text:
\section{Introduction}
Black holes are one of the most exciting predictions of general relativity. There have been accumulated evidences to support their existence  in our universe. In particular, in recent years the successful detections of the gravitational waves by LIGO and Virgo \cite{LIGOScientific:2016aoc, LIGOScientific:2016sjg, LIGOScientific:2017vwq, lipunov2017master} and the black hole images photographed by the Event Horizon Tecescope (EHT) collaboration\cite{EventHorizonTelescope:2019dse, EventHorizonTelescope:2019jan, EventHorizonTelescope:2019ths, EventHorizonTelescope:2019pgp, EventHorizonTelescope:2019ggy, EventHorizonTelescope:2022xnr} have given us more confidence in the correctness of general relativity and the existence of black holes. General relativity and other theories of gravity admit various black holes which can be neutral, charged, magnetic or both charged and magnetic. Generally speaking, the astrophysical black holes are expected to be neutral due to the selective accretion of the ambient matter. However, in fact it is not impossible to have a weakly charged black hole in our real universe. One charging mechanism is that a rotating black hole immersed in a magnetic field can accrete ambient free charged particles to neutralize the magnetically induced electric field \cite{wald}.

In the complex environment around an astrophysical black hole, there always presents a plasma, part of which forms an accretion disk and generate magnetic fields. Up to now, many observations suggesting magnetic fields outside black holes include but are not limited to bright "flares", polarized broad H$\alpha$ lines and the polarization of the accretion disk radiation in the vicinity of active galactic nuclei  \cite{dovvciak2004polarization, Zajacek:2018vsj, Daly_2019,2010arXiv1002.4948P}. In particular, the recent polarized image of M87 black hole published by the EHT collaboration also reveals a magnetic field around the black hole \cite{EventHorizonTelescope:2021bee, EventHorizonTelescope:2021srq, narayan2021polarized}.

It is essential to study the effects of the magnetic field around a black hole, in order to read the information from polarized images of the black hole. Among various effects,  the influence of Lorentz force on the charged particles' innermost stable circular orbits (ISCOs) and the polarized image of synchrotron radiations in the presence of magnetic fields are important. The ISCOs of charged particles around a Schwarzschild and Kerr black hole immersed in a weak, axially symmetric magnetic field have been widely studied and many interesting results have been found \cite{frolov, Zahrani:2022fdd, Shiose:2014bqa, AlZahrani:2014dfi}. As for a strong magnetic field, the spacetimes can be modeled by the Schwarzschild and Kerr black holes embedded in the Melvin universe \cite{ernst, kerrmel}, and the corresponding ISCOs for massive particles have also been studied in the literatures \cite{Lim:2015oha, Zhu:2022amy, Li:2018wtz}. On the other hand, the polarized image of the synchrotron radiations originated by circularly orbiting charged particles acted upon by Lorentz force had also been investigated in the Schwarzschild and Kerr black holes with weak or strong magnetic fields \cite{stab1289, Lupsasca:2018tpp, EventHorizonTelescope:2021btj, Gelles:2021kti, Qin:2021xvx, Zhang:2021hit, hu2022polarization, Zhu:2022amy, Qin:2022kaf}.

In this work, we revisit the ISCOs of the charged particles around a Kerr black hole with a weak magnetic field, which was first found by Wald in \cite{wald}. Different from the existing researches, we focus on the case $\mathcal{B}=\frac{qB}{2m}\ll1$, where $q$, $m$ and $B$ are the charge, mass and strength of the magnetic field respectively, so that we are allowed to find an analytical expression for the radius of the ISCO which involves the energy $E$ and angular momentum $L$ of charged particles. As many astrophysical dusts or fluids in our universe do not have an extreme charge-mass ratio, the condition $\mathcal{B}\ll1$ can be satisfied ubiquitously  and our results will be very useful in many practical situations. In addition, we study the polarized image of charged particles in circular motions on the equatorial plane observed by an observer at infinity. Different from the usual treatment on the synchrotron radiations of charged particle around black hole, which relies on the study of radiation in the local inertial frame, we apply the covariant formulas of the electromagnetic radiations encoding the intensity and polarization direction of synchrotron radiations in curved spacetime, developed recently in\footnote{The earlier related discussion can be found in \cite{Quinn:1996am}.} \cite{hu2022polarization}. In particular we show explicitly the polarization directions and total flux of the synchrotron radiations emitted from these charged particles for various spins, observational angles and orbits. Our numerical studies on the polarized images are not limited to the case $\mathcal{B}\ll1$,  though we mainly give a careful study on the ISCOs in this approximation

The remaining parts of this paper are organized as follows. In Sec. \ref{sec2}, we study the circular orbits of the charged particles around a Kerr black hole with a Wald magenetic field. In Sec. \ref{sec3}, we review the synchrotron radiations in curved spacetime\cite{hu2022polarization},  and calculate the polarized images of the charged particles in circular motions around a Kerr black hole. The main conclusions are summarized in Sec. \ref{summary}. In this work, we have set the fundamental constants $c$, $G$ and the vacuum permittivity $\varepsilon_0$ to unity, and we will work in the signature convention $(-, +, +, +)$ for the spacetime metric.

\section{Circular orbits of a charged particle around a Kerr black hole with a Wald magnetic field}\label{sec2}
In this section, we would like to focus on circular orbits of a charged particle around a Kerr black hole with Wald magnetic field.

\subsection{Kerr black hole with a Wald magnetic field}
The spacetime of a neutral and rotating black hole in general relativity is described by the Kerr black hole metric, which is a stationary and axisymmetric solution of the Einstein equations and takes,
\setlength\arraycolsep{2pt} \begin{eqnarray}
\mathrm{d}s^2=-\left(1-\frac{2Mr}{\Sigma}\right)\mathrm{d}t^2+\frac{\Sigma}{\Delta}\mathrm{d}r^2+\Sigma\mathrm{d}\theta^2+\left(r^2+a^2+\frac{2Mra^2}{\Sigma}\mathrm{sin}^2\theta\right)\mathrm{d}\phi^2-\frac{4Mra}{\Sigma}\mathrm{sin}^2\theta\mathrm{d}t\mathrm{d}\phi\,,\nonumber\\
\end{eqnarray}
in the Boyer-Lindquist coordinates, where
\setlength\arraycolsep{2pt} \begin{eqnarray}
\Sigma=r^2+a^2\mathrm{cos}^2\theta,\quad \Delta = r^2-2Mr+a^2\,,
\end{eqnarray}
with $M$ and $a$ being the mass and spin of the Kerr black hole, respectively. And the angular momentum of the Kerr black hole is $J=Ma$. The horizons are located at $r_\pm=M\pm\sqrt{M^2-a^2}$ which are given by the equation $\Delta=0$. The outer one is the event horizon of the Kerr black hole, that is, $r_h=r_+$. Next, we would like to assume that there is a vertical and uniform magnetic field outside the Kerr black hole which is described by the Wald solution to the source-free Maxwell equations, that is, $\nabla_\mu F^{\mu\nu}=0$. The magnetic field is aligned with the rotation axis of the black hole and independent of the coordinates $t$ and $\phi$. Considering the two Killing vectors $\eta^a=\left(\frac{\partial}{\partial t}\right)^a$ and $\psi^a=\left(\frac{\partial}{\partial \phi}\right)^a$, Wald  proved that the field
\begin{equation}
A^a=\frac{1}{2}B(\psi^a+\frac{2J}{M}\eta^a)-\frac{Q}{2M}\eta^a \label{A}
\end{equation}
gives a unique field that is an asymptotically uniform magenetic field of strength $B_0$, with $Q$ being the charge of the Kerr black hole. Next, we inroduce the electrostatic injection energy, defined as the difference between the electric potential at the horizon and infinity, i.e.,
\setlength\arraycolsep{2pt} \begin{eqnarray}
\epsilon=e A_\mu\eta^\mu|_{\text{horizon}}-eA_\mu\eta^\mu|_\infty
\end{eqnarray}
which is a constant over the black hole\cite{wald}. For a pure Kerr spacetime, it seems that we should have $Q=0$. However, due to non-vanishing electrostatic injection energy $\epsilon$, an electric field is induced by the rotation of the black hole in the presence of the magnetic field and cause charged particles to be attracted by the black hole, which is a process called charged accretion. As a result, to reach a steady state the black hole has to gain a nonzero net charge to reach equilibrium, that is, $\epsilon=0$, which gives
\begin{equation}\label{ec}
Q=2BJ\,.
\end{equation}
In the following, we would like to mainly focus on this equilibrium situation. Note that in \cite{al2022charged}, they also paid attention to the nonequilibrium situations. It is worth mentioning that the Eq. (\ref{ec}) holds for an arbitrary stationary, axisymmetric black hole, that is, the condition is not limited to Kerr black holes. In principle, the charge and magnetic field would react upon the background spacetime. If we want to ignore the contribution of the magetic field to the background spacetime for simplicity, we need to consider the weak field approximation,
\setlength\arraycolsep{2pt} \begin{eqnarray}
BM\ll 1\,.
\end{eqnarray}
On the other hand, the characteristic length scale given by the charge of the black hole is comparable with the gravitational radius when
\setlength\arraycolsep{2pt} \begin{eqnarray}
\sqrt{\frac{Q_G^2G}{c^4}}=\frac{2GM}{c^2}\,,
\end{eqnarray}
in the International System of Units, so that the gravitational effect of the charge $Q$ on the background can be ignored if
\setlength\arraycolsep{2pt} \begin{eqnarray}
Q\ll Q_G\,.
\end{eqnarray}
In addition, considering $J\le M^2$, from Eq. (\ref{ec}) we have
\setlength\arraycolsep{2pt} \begin{eqnarray}
\frac{Q}{Q_G}=\frac{Q}{2M}=Ba\le BM\ll1\,,
\end{eqnarray}
which means we are allowed to omit the influence of the charge on the metric of the spacetime although there is a nonzero net charge, so that we can stick to the Kerr metric under the weak field approximation. In fact, in the Gauss units we find
\setlength\arraycolsep{2pt} \begin{eqnarray}\label{btg}
B_{\text{Gauss}}=\frac{c^4}{G^{3/2}M}(BM)\simeq2.36\times 10^{19}\frac{M_{\odot}}{M}(BM)\quad \text{Gauss}\,.
\end{eqnarray}
Here (BM) is a dimensionless number and can also be seen as the value of $B$ in our simplified unit after setting $c$, $G$, $\epsilon_0$ and $M$ to 1, and  $M_\odot$ is the mass of sun. For a supermassive black holes $M\sim 10^9 M_\odot$, the dimensionless $BM=0.01\ll 1$ corresponds to $B_{\text{Gauss}}\sim 10^8 \text{Gauss}$, which is already a very big value for astrophysically reasonable magnetic field around a astronomical  black hole. Thus, the weak field approximation $BM\ll 1$ can be imposed for most real astronomical black holes with magnetic fields and deserves careful study.

