Charmonium properties in deconfinement phase in anisotropic lattice QCD

$J/\Psi$ and ${\eta }_c$ above the QCD critical temperature $T_c$ are studied in anisotropic quenched lattice QCD, considering whether the $c\overline{c}$ systems above $T_c$ are spatially compact (quasi-)bound states or scattering states. We adopt the standard Wilson gauge action and $O(a)$-improved Wilson quark action with renormalized anisotropy $a_s/a_t=4.0$ at $\beta =6.10$ on ${16}^3\times (14–26)$ lattices, which correspond to the spatial lattice volume $V\equiv L^3\simeq (1.55\mathrm{fm}{)}^3$ and temperatures $T\simeq (1.11–2.07)T_c$. We investigate the $c\overline{c}$ system above $T_c$ from the temporal correlators with spatially extended operators, where the overlap with the ground state is enhanced. To clarify whether compact charmonia survive in the deconfinement phase, we investigate spatial boundary-condition dependence of the energy of $c\overline{c}$ systems above $T_c$. In fact, for low-lying $S$-wave $c\overline{c}$ scattering states, it is expected that there appears a significant energy difference $\Delta E\equiv E(\mathrm{APBC})-E(\mathrm{PBC})\simeq 2\sqrt{m_c^2+3{\pi }^2/L^2}-2m_c$ ($m_c$: charm quark mass) between periodic and antiperiodic boundary conditions on the finite-volume lattice. In contrast, for compact charmonia, there is no significant energy difference between periodic and antiperiodic boundary conditions. As a lattice QCD result, almost no spatial boundary-condition dependence is observed for the energy of the $c\overline{c}$ system in $J/\Psi$ and ${\eta }_c$ channels for $T\simeq (1.11–2.07)T_c$. This fact indicates that $J/\Psi$ and ${\eta }_c$ would survive as spatially compact $c\overline{c}$ (quasi-)bound states below $2T_c$. We also investigate a $P$-wave channel at high temperature with maximal entropy method and find no low-lying peak structure corresponding to ${\chi }_{c1}$ at $1.62T_c$.

Figure 1

Schematic figures for boundary-condition dependence of $c\overline{c}$ wave functions: (a) the bound-state case and (b) the scattering-state case in terms of PBC and APBC. (a) For the bound state, the wave function is spatially localized, and it is insensitive to the spatial boundary condition. Hence, the basic properties of the spatially localized bound state are almost the same between PBC and APBC. (b) For the scattering state, the wave function is spatially spread, and therefore it is sensitive to the spatial boundary condition in a finite-size box. In particular, a drastic change occurs for the wave function of the low-lying scattering state between PBC and APBC cases.

Figure 2

Pictorial expression of minimum momentum of the (anti-)quark on PBC and APBC in the case of $c\overline{c}$ scattering state in the center of mass flame. $c$ and $\overline{c}$ are in the finite-size box with the spatial volume $L^3$. On PBC, both particles have zero lowest momentum in the lowest state. On the other hand, on APBC, quark and antiquark have an opposite nonzero momentum.

Figure 3

Schematic figures for the radial wave functions of $c\overline{c}$ compact bound states for (a) the $S$-wave state and (b) the $P$-wave state. The $S$-wave state can be approximated with a Gaussian form $e^{-r^2/2{\rho }^2}$. In contrast, the wave function of $P$-wave state should be zero at $r=0$ and spatially spreads due to the centrifugal potential.

Figure 4

The effective-mass $m_{\mathrm{eff}}$ is plotted against $t$ for $J/\Psi$ at (a) $1.11T_c$, (b) $1.32T_c$, (c) $1.61T_c$, and (d) $2.07T_c$ in the lattice unit with $a_t=(8.12\mathrm{GeV}{)}^{-1}$. The circles and the triangles denote the results on PBC and APBC, respectively. The dashed and dotted lines denote $E(\mathrm{PBC})$ and $E(\mathrm{APBC})$ obtained from the best-fit analysis, respectively.

