# Halo Orbits around $L_{1}$ and $L_{2}$ in the Photogravitational Sun-Earth
System with Oblateness

Dhwani Sheth11affiliationmark<EMAIL_ADDRESS>V. O.
Thomas11affiliationmark<EMAIL_ADDRESS>
###### Abstract

The Photogravitational Restricted Three Body Problem with oblateness has been
studied to obtain halo orbits around the Lagrangian points $L_{1}$ and $L_{2}$
of the Sun-Earth system in which the Sun is taken as radiating and the Earth
as an oblate spheroid. The halo orbits corresponding to fourth and fifth order
approximations around $L_{1}$ and $L_{2}$ for actual oblateness of the Earth
and for different radiation pressures for the Sun are displayed graphically.
The time period of halo orbits around $L_{1}$ decreases with increase in
oblateness and increases with increase in radiation pressure. A reverse effect
is observed due to increase in oblateness and radiation pressure on time
period of orbits around $L_{2}$. It is also observed that halo orbits around
$L_{1}$ shifts towards the source of radiation due to increase in both
radiation pressure and oblateness. However, halo orbits around $L_{2}$ shifts
towards the source of radiation due to increase in radiation but recedes with
increase in oblateness.

00footnotetext: Department of Mathematics, Faculty of Science, The Maharaja
Sayajirao University of Baroda, Vadodara-390002, India
email<EMAIL_ADDRESS>

Keywords Restricted Three Body Problem, Photogravitational Sun-Earth System,
Oblateness, Halo orbits

##  1 Introduction

Restricted Three Body Problem(RTBP) deals with the motion of an infinitesimal
body which moves under the gravitational influence of two massive bodies
called the primaries. The infinitesimal body is called the secondary body. The
only force acting on this system is the gravitational attraction force between
the primaries. The mass of the secondary body is negligible compared to the
primary masses and it does not influence the motion of the primaries. RTBP is
very useful for describing the motion of planets, asteroids, comets and
satellites(Plummer, 1919; Winter, 1941; Brouwer and Clemence, 1961; Danby,
1964; Pollard, 1966; Murray and Dermot, 1999). It plays an important role in
space dynamics, celestial dynamics and analytic dynamics. It has applications
in the fields of mathematics, theoretical physics and quantum physics. In
Circular Restricted Three Body Problem(CRTBP), the primaries move in a
circular path around their common centre of mass. This is a particular case of
RTBP(Moulton, 1914; McCuskey, 1963; Szebehely, 1967; Roy, 2005; Fitzpatrick,
2012; Vallado, 2013). Most of the celestial bodies are radiating and hence
study of RTBP incorporating radiation, usually called photogravitational RTBP,
is pertinent. The solar radiation pressure force changes with the distance in
a similar law as the gravitational attraction force but acts in an opposite
direction to it. This reduces the effective mass of the Sun(Poynting, 1903;
Robertson and Russell, 1937; Schuerman, 1980; Simmons et al., 1985;
Abouelmagd, 2013; Pathak et al., 2016). In the case of planar CRTBP, there
exist five equilibrium points known as Libration points or Lagrangian points.
Among these, three points, denoted by $L_{1},L_{2}$ and $L_{3}$ are collinear
with $L_{1}$ lying between the primaries.The remaining Lagrangian points
$L_{4}$ and $L_{5}$ lie opposite sides of the joining the primaries. The three
dimensional periodic orbits around Lagrangian points are called halo orbits.
Halo orbits were introduced by Farquhar (1968). He discovered the trajectories
around the Earth-Moon $L_{2}$ which could be used to place a communication
satellite that would continuously link between the Earth and the Moon. Other
researchers (Breakwell and Brown, 1979; Howell and V.Breakwell, 1984; Howell,
1984) have studied halo orbit families for the Earth-Moon system. ISEE-3 was
the first halo orbit mission. A third order approximation was introduced by
Richardson (1980) to represent halo orbits in the Sun-Earth system. Tiwary and
Kushvah (2015) have computed a first guess of halo orbits upto fourth order
approximation using the Lindstedt-Poincar$\acute{e}$ method in the
photogravitational RTBP with oblateness.
In this paper we have computed halo orbits around the Lagrangian points
$L_{1}$ and $L_{2}$ in the Sun-Earth system considering the Sun as radiating
body and the Earth as oblate spheroid. The fourth order approximations to
solutions given by (Tiwary and Kushvah, 2015) have been improved incorporating
fifth order approximation. The comparison of the orbits obtained by fourth and
fifth order approximations are shown graphically. Equations of motion of an
infinitesimal body in a synodic system are described in Section 2. Equations
of motion around $L_{1}$ and $L_{2}$ and the procedure to obtain halo orbits
are described in Section 3. Section 4 has discussion of effects of solar
radiation pressure and oblateness on various parameters of halo orbits. In
Section 5, the conclusions are given.

##  2 Equations of Motion

Consider the photogravitational CRTBP with oblateness and assume that the
bigger primary, the Sun, is the source of radiation and the smaller primary,
the Earth, is an oblate spheroid. The mass reduction factor $q$, the perturbed
mean motion $n$ and the oblateness coefficient $A_{2}$ are given as (McCuskey,
1963; Sharma, 1987)

$q=\left(1-\frac{F_{p}}{F_{g}}\right),n=\sqrt{\left(1+\frac{3}{2}A_{2}\right)},A_{2}=\frac{R_{e}^{2}-R_{p}^{2}}{5R^{2}}$

where $F_{p}$ is the solar radiation pressure force, $F_{g}$ is the
gravitational attraction force, $R_{e}$ and $R_{p}$ are, respectively, the
equatorial and polar radii of the smaller primary and $R$ is the distance
between the two primaries. The expression for $q$ shows that as solar
radiation pressure increases, $q$ decreases. We use synodic coordinates with
origin at the centre of mass of the primaries for describing the motion of the
system. Choose the unit of length as the distance between the primaries. Let
$(x,y,z)$ be the coordinates of the infinitesimal body and $(-\mu,0,0)$ and
$(1-\mu,0,0)$ be the coordinates of the bigger and smaller primaries,
respectively, in the dimensionless synodic coordinate system, where
$\mu=\text@frac{m_{2}}{m_{1}+m_{2}}$, $m_{1},m_{2}$ being the masses of the
bigger and smaller primaries. The equations of motion of infinitesimal body
with oblateness and solar radiation pressure are given by (Szebehely, 1967;
Sharma, 1987; Tiwary and Kushvah, 2015)

$\displaystyle\ddot{x}-2n\dot{y}$ $\displaystyle=$
$\displaystyle\frac{\partial\Omega^{*}}{\partial x},$ (1)
$\displaystyle\ddot{y}+2n\dot{x}$ $\displaystyle=$
$\displaystyle\frac{\partial\Omega^{*}}{\partial y},$ (2)
$\displaystyle\ddot{z}$ $\displaystyle=$
$\displaystyle\frac{\partial\Omega^{*}}{\partial z},$ (3)

where

$\Omega^{*}=n^{2}\frac{(x^{2}+y^{2})}{2}+\frac{(1-\mu)q}{r_{1}}+\frac{\mu}{r_{2}}+\frac{\mu
A_{2}}{2r_{2}^{3}},$ (4)

and

$\displaystyle r_{1}$ $\displaystyle=$
$\displaystyle\sqrt{(x+\mu)^{2}+y^{2}+z^{2}},$ (5) $\displaystyle r_{2}$
$\displaystyle=$ $\displaystyle\sqrt{(x+\mu-1)^{2}+y^{2}+z^{2}}$ (6)

are the distances of the infinitesimal body from the bigger and smaller
primaries, respectively.

##  3 Computation of Halo orbits

Lindstedt-Poincar$\acute{e}$ method(Koon et al., 2011) is used to compute the
halo orbits around the libration points $L_{1}$ and $L_{2}$. It is used for
solving non-linear ordinary differential equation when the regular
perturbation method fails by removing secular terms and thereby converting to
weakly non-linear equation with finite oscillatory solutions.

###  3.1 Equations of motion near $L_{1}$ and $L_{2}$

To obtain the halo orbits around the Lagrangian point the origin is shifted to
the location of the Lagrangian point. Then the new coordinates are given by
(Koon et al., 2011)

$\displaystyle X$ $\displaystyle=$
$\displaystyle\frac{1}{\gamma}(x+\mu-1\pm\gamma),$ (7) $\displaystyle Y$
$\displaystyle=$ $\displaystyle\frac{1}{\gamma}y,$ (8) $\displaystyle Z$
$\displaystyle=$ $\displaystyle\frac{1}{\gamma}z,$ (9)

where $\gamma$ is the distance between the Lagrangian point and the smaller
primary. In (7), upper sign corresponds to $L_{1}$ and lower sign corresponds
to $L_{2}$. The variables $X,Y$ and $Z$ are normalized so that the distance
between the Lagrangian point and the smaller primary is $1$. Using the above
transformation in the equations of motion (1)-(3), we obtain

$\displaystyle\gamma(\ddot{X}-2n\dot{Y})$ $\displaystyle=$
$\displaystyle\frac{1}{\gamma}\frac{\partial\Omega}{\partial X},$ (10)
$\displaystyle\gamma(\ddot{Y}+2n\dot{X})$ $\displaystyle=$
$\displaystyle\frac{1}{\gamma}\frac{\partial\Omega}{\partial Y},$ (11)
$\displaystyle\gamma\ddot{Z}$ $\displaystyle=$
$\displaystyle\frac{1}{\gamma}\frac{\partial\Omega}{\partial Z},$ (12)

where

$\Omega=\frac{n^{2}}{2}[(\gamma X+1-\mu\mp\gamma)^{2}+(\gamma
Y)^{2}]+\frac{(1-\mu)q}{R_{1}}+\frac{\mu}{R_{2}}+\frac{\mu A_{2}}{2R_{2}^{3}}$
(13)

and

$\displaystyle R_{1}$ $\displaystyle=$ $\displaystyle\sqrt{(\gamma
X+1\mp\gamma)^{2}+(\gamma Y)^{2}+(\gamma Z)^{2}},$ $\displaystyle R_{2}$
$\displaystyle=$ $\displaystyle\sqrt{(\gamma X\mp\gamma)^{2}+(\gamma
Y)^{2}+(\gamma Z)^{2}}.$

Expanding the nonlinear terms
$\frac{(1-\mu)q}{R_{1}}+\frac{\mu}{R_{2}}+\frac{\mu A_{2}}{2R_{2}^{3}}$ of
(13) using Legendre polynomials , equations of motion can be written as (Koon
et al., 2011; Tiwary and Kushvah, 2015)

$\displaystyle\ddot{X}-2n\dot{Y}-(n^{2}+2C_{2})X=\frac{\partial}{\partial
X}\sum_{k\geqslant 3}C_{k}\rho^{k}P_{k}\left(\frac{X}{\rho}\right),$ (14)
$\displaystyle\ddot{Y}+2n\dot{X}+(C_{2}-n^{2})Y=\frac{\partial}{\partial
Y}\sum_{k\geqslant 3}C_{k}\rho^{k}P_{k}\left(\frac{X}{\rho}\right),$ (15)
$\displaystyle\ddot{Z}+C_{2}Z=\frac{\partial}{\partial Z}\sum_{k\geqslant
3}C_{k}\rho^{k}P_{k}\left(\frac{X}{\rho}\right).$ (16)

In above equations, the left hand side contains the linear terms and the right
hand side contains the non-linear terms. The coefficients $C_{k}$ are given by

$C_{k}=\frac{1}{\gamma^{3}}\hskip-3.0pt\left[\frac{(-1)^{k}q(1-\mu)\gamma^{k+1}}{(1\mp\gamma)^{k+1}}\,+(\pm
1)^{k}\left(\mu\,+\frac{3\mu A_{2}}{2\gamma^{2}}\right)\right],$ (17)

for $k\geqslant 1$. Considering only linear terms in equations (14)-(16), the
solution of the linearized equations is

$X(t)=A_{1}e^{\alpha t}+A_{2}e^{-\alpha t}+A_{3}\cos\lambda t+A_{4}\sin\lambda
t,$ $Y(t)=-k_{1}A_{1}e^{\alpha t}+k_{1}A_{2}e^{-\alpha
t}-k_{2}A_{3}\sin\lambda t+k_{2}A_{4}\cos\lambda t,$
$Z(t)=A_{5}\cos\sqrt{C_{2}}t+A_{6}\sin\sqrt{C_{2}}t,$

where $A_{1},A_{2},A_{3},A_{4},A_{5}$ and $A_{6}$ are arbitrary constants,

$\displaystyle\alpha=$
$\displaystyle\sqrt{\frac{-(2n^{2}-C_{2})+\sqrt{9C_{2}^{2}-8n^{2}C_{2}}}{2}},$
$\displaystyle\lambda=$
$\displaystyle\sqrt{\frac{2n^{2}-C_{2}+\sqrt{9C_{2}^{2}-8n^{2}C_{2}}}{2}},$
$\displaystyle k_{1}=$
$\displaystyle\frac{(2C_{2}+n^{2})-\alpha^{2}}{2n\alpha},$ $\displaystyle
k_{2}=$ $\displaystyle\frac{(2C_{2}+n^{2})+\lambda^{2}}{2n\lambda}.$

Linearized equations corresponding to equations (14)-(16) have two real roots
which are equal in magnitude and opposite in sign. If the initial conditions
are chosen arbitrarily, then these roots give rise to unbounded solutions.To
avoid this, we take $A_{1}=A_{2}=0$ and
$A_{3}=-A_{X}\cos\phi,A_{4}=A_{X}\sin\phi,A_{5}=A_{Z}\sin\psi$ and
$A_{6}=A_{Z}\cos\psi$ and get the bounded solution in the following form:
(Koon et al., 2011)

$\displaystyle X(t)=$ $\displaystyle-A_{X}\cos(\lambda t+\phi),$
$\displaystyle Y(t)=$ $\displaystyle kA_{X}\sin(\lambda t+\phi),$
$\displaystyle Z(t)=$ $\displaystyle A_{Z}\sin(\sqrt{C_{2}}t+\psi),$

where $A_{X}$ and $A_{Z}$ are amplitudes; $\lambda$ and $\sqrt{C_{2}}$ are the
frequencies; $k=k_{2}$; $\phi$ and $\psi$ are phases of the in-plane and out
of plane motions respectively. The ratio of $\lambda$ and $\sqrt{C_{2}}$ is
irrational. This gives Lissajous(quasi periodic) orbits.

###  3.2 Lindstedt-Poincar$\acute{e}$ Method for the Halo Orbits

Halo orbits are important for spacecraft mission design. Many researchers have
obtained the halo orbits upto third order approximation(Richardson, 1980;
Howell, 1984; Breakwell and Brown, 1979; Koon et al., 2011; Chidambararaj and
Sharma, 2016; Pushparaj and Sharma, 2016; Ghotekar and Sharma, 2019). Tiwary
and Kushvah (2015) have computed halo orbits upto fourth order approximation
with the Sun as a radiating body and the Earth as an oblate spheroid using
Lindstedt-Poincar$\acute{e}$ method. Here, we have computed halo orbits upto
fifth order approximation with radiation pressure and oblateness using
Lindstedt-Poincar$\acute{e}$ method. The non-linear terms in (14)-(16) change
the frequency of the linearized system. Due to this secular terms appear in
successive approximations. To change the frequency, we take a new independent
variable $\tau=\omega t$, where $\omega$ is a frequency connection. Then the
equations of motion (14)-(16) in terms of $\tau$ truncated at degree $5$ are:

$\omega^{2}X^{\prime\prime}-2n\omega Y^{\prime}-(n^{2}+2C_{2})X\\\
=\frac{3}{2}C_{3}(2X^{2}-Y^{2}-Z^{2})+2C_{4}X(2X^{2}-3Y^{2}-3Z^{2})\\\
\quad+\frac{5}{8}C_{5}[8X^{2}\\{X^{2}-3(Y^{2}+Z^{2})\\}+3(Y^{2}+Z^{2})^{2}]\\\
\quad+3C_{6}[2X^{3}\\{X^{2}-5(Y^{2}+Z^{2})\\}+\frac{15}{4}X(Y^{2}+Z^{2})^{2}],$
(18) $\omega^{2}Y^{\prime\prime}+2n\omega X^{\prime}+(C_{2}-n^{2})Y\\\
=-3C_{3}XY-\frac{3}{2}C_{4}Y(4X^{2}-Y^{2}-Z^{2})\\\
\quad-\frac{5}{2}C_{5}XY(4X^{2}-3Y^{2}-3Z^{2})\\\
\quad+\frac{15}{2}C_{6}[X^{2}Y\\{-2X^{2}+3(Y^{2}+Z^{2})\\}-\frac{1}{4}Y(Y^{2}+Z^{2})^{2}],$
(19) $\omega^{2}Z^{\prime\prime}+\lambda^{2}Z\\\
=-3C_{3}XZ-\frac{3}{2}C_{4}Z(4X^{2}-Y^{2}-Z^{2})\\\
\quad-\frac{5}{2}C_{5}XZ(4X^{2}-3Y^{2}-3Z^{2})\\\
\quad+\frac{15}{2}C_{6}[X^{2}Z\\{-2X^{2}+3(Y^{2}+Z^{2})\\}-\frac{1}{4}Z(Y^{2}+Z^{2})^{2}]\\\
+\Delta Z,\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\,\,\,\,\,\,$ (20)

where $\Delta=\lambda^{2}-C_{2}$ is the frequency correction term to obtain
halo orbit and $\Delta=\mathcal{O}(\epsilon^{2})$.
The solutions of (18)-(20) are assumed in the perturbations form as (Thurman
and Worfolk, 1996):

$\displaystyle X(\tau)=$ $\displaystyle\epsilon
X_{1}(\tau)+\epsilon^{2}X_{2}(\tau)+\epsilon^{3}X_{3}(\tau)+\epsilon^{4}X_{4}(\tau)$
$\displaystyle+\epsilon^{5}X_{5}(\tau)+...\,,$ (21) $\displaystyle Y(\tau)=$
$\displaystyle\epsilon
Y_{1}(\tau)+\epsilon^{2}Y_{2}(\tau)+\epsilon^{3}Y_{3}(\tau)+\epsilon^{4}Y_{4}(\tau)$
$\displaystyle+\epsilon^{5}Y_{5}(\tau)+...\,,$ (22) $\displaystyle Z(\tau)=$
$\displaystyle\epsilon
Z_{1}(\tau)+\epsilon^{2}Z_{2}(\tau)+\epsilon^{3}Z_{3}(\tau)+\epsilon^{4}Z_{4}(\tau)$
$\displaystyle+\epsilon^{5}Z_{5}(\tau)+...,$ (23)

and let

$\omega=1+\epsilon\omega_{1}+\epsilon^{2}\omega_{2}+\epsilon^{3}\omega_{3}+\epsilon^{4}\omega_{4}+...$
(24)

Substituting the solutions (21)-(24) into equations of motion (18)-(20) and
equating the coefficients of the same order of
$\epsilon,\epsilon^{2},\epsilon^{3},$ and $\epsilon^{4}$, we obtain the first,
second, third and fourth order equations, respectively (Thurman and Worfolk,
1996; Tiwary and Kushvah, 2015). For obtaining more accurate solutions of the
equations we have collected the coefficients of $\epsilon^{5}$ and obtained
the fifth order equations.

