# Theoretical interpretation of the $\Xi(1620)$ and $\Xi(1690)$ resonances
seen in $\Xi_{c}^{+}\to\Xi^{-}\pi^{+}\pi^{+}$ decay

Hai-Peng Li  Department of Physics, Guangxi Normal University, Guilin 541004,
China  Gong-Jie Zhang  Department of Physics, Guangxi Normal University,
Guilin 541004, China  Wei-Hong Liang  Department of Physics, Guangxi Normal
University, Guilin 541004, China  Guangxi Key Laboratory of Nuclear Physics
and Technology, Guangxi Normal University, Guilin 541004, China  E. Oset
Department of Physics, Guangxi Normal University, Guilin 541004, China
Departamento de Física Teórica and IFIC, Centro Mixto Universidad de Valencia-
CSIC Institutos de Investigación de Paterna, Aptdo.22085, 46071 Valencia,
Spain

###### Abstract

We study the Belle reaction $\Xi_{c}^{+}\to\Xi^{-}\pi^{+}\pi^{+}$ reaction
looking at the mass distribution of $\pi^{+}\Xi$, where clear signals for the
$\Xi(1620)$ and $\Xi(1690)$ resonances are seen. These two resonances are
generated dynamically from the interaction in coupled channels of
$\pi\Xi,\bar{K}\Lambda,\bar{K}\Sigma$ and $\eta\Xi$ within the chiral unitary
approach. Yet, the weak decay process at the quark level, together with the
hadronization to produce pairs of mesons, does not produce the $\pi\pi\Xi$
final state. In order to produce this state one must make transitions from the
$\bar{K}\Lambda,\bar{K}\Sigma$ and $\eta\Xi$ components to $\pi\Xi$, and this
interaction is what produces the resonances. So, the reaction offers a good
test for the molecular picture of these resonances. Adding the contribution of
the $\Xi^{*}(1530)$ and some background we are able to get a good reproduction
of the mass distribution showing the signatures of the two resonances as found
in the experiment.

##  I Introduction

The $\Xi$ states are still relatively poorly known and only a few of them have
been observed pdg . One of them, the $\Xi(1690)$, has a three star status, and
another one, the $\Xi(1620)$, is rated with only one star, and both of them
appear with unknown spin-parity. The $\Xi(1620)$ is observed in
$\Xi(1620)\to\pi\Xi$ decay with large statistical uncertainties Briefel ;
bellefon ; ross  and is not found in Ref. Hassall . From the theoretical point
of view the quark models systematically give the first excited state, after
the $\Xi(\frac{1}{2}^{+})$, $\Xi(\frac{3}{2}^{+})$ ground states, at about
$1750$ MeV Isgur ; Metsz ; Riska ; Roberts ; Xiao  and so does the algebraic
model of Ref. Bijker . One exception is the Skyrme model of Ref. Oh , assuming
a soliton-$h$ system, where two states of $\frac{1}{2}^{-}$ with $1616$ MeV
and $1658$ MeV are obtained. The situation with the $\Xi(1620)$ state is
similar to that of the $\Lambda(1405)$, where quark models overestimate the
energy. In both cases, the advent of chiral dynamics, implemented with coupled
channels and unitarity, came to the rescue and produced a global picture where
these states appear naturally review . In particular, in Refs. Bennhold ;
Carmen  the $\Xi(1620)$ and $\Xi(1690)$ appeared both with
$J^{P}=\frac{1}{2}^{-}$, coming from the interaction of the coupled channels
$\pi\Xi$, $\bar{K}\Lambda$, $\bar{K}\Sigma,\eta\Xi$.

The advent of new facilities as Belle, LHCb, BES has added a new dimension to
hadron spectroscopy and new states or old ones with excellent statistics are
being observed Olsen ; Karliner ; slzhurep . In particular weak decays of
heavy hadrons have added a new dimension to the information available on
dynamically generated resonances, stemming from the interaction of hadron
states weakrev . In that line, it was suggested in Ref. Miyahara  to study the
$\Xi_{c}\to\pi\pi\Xi$ reaction and look for the $\Xi(1620)$ and $\Xi(1690)$
resonances in the $\pi\Xi$ spectrum. Such reaction has been recently measured
at Belle Belleexp  and excellent signals for these resonances have been
observed. The purpose of the present work is to use this experimental
information to further dig into the nature of these resonances.

