# The possible assignments of the scalar $K_{0}^{*}(1950)$ and
$K_{0}^{*}(2130)$ within the ${}^{3}P_{0}$ model

Tian-Ge Li  Joint Research Center for Theoretical Physics, School of Physics
and Electronics, Henan University, Kaifeng 475004, China  Zhuo Gao  Joint
Research Center for Theoretical Physics, School of Physics and Electronics,
Henan University, Kaifeng 475004, China  Guan-Ying Wang
<EMAIL_ADDRESS>Joint Research Center for Theoretical Physics, School
of Physics and Electronics, Henan University, Kaifeng 475004, China  De-Min Li
<EMAIL_ADDRESS>School of Physics and Microelectronics, Zhengzhou University,
Zhengzhou, Henan 450001, China  En Wang<EMAIL_ADDRESS>School of Physics
and Microelectronics, Zhengzhou University, Zhengzhou, Henan 450001, China
Jingya Zhu<EMAIL_ADDRESS>Joint Research Center for Theoretical Physics,
School of Physics and Electronics, Henan University, Kaifeng 475004, China

###### Abstract

We have evaluated the strong decays of the $K_{0}^{*}(1950)$ and
$K_{0}^{*}(2130)$ within the ${}^{3}P_{0}$ model, by employing the meson wave
functions from the relativized quark model. By comparing with the experimental
measurements, the $K_{0}^{*}(2130)$ could be assigned as
$K_{0}^{*}(3^{3}P_{0})$, while the $K_{0}^{*}(1950)$ seems like an exotic
state, because its width can not be reasonably reproduced within the
${}^{3}P_{0}$ model. We also predict that the $K_{0}^{*}(2^{3}P_{0})$ state
has a mass of about $1811$ MeV and a width of about $656$ MeV, while the
$K_{0}^{*}(4^{3}P_{0})$ state has a mass of about $2404$ MeV and a width of
about $180$ MeV.

###### pacs:

##  I Introduction

According to the theory of quantum chromodynamics (QCD), in addition to the
conventional $q\bar{q}$ mesons, the so-called exotic states are also
permitted, such as tetraquarks, molecules, glueballs, and hybrids
Richard:2016eis ; Jaffe:2004ph ; Amsler:2004ps . Although many exotic states
have been observed experimentally, such as $X(3872)$, $Z_{c}(3900)$,
$Z_{c}(4025)$, $P_{c}$, $T_{cc}$, it is still difficult to distinguish between
the exotic states with conventional quantum numbers and the ordinary
$q\bar{q}$ mesons.

One puzzle in hadron spectra is the scalar mesons, since there are too many
states to be accommodated within the quark model without difficulty
tHooft:2008rus . For example, ${K_{0}^{*}}(700)$ state (also known as
$\kappa$), together with its multiple partners $a_{0}(980)$,
$f_{0}(500)(\sigma)$, and $f_{0}(980)$, does not fit well into the predictions
of the quark model, since the observed mass ordering of these lowest scalar
states is $m_{\sigma}<m_{\kappa}<m_{a_{0},f_{0}}$ Zyla:2020zbs , while in the
conventional quark model, by a naive counting of the quark mass, the mass
ordering of the scalar $q\bar{q}$ nonet should be $m_{\sigma}\sim
m_{a_{0}}<m_{\kappa}<m_{f_{0}}$. These scalar states below 1 GeV are generally
believed not to be $q\bar{q}$ states Close:2002zu ; Maiani:2004uc ;
Amsler:2004ps ; Pelaez:2003dy ; Jaffe:2004ph ; Eichmann:2015cra .

Within the naive quark model, it is natural to assume that the $a_{0}(1450)$,
${K_{0}^{*}}(1430)$, $f_{0}(1710)$, and $f_{0}(1370)$ are the $1{{}^{3}P_{0}}$
members of the SU(3) flavor nonet Zyla:2020zbs . The isovector scalar mesons
$a_{0}(2020)/a_{0}(1950)$ are suggested to be the good candidates of the
$a_{0}(3{{}^{3}P_{0}})$ in our previous work Wang:2017pxm . However, the
assignments of the excited scalar ${K_{0}^{*}}$ states are still unclear. Up
to now, above the ${K_{0}^{*}}(1430)$ mass, only two scalar states
${K_{0}^{*}}(1950)$ and ${K_{0}^{*}}(2130)$ are reported. Their masses and
widths are listed in Table 1 .

