Investigating the critical properties of beyond-QCD theories using Monte Carlo renormalization group matching

Monte Carlo renormalization group (MCRG) methods were designed to study the nonperturbative phase structure and critical behavior of statistical systems and quantum field theories. I adopt the 2-lattice matching method used extensively in the 1980’s and show how it can be used to predict the existence of nonperturbative fixed points and their related critical exponents in many flavor SU(3) gauge theories. This work serves to test the method and I study relatively well understood systems: the $N_f=0$, 4 and 16 flavor models. The pure gauge and $N_f=4$ systems are confining and chirally broken and the MCRG method can predict their bare step scaling functions. Results for the $N_f=16$ model indicate the existence of an infrared fixed point with nearly marginal gauge coupling. I present preliminary results for the scaling dimension of the mass at this new fixed point.

Figure 1

Sketch of the RG flow around a FP with one relevant operator. The coupling pair $(K,K^{\prime })$ indicates matched couplings whose correlation length differ by a factor of $s$.

Figure 2

The matching of the plaquette for pure gauge SU(3) theory. The simulations were done on ${32}^4$ volumes at $\beta =7.0$ (symbols) and ${16}^4$ volumes at many coupling values (solid interpolating lines). The configurations were blocked with $s=2$, $\alpha =0.65$ parameter block transformation 2(1) (cyan), 3(2) (blue) and 4(3) times (red).

Figure 3

Matching at $\beta =7.0$ from ${32}^4$ to ${16}^4$ lattices. The matching values $\Delta \beta$ as the function of the blocking parameter $\alpha$ are shown for the 5 different operators measured. Left panel: blocking level $n_b=2(1)$; middle panel $n_b=3(2)$; right panel $n_b=4(3)$.

Figure 4

Matching at $\beta =7.0$ from ${32}^4$ to ${16}^4$ lattices. The average matching values $\Delta \beta$ as the function of the blocking parameter for the last 3 blocking levels are shown. Note that the “error bars” denote the spread of the predictions for the 5 different operators used and thus represent systematical errors.

Figure 5

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

Figure 6

The condensate and the disconnected chiral condensate on $8^3\times 4$ volumes at $m=0.04$ in the $N_f=4$ theory.

Figure 7

The condensate and the disconnected chiral condensate on ${12}^3\times 6$ volumes at $m=0.05$ in the $N_f=4$ theory.

Figure 8

Matching of the plaquette in the $N_f=4$ model. The blue and red symbols show the plaquette value after 2 and 3 blocking steps on the ${16}^4$ volumes at $\beta =6.0$, $m=0.01$. The lines interpolate the plaquette on the $8^4$ volumes at several couplings and $m=0.015$ (dashed) and $m=0.025$ (dotted) after 1 and 2 blocking levels. The black line is the unblocked plaquette on the $8^4$ lattices. The dashed and dotted curves are barely distinguishable.

Figure 9

Matching at $\beta =6.0$ from ${16}^4$ to $8^4$ lattices in the $N_f=4$ model. The matching values $\Delta \beta$ as the function of the blocking parameter for the last 2 blocking levels are shown. Note that the error bars denote the spread of the predictions for the 5 different operators used and thus represent systematical errors.

Figure 10

The bare step scaling function for 4 flavor SU(3) theory. The dashed line indicates the 1-loop perturbative value.

Figure 11

The dependence of the plaquette on the mass after 1 (red diamonds and solid line) and 2 (blue diamonds and solid line) blocking steps on $8^4$ volumes at $\beta =5.8$. The bursts at $m=0.15$ show the plaquette after 2 and 3 blocking steps starting form ${16}^4$ volumes at the same $\beta$ value. The dashed lines indicate matching in the mass. The blocking parameter is $\alpha =0.75$, close to the optimal value at $\beta =5.8$.

Figure 12

The matching of the plaquette for $N_f=16$ flavor. The individual data points are at $\beta =5.6$ on ${16}^4$ lattices after 2 and 3 levels of blocking. They are compared to the 1 and 2 times blocked values as measured on the $8^4$ lattices (solid lines). The matching values $\Delta \beta \approx 0$ indicate a nearly marginal flow. Note that the plaquette increases with $n_b$ implying that the RG flow is to an IRFP and not the $\beta =0$ trivial FP.

Figure 13

Matching at $\beta =5.4$ (left panel) and $\beta =6.6$ (right panel) from ${16}^4$ to $8^4$ lattices in the $N_f=16$ flavor theory.

Figure 14

Matched $(m_1,m_2)$ pairs in the $N_f=16$ system. Red diamond: $\beta =5.8$, blue bursts: $\beta =6.6$. The linear fit predicts $y_m=1.02(7)$.