\subsection{Motions of charged particles on the equatorial plane}
In this subsection, we move to study the motions of charged particles on the equatorial plane under the weak field approximation employing the standard approach. In the Boyer-Lindquist coordinates, the generalized four momentum $P^\mu$ can be written as
\begin{equation}
P^\mu=p^\mu+qA^\mu\,,
\end{equation}
where $p^\mu=mu^\mu$ is the four momentum for the charged particle with $m$ being the mass, $A^\mu$ is the electric vector potential and $q$ is the charge of the particle. From \eqref{A} we have
\begin{equation}
A^\mu=(aB-\frac{Q}{2M},0,0,\frac{B}{2})\,,
\end{equation}
Note that the Lie derivatives of $A_\mu$ with respect to the Killing vectors $\eta^\mu$ and $\psi^\mu$ are zero, that is,
\setlength\arraycolsep{2pt} \begin{eqnarray}
\mathcal{L}_\eta A_\mu=0\,,\quad \mathcal{L}_\psi A_\mu=0\,,
\end{eqnarray}
thus, we can construct two conserved quantities along the motions of charged particles
\setlength\arraycolsep{2pt} \begin{eqnarray}\label{ccs}
E&=&-\frac{P^\mu}{m}g_{\mu\nu}\eta^\nu=-P_t/m\,,\nonumber\\
L&=&\frac{P^\mu}{m}g_{\mu\nu}\psi^\nu=P_\phi/m\,.
\end{eqnarray}
Note that the energy can be explicitly written out as
\begin{gather}
E=\left(1-\frac{2Mr}{\Sigma}\right)\left[\frac{q\left(aB-\frac{Q}{2M}\right)}{m}+\frac{\mathrm{d}t}{\mathrm{d}\tau}\right]+\frac{aMr\mathrm{sin}^2\theta\left(Bq+2m\frac{\mathrm{d}\phi}{\mathrm{d}\tau}\right)}{m\Sigma}\,,
\end{gather}
so it is not necessarily positive. Similarly, $L$ wouldn't be necessarily positive even if $\frac{\partial \phi}{\partial \tau}>0$.

\begin{figure}
\centering
\includegraphics[width=8cm]{setup.jpg}
\caption{A diagram of charged particles moving in circular orbits on the equatorial plane of the Kerr spacetime with a vertical and uniform magnetic field. We assume the spin of the Kerr black hole and the magnetic field are in the same direction when $a$ and $\mathcal{B}$ have the same sign.}\label{setup}
\end{figure}

Since we are interested in the circular motions confined to the equatorial plane as illustrated in Fig. \ref{setup}, the four velocity of the charged particle takes
\setlength\arraycolsep{2pt} \begin{eqnarray}
u^\mu=(\dot{t}, \dot{r}, 0, \dot{\phi})\,,
\end{eqnarray}
where `` $\cdot{}$ '' denotes derivative with respect to the proper time $\tau$. From the Eq. (\ref{ccs}), we can obtain
\begin{gather}
\frac{\mathrm{d}t}{\mathrm{d}\tau}=\frac{g_{\phi\phi}(E+\frac{q}{m}A_t)+g_{t\phi}(L-\frac{q}{m}A_\phi)}{g^2_{t\phi}-g_{tt}g_{\phi\phi}}\,, \\
\frac{\mathrm{d}\phi}{\mathrm{d}\tau}=\frac{g_{tt}(-L+\frac{q}{m}A_t)+g_{t\phi}(-E-\frac{q}{m}A_t)}{g_{t\phi}^2-g_{tt}g_{\phi\phi}}
\end{gather}
Note that the four velocity of a charged particle is future-directed, thus we have $\dot{t}>0$ which gives us
\setlength\arraycolsep{2pt} \begin{eqnarray}\label{lle}
E>\frac{2aL}{r^3+a^2r+2a^2}\,,
\end{eqnarray}
where we have used the equilibrium condition, that is, $Q=2BJ$ and set the mass of the black hole $M=1$ here and hereafter.Moreover, the four velocity of the charged particle is normalized, that is, $u^\mu u_\mu=-1$.  Then, combining the Eq. (\ref{ccs}) and the normalization condition of the four velocity, we can obtain
\setlength\arraycolsep{2pt} \begin{eqnarray}
\dot{r}^2+V_{eff}=0\,,
\end{eqnarray}
where we have defined the effective potential $V_{eff}$ as
\setlength\arraycolsep{2pt} \begin{eqnarray}\label{pre}
V_{eff}=\frac{\mathcal{B}^2\Delta(a^2r+r^3+2a^2)-2Lr\Delta\mathcal{B}-(2a^2+a^2r+r^3)E^2+4aLE +r\Delta -2L^2+L^2r}{r^3}\,.\nonumber\\
\end{eqnarray}
Here, we have introduced a new parameter
\setlength\arraycolsep{2pt} \begin{eqnarray}
\mathcal{B}=\frac{qB}{2m}\,,
\end{eqnarray}
It can be explicitly written as $V_{eff}=\frac{P_5(r)}{r^3}$, where $P_5(r)$ is a fifth-order polynomial of $r$. To write out the coefficients explicitly, we have
\begin{gather}
    P_5(r)=c_5 r^5+c_4 r^4+c_3 r^3+c_2 r^2+c_1 r +c_0\,,\nonumber\\ \label{p5}
    c_0=-2(L-aE)^2+2a^4 \mathcal{B}^2 \,,\nonumber \\
    c_1=a^2+(-4a^2+a^4)\mathcal{B}^2-2a^2\mathcal{B}L+L^2-a^2E^2 \,,\nonumber\\
    c_2=-2+4\mathcal{B}L \, ,\,
    c_3=1+2a^2\mathcal{B}^2-2\mathcal{B}L-E^2\,,\nonumber\\
    c_4=-2\mathcal{B}^2 \,,\,
    c_5=2\mathcal{B}^2
\end{gather}

In addition, it will useful to evaluate $V_{eff}$ at the horizon $x=\frac{1}{r_h}$ and the result reads
\setlength\arraycolsep{2pt} \begin{eqnarray}
-r_+^3V_{eff}(r=r_h)=4E^2r_++L^2r_--4aEL
\end{eqnarray}
Here the terms regarding $\mathcal{B}$ have become zero.  This also implies that at around the horizon, the impact of the magnetic field on particles would become small. Also,
\begin{equation}
 4E^2r_++L^2r_--4aEL\ge2\sqrt{4E^2L^2r_+r_-}-4aEL=0\,,
\end{equation}
therefore, we can see that the effective potential is always non-negative at the horizon. The equation holds if and only if $4E^2r_+=L^2r_-$, which gives $\frac{L}{E}=\pm\frac{2r_+}{a}$.

\subsection{Circular orbits of charged particles}
Next, we turn to calculate the circular orbits of charged particles in the Kerr spacetime with a Wald magnetic field. For simplicity, we would like to set $\mathcal{B}>0$ and let the spin $a$ vary from $-1$ to $1$. In order to find the energy $E$ and angular momentum $L$ at a radius $r$, we need to solve the following equations
\setlength\arraycolsep{2pt} \begin{eqnarray}\label{coe}
V_{eff}=0\,\quad\text{and} \quad\,\partial_rV_{eff}=0\,,
\end{eqnarray}
and the ISCO can be determined by further requiring
\setlength\arraycolsep{2pt} \begin{eqnarray}\label{iscoe}
\partial_r^2 V_{eff}=0\,.
\end{eqnarray}
It's easy to see that $V_{eff}=0$ can be reduced to a quintic equation for $r$, which cannot be solved analytically in general. And thus, one have to solve Eqs (\ref{coe}) and (\ref{iscoe}) numerically to gain insights of the ISCOs of charged particles in the Kerr spacetime with a Wald magnetic field, see examples in \cite{al2022charged}.

However, one can see that $V_{eff}=0$ reduces to a cubic equation for $r$ if the term including $\mathcal{B}^2$ can be dropped, which is legal when $\mathcal{B}\ll1$. Considering that a cubic equation can be solved analytically and an analytical expression is useful for us to better understand the circular orbits of charged particles in the Kerr spacetime with a Wald magnetic field, we would like to carefully study the case $\mathcal{B}\ll1$. Let us start with the discussion on when the approximate condition holds in the Gauss units. Note that in the present convention $\mathcal{B}=k$ means $\mathcal{B}=kM^{-1}$, where $k$ is a dimensionless number,  we can recover the full expression of $\mathcal{B}=k$ in the Gauss units,
\setlength\arraycolsep{2pt} \begin{eqnarray}
\left(\frac{qB}{2m}\right)_{\text{Gauss}}= k\frac{c^4}{GM}\,,
\end{eqnarray}
and this gives
\setlength\arraycolsep{2pt} \begin{eqnarray}
\left(\frac{q}{e}\right)\left(\frac{m_p}{m}\right)=\frac{43}{B_{\text{Gauss}}}\frac{M_\odot}{M}k=1.8\times10^{-18}\frac{(\mathcal{B}M)}{(BM)}\,,
\end{eqnarray}
where $m_p$ is the proton mass, $e$ is the unit charge and we have plunged Eq. (\ref{btg}) in the last `` $=$ ''. Thus, the approximate condition $\mathcal{B}\ll1$ corresponds to
\setlength\arraycolsep{2pt} \begin{eqnarray}
\left(\frac{q}{e}\right)\left(\frac{m_p}{m}\right)(BM)\ll 1.8\times 10^{-18}\,.
\end{eqnarray}
 As a result, we can see that the approximate condition $\mathcal{B}\ll1$ is applicable for an object with a unit charge whose mass is much larger than $10^{18}(BM)m_p$. For example, from Eq. (\ref{btg}), we find $BM=10^{-9}$, if we consider a supermassive black hole with $M\sim10^9M_{\odot}$ immersed in a magnetic field with $B=10\, \text{Gauss}$, then the mass of the object with a unit charge should be much larger than $m=10^{9}m_p=10^{-18}\,\text{kg}$ in the International System of Units if we require that the approximate condition $\mathcal{B}\ll1$ holds in our present convention. More precisely, an electron would have $\mathcal{B}=8.6\times10^{10}$, while, the mass-charge ratio of an electron is $1.7\times10^{11}$ C/kg which is too extreme. So it clearly does not fall within our approximation. However, for ionized gas mocules, the charge-mass ratio is already much lower, to the scale of $10^6-10^8$. For charged astrophysical dusts, the approximation can be applied, especially for low mass and weakly magnetized black holes.

Now, we move to study the orbits of charged particles in the case $\mathcal{B}\ll1$. We dropped the second order terms for $\mathcal{B}$ in the effective potential, and the Eq. (\ref{pre}) can be simplified as
\setlength\arraycolsep{2pt} \begin{eqnarray}
-V_{eff}=2(L-2aE)^2x^3+(-a^2+2a^2\mathcal{B}L-L^2+a^2E^2)x^2+(2-4\mathcal{B}L)x+E^2+2\mathcal{B}L-1\,,\nonumber\\
\end{eqnarray}
where a new parameter $x=1/r$ is introduced for simplicity. The equations for a circular orbit
\setlength\arraycolsep{2pt} \begin{eqnarray}
V_{eff}=\partial_r V_{eff}=\partial_r^2V_{eff}=0
\end{eqnarray}
correspond to the equations
\setlength\arraycolsep{2pt} \begin{eqnarray}\label{epx}
V_{eff}=\partial_x V_{eff}=\partial_x^2V_{eff}=0\,,
\end{eqnarray}
This would be true except for two special points: $r=0$, which is inside the horizon and $r=\infty$. So that we can obtain the ISCOs by solving the Eq. (\ref{epx}).

In the following subsections, we will analyze the form and the root structures of the effective potential under this approximation.

\subsubsection{Quadratic form}
Next, let us begin with the first equation $V_{eff}=0$. It's easy to see that the cubic equation is simplified into a quadratic equation when $L=aE$. Then the effective potential function becomes \
\begin{align}
-V_{eff}&=(-1+2\mathcal{B}L)a^2x^2+2(1-2\mathcal{B}L)x+E^2+2\mathcal{B}L-1 \\
&=(-1+2\mathcal{B}L)\left(ax-\frac{1}{a}\right)^2+E^2+(2\mathcal{B}L-1)\left(1+\frac{1}{a^2}\right) \, .
\end{align}
Obviously, when $-1+2\mathcal{B}L=0$, that is, $L=\frac{1}{2\mathcal{B}}$, we find $V_{eff}=-E^2$, which implies no stable circular orbits of charged particles exist in this situation.

When $-1+2\mathcal{B}L>0$, we find
\setlength\arraycolsep{2pt} \begin{eqnarray}
-V_{eff}(x=0)=E^2+2\mathcal{B}L-1>0\,, -V_{eff}(x=\frac{1}{r_+})>0\,,\frac{1}{a^2}\ge1\ge\frac{1}{r_+}\,,
\end{eqnarray}
so that, $-V_{eff}$ is a monotonic decreasing function and $-V_{eff}>0$ is always true in the region $x\in(0, 1/r_+)$ which implies there are no circular orbits of charged particles.

At last, we focus on the case $2\mathcal{B}L-1<0$. In addition, combining the Eq. (\ref{lle}) with $L=aE$, we can find the energy satisfies
\setlength\arraycolsep{2pt} \begin{eqnarray}
0<E<\frac{1}{2\mathcal{B}a}\quad\text{with}\quad 0<a\le1\,,
\end{eqnarray}
thus, we can obtain
\setlength\arraycolsep{2pt} \begin{eqnarray}
-1<-V_{eff}(x=0)=E^2+2\mathcal{B}aE-1<\frac{1}{4\mathcal{B}^2a^2}\,.
\end{eqnarray}
On the other hand, we find $-V_{eff}$ is a monotonic increasing function in the region $x\in(0, 1/r_+)$ in the case $2\mathcal{B}L-1<0$, and so
\setlength\arraycolsep{2pt} \begin{eqnarray}
-V_{eff}(x=0)<-V_{eff}<-V_{eff}(x=1/r_+)
\end{eqnarray}
in the region $x\in(0, 1/r_+)$. Thus, when $0<E<\sqrt{1+a^2\mathcal{B}^2}-a^2\mathcal{B}^2 $ we have $-V_{eff}(x=0)<0$ and there must be a $r_0$, such that $V_{eff}(r_0)=0$. This implies there's a forbidden zone $(r_0, +\infty)$ for charged particles when $0<E<\sqrt{1+a^2\mathcal{B}^2}-a^2\mathcal{B}^2$. When $\sqrt{1+a^2\mathcal{B}^2}-a^2\mathcal{B}^2\le E<\frac{1}{2\mathcal{B}a}$, we have $-V_{eff}(x=0)>0$ and there's no forbidden zone for charged particles. In particular, in both cases, there are no circular orbits for charged particles.

