Abstract

We show that for four dimensional gauge theories in the conformal window, the Euler anomaly, known as the $a$-function, can be computed from a $2$-point function of the trace of the energy momentum tensor making it more amenable to lattice simulations. Concretely, we derive an expression for the $a$-function as an integral over the renormalisation scale of quantities related to $2$- and $3$-point functions of the trace of the energy momentum tensor.The crucial ingredients are that the square of the field strength tensor is an exactly marginal operator at the Gaussian fixed point and that the relevant $3$-point correlation function is finite when resummed to all orders. This allows us to define a scheme for which the $3$-point contribution vanishes, thereby explicitly establishing the strong version of the $a$-theorem for this class of theories.

Highlights

  • The conformal anomaly, first known as the central charge c of the Virasoro algebra, is key to the physics of conformal field theories (CFTs) as it is a measure of the number of degrees of freedom

  • The crucial ingredients are that the square of the field strength tensor is an exactly marginal operator at the Gaussian fixed point and that the relevant three-point correlation function is finite when resummed to all orders

  • Our starting point was the derivation of a formula for the Euler anomaly or a function as an integral over the renormalization group (RG) scale of a two- and three-point functions of the trace of the energy momentum tensor (5), valid for theories which are governed by β functions at both fixed points (4)

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Summary

INTRODUCTION

The conformal anomaly, first known as the central charge c of the Virasoro algebra, is key to the physics of conformal field theories (CFTs) as it is a measure of the number of degrees of freedom. A representation similar to (1) has been proposed involving two- and three-point functions [11], 1In curved space, βa can be assessed from a two-point function [8] This expression is derived in Appendix B using conformal anomaly matching. We will show that for gauge theories with gauge couplings only, the three-point function term drops for theories in the conformal window cf Fig. 1. This establishes the positivity with Euclidean methods and makes the evaluation more amenable to lattice simulations

Executive summary
THE THREE METRIC χ ggg IN GAUGE THEORIES
Finiteness of the three-point function
Constructing the R3χ scheme for the three metric χ ggg
Nf the is a one-loop exact β
CONCLUSIONS AND DISCUSSIONS
Renormalization in curved space and βUa V
Sum rule from the dilaton effective action

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