####  3.2.1 Fifth Order Equations

Collecting the coefficients of $\epsilon^{5}$ and incorporating all the
solutions and conditions used upto fourth order approximations(Tiwary and
Kushvah, 2015), we get the fifth order equations as :

$\displaystyle
X_{5}^{\prime\prime}-2nY_{5}^{\prime}-(n^{2}+2C_{2})X_{5}=\gamma_{51}$ (25)
$\displaystyle
Y_{5}^{\prime\prime}+2nX_{5}^{\prime}+(C_{2}-n^{2})Y_{5}=\gamma_{52}\,\,\,$
(26) $\displaystyle
Z_{5}^{\prime\prime}+\lambda^{2}Z_{5}=\left\\{\begin{array}[]{c}f_{3},\quad
p=0,2\\\ f_{4},\quad p=1,3\end{array}\right.\quad$ (29)

where

$\gamma_{51}=[v_{4}+2\lambda
A_{X}\omega_{4}(nk-\lambda)]\cos\tau_{1}+\gamma_{8}\cos
3\tau_{1}+\gamma_{9}\cos 5\tau_{1},$ $\gamma_{52}=[v_{5}+2\lambda
A_{X}\omega_{4}(\lambda k-n)]\sin\tau_{1}+\beta_{9}\sin
3\tau_{1}+\beta_{10}\sin 5\tau_{1},$ $f_{3}=[v_{6}\pm
2\omega_{4}\lambda^{2}A_{Z}]\sin\tau_{1}+\delta_{8}\sin
3\tau_{1}+\delta_{9}\sin 5\tau_{1},$ $f_{4}=[v_{6}\pm
2\omega_{4}\lambda^{2}A_{Z}]\cos\tau_{1}+\delta_{8}\cos
3\tau_{1}+\delta_{9}\cos 5\tau_{1}$

and the remaining coefficients are given in Appendix. In $f_{3}$, upper sign
corresponds to $p=0$ and lower sign corresponds to $p=2$. Similarly, in
$f_{4}$, upper sign corresponds to $p=1$ and lower corresponds to $p=3$.
The secular term can be removed from (29) if

$v_{6}\pm 2\omega_{4}\lambda^{2}A_{Z}=0,$ (30)

where the upper sign corresponds to $p=0,1$ and the lower sign corresponds to
$p=2,3$.

To remove the secular terms from (25) and (26), we use a single condition from
their particular solution (Thurman and Worfolk, 1996; Tiwary and Kushvah,
2015)

$[v_{4}+2\lambda A_{X}\omega_{4}(nk-\lambda)]-k[v_{5}+2\lambda
A_{X}\omega_{4}(\lambda k-n)]=0.$ (31)

From equation (31), we get

$\omega_{4}=\dfrac{v_{4}-kv_{5}}{2\lambda A_{x}(\lambda(k^{2}+1)-2nk)}.$ (32)

Using conditions (30) and (32) in equations (25)-(29), the equations of motion
take the following form:

$X_{5}^{\prime\prime}-2nY_{5}^{\prime}-(n^{2}+2C_{2})X_{5}\\\
=k\beta_{11}\cos\tau_{1}+\gamma_{8}\cos 3\tau_{1}+\gamma_{9}\cos
5\tau_{1},\qquad\qquad$ (33)
$Y_{5}^{\prime\prime}+2nX_{5}^{\prime}+(C_{2}-n^{2})Y_{5}\\\
=\beta_{11}\sin\tau_{1}+\beta_{9}\sin 3\tau_{1}+\beta_{10}\sin
5\tau_{1},\qquad\qquad$ (34)
$Z_{5}^{\prime\prime}+\lambda^{2}Z_{5}=\left\\{\begin{array}[]{c}\delta_{8}\sin
3\tau_{1}+\delta_{9}\sin 5\tau_{1},\quad p=0,2,\\\ \delta_{8}\cos
3\tau_{1}+\delta_{9}\cos 5\tau_{1},\quad p=1,3,\end{array}\right.$ (35)

where $\beta_{11}=v_{5}+2\lambda A_{X}\omega_{4}(\lambda k-n)$. The solution
of equations (33)-(35) is given by

$\displaystyle X_{5}(\tau)=$ $\displaystyle\rho_{51}\cos
3\tau_{1}+\rho_{52}\cos 5\tau_{1},$ (36) $\displaystyle Y_{5}(\tau)=$
$\displaystyle\sigma_{51}\sin\tau_{1}+\sigma_{52}\sin
3\tau_{1}+\sigma_{53}\sin 5\tau_{1},$ (37) $\displaystyle Z_{5}(\tau)=$
$\displaystyle\left\\{\begin{array}[]{c}k_{51}\sin 3\tau_{1}+k_{52}\sin
5\tau_{1},\quad p=0,2,\\\ k_{51}\cos 3\tau_{1}+k_{52}\cos 5\tau_{1},\quad
p=1,3,\end{array}\right.$ (40)

where the coefficients are given in the Appendix.

####  3.2.2 Final Approximation

Final approximation is obtained by removing $\epsilon$ from all the equations.
For that we take the mapping $A_{X}\rightarrow\frac{A_{X}}{\epsilon}$ and
$A_{Z}\rightarrow\frac{A_{Z}}{\epsilon}$. Combining the solutions component
wise in (21)-(23), we get (Tiwary and Kushvah, 2015)

$\displaystyle X(\tau)=$
$\displaystyle(\rho_{20}+\rho_{40})-A_{X}\cos\tau_{1}$
$\displaystyle+(\rho_{21}+\zeta\rho_{22}+\rho_{41})\cos 2\tau_{1}$
$\displaystyle+(\rho_{31}+\rho_{51})\cos 3\tau_{1}+\rho_{42}\cos
4\tau_{1}+\rho_{52}\cos 5\tau_{1},$ (41) $\displaystyle Y(\tau)=$
$\displaystyle(kA_{X}+\sigma_{32}+\sigma_{51})\sin\tau_{1}$
$\displaystyle+(\sigma_{21}+\sigma_{41}+\zeta\sigma_{22})\sin 2\tau_{1}$
$\displaystyle+(\sigma_{31}+\sigma_{52})\sin 3\tau_{1}+\sigma_{42}\sin
4\tau_{1}+\sigma_{53}\sin 5\tau_{1},$ (42) $\displaystyle Z(\tau)=$
$\displaystyle\left\\{\begin{array}[]{c}f_{5},\quad p=0,2\\\ f_{6},\quad
p=1,3\end{array}\right.$ (45)

where

$f_{5}=(-1)^{\frac{p}{2}}(A_{Z}\sin\tau_{1}+k_{21}\sin 2\tau_{1}+k_{31}\sin
3\tau_{1})+k_{41}\sin 2\tau_{1}+k_{42}\sin 4\tau_{1}+k_{51}\sin
3\tau_{1}+k_{52}\sin 5\tau_{1},$
$f_{6}=(-1)^{\frac{p-1}{2}}(A_{Z}\cos\tau_{1}+k_{21}\cos
2\tau_{1}+k_{22}+k_{32}\cos 3\tau_{1})+k_{40}+k_{41}\cos 2\tau_{1}+k_{42}\cos
4\tau_{1}+k_{51}\cos 3\tau_{1}+k_{52}\cos 5\tau_{1}.$

Using equations (41)-(45), we can get the first guess of halo orbits.

##  4 Discussion

The halo orbits in the photogravitational Sun-Earth system with oblateness
upto fourth order approximations using Lindstedt-Poincar$\acute{e}$ method are
obtained by Tiwary and Kushvah (2015). Here, the first guess of the halo orbit
in the same system is obtained upto fifth order approximation using Lindstedt-
Poincar$\acute{e}$ method. Equations (41)-(45) are used with the amplitudes
$A_{X}=206000$ km and $A_{Z}=110000$ km from the ISEE-$3$ mission.

The orbits are plotted for different values of phases. Fig.1 to Fig.4 show
halo orbits around $L_{1}$ for different values of $q$, mass reduction factor.
Orbits coloured in blue represents fourth order orbits and red corresponds to
fifth order orbits.

The effects of radiation pressure on the position of halo orbits are given in
Fig.5. Fig.5 shows the positions of halo orbits for $q=0.9995,0.9945,0.9895$
and $0.9845$ labeled as $1,2,3$ and $4$, respectively, with the actual
oblateness of Earth. As radiation pressure increases, the halo orbits move
towards the source of radiation. This agrees with conclusions of Eapen and
Sharma (2014).

Fig.$\ref{fig:8}-\ref{fig:11}$ represent halo orbits around $L_{2}$
corresponding to mass reduction factor $0.9995,0.9945,0.9895$ and $0.9845$,
respectively, with oblateness $A_{2}=2.4\times 10^{-12}$, the obalteness of
the Earth. Fig.10 shows the variation in position of halo orbits due to
radiation pressure. Here, the orbits labeled as $1,2,3$ and $4$ correspond to
$q=0.9995,0.9945,0.9895,0.9845$ and oblateness is $2.4\times 10^{-12}$. Halo
orbits move towards the source of radiation with the increase in radiation
pressure. Table1 , Table2, Table3 and Table4 show the variation in
coefficients, the position of Lagrangian points, $\Delta$ and time period
$\tau$ due to variation in radiation pressure and oblateness.
$\Delta=\lambda^{2}-C_{2}$ is the frequency correction term to obtain the halo
orbits. $\tau$ is the time taken by the infinitesimal body to complete one
rotation about the Lagrangian point. Table1 shows the effect of radiation
pressure on parameters of orbits around $L_{1}$. It can be observed that as
the radiation pressure increases, that is, $q$ decreases, $L_{1}$ move towards
the source of radiation, the Sun. Also, the time period of orbits increase
with the increase in radiation pressure. Table2 represents the effect of
oblateness on various parameters of orbits around $L_{1}$. With the increase
in radiation pressure, orbits move towards the Sun and their time period is
decreased. From Table3, it can be observed that due to increase in radiation
pressure, the orbits around $L_{2}$ move towards the Sun and their time period
is decreased. Effect of oblateness on position of orbits and time period can
be observed from Table 4. Halo orbits around $L_{2}$ move away from the source
of radiation and also time period of orbits increase due to increase in
oblateness. Fig.11 represents the effect of oblateness on position of $L_{1}$.
As oblateness increases, $L_{1}$ moves towards the source of radiation, the
Sun. In Fig.12, the reverse effect of oblateness is observed on the position
of $L_{2}$. That is, as oblateness increases, $L_{2}$ moves away from the Sun.
Fig.13 and Fig.14 show the variation in position of $L_{1}$ and $L_{2}$ due to
radiation pressure, respectively. With the increase in radiation pressure,
$L_{1}$ and $L_{2}$ both move towards the Sun. The effect of radiation
pressure and oblateness on time period is graphically shown in
Fig.$\ref{fig:17}-\ref{fig:20}$. Time periods of halo orbits decrease with the
increase in oblateness around $L_{1}$ while they increase with the increase in
oblateness around $L_{2}$. With the increase in radiation pressure, time
period of orbits around $L_{1}$ increases and decreases around $L_{2}$.

##  5 Conclusion

Photogravitational RTBP with oblateness, where the Sun is radiating and the
Earth an oblate spheroid, is studied for halo orbits. We have improved the
fourth order equations obtained by (Tiwary and Kushvah, 2015) using Lindstedt-
Poincar$\acute{e}$ method to fifth order and to obtain halo orbits around
$L_{1}$ and $L_{2}$. The deviations of the orbits around $L_{1}$ and $L_{2}$
obtained from fourth order and fifth order equations are shown graphically.
The variations in position and time of halo orbits around $L_{1}$ and $L_{2}$
due to radiation pressure and oblateness are studied. It is found that the
halo orbits around $L_{1}$ shift towards the source of radiation (Sun) as the
radiation pressure and oblateness increase. However, the time period of halo
orbits around $L_{1}$ increases with the increase in radiation pressure but
decreases with the increase in the oblateness. Halo orbits around $L_{2}$
approaches the source of radiation with increase in the radiation pressure but
recedes from the source of radiation due to increase in the oblateness. The
period of halo orbits around $L_{2}$ decreases with increase in the radiation
pressure but increases with increase in oblateness.

Acknowledgements
One of the authors(DS) would like to thank Council of Scientific and
Industrial Research (CSIR) for financial support through JRF(File No.
09/114(0218)/2019-EMR-I).

## Compliance with Ethical Standards

Conflict of Interest : Author Dhwani Sheth has received Junior Research
Fellowship(JRF) from CSIR (File No. 09/114(0218)/2019-EMR-I).

