# Magnetic field induced hair structure in charmonium gluon-dissociation

Jin Hu1 Shuzhe Shi2,3 Zhe Xu1 Jiaxing Zhao1 Pengfei Zhuang1 1Physics
Department, Tsinghua University, Beijing 100084, China
2Department of Physics, McGill University, Montreál, QC H3A 2T8, Canada
3Center for Nuclear Theory, Department of Physics and Astronomy, Stony Brook
University, Stony Brook, New York, 11784, USA

###### Abstract

We study electromagnetic field effect on charmonium gluon-dissociation in
quark-gluon plasma. With the effective Hamiltonian derived from QCD multipole
expansion under an external electromagnetic field, we first solve the two-body
Schrödinger equation for a pair of charm quarks with mean field potentials for
color and electromagnetic interactions and obtain the charmonium binding
energies and wave functions, and then calculate the gluon-dissociation cross-
section and decay width by taking the color electric and magnetic dipole
interactions as perturbations above the mean field and employing Fermi’s
Golden Rule. Considering the charmonium deformation in magnetic field, the
discrete Landau energy levels make the dissociation cross-section grow hair,
and the electric dipole channel is significantly changed, especially for the
$P-$wave states $\chi_{c0}$ and $\chi_{c\pm}$. From our numerical calculation,
the magnetic field strength $eB=5~{}m_{\pi}^{2}$ already changes the gluon
dissociation strongly, which may indicate measurable effects in high-energy
nuclear collisions.

##  I Introduction

It is widely accepted that the strongest electromagnetic field in nature can
be created in non-central relativistic heavy-ion collisions Skokov:2009qp ;
Voronyuk:2011jd ; WTDeng:2012prc ; Tuchin:2013ie . In Au-Au collisions at
Relativistic Heavy Ion Collider (RHIC) the peak value of the magnetic field is
around $eB\sim 5~{}m_{\pi}^{2}$, and in Pb-Pb collisions at Large Hadron
Collider (LHC) the value even reaches $eB\sim 70~{}m_{\pi}^{2}$ WTDeng:2012prc
, where $m_{\pi}$ is the pion mass in vacuum. While such strong
electromagnetic field can bring us many fantastic topics in quantum
chromodynamics (QCD) physics, such as chiral magnetic effect Kharzeev:2008npa
; Fukushima:2008xe  and inverse magnetic catalysis Shovkovy:2012zn ;
Bruckmann:2013oba , the initially produced field decays very fast and survives
only in the very beginning of the collisions, although the attenuation is
delayed slightly as quark-gluon plasma (QGP) appears afterward Tuchin:2013ie ;
Gursoy:2014aka ; Yan:2021zjc ; Chen:2021nxs ; Wang:2021oqq .

Heavy quarks are probably an ideal probe of the short-lived electromagnetic
field due to the fact that they are produced at the very early stage of heavy-
ion collisions too. The difference in the directed flow between $D^{0}$ and
$\bar{D}^{0}$ may come from the electromagnetic field Adam:2019wnk ;
Acharya:2019ijj ; Das:2016cwd , and the quarkonium static properties such as
the mass and shape are changed sizeably in the field Marasinghe:2011bt ;
Alford:2013jva ; Machado:2013rta ; Cho:2014exa ; Guo:2015nsa ; Bonati:2015dka
; Bonati:2017uvz ; Yoshida:2016xgm ; Zhao:2020jqu ; Mishra:2020kts ;
Chen:2020xsr ; Iwasaki:2021nrz . The field affects also the quarkonium
dissociation in hot medium Singh:2017nfa ; Hasan:2018kvx ; Hasan:2017fmf ;
Hasan:2020iwa . Different from the color screening picture Karsch:1987pv
based on calculations at mean field level, the dissociation processes which
originate from the scattering between quarkonia and thermal partons might be
realistic dynamics for quarkonium suppression in high energy nuclear
collisions. There are two kinds of dissociation processes, one is gluon
dissociation ($g+\Psi\to Q+\bar{Q}$), the other is inelastic parton scattering
($p+\Psi\to Q+\bar{Q}+p$), where $g$ and $p$ represent gluons and partons. The
former is dominant in the temperature region where the Debye mass is much
smaller than the binding energy, and the latter is essential when the
quarkonium becomes a loosely bound state Brambilla:2013dpa ; Brambilla:2011sg
. When the external electromagnetic field is turned on, the Landau-damping
leads to an increasing decay width in the inelastic scattering processes
Singh:2017nfa ; Hasan:2018kvx ; Hasan:2017fmf ; Hasan:2020iwa .

The gluon dissociation describes the process of a color-singlet state
converting to a color-octet state by absorbing a gluon Brambilla:2011sg . The
cross section in vacuum neglecting the color-octet interaction in the final
state was calculated firstly by Bhanot and Peskin via the operator-product-
expansion (OPE) method Peskin:1979va ; Bhanot:1979vb . Peskin’s perturbative
analysis can be represented by a gauge-invariant effective action from which
one can get a non-relativistic Hamiltonian for heavy quark systems via QCD
multipole expansion Yan:1980uh ; Kuang:1981se ; Kuang:2006me . Based on this
effective Hamiltonian, the cross section of gluon dissociation in hot medium
are derived in the frame of perturbation theory of quantum mechanics
Liu:2013kkg ; Chen:2017jje ; Chen:2018dqg . The result in the Coulomb
approximation is in consistent with the OPE .

The goal of this paper is to study the electromagnetic field effect on the
gluon dissociation process and the charmonium decay width in QGP. We first
introduce in Section II the framework of QCD multipole expansion, including an
external electromagnetic field. We then systematically solve the two-body
Schrödinger equation for a pair of charm quarks at finite temperature. At mean
field level the solution of the equation gives the magnetic field dependence
of the static properties of the $c\bar{c}$ bound states, shown in Section III.
Above the mean field we focus in Section IV on the magnetic field effect on
the gluon-dissociation cross-section and calculate the corresponding decay
width, by taking the color electric and magnetic dipole interactions as
perturbations and employing Fermi’s Golden Rule. We summarize in Section V.

##  II QCD multipole expansion

Multipole expansion is widely used for studying radiation processes in
classical electrodynamics Bhanot:1979af ; Pineda:1997ie ; Brambilla:2017uyf .
Considering the large mass and slowly moving of heavy quarks, a heavy flavor
system can be treated non-relativistically, and a multipole expansion of the
changing gluon field converges rapidly Gottfried:1977gp . The method has been
successfully used to calculate hadronic transition rates for both charm and
bottom systems Gottfried:1977gp ; Yan:1980uh ; Kuang:1981se . Including an
external electromagnetic field, we start from the gauge-invariant effective
Lagrangian density for heavy quarks, which represents the result of partial
summation of the perturbation series Peskin:1979va ; Yan:1980uh  in the
absence of electromagnetic field,

$\displaystyle\mathcal{L}$ $\displaystyle=$ $\displaystyle\int
d^{3}{\bm{x}}\bar{\psi}^{\prime}(x)\left(i\gamma^{\mu}D_{\mu}-m_{Q}\right)\psi^{\prime}(x)$
(1) $\displaystyle-{1\over 2}{g^{2}\over 4\pi}\sum_{a=0}^{8}\int
d^{3}{\bm{x}}_{1}d^{3}{\bm{x}}_{2}\rho_{a}(x_{1}){1\over|{\bm{r}}|}\rho_{a}(x_{2}),$

where $m_{Q}$ is the heavy quark mass,
$D_{\mu}=\partial_{\mu}+ig\mathcal{A}_{\mu}^{\prime a}+iqA_{\mu}$ is the
covariant derivative with electric charge $q$, strong coupling constant $g$
and two gauge fields, namely gluon field $\mathcal{A}_{\mu}^{a}$ and photon
field $A_{\mu}$. The interaction among heavy quarks here is perturbatively
described by a Coulomb potential between a pair of heavy quarks located at
${\bm{x}}_{1}$ and ${\bm{x}}_{2}$ with relative coordinate
${\bm{r}}={\bm{x}}_{1}-{\bm{x}}_{2}$. To guarantee the gauge invariance, the
heavy quark field $\psi(x_{i})$ and gluon field $\mathcal{A}_{\mu}^{a}(x_{i})$
are transformed to be $\psi^{\prime}(x_{i})=U^{-1}\psi(x_{i})$ and
$\mathcal{A}^{\prime
a}_{\mu}(x_{i})=U^{-1}\mathcal{A}^{a}_{\mu}(x_{i})U-(i/g)U^{-1}\partial_{\mu}U$
through the equal-time gauge link operator
$U(x_{i})=\mathcal{P}e^{ig\int_{\bm{X}}^{{\bm{x}}_{i}}d{\bm{y}}\cdot{\bm{\mathcal{A}}}^{a}(y)}$,
where $\mathcal{P}$ is the path-ordering operator and the line integral is
along the straight-line segment from the center-of-mass coordinate
${\bm{X}}=({\bm{x}}_{1}+{\bm{x}}_{2})/2$ of the pair to the quark (anti-quark)
coordinate ${\bm{x}}_{i}$. Note that, the external electromagnetic field
$A_{\mu}(x)$ does not experience such a transformation because it commutates
with the link operator $U$. The color charge density (vertex factor)
$\rho_{a}$ is defined as
$\rho_{a}(x_{i})=\psi^{\prime{\dagger}}(x_{i})(\lambda_{a}/2)\psi^{\prime}(x_{i})$
with Gell-Mann matrix $\lambda_{a}\ (a=1,...,8$ and $\lambda_{0}/2=1$) . If
the electromagnetic field $A_{\mu}$ is turned off, the effective Lagrangian
becomes the original one in Refs. Yan:1980uh ; Kuang:1981se .

The Coulomb potential in the Lagrangian is only the leading term of the color
interaction between a pair of heavy quarks. Aiming to go beyond the
perturbation theory, one assumes that the heavy quark interaction can be
described by a non-relativistic potential and generalizes the Coulomb
interaction to including the color confinement (Cornell) part in color singlet
state Yan:1980uh ; Kuang:1981se . With this consideration, we replace the
Coulomb potential $g^{2}/(4\pi)/|{\bm{r}}|$ in the above Lagrangian by a
general and radial symmetric potential

$V_{a}(|{\bm{r}}|)=V_{1}(|{\bm{r}}|)\delta_{a0}+V_{2}(|{\bm{r}}|)(1-\delta_{a0}),$
(2)

where $V_{1}$ and $V_{2}$ are the interaction potentials between $Q$ and
$\bar{Q}$ in color-singlet state and color-octet state.

Using the expression for the gauge link operator $U$, the transformed gluon
field can be explicitly expressed as Yan:1980uh

$\displaystyle\mathcal{A}_{0}^{\prime a}(x_{i})$ $\displaystyle=$
$\displaystyle\mathcal{A}_{0}^{a}(x_{i})+\int_{\bm{X}}^{{\bm{x}}_{i}}d{\bm{y}}\cdot{\partial{\bm{\mathcal{A}}}^{a}(y)\over\partial
t},$ $\displaystyle{\bm{\mathcal{A}}}^{\prime a}(x_{i})$ $\displaystyle=$
$\displaystyle{\bm{\mathcal{A}}}^{a}(x_{i})-{\bm{\nabla}}\int_{\bm{X}}^{{\bm{x}}_{i}}d{\bm{y}}\cdot{\bm{\mathcal{A}}}^{a}(y),$
(3)

and expanding further the original field $\mathcal{A}_{\mu}^{a}$ in Taylor
series of ${\bm{x}}_{i}-{\bm{X}}$ at the center-of-mass coordinate ${\bm{X}}$,
one obtains the perturbative expression of $\mathcal{A}_{\mu}^{\prime a}$ in
terms of the color-electric and color-magnetic fields
${\bm{\mathcal{E}}}^{a}=\partial{\bm{\mathcal{A}}}^{a}/\partial t$ and
${\bm{\mathcal{B}}}^{a}={\bm{\nabla}}\times{\bm{\mathcal{A}}}^{a}$,

$\displaystyle\mathcal{A}_{0}^{\prime a}(x_{i})$ $\displaystyle=$
$\displaystyle\mathcal{A}_{0}^{a}(X)-({\bm{x}}_{i}-{\bm{X}})\cdot{\bm{\mathcal{E}}}^{a}(X)+\cdots,$
$\displaystyle{\bm{\mathcal{A}}}^{\prime a}(x_{i})$ $\displaystyle=$
$\displaystyle-({\bm{x}}_{i}-{\bm{X}})\times{\bm{\mathcal{B}}}^{a}(X)/2+\cdots.$
(4)

The effective Lagrangian (1) with the non-perturbative interaction (2) is the
potential version of QCD to treat heavy quark systems and the foundation for
us to calculate the quarkonium gluon-dissociation. If neglecting the color
degrees of freedom and the external electromagnetic field, it returns to the
QED multipole expansion Bhanot:1979af ; Pineda:1997ie ; Brambilla:2017uyf . To
solve the Schrödinger equation for a $Q\bar{Q}$ system, we transfer the
Lagrangian to the Hamiltonian in coordinate representation,

$\displaystyle\hat{H}$ $\displaystyle=$
$\displaystyle\hat{H}_{0}+\hat{H}_{I},$ $\displaystyle\hat{H}_{0}$
$\displaystyle=$ $\displaystyle{(\hat{\bm{p}}_{1}-q{\bm{A}}(x_{1}))^{2}\over
2m_{Q}}+{(\hat{\bm{p}}_{2}+q{\bm{A}}(x_{2}))^{2}\over 2m_{Q}}$ $\displaystyle-
A_{0}(x_{1})-A_{0}(x_{2})+V_{1}(|{\bm{r}}|)+\sum_{a=1}^{8}{\lambda_{a}\over
2}{\bar{\lambda}_{a}\over 2}V_{2}(|{\bm{r}}|),$ $\displaystyle\hat{H}_{I}$
$\displaystyle=$ $\displaystyle
q_{a}\mathcal{A}^{a}_{0}(X)-{\bm{d}}_{a}\cdot{\bm{\mathcal{E}}}^{a}(X)-{\bm{m}}_{a}\cdot{\bm{\mathcal{B}}}^{a}(X)+\cdots,$
(5)

where $\hat{\bm{p}}_{i}=-i{\bm{\nabla}}_{i}$ is the heavy quark (anti-quark)
momentum operator, and

$\displaystyle q_{a}$ $\displaystyle=$ $\displaystyle
g(\lambda_{a}+\bar{\lambda}_{a})/2,$ $\displaystyle{\bm{d}}_{a}$
$\displaystyle=$ $\displaystyle
g({\bm{x}}_{1}-{\bm{x}}_{2})(\lambda_{a}-\bar{\lambda}_{a})/4,$
$\displaystyle{\bm{m}}_{a}$ $\displaystyle=$ $\displaystyle
g/m_{Q}(\lambda_{a}-\bar{\lambda}_{a})({\bm{\sigma}}_{1}-{\bm{\sigma}}_{2})/8$
(6)

are the color monopole, electric dipole and magnetic dipole moments of the
$Q\bar{Q}$ system with the Pauli matrix ${\bm{\sigma}}_{i}$ for heavy quark
and anti-quark. It is clear that, $\hat{H}_{0}$ describes a pair of heavy
quarks moving in a mean field which contains two parts: the strong potentials
$V_{1}$ and $V_{2}$ and electromagnetic potential $\mathcal{A}_{\mu}$, and
$\hat{H}_{I}$ is considered as a perturbation above the mean field. The former
controls the static properties of the $Q\bar{Q}$ bound states, and the latter
characterizes the quarkonium gluon-dissociation into a color-octet state.