##  II Generation of the $\Xi(1620)$ and $\Xi(1690)$ states

Using the chiral unitary approach, we follow closely the work of Ref. Bennhold
and consider the $\pi\Xi$ scattering with the
$\pi\Xi,\bar{K}\Lambda,\bar{K}\Sigma,\eta\Xi$ coupled channels. The $s$-wave
scattering amplitudes in matrix form are given by means of the Bethe-Salpeter
(BS) equation

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

where $V_{ij}$, the transition potential between channels, is obtained from
the chiral Lagrangians and given explicitly in Ref. Bennhold . The $G$
function is the diagonal loop function of the meson-baryon propagators. Here
we differ a bit from Ref. Bennhold  and use the cutoff method to regularize
the loop function, given by

$G_{l}=\int_{|\vec{q}\,|<q_{\rm max}}\frac{{\rm
d}^{3}q}{(2\pi)^{3}}\,\frac{1}{2\,w_{l}(\vec{q}\,)}\,\frac{M_{l}}{E_{l}(\vec{q}\,)}\,\frac{1}{\sqrt{s}-w_{l}(\vec{q}\,)-E_{l}(\vec{q}\,)+i\epsilon},$
(2)

where $w_{l}(\vec{q}\,)=\sqrt{m_{l}^{2}+\vec{q}\,^{2}}$,
$E_{l}(\vec{q}\,)=\sqrt{M_{l}^{2}+\vec{q}\,^{2}}$, and $m_{l},M_{l}$ are the
meson, baryon masses of the channel $l$. $G_{l}$ is regularized cutting with a
three momentum $q_{\rm max}$. This is the only parameter in the theory, and
input from experiment is needed to determine it. The results from the Belle
experiment Belleexp  provide this information. Then the BS equation of Eq. (1)
is solved, and poles of the $T$ matrix in the second Riemann sheet are
searched for.

There exist two poles found in the $T$ amplitude, which are shown in Table 1
with different values of the cutoff $q_{\text{max}}$.

Table 1: Poles of the $T$ matrix with different values of $q_{\text{max}}$. (in MeV) $q_{\text{max}}$ | $630$ | $700$ | $750$ | $770$
---|---|---|---|---
poles | $1569.4+125.7i$ | $1563.7+106.1i$ | $1558.0+94.0i$ | $1555.6+89.7i$
$1687.9+0.7i$ | $1681.8+1.8i$ | $1674.8+2.3i$ | $1671.5+2.4i$

The two poles correspond to the $\Xi^{*}$ resonances generated dynamically
from the coupled-channel $\pi\Xi$ interaction, having the quantum numbers of
isospin $I=\frac{1}{2}$ and spin-parity $J^{P}=\frac{1}{2}^{-}$. The first
pole locates below the $\bar{K}\Lambda$ threshold ($\sim 1611$ MeV) and has a
rather wide width. While the second pole appears near the $\bar{K}\Xi$
threshold ($\sim 1689$ MeV), with a narrow width. We identify the first pole
and the second pole as the $\Xi(1620)$ and $\Xi(1690)$ states, respectively.
According to Table 1, as the cutoff $q_{\text{max}}$ increases from $630$ to
$770$ MeV, the positions of the two poles decrease by less than $17$ MeV. This
means that the masses of the $\Xi(1620)$ and $\Xi(1690)$ states are not very
much dependent on $q_{\text{max}}$. But their widths are more sensitive to the
cutoff.

To further understand the molecular components of the $\Xi(1620)$ and
$\Xi(1690)$ states, we calculated the couplings of the two resonances to
different channels, which are showed in Table 2 when taking
$q_{\text{max}}=630$ MeV.

