# Canonical interpretation of the $D_{s0}(2590)^{+}$ resonance

Zhuo Gao  School of Physics and Electronics, Henan University, Kaifeng 475004,
China  Guan-Ying Wang111Corresponding author<EMAIL_ADDRESS>School
of Physics and Electronics, Henan University, Kaifeng 475004, China  Qi-Fang
Lü222Corresponding author<EMAIL_ADDRESS>Department of Physics, Hunan
Normal University, Changsha 410081, China  Synergetic Innovation Center for
Quantum Effects and Applications (SICQEA), Changsha 410081,China  Key
Laboratory of Low-Dimensional Quantum Structures and Quantum Control of
Ministry of Education, Changsha 410081, China  Jingya Zhu  School of Physics
and Electronics, Henan University, Kaifeng 475004, China  Gao-Feng
Zhao333Corresponding author<EMAIL_ADDRESS>School of Physics and
Electronics, Henan University, Kaifeng 475004, China

###### Abstract

The $D_{s0}(2590)^{+}$ resonance observed by LHCb Collaboration is a strong
candidate of the $D_{s}(2^{1}S_{0})$ state according to its spin parity and
strong decay mode. However, the measured mass seems relatively lower than the
previous theoretical predictions, which interests the coupled channel
interpretations in the literature. In this work, we adopt an alternate
approach, taking into account the screening effects in the potential model, to
describe the $D_{s0}(2590)^{+}$ resonance. The mass spectrum and strong decays
of the excited charmed-strange mesons are investigated within the modified
relativized quark model and ${}^{3}P_{0}$ model. The calculated mass and width
of the $D_{s0}(2590)^{+}$ are consistent with the experimental observations,
which indicate that it can be reasonably interpreted as the
$D_{s}(2^{1}S_{0})$ state.

###### pacs:

##  I Introduction

A heavy-light meson is composed of one heavy quark and one light antiquark,
and acts as a hydrogen atom. Understanding the heavy-light meson spectrum and
searching for the missing resonances are important tasks in hadronic physics,
which provide us a good opportunity to deepen our understanding of the
complicated non-perturbative behavior of QCD in the low energy regime. Among
of them, the charmed-strange sector is particularly interesting and has gained
wide attentions, since the mysterious $D_{s0}^{\ast}(2317)$ and $D_{s1}(2460)$
states observed by BaBar and CLEO Collaborations BaBar:2003oey ; CLEO:2003ggt
have rather lower masses compared with the theoretical predictions in
conventional quark models Godfrey:2015dva ; Ebert:2009ua ; Zeng:1994vj ;
Lahde:1999ih ; DiPierro:2001dwf ; Li:2010vx ; Ni:2021pce .

From the Review of Particle Physics ParticleDataGroup:2021 , there exist
twelve states in the charmed-strange sector. The $D_{s}$ and $D_{s}^{*}$ are
the ground states, and $D_{s1}(2536)$ and $D_{s2}^{*}(2573)$ can be well
understood as the $P-$wave states. The $D_{s1}^{*}(2700)$, $D_{s1}^{*}(2860)$,
and $D_{s3}^{*}(2860)$ can be assigned as the $D_{s}(2^{3}S_{1})$,
$D_{s}(1^{3}D_{1})$, and $D_{s}(1^{3}D_{3})$ states, respectively, where the
$2S-1D$ mixing effect may be also significant for the vector mesons. The
canonical interpretations of $D_{s0}^{\ast}(2317)$ and $D_{s1}(2460)$ are
problematic in the traditional quark model, while the $X_{0}(2900)$ and
$X_{1}(2900)$ observed by LHCb Collaboration are undoubtedly exotic
LHCb:2020pxc ; LHCb:2020bls . Moreover, information on the $D_{sJ}(3040)$
state is quite limited, which prevents us to reach a definite conclusion. It
can be seen that the low-lying charmed-strange spectrum is far from being
established.

Recently, the LHCb Collaboration observed a new excited resonance
$D_{s0}(2590)^{+}$ in the $D^{+}K^{+}\pi^{-}$ mass distribution of the
$B^{0}\rightarrow D^{-}D^{+}K^{+}\pi^{-}$ decay LHCb:2020gnv . Its mass,
width, and spin parity are determined to be $m=2591\pm 6\pm 7$ MeV,
$\Gamma=89\pm 16\pm 12$ MeV, and $J^{P}=0^{-}$, respectively. Based on these
properties, the LHCb Collaboration suggested that this state is a strong
candidate for the radial excited $D_{s}(2^{1}S_{0})$ state. However, the
measured mass seems relatively lower than the previous theoretical predictions
in the literature Godfrey:2015dva ; Ebert:2009ua ; Zeng:1994vj ; Lahde:1999ih
; DiPierro:2001dwf ; Li:2010vx ; Ni:2021pce ; Godfrey:1985xj , which leads to
different interpretations on the theoretical side. In Ref. Wang:2021orp , the
authors investigated the mass and width of $D_{s0}(2590)^{+}$ by solving the
Bethe-Salpeter equation, and concluded that it can be hardly explained as
$D_{s}(2^{1}S_{0})$ state. Within the semi-relativistic potential model and
chiral quark model, the mass and width of $D_{s0}(2590)^{+}$ are not
consistent with that of $D_{s}(2^{1}S_{0})$ state Ni:2021pce . In Ref.
Xie:2021dwe , the authors employ the unquenched quark model to describe the
mass of $D_{s0}(2590)^{+}$ by considering the mixture of the
$D_{s}(2^{1}S_{0})$ state and $D^{\ast}K$ component. Also, the authors
performed a coupled-channel calculation including the $D^{(\ast)}K^{(\ast)}$,
$D_{s}^{(\ast)}\omega$ and $D_{s}^{(\ast)}\eta$ channels, and found that the
$D_{s0}(2590)$ can be regarded as a bare $D_{s}(2^{1}S_{0})$ state plus
dominant $D^{\ast}K$ part Ortega:2021fem . These theoretical works suggest
that the $D_{s0}(2590)^{+}$ may be not a pure $D_{s}(2^{1}S_{0})$ state and
the $D^{\ast}K$ component should be significant.