Focusing on charmonia (bottom quarks are too heavy and probably not so
sensitive to the electromagnetic field) and taking the standard perturbative
calculation in quantum mechanics, the $c\bar{c}$ transition rate from a
charmonium state into a color octet state via absorbing a gluon at leading
order can be given by Fermi’s Golden Rule, $\Gamma=2\pi|_{8}\langle
c\bar{c}|\hat{H}_{I}|\Psi\rangle|^{2}\rho(E_{c\bar{c}})$, where $|\Psi\rangle$
and $|c\bar{c}\rangle_{8}$ are the initial charmonium bound state and final
octet scattering state, and $\rho(E_{c\bar{c}})$ is the phase-space volume of
the final state with energy $E_{c\bar{c}}$. The transition can be divided into
the color-electric dipole ($E1$) and color-magnetic dipole ($M1$) parts.
Dividing the transition rate by the flux of the incident gluons, one can
obtain the corresponding cross section. Following the procedure in Refs.
Chen:2017jje ; Chen:2018dqg , the cross sections via transition processes $E1$
and $M1$ read

$\displaystyle\sigma_{E1}$ $\displaystyle=$ $\displaystyle{\pi g^{2}E_{g}\over
18}\sum_{n,m,k}|\langle
nmk|{\bm{r}}|\Psi\rangle|^{2}\delta(E_{g}-E_{B}-E_{nmk}),$
$\displaystyle\sigma_{M1}$ $\displaystyle=$ $\displaystyle{\pi g^{2}E_{g}\over
6m_{c}^{2}}\sum_{n,m,k}|\langle
nmk|\Psi\rangle|^{2}\delta(E_{g}-E_{B}-E_{nmk})$ (7)

with the explicit transition matrix elements

$\displaystyle\langle nmk|{\bm{r}}|\Psi\rangle$ $\displaystyle=$
$\displaystyle\int
d^{3}{\bm{r}}\Phi^{*}_{nmk}({\bm{r}}){\bm{r}}\Psi({\bm{r}}),$
$\displaystyle\langle nmk|\Psi\rangle$ $\displaystyle=$ $\displaystyle\int
d^{3}{\bm{r}}\Phi^{*}_{nmk}({\bm{r}})\Psi({\bm{r}}),$ (8)

where $E_{g}$ is the incident gluon energy, $E_{B}$ and $\Psi({\bm{r}})$ are
the binding energy and wave function of the charmonium state $|\Psi\rangle$,
and $E_{nmk}$ and $\Phi_{nmk}({\bm{r}})$ are the relative energy and wave
function of the $c\bar{c}$ pair in color-octet state. The $\delta-$function
guarantees the energy conservation in the transition processes.

Before we solve the relative motion for the charmonium state and octet state
in Section III and then calculate the charmonium dissociation cross section in
Section IV, we simply point out the external electromagnetic field effect on
the cross section. While the perturbative Hamiltonian $\hat{H}_{I}$ is
electromagnetic field independent, the initial and final states $|\Psi\rangle$
and $|c\bar{c}\rangle_{8}$ of the transition are both the field dependent.
Especially, for the color octet state $|c\bar{c}\rangle_{8}$, it is no longer
a bound state of strong interaction, but probably a bound state of
electromagnetic interaction in the plane perpendicular to the magnetic field
Chen:2020xsr . That is the reason why we describe the octet state
$|c\bar{c}\rangle_{8}=|nmk\rangle$ with two discrete quantum numbers $n$ and
$m$ for the transverse bound state and a continuous momentum $k$ for the
longitudinal motion. Therefore, the summation over the final state energy
means a summation over $n$ and $m$ and an integration over $k$,
$\sum_{n,m,k}=\sum_{n,m}\int dk$.

##  III Static properties of $c\bar{c}$ pairs

Both the charmonium state $|\Psi\rangle$ and octet state $|nmk\rangle$ are
determined by the main Hamiltonian $\hat{H}_{0}$. We first consider the
Schröedinger equation for the charmonium state $|\Psi\rangle$ at finite
temperature $T$ and under external magnetic field $B$,

$\hat{H}_{0}|\Psi\rangle=E|\Psi\rangle.$ (9)

Taking the symmetric gauge for electromagnetic field
$A_{\mu}=(-{\bm{E}}\cdot{\bm{x}},({\bm{B}}\times{\bm{x}})/2)$, and making
transformation from the coordinates ${\bm{x}}_{1}$ and ${\bm{x}}_{2}$ to the
center-of-mass and relative coordinates ${\bm{X}}$ and ${\bm{r}}$ and from the
quark momenta ${\bm{p}}_{1}$ and ${\bm{p}}_{2}$ to their total and relative
momenta ${\bm{P}}={\bm{p}}_{1}+{\bm{p}}_{2}$ and
${\bm{p}}=({\bm{p}}_{1}-{\bm{p}}_{2})/2$, the total kinetic energy in
$\hat{H}_{0}$ becomes

${(\hat{\bm{p}}_{1}-q{\bm{A}}(x_{1}))^{2}\over
2m_{c}}+{(\hat{\bm{p}}_{2}+q{\bm{A}}(x_{2}))^{2}\over
2m_{c}}={\hat{\bm{P}}_{kin}^{2}\over 4m_{c}}+{\hat{\bm{p}}^{\prime 2}\over
m_{c}}$ (10)

with kinetic momentum ${\bm{P}}_{kin}={\bm{P}}-q{\bm{B}}\times{\bm{r}}/2$ and
modified relative momentum
${\bm{p}}^{\prime}={\bm{p}}-q{\bm{B}}\times{\bm{X}}/2$. While the kinetic
momentum ${\bm{P}}_{kin}$ and total momentum ${\bm{P}}$ are not conserved in
electromagnetic field with $\left[\hat{\bm{P}},\hat{H}_{0}\right]\neq 0$ and
$\left[\hat{\bm{P}}_{kin},\hat{H}_{0}\right]\neq 0$, the pseudo-momentum
${\bm{P}}_{ps}={\bm{P}}+q{\bm{B}}\times{\bm{r}}/2$ is a conserved quantity
with $\left[\hat{\bm{P}}_{ps},\hat{H}_{0}\right]=0$ Alford:2013jva . Keeping
this in mind, one factorizes the total wave function as
$e^{i({\bm{P}}_{ps}-q{\bm{B}}\times{\bm{r}}/2)\cdot{\bm{X}}}\Psi({\bm{r}})$.
Substituting this factorization into the Schrödinger equation (9), one derives
the equation controlling the relative energy
$E_{\Psi}=E-{\bm{P}}_{ps}^{2}/(4m_{c})$ and wave function $\Psi({\bm{r}})$,

$\displaystyle\bigg{[}{\hat{\bm{p}}^{2}\over
m_{c}}+{q^{2}({\bm{B}}\times{\bm{r}})^{2}-2q({\bm{P}}_{ps}\times{\bm{B}})\cdot{\bm{r}}\over
4m_{c}}$ $\displaystyle\ \
-{\bm{E}}\cdot{\bm{r}}+V_{1}({\bm{r}})\bigg{]}\Psi({\bm{r}})=E_{\Psi}\Psi({\bm{r}}).$
(11)

The equation has been solved in previous studies for both charmonium and
bottomonium systems Marasinghe:2011bt ; Alford:2013jva ; Machado:2013rta ;
Cho:2014exa ; Guo:2015nsa ; Bonati:2015dka ; Bonati:2017uvz ; Yoshida:2016xgm
; Zhao:2020jqu ; Mishra:2020kts ; Chen:2020xsr ; Iwasaki:2021nrz . Considering
the fact that, the electromagnetic field breaks down the central symmetry, the
orbital angular momentum is no longer conserved even the strong potentials
$V_{1}$ and $V_{2}$ are radial symmetric. Therefore, one can not further
separate the relative wave function into a radial part and the eigen state
$Y_{lm}(\theta,\varphi)$ of the orbital angular momentum. In this case a
straightforward way to solve the relative equation is to expand the wave
function in terms of $Y_{lm}$,

$r\Psi({\bm{r}})=\sum_{l,m}\phi_{lm}(r)Y_{lm}(\theta,\varphi).$ (12)

To simplify the calculation, we consider in the following only magnetic field
and neglect the electric field. For convenience, we take the magnetic field to
be in the $z$-direction ${\bm{B}}=B{\bm{e}}_{z}$ and the transverse pseudo-
momentum in the $y$-direction
${\bm{P}}_{ps}^{\perp}=P_{ps}^{\perp}{\bm{e}}_{y}$. Under this choice, the
Lorentz potential and the quadratic term in the relative equation become
$-q({\bm{P}}_{ps}\times{\bm{B}})\cdot{\bm{r}}/(2m_{c})=qBP_{ps}^{\perp}r\sin\theta\sin\varphi/(2m_{c})$
and
$q^{2}({\bm{B}}\times{\bm{r}})^{2}/(4m_{c})=q^{2}B^{2}r^{2}\sin^{2}\theta/(4m_{c})$.
Expanding the functions $\sin^{2}\theta Y_{lm}$ and $\sin\theta\sin\varphi
Y_{lm}$ in terms of $Y_{lm}$, one obtains the equations for the radial
functions $\phi_{lm}(r)$,

$\displaystyle\bigg{[}-{d^{2}\over dr^{2}}+m_{c}V_{1}(r)+{U\over
r^{2}}+{q^{2}B^{2}V\over 4}r^{2}+{qBP_{ps}^{\perp}W\over 2}r$ $\displaystyle\
\ -m_{c}E_{\Psi}\bigg{]}R(r)=0$ (13)

with the coefficient matrices

$\displaystyle U$ $\displaystyle=$ $\displaystyle
l(l+1)\delta_{ll^{\prime}}\delta_{lm^{\prime}},$ $\displaystyle V$
$\displaystyle=$ $\displaystyle
u_{lm}\delta_{ll^{\prime}}\delta_{mm^{\prime}}-v_{lm}\delta_{l+2,l^{\prime}}\delta_{mm^{\prime}}-v_{l-2,m}\delta_{l-2,l^{\prime}}\delta_{mm^{\prime}},$
$\displaystyle W$ $\displaystyle=$ $\displaystyle
w_{l-1,-m-1}\delta_{l-1,l^{\prime}}\delta_{m+1,m^{\prime}}-w_{lm}\delta_{l+1,l^{\prime}}\delta_{m+1,m^{\prime}}$
$\displaystyle+w_{l-1,m-1}\delta_{l-1,l^{\prime}}\delta_{m-1,m^{\prime}}-w_{l,-m}\delta_{l+1,l^{\prime}}\delta_{m-1,m^{\prime}},$
$\displaystyle u_{lm}$ $\displaystyle=$
$\displaystyle{2(l^{2}+l-1+m^{2})\over(2l-1)(2l+3)},$ $\displaystyle v_{lm}$
$\displaystyle=$ $\displaystyle{1\over
2l+3}{\sqrt{((l+1)^{2}-m^{2})((l+2)^{2}-m^{2})}\over\sqrt{(2l+1)(2l+5)}},$
$\displaystyle w_{lm}$ $\displaystyle=$
$\displaystyle{\sqrt{(l+m+1)(l+m+2)}\over 2i\sqrt{(2l+1)(2l+3)}}$ (14)

and the radial wave function vector

$R(r)=(\phi_{00}(r),\phi_{1,-1}(r),\phi_{10}(r),\phi_{11}(r),...)^{T}.$ (15)

Since the matrices $V$ and $W$ are with off-diagonal elements, this is a group
of coupled equations for, in principle, all the radial functions. In a
realistic calculation, a cut-off of the orbital angular momentum is needed,
$l\leq l_{max}$. We choose $l_{max}=7$ and solve the radial equation via the
inverse power method H.W. Crater .

Like usual treatment Liu:2013kkg ; Chen:2017jje ; Chen:2018dqg , we have
neglected in the relative equation the potential $V_{2}$ in color octet state.
In vacuum the potential $V_{1}$ in color singlet state is often taken as the
Cornell form,

$V_{1}(|{\bm{r}}|)=-{\alpha\over|{\bm{r}}|}+\sigma|{\bm{r}}|.$ (16)

The eigen value of the radial equation (III) determines the charmonium mass
$M_{\Psi}=2m_{c}+E_{\Psi}$ at zero temperature. Taking the charm quark mass
$m_{c}=1.29$ GeV, by fitting the experimentally measured charmonium masses at
vanishing magnetic field, the two parameters in the potential are fixed to be
$\alpha=0.4105$ and $\sigma=0.2~{}\text{GeV}^{2}$ Zhao:2020jqu . When the
magnetic field is turned on, the central symmetry is broken by the field, the
energy levels of the P-wave states with different magnetic quantum number $m$
will no longer degenerate. For instance, the $\chi_{c}$ state split into three
states $\chi_{c0}$, $\chi_{c+}$, and $\chi_{c-}$, corresponding to the
magnetic quantum number $m=0,1,-1$. On the other hand, if we take the
conserved pseudo-momentum ${\bm{P}}_{ps}=0$, the rotational symmetry around
the $z$-axis is restored, which leads to the degeneration of the two states
$\chi_{c+}$ and $\chi_{c-}$. The masses of $J/\psi$, $\psi(2S)$, $\chi_{c0}$,
and $\chi_{c\pm}$ states are shown in Fig. 1. It is clear that all the
charmonium masses increase with the magnetic field, due to the attractive
quadratic potential in the relative equation (at ${\bm{P}}_{ps}=0$ this is the
only electromagnetic potential). The result is similar to the previous study
Alford:2013jva .

Figure 1: The charmonium mass $M_{\Psi}$ as a function of magnetic field $eB$
at vanishing temperature and pseudo-momentum $T=0$ and ${\bm{P}}_{ps}=0$.

We now turn to the calculation at finite temperature. Due to the many-body
interaction in hot medium, the potential between $c$ and $\bar{c}$ is
screened. When the screening length (screening mass) is short (large) enough,
the charmonium state is melted by the medium. At very high temperature, the
hard-thermal loop (HTL) calculation shows that the potential is modified by a
screening factor $e^{-m_{D}r}$ with the Debye mass $m_{D}$ Laine:2006ns . For
the QGP at finite temperature the potential is simulated by lattice QCD
Burnier:2014ssa ; Burnier:2015tda . Based on the Gauss-law approach by using
the permittivity obtained from the HTL approximation to modify the non-
perturbative vacuum potential, one takes the finite temperature potential
$V_{1}$ as Lafferty:2019jpr ,

$\displaystyle V_{1}(T,r)$ $\displaystyle=$
$\displaystyle-\alpha\left[m_{D}+{e^{-m_{D}r}\over r}\right]$ (17)
$\displaystyle+{\sigma\over m_{D}}\left[2-(2+m_{D}r)e^{-m_{D}r}\right],$

and the temperature dependent Debye mass $m_{D}(T)$ is obtained by fitting the
lattice data Burnier:2014ssa ; Burnier:2015tda . The influence of the magnetic
field on Debye mass is neglected here, since the change is very small
Singh:2017nfa ; Hasan:2018kvx ; Hasan:2017fmf .