Table 2: Couplings of the two generated states to different channels. (with $q_{\text{max}}=630\;{\rm MeV}$) $\bm{1569.4+i125.7}$ | $\pi\Xi$ | $\bar{K}\Lambda$ | $\bar{K}\Sigma$ | $\eta\Xi$
---|---|---|---|---
$g_{i}$ | $-2.0-1.6i$ | $1.9+0.9i$ | $0.7+0.5i$ | $-0.1-0.4i$
$|g_{i}|^{2}$ | $6.6$ | $4.5$ | $0.6$ | $0.1$
$\bm{1687.9+i0.7}$ | $\pi\Xi$ | $\bar{K}\Lambda$ | $\bar{K}\Sigma$ | $\eta\Xi$
$g_{i}$ | $0.1-0.1i$ | $0.2+0.1i$ | $-1.2+0.2i$ | $-0.8+0.1i$
$|g_{i}|^{2}$ | $0.01$ | $0.04$ | $1.5$ | $0.6$

One can see that the $\Xi(1620)$ state couples mainly to the $\pi\Xi$ and
$\bar{K}\Lambda$ channels, while the $\Xi(1690)$ state couples strongly to the
$\bar{K}\Sigma$ channel. In Ref. Bennhold  where dimensional regularization is
used to regularize the loop function, only one exited $\Xi$ state with mass
around $1600$ MeV, which is identified as the $\Xi(1620)$, is generated from
the $\pi\Xi$ interaction in coupled channel. The reason why the $\Xi(1690)$
state is not found might be that its mass is very much close to the
$\bar{K}\Sigma$ threshold. After switching to the cutoff method to deal with
the divergence of the loop function, both the $\Xi(1620)$ and $\Xi(1690)$
states can be found, as shown in Table 1. Through the experimental data
provided by the Belle experiment Belleexp , their properties can be further
checked.

##  III The $\Xi_{c}^{+}\to\pi^{+}\pi^{+}\Xi^{-}$ reaction

Here we follow closely the formalism of Ref. Miyahara . At the quark level the
reaction to produce the $\Xi(1620)$ and $\Xi(1690)$ proceeds as depicted in
Fig. 1, followed by the hadronization of the final quarks, where a $\bar{q}q$
state with the quantum numbers of the vacuum is created to produce a meson-
baryon pair.

Figure 1: Feynman diagram at quark level for the first step of the
$\Xi_{c}^{+}\to\Xi^{-}\pi^{+}\pi^{+}$ decay.

The original $s,u$ quarks are spectators in the weak process and are in a
state $\frac{1}{\sqrt{2}}(su-us)$ Roberts2 . The weak process is doubly
Cabibbo favored and the hadronization must necessarily involve the created $s$
quark after the weak vertex because in order to have the $\Xi(1620)$ state
with negative parity in
$\Xi_{c}^{+}\to\pi^{+}\,\Xi^{0}(1620)(\frac{1}{2}^{-})$, the created $s$ quark
must have orbital angular momentum $L=1$. Yet, when the pseudoscalar-baryon
coupled channels of section II are created in the hadronization, all quarks
are in their ground state, which forces this $s$ quark to participate in the
hadronization process to get deexcited. Bookkeeping the meson and baryon
states coming from the hadronization, it is found in Ref. Miyahara  that the
hadronic state coming after this process is given by

$H=K^{-}\,\Sigma^{+}-\frac{1}{\sqrt{2}}\,\bar{K}^{0}\,\Sigma^{0}+\frac{1}{\sqrt{6}}\,\bar{K}^{0}\,\Lambda-\frac{1}{\sqrt{3}}\,\eta\,\Xi^{0},$
(3)

where the $\eta^{\prime}$ component, which plays no role in the $\Xi(1620)$
and $\Xi(1690)$ formation, has been neglected.

It is surprising that the $\pi\,\Xi$ component does not appear in Eq. (3), but
it is precisely this feature what stresses the production of a dynamically
generated resonance, because the $\pi\Xi$ final state comes after rescattering
of the components of Eq. (3) to give $\pi\Xi$ in the final state, and it is
precisely this interaction the one that produces the resonances.
Diagrammatically, this is depicted in Fig. 2,

Figure 2: The rescattering mechanism for $\Xi_{c}^{+}\to\pi^{+}\pi^{+}\Xi^{-}$
decay.

and analytically the decay amplitude corresponding to the mechanism of Fig. 2
is given by

