z_{\min}\left( 0\right) $. Below the temperature $T_{\min}\left( 0\right) $, there is no black hole solution and we expect a Hawking-Page phase transition happens at a temperature $T_{\rm HP}\left( 0\right) \gtrsim T_{\min}\left( 0\right) $ where the black hole dissolves to a thermal gas background. For $0<\mu<\mu_{c}$, the temperature has a local minimum/maximum $T_{\min}\left( \mu\right) /T_{\max }\left( \mu\right) $\ at $z_{H}=z_{\min}\left( \mu\right) /z_{\max}\left( \mu\right) $ and decreases to zero at a finite size of horizon. The black holes between $z_{\min}\left( \mu\right) $ and $z_{\max}\left( \mu\right) $ are thermodynamically unstable. There are two sections that are stable with $z_{H}z_{\max}\left( \mu\right) $. We expect a similar Hawking-Page phase transition happens at a temperature $T_{\rm HP}\left( \mu\right) \gtrsim T_{\min}\left( \mu\right) $. Nevertheless, since the thermodynamically stable black hole solutions exist even when the temperature below $T_{\min}\left( \mu\right) $ for the section $z_{H}>z_{\max}\left( \mu\right) $, we also expect a black hole to black hole phase transition happening at a temperature $T_{BB}\left( \mu\right) $ between $T_{\min}\left( \mu\right) $ and $T_{\max}\left( \mu\right) $, where a large black hole with the horizon $z=z_{Hl}\left( \mu\right) $ collapses to a small black hole with the horizon $z=z_{Hs}\left( \mu\right) $ as showed in figure~\ref{horizon}. Finally, for $\mu>\mu_{c}$, the temperature monotonously decreases to zero and there is no black hole to black hole phase transition anymore.\footnote{There could still be a Hawking-Page phase transition at some temperature for the case of $\mu>\mu_{c}$, but we will show later that the black hole solution is always thermodynamically favored in this case.} \begin{figure} \centering \includegraphics[ height=1.7in, width=3.74in ]{horizon} ]]>

\mu_{c}$, the curve of the free energy increases smoothly from higher temperature to lower temperature. The phase diagram of the background is plotted in (b) of figure~\ref{phase diagram}. At $\mu=0$, the system undergoes a black hole to thermal gas phase transition at $T=T_{\rm HP}\left( 0\right) $. For $0<\mu<\mu_{c}$, the system undergoes a large black hole to small black hole phase transition at $T_{BB}\left( \mu\right) $. The phase transition temperature $T_{BB}\left( \mu\right) $ approaches to $T_{\rm HP}$ at $\mu\rightarrow0$ that makes the phase diagram continuous at $\mu=0$. The phase transition stops at $\mu=\mu_{c}$ and reduces to a crossover for $\mu>\mu_{c}$. The phase diagram we obtained here in figure~\ref{phase diagram} is different from the conventional QCD phase diagram, in which crossover happens for small chemical potential and phase transition happens for large chemical potential. In~\cite{bib1301.0385}, by comparing with the phase structure in lattice QCD simulation, the authors argued that this `reversed' phase diagram should be interpreted as confinement-deconfinement phase transition of heavy quarks in QCD. In this paper, we consider the same background as in~\cite{bib1301.0385} to study pure gluon QCD with one additional heavy flavour, and not light quarks. Since our model describes heavy quarks system in QCD, the flavour field ${A}_{\mu}^{V}$ in the matter action~\ref{action-m} should be associated to the mesons make up of heavy quarks, i.e.\ quarkonium states. By fitting the lowest two spectrum of quarkonium states $m_{J/\psi}=3.096$\,GeV and $m_{\psi^{\prime}}=3.685$\,GeV, we can fix $c\simeq1.16GeV^{2}$ in eq.~(\ref{A}). Nevertheless, there left a problem that, in the gravity side, it is commonly believed that the confinement-deconfinement phase transition in the field theory side is dual to the Hawking-Page phase transition. Hawking-Page phase transition is the transition between black hole and thermal gas backgrounds. However, in our gravity background, the phase transition is between two black holes for a non-zero chemical potential. Thus it is not consistent to consider a black hole to black hole phase transition in the gravity side to be dual to the confinement-deconfinement phase transition in QCD. In the following of this paper, by adding open strings in the background, we will study this issue more carefully to gain a more reasonable physical picture. ]]>

r_{M}$. Thus when a large black hole shrinks to a small one, we expect that a dynamical wall will appear when the black hole horizon equal to a critical value $z_{H\mu}$ for each chemical potential $\mu$, showed in figure~\ref{phase-cross}. \begin{figure} \centering \includegraphics[ height=1.7409in, width=3.755in ]{phase-cross} ]]>

z_{H\mu}$, there exists a dynamical wall at $z=z_{m}

z_{H\mu}$, the potential is linear to $r$ for $r\rightarrow\infty$. While for large black hole with $z_{H}