Based on the $SU(3)$ light quark flavor symmetry, the $P-$wave charmed-strange
mesons is supposed to be higher than their charmed partners. The violation of
$SU(3)$ flavor symmetry for $D_{s0}^{\ast}(2317)$ and $D_{s1}(2460)$
resonances suggest that they are not pure $P-$wave $c\bar{s}$ states and the
coupled-channel effects are essential. Actually, in the literature, the
coupled-channel approach, meson-loop effects, or the unquenched quark model
has been widely discussed in the charmed-strange sector to reduce the
theoretically predicted masses Xie:2021dwe ; Ortega:2021fem ; Tan:2021bvl ;
Browder:2003fk ; Hwang:2005tm ; Lu:2006ry ; Bicudo:2005de ; Mohler:2011ke ;
MartinezTorres:2011pr ; Mohler:2013rwa ; Lang:2014yfa ; Ortega:2016mms ;
MartinezTorres:2017bdo ; Bali:2017pdv ; Albaladejo:2018mhb , which mainly
focused on the $D_{s0}^{\ast}(2317)$ and $D_{s1}(2460)$ resonances. However,
the situation of the $D_{s0}(2590)^{+}$ is better. The mass gap between
$D_{s1}^{*}(2700)$ and $D_{s0}(2590)$ is

$\displaystyle m[D_{s1}^{*}(2700)]-m[D_{s0}(2590)]=123~{}\rm{MeV},$ (1)

and the mass gap between two $2S$ charmed states $D_{1}^{*}(2600)$ and
$D_{0}(2550)$ with the latest measurements of LHCb Collaboration is
LHCb:2019juy

$\displaystyle m[D_{1}^{*}(2600)]-m[D_{0}(2550)]=124~{}\rm{MeV}.$ (2)

The approximately equal mass splittings of charmed and charmed-strange sectors
strongly suggest that the $D_{s0}(2590)$ should be the partner of
$D_{0}(2550)$ and can be assigned as the $D_{s}(2^{1}S_{0})$ state as the LHCb
Collaboration suggested.

Instead of the unquenched approaches with higher Fock states, the potential
model including screening effects is an alternate approach to lower the mass
spectrum, which has been extensively employed to study the properties of
conventional mesons and achieved significant success. The advantage of the
screening potential is that one can bring down the masses of excited states
while avoiding involving higher Fock components. Hence, we expect that the
potential model including screening effects may relieve the tension between
measured mass and theoretical predictions under the assignment of
$D_{s0}(2590)$ as $D_{s}(2^{1}S_{0})$ state. Moreover, it is natural and
necessary to explore the possible conventional descriptions for a newly
observed particle before introducing more complicated and exotic
configurations. In this work, we apply the Godfrey-Isgur’s relativized quark
model including screening effects to revisit the mass spectrum of the charmed-
strange mesons, and then adopt the obtained wave functions to study their
strong decay behaviors in the ${{}^{3}\\!P_{0}}$ model. Our results show that
the calculated mass and width of the $D_{s0}(2590)^{+}$ are consistent with
the experimental observations, which suggest that it can be reasonably
interpreted as the $D_{s}(2^{1}S_{0})$ state.

This article is organized as follows. In Sec. II, we briefly introduce the
relativized quark models and ${{}^{3}\\!P_{0}}$ models. The results and
discussions of charmed-strange mesons are presented in Sec. III. Finally, a
summary is given in the last section.

##  II Models

###  II.1 The relativized quark model

In this subsection, we will give a brief introduction of the relativized quark
model proposed by Godfrey and Isgur (GI model) Godfrey:1985xj . This model has
been extensively adopted to investigate the properties of conventional hadrons
Godfrey:1985xj ; Godfrey:2014fga ; Godfrey:2015dva ; Li:2021qod ;
Godfrey:2004ya ; Capstick:1985xss ; Barnes:2005pb ; Sun:2014wea ;
Godfrey:2016nwn  and tetraquarks Lu:2021kut ; Lu:2020cns ; Lu:2020rog ;
Lu:2020qmp ; Lu:2016cwr ; Lu:2016zhe ; Lu:2019ira ; Anwar:2017toa ;
Anwar:2018sol ; Bedolla:2019zwg ; Ferretti:2020ewe , and give a unified
description of different flavor sectors. In particular, for the low-lying
states, the relativized quark model plays an important role in studying their
mass spectra and provides an effective criterion to distinguish conventional
mesons from exotics.

For a two-body system, the relevant Hamiltonian can be written as

$H=H_{0}+V^{\rm oge}+V^{\rm conf},$ (3)

where

$H_{0}=\sqrt{p^{2}+m_{1}^{2}}+\sqrt{p^{2}+m_{2}^{2}}$ (4)

is the relativistic kinetic energy, $V^{\rm oge}$ is the one gluon exchange
potential, and $V^{\rm conf}$ corresponds to the confining potential. The
induced spin-dependent interactions are also included in the $V^{\rm oge}$ and
$V^{\rm conf}$.