$t=V_{P}\,\sum_{i=1}^{4}h_{i}\,G_{i}(M_{\rm inv})\,t_{i,\,\pi^{+}\,\Xi^{-}},$
(4)

where $V_{P}$ is a global factor, $t_{i,\,\pi^{+}\,\Xi^{-}}$ are the
transition amplitudes from channel $i$ (each state of Eq. (3)) to
$\pi^{+}\,\Xi^{-}$, and $h_{i}$ are the weights of each of these channels in
Eq. (3),

$h_{K^{-}\Sigma^{+}}=1;~{}~{}~{}h_{\bar{K}^{0}\Sigma^{0}}=-\frac{1}{\sqrt{2}};~{}~{}~{}h_{\bar{K}^{0}\Lambda}=\frac{1}{\sqrt{6}};~{}~{}~{}h_{\eta\Xi^{0}}=-\frac{1}{\sqrt{3}}.$
(5)

Since the $I=\frac{3}{2}$ sector does not contribute to generating the
$\Xi(1620)$ and $\Xi(1690)$, we neglect it. Considering the phase convention
of the isospin multiplets $(-\Sigma^{+},\Sigma^{0},\Sigma^{-})$,
$(\bar{K}^{0},-K^{-})$, $(-\pi^{+},\pi^{0},\pi^{-})$, and
$(\Xi^{0},-\Xi^{-})$, the $t_{i,\,\pi^{+}\,\Xi^{-}}$ amplitudes are obtained
from the amplitudes generated in Eq. (1) in isospin basis by means of

$\displaystyle t_{K^{-}\Sigma^{+},\,\pi^{+}\Xi^{-}}$ $\displaystyle=$
$\displaystyle-\frac{2}{3}\;t_{\bar{K}\Sigma,\,\pi\Xi}^{I=1/2},$
$\displaystyle t_{\bar{K}^{0}\Sigma^{0},\,\pi^{+}\Xi^{-}}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{2}}{3}\;t_{\bar{K}\Sigma,\,\pi\Xi}^{I=1/2},$
$\displaystyle t_{\bar{K}^{0}\Lambda,\,\pi^{+}\Xi^{-}}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{2}{3}}\;t_{\bar{K}\Lambda,\,\pi\Xi}^{I=1/2},$ (6)
$\displaystyle t_{\eta\Xi^{0},\,\pi^{+}\Xi^{-}}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{2}{3}}\;t_{\eta\Xi,\,\pi\Xi}^{I=1/2}.$

In the experiment of Ref. Belleexp , apart from the signals of the $\Xi(1620)$
and $\Xi(1690)$ one can see a clean peak for the
$\Xi^{*}(1530)(\frac{3}{2}^{+})$. It is interesting to see how this is
produced. It cannot be produced with the mechanism of external emission of
Fig. 1 because the final wave function is $\frac{1}{\sqrt{2}}s\,(su-us)$,
which is mixed antisymmetric and orthogonal to the fully symmetric
$(ssu+sus+uss)$ of the $\Xi^{*}(1530)(\frac{3}{2}^{+})$. Then one needs
internal emission with quark rearrangement as depicted in Fig. 3.

Figure 3: Mechanism for $\Xi^{*}(1530)(\frac{3}{2}^{+})$ production.

This observation is relevant because the external emission process has bigger
strength than internal emission, but the hadronization also leads to a
reduction with respect to a process that requires no hadronization, as the one
in Fig. 3. As a consequence of this, one can qualitatively understand that the
integrated strengths of the excitation of the $\Xi^{*}(1530)$ and $\Xi(1620)$
are similar, as one can observe in the experiment. We are only concerned here
about the $\Xi(1620)$ and $\Xi(1690)$, but to compare our results with
experiment we will account for the contribution of the $\Xi^{*}(1530)$
empirically.

