma_{\rho}),$ (218)

where $\Gamma_{\omega,\rho}$ is the $\omega$ or $\rho$ decay width,
respectively. The $\rho$ meson width is taken to be energy dependent as
discussed in [38]. The widths of $D^{*}$ mesons are neglected. The width
$\Gamma$ is ignored when the bound state energy or the component is calculated
above. It, however, is essential to include them when one investigates the
decay spectrum.

In order to obtain the decay spectrum of each final two-meson channel
separately, we have rewritten the right-hand side of (213) as follows. Since
the system has only one $c\bar{c}$ state in the present model, the above
${G^{(c\bar{c})}}(E)$ and $\Sigma_{c\bar{c}}(E)$ are single channel functions
of the energy $E$. They become matrices when more than one $c\bar{c}$ states
are introduced, but the following procedure can be extended in a
straightforward way. As seen from (214), the imaginary part of
${G^{(c\bar{c})}}(E)$ comes only from the imaginary part of
$\Sigma_{c\bar{c}}(E)$. Therefore,

$\displaystyle{\rm Im}\,{G^{(c\bar{c})}}$ $\displaystyle=$ $\displaystyle{\rm
Im}\,({G^{(c\bar{c})}}^{*}\,({G^{(c\bar{c})}}^{*})^{-1}\,{G^{(c\bar{c})}})={\rm
Im}\,({G^{(c\bar{c})}}^{*}\,\Sigma_{c\bar{c}}\,{G^{(c\bar{c})}})$ (219)
$\displaystyle=$ $\displaystyle{\rm
Im}\,({G^{(c\bar{c})}}^{*}\,U^{(mc)}{}^{\dagger}{G^{(mm)}}U^{(mc)}\,{G^{(c\bar{c})}}),$

where ∗ stands for the complex conjugate. Using the following relation for a
real potential

$\displaystyle{\rm Im}\,{G^{(mm)}}^{-1}$ $\displaystyle=$ $\displaystyle{\rm
Im}\,{G^{(mm)}_{0}}^{-1}$ (220)

and Lippmann Schwinger equation for the propagator, $G=(1+GV)G_{0}$, we have
for Im${G^{(mm)}}$ on the right-hand side of (219)

$\displaystyle{\rm Im}\,{G^{(mm)}}$ $\displaystyle=$ $\displaystyle{\rm
Im}\,({G^{(mm)}}({G^{(mm)}}{}^{*})^{-1}{G^{(mm)}}{}^{*})$ (221)
$\displaystyle=$ $\displaystyle{\rm
Im}\,({G^{(mm)}}({G^{(mm)}_{0}}{}^{*})^{-1}{G^{(mm)}}{}^{*})$ $\displaystyle=$
$\displaystyle{\rm
Im}\,\Big{(}(1+{G^{(mm)}}V^{(mm)}){G^{(mm)}_{0}}(1+{G^{(mm)}}{}^{*}V^{(mm)})\Big{)}.$

Thus, (219) can be rewritten as

$\displaystyle{\rm Im}\,{G^{(c\bar{c})}}$ $\displaystyle=$ $\displaystyle{\rm
Im}\,\Big{(}{G^{(c\bar{c})}}^{*}U^{(mc)}{}^{\dagger}(1+{G^{(mm)}}V^{(mm)}){G^{(mm)}_{0}}(1+{G^{(mm)}}{}^{*}V^{(mm)})U^{(mc)}{G^{(c\bar{c})}}\Big{)}.$

When we apply the plain wave expansion for the ${G^{(mm)}_{0}}$ in
(LABEL:eq:ImG-3), we have

$\displaystyle{\rmd W(c\bar{{c}}\rightarrow f;E)\over\rmd E}$ $\displaystyle=$
$\displaystyle-{1\over\pi}{\rm Im}\int{\rm d}{\bi k}\langle f{\bi
k}|{G^{(mm)}_{0}}(E)|f{\bi k}\rangle\Big{|}\sum_{f^{\prime}}\int\rmd{{\bi
k}^{\prime}}\langle\\!\langle f^{\prime}{\bi k}^{\prime}(f{\bi
k})|U^{(mc)}({\bi k}^{\prime}){G^{(c\bar{c})}}(E)|c\bar{c}\rangle\Big{|}^{2}$
$\displaystyle=$ $\displaystyle{2\over\pi}\mu_{f}\int{k^{2}{\rm
d}k\;\mu_{f}\Gamma_{f}\over(k_{f}^{2}-k^{2})^{2}+(\mu_{f}\Gamma_{f})^{2}}\Big{|}\sum_{f^{\prime}}\int\rmd{{\bi
k}^{\prime}}\langle\\!\langle f^{\prime}{\bi k}^{\prime}(f{\bi
k})|U^{(mc)}({\bi k}^{\prime}){G^{(c\bar{c})}}(E)|c\bar{c}\rangle\Big{|}^{2},$

where $k_{f}$ and $\mu_{f}$ stand for the three-momentum and the reduced mass
of the final two-meson state where $E=m_{1f}+m_{2f}+k_{f}^{2}/(2\mu_{f})$.
$\Gamma_{f}$ is the the decay width of mesons in the final state $f$, i.e. 0
if $f$ is $D\bar{{D}}{}^{*}$, $\Gamma_{\omega}$ or $\Gamma_{\rho}$ when $f$ is
$J/\psi\omega$ or $J/\psi\rho$. $|f{\bi k}\rangle$ stands for the plain wave
of the channel $f$ with the momentum ${\bi k}$. $|f^{\prime}{\bi
k}^{\prime}(f{\bi k})\rangle\\!\rangle$ stands for the distorted wave function
of the channel $f^{\prime}$ with the momentum ${\bi k}^{\prime}$ which is
generated from $|f{\bi k}\rangle$. This can be obtained by the Lippmann
Schwinger equation as

$\displaystyle|f^{\prime}{\bi k}^{\prime}(f{\bi k})\rangle\\!\rangle$
$\displaystyle=$ $\displaystyle(1+{G^{(mm)}}V^{(mm)})|f{\bi k}\rangle.$ (224)

In the present model, only the $D\bar{D}{}^{*}$ channels couple directly to
$c\bar{c}$. The summation over $f^{\prime}$ in (LABEL:eq:X3872-width) means
summation over $D^{0}\bar{{D}}{}^{*0}$ and $D^{+}D^{-*}$. The final two-meson
fraction expressed by $f$ in the above equations can be $J\\!/\\!\psi\omega$
or $J\\!/\\!\psi\rho$ as well as $D\bar{D}{}^{*}$. For the channels where
$\Gamma_{f}$ is small, the transfer strength becomes

$\displaystyle{\rmd W(c\bar{{c}}\rightarrow f;E)\over\rmd E}$
$\displaystyle\rightarrow$
$\displaystyle\mu_{f}k_{f}\Big{|}\sum_{f^{\prime}}\int\rmd{{\bi
k}^{\prime}}\langle\\!\langle f^{\prime}{\bi k}^{\prime}(fk_{f})|U^{(mc)}({\bi
k}^{\prime}){G^{(c\bar{c})}}(E)|c\bar{c}\rangle\Big{|}^{2}~{}~{}~{}(\Gamma_{f}\rightarrow
0).$

The calculated $W$ spectrum is shown for each final state in figure 14. The
narrow peak at around the $D^{0}\bar{{D}}{}^{*0}$ threshold as well as the
large isospin mixing are successfully reproduced by this picture of the
$c\bar{{c}}$, $D^{0}\bar{{D}}{}^{*0}$, $D^{+}D^{-*}$, $J\\!/\\!\psi\omega$ and
$J\\!/\\!\psi\rho$. The spectrum shape depends on the $X$(3872) binding
energy, but the qualitative feature does not. The $J\\!/\\!\psi\omega$ and
$J\\!/\\!\psi\rho$ peaks in the decay spectrum remain sharp when the $X$(3872)
is around the threshold[38].

Figure 14: The transfer strength from the $c\bar{{c}}$ state to the final two-
meson states. The energy $E$ is the center of mass energy of the two-meson
states. Figure (b) is the same as (a) but magnified at around the
$D^{0}\bar{{D}}{}^{*0}$ threshold. Taken from [38].

Let us discuss the mechanism to have the small width of the peak and the
isospin mixing in the final fractions, the two notable features of the final
$J\\!/\\!\psi\omega(\rho)$ spectrum mentioned before. The small width of
$X$(3872) means that the corresponding pole of ${G^{(c\bar{c})}}$ is close to
the real axis. The imaginary part in the denominator of ${G^{(c\bar{c})}}$
comes from the imaginary part of $\Sigma_{c\bar{c}}$ as shown (214), which can
be expanded by using (215) as

$\displaystyle\Sigma_{c\bar{c}}$ $\displaystyle=$ $\displaystyle
u^{\dagger}G^{(D\bar{D}{}^{*})}u+u^{\dagger}G^{(D\bar{D}{}^{*})}vG^{(J/\psi\omega,\rho)}_{0}vG^{(D\bar{D}{}^{*})}u+\cdots,$
$\displaystyle=$ $\displaystyle
u^{\dagger}G^{(D\bar{D}{}^{*})}u\Big{(}1+{g_{c\bar{c}}}^{-1}v_{0}^{2}(\Lambda_{q}^{-2}L_{\Lambda_{q}}G^{(J/\psi\omega,\rho)}_{0}L_{\Lambda_{q}})(\Lambda_{q}^{-3/2}L_{\Lambda_{q}}G^{(D\bar{D}{}^{*})}u)+\cdots\Big{)},$

where $G^{(D\bar{D}{}^{*})}$ stands for the full propagator obtained within
the $D\bar{{D}}{}^{*}$ space, and the second equation can be obtained for the
separable potential, (212) and (211).