Table 1: The masses and widths of the excited scalar $K_{0}^{\ast}$ states (in MeV). state | mass | width | ref
---|---|---|---
${K_{0}^{*}}(1950)$ | 1945$\pm$10$\pm$20 | 201$\pm$34$\pm$79 | Zyla:2020zbs
${K_{0}^{*}}(2130)$ | 2128$\pm$31$\pm$9 | 95$\pm$42$\pm$76 | Lees:2021cww

Since ${K_{0}^{*}}(1950)$ was first reported in the $K\pi$ invariant mass
distribution of the $K^{-}p\to K^{-}\pi^{+}n$ reaction by LASS in 1988
Aston:1987ir , it is difficult to interpret its properties within the quark
model. The ${K_{0}^{*}}(1950)$ mass is close to the
${K_{0}^{*}}(2{{}^{3}P_{0}})$ mass of about 1890 MeV predicted by the Godfrey-
Isgur (GI) quark model Godfrey:1985xj . However, it is expected that the
${K_{0}^{*}}(2{{}^{3}P_{0}})$ with a mass of 1850 MeV has a width of about 450
MeV within the ${{}^{3}P_{0}}$ decay model Barnes:2002mu , larger than the
${K_{0}^{*}}(1950)$ width. In addition, Ref. Pang:2017dlw  recently analyzed
the kaon family within the modified GI model involving the color screening
effect, and predicted the mass and width of the ${K_{0}^{*}}(2{{}^{3}P_{0}})$
to be $M=1829$ MeV and $\Gamma=1000$ MeV, respectively, both of which disfavor
the assignment of ${K_{0}^{*}}(1950)$ as the candidate the
${K_{0}^{*}}(2{{}^{3}P_{0}})$ state.

The ${K_{0}^{*}}(2130)$ was recently observed in the ${\eta}_{c}$ decays by
the BABAR Collaboration Lees:2021cww , and its mass is close to the
${K_{0}^{*}}(3{{}^{3}P_{0}})$ mass of 2176 MeV predicted by the modified GI
model Pang:2017dlw . In addition, we have estimated that the
$n\bar{n}(3^{3}P_{0})$ mass is about $1.9\sim 2.0$ GeV Wang:2017pxm , thus one
can naturally expect the ${K_{0}^{*}}(3{{}^{3}P_{0}})$ mass should be about
$100\sim 200$ MeV larger than the $a_{0}(3{{}^{3}P_{0}})$ mass. Based on its
mass information, the ${K_{0}^{*}}(2130)$ seems a good candidate of the
${K_{0}^{*}}(3{{}^{3}P_{0}})$.

The mass information alone is insufficient to identify the ${K_{0}^{*}}(2130)$
as the ${K_{0}^{*}}(3{{}^{3}P_{0}})$ state. We shall discuss the possibility
of the ${K_{0}^{*}}(2130)$ as the ${K_{0}^{*}}(3{{}^{3}P_{0}})$ state by
studying its strong decay properties.

In this work, we will investigate the possible assignment of
${K_{0}^{*}}(2130)$ by analyzing the strong decay behaviors within the
${{}^{3}P_{0}}$ decay model. For completeness, we also check the possibility
of the ${K_{0}^{*}}(1950)$ as the ordinary scalar mesons, since it is natural
and necessary to exhaust the possible $q\bar{q}$ descriptions of a newly
observed state before restoring to the more exotic assignments. This paper is
organized as follows. In Sec. II, we introduce the ${{}^{3}P_{0}}$ strong
decay model used in our calculations, and the results and discussions are
given in Sec. III. Finally, a summary is given in Sec. IV.

