Holographically emulating sequential versus instantaneous disappearance of vector mesons in a hot environment

Minor extensions of the soft-wall model are used to accommodate two variants of Regge trajectories of vector meson excitations. At nonzero temperatures, various options for either sequential or instantaneous disappearance of vector mesons as normalizable modes are found, thus emulating deconfinement at a certain temperature in the order of the (pseudo)critical temperature of QCD. The crucial role of the blackness function, which steers the thermodynamic properties of the considered system, is highlighted.

Figure 1

Scaled Schrödinger equivalent potential $U_T/c^2$ from Eq. (4) (solid curves, labeled by the temperature in units of MeV) as a function of $cz$ (left panel: set 1.2, $c{z}_H=2$ and 4, right panel: set 2.0, $c{z}_H=1.75$ and 3.5; the sets are defined in Table 1 below). The dashed curve shows $U_0$, i.e., $U_T(z_H\to \infty )$ corresponding to $T=0$, with the first three normalizable states (horizontal dotted lines).

Figure 2

Squared vector meson masses (crosses) from (8) at $T=0$ in comparison with data (circles) for set 1.2 (left panel, [54]) and set 2.0 (right panel, [55]) from Table 1.

Figure 3

Contour plot (gray curves) of the disappearance temperature $T_{\mathrm{dis}}^{\mathrm{g}.\mathrm{s}.}/c$ of the ground state over the $p\text{−}\mu$ plane when employing the potential (14). The black dot at $p=1.99$ and $\mu =0.5$ with $c=443\mathrm{MeV}$ belongs to the set 2.0. The other optimized parameter points for sets 1.1 and 1.2 are outside the plot on the $T_{\mathrm{dis}}^{\mathrm{g}.\mathrm{s}.}/c=0.11$ curve. The black curves limit the corridor wherein $\left|{\beta }_2\right|\le 0.01\left|{\beta }_1\right|$.

Figure 4

Schematic plot of the temperature $T$ of the ambient medium as a function of the horizon position $z_H$ (bottom panels, unstable sections are dotted) according to Eqs. (15), (16) [left panels: option (ii) for a small value of $T_{min}$, middle and right panels: option (i)] and the temperature dependence of states (top panels, thermal gas sections are dashed).

Figure 5

Contour plot of the disappearance temperature $T_{\mathrm{dis}}^{\mathrm{g}.\mathrm{s}.}$ in units of $\mathrm{MeV}$ of the ground state for the parameters of set 1.2 (left panel) and set 2.0 (right panel). The thick gray curve highlights $T_{\mathrm{dis}}^{\mathrm{g}.\mathrm{s}.}=150\mathrm{MeV}$. If $z_{min}$ is too small no normalizable states can be found (left part of the panels), i.e., the contour curves end at $T_{min}=T_{\mathrm{dis}}^{\mathrm{g}.\mathrm{s}.}$.

Figure 6

Disappearance temperature $T_{\mathrm{dis}}$ of the first three states (solid lines: g.s., dashed lines: first excited state, dotted lines: second excited state) as a function of the parameter $T_{min}$ for various values of $z_{min}$. Left panel: $c{z}_{min}=6$ displays again the case of sequential disappearance, i.e., at given value $T_{min},T_{\mathrm{dis}}^{\mathrm{g}.\mathrm{s}.}>T_{\mathrm{dis}}^{1^{\mathrm{st}}}>T_{\mathrm{dis}}^{2^{\mathrm{nd}}}$. Middle panel: $c{z}_{min}=4$; up to $T_{min}=70\mathrm{MeV}$, the second and the third mode disappear simultaneously at $T=T_{min}$, but for greater values of $T_{min}$ the modes disappear sequentially. Right panel: $c{z}_{min}=3$; all states disappear instantaneously up to $T_{min}=70\mathrm{MeV}$; for greater values of $T_{min}$ the ground state survives. The chosen values of $p,\mu$, and $c$ belong to set 2.0 of Table 1, i.e., the plots display a cross section through Fig. 5 (right panel) with respect to the ground state.

Figure 7

Scaled Schrödinger equivalent potentials $U_T/c^2$ from Eq. (9) (solid curves, labeled by the temperature in units of MeV) and $U_0$ (dashed curves) as a function of $cz$, parameter set 2.0. Left panel: $c{z}_{min}=6,T_{min}=120\mathrm{MeV}$, the first three states disappear sequentially. Middle panel: $c{z}_{min}=4,T_{min}=130\mathrm{MeV}$, the second excited state and all higher ones disappear instantaneously at $T=T_{min}$ and the remaining two states disappear sequentially. Right panel: $c{z}_{min}=3,T_{min}=140\mathrm{MeV}$, all excited states disappear instantaneously.

Figure 8

Temperature behavior of the first three states (set 2.0) for the blackness function $f$ from (15) with $T$ taken from Eq. (16) with $T_{min}=150\mathrm{MeV}$ and $c{z}_{min}=2$ (left panel) and $T_{min}=120\mathrm{MeV}$ and $c{z}_{min}=6$ (right panel). The dashed lines represent the range of $T$ where the thermal gas solution is valid, and the solid curves belong to the black hole solution. This figure provides the quantitative results corresponding to Fig. 4, middle and right top panels.

Figure 9

Disappearance temperature $T_{\mathrm{dis}}$ of the ground state (solid lines), the first excited state (dashed lines), and the second excited state (dotted lines) versus the parameter $T_{min}$ for set 1.2 (left panel) and for set 2.0 (right panel) for $z_{min}\to \infty$ in ((15), (16)), i.e., option (ii). Choosing for instance $T_{min}=55\mathrm{MeV}$ (left) or $T_{min}=95\mathrm{MeV}$ (right) the ground state disappears at $T=150\mathrm{MeV}$, the first excited state at $T=98\mathrm{MeV}$ or $T=132\mathrm{MeV}$, respectively.