##  IV Results

The differential width distribution in terms of the $\pi^{+}\Xi^{-}$ invariant
mass is given by

$\frac{{\rm d}\Gamma}{{\rm d}M_{\rm
inv}}=\frac{1}{(2\pi)^{3}}\,\frac{1}{4\,M^{2}_{\Xi_{c}}}\,p_{\pi^{+}}\,\tilde{p}_{\pi^{+}}\,|t|^{2},$
(7)

where $t$ is given by Eq. (4) for $\Xi(1620)$ and $\Xi(1690)$. The amplitudes
of Eq. (III) contain the contribution of both resonances. In Eq. (7)
$p_{\pi^{+}},\,\tilde{p}_{\pi^{+}}$ are the momentum of the $\pi^{+}$ produced
in the weak process, Fig. 1, and the momentum of the $\pi^{+}$ produced in the
final $\pi^{+}\Xi^{-}$ state, in the $\Xi_{c}$ and $\pi^{+}\Xi^{-}$ rest
frames, respectively,

$\displaystyle p_{\pi^{+}}$ $\displaystyle=$
$\displaystyle\frac{\lambda^{1/2}(M^{2}_{\Xi_{c}},\,m^{2}_{\pi},\,M^{2}_{\rm
inv})}{2\,M_{\Xi_{c}}},$ $\displaystyle\tilde{p}_{\pi^{+}}$ $\displaystyle=$
$\displaystyle\frac{\lambda^{1/2}(M^{2}_{\rm
inv},\,m^{2}_{\pi},\,M^{2}_{\Xi^{-}})}{2\,M_{\rm inv}}.$

The existence of two $\pi^{+}$ in the final state of the decay,
$\pi^{+}\pi^{+}\Xi^{-}$, poses in principle problems in the identification
since they are identical particles. However, in the analysis of the
experiment, a most useful strategy is followed, distinguishing the $\pi_{L}$
and $\pi_{H}$ where $\pi_{L}$ corresponds to the pion with lowest momentum and
$\pi_{H}$ to the pion with highest momentum. It is an interesting kinematical
exercise to see that both for $\Xi(1620)$ and $\Xi(1690)$, as well as
$\Xi(1530)$ production, $\pi_{H}$ corresponds to the pion produced in the weak
vertex (Fig. 1 and Fig. 3), while $\pi_{L}$ corresponds to the one of
$\pi^{+}\Xi^{-}$ decay of the $\Xi(1620)$, $\Xi(1690)$, $\Xi(1530)$ states.
The formula of Eq. (7), with $\pi^{+}$ corresponding to $\pi_{H}$ and
$\tilde{p}_{\pi^{+}}$ to $\pi_{L}$, has $M_{\rm inv}$ formed from the
$\pi^{+}_{L}$ and $\Xi^{-}$, consistently with the former discussion. Hence, a
measurement of $\frac{{\rm d}\Gamma}{{\rm d}M_{\rm inv}}$ with $M^{2}_{\rm
inv}=(p_{\pi^{+}_{L}}+p_{\Xi^{-}})^{2}$ provides the magnitude to be compared
to our results. This magnitude is actually provided in the analysis of Ref.
Belleexp , allowing a direct comparison of our results with the data. Note
that this distinction was not done in Ref. Miyahara , where the
$\Xi^{0}\pi^{0}$ decay mode was suggested to avoid this problem.

Yet, the data contain also the signal for $\Xi(1530)$ production and some
background. We parametrize them in the following way. The
$\Xi^{*}(1530)(\frac{3}{2}^{+})$ production can be depicted as in Fig. 2
substituting the loop function by the $\Xi^{*}(1530)$ propagator. But the
$\Xi_{c}(\frac{1}{2}^{+})\,\pi^{+}\,\Xi(\frac{3}{2}^{+})$ vertex requires
$p$-wave by angular momentum conservation and the $\Xi(\frac{3}{2}^{+})$ also
decays in $p$-wave into $\pi\Xi$. Thus we must introduce an extra
$|t^{\prime}|^{2}$ of the type

$|t^{\prime}|^{2}=B^{2}\,|\vec{p}_{\pi^{+}}|^{2}\,|\tilde{p}_{\pi^{+}}|^{2}\;\big{|}\frac{1}{M_{\rm
inv}-M_{\Xi(1530)}+i\Gamma_{\Xi(1530)}/2}\big{|}^{2}.$ (8)