##  II Model and Parameters

The ${{}^{3}P_{0}}$ model has been widely used to study the Okubo-Zweig-Iizuka
(OZI)-allowed open flavor two-body strong decays, it was originally introduced
by Micu Micu:1968mk  and further developed by Le Yaouanc $et$ $al.$
LeYaouanc:1972vsx ; LeYaouanc:1973ldf ; LeYaouanc:alo . In the ${{}^{3}P_{0}}$
model, the meson strong decay takes place by producing a quark-antiquark pair
with vacuum quantum number $J^{PC}=0^{++}$. The newly produced quark-antiquark
pair, together with the $q\bar{q}$ within the initial meson, regroups into two
outgoing mesons in two possible quark rearrangement ways, as shown in Fig. 1.
The ${}^{3}P_{0}$ model has been widely applied to study strong decays of
hadrons with considerable success Roberts:1992js ; Blundell:1996as ;
Barnes:1996ff ; Barnes:2002mu ; Close:2005se ; Barnes:2005pb ; Zhang:2006yj ;
Li:2009rka ; Li:2010vx ; Lu:2014zua ; Pan:2016bac ; Lu:2016bbk ; Wang:2017pxm
; Li:2021qgz ; Hao:2019fjg ; Xue:2018jvi .

Figure 1: Two-body-decay diagrams of $A\to BC$ according to the
${{}^{3}P_{0}}$ model, where a pair of quark-antiquark are created to form the
final two mesons. In the left diagram, Meson $B$ is formed by quark in Meson
$A$ combined with the created antiquark, and Meson $C$ is formed by antiquark
in Meson $A$ combined with the created quark. In the right diagram, Meson $B$
is formed by antiquark in Meson $A$ combined with the created quark, and Meson
$C$ is formed by quark in Meson $A$ combined with the created antiquark.

Following the conventions in Refs. Roberts:1992js ; Blundell:1996as , the
transition operator $T$ of the decay $A\rightarrow BC$ in the ${{}^{3}P_{0}}$
model is given by

$\displaystyle T$ $\displaystyle=$
$\displaystyle-3\gamma\sum_{m}\left<1m1-m|00\right>\int
d^{3}{\bm{p}}_{3}d^{3}{\bm{p}}_{4}\delta^{3}({\bm{p}}_{3}+{\bm{p}}_{4})$ (1)
$\displaystyle{\cal{Y}}^{m}_{1}\left(\frac{{\bm{p}}_{3}-{\bm{p}}_{4}}{2}\right)\chi^{34}_{1-m}\phi^{34}_{0}\omega^{34}_{0}b^{\dagger}_{3}({\bm{p}}_{3})d^{\dagger}_{4}({\bm{p}}_{4}),$

where the $\gamma$ is a dimensionless parameter corresponding to the
production strength of the quark-antiquark pair $q_{3}\bar{q}_{4}$ with
quantum number $J^{PC}=0^{++}$. ${\bm{p}}_{3}$ and ${\bm{p}}_{4}$ are the
momenta of the created quark $q_{3}$ and antiquark $\bar{q}_{4}$,
respectively. $\chi^{34}_{1,-m}$, $\phi^{34}_{0}$, and $\omega^{34}_{0}$ are
the spin, flavor, and color wave functions of $q_{3}\bar{q}_{4}$ system,
respectively. The solid harmonic polynomial
${\cal{Y}}^{m}_{1}({\bm{p}})\equiv|{\bm{p}}|^{1}Y^{m}_{1}(\theta_{p},\phi_{p})$
reflects the momentum-space distribution of the $q_{3}\bar{q_{4}}$.

The $S$ matrix of the process $A\rightarrow BC$ is defined by

$\displaystyle\langle BC|S|A\rangle=I-2\pi i\delta(E_{A}-E_{B}-E_{C})\langle
BC|T|A\rangle,$ (2)

where $|A\rangle$ ($|B\rangle$, $|C\rangle$) are the wave functions of the
mock mesons defined by Ref. Hayne:1981zy .

The transition matrix element $\langle BC|T|A\rangle$ can be written as

$\displaystyle\langle
BC|T|A\rangle=\delta^{3}({\bm{p}}_{A}-{\bm{p}}_{B}-{\bm{p}}_{C}){\cal{M}}^{M_{J_{A}}M_{J_{B}}M_{J_{C}}}({\bm{p}}),$
(3)

where ${\cal{M}}^{M_{J_{A}}M_{J_{B}}M_{J_{C}}}({\bm{p}})$ is the helicity
amplitude.