On the right-hand side of (LABEL:eq:Sigcc-1), the first term is real for the
energy below the threshold. It is because there is no bound state without the
$c\bar{c}$ core, and also because the decay width of $D^{*}$ is very small,
$83.4\pm 1.8$ keV for $D^{*\pm}$ and $<2.1$ MeV for $D^{*0}$ [5], and can be
neglected. The first term gives most of the size of the real part of the self
energy, $\Sigma_{c\bar{c}}$, which gives the mass difference between the
$c\bar{c}$ core and the $X$(3872).

The second term has an imaginary part which comes from the width of the
$\omega$ and $\rho$ in the propagator $G^{(J/\psi\omega,\rho)}_{0}$. This term
corresponds to the decay with the rearrangement, $D\bar{D}{}^{*}\to
J\\!/\\!\psi\omega$, $J\\!/\\!\psi\rho$. The second term is very much
suppressed. One suppression comes from the color factor. Probability to find a
color-singlet $c\bar{c}$ pair in the $D\bar{D}{}^{*}$ channel has a factor 1/9
in the color space. In the present model this effect is taken into account by
employing a small coupling constant $v_{0}$ in the transfer potential $v$.
Another suppression comes from the range of the rearrangement. It occurs with
charm quark exchange, which means that the reaction occurs in the hadron size
region of the very large object, $X$(3872). The factor
$G^{(D\bar{D}{}^{*})}u|c\bar{c}\rangle$ can be regarded as the
${D\bar{D}{}^{*}}$ wave function generated from $|c\bar{c}\rangle$. So, one of
the factors of the second term becomes a overlap:

$\displaystyle(\Lambda_{q}^{-3/2}L_{\Lambda_{q}}G^{(D\bar{D}{}^{*})}u)$
$\displaystyle\propto 4\pi\sqrt{2}\Lambda_{q}^{-3/2}\int{q^{2}\rmd
q\over(2\pi)^{3}}L_{\Lambda_{q}}(q)\psi_{X(3872)}(q)|_{D\bar{D}{}^{*}}$ (227)

which is 0.109 for $I=0$, and 0.017 for $I=1$ when the wave function plotted
in figure 11 is used. These suppressions, with kinematical factor for the
final $J/\psi\omega$ channel discussed below, bring about the small imaginary
part for the $X$(3872).

The isospin ratio of (227), 0.15 (=0.017/0.109), is a rough size of the
isospin symmetry breaking in the short range part of the $X$(3872) wave
function. This ratio has been calculated in other literatures, $(11.5\pm
5.7)^{-1}$[126], or $0.27^{2}$ in (177) [127]. In order to compare this ratio
to the experimental branching fractions (176), the kinematical factor, another
factor in the second term, should be considered. For the final $J/\psi\omega$
channel, the kinematical suppression appears because the charged and the
neutral $D\bar{D}{}^{*}$ threshold difference is comparable in size to the
$\omega$ decay width.

$\displaystyle{\rm Im}\int\rmd
q\;\Lambda_{q}^{-2}L_{\Lambda_{q}}^{2}(q)G^{(J/\psi\omega)}_{0}(E=m_{X(3872)})\Big{/}(\omega\leftrightarrow\rho)=0.15~{}.$
(228)

The corresponding value with a careful treatment for the off-shell $\omega$
meson is found to be 0.0870 [126], or 1/13.3 [127]. When both of the factors,
(227) and (228), are considered, the ratio of the branching fractions becomes
of the order of 1, that is consistent with the observed one as discussed in
section 4. The above estimate gives a rough size of the isospin violation in
the $X(3872)$. The new experiment, the $X(3872)\to\chi_{c1}\pi^{0}$ decay,
which we have not discussed here, will also support this value because that
decay fraction has a similar size of that of the $\pi\pi J/\psi$ [130].

In figure 15, the $J\\!/\\!\psi\omega$ and $J\\!/\\!\psi\rho$ peaks are
plotted for the $c\bar{c}$ coupling with four different values of the strength
$g_{c\bar{c}}$. The corresponding binding energies are shown by the arrows,
which, as mentioned before, are calculated without the $\rho$ or $\omega$
meson widths. The left two peaks correspond to the ones with stronger coupling
than that of the reference [38]. The third one is that of the original value
with a binding energy of 0.16 MeV. The most right one corresponds to the one
where the coupling $g_{c\bar{c}}^{2}$ is reduced by a factor 0.95; there is no
bound state but a virtual state appears. As the coupling $g_{c\bar{c}}^{2}$ is
weakened, the peak moves as the bound state moves to the threshold, but stops
at the threshold when the pole moves to the second Riemann sheet.

Figure 15: The transfer strength from the $c\bar{{c}}$ state to the final
$J\\!/\\!\psi\omega$ and $J\\!/\\!\psi\rho$ states. Note that the energy
abscissa is taken to be from 3870.8 to 3872 MeV. Final states are
$J\\!/\\!\psi\omega$ (Solid lines), and $J\\!/\\!\psi\rho$ (dashed lines). The
coupling $g_{c\bar{c}}^{2}$ is factored by 1.1, 1.05, 1, 0.95. The
corresponding bound state energies calculated without the decay widths are
shown by the left three vertical lines: 3871.25, 3871.51, and 3871.69 MeV,
respectively. The most right vertical line corresponds to the
$D^{0}\bar{{D}}{}^{*0}$ threshold. The one with the weakest coupling,
0.95$g_{c\bar{c}}^{2}$, does not have a bound state.

Namely, the peak energy of the final $J\\!/\\!\psi\pi^{n}$ is kept lower than
or equal to the $D^{0}\bar{{D}}{}^{*0}$ threshold energy. On the other hand,
the peak energy of the final $D^{0}\bar{{D}}{}^{*0}$ fraction is higher than
threshold by definition. This means that the $X$(3872) mass is higher when
measured by the final $D\bar{{D}}{}^{*}$ fraction, which is consistent with
the observation. Experimentally the $X$(3872) mass observed by the $J/\psi$
and anything is 3871.69$\pm$0.17 MeV with a width of $<$1.2 MeV [5], while
that of the final $D\bar{D}{}^{*}$ is 3872.9${+0.6\atop-0.4}{+0.4\atop-0.5}$
MeV [74] or 3875.1${+0.7\atop-0.5}\pm 0.5$ MeV[75]. The observed threshold
mass is 3871.68$\pm$0.07 MeV [5]. Thus the mass observed by the $J/\psi$ and
anything is consistent with the heavier two peaks in figure 15. This suggests
that the $X$(3872) is either a virtual state or a very shallow bound state.
These two states are very hard to distinguish from each other experimentally
when one considers the width of the component, such as the $\rho$ meson. In
literature, line shapes were studied by using amplitudes parametrized by
effective range method [147] or by the Flatte parametrization [185, 186, 187].
It, however, seems difficult to determine the position of the resonance poles
just from the shape of the decay spectrum. To discuss this subject, it will
probably be necessary to treat the $\rho$ meson as a resonance of the $\pi\pi$
continuum and perform dynamical analyses.

In order to discuss possibilities of the other mechanism for the $X$(3872)
peak as well as the effects of the meson decay width, let us ignore the OZI
rule just for now. In strong decay, where the isospin is conserved,
$J\\!/\\!\psi\rho$ does not directly decay from the $c\bar{c}$ core, but
$J\\!/\\!\psi\omega$ may. We estimate the effects of existence of such a
process by introducing a direct coupling between
$J\\!/\\!\psi\omega$-$c\bar{c}$. Suppose the $J\\!/\\!\psi\omega$-$c\bar{c}$
coupling occurs by the same potential, $u$, there is a bound state with a
binding energy of 10 MeV below the $D^{0}\bar{{D}}{}^{*0}$ threshold. But as
seen in figure 16 and as expected, almost all the decay fraction is
$J\\!/\\!\psi\omega$ and not $J\\!/\\!\psi\rho$ in that case, which is
excluded by experiment. As far as $X$(3872) is concerned, no direct coupling
seems to occur between the $c\bar{c}$ core and the $J/\psi\omega$ channel. One
interesting point of this trial calculation is that the peak energy approaches
the threshold as a less pronounced peak when the width of the $\omega$ meson
is enlarged by hand. Therefore, exotic hadrons which appears at around a two-
meson threshold and which contain meson(s) with a large decay width should be
examined carefully. Let us make one more comment on the direct decay from the
$c\bar{c}$ core. The $\chi_{c1}(2P)$ peak may not be seen in the
$D\bar{{D}}{}^{*}$ decay spectrum, but it may be seen from the radiative
decay. There is a selection rule of $E1$ transition that reduces the fraction
$\chi_{c1}(2P)\to J/\psi\gamma$ but not $\to\psi(2S)\gamma$, which may show
clearly the existence of $\chi_{c1}(2P)$ [188, 173], and if so, support the
$c\bar{c}$ admixture of $X$(3872) discussed in this section.

Figure 16: Trial calculation to estimate the effects of constructing mesons’
width. The $J\\!/\\!\psi\omega$ channel is assumed to couple to the $c\bar{c}$
core, and the width is enhanced by hand. (See text.) The solid line is for the
final $J\\!/\\!\psi\omega$ fraction, the dashed line is for the final
$J\\!/\\!\psi\rho$ fraction. The latter is shown only for $\Gamma_{\omega}$ =
8 MeV. The vertical lines correspond to the bound state energy and the
${D^{0}\bar{{D}}{}^{*0}}$ threshold.