Figure 2: The charmonium binding energy $\epsilon$ and longitudinal and
transverse mean square radii $\langle z^{2}\rangle$ and
$\langle\rho^{2}\rangle$ as functions of temperature $T$ at vanishing pseudo-
momentum ${\bm{P}}_{ps}=0$. The temperature and radii are scaled by their
values at the deconfinement phase transition temperature $T_{c}$. The thin and
thick solid lines are $\epsilon$ and $\langle z^{2}\rangle$ at $eB=0$ and
$5~{}m_{\pi}^{2}$, and the dashed line is $\langle\rho^{2}\rangle$ at
$eB=5~{}m_{\pi}^{2}$.

At finite temperature, the long-distance part of the potential is suppressed
by the hot medium and becomes saturated with the value
$V_{1}(T,\infty)=-\alpha m_{D}+2\sigma/m_{D}$. Therefore, the charmonium
binding energy relative to the saturated potential is redefined as
$\epsilon=E_{\Psi}-V_{1}(T,\infty)$. The temperature and magnetic field
dependence of the binding energy and mean square radii $\langle z^{2}\rangle$
in longitudinal direction and $\langle\rho^{2}\rangle=\langle
x^{2}\rangle+\langle y^{2}\rangle$ in transverse plane are shown in Fig. 2,
again the conserved pseudo-momentum is taken to be zero ${\bm{P}}_{ps}=0$.
Since what we are interested in is the charmonium behavior in the QGP phase,
the temperature we considered here is above the critical temperature
$T_{c}=172$ MeV Lafferty:2019jpr  of deconfinement phase transition. Let’s
first consider the pure temperature effect, see the thin solid lines. The
binding energy, which is negative, approaches to zero gradually and becomes
saturated at the melting temperature $T_{m}$ with $\epsilon(T_{m})=0$.
Correspondingly, the mean square radii $\langle z^{2}\rangle$ and
$\langle\rho^{2}\rangle$, which are the same due to the radial symmetry of the
system in the absence of magnetic field, increase with temperature and go to
infinity at $T_{m}$. Obviously, the excited states $\psi(2S)$ and $\chi_{c}$
are easier to be melted than the ground state $J/\psi$, and the three $P-$wave
states $\chi_{c0}$ and $\chi_{c\pm}$ are degenerate in the absence of magnetic
field.

Different from the strong interaction ($V_{1}$) which is suppressed by the hot
medium, the external magnetic field is temperature independent, and its effect
on the $c\bar{c}$ pair above the melting temperature $T_{m}$ becomes the
dominant interaction. When the magnetic field is turned on, while the mean
square radius $\langle z^{2}\rangle$ still goes to infinity at high enough
temperature, the magnetic interaction confines the pair motion in the
transverse plane and makes the mean square radius $\langle\rho^{2}\rangle$
finite at any temperature Chen:2020xsr . Therefore, the melting temperature
$T_{m}$ is in fact a transition temperature for the $c\bar{c}$ pair to change
from a bound state of strong interaction to a transverse bound state of
electromagnetic interaction. The melting temperature (transition temperature)
$T_{m}$ can then be defined through the divergence of the longitudinal size
$\langle z^{2}\rangle(T_{m})\to\infty$ and the saturation of binding energy
and transverse size $\epsilon(T\geq T_{m})=const$ and
$\langle\rho^{2}\rangle(T\geq T_{m})=const$, see the horizontal lines in Fig.
2.

To determine the saturation values, we now turn to calculate the relative
energy and wave function $E_{nmk}$ and $\Phi_{nmk}$ for the octet state of
$c\bar{c}$ pairs. When the strong interaction potential $V_{1}$ disappears,
the $c\bar{c}$ pair is controlled only by the magnetic field. The relative
Hamiltonian can be written as

${p_{x}^{2}\over m_{c}}+{q^{2}B^{2}\over 4m_{c}}x^{2}+{p_{y}^{2}\over
m_{c}}+{q^{2}B^{2}\over 4m_{c}}\left(y-{P_{ps}^{\perp}\over
qB}\right)^{2}+{p_{z}^{2}\over m_{c}}.$ (18)

It is clear that the relative motion can be separated into a two dimensional
harmonic oscillator in the $x-y$ plane and a plain wave in the $z$ direction.
The eigen value $E_{nmk}$ and eigen function $\Phi_{nmk}({\bm{r}})$ of the
Hamiltonian can analytically be expressed as

$\displaystyle E_{nmk}=(2n+|m|+1){qB\over m_{c}}+{k^{2}\over m_{c}},$ (19)
$\displaystyle\Phi_{nmk}({\bm{r}})=N_{nm}{e^{ikz}\over\sqrt{2\pi}}\rho^{|m|}L_{n}^{(|m|)}(qB\rho^{2}/2)e^{-qB\rho^{2}/4}e^{im\varphi}$

with the normalization factor $N_{nm}=\sqrt{n!(qB/2)^{|m|+1}/(n+|m|)!/\pi}$,
where $k$ is the continuous momentum describing the plane wave in the
$z-$direction, the main and magnetic quantum numbers $n$ and $m$ characterize
the transverse wave function, and $L_{n}^{(|m|)}$ is the associated Laguerre
polynomials. The transverse radius $\rho$ and azimuth angle $\varphi$ are
defined through $x=\rho\cos\varphi$ and
$y=P_{ps}^{\perp}/(qB)+\rho\sin\varphi$, and the wave function satisfies the
orthogonal condition

$\int
d^{3}{\bm{r}}\Phi^{*}_{nmk}({\bm{r}})\Phi_{n^{\prime}m^{\prime}k^{\prime}}({\bm{r}})=\delta_{nn^{\prime}}\delta_{mm^{\prime}}\delta(k-k^{\prime}).$
(20)

With the relative energy level $E_{nmk}$, one can determine the saturation
values of the binding energy $\epsilon$ and transverse mean squared radius
$\langle\rho^{2}\rangle$ of the charmonium state. They are controlled by the
corresponding lowest Landau energy level,

$\displaystyle\epsilon(T_{m})=(1+|m|){qB\over m_{c}},$
$\displaystyle\langle\rho^{2}\rangle(T_{m})=(1+|m|){2\over qB}.$ (21)

The saturated binding energy increases linearly with the magnetic field, and
self-consistently the saturated transverse size decreases linearly with the
field, which mean a more and more tight $c\bar{c}$ bound state of
electromagnetic interaction in the transverse plane.

##  IV Charmonium gluon-dissociation

To calculate the gluon dissociation cross sections (II), we need the wave
functions $\Psi$ and $\Phi_{nmk}$ and the binding energies $E_{B}$ and
$E_{nmk}$ for the initial charmonium and final octet states. $\Psi,\Phi_{nmk}$
and $E_{nmk}$ are calculated in the last section. The charmonium binding
energy at finite temperature $\epsilon(T)=E_{\Psi}(T)-V_{1}(T,\infty)$ is
relative to the saturated strong potential. Considering the fact that the
electromagnetic interaction makes the binding energy nonzero above the
dissociation temperature, the charmonium binding energy $E_{B}$ defined
through the energy conservation in dissociation cross sections (II) should be

$\displaystyle E_{B}(T)$ $\displaystyle=$
$\displaystyle-\left[\epsilon(T)-\epsilon(T_{m})\right]$ (22) $\displaystyle=$
$\displaystyle-\left[E_{\Psi}(T)-V_{1}(T,\infty)-\epsilon(T_{m})\right],$

when both the strong and electromagnetic interactions are taken into account.
In this case the binding energy $E_{B}$ satisfies the physics: it vanishes
above the dissociation temperature, $E_{B}(T>T_{m})=0$.

When the charmonia are at rest with ${\bm{P}}_{ps}=0$, the expansion for the
relative wave function (12) is reduced to

$\Psi({\bm{r}})=\sum_{l=0}^{\infty}\sqrt{{2l+1\over
4\pi}}\phi_{l}(r)P_{l}(\cos\theta)$ (23)

for the $S$-wave states $J/\psi$ and $\psi(2S)$ with even $l$ and $P-$wave
state $\chi_{c0}$ with odd $l$, and

$\Psi({\bm{r}})=\sum_{l=1}^{\infty}\sqrt{{2l+1\over 4\pi
l(l+1)}}\phi_{l}(r)P_{l}^{(1)}(\cos\theta)(e^{-i\varphi}\pm e^{i\varphi})$
(24)

for the $P-$wave states $\chi_{c\pm}$ with odd $l$.

Substituting the expansion (23) for $J/\psi$, $\psi(2S)$ and $\chi_{c0}$ into
the transition elements (II) and using the explicit expression for the octet
state $\Phi_{nmk}({\bm{r}})$ (19) with $\rho=r\sin\theta$ and $z=r\cos\theta$,
the integration over the azimuth angle $\varphi$ leads to the selection rules:
the transition elements $\langle nmk|z|\Psi\rangle$ and $\langle
nmk|\Psi\rangle$ are always zero unless $m=0$, and the elements $\langle
nmk|x|\Psi\rangle$ and $\langle nmk|y|\Psi\rangle$ are always zero unless
$m=\pm 1$. Since gluon carries spin $1$ and its $z$ component is $1$, $0$ and
$-1$, the physics behind the selection rules is the conservation of the $z$
component of total angular momentum for charmonium states with zero $z$
component of orbital angular momentum. From the $m-$dependence of the wave
function $\Phi_{nmk}$, the nonzero transition elements depend only on $|m|$.
This means that there are only one independent transition element $T_{n}$ for
channel $M1$ and two independent elements $T_{nz}$ and $T_{n\rho}$ for channel
$E1$,

$\displaystyle T_{n}(k)=N_{n0}\sum_{l,r,x}r^{2}G_{nl}^{(0)}(r,x,k),$ (25)
$\displaystyle T_{nz}(k)=N_{n0}\sum_{l,r,x}r^{3}xG_{nl}^{(0)}(r,x,k),$
$\displaystyle
T_{n\rho}(k)=N_{n1}\sum_{l,r,x}r^{4}(1-x^{2})G_{nl}^{(1)}(r,x,k)$

with the definition of
$\sum_{l,r,x}=\sum_{l}\int_{0}^{\infty}dr\int_{-1}^{1}dx$ and

$\displaystyle G_{nl}^{(i)}(r,x,k)$ $\displaystyle=$
$\displaystyle\sqrt{2l+1\over 2}\phi_{l}(r)e^{-qBr^{2}(1-x^{2})/4}$ (26)
$\displaystyle\times L_{n}^{(i)}(qBr^{2}(1-x^{2})/2)P_{l}(x)e^{ikrx}$

for $i=0,1$.

We take then the integration over the longitudinal momentum $k$ in the
dissociation cross sections. By employing the relation for the
$\delta-$function,

$\displaystyle\int dkF(k)\delta(E_{g}-E_{B}-E_{nmk})$ (27) $\displaystyle=$
$\displaystyle{m_{c}\over 2k_{nm}}\left[F(k_{nm})+F(-k_{nm})\right]$

with

$k_{nm}=\sqrt{m_{c}(E_{g}-E_{B}-E_{nm0})}$ (28)

for any function $F(k)$, the cross sections (II) for charmonium states
$J/\psi$, $\psi(2S)$ and $\chi_{c0}$ are simplified as

$\displaystyle\sigma_{E1}$ $\displaystyle=$ $\displaystyle{\pi g^{2}E_{g}\over
18}\sum_{n}\left[{m_{c}\over k_{n0}}|T_{nz}(k_{n0})|^{2}+{m_{c}\over
k_{n1}}|T_{n\rho}(k_{n1})|^{2}\right],$ $\displaystyle\sigma_{M1}$
$\displaystyle=$ $\displaystyle{\pi g^{2}E_{g}\over
6m_{c}^{2}}\sum_{n}{m_{c}\over k_{n0}}|T_{n}(k_{n0})|^{2}.$ (29)

For the $P-$wave states $\chi_{c\pm}$, similar calculations can be done.
Substituting the expansion (24) into the transition elements (II), the
integration over the azimuth angle $\varphi$ is controlled by the selection
rules: only for the quantum number $m=\pm 1$ the transition elements $\langle
nmk|z|\Psi\rangle$ and $\langle nmk|\Psi\rangle$ are not zero, and only for
$m=0$ and $\pm 2$ the elements $\langle nmk|x|\Psi\rangle$ and $\langle
nmk|y|\Psi\rangle$ are not zero. The physics is again the conservation of the
$z$ component of total angular momentum for charmonium states with $z$
component of orbital angular momentum $\pm 1$. Again the nonzero transition
elements are $|m|-$dependent, there are only one independent transition
element $\mathcal{T}_{n}$ for channel $M1$ and three independent elements
$\mathcal{T}_{nz},\mathcal{T}_{n\rho}^{(0)}$ and $\mathcal{T}_{n\rho}^{(2)}$
for channel $E1$,

$\displaystyle\mathcal{T}_{n}(k)=N_{n1}\sum_{l,r,x}r^{3}(1-x^{2})^{1/2}\mathcal{G}_{nl}^{(1)}(r,x,k),$
$\displaystyle\mathcal{T}_{nz}(k)=N_{n1}\sum_{l,r,x}r^{4}(1-x^{2})^{1/2}\mathcal{G}_{nl}^{(1)}(r,x,k),$
$\displaystyle\mathcal{T}_{n\rho}^{(0)}(k)=N_{n0}\sum_{l,r,x}r^{3}(1-x^{2})^{1/2}\mathcal{G}_{nl}^{(0)}(r,x,k),$
$\displaystyle\mathcal{T}_{n\rho}^{(2)}(k)=N_{n2}\sum_{l,r,x}r^{5}(1-x^{2})^{3/2}\mathcal{G}_{nl}^{(2)}(r,x,k)$
(30)

with

$\displaystyle\mathcal{G}_{nl}^{(i)}(r,x,k)$ $\displaystyle=$
$\displaystyle\sqrt{2l+1\over 2l(l+1)}\phi_{l}(r)e^{-qBr^{2}(1-x^{2})/4}$
$\displaystyle\times L_{n}^{(i)}(qBr^{2}(1-x^{2})/2)P_{l}^{(1)}(x)e^{ikrx}$

for $i=0,1,2$.