The partial wave amplitude ${\cal{M}}^{LS}({\bm{p}})$ can be given by
Jacob:1959at ,

$\displaystyle{\cal{M}}^{LS}({\bm{p}})$ $\displaystyle=$
$\displaystyle\sum_{\begin{array}[t]{l}\scriptstyle M_{J_{B}},M_{J_{C}},\\\
\scriptstyle M_{S},M_{L}\end{array}}\\!\\!\langle
LM_{L}SM_{S}|J_{A}M_{J_{A}}\rangle$ (7) $\displaystyle\langle
J_{B}M_{J_{B}}J_{C}M_{J_{C}}|SM_{S}\rangle$ $\displaystyle\times\int
d\Omega\,\mbox{}Y^{\ast}_{LM_{L}}{\cal{M}}^{M_{J_{A}}M_{J_{B}}M_{J_{C}}}({\bm{p}}).$

Various ${{}^{3}P_{0}}$ models exist in literature and typically differ in the
choices of the pair-production vertex, the phase space conventions, and the
meson wave functions employed. In this work, we restrict to the simplest
vertex as introduced originally by Micu Micu:1968mk  which assumes a spatially
constant pair-production strength $\gamma$, adopt the relativistic phase
space, and employ the relativized quark model (RQM) wave functions
Godfrey:1985xj .

With the relativistic phase space, the decay width $\Gamma(A\rightarrow BC)$
can be expressed in terms of the partial wave amplitude

$\displaystyle\Gamma(A\rightarrow
BC)=\frac{\pi|{\bm{p}}|}{4M^{2}_{A}}\sum_{LS}|{\cal{M}}^{LS}({\bm{p}})|^{2},$
(8)

where
$|{\bm{p}}|=\sqrt{[M^{2}_{A}-(M_{B}+M_{C})^{2}][M^{2}_{A}-(M_{B}-M_{C})^{2}]}/(2M_{A})$,
and $M_{A}$, $M_{B}$, and $M_{C}$ are the masses of the mesons $A$, $B$, and
$C$, respectively.

##  III RESULTS

In our calculations, the parameters involve the $q\bar{q}$ pair production
strength $\gamma$, and the ones in relativized quark model, as used in the
work of Godfrey and Isgur Godfrey:1985xj . The flavor wave functions for the
mesons are adopted by following the conventions of Refs. Barnes:2002mu ;
Godfrey:1985xj  except for (1) $f_{1}(1285)=-0.28n\bar{n}+0.96s\bar{s}$ and
$f_{1}(1420)=-0.96n\bar{n}-0.28s\bar{s}$ as Ref. Li:2000dy , (2)
$\eta(1295)=(n\bar{n}-s\bar{s})/\sqrt{2}$ and
$\eta(1475)=(n\bar{n}+s\bar{s})/\sqrt{2}$ as Ref. Yu:2011ta , where
$n\bar{n}=(u\bar{u}+d\bar{d})/\sqrt{2}$. The masses of the final mesons are
taken from the Review of Particle Physics Zyla:2020zbs .

We take $\gamma=0.52$ by fitting to the total width of ${K_{0}^{*}}(1430)$ as
the $1{{}^{3}P_{0}}$ state. The decay widths of ${K_{0}^{*}}(1430)$ as the
${K_{0}^{*}}(1{{}^{3}P_{0}})$ state are listed in Table 2. According to our
results, the dominant decay mode of ${K_{0}^{*}}(1430)$ is $K\pi$, which is
consistent with the experimental data Zyla:2020zbs .

Table 2: Decay widths of ${K_{0}^{*}}(1430)$ as $1{{}^{3}P_{0}}$ state (in MeV). Channel | Mode | $\Gamma_{i}(1{{}^{3}P_{0}})$
---|---|---
$0^{+}\rightarrow 0^{-}+0^{-}$ | $K\pi$ | 262.53
| $K\eta$ | 9.42
$0^{+}\rightarrow 0^{-}+1^{+}$ | $\pi K_{1B}$ | $<0.1$
| Total width | 271.95
| Experiment | $270\pm 80$

With the parameter $\gamma=0.52$, we have calculated the decay widths of
${K_{0}^{*}}(1950)$ as the ${K_{0}^{*}}(2{{}^{3}P_{0}})$ and
${K_{0}^{*}}(3{{}^{3}P_{0}})$ states, respectively, which are listed in Table
3. The total widths of the ${K_{0}^{*}}(2{{}^{3}P_{0}})$ and
${K_{0}^{*}}(3{{}^{3}P_{0}})$ states with a mass of 1945 MeV are expected to
be $1308$ MeV and $74$ MeV, respectively. By comparing with the
$K^{\ast}_{0}(1950)$ width $\Gamma=201\pm 34\pm 79$ MeV Zyla:2020zbs , it is
hard to assign the ${K_{0}^{*}}(1950)$ as the ordinary scalar $q\bar{q}$
meson.