More explicitly, the potentials $V^{\rm oge}$ and $V^{\rm conf}$ can be
expressed as

$\displaystyle V^{\rm{oge}}$ $\displaystyle=$
$\displaystyle\beta_{12}^{1/2}\tilde{G}(r)\beta_{12}^{1/2}+\delta_{12}^{1/2+\epsilon_{\rm
c}}\frac{2\bm{S_{1}}\cdot\bm{S_{2}}}{3m_{1}m_{2}}\nabla^{2}\tilde{G}(r)\delta_{12}^{1/2+\epsilon_{\rm
c}}$ (5)
$\displaystyle+\delta_{12}^{1/2+so(v)}\frac{(\bm{S_{1}}+\bm{S_{2}})\cdot\bm{L}}{m_{1}m_{2}}\frac{1}{r}\frac{\partial\tilde{G}(r)}{\partial
r}\delta_{12}^{1/2+so(v)}$
$\displaystyle+\delta_{11}^{1/2+so(v)}\frac{\bm{S_{1}}\cdot\bm{L}}{2m_{1}^{2}}\frac{1}{r}\frac{\partial\tilde{G}(r)}{\partial
r}\delta_{11}^{1/2+so(v)}$
$\displaystyle+\delta_{22}^{1/2+so(v)}\frac{\bm{S_{2}}\cdot\bm{L}}{2m_{2}^{2}}\frac{1}{r}\frac{\partial\tilde{G}(2)}{\partial
r}\delta_{22}^{1/2+so(v)}$ $\displaystyle+\delta_{12}^{1/2+\epsilon_{\rm
t}}\Bigg{(}\frac{\bm{S_{1}}\cdot\hat{\bm{r}}\bm{S_{2}}\cdot\hat{\bm{r}}-\bm{S_{1}}\cdot\bm{S_{2}}/3}{m_{1}m_{2}}\Bigg{)}$
$\displaystyle\times\Bigg{(}\frac{1}{r}\frac{\partial}{\partial
r}-\frac{\partial^{2}}{\partial
r^{2}}\Bigg{)}\tilde{G}(r)\delta_{12}^{1/2+\epsilon_{\rm t}},$

and

$\displaystyle V^{\rm{conf}}$ $\displaystyle=$
$\displaystyle\tilde{S}(r)-\delta_{11}^{1/2+so(s)}\frac{\bm{S_{1}}\cdot\bm{L}}{2m_{1}^{2}}\frac{1}{r}\frac{\partial\tilde{S}(r)}{\partial
r}\delta_{11}^{1/2+so(s)}$ (6)
$\displaystyle-\delta_{22}^{1/2+so(s)}\frac{\bm{S_{2}}\cdot\bm{L}}{2m_{2}^{2}}\frac{1}{r}\frac{\partial\tilde{S}}{\partial
r}\delta_{22}^{1/2+so(s)}.$

Here, the $\tilde{G}(r)$ and $\tilde{S}(r)$ are the smeared potentials, and
can be written as

$\tilde{G}(r)=-\sum_{k=1}^{3}\frac{4\alpha_{k}}{3r}{\rm erf}(\tau_{k12}r)$ (7)

and

$\displaystyle\tilde{S}(r)$ $\displaystyle=$ $\displaystyle
br\left[\frac{e^{-\sigma_{12}^{2}r^{2}}}{\sqrt{\pi}\sigma_{12}r}+\left(1+\frac{1}{2\sigma_{12}^{2}r^{2}}\right){\rm
erf}(\sigma_{12}r)\right]+c$ (8)

with

$\frac{1}{\tau_{k12}^{2}}=\frac{1}{\gamma_{k}^{2}}+\frac{1}{\sigma_{12}^{2}}$
(9)

and

$\sigma_{12}^{2}=\sigma_{0}^{2}\left[\frac{1}{2}+\frac{1}{2}\left(\frac{4m_{1}m_{2}}{(m_{1}+m_{2})^{2}}\right)^{4}\right]+s^{2}\left(\frac{2m_{1}m_{2}}{m_{1}+m_{2}}\right)^{2}.$
(10)

The definition of $\delta_{11}$, $\delta_{12}$, $\delta_{22}$, and
$\beta_{12}$ are

$\delta_{ij}=\frac{m_{i}m_{j}}{(p^{2}+m_{i}^{2})^{1/2}(p^{2}+m_{j}^{2})^{1/2}}$
(11)

and

$\beta_{12}=1+\frac{p^{2}}{(p^{2}+m_{1}^{2})^{1/2}(p^{2}+m_{2}^{2})^{1/2}}.$
(12)

The $p$ is the magnitude of the relative momentum between the quark and
antiquark. The $m_{1}$ and $m_{2}$ are masses of the quark and antiquark,
respectively. The $\alpha_{k}$, $\gamma_{k}$, $b$, $c$, $\sigma_{0}$, $s$ and
$\epsilon_{i}$ are the parameters introduced in the relativized quark model.

###  II.2 Screened potential

For high excited states, it is necessary to introduce the screening effects to
the relativized model, because the linear confining potential will be screened
and softened by the vacuum polarization effects at a large distance Li:2009zu
; Chao:1992et ; Ding:1993uy . Also, the modified relativized model (MGI model)
including screen effects turns out to be able to give a better description of
the mass spectra for the radial and orbital excitations Song:2015fha ;
Song:2015nia ; Lu:2016cwr ; Wang:2018rjg ; Pang:2017dlw ; Pang:2018gcn ;
Pang:2019ttv ; Hao:2019fjg .

To incorporate the screen effects in the relatived quark model, we should
replace the confining potential $\tilde{S}(r)$ with a screened potential. The
$\tilde{S}(r)$ actually arises from the liner confinement according to the
smearing transformation. For an arbitrary potential $f(r)$, the smeared ones
$\tilde{f}(r)$ can be expressed as

$\displaystyle\tilde{f}(r)=\int
d^{3}r^{\prime}\rho_{12}(\mathbf{r}-\mathbf{r^{\prime}})f(r^{\prime})$ (13)

with

$\displaystyle\rho_{12}\left(\mathbf{r}-\mathbf{r^{\prime}}\right)=\frac{\sigma_{12}^{3}}{\pi^{3/2}}e^{-\sigma_{12}^{2}\left(\mathbf{r}-\mathbf{r^{\prime}}\right)^{2}}.$
(14)

It can be noticed that the linear confining potential $S(r)=br+c$ indeed leads
to the $\tilde{S}(r)$ through the above smearing transformation. Here, the
constant $c$ always attaches to the confining potential for the same
convention as Ref. Godfrey:1985xj , which can be fixed by the mass of the
ground state.