In conclusion, now we can make sure that there are no circular orbits for charged particles when $L=aE$.

\subsubsection{Cubic form}
Then let's pay attention to the more general situation in which $L\neq aE$, and the effective potential function is in a cubic form. In order to investigate the structure of the potential, it's convenient to define a new variable $y$ by
\setlength\arraycolsep{2pt} \begin{eqnarray}
x=y+\frac{d}{3\tilde{a}}\,,
\end{eqnarray}
where we set
\setlength\arraycolsep{2pt} \begin{eqnarray}
d&=&L^2+a^2(1-2\mathcal{B}L-E^2)\,,\nonumber\\
\tilde{a} &=& 2(L-aE)^2\,,
\end{eqnarray}
and then we get a compact form
\setlength\arraycolsep{2pt} \begin{eqnarray}
f(y)=-V_{eff}=\tilde{a}y^3+by+c\,,
\end{eqnarray}
where
\setlength\arraycolsep{2pt} \begin{eqnarray}
b &=& 2-4\mathcal{B}L-\frac{d^2}{3\tilde{a}}\,,\nonumber\\
c &=& -1+2\mathcal{B}L+E^2+\frac{2(1-2\mathcal{B}L)d}{3\tilde{a}}+\frac{d^3}{27\tilde{a}}\,.
\end{eqnarray}
Then, we find
\setlength\arraycolsep{2pt} \begin{eqnarray}
-\partial_xV_{eff}(x)=\partial_yf(y)=3\tilde{a}t^2+b\,.
\end{eqnarray}
A circular orbit has to satisfy $\partial_xV_{eff}(x)=0$, such that we have $b<0$ since $\tilde{a}$ is always positive. Furthermore, $y=\sqrt{\frac{-b}{3\tilde{a}}}$ gives us a local maximum of the effective potential $V_{eff}$ while $y=-\sqrt{\frac{-b}{3\tilde{a}}}$ corresponds to a local maximum. Then the necessary and sufficient conditions of the existence of a future-directed stable circular orbit are
\begin{equation}\label{cd2}
0<y_0=\sqrt{\frac{-b}{3\tilde{a}}}+\frac{d}{3\tilde{a}}<\frac{1}{r_+}\,,
\end{equation}
and
\begin{equation} \label{ss}
f(\sqrt{\frac{-b}{3\tilde{a}}})=0\,,
\end{equation}
as well as the Eq. (\ref{lle}). In addition, the condition $f(\sqrt{\frac{-b}{3\tilde{a}}})=0$ is equivalent to
\setlength\arraycolsep{2pt} \begin{eqnarray}\label{cd1}
\frac{c^2}{4}+\frac{b^3}{27\tilde{a}}=0\,,
\end{eqnarray}
which gives a constraint for the conserved quantities $L$ and $E$. That is to say, when the Eqs. (\ref{cd2}), (\ref{cd1}) and (\ref{lle}) hold, a stable future directed circular orbit for a charged particle must exist and the radius is given by
\setlength\arraycolsep{2pt} \begin{eqnarray}
r_0=1/y_0\,,
\end{eqnarray}
where $y_0$ is given in Eq. (\ref{cd2}).

We can push our calculations to determine the ISCO of the charged particle. Considering the equation $\partial_x^2V_{eff}(x)=\partial_y^2f(y)=0$, we can easily find the following conditions
\setlength\arraycolsep{2pt} \begin{eqnarray}\label{bcz}
b&=&c=0\,,\\
r_{ISCO}&=&1/y_{ISCO} =\frac{3\tilde{a}}{d}=\frac{6(L-aE)^2}{a^2-2a^2\mathcal{B}L+L^2-a^2E^2}\,,\label{iscor}
\end{eqnarray}
are true if there's a ISCO of the charged particle on the equtorial plane in the Kerr spacetime with a weak magnetic field under the approximate condition $\mathcal{B}\ll1$. Here, it is worth emphasizing that the energy $E$ and angular momentum $L$ can be obtained by solving the Eq. (\ref{bcz}), and then the Eq. (\ref{iscor}) can give us the value of the radius of the ISCO. While, for $\mathcal{B}\gg1$ and $\mathcal{B}\sim\mathcal{O}(1)$, we have to obtain the ISCO and orbits of charged particles numerically. Note that in numerical calculations we can make use of the fact that if Eqs. (\ref{coe}) and (\ref{iscoe}) hold, then the fifth order polynomial \ref{p5} that we mentioned should also satisfy
\setlength\arraycolsep{2pt} \begin{eqnarray}
P_5(r)=0\, , \, \partial_rP_5(r)=0
\end{eqnarray}
This will greatly simplify the computation.

\section{Polarized images of charged particles in circular orbits}
\label{sec3}
In this section, we move to study polarzied images of charged particles which move in circular motions on the equatorial plane of the Kerr spacetime with a Wald magnetic field. The observer is located at infinity with an inclination angle $\theta_o$ as shown in Fig. \ref{setup}.

\subsection{Polarization of synchrotron radiations and propagation of light}
In this subsection, we would like to introduce necessary formulas to calculate the polarization of synchrotron radiations originated from the charged particle and propagation of light following our previous work \cite{hu2022polarization}. When charged particles are affected by Lorentz forces, they are obliged to to move in non-geodesic trajectories and generate synchrotron radiations which possess nice polarization properties. The polarization vector of the synchrotron radiation takes
\setlength\arraycolsep{2pt} \begin{eqnarray}\label{pve}
f^\mu=N^{-1}\left[\delta^\mu_\alpha\left(k_\beta D_\tau u^{[\beta}u^{\alpha]}\right)\right]\,,
\end{eqnarray}
where $u^\alpha$ has been introduced to be the four velocity of the charged particle in the last section, $k_\beta$ is the wave vector of the synchrotron radiation and normalized by $k^\beta u_\beta=1$, $D_\tau$ is derivative operator along the vector $u^\beta$, that is, $D_\tau\equiv u^\alpha\nabla_\alpha$ with $\tau$ being the proper time of $u^\alpha$, and $N^{-1}$ is a normalization factor to ensure $f_\mu f^\mu=1$. In addition, the luminosity of the synchrotron radiation is given by
\setlength\arraycolsep{2pt} \begin{eqnarray}
L=4\pi|k_\beta D_\tau u^\beta u^\alpha-D_\tau u^\alpha|^2\,.
\end{eqnarray}
Next, we consider the position of the photon that hits the screen of the observer with coordinates $(t_o, r_o, \theta_o, \phi_o)$. In terms of celestial coordinates, the position can be described by
\setlength\arraycolsep{2pt} \begin{eqnarray}
\alpha=-\frac{\lambda}{\sin\theta_o}\,,\quad\beta=\pm_o\sqrt{\Theta(\theta_o)}\,,
\end{eqnarray}
where $\pm_o=\text{sign}(k_o^\theta)$ represents the sign of $k^\theta$ in the observer's frame, $\lambda=l/\omega$ is a impact parameter with $l$ and $\omega$ being the angular momentum and energy of the photon, respectively. $\Theta(\theta)$ is the angular potential of the photon which reads
\setlength\arraycolsep{2pt} \begin{eqnarray}
\Theta(\theta)=\eta+a^2\cos^2\theta-\lambda^2\cot^2\theta\,,
\end{eqnarray}
where $\eta=\mathcal{Q}/\omega^2$ is the other impact parameter with $\mathcal{Q}$ being the Carter constant in the Kerr spacetime. On the other hand, charged particles as the light source are located on the equatorial plane with coordinates $(t_s, r_s, \pi/2, \phi_s)$. In principle, with the help of null geodesic equations in Kerr spacetime, one can obtain the  trajectories of photons connecting the source and the observer, and then we can find the images of the source, that is, we know the coordinates $(\alpha, \beta)$ on the screen of the observer. We would like to stress that there is more than one trajectory of lights starting at the source and ending at the observer since the lights may turn around the black hole more than once before reaching the observer. We use $m$ to denote the number of times that the trajectory crosses the equatorial plane between the source and the observer, different values of $m$ correspond to the $(m+1)$th image on the observers' screen. In particular, the image formed by the trajectories with $m=0$ is offen called the primary image, the image with $m=1$ is called the secondary image and so on. For more details, one is suggested to refer \cite{hou2022multi}.

Furthermore, we also need obtain the polarization information of the image. Following \cite{hu2022polarization}, the polarization information can be transmitted from the source to the observer by using the Penrose-Walker (PW) constant
\setlength\arraycolsep{2pt} \begin{eqnarray}\label{kappa}
\kappa=2k^\mu f^{\nu}\left(\hat{l}_{[\mu}\hat{n}_{\nu]}-\hat{m}_{[\mu}\hat{\bar{m}}_{\nu]}\right)(r-ia\cos\theta)\equiv\omega(\kappa_1+i\kappa_2)\,,
\end{eqnarray}
which is a conserved quantity along the null geodesics for Kerr spacetime. In the above formula, $\kappa_{1,2}$ are the rescaled real part and imaginary part of PW constant, respectively. And ${\hat{l}, \hat{n}, \hat{m}, \hat{\bar{m}}}$ are the Neumann-Penrose tetrads which can be chosen as
\setlength\arraycolsep{2pt} \begin{eqnarray}
\hat{l}&=&\frac{1}{\sqrt{2\Delta}\Sigma}\left[(r^2+a^2)\partial_t+\Delta\partial_r+a\partial_\phi\right]\,\nonumber\\
\hat{n}&=&\frac{1}{\sqrt{2\Delta}\Sigma}\left[(r^2+a^2)\partial_t-\Delta\partial_r+a\partial_\phi\right]\,\nonumber\\
\hat{m}&=&\frac{1}{\sqrt{2}(r+ia\cos\theta)}\left(ia\sin\theta\partial_t+\partial_\theta+\frac{i}{\sin\theta}\partial_\phi\right)\,,\nonumber\\
\hat{\bar{m}}&=&\frac{1}{\sqrt{2}(r-ia\cos\theta)}\left(-ia\sin\theta\partial_t+\partial_\theta+\frac{i}{\sin\theta}\partial_\phi\right)\,.
\end{eqnarray}
Thus, employing the Eq. (\ref{kappa}), at the observer we can decode the information of $f^\mu$ given at the source and on the polarization vector can be read from the PW constant as \cite{chandrasekhar}
\begin{equation}
\overrightarrow{\mathcal{E}}=\left(\mathcal{E}_\alpha, \mathcal{E}_\beta\right)=\frac{1}{\omega\left(\beta^2+\gamma^2\right)}\left(\beta \kappa_2-\gamma \kappa_1, \beta \kappa_1+\gamma \kappa_2\right), \quad \gamma=-\left(\alpha+a \sin \theta_o\right),
\end{equation}

On the other hand, we still need to know the total fluxes of the image considering charged particles have a finite size. We omit the details and give the result of the total fluxes instead, which takes
\setlength\arraycolsep{2pt} \begin{eqnarray}
F_o=\frac{g^4L}{4\pi r_o^2|\hat{k}|\cdot X}\left|\frac{\partial(\alpha, \beta)}{\partial(Y_s, Z_s)}\right|\,,
\end{eqnarray}
where $\left|\frac{\partial(\alpha, \beta)}{\partial(Y_s, Z_s)}\right|$ denotes the Jacobian determinant between the coordinates $(\alpha, \beta)$ and $(Y_s, Z_s)$. Thereinto, $(Y_s, Z_s)$ are coordinates defined on the `` source screen '' with $T=X=0$. $\{T, X, Y, Z\}$ is a local Minkowski coordinate system in the neighborhood of the source, which is defined as
\setlength\arraycolsep{2pt} \begin{eqnarray}
Te_{(t)}+Xe_{(r)}+Ye_{(\phi)}-Ze_{(\theta)}=(x^\mu-x_\ast^\mu)\partial_\mu\,,
\end{eqnarray}
where $x_\ast^\mu$ are the coordinates of the source and $e_{(\mu)}$ are the tetrad of the source. For a Kerr metric, we would like to choose
\setlength\arraycolsep{2pt} \begin{eqnarray}
e_{(t)}&=&\gamma\sqrt{\frac{\Xi}{\Delta\Sigma}}(\partial_t+\Omega_s\partial_\phi)\,,\quad e_{(r)}=\frac{\Delta}{\Sigma}\partial_r\,,\quad e_{(\theta)}=\frac{1}{\sqrt{\Sigma}}\partial_\theta\,,\nonumber\\
e_{(\phi)}&=&\gamma v_s\sqrt{\frac{\Xi}{\Delta\Sigma}}\left[\partial_t+\frac{2aMr}{\Xi}\right]+\gamma\sqrt{\frac{\Sigma}{\Xi}}
\end{eqnarray}
where we have introduced
\setlength\arraycolsep{2pt} \begin{eqnarray}
\Xi=(r^2+a^2)^2-\Delta a^2\sin^2\theta\,,
\end{eqnarray}
and
\setlength\arraycolsep{2pt} \begin{eqnarray}
v_s=\frac{\Xi}{\Sigma\sqrt{\Delta}}(\Omega_s-\frac{2aMr}{\Xi})\,,\quad\gamma=\frac{1}{\sqrt{1-v_s^2}}\,,
\end{eqnarray}
with $\Omega_s=\frac{d\phi_s}{dt_s}$ being the angular velocity of the source. In addition, the unit vector $\hat{k}$ is given by
\setlength\arraycolsep{2pt} \begin{eqnarray}
\hat{k}=\frac{1}{k^{(t)}}\left(k^{(r)}X+k^{(\phi)}Y-k^{(\theta)}Z\right)\,,
\end{eqnarray}
and $g$ is the redshift factor which is given by
\setlength\arraycolsep{2pt} \begin{eqnarray}
g=\frac{\omega_o}{\omega_s}\,,
\end{eqnarray}
where $\omega_{o,s}$ are the frequencies of the photon in the frame of the observer and the source, respectively. Now, we are ready to calculate some specific polarized images of charged particles orbiting the Kerr black hole with a Wald magnetic field.