## Appendix

$\displaystyle
v_{4}=\left\\{\begin{array}[]{c}v_{41},\quad\text{when}\,\,p=0,2,\\\
v_{42},\quad\text{when}\,\,p=1,3.\end{array}\right.$
$v_{41}=\lambda\omega_{2}(2n\sigma_{32}-\lambda\omega_{2}A_{X})+\frac{3}{2}C_{3}(-2A_{X}(2\rho_{40}+\rho_{41})+2\rho_{31}(\rho_{21}+\rho_{22})-kA_{X}\sigma_{41}-(\sigma_{21}+\sigma_{22})(\sigma_{31}+\sigma_{32})-(-1)^{\frac{p}{2}}A_{Z}k_{41}-k_{21}k_{31})+\frac{3}{2}C_{4}(2A_{X}^{2}\rho_{31}-4A_{X}((\rho_{20}+\rho_{21}+\rho_{22})^{2}+\rho_{20}^{2})+2kA_{X}^{2}(\sigma_{31}+\sigma_{32})+2A_{X}(\sigma_{21}+\sigma_{22})^{2}-4kA_{X}\rho_{20}(\sigma_{21}+\sigma_{22})+k^{2}A_{X}^{2}\rho_{31}+2A_{X}A_{Z}k_{31}+2A_{X}k_{21}^{2}-4A_{Z}k_{21}\rho_{20}+A_{Z}^{2}\rho_{31})+\frac{5}{2}C_{5}(-2A_{X}^{3}(3\rho_{20}+2(\rho_{21}+\rho_{22}))-3kA_{X}^{3}(\sigma_{21}+\sigma_{22})+3k^{2}A_{X}^{3}\rho_{20}-3A_{X}^{2}A_{Z}k_{21}+3A_{X}A_{Z}^{2}\rho_{20}+\frac{3}{4}k^{3}A_{X}^{3}(\sigma_{21}+\sigma_{22})+\frac{3}{4}k^{2}A_{X}^{2}A_{Z}k_{21}+\frac{3}{4}kA_{X}A_{Z}^{2}(\sigma_{21}+\sigma_{22})+\frac{3}{4}A_{Z}^{3}k_{21})+\frac{15}{32}C_{6}(-8A_{X}^{5}+8k^{2}A_{X}^{5}+8A_{X}^{3}A_{Z}^{2}-3k^{4}A_{X}^{5}-6k^{2}A_{X}^{3}A_{Z}^{2}-3A_{X}A_{Z}^{4}),$
$v_{42}=\lambda\omega_{2}(2n\sigma_{32}-\lambda\omega_{2}A_{X})+\frac{3}{2}C_{3}(-2A_{X}(2\rho_{40}+\rho_{41})+2\rho_{31}(\rho_{21}-\rho_{22})-kA_{X}\sigma_{41}-(\sigma_{21}-\sigma_{22})(\sigma_{31}+\sigma_{32})-(-1)^{\frac{p-1}{2}}A_{Z}(2k_{40}+k_{41})-k_{21}k_{32})+\frac{3}{2}C_{4}(2A_{X}^{2}\rho_{31}-4A_{X}((\rho_{20}+\rho_{21}-\rho_{22})^{2}+\rho_{20}^{2})+2kA_{X}^{2}(\sigma_{31}+\sigma_{32})+2A_{X}(\sigma_{21}-\sigma_{22})^{2}-4kA_{X}\rho_{20}(\sigma_{21}-\sigma_{22})+k^{2}A_{X}^{2}\rho_{31}+2A_{X}A_{Z}k_{32}+2A_{X}(k_{21}^{2}+2k_{21}k_{22}+2k_{22}^{2})-4A_{Z}(k_{21}+k_{22})(\rho_{20}+\rho_{21}-\rho_{22})-A_{Z}^{2}\rho_{31})+\frac{5}{2}C_{5}(-2A_{X}^{3}(3\rho_{20}+2(\rho_{21}-\rho_{22}))-3kA_{X}^{3}(\sigma_{21}-\sigma_{22})+3k^{2}A_{X}^{3}\rho_{20}-3A_{X}^{2}A_{Z}(2k_{21}+3k_{22})+3A_{X}A_{Z}^{2}(3\rho_{20}+2(\rho_{21}-\rho_{22}))+\frac{3}{4}k^{3}A_{X}^{3}(\sigma_{21}-\sigma_{22})+\frac{3}{4}k^{2}A_{X}^{2}A_{Z}k_{22}+\frac{3}{4}kA_{X}A_{Z}^{2}(\sigma_{21}-\sigma_{22})+\frac{3}{4}A_{Z}^{3}(2k_{21}+3k_{22}))+\frac{15}{32}C_{6}(-8A_{X}^{5}+8k^{2}A_{X}^{5}+40A_{X}^{3}A_{Z}^{2}-3k^{4}A_{X}^{5}-6k^{2}A_{X}^{3}A_{Z}^{2}-15A_{X}A_{Z}^{4}).$
$\displaystyle
v_{5}=\left\\{\begin{array}[]{c}v_{51},\quad\text{when}\,\,p=0,2,\\\
v_{52},\quad\text{when}\,\,p=1,3.\end{array}\right.$
$v_{51}=\omega_{2}\lambda^{2}(2\sigma_{32}+\omega_{2}kA_{X})-\frac{3}{2}C_{3}(-A_{X}\sigma_{41}+2\rho_{20}\sigma_{32}+(\rho_{21}+\rho_{22})(\sigma_{31}-\sigma_{32})-\rho_{31}(\sigma_{21}+\sigma_{22})+kA_{X}(2\rho_{40}-\rho_{41}))-\frac{3}{8}C_{4}(4A_{X}^{2}(\sigma_{31}+\sigma_{32})-16A_{X}\rho_{20}(\sigma_{21}+\sigma_{22})+8kA_{X}^{2}\rho_{31}+8kA_{X}(2\rho_{20}^{2}-2\rho_{20}(\rho_{21}+\rho_{22})+(\rho_{21}+\rho_{22})^{2})+3k^{2}A_{X}^{2}(\sigma_{31}-3\sigma_{32})-6kA_{X}(\sigma_{21}+\sigma_{22})^{2}+2kA_{X}A_{Z}k_{31}-2kA_{X}k_{21}^{2}-4A_{Z}k_{21}(\sigma_{21}+\sigma_{22})+A_{Z}^{2}(\sigma_{31}-3\sigma_{32}))+\frac{5}{8}C_{5}(4A_{X}^{3}(\sigma_{21}+\sigma_{22})+3k^{3}A_{X}^{3}(3\rho_{20}-2(\rho_{21}+\rho_{22}))-9k^{2}A_{X}^{3}(\sigma_{21}+\sigma_{22})-12kA_{X}^{3}\rho_{20}-6kA_{X}^{2}A_{Z}k_{21}-3A_{X}A_{Z}^{2}(\sigma_{21}+\sigma_{22})+3kA_{X}A_{Z}^{2}(3\rho_{20}-2(\rho_{21}+\rho_{22})))+\frac{15}{64}C_{6}(-8kA_{X}^{5}+12k^{3}A_{X}^{5}+12kA_{X}^{3}A_{Z}^{2}-5k^{5}A_{X}^{5}-10k^{3}A_{X}^{3}A_{Z}^{2}-5kA_{X}A_{Z}^{4}),$
$v_{52}=\omega_{2}\lambda^{2}(2\sigma_{32}+\omega_{2}kA_{X})-\frac{3}{2}C_{3}(-A_{X}\sigma_{41}+2\rho_{20}\sigma_{32}+(\rho_{21}-\rho_{22})(\sigma_{31}-\sigma_{32})-\rho_{31}(\sigma_{21}-\sigma_{22})+kA_{X}(2\rho_{40}-\rho_{41}))-\frac{3}{8}C_{4}(4A_{X}^{2}(\sigma_{31}+\sigma_{32})-16A_{X}\rho_{20}(\sigma_{21}-\sigma_{22})+8kA_{X}^{2}\rho_{31}+8kA_{X}(2\rho_{20}^{2}-2\rho_{20}(\rho_{21}-\rho_{22})+(\rho_{21}-\rho_{22})^{2})+3k^{2}A_{X}^{2}(\sigma_{31}-3\sigma_{32})-6kA_{X}(\sigma_{21}-\sigma_{22})^{2}+2kA_{X}A_{Z}k_{32}-2kA_{X}(k_{21}^{2}-2k_{21}k_{22}+2k_{22}^{2})-4A_{Z}k_{22}(\sigma_{21}-\sigma_{22})-A_{Z}^{2}(\sigma_{31}+\sigma_{32}))+\frac{5}{8}C_{5}(4A_{X}^{3}(\sigma_{21}-\sigma_{22})+3k^{3}A_{X}^{3}(3\rho_{20}-2(\rho_{21}-\rho_{22}))-9k^{2}A_{X}^{3}(\sigma_{21}-\sigma_{22})-12kA_{X}^{3}\rho_{20}-6kA_{X}^{2}A_{Z}k_{22}-3A_{X}A_{Z}^{2}(\sigma_{21}-\sigma_{22})+3kA_{X}A_{Z}^{2}\rho_{20})+\frac{15}{64}C_{6}(-8kA_{X}^{5}+12k^{3}A_{X}^{5}+12kA_{X}^{3}A_{Z}^{2}-5k^{5}A_{X}^{5}-2k^{3}A_{X}^{3}A_{Z}^{2}-kA_{X}A_{Z}^{4}).$
$\displaystyle
v_{6}=\left\\{\begin{array}[]{c}v_{60},\quad\text{when}\,\,p=0,\\\
v_{61},\quad\text{when}\,\,p=1,\\\ v_{62},\quad\text{when}\,\,p=2,\\\
v_{63},\quad\text{when}\,\,p=3.\end{array}\right.$
$v_{60}=\omega_{2}^{2}\lambda^{2}A_{Z}-\frac{3}{2}C_{3}(-A_{X}k_{41}+k_{31}(\rho_{21}+\rho_{22})-\rho_{31}k_{21}+A_{Z}(2\rho_{40}-\rho_{41}))-\frac{3}{2}C_{4}(A_{X}^{2}k_{31}-4A_{X}\rho_{20}k_{21}+2A_{X}A_{Z}\rho_{31}+2A_{Z}\\{2\rho_{20}(\rho_{20}-\rho_{21}-\rho_{22})+(\rho_{21}+\rho_{22})^{2}\\}+\frac{1}{4}k^{2}A_{X}^{2}k_{31}-kA_{X}k_{21}(\sigma_{21}+\sigma_{22})+\frac{1}{2}kA_{X}A_{Z}(\sigma_{31}-3\sigma_{32})+\frac{3}{4}A_{Z}^{2}k_{31}-\frac{1}{2}A_{Z}(\sigma_{21}+\sigma_{22})^{2}-\frac{3}{2}A_{Z}k_{21}^{2})-\frac{5}{8}C_{5}(-4A_{X}^{3}k_{21}+12A_{X}^{2}A_{Z}\rho_{20}+9A_{X}A_{Z}^{2}k_{21}-3A_{Z}^{3}(3\rho_{20}-2(\rho_{21}+\rho_{22}))+3k^{2}A_{X}^{3}k_{21}-6kA_{X}^{2}A_{Z}(\sigma_{21}+\sigma_{22})-3k^{2}A_{X}^{2}A_{Z}\rho_{20})+\frac{15}{64}C_{6}(-8A_{X}^{4}A_{Z}+12k^{2}A_{X}^{4}A_{Z}+12A_{X}^{2}A_{Z}^{3}-5k^{4}A_{X}^{4}A_{Z}-10k^{2}A_{X}^{2}A_{Z}^{3}-5A_{Z}^{5}),$
$v_{61}=\omega_{2}^{2}\lambda^{2}A_{Z}-\frac{3}{2}C_{3}(-A_{X}(2k_{40}+k_{41})+k_{32}(\rho_{21}-\rho_{22})+\rho_{31}k_{21}+A_{Z}(2\rho_{40}+\rho_{41}))-\frac{3}{2}C_{4}(A_{X}^{2}k_{32}-4A_{X}(\rho_{20}(k_{21}+2k_{22})+(\rho_{21}-\rho_{22})(k_{21}+k_{22}))-2A_{X}A_{Z}\rho_{31}+2A_{Z}\\{2\rho_{20}(\rho_{20}+\rho_{21}-\rho_{22})+(\rho_{21}-\rho_{22})^{2}\\}+\frac{1}{4}k^{2}A_{X}^{2}k_{32}-kA_{X}k_{22}(\sigma_{21}-\sigma_{22})-\frac{1}{2}kA_{X}A_{Z}(\sigma_{31}+\sigma_{32})-\frac{3}{4}A_{Z}^{2}k_{32}-\frac{1}{2}A_{Z}(\sigma_{21}-\sigma_{22})^{2}-\frac{3}{2}A_{Z}(k_{21}^{2}+2k_{21}k_{22}+2k_{22}^{2}))-\frac{5}{8}C_{5}(-4A_{X}^{3}(2k_{21}+3k_{22})+12A_{X}^{2}A_{Z}(3\rho_{20}+2(\rho_{21}-\rho_{22}))+9A_{X}A_{Z}^{2}(2k_{21}+3k_{22})-3A_{Z}^{3}(3\rho_{20}+2(\rho_{21}-\rho_{22}))+3k^{2}A_{X}^{3}k_{22}+6kA_{X}^{2}A_{Z}(\sigma_{21}-\sigma_{22})-3k^{2}A_{X}^{2}A_{Z}\rho_{20})+\frac{15}{64}C_{6}(-40A_{X}^{4}A_{Z}+12k^{2}A_{X}^{4}A_{Z}+60A_{X}^{2}A_{Z}^{3}-k^{4}A_{X}^{4}A_{Z}-2k^{2}A_{X}^{2}A_{Z}^{3}-5A_{Z}^{5}),$
$v_{62}=-\omega_{2}^{2}\lambda^{2}A_{Z}-\frac{3}{2}C_{3}(-A_{X}k_{41}-k_{31}(\rho_{21}+\rho_{22})+\rho_{31}k_{21}-A_{Z}(2\rho_{40}-\rho_{41}))-\frac{3}{2}C_{4}(-A_{X}^{2}k_{31}+4A_{X}\rho_{20}k_{21}-2A_{X}A_{Z}\rho_{31}-2A_{Z}\\{2\rho_{20}(\rho_{20}-\rho_{21}-\rho_{22})+(\rho_{21}+\rho_{22})^{2}\\}-\frac{1}{4}k^{2}A_{X}^{2}k_{31}+kA_{X}k_{21}(\sigma_{21}+\sigma_{22})-\frac{1}{2}kA_{X}A_{Z}(\sigma_{31}-3\sigma_{32})-\frac{3}{4}A_{Z}^{2}k_{31}+\frac{1}{2}A_{Z}(\sigma_{21}+\sigma_{22})^{2}+\frac{3}{2}A_{Z}k_{21}^{2})-\frac{5}{8}C_{5}(4A_{X}^{3}k_{21}-12A_{X}^{2}A_{Z}\rho_{20}-9A_{X}A_{Z}^{2}k_{21}+3A_{Z}^{3}(3\rho_{20}-2(\rho_{21}+\rho_{22}))-3k^{2}A_{X}^{3}k_{21}+6kA_{X}^{2}A_{Z}(\sigma_{21}+\sigma_{22})+3k^{2}A_{X}^{2}A_{Z}\rho_{20})+\frac{15}{64}C_{6}(8A_{X}^{4}A_{Z}-12k^{2}A_{X}^{4}A_{Z}-12A_{X}^{2}A_{Z}^{3}+5k^{4}A_{X}^{4}A_{Z}+10k^{2}A_{X}^{2}A_{Z}^{3}+5A_{Z}^{5}),$
$v_{63}=-\omega_{2}^{2}\lambda^{2}A_{Z}-\frac{3}{2}C_{3}(-A_{X}(2k_{40}+k_{41})-k_{32}(\rho_{21}-\rho_{22})-\rho_{31}k_{21}-A_{Z}(2\rho_{40}+\rho_{41}))-\frac{3}{2}C_{4}(-A_{X}^{2}k_{32}+4A_{X}(\rho_{20}(k_{21}+2k_{22})+(\rho_{21}-\rho_{22})(k_{21}+k_{22}))+2A_{X}A_{Z}\rho_{31}-2A_{Z}\\{2\rho_{20}(\rho_{20}+\rho_{21}-\rho_{22})+(\rho_{21}-\rho_{22})^{2}\\}-\frac{1}{4}k^{2}A_{X}^{2}k_{32}+kA_{X}k_{22}(\sigma_{21}-\sigma_{22})+\frac{1}{2}kA_{X}A_{Z}(\sigma_{31}+\sigma_{32})+\frac{3}{4}A_{Z}^{2}k_{32}+\frac{1}{2}A_{Z}(\sigma_{21}-\sigma_{22})^{2}+\frac{3}{2}A_{Z}(k_{21}^{2}+2k_{21}k_{22}+2k_{22}^{2}))-\frac{5}{8}C_{5}(4A_{X}^{3}(2k_{21}+3k_{22})-12A_{X}^{2}A_{Z}(3\rho_{20}+2(\rho_{21}-\rho_{22}))-9A_{X}A_{Z}^{2}(2k_{21}+3k_{22})+3A_{Z}^{3}(3\rho_{20}+2(\rho_{21}-\rho_{22}))-3k^{2}A_{X}^{3}k_{22}-6kA_{X}^{2}A_{Z}(\sigma_{21}-\sigma_{22})+3k^{2}A_{X}^{2}A_{Z}\rho_{20})+\frac{15}{64}C_{6}(40A_{X}^{4}A_{Z}-12k^{2}A_{X}^{4}A_{Z}-60A_{X}^{2}A_{Z}^{3}+k^{4}A_{X}^{4}A_{Z}+2k^{2}A_{X}^{2}A_{Z}^{3}+5A_{Z}^{5}).$
$\displaystyle\gamma_{8}=\left\\{\begin{array}[]{c}\gamma_{81},\quad\text{when}\,\,p=0,2,\\\
\gamma_{82},\quad\text{when}\,\,p=1,3.\end{array}\right.$
$\gamma_{81}=6\lambda\omega_{2}(3\lambda\rho_{31}+n\sigma_{31})+\frac{3}{2}C_{3}(-2A_{X}(\rho_{41}+\rho_{42})+4\rho_{20}\rho_{31}+kA_{X}(\sigma_{41}-\sigma_{42})+\sigma_{32}(\sigma_{21}+\sigma_{22})+(-1)^{\frac{p}{2}}A_{Z}(k_{41}-k_{42}))+\frac{3}{2}C_{4}(4A_{X}^{2}\rho_{31}-2A_{X}(\rho_{21}+\rho_{22})(4\rho_{20}+\rho_{21}+\rho_{22})-2kA_{X}^{2}\sigma_{32}-A_{X}(\sigma_{21}+\sigma_{22})^{2}+2kA_{X}(\sigma_{21}+\sigma_{22})(2\rho_{20}-(\rho_{21}+\rho_{22}))-2k^{2}A_{X}^{2}\rho_{31}-A_{X}k_{21}^{2}+2A_{Z}k_{21}(2\rho_{20}-(\rho_{21}+\rho_{22}))-2A_{Z}^{2}\rho_{31})+\frac{5}{16}C_{5}(-8A_{X}^{3}(2\rho_{20}+3(\rho_{21}+\rho_{22}))+12kA_{X}^{3}(\sigma_{21}+\sigma_{22})-12k^{2}A_{X}^{3}(2\rho_{20}-(\rho_{21}+\rho_{22}))+12A_{X}^{2}A_{Z}k_{21}-12A_{X}A_{Z}^{2}(2\rho_{20}-(\rho_{21}+\rho_{22}))-9k^{3}A_{X}^{3}(\sigma_{21}+\sigma_{22})-9k^{2}A_{X}^{2}A_{Z}k_{21}-9kA_{X}A_{Z}^{2}(\sigma_{21}+\sigma_{22})-9A_{Z}^{3}k_{21})+\frac{15}{64}C_{6}(-8A_{X}^{5}-8k^{2}A_{X}^{5}-8A_{X}^{3}A_{Z}^{2}+9k^{4}A_{X}^{5}+18k^{2}A_{X}^{3}A_{Z}^{2}+9A_{X}A_{Z}^{4}),$
$\gamma_{82}=6\lambda\omega_{2}(3\lambda\rho_{31}+n\sigma_{31})+\frac{3}{2}C_{3}(-2A_{X}(\rho_{41}+\rho_{42})+4\rho_{20}\rho_{31}+kA_{X}(\sigma_{41}-\sigma_{42})+\sigma_{32}(\sigma_{21}-\sigma_{22})-(-1)^{\frac{p-1}{2}}A_{Z}(k_{41}+k_{42})-2k_{22}k_{32})+\frac{3}{2}C_{4}(4A_{X}^{2}\rho_{31}-2A_{X}(\rho_{21}-\rho_{22})(4\rho_{20}+\rho_{21}-\rho_{22})-2kA_{X}^{2}\sigma_{32}-A_{X}(\sigma_{21}-\sigma_{22})^{2}+2kA_{X}(\sigma_{21}-\sigma_{22})(2\rho_{20}-(\rho_{21}-\rho_{22}))-2k^{2}A_{X}^{2}\rho_{31}+4A_{X}A_{Z}k_{32}+A_{X}(k_{21}^{2}+4k_{21}k_{22})-2A_{Z}(k_{21}(2\rho_{20}+\rho_{21}-\rho_{22})+2k_{22}(\rho_{21}-\rho_{22}))-2A_{Z}^{2}\rho_{31})+\frac{5}{16}C_{5}(-8A_{X}^{3}(2\rho_{20}+3(\rho_{21}-\rho_{22}))+12kA_{X}^{3}(\sigma_{21}-\sigma_{22})-12k^{2}A_{X}^{3}(2\rho_{20}-(\rho_{21}-\rho_{22}))-12A_{X}^{2}A_{Z}(3k_{21}+2k_{22})+12A_{X}A_{Z}^{2}(2\rho_{20}+3(\rho_{21}-\rho_{22}))-9k^{3}A_{X}^{3}(\sigma_{21}-\sigma_{22})+3k^{2}A_{X}^{2}A_{Z}(k_{21}-2k_{22})-3kA_{X}A_{Z}^{2}(\sigma_{21}-\sigma_{22})+3A_{Z}^{3}(3k_{21}+2k_{22}))+\frac{15}{64}C_{6}(-8A_{X}^{5}-8k^{2}A_{X}^{5}+40A_{X}^{3}A_{Z}^{2}+9k^{4}A_{X}^{5}+6k^{2}A_{X}^{3}A_{Z}^{2}-15A_{X}A_{Z}^{4}).$
$\displaystyle\gamma_{9}=\left\\{\begin{array}[]{c}\gamma_{91},\quad\text{when}\,\,p=0,2,\\\
\gamma_{92},\quad\text{when}\,\,p=1,3.\end{array}\right.$
$\gamma_{91}=\frac{3}{2}C_{3}(-2A_{X}\rho_{42}+2\rho_{31}(\rho_{21}+\rho_{22})+kA_{X}\sigma_{42}+\sigma_{31}(\sigma_{21}+\sigma_{22})+(-1)^{\frac{p}{2}}A_{Z}k_{42}+k_{21}k_{31})+\frac{3}{2}C_{4}(2A_{X}^{2}\rho_{31}-2A_{X}(\rho_{21}+\rho_{22})^{2}-2kA_{X}^{2}\sigma_{31}-A_{X}(\sigma_{21}+\sigma_{22})^{2}+2kA_{X}(\sigma_{21}+\sigma_{22})(\rho_{21}+\rho_{22})+k^{2}A_{X}^{2}\rho_{31}-2A_{X}A_{Z}k_{31}-A_{X}k_{21}^{2}+2A_{Z}k_{21}(\rho_{21}+\rho_{22})+A_{Z}^{2}\rho_{31})+\frac{5}{16}C_{5}(-8A_{X}^{3}(\rho_{21}+\rho_{22})+12kA_{X}^{3}(\sigma_{21}+\sigma_{22})-12k^{2}A_{X}^{3}(\rho_{21}+\rho_{22})+12A_{X}^{2}A_{Z}k_{21}-12A_{X}A_{Z}^{2}(\rho_{21}+\rho_{22})+3k^{3}A_{X}^{3}(\sigma_{21}+\sigma_{22})+3k^{2}A_{X}^{2}A_{Z}k_{21}+3kA_{X}A_{Z}^{2}(\sigma_{21}+\sigma_{22})+3A_{Z}^{2}k_{21})+\frac{3}{64}C_{6}(-8A_{X}^{5}-40k^{2}A_{X}^{5}-40A_{X}^{3}A_{Z}^{2}-15k^{4}A_{X}^{5}-30k^{2}A_{X}^{3}A_{Z}^{2}-15A_{X}A_{Z}^{4}),$
$\gamma_{92}=\frac{3}{2}C_{3}(-2A_{X}\rho_{42}+2\rho_{31}(\rho_{21}-\rho_{22})+kA_{X}\sigma_{42}+\sigma_{31}(\sigma_{21}-\sigma_{22})-(-1)^{\frac{p-1}{2}}A_{Z}k_{42}-k_{21}k_{31})+\frac{3}{2}C_{4}(2A_{X}^{2}\rho_{31}-2A_{X}(\rho_{21}-\rho_{22})^{2}-2kA_{X}^{2}\sigma_{31}-A_{X}(\sigma_{21}-\sigma_{22})^{2}+2kA_{X}(\sigma_{21}-\sigma_{22})(\rho_{21}-\rho_{22})+k^{2}A_{X}^{2}\rho_{31}+2A_{X}A_{Z}k_{32}+A_{X}k_{21}^{2}-2A_{Z}k_{21}(\rho_{21}-\rho_{22})-A_{Z}^{2}\rho_{31})+\frac{5}{16}C_{5}(-8A_{X}^{3}(\rho_{21}-\rho_{22})+12kA_{X}^{3}(\sigma_{21}-\sigma_{22})-12k^{2}A_{X}^{3}(\rho_{21}-\rho_{22})-12A_{X}^{2}A_{Z}k_{21}+12A_{X}A_{Z}^{2}(\rho_{21}-\rho_{22})+3k^{3}A_{X}^{3}(\sigma_{21}-\sigma_{22})-3k^{2}A_{X}^{2}A_{Z}k_{21}-3kA_{X}A_{Z}^{2}(\sigma_{21}-\sigma_{22})+3A_{Z}^{2}k_{21})+\frac{3}{64}C_{6}(-8A_{X}^{5}-40k^{2}A_{X}^{5}+40A_{X}^{3}A_{Z}^{2}-15k^{4}A_{X}^{5}+30k^{2}A_{X}^{3}A_{Z}^{2}-15A_{X}A_{Z}^{4}).$
