Quantum critical behavior of semisimple gauge theories

We study the perturbative phase diagram of semisimple fermionic gauge theories resembling the Standard Model. We investigate an $SU(N)$ gauge theory with $M$ Dirac flavors where we gauge first an $SU(M{)}_L$ and then an $SU(2{)}_L\subset SU(M{)}_L$ of the original global symmetry $SU(M{)}_L\times SU(M{)}_R\times U(1)$ of the theory. To avoid gauge anomalies we add leptonlike particles. At the two-loop level an intriguing phase diagram appears. We uncover phases in which one, two or three fixed points exist and discuss the associated flows of the coupling constants. We discover a phase featuring complete asymptotic freedom and simultaneously an interacting infrared fixed point in both couplings. The analysis further reveals special renormalization group trajectories along which one coupling displays asymptotic freedom and the other asymptotic safety, while both flowing in the infrared to an interacting fixed point. These are safety free trajectories. We briefly sketch out possible phenomenological implications, among which an independent way to generate near-conformal dynamics à la walking is investigated.

Figure 1

Triangle diagrams that potentially can give rise to anomalies.

Figure 2

The three regions where (I) both couplings ${\alpha }_N$ and ${\alpha }_M$ are asymptotically free (red), (II) where ${\alpha }_M$ (${\alpha }_N$) is asymptotically (infrared) free (blue), (III) where ${\alpha }_N$ (${\alpha }_M$) is asymptotically (infrared) free (green). The hatched regions indicate where ${\mathrm{FP}}_1$ (the upper region) and ${\mathrm{FP}}_2$ (the lower region) are physical.

Figure 3

The left (right) panel shows the phase diagram of the theory with $N=7$ and $M=25$ ($N=25$ and $M=7$) in the region where only ${\mathrm{FP}}_1$ (${\mathrm{FP}}_2$) exists. The violet point is the ultraviolet free fixed point while the red point is ${\mathrm{FP}}_1$ (left) or ${\mathrm{FP}}_2$ (right).

Figure 4

The four regions where in region I (IV) both couplings are asymptotically (infrared) free and in region II (III) ${\alpha }_N$ (${\alpha }_2$) is infrared free while ${\alpha }_2$ (${\alpha }_N$) is asymptotically free. The hatched regions mark where the fixed points ${\mathrm{FP}}_1$ (horizontal lines), ${\mathrm{FP}}_2$ (diagonal lines) and ${\mathrm{FP}}_3$ (vertical lines) exist.

Figure 5

Coupling flows of theories in region I. Left: The $N=9$ and $M=40$ theory with fixed points ${\mathrm{FP}}_1$, ${\mathrm{FP}}_2$ and ${\mathrm{FP}}_3$. Right: The $N=8$ and $M=30$ theory with fixed points ${\mathrm{FP}}_1$ and ${\mathrm{FP}}_3$.

Figure 6

Coupling flows of theories in region I. Left: The $N=6$, $M=24$ theory with fixed point ${\mathrm{FP}}_1$. Center: The $N=9$, $M=18$ theory with fixed point ${\mathrm{FP}}_2$. Right: The $N=10$ and $M=34$ theory with fixed points ${\mathrm{FP}}_1$ and ${\mathrm{FP}}_2$.

Figure 7

Coupling flows of theories in regions II and III. Left: The $N=10$ and $M=60$ theory with fixed point ${\mathrm{FP}}_2$. Right: The $N=13$ and $M=50$ theory with fixed point ${\mathrm{FP}}_2$.

Figure 8

The running of ${\alpha }_N$ and ${\alpha }_M$ in the $SU(N)\times SU(M{)}_L$ theory for $N=7$ and $M=25$ along a trajectory in the left plot of Fig. 3 close to the fixed point on the ${\alpha }_N$ axis.

Figure 9

The running of the coupling of ${\alpha }_N$ and ${\alpha }_2$ in the $SU(N)\times SU(2{)}_L$ theory for $N=9$ and $M=40$ along a trajectory in the left plot of Fig. 5 from the UV Gaussian fixed point to the central infrared interacting fixed point.

Figure 10

Left: The running of ${\alpha }_N$ and ${\alpha }_2$ in the $SU(N)\times SU(2{)}_L$ theory for $N=9$ and $M=40$ along the fine-tuned safety free trajectory from the central infrared fixed point to the ultraviolet fixed point on the ${\alpha }_2$ axis. Right: The running of ${\alpha }_N$ and ${\alpha }_2$ of the same theory but now along the fine-tuned safety free trajectory from the central infrared fixed point to the ultraviolet fixed point on the ${\alpha }_N$ axis.

Figure 11

Left: The running of ${\alpha }_N$ and ${\alpha }_2$ in the $SU(N)\times SU(2{)}_L$ theory for $N=9$ and $M=40$ along a trajectory in the left plot of Fig. 5. The flow starts off near the Gaussian ultraviolet fixed point and then flows close to the interacting fixed point along the ${\alpha }_2$ axis but settling at the central infrared interacting fixed point. Right: The running of ${\alpha }_N$ and ${\alpha }_2$ of the same theory but now along a trajectory flowing from the Gaussian ultraviolet fixed point, then flowing close to the infrared interacting fixed point along the ${\alpha }_N$ axis but settling at the central infrared interacting fixed point.