\subsection{Results}
Based on our model, in this subsection we would like to show some results for our calculations with the observation angle at $\theta_o=17^\circ$  and $\theta_o=80^\circ$. In our computations of the polarized images, we use the full form of the effective potential to determine the parameters for stable circular orbits numerically, without any approximation. According to the previous analysis, when $\mathcal{B}$ and $a$ are given, there is only one degree of freedom left among $r$, $E$ and $L$ for stable circular orbits. In our following calculations, we usually set $r_s=6$ and compare the results with other different parameters, that is, we fix the radius of the orbit for charged particles and discuss the influence of the other paramers. And the reason why we choose $r_s=6$ is because the ISCO for a neutral particle in a Schwarzschild black hole spacetime is at $r_s=6$ and we still hope to focus on polarized images of the source in this region. Our main interest is in the effects of the spin $a$, the observational angle $\theta_o$ on the polarized images and the difference between the
retrograde or prograde orbiting source \footnote{Note that prograde (retrograde) is defined by $\dot{\phi}>0$ ($\dot{\phi}<0$) instead of $L>0$($L<0$), since the sign of the angular momentum $L$ is not necessarily as same as the sign of $\dot{\phi}$, when there is an external magnetic field.}.
We will vary $a$ and $\theta_o$, and make plots in the retrograde or prograde cases in the following studies. In addition, as pointed out in \cite{hu2022polarization}, empirically, the strength of $\mathcal{B}$ doesn't affect the direction of the polarization vector $f^\mu$ and only changes the overall intensity of flux. It's also the case with Kerr space-time. So we also fix $\mathcal{B}=2$ in our work.

\begin{figure}[h!]
\centering
\includegraphics[width=15cm]{varyth.pdf}
\caption{Comparisons of polarized images between the prograde and retrograde orbits with $\theta_o=17^\circ$, $a=0.94$, $r_s=6$ and $\mathcal{B}=2$. The left plot is for prograde prbit while the right plot is retrograde. The corresponding energy $E$ and angular momentun $L$ for stable circular orbits are also shown in each plot. Blue circles give shapes of the images of charged particles and red lines denote the polarization information. In particular, the length of the red line signifies the magnitude of the total flux. Note the lengths of the flux are scaled in a different way to make the lengths of lines in the two plots are of the same order of magnitude.}\label{figure:rp}
\end{figure}

At first, we look at the difference of the polarized images between the prograde and retrograde orbits. We would like to fix $a=0.94$ which is a potential value of the spin of the supermassive black hole in M87 predicted by EHT \cite{EventHorizonTelescope:2019dse, EventHorizonTelescope:2019jan, EventHorizonTelescope:2019ths, EventHorizonTelescope:2019pgp, EventHorizonTelescope:2019ggy}. The results are shown in Fig. \ref{figure:rp}, in which blue circles are the images of charged particles and red lines denote the polarization directions and total flux with the length. From the Fig. \ref{figure:rp}, one can see that for the prograde case in the left pannel, there is barely any difference between the lengths of red lines on the left and right side. However, for the retrograde case in the right panel of Fig. \ref{figure:rp}, we can find that the fluxes are much stronger on the right side than those on the left side. The main reason for this phenomenon is the so-called headlight effect, that is, the radiations from moving particles concentrate upon the vicinity along the direction of the motion of particles, which becomes more significant when the velocity of moving particles are higher. Qualitatively, when $\mathcal{B}$ and $L$ are in the opposite directions, charged particles would have higher energies to stay in a circular orbit, so that the headlight effect is stronger in the retrograde case. In addition, we can infer that the phenomenon would be more distinct when the observer is closer to the equatorial plane.

\begin{figure}[h!]
\centering
\includegraphics[width=15cm]{varya.pdf}
\caption{Polarized images of charged particles in retrograde orbits are produced with $\theta_o=17^\circ$, $\mathcal{B}=2$ and $r_s=6$. We let the spin $a$ vary from $-0.99$ to $0.99$. The energy $E$ and the angular momentum $L$ are directly calculated from the conditions of the stable circular orbit. Blue lines show the shapes of the images of the stable orbits. The inclinations of red lines denote the polarization directions of radiations, while the lengths represent the total flux. In this set of plots, we set length/flux = 4000 to make the plots more ornamental.}\label{figure:a}
\end{figure}

Next, we turn to study the influence of the spin on the polarized images. In Fig. \ref{figure:a}, we fix $\theta_o=17^\circ$, $r_s=6$ and vary $a$ for retrograde orbits. The ratio of the length to the flux is set to be $200$ for the five plots. We can see that there's evident changes for the magnitude of the flux while the corresponding polarized directions show no significant changes as the spin $a$ increases. In addition, we also checked the prograde orbits and find that when we vary $a$ the changes in polarization and intensity are very minor. So we would not like to present these results here.

\begin{figure}[h!]
\centering
\includegraphics[width=15cm]{varya2.pdf}
\caption{Polarized images of charged particles in prograde orbits are produced with $\theta_o=80^\circ$, $\mathcal{B}=2$ and $r_s=6$. We let the spin $a$ vary from $-0.99$ to $0.99$. In this set of plots, we set length/flux = 4000.}\label{figure:80-pro}
\end{figure}

Now we move on to the case that the observational angle is fixed at $\theta_o=80^\circ$, that is, the observer is close to the equatorial plane, and the spin $a$ is varied from $-0.99$ to $0.99$. The images of orbits deviate from circles a lot in this case. The results for the prograde case are shown in Fig. \ref{figure:80-pro}, where we have scaled the length of the polarization flux by $\text{ length/flux}=4000$. From the plot with $a=0$, it seems that the polarizations of both sides centered on $\alpha=0$ are approximately symmetric which means that the Doppler effect or headlight effect is not obvious. However, there are significant difference of the flux between the upper and lower part of the plot which is considered to be caused by the curvature of the strongly curved spacetime. Meanwhile, at the top of orbit, where $\alpha=0$, the flux of the polarization is much smaller than those in neighboring places. The reason is that the position $(\alpha=0, \beta=\beta_{max})$ on the screen corresponds to the lights that radiate from the source with the four momentum perpendicular to the velocity of charged particles in the local frame of the source, and they are much weaker than radiations going in other directions. This phenomenon can be usually observed when $\theta_o$ is bigger. Moreover, the spin of the Kerr black hole also has large impact on the image. When $a$ is positive, the right side of the image is brighter than the left side and vice versa.

\begin{figure}[h!]
\centering
\includegraphics[width=15cm]{varya3.pdf}
\caption{Polarized images of charged particles in retrograde orbits are produced with $\theta_o=80^\circ$, $r=6$ and $\mathcal{B}=2$. We also let the spin $a$ vary from $-0.99$ to $0.99$. The red line is scaled by taking the logarithm of flux for the sake of visualization, since the Doppler effect is very strong in this case.}\label{figure:80-retro}
\end{figure}

The results for the retrograd case are shown in Figure \ref{figure:80-retro}, where we let $\theta_o=80^\circ$, $r=6$ and $\mathcal{B}=2$ and the spin $a$ go from $-0.99$ to $0.99$. In this case, one of the most obvious phenomena is that the Doppler effect is very strong due to that both the observational angle $\theta_o$ and the energy of the charged particles $E$ are large. Thus for the sake of visualization, we take a logarithm of the flux for the plots. In addition, it is clear that the right side of the plot, where the particle moves towards the observer has much stronger radiations. Also, in this case the black hole spin $a$ has much larger impact on the polarization. Furthermore, in these plots we can see a pronounced clockwise rotation of the polarizations as the spin $a$ increases, while this rotation is unapparent for prograde orbits. Besides, we also find that this rotation happens but it's very small for both prograde and  retrograd orbits when $\theta_o$ is small such as $\theta_o=17^\circ$.

\section{Summary}\label{summary}

In this work, we studied the circular motions and the polarized images of charged particles in the Kerr spacetime with a weak magnetic field. We focused on the case that both the magnetic field and the charge-to-mass ratio are not large, which apply to many charged astrophysical dusts and objects. In this case, the effective potential gets simplified and allows analytical treatment. We found an analytical expression of the radius of the ISCOs in terms of the energy density $E$ and the angular momentum density $L$, and obtained the constraints that $E$ and $L$ should satisfy. It is worth emphasizing that astrophysical objects which satisfy the condition $\mathcal{B}\ll1$ widely exist in the universe and thus our simplified treatment and analytical results are very useful to relevant studies.

Moreover we revisited how to calculate the polarized images including the orbits of charged particles, polarization direction and flux of the synchronize radiations following our previous work \cite{hu2022polarization} to prepare for numerical computations. We obtained the images of circular orbits  through ray-tracing method, and we presented several figures to show  the polarized images of charged particles at fixed $\mathcal{B}$ and $r_s$, where $r_s$ was the orbit radius of the source. We would like to stress that in studying the polarized images with the numerical method we did not assume the approximate condition $\mathcal{B}\ll1$,  and our polarized images were not limited to the case $\mathcal{B}\ll1$.

From our study, we read the influences of the orbit directions, the spins of the black hole and the observational angles on the polarized images. The images for the prograde and retrograde orbits are very different for the same orbit radius. For example, we found that the so-called headlight effect was pretty obvious for the retrograde orbits. In addition, we observed a significant influence of the spin $a$ on the polarized images. More precisely, when $a$ increases,  the angle of polarization rotates clockwise.  This phenomenon becomes especially visible   for the retrograde orbits. The spin $a$ has a non-negligible effect on the magnitudes of the polarization fluxes as well. Moreover, we showed the impacts of the observational angle on the polarized images.

It would be illuminating to compare our results with the ones in in \cite{Gelles:2021kti}, which also studied the polarized images of equatorial emission in the Kerr geometry. In our study, we focused on a vertical magnetic field corresponding to $(B^{r}=0, B^{\phi}=0, B^{(\theta)})$ in \cite{Gelles:2021kti}, however, our results were closer to the ones in their case $(B^{r}=0, B^{\phi}, B^{(\theta)}=0)$. The main reason is that the synchrotron radiations in our model are purely from charged particles without self-interactions,  while \cite{Gelles:2021kti} assumed that the synchrotron radiations were from charged fluids, and thus the formulas of the polarization vector of the synchrotron radiation are different.
Moreover, the formula Eq. (\ref{pve}) for the polarization vector, which was directly derived from the Maxwell equation in a curved spacetime, takes a covariant form, and can be used in the numerical studies easily. In contrast, the polarization vector in \cite{Gelles:2021kti} is determined in a local frame.

\section*{Acknowledgments}
We would like to thank Y.H. Hou, Y.Shen, H.P. Yan and Z.Y. Zhang for valuable discussions. The work is in part supported by NSFC Grant  No. 12275004, 11775022 and 11873044. MG is also supported by ``the Fundamental Research Funds for the Central Universities'' with Grant No. 2021NTST13.