Table 3: Decay widths of ${K_{0}^{*}}(1950)$ as $2{{}^{3}P_{0}}$ and $3{{}^{3}P_{0}}$ states (in MeV). The initial state mass is $1945$ MeV. Channel | Mode | $\Gamma_{i}(2^{3}P_{0})$ | $\Gamma_{i}(3^{3}P_{0})$
---|---|---|---
$0^{+}\rightarrow 0^{-}+0^{-}$ | $\pi K$ | 78.26 | 1.64
| $K\eta$ | 1.74 | $<0.01$
| $K\eta^{\prime}$ | 11.39 | 1.37
| $\pi(1300)K$ | 170.82 | 6.79
| $K\eta(1295)$ | 147.80 | 4.15
| $\pi K(1460)$ | 91.15 | 0.34
$0^{+}\rightarrow 0^{-}+1^{+}$ | $Ka_{1}(1260)$ | 57.37 | 0.03
| $Kb_{1}(1235)$ | 106.81 | 10.46
| $h_{1}(1415)K$ | 11.70 | 2.77
| $h_{1}(1170)K$ | 5.87 | $<0.01$
| $Kf_{1}(1420)$ | 2.05 | 0.02
| $Kf_{1}(1285)$ | 4.59 | $<0.01$
| $\pi K_{1A}$ | 132.10 | 1.13
| $\pi K_{1B}$ | 25.27 | 2.08
| $\eta K_{1B}$ | 54.80 | 6.70
$0^{+}\rightarrow 1^{-}+1^{-}$ | $K^{*}(892)\rho$ | 204.87 | 18.60
| $K^{*}(892)\phi(1020)$ | 9.17 | 0.04
| $K^{*}(892)\omega$ | 192.59 | 17.84
$0^{+}\rightarrow 0^{-}+2^{-}$ | $\pi K_{2}(1770)$ | 0.02 | 0.03
| Total width | 1308.38 | 74.01
| Experiment | 201$\pm$34$\pm$79

Then we will discuss the possible assignment of the ${K_{0}^{*}}(2130)$ state.
We have shown the decay widths of ${K_{0}^{*}}(2130)$ as the
${K_{0}^{*}}(3{{}^{3}P_{0}})$ and ${K_{0}^{*}}(4{{}^{3}P_{0}})$ states in
Table 4, and total width is expected to be about $105$ MeV and $20$ MeV,
respectively for the cases of the ${K_{0}^{*}}(3{{}^{3}P_{0}})$ and
${K_{0}^{*}}(4{{}^{3}P_{0}})$. The dependences of the total widths of
${K_{0}^{*}}(3{{}^{3}P_{0}})$ and ${K_{0}^{*}}(4{{}^{3}P_{0}})$ on the initial
state mass are shown in Fig. 2 and Fig. 3, respectively. Within the
uncertainties of the $K^{*}_{0}(2130)$ mass, the total width of
${K_{0}^{*}}(3{{}^{3}P_{0}})$ is in agreement with the experimental data
$\Gamma=95\pm 42\pm 76$ MeV Zyla:2020zbs , which implies that the
$K^{\ast}_{0}(2130)$ should be the good candidate of the
${K_{0}^{*}}(3{{}^{3}P_{0}})$ state.