Obviously this term does not interfere with the $s$-wave production of the
$\Xi(1620)$ and $\Xi(1690)$. We can consider the effect of $\frac{3}{2}^{+}$
background by adding a term $C^{2}$ to the propagator square in Eq. (8). In
addition we can also add a $\frac{1}{2}^{+}$ background, which will have
$s$-wave in the weak vertex and $p$-wave in the $\pi^{+}\Xi^{-}$ final state.
This term also does not interfere with the other two. Altogether we can
calculate $\frac{{\rm d}\Gamma}{{\rm d}M_{\rm inv}}$ using Eq. (7) and
substituting

$|t|^{2}\to|t|^{2}+B^{2}\,|\vec{p}_{\pi^{+}}|^{2}\,|\tilde{p}_{\pi^{+}}|^{2}\;\left[\big{|}\frac{1}{M_{\rm
inv}-M_{\Xi(1530)}+i\Gamma_{\Xi(1530)}/2}\big{|}^{2}+C^{2}\right]+D^{2}|\tilde{p}_{\pi^{+}}|^{2}.$
(9)

As to possible background from $\frac{1}{2}^{-}$ we do not need to take it
into account. We have seen that $\frac{1}{2}^{-}$ comes from rescattering and
the chiral amplitudes are good in a reasonable range of energies, containing
much more information than just the poles of the resonances. There are five
parameters in total,

$q_{\text{max}},\quad V_{p}^{2},\quad B^{2},\quad C^{2},\quad D^{2},$

which are used to fit the experimental data from Belle Belleexp . The
parameter $V_{P}$ regulates the strength of the $\Xi(1620)$ and the parameter
$q_{\rm max}$ determines the position and relative strength of the $\Xi(1620)$
to the $\Xi(1690)$. The other 3 parameters fit the $\Xi(1530)$ signal and the
background.

With the aim of seeing more clearly the effect of the individual parameters on
the final result, we present the results with different sets of parameters. In
Figs. 4-8, we show the $\Xi^{-}\pi_{L}^{+}$ invariant mass distribution for
$\Xi_{c}^{+}\to\pi^{+}_{H}\pi^{+}_{L}\Xi^{-}$ decay with different sets of
parameters, and the comparison with the experimental data from Belle Belleexp
, where the blue points with error bar are Belle data and the black lines are
our theoretical results. One finds that the results are inconsistent with
experimental data below 1520 MeV. The reason may be that this is the tail of
other low mass resonance states. It can be seen that the parameter $B$ mainly
determines the strength of the $\Xi(1530)$, from Figs. 4-5. Likewise, one can
know that the parameters $C^{2}$ and $D^{2}$ affect the global strength and
the parameter $D^{2}$ mainly affects the high energy region, when comparing
Fig. 4 and Figs. 6-7. The position and relative strength of the $\Xi(1620)$ to
the $\Xi(1690)$ change when $q_{\text{max}}$ and $V_{p}^{2}$ change as shown
in Fig. 4 and Fig. 8. We calculate the $\chi^{2}$ in the invariant mass of
$[1530,1750]$ MeV, so as to pick out better sets of parameters. The
$\chi^{2}/n$ is shown in the figure.

Figure 4: $\Xi^{-}\pi_{L}^{+}$ invariant mass distribution for
$\Xi_{c}^{+}\to\pi^{+}_{H}\pi^{+}_{L}\Xi^{-}$ decay. [parameters:
$q_{\text{max}}=750\;{\rm MeV}$, $V_{P}^{2}=8.1\times 10^{7}$, $B^{2}=9\times
10^{-2}$, $C^{2}=2.2\times 10^{-5}$, $D^{2}=22$] Figure 5:
$\Xi^{-}\pi_{L}^{+}$ invariant mass distribution for
$\Xi_{c}^{+}\to\pi^{+}_{H}\pi^{+}_{L}\Xi^{-}$ decay. [parameters:
$q_{\text{max}}=750\;{\rm MeV}$, $V_{P}^{2}=8.1\times 10^{7}$,
$B^{2}=8.3\times 10^{-2}$, $C^{2}=2.4\times 10^{-5}$, $D^{2}=22$] Figure 6:
$\Xi^{-}\pi_{L}^{+}$ invariant mass distribution for
$\Xi_{c}^{+}\to\pi^{+}_{H}\pi^{+}_{L}\Xi^{-}$ decay. [parameters:
$q_{\text{max}}=750\;{\rm MeV}$, $V_{P}^{2}=8.1\times 10^{7}$, $B^{2}=9\times
10^{-2}$, $C^{2}=1.3\times 10^{-4}$, $D^{2}=22$] Figure 7:
$\Xi^{-}\pi_{L}^{+}$ invariant mass distribution for
$\Xi_{c}^{+}\to\pi^{+}_{H}\pi^{+}_{L}\Xi^{-}$ decay. [parameters:
$q_{\text{max}}=750\;{\rm MeV}$, $V_{P}^{2}=8.1\times 10^{7}$,
$B^{2}=9.0\times 10^{-2}$, $C^{2}=2.2\times 10^{-5}$, $D^{2}=26$]