\section{Introduction}
Black holes are one of the most exciting predictions of general relativity. There have been accumulated evidences to support their existence  in our universe. In particular, in recent years the successful detections of the gravitational waves by LIGO and Virgo \cite{LIGOScientific:2016aoc, LIGOScientific:2016sjg, LIGOScientific:2017vwq, lipunov2017master} and the black hole images photographed by the Event Horizon Tecescope (EHT) collaboration\cite{EventHorizonTelescope:2019dse, EventHorizonTelescope:2019jan, EventHorizonTelescope:2019ths, EventHorizonTelescope:2019pgp, EventHorizonTelescope:2019ggy, EventHorizonTelescope:2022xnr} have given us more confidence in the correctness of general relativity and the existence of black holes. General relativity and other theories of gravity admit various black holes which can be neutral, charged, magnetic or both charged and magnetic. Generally speaking, the astrophysical black holes are expected to be neutral due to the selective accretion of the ambient matter. However, in fact it is not impossible to have a weakly charged black hole in our real universe. One charging mechanism is that a rotating black hole immersed in a magnetic field can accrete ambient free charged particles to neutralize the magnetically induced electric field \cite{wald}.

In the complex environment around an astrophysical black hole, there always presents a plasma, part of which forms an accretion disk and generate magnetic fields. Up to now, many observations suggesting magnetic fields outside black holes include but are not limited to bright "flares", polarized broad H$\alpha$ lines and the polarization of the accretion disk radiation in the vicinity of active galactic nuclei  \cite{dovvciak2004polarization, Zajacek:2018vsj, Daly_2019,2010arXiv1002.4948P}. In particular, the recent polarized image of M87 black hole published by the EHT collaboration also reveals a magnetic field around the black hole \cite{EventHorizonTelescope:2021bee, EventHorizonTelescope:2021srq, narayan2021polarized}.

It is essential to study the effects of the magnetic field around a black hole, in order to read the information from polarized images of the black hole. Among various effects,  the influence of Lorentz force on the charged particles' innermost stable circular orbits (ISCOs) and the polarized image of synchrotron radiations in the presence of magnetic fields are important. The ISCOs of charged particles around a Schwarzschild and Kerr black hole immersed in a weak, axially symmetric magnetic field have been widely studied and many interesting results have been found \cite{frolov, Zahrani:2022fdd, Shiose:2014bqa, AlZahrani:2014dfi}. As for a strong magnetic field, the spacetimes can be modeled by the Schwarzschild and Kerr black holes embedded in the Melvin universe \cite{ernst, kerrmel}, and the corresponding ISCOs for massive particles have also been studied in the literatures \cite{Lim:2015oha, Zhu:2022amy, Li:2018wtz}. On the other hand, the polarized image of the synchrotron radiations originated by circularly orbiting charged particles acted upon by Lorentz force had also been investigated in the Schwarzschild and Kerr black holes with weak or strong magnetic fields \cite{stab1289, Lupsasca:2018tpp, EventHorizonTelescope:2021btj, Gelles:2021kti, Qin:2021xvx, Zhang:2021hit, hu2022polarization, Zhu:2022amy, Qin:2022kaf}.

In this work, we revisit the ISCOs of the charged particles around a Kerr black hole with a weak magnetic field, which was first found by Wald in \cite{wald}. Different from the existing researches, we focus on the case $\mathcal{B}=\frac{qB}{2m}\ll1$, where $q$, $m$ and $B$ are the charge, mass and strength of the magnetic field respectively, so that we are allowed to find an analytical expression for the radius of the ISCO which involves the energy $E$ and angular momentum $L$ of charged particles. As many astrophysical dusts or fluids in our universe do not have an extreme charge-mass ratio, the condition $\mathcal{B}\ll1$ can be satisfied ubiquitously  and our results will be very useful in many practical situations. In addition, we study the polarized image of charged particles in circular motions on the equatorial plane observed by an observer at infinity. Different from the usual treatment on the synchrotron radiations of charged particle around black hole, which relies on the study of radiation in the local inertial frame, we apply the covariant formulas of the electromagnetic radiations encoding the intensity and polarization direction of synchrotron radiations in curved spacetime, developed recently in\footnote{The earlier related discussion can be found in \cite{Quinn:1996am}.} \cite{hu2022polarization}. In particular we show explicitly the polarization directions and total flux of the synchrotron radiations emitted from these charged particles for various spins, observational angles and orbits. Our numerical studies on the polarized images are not limited to the case $\mathcal{B}\ll1$,  though we mainly give a careful study on the ISCOs in this approximation

The remaining parts of this paper are organized as follows. In Sec. \ref{sec2}, we study the circular orbits of the charged particles around a Kerr black hole with a Wald magenetic field. In Sec. \ref{sec3}, we review the synchrotron radiations in curved spacetime\cite{hu2022polarization},  and calculate the polarized images of the charged particles in circular motions around a Kerr black hole. The main conclusions are summarized in Sec. \ref{summary}. In this work, we have set the fundamental constants $c$, $G$ and the vacuum permittivity $\varepsilon_0$ to unity, and we will work in the signature convention $(-, +, +, +)$ for the spacetime metric.

\section{Circular orbits of a charged particle around a Kerr black hole with a Wald magnetic field}\label{sec2}
In this section, we would like to focus on circular orbits of a charged particle around a Kerr black hole with Wald magnetic field.

\subsection{Kerr black hole with a Wald magnetic field}
The spacetime of a neutral and rotating black hole in general relativity is described by the Kerr black hole metric, which is a stationary and axisymmetric solution of the Einstein equations and takes,
\setlength\arraycolsep{2pt} \begin{eqnarray}
\mathrm{d}s^2=-\left(1-\frac{2Mr}{\Sigma}\right)\mathrm{d}t^2+\frac{\Sigma}{\Delta}\mathrm{d}r^2+\Sigma\mathrm{d}\theta^2+\left(r^2+a^2+\frac{2Mra^2}{\Sigma}\mathrm{sin}^2\theta\right)\mathrm{d}\phi^2-\frac{4Mra}{\Sigma}\mathrm{sin}^2\theta\mathrm{d}t\mathrm{d}\phi\,,\nonumber\\
\end{eqnarray}
in the Boyer-Lindquist coordinates, where
\setlength\arraycolsep{2pt} \begin{eqnarray}
\Sigma=r^2+a^2\mathrm{cos}^2\theta,\quad \Delta = r^2-2Mr+a^2\,,
\end{eqnarray}
with $M$ and $a$ being the mass and spin of the Kerr black hole, respectively. And the angular momentum of the Kerr black hole is $J=Ma$. The horizons are located at $r_\pm=M\pm\sqrt{M^2-a^2}$ which are given by the equation $\Delta=0$. The outer one is the event horizon of the Kerr black hole, that is, $r_h=r_+$. Next, we would like to assume that there is a vertical and uniform magnetic field outside the Kerr black hole which is described by the Wald solution to the source-free Maxwell equations, that is, $\nabla_\mu F^{\mu\nu}=0$. The magnetic field is aligned with the rotation axis of the black hole and independent of the coordinates $t$ and $\phi$. Considering the two Killing vectors $\eta^a=\left(\frac{\partial}{\partial t}\right)^a$ and $\psi^a=\left(\frac{\partial}{\partial \phi}\right)^a$, Wald  proved that the field
\begin{equation}
A^a=\frac{1}{2}B(\psi^a+\frac{2J}{M}\eta^a)-\frac{Q}{2M}\eta^a \label{A}
\end{equation}
gives a unique field that is an asymptotically uniform magenetic field of strength $B_0$, with $Q$ being the charge of the Kerr black hole. Next, we inroduce the electrostatic injection energy, defined as the difference between the electric potential at the horizon and infinity, i.e.,
\setlength\arraycolsep{2pt} \begin{eqnarray}
\epsilon=e A_\mu\eta^\mu|_{\text{horizon}}-eA_\mu\eta^\mu|_\infty
\end{eqnarray}
which is a constant over the black hole\cite{wald}. For a pure Kerr spacetime, it seems that we should have $Q=0$. However, due to non-vanishing electrostatic injection energy $\epsilon$, an electric field is induced by the rotation of the black hole in the presence of the magnetic field and cause charged particles to be attracted by the black hole, which is a process called charged accretion. As a result, to reach a steady state the black hole has to gain a nonzero net charge to reach equilibrium, that is, $\epsilon=0$, which gives
\begin{equation}\label{ec}
Q=2BJ\,.
\end{equation}
In the following, we would like to mainly focus on this equilibrium situation. Note that in \cite{al2022charged}, they also paid attention to the nonequilibrium situations. It is worth mentioning that the Eq. (\ref{ec}) holds for an arbitrary stationary, axisymmetric black hole, that is, the condition is not limited to Kerr black holes. In principle, the charge and magnetic field would react upon the background spacetime. If we want to ignore the contribution of the magetic field to the background spacetime for simplicity, we need to consider the weak field approximation,
\setlength\arraycolsep{2pt} \begin{eqnarray}
BM\ll 1\,.
\end{eqnarray}
On the other hand, the characteristic length scale given by the charge of the black hole is comparable with the gravitational radius when
\setlength\arraycolsep{2pt} \begin{eqnarray}
\sqrt{\frac{Q_G^2G}{c^4}}=\frac{2GM}{c^2}\,,
\end{eqnarray}
in the International System of Units, so that the gravitational effect of the charge $Q$ on the background can be ignored if
\setlength\arraycolsep{2pt} \begin{eqnarray}
Q\ll Q_G\,.
\end{eqnarray}
In addition, considering $J\le M^2$, from Eq. (\ref{ec}) we have
\setlength\arraycolsep{2pt} \begin{eqnarray}
\frac{Q}{Q_G}=\frac{Q}{2M}=Ba\le BM\ll1\,,
\end{eqnarray}
which means we are allowed to omit the influence of the charge on the metric of the spacetime although there is a nonzero net charge, so that we can stick to the Kerr metric under the weak field approximation. In fact, in the Gauss units we find
\setlength\arraycolsep{2pt} \begin{eqnarray}\label{btg}
B_{\text{Gauss}}=\frac{c^4}{G^{3/2}M}(BM)\simeq2.36\times 10^{19}\frac{M_{\odot}}{M}(BM)\quad \text{Gauss}\,.
\end{eqnarray}
Here (BM) is a dimensionless number and can also be seen as the value of $B$ in our simplified unit after setting $c$, $G$, $\epsilon_0$ and $M$ to 1, and  $M_\odot$ is the mass of sun. For a supermassive black holes $M\sim 10^9 M_\odot$, the dimensionless $BM=0.01\ll 1$ corresponds to $B_{\text{Gauss}}\sim 10^8 \text{Gauss}$, which is already a very big value for astrophysically reasonable magnetic field around a astronomical  black hole. Thus, the weak field approximation $BM\ll 1$ can be imposed for most real astronomical black holes with magnetic fields and deserves careful study.

