Black hole in the throat: Thermodynamics of strongly coupled cascading gauge theories

We numerically construct black hole solutions corresponding to the deconfined, chirally symmetric phase of strongly coupled cascading gauge theories at various temperatures. We compute the free energy as a function of the temperature, and we show that it becomes positive below some critical temperature, indicating the possibility of a first order phase transition at which the theory deconfines and restores the chiral symmetry.

Figure 1

Mismatch of $h_b(x)$ (for $x<0.5$) and $h_h(y\equiv 1-x)$ (for $x>0.5$) for different values of the parameters, for $k_s=0.4$. The solid (blue) curves correspond to “correct” values of the parameters (5.7, 5.8), with $||{\overrightarrow{v}}_{\mathrm{mismatch}}||\approx 9\times {10}^{-6}$. The dotted (green) curves corresponding to all values of parameters 10% larger than the correct ones, produce $||{\overrightarrow{v}}_{\mathrm{mismatch}}||\approx 3\times {10}^{-1}$. The dashed (red) curves correspond to all values of parameters 20% smaller than the correct ones, giving $||{\overrightarrow{v}}_{\mathrm{mismatch}}||\approx 8\times {10}^{-1}$.

Figure 2

Values of the UV parameters ${\widehat{a}}_{2,0}$ and ${\widehat{a}}_{3,0}$ as a function of $k_s$ (blue points). The dashed (dotted) red (green) curves represent the perturbative $\mathcal{O}(k_s^{-1})/\mathcal{O}(k_s^{-2})$ asymptotics of the parameters, given by (5.5).

Figure 3

Values of the UV parameters ${\widehat{a}}_{4,0}$ and $g_{2,0}$ as a function of $k_s$ (blue points). The dashed (dotted) red (green) curves represent the perturbative $\mathcal{O}(k_s^{-1})/\mathcal{O}(k_s^{-2})$ asymptotics of the parameters, given by (5.5).

Figure 4

The values of the UV parameters ${\widehat{a}}_{2,0}$ and ${\widehat{a}}_{3,0}$ as a function of $k_s$.

Figure 5

The values of the UV parameters ${\widehat{a}}_{4,0}$ and $g_{2,0}$ as a function of $k_s$.

Figure 6

The values of the IR parameters $a_0^h$ and $a_1^h$ as a function of $k_s$.

Figure 7

The values of the IR parameters $b_0^h$ and $g_0^h$ as a function of $k_s$.

Figure 8

The values of the IR parameters $k_0^h$ and $h_0^h$ as a function of $k_s$.

Figure 9

The relation between $k_s$ and the temperature $T$.

Figure 10

The free energy density $\mathcal{F}$, divided by $sT$, as a function of $\frac{T}{\Lambda }$. On the left we plot temperatures at and slightly above the deconfinement transition, and on the right much higher temperatures.

Figure 11

The energy density $\mathcal{E}$, divided by $sT$, as a function of $\frac{T}{\Lambda }$.

Figure 12

The temperature dependence of the effective number of degrees of freedom in the strongly coupled cascading gauge theory, as defined by $N_{\mathrm{eff}}^2\propto s/T^3$.

Figure 13

The temperature dependence of the vacuum expectation values of the dimension 4 operators $⟨{\mathcal{O}}_{K_0}⟩$ and $⟨{\mathcal{O}}_{p_0}⟩$. The operators are normalized such that they are invariant under the scaling transformation (5.4).