After the integration over the longitudinal momentum $k$, the dissociation
cross sections for charmonium states $\chi_{c\pm}$ are expressed as

$\displaystyle\sigma_{E1}$ $\displaystyle=$ $\displaystyle{\pi g^{2}E_{g}\over
18}\sum_{n}\bigg{[}{2m_{c}\over
k_{n1}}|\mathcal{T}_{nz}(k_{n1})|^{2}+{m_{c}\over
k_{n2}}|\mathcal{T}_{n\rho}^{(2)}(k_{n2})|^{2}$ $\displaystyle+{m_{c}\over
k_{n0}}|\mathcal{T}_{n\rho}^{(0)}(k_{n0})|^{2}\bigg{]},$
$\displaystyle\sigma_{M1}$ $\displaystyle=$ $\displaystyle{\pi g^{2}E_{g}\over
6m_{c}^{2}}\sum_{n}{2m_{c}\over k_{n1}}|\mathcal{T}_{n}(k_{n1})|^{2}.$ (32)

|  $J/\psi$, $\psi(2S)$ | $\chi_{c0}$ | $\chi_{c\pm}$
---|---|---|---
$\sigma_{E1}\propto$ | $1/k_{n1}$ | $1/k_{n0}$ | $1/k_{n0},\ 1/k_{n2}$
$\sigma_{M1}\propto$ | $1/k_{n0}$ | $k_{n0}$ | $1/k_{n1}$
Table 1: The charmonium gluon-dissociation cross-sections in channels $E1$ and
$M1$ around the maximum Landau energy level.

We now analyze the infrared divergence of the transition elements $T$ and
$\mathcal{T}$ in the limit of longitudinal momentum $k_{nm}=0$. Let’s consider
the $S-$wave states $J/\psi$ and $\psi(2S)$ as an example. In this case, $l$
is even, $P_{l}(x)$ is an even function, the requirement that the integrated
function in any $T$ should be an even function of $x$ leads to the replacement
of $e^{ik_{nm}rx}$ by $\cos(k_{nm}rx)$ in $T_{n}$ and $T_{n\rho}$ and by
$i\sin(k_{nm}rx)$ in $T_{nz}$. Around $k_{nm}=0$, by taking the expansions
$\cos(k_{nm}rx)=1+\mathcal{O}(k_{nm}^{2})$ and
$\sin(k_{nm}rx)=k_{nm}rx+\mathcal{O}(k_{nm}^{3})$, $\sigma_{E1}$ is
proportional to $1/k_{n1}$ and becomes divergent at $k_{n1}=0$, and
$\sigma_{M1}$ to $1/k_{n0}$ and divergent at $k_{n0}=0$. Now the only thing
left is the condition for the limit $k_{nm}=0$. For a given incident gluon
energy $E_{g}$, the limit is realized only when the maximum Landau energy
level $E_{n_{max}m0}=(2n_{max}+|m|+1)qB/m_{c}$ satisfies the energy
conservation,

$E_{g}-E_{B}-E_{n_{max}m0}=0.$ (33)

The conclusion is therefore the following: When the maximum Landau energy
level $E_{n_{max}m0}(E_{g})$ satisfies the conservation law, the cross section
is divergent at the corresponding $E_{g}$; If not, the cross section is finite
but still peaks at $E_{g}$. The similar analysis can be done for the $P-$wave
states $\chi_{c0}$ and $\chi_{c\pm}$. The behavior of the cross sections
around the maximum Landau energy level for all the charmonium states is shown
in Table 1. Except for channel $M1$ for $\chi_{0}$, all the other cross
sections are divergent at $k_{n0}=0$ or $k_{n1}=0$ or $k_{n2}=0$.

Figure 3: The charmonium gluon-dissociation cross-sections in channels $E1$
and $M1$ at vanishing temperature and conserved momentum $T=0$ and
${\bm{P}}_{ps}=0$. The bottom label $E_{g}-E_{B}$ is the energy difference
between the initial gluon and charmonium, and the top label is $2n+|m|$
characterizing the Landau energy level. Solid and dashed lines are the
calculations with and without magnetic field.

The cross sections in channels $E1$ and $M1$ for different charmonium states
at vanishing temperature and conserved momentum are shown in Fig. 3 as
functions of incident gluon energy $E_{g}$. The dashed lines are the result
without magnetic field, which were calculated in Ref. Chen:2017jje . When the
magnetic field is turned on, while the global trend of the cross section is
similar to the one without the field, a significant change is the field
induced hair structure. Let us consider $J/\psi$ as an example. As analyzed
above, the cross section $\sigma_{E1}\ (\sigma_{M1})$ goes to infinity when
the energy difference $E_{g}-E_{B}$ between the initial gluon and $J/\psi$
reaches some Landau energy level $2(n+1)qB/m_{c}\ ((2n+1)qB/m_{c})$
characterized by the main quantum number $n$. Therefore, with increasing gluon
energy, the cross sections become divergent at the Landau levels and are
continuous between two neighboured levels. This indicates that the magnetic
field makes the cross sections grow hair! The behavior of the cross sections
for $\psi(2S)$ and $\chi_{c\pm}$ are very similar to $J/\psi$. The only
exception is $\chi_{c0}$. As shown in Table 1, there is no infrared divergence
for the cross section in channel $M1$, $\sigma_{M1}$ is continuous at any
incident gluon energy. Note that, the cross sections for the three $P-$wave
states $\chi_{c0}$ and $\chi_{c\pm}$ are the same in the absence of magnetic
field but separated by the field.

We finally calculate the charmonium decay width through gluon dissociation at
finite temperature and magnetic field. For a charmonium at rest in hot medium,
the width is the integration of the weighted cross-section over the gluon
momentum,

$\Gamma(T,B)=d_{g}\int{d^{3}{\bm{p}}\over(2\pi)^{3}}\sigma(E_{g},T,B)f_{g}(E_{g},T),$
(34)

where $d_{g},\ {\bm{p}},\ E_{g}$ and $f_{g}$ are the gluon degeneracy,
momentum, energy and phase-space distribution. Gluons are massless in vacuum
with energy $E_{g}=|{\bm{p}}|$ but obtain thermal mass at finite temperature
$m_{g}(T)=\sqrt{(2N_{c}+N_{f})/12}\,gT$ FTFT  with energy
$E_{g}=\sqrt{{\bm{p}}^{2}+m_{g}^{2}}$. We take in the calculation the
degeneracy $d_{g}=16$ and coupling constant $g\approx 2$ for $N_{c}=N_{f}=3$,
as used in Ref. Riek:2010fk . Since gluons do not carry electric charge, the
mass and in turn the energy and distribution function are magnetic field
independent at leading order (in general the field can change the gluon
properties through modifications from quark loops). Therefore, the gluon
distribution can be taken as the Bose–Einstein function in the local rest
frame of the medium $f_{g}(E_{g},T)=1/(e^{E_{g}/T}-1)$.

The charmonium decay widths for channels $E1$ and $M1$ are shown in Fig. 4 as
functions of temperature in the deconfined phase with $T>T_{c}$. From the
picture of color screening, the shape of a width is exactly a
$\delta-$function located at the melting temperature $T_{m}$. Considering
realistic collision processes, the $\delta-$function is expanded to be a
distribution covering both $T<T_{m}$ and $T>T_{m}$. While $T_{m}$ is very
different for the ground and excited states, for instance at
$eB=5~{}m_{\pi}^{2}$ there are from Fig.2 $T_{m}/T_{c}\sim 1.4$ for $\psi(2S)$
and $\chi_{c0}$, $2.1$ for $\chi_{c\pm}$ and $3$ for $J/\psi$, all the decay
widths peak at about $T/T_{c}\sim 1.2$. For any charmonium state and in any
case with and without magnetic field, the channel $E1$ always dominants both
the cross section and the decay width, in comparison with the channel $M1$.
This is mainly due to the $M1$ suppression by the mass factor $m_{c}^{2}$ in
the denominator of the cross sections, see Eqs. (IV) and (IV). It is also easy
to understand that the loosely bound states $\psi(2S)$ and $\chi_{c}$ are
easier to decay than the tightly bound state $J/\psi$.

Figure 4: The charmonium decay width through gluon dissociation in channels
$E1$ and $M1$ as functions of scaled temperature $T/T_{c}$ at vanishing
pseudo-momentum ${\bm{P}}_{ps}=0$. Solid and dashed lines are the calculations
with and without magnetic field.

Now we focus on the magnetic field effect on the decay width. Considering the
fact that $k$ is the magnitude of the charmonium longitudinal momentum,
$k_{nm}=\sqrt{m_{c}(E_{g}(p)-E_{B}-E_{nm0})}$ should be positive, and the
momentum integration (34) around a divergence is proportional to

$\int dp{1\over k_{nm}}\theta(k_{nm})=2\int_{0}^{\delta}dk_{nm}=2\delta$ (35)

and is finite. Therefore, the integrated decay width is convergent at any
temperature $T$. Secondly, the radial symmetry breaking deforms the charmonium
and octet states, the change in the transition element $\langle
nmk|{\bm{r}}|\Psi\rangle$ by the magnetic field should be stronger than the
element $\langle nmk|\Psi\rangle$. This means that the cross section and decay
width in channel $E1$ are more sensitive to the field than that in channel
$M1$. Due to the larger deformation of the $P-$wave states, the magnetic field
effect on $\chi_{c0}$ and $\chi_{c\pm}$ is more important than the $S-$wave
states $J/\psi$ and $\psi(2S)$. This is clearly shown in Fig. 4.

##  V Summary

A typical quantum mechanics problem is the particle motion in a magnetic
field, which leads to the famous Landau energy levels. While the magnetic
field effect is recently widely discussed in high energy physics, like the
influence on QCD phase transitions and static particle properties, it is
rarely introduced in the calculation of particle collisions. In this paper we
investigated the gluon dissociation process $g+\Psi\to c+\bar{c}$ in a strong
magnetic field and found that the Landau energy levels make the cross section
grow hair.

We extended the QCD multipole expansion for a pair of heavy quarks to
including an external electromagnetic field. By solving the two-body
Schrödinger equation with mean field potentials for strong and electromagnetic
interactions we firstly determined the charmonium static properties, including
the binding energy and wave function. Taking then the color dipole
interactions as perturbations above the mean field and employing Fermi’s
Golden Rule, we focused on the magnetic field effect on the gluon-dissociation
process in the quark-gluon plasma. In general case the dissociation cross-
section becomes divergent when the energy difference between the initial gluon
and charmonium reaches a Landau energy level for the final octet state. These
divergences at different Landau levels look like hairs of the cross section.
However, the gluon energy integrated decay width is always continuous at any
temperature. Considering the deformation of the charmonium states, especially
for the loosely bound states, the magnetic field effect on the color-electric
dipole channel and the excited states is significantly important. In our
numerical calculation the difference between the decay widths with and without
magnetic field is already large enough at $eB=5~{}m_{\pi}^{2}$. This indicates
that the magnetic field effect on charmonium dissociation in high energy
nuclear collisions at RHIC and LHC energies might be sizeable and considered
as a probe of the initially produced electromagnetic field.

Acknowledgement: We thank very much Shile Chen and Kai Zhou for helpful
discussions. The work is supported by NSFC grant Nos. 11890712, 12035006,
12047535 and 12075129 and Guangdong Major Project of Basic and Applied Basic
Research No. 2020B0301030008. S.S. is grateful to supports from Natural
Sciences and Engineering Research Council of Canada, the Bourses d’excellence
pour étudiants étrangers (PBEEE) from Le Fonds de Recherche du Québec - Nature
et technologies (FRQNT), and the U.S. Department of Energy, Office of Science,
Office of Nuclear Physics under grant No. DEFG88ER40388.

## References

  * (1) V. Skokov, A. Y. Illarionov and V. Toneev, Int. J. Mod. Phys. A 24, 5925-5932 (2009).
  * (2) V. Voronyuk, V. D. Toneev, W. Cassing, E. L. Bratkovskaya, V. P. Konchakovski and S. A. Voloshin, Phys. Rev. C 83, 054911 (2011).
  * (3) W. T. Deng and X. G. Huang, Phys. Rev. C 85, no. 5, 044907 (2012).
  * (4) K. Tuchin, Adv. High Energy Phys. 2013, 490495 (2013).
  * (5) D. E. Kharzeev, L. D. McLerran and H. J. Warringa, Nucl. Phys. A 803, 227 (2008).
  * (6) K. Fukushima, D. E. Kharzeev and H. J. Warringa, Phys. Rev. D 78, 074033 (2008).
  * (7) I. A. Shovkovy, Lect. Notes Phys. 871, 13 (2013).
  * (8) F. Bruckmann, G. Endrodi and T. G. Kovacs, JHEP 1304, 112 (2013).
  * (9) U. Gursoy, D. Kharzeev and K. Rajagopal, Phys. Rev. C 89, no.5, 054905 (2014).
  * (10) L. Yan and X. G. Huang, [arXiv:2104.00831 [nucl-th]].
  * (11) Y. Chen, X. L. Sheng and G. L. Ma, Nucl. Phys. A 1011, 122199 (2021).
  * (12) Z. Wang, J. Zhao, C. Greiner, Z. Xu and P. Zhuang, [arXiv:2110.14302 [hep-ph]].
  * (13) S. K. Das, S. Plumari, S. Chatterjee, J. Alam, F. Scardina and V. Greco, Phys. Lett. B 768, 260-264 (2017).
  * (14) J. Adam et al. [STAR], Phys. Rev. Lett. 123, no.16, 162301 (2019).
  * (15) S. Acharya et al. [ALICE], Phys. Rev. Lett. 125, no.2, 022301 (2020).
  * (16) K. Marasinghe and K. Tuchin, Phys. Rev. C 84, 044908 (2011).
  * (17) J. Alford and M. Strickland, Phys. Rev. D 88, 105017 (2013).
  * (18) C. S. Machado, F. S. Navarra, E. G. de Oliveira, J. Noronha and M. Strickland, Phys. Rev. D 88, 034009 (2013).
  * (19) S. Cho, K. Hattori, S. H. Lee, K. Morita and S. Ozaki, Phys. Rev. Lett. 113, no.17, 172301 (2014).
  * (20) X. Guo, S. Shi, N. Xu, Z. Xu and P. Zhuang, Phys. Lett. B 751, 215-219 (2015).
  * (21) C. Bonati, M. D’Elia and A. Rucci, Phys. Rev. D 92, no.5, 054014 (2015).
  * (22) T. Yoshida and K. Suzuki, Phys. Rev. D 94, 074043 (2016).
  * (23) C. Bonati, M. D’Elia, M. Mariti, M. Mesiti, F. Negro, A. Rucci and F. Sanfilippo, Phys. Rev. D 95, no.7, 074515 (2017).
  * (24) J. Zhao, K. Zhou, S. Chen and P. Zhuang, Prog. Part. Nucl. Phys. 114, 103801 (2020).
  * (25) A. Mishra and S. P. Misra, Phys. Rev. C 102, no.4, 045204 (2020).
  * (26) S. Chen, J. Zhao and P. Zhuang, Phys. Rev. C 103, no.3, L031902 (2021).
  * (27) S. Iwasaki, M. Oka and K. Suzuki, Eur. Phys. J. A 57, no.7, 222 (2021).
  * (28) B. Singh, L. Thakur and H. Mishra, Phys. Rev. D 97, no.9, 096011 (2018).
  * (29) M. Hasan, B. K. Patra, B. Chatterjee and P. Bagchi, Nucl. Phys. A 995, 121688 (2020).
  * (30) M. Hasan, B. Chatterjee and B. K. Patra, Eur. Phys. J. C 77, no.11, 767 (2017).
  * (31) M. Hasan and B. K. Patra, Phys. Rev. D 102, no.3, 036020 (2020).
  * (32) F. Karsch, M. T. Mehr and H. Satz, Z. Phys. C 37, 617 (1988).
  * (33) N. Brambilla, M. A. Escobedo, J. Ghiglieri and A. Vairo, JHEP 12, 116 (2011).
  * (34) N. Brambilla, M. A. Escobedo, J. Ghiglieri and A. Vairo, JHEP 05, 130 (2013)
  * (35) M. E. Peskin, Nucl. Phys. B 156, 365-390 (1979);
  * (36) G. Bhanot and M. E. Peskin, Nucl. Phys. B 156, 391-416 (1979).
  * (37) T. M. Yan, Phys. Rev. D 22, 1652 (1980).
  * (38) Y. P. Kuang and T. M. Yan, Phys. Rev. D 24, 2874 (1981).
  * (39) Y. P. Kuang, Front. Phys. China 1, 19-37 (2006).
  * (40) Y. Liu, C. M. Ko and T. Song, Phys. Rev. C 88, no.6, 064902 (2013).
  * (41) S. Chen and M. He, Phys. Rev. C 96, no.3, 034901 (2017).
  * (42) S. Chen and M. He, Phys. Lett. B 786, 260-267 (2018).
  * (43) G. Bhanot, W. Fischler and S. Rudaz, Nucl. Phys. B 155, 208-236 (1979).
  * (44) A. Pineda and J. Soto, Phys. Lett. B 420, 391-396 (1998).
  * (45) N. Brambilla, G. Krein, J. Tarrús Castellà and A. Vairo, Phys. Rev. D 97, no.1, 016016 (2018).
  * (46) K. Gottfried, Phys. Rev. Lett. 40, 598 (1978).
  * (47) H.W. Crater, J. Comput. Phys. 115 (1994) 470.
  * (48) M. Laine, O. Philipsen, P. Romatschke and M. Tassler, JHEP 03, 054 (2007).
  * (49) Y. Burnier, O. Kaczmarek and A. Rothkopf, Phys. Rev. Lett. 114, no.8, 082001 (2015).
  * (50) Y. Burnier, O. Kaczmarek and A. Rothkopf, JHEP 12, 101 (2015).
  * (51) D. Lafferty and A. Rothkopf, Phys. Rev. D 101, no.5, 056010 (2020).
  * (52) J. I. Kapusta and C. Gale, Finite-Temperature Field Theory Principles and Applications, Cambridge University Press.
  * (53) F. Riek and R. Rapp, Phys. Rev. C 82, 035201 (2010).