\subsection{Motions of charged particles on the equatorial plane}
In this subsection, we move to study the motions of charged particles on the equatorial plane under the weak field approximation employing the standard approach. In the Boyer-Lindquist coordinates, the generalized four momentum $P^\mu$ can be written as
\begin{equation}
P^\mu=p^\mu+qA^\mu\,,
\end{equation}
where $p^\mu=mu^\mu$ is the four momentum for the charged particle with $m$ being the mass, $A^\mu$ is the electric vector potential and $q$ is the charge of the particle. From \eqref{A} we have
\begin{equation}
A^\mu=(aB-\frac{Q}{2M},0,0,\frac{B}{2})\,,
\end{equation}
Note that the Lie derivatives of $A_\mu$ with respect to the Killing vectors $\eta^\mu$ and $\psi^\mu$ are zero, that is,
\setlength\arraycolsep{2pt} \begin{eqnarray}
\mathcal{L}_\eta A_\mu=0\,,\quad \mathcal{L}_\psi A_\mu=0\,,
\end{eqnarray}
thus, we can construct two conserved quantities along the motions of charged particles
\setlength\arraycolsep{2pt} \begin{eqnarray}\label{ccs}
E&=&-\frac{P^\mu}{m}g_{\mu\nu}\eta^\nu=-P_t/m\,,\nonumber\\
L&=&\frac{P^\mu}{m}g_{\mu\nu}\psi^\nu=P_\phi/m\,.
\end{eqnarray}
Note that the energy can be explicitly written out as
\begin{gather}
E=\left(1-\frac{2Mr}{\Sigma}\right)\left[\frac{q\left(aB-\frac{Q}{2M}\right)}{m}+\frac{\mathrm{d}t}{\mathrm{d}\tau}\right]+\frac{aMr\mathrm{sin}^2\theta\left(Bq+2m\frac{\mathrm{d}\phi}{\mathrm{d}\tau}\right)}{m\Sigma}\,,
\end{gather}
so it is not necessarily positive. Similarly, $L$ wouldn't be necessarily positive even if $\frac{\partial \phi}{\partial \tau}>0$.

\begin{figure}
\centering
\includegraphics[width=8cm]{setup.jpg}
\caption{A diagram of charged particles moving in circular orbits on the equatorial plane of the Kerr spacetime with a vertical and uniform magnetic field. We assume the spin of the Kerr black hole and the magnetic field are in the same direction when $a$ and $\mathcal{B}$ have the same sign.}\label{setup}
\end{figure}

Since we are interested in the circular motions confined to the equatorial plane as illustrated in Fig. \ref{setup}, the four velocity of the charged particle takes
\setlength\arraycolsep{2pt} \begin{eqnarray}
u^\mu=(\dot{t}, \dot{r}, 0, \dot{\phi})\,,
\end{eqnarray}
where `` $\cdot{}$ '' denotes derivative with respect to the proper time $\tau$. From the Eq. (\ref{ccs}), we can obtain
\begin{gather}
\frac{\mathrm{d}t}{\mathrm{d}\tau}=\frac{g_{\phi\phi}(E+\frac{q}{m}A_t)+g_{t\phi}(L-\frac{q}{m}A_\phi)}{g^2_{t\phi}-g_{tt}g_{\phi\phi}}\,, \\
\frac{\mathrm{d}\phi}{\mathrm{d}\tau}=\frac{g_{tt}(-L+\frac{q}{m}A_t)+g_{t\phi}(-E-\frac{q}{m}A_t)}{g_{t\phi}^2-g_{tt}g_{\phi\phi}}
\end{gather}
Note that the four velocity of a charged particle is future-directed, thus we have $\dot{t}>0$ which gives us
\setlength\arraycolsep{2pt} \begin{eqnarray}\label{lle}
E>\frac{2aL}{r^3+a^2r+2a^2}\,,
\end{eqnarray}
where we have used the equilibrium condition, that is, $Q=2BJ$ and set the mass of the black hole $M=1$ here and hereafter.Moreover, the four velocity of the charged particle is normalized, that is, $u^\mu u_\mu=-1$.  Then, combining the Eq. (\ref{ccs}) and the normalization condition of the four velocity, we can obtain
\setlength\arraycolsep{2pt} \begin{eqnarray}
\dot{r}^2+V_{eff}=0\,,
\end{eqnarray}
where we have defined the effective potential $V_{eff}$ as
\setlength\arraycolsep{2pt} \begin{eqnarray}\label{pre}
V_{eff}=\frac{\mathcal{B}^2\Delta(a^2r+r^3+2a^2)-2Lr\Delta\mathcal{B}-(2a^2+a^2r+r^3)E^2+4aLE +r\Delta -2L^2+L^2r}{r^3}\,.\nonumber\\
\end{eqnarray}
Here, we have introduced a new parameter
\setlength\arraycolsep{2pt} \begin{eqnarray}
\mathcal{B}=\frac{qB}{2m}\,,
\end{eqnarray}
It can be explicitly written as $V_{eff}=\frac{P_5(r)}{r^3}$, where $P_5(r)$ is a fifth-order polynomial of $r$. To write out the coefficients explicitly, we have
\begin{gather}
    P_5(r)=c_5 r^5+c_4 r^4+c_3 r^3+c_2 r^2+c_1 r +c_0\,,\nonumber\\ \label{p5}
    c_0=-2(L-aE)^2+2a^4 \mathcal{B}^2 \,,\nonumber \\
    c_1=a^2+(-4a^2+a^4)\mathcal{B}^2-2a^2\mathcal{B}L+L^2-a^2E^2 \,,\nonumber\\
    c_2=-2+4\mathcal{B}L \, ,\,
    c_3=1+2a^2\mathcal{B}^2-2\mathcal{B}L-E^2\,,\nonumber\\
    c_4=-2\mathcal{B}^2 \,,\,
    c_5=2\mathcal{B}^2
\end{gather}

In addition, it will useful to evaluate $V_{eff}$ at the horizon $x=\frac{1}{r_h}$ and the result reads
\setlength\arraycolsep{2pt} \begin{eqnarray}
-r_+^3V_{eff}(r=r_h)=4E^2r_++L^2r_--4aEL
\end{eqnarray}
Here the terms regarding $\mathcal{B}$ have become zero.  This also implies that at around the horizon, the impact of the magnetic field on particles would become small. Also,
\begin{equation}
 4E^2r_++L^2r_--4aEL\ge2\sqrt{4E^2L^2r_+r_-}-4aEL=0\,,
\end{equation}
therefore, we can see that the effective potential is always non-negative at the horizon. The equation holds if and only if $4E^2r_+=L^2r_-$, which gives $\frac{L}{E}=\pm\frac{2r_+}{a}$.

\subsection{Circular orbits of charged particles}
Next, we turn to calculate the circular orbits of charged particles in the Kerr spacetime with a Wald magnetic field. For simplicity, we would like to set $\mathcal{B}>0$ and let the spin $a$ vary from $-1$ to $1$. In order to find the energy $E$ and angular momentum $L$ at a radius $r$, we need to solve the following equations
\setlength\arraycolsep{2pt} \begin{eqnarray}\label{coe}
V_{eff}=0\,\quad\text{and} \quad\,\partial_rV_{eff}=0\,,
\end{eqnarray}
and the ISCO can be determined by further requiring
\setlength\arraycolsep{2pt} \begin{eqnarray}\label{iscoe}
\partial_r^2 V_{eff}=0\,.
\end{eqnarray}
It's easy to see that $V_{eff}=0$ can be reduced to a quintic equation for $r$, which cannot be solved analytically in general. And thus, one have to solve Eqs (\ref{coe}) and (\ref{iscoe}) numerically to gain insights of the ISCOs of charged particles in the Kerr spacetime with a Wald magnetic field, see examples in \cite{al2022charged}.

However, one can see that $V_{eff}=0$ reduces to a cubic equation for $r$ if the term including $\mathcal{B}^2$ can be dropped, which is legal when $\mathcal{B}\ll1$. Considering that a cubic equation can be solved analytically and an analytical expression is useful for us to better understand the circular orbits of charged particles in the Kerr spacetime with a Wald magnetic field, we would like to carefully study the case $\mathcal{B}\ll1$. Let us start with the discussion on when the approximate condition holds in the Gauss units. Note that in the present convention $\mathcal{B}=k$ means $\mathcal{B}=kM^{-1}$, where $k$ is a dimensionless number,  we can recover the full expression of $\mathcal{B}=k$ in the Gauss units,
\setlength\arraycolsep{2pt} \begin{eqnarray}
\left(\frac{qB}{2m}\right)_{\text{Gauss}}= k\frac{c^4}{GM}\,,
\end{eqnarray}
and this gives
\setlength\arraycolsep{2pt} \begin{eqnarray}
\left(\frac{q}{e}\right)\left(\frac{m_p}{m}\right)=\frac{43}{B_{\text{Gauss}}}\frac{M_\odot}{M}k=1.8\times10^{-18}\frac{(\mathcal{B}M)}{(BM)}\,,
\end{eqnarray}
where $m_p$ is the proton mass, $e$ is the unit charge and we have plunged Eq. (\ref{btg}) in the last `` $=$ ''. Thus, the approximate condition $\mathcal{B}\ll1$ corresponds to
\setlength\arraycolsep{2pt} \begin{eqnarray}
\left(\frac{q}{e}\right)\left(\frac{m_p}{m}\right)(BM)\ll 1.8\times 10^{-18}\,.
\end{eqnarray}
 As a result, we can see that the approximate condition $\mathcal{B}\ll1$ is applicable for an object with a unit charge whose mass is much larger than $10^{18}(BM)m_p$. For example, from Eq. (\ref{btg}), we find $BM=10^{-9}$, if we consider a supermassive black hole with $M\sim10^9M_{\odot}$ immersed in a magnetic field with $B=10\, \text{Gauss}$, then the mass of the object with a unit charge should be much larger than $m=10^{9}m_p=10^{-18}\,\text{kg}$ in the International System of Units if we require that the approximate condition $\mathcal{B}\ll1$ holds in our present convention. More precisely, an electron would have $\mathcal{B}=8.6\times10^{10}$, while, the mass-charge ratio of an electron is $1.7\times10^{11}$ C/kg which is too extreme. So it clearly does not fall within our approximation. However, for ionized gas mocules, the charge-mass ratio is already much lower, to the scale of $10^6-10^8$. For charged astrophysical dusts, the approximation can be applied, especially for low mass and weakly magnetized black holes.

Now, we move to study the orbits of charged particles in the case $\mathcal{B}\ll1$. We dropped the second order terms for $\mathcal{B}$ in the effective potential, and the Eq. (\ref{pre}) can be simplified as
\setlength\arraycolsep{2pt} \begin{eqnarray}
-V_{eff}=2(L-2aE)^2x^3+(-a^2+2a^2\mathcal{B}L-L^2+a^2E^2)x^2+(2-4\mathcal{B}L)x+E^2+2\mathcal{B}L-1\,,\nonumber\\
\end{eqnarray}
where a new parameter $x=1/r$ is introduced for simplicity. The equations for a circular orbit
\setlength\arraycolsep{2pt} \begin{eqnarray}
V_{eff}=\partial_r V_{eff}=\partial_r^2V_{eff}=0
\end{eqnarray}
correspond to the equations
\setlength\arraycolsep{2pt} \begin{eqnarray}\label{epx}
V_{eff}=\partial_x V_{eff}=\partial_x^2V_{eff}=0\,,
\end{eqnarray}
This would be true except for two special points: $r=0$, which is inside the horizon and $r=\infty$. So that we can obtain the ISCOs by solving the Eq. (\ref{epx}).

In the following subsections, we will analyze the form and the root structures of the effective potential under this approximation.

\subsubsection{Quadratic form}
Next, let us begin with the first equation $V_{eff}=0$. It's easy to see that the cubic equation is simplified into a quadratic equation when $L=aE$. Then the effective potential function becomes \
\begin{align}
-V_{eff}&=(-1+2\mathcal{B}L)a^2x^2+2(1-2\mathcal{B}L)x+E^2+2\mathcal{B}L-1 \\
&=(-1+2\mathcal{B}L)\left(ax-\frac{1}{a}\right)^2+E^2+(2\mathcal{B}L-1)\left(1+\frac{1}{a^2}\right) \, .
\end{align}
Obviously, when $-1+2\mathcal{B}L=0$, that is, $L=\frac{1}{2\mathcal{B}}$, we find $V_{eff}=-E^2$, which implies no stable circular orbits of charged particles exist in this situation.

When $-1+2\mathcal{B}L>0$, we find
\setlength\arraycolsep{2pt} \begin{eqnarray}
-V_{eff}(x=0)=E^2+2\mathcal{B}L-1>0\,, -V_{eff}(x=\frac{1}{r_+})>0\,,\frac{1}{a^2}\ge1\ge\frac{1}{r_+}\,,
\end{eqnarray}
so that, $-V_{eff}$ is a monotonic decreasing function and $-V_{eff}>0$ is always true in the region $x\in(0, 1/r_+)$ which implies there are no circular orbits of charged particles.

At last, we focus on the case $2\mathcal{B}L-1<0$. In addition, combining the Eq. (\ref{lle}) with $L=aE$, we can find the energy satisfies
\setlength\arraycolsep{2pt} \begin{eqnarray}
0<E<\frac{1}{2\mathcal{B}a}\quad\text{with}\quad 0<a\le1\,,
\end{eqnarray}
thus, we can obtain
\setlength\arraycolsep{2pt} \begin{eqnarray}
-1<-V_{eff}(x=0)=E^2+2\mathcal{B}aE-1<\frac{1}{4\mathcal{B}^2a^2}\,.
\end{eqnarray}
On the other hand, we find $-V_{eff}$ is a monotonic increasing function in the region $x\in(0, 1/r_+)$ in the case $2\mathcal{B}L-1<0$, and so
\setlength\arraycolsep{2pt} \begin{eqnarray}
-V_{eff}(x=0)<-V_{eff}<-V_{eff}(x=1/r_+)
\end{eqnarray}
in the region $x\in(0, 1/r_+)$. Thus, when $0<E<\sqrt{1+a^2\mathcal{B}^2}-a^2\mathcal{B}^2 $ we have $-V_{eff}(x=0)<0$ and there must be a $r_0$, such that $V_{eff}(r_0)=0$. This implies there's a forbidden zone $(r_0, +\infty)$ for charged particles when $0<E<\sqrt{1+a^2\mathcal{B}^2}-a^2\mathcal{B}^2$. When $\sqrt{1+a^2\mathcal{B}^2}-a^2\mathcal{B}^2\le E<\frac{1}{2\mathcal{B}a}$, we have $-V_{eff}(x=0)>0$ and there's no forbidden zone for charged particles. In particular, in both cases, there are no circular orbits for charged particles.

