#  Relativistic Correction to Charmonium Dissociation Temperature

Xingyu Guo  Shuzhe Shi  Pengfei Zhuang  Physics Department, Tsinghua
University, Beijing 100084, China

###### Abstract

By solving the covariant relativistic Schrödinger equations for a pair of
heavy quarks, we obtained the wave functions for the ground and excited
quarkonium states at finite temperature. In comparison with the non-
relativistic calculation, the $J/\psi$ dissociation temperature determined by
the infinity size or zero binding energy of the system increases $7\%-13\%$,
when the central potential varies between the free energy and internal energy.

It is widely accepted that there exists a quantum chromodynamics (QCD) phase
transition from hadron gas to a new state of matter, the quark-gluon plasma
(QGP) at finite temperature and baryon density. To probe the realization of
such a phase transition in relativistic heavy ion collisions, some signatures
of the new state of matter have been discussed for decadesqgp , and among
which the quarkonium suppression is considered as a smoking gun of the
formation of QGPsatz1 .

A quarkonium is a deeply bound state of a pair of heavy quarks, its
dissociation temperature $T_{D}$ in a hot medium should be higher than the
critical temperature $T_{c}$ for the deconfinement of light hadrons. Since a
quarkonium is so heavy, one normally uses non-relativistic Schrödinger
equation to describe its dynamical evolution at finite temperature. The free
energy $F$ between a pair of heavy quarks can be extracted from the lattice
QCD simulationsdigal ; kaczmarek . Taking $F$ and the internal energy $U$ as
the two limits of the heavy quark potential $V$, which correspond respectively
to a slow and a rapid quarkonium dissociation in the hot medium, the $J/\psi$
dissociation temperature $T_{D}$ determined by the infinity size or the zero
binding energy of the system is in between $1.2T_{c}$ and $2T_{c}$digal ;
satz2 .

A nature question we ask ourselves is the relativistic correction to the
dynamical evolution of a quarkonium in the hot medium. The correction to a
bottonium is expected to be neglected safely, but for a lighter quarkonium
like $J/\psi$, the correction might be remarkable. The two-body Dirac equation
(TBDE)t1 ; t2 ; t3 ; t4 ; t5 ; long  of constrained dynamics was successfully
applied to the relativistic description of light meson spectras1 ; s2 ; s3 ;
s4 ; crater2 ; crater1  in vacuum. In this Letter we take the TBDE to
calculate the charmonium wave functions at finite temperature and see the
relativistic correction to the dissociation temperature.

We can qualitatively estimate the relativistic effect on the quarkonium
dissociation before a strict calculation. Neglecting the quark spin, the
relative part of the Hamiltonian for a pair of heavy quarks can be
approximately written as a non-relativistic form

$H=\sqrt{\mu^{2}+p^{2}}-\mu+V(r)\simeq{p^{2}\over 2\mu}+V_{eff}$ (1)

with an effective potential

$V_{eff}=V-{p^{4}\over 8\mu^{3}},$ (2)

where $\mu=m_{Q}/2$ is the reduced mass with $m_{Q}$ being the heavy quark
mass. Since the relativistic correction leads to a deeper potential well,
$V_{eff}<V$, the quarkonium becomes a more deeply bound state and the
temperature needed to dissociate the quarkonium should be higher.

We now calculate the wave functions for a pair of heavy quarks in the frame of
TBDE. Taking Pauli reduction and scale transformation333 , the Dirac equation
can be expressed as a covariant relativistic Schrödinger equation for a four-
component spinorlong . Explicitly, the radial motion relative to the center of
mass is controlled by the following four equationscrater2 ,