Let us summarize the features of $X$(3872) and the $B$-decay to the final two-
meson spectrum obtained in section 4. The two-meson and the $c\bar{c}$ hadron
model which consists of the $c\bar{c}(1^{++})$ core,
$D^{(*)0}\bar{{D}}{}^{(*)0}$ $({}^{3}S,{}^{3}D,{}^{5}D)$,
$D^{(*)\pm}D^{(*)\mp}({}^{3}S,{}^{3}D,{}^{5}D)$, $J\\!/\\!\psi\omega$ and
$J\\!/\\!\psi\rho^{0}$, the following features are obtained:

  * •

$X$(3872) is a very shallow bound state or a virtual state which is close to
the $D^{0}\bar{D}{}^{*0}$ threshold, which are very difficult to distinguish
each other.

  * •

The state is molecular, mostly $D^{0}\bar{D}{}^{*0}$ in the long range region,
but has a considerable $D^{\pm}\bar{D}{}^{*\mp}$ component at the short
distance.

  * •

Two kinds of channel couplings provides attraction for $X$(3872): one is OPEP
tensor, which mixes $D^{(*)}\bar{{D}}^{(*)}$ $S$-$D$-waves, the other is the
$c\bar{c}$-$D\bar{D}{}^{*}$ coupling. These two effects are comparable in
size.

  * •

The amount of the $c\bar{c}$ component is found to be about 6 % in the model
which contains both of the OPEP and the $c\bar{c}$-couplings, which meets the
requirement from the production rate in the $p\bar{p}$ experiments.

  * •

When considering the whole energy spectrum of the $B$ weak decay, there is one
very narrow peak at the $D^{0}\bar{D}{}^{*0}$ threshold, but not around the
$D^{\pm}\bar{D}{}^{*\mp}$ threshold, nor around the $c\bar{c}(1^{++})$ bare
mass.

  * •

Among the final products, amounts of $D\bar{{D}}^{*}$’s is the largest, which
are produced directly from $c\bar{{c}}$ core. There are small amount of
$J\\!/\\!\psi\omega$ and $J\\!/\\!\psi\rho$ final product, which are
comparable to each other.

  * •

The spectrum of the $J\\!/\\!\psi\omega$ and the $J\\!/\\!\psi\rho$ final
products makes a very narrow peak at the bound state energy, if a bound state
exists, or at the $D^{0}\bar{D}{}^{*0}$ threshold, if not. For instance,
according to figure 15, the width is around 0.1 MeV when the binding energy is
0.5 MeV.

##  5 Charged heavy mesons

###  5.1 Brief overview in experimental status

Here we overview current experimental status about charged exotic mesons. We
have summarized major experimental results of the charged charmonium-like and
bottomonium-like exotic candidates in Table 11.

State | Mass [MeV] | Width [MeV] | Reaction | Exp. (Year)
---|---|---|---|---
$Z_{c}(3900)^{\pm}$ | $3899.0\pm 3.6\pm 4.9$ | $46\pm 10\pm 20$ | $Y(4260)\to\pi^{+}\pi^{-}J/\psi$ | BESIII(2013)
| $3894.5\pm 6.6\pm 4.5$ | $63\pm 24\pm 26$ | $Y(4260)\to\pi^{+}\pi^{-}J/\psi$ | Belle(2013)
| $3886\pm 4\pm 2$ | $37\pm 4\pm 8$ | $\psi(4160)\to\pi^{+}\pi^{-}J/\psi$ | CLEO-c(2013)
| $3883.9\pm 1.5\pm 4.2$ | $24.8\pm 3.3\pm 11.0$ | $Y(4260)\to(D\bar{D}^{\ast})^{\pm}\pi^{\mp}$ | BESIII(2014)
| $3895.0\pm 5.2\pm^{+4.0}_{-2.7}$ |  | $H_{b}\to J/\psi\pi^{+}\pi^{-}+X$ | D0(2018)
$Z_{c}(3900)^{0}$ | $3894.8\pm 2.3\pm 3.2$ | $29.6\pm 8.2\pm 8.2$ | $Y(4260)\to\pi^{0}\pi^{0}J/\psi$ | BESIII(2015)
| $3885.7^{+4.3}_{-5.7}\pm 8.4$ | $35^{+11}_{-12}\pm 15$ | $Y(4260)\to(D\bar{D}^{\ast})^{0}\pi^{0}$ | BESIII(2015)
$Z_{c}(4020)^{\pm}$ | $4022.9\pm 0.8\pm 2.7$ | $7.9\pm 2.7\pm 2.6$ | $e^{+}e^{-}\to\pi^{+}\pi^{-}h_{c}$ | BESIII(2013)
$Z_{c}(4025)^{\pm}$ | $4026.3\pm 2.6\pm 3.7$ | $24.8\pm 2.6\pm 7.7$ | $Y(4260)\to(D^{\ast}\bar{D}^{\ast})^{\pm}\pi^{\mp}$ | BESIII(2014)
$Z_{c}(4025)^{0}$ | $4025.5^{+2.0}_{-4.7}\pm 3.1$ | $23.0\pm 6.0\pm 1.0$ | $Y(4260)\to(D^{\ast}\bar{D}^{\ast})^{0}\pi^{0}$ | BESIII(2015)
$Z_{c}(4050)^{+}$ | $4051\pm 14^{+20}_{-41}$ | $82^{+21}_{-17}{}^{+47}_{-22}$ | $\bar{B}^{0}\to\chi_{c1}(1P)K^{-}\pi^{+}$ | Belle(2008)
$Z_{c}(4200)^{+}$ | $4196^{+31}_{-29}{}^{+17}_{-13}$ | $370^{+70}_{-70}{}^{+70}_{-132}$ | $\bar{B}^{0}\to J/\psi K^{-}\pi^{+}$ | Belle(2014)
$Z_{c}(4250)^{+}$ | $4248^{+44}_{-29}{}^{+180}_{-35}$ | $177^{+54}_{-39}{}^{+316}_{-61}$ | $\bar{B}^{0}\to\chi_{c1}(1P)K^{-}\pi^{+}$ | Belle(2008)
$Z_{c}(4430)^{+}$ | $4433\pm 4\pm 2$ | $45^{+18}_{-13}{}^{+30}_{-13}$ | $B\to K\pi^{+}\psi(2S)$ | Belle(2008)
$Z_{c}(4430)^{-}$ | $4485^{+22+28}_{-22-11}$ | $200^{+41+26}_{-46-35}$ | $B^{0}\to K^{+}\pi^{-}\psi(2S)$ | Belle(2013)
$Z_{c}(4430)^{-}$ | $4475\pm 7^{+15}_{-25}$ | $172\pm 13^{+37}_{-34}$ | $B^{0}\to K^{+}\pi^{-}\psi(2S)$ | LHCb(2014)
$Z_{b}(10610)^{\pm}$ | $10607.2\pm 2.0$ | $18.4\pm 2.4$ | $\Upsilon(5S)\to\pi^{+}\pi^{-}\Upsilon(1,2,3S)$ | Belle(2012)
|  |  | $\Upsilon(5S)\to\pi^{+}\pi^{-}h_{b}(1,2P)$ |
$Z_{b}(10610)^{0}$ | $10609\pm 4\pm 4$ | 18.4 (input) | $\Upsilon(5S)\to\pi^{0}\pi^{0}\Upsilon(2,3S)$ | Belle(2013)
$Z_{b}(10650)^{\pm}$ | $10652.2\pm 1.5$ | $11.5\pm 2.2$ | $\Upsilon(5S)\to\pi^{+}\pi^{-}\Upsilon(1,2,3S)$ | Belle(2012)
|  |  | $\Upsilon(5S)\to\pi^{+}\pi^{-}h_{b}(1,2P)$ |
Table 11:  Experimental status of the charged charmonium-like and bottomonium-
like exotic candidates with the neutral isospin partners.

Below, we pickup $Z_{c}(3900)$, $Z_{c}(4200)$, $Z_{c}(4430)$, $Z_{b}(10610)$
and $Z_{b}(10650)$, as well-known candidates. All these states have the
quantum numbers $I^{G}(J^{PC})=1^{+}(1^{+-})$.

$Z_{c}(3900)$ with mass $3887.2\pm 2.3$ MeV and decay width $28.2\pm 2.6$ MeV
was found in the decay of $Y(4260)$:
$Y(4260)\rightarrow\pi^{\mp}Z_{c}(3900)^{\pm}$,
$Z_{c}(3900)^{\pm}\rightarrow\pi^{\pm}J/\psi$ in BESIII [189] and was also
confirmed by Belle [190] and CLEO [191]. The mass lies about 11 MeV above the
$D^{+}\bar{D}^{\ast 0}$ (or 12 MeV above the $D^{\ast+}\bar{D}^{0}$)
threshold. The process $e^{+}e^{-}\to\pi^{\pm}(D\bar{D}^{\ast})^{\mp}$ at
$\sqrt{s}=4.26$ GeV was studied and the strong threshold enhancement,
$Z_{c}(3885)$ with mass $3883.9\pm 1.5\pm 4.2$ MeV and width $24.8\pm 3.3\pm
11.0$ MeV was observed in [192]. In PDG [5], $Z_{c}(3885)$ is assumed to be
related to $Z_{c}(3900)$. Its quantum numbers, $I^{G}(J^{P})=1^{+}(1^{+})$,
have been determined in a partial wave analysis of the process
$e^{+}e^{-}\to\pi^{+}\pi^{-}J/\psi$ by BESIII [193]. The decay
$Z_{c}(3900)\rightarrow\pi^{\pm}h_{c}$ was not seen in BESIII [194].