In conclusion, now we can make sure that there are no circular orbits for charged particles when $L=aE$.

\subsubsection{Cubic form}
Then let's pay attention to the more general situation in which $L\neq aE$, and the effective potential function is in a cubic form. In order to investigate the structure of the potential, it's convenient to define a new variable $y$ by
\setlength\arraycolsep{2pt} \begin{eqnarray}
x=y+\frac{d}{3\tilde{a}}\,,
\end{eqnarray}
where we set
\setlength\arraycolsep{2pt} \begin{eqnarray}
d&=&L^2+a^2(1-2\mathcal{B}L-E^2)\,,\nonumber\\
\tilde{a} &=& 2(L-aE)^2\,,
\end{eqnarray}
and then we get a compact form
\setlength\arraycolsep{2pt} \begin{eqnarray}
f(y)=-V_{eff}=\tilde{a}y^3+by+c\,,
\end{eqnarray}
where
\setlength\arraycolsep{2pt} \begin{eqnarray}
b &=& 2-4\mathcal{B}L-\frac{d^2}{3\tilde{a}}\,,\nonumber\\
c &=& -1+2\mathcal{B}L+E^2+\frac{2(1-2\mathcal{B}L)d}{3\tilde{a}}+\frac{d^3}{27\tilde{a}}\,.
\end{eqnarray}
Then, we find
\setlength\arraycolsep{2pt} \begin{eqnarray}
-\partial_xV_{eff}(x)=\partial_yf(y)=3\tilde{a}t^2+b\,.
\end{eqnarray}
A circular orbit has to satisfy $\partial_xV_{eff}(x)=0$, such that we have $b<0$ since $\tilde{a}$ is always positive. Furthermore, $y=\sqrt{\frac{-b}{3\tilde{a}}}$ gives us a local maximum of the effective potential $V_{eff}$ while $y=-\sqrt{\frac{-b}{3\tilde{a}}}$ corresponds to a local maximum. Then the necessary and sufficient conditions of the existence of a future-directed stable circular orbit are
\begin{equation}\label{cd2}
0<y_0=\sqrt{\frac{-b}{3\tilde{a}}}+\frac{d}{3\tilde{a}}<\frac{1}{r_+}\,,
\end{equation}
and
\begin{equation} \label{ss}
f(\sqrt{\frac{-b}{3\tilde{a}}})=0\,,
\end{equation}
as well as the Eq. (\ref{lle}). In addition, the condition $f(\sqrt{\frac{-b}{3\tilde{a}}})=0$ is equivalent to
\setlength\arraycolsep{2pt} \begin{eqnarray}\label{cd1}
\frac{c^2}{4}+\frac{b^3}{27\tilde{a}}=0\,,
\end{eqnarray}
which gives a constraint for the conserved quantities $L$ and $E$. That is to say, when the Eqs. (\ref{cd2}), (\ref{cd1}) and (\ref{lle}) hold, a stable future directed circular orbit for a charged particle must exist and the radius is given by
\setlength\arraycolsep{2pt} \begin{eqnarray}
r_0=1/y_0\,,
\end{eqnarray}
where $y_0$ is given in Eq. (\ref{cd2}).

We can push our calculations to determine the ISCO of the charged particle. Considering the equation $\partial_x^2V_{eff}(x)=\partial_y^2f(y)=0$, we can easily find the following conditions
\setlength\arraycolsep{2pt} \begin{eqnarray}\label{bcz}
b&=&c=0\,,\\
r_{ISCO}&=&1/y_{ISCO} =\frac{3\tilde{a}}{d}=\frac{6(L-aE)^2}{a^2-2a^2\mathcal{B}L+L^2-a^2E^2}\,,\label{iscor}
\end{eqnarray}
are true if there's a ISCO of the charged particle on the equtorial plane in the Kerr spacetime with a weak magnetic field under the approximate condition $\mathcal{B}\ll1$. Here, it is worth emphasizing that the energy $E$ and angular momentum $L$ can be obtained by solving the Eq. (\ref{bcz}), and then the Eq. (\ref{iscor}) can give us the value of the radius of the ISCO. While, for $\mathcal{B}\gg1$ and $\mathcal{B}\sim\mathcal{O}(1)$, we have to obtain the ISCO and orbits of charged particles numerically. Note that in numerical calculations we can make use of the fact that if Eqs. (\ref{coe}) and (\ref{iscoe}) hold, then the fifth order polynomial \ref{p5} that we mentioned should also satisfy
\setlength\arraycolsep{2pt} \begin{eqnarray}
P_5(r)=0\, , \, \partial_rP_5(r)=0
\end{eqnarray}
This will greatly simplify the computation.

\section{Polarized images of charged particles in circular orbits}
\label{sec3}
In this section, we move to study polarzied images of charged particles which move in circular motions on the equatorial plane of the Kerr spacetime with a Wald magnetic field. The observer is located at infinity with an inclination angle $\theta_o$ as shown in Fig. \ref{setup}.

\subsection{Polarization of synchrotron radiations and propagation of light}
In this subsection, we would like to introduce necessary formulas to calculate the polarization of synchrotron radiations originated from the charged particle and propagation of light following our previous work \cite{hu2022polarization}. When charged particles are affected by Lorentz forces, they are obliged to to move in non-geodesic trajectories and generate synchrotron radiations which possess nice polarization properties. The polarization vector of the synchrotron radiation takes
\setlength\arraycolsep{2pt} \begin{eqnarray}\label{pve}
f^\mu=N^{-1}\left[\delta^\mu_\alpha\left(k_\beta D_\tau u^{[\beta}u^{\alpha]}\right)\right]\,,
\end{eqnarray}
where $u^\alpha$ has been introduced to be the four velocity of the charged particle in the last section, $k_\beta$ is the wave vector of the synchrotron radiation and normalized by $k^\beta u_\beta=1$, $D_\tau$ is derivative operator along the vector $u^\beta$, that is, $D_\tau\equiv u^\alpha\nabla_\alpha$ with $\tau$ being the proper time of $u^\alpha$, and $N^{-1}$ is a normalization factor to ensure $f_\mu f^\mu=1$. In addition, the luminosity of the synchrotron radiation is given by
\setlength\arraycolsep{2pt} \begin{eqnarray}
L=4\pi|k_\beta D_\tau u^\beta u^\alpha-D_\tau u^\alpha|^2\,.
\end{eqnarray}
Next, we consider the position of the photon that hits the screen of the observer with coordinates $(t_o, r_o, \theta_o, \phi_o)$. In terms of celestial coordinates, the position can be described by
\setlength\arraycolsep{2pt} \begin{eqnarray}
\alpha=-\frac{\lambda}{\sin\theta_o}\,,\quad\beta=\pm_o\sqrt{\Theta(\theta_o)}\,,
\end{eqnarray}
where $\pm_o=\text{sign}(k_o^\theta)$ represents the sign of $k^\theta$ in the observer's frame, $\lambda=l/\omega$ is a impact parameter with $l$ and $\omega$ being the angular momentum and energy of the photon, respectively. $\Theta(\theta)$ is the angular potential of the photon which reads
\setlength\arraycolsep{2pt} \begin{eqnarray}
\Theta(\theta)=\eta+a^2\cos^2\theta-\lambda^2\cot^2\theta\,,
\end{eqnarray}
where $\eta=\mathcal{Q}/\omega^2$ is the other impact parameter with $\mathcal{Q}$ being the Carter constant in the Kerr spacetime. On the other hand, charged particles as the light source are located on the equatorial plane with coordinates $(t_s, r_s, \pi/2, \phi_s)$. In principle, with the help of null geodesic equations in Kerr spacetime, one can obtain the  trajectories of photons connecting the source and the observer, and then we can find the images of the source, that is, we know the coordinates $(\alpha, \beta)$ on the screen of the observer. We would like to stress that there is more than one trajectory of lights starting at the source and ending at the observer since the lights may turn around the black hole more than once before reaching the observer. We use $m$ to denote the number of times that the trajectory crosses the equatorial plane between the source and the observer, different values of $m$ correspond to the $(m+1)$th image on the observers' screen. In particular, the image formed by the trajectories with $m=0$ is offen called the primary image, the image with $m=1$ is called the secondary image and so on. For more details, one is suggested to refer \cite{hou2022multi}.

Furthermore, we also need obtain the polarization information of the image. Following \cite{hu2022polarization}, the polarization information can be transmitted from the source to the observer by using the Penrose-Walker (PW) constant
\setlength\arraycolsep{2pt} \begin{eqnarray}\label{kappa}
\kappa=2k^\mu f^{\nu}\left(\hat{l}_{[\mu}\hat{n}_{\nu]}-\hat{m}_{[\mu}\hat{\bar{m}}_{\nu]}\right)(r-ia\cos\theta)\equiv\omega(\kappa_1+i\kappa_2)\,,
\end{eqnarray}
which is a conserved quantity along the null geodesics for Kerr spacetime. In the above formula, $\kappa_{1,2}$ are the rescaled real part and imaginary part of PW constant, respectively. And ${\hat{l}, \hat{n}, \hat{m}, \hat{\bar{m}}}$ are the Neumann-Penrose tetrads which can be chosen as
\setlength\arraycolsep{2pt} \begin{eqnarray}
\hat{l}&=&\frac{1}{\sqrt{2\Delta}\Sigma}\left[(r^2+a^2)\partial_t+\Delta\partial_r+a\partial_\phi\right]\,\nonumber\\
\hat{n}&=&\frac{1}{\sqrt{2\Delta}\Sigma}\left[(r^2+a^2)\partial_t-\Delta\partial_r+a\partial_\phi\right]\,\nonumber\\
\hat{m}&=&\frac{1}{\sqrt{2}(r+ia\cos\theta)}\left(ia\sin\theta\partial_t+\partial_\theta+\frac{i}{\sin\theta}\partial_\phi\right)\,,\nonumber\\
\hat{\bar{m}}&=&\frac{1}{\sqrt{2}(r-ia\cos\theta)}\left(-ia\sin\theta\partial_t+\partial_\theta+\frac{i}{\sin\theta}\partial_\phi\right)\,.
\end{eqnarray}
Thus, employing the Eq. (\ref{kappa}), at the observer we can decode the information of $f^\mu$ given at the source and on the polarization vector can be read from the PW constant as \cite{chandrasekhar}
\begin{equation}
\overrightarrow{\mathcal{E}}=\left(\mathcal{E}_\alpha, \mathcal{E}_\beta\right)=\frac{1}{\omega\left(\beta^2+\gamma^2\right)}\left(\beta \kappa_2-\gamma \kappa_1, \beta \kappa_1+\gamma \kappa_2\right), \quad \gamma=-\left(\alpha+a \sin \theta_o\right),
\end{equation}

On the other hand, we still need to know the total fluxes of the image considering charged particles have a finite size. We omit the details and give the result of the total fluxes instead, which takes
\setlength\arraycolsep{2pt} \begin{eqnarray}
F_o=\frac{g^4L}{4\pi r_o^2|\hat{k}|\cdot X}\left|\frac{\partial(\alpha, \beta)}{\partial(Y_s, Z_s)}\right|\,,
\end{eqnarray}
where $\left|\frac{\partial(\alpha, \beta)}{\partial(Y_s, Z_s)}\right|$ denotes the Jacobian determinant between the coordinates $(\alpha, \beta)$ and $(Y_s, Z_s)$. Thereinto, $(Y_s, Z_s)$ are coordinates defined on the `` source screen '' with $T=X=0$. $\{T, X, Y, Z\}$ is a local Minkowski coordinate system in the neighborhood of the source, which is defined as
\setlength\arraycolsep{2pt} \begin{eqnarray}
Te_{(t)}+Xe_{(r)}+Ye_{(\phi)}-Ze_{(\theta)}=(x^\mu-x_\ast^\mu)\partial_\mu\,,
\end{eqnarray}
where $x_\ast^\mu$ are the coordinates of the source and $e_{(\mu)}$ are the tetrad of the source. For a Kerr metric, we would like to choose
\setlength\arraycolsep{2pt} \begin{eqnarray}
e_{(t)}&=&\gamma\sqrt{\frac{\Xi}{\Delta\Sigma}}(\partial_t+\Omega_s\partial_\phi)\,,\quad e_{(r)}=\frac{\Delta}{\Sigma}\partial_r\,,\quad e_{(\theta)}=\frac{1}{\sqrt{\Sigma}}\partial_\theta\,,\nonumber\\
e_{(\phi)}&=&\gamma v_s\sqrt{\frac{\Xi}{\Delta\Sigma}}\left[\partial_t+\frac{2aMr}{\Xi}\right]+\gamma\sqrt{\frac{\Sigma}{\Xi}}
\end{eqnarray}
where we have introduced
\setlength\arraycolsep{2pt} \begin{eqnarray}
\Xi=(r^2+a^2)^2-\Delta a^2\sin^2\theta\,,
\end{eqnarray}
and
\setlength\arraycolsep{2pt} \begin{eqnarray}
v_s=\frac{\Xi}{\Sigma\sqrt{\Delta}}(\Omega_s-\frac{2aMr}{\Xi})\,,\quad\gamma=\frac{1}{\sqrt{1-v_s^2}}\,,
\end{eqnarray}
with $\Omega_s=\frac{d\phi_s}{dt_s}$ being the angular velocity of the source. In addition, the unit vector $\hat{k}$ is given by
\setlength\arraycolsep{2pt} \begin{eqnarray}
\hat{k}=\frac{1}{k^{(t)}}\left(k^{(r)}X+k^{(\phi)}Y-k^{(\theta)}Z\right)\,,
\end{eqnarray}
and $g$ is the redshift factor which is given by
\setlength\arraycolsep{2pt} \begin{eqnarray}
g=\frac{\omega_o}{\omega_s}\,,
\end{eqnarray}
where $\omega_{o,s}$ are the frequencies of the photon in the frame of the observer and the source, respectively. Now, we are ready to calculate some specific polarized images of charged particles orbiting the Kerr black hole with a Wald magnetic field.