Table 4: Decay widths of ${K_{0}^{*}}(2130)$ as $3{{}^{3}P_{0}}$ and $4{{}^{3}P_{0}}$ states (in MeV). The initial state mass is $2128$ MeV. Channel | Mode | $\Gamma_{i}(3^{3}P_{0})$ | $\Gamma_{i}(4^{3}P_{0})$
---|---|---|---
$0^{+}\rightarrow 0^{-}+0^{-}$ | $\pi K$ | 8.58 | 0.08
| $K\eta$ | 0.07 | $<0.01$
| $K\eta^{\prime}$ | 0.21 | 0.29
| $\pi(1300)K$ | 4.35 | 4.09
| $K\eta(1295)$ | 7.38 | 3.55
| $\pi K(1460)$ | 8.83 | 2.83
| $K\eta(1475)$ | $<0.01$ | 0.05
| $K(1460)\eta$ | 0.87 | $<0.01$
$0^{+}\rightarrow 0^{-}+1^{+}$ | $Ka_{1}(1260)$ | 3.58 | 0.09
| $Kb_{1}(1235)$ | 1.63 | 1.52
| $h_{1}(1415)K$ | 12.41 | 0.02
| $h_{1}(1170)K$ | 0.84 | 0.11
| $Kf_{1}(1420)$ | 0.04 | 0.10
| $Kf_{1}(1285)$ | 0.16 | $<0.01$
| $\pi K_{1A}$ | 2.01 | 0.19
| $\pi K_{1B}$ | 0.06 | 3.02
| $\eta K_{1B}$ | 2.67 | 0.32
| $\eta K_{1A}$ | 3.26 | 0.04
$0^{+}\rightarrow 1^{-}+1^{-}$ | $K^{*}(892)\rho$ | 14.27 | 0.61
| $K^{*}(892)\phi(1020)$ | 1.42 | 0.08
| $K^{*}(892)\omega$ | 14.97 | 0.41
$0^{+}\rightarrow 1^{+}+1^{-}$ | $K^{*}(892)a_{1}(1260)$ | 1.20 | 0.16
| $K^{*}(892)b_{1}(1235)$ | 0.15 | 0.02
| $K^{*}(892)h_{1}(1170)$ | 2.33 | 0.26
| $\rho K_{1B}$ | 2.56 | 0.11
| $\omega K_{1B}$ | 2.87 | 0.30
$0^{+}\rightarrow 0^{-}+2^{-}$ | $\pi K_{2}(1820)$ | 1.28 | 0.11
| $\pi K_{2}(1770)$ | 7.22 | 1.69
| $\eta_{2}(1645)K$ | 0.05 | 0.02
| Total width | 105.33 | 20.07
| Experiment | 95$\pm$42$\pm$76
Figure 2: The dependence of the total width of ${K_{0}^{*}}(3{{}^{3}P_{0}})$
on the initial state mass. The experimental total width of the
${K_{0}^{*}}(2130)$ is denoted by the dashed line with a green band. Figure 3:
The dependence of the total width of ${K_{0}^{*}}(4{{}^{3}P_{0}})$ on the
initial state mass.

Phenomenologically, it is suggested that the light mesons could be grouped
into the following Regge trajectories Anisovich:2000kxa ; Pan:2016bac ;
Xue:2018jvi ,

$M_{n}^{2}=M_{0}^{2}+(n-1)\mu^{2},$ (9)

where $M_{0}$ is the lowest-lying meson mass, $n$ is the radial quantum
number, and $\mu^{2}$ is the slope parameter of the corresponding trajectory.
In the presence of ${K_{0}^{*}}(1430)$ and ${K_{0}^{*}}(2130)$ being the
${K_{0}^{*}}(1{{}^{3}P_{0}})$ and ${K_{0}^{*}}(3{{}^{3}P_{0}})$ states, we can
roughly estimate the ${K_{0}^{*}}(2{{}^{3}P_{0}})$ mass to be about 1811 MeV
and ${K_{0}^{*}}(4{{}^{3}P_{0}})$ mass to be about 2404 MeV as shown in Fig.4.

Figure 4: The Regge trajectory of the ${K_{0}^{*}}(n{{}^{3}P_{0}})$
($n=1,2,3,4$) states, where $M_{n}$ is its mass, and the red line is the
fitting result.

It should be pointed out that, in our previous work Wang:2017pxm , the mass
scale for the $n\bar{n}(2^{3}P_{0})$ nonets is expected to be $1700\sim 1800$
MeV. In addition, the mass of ${K_{0}^{*}}(2{{}^{3}P_{0}})$ state is predicted
to be 1890 MeV by the GI model Godfrey:1985xj  and to be 1829 MeV by the
modified GI model Pang:2017dlw . The prediction of the
${K_{0}^{*}}(2{{}^{3}P_{0}})$ mass from the Regge trajectories is consistent
with these previous predictions. The strong decays of
${K_{0}^{*}}(2{{}^{3}P_{0}})$ with a mass of 1811 MeV are presented in Table
5. The dependence of the total width of ${K_{0}^{*}}(2{{}^{3}P_{0}})$ on the
initial state mass is shown in Fig. 5. When the initial state mass varies from
1700 MeV to 1900 MeV, the total width of the ${K_{0}^{*}}(2{{}^{3}P_{0}})$
varies from about 215 MeV to 1111 MeV. The decay width varies greatly, since
some decay modes are open gradually. The total width of
${K_{0}^{*}}(2{{}^{3}P_{0}})$ is expected to be about 656 MeV. The dominant
decay modes of ${K_{0}^{*}}(2{{}^{3}P_{0}})$ are $\pi(1300)K$, $K\eta(1295)$,
and $\pi K(1460)$. The ${K_{0}^{*}}(2{{}^{3}P_{0}})$ is predicted to be a
broad state, which could be the reason that the ${K_{0}^{*}}(2{{}^{3}P_{0}})$
candidate is not yet observed experimentally.