$\displaystyle\left[-{d^{2}\over dr^{2}}+{J(J+1)\over
r^{2}}+2m_{w}B+B^{2}-A^{2}+2\epsilon_{w}A+\Phi_{D}-2\Phi_{SO}+\Phi_{SS}+2\Phi_{T}-2\Phi_{SOT}\right]u_{1}^{0}=b^{2}u_{1}^{0},$
(3) $\displaystyle\left[-{d^{2}\over dr^{2}}+{J(J-1)\over
r^{2}}+2m_{w}B+B^{2}-A^{2}+2\epsilon_{w}A+\Phi_{D}+2(J-1)\Phi_{SO}+\Phi_{SS}+{2(J-1)\over
2J+1}(\Phi_{SOT}-\Phi_{T})\right]u_{1}^{+}$ $\displaystyle+$
$\displaystyle{2\sqrt{J(J+1)}\over
2J+1}\left(3\Phi_{T}-2(J+2)\Phi_{SOT}\right)u_{1}^{-}=b^{2}u_{1}^{+},$
$\displaystyle\left[-{d^{2}\over dr^{2}}+{(J+1)(J+2)\over
r^{2}}+2m_{w}B+B^{2}-A^{2}+2\epsilon_{w}A+\Phi_{D}-2(J+2)\Phi_{SO}+\Phi_{SS}+{2(J+2)\over
2J+1}\left(\Phi_{SOT}-\Phi_{T}\right)\right]u_{1}^{-}$ $\displaystyle+$
$\displaystyle{2\sqrt{J(J+1)}\over
2J+1}\left(3\Phi_{T}+2(J-1)\Phi_{SOT}\right)u_{1}^{+}=b^{2}u_{1}^{-}$

for the spin triplet $u_{1}^{0},u_{1}^{+}$ and $u_{1}^{-}$ with quantum
numbers $n^{2s+1}L_{J}=n^{3}L_{L},\ n^{3}L_{L+1}$ and $n^{3}L_{L-1}$, and

$\left[-{d^{2}\over dr^{2}}+{J(J+1)\over
r^{2}}+2m_{w}B+B^{2}-A^{2}+2\epsilon_{w}A+\Phi_{D}-3\Phi_{SS}\right]u_{0}=b^{2}u_{0}$
(4)

for the spin singlet $u_{0}$ with quantum numbers $n^{1}L_{L}$, where
$b^{2}=(m_{m}^{2}-4m_{Q}^{2})/4$ is the energy eigenvalue in the meson rest
frame with $m_{m}$ being the meson mass, $n$ is the principal quantum number,
and $L,s$ and $J$ are the orbital, spin and total angular momentum numbers.
Note that the components $u_{1}^{+}$ with $J=L+1$ and $u_{1}^{-}$ with $J=L-1$
are coupled to each other. Following the notations in crater2 , we have
separated the central potential into two parts,

$V(r)=A(r)+B(r),$ (5)

and the abbreviations for the Darwin, spin-spin, spin-orbit and tensor terms
introduced in the dynamical equations are defined as crater2