In the literature, the following replacement is often employed to modify the
linear confining potential in the quark model Chao:1992et ; Ding:1993uy ,

$\displaystyle S(r)=br+c\rightarrow V^{scr}(r)=\frac{b(1-e^{-\mu r})}{\mu}+c.$
(15)

If $r$ is small enough, one has $V^{scr}(r)=V(r)$. Therefore, this replacement
will minimally affect the ground states, and reduce the excited states
significantly. The parameter $\mu$ is related to the strength of the screening
effects, and one can roughly understand that the screening effects begin to
work from $r\sim 1/\mu$. With the smearing transformation, one have

$\displaystyle\tilde{V}^{scr}(r)$ $\displaystyle=$ $\displaystyle\frac{b}{\mu
r}\Bigg{[}e^{\frac{\mu^{2}}{4\sigma_{12}^{2}}+\mu
r}\Bigg{(}\frac{1}{\sqrt{\pi}}\int_{0}^{\frac{\mu+2r\sigma_{12}^{2}}{2\sigma_{12}}}e^{-x^{2}}dx-\frac{1}{2}\Bigg{)}$
(16)
$\displaystyle\times\frac{\mu+2r\sigma_{12}^{2}}{2\sigma_{12}^{2}}+r-e^{\frac{\mu^{2}}{4\sigma_{12}^{2}}-\mu
r}\frac{\mu-2r\sigma_{12}^{2}}{2\sigma_{12}^{2}}$
$\displaystyle\times\Bigg{(}\frac{1}{\sqrt{\pi}}\int_{0}^{\frac{\mu-2r\sigma_{12}^{2}}{2\sigma_{12}}}e^{-x^{2}}dx-\frac{1}{2}\Bigg{)}\Bigg{]}+c.$

Finally, by replacing the $\tilde{S}(r)$ with $\tilde{V}^{scr}$ in the
original relativized quark model, we obtain the modified relativized quark
model including the screening effects. The mass spectrum and wave functions of
the mesons can be obtained by solving the relativized Hamiltonian, and the
wave functions are used as inputs to investigate the subsequent strong decays
for mesons.

###  II.3  The ${{}^{3}\\!P_{0}}$ model

In addition to the mass spectrum, the decay widths are crucial to identify the
assignments for mesons. Here, we give a brief introduction of the
${{}^{3}\\!P_{0}}$ model which is widely used in studying two-body OZI-allowed
strong decays of mesons Barnes:2002mu ; Wang:2017pxm ; Roberts:1992js ;
Blundell:1996as ; Barnes:1996ff ; Close:2005se ; Barnes:2005pb ; Zhang:2006yj
; Ding:2007pc ; Li:2008mza ; Li:2008we ; Li:2008et ; Li:2008xy ; Li:2009rka ;
Li:2009qu ; Li:2010vx ; Lu:2014zua ; Pan:2016bac ; Lu:2016bbk . In the
${{}^{3}\\!P_{0}}$ model, the strong decay of a meson takes place by producing
a quark-antiquark pair with vacuum quantum number $J^{PC}=0^{++}$. The newly
created quark-antiquark pair, together with the $q\bar{q}$ in the initial
meson, regroups into two outgoing mesons in all possible quark rearrangements.
Some detailed reviews on the ${{}^{3}\\!P_{0}}$ model can be found in Refs.
Roberts:1992js ; Blundell:1996as ; Li:2008mza ; Li:2008we ; Li:2008et .

The transition operator $T$ of the decay $A\rightarrow BC$ in the
${{}^{3}\\!P_{0}}$ model is given by

$\displaystyle T$ $\displaystyle=$
$\displaystyle-3\gamma\sum_{m}\left<1,m;1,-m|0,0\right>\int\\!\\!{\rm{d}^{3}}{\bm{p}}_{3}{\rm{d}^{3}}{\bm{p}}_{4}\delta^{3}({\bm{p}}_{3}\\!+\\!{\bm{p}}_{4})$
(17)
$\displaystyle\qquad{\cal{Y}}_{1}^{m}\left(\frac{{\bm{p}}_{3}\\!-\\!{\bm{p}}_{4}}{2}\right)\chi_{1\\!-\\!m}^{34}\phi_{0}^{34}\omega_{0}^{34}b_{3}^{\dagger}({\bm{p}}_{3})d_{4}^{\dagger}({\bm{p}}_{4}),$

where the $\gamma$ is a dimensionless parameter denoting 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 $q_{3}\bar{q}_{4}$, 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

$\displaystyle\langle BC|S|A\rangle=I-2\pi i\delta(E_{A}-E_{B}-E_{C})\langle
BC|T|A\rangle,$ (18)

where $|A\rangle$ ($|B\rangle$,$|C\rangle$) is the mock meson defined by
Hayne:1981zy

$\displaystyle|A(n^{2S_{A}+1}_{A}L_{A}\,\mbox{}_{J_{A}M_{J_{A}}})({\bm{p}}_{A})\rangle\equiv$
$\displaystyle\sqrt{2E_{A}}\sum_{M_{L_{A}},M_{S_{A}}}\langle
L_{A}M_{L_{A}}S_{A}M_{S_{A}}|J_{A}M_{J_{A}}\rangle$ $\displaystyle\times\int
d^{3}{\bm{p}}_{A}\psi_{n_{A}L_{A}M_{L_{A}}}({\bm{p}}_{A})\chi^{12}_{S_{A}M_{S_{A}}}\phi^{12}_{A}\omega^{12}_{A}$
$\displaystyle\times\left|q_{1}\left({\scriptstyle\frac{m_{1}}{m_{1}+m_{2}}}{\bm{p}}_{A}+{\bm{p}}_{A}\right)\bar{q}_{2}\left({\scriptstyle\frac{m_{2}}{m_{1}+m_{2}}}{\bm{p}}_{A}-{\bm{p}}_{A}\right)\right\rangle.$
(19)