#  Three-dimensional charged Einstein-aether black holes and Smarr formula

Chikun Dinga,c<EMAIL_ADDRESS><EMAIL_ADDRESS>Changqing Liua
Anzhong Wangb<EMAIL_ADDRESS>Jiliang Jingc<EMAIL_ADDRESS>a
Department of Physics, Hunan University of Humanities, Science and Technology,
Loudi, Hunan 417000, P. R. China
b GCAP-CASPER, Physics Department, Baylor University, Waco, Texas 76798-7316,
USA
cKey Laboratory of Low Dimensional Quantum Structures and Quantum Control of
Ministry of Education, and Synergetic Innovation Center for Quantum Effects
and Applications, Hunan Normal University, Changsha, Hunan 410081, P. R. China

###### Abstract

Abstract

We investigate behaviors of the three-dimensional gravity coupled to a
dynamical unit timelike vector: the aether, and present two new classes of
exact charged solutions. When $c_{13}=0,\Lambda^{\prime}=0$, we find the
solutions is the usual BTZ black hole but now with an universal horizon. In
the frame of black hole chemistry, we then calculate the temperature of the
universal horizons and, construct the Smarr formulas and first law in the
three cases: (quasi)asymptotically flat, aether asymptotically flat and quasi-
BTZ black hole spacetime. We found these universal horizons obey an exact (or
slightly modified) first law of black hole mechanics and may have an entropy
and, black hole mass can be interpreted as enthalpy of spacetime. Then the
holography may be extended to these horizons under violating Lorentz symmetry.

###### pacs:

04.50.Kd, 04.20.Jb, 04.70.Dy

##  I Introduction

Lorentz invariance is one of the fundamental principles of Einstein’s general
relativity (GR) and modern physics. However, Lorentz invariance may not be an
exact symmetry at all energies mattingly . Any effective description must
break down at a certain cutoff scale signaling the emergence of new physical
degrees of freedom beyond that scale. For example, the hydrodynamics, Fermi’s
theory of beta decay bhattacharyya2  and quantization of GR shomer  at
energies beyond the Planck energy. Lorentz invariance also leads to
divergences in quantum field theory which can be cured with a short distance
of cutoff that breaks it jacobson . Astrophysical observations on high-energy
cosmic rays seem to have proven this conjectureabdo . Einstein-aether theory
can be considered as an effective description of Lorentz symmetry breaking in
the gravity sector and has been extensively used in order to obtain
quantitative constraints on Lorentz-violating gravityjacobson2 . And
violations of Lorentz symmetry have been used to construct modified-gravity
theories that account for Dark-Matter phenomenology without any actual Dark
Mater bekenstein2004 .

In Einstein-aether theory, the Lorentz symmetry is broken only down to a
rotation subgroup by the existence of a preferred time direction at every
point of spacetime, i.e., existing a preferred frame of reference established
by aether vector $u^{a}$. This timelike unit vector field $u^{a}$ can be
interpreted as a velocity four-vector of some medium substratum (aether,
vacuum or dark fluid), bringing into consideration of non-uniformly moving
continuous media and their interaction with other fields. Meanwhile, this
theory can be also considered as a realization of dynamic self-interaction of
complex systems moving with a spacetime dependant macroscopic velocity. As to
an accelerated expansion of the universe, this dynamic self-interaction can
produce the same cosmological effects as the dark energy balakin2 .

The introduction of the aether vector allows for some novel effects, e.g.,
matter fields can travel faster than the speed of light jacobson3 , new
gravitational wave polarizations can spread at different speeds jacobson4 .
Gravitational theories with breaking Lorentz invariance still allow the
existence of black holes blas2 ; barausse ; lin ; UHs  which have an universal
horizon. Instead of Killing horizon, it can trap excitations traveling at
arbitrarily high velocities. So it attracts many concerns in many fields
recently, e.g., Einstein-aether perfect fluid modelslatta , its axionic
extensionbalakin2016 , null aether theorygurses , analogue black
holescropp2016 , etc.

Here we would fix our attention upon deriving a black hole solution of
Einstein-aether theory. In 4-dimensions, the neutral black hole
solutionsberglund ; bhattacharyya , charged solutionsding , slowly rotating
solutions barausse2016  are found. On the other hand, there has been much
interest in lower dimensional theories of gravity which there is no
dynamically propagating degrees of freedom: curvature is algebraically fixed
by the matter content, in contrast to the four dimensional counterparts. The
BTZ solution is the unique black hole of general relativity in
3-dimensionsbanados . It generated a considerable amount of attention due to
that it can be holographically described by a two-dimensional conformal field
theory(AdS${}_{3}/$CFT2donnay ), whose foreseen applications in addressing
conceptual issues of quantum gravity that become more tractablesotiriou . So
many lower dimensional solutions are found, such as, solutions in two-
dimensionseling06 , numerical solutions of asymptotically Lifshitz
spacetimebasu  in three-dimension are found.

In another side, the relations between black hole physics and thermodynamics
has been recognized for decades. Inspired by holography, novel studies of
black holes with cosmological constant provide some connections between
seemingly disparate theoretical concepts. Recently, black hole chemistry has
emerged due to this thermodynamic connectionfrassino . It tries to associate
to each black hole parameter a chemical equivalent in representations of the
first law. The present identifications are mass $M$ to thermal energy $E$,
surface gravity $\kappa$ to temperature $T$, horizon area $A$ to entropy $S$.
However, one quantity—the pressure-volume term $PV$— has no gravitational
analogue. So the black hole chemistry regards the cosmological constant
$\Lambda$ of AdS spacetime as a thermodynamic pressure, and the mass $M$ as
chemical enthalpy.

In this paper, we firstly derive some new Lorentz-violating versions of the
BTZ black hole in Einstein-aether theory. Nextly by using Komar integral
method, seek for the Smarr relation and the first law in this black hole
spacetime via the black hole chemistry.

The rest of the paper is organized as follows. In Sec. II we provide the
background for the Einstein-Maxwell-aether theory studied in this paper. In
Sec. III we review the first law and Smarr formula of BTZ black hole, and then
show the procedure of how to construct a Smarr formula for spherically
symmetric solutions. In Sec. IV, we first construct two new classes of exact
charged solutions, and then use them as examples to study the Smarr formula
and first law. In Sec. V, we present our main conclusions.

##  II Einstein-Maxwell-aether theory

The general action for the Einstein-Maxwell-aether theory can be constructed
by assuming that: (1) it is general covariant; and (2) it is a functional of
only the spacetime metric $g_{ab}$, an unit timelike vector $u^{a}$ and
Maxwell field $\mathcal{A}^{a}$, and involves no more than two derivatives of
them. So that the resulting field equations are second-order differential
equations of $g_{ab}$, $u^{a}$ and $\mathcal{A}^{a}$. To simplify the problem,
the couple between aether field and Maxwell one is ignored. Then, the
Einstein-Maxwell-aether theory to be studied in this paper is described by the
action,

$\displaystyle\mathcal{S}=\int d^{4}x\sqrt{-g}\Big{[}\frac{1}{16\pi
G_{\ae}}(\mathcal{R}-2\Lambda+\mathcal{L}_{\ae}+\mathcal{L}_{M})\Big{]}\,.$
(1)

In terms of the tensor $Z^{ab}_{~{}~{}cd}$ defined as eling ; garfinkle ,

$\displaystyle
Z^{ab}_{~{}~{}cd}=c_{1}g^{ab}g_{cd}+c_{2}\delta^{a}_{~{}c}\delta^{b}_{~{}d}+c_{3}\delta^{a}_{~{}d}\delta^{b}_{~{}c}-c_{4}u^{a}u^{b}g_{cd}\,,$
(2)

the aether Lagrangian $\mathcal{L}_{\ae}$ is given by

$\displaystyle-\mathcal{L}_{\ae}=Z^{ab}_{~{}~{}cd}(\nabla_{a}u^{c})(\nabla_{b}u^{d})-\lambda(u^{2}+1),$
(3)

where $c_{i}(i=1,2,3,4)$ are coupling constants of the theory. The aether
Lagrangian is therefore the sum of all possible terms for the aether field
$u^{a}$ up to mass dimension two, and the constraint term $\lambda(u^{2}+1)$
with the Lagrange multiplier $\lambda$ implementing the normalization
condition $u^{2}=-1$. The source-free Maxwell Lagrangian $\mathcal{L}_{M}$ is
given by

$\mathcal{L}_{M}=-\frac{1}{4}\mathcal{F}_{ab}\mathcal{F}^{ab},~{}\mathcal{F}_{ab}=\nabla_{a}\mathcal{A}_{b}-\nabla_{b}\mathcal{A}_{a},$
(4)

where $\mathcal{A}_{a}$ is the electromagnetic potential four-vector.

The equations of motion, obtained by varying the action (1) with respect to
$g_{ab}$, $u^{a}$, $\mathcal{A}^{a}$ and $\lambda$ are

$\displaystyle\mathcal{G}_{ab}+\Lambda g_{ab}=\mathcal{T}^{\ae}_{ab}+8\pi
G_{\ae}\mathcal{T}^{M}_{ab},\quad{\AE}_{a}=0,\quad\nabla^{a}\mathcal{F}_{ab}=0,\quad
u^{2}=-1,$ (5)

respectively, where the aether and Maxwell energy-momentum stress tensors
$\mathcal{T}^{\ae}_{ab}$ and $\mathcal{T}^{M}_{ab}$ are given by

$\displaystyle\mathcal{T}^{\ae}_{ab}=\lambda
u_{a}u_{b}+c_{4}a_{a}a_{b}-\frac{1}{2}g_{ab}Y^{c}_{~{}~{}d}\nabla_{c}u^{d}+\nabla_{c}X^{c}_{~{}~{}ab}+c_{1}[(\nabla_{a}u_{c})(\nabla_{b}u^{c})-(\nabla^{c}u_{a})(\nabla_{c}u_{b})],$
$\displaystyle\mathcal{T}_{ab}^{M}=\frac{1}{16\pi
G_{\ae}}\Big{[}-\frac{1}{4}g_{ab}\mathcal{F}_{mn}\mathcal{F}^{mn}+\mathcal{F}_{am}\mathcal{F}_{b}^{~{}m}\Big{]},$
(6)

with

$\displaystyle{\AE}_{a}=\nabla_{b}Y^{b}_{~{}~{}a}+\lambda
u_{a}+c_{4}(\nabla_{a}u^{b})a_{b},\quad
Y^{a}_{~{}~{}b}=Z^{ac}_{~{}~{}~{}bd}\nabla_{c}u^{d},\quad
X^{c}_{~{}~{}ab}=Y^{c}_{~{}~{}(a}u_{b)}-u_{(a}Y^{~{}~{}c}_{b)}+u^{c}Y_{(ab)}.$
(7)

The acceleration vector $a^{a}$ appearing in the expression for the aether
energy-momentum stress tensor is defined as the parallel transport of the
aether field along itself, $a^{a}\equiv\nabla_{u}u^{a},$ where
$\nabla_{X}\equiv X^{b}\nabla_{b}$.

Following berglund , in spherically symmetry spacetime, the symmetry enforces
$u^{a}$ hypersurface-orthogonal and becoming normal to one or more constant-
radius hyperfurface that lies inside the Killing horizon. So one can let
$\Sigma_{U}$ denote a surface orthogonal to the aether vector $u^{a}$, then
$U$ is the aether time generated by $u^{a}$ that specifies each hypersurface
in a foliation. As one moves in toward the origin, each $\Sigma_{U}$
hypersurface bends down to the infinite past, and asymptoting to a three-
dimensional spacelike hypersurface on which $(u\cdot\chi)=0$, which implies
that the Killing vector becomes tangent to $\Sigma_{U}$. This hypersurface is
the universal horizon. Therefore, we are going to reduce these equations to a
spherical symmetry case.