Table 5: Decay widths of ${K_{0}^{*}}(2{{}^{3}P_{0}})$ (in MeV). The initial state mass is $1811$ MeV.  Channel | Mode | $\Gamma_{i}(2{{}^{3}P_{0}})$
---|---|---
$0^{+}\rightarrow 0^{-}+0^{-}$ | $\pi K$ | 57.13
| $\pi(1300)K$ | 96.06
| $K\eta$ | 1.01
| $K\eta^{\prime}$ | 0.77
| $K\eta(1295)$ | 100.95
| $\pi K(1460)$ | 99.91
$0^{+}\rightarrow 0^{-}+1^{+}$ | $h_{1}(1170)K$ | 4.42
| $Ka_{1}(1260)$ | 16.35
| $Kf_{1}(1285)$ | 0.42
| $Kb_{1}(1235)$ | 33.29
| $\pi K_{1A}$ | 64.31
| $\pi K_{1B}$ | 24.37
$0^{+}\rightarrow 1^{-}+1^{-}$ | $K^{*}(892)\rho$ | 81.42
| $K^{*}(892)\omega$ | 75.93
| Total width | 656.34
Figure 5: The dependence of the total width of ${K_{0}^{*}}(2{{}^{3}P_{0}})$
on the initial state mass.

We show the partial decay widths and the total decay width of the
${K_{0}^{*}}(4{{}^{3}P_{0}})$ state with a mass of 2404 MeV in Table 6. The
total width of ${K_{0}^{*}}(4{{}^{3}P_{0}})$ is expected to be about 180 MeV.
The dominant decay modes of ${K_{0}^{*}}(4{{}^{3}P_{0}})$ include
$K^{*}(1410)^{+}\omega$, $K^{*}(1410)^{+}\rho$, $K^{*}(892)\omega(1420)$. The
dependence of the total width of the ${K_{0}^{*}}(4{{}^{3}P_{0}})$ state on
the initial state mass is shown in Fig. 3. When the initial state mass varies
from 2300 MeV to 2500 MeV, the total width of the
${K_{0}^{*}}(4{{}^{3}P_{0}})$ varies from about $88\sim 246$ MeV.