Here, $m_{1}$ and $m_{2}$ (${\bm{p}}_{1}$ and ${\bm{p}}_{2}$) are the masses
(momenta) of the quark $q_{1}$ and the antiquark $\bar{q}_{2}$, respectively;
${\bm{p}}_{A}={\bm{p}}_{1}+{\bm{p}}_{2}$,
${\bm{p}}_{A}=\frac{m_{2}{\bm{p}}_{1}-m_{1}{\bm{p}}_{2}}{m_{1}+m_{2}}$;
$\chi^{12}_{S_{A}M_{S_{A}}}$, $\phi^{12}_{A}$, $\omega^{12}_{A}$, and
$\psi_{n_{A}L_{A}M_{L_{A}}}({\bm{p}}_{A})$ are the spin, flavor, color, and
space wave functions of the meson $A$ composed of $q_{1}\bar{q}_{2}$ with
total energy $E_{A}$, respectively. $n_{A}$ is the radial quantum number of
the meson $A$. $\bm{S}_{A}=\bm{s}_{q_{1}}+\bm{s}_{\bar{q}_{2}}$,
$\bm{J}_{A}=\bm{L}_{A}+\bm{S}_{A}$, $\bm{s}_{q_{1}}(\bm{s}_{\bar{q}_{2}})$ is
the spin of $q_{1}(\bar{q}_{2})$, and $\bm{L}_{A}$ is the relative orbital
angular momentum between $q_{1}$ and $\bar{q}_{2}$.

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

$\displaystyle\langle
BC|T|A\rangle=\delta^{3}({\bm{p}}_{A}-{\bm{p}}_{B}-{\bm{p}}_{C}){\cal{M}}^{M_{J_{A}}M_{J_{B}}M_{J_{C}}}({\bm{p}}),$
(20)

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

$\displaystyle{{\cal{M}}}^{M_{J_{A}}M_{J_{B}}M_{J_{C}}}({\bm{p}})=\gamma\sqrt{8E_{A}E_{B}E_{C}}\sum_{M_{L_{A}},M_{S_{A}}}$
$\displaystyle\times\sum_{M_{L_{B}},M_{S_{B}}}\sum_{M_{L_{C}},M_{S_{C}}}\sum_{m}\langle
L_{A}M_{L_{A}}S_{A}M_{S_{A}}|J_{A}M_{J_{A}}\rangle$
$\displaystyle\times\langle
L_{B}M_{L_{B}}S_{B}M_{S_{B}}|J_{B}M_{J_{B}}\rangle\langle
L_{C}M_{L_{C}}S_{C}M_{S_{C}}|J_{C}M_{J_{C}}\rangle$
$\displaystyle\times\langle
1m1-m|00\rangle\langle\chi^{14}_{S_{B}M_{S_{B}}}\chi^{32}_{S_{C}M_{S_{C}}}|\chi^{12}_{S_{A}M_{S_{A}}}\chi^{34}_{1-m}\rangle$
$\displaystyle\times[f_{1}I({\bm{p}},m_{1},m_{2},m_{3})$
$\displaystyle+(-1)^{1+S_{A}+S_{B}+S_{C}}f_{2}I(-{\bm{p}},m_{2},m_{1},m_{3})]$
(21)

with
$f_{1}=\langle\phi^{14}_{B}\phi^{32}_{C}|\phi^{12}_{A}\phi^{34}_{0}\rangle$
and
$f_{2}=\langle\phi^{32}_{B}\phi^{14}_{C}|\phi^{12}_{A}\phi^{34}_{0}\rangle$,
and

$\displaystyle I({\bm{p}},m_{1},m_{2},m_{3})=$ $\displaystyle\int
d^{3}{\bm{p}}\psi^{*}_{n_{B}L_{B}M_{L_{B}}}\left({\scriptstyle\frac{m_{3}}{m_{1}+m_{3}}}{\bm{p}}_{B}+{\bm{p}}\right)$
(22)
$\displaystyle\times\psi^{*}_{n_{C}L_{C}M_{L_{C}}}\left({\scriptstyle\frac{m_{3}}{m_{2}+m_{3}}}{\bm{p}}_{B}+{\bm{p}}\right)$
$\displaystyle\times\psi_{n_{A}L_{A}M_{L_{A}}}\left({\bm{p}}_{B}+{\bm{p}}\right){{\cal{Y}}}^{m}_{1}({\bm{p}}),$

where ${\bm{p}}={{\bm{p}}}_{B}=-{{\bm{p}}}_{C}$, ${\bm{p}}={\bm{p}}_{3}$,
$m_{3}$ is the mass of the created quark $q_{3}$. Also, the helicity amplitude
can be transform into the partial wave amplitude ${{\cal{M}}}^{LS}({\bm{p}})$
Jacob:1959at ,

$\displaystyle{{\cal{M}}}^{LS}({\bm{p}})$ $\displaystyle=$
$\displaystyle\sum_{\begin{array}[t]{l}\scriptstyle M_{J_{B}},M_{J_{C}},\\\
\scriptstyle M_{S},M_{L}\end{array}}\\!\\!\langle
LM_{L}SM_{S}|J_{A}M_{J_{A}}\rangle$ (26) $\displaystyle\langle
J_{B}M_{J_{B}}J_{C}M_{J_{C}}|SM_{S}\rangle$ $\displaystyle\times\int
d\Omega\,\mbox{}Y^{\ast}_{LM_{L}}{{\cal{M}}}^{M_{J_{A}}M_{J_{B}}M_{J_{C}}}({\bm{p}}).$

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 creation strength $\gamma$ for the $u\bar{u}$ and $d\bar{d}$
pairs. For the $s\bar{s}$ pair, the creation strength is multiplied by a
factor $m_{u}/m_{s}$. The wave functions can be obtained from the modified
relativized quark model including the screening effects. With the relativistic
phase space, the decay width $\Gamma(A\rightarrow BC)$ can be expressed in
terms of the partial wave amplitude

$\displaystyle\Gamma(A\rightarrow
BC)=\frac{\pi|{\bm{p}}|}{4M^{2}_{A}}\sum_{LS}|{{\cal{M}}}^{LS}({\bm{p}})|^{2},$
(27)