We first define a set of basis vectors at every point in the spacetime, so
that we can project out various components of the equations of motion. Let us
first take the aether field $u^{a}$ to be the basis vector. Then, pick up two
spacelike unit vectors, denoted, respectively, by $m^{a}$ and $n^{a}$, both of
which are normalized to unity, mutually orthogonal, and lie on the tangent
plane of the two-spheres $\mathcal{B}$ that foliate the hypersurface
$\Sigma_{U}$. Finally, let us pick up $s^{a}$, a spacelike unit vector that is
orthogonal to $u^{a}$, $m^{a}$, $n^{a}$, and points “outwards” along a
$\Sigma_{U}$ hypersurface, so we have the four tetrad,
$\left(u^{a},s^{a},m^{a},n^{a}\right)$, with the metric

$\displaystyle ds^{2}=-u_{a}u_{b}+s_{a}s_{b}+\hat{g}_{ab},$ (8)

where $\hat{g}_{ab}$ is

$\displaystyle\hat{g}_{ab}=r^{2}(d\theta,d\phi)(d\theta,d\phi)\left(\begin{array}[]{cc}1&0\\\
0&\sin^{2}\theta\end{array}\right)=r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\phi^{2},$ (11)

which are two-spheres $\mathcal{B}$ of 4D spherically spacetime. As for 3D
case, we can let the ansatz of $u^{a},s^{a}$ unchanged and now, $\mathcal{B}$
is one-spheres111 For $(n+2)$ dimensional spherically spacetime, the
transverse space could be a $n$-dimensional flat space. Then we can pick $n$
coordinates $y^{i}$ to describe this space, so that
$\hat{g}_{ab}=b(r)q_{ij}(y)dy^{i}dy^{j}$bhattacharyya2 . Any physical vector
$X^{a}$ must be orthogonal to the transverse space, $\hat{g}_{ab}X^{b}=0$,
hence we can let the ansatz of $u^{a}$ and $s^{a}$ unchanged and, all the
following equations can be calculated directly from the 3D metric..

By spherical symmetry, any physical vector $A^{a}$ has at most two non-
vanishing components along, respectively, $u^{a}$ and $s^{a}$, i.e.,
$A^{a}=A_{1}u^{a}+A_{2}s^{a}$. In particular, the acceleration $a^{a}$ has
only one component along $s^{a}$, namely, $a^{a}=(a\cdot s)s^{a}$. Similarly,
any rank-two tensor $F_{ab}$ may have components along the directions of the
bi-vectors
$u_{a}u_{b},~{}u_{(a}s_{b)},~{}u_{[a}s_{b]},~{}s_{a}s_{b},~{}\hat{g}_{ab}$. In
the following, we study the expansion of the Maxwell field $\mathcal{F}^{ab}$,
Killing vector $\chi^{a}$, surface gravity $\kappa$, energy-momentum stress
tensors $\mathcal{T}_{ab}^{\ae}$ and $\mathcal{T}_{ab}^{M}$, and Ricci tensor
$\mathcal{R}_{ab}$. The given source-free Maxwell field $\mathcal{F}^{ab}$ can
be formulated in terms of four-vectors representing physical fields. They are
the electric field $E^{a}$ and magnetic excitation $B^{a}$ asbalakin ,

$\displaystyle E^{a}=\mathcal{F}^{ab}u_{b},\quad
B^{a}=\frac{e^{abmn}}{2\sqrt{-g}}\mathcal{F}_{mn}u_{b},$ (12)

where $e^{abmn}$ is the Levi-Civita tensor. For source free Maxwell field or
from Eq.(5), it can be shown $B^{a}=0$, that is, there is no need to consider
$e^{abmn}$. Then, we find

$\displaystyle\mathcal{F}^{ab}=-E^{a}u^{b}+E^{b}u^{a}.$ (13)

On the other hand, the electric field is spacelike, since $E^{a}u_{a}=0$. So,
we have $E^{a}=(E\cdot s)s^{a}$. Thus, $\mathcal{F}_{ab}=(E\cdot
s)\varepsilon^{II}_{ab}$, where
$\varepsilon^{II}_{ab}=(-s_{a}u_{b}+s_{b}u_{a})$.

Using Gauss law, there has bhattacharyya2

$\displaystyle\nabla^{b}[F(r)\varepsilon_{ab}^{II}]=0\quad\Longrightarrow\quad
F(r)=\frac{F_{0}}{b(r)^{n/2}},$ (14)

where $F_{0}$ is a constant, $b(r)=r^{2}$ for 4D or 3D spherically spacetime,
$n$ is the number of dimension of sphere $\mathcal{B}$, i.e., $n$-sphere. Here
$\mathcal{B}$ is one-sphere for 3D spherically spacetime,

$\displaystyle ds^{2}=-e(r)dt^{2}+e^{-1}(r)dr^{2}+r^{2}d\phi^{2},$ (15)

then $n=1$. From Eq. (5), we can see $(E\cdot s)=Q/r$, where $Q$ is an
integral constant, representing the total charge of the space-time. Therefore,
we have

$\displaystyle\mathcal{F}_{ab}=\frac{Q}{r}(u_{a}s_{b}-u_{b}s_{a}).$ (16)

The Einstein, aether and Maxwell equations of motion (5) can be decomposed by
using the tetrad $u^{a}$, $s^{a}$ and $\hat{g}^{ab}$ defined above. In
particular, the aether and electromagnetic energy-momentum stress tensors and
the Ricci tensor can be cast, respectively, in the forms according directly to
the 3D metric (15),

$\displaystyle\mathcal{T}^{\ae}_{ab}=\mathcal{T}^{\ae}_{uu}u_{a}u_{b}-2\mathcal{T}^{\ae}_{us}u_{(a}s_{b)}+\mathcal{T}^{\ae}_{ss}s_{a}s_{b}+\hat{\mathcal{T}}_{\ae}\hat{g}_{ab},$
$\displaystyle\mathcal{R}_{ab}=\mathcal{R}_{uu}u_{a}u_{b}-2\mathcal{R}_{us}u_{(a}s_{b)}+\mathcal{R}_{ss}s_{a}s_{b}+\mathcal{\hat{R}}\hat{g}_{ab},$
$\displaystyle\mathcal{T}^{M}_{ab}=\mathcal{T}^{M}_{uu}u_{a}u_{b}-2\mathcal{T}^{M}_{us}u_{(a}s_{b)}+\mathcal{T}^{M}_{ss}s_{a}s_{b}+\hat{\mathcal{T}}_{M}\hat{g}_{ab}.$
(17)

The coefficients of $\mathcal{T}^{\ae}_{ab}$ and $\mathcal{T}^{M}_{ab}$ in
(II) can be computed from the general expression (II). The corresponding
coefficients for $\mathcal{R}_{ab}$, on the other hand, are computed from the
definition
$[\nabla_{a},~{}\nabla_{b}]X^{c}\equiv-\mathcal{R}^{c}_{~{}abd}X^{d}$ by
choosing $X^{a}=u^{a}$ or $s^{a}$, and then contracting the resulting
expressions again with $u^{a}$ and/or $s^{a}$ appropriately. The coefficients
for the three $(u,s)$ cross terms are

$\displaystyle\mathcal{T}^{\ae}_{us}=c_{14}\left[\hat{K}(a\cdot
s)+\nabla_{u}(a\cdot
s)\right],\quad\mathcal{T}^{M}_{us}=0,\quad\mathcal{R}_{us}=(K_{0}-\hat{K})\hat{k}-\nabla_{s}\hat{K},$
(18)

where

$\displaystyle\nabla_{[a}s_{b]}\equiv-
K_{0}u_{[a}s_{b]},\;\;\;\hat{k}\equiv\frac{1}{2}g^{ab}{\cal{L}}_{s}\hat{g}_{ab},\;\;\;\hat{K}\equiv\frac{1}{2}g^{ab}{\cal{L}}_{u}\hat{g}_{ab},$
(19)

with $K\;(\equiv K_{0}+\hat{K})$ being the trace of the extrinsic curvature of
the hypersurface $\Sigma_{U}$. The aether equation $s\cdot{\AE}=0$ and the
$us$-component $\mathcal{R}_{us}=\mathcal{T}^{\ae}_{us}+8\pi
G_{\ae}\mathcal{T}^{M}_{us}$ yield

$\displaystyle
c_{123}\nabla_{s}K_{0}-(1-c_{13})(K_{0}-\hat{K})\hat{k}+(1+c_{2})\nabla_{s}\hat{K}=0,$
(20) $\displaystyle c_{123}\nabla_{s}K-(1-c_{13})\mathcal{T}^{\ae}_{us}=0.$
(21)

After rewriting the motion Eq. (5) as

$\displaystyle\mathcal{R}_{ab}-2\Lambda
g_{ab}=\mathcal{T}^{\ae}_{ab}-g_{ab}\mathcal{T}^{\ae}+8\pi
G_{\ae}\big{[}\mathcal{T}^{M}_{ab}-g_{ab}\mathcal{T}^{M}\big{]},$ (22)

where $T=g^{ab}T_{ab}$, then the $uu$-, $ss$\- and $\hat{g}$-components of the
gravitational field equations give

$\displaystyle\nabla\cdot a-(1-c_{13})(K_{0}^{2}+\hat{K}^{2})+2\Lambda-
c_{123}\nabla_{c}(Ku^{c})-(1+c_{2})\nabla_{u}K=0,$ (23)
$\displaystyle\nabla_{c}\big{[}(K_{0}+c_{13}\hat{K}+c_{2}K)u^{c}\big{]}-\big{[}a^{2}+\hat{k}^{2}+\nabla_{s}(a\cdot
s+\hat{k})+2\Lambda\big{]}-c_{14}(\nabla\cdot a-a^{2})=0,$ (24)
$\displaystyle\mathcal{R}+(1-c_{14})\nabla\cdot
a+[a^{2}+\hat{k}^{2}+\nabla_{s}(a\cdot s+\hat{k})]$
$\displaystyle-\nabla_{c}[\big{(}(1-c_{13})K_{0}-c_{2}K\big{)}u^{c}]-\nabla_{u}K-(K_{0}^{2}+\hat{K}^{2})-2\Lambda-\frac{Q^{2}}{2r^{2}}=0,$
(25)

where
$\mathcal{R}=g^{ab}\mathcal{R}_{ab}=-e^{\prime\prime}(r)-2e^{\prime}(r)/r$. In
the next sections, we will use these equations to obtain new black holes
solutions.

##  III Smarr formula

In this section we firstly review the Smarr formula and first law of BTZ black
hole spacetime via black hole chemistry. And then using Komar integral method,
we show the procedure of deriving the Smarr formula in 3-dimensional Einstein-
Maxwell-aether theory.

In three dimensions, the static BTZ black hole readsbanados

$\displaystyle
ds^{2}=-fdt^{2}+\frac{dr^{2}}{f}+r^{2}d\phi^{2},\;f=-m+r^{2}/l^{2},$ (26)

where $1/l^{2}=-\Lambda$. The first law of black hole mechanics and the Smarr
formula arekubiznak

$\displaystyle dM=TdS+VdP,~{}~{}~{}~{}0=TS-2PV,$ (27)

where the temperature $T=f^{\prime}(r_{+})/4\pi=r_{+}/2\pi l^{2}$, the entropy
$S=A/4=\pi r_{+}/2$, the black hole mass $M=m/8=r_{+}^{2}/8l^{2}$ and,
$P=1/8\pi l^{2}$ is thermodynamic pressure, its thermodynamic conjugate volume
$V=\partial M/\partial P|_{S}=\pi r_{+}^{2}$, $r_{+}$ is the horizon radius.
The pressure-volume term reinterprets $M$ as the enthalpy of the black
holekastor : the energy required to both form a black hole and place it into
its cosmological environment. In the next section, these exact or slightly
modified first law and Smarr formulas will be constructed for those Lorentz-
violating BTZ-like black holes by the following procedure.

Now we shall present the process of deriving Smarr formulas of the universal
horizons for general $2+1$ static and spherically symmetric Einstein-Maxwell-
aether black holes. Let us first consider the geometric identity BCH ,

$\displaystyle\mathcal{R}_{ab}\chi^{b}=\nabla^{b}(\nabla_{a}\chi_{b}).$ (28)

The derivative of the Killing vector
$\chi^{a}=-(u\cdot\chi)u^{a}+(s\cdot\chi)s^{a}$ is given by

$\displaystyle\nabla^{a}\chi^{b}=-2\kappa u^{[a}s^{b]},\quad$ (29)

where $\kappa$ denotes the surface gravity usually defined in GR, and is given
by

$\displaystyle\kappa=\sqrt{-\frac{1}{2}(\nabla_{a}\chi_{b})(\nabla^{a}\chi^{b})}=-(a\cdot
s)(u\cdot\chi)+K_{0}(s\cdot\chi).$ (30)

From the Einstein field equations (22), we find that

$\displaystyle 2\Lambda g_{ab}\chi^{b}=\nabla^{b}\Big{(}\Lambda
r\;u_{[a}s_{b]}\Big{)},$ $\displaystyle 8\pi
G_{\ae}(\mathcal{T}^{M}_{ab}-g_{ab}\mathcal{T}^{M})\chi^{b}=0,$
$\displaystyle\left(\mathcal{T}^{\ae}_{ab}-g_{ab}\mathcal{T}^{\ae}\right)\chi^{b}=\nabla^{b}\big{[}\left(c_{123}K-c_{13}K_{0}\right)\left(s\cdot\chi\right)u_{[a}s_{b]}\big{]}.$
(31)

Then Eq.(28) can be cast in the form,

$\displaystyle\nabla_{b}F^{ab}=0,\;\;\;F^{ab}\equiv 2q(r)u^{[a}s^{b]},$ (32)

where

$\displaystyle q(r)\equiv\Lambda r-(a\cdot
s)(u\cdot\chi)+\left[(1-c_{13})K_{0}+c_{123}K\right](s\cdot\chi).$ (33)

On the other hand, comparing Eq.(32) with the soucre-free Maxwell equations
(5), we find that its solution must also take the form (16), that is,
$q(r)=q_{0}/r$. On the other hand, using Gauss’ law, from Eq.(32) we find that

$\displaystyle
0=\int_{\Sigma}{\left(\nabla_{b}F^{ab}\right)d\Sigma_{a}}=\int_{{\cal{B}}_{\infty}}{F^{ab}d\Sigma_{ab}}-\int_{{\cal{B}}_{H}}{F^{ab}d\Sigma_{ab}}=\int_{{\cal{B}}_{\infty}}{qdA}-\int_{{\cal{B}}_{H}}{qdA}.$
(34)

Here $d\Sigma_{a}$ is the surface element of a spacelike hypersurface
$\Sigma$, and $dA=rd\phi$ is the area element of 1-dimensional sphere. The
boundary $\partial\Sigma$ of $\Sigma$ consists of the boundary at spatial
infinity ${\cal{B}}_{\infty}$, and the horizon ${\cal{B}}_{H}$, either the
Killing or the universal. Note that Eq.(34) is nothing but the conservation
law of the flux of $F^{ab}$. Comparing the above expression and Eq.(33), we
find the following Smarr formula in 3-dimensionskastor ,

$\displaystyle 0\cdot MG_{\ae}=\frac{q_{H}A_{H}-q_{\infty}A_{\infty}}{8\pi},$
(35)

where $A\infty$ and $A_{H}$ are the area of infinite boundary and the
universal or Killing horizon, however now $M$ is the enthalpy instead of the
total energy of spacetime defined in the asymptotic aether rest frame. The
$q_{\infty}$ and $q_{H}$ are the value of $q$ in (33) at the infinite and
universal horizon or Killing horizon.