Figure 5

Temperature dependence of the pole-mass (energy) $M_{J/\Psi }(T)$ of $J/\Psi$ for $(1.11–2.07)T_c$ on PBC (circles) and APBC (triangles). The circles and the triangles correspond to $E(\mathrm{PBC})$ and $E(\mathrm{APBC})$ of the $c\overline{c}$ system, respectively. The energy difference $\Delta E\equiv E(\mathrm{APBC})-E(\mathrm{PBC})\simeq (0.02–0.04)\mathrm{GeV}$ between PBC and APBC are considerably smaller than that of the $c\overline{c}$ scattering state, $\Delta E\simeq 0.35\mathrm{GeV}$.

Figure 6

The effective-mass $m_{\mathrm{eff}}(t)$ is plotted against $t$ for ${\eta }_c$ at (a) $1.11T_c$, (b) $1.32T_c$, (c) $1.61T_c$, and (d) $2.07T_c$ in the lattice unit with $a_t\simeq (8.12\mathrm{GeV}{)}^{-1}$. The circles and the triangles denote the results on PBC and APBC, respectively. The dashed and the dotted lines denote $E(\mathrm{PBC})$ and $E(\mathrm{APBC})$ obtained from the best-fit analysis, respectively.

Figure 7

Temperature dependence of the pole mess (energy) $M_{{\eta }_c}(T)$ of ${\eta }_c$ for $(1.11–2.07)T_c$ on PBC (circles) and APBC (triangles). The energy difference $\Delta E\simeq 0$ between PBC and APBC are also considerably smaller than that of the $c\overline{c}$ scattering state as well as the $J/\Psi$ case.

Figure 8

Temperature dependence of the pole mass (on PBC) of $J/\Psi$ and ${\eta }_c$ for $(1.11–2.07)T_c$. The squares denote $M_{J/\Psi }(T)$ and the inverse triangles denote $M_{{\eta }_c}(T)$. There occurs the level inversion of $J/\Psi$ and ${\eta }_c$ above $1.3T_c$.

Figure 9

The spectral function $A(\omega )$ of the $c\overline{c}$ state in vector ($J^P=1^{-}$) channel on (a) PBC and (b) APBC at $1.62T_c$ extracted with MEM from lattice QCD data of the temporal correlator. $m(\omega )$ denotes the default function of MEM [23, 24, 25]. The statistical error is denoted by the three solid lines. There is a low-lying peak which corresponds to $J/\Psi (m_{J/\Psi }\simeq 3.1\mathrm{GeV})$ even above $T_c$. We add (c) as the comparison between PBC (dotted line) and APBC (solid line). The results almost coincide between PBS and APBC cases.

Figure 10

The spectral function $A(\omega )$ of the $c\overline{c}$ state in pseudoscalar ($J^P=0^{-}$) channel on (a) PBC and (b) APBC at $1.62T_c$ extracted with MEM from lattice QCD data of the temporal correlator. $m(\omega )$ denotes the default function of MEM [23, 24, 25]. The statistical error is denoted by the three solid lines. There is a low-lying peak which corresponds to ${\eta }_c(m_{{\eta }_c}\simeq 3.0\mathrm{GeV})$ even above $T_c$. We add (c) as the comparison between PBC (dotted line) and APBC (solid line). The results almost completely coincide between PBC and APBC cases.

Figure 11

The spectral function $A(\omega )$ of the $c\overline{c}$ state in axial-vector ($J^P=1^{+}$) channel on (a) PBC and (b) APBC at $1.62T_c$ extracted with MEM from lattice QCD data of the temporal correlator. $m(\omega )$ denotes the default function of MEM [23, 24, 25]. The statistical error is denoted by the three solid lines. In contrast to the $J/\Psi$ and ${\eta }_c$ cases, there is no low-lying peak which corresponds to ${\chi }_{c1}(m_{{\chi }_{c1}}\simeq 3.5\mathrm{GeV})$. The peaks in high-energy region ($\omega >5\mathrm{GeV}$) are considered as lattice artifacts. We add (c) as the comparison between PBC (dotted line) and APBC (solid line). These results almost coincide.