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 CALCULATION AND RESULTS

###  III.1  Mass spectrum

Table 1:  Parameters in the Godfrey-Isgur’s relativized quark model Godfrey:1985xj .  Parameter | value | Parameter | value | Parameter | value
---|---|---|---|---|---
$m_{u}$ (GeV) | 0.22 |  $b$ (GeV2) | 0.18 | $\epsilon_{\rm c}$ | -0.168
$m_{d}$ (GeV) | 0.22 |  $c$ (GeV) | -0.253 | $\epsilon_{\rm t}$ | +0.025
$m_{s}$ (GeV) | 0.419 |  $\sigma_{0}$ (GeV) | 1.8 | $\epsilon_{\rm{so(v)}}$ | -0.035
$m_{c}$ (GeV) | 1.628 | $s$ | 1.55 | $\epsilon_{\rm so(s)}$ | +0.055
Table 2: Comparison of the experimental data and theoretical results with different $\mu$. We take $\mu$=0.04, 0.045, 0.05 GeV to show the results with the modified relativized quark model with screened potential. We also list the $\chi^{2}$ values for different models.  | $n^{2s+1}L_{J}$ | Experimental values | GI model |  | Modified GI model |  |
---|---|---|---|---|---|---|---
|  |  |  | $\mu=0.04$ | $\mu=0.045$ | $\mu=0.05$ |
$D_{s}^{\pm}$ | $1^{1}S_{0}$ |  1968.34$\pm$0.07 | 1979 | 1968 | 1968 | 1968 |
$D_{s}^{\ast\pm}$ | $1^{3}S_{1}$ |  2112.2$\pm$0.4 | 2129 | 2114 | 2114 | 2113 |
$D_{s2}^{\ast}(2573)$ | $1^{3}P_{2}$ |  2569.1$\pm$0.8 | 2592 | 2559 | 2556 | 2553 |
$D_{s1}^{\ast}(2700)^{\pm}$ | $2^{3}S_{1}$ |  2708${}^{+4.0}_{-3.4}$ | 2732 | 2681 | 2675 | 2670 |
$D_{s1}^{\ast}(2860)^{\pm}$ | $1^{3}D_{1}$ |  2859$\pm$12$\pm$24 | 2899 | 2839 | 2833 | 2827 |
$D_{s3}^{\ast}(2860)^{\pm}$ | $1^{3}D_{3}$ |  2860.5$\pm$2.6$\pm$6.5 | 2917 | 2858 | 2852 | 2846 |
$\chi^{2}$ |  |  | 666 | 55.17 | 86.23 | 119.39 |
Figure 1: Mass spectrum of the charmed-strange mesons in units of MeV. The
black lines show the MGI model with $\mu=0.045$ GeV and the shaded regions
stand for the theoretical uncertainties with $\mu=0.04-0.05$ GeV. The dark
blue dot denote the experimental data Zyla:2020zbs  and the vertical lines
represent the errors.

The relevant parameters used in the original relativized quark model are
listed in Table 1 Godfrey:1985xj . When the screening effects are included,
and extra parameter $\mu$ is introduced, which reflects the strength of
screening effects. In present work, we can get the parameter $\mu$ by
reproducing the experimental data of low-lying states. As mentioned in the
Introduction, seven states, $D_{s}$, $D_{s}^{*}$, $D_{s1}(2536)$,
$D_{s2}^{*}(2573)$, $D_{s1}^{*}(2700)$, $D_{s1}^{*}(2860)$, and
$D_{s3}^{*}(2860)$, can be reasonably classified in the conventional charmed-
strange mesons. Since the $D_{s1}(2536)$ is a mixture of the
$D_{s}(1^{1}P_{1})$ and $D_{s}(1^{3}P_{1})$ states, we do not include it when
determining the parameter $\mu$ . Also, the overall constant $c$ is readjusted
by fixing the mass of $D_{s}(1^{1}S_{0})$ to 1968 MeV when the $\mu$ varies.

