Holographic gauged NJL model: The conformal window and ideal walking

We study the holographic dynamic anti-de Sitter/QCD description of a $\mathrm{SU}(N_c)$ non-Abelian gauge theory with $N_f$ fermions in the fundamental representation which also have Nambu–Jona-Lasinio (NJL) interactions included using Witten’s multitrace prescription. In particular, here, we study aspects of the dynamics in and near the conformal window of the gauge theory as described by the two-loop running of the gauge theory. If the number of flavors is such that the IR fixed point lies with the anomalous dimension, $\gamma$, of the quark bilinear above one, then chiral symmetry breaking occurs. Here, we display a spiral in the quark mass/condensate plane describing a sequence of unstable excited states of the vacuum. An attractive NJL operator enhances the vacuum condensate, but only an infinitely repulsive NJL interaction switches off the condensation completely. When $N_f$ changes so that the IR fixed point falls below 1 (the conformal window region), there is a numerical discontinuity in the phase structure with condensation only occurring with a supercritical NJL interaction. In the conformal window, the running of $\gamma$ to a nontrivial IR fixed point is similar to walking dynamics, although chiral symmetry breaking is not triggered. In the “ideal walking” scenario, chiral symmetry is broken in that IR conformal regime by the NJL interaction, but the change in $\gamma$ enhances the UV condensate. That enhancement of the condensate is shown in an analytic model with a sharp change in $\gamma$, and we show equivalent numerical results for the case of the two-loop running. In the model, the $\sigma$ becomes massless as the gauge theory running becomes near conformal, and we show it is possible to realize a light Higgs-like state in ideal walking models.

Figure 1

An example plot of the running of $\gamma$, calculated from two loops, from the IR fixed point to the asymptotically free UV. ${\gamma }_{*}$ is the value in the IR. Here, we have $N_c=3$ and $N_f=13$.

Figure 2

The functions $L(\rho )$ with $m=0$ in the far UV for $N_f=9$. In the IR, we cut off scales below which the quarks become on mass shell when $L(\rho =m_{\mathrm{IR}})=m_{\mathrm{IR}}$. Here, the BF bound is violated at $r=11$.

Figure 3

The regular embeddings $L(\rho )$ plotted in the $m_{\mathrm{UV}}-c_{\mathrm{UV}}$ plane for (from top to bottom) $N_f=9$, 11 showing the spiral structure and how the scale of chiral symmetry breaking shrinks as one approaches the BKT transition at $N_f\simeq 12$. Here, both theories have $\gamma =0.3$ at the same UV scale.

Figure 4

Above: a sketch of the low energy potential against the $\sigma$ and ${\sigma }_{*}$ fields showing mimina on each axis but only a single true minima on the $\sigma$ axis. Below: the $\sigma$’s mass against the UV quark mass as we move along the spiral of Fig. 3 with $N_f=9$ showing the instability of the excited states at $m_{\mathrm{UV}}=0$.

Figure 5

Plot of $c_{\mathrm{UV}}$ against $g^2$ for the $N_f=9$ theory.

Figure 6

The top two plots show the effective action ($-\mathrm{S}$ evaluated on the vacuum solutions) for the solutions from Fig. 3 for the $N_f=9$ theory as a function of each of $m_{\mathrm{UV}}$ and $c_{\mathrm{UV}}$. The lower plot shows the same potential against $m_{\mathrm{UV}}$ but including the NJL interaction term. The solid lines are for attractive NJL interactions, and dashed lines are for repulsive NJL interactions.

Figure 7

The $N_f-g^2$ phase diagram—the left-hand region has chiral symmetry breaking, and the right-hand region has restored chiral symmetry.

Figure 8

Plot of Log $m_{\mathrm{UV}}/m_{\mathrm{IR}}$ against Log $m_{\mathrm{IR}}/{\lambda }_{\mathrm{UV}}$ for $N_f=13$, $\gamma ({\mathrm{\Lambda }}_{\mathrm{UV}})=0.05$.

Figure 9

The $N_f=13$ theory with $m_{\mathrm{IR}}$ lying in the fixed point regime. The $\sigma$’s mass is plotted against Log $\mathrm{\Lambda }/m_{\mathrm{IR}}$ for different separations between the IR and UV cutoffs.

Figure 10

This is a plot in the $N_f=12$ theory where the IR fixed point is ${\gamma }_{\mathrm{IR}}=0.48$. Here, we have a separation of 7.5 between the $m_{\mathrm{IR}}$ and $\mathrm{\Lambda }$. We vary $m_{\mathrm{IR}}$ to scales with different values of ${\gamma }_{\mathrm{IR}}$ and compute the $\sigma$ mass in units of $f_{\pi }$.