Figure 8: $\Xi^{-}\pi_{L}^{+}$ invariant mass distribution for
$\Xi_{c}^{+}\to\pi^{+}_{H}\pi^{+}_{L}\Xi^{-}$ decay. [parameters:
$q_{\text{max}}=770\;{\rm MeV}$, $V_{P}^{2}=7.6\times 10^{7}$,$B^{2}=9\times
10^{-2}$, $C^{2}=2.2\times 10^{-5}$, $D^{2}=23$]

In the work we consider that $\Xi(1620)$ and $\Xi(1690)$ are molecular states,
with $I(J^{P})=\frac{1}{2}(\frac{1}{2}^{-})$. The theoretical results are in
good agreement with the experimental data in the range of $[1530,1750]$ MeV,
and support the molecular state picture of the excited hyperon resonances
$\Xi(1620)$ and $\Xi(1690)$.

It is interesting to remark that the peak originated by the calculation in
$d\Gamma/dM_{\rm inv}$ in the figures does not exactly correspond to the pole
position of the resonance in the second Riemann sheet shown in Table 1. This
is a quite common feature in the PDG pdg  for resonances which are not narrow.
It is also worth mentioning that the $\Xi(1690)$ signal has appeared with a
clear interference with the $\Xi(1620)$ and does not have a Breit-Wigner (BW)
shape. It rather looks like ${\rm Im}(iBW)$, with a fast change of sign in the
region of the pole. It is interesting to note that the data with high
precision also show a fast jump from one bin to the next in that region. The
molecular picture for the $\Xi(1620)$ and $\Xi(1690)$ that we have used in
this work is supported by a large number of papers in the literature in
addition to Refs. Bennhold ; Carmen  mentioned in the Introduction. The
$\Xi(1620)$ can be interpreted as $\bar{K}\Lambda$ molecular state with
$I(J^{P})=\frac{1}{2}(\frac{1}{2}^{-})$ in Ref. Chen:2019uvv , under the
framework of the one-boson-exchange (OBE) model. This work gives the binding
energy for the $\bar{K}\Lambda$ system as $-2.9$ MeV. The $\Xi(1620)$ is also
assumed to be a $\bar{K}\Lambda$-$\bar{K}\Sigma$ molecular state in Ref.
Huang:2020taj , and the decay widths of this state decaying into $\pi\Xi$ and
$\pi\pi\Xi$ are calculated through triangle diagrams in an effective
Lagrangian approach, showing that the $\bar{K}\Lambda$ channel provides about
50-68% of the total decay width, while the $\bar{K}\Sigma$ channel provides
the rest. Similarly, $\Xi(1620)$ is considered a $s$-wave $\bar{K}\Lambda$ or
$\bar{K}\Sigma$ bound states, based on the Bethe-Salpeter equation, in Ref.
Wang:2019krq , and the decay widths of $\Xi(1620)\to\pi\Xi$ are $36.94$ MeV
and $9.35$ MeV for the $\bar{K}\Lambda$ and $\bar{K}\Sigma$ bound states,
respectively. However, the $\Xi(1690)$ is not discussed in Refs. Chen:2019uvv
; Huang:2020taj ; Wang:2019krq . In our work, the $\Xi(1620)$ is considered as
a molecular state with $\pi\Xi$ and $\bar{K}\Lambda$ as its main components,
and the $\Xi(1690)$ is considered as mostly a molecular state of
$\bar{K}\Sigma$, very close to threshold, which could be equally considered as
a virtual state, without changing the features of the mass distribution. An
advantage of our work is that we can discuss the $\Xi(1620)$ and $\Xi(1690)$
states at the same time.