Table 3: Our predicted masses of charmed-strange mesons compared with the experimental data and other quark model predictions. The mixing angles of $D_{s}$–$D_{s}^{\prime}$ obtained in this work are $\theta_{1P}$=$-42.7^{\circ}$, $\theta_{2P}$=$-31.4^{\circ}$, $\theta_{1D}$=$-39.4^{\circ}$, $\theta_{2D}$=$-38.4^{\circ}$, $\theta_{1F}$=$-39.9^{\circ}$. Horizontal lines indicate that the corresponding masses are not calculated in the references. The units are in MeV. State | $J^{P}$ | Ours | NLZ Ni:2021pce  | EFG Ebert:2009ua  | ZVR Zeng:1994vj  | GM Godfrey:2015dva  | LNR Lahde:1999ih  | DE DiPierro:2001dwf  | LJM Li:2010vx  | GI Godfrey:1985xj  | Exp Zyla:2020zbs
---|---|---|---|---|---|---|---|---|---|---|---
$D_{s}(1^{1}S_{0})$ | $0^{-}$ | 1968 | 1969 | 1969 | 1940 | 1979 | 1975 | 1965 | 1969 | 1979 | 1968.34$\pm$0.07
$D_{s}(1^{3}S_{1})$ | $1^{-}$ | 2114 | 2112 | 2111 | 2130 | 2129 | 2180 | 2113 | 2107 | 2129 | 2112.2$\pm$0.4
$D_{s}(2^{1}S_{0})$ | $0^{-}$ | 2620 | 2649 | 2688 | 2610 | 2673 | 2659 | 2700 | 2640 | 2673 | 2591$\pm$6$\pm$7
$D_{s}(2^{3}S_{1})$ | $1^{-}$ | 2675 | 2737 | 2731 | 2730 | 2732 | 2722 | 2806 | 2714 | 2732 | 2708${}^{+4.0}_{-3.4}$
$D_{s}(3^{1}S_{0})$ | $0^{-}$ | 3072 | 3126 | 3219 | 3090 | 3154 | 3044 | 3259 | - | - | -
$D_{s}(3^{3}S_{1})$ | $1^{-}$ | 3036 | 3196 | 3242 | 3190 | 3193 | 3087 | 3345 | - | - | -
$D_{s}(1^{3}P_{0})$ | $0^{+}$ | 2450 | 2409 | 2509 | 2380 | 2484 | 2455 | 2487 | 2344 | 2484 | 2317.8$\pm$0.5
$D_{s}(1P)$ | $1^{+}$ | 2515 | 2528 | 2536 | 2510 | 2549 | 2502 | 2535 | 2488 | - | -
$D_{s}(1P^{\prime})$ | $1^{+}$ | 2519 | 2545 | 2574 | 2520 | 2556 | 2522 | 2605 | 2510 | - | -
$D_{s}(1^{3}P_{2})$ | $2^{+}$ | 2556 | 2575 | 2571 | 2580 | 2592 | 2586 | 2581 | 2559 | 2592 | 2569.1$\pm$0.8
$D_{s}(2^{3}P_{0})$ | $0^{+}$ | 2913 | 2940 | 3054 | 2900 | 3005 | 2901 | 3067 | 2830 | 3005 |
$D_{s}(2P)$ | $1^{+}$ | 2930 | 3002 | 3067 | 3000 | 3018 | 2928 | 3114 | 2958 | - | -
$D_{s}(2P^{\prime})$ | $1^{+}$ | 2944 | 3026 | 3154 | 3010 | 3038 | 2942 | 3165 | 2995 | - | -
$D_{s}(2^{3}P_{2})$ | $2^{+}$ | 2957 | 3053 | 3142 | 3060 | 3048 | 2988 | 3157 | 3040 | 3048 | -
$D_{s}(1^{3}D_{1})$ | $1^{-}$ | 2833 | 2843 | 2913 | 2820 | 2899 | 2845 | 2913 | 2804 | 2899 | 2859$\pm$12$\pm$24
$D_{s}(1D)$ | $2^{-}$ | 2837 | 2851 | 2931 | 2860 | 2900 | 2838 | 2900 | 2788 | - | -
$D_{s}(1D^{\prime})$ | $2^{-}$ | 2859 | 2911 | 2961 | 2880 | 2926 | 2856 | 2953 | 2849 | - | -
$D_{s}(1^{3}D_{3})$ | $3^{-}$ | 2852 | 2882 | 2971 | 2900 | 2917 | 2857 | 2925 | 2811 | 2917 | 2860.5$\pm$2.6$\pm$6.5
$D_{s}(2^{3}D_{1})$ | $1^{-}$ | 3176 | 3233 | 3383 | 3250 | 3306 | 3172 | - | 3217 | 3306 | -
$D_{s}(2D)$ | $2^{-}$ | 3173 | 3267 | 3403 | 3280 | 3298 | 3144 | - | 3217 | - | -
$D_{s}(2D^{\prime})$ | $2^{-}$ | 3192 | 3306 | 3456 | 3290 | 3323 | 3167 | - | 3260 | - | -
$D_{s}(2^{3}D_{3})$ | $3^{-}$ | 3185 | 3299 | 3469 | 3310 | 3311 | 3157 | - | 3240 | 3311 | -
$D_{s}(1^{3}F_{2})$ | $2^{+}$ | 3107 | 3176 | 3230 | 3120 | 3208 | - | 3224 | - | 3208 | -
$D_{s}(1F)$ | $3^{+}$ | 3078 | 3123 | 3254 | 3130 | 3186 | - | - | - | - | -
$D_{s}(1F^{\prime})$ | $3^{+}$ | 3104 | 3205 | 3266 | 3150 | 3218 | - | - | - | - | -
$D_{s}(1^{3}F_{4})$ | $4^{+}$ | 3094 | 3134 | 3300 | 3160 | 3190 | - | 3220 | - | 3190 | -

The mass spectrum of the charmed-strange meson with $\mu$ from 0.04 to 0.05
GeV is listed in Table 2. For comparison, the experiment data and predictions
of the original relativized quark model are also presented. It can be seen
that the measured masses of the low-lying states can be well reproduced and
the predicted spectrum in the screened potential is improved significantly.
Moreover, we can estimate the corresponding $\chi^{2}$ and present them in
Table 2. Here the $\chi^{2}$ can be defined as

$\displaystyle\chi^{2}=\sum_{i}\left(\frac{\mathcal{A}_{\mathrm{Th}}(i)-\mathcal{A}_{\mathrm{Exp}}(i)}{\mathrm{Error}(i)}\right)^{2},$
(28)

where$\mathcal{A}_{\mathrm{Th}}(i)$, $\mathcal{A}_{\mathrm{Exp}}(i)$, and
$\mathrm{Error}(i)$ are theoretical values, experimental values, and
experimental errors, respectively. With the reasonable range of $\mu$, the
$\chi^{2}$ of screened potential is significantly smaller than that of
original relativized quark model.

It should be mentioned that the $\chi^{2}$ is not the only criterion of the
performances for different predictions in quark models. From Eq. (28), if the
experimental accuracies of several states are high enough, the model with the
smallest $\chi^{2}$ may only reproduce these few states and fail to describe
the whole mass spectrum. Phenomenologically, we also expect the absolute value
$|\mathcal{A}_{\mathrm{Th}}(i)-\mathcal{A}_{\mathrm{Exp}}(i)|$ for each state
is not too large, such that these states can be interpreted in the
conventional $c\bar{s}$ picture. In the range of $\mu=0.04\sim 0.05$ GeV, the
results meet the above requirements.

Hence, we prefer to choose $\mu=0.045$ GeV to calculate the mass spectrum of
charmed-strange mesons, and take the masses with $\mu=0.04$ and 0.05 GeV as
the theoretical uncertainties. With $\mu=0.045$ GeV, one can obtain the
constant $c$ equals to $-0.243$. The theoretical predictions together with
experimental data are shown in Figure 1. It can be seen that the
$D_{s0}(2590)^{+}$ can be assigned as the $D_{s}(2^{1}S_{0})$ state according
to its mass. Moreover, we compare the predictions of different models in Table
3, and find that they give rather different predictions for the higher states.
Also, the screening effects become increasingly important as the masses go up.
The information on highly excited states is crucial to distinguish these
different models and test our screened potential.