Table 6: Decay widths of ${K_{0}^{*}}(4{{}^{3}P_{0}})$ (in MeV). The initial state mass is $2404$ MeV.  Channel | Mode | $\Gamma_{i}(4{{}^{3}P_{0}})$
---|---|---
$0^{+}\rightarrow 0^{-}+0^{-}$ | $\pi K$ | 0.60
| $K\eta$ | $<0.01$
| $K\eta^{\prime}$ | 0.28
| $\pi(1300)K$ | 0.16
| $K\eta(1295)$ | 0.07
| $\pi K(1460)$ | 0.12
| $K\eta(1475)$ | $<0.01$
| $K(1460)\eta$ | 0.31
$0^{+}\rightarrow 0^{-}+1^{+}$ | $Ka_{1}(1260)$ | 0.70
| $Kb_{1}(1235)$ | 4.50
| $h_{1}(1415)K$ | 4.60
| $h_{1}(1170)K$ | 0.06
| $Kf_{1}(1420)$ | 0.21
| $Kf_{1}(1285)$ | 0.03
| $\pi K_{1A}$ | 0.02
| $\pi K_{1B}$ | 1.69
| $\eta K_{1B}$ | 2.74
| $\eta K_{1A}$ | 0.98
| $\eta^{\prime}K_{1B}$ | 0.08
| $\eta^{\prime}K_{1A}$ | 0.01
| $Ka_{1}(1640)$ | 0.23
$0^{+}\rightarrow 0^{+}+1^{-}$ | $K_{0}^{*}(1430)\rho$ | 0.09
$0^{+}\rightarrow 1^{-}+1^{-}$ | $K^{*}(892)\rho$ | 8.76
| $K^{*}(892)\rho(1450)$ | 6.34
| $K^{*}(892)\omega(1420)$ | 19.83
| $K^{*}(892)\phi(1020)$ | 0.02
| $K^{*}(892)\omega$ | 8.15
| $K^{*}(1410)\rho$ | 32.46
| $K^{*}(1410)\omega$ | 36.79
$0^{+}\rightarrow 1^{+}+1^{-}$ | $K^{*}(892)a_{1}(1260)$ | 2.50
| $K^{*}(892)b_{1}(1235)$ | 0.03
| $K^{*}(892)h_{1}(1170)$ | 0.13
| $K^{*}(892)h_{1}(1415)$ | $<0.01$
| $K^{*}(892)f_{1}(1420)$ | 0.69
| $K^{*}(892)f_{1}(1285)$ | 0.34
| $\rho K_{1A}$ | $<0.01$
| $\omega K_{1A}$ | $<0.01$
| $\phi K_{1B}$ | 0.25
| $\rho K_{1B}$ | 1.29
| $\omega K_{1B}$ | 0.22
$0^{+}\rightarrow 0^{-}+2^{-}$ | $\pi K_{2}(1820)$ | 0.17
| $\pi K_{2}(1770)$ | 2.03
| $K\pi_{2}(1670)$ | 7.15
| $\eta_{2}(1645)K$ | 4.02
| $\eta_{2}(1870)K$ | 0.07
| $\eta K_{2}(1820)$ | 0.05
| $\eta K_{2}(1770)$ | $<0.01$
$0^{+}\rightarrow 1^{-}+2^{+}$ | $K^{*}(892)f_{2}(1270)$ | 0.93
| $K^{*}(892)a_{2}(1320)$ | 12.56
| $K_{2}^{*}(1430)\rho$ | 17.92
| Total width | 180.23

##  IV SUMMARY

In this work, we have discussed the possible assignments of
${K_{0}^{*}}(1950)$ and ${K_{0}^{*}}(2130)$ by calculating the strong decay
widths within the ${{}^{3}P_{0}}$ strong decay model.

We suggest that the $K_{0}^{*}(2130)$ could be assigned as
$K_{0}^{*}(3^{3}P_{0})$ based on its mass and width. However, the
$K_{0}^{*}(1950)$ seems like an exotic state, because its width can not be
reasonably reproduced within the ${}^{3}P_{0}$ model.

With the assignment of the ${K_{0}^{*}}(2130)$ as the
${K_{0}^{*}}(3{{}^{3}P_{0}})$ state, we have roughly estimated the masses of
${K_{0}^{*}}(2{{}^{3}P_{0}})$ and ${K_{0}^{*}}(4{{}^{3}P_{0}})$ to be about
1811 MeV and 2404 MeV, respectively, within the Regge trajectories. The total
width of ${K_{0}^{*}}(2{{}^{3}P_{0}})$ is predicted to be about 656 MeV, which
implies that this state is not easy to be observed experimentally. The total
width of ${K_{0}^{*}}(4{{}^{3}P_{0}})$ is predicted to be about 180 MeV, which
could be helpful to search for the ${K_{0}^{*}}(4{{}^{3}P_{0}})$ state in
future.

##  V Acknowledgements

This work is partly supported by the China Postdoctoral Science Foundation
Funded Project under Grand No.2021M701086, and supported by the Natural
Science Foundation of Henan under Grand No.212300410123 and No.222300420554.
It is also supported by the National Natural Science Foundation of China under
Grant No.12192263. It is also supported by the Key Research Projects of Henan
Higher Education Institutions under No.20A140027, the Project of Youth
Backbone Teachers of Colleges and Universities of Henan Province
(2020GGJS017), the Youth Talent Support Project of Henan (2021HYTP002), the
Fundamental Research Cultivation Fund for Young Teachers of Zhengzhou
University (JC202041042), and the Open Project of Guangxi Key Laboratory of
Nuclear Physics and Nuclear Technology, No.NLK2021-08.

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