Constructing a c-function for SUSY gauge theories

Abstract

Recently a non-perturbative formula for the RG flow between UV and IR fixed points of the coefficient in the trace of the energy momentum tensor of the Euler density has been obtained for N = 1 SUSY gauge theories by relating the trace and R-current anomalies. This result is compared here with an earlier perturbation theory analysis based on a naturally defined metric on the space of couplings for general renormalisable quantum field theories. This approach is specialised to N = 1 supersymmetric theories and extended, using consistency arguments, to obtain the Euler coefficient at fixed points to 4-loops. The result agrees completely, to this order, with the exact formula. emails: dzf@math.mit.edu and ho@damtp.cam.ac.uk

The a-function for gauge theories

“…As a further check, we also computeà using the dimensional reduction version of the Yukawa β-function and show that in the supersymmetric case, this reduces to the version ofà presented in ref. [25] (and is hence compatible with ref. [20][21][22][23][24]).…”

“…Moreover, here it was possible to show that an exact formula for the a-function conjectured in refs. [20][21][22][23][24] was valid at this order; in previous work [25] this had been checked at the level of the two-loop β-functions for a general, gauged N = 1 supersymmetric theory (in the rest of this paper we shall describe this as a "two-loop check" although the corresponding a-function is actually a four-loop quantity). A sufficient condition on the chiral field anomalous dimension for this exact a-function to be viable was presented and shown (using the results of ref.…”

“…This exact form was verified up to two loops in ref. [25] for a general supersymmetric gauge theory, and up to three loops [18] in the case of the Wess-Zumino model. Moreover in ref.…”

“…(4.13) was verified up to twoloop order (for the anomalous dimension) in ref. [25]. Λ may be constrained by imposing eq.…”

N=1 Euler Anomaly from RG-dependent metric-Background

“…Ever since it has served as a fruitful laboratory for testing different techniques by rederiving the result. Examples include four-loop perturbation theory [4], the local renormalisation group (RG) [5] and employing superspace techniques assuming a gradient flow equation [6] conforming an older conjecture [7].…”