\mu_{c}$ in (b) of figure~\ref{meson potential}. For $T\leq T_{\mu}$, i.e.\ $z_{H}>z_{H\mu}$, the heavy quark potential is linear at large separate distance $r$ with the slopes decrease with the temperature increasing. The linear potential implies that the system is in the confinement phase. While for $T>T_{\mu}$, i.e.\ $z_{H}

z_{H\mu}$, open strings can not exceed a dynamical wall at $z=z_{m}$\ even the distance $r$ between the quark-antiquark pair goes to infinity. This means that both ends of the open strings have to touch the boundary at $z=0$, and the quark-antiquark pair is always connected by an open string in the U-shape to form a bound state, which corresponds to a meson state in the dual holographic QCD, as in (a) of figure~\ref{conf-deconf}. We interpret this phase as the confinement phase in QCD. On the other hand, for a large black hole with $z_{H}r_{M}$, an open string of U-shape\ would break to two straight open strings as showed in (b) of figure~\ref{conf-deconf}. Thus the meson state would decay to a pair of free quark and antiquark. We interpret this phase as the deconfinement phase in QCD. We should remark that for a small black hole with $z_{H}>z_{H\mu}$, it is impossible for an open string to break up due to the dynamical wall at $z=z_{m}$. Thus even in the black hole background, the holographic QCD could still be in the confinement phase. This clarifies the puzzle in~\cite{bib1301.0385} that a black hole background does not correspond to the confinement phase in QCD in the original AdS/QCD correspondence. The black hole phases, open string configurations and QCD phases are summarized in table~\ref{table}. \begin{table} \centering\renewcommand\arraystretch{1.2} \begin{tabular} [c]{|c|c|c|}\hline Black hole & String configurations for $r\rightarrow\infty$ & Phase in QCD\\\hline Small $\left( z_{H}>z_{H\mu}\right) $ & U-shape & Confinement\\\hline Large $\left( z_{H}

z_{H\mu}$ suddenly jumps to a large black hole with horizon $z_{Hl}\mu_{c}$, when the temperature increases from zero, the black hole horizon increases gradually with the temperature and continuously passes $z_{H\mu}$ at $T=T_{\mu}\mu_{c}$, the confinement-deconfinement phase transition reduces to a smooth crossover. The critical point is located at $\mu _{c}=0.678$\,GeV and $T_{c}=0.536$\,GeV. This result is consistent with the conclusion from the lattice QCD for the heavy quarks~\cite{bib1111.4953}. \begin{figure} \centerline{ \scalebox{.9}{ \subfigure[]{\includegraphics[height=2in,width=3in]{both}} \qquad \subfigure[]{\includegraphics[height=2in,width=3in]{final}} } } ]]>

r_{M}$, the energy of the two straight strings is less than the free energy of an open string in the U-shape and the U-shape open string will break up to two straight strings, corresponding to that a meson state melts to a pair of free quark and antiquark.\ The phenomenon of mesons melting has been previously studied in~\cite{bib1006.0055,bib1108.0684,bib1102.2289,bib1203.3942}. In this work, we define the screening length as the maximum length $r_{M}$, achieved by a pair of quark and antiquark in the bound state at a temperature $T>T_{BB}(\mu)$ for $\mu \leq\mu_{c}$ or $T>T_{\mu}$ for $\mu>\mu_{c}$. The screening length at a fixed chemical potential and temperature can be determined by the equation $V-2F_{q}=0$, where $V$ is the heavy quark potential energy defined in~(\ref{quark potential}) and $F_{q}$ is the free energy of a straight string defined as\begin{equation} F_{q}=\int_{0}^{z_{H}}dz\frac{e^{2A}}{z^{2}}. \end{equation} We plot the `melting lines' of screening length versus chemical potential in figure~\ref{melt}. The screening length is a possible signal form Quark Gluon Plasma (QGP). Right after the collision QGP is formed and the temperature is high enough to be in the deconfinement phase. As temperature decreases, heavy quarks form bound states at melting temperatures higher than the deconfinement temperature. This means heavy quark bound sates can coexist with QGP. \begin{figure} \centerline{ \scalebox{.9}{ \subfigure[]{\includegraphics[height=2in,width=3in]{melt1}} \quad \subfigure[]{\includegraphics[height=2in,width=3in]{melt2}} } } ]]>