The $\pm 1$ electric charge of the $Z_{c}(3900)^{\pm}$, indicating the quark
contents $u\bar{d}c\bar{c}$ and $d\bar{u}c\bar{c}$, suggests the existence of
the isospin partner, $(u\bar{u}-d\bar{d})c\bar{c}$. Indeed, a 3.5 $\sigma$
level significance evidence of a neutral $Z_{c}(3900)^{0}$ was reported in
[191]. Later, $Z_{c}(3900)^{0}$ was also observed in
$e^{+}e^{-}\rightarrow\pi^{0}Z_{c}(3900)^{0}\rightarrow\pi^{0}\pi^{0}J/\psi$
(10.4 $\sigma$) [195] and in
$e^{+}e^{-}\rightarrow\pi^{0}Z_{c}(3900)^{0}\rightarrow\pi^{0}(D\bar{D}^{\ast})^{0}$
(greater than 10$\sigma$) [196].

Belle observed $Z_{c}(4200)$ via $Z_{c}(4200)\rightarrow J/\psi\pi^{+}$ in
$\bar{B}^{0}\rightarrow K^{-}\pi^{+}J/\psi$, and it has mass
$4196^{+31}_{-29}{}^{+17}_{-13}$ MeV and decay width
$370^{+70}_{-70}{}^{+70}_{-132}$ MeV with the quantum numbers
$I^{G}(J^{P})=1^{+}(1^{+-})$ [197]. LHCb found the evidence for $Z_{c}(4200)$
contributions to $\Lambda_{b}\to J/\psi p\pi^{-}$ decays [198], and found some
structure where the invariant mass of $J/\psi\pi^{-}$ is near 4200 MeV in
$B^{0}\to J/\psi K^{+}\pi^{-}$ decays [199].

$Z_{c}(4430)$ was first observed in Belle with the mass of $4433\pm 4\pm 2$
MeV [200]. The present world average of the mass, $4478^{+15}_{-18}$ MeV [5],
is about 50 MeV higher than the original value, though the name $Z_{c}(4430)$
is still used. An interesting point is that $Z_{c}(4430)$ is first observed in
the channel including the radially excited state of the charmonium $\psi(2S)$,
not the ground state charmonium $J/\psi$. Dalitz analysis of $B\rightarrow
K\pi\psi(2S)$ decays was performed in [201] and the full amplitude analysis of
$B^{0}\rightarrow K^{+}\pi^{-}\psi(2S)$ decays was done in [202]. As for the
quantum numbers of the $Z_{c}(4430)$, $J^{P}=1^{+}$ were favored in [202] and
confirmed by LHCb [203]. LHCb also performed Argand diagram analysis and
showed its resonance character.

The decay patterns of $Z_{c}(4430)$ exhibit interesting features:

$\displaystyle{\cal B}(B^{0}\rightarrow K^{+}Z_{c}^{-}(4430))\times{\cal
B}(Z_{c}^{-}(4430)\rightarrow\pi^{-}\psi(2S))=(6.0^{+1.7+2.5}_{-2.0-1.4})\times
10^{-5}$ (229)

from Ref. [202], and

$\displaystyle{\cal B}(\bar{B}^{0}\rightarrow K^{-}Z_{c}^{+}(4430))\times{\cal
B}(Z_{c}^{+}(4430)\rightarrow\pi^{+}J/\psi)=(5.4^{+4.0+1.1}_{-1.0-0.9})\times
10^{-6}$ (230)

from Ref. [197]. From the above two results, we obtain the branching ratio:

$\displaystyle\frac{{\cal B}(Z_{c}^{-}(4430)\rightarrow\pi^{-}\psi(2S))}{{\cal
B}(Z_{c}^{+}(4430)\rightarrow\pi^{+}J/\psi)}\simeq 10,$ (231)

which indicates that the decay to $\pi\psi(2S)$ is enhanced relative to the
decay to $\pi J/\psi$. This ordering is against the naive intuition; the decay
rate to $\pi\psi(2S)$ should be smaller than the decay to $\pi J/\psi$ if only
their phase space is considered. Further investigations will be useful for
understanding the internal structure of the $Z_{c}(4430)$.

There are charged bottomonium-like states. The $Z_{b}^{\pm}(10610)$ and
$Z_{b}^{\pm}(10650)$ were found in
$\Upsilon(5S)\rightarrow\pi^{+}\pi^{-}\Upsilon(nS)$ ($n=1,2,3$) and
$\Upsilon(5S)\rightarrow\pi^{+}\pi^{-}h_{b}(mP)$ ($m=1,2$) by Belle [204,
205]¶¶¶$\Upsilon(10860)$ was regarded as $\Upsilon(5S)$.. $Z_{b}^{\pm}(10610)$
has the mass $10607.2\pm 2.0$ MeV and the decay width $18.4\pm 2.4$ MeV, and
$Z_{b}^{\pm}(10650)$ has the mass $10652.2\pm 1.5$ MeV and the decay width
$11.5\pm 2.2$ MeV. It is noted that their masses are very close to the
$B\bar{B}^{\ast}$ and $B^{\ast}\bar{B}^{\ast}$ thresholds. The fact that
$Z_{b}^{\pm}(10610)$ and $Z_{b}^{\pm}(10650)$ were found in
$e^{+}e^{-}\rightarrow\Upsilon(nS)\pi^{+}\pi^{-}$ favors
$I^{G}(J^{P})=1^{+}(1^{+})$, as it is expected from the final state when
$S$-waves are assumed [205].

It is interesting that $Z_{b}^{\pm}(10610)$ and $Z_{b}^{\pm}(10650)$ have a
larger probability in the decay to open heavy mesons rather than the decay to
bottomonia; $Z_{b}^{\pm}(10610)\rightarrow B\bar{B}^{\ast}$ and
$Z_{b}^{\pm}(10650)\rightarrow B^{\ast}\bar{B}^{\ast}$ are the dominant
channels with the fractions about 86 % and 74 %, respectively [206]. For a
charged state, $Z_{b}^{\pm}(10610)$ decays to $\pi\Upsilon(nS)$ ($n=1,2,3$)
and to $\pi h_{b}(mP)$ ($n=1,2$) with the fractions whose values are of the
same order. Situation is the same for the $Z_{b}^{\pm}(10650)$. This indicates
that the heavy quark spin symmetry is violated largely for
$Z_{b}^{\pm}(10610)$ and $Z_{b}^{\pm}(10650)$ [207]. $Z_{b}^{\pm}(10610)$ and
$Z_{b}^{\pm}(10650)$ were also found in $\Upsilon(11020)\rightarrow
h_{b}(nP)\pi^{+}\pi^{-}$ [208], where the mass of $\Upsilon(11020)$ lies above
$\Upsilon(5S)$. The neutral partner, $Z_{b}^{0}(10610)$, was found in
$\Upsilon(5S)\rightarrow\pi^{0}\pi^{0}\Upsilon(nS)$ ($n=1,2,3$) by Belle[209].
However, no significance signal was obtained for $Z_{b}^{0}(10650)$.

The structures of the charged charmonium-like and bottomoniumu-like states
discussed above have been studied theoretically in the hadronic-molecular
approaches [210, 211, 212, 213, 214, 59, 15, 215, 216, 217, 218, 219, 220,
221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231], in the hadrocharmonium
approach [232], in the QCD sum rule approaches [233, 234, 235, 236, 237, 105,
238, 239, 240, 241, 242, 243, 244], in the tetraquark approaches [245, 246,
247, 248, 249, 250, 251, 252, 253, 254], in the heavy quark spin symmetry
approaches [207, 100, 255, 256] and in the lattice QCD approaches [257, 258,
259, 260, 261, 262, 263]. The production and decay processes of the charged
charmonium-like and bottomoniumu-like states have been studied in [264, 265,
266, 267, 268, 269, 270, 271, 272, 273, 274, 275, 276, 277, 278, 279, 280,
281, 282, 283, 284, 285, 286, 287, 288, 289, 290, 291, 292, 293, 294, 295,
296, 297, 298, 299, 300, 301, 302, 303, 304].

###  5.2 Isovector $P^{(\ast)}\bar{P}^{(\ast)}$ molecule with OPEP

In this subsection, we study how the $Z_{c}$ and $Z_{b}$ states are generated
as isovector $P^{(\ast)}\bar{P}^{(\ast)}$ molecular state with the OPEP. We
focus on the states with $J^{PC}=0^{++},$ $1^{+-}$ and $1^{++}$, where the
$S$-wave $P^{(\ast)}\bar{P}^{(\ast)}$ component is included. Among them,
$J^{PC}=1^{+-}$ is assigned as the quantum number of $Z_{c}(3900)$,
$Z_{c}(4200)$, and $Z_{c}(4430)$. The $J^{PC}=1^{++}$ state has not been
reported, but it is the isospin partner of $X(3872)$.

The components of the isovector $P^{(\ast)}\bar{P}^{(\ast)}$ states for
$J^{PC}=0^{++},1^{++}$, and $1^{+-}$ are given by [12, 57]

$\displaystyle 0^{++}:$ $\displaystyle
P\bar{P}(^{1}S_{0}),P^{\ast}\bar{P}^{\ast}(^{1}S_{0},{{}^{5}D_{0}}),$ (232)
$\displaystyle 1^{++}:$
$\displaystyle\frac{1}{\sqrt{2}}\left(P\bar{P}^{\ast}-P^{\ast}\bar{P}\right)(^{3}S_{1},{{}^{3}D_{1}}),P^{\ast}\bar{P}^{\ast}(^{5}D_{1}),$
(233) $\displaystyle 1^{+-}:$
$\displaystyle\frac{1}{\sqrt{2}}\left(P\bar{P}^{\ast}+P^{\ast}\bar{P}\right)(^{3}S_{1},{{}^{3}D_{1}}),P^{\ast}\bar{P}^{\ast}(^{3}S_{1},{{}^{3}D_{1}}).$
(234)

The lowest threshold of the $J^{PC}=0^{++}$ state is $P\bar{P}$, while
$P\bar{P}^{\ast}$ and $P^{\ast}\bar{P}$ are the lowest thresholds for
$J^{PC}=1^{++}$ and $1^{+-}$. We note that the phase convention in (232)-(234)
is different from the one in the literatures [12, 57] as discussed in section
3.7.