$\displaystyle\Phi_{D}$ $\displaystyle=M+F^{\prime 2}+K^{\prime
2}-\nabla^{2}F+2K^{\prime}P-2\left(F^{\prime}+{1\over r}\right)Q,$ (6)
$\displaystyle\Phi_{T}$ $\displaystyle={1\over
3}\left[N+2F^{\prime}K^{\prime}-\nabla^{2}K+\left(3F^{\prime}-K^{\prime}+{3\over
r}\right)P+\left(F^{\prime}-3K^{\prime}+{1\over r}\right)Q\right]$
$\displaystyle\Phi_{SO}$ $\displaystyle=-{F^{\prime}\over
r}+K^{\prime}P-\left(F^{\prime}+{1\over r}\right)Q,$ $\displaystyle\Phi_{SS}$
$\displaystyle=O+{2\over 3}F^{\prime}K^{\prime}-{1\over 3}\nabla^{2}K+{2\over
3}K^{\prime}P-2\left(F^{\prime}+{1\over 3r}\right)Q,$
$\displaystyle\Phi_{SOT}$ $\displaystyle=-{K^{\prime}\over
r}+\left(F^{\prime}+{1\over r}\right)P-K^{\prime}Q,$ $\displaystyle F$
$\displaystyle={1\over 2}L-{3\over 2}G,$ $\displaystyle G$
$\displaystyle=-{1\over 2}\ln\left(1-2{A\over m_{m}}\right),$ $\displaystyle
K$ $\displaystyle={1\over 2}L+{1\over 2}G,$ $\displaystyle L$
$\displaystyle=\ln\sqrt{1+{2m_{w}B+B^{2}\over
m_{Q}^{2}\left(1-2A/m_{m}\right)}},$ $\displaystyle M$ $\displaystyle=-{1\over
2}\nabla^{2}G+{3\over 4}G^{\prime 2}+G^{\prime}F^{\prime}-K^{\prime 2},$
$\displaystyle N$ $\displaystyle={1\over 3}\left[\nabla^{2}K-{1\over
2}\nabla^{2}G+{3\over 2}{G^{\prime}-2K^{\prime}\over
r}+F^{\prime}(G^{\prime}-2K^{\prime})\right]$ $\displaystyle O$
$\displaystyle={1\over 3}\nabla^{2}(K+G)-{1\over
3}F^{\prime}(G^{\prime}+K^{\prime})-{1\over 2}G^{\prime 2},$ $\displaystyle P$
$\displaystyle={\sinh 2K\over r},$ $\displaystyle Q$ $\displaystyle={\cosh
2K-1\over r}$

and
$m_{w}=m_{Q}^{2}/m_{m},\epsilon_{w}=(m_{m}^{2}-2m_{Q}^{2})/(2m_{m}),F^{\prime}=dF/dr,G^{\prime}=dG/dr$
and $K^{\prime}=dK/dr$.

We now focus on the charmonium states. The $J/\psi$ state includes two
components $1^{3}S_{1}$ and $1^{3}D_{1}$ determined simultaneously by the two
coupled equations of (3) with energy eigenvalue
$b^{2}=(m_{J/\psi}^{2}-4m_{c}^{2})/4$. Similarly, the two components
$2^{3}S_{1}$ and $2^{3}D_{1}$ for the state $\psi^{\prime}$ are controlled by
the two coupled equations but with eigenvalue
$b^{2}=(m_{\psi^{\prime}}^{2}-4m_{c}^{2})/4$. The $\chi_{0}$ state
$1^{3}P_{0}$ is described by the last equation of (3) with disappeared
component $u_{1}^{+}$, and the $\chi_{1}$ state $1^{3}P_{1}$ is characterized
by the independent equation (4) for $u_{1}^{0}$.

In vacuum, the potential between two quarks is usually taken as the Cornell
form, including a Coulomb-like part which dominates the wave functions around
$r=0$ and a linear part which leads to the quark confinement,

$\displaystyle A(r)$ $\displaystyle=-{\alpha\over r},$ (7) $\displaystyle
B(r)$ $\displaystyle=\sigma r.$

The three parameters in the model, namely the charm quark mass $m_{c}$ in the
Schrödinger equation and the two coupling constants $\alpha$ and $\sigma$ in
the potential, can be fixed by fitting the charmonium masses in vacuum. By
taking $m_{c}=1.422$ GeV, $\alpha=0.492$ and $\sigma=0.186$ GeV2, we obtain
the charmonium masses $m_{J/\psi}=3.113$ GeV, $m_{\psi^{\prime}}=3.692$ GeV,
$m_{\chi_{0}}=3.404$ GeV and $m_{\chi_{1}}=3.504$ GeV, which are very close to
the experimental values $m_{J/\psi}=3.097$ GeV, $m_{\psi^{\prime}}=3.686$ GeV,
$m_{\chi_{0}}=3.415$ GeV and $m_{\chi_{1}}=3.511$ GeV.