$\displaystyle\beta_{9}=\left\\{\begin{array}[]{c}\beta_{91},\quad\text{when}\,\,p=0,2,\\\
\beta_{92},\quad\text{when}\,\,p=1,3.\end{array}\right.$
$\beta_{91}=6\lambda\omega_{2}(3\lambda\sigma_{31}+n\rho_{31})-\frac{3}{2}C_{3}(-A_{X}(\sigma_{41}+\sigma_{42})+2\rho_{20}\sigma_{31}+\sigma_{32}(\rho_{21}+\rho_{22})+kA_{X}(\rho_{41}-\rho_{42}))-\frac{3}{8}C_{4}(4A_{X}^{2}(2\sigma_{31}+\sigma_{32})-8A_{X}(\sigma_{21}+\sigma_{22})(2\rho_{20}+\rho_{21}+\rho_{22})+4kA_{X}(\rho_{21}+\rho_{22})(4\rho_{20}-(\rho_{21}+\rho_{22}))-3k^{2}A_{X}^{2}(2\sigma_{31}-\sigma_{32})-3kA_{X}(\sigma_{21}+\sigma_{22})^{2}-4kA_{X}A_{Z}k_{31}-kA_{X}k_{21}^{2}-2A_{Z}k_{21}(\sigma_{21}+\sigma_{22})-A_{Z}^{2}(2\sigma_{31}-\sigma_{32}))+\frac{5}{16}C_{5}(12A_{X}^{3}(\sigma_{21}+\sigma_{22})-12kA_{X}^{3}(2\rho_{20}+\rho_{21}+\rho_{22})+3k^{3}A_{X}^{3}(-2\rho_{20}+3(\rho_{21}+\rho_{22}))-9k^{2}A_{X}^{3}(\sigma_{21}+\sigma_{22})-6kA_{X}^{2}A_{Z}k_{21}-3A_{X}A_{Z}^{2}(\sigma_{21}+\sigma_{22})+3kA_{X}A_{Z}^{2}(-2\rho_{20}+\rho_{21}+\rho_{22}))+\frac{15}{128}C_{6}(-24kA_{X}^{5}+12k^{3}A_{X}^{5}+12kA_{X}^{3}A_{Z}^{2}+5k^{5}A_{X}^{5}+10k^{3}A_{X}^{3}A_{Z}^{2}+5kA_{X}A_{Z}^{4}),$
$\beta_{92}=6\lambda\omega_{2}(3\lambda\sigma_{31}+n\rho_{31})-\frac{3}{2}C_{3}(-A_{X}(\sigma_{41}+\sigma_{42})+2\rho_{20}\sigma_{31}+\sigma_{32}(\rho_{21}-\rho_{22})+kA_{X}(\rho_{41}-\rho_{42}))-\frac{3}{8}C_{4}(4A_{X}^{2}(2\sigma_{31}+\sigma_{32})-8A_{X}(\sigma_{21}-\sigma_{22})(2\rho_{20}+\rho_{21}-\rho_{22})+4kA_{X}(\rho_{21}-\rho_{22})(4\rho_{20}-(\rho_{21}-\rho_{22}))-3k^{2}A_{X}^{2}(2\sigma_{31}+\sigma_{32})-3kA_{X}(\sigma_{21}-\sigma_{22})^{2}-kA_{X}(4k_{21}k_{22}-k_{21}^{2})-2A_{Z}(k_{21}+2k_{22})(\sigma_{21}-\sigma_{22})-A_{Z}^{2}(2\sigma_{31}+\sigma_{32}))+\frac{5}{16}C_{5}(12A_{X}^{3}(\sigma_{21}-\sigma_{22})-12kA_{X}^{3}(2\rho_{20}+\rho_{21}-\rho_{22})+3k^{3}A_{X}^{3}(-2\rho_{20}+3(\rho_{21}-\rho_{22}))-9k^{2}A_{X}^{3}(\sigma_{21}-\sigma_{22})-6kA_{X}^{2}A_{Z}(k_{21}+2k_{22})-9A_{X}A_{Z}^{2}(\sigma_{21}-\sigma_{22})+3kA_{X}A_{Z}^{2}(2\rho_{20}+\rho_{21}-\rho_{22}))+\frac{15}{128}C_{6}(-24kA_{X}^{5}+12k^{3}A_{X}^{5}\\\
+36kA_{X}^{3}A_{Z}^{2}+5k^{5}A_{X}^{5}-2k^{3}A_{X}^{3}A_{Z}^{2}-3kA_{X}A_{Z}^{4}).$
$\displaystyle\beta_{10}=\left\\{\begin{array}[]{c}\beta_{101},\quad\text{when}\,\,p=0,2,\\\
\beta_{102},\quad\text{when}\,\,p=1,3.\end{array}\right.$
$\beta_{101}=-\frac{3}{2}C_{3}(-A_{X}\sigma_{42}+\sigma_{31}(\rho_{21}+\rho_{22})+\rho_{31}(\sigma_{21}+\sigma_{22})+kA_{X}\rho_{42})-\frac{3}{8}C_{4}(4A_{X}^{2}\sigma_{31}-8A_{X}(\sigma_{21}+\sigma_{22})(\rho_{21}+\rho_{22})-8kA_{X}^{2}\rho_{31}+4kA_{X}(\rho_{21}+\rho_{22})^{2}+3k^{2}A_{X}^{2}\sigma_{31}+3kA_{X}(\sigma_{21}+\sigma_{22})^{2}+2kA_{X}A_{Z}k_{31}+kA_{X}k_{21}^{2}+2A_{Z}k_{21}(\sigma_{21}+\sigma_{22})+A_{Z}^{2}\sigma_{31})+\frac{5}{16}C_{5}(4A_{X}^{3}(\sigma_{21}+\sigma_{22})-12kA_{X}^{3}(\rho_{21}+\rho_{22})-3k^{3}A_{X}^{3}(\rho_{21}+\rho_{22})+9k^{2}A_{X}^{3}(\sigma_{21}+\sigma_{22})+6kA_{X}^{2}A_{Z}k_{21}+3A_{X}A_{Z}^{2}(\sigma_{21}+\sigma_{22})-3kA_{X}A_{Z}^{2}(\rho_{21}+\rho_{22}))+\frac{15}{128}C_{6}(-8kA_{X}^{5}-12k^{3}A_{X}^{5}-12kA_{X}^{3}A_{Z}^{2}-k^{5}A_{X}^{5}-2k^{3}A_{X}^{3}A_{Z}^{2}-kA_{X}A_{Z}^{4}),$
$\beta_{102}=-\frac{3}{2}C_{3}(-A_{X}\sigma_{42}+\sigma_{31}(\rho_{21}-\rho_{22})+\rho_{31}(\sigma_{21}-\sigma_{22})+kA_{X}\rho_{42})-\frac{3}{8}C_{4}(4A_{X}^{2}\sigma_{31}-8A_{X}(\sigma_{21}-\sigma_{22})(\rho_{21}-\rho_{22})-8kA_{X}^{2}\rho_{31}+4kA_{X}(\rho_{21}-\rho_{22})^{2}+3k^{2}A_{X}^{2}\sigma_{31}+3kA_{X}(\sigma_{21}-\sigma_{22})^{2}-2kA_{X}A_{Z}k_{32}-kA_{X}k_{21}^{2}-2A_{Z}k_{21}(\sigma_{21}-\sigma_{22})-A_{Z}^{2}\sigma_{31})+\frac{5}{16}C_{5}(4A_{X}^{3}(\sigma_{21}-\sigma_{22})-12kA_{X}^{3}(\rho_{21}-\rho_{22})-3k^{3}A_{X}^{3}(\rho_{21}-\rho_{22})+9k^{2}A_{X}^{3}(\sigma_{21}-\sigma_{22})-6kA_{X}^{2}A_{Z}k_{21}-3A_{X}A_{Z}^{2}(\sigma_{21}-\sigma_{22})+3kA_{X}A_{Z}^{2}(\rho_{21}-\rho_{22}))+\frac{15}{128}C_{6}(-8kA_{X}^{5}-12k^{3}A_{X}^{5}+12kA_{X}^{3}A_{Z}^{2}-k^{5}A_{X}^{5}-2k^{3}A_{X}^{3}A_{Z}^{2}-kA_{X}A_{Z}^{4}).$
$\displaystyle\delta_{8}=\left\\{\begin{array}[]{c}\delta_{80},\quad\text{when}\,\,p=0,\\\
\delta_{81},\quad\text{when}\,\,p=1,\\\
\delta_{82},\quad\text{when}\,\,p=2,\\\
\delta_{83},\quad\text{when}\,\,p=3.\end{array}\right.$
$\delta_{80}=18\omega_{2}\lambda^{2}k_{31}-\frac{3}{2}C_{3}(-A_{X}(k_{41}+k_{42})+A_{Z}(\rho_{41}-\rho_{42})+2\rho_{20}k_{31})-\frac{3}{8}C_{4}(8A_{X}^{2}k_{31}-8A_{X}k_{21}(2\rho_{20}+\rho_{21}+\rho_{22})\\\
+4A_{Z}(\rho_{21}+\rho_{22})(4\rho_{20}-\rho_{21}-\rho_{22})-2k^{2}A_{X}^{2}k_{31}-2kA_{X}k_{21}(\sigma_{21}+\sigma_{22})-2kA_{X}A_{Z}(2\sigma_{31}-\sigma_{32})-6A_{Z}^{2}k_{31}-A_{Z}(\sigma_{21}+\sigma_{22})^{2}-3A_{Z}k_{21}^{2})-\frac{5}{16}C_{5}(-12A_{X}^{3}k_{21}+12A_{X}^{2}A_{Z}(2\rho_{20}+\rho_{21}+\rho_{22})+9A_{X}A_{Z}^{2}k_{21}+3A_{Z}^{3}(2\rho_{20}-3(\rho_{21}+\rho_{22}))+3k^{2}A_{X}^{3}k_{21}-6kA_{X}^{2}A_{Z}(\sigma_{21}+\sigma_{22})+3k^{2}A_{X}^{2}A_{Z}(2\rho_{20}-3(\rho_{21}+\rho_{22})))+\frac{15}{128}C_{6}(-24A_{X}^{4}A_{Z}+12k^{2}A_{X}^{4}A_{Z}+12A_{X}^{2}A_{Z}^{3}+5k^{4}A_{X}^{4}A_{Z}+10k^{2}A_{X}^{2}A_{Z}+5A_{Z}^{5})+\frac{\Delta}{\epsilon^{2}}k_{31},$
$\delta_{81}=18\omega_{2}\lambda^{2}k_{32}-\frac{3}{2}C_{3}(-A_{X}(k_{41}+k_{42})+A_{Z}(\rho_{41}+\rho_{42})+2\rho_{31}k_{22}+2\rho_{20}k_{32})-\frac{3}{8}C_{4}(8A_{X}^{2}k_{32}-8A_{X}(2\rho_{20}k_{21}+(\rho_{21}-\rho_{22})(k_{21}+2k_{22}))-16A_{X}A_{Z}\rho_{31}+4A_{Z}(\rho_{21}-\rho_{22})(4\rho_{20}+\rho_{21}-\rho_{22})\\\
-2k^{2}A_{X}^{2}k_{32}-2kA_{X}(k_{21}-2k_{22})(\sigma_{21}-\sigma_{22})+2kA_{X}A_{Z}\sigma_{32}-6A_{Z}^{2}k_{32}+A_{Z}(\sigma_{21}-\sigma_{22})^{2}-3A_{Z}k_{21}(k_{21}+4k_{22}))-\frac{5}{16}C_{5}(-4A_{X}^{3}(3k_{21}+2k_{22})\\\
+12A_{X}^{2}A_{Z}(2\rho_{20}+3(\rho_{21}-\rho_{22}))+9A_{X}A_{Z}^{2}(3k_{21}+2k_{22})-3A_{Z}^{3}(2\rho_{20}+3(\rho_{21}-\rho_{22}))+3k^{2}A_{X}^{3}(k_{21}-2k_{22})-6kA_{X}^{2}A_{Z}(\sigma_{21}-\sigma_{22})+3k^{2}A_{X}^{2}A_{Z}(2\rho_{20}-(\rho_{21}-\rho_{22})))+\frac{15}{128}C_{6}(-40A_{X}^{4}A_{Z}-12k^{2}A_{X}^{4}A_{Z}+60A_{X}^{2}A_{Z}^{3}+3k^{4}A_{X}^{4}A_{Z}+2k^{2}A_{X}^{2}A_{Z}^{3}-A_{Z}^{5})+\frac{\Delta}{\epsilon^{2}}k_{32},$
$\delta_{82}=-18\omega_{2}\lambda^{2}k_{31}-\frac{3}{2}C_{3}(-A_{X}(k_{41}+k_{42})-A_{Z}(\rho_{41}-\rho_{42})-2\rho_{20}k_{31})-\frac{3}{8}C_{4}(-8A_{X}^{2}k_{31}+8A_{X}k_{21}(2\rho_{20}+\rho_{21}+\rho_{22})-4A_{Z}(\rho_{21}+\rho_{22})(4\rho_{20}-\rho_{21}-\rho_{22})+2k^{2}A_{X}^{2}k_{31}+2kA_{X}k_{21}(\sigma_{21}+\sigma_{22})+2kA_{X}A_{Z}(2\sigma_{31}-\sigma_{32})+6A_{Z}^{2}k_{31}+A_{Z}(\sigma_{21}+\sigma_{22})^{2}+3A_{Z}k_{21}^{2})-\frac{5}{16}C_{5}(12A_{X}^{3}k_{21}-12A_{X}^{2}A_{Z}(2\rho_{20}+\rho_{21}+\rho_{22})-9A_{X}A_{Z}^{2}k_{21}-3A_{Z}^{3}(2\rho_{20}-3(\rho_{21}+\rho_{22}))-3k^{2}A_{X}^{3}k_{21}+6kA_{X}^{2}A_{Z}(\sigma_{21}+\sigma_{22})-3k^{2}A_{X}^{2}A_{Z}(2\rho_{20}-3(\rho_{21}+\rho_{22})))+\frac{15}{128}C_{6}(24A_{X}^{4}A_{Z}-12k^{2}A_{X}^{4}A_{Z}-12A_{X}^{2}A_{Z}^{3}-5k^{4}A_{X}^{4}A_{Z}-10k^{2}A_{X}^{2}A_{Z}^{3}-5A_{Z}^{5})-\frac{\Delta}{\epsilon^{2}}k_{31},$
$\delta_{83}=-18\omega_{2}\lambda^{2}k_{32}-\frac{3}{2}C_{3}(-A_{X}(k_{41}+k_{42})-A_{Z}(\rho_{41}+\rho_{42})-2\rho_{31}k_{22}-2\rho_{20}k_{32})-\frac{3}{8}C_{4}(-8A_{X}^{2}k_{32}+8A_{X}(2\rho_{20}k_{21}+(\rho_{21}-\rho_{22})(k_{21}+2k_{22}))+16A_{X}A_{Z}\rho_{31}-4A_{Z}(\rho_{21}-\rho_{22})(4\rho_{20}+\rho_{21}-\rho_{22})+2k^{2}A_{X}^{2}k_{32}+2kA_{X}(k_{21}-2k_{22})(\sigma_{21}-\sigma_{22})-2kA_{X}A_{Z}\sigma_{32}+6A_{Z}^{2}k_{32}-A_{Z}(\sigma_{21}-\sigma_{22})^{2}+3A_{Z}k_{21}(k_{21}+4k_{22}))-\frac{5}{16}C_{5}(4A_{X}^{3}(3k_{21}+2k_{22})\\\
-12A_{X}^{2}A_{Z}(2\rho_{20}+3(\rho_{21}-\rho_{22}))-9A_{X}A_{Z}^{2}(3k_{21}+2k_{22})+3A_{Z}^{3}(2\rho_{20}+3(\rho_{21}-\rho_{22}))-3k^{2}A_{X}^{3}(k_{21}-2k_{22})+6kA_{X}^{2}A_{Z}(\sigma_{21}-\sigma_{22})-3k^{2}A_{X}^{2}A_{Z}(2\rho_{20}-(\rho_{21}-\rho_{22})))+\frac{15}{128}C_{6}(40A_{X}^{4}A_{Z}+12k^{2}A_{X}^{4}A_{Z}-60A_{X}^{2}A_{Z}^{3}-3k^{4}A_{X}^{4}A_{Z}-2k^{2}A_{X}^{2}A_{Z}^{3}+5A_{Z}^{5})-\frac{\Delta}{\epsilon^{2}}k_{32}.$
$\displaystyle\delta_{9}=\left\\{\begin{array}[]{c}\delta_{90},\quad\text{when}\,\,p=0,\\\
\delta_{91},\quad\text{when}\,\,p=1,\\\
\delta_{92},\quad\text{when}\,\,p=2,\\\
\delta_{93},\quad\text{when}\,\,p=3.\end{array}\right.$
$\delta_{90}=-\frac{3}{2}C_{3}(-A_{X}k_{42}+k_{31}(\rho_{21}+\rho_{22})+\rho_{31}k_{21}+A_{Z}\rho_{42})-\frac{3}{8}C_{4}(4A_{X}^{2}k_{31}-8A_{X}k_{21}(\rho_{21}+\rho_{22})-8A_{X}A_{Z}\rho_{31}+4A_{Z}(\rho_{21}+\rho_{22})^{2}+k^{2}A_{X}^{2}k_{31}+2kA_{X}k_{21}(\sigma_{21}+\sigma_{22})+2kA_{X}A_{Z}\sigma_{31}+3A_{Z}^{2}k_{31}+A_{Z}(\sigma_{21}+\sigma_{22})^{2}+3A_{Z}k_{21}^{2})-\frac{5}{16}C_{5}(-4A_{X}^{3}k_{21}+12A_{X}^{2}A_{Z}(\rho_{21}+\rho_{22})-9A_{X}A_{Z}^{2}k_{21}+3A_{Z}^{3}(\rho_{21}+\rho_{22})-3k^{2}A_{X}^{3}k_{21}+6kA_{X}^{2}A_{Z}(\sigma_{21}+\sigma_{22})+3k^{2}A_{X}^{2}A_{Z}(\rho_{21}+\rho_{22}))+\frac{15}{128}C_{6}(-8A_{X}^{4}A_{Z}-12k^{2}A_{X}^{4}A_{Z}-12A_{X}^{2}A_{Z}^{3}-k^{4}A_{X}^{4}A_{Z}-2k^{2}A_{X}^{2}A_{Z}^{3}-A_{Z}^{5}),$
$\delta_{91}=-\frac{3}{2}C_{3}(-A_{X}k_{42}+k_{32}(\rho_{21}-\rho_{22})+\rho_{31}k_{21}+A_{Z}\rho_{42})-\frac{3}{8}C_{4}(4A_{X}^{2}k_{32}-8A_{X}k_{21}(\rho_{21}-\rho_{22})-8A_{X}A_{Z}\rho_{31}+4A_{Z}(\rho_{21}-\rho_{22})^{2}+k^{2}A_{X}^{2}k_{32}+2kA_{X}k_{21}(\sigma_{21}-\sigma_{22})+2kA_{X}A_{Z}\sigma_{31}-3A_{Z}^{2}k_{32}+A_{Z}(\sigma_{21}-\sigma_{22})^{2}-3A_{Z}k_{21}^{2})-\frac{5}{16}C_{5}(-4A_{X}^{3}k_{21}+12A_{X}^{2}A_{Z}(\rho_{21}-\rho_{22})+9A_{X}A_{Z}^{2}k_{21}-3A_{Z}^{3}(\rho_{21}-\rho_{22})-3k^{2}A_{X}^{3}k_{21}-6kA_{X}^{2}A_{Z}(\sigma_{21}-\sigma_{22})+3k^{2}A_{X}^{2}A_{Z}(\rho_{21}-\rho_{22}))+\frac{15}{128}C_{6}(-8A_{X}^{4}A_{Z}-12k^{2}A_{X}^{4}A_{Z}+12A_{X}^{2}A_{Z}^{3}-k^{4}A_{X}^{4}A_{Z}+2k^{2}A_{X}^{2}A_{Z}^{3}-A_{Z}^{5}),$
$\delta_{92}=-\frac{3}{2}C_{3}(-A_{X}k_{42}-k_{31}(\rho_{21}+\rho_{22})-\rho_{31}k_{21}-A_{Z}\rho_{42})-\frac{3}{8}C_{4}(-4A_{X}^{2}k_{31}+8A_{X}k_{21}(\rho_{21}+\rho_{22})+8A_{X}A_{Z}\rho_{31}-4A_{Z}(\rho_{21}+\rho_{22})^{2}-k^{2}A_{X}^{2}k_{31}-2kA_{X}k_{21}(\sigma_{21}+\sigma_{22})-2kA_{X}A_{Z}\sigma_{31}-3A_{Z}^{2}k_{31}-A_{Z}(\sigma_{21}+\sigma_{22})^{2}-3A_{Z}k_{21}^{2})-\frac{5}{16}C_{5}(4A_{X}^{3}k_{21}-12A_{X}^{2}A_{Z}(\rho_{21}+\rho_{22})+9A_{X}A_{Z}^{2}k_{21}-3A_{Z}^{3}(\rho_{21}+\rho_{22})+3k^{2}A_{X}^{3}k_{21}-6kA_{X}^{2}A_{Z}(\sigma_{21}+\sigma_{22})-3k^{2}A_{X}^{2}A_{Z}(\rho_{21}+\rho_{22}))+\frac{15}{128}C_{6}(8A_{X}^{4}A_{Z}+12k^{2}A_{X}^{4}A_{Z}+12A_{X}^{2}A_{Z}^{3}+k^{4}A_{X}^{4}A_{Z}+2k^{2}A_{X}^{2}A_{Z}^{3}+A_{Z}^{5}),$
$\delta_{93}=-\frac{3}{2}C_{3}(-A_{X}k_{42}-k_{32}(\rho_{21}-\rho_{22})-\rho_{31}k_{21}-A_{Z}\rho_{42})-\frac{3}{8}C_{4}(-4A_{X}^{2}k_{32}+8A_{X}k_{21}(\rho_{21}-\rho_{22})+8A_{X}A_{Z}\rho_{31}-4A_{Z}(\rho_{21}-\rho_{22})^{2}-k^{2}A_{X}^{2}k_{32}-2kA_{X}k_{21}(\sigma_{21}-\sigma_{22})-2kA_{X}A_{Z}\sigma_{31}+3A_{Z}^{2}k_{32}-A_{Z}(\sigma_{21}-\sigma_{22})^{2}+3A_{Z}k_{21}^{2})-\frac{5}{16}C_{5}(4A_{X}^{3}k_{21}-12A_{X}^{2}A_{Z}(\rho_{21}-\rho_{22})-9A_{X}A_{Z}^{2}k_{21}+3A_{Z}^{3}(\rho_{21}-\rho_{22})+3k^{2}A_{X}^{3}k_{21}+6kA_{X}^{2}A_{Z}(\sigma_{21}-\sigma_{22})-3k^{2}A_{X}^{2}A_{Z}(\rho_{21}-\rho_{22}))+\frac{15}{128}C_{6}(8A_{X}^{4}A_{Z}+12k^{2}A_{X}^{4}A_{Z}-12A_{X}^{2}A_{Z}^{3}+k^{4}A_{X}^{4}A_{Z}-2k^{2}A_{X}^{2}A_{Z}^{3}+A_{Z}^{5}).$
$\displaystyle\rho_{51}=$
$\displaystyle\frac{6n\lambda\beta_{9}-(9\lambda^{2}+n^{2}-C_{2})\gamma_{8}}{(n^{2}-9\lambda^{2})^{2}+C_{2}(n^{2}-2C_{2}+9\lambda^{2})}_{,}$
$\displaystyle\rho_{52}=$
$\displaystyle\frac{10n\lambda\beta_{10}-(25\lambda^{2}+n^{2}-C_{2})\gamma_{9}}{(n^{2}-25\lambda^{2})^{2}+C_{2}(n^{2}-2C_{2}+25\lambda^{2})}_{,}$
$\displaystyle\sigma_{51}=$ $\displaystyle-\frac{k\beta_{11}}{2\lambda n}_{,}$
$\displaystyle\sigma_{52}=$
$\displaystyle\frac{6n\lambda\gamma_{8}-(9\lambda^{2}+n^{2}+2C_{2})\beta_{9}}{(n^{2}-9\lambda^{2})^{2}+C_{2}(n^{2}-2C_{2}+9\lambda^{2})}_{,}$
$\displaystyle\sigma_{53}=$
$\displaystyle\frac{10n\lambda\gamma_{9}-(25\lambda^{2}+n^{2}+2C_{2})\beta_{10}}{(n^{2}-25\lambda^{2})^{2}+C_{2}(n^{2}-2C_{2}+25\lambda^{2})}_{,}$
$\displaystyle k_{51}=$ $\displaystyle-\frac{\delta_{8}}{8\lambda^{2}}_{,}$
$\displaystyle k_{52}=$ $\displaystyle-\frac{\delta_{9}}{24\lambda^{2}}_{.}$