In the basis (232)-(234), the matrix elements of the OPEP in (92)-(95) are
given by [12, 57]

$\displaystyle V^{0^{++}}_{\pi}(r)$ $\displaystyle=$
$\displaystyle\left(\frac{g_{A}}{2f_{\pi}}\right)^{2}\frac{1}{3}\left(\begin{array}[]{ccc}0&-\sqrt{3}{\cal
C}_{\pi}&\sqrt{6}{\cal T}_{\pi}\\\ -\sqrt{3}{\cal
C}_{\pi}&2C_{\pi}&\sqrt{2}T_{\pi}\\\ \sqrt{6}{\cal
T}_{\pi}&\sqrt{2}T_{\pi}&-C_{\pi}+2T_{\pi}\\\ \end{array}\right),$ (238)
$\displaystyle V^{1^{++}}_{\pi}(r)$ $\displaystyle=$
$\displaystyle\left(\frac{g_{A}}{2f_{\pi}}\right)^{2}\frac{1}{3}\left(\begin{array}[]{ccc}-{\cal
C}_{\pi}&\sqrt{2}{\cal T}_{\pi}&\sqrt{6}T_{\pi}\\\ \sqrt{2}{\cal
T}_{\pi}&-{\cal C}_{\pi}-{\cal T}_{\pi}&\sqrt{3}T_{\pi}\\\
\sqrt{6}T_{\pi}&\sqrt{3}T_{\pi}&-C_{\pi}+T_{\pi}\\\ \end{array}\right),$ (242)
$\displaystyle V^{1^{+-}}_{\pi}(r)$ $\displaystyle=$
$\displaystyle\left(\frac{g_{A}}{2f_{\pi}}\right)^{2}\frac{1}{3}\left(\begin{array}[]{cccc}{\cal
C}_{\pi}&-\sqrt{2}{\cal T}_{\pi}&-2C_{\pi}&-\sqrt{2}T_{\pi}\\\ -\sqrt{2}{\cal
T}_{\pi}&{\cal C}_{\pi}+{\cal T}_{\pi}&-\sqrt{2}T_{\pi}&-2C_{\pi}+T_{\pi}\\\
-2C_{\pi}&-\sqrt{2}T_{\pi}&C_{\pi}&-\sqrt{2}T_{\pi}\\\
-\sqrt{2}T_{\pi}&-2C_{\pi}+T_{\pi}&-\sqrt{2}T_{\pi}&C_{\pi}+T_{\pi}\\\
\end{array}\right),$ (247)

where the functions ${\cal C}_{\pi}$, ${\cal T}_{\pi}$, $C_{\pi}$ and
$T_{\pi}$ are given in (182). The potential form in the isovector channel is
almost the same as the one in the isoscalar channel, while the isospin factors
are different, ${\boldsymbol{\tau}}_{1}\cdot{\boldsymbol{\tau}}_{2}=+1$ for
$I=1$ and ${\boldsymbol{\tau}}_{1}\cdot{\boldsymbol{\tau}}_{2}=-3$ for $I=0$,
compare (247) and (182).

As in section 4.2, the $P^{(\ast)}\bar{P}^{(\ast)}$ state is studied by
solving the coupled channel Schrödinger equation with the Hamiltonian

$\displaystyle H^{J^{PC}}$ $\displaystyle=$ $\displaystyle
K^{J^{PC}}+V^{J^{PC}}_{\pi}.$ (248)

The kinetic term is given by

$\displaystyle K^{0^{++}}$ $\displaystyle=$ $\displaystyle{\rm
diag}\left(-\frac{1}{2\mu_{P\bar{P}}}\Delta_{0},-\frac{1}{2\mu_{P^{\ast}\bar{P}^{\ast}}}\Delta_{0}+2\Delta
m_{PP^{\ast}},-\frac{1}{2\mu_{P^{\ast}\bar{P}^{\ast}}}\Delta_{2}+2\Delta
m_{PP^{\ast}},\right)$ (249) $\displaystyle K^{1^{++}}$ $\displaystyle=$
$\displaystyle{\rm
diag}\left(-\frac{1}{2\mu_{P\bar{P}^{\ast}}}\Delta_{0},-\frac{1}{2\mu_{P\bar{P}^{\ast}}}\Delta_{2},-\frac{1}{2\mu_{P^{\ast}\bar{P}^{\ast}}}\Delta_{2}+\Delta
m_{PP^{\ast}},\right)$ (250) $\displaystyle K^{1^{+-}}$ $\displaystyle=$
$\displaystyle{\rm
diag}\left(-\frac{1}{2\mu_{P\bar{P}^{\ast}}}\Delta_{0},-\frac{1}{2\mu_{P\bar{P}^{\ast}}}\Delta_{2},-\frac{1}{2\mu_{P^{\ast}\bar{P}^{\ast}}}\Delta_{0}+\Delta
m_{PP^{\ast}},\right.$ (251)
$\displaystyle\left.-\frac{1}{2\mu_{P^{\ast}\bar{P}^{\ast}}}\Delta_{2}+\Delta
m_{PP^{\ast}}\right),$

where $\mu_{P^{(\ast)}\bar{P}^{(\ast)}}$, $\Delta_{\ell}$ and $\Delta
m_{P^{(\ast)}\bar{P}^{(\ast)}}$ are defined in (185)-(187), and the masses of
$P^{(\ast)}=D^{(\ast)},B^{(\ast)}$ are given in table 7.

As studied in section 4.2, we search the parameter region which gives a bound
state by varying the parameters $g_{A}$ and $\Lambda$. As a result, the
boundary of the isovector $D^{(\ast)}\bar{D}^{(\ast)}$ and
$B^{(\ast)}\bar{B}^{(\ast)}$ bound states in the $(g_{A},\Lambda)$ plane is
shown in figure 17. The results for $J^{PC}=0^{++}$ and $1^{+-}$ are similar,
while we note that the lowest thresholds are different, $P\bar{P}$ for
$J^{PC}=0^{++}$, and $P\bar{P}^{\ast}$ $(P^{\ast}\bar{P})$ for
$J^{PC}=1^{+-}$. The bound region of the $J^{PC}=1^{++}$ is slightly larger
than the others, and hence the attraction for $J^{PC}=1^{++}$ is larger than
that for $J^{PC}=0^{++}$ and $1^{+-}$. Comparing the results of the
$D^{(\ast)}\bar{D}^{(\ast)}$ and $B^{(\ast)}\bar{B}^{(\ast)}$ states, the
$B^{(\ast)}\bar{B}^{(\ast)}$ bound region is wider than the
$D^{(\ast)}\bar{D}^{(\ast)}$ one. This is because the heavier mass suppresses
the kinetic energy, and because the the small $BB^{\ast}$ mass splitting
enhances the attraction from the coupled channel effect. For the reference
point $(g_{A},\Lambda)=(0.55,1.13\,{\rm GeV})$, no bound state is found for
the isovector channels of both the charm and bottom sectors. To accommodate
bound states, we need larger $g_{A}$ and/or $\Lambda$. In fact, our previous
choice of the overestimated coupling strength corresponds to the vertical line
$g_{A}\sim 0.83=\sqrt{2}\times 0.59$ [57] (see the footnote§ ‣ 3.4 in page § ‣
3.4) which allowed a shallow bound state for the isovector
$B^{(\ast)}\bar{B}^{(\ast)}$ channel. When we have only the OPEP, larger
$g_{A}$ or $\Lambda$ is needed to produce a isovector
$P^{(\ast)}\bar{P}^{\ast}$ bound state.

$J^{PC}=0^{++}$

$J^{PC}=1^{++}$ | $J^{PC}=1^{+-}$
---|---
|

Figure 17:  The boundary lines of the isovector $P^{(\ast)}\bar{P}^{(\ast)}$
bound states in the $(g_{A},\Lambda)$ plane for $J^{PC}=0^{++}$, $1^{++}$ and
$1^{+-}$. The results of the $D^{(\ast)}\bar{D}^{(\ast)}$ and
$B^{(\ast)}\bar{B}^{(\ast)}$ states are shown by the solid and dashed lines,
respectively. The right side beyond the line is the bound region, while the
left side is the unbound region. The vertical solid lines is the values at
$g_{A}=0.55$, while the horizontal solid and dashed lines are the values at
$\Lambda=\Lambda_{D}=1.13$ GeV and $\Lambda=\Lambda_{B}=1.08$ GeV,
respectively.