In GR, from Eq. (29) and (28), the $q_{H}$ is just the surface gravity
$\kappa_{H}$ on the usual Killing horizon. Now in the presence of aether
field, it contains the aether contribution and becomes complicated. However,
the first law for the aether black hole may still be obtained via a variation
of these Smarr relations. In the next section we consider it for two new
classes of exact charged aether black hole solutions.

For the surface gravity at the universal horizon, when one considers the
peeling behavior of particles moving at any speed, i.e., capturing the role of
the aether in the propagation of the physical rays, one finds that the surface
gravity at the universal horizon is cropp ; jacobson5 ; lin

$\displaystyle\kappa_{UH}\equiv\frac{1}{2}\nabla_{u}(u\cdot\chi)=\left.\frac{1}{2}\left(a\cdot
s\right)\left(s\cdot\chi\right)\right|_{r=r_{UH}},$ (36)

where in the last step we used the fact that $\chi_{a}$ is a Killing vector,
$\nabla_{(a}\chi_{b)}=0$. It must be noted that this is different from the
surface gravity defined in GR by Eq.(30). In particular, at the universal
horizon we have $u\cdot\chi=0$, and Eq.(30) yields,

$\displaystyle\kappa\left(r_{UH}\right)=\left.K_{0}(s\cdot\chi)\right|_{r=r_{UH}}.$
(37)

##  IV Exact Solutions of charged aether black holes

To construct exact solutions of charged aether black holes, let us first
choose the Eddington-Finklestein coordinate system, in which the metric takes
the form

$\displaystyle ds^{2}=-e(r)dv^{2}+2dvdr+r^{2}d\phi^{2},$ (38)

and the corresponding timelike Killing and aether vectors are

$\chi^{a}=(1,0,0),\quad u^{a}=\big{(}\alpha,~{}\beta,~{}0,\big{)},\quad
u_{a}dx^{a}=\big{(}-e\alpha+\beta,~{}\alpha,~{}0,\big{)}\left(\begin{array}[]{c}dv\\\
dr\\\ d\phi\end{array}\right),$ (39)

where $\alpha(r)$ and $\beta(r)$ are functions of $r$ only. Then, the metric
can be written as $g_{ab}=-u_{a}u_{b}+s_{a}s_{b}+\hat{g}_{ab}$, where we have
the constraints $u^{2}=-1,~{}s^{2}=1,~{}u\cdot s=0$.

Some quantities that explicitly appear in Eqs.(20)-(II) are bhattacharyya

$\displaystyle(a\cdot s)=-(u\cdot\chi)^{\prime},\quad
K_{0}=-(s\cdot\chi)^{\prime},\quad\hat{K}=-\frac{(s\cdot\chi)}{r},\quad\hat{k}=-\frac{(u\cdot\chi)}{r},$
(40)

where a prime $(^{\prime})$ denotes a derivative with respect to $r$. And
$\alpha(r),~{}\beta(r)$ and $e(r)$ are

$\displaystyle\alpha(r)=\frac{1}{(s\cdot\chi)-(u\cdot\chi)},\quad\beta(r)=-(s\cdot\chi),\quad
e(r)=(u\cdot\chi)^{2}-(s\cdot\chi)^{2}.$ (41)

Then, from Eqs.(36) and (37) we obtain

$\displaystyle\kappa_{UH}=-\left.\frac{1}{2}\left(u\cdot\chi\right)^{\prime}(s\cdot\chi)\right|_{UH},\quad\kappa(r_{UH})=-\left.\left(s\cdot\chi\right)^{\prime}(s\cdot\chi)\right|_{UH}.$
(42)

Clearly, in general $\kappa_{UH}\not=\kappa(r_{UH})$.

Substituting Eq.(40) into (18), a straightforward calculation yields

$\displaystyle\mathcal{R}_{us}=0.$ (43)

So that the $us$-component gravitational field motion equation
$\mathcal{R}_{us}=\mathcal{T}^{\ae}_{us}+8\pi G_{\ae}\mathcal{T}^{M}_{us}$
yields

$\displaystyle\mathcal{T}^{\ae}_{us}=0,$ (44)

and the aether field motion Eq.(21) gives

$\displaystyle c_{123}\nabla_{s}K=0.$ (45)

They both together with Eqs. (18) and (40) lastly give

$\displaystyle\mathcal{T}^{\ae}_{us}=c_{14}\frac{(s\cdot\chi)}{r^{2}}[r(u\cdot\chi)^{\prime}]^{\prime}=0,$
$\displaystyle
c_{123}\nabla_{s}K=c_{123}(u\cdot\chi)\big{[}\frac{1}{r}\big{(}r(s\cdot\chi)\big{)}^{\prime}\big{]}^{\prime}=0.$
(46)

It is easy to see that there are many ways for satisfying these two equations,
in the following, we shall consider only two special cases
$c_{14}=0,~{}c_{123}\neq 0$ and $c_{123}=0,~{}c_{14}\neq 0$ to obtain both
classes of exact solutions.

###  IV.1 Exact solutions for $c_{14}=0$

When the coupling constant $c_{14}$ is set to zero and $c_{123}\neq 0$, from
Eqs.(21) and (IV) one can see the quantity $\nabla_{s}K$ has to be vanished,
i.e., $\nabla_{s}K=0$. So, the trace of the extrinsic curvature $K$ of the
$\Sigma_{U}$ hypersurface is a constant and, Eq. (IV) gives

$\displaystyle(s\cdot\chi)=\Lambda^{\prime}r+\frac{r_{\ae}}{r},$ (47)

where $\Lambda^{\prime},r_{\ae}$ are some constants. Since a vanishing
$(s\cdot\chi)$ signifies of the aether $u^{a}$ aligning with Killing vector
$\chi^{a}$, so both constants measure the misalignment of the aether. When
$\Lambda^{\prime}=0$, then the aether aligns Killing vector at infinite, say,
aether asymptotically flat. Substituting Eq.(47) into (II), we obtain

$\displaystyle(u\cdot\chi)=-\sqrt{-m+2\Lambda^{\prime}r_{\ae}+\bar{\Lambda}r^{2}-\frac{Q^{2}}{2}\ln\frac{r}{l_{eff}}+(1-c_{13})\frac{r^{2}_{\ae}}{r^{2}}}\;,$
(48) $\displaystyle e(r)=-m+(\bar{\Lambda}-\Lambda^{\prime
2})r^{2}-\frac{Q^{2}}{2}\ln\frac{r}{l_{eff}}-\frac{c_{13}r^{2}_{\ae}}{r^{2}},$
(49)

where $m$ is an integral constant which then shows the mass of the black hole,
$1/l_{eff}=\sqrt{\bar{\Lambda}}$ and,
$\bar{\Lambda}=-\Lambda+(1+c_{123}+c_{2})\Lambda^{\prime 2}$. It is BTZ-like
black hole, and when $Q=0$, it can reduce to the static solution given in Ref.
sotiriou  via Hořava-Lifshitz gravity. If $c_{13}=0$ and $\Lambda^{\prime}=0$,
it is the usual BTZ black hole, but now with an universal horizon, which is
very interesting.

The metric (49) can be (quasi) asymptotically dS, flat or AdS, when
$(\bar{\Lambda}-\Lambda^{\prime 2})<0,=0$, or $>0$, respectively. And there is
a condition that $(u\cdot\chi)^{2}\geqslant 0$. From (48) and at large $r$,
the term $\bar{\Lambda}r^{2}$ dominates $(u\cdot\chi)^{2}$; at little $r$, the
term $(1-c_{13})$ dominates, therefore $\bar{\Lambda}$ should be positive,
$\bar{\Lambda}>0$ and $c_{13}<1$. Hence for AdS asymptotics, it is
$(c_{2}+c_{123})\Lambda^{\prime 2}>\Lambda$ and $m>0$; for quasi-flat, it is
$(c_{2}+c_{123})\Lambda^{\prime 2}=\Lambda$ and $m<0$($m=-1$ for flat); for
dS, it is $(c_{2}+c_{123})\Lambda^{\prime
2}<\Lambda<(1+c_{2}+c_{123})\Lambda^{\prime 2}$ and $m<0$. If
$\Lambda^{\prime}=0$, there are also three asymptotics, dS, flat or AdS for
$\Lambda>0,=0$ or $<0$, respectively.

The location of the universal horizon $r_{UH}$ is the largest root of equation
$u\cdot\chi=0$. Meanwhile, $u\cdot\chi$ is a physical component of the aether,
and should be regular and real everywhere. However, from Eq.(48) one can see
that in the region $r_{-}<r<r_{UH}$, this term becomes purely imaginary, where
$r_{-}$ is another root of $u\cdot\chi=0$, unless the two real roots coincide.
Then, $r_{\ae}$ becomes a function of $m$. That is, the global existence of
the aether reduces the number of four independent constants
($m,\Lambda^{\prime},r_{\ae},Q$) to three, ($m,\Lambda^{\prime},Q$). Thus,
from $(u\cdot\chi)^{2}=0$ and $d(u\cdot\chi)^{2}/dr=0$ lin , we find

$\displaystyle
r^{2}_{\ae}=\frac{1}{1-c_{13}}\left(\bar{\Lambda}r_{UH}^{4}-\frac{Q^{2}}{4}r_{UH}^{2}\right),\quad
m=2\bar{\Lambda}r_{UH}^{2}-\frac{Q^{2}}{2}\Big{(}\ln\frac{r_{UH}}{l}+\frac{1}{2}\Big{)}+2\Lambda^{\prime}r_{\ae}.$
(50)

Note that from (50), if the black hole and the aether both asymptotically
flat, i.e., $\Lambda^{\prime}=0,\bar{\Lambda}=0$, there is no universal
horizon which has been showed in Ref.sotiriou . However, if $r_{\ae}=0$, there
still exists an universal horizon due to presence of electric charge $Q$, and
we will call it quasi-BTZ black hole.

Now let us derive the Smarr formula and the first law at the universal
horizon. The surface gravity at the universal horizon can be computed via (36)
and given by

$\displaystyle\kappa_{UH}=\frac{1}{2}\nabla_{u}(u\cdot\chi)|_{r_{UH}}=\frac{1}{2r_{UH}}\sqrt{\bar{\Lambda}r_{UH}^{2}-\frac{Q^{2}}{8}}\left[\sqrt{\frac{4\bar{\Lambda}r_{UH}^{2}-Q^{2}}{1-c_{13}}}+2\Lambda^{\prime}r_{UH}\right].$
(51)

If one uses definition (37), then this surface gravity is

$\displaystyle\kappa(r_{UH})=K_{0}(s\cdot\chi)|_{r_{UH}}=-\Lambda^{\prime
2}r_{UH}+\frac{1}{(1-c_{13})r_{UH}}\left(\bar{\Lambda}r_{UH}^{2}-\frac{Q^{2}}{4}\right),$
(52)

which is different.

From (33) and (34), one get the Komar integral relation

$\displaystyle 0\cdot MG_{\ae}=2\pi r_{UH}[\Lambda
r_{UH}+(1-c_{13})K_{0}(s\cdot\chi)+c_{123}K(s\cdot\chi)]+4\pi
c_{123}\Lambda^{\prime}r_{\ae}+\frac{\pi Q^{2}}{2}.$ (53)

Then by using surface gravity (51) and the knowledge of black hole chemistry,
we can derive the Smarr formula and first law of these Lorentz-violating black
holes. In the following, we study on these thermodynamic properties of the
universal horizons in three cases.

####  IV.1.1 (Quasi)asymptotically flat black hole

Consider firstly uncharged and quasi-asymptotically flat spacetime, i.e.,
$Q=0$ and $\Lambda^{\prime 2}=\Lambda/(c_{2}+c_{123})$. From (49), this metric
is

$\displaystyle e(r)=-m-\frac{c_{13}r^{2}_{\ae}}{r^{2}}.$ (54)

To ensure there exists a black hole, it requires that $m<0$ when $0<c_{13}<1$;
$m>0$ when $c_{13}<0$. If $m=-1$, it is exact asymptotically flat spacetime,
therefore these black holes are termed (quasi)asymptotically flat. From (50),
then the mass $m$ is

$\displaystyle
m=\left\\{\begin{array}[]{c}2\bar{\Lambda}r_{UH}^{2}\left(1-\frac{1}{\sqrt{1-c_{13}}}\right),\quad(0<c_{13}<1),\\\
2\bar{\Lambda}r_{UH}^{2}\left(1+\frac{1}{\sqrt{1-c_{13}}}\right),\quad(c_{13}<0).\end{array}\right.$
(57)

In the following, we let $m>0$.

Dividing both sides of Eq. (53) by $8\pi
c_{13}\sqrt{1-c_{13}}/(1+\sqrt{1-c_{13}})$, we can obtain the Smarr formula

$\displaystyle 0=TS-2PV,$ (58)

where

$\displaystyle
T=\frac{\kappa_{UH}}{2\pi}=\Big{[}1+\frac{1}{\sqrt{1-c_{13}}}\Big{]}\frac{r_{UH}}{2\pi
l_{eff}^{2}},\quad S=A/4=\frac{\pi r_{UH}}{2},\quad
M=\frac{m}{16}=\Big{[}1+\frac{1}{\sqrt{1-c_{13}}}\Big{]}\frac{r_{UH}^{2}}{8l_{eff}^{2}},$
$\displaystyle P=\frac{1}{8\pi l_{eff}^{2}},\quad V=\frac{\partial M}{\partial
P}\Big{|}_{S}=\Big{[}1+\frac{1}{\sqrt{1-c_{13}}}\Big{]}\pi
r_{UH}^{2},\quad(\frac{1}{l_{eff}}=\sqrt{\bar{\Lambda}}).$ (59)

Based on these definitions, it is straightforward to verify the first law

$\displaystyle dM=TdS+VdP,$ (60)

which is the same as (27).

####  IV.1.2 Aether asymptotically flat black hole

Secondly the uncharged black hole and the aether aligning with Killing vector
at infinite, i.e., $Q=0$ and $\Lambda^{\prime}=0$. From (49), this metric is

$\displaystyle e(r)=-m-\Lambda r^{2}-\frac{c_{13}r^{2}_{\ae}}{r^{2}}.$ (61)

There are both Killing horizons

$\displaystyle
r^{2}_{\pm}=-\frac{m}{2\Lambda}\left[1\pm\sqrt{1-\frac{\Lambda}{m^{2}}c_{13}r^{2}_{\ae}}\right],$
(62)

and have three asymptotics, i.e., AdS when $(\Lambda<0,m>0,c_{13}<0)$; dS when
$(\Lambda>0,m<0,0<c_{13}<1)$; (quasi)flat when $\Lambda=0$.