Fig. 1 : $4^{th}$ and $5^{th}$ order halo orbits around $L_{1}$ corresponding to $A_{2}=2.4\times 10^{-12},q=0.9995$ Fig. 2 : $4^{th}$ and $5^{th}$ order halo orbits around $L_{1}$ corresponding to $A_{2}=2.4\times 10^{-12},q=0.9945$ Fig. 3 : $4^{th}$ and $5^{th}$ order halo orbits around $L_{1}$ corresponding to $A_{2}=2.4\times 10^{-12},q=0.9895$ Fig. 4 : $4^{th}$ and $5^{th}$ order halo orbits around $L_{1}$ corresponding to $A_{2}=2.4\times 10^{-12},q=0.9845$ Fig. 5 : Effect of radiation pressure on the position of halo orbits around $L_{1}$ Fig. 6 : $4^{th}$ and $5^{th}$ order halo orbits around $L_{2}$ corresponding to $A_{2}=2.4\times 10^{-12},q=0.9995$ Fig. 7 : $4^{th}$ and $5^{th}$ order halo orbits around $L_{2}$ corresponding to $A_{2}=2.4\times 10^{-12},q=0.9945$ Fig. 8 : $4^{th}$ and $5^{th}$ order halo orbits around $L_{2}$ corresponding to $A_{2}=2.4\times 10^{-12},q=0.9895$ Fig. 9 : $4^{th}$ and $5^{th}$ order halo orbits around $L_{2}$ corresponding to $A_{2}=2.4\times 10^{-12},q=0.9845$ Fig. 10 : Effect of radiation pressure on the position of halo orbits around $L_{2}$ Fig. 11 : Effect of oblateness on the position of $L_{1}$ Fig. 12 : Effect of oblateness on the position of $L_{2}$ Fig. 13 : Effect of radiation pressure on the position of $L_{1}$ Fig. 14 : Effect of radiation pressure on the position of $L_{2}$ Fig. 15 : Effect of oblateness on time period of halo orbits around $L_{1}$ Fig. 16 : Effect of oblateness on time period of halo orbits around $L_{2}$ Fig. 17 : Effect of radiation pressure on time period of halo orbits around $L_{1}$ Fig. 18 : Effect of radiation pressure on time period of halo orbits around $L_{2}$ Table 1 : Effect of radiation pressure on different parameters of orbits around $L_{1}$ when $A_{2}=2.4\times 10^{-12}$ $q$ | 1 | 0.9995 | 0.9945 | 0.9895 | 0.9845
---|---|---|---|---|---
$\gamma$ | 0.009966562831474 | 0.010022806042997 | 0.010621857046575 | 0.011292634839768 | 0.012042049799192
$L_{1}$ | 0.990030433658191 | 0.989974190565718 | 0.989375139443090 | 0.988704361649897 | 0.987954946690473
$C_{2}$ | 4.064344359358607 | 4.013217307075904 | 3.533142166508352 | 3.109449638463373 | 2.740934989090651
$C_{3}$ | 3.023467759599181 | 2.972624703974638 | 2.495245216485259 | 2.073965581138543 | 1.707553745778144
$C_{4}$ | 3.033946155218810 | 2.983159954578718 | 2.506387966212186 | 2.085792481345639 | 1.720149452296818
$C_{5}$ | 3.033840670308508 | 2.983053292754523 | 2.506268338852770 | 2.085657399045222 | 1.719995925393043
$C_{6}$ | 3.033841732214042 | 2.983054372628669 | 2.506269623159199 | 2.085658941903247 | 1.719997796706109
$\lambda$ | 2.087246092417118 | 2.074819527399089 | 1.954036850904442 | 1.840326271553855 | 1.734687871104357
$\Delta$ | 0.292251890951922 | 0.291658764200673 | 0.285117848184197 | 0.277351147307939 | 0.268207021065916
$k$ | 3.230401297192164 | 3.212643439431957 | 3.041023597428729 | 2.881472765622095 | 2.735654104240122
$\tau$ | 3.010275276119164 | 3.028304497912615 | 3.215489669128482 | 3.414169217871602 | 3.622084071631579
Table 2 : Effect of oblateness on different parameters of orbits around $L_{1}$ when $q=0.9995$ $A_{2}$ | 0 | $2.4\times 10^{-12}$ | $3\times 10^{-12}$ | $3.5\times 10^{-12}$ | $4\times 10^{-12}$
---|---|---|---|---|---
$\gamma$ | 0.010022805923947 | 0.010022806042997 | 0.010022806072760 | 0.010022806097562 | 0.010022806122364
$L_{1}$ | 0.989974190565718 | 0.989974190446668 | 0.989974190416905 | 0.989974190392103 | 0.989974190367301
$C_{2}$ | 4.013217306099824 | 4.013217307075904 | 4.013217307319478 | 4.013217307522901 | 4.013217307726325
$C_{3}$ | 2.972624703499105 | 2.972624703974638 | 2.972624704093073 | 2.972624704192215 | 2.972624704291359
$C_{4}$ | 2.983159953971714 | 2.983159954578718 | 2.983159954730022 | 2.983159954856554 | 2.983159954983087
$C_{5}$ | 2.983053292150130 | 2.983053292754523 | 2.983053292905174 | 2.983053293031162 | 2.983053293157152
$C_{6}$ | 2.983054372024236 | 2.983054372628669 | 2.983054372779329 | 2.983054372905326 | 2.983054373031323
$\lambda$ | 2.074819527160888 | 2.074819527399089 | 2.074819527458530 | 2.074819527508173 | 2.074819527557816
$\Delta$ | 0.291658764188306 | 0.291658764200673 | 0.291658764203759 | 0.291658764206337 | 0.291658764208914
$k$ | 3.212643439097060 | 3.212643439431957 | 3.212643439515526 | 3.212643439585321 | 3.212643439655118
$\tau$ | 3.028304498260281 | 3.028304497912615 | 3.028304497825857 | 3.028304497753401 | 3.028304497680945
Table 3 : Effect of radiation pressure on different parameters of orbits around $L_{2}$ when $A_{2}=2.4\times 10^{-12}$ $q$ | 1 | 0.9995 | 0.9945 | 0.9895 | 0.9845
---|---|---|---|---|---
$\gamma$ | 0.010033228531910 | 0.009978343639533 | 0.009461048204950 | 0.008996749353809 | 0.008579422887427
$L_{2}$ | 1.010030225021575 | 1.009975340129198 | 1.009458044694615 | 1.008993745843474 | 1.008576419377092
$C_{2}$ | 3.944259093871529 | 3.993273144291635 | 4.513389113295589 | 5.087772984797185 | 5.715729697751704
$C_{3}$ | -2.983408088154385 | -3.032693790758545 | -3.555654417985380 | -4.133098257601873 | -4.764305916840534
$C_{4}$ | 2.973863414322725 | 3.023203497285466 | 3.546678168593691 | 4.124585872165059 | 4.756212685136259
$C_{5}$ | -2.973768601705565 | -3.023109735463630 | -3.546594039811956 | -4.124509971228582 | -4.756143840525967
$C_{6}$ | 2.973767659878476 | 3.023108809119334 | 3.546593251325406 | 4.124509294455637 | 4.756143254903247
$\lambda$ | 2.057933451632755 | 2.069950687116124 | 2.193217663095480 | 2.321449398886311 | 2.453753890572465
$\Delta$ | 0.290830997477577 | 0.291422702800878 | 0.296814604418408 | 0.301354326792431 | 0.305178457747805
$k$ | 3.188540491593625 | 3.205690410467006 | 3.382469098698882 | 3.567743774450670 | 3.760028180060947
$\tau$ | 3.053152813175049 | 3.035427532785036 | 2.864825235043738 | 2.706578618596585 | 2.560642015207853
Table 4 : Effect of oblateness on different parameters of orbits around $L_{2}$ when $q=0.9995$ $A_{2}$ | 0 | $2.4\times 10^{-12}$ | $3\times 10^{-12}$ | $3.5\times 10^{-12}$ | $4\times 10^{-12}$
---|---|---|---|---|---
$\gamma$ | 0.009978343518616 | 0.009978343639533 | 0.009978343669762 | 0.009978343694953 | 0.009978343720144
$L_{2}$ | 1.009975340008281 | 1.009975340129198 | 1.009975340159427 | 1.009975340184618 | 1.009975340209809
$C_{2}$ | 3.993273145236837 | 3.993273144291635 | 3.993273144055562 | 3.993273143858681 | 3.993273143661800
$C_{3}$ | -3.032693791243736 | -3.032693790758545 | -3.032693790637474 | -3.032693790536429 | -3.032693790435384
$C_{4}$ | 3.023203497879979 | 3.023203497285466 | 3.023203497137065 | 3.023203497013244 | 3.023203496889423
$C_{5}$ | -3.023109736060348 | -3.023109735463630 | -3.023109735314677 | -3.023109735190398 | -3.023109735066117
$C_{6}$ | 3.023108809716084 | 3.023108809119334 | 3.023108808970372 | 3.023108808846086 | 3.023108808721798
$\lambda$ | 2.069950687346945 | 2.069950687116124 | 2.069950687058475 | 2.069950687010396 | 2.069950686962317
$\Delta$ | 0.291422702811252 | 0.291422702800878 | 0.291422702798289 | 0.291422702796127 | 0.291422702793967
$k$ | 3.205690410801889 | 3.205690410467006 | 3.205690410383364 | 3.205690410313609 | 3.205690410243853
$\tau$ | 3.035427532446555 | 3.035427532785036 | 3.035427532869575 | 3.035427532940079 | 3.035427533010583

## References

  * Abouelmagd (2013) Abouelmagd, E.I.: Astrophysics and Space Science 346(1), 51 (2013). doi:10.1007/s10509-013-1439-9
  * Breakwell and Brown (1979) Breakwell, J.V., Brown, J.V.: Celestial Mechanics 20, 389 (1979)
  * Brouwer and Clemence (1961) Brouwer, D., Clemence, G.M.: Planets and Satellites 31 (1961)
  * Chidambararaj and Sharma (2016) Chidambararaj, P., Sharma, R.K.: International Journal of Astronomy and Astrophysics 6, 293 (2016). doi:10.4236/ijaa.2016.63025
  * Danby (1964) Danby, J.M.A.: Fundamentals of Celestial Mechanics. Macmillan Company, New York (1964)
  * Eapen and Sharma (2014) Eapen, R.T., Sharma, R.K.: Astrophysics and Space Science 352(2), 437 (2014). doi:10.1007/s10509-014-1951-6
  * Farquhar (1968) Farquhar, R.W.: The control and use of libration-point satellites. PhD thesis, Department of Aeronautics and Astronautics, Stanford University, Stanford (1968)
  * Fitzpatrick (2012) Fitzpatrick, R.: An Introduction to Celestial Mechanics. Cambridge University Press, New York (2012)
  * Ghotekar and Sharma (2019) Ghotekar, S., Sharma, R.K.: International Journal of Astronomy and Astrophysics 9, 274 (2019). doi:10.4236/ijaa.2019.93020
  * Howell (1984) Howell, K.C.: Celestial Mechanics 32, 53 (1984)
  * Howell and V.Breakwell (1984) Howell, K.C., V.Breakwell, J.: Celestial Mechanics 32, 29 (1984)
  * Koon et al. (2011) Koon, W.S., W.Lo, M., Marsden, J.E., Ross, S.D.: Dynamical Systems, the Three-body Problem and Space Mission Design. Interdisplinary Applied Mathemstics, Springer, Berlin (2011)
  * McCuskey (1963) McCuskey, S.W.: Introduction to Celestial Mechanics. Addison-Wesley, London (1963)
  * Moulton (1914) Moulton, F.R.: An Introduction to Celestial Mechanics. Dover Publications, New York (1914)
  * Murray and Dermot (1999) Murray, C.D., Dermot, S.F.: Solar System Dynamics. Cambridge University Press, Cambridge (1999)
  * Pathak et al. (2016) Pathak, N., Sharma, R.K., Thomas, V.O.: International Journal of Astronomy and Astrophysics 6, 175 (2016). doi:10.4236/ijaa.2016.62015
  * Plummer (1919) Plummer, H.C.: The American Mathematical Monthly 26(6) (1919). doi:10.2307/2973529
  * Pollard (1966) Pollard, H.: Mathematical Introduction to Celestial Mechanics. Prentice Hall, New Jersey (1966)
  * Poynting (1903) Poynting, J.H.: Monthly Notices of the Royal Astronomical Society 64(1), 1 (1903). doi:10.1093/mnras/64.1.1a
  * Pushparaj and Sharma (2016) Pushparaj, N., Sharma, R.K.: International Journal of Astronomy and Astrophysics 6, 347 (2016). doi:10.4236/ijaa.2016.64029
  * Richardson (1980) Richardson, D.L.: Celestial Mechanics 22, 231 (1980). doi:10.1007/BF01229511
  * Robertson and Russell (1937) Robertson, H.P., Russell, H.N.: Monthly Notices of the Royal Astronomical Society 97(6), 423 (1937). doi:10.1093/mnras/97.6.423
  * Roy (2005) Roy, A.E.: Orbital Motion. Institute of Physics Publishing, UK (2005)
  * Schuerman (1980) Schuerman, D.W.: The Astrophysical Journal 238, 337 (1980). doi:10.1086/157989
  * Sharma (1987) Sharma, R.K.: Astrophysics and Space Science 135(2), 271 (1987)
  * Simmons et al. (1985) Simmons, J.F.L., McDonald, A.J.C., Brown, J.C.: Celestial Mechanics 35(145), 146 (1985). doi:10.1007/BF01227667
  * Szebehely (1967) Szebehely, V.: Theory of Orbits. The Restricted Problem of Three Bodies. Academic Press, New York (1967)
  * Thurman and Worfolk (1996) Thurman, R., Worfolk, P.A.: The Geometry of Halo Orbits in the Circular Restricted Three-body Problem. Technical Report, Minneapolis (1996)
  * Tiwary and Kushvah (2015) Tiwary, R.D., Kushvah, B.S.: Astrophysics and Space Science 357(1), 73 (2015). doi:10.1007/s10509-015-2243-5
  * Vallado (2013) Vallado, D.A.: Fundamentals of Astrodynamics and Applications. Microcosm, Hawthorne (2013)
  * Winter (1941) Winter, A.: The Analytical Foundations of Celestial Mechanics. Princeton University Press, New Jersey (1941)