The charmonium dissociation temperature in a hot medium can be determined by
solving the corresponding dynamical equation for the $c\bar{c}$ system with
potential $V$ between the two heavy quarks at finite temperature. The
potential depends on the dissociation process in the medium. At the moment, we
know only its two limits. For a rapid dissociation where there is no heat
exchange between the heavy quarks and the medium, the potential is just the
internal energy $U$, while for a slow dissociation, there is enough time for
the heavy quarks to exchange heat with the medium, the free energy $F$ which
can be extracted from the lattice calculations is taken as the
potentialshuryak . From the thermodynamic relation $F=U-T{\cal S}$ where
${\cal S}$ is the entropy, the potential well is deeper for $V=U$ and
therefore the dissociation temperature of charmonium states with potential
$V=U$ is higher than that with $V=F$. In the literatures, a number of
effective potentials in between $F$ and $U$ have been used to evaluate the
charmonium evolution in QCD mediumdigal ; satz2 ; shuryak ; wong . In non-
relativistic case, H.Satz and his collaboratorsdigal ; satz2  solved the
Schrödinger equation and found the dissociation temperatures $T_{d}/T_{c}=2.1$
and $1.26$ for $V=U$ and $V=F$ respectively, where $T_{c}=165$ MeVdigal ;
satz2  is the critical temperature for the deconfinement.

Considering the Debye screening at finite temperature, the potential $V=F=A+B$
can be written asdigal ; satz2

$\displaystyle A(r,T)$ $\displaystyle=$ $\displaystyle-{\alpha\over r}e^{-\mu
r},$ $\displaystyle B(r,T)$ $\displaystyle=$
$\displaystyle{\sigma\over\mu}\left[{\Gamma\left({1\over 4}\right)\over
2^{3\over 2}\Gamma\left({3\over 4}\right)}-{\sqrt{\mu r}\over 2^{3\over
4}\Gamma\left({3\over 4}\right)}K_{1\over
4}\left(\mu^{2}r^{2}\right)\right]-\alpha\mu,$ (8)

where $\Gamma$ is the Gamma function, $K$ is the modified Bessel function of
the second kind, and the temperature dependent parameter $\mu(T)$, namely the
screening mass or the inverse screening radius, can be extracted from fitting
the lattice simulated free energy digal ; kaczmarek . From the known free
energy $F$, one can then obtain the other limit of the potential,
$V=U=F+T{\cal S}$ by taking ${\cal S}=-\partial F/\partial T$.

We use the inverse power methodcrater3  to solve the Schrödinger equations (3)
and (4) and obtain the radial wave functions

$\Psi(r,T)={u(r,T)\over r}$ (9)

for the charmonium states $J/\psi$, $\psi^{\prime}$ and $\chi_{c}$ at finite
temperature.

Figure 1:  The $S$-wave (solid lines) and $D$-wave (dashed lines) functions
for the $J/\psi$ meson at three temperatures in the limit of quark potential
$V=F$. From the top down the temperature $T$ is zero, critical temperature
$T_{c}$ and relativistic dissociation temperature $T_{d}$.

The $S-$ and $D-$wave functions for the meson $J/\psi$ at different
temperature are shown in Fig.1 in the limit of $V=F$. The wave functions in
vacuum are very similar to the results obtained in crater2 , and the component
$1^{3}S_{1}$ dominates the state at any temperature. With increasing
temperature, both the $S$\- and $D$-wave functions expand continuously. At the
critical temperature of deconfinement $T_{c}$, while the peak values of the
wave functions drop down a little, their behavior is similar to the one in
vacuum. This means that the $J/\psi$ meson is still a bound state at the
deconfinement phase transition where the light mesons start to melt in the hot
medium, and therefore the observed $J/\psi$s in the final state of heavy ion
collisions can signal the QGP formation in the early stage. However, with
further increasing temperature, the wave functions expand rapidly around
$T_{d}=1.35T_{c}$. By calculating the average distance between the $c$ and
$\bar{c}$,

$\langle r\rangle(T)={\int drr^{3}\left|\psi(r,T)\right|^{2}\over\int
drr^{2}\left|\psi(r,T)\right|^{2}},$ (10)

we found that at $T_{d}$ the average size becomes infinite, see Fig.2, which
indicates the $J/\psi$ dissociation. Therefore, $T_{d}$ is called the
relativistic dissociation temperature.