Dividing both sides of Eq. (53) by $8\pi$, we can obtain the slightly modified
Smarr formula

$\displaystyle 0=\sqrt{1-c_{13}}TS-2PV,$ (63)

where

$\displaystyle
T=\frac{\kappa_{UH}}{2\pi}=\frac{1}{\sqrt{1-c_{13}}}\frac{r_{UH}}{2\pi
l^{2}},\quad S=A/4=\frac{\pi r_{UH}}{2},\quad
M=\frac{m}{16}=\frac{r_{UH}^{2}}{8l^{2}},$ $\displaystyle P=\frac{1}{8\pi
l^{2}},\quad V=\frac{\partial M}{\partial P}\Big{|}_{S}=\pi
r_{UH}^{2},\quad(\frac{1}{l}=\sqrt{|\Lambda|}).$ (64)

Based on these definitions, it is straightforward to verify the slightly
modified first law

$\displaystyle dM=\sqrt{1-c_{13}}TdS+VdP.$ (65)

####  IV.1.3 Quasi-BTZ black hole

Thirdly, the case of $r_{\ae}=0$. From (49), this metric is

$\displaystyle e(r)=-m+(\bar{\Lambda}-\Lambda^{\prime
2})r^{2}-\frac{Q^{2}}{2}\ln\frac{r}{l_{eff}},$ (66)

which is only different cosmological constant from BTZ metric banados , so
termed quasi-BTZ black hole. Its universal horizon is

$\displaystyle r_{UH}=\frac{Q}{2\sqrt{\bar{\Lambda}}}.$ (67)

Dividing both sides of Eq. (53) by $8\pi$, we can obtain the slightly modified
Smarr formula

$\displaystyle 0=-\frac{\sqrt{2}}{\Lambda^{\prime}l_{eff}}TS-2PV,$ (68)

where

$\displaystyle
T=\frac{\kappa_{UH}}{2\pi}=\frac{\Lambda^{\prime}}{2\sqrt{2}\pi}\frac{r_{UH}}{l_{eff}},\quad
S=A/4=\frac{\pi r_{UH}}{2},\quad
M=\frac{m}{8}=\frac{Q^{2}}{16}(\frac{1}{2}-\ln\frac{r_{UH}}{l_{eff}}),$
$\displaystyle P=\frac{1}{8\pi l_{eff}^{2}},\quad V=\frac{\partial M}{\partial
P}\Big{|}_{S,Q}=-\frac{\pi Q^{2}}{4}l_{eff}^{2},\quad\Phi=\frac{\partial
M}{\partial
Q}\Big{|}_{S,P}=\frac{Q}{8}(\frac{1}{2}-\ln\frac{r_{UH}}{l_{eff}}),$ (69)

where $\frac{1}{l_{eff}}=\sqrt{\bar{\Lambda}}$. Based on these definitions, it
is straightforward to verify the slightly modified first law

$\displaystyle dM=-\frac{\sqrt{2}}{\Lambda^{\prime}l_{eff}}TdS+VdP+\Phi dQ.$
(70)

In summary, for these three cases, the exact or the slightly modified first
law of black hole mechanics can be construct, which show the corresponding
universal horizons would have a thermodynamic interpretation. For the general
cases of nonzero $Q,\Lambda^{\prime},r_{\ae}$, and $\Lambda^{\prime
2}\neq\Lambda/(c_{2}+c_{123})$, instead of above three cases, one cannot build
the Smarr formula and first law. As for causes, there may be the black hole
mass need to be modified, see Ref.ding ; ding2  for 4-dimensions, or couples
between the charge $Q$ and aether field should be considered togetherbalakin ;
balakin2016 . These open issues are our future works.

###  IV.2 Exact solutions for $c_{123}=0$

When the coupling constant $c_{123}$ is set to zero and $c_{14}\neq 0$, Eq.
(IV) gives

$\displaystyle(u\cdot\chi)=u_{0}\ln\frac{r}{r_{UH}},$ (71)

where $u_{0}$ is a constant and $r_{UH}$ shows the location of universal
horizon. Substituting it into Eqs.(II), we obtain

$\displaystyle(s\cdot\chi)=\frac{1}{\sqrt{1-c_{13}}}\sqrt{(1-c_{13})m+\Lambda
r^{2}+(c_{14}u_{0}^{2}+\frac{Q^{2}}{2})\ln\frac{r}{l}+u_{0}^{2}\ln^{2}\frac{r}{r_{UH}}}\;,$
(72) $\displaystyle e(r)=-m-\frac{1}{1-c_{13}}\Big{[}\Lambda
r^{2}+(c_{14}u_{0}^{2}+\frac{Q^{2}}{2})\ln\frac{r}{l}+c_{13}u_{0}^{2}\ln^{2}\frac{r}{r_{UH}}\Big{]},$
(73)

where $m$ is an integral constant. Since $r^{2}\gg\ln r$ and
$(s\cdot\chi)\sim\sqrt{\Lambda}r$ as $r\rightarrow\infty$, the cosmological
constant would be non-negative. Therefore this solution can be asymptotically
dS. It is easy to see that there are four independent constants
$(m,u_{0},r_{UH},Q)$, i.e., the universal horizon $r_{UH}$ cannot be showed
via mass parameter $m$, which is unacceptable.

When $c_{13}=0$, it also reduces quasi-BTZ black hole

$\displaystyle e(r)=-m-\Big{[}\Lambda
r^{2}+(c_{14}u_{0}^{2}+\frac{Q^{2}}{2})\ln\frac{r}{l}\Big{]},$ (74)

which is similar to (66).

The surface gravity at the universal horizon can be computed and given by

$\displaystyle\kappa_{UH}=\frac{u_{0}}{2r_{UH}}\sqrt{m+\frac{1}{1-c_{13}}\big{[}\Lambda
r_{UH}^{2}+(\frac{Q^{2}}{2}+c_{14}u_{0}^{2})\frac{r_{UH}}{l}\big{]}}.$ (75)

From (33) and (34), one get the Komar integral

$\displaystyle 0\cdot MG_{\ae}-\frac{\pi Q^{2}}{2}-\pi c_{14}u_{0}^{2}=2\pi
r_{UH}[\Lambda r_{UH}+(1-c_{13})K_{0}(s\cdot\chi)|_{UH}].$ (76)

However the relation between $m$ and $r_{UH}$ is unknown, so that we cannot
derive the Smarr formula and the first law at universal horizon. In Ref.
sotiriou , the authors used Brown-Henneaux AdS boundary conditions to show
that $c_{14}(\sim\eta)$ should be zero. So in the case of $c_{14}\neq 0$, it
is dS asymptotics, instead of AdS asymptotics. But neither its universal
horizon nor black hole mass is under determined.

##  V Conclusions

In this paper, we have studied the Einstein-Maxwell-aether theory in
3-dimensions, and found two new classes of charged black hole solutions for
the special choices of the coupling constants: (1) $c_{14}=0,~{}c_{123}\neq
0$, and (2) $c_{14}\neq 0,~{}c_{123}=0$. There have three asymptotic forms,
dS, flat or AdS, unlike general relativity, dependant on the bare(effective)
cosmological constant $\Lambda(\bar{\Lambda})$.

In the first case $c_{14}=0,~{}c_{123}\neq 0$, the universal horizon depends
on its electric charge $Q$, cosmological constant $\Lambda$ and aether
constant $\Lambda^{\prime}$. When $c_{13}\;(\equiv c_{1}+c_{3})$ and
$\Lambda^{\prime}$ are very small and approach to zero, the solutions reduce
to the usual charged BTZ black hole solution but now with an universal
horizon, which is very interesting. In the second case $c_{14}\neq
0,~{}c_{123}=0$, the universal horizon may not exist though there have
solutions with Killing horizon.

To study the solutions further, we have considered their surface gravity and
constructed the Smarr formula at the universal horizons in three cases:
(quasi)asymptotically flat $Q=0,\Lambda^{\prime 2}=\Lambda/(c_{2}+c_{123})$,
aether asymptotically flat $Q=0,\Lambda^{\prime}=0$ and quasi-BTZ black hole
spacetime $Q\neq 0,r_{\ae}=0$. As for (quasi)asymptotically flat and uncharged
black hole, the exact Smarr formula and the first law of black hole mechanics
have been constructed with the knowledge of black hole chemistry. It is showed
that the universal horizon may have an entropy and, the black hole mass is
interpreted as an enthalpy of spacetime, the cosmological constant as a vacuum
pressure. As for aether asymptotically flat and uncharged black hole, the
slightly modified Smarr formula and the first law have been constructed. As
for quasi-BTZ black hole, also the slightly modified Smarr formula and the
first law have been constructed in the presence of electrical charge $Q$.

For the general cases of nonzero $Q,\Lambda^{\prime},r_{\ae}$, and
$\Lambda^{\prime 2}\neq\Lambda/(c_{2}+c_{123})$, instead of above three
special cases, one cannot build the Smarr formula and first law. However, if
one modifies the black hole mass via universal horizon temperature, or
considers together with the couples between electromagnetical field and aether
field, the first law would be constructed. Therefore, the universal horizon,
as well as Killing horizon, can have an thermodynamical interpretation and,
holography may be extended to it.

###### Acknowledgements.

C.D. was supported by NNSFC No. 11247013, Hunan Provincial NSFC No. 2015JJ2085
and QSQC1203. C.L. was supported by special fund of NNSFC No. 11447168. A.W.
was supported in part by Ciência Sem Fronteiras, No. A045/2013 CAPES, Brazil
and NNSFC No. 11375153, China. J.J. was supported by NNSFC No. 11475061; the
SRFDP No. 20114306110003.

## References

  * (1) D. Mattingly, Living Rev. Rel. 8, 5 (2005).
  * (2) J. Bhattacharyya, Ph.D dessertation, University of New Hamspshire, Durham, NH, USA (2013).
  * (3) A. Shomer, arXiv:0709.3555.
  * (4) T. Jacobson and D. Mattingly, Phys. Rev. D 64, 024028 (2001).
  * (5) A. A. Abdo et al., Nature 462, 331 (2009).
  * (6) T. Jacobson and D. Mattingly, Phys. Rev. D 64, 024028 (2001); T. Jacobson, Proc. Sci. QG-PH, 020 (2007) [arXiv:0801.1547].
  * (7) J. D. Bekenstein, Phys. Rev. D 70, 083509 (2004), [Erratum: Phys. Rev.D 71,069901(2005)]; T. G. Zlosnik, P. G. Ferreira, and G. D. Starkman, Phys. Rev.D 74, 044037 (2006); T. G. Zlosnik, P. G. Ferreira, and G. D. Starkman, Phys. Rev. D 75, 044017 (2007); L. Blanchet and S. Marsat, Phys. Rev. D 84, 044056 (2011); M. Bonetti and E. Barausse, Phys. Rev. D 91, 084053 (2015).
  * (8) A. B. Balakin and H. Dehnen, Phys. Lett. B 681, 113 (2009).
  * (9) T. Jacobson and D. Mattingly, Phys. Rev. D 63, 041502 (2001).
  * (10) T. Jacobson and D. Mattingly, Phys. Rev. D 70, 024003 (2004).
  * (11) D. Blas and S. Sibiryakov, Phys. Rev. D 84, 124043 (2011).
  * (12) K. Lin, O. Goldoni, M. F. da Silva and A. Wang, Phys. Rev. D 91, 024047 (2015).
  * (13) K. Lin, F.-W. Shu, A. Wang, and Q. Wu, Phys. Rev. D 91, 044003 (2015);F.-W. Shu, K. Lin, A. Wang, and Q. Wu, JHEP 04, 056 (2014); K. Lin, E. Abdalla, R.-G. Cai, and A. Wang, Inter. J. Mod. Phys. D 23, 1443004 (2014); P. Horava, A. Mohd, C. M. Melby-Thompson, P. Shawhan, Gen. Rel. Grav. 46, 1720 (2014); S. Janiszewski, A. Karch, B. Robinson, and D. Sommer, JHEP 04, 163 (2014); C. Eling and Y. Oz, JHEP 11, 067 (2014); M. Saravani, N. Afshordi, and R.B. Mann, Phys. Rev. D 89, 084029 (2014); A. Mohd, arXiv:1309.0907; B. Cropp, S. Liberati, and M. Visser, Class. Quantum Grav. 30, 125001 (2013).
  * (14) E. Barausse, T. Jacobson and T. P. Sotiriou, Phys. Rev. D 83, 124043 (2011).
  * (15) J. Latta and G. Leon, arXiv: 1606.08586.
  * (16) A. B. Balakin, Phys. Rev. D 94, 024021 (2016).
  * (17) M. Gürses and Ç. Şentürk, arXiv: 1604.02266.
  * (18) B. Cropp, S. Liberati and R. Turcati, Phys. Rev. D 94, 063003 (2016).
  * (19) P. Berglund, J. Bhattacharyya and D. Mattingly, Phys. Rev. D 85, 124019 (2012).
  * (20) J. Bhattacharyya and D. Mattingly, Int. J. Mod. Phys. D 23, 1443005 (2014).
  * (21) C. Ding, A. Wang and X. Wang, Phys. Rev. D 92, 084055 (2015).
  * (22) E. Barausse, T. P. Sotiriou and I. Vega, Phys. Rev. D 93, 044044 (2016).
  * (23) M. Banãdos, C. Teitelboim and J. Zanelli, Phys. Rev. Lett. 69, 1849 (1992); M. Banãdos, M. Henneaux, C. Teitelboim and J. Zanelli, Phys. Rev. D 48, 1506 (1993).
  * (24) L. Donnay, arXiv: 1602.09021.
  * (25) T. Sotiriou, I. Vega, and D. Vernieri, Phys. Rev. D 90, 044046 (2014).
  * (26) C. Eling and T. Jacobson, Phys. Rev. D 74, 084027 (2006).
  * (27) S. Basu, J. Bhattacharyya, D. Mattingly and M. Roberson, Phys. Rev. D 93, 064072 (2016).
  * (28) A. M. Frassino, R. B. Mann and J. R. Mureika, Phys. Rev. D 92, 124069 (2015).
  * (29) C. Eling, Phys. Rev. D 73, 084026 (2006).
  * (30) D. Garfinkle, T. Jacobson, Phys. Rev. Lett. 107, 191102 (2011).
  * (31) A. B. Balakin and J. P. S. Lemos, Ann. Phys. 350, 454 (2014); S. Janiszewski, A. Karch, B. Robinson, and D. Sommer, JHEP 04, 163 (2014).
  * (32) D. Kastor, S. Ray and J. Traschen, Class. Quantum Gravi. 26, 195011 (2009).
  * (33) D. Kubiznak and R. B. Mann, Can. J. Phys. 93, 999 (2015).
  * (34) J. M. Bardeen, B. Carter, and S. W. Hawking, Commun. Math. Phys. 31, 161 (1973).
  * (35) T. Jacobson, in CPT and Lorentz Symmetry: Proceedings of the Fourth Meeting, Bloomington, USA, 11 August 2007, edited by V. A. Kostelecky (World Scientific, Singapore, 2008), p. 92, arXiv:0711.3822.
  * (36) B. Cropp, S. Liberati and A. Mohd, Phys. Rev. D 89, 064061 (2014).
  * (37) C. Ding, A. Wang, X. Wang and T. Zhu, Nucl. Phys. B 913, 694 (2016).