# Description of the newly observed $\Omega^{*}_{c}$ states as molecular
states

Jingwen Feng  School of Physics and Electronic Engineering, Sichuan Normal
University, Chengdu 610101, China  Feng Yang  School of Physical Science and
Technology, Southwest Jiaotong University, Chengdu 610031,China  Cai Cheng
<EMAIL_ADDRESS>School of Physics and Electronic Engineering, Sichuan
Normal University, Chengdu 610101, China  Yin Huang<EMAIL_ADDRESS>School of Physical Science and Technology, Southwest Jiaotong University,
Chengdu 610031,China

###### Abstract

In this work, we study the strong decays of the newly observed
$\Omega^{*}_{c}(3185)$ and $\Omega^{*}_{c}(3327)$ assuming that
$\Omega^{*}_{c}(3185)$ and $\Omega^{*}_{c}(3327)$ as $S$-wave $D\Xi$ and
$D^{*}\Xi$ molecular state, respectively. Since the $\Omega_{c}^{*}$ was
observed in the $\Xi_{c}^{+}K^{-}$ invariant mass distributions, the partial
decay width of $\Omega^{*}_{c}(3185)$ and $\Omega^{*}_{c}(3327)$ into
$\Xi_{c}^{+}K^{-}$ through hadronic loops are evaluated with the help of the
effective Lagrangians. Moreover, the decay channel of
$\Xi_{c}^{{}^{\prime}}\bar{K}$ is also included. The decay process is
described by the $t$-channel $\Lambda$, $\Sigma$ baryons and $D_{s}$,
$D_{s}^{*}$ mesons exchanges, respectively. By comparison with the LHCb
observation, the current results support the $\Omega^{*}_{c}(3327)$
with$J^{P}=3/2^{-}$ as pure $D^{*}\Xi$ molecule while the
$\Omega^{*}_{c}(3327)$ with $J^{P}=1/2^{-}$ can not be well reproduced in the
molecular state picture. In addition, the spin-parity $J^{P}=1/2^{-}$ $D\Xi$
molecular assumptions for the $\Omega^{*}_{c}(3185)$ can’t be conclusively
determined. It may be a meson-baryon molecule with a big $D\Xi$ component.
Although the decay width of the
$\Omega_{c}^{*}\to{}\bar{K}\Xi_{c}^{{}^{\prime}}$ is of the order several MeV,
it can be well employed to test the molecule interpretations of
$\Omega^{*}_{c}(3185)$ and $\Omega^{*}_{c}(3327)$.

##  I Introduction

After the discovery of the five $\Omega^{*}_{c}$ states in one observation
simultaneously back in 2017 LHCb:2017uwr , the latest experimental results of
the LHCb Collaboration reported the existence of an additional two
$\Omega_{c}$ states in the $\Xi_{c}^{+}K^{-}$ invariant mass spectrum,
originating from $pp$ collisions LHCb:2023rtu . The measured parameters of
these newly states are as follows:

$\displaystyle M_{\Omega_{c}(3185)}$
$\displaystyle=3185.1\pm{}1.7^{+7.4}_{-0.9}\pm{}0.2~{}~{}~{}~{}~{}{\rm MeV},$
$\displaystyle\Gamma_{\Omega_{c}(3185)}$ $\displaystyle=50\pm
7^{+10}_{-20}~{}~{}~{}~{}~{}{\rm MeV},$ (1) $\displaystyle
M_{\Omega_{c}(3327)}$
$\displaystyle=3327.1\pm{}1.2^{+0.1}_{-1.3}\pm{}0.2~{}~{}~{}~{}~{}{\rm MeV},$
$\displaystyle\Gamma_{\Omega_{c}(3327)}$ $\displaystyle=20\pm
5^{+13}_{-1}~{}~{}~{}~{}~{}{\rm MeV}.$

Compared to the earlier discovery of two ground states $\Omega^{*}_{c}(2695)$
and $\Omega^{*}_{c}(2770)$ Workman:2022ynf , the findings presented in not
only expand our understanding of the charmed baryon with quantum numbers $C=1$
and $S=2$, which is composed of one charm quark and two strange quarks, but
also help us to strange quarks, but also help us to understand the formation
mechanism of exotic hadron states. This is mainly because the newly observed
$\Omega^{*}_{c}$ states may display a more complex internal structure than the
ground states $\Omega^{*}_{c}(2695)$ and $\Omega^{*}_{c}(2770)$, which are
made up of three quarks.

Indeed, the $\Omega^{*}_{c}(3050)$ and $\Omega_{c}^{*}(3119)$ were suggested
to be the exotic pentaquarks in the chiral quark-soliton model Kim:2017jpx ;
Kim:2017khv . Similar qualitative results can also be found in Refs.
Yang:2017rpg ; An:2017lwg . In Refs. Debastiani:2017ewu ; Montana:2017kjw ,
based on the analysis of the mass spectrum, the $\Omega^{*}_{c}(3050)$ was
identified as a meson-baryon molecule with $J^{P}=1/2^{-}$. This is the same
with the results in Ref. Huang:2018wgr  that the total decay widths of the
$\Omega^{*}_{c}(3050)$ can be well reproduced with the assumption that
$\Omega^{*}_{c}(3050)$ as $S$-wave $\Xi{}D$ bound state with $J^{P}=1/2^{-}$.
By solving a coupled-channel Bethe Salpeter equation using a
$SU(6)_{lsf}\times$ HQSS extended WT interaction as a kernel, three meson-
baryon molecular states were obtained Nieves:2017jjx  that can be associated
to the experimental states $\Omega^{*}_{c}(3000)$, $\Omega^{*}_{c}(3050)$, and
$\Omega^{*}_{c}(31119)$ or $\Omega^{*}_{c}(3090)$.

There also exist some clues support the newly observed $\Omega^{*}_{c}(3185)$
and $\Omega^{*}_{c}(3227)$ as molecular states. It was suggested in Ref.
Yu:2023bxn  that $\Omega^{*}_{c}(3185)$ and $\Omega^{*}_{c}(3327)$ were
proposed to be the $2S(3/2^{+})$ and $1D(3/2^{+})$ states, respectively.
However, a completely different conclusion was drawn from Ref. luo:2023bxn
that the $\Omega^{*}_{c}(3327)$ is a good candidate of $\Omega^{*}_{c}(1D)$
state with $J^{P}=5/2^{+}$. There exists spin-parity puzzle for
$\Omega^{*}_{c}(3327)$ indicate it difficult to put $\Omega^{*}_{c}(3327)$
into the conventional $css$ states. Since the mass of $\Omega^{*}_{c}(3327)$
is very close to the threshold of $D^{*}\Xi$, the hadronic molecular
configuration for $\Omega^{*}_{c}(3327)$ is possible. Although the
$\Omega^{*}_{c}(3185)$ can be considered as a conventional charmed baryon
Yu:2023bxn , the hadronic molecule interpretations cannot be excluded due to
the mass of $\Omega^{*}_{c}(3185)$ is about 6.37 MeV below the threshold of
$D\Xi$.

However, no work has been conducted to study whether the newly observed
$\Omega^{*}_{c}(3185)$ and $\Omega^{*}_{c}(3327)$ can be explained as $D\Xi$
and $D^{*}\Xi$ molecular state, respectively. In this work, we estimate
possible strong decay modes by assuming $\Omega^{*}_{c}(3185)$ and
$\Omega^{*}_{c}(3327)$ as molecular states. By comparing the calculated decay
widths with experimental data, we can evaluate the validity of the proposed
molecular explanations for the structures of $\Omega^{*}_{c}(3185)$ and
$\Omega^{*}_{c}(3327)$.

This paper is organized as follows. In Sec. II, we will present the
theoretical formalism. In Sec. III, the numerical result will be given,
followed by discussions and conclusions in last section.

##  II FORMALISM AND INGREDIENTS

Figure 1: Feynman diagrams for the process
$\Omega^{0}_{c}(3185,3327)\to{}K^{-}\Xi^{+}_{c}$ and
$K^{-}\Xi^{{}^{\prime}+}_{c}$.

With assuming the newly observed $\Omega^{*}_{c}(3185)$ and
$\Omega^{*}_{c}(3227)$ as $S$-wave molecular states, the decay of
$\Omega^{*}_{c}(3185)$ and $\Omega^{*}_{c}(3227)$ into $\bar{K}\Xi_{c}$ and
$\bar{K}\Xi_{c}^{{}^{\prime}}$ are allowed. The corresponding Feynman diagrams
are illustrated in Fig. 1, which includes the $t$-channel $\Sigma$, $\Lambda$
baryons exchange and $D_{s}$, $D_{s}^{*}$ mesons exchange. To evaluate these
diagrams, the Lagrangian for the coupling between the mesons and the baryons
are obtained using the $SU(4)$ invariant interaction Lagrangians Liu:2001ce

$\displaystyle\mathcal{L}_{PBB}$
$\displaystyle=ig_{p}(a\phi^{*\alpha\nu\nu}\gamma^{5}P^{\beta}_{\alpha}\phi_{\beta\mu\nu}+b\phi^{*\alpha\mu\nu}\gamma^{5}P^{\beta}_{\alpha}\phi_{\beta\mu\nu}),$
(2) $\displaystyle\mathcal{L}_{VBB}$
$\displaystyle=ig_{v}(c\phi^{*\alpha\nu\nu}\gamma
P^{\beta}_{\alpha}\phi_{\beta\mu\nu}+d\phi^{*\alpha\nu\nu}\gamma
P^{\beta}_{\alpha}\phi_{\beta\mu\nu}),$

where $V_{\mu}$ and $P$ are the $SU(4)$ vector meson and pseudoscalar meson
matrix, respectively. The meson matrices are

$\displaystyle
P=\begin{pmatrix}\frac{\pi^{0}}{\sqrt{2}}+\frac{\eta}{\sqrt{6}}+\frac{\eta_{c}}{\sqrt{12}}&\pi^{+}&K^{+}&\bar{D}^{0}\\\
\pi^{-}&-\frac{\pi^{0}}{\sqrt{2}}+\frac{\eta}{\sqrt{6}}+\frac{\eta_{c}}{\sqrt{12}}&K^{0}&D^{-}\\\
K^{-}&\bar{K}^{0}&-\frac{2\eta}{\sqrt{6}}+\frac{\eta_{c}}{\sqrt{12}}&D_{s}^{-}\\\
D^{0}&D^{+}&D_{s}^{+}&-\frac{3\eta_{c}}{\sqrt{12}}\end{pmatrix}$ (3)

and

$\displaystyle
V=\begin{pmatrix}\frac{\rho^{0}}{\sqrt{2}}+\frac{\omega_{8}}{\sqrt{6}}+\frac{J/\Psi}{\sqrt{12}}&\rho^{+}&K^{*+}&\bar{D}^{*0}\\\
\rho^{-}&-\frac{\rho^{0}}{\sqrt{2}}+\frac{\omega_{8}}{\sqrt{6}}+\frac{J/\Psi}{\sqrt{12}}&K^{*0}&D^{*-}\\\
K^{*-}&\bar{K}^{*0}&-\frac{2\omega_{8}}{\sqrt{6}}+\frac{J/\Psi}{\sqrt{12}}&D_{s}^{*-}\\\
D^{*0}&D^{*+}&D_{s}^{*+}&-\frac{3J/\Psi}{\sqrt{12}}\end{pmatrix}.$ (4)

with $\omega_{8}=\omega\cos\theta+\phi\sin\theta$ and $\sin\theta=-0.761$. The
tensors $\phi_{\beta\mu\nu}$ in above equation represents 20-plet of the
proton Liu:2001ce ,

$\displaystyle
p=\phi_{112},n=\phi_{221},\Lambda=\sqrt{\frac{2}{3}}(\phi_{321}-\phi_{312}),$
$\displaystyle\Sigma^{+}=\phi_{113},\Sigma^{0}=\sqrt{2}\phi_{123},\Sigma^{-}=\phi_{233},$
$\displaystyle\Xi^{0}=\phi_{311},\Xi^{-}=\phi_{332},\Sigma^{++}_{c}=\phi_{114},$
$\displaystyle\Sigma^{+}_{c}=\phi_{124},\Sigma^{0}_{c}=\phi_{224},\Xi_{c}^{+}=\phi_{134},$
$\displaystyle\Xi_{c}^{0}=\phi_{234},\Xi_{c}^{+^{\prime}}=\sqrt{\frac{2}{3}}(\phi_{413}-\phi_{431}),\Xi_{c}^{0^{\prime}}=\sqrt{\frac{2}{3}}(\phi_{423}-\phi_{432})$
$\displaystyle\Lambda_{c}^{+}=\sqrt{\frac{2}{3}}(\phi_{421}-\phi_{412}),\Omega_{c}^{0}=\phi_{334},\Xi^{++}_{cc}=\phi_{441}$
$\displaystyle\Xi_{cc}^{+}=\phi_{442},\Omega_{cc}^{+}=\phi_{443},$ (5)

where the indices $\beta$, $\mu$, $\nu$ denote the quark content of the baryon
fields with the identification $1\leftrightarrow{}u$, $2\leftrightarrow{}d$,
$3\leftrightarrow{}s$, and $4\leftrightarrow{}c$. Hence, the tensors
$\phi_{\beta\mu\nu}$ satisfy following condition

$\displaystyle\phi_{\mu\nu\lambda}+\phi_{\nu\lambda\mu}+\phi_{\lambda\mu\nu}=0,\phi_{\mu\nu\lambda}=\phi_{\nu\mu\lambda}.$
(6)

By using the exact form of the matrix and the above relationship, the
interaction vertices between baryons and pseudoscalar mesons can be estimated
by expanding the SU(4) invariant interaction Lagrangians. The values of the
coupling constants adopted in this work can be computed by comparing with the
coefficients of the interaction Lagrangians ${\cal{L}}_{\pi{}NN}$ and
${\cal{L}}_{\rho{}NN}$ and determining the constants $g_{V}$, $g_{P}$, $a$,
$b$, $c$, and $d$ in terms of $g_{\pi{}NN}=13.5$ and $g_{\rho{}NN}=3.25$
Liu:2001ce . Then the values of the coupling constants can be compute and are
listed in Tab. 1.

Table 1: Values of the effective couplings constants. $g_{\Xi_{c}^{+}D^{+}\Sigma^{0}}$ | $g_{\Xi^{-}K^{-}\Sigma^{0}}$ | $g_{D^{+}\Xi_{c}^{+}\Lambda^{0}}$ | $g_{\Xi^{-}K^{-}\Lambda^{0}}$ | $g_{D^{0}\Xi_{c}^{+}\Sigma^{+}}$ | $g_{\Xi^{0}\Sigma^{+}K^{-}}$ | $g_{\Xi^{0}D^{*-}_{s}\Xi_{c}^{+}}$
---|---|---|---|---|---|---
3.78 | 13.5 | 6.55 | 3.43 | 5.35 | -19.09 | 4.60
$g_{D^{*+}\Sigma^{0}\Xi_{c}^{+}}$ | $g_{D^{*+}\Lambda^{0}\Xi_{c}^{+}}$ | $g_{D^{*0}\Sigma^{+}\Xi_{c}^{+}}$ | $g_{\Xi^{0}D_{s}^{-}\Xi_{c}^{+}}$ | $g_{D^{+}\Sigma^{0}\Xi_{c}^{{}^{\prime}+}}$ | $g_{D^{+}\Lambda^{0}\Xi_{c}^{{}^{\prime}+}}$ | $g_{D^{0}\Sigma^{+}\Xi_{c}^{{}^{\prime}+}}$
3.25 | -5.63 | -4.60 | 5.35 | 9.48 | 5.47 | 13.41
$g_{\Xi_{c}^{{}^{\prime}+}D^{*-}_{s}\Xi^{0}}$ | $g_{D^{*+}\Sigma^{0}\Xi_{c}^{{}^{\prime}+}}$ | $g_{D^{*+}\Lambda^{0}\Xi_{c}^{{}^{\prime}+}}$ | $g_{D^{*0}\Sigma^{+}\Xi_{c}^{{}^{\prime}+}}$ | $g_{\Xi^{0}D_{s}^{-}\Xi_{c}^{{}^{\prime}+}}$ | $g_{D^{*0}D^{-}_{s}K^{-}}$ | $g_{D^{0}D^{*-}_{s}K^{-}}$
-5.63 | 3.98 | -2.30 | 5.63 | -13.40 | 12.00 | 12.00

In addition to the Lagrangian shown in Eq.2, the effective Lagrangians of
relevant interaction vertices are also needed Yang:2021pio

$\displaystyle\mathcal{L}_{PPV}$
$\displaystyle=\frac{iG}{2\sqrt{2}}\langle\partial^{\mu}P(PV_{\mu}-V_{\mu}P)\rangle,$
(7) $\displaystyle\mathcal{L}_{VVP}$
$\displaystyle=\frac{G^{\prime}}{\sqrt{2}}\epsilon^{\mu\nu\alpha\beta}\langle\partial_{\mu}V_{\nu}\partial_{\alpha}V_{\beta}P\rangle,$
(8) $\displaystyle\mathcal{L}_{VVV}$
$\displaystyle=\frac{iG}{2\sqrt{2}}\langle\partial^{\mu}V^{\nu}(V_{\mu}V_{\nu}-V_{\nu}V_{\mu})\rangle,$
(9)

where the coupling constants $G=12.0$ and $G^{{}^{\prime}}=55.51$ Yang:2021pio
.