\subsection{Results}
Based on our model, in this subsection we would like to show some results for our calculations with the observation angle at $\theta_o=17^\circ$  and $\theta_o=80^\circ$. In our computations of the polarized images, we use the full form of the effective potential to determine the parameters for stable circular orbits numerically, without any approximation. According to the previous analysis, when $\mathcal{B}$ and $a$ are given, there is only one degree of freedom left among $r$, $E$ and $L$ for stable circular orbits. In our following calculations, we usually set $r_s=6$ and compare the results with other different parameters, that is, we fix the radius of the orbit for charged particles and discuss the influence of the other paramers. And the reason why we choose $r_s=6$ is because the ISCO for a neutral particle in a Schwarzschild black hole spacetime is at $r_s=6$ and we still hope to focus on polarized images of the source in this region. Our main interest is in the effects of the spin $a$, the observational angle $\theta_o$ on the polarized images and the difference between the
retrograde or prograde orbiting source \footnote{Note that prograde (retrograde) is defined by $\dot{\phi}>0$ ($\dot{\phi}<0$) instead of $L>0$($L<0$), since the sign of the angular momentum $L$ is not necessarily as same as the sign of $\dot{\phi}$, when there is an external magnetic field.}.
We will vary $a$ and $\theta_o$, and make plots in the retrograde or prograde cases in the following studies. In addition, as pointed out in \cite{hu2022polarization}, empirically, the strength of $\mathcal{B}$ doesn't affect the direction of the polarization vector $f^\mu$ and only changes the overall intensity of flux. It's also the case with Kerr space-time. So we also fix $\mathcal{B}=2$ in our work.

\begin{figure}[h!]
\centering
\includegraphics[width=15cm]{varyth.pdf}
\caption{Comparisons of polarized images between the prograde and retrograde orbits with $\theta_o=17^\circ$, $a=0.94$, $r_s=6$ and $\mathcal{B}=2$. The left plot is for prograde prbit while the right plot is retrograde. The corresponding energy $E$ and angular momentun $L$ for stable circular orbits are also shown in each plot. Blue circles give shapes of the images of charged particles and red lines denote the polarization information. In particular, the length of the red line signifies the magnitude of the total flux. Note the lengths of the flux are scaled in a different way to make the lengths of lines in the two plots are of the same order of magnitude.}\label{figure:rp}
\end{figure}

At first, we look at the difference of the polarized images between the prograde and retrograde orbits. We would like to fix $a=0.94$ which is a potential value of the spin of the supermassive black hole in M87 predicted by EHT \cite{EventHorizonTelescope:2019dse, EventHorizonTelescope:2019jan, EventHorizonTelescope:2019ths, EventHorizonTelescope:2019pgp, EventHorizonTelescope:2019ggy}. The results are shown in Fig. \ref{figure:rp}, in which blue circles are the images of charged particles and red lines denote the polarization directions and total flux with the length. From the Fig. \ref{figure:rp}, one can see that for the prograde case in the left pannel, there is barely any difference between the lengths of red lines on the left and right side. However, for the retrograde case in the right panel of Fig. \ref{figure:rp}, we can find that the fluxes are much stronger on the right side than those on the left side. The main reason for this phenomenon is the so-called headlight effect, that is, the radiations from moving particles concentrate upon the vicinity along the direction of the motion of particles, which becomes more significant when the velocity of moving particles are higher. Qualitatively, when $\mathcal{B}$ and $L$ are in the opposite directions, charged particles would have higher energies to stay in a circular orbit, so that the headlight effect is stronger in the retrograde case. In addition, we can infer that the phenomenon would be more distinct when the observer is closer to the equatorial plane.

\begin{figure}[h!]
\centering
\includegraphics[width=15cm]{varya.pdf}
\caption{Polarized images of charged particles in retrograde orbits are produced with $\theta_o=17^\circ$, $\mathcal{B}=2$ and $r_s=6$. We let the spin $a$ vary from $-0.99$ to $0.99$. The energy $E$ and the angular momentum $L$ are directly calculated from the conditions of the stable circular orbit. Blue lines show the shapes of the images of the stable orbits. The inclinations of red lines denote the polarization directions of radiations, while the lengths represent the total flux. In this set of plots, we set length/flux = 4000 to make the plots more ornamental.}\label{figure:a}
\end{figure}

Next, we turn to study the influence of the spin on the polarized images. In Fig. \ref{figure:a}, we fix $\theta_o=17^\circ$, $r_s=6$ and vary $a$ for retrograde orbits. The ratio of the length to the flux is set to be $200$ for the five plots. We can see that there's evident changes for the magnitude of the flux while the corresponding polarized directions show no significant changes as the spin $a$ increases. In addition, we also checked the prograde orbits and find that when we vary $a$ the changes in polarization and intensity are very minor. So we would not like to present these results here.

\begin{figure}[h!]
\centering
\includegraphics[width=15cm]{varya2.pdf}
\caption{Polarized images of charged particles in prograde orbits are produced with $\theta_o=80^\circ$, $\mathcal{B}=2$ and $r_s=6$. We let the spin $a$ vary from $-0.99$ to $0.99$. In this set of plots, we set length/flux = 4000.}\label{figure:80-pro}
\end{figure}

Now we move on to the case that the observational angle is fixed at $\theta_o=80^\circ$, that is, the observer is close to the equatorial plane, and the spin $a$ is varied from $-0.99$ to $0.99$. The images of orbits deviate from circles a lot in this case. The results for the prograde case are shown in Fig. \ref{figure:80-pro}, where we have scaled the length of the polarization flux by $\text{ length/flux}=4000$. From the plot with $a=0$, it seems that the polarizations of both sides centered on $\alpha=0$ are approximately symmetric which means that the Doppler effect or headlight effect is not obvious. However, there are significant difference of the flux between the upper and lower part of the plot which is considered to be caused by the curvature of the strongly curved spacetime. Meanwhile, at the top of orbit, where $\alpha=0$, the flux of the polarization is much smaller than those in neighboring places. The reason is that the position $(\alpha=0, \beta=\beta_{max})$ on the screen corresponds to the lights that radiate from the source with the four momentum perpendicular to the velocity of charged particles in the local frame of the source, and they are much weaker than radiations going in other directions. This phenomenon can be usually observed when $\theta_o$ is bigger. Moreover, the spin of the Kerr black hole also has large impact on the image. When $a$ is positive, the right side of the image is brighter than the left side and vice versa.

\begin{figure}[h!]
\centering
\includegraphics[width=15cm]{varya3.pdf}
\caption{Polarized images of charged particles in retrograde orbits are produced with $\theta_o=80^\circ$, $r=6$ and $\mathcal{B}=2$. We also let the spin $a$ vary from $-0.99$ to $0.99$. The red line is scaled by taking the logarithm of flux for the sake of visualization, since the Doppler effect is very strong in this case.}\label{figure:80-retro}
\end{figure}

The results for the retrograd case are shown in Figure \ref{figure:80-retro}, where we let $\theta_o=80^\circ$, $r=6$ and $\mathcal{B}=2$ and the spin $a$ go from $-0.99$ to $0.99$. In this case, one of the most obvious phenomena is that the Doppler effect is very strong due to that both the observational angle $\theta_o$ and the energy of the charged particles $E$ are large. Thus for the sake of visualization, we take a logarithm of the flux for the plots. In addition, it is clear that the right side of the plot, where the particle moves towards the observer has much stronger radiations. Also, in this case the black hole spin $a$ has much larger impact on the polarization. Furthermore, in these plots we can see a pronounced clockwise rotation of the polarizations as the spin $a$ increases, while this rotation is unapparent for prograde orbits. Besides, we also find that this rotation happens but it's very small for both prograde and  retrograd orbits when $\theta_o$ is small such as $\theta_o=17^\circ$.

\section{Summary}\label{summary}

In this work, we studied the circular motions and the polarized images of charged particles in the Kerr spacetime with a weak magnetic field. We focused on the case that both the magnetic field and the charge-to-mass ratio are not large, which apply to many charged astrophysical dusts and objects. In this case, the effective potential gets simplified and allows analytical treatment. We found an analytical expression of the radius of the ISCOs in terms of the energy density $E$ and the angular momentum density $L$, and obtained the constraints that $E$ and $L$ should satisfy. It is worth emphasizing that astrophysical objects which satisfy the condition $\mathcal{B}\ll1$ widely exist in the universe and thus our simplified treatment and analytical results are very useful to relevant studies.

Moreover we revisited how to calculate the polarized images including the orbits of charged particles, polarization direction and flux of the synchronize radiations following our previous work \cite{hu2022polarization} to prepare for numerical computations. We obtained the images of circular orbits  through ray-tracing method, and we presented several figures to show  the polarized images of charged particles at fixed $\mathcal{B}$ and $r_s$, where $r_s$ was the orbit radius of the source. We would like to stress that in studying the polarized images with the numerical method we did not assume the approximate condition $\mathcal{B}\ll1$,  and our polarized images were not limited to the case $\mathcal{B}\ll1$.

From our study, we read the influences of the orbit directions, the spins of the black hole and the observational angles on the polarized images. The images for the prograde and retrograde orbits are very different for the same orbit radius. For example, we found that the so-called headlight effect was pretty obvious for the retrograde orbits. In addition, we observed a significant influence of the spin $a$ on the polarized images. More precisely, when $a$ increases,  the angle of polarization rotates clockwise.  This phenomenon becomes especially visible   for the retrograde orbits. The spin $a$ has a non-negligible effect on the magnitudes of the polarization fluxes as well. Moreover, we showed the impacts of the observational angle on the polarized images.

It would be illuminating to compare our results with the ones in in \cite{Gelles:2021kti}, which also studied the polarized images of equatorial emission in the Kerr geometry. In our study, we focused on a vertical magnetic field corresponding to $(B^{r}=0, B^{\phi}=0, B^{(\theta)})$ in \cite{Gelles:2021kti}, however, our results were closer to the ones in their case $(B^{r}=0, B^{\phi}, B^{(\theta)}=0)$. The main reason is that the synchrotron radiations in our model are purely from charged particles without self-interactions,  while \cite{Gelles:2021kti} assumed that the synchrotron radiations were from charged fluids, and thus the formulas of the polarization vector of the synchrotron radiation are different.
Moreover, the formula Eq. (\ref{pve}) for the polarization vector, which was directly derived from the Maxwell equation in a curved spacetime, takes a covariant form, and can be used in the numerical studies easily. In contrast, the polarization vector in \cite{Gelles:2021kti} is determined in a local frame.

\section*{Acknowledgments}
We would like to thank Y.H. Hou, Y.Shen, H.P. Yan and Z.Y. Zhang for valuable discussions. The work is in part supported by NSFC Grant  No. 12275004, 11775022 and 11873044. MG is also supported by ``the Fundamental Research Funds for the Central Universities'' with Grant No. 2021NTST13.

redpajama_set_name: RedPajamaArXiv


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