a -function for N=2 supersymmetric gauge theories in three dimensions

Abstract

Recently, the existence of a candidate a-function for renormalisable 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 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.

Supersymmetric tensor model at large N and small ε

“…The beta-function for a general quartic superpotential was considered in the paper [47]. The beta-function receives corrections only from the field renormalizations, meaning that it has the following form…”

Scheme invariants in ϕ4 theory in four dimensions

“…From our point of view, as mentioned in the introduction, the crucial development is the demonstration that the β-functions in theories in four and six dimensions obey a gradient flow equation similar to one which plays a critical role in the derivation of the c-theorem [22][23][24]26]. These gradient flow equations often place constraints relating the β-function coefficients, as has been shown for fourdimensional gauge theories [4] and six-dimensional φ 3 theories [5] (similar gradient flows have been demonstrated in three dimensions [1][2][3] though here the theoretical underpinning has not yet been provided). Our purpose in this section is to apply the same considerations to our four-dimensional φ 4 theory where we are able to confirm our results using the explicit calculations available to a high loop order.…”

“…There is also one four-loop vacuum diagram corresponding to the freedom in Eq. (6.18) (diagrammatically corresponding to that appearing in (6.9), but with insertions of β (1) , β (2) replacing the diamond and cross respectively). Finally there is a three-loop vacuum diagram corresponding to the freedom in Eq.…”

“…Finally there is a three-loop vacuum diagram corresponding to the freedom in Eq. (6.10), with insertions of β (1) , β (3) or β (2) , β (2) . Therefore the number of unknowns which are solved for is only 40 − 6 − 1 − 1 = 32.…”

“…a function which generates the β-functions through a gradient-flow equation. For this to be feasible, the β-function coefficients must satisfy a set of consistency conditions, which must clearly be scheme-invariant; as has been verified for various field theories in three [1][2][3], four [4] and six [5] dimensions. The number of scheme-independent combinations at each loop order would naively be expected to be given by the difference of the number of β-function coefficients and the number of independent variations of coefficients; however the number of independent invariants actually found is considerably larger.…”