Since the $\Omega^{*}_{c}(3185)$ and $\Omega^{*}_{c}(3227)$ states are
considered as $S$-wave bound states of $D^{(*)}\Xi$, the couplings of
$\Omega^{*}_{c}(3185)$ and $\Omega^{*}_{c}(3227)$ to their components are
written as

$\displaystyle{\cal{L}}^{1/2^{-}}_{\Omega^{*}_{c}(3185)}$
$\displaystyle=\sum_{j=\Xi^{-}D^{+},\Xi^{0}D^{0}}C_{j}g^{1/2^{-}}_{\Omega_{c}(3185)\Xi
D}\Omega_{c}(3185)(x)$ $\displaystyle\times\int
dy\Phi(y^{2})\Xi(x+\omega_{D}y)D(x-\omega_{\Xi}y),$
$\displaystyle{\cal{L}}^{1/2^{-}}_{\Omega^{*}_{c}(3327)}=$
$\displaystyle\sum_{j=\Xi^{-}D^{*+},\Xi^{0}D^{*0}}C_{j}g^{1/2^{-}}_{\Omega^{*}_{c}(3327)\Xi
D^{*}}\Omega^{*}_{c}(3327)(x)\gamma^{\mu}\gamma^{5}$ $\displaystyle\times\int
dy\Phi(y^{2})\Xi(x+\omega_{D^{*}}y)D^{*}_{\mu}(x-\omega_{\Xi}y),$
$\displaystyle{\cal{L}}^{3/2^{-}}_{\Omega^{*}_{c}(3327)}=$
$\displaystyle-i\sum_{j=\Xi^{-}D^{*+},\Xi^{0}D^{*0}}C_{j}g^{3/2^{-}}_{\Omega_{c}(3327)\Xi
D^{*}}\Omega_{c}(3327)^{\mu}(x)$ $\displaystyle\int
dy\Phi(y^{2})\Xi(x+\omega_{D^{*}}y)D^{*}_{\mu}(x-\omega_{\Xi}y),$ (10)

where $\omega_{\Xi}=m_{\Xi}/(m_{\Xi}+m_{D^{(*)}})$ and
$\omega_{D^{(*)}}=m_{D^{(*)}}/(m_{\Xi}+m_{D^{(*)}})$ with $m_{\Xi}$ and
$m_{D^{(*)}}$ are the masses of the $\Xi$ and $D^{(*)}$, respectively.
$C_{j}=1/\sqrt{2}$ is the isospin coefficient, which is calculated from the
following isospin assignments for the $\Xi$ and $D^{(*)}$

$\begin{pmatrix}\Xi^{-}\\\
\Xi^{0}\end{pmatrix}\sim\begin{pmatrix}-|\frac{1}{2},-\frac{1}{2}\rangle\\\
|\frac{1}{2},+\frac{1}{2}\rangle\end{pmatrix};~{}~{}~{}\begin{pmatrix}D^{(*)+}\\\
D^{(*)0}\end{pmatrix}\sim\begin{pmatrix}|\frac{1}{2},+\frac{1}{2}\rangle\\\
|\frac{1}{2},-\frac{1}{2}\rangle\end{pmatrix}.$

In the above equation, $\Phi(y^{2})$ is the effective correlation function and
serves the following two roles: 1) it shows the distribution of the components
in the hadronic molecule, 2) it has the same role with the form factor that
can avoid the Feynman diagram’s ultraviolet divergence. Usually, the
correlation function can vanish quickly in the ultraviolet region. Here we
choose the Fourier transformation of the correlation function to have a
Gaussian form

$\displaystyle\Phi(-p_{E}^{2})\doteq{}\exp(-p_{E}^{2}/\Lambda^{2}),$ (11)

where $p_{E}$ is the Euclidean Jacobi momentum and the parameter $\Lambda$ is
taken as a parameter, which will be discussed later. The coupling constants of
$g^{1/2^{-}}_{\Omega^{*}_{c}(3185)\Xi D}$,
$g^{1/2^{-}}_{\Omega^{*}_{c}(3327)\Xi D^{*}}$,
$g^{3/2^{-}}_{\Omega^{*}_{c}(3327)\Xi{}D^{*}}$ appearing in Eq. (10) are given
in Eq. (12)

$\displaystyle\Sigma^{1/2^{-}}_{\Omega^{*}_{c}(3185)}(k_{0})=$
$\displaystyle(g^{1/2^{-}}_{\Omega_{c}\Xi
D})^{2}\int_{0}^{\infty}d\alpha\int_{0}^{\infty}d\beta\sum_{j}C_{j}{\cal{Y}}(\omega_{D},m_{D})$
$\displaystyle\times{\cal{Z}}(\omega_{D},m_{D}),$
$\displaystyle\Sigma^{1/2^{-}}_{\Omega^{*}_{c}(3327)}(k_{0})=$
$\displaystyle(g^{1/2^{-}}_{\Omega_{c}\Xi
D^{*}})^{2}\int_{0}^{\infty}d\alpha\int_{0}^{\infty}d\beta\sum_{j}C_{j}{\cal{Y}}(\omega_{D^{*}},m_{D^{*}})$
$\displaystyle\times{}[2{\cal{Z}}(\omega_{D^{*}},m_{D^{*}})+\frac{k^{3}_{0}(-4\omega_{D^{*}}-2\beta)^{3}}{8m^{2}_{D^{*}}z^{3}}$
$\displaystyle-\frac{3k_{0}\Lambda^{2}(-4\omega_{D^{*}}-2\beta)}{2m^{2}_{D^{*}}z^{2}}+\frac{k_{0}^{3}(-4\omega_{D^{*}}-2\beta)}{4m^{2}_{D^{*}}z^{2}}$
$\displaystyle-\frac{k^{2}_{0}(-4\omega_{D^{*}}-2\beta)^{2}m_{\Xi}}{4m^{2}_{D^{*}}z^{2}}+\frac{k_{0}\Lambda^{2}}{m^{2}_{D^{*}}z}+\frac{2\Lambda^{2}m_{\Xi}}{m^{2}_{D^{*}}z}],$
$\displaystyle\Sigma^{T3/2^{-}}_{\Omega^{*}_{c}(3327)}(k_{0})=$ $\displaystyle
g_{\Omega_{c}\Xi
D^{*}}^{2}\int_{0}^{\infty}d\alpha\int_{0}^{\infty}d\beta\sum_{j}C_{j}{\cal{Y}}(\omega_{D^{*}},m_{D^{*}})$
$\displaystyle\times[{\cal{Z}}(\omega_{D^{*}},m_{D^{*}})+\frac{k_{0}\Lambda^{2}(-4\omega_{D^{*}}-2\beta)}{4m^{2}_{D^{*}}z^{2}}$
$\displaystyle+\frac{k_{0}\Lambda^{2}}{2m^{2}_{D^{*}}z}+\frac{\Lambda^{2}m_{\Xi}}{2m^{2}_{D^{*}}z}],$
(12)

with

$\displaystyle{\cal{Y}}(\omega_{D^{(*)}},m_{D^{(*)}})$
$\displaystyle=\frac{1}{16\pi^{2}z^{2}}\exp\\{-\frac{1}{\Lambda^{2}}[-2k_{0}^{2}\omega_{D}^{2}+\alpha
m_{D}^{2}$
$\displaystyle+\beta(-k_{0}^{2}+m^{2}_{\Xi})+\frac{(-4\omega_{D}-2\beta)^{2}k_{0}^{2}}{4z}]\\}$
$\displaystyle{\cal{Z}}(\omega_{D^{(*)}},m_{D^{(*)}})$
$\displaystyle=k_{0}+m_{\Xi}+\frac{k_{0}(-4\omega_{D}-2\beta)}{2z},$ (13)

where $Z=2+\alpha+\beta$ and $k_{0}^{2}=m^{2}_{\Omega^{*}_{c}}$ with $k_{0}$,
$m_{\Omega^{*}_{c}}$ are the four momenta and the mass of the newly observed
$\Omega^{*}_{c}$, respectively. If the $\Omega_{c}^{*}$ is a $D^{(*)}\Xi$
molecular states with $J^{P}=1/2^{-}$, the $\Sigma^{1/2^{-}}$ is the self-
energy operator of the hadronic molecule $\Omega_{c}^{*}$. However,
$\Sigma^{T3/2^{-}}$ is the transverse part of the self-energy operator
$\Sigma^{\mu\nu}_{\Omega_{c}^{*}}$ when we assuming $\Omega_{c}^{*}$ as
$D^{(*)}\Xi$ molecular states with $J^{P}=3/2^{-}$. Once the self-energy
operator and its transverse part are obtained, the coupling constants of the
hadronic molecule $\Omega_{c}^{*}$ to its constituents $D^{(*)}\Xi$ can be
determined using the compositeness condition based on the work in Ref.
Weinberg:1962hj . This condition requires that the renormalization constant of
the hadronic molecular wave function is equal to zero

$\displaystyle
1-\frac{d\Sigma_{\Omega^{*}_{c}(3185/3327)}}{dk_{0}}=0,~{}~{}~{}~{}~{}J=\frac{1}{2}$
$\displaystyle
1-\frac{d\Sigma^{T}_{\Omega^{*}_{c}(3327)}}{dk_{0}}=0.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}J=\frac{3}{2}$
(14)

With the above prepared, we can obtain the general expressions of the
amplitudes corresponding to the Feynman diagrams Fig. 1

$\displaystyle\mathcal{M}_{a}^{1/2^{-}}$
$\displaystyle=\mu(p_{2})[\frac{1}{\sqrt{2}}g_{\Omega
11}g_{\Xi^{(^{\prime})+}_{c}D^{+}\Sigma^{0}}g_{\Xi^{-}K^{-}\Sigma^{0}}\int\frac{d^{4}k_{1}}{(2\pi)^{4}}$
$\displaystyle\times\Phi[(p\omega_{D^{+}}-q\omega_{\Xi^{-}})^{2}]\gamma^{5}\frac{i(k\\!\\!\\!/_{1}+m_{\Sigma^{0}})}{k_{1}^{2}-m_{\Sigma^{0}}^{2}}\gamma^{5}$
$\displaystyle\times\frac{i(p\\!\\!\\!/+m_{\Xi^{-}})}{p^{2}-m_{\Xi^{-}}^{2}}\frac{i}{q^{2}-m_{D^{+}}^{2}}+\frac{1}{\sqrt{2}}g_{\Omega
11}g_{\Xi^{(^{\prime})+}_{c}D^{+}\Lambda^{0}}$
$\displaystyle\times{}g_{\Xi^{-}K^{-}\Lambda^{0}}\int\frac{d^{4}k_{1}}{(2\pi)^{4}}\Phi[(p\omega_{D^{+}}-q\omega_{\Xi^{-}})^{2}]\gamma^{5}$
$\displaystyle\times\frac{i(k\\!\\!\\!/_{1}+m_{\Lambda^{0}})}{k_{1}^{2}-m_{\Lambda^{0}}^{2}}\gamma^{5}\frac{i(p\\!\\!\\!/+m_{\Xi^{-}})}{p^{2}-m_{\Xi^{-}}^{2}}\frac{i}{q^{2}-m_{D^{+}}^{2}}]\mu(k_{0}),$
(15) $\displaystyle\mathcal{M}_{b}^{1/2^{-}}$
$\displaystyle=\frac{1}{\sqrt{2}}g_{\Omega
12}g_{D^{0}\Xi_{c}^{(^{\prime})+}\Sigma^{+}}g_{\Xi^{0}K^{-}\Sigma^{+}}\int\frac{d^{4}k_{1}}{(2\pi)^{4}}$
$\displaystyle\times\phi[(p\omega_{D^{0}}-q\omega_{\Xi^{0}})^{2}]\mu(p_{2})\gamma^{5}\frac{i(k\\!\\!\\!/_{1}+m_{\Sigma^{+}})}{k_{1}^{2}-m_{\Sigma^{+}}^{2}}\gamma^{5}$
$\displaystyle\times\frac{i(p\\!\\!\\!/+m_{\Xi^{0}})}{p^{2}-m_{\Xi^{0}}^{2}}\mu(k_{0})\frac{i}{q^{2}-m_{D^{0}}^{2}},$
(16) $\displaystyle\mathcal{M}_{c}^{1/2^{-}}$
$\displaystyle=i\frac{1}{2}g_{\Omega
12}g_{D^{0}D^{*-}_{s}K^{-}}g_{\Xi^{0}\Xi_{c}^{(^{\prime})+}D^{*-}_{s}}\int\frac{d^{4}k_{1}}{(2\pi)^{4}}$
$\displaystyle\times\Phi[(p\omega_{D^{0}}-q\omega_{\Xi^{0}})^{2}]\mu({p_{2}})\gamma^{\nu}\frac{i(-g^{\mu\nu}+\frac{k_{1}^{\mu}k_{1}^{\nu}}{m_{D^{*-}_{s}}^{2}})}{k^{2}_{1}-m^{2}_{D^{*-}_{s}}}$
$\displaystyle\times(iq^{\mu}-ip_{1}^{\mu})\frac{i}{q^{2}-m_{D^{0}}^{2}}\mu(k_{0})\frac{i(p\\!\\!\\!/+m_{\Xi^{0}})}{p^{2}-m^{2}_{\Xi^{0}}},$
(17) $\displaystyle\mathcal{M}_{d}^{1/2^{-}}$
$\displaystyle=\mu(p_{2})[\frac{1}{\sqrt{2}}g_{\Omega
21}g_{\Xi^{(^{\prime})+}_{c}D^{*+}\Sigma^{0}}g_{\Xi^{-}K^{-}\Sigma^{0}}\int\frac{d^{4}k_{1}}{(2\pi)^{4}}$
$\displaystyle\times\Phi[(p\omega_{D^{*+}}-q\omega_{\Xi^{-}})^{2}]\gamma^{\mu}\frac{i(k\\!\\!\\!/_{1}+m_{\Sigma^{0}})}{k_{1}^{2}-m_{\Sigma^{0}}^{2}}\gamma^{5}$
$\displaystyle\times\frac{i(p\\!\\!\\!/+m_{\Xi^{-}})}{p^{2}-m_{\Xi^{-}}^{2}}\gamma^{\nu}\gamma^{5}\frac{i(-g^{\mu\nu}+\frac{q^{\mu}q^{\nu}}{m^{2}_{D^{*+}}})}{q^{2}-m_{D^{*+}}^{2}}$
$\displaystyle+\frac{1}{\sqrt{2}}g_{\Omega
21}g_{\Xi^{(^{\prime})+}_{c}D^{*+}\Lambda^{0}}g_{\Xi^{-}K^{-}\Lambda^{0}}\int\frac{d^{4}k_{1}}{(2\pi)^{4}}$
$\displaystyle\times\Phi[(p\omega_{D^{*+}}-q\omega_{\Xi^{-}})^{2}]\gamma^{\alpha}\frac{i(k\\!\\!\\!/_{1}+m_{\Lambda^{0}})}{k_{1}^{2}-m_{\Lambda^{0}}^{2}}\gamma^{5}$
$\displaystyle\times\frac{i(p\\!\\!\\!/+m_{\Xi^{-}})}{p^{2}-m_{\Xi^{-}}^{2}}\gamma^{\beta}\gamma^{5}\frac{i(-g^{\alpha\beta}+\frac{q^{\alpha}q^{\beta}}{m^{2}_{D^{*+}}})}{q^{2}-m_{D^{*+}}^{2}}]\mu(k_{0}),$
(18) $\displaystyle\mathcal{M}_{d}^{3/2^{-}}$
$\displaystyle=\mu(p_{2})[-i\frac{1}{\sqrt{2}}g_{\Omega
31}g_{\Xi^{(^{\prime})+}_{c}D^{*+}\Sigma^{0}}g_{\Xi^{-}K^{-}\Sigma^{0}}\int\frac{d^{4}k_{1}}{(2\pi)^{4}}$
$\displaystyle\times\Phi[(p\omega_{D^{*+}}-q\omega_{\Xi^{-}})^{2}]\gamma^{\mu}\frac{i({k\\!\\!\\!/_{1}}+m_{\Sigma^{0}})}{k_{1}^{2}-m_{\Sigma^{0}}^{2}}\gamma^{5}$
$\displaystyle\times\frac{i({p\\!\\!\\!/}+m_{\Xi^{-}})}{p^{2}-m_{\Xi^{-}}^{2}}g^{\nu\lambda}\frac{i(-g^{\mu\nu}+\frac{q^{\mu}q^{\nu}}{m^{2}_{D^{*+}}})}{q^{2}-m_{D^{*+}}^{2}}$
$\displaystyle-i\frac{1}{\sqrt{2}}g_{\Omega
31}g_{\Xi^{(^{\prime})+}_{c}D^{*+}\Lambda^{0}}g_{\Xi^{-}K^{-}\Lambda^{0}}\int\frac{d^{4}k_{1}}{(2\pi)^{4}}$
$\displaystyle\times\Phi[(p\omega_{D^{*+}}-q\omega_{\Xi^{-}})^{2}]\gamma^{\alpha}\frac{i({k\\!\\!\\!/_{1}}+m_{\Lambda^{0}})}{k_{1}^{2}-m_{\Lambda^{0}}^{2}}\gamma^{5}$
$\displaystyle\times\frac{i({p\\!\\!\\!/}+m_{\Xi^{-}})}{p^{2}-m_{\Xi^{-}}^{2}}g^{\beta\lambda}\frac{i(-g^{\alpha\beta}+\frac{q^{\alpha}q^{\beta}}{m^{2}_{D^{*+}}})}{q^{2}-m_{D^{*+}}^{2}}]\mu_{\lambda}(k_{0}),$
(19) $\displaystyle\mathcal{M}_{e}^{1/2^{-}}$
$\displaystyle=\frac{1}{\sqrt{2}}g_{\Omega
22}g_{\Xi^{(^{\prime})+}_{c}D^{*0}\Sigma^{0}}g_{\Xi^{0}K^{-}\Sigma^{0}}\int\frac{d^{4}k_{1}}{(2\pi)^{4}}$
$\displaystyle\times\Phi[(p\omega_{D^{*0}}-q\omega_{\Xi^{0}})^{2}]\mu(p_{2})\gamma^{\mu}\frac{i({k\\!\\!\\!/_{1}}+m_{\Sigma^{+}})}{k_{1}^{2}-m_{\Sigma^{+}}^{2}}\gamma^{5}$
$\displaystyle\times\frac{i({p\\!\\!\\!/}+m_{\Xi^{0}})}{p^{2}-m_{\Xi^{0}}^{2}}\mu(k_{0})\gamma^{\nu}\gamma^{5}\frac{i(-g^{\mu\nu}+\frac{q^{\mu}q^{\nu}}{m^{2}_{D^{*0}}})}{q^{2}-m_{D^{*0}}^{2}},$
(20) $\displaystyle\mathcal{M}_{e}^{3/2^{-}}$
$\displaystyle=-\frac{i}{\sqrt{2}}g_{\Omega
22}g_{\Xi^{(^{\prime})+}_{c}D^{*0}\Sigma^{0}}g_{\Xi^{0}K^{-}\Sigma^{0}}\int\frac{d^{4}k_{1}}{(2\pi)^{4}}$
$\displaystyle\times\Phi[(p\omega_{D^{*0}}-q\omega_{\Xi^{0}})^{2}]\mu(p_{2})\gamma^{\mu}\frac{i({k\\!\\!\\!/_{1}}+m_{\Sigma^{+}})}{k_{1}^{2}-m_{\Sigma^{+}}^{2}}\gamma^{5}$
$\displaystyle\times\frac{i({p\\!\\!\\!/}+m_{\Xi^{0}})}{p^{2}-m_{\Xi^{0}}^{2}}\mu^{\nu}(k_{0})\frac{i(-g^{\mu\nu}+\frac{q^{\mu}q^{\nu}}{m^{2}_{D^{*0}}})}{q^{2}-m_{D^{*0}}^{2}},$
(21) $\displaystyle\mathcal{M}_{f}^{1/2^{-}}$
$\displaystyle=\frac{i}{2}g_{\Omega
31}g_{D^{*0}D^{-}_{s}K^{-}}g_{\Xi^{0}D_{s}^{-}\Xi_{c}^{(^{\prime})+}}\int\frac{d^{4}k_{1}}{(2\pi)^{4}}$
$\displaystyle\times\Phi[(p\omega_{D^{*0}}-q\omega_{\Xi^{0}})^{2}]\mu(p_{2})\gamma^{5}\frac{1}{k_{1}^{2}-m_{D_{s}^{-}}^{2}}$
$\displaystyle\times(p_{1}^{\mu}-k_{1}^{\mu})\frac{(-g^{\mu\nu}+\frac{q^{\mu}q^{\nu}}{m^{2}_{D^{*0}}})}{q^{2}-m_{D^{*0}}^{2}}\mu(k_{0})\gamma^{\nu}$
$\displaystyle\times\gamma^{5}\frac{({p\\!\\!\\!/}+m_{\Xi^{0}})}{p^{2}-m_{\Xi^{0}}^{2}},$
(22) $\displaystyle\mathcal{M}_{f}^{3/2^{-}}$
$\displaystyle=\frac{1}{2}g_{\Omega
32}g_{D^{*0}D^{-}_{s}K^{-}}g_{\Xi^{0}D_{s}^{-}\Xi_{c}^{(^{\prime})+}}\int\frac{d^{4}k_{1}}{(2\pi)^{4}}$
$\displaystyle\Phi[(p\omega_{D^{*0}}-q\omega_{\Xi^{0}})^{2}]\mu(p_{2})\gamma^{5}\frac{1}{k_{1}^{2}-m_{D_{s}^{-}}^{2}}$
$\displaystyle\times(p_{1}^{\mu}-k_{1}^{\mu})\frac{(-g^{\mu\nu}+\frac{q^{\mu}q^{\nu}}{m^{2}_{D^{*0}}})}{q^{2}-m_{D^{*0}}^{2}}\mu^{\nu}(k_{0})\frac{({p\\!\\!\\!/}+m_{\Xi^{0}})}{p^{2}-m_{\Xi^{0}}^{2}},$
(23)