Figure 2:  The average size of the $J/\psi$ meson in the quark-gluon plasma in
the limit of potential $V=F$. The temperature and average size are scaled by
their corresponding values at the critical point of the deconfinement.

The other quantity which can be used to characterize the quarkonium
dissociation is the heavy quark binding energy. For the Dirac equation or
equivalently the Schrödinger equations (3) and (4), the usual binding energy
defined as $\epsilon=V(\infty)+2m_{c}-m_{m}$ is no longer valid to describe
the quarkonium dissociation. In the limit of $r\to\infty$, the four equations
for the spin singlet and triplet degenerate and the asymptotic equation is
simplifies as

$\left(-{d^{2}\over dr^{2}}+2m_{w}V(\infty)+V^{2}(\infty)\right)u=b^{2}u$ (11)

with $V(\infty)=B(\infty)$, and the energy for the scattering state is

$2m_{w}V(\infty)+V^{2}(\infty)=b^{2},$ (12)

which leads to the binding energy for the $c\bar{c}$ bound state

$\epsilon(T)=V(\infty,T)+\sqrt{V^{2}(\infty,T)+4m_{c}^{2}}-m_{m}.$ (13)

Figure 3:  The binding energy of the $J/\psi$ meson in the quark-gluon plasma
and in the limit of potential $V=F$. The temperature and binding energy are
scaled by their corresponding values at the critical point of the
deconfinement.

Fig.3 shows the temperature dependence of the binding energy in the QGP phase
and in the limit of potential $V=F$. It drops down monotonously with
increasing temperature and reaches zero at $T_{d}$. The result is consistent
with the calculation of the charmonium average size. The infinite size and
zero binding energy of the $c\bar{c}$ system define the unique dissociation
temperature. In comparison with the non-relativistic calculation, the $J/\psi$
dissociation temperature increases from $1.26T_{c}$ to $1.35T_{c}$, the
relativistic correction is $7\%$. We also calculated the $J/\psi$ wave
functions at finite temperature in the other limit of potential $V=U$ and
found that the dissociation temperature $T_{d}$ goes up from the non-
relativistic value $2.1T_{c}$ to $2.38T_{c}$. The relativistic correction is
$13\%$.

Figure 4:  The wave function $u(r)=r\Psi(r)$ for the $\chi_{0}$ meson at zero
temperature and critical temperature of deconfinement $T_{c}$ in the limit of
quark potential $V=F$.

The wave function for the excited state $\chi_{0}$ is shown in Fig.4 in the
limit of potential $V=F$. Considering the fact that the radial function $\Psi$
is divergent at the origincrater2 , we plot $u=r\Psi$ directly from the
Schrödinger equation (3). In comparison with the ground state $J/\psi$,
$\chi_{0}$ wave function distributes in a wider region, the average size is
larger and the binding energy is smaller. The corresponding dissociation
temperature defined by $\langle r\rangle(T_{d})=\infty$ and
$\epsilon(T_{d})=0$ is around the critical temperature $T_{c}$ and a little
bit higher than the non-relativistic value. It is easy to understand that the
relativistic correction for the excited states should be smaller than the one
for the ground state.

In summary, we calculated the charmonium wave functions at finite temperature
by solving the covariant relativistic Schrödinger equations for the $c\bar{c}$
spin singlet and triplet states. The relativistic effect makes the quark
potential well more deep, and charmonia can survive in a more hot medium. By
considering the two limits of the central potential, the relativistic
correction to the $J/\psi$ dissociation temperature in the QGP phase is in
between $7\%$ and $13\%$.

Acknowledgement: The work is supported by the NSFC (Grant Nos. 10975084 and
11079024), RFDP (Grant No.20100002110080 ) and MOST (Grant No.2013CB922000).

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