Conformal or walking? Monte Carlo renormalization group studies of $SU(3)$ gauge models with fundamental fermions

Strongly coupled gauge systems with many fermions are important in many phenomenological models. I use the 2-lattice matching Monte Carlo renormalization group method to study the fixed point structure and critical indexes of $SU(3)$ gauge models with 8 and 12 flavors of fundamental fermions. With an improved renormalization group block transformation I am able to connect the perturbative and confining regimes of the $N_f=8$ flavor system, thus verifying its QCD-like nature. With $N_f=12$ flavors the data favor the existence of an infrared fixed point and conformal phase, though the results are also consistent with very slow walking. I measure the anomalous mass dimension in both systems at several gauge couplings and find that they are barely different from the free-field value.

Figure 1

The schematic phase diagram of QCD-like and conformal systems.

Figure 2

The bare step scaling function $s_b(\beta ;s=2)$ for the pure gauge $SU(3)$ system as predicted by different methods. This figure is from Ref. 25.

Figure 3

The bare step scaling function $s_b(\beta ;s=2)$ for the pure gauge $SU(3)$ system as predicted by different RG transformations. The dashed line is the 1-loop perturbative prediction $s_b^{(\mathrm{pert})}(s=2)=0.59$.

Figure 4

The bare step scaling function $s_b(\beta ;s=2)$ for the 16 flavor system. The octagons and crosses correspond to the original and HYP RG transformations.

Figure 5

The series of red lines indicate possible RG $\beta$ functions that differ in the nonuniversal coefficient of the 3-loop term, $b_2$. An optimized MCRG could pick a different RG transformation at each gauge coupling value, thus predicting the thick blue line as an “RG $\beta$ function.”

Figure 6

The bare step scaling function $s_b(\beta ;s=2)$ for the 8 flavor system. The dashed line indicates the 1-loop perturbative prediction.

Figure 7

$r_0/a$ and $a\sqrt{\sigma }$ as the function of the quark mass at $\beta =4.8$ for the $N_f=8$ flavor model.

Figure 8

Matching of the plaquette in the mass at ($\beta =5.0$, ${\beta }^{\prime }=4.81$). The upper red and lower blue curves show the plaquette after $n_b=1$ and 2 blocking steps on the $8^4$, $\beta =4.81$ lattices. These are matched to the $n_b=2$ and 3 times blocked ${16}^4$, $\beta =5.0$ plaquette values (red diamonds and blue bursts). The horizontal lines indicate the matching.

Figure 9

Matching masses for the 8 flavor system. In principle the slope of the matched mass pairs could be different at each gauge coupling, the result indicates that is not the case here. The data predicts, within error, a vanishing anomalous dimension.

Figure 10

The bare step scaling function $s_b(\beta ;s=2)$ for the 12 flavor system.

Figure 11

Matching masses for the 12 flavor system. Red diamonds: $\beta =5.0$; blue bursts: $\beta =5.1$.