Hyper-scaling relations in the conformal window from dynamic AdS/QCD

Dynamic AdS/QCD is a holographic model of strongly coupled gauge theories with the dynamics included through the running anomalous dimension of the quark bilinear, $\gamma$. We apply it to describe the physics of massive quarks in the conformal window of $\mathrm{SU}(N_c)$ gauge theories with $N_f$ fundamental flavors, assuming the perturbative two-loop running for $\gamma$. We show that to find regular, holographic renormalization group flows in the infrared, the decoupling of the quark flavors at the scale of the mass is important, and enact it through suitable boundary conditions when the flavors become on shell. We can then compute the quark condensate and the mesonic spectrum $(M_{\rho },M_{\pi },M_{\sigma })$ and decay constants. We compute their scaling dependence on the quark mass for a number of examples. The model matches perturbative expectations for large quark mass and naïve dimensional analysis (including the anomalous dimensions) for small quark mass. The model allows study of the intermediate regime where there is an additional scale from the running of the coupling, and we present results for the deviation of scalings from assuming only the single scale of the mass.

Figure 1

Plots for the theory $N_c=3$ and $N_f=12$: (a) Log $c$ against Log $m_{\text{bare}}$. (b) Numerical points for $\gamma$ extracted from $b=\frac{3-\gamma }{1+\gamma }$ against Log $m_{\text{bare}}$ (the solid line is the input $\gamma$ from the two-loop running). (c) Percentage difference between the extracted form of $\gamma$ and the input form (solid line in b).

Figure 2

Plots for the running of mass and condensate at $N_c=3$, $N_f=12$: (a) The running of the mass shown against RG scale for different values of $L_0$. (b) The running of the condensate parameter $c$ against RG scale $\mu$ at $L_0={10}^{-20}$.

Figure 3

$N_c=3$: (a) $\gamma$ versus $m_{\text{bare}}$ from $c$ for $N_f=13$. (b) $\gamma$ versus $m_{\text{bare}}$ from $c$ for $N_f=15$.

Figure 4

$N_c=3$, $N_f=12$: (a) $\rho$-meson mass against quark mass. (b) Extracted value $\gamma$ versus $m_{\text{bare}}$ from $\rho$-meson mass spectrum. The solid line shows the holographic input of $\gamma$ from the two-loop running. (c) The percentage difference seen between the input $\gamma$ running and the extracted $\gamma$ running.

Figure 5

$N_c=3$, $N_f=12$: (a) $\sigma$-meson mass against quark mass. (b) Extracted value $\gamma$ versus $m_{\text{bare}}$ from $\sigma$-meson mass spectrum. The solid line shows the holographic input of $\gamma$ from the two-loop running.

Figure 6

$N_c=3$, $N_f=12$: (a) $\pi$-meson mass against bare quark mass. (b) Extracted value $\gamma$ versus $m_{\text{bare}}$ from $\pi$-meson mass spectrum. The solid line shows the holographic input of $\gamma$ from the two-loop running.

Figure 7

$N_c=3$, $N_f=12$: (a) $\rho$ mass versus $\pi$ mass. (b) $b$ versus $M_{\pi }$, where we have assumed $M_{\rho }\propto M_{\pi }^b$.

Figure 8

$N_c=3$, $N_f=12$: (a) $f_{\rho }$ versus the bare quark mass. (b) The extracted $\gamma$ versus $m_{\text{bare}}$.

Figure 9

$N_c=3$, $N_f=12$: (a) $f_{\sigma }$ versus the bare quark mass. (b) The extracted $\gamma$ versus $m_{\text{bare}}$.

Figure 10

$N_c=3$: (a) $\gamma$ extracted from $M_{\rho }$ against the bare quark mass at $N_f=13$, $N_c=3$. (b) The same for $N_f=15$.

Figure 11

$N_c=3$: (a) $M_{\rho }$ versus $M_{\pi }$ for $N_f=13$. (b) Extracted $\gamma$ for $N_f=13$. (c) $M_{\rho }$ versus $M_{\pi }$ for $N_f=15$. (d) Extracted $\gamma$ for $N_f=15$.