Finally, we compare the results for the isovector and isoscalar channels. In
figure 18, the boundary lines of the $D^{(\ast)}\bar{D}^{(\ast)}$ bound states
for $I(J^{PC})=I(1^{++})$ ($I=0,1$) are shown, which were obtained in figure
17 for $I=1$ and in figure 7 for $I=0$. As seen in figure 18, the bound region
for $I=0$ is obviously larger than that for $I=1$. This fact indicates that
the attraction in the $I=0$ channel is stronger than that in the $I=1$
channel. The difference between them comes from the isospin dependence of the
OPEP, which is given by the isospin factor
$\boldsymbol{\tau}_{1}\cdot\boldsymbol{\tau}_{2}$ in (92)-(95);
${\boldsymbol{\tau}}_{1}\cdot{\boldsymbol{\tau}}_{2}=-3$ for the isoscalar
channel, while ${\boldsymbol{\tau}}_{1}\cdot{\boldsymbol{\tau}}_{2}=+1$ for
the isovector channel. For the OPEP, the tensor force in the off-diagonal term
has the dominant role to produce an attraction rather than the diagonal term.
For the off-diagonal term, the sign of the potential is not important, but the
magnitude is important because the off-diagonal term contributes as the second
order of the perturbation. Thus, the attraction in the isoscalar channel with
$|{\boldsymbol{\tau}}_{1}\cdot{\boldsymbol{\tau}}_{2}|=3$ is larger than that
in the isovector one with
$|{\boldsymbol{\tau}}_{1}\cdot{\boldsymbol{\tau}}_{2}|=1$ by about a factor
$9$.

Figure 18:  The boundary lines of the $D^{(\ast)}\bar{D}^{(\ast)}$ bound
states $J^{PC}=1^{++}$ in the $(g_{A},\Lambda)$ plane for the isospin $I=1$
and $0$. The solid and dashed lines show the results for $I=1$ and for $I=0$,
respectively, which are obtained in figure 17 and figure 7. The right side
beyond the line is the bound region, while the left side is the unbound
region. The vertical and horizontal solid lines show the values at
$g_{A}=0.55$ and at $\Lambda=\Lambda_{D}=1.13$ GeV, respectively.

###  5.3 $P^{(\ast)}\bar{P}^{(\ast)}$ molecule with OPEP and $\sigma$ exchange
potential

In the previous subsection, we have seen that the OPEP contribution is rather
small in the isovector channel. In such a situation, the short and middle
range interactions may become important. Such interaction includes $\sigma$,
$\rho$ and $\omega$ exchange interactions [305]. In the isoscalar
$P^{(\ast)}\bar{P}^{(\ast)}$ channel for $X(3872)$, we have not considered
them because of a reason discussed below. In the isovector channel, there is
another reason that we expect that the vector meson exchanges are not
important; $\rho$ and $\omega$ exchange interactions have opposite signs and
mostly cancel.

The $\sigma$ exchange potential may be effective not only for the isovector
channel but also for the $X(3872)$ isoscalar channel. One of the reasons that
we have considered only OPEP in the previous sections is that the effect of
the short range interaction including the $\sigma$ exchange has been
effectively taken care of by the suitable choice of the cutoff parameter
$\Lambda$. To determine the reference value of $\Lambda$, we have used the
binding energy of the deuteron. The OPEP thus determined for the nucleon-
nucleon interaction is extrapolated to that of $D\bar{D}^{\ast}$ by assuming
hadron structures by constituent quarks and the pion coupling to the light
quarks. Strictly, however, we do not know to what extent such extrapolation
works. Therefore, in this subsection we consider the role of $\sigma$-exchange
interaction in some detail. Analysis here also provides an estimate of
ambiguities coming from short range interactions.

The $\sigma$ exchange potential is derived by the effective Lagrangian of the
heavy and $\sigma$ mesons [306],

$\displaystyle{\cal L}_{\sigma HH}=g_{\sigma}{\rm
Tr}\left[H\sigma\bar{H}\right].$ (252)

From this Lagrangian, the $\sigma$ exchange potential is obtained as

$\displaystyle V^{\sigma}_{P\bar{P}{\mathchar
45\relax}P\bar{P}}(r)=-\left(\frac{g_{\sigma}}{m_{\sigma}}\right)^{2}C(r;m_{\sigma},\Lambda),$
(253) $\displaystyle V^{\sigma}_{P\bar{P}^{\ast}{\mathchar
45\relax}P\bar{P}^{\ast}}(r)=-\left(\frac{g_{\sigma}}{m_{\sigma}}\right)^{2}C(r;m_{\sigma},\Lambda),$
(254) $\displaystyle V^{\sigma}_{P^{\ast}\bar{P}^{\ast}{\mathchar
45\relax}P^{\ast}\bar{P}^{\ast}}(r)=-\left(\frac{g_{\sigma}}{m_{\sigma}}\right)^{2}C(r;m_{\sigma},\Lambda),$
(255)

where the $\sigma$ mass $m_{\sigma}=550$ MeV is used. The factor
$1/m^{2}_{\sigma}$ is multiplied because the function
$C(r;m_{\sigma},\Lambda)$ includes $m^{2}_{\sigma}$ (see (87)).

There is ambiguity in the value of the coupling constant $g_{\sigma}$. In
[305], $g_{\sigma}=3.65$ is taken, which is determined by a quark model
estimation. This value is one-third of the value of the $\sigma NN$ coupling
$g_{\sigma NN}$ according to the quark number counting, because the $\sigma$
meson couples to the scalar charge of hadrons which is additive. Another way
to estimate $g_{\sigma}$ is to use a chiral theory for quarks such as the
Nambu-Jona-Lasinio model [25, 26, 307]. By using the equality of the $\sigma
qq$ and $\pi qq$ couplings, and the Goldberger-Treiman relation for the
constituent quark, we have the relation

$\displaystyle\frac{g_{\sigma}}{m_{q}}=\frac{g_{A}^{q}}{f_{\pi}}.$ (256)

By using the parameter values $m_{q}\sim 300$ MeV, $f_{\pi}=93$ MeV,
$g_{A}^{q}\sim 0.55$, we find $g_{\sigma}\sim 1.8$. Yet, in [306], an even
smaller value $g_{\sigma}=g_{\pi}/2\sqrt{6}\sim 0.76$ is obtained, where
$g_{\pi}=\Delta M/f_{\pi}$ and $\Delta M$ is the mass difference between the
$0^{+}$ and $0^{-}$ heavy-light mesons. $\Delta M=349$ MeV is used in [306],
which is the mass difference between $D^{\ast+}_{s0}$ and $D^{+}_{s}$. In this
subsection, we present the results for $g_{\sigma}=0.76$ and $3.65$, which are
regarded as the lower and upper limits of the attractive contribution due to
the sigma meson exchange potential.

Using $\Lambda_{N}=681$ MeV for the $\pi\sigma$ potential for the nucleon as
shown in table 4, $\Lambda_{D}$ and $\Lambda_{B}$ are obtained by $919$ MeV
and $878$ MeV, respectively. In the basis of (232)-(233), the matrix elements
of the $\sigma$ exchange potential for the given $J^{PC}$ are obtained by

$\displaystyle V^{0^{++}}_{\sigma}(r)$ $\displaystyle=$
$\displaystyle-\left(\frac{g_{\sigma}}{m_{\sigma}}\right)^{2}\left(\begin{array}[]{ccc}C_{\sigma}&0&0\\\
0&C_{\sigma}&0\\\ 0&0&C_{\sigma}\\\ \end{array}\right),$ (260) $\displaystyle
V^{1^{++}}_{\sigma}(r)$ $\displaystyle=$
$\displaystyle-\left(\frac{g_{\sigma}}{m_{\sigma}}\right)^{2}\left(\begin{array}[]{ccc}C_{\sigma}&0&0\\\
0&C_{\sigma}&0\\\ 0&0&C_{\sigma}\\\ \end{array}\right),$ (264) $\displaystyle
V^{1^{+-}}_{\sigma}(r)$ $\displaystyle=$
$\displaystyle-\left(\frac{g_{\sigma}}{m_{\sigma}}\right)^{2}\left(\begin{array}[]{cccc}C_{\sigma}&0&0&0\\\
0&C_{\sigma}&0&0\\\ 0&0&C_{\sigma}&0\\\ 0&0&0&C_{\sigma}\\\
\end{array}\right),$ (269)

with the function $C_{\sigma}=C(r;m_{\sigma},\Lambda)$.

(i) $J^{PC}=0^{++}$

(ii) $J^{PC}=1^{++}$ | (iii) $J^{PC}=1^{+-}$
---|---
|

Figure 19:  The boundary of the isovector $D^{(\ast)}\bar{D}^{(\ast)}$ bound
state with $V_{\pi}$ and $V_{\sigma}$ for (i) $J^{PC}=0^{++}$, (ii) $1^{++}$
and (iii) $1^{+-}$ in the $(g_{A},\Lambda)$ plane. The solid line shows the
result for $g_{\sigma}=0$, namely only with $V_{\pi}$, while the dashed and
dashed-dot lines are the results for $g_{\sigma}=0.76$ and $3.65$,
respectively. The right side beyond the boundary line is the bound region,
while the left side is the unbound region. The vertical and horizontal solid
lines show the values at $g_{A}=0.55$ and at $\Lambda=\Lambda_{D}=0.92$ GeV,
respectively.

In figure 19, the boundaries of the isovector $D^{(\ast)}\bar{D}^{(\ast)}$
bound states are shown for the case only with OPEP ($V_{\pi}$) is used and for
the case of the $\pi\sigma$ exchange potential ($V_{\sigma}$) used in the
$(g_{A},\Lambda)$ plane. The result with $g_{\sigma}=0$ is corresponds to the
one only with the OPEP as shown in figure 17. Since $V_{\sigma}$ is
attractive, the bound region for $g_{\sigma}\neq 0$ is larger than that for
$g_{\sigma}=0$. For the small coupling $g_{\sigma}=0.76$, the boundary is
close to the one for $g_{\sigma}=0$. Thus the $V_{\sigma}$ contribution is
small, and the OPEP plays the dominant role. For $g_{\sigma}=3.65$, however,
the bound region is much larger than that for $g_{\sigma}=0$ and $0.76$.

