Conformal behavior at four loops and scheme (in)dependence

We search for infrared zeros of the beta function and evaluate the anomalous dimension of the mass at the associated fixed point for asymptotically free vectorlike fermionic gauge theories with gauge group $SU(N)$. The fixed points of the beta function are studied at the two, three and four loop level in two different explicit schemes. These are the modified regularization invariant scheme and the minimal momentum subtraction scheme. The search is performed in Landau gauge where the beta function of the gauge parameter vanishes. We then compare our findings to earlier identical investigations performed in the modified minimal subtraction scheme. It is found that the value of the anomalous dimension of the mass is smaller at three and four loops than at two loops. This seems to be a generic pattern that is observed in all three different schemes. We then estimate the value of the anomalous dimension to be $\gamma \sim 0.225–0.375$ for twelve fundamental flavors and three colors, $\gamma \sim 0.500–0.593$ for two adjoint flavors and two colors and finally $\gamma \sim 1.12–1.70$ for two two-indexed flavors and three colors with the lower and upper bound set by the minimum and maximum value respectively over all three schemes and at three and four loops. Our analysis suggests that the former two theories lie in the conformal window while the latter belongs to the chirally broken phase.

Figure 1

In all the plots below blue curves correspond to two loops, red curves correspond to three loops and green curves correspond to four loops. In addition if the fixed point value ${\alpha }_0<1$ then the curve is solid while if ${\alpha }_0>1$ then the curve is dashed.

Figure 2

The anomalous dimension at an infrared fixed point as a function of the number of flavors. The fermions are in the fundamental representation, F, and the number of colors is $N=2$.

Figure 3

The anomalous dimension at an infrared fixed point as a function of the number of flavors. The fermions are in the fundamental representation, F, and the number of colors is $N=3$.

Figure 4

The anomalous dimension at an infrared fixed point as a function of the number of flavors. The fermions are in the fundamental representation, F, and the number of colors is $N=4$.

Figure 5

The anomalous dimension at an infrared fixed point as a function of the number of flavors. The fermions are in the adjoint representation, G, and the number of colors is $N=2$.

Figure 6

The anomalous dimension at an infrared fixed point as a function of the number of flavors. The fermions are in the two-indexed symmetric representation, 2S, and the number of colors is $N=3$.

Figure 7

The anomalous dimension at an infrared fixed point as a function of the number of flavors. The fermions are in the two-indexed symmetric representation, 2S, and the number of colors is $N=4$.

Figure 8

The anomalous dimension at an infrared fixed point as a function of the number of flavors. The fermions are in the two-indexed antisymmetric representation, 2A, and the number of colors is $N=4$.