$a$-function for $\mathcal{N}=2$ supersymmetric gauge theories in three dimensions

Recently, the existence of a candidate $a$-function for renormalizable theories in three dimensions was demonstrated for a general theory at leading order and for a scalar-fermion theory at next-to-leading order. Here we extend this work by constructing the $a$-function at next-to-leading order for an $\mathcal{N}=2$ supersymmetric Chern-Simons theory. This increase in precision for the $a$-function necessitated the evaluation of the underlying renormalization-group functions at four loops.

Figure 1

Contributions to $A^{(7)}$ (for notation, see Sec. 3).

Figure 2

Contributions to $T^{(5)}$.

Figure 3

Contributions to $K^{(5)}$.

Figure 4

Classes of diagrams that do not contribute.

Figure 5

Four-loop diagrams contributing to the Yukawa-dependent part of the anomalous dimension, with pole terms listed in Table 1.

Figure 6

The basis of massless two-point momentum integrals. For definitions and results, see Appendix pp2 as well as the main text. The dotted propagator is squared, and the arrows on the last integral denote numerator factors of $\cancel{k}$.