The work of Ref. Bennhold  is repeated in Ref. Sekihara:2015qqa , by using the
chiral unitary approach. When selecting an appropriate set of subtraction
constants, a pole of $1682-0.8i$ MeV is found, corresponding to $\Xi(1690)$.
The coupling constants appears to be dominated by $\bar{K}\Sigma$, which is in
line with our results obtained by the cutoff method to regularize the loop
function. More recently, two more works address the nature of these resonances
from the molecular perspective. In Ref. Nishibuchi , the $\Xi(1620)$ is
obtained using the chiral unitary approach and the $\bar{K}\Lambda$ scattering
length is evaluated. In Ref. Feijoo , the chiral unitary approach is again
used to stdy the meson-baryon interaction in the $S=-2$ sector including next
to leading order and Born terms in the interaction. In that work both the
$\Xi(1620)$ and $\Xi(1690)$ are generated with similar results and conclusions
regarding the nature of the states, as exposed in the present work. In our
work we dynamically generate the $\Xi(1620)$ and $\Xi(1690)$ states, with only
one parameter $q_{\text{max}}$. We must stress that the only parameter of the
chiral unitary approach at this leading order already determines at the same
time the position of the two resonances, their widths and the relative
strength of the production rates of the two resonances, a real challenge for
any theory.

##  V Conclusions

In this work, we have studied $\Xi^{-}\pi^{+}_{L}$ invariant mass distribution
for $\Xi^{+}_{c}\to\pi_{H}^{+}\pi^{+}_{L}\Xi^{-}$ decay, considering the
$\Xi(1620)$ and $\Xi(1690)$ as meson-baryon molecular states, under the
framework of the chiral unitary approach. We can dynamically generate double
resonance states simultaneously – $\Xi(1620)$ and $\Xi(1690)$, from the s-wave
interaction of $\pi\Xi$ and other coupled channels, when using the cutoff
method to regularize the $G$ functions. We provide the coupling constants of
$\Xi(1620)$ and $\Xi(1690)$ to different channels in Table 2. From the
coupling constants, it can be known that $\Xi(1620)$ is a molecular state with
main components $\pi\Xi$ and $\bar{K}\Lambda$, and $\Xi(1690)$ is a molecular
state with main component $\bar{K}\Sigma$. The mechanism of
$\Xi^{+}_{c}\to\pi^{+}MB$ decay was analyzed in Ref. Miyahara . We follow
closely this work and find that the meson and baryon states from hadronization
after the weak process do not contain the final state $\pi^{+}\Xi^{-}$, so it
must come from rescattering. But what might be a problem to describe the
reaction becomes actually a strong point to look into the nature of the
$\Xi(1620)$ and $\Xi(1690)$ resonances, which are produced by this final state
interaction in the chiral unitary approach.

In order to compare with the experimental data from Belle, we take into
account contributions from $\Xi(1530)$ and other backgrounds, and get the
$\Xi^{-}\pi^{+}_{L}$ invariant mass distribution for
$\Xi_{c}^{+}\to\pi^{+}_{H}\pi^{+}_{L}\Xi^{-}$ decay. Results in comparison
with experimental data strongly support that the $\Xi(1620)$ and $\Xi(1690)$
are molecular states, with spin-parity being both $\frac{1}{2}^{-}$. Other
reactions requiring the need of final state interaction to produce these
resonances would be most welcome to further support this picture.

##  VI ACKNOWLEDGEMENT

This work is partly supported by the National Natural Science Foundation of
China under Grant No. 11975083, and by the Central Government Guidance Funds
for Local Scientific and Technological Development, China (No. Guike
ZY22096024). This work is also partly supported by the Spanish Ministerio de
Economia y Competitividad and European FEDER funds under Contracts No.
FIS2017- 84038-C2-1-P B and No. FIS2017-84038-C2-2-P B, and the Generalitat
Valenciana in the program Prometeo II-2014/068, and the project Severo Ochoa
of IFIC, SEV-2014-0398.

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