(i) $J^{PC}=0^{++}$

(ii) $J^{PC}=1^{++}$ | (iii) $J^{PC}=1^{+-}$
---|---
|

Figure 20:  The boundary of the isovector $B^{(\ast)}\bar{B}^{(\ast)}$ bound
state for (i) $J^{PC}=0^{++}$, (ii) $1^{++}$ and (iii) $1^{+-}$ in the
$(g_{A},\Lambda)$ plane. The same convention is used as figure 19, while the
vertical and horizontal solid lines show the values at $g_{A}=0.55$ and
$\Lambda=\Lambda_{B}=0.88$ GeV, respectively.

The results for the isovector $B^{(\ast)}\bar{B}^{(\ast)}$ state are
summarized in figure 20 for $J^{PC}=0^{++},1^{++}$ and $1^{+-}$. As seen in
the $D^{(\ast)}\bar{D}^{(\ast)}$ state, the bound region of the
$B^{(\ast)}\bar{B}^{(\ast)}$ state becomes large as the coupling $g_{\sigma}$
increases. The result for $g_{\sigma}=0.76$ is the similar to the one for
$g_{\sigma}=0$. For $g_{\sigma}=3.65$, the bound region is much larger than
those for $g_{\sigma}=0$ and $0.76$. For $(g_{A},\Lambda)=(0.55,0.88\,{\rm
GeV})$, there are bound states for $J^{PC}=0^{++},1^{++}$ and $1^{+-}$. The
binding energies are $0.92$ MeV for $0^{++}$, $1.51$ MeV for $1^{++}$, and
$0.76$ MeV for $1^{+-}$. In experiments, however, the charged
$Z^{(\prime)}_{b}$ states have been found above the
$B^{(\ast)}\bar{B}^{(\ast)}$ threshold so far. The attraction generated by the
interaction parameters $(g_{A},\Lambda)=(0.55,0.88\,{\rm GeV})$ could be
overestimated. In the study of the hadronic molecules, the uncertainty of the
parameters remains a problem, which should be addressed.

(i) $D^{(\ast)}\bar{D}^{(\ast)}$ | (ii) $B^{(\ast)}\bar{B}^{(\ast)}$
---|---
|
Figure 21:  The boundary of the isovector (i) $D^{(\ast)}\bar{D}^{(\ast)}$ and
(ii) $B^{(\ast)}\bar{B}^{(\ast)}$ bound states in the $(g_{\sigma},\Lambda)$
plane. The vertical lines show the values at $g_{s}=0.76$ and $g_{s}=3.65$.
The horizontal lines show the values at $\Lambda=\Lambda_{D}=0.92$ GeV for (i)
$D^{(\ast)}\bar{D}^{(\ast)}$ and at $\Lambda=\Lambda_{B}=0.88$ GeV for (ii)
$B^{(\ast)}\bar{B}^{(\ast)}$.

In the end, we show $g_{\sigma}$-$\Lambda$ plots to see continuously the
change in the role of $\sigma$ exchange as $g_{\sigma}$ is varied figure 21.
For the charm sector, the left figure indicates an unlikely situation for the
molecular states to be generated at the mean value $g_{\sigma}\lesssim$ 1.8,
where a very large cutoff is needed, $\Lambda\sim 4$ GeV. For bottom sector,
the molecular states are not yet generated there, but only slight increase of
$g_{\sigma}$ will do.

##  6 Pentaquark baryons

The discussion of exotic hadrons was activated when two new observations were
reported in 2003; one is the $X(3872)$ which we have discussed in detail in
this review, and the other the pentaquark $\Theta^{+}$ [308]. While the
observation of the $X(3872)$ was rather accidental in the study of charmonium
spectrum, that of $\Theta^{+}$ was motivated by the theoretical prediction by
Diakonov et al [309]. The expected flavor quantum number $S=+1$ of the
$\Theta^{+}$ requires the minimal quark content $uudd\bar{s}$, and hence the
genuine multiquark exotic state. Various theoretical models have shown very
different features, which was thought as an indication that we understood
hadrons beyond the ground states only poorly. As in Ref. [309], a model of
chiral symmetry with strong pionic soliton correlations predicts a positive
parity state with a relatively low mass at around 1.5 GeV. The positive parity
is also explained due to the strong pionic spin and isospin correlation [310].
One unique feature of this model is that the decay width is very narrow around
10 MeV or less. A model with strong diquark correlations can also make similar
predictions, though their mechanisms are very much different [311]. Contrary,
conventional quark model predicts a negative parity state at relatively large
mass around 1.7 GeV or more and with a wide decay width [312]. Therefore, the
experimental signals of mass at around 1.5 GeV with a narrow width seemed to
support the chiral or diquark model.

After the first observation many data appeared supporting the $\Theta^{+}$,
which are, however, followed by a number of experiments with no evidence
[313]. There, more experimental data were taken and higher statistics analyses
were done. In Ref. [314] how statistical fluctuations would have lead no
evidence is discussed. One cannot, however, definitely conclude that these
results have proven that the $\Theta^{+}$ does not exist. There are also
discussions how multiquark exotics can be seen in some processes while not in
the others [315]. It is particularly so when many experiments utilizes
indirect processes induced by photons, leptons and hadrons (protons or pions)
because the direct formation experiments, for instance, $K^{+}+n\to\Theta^{+}$
with a free neutron, is not possible. Therefore, we may expect further
analyses with improved signal to noise ratio or direct experiments [316].

Aside from the $\Theta^{+}$, $\Lambda(1405)$ has long been an exotic baryon
candidate that is described as a $\bar{K}N$ molecule [317, 318, 319, 5]. Its
existence was first inferred by the analysis of $\bar{K}N$ scattering [11]. It
is the lowest negative parity hyperon with an excitation energy of about 300
MeV. This amount is significantly small as compared to the others. For
instance, the lowest negative parity nucleon $N(1535)$ is about 600 MeV above
the ground state nucleon. This feature is not easy to be explained by the
conventional quark model where baryons are made from three valence quarks. In
Ref. [320], by introducing five quark states corresponding to the molecular
channels such as $\bar{K}N$ coupled to the three-quark states, they have shown
that the resonance structure at around 1405 MeV was generated.

Whether the picture of hadronic molecule $\bar{K}N$ crucially depends on the
interaction between them. In the low energy theorem of chiral symmetry, the
$\bar{K}N$ interaction appears attractive as given by the Weinberg-Tomozawa
theorem. Physically, much of attraction strength is provided by $\omega$ meson
exchange. At the quark level, it originates from the interaction between the
antiquark $\bar{q}$ in the antikaon $\bar{K}$ and the quarks $q$ in the
nucleon $N$. Due to charge conjugation, the sign of the vector type
interaction flips from the one between quarks. Such a picture has been shown
in the Skyrme model, which is one of successful chiral models for baryons
[321, 322, 323, 324]. As a result, a $\bar{K}N$ bound state appears near the
$\bar{K}N$ threshold. By coupling to the lower $\pi\Sigma$ channel, the bound
state turns into a resonance, which is a typical mechanism of a Feshbach
resonance. An experimental study of the $\bar{K}N$ molecule is going at J-PARC
and analysis is underway [325, 326, 327].

Another important topic is the hidden charm $P_{c}$ baryons observed by LHCb
in the weak decay $\Lambda_{b}\to J/\\!\psi p\bar{K}$. Generally speaking,
heavier constituents are more likely to be bound or resonate due to the
suppression of their kinetic energies. Thus we expect more chances for exotic
baryons. The first observation was reported in 2015 with a prominent peak
structure in the invariant mass plot of $J/\\!\psi p$ [328]. In their detailed
analysis, they claimed that the peak was generated by two resonant states, one
at 4380 MeV and the other at 4450 MeV. The analysis has been further performed
with more statistics data in 2019, and they have reported three narrow peaks
[19]; two at 4440 MeV and 4457 MeV that seem to split the strength of the
former prominent peak, and one at 4312 MeV that was not seen in the former
analysis. Thus they are denoted as $P_{c}(4312)^{+},P_{c}(4440)^{+}$ and
$P_{c}(4457)^{+}$. Interestingly these three peaks are just below the
thresholds; the higher two below $\Sigma_{c}\bar{D}^{*}$ threshold, and the
lower one below the $\Sigma_{c}\bar{D}$ threshold.

The new observation has lead to a number of theory discussions of heavy quark
multiplets formed by the four combinations of $\Sigma_{c},\Sigma_{c}^{*}$ and
$\bar{D},\bar{D}^{*}$ [329, 330, 331, 332]. In the heavy quark limit, the pair
of $\bar{D}$ and $\bar{D}^{*}$, and the pair of $\Sigma_{c}$ and
$\Sigma_{c}^{*}$ are considered as a spin doublet of $J=0$ and 1, and the one
of $J=1/2$ and 3/2, respectively. Their hadronic molecules also form
multiplets of heavy quark spin symmetry. In connection with the present
discussions, in the formation of these multiplets the tensor interaction of
OPEP seem to play an important role [333]. It not only binds the constituent
hadrons of these states but also splits the above two $P_{c}(4440)^{+}$,
$P_{c}(4457)^{+}$ as a spin doublet.

Because the LHCb observed these states in the $J/\\!\psi p$ final state, the
isospin of these $P_{c}$ states is $I=1/2$. Therefore their masses are,
respectively, 23 MeV and 6 MeV below the isospin averaged threshold of
$\Sigma_{c}\bar{D}^{*}$, 4463 MeV. A relevant question is the origin of the
masses, decay widths and quantum numbers of these states. Assuming that the
orbital motion of the molecules is dominated by $S$-waves, possible total spin
values are $J=1/2$ and 3/2. Now the crucial observation is that the tensor
interaction can be effective for both states because the $S$-$D$ couplings
survive for both channels. This is understood by the fact that the sum of
$S(\Sigma_{c})=1/2,S(\bar{D}^{*})=1,L=0$ and the sum of
$S(\Sigma_{c})=1/2,S(\bar{D}^{*})=1,L=2$ can both make the total spin $J=1/2$
and 3/2. This contrasts with the two nucleon system, where two states of $J=1$
and 0 are possible while the tensor force is effective only for the $J=1$
state (corresponding to the deuteron). Therefore, the two $P_{c}$’s provide an
interesting opportunity to study the role of the tensor force in the OPEP.