where the $p_{1}$, $p_{2}$, $p$, $q$, and $k_{1}$ are the four momenta of the
$K$, $\Xi_{c}^{(^{\prime})}$, $\Xi$, $D^{(*)}$, and $t$-channel exchanged
particles, respectively.

Once the amplitudes are determined, the corresponding decay widths can be
obtained, which read,

$\displaystyle\Gamma(\Xi_{c}^{*}\to)=\frac{1}{2J+1}\frac{1}{8\pi}\frac{|\mathbf{p}_{1}|}{m_{\Omega_{c}^{*2}}}\overline{{\cal{M}}^{2}},$
(24)

where $J$ is the total angular momentum of the initial state $\Omega_{c}^{*}$,
the overline indicates the sum over the polarization vectors of final hadrons.
Here $\mathbf{p}_{1}$ is the 3-momenta of the decay products in the center of
mass frame.

##  III RESULTS AND DISCUSSIONS

In this work, we study the strong decay pattern of $S$-wave $D^{(*)}\Xi$
molecular states within the effective Lagrangians approach, and find the
relation between the $D^{(*)}\Xi$ molecular state and the newly observed
$\Omega^{*}_{c}(3185)$ and $\Omega^{*}_{c}(3327)$ states. If the estimated
strong decay width matches well with the LHCb observation, we can judge the
molecule explanations for the structure of $\Omega^{*}_{c}(3185)$ and
$\Omega^{*}_{c}(3327)$. To make a reliable prediction for the strong decay
width, the parameter $\Lambda$ that affects our accurate calculation must be
clarified. Since the $\Lambda$ could not be well determined from first
principles, it has to be determined by fitting to the experimental data. We
especially hope the value of the $\Lambda$ can be determined within same
theoretical framework adopted in this work.

Fortunately, within the same theoretical framework adopted in current work in
Ref Dong:2017rmg , the parameter $\Lambda$ was constrained as
$\Lambda=0.91-1.00$ GeV by comparing the sum of the partial decay modes of the
$\eta(2225)$ and $\phi(2170)$ with the total width. We also note that many
exotic states can be well considered as molecules with $\Lambda=0.90-1.10$
GeV, and we refer the reader to review the Refs. Yang:2021pio ; Dong:2008gb ;
Huang:2020taj  and their reference. Therefore, we take $\Lambda$ in the range
of $\Lambda=0.90-1.10$ GeV to study whether the $\Omega^{*}_{c}(3185)$ and
$\Omega^{*}_{c}(3327)$ can be interpreted as molecule composed of $D^{(*)}\Xi$
molecular components.

Figure 2: Coupling constants for the newly observed $\Omega_{c}^{*}$ with
different spin-parity as a function of the parameter $\Lambda$.

Considering the $\Lambda$ values adopted in this work, we will first discuss
the results of the calculative coupling constants. Substituting Eq. (12) into
Eq. (14), the dependence of the coupling constants on the parameter $\Lambda$
is solved. By performing the integration of parameters $\alpha$ and $\beta$
from $0$ to infinite, the numerical results are presented in Fig. 2, which
shows the variation of the coupling constants with respect to the $\Lambda$ in
the range of 0.90 to 1.10 GeV. From the results, we can find that the value of
the coupling constants decreases with the increase of $\Lambda$, and they are
not very sensitive to the $\Lambda$. It is important to note that the coupling
constant $g_{\Omega^{*}_{c}(3327)\Xi{}D^{*}}$ decreases sharply at a small
$\Lambda$ value. Due to the presence of ultraviolet divergences in the
calculation, the average value of the molecular mass was used. Here, the
divergences originate from the threshold of $D^{0}\Xi^{0}$ and $D^{*0}\Xi^{0}$
channels being slightly below the masses of the $\Omega^{*}_{c}(3185)$ and
$\Omega^{*}_{c}(3327)$, respectively.

After obtaining the coupling constants, we compute the partial decay widths
for the transitions $\Omega^{*}_{c}(3185)\to{}\bar{K}\Xi^{(^{\prime})}_{c}$
and $\Omega^{*}_{c}(3327)\to\bar{K}\Xi_{c}^{(^{\prime})}$, and plot the
results in Fig. 3, which vary with the parameter $\Lambda=0.9-1.1$ GeV. Note
that the partial decay widths in the channels
$\Omega^{*}_{c}(3185)\to{}K^{-}\Xi^{(^{\prime})+}_{c}$
$\Omega^{*}_{c}(3185)\to{}K^{-}\Xi^{(^{\prime})+}_{c}$ and
$\Omega^{*}_{c}(3327)\to{}K^{-}\Xi_{c}^{(^{\prime})+}$ are only estimated
here, and the other channels
$\Omega^{*}_{c}(3185)\to{}\bar{K}^{0}\Xi^{(^{\prime})0}_{c}$ and
$\Omega^{*}_{c}(3327)\to{}\bar{K}^{0}\Xi_{c}^{(^{\prime})0}$ can be obtained
by isospin symmetry. The sum of these partial decay widths gives the total
decay width of the $\Omega^{*}_{c}(3185)$ or total decay width of the
$\Omega^{*}_{c}(3185)$ or $\Omega^{*}_{c}(3327)$, which are also shown in Fig.
3.

From the results of Fig. 3(a), we can find that the total decay width for the
transitions $\Omega^{*}_{c}(3185)\to{}\bar{K}\Xi^{(^{\prime})}_{c}$ increases
with the increase of $\Lambda$, while it decreases when $\Lambda$ varies from
0.940 to 0.945 GeV. Comparing with the total decay width for the
$\Omega^{*}_{c}(3185)\to{K}\Xi_{c}^{(^{\prime})}$ reactions, we found the line
shape for the total decay width of the
$\Omega^{*}_{c}(3327)\to{}\bar{K}\Xi_{c}^{{}^{\prime}}$ reactions are very
different. To see how different, we take $\Omega^{*}_{c}(3327)$ at
$J^{P}=1/2^{-}$ case as an example. The obtained total decay width for the
$\Omega^{*}_{c}(3327)\to{}\bar{K}\Xi_{c}^{{}^{\prime}}$ reaction decreases,
then it begin increases at $\Lambda=0.935$ GeV.

Figure 3: Partial decay widths of the $\Omega_{c}^{*}\to{}\bar{K}\Xi_{c}$ (red
dash line), $\Omega_{c}^{*}\to{}\bar{K}\Xi_{c}^{{}^{\prime}}$ (magenta dash
dot line), and the total decay width with different $\Omega_{c}^{*}$ states
depending on the parameter $\Lambda$. The oycn error bands correspond to the
LHCb observed LHCb:2023rtu .

Now, let me give a clear discussion about whether the $\Omega^{*}_{c}(3185)$
and $\Omega^{*}_{c}(3327)$ can be interpreted as molecules composed of
$D^{(*)}\Xi$ molecular components. From Fig. 3, we observe that the total
decay width for the
$\Omega^{J^{P}=1/2^{-}}_{c}(3327)\to\bar{K}\Xi_{c}^{(^{\prime})}$ and
$\Omega^{J^{P}=3/2^{-}}_{c}(3327)\to\bar{K}\Xi_{c}^{(^{\prime})}$ is estimated
to be about 104.39-167.54 MeV and 15.55-21.04 MeV, respectively, where the
theoretical value 15.55-21.04 MeV is close to the experimental data $20\pm
5^{+13}_{-1}$. This suggests if the spin-parity of the $\Omega^{*}_{c}(3327)$
is $J^{P}=3/2^{-}$, the assignment as an $S$-wave pure $D^{*}\Xi$ molecular
state for the $\Omega^{*}_{c}(3327)$ is supported. However, if the
$\Omega^{*}_{c}(3327)$ has a spin-parity of $1/2^{-}$, the predicted total
decay width is much bigger than the experimental total width, which disfavors
such a spin-parity assignment for the $\Omega^{*}_{c}(3327)$ in the$D^{*}\Xi$
molecular picture. Moreover, the spin-parity $J^{P}=1/2^{-}$ $D\Xi$ molecular
assumptions for the $\Omega^{*}_{c}(3185)$ cannot be conclusively determined.
This is because the obtained total decay width for the $\Omega^{*}_{c}(3185)$
with $J^{P}=1/2^{-}$ $D\Xi$ assignment is comparable with that of the
experimental total width in the range of $\Lambda=0.900-0.915$ and
$\Lambda=0.945-0.990$ GeV. It may be a meson-baryon molecule contain a big
$D\Xi$ component.

Indeed, there exist several coupled states composed of $D\Xi$ and $D^{*}\Xi$
components Zhu:2022fyb . However, the authors in Ref. Zhu:2022fyb  claim that
the interaction between $D^{(*)}$ meson and $\Xi$ baryon is not strong enough
to form a pure bound state with quantum numbers considered in the current
work. A possible reason for this is that the potential kernels they obtained
only consider light meson exchanges and do not include the contact term that
describes the short distance interaction between the $\Xi$ and charmed meson.

Fig. 3 also tells us that the decay width of $\Omega^{*}_{c}(3185)$ into
$\bar{K}\Xi_{c}$ is about 64.60-82.98 MeV, which almost fully accounts for the
total width of $\Omega_{c}^{*}(3185)$. In other words, the transition from
$\Omega^{*}_{c}(3185)$ to $\bar{K}\Xi_{c}$, which is the experimental
observation channel, provides a dominant contribution to the total decay
width. However, the decay width of $\Omega_{c}^{*}(3185)\to\bar{K}\Xi_{c}$ is
up to several MeV, which accounts for 9.86%-10.70% of its total width. That
means the transition $\Omega_{c}^{*}(3185)\to\bar{K}\Xi^{{}^{\prime}}_{c}$
gives a minor contribution. A possible explanation for this may be that the
phase space for the transition
$\Omega_{c}^{*}(3185)\to\bar{K}\Xi^{{}^{\prime}}_{c}$ is smaller than that of
the $\Omega_{c}^{*}(3185)\to\bar{K}\Xi_{c}$ reaction. The same conclusion can
also be drawn for the $\Omega_{c}(3327)\to\bar{K}\Xi_{c}^{(^{\prime})}$
reactions.

It should be noted that the authors in Ref. Yu:2023bxn  only compute the decay
widths of $\Omega_{c}^{*}(3185)$ and $\Omega_{c}^{*}(33327)$ into
$\bar{K}\Xi_{c}$ by assigning $\Omega_{c}^{*}(3185)$ and
$\Omega_{c}^{*}(3327)$ as state $2S(3/2^{+})$ and $1D(3/2^{+})$, respectively.
The predicted decay widths can reach 78.16 MeV and 25.60 MeV, respectively,
which are both larger than the experimental cental value. This suggest that if
the newly observed $\Omega_{c}^{*}$ is conventional three quark state, the
transition $\Omega_{c}^{*}\to\bar{K}\Xi^{{}^{\prime}}_{c}$ is very small and
can be ignored. Thus, we propose the experimental search for $\Omega_{c}^{*}$
in the $\Omega_{c}^{*}\to\bar{K}\Xi^{{}^{\prime}}_{c}$ reaction that offers a
nice channel to test the molecular nature of the $\Omega_{c}^{*}(3185)$ and
$\Omega_{c}^{*}(33327)$.

##  IV Summary

Inspired by the newly observed baryons $\Omega_{c}^{*}(3185)$ and
$\Omega_{c}^{*}(33327)$, the strong decay widths of the
$\Omega_{c}^{*}\to{}\bar{K}\Xi_{c}$ and
$\Omega_{c}^{*}\to{}\bar{K}\Xi_{c}^{{}^{\prime}}$ was studied in an effective
Lagrangian approach. Our theoretical approach is based on the assuming that
the newly observed $\Omega^{*}_{c}(3185)$ and $\Omega^{*}_{c}(3327)$ can be
explained as $S$-wave $D\Xi$ and $D^{*}\Xi$ molecular state, respectively.
With the $D^{(*)}\Xi$ assignment, the partial decay widths of the
$\Omega_{c}^{*}(3185)$ and $\Omega_{c}^{*}(33327)$ into the
$\bar{K}\Xi_{c}^{(^{\prime})}$ final states through hadronic loop are
calculated. The decay process is described by the $t$-channel $\Lambda$,
$\Sigma$ baryons and $D_{s}$, $D_{s}^{*}$ mesons exchanges, respectively.

The current results support the $\Omega^{*}_{c}(3327)$ with$J^{P}=3/2^{-}$ as
pure $D^{*}\Xi$ molecule while the $\Omega^{*}_{c}(3327)$ with $J^{P}=1/2^{-}$
can not be well reproduced in the molecular state picture. In addition, the
spin-parity $J^{P}=1/2^{-}$ $D\Xi$ molecular assumptions for the
$\Omega^{*}_{c}(3185)$ can’t be conclusively determined. It may be a meson-
baryon molecule with a big $D\Xi$ component. We also find that the transition
$\Omega^{*}_{c}(3185)\to\bar{K}\Xi_{c}$, which is the experimental observation
channel, provides a dominant contribution to the total decay width while the
transition $\Omega_{c}^{*}(3185)\to\bar{K}\Xi^{{}^{\prime}}_{c}$ gives a minor
contribution. However, the decay channel
$\Omega_{c}^{*}\to{}\bar{K}\Xi_{c}^{{}^{\prime}}$ can be well employed to test
the molecule interpretations of $\Omega^{*}_{c}(3185)$ and
$\Omega^{*}_{c}(3327)$ by comparing our results with this in Ref. Yu:2023bxn .

## Acknowledgments

Yin Huang acknowledges the YST Program of the APCTP and the support from the
National Natural Science Foundation of China under Grant No.12005177.

## References

  * (1) R. Aaij et al. [LHCb], Phys. Rev. Lett. 118, 182001 (2017).
  * (2) [LHCb], [arXiv:2302.04733 [hep-ex]].
  * (3) R. L. Workman [Particle Data Group], PTEP 2022, 083C01 (2022).
  * (4) H. C. Kim, M. V. Polyakov and M. Praszałowicz, Phys. Rev. D 96,014009 (2017).
  * (5) H. C. Kim, M. V. Polyakov, M. Praszalowicz and G. S. Yang, Phys. Rev. D 96, 094021 (2017) [erratum: Phys. Rev. D 97,039901 (2018)].
  * (6) G. Yang and J. Ping, Phys. Rev. D 97, 034023 (2018).
  * (7) C. S. An and H. Chen, Phys. Rev. D 96, 034012 (2017).
  * (8) V. R. Debastiani, J. M. Dias, W. H. Liang and E. Oset, Phys. Rev. D 97, 094035 (2018).
  * (9) G. Montaña, A. Feijoo and À. Ramos, Eur. Phys. J. A 54, 64 (2018).
  * (10) Y. Huang, C. j. Xiao, Q. F. Lü, R. Wang, J. He and L. Geng, Phys. Rev. D 97,094013 (2018).
  * (11) J. Nieves, R. Pavao and L. Tolos, Eur. Phys. J. C 78, 114 (2018).
  * (12) G. L. Yu, Y. Meng, Z. Y. Li, Z. G. Wang and L. Jie, [arXiv:2302.11758 [hep-ph]].
  * (13) Si-Qiang Luo, Xiang Liu, [arXiv:2303.04022 [hep-ph]].
  * (14) W. Liu, C. M. Ko and Z. W. Lin, Phys. Rev. C 65 (2002), 015203.
  * (15) F. Yang, Y. Huang and H. Q. Zhu, Sci. China Phys. Mech. Astron. 64 (2021), 121011.
  * (16) S. Weinberg, Phys. Rev. 130 (1963), 776-783.
  * (17) Y. Dong, A. Faessler, T. Gutsche, Q. Lü and V. E. Lyubovitskij, Phys. Rev. D 96 (2017) 074027.
  * (18) Y. b. Dong, A. Faessler, T. Gutsche and V. E. Lyubovitskij, Phys. Rev. D 77 (2008), 094013.
  * (19) Y. Huang and L. Geng, Eur. Phys. J. C 80 (2020) 837.
  * (20) J. T. Zhu, S. Y. Kong, L. Q. Song and J. He, Phys. Rev. D 105 (2022), 094036.