$a$-function in three dimensions: Beyond the leading order

Recently, evidence was provided for the existence of an $a$-function for renormalizable quantum field theories in three dimensions. An explicit expression was given at lowest order for general theories involving scalars and fermions and shown to be related to the $\beta$-functions by a gradient flow equation with positive-definite metric as in four dimensions. Here, we extend this lowest-order calculation to a general Abelian Chern-Simons gauge theory coupled to fermions and scalars and derive a prediction for part of the four-loop Yukawa $\beta$-function. We also compute the complete four-loop Yukawa $\beta$-function for the scalar-fermion theory and show that it is entirely consistent with the gradient flow equations at next-to-leading order.

Figure 1

Two-loop tensor structures appearing in ${\beta }_Y^{(2)}$.

Figure 2

Two-loop tensor structures appearing in ${\beta }_h^{(2)}$.

Figure 3

Contributions to $A^{(5)}$ from Yukawa couplings.

Figure 4

Contributions to $A_h^{(7)}$ and $A_{hY}^{(7)}$.

Figure 5

Contributions to $A^{(7)}$, terms 1–30.

Figure 6

Contributions to $A^{(7)}$, terms 31–52.

Figure 7

Contributions to $T_{YY}^{(5)}$.

Figure 8

Frequently occurring Feynman integrals.

Figure 9

Feynman integrals for terms undetermined by (3.2).