A Finite Energy-Momentum Tensor for the $\phi^3$ theory in $6$ dimensions