###  III.2 Strong decays

Besides the mass spectrum, the strong decay behaviors are essential to clarify
the internal structure of a new resonance. In this work, the
${{}^{3}\\!P_{0}}$ model is adopted to investigate the strong decays of the
$D_{s0}(2590)$. While we calculate the mass spectrum, the corresponding wave
functions of mesons are also obtained. Then, only one parameter $\gamma$ in
the ${{}^{3}\\!P_{0}}$ model needs to determine. We can assume that the
charmed-strange mesons share the same $\gamma$, and fit this parameter from
the known states. Among the seven reasonably classified states, $D_{s}$ only
decays though weak processes, $D_{s}^{*}$ has no OZI-allowed strong decay, and
the strong decays of $D_{s1}(2536)$ depend on the mixing angle sensitively.
Hence, we adopt the remaining resonances, $D_{s2}^{*}(2573)$,
$D_{s1}^{*}(2700)$, $D_{s1}^{*}(2860)$, and $D_{s3}^{*}(2860)$, to fit the
parameter $\gamma$.

According to the fitting process, the $\gamma=9.32$ is obtained, and the
strong decay behaviors of the $D_{s2}^{*}(2573)$, $D_{s1}^{*}(2700)$,
$D_{s1}^{*}(2860)$, and $D_{s3}^{*}(2860)$ are listed in Table 4. It can be
seen that the calculated widths of $D_{s2}^{\ast}(2573)$ and
$D_{s3}^{\ast}(2860)$ are consistent with the experimental data within errors,
and the theoretical width of $D_{s1}^{\ast}(2700)$ and $D_{s1}^{\ast}(2860)$
seems a little bit larger. These differences may arise from the theoretical
uncertainties of the ${{}^{3}\\!P_{0}}$ model or the possible complicated
$S-D$ mixing mechanism for the $D_{s1}^{\ast}(2700)$ and $D_{s1}^{\ast}(2860)$
states. Hence, with the $\gamma=9.32$, the strong decay behaviors of these
four states are fairly described. We employ this value to investigate the
strong decays of $D_{s0}(2590)$.

Table 4: Decay widths of $D_{s2}^{\ast}(2573)$, $D_{s1}^{\ast}(2700)^{+}$, $D_{s1}^{\ast}(2860)^{+}$ and $D_{s3}^{\ast}(2860)^{+}$ with fitted $\gamma=9.32$ (in MeV). Mode | $D_{s2}^{\ast}(2573)$ | $D_{s1}^{\ast}(2700)^{+}$ | $D_{s1}^{\ast}(2860)^{+}$ | $D_{s3}^{\ast}(2860)^{+}$
---|---|---|---|---
$DK$ | 12.07 | 61.13 | 150.72 | 25.65
$D^{\ast}K$ | 1.27 | 116.05 | 76.36 | 17.27
$DK^{\ast}$ | 0.03 | 3.69 | 45.22 | 1.32
$D_{s}\eta$ | - | 1.61 | 10.78 | 0.52
$D_{s}^{\ast}\eta$ | - | - | 3.72 | 0.16
Total width | 13.37 | 182.48 | 286.80 | 44.92
Experiment | $16.9\pm 0.7$ | $122\pm 10$ | $159\pm 23\pm 77$ | $53\pm 7\pm 7$
Table 5: Decay widths of $D_{s0}(2590)^{+}$ as the $D_{s}(2^{1}S_{0})$ state with fitted $\gamma=9.32$ (in MeV). Mode | $D_{s0}(2590)^{+}$
---|---
$D^{\ast+}K^{0}$ | 35.52
$D^{\ast 0}K^{+}$ | 39.38
Total width | 74.90
Experiment | $89\pm 16\pm 12$
Figure 2: The dependence of the total width of $D_{s}(2^{1}S_{0})$ on the
initial state mass.

The results of $D_{s0}(2590)$ as the $D_{s}(2^{1}S_{0})$ state are listed in
Table 5. The calculated width is about 75 MeV, which agrees well with the
experimental data $89\pm 16\pm 12$ MeV. Also, the dependence on the mass of
initial state is shown in Fig.2. When the mass of initial $D_{s}(2^{1}S_{0})$
state varies from 2570 to 2610 MeV, the total width lies in the range of 51 to
98 MeV. Our results indicate that the $D_{s0}(2590)^{+}$ observed by LHC
Collaboration can be interpreted as the conventional $D_{s}(2^{1}S_{0})$
state.

##  IV SUMMARY AND CONCLUSION

In this work, we investigate the mass spectrum of charmed-strange mesons with
the modified relativized quark model including the screening effects. With
reasonable strength of screening effects, the calculated mass spectrum can
explain the $D_{s0}(2590)^{+}$ as well as other known charmed-strange mesons.
The information on highly excited states is crucial to distinguish various
predictions and test our results with screened potential.

Besides the mass spectrum, the strong decays of $D_{s0}(2590)^{+}$ as
$D_{s}(2^{1}S_{0})$ state are also investigated in the ${{}^{3}\\!P_{0}}$
model with the obtained relativistic wave functions. The calculated width is
about 75 MeV, which agrees well with the experimental data $89\pm 16\pm 12$
MeV. Our results indicate that the $D_{s0}(2590)^{+}$ can be interpreted as
the conventional $D_{s}(2^{1}S_{0})$ state.

##  V Acknowledgements

This work is supported by the China Postdoctoral Science Foundation Funded
Project (Project No.:2021M701086), Natural Science Foundation of Henan
(212300410123). Qi-Fang Lü is supported by the National Natural Science
Foundation of China under Grants No. 11705056 and No. U1832173, by the Key
Project of Hunan Provincial Education Department under Grant No. 21A0039, and
by the State Scholarship Fund of China Scholarship Council under Grant No.
202006725011.

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