Having said this much, in [333] the role of the tensor force in the OPEP has
been discussed in a coupled channel model of $\Sigma_{c}^{(*)}\bar{D}^{(*)}$,
$\Lambda_{c}\bar{D}^{(*)}$ with OPEP supplemented by the short range
interaction that is brought about by the coupling of the molecular states with
compact five quark states [334]. An interesting observation there is that by
adjusting one most important model parameter for the short range interaction,
they have made predictions of ten states. Three of them correspond to the
$P_{c}$ states of the LHCb with good agreement with the observed masses and
decay widths. The quantum numbers for the would be doublet $P_{c}(4440)$ and
$P_{c}(4457)$ are then identified with $J=3/2$ and 1/2, respectively. It turns
out that the mechanism of lowering the $J=3/2$ state is mostly due to the
tensor force. The assignment of these quantum numbers is unique because the
tensor force acts in the second order and the sign of the interaction does not
matter. In general spin-spin interaction with heavy quark symmetry is employed
to explain the splitting. However, the quantum numbers are not uniquely
determined. Therefore, in Ref. [331], two options were investigated for the
spin assignments. The spins of the $P_{c}$ states are not yet known, and hence
the determination of them is very important for the further understanding of
these states.

##  7 Summary and complements

###  7.1 Brief summary

In this article we have discussed the hadronic molecule as one of the exotic
structures of hadrons. It has become possible experimentally to observe
various exotic phenomena long after the predictions made more than half
century ago, which have stimulated a diverse body of theoretical work. The
ingredients of hadronic molecules are constituent hadrons and their
interactions. The constituent hadrons also couple to compact structures.
Therefore, we have discussed in detail how the admixture model has been
applied to the $X(3872)$.

From a first-principle point of view, such a picture should effectively and
conveniently replace the direct but complicated approach of QCD. In other
words, the model should be economized [33, 335, 34], such that its work
region, where and how, is under control. As emphasized in the introduction,
hadronic molecules are expected to appear near threshold regions. Their
formation is a consequence of finely tuned hadron dynamics, as their binding
or resonant energies are of order MeV which is much smaller than the scale of
low energy QCD, $\Lambda_{QCD}\sim$ some hundreds MeV. We have discussed why
such conditions are likely to be realized for heavy hadrons. Their kinetic
energies are suppressed and relatively weak force is sufficient to generate
hadronic molecules.

For the interaction, we have emphasized the role of the one-pion exchange
interaction. Pion dynamics are well established because the pion is the Nambu-
Goldstone boson of spontaneous breaking of chiral symmetry, where the pion
interaction is dictated by the low energy theorems. In the constituent quark
picture the pion interacts with the light $u,d$ quarks by the pseudoscalar
Yukawa coupling of $\boldsymbol{\sigma}\cdot\bi q$ type, whose strength is
extracted from the empirically known axial coupling constants of hadrons.

The long range part of hadron interactions is provided by the OPEP, which we
have discussed in detail in this paper. Because of the spin structure of the
Yukawa coupling, the OPEP contributes to the transitions $D\to D^{*}$ and
$D^{*}\to D^{*}$. It turns out that they are effective in the formation of a
$D\bar{D}^{*}$ molecule for the $X(3872)$ channel . Therefore, an emphasis has
been put on the role of the tensor component of the OPEP that causes mixing of
states (channels) differing in angular momenta by 2$\hbar$. Because this
transition leads to second order process, the resulting interaction for the
relevant low lying channel must be attractive, with more attraction with more
channel couplings.

There still remains a question for the short-range part, because the bare OPEP
is singular. To avoid this we introduce a form factor. Such a prescription is
known to work well for low energy properties of the deuteron [53] by employing
a cutoff parameter around $\Lambda\sim 800$ MeV (see Table 4). To extend such
a prescription to $DD^{*}$ molecule, in particular to determine the coupling
strength and cutoff values for $DD^{*}$, we have employed a counting of quarks
and the sizes of the nucleon and $D^{(*)}$ using the quark model.

It is interesting to note that the employed cutoff, $\Lambda\sim 800$ MeV for
the nucleon, is consistent with the size of the nucleon core
$\sqrt{6}/\Lambda\sim 0.5$ fm. Discussions of the nucleon core have a long
history. It is recognized as the repulsive core of the $NN$ interaction, which
has been explicitly shown in the study by the quark cluster model [336, 337].
It was also discussed in the chiral bag model where the nucleon is expressed
as a quark core with pion clouds [338, 44].

Having this construction of the interactions, the OPEP provides a non-
negligible amount of attraction particularly for hidden heavy hadrons such as
$X(3872)$ with isospin 0, where the tensor force plays the dominant role,
while the central component plays little. Therefore, the inclusion of the
channel coupling of $SD$ waves (generally, states that differ in angular
momenta by 2$\hbar$) is very important. The resulting strength of the
attraction, however, turns out not to be sufficient to generate the $X(3872)$
as a molecular state of $D\bar{D}^{*}$. The coupling/mixing with a compact
state of $c\bar{c}$ supplies additional attraction, if the compact state has a
larger mass such as that theoretically expected for $\chi_{c1}(2P)$ charmonium
meson. The mixing is also required to explain the large production rates of
the $X(3872)$ in high energy hadron processes. Quantitative estimates of the
production rate, however, have to be done carefully [133, 223, 144, 131, 132].
In the present analysis of the $X(3872)$, the OPEP and the short-range
coupling play roughly equal roles. In general, however, their relative
importance depends on the system under study.

Another possible molecule that we have discussed is the $Z_{c}(3900)$.
However, the strength of the OPEP for the $Z_{c}(3900)$ of isospin one is
smaller than that for the $X(3872)$ of isospin zero by factor three. As a
result, the attraction is reduced and the formation of molecular state is less
likely. In the remaining part of this article, we have also discussed the
above features for the bottom sector.

###  7.2 Resonances or cusps

Here, we briefly mention a question which one would yet like to ask; whether
the observed exotic phenomena imply physical resonant states or cusps of
virtual states. At this moment, we have no decisive answer to this question
for the observed signals, while there are many articles discussing the nature
of the signals theoretically. Here we just refer only to a few of them in
relation with the $X(3872)$ [185, 147, 186, 187, 339]. There, amplitude
analyses are performed by using parametrizations of Flattè or effective range
expansion types. Then an observation was made that by suitably choosing
parameters, the line shape $X(3872)$ was shown to emerge as a virtual state
cusp at the threshold [147].

To reproduce a very narrow or sharp peak at the $D^{0}\bar{D}^{*0}$ threshold,
however, there must be sufficient amount of attraction between them. If the
attraction is larger than the critical value, the peak appears as a resonant
state, otherwise as a virtual state cusp. The difference between them is
subtle because only a small change in the interaction strength may change the
nature of the peak. Moreover, in such a situation it is difficult to
differentiate them experimentally. But then the important question is; what
would be the mechanism to provide that suitable amount of the attraction? In
this paper, we have tried to offer an option that a model with the pion
exchange interaction does it, supplemented by a coupling to a short distance
structure. This is a dynamical approach for the construction of amplitudes
that we discuss shortly below.

###  7.3 Hadron interactions and exotics

The last issue that we would like to mention is the dynamical approach for the
construction of amplitudes from reliable hadron interactions. For heavy
hadrons including charm or bottom quarks, it is formidably difficult to derive
interactions from experiments. This is the reason that we have resorted to a
model for the study of the the $X(3872)$ in this paper.

Yet another powerful and promising method is lattice QCD, which is, in
principle, the first principle method for the strong interaction. In the so
called HAL QCD method, hadron interactions are obtained by using the Nambu-
Bethe-Salpeter amplitude [340, 341]. To obtain hadron interactions, this
method is practically more powerful than the widely used Luscher’s method
[342, 343]. An attempt was made for the $Z_{c}(3900)$ with coupled channels of
$D\bar{D}^{*}$, $\eta_{c}\rho$, $J/\\!\psi\pi$, where they have derived the
interactions between these channels and solved the coupled channel problem
[262, 263]. Unexpectedly it was found that there is a rather strong coupling
between $J/\\!\psi\pi$ and $D\bar{D}^{*}$ channels, which effectively causes
an attraction in the $J/\\!\psi\pi$ channel. As a consequence, they have found
rather than a resonance, a virtual state pole that contributes to an
enhancement near the $D\bar{D}^{*}$ threshold corresponding to $Z_{c}(3900)$.

For the study of exotic hadrons, an approach based on the coupled channel
method with suitable hadron interactions is highly desired. It is a non-
trivial program because many channels may couple, including those with more
than two particles. With complementary approaches of experiments, effective
theories and lattice simulations, such an approach can be further elaborated,
thereby enabling elucidation of the nature of exotic hadrons.

## Acknowledgments

The authors thank S. Yasui for fruitful discussions and useful comments. This
work is supported in part by the Special Postdoctoral Researcher Program
(SPDR) of RIKEN (Y.Y.), and by JSPS KAKENHI Grant Numbers JP16K05361 (S.T. and
M.T.), JP17K05441, and Grants-in-Aid for Scientific Research on Innovative
Areas (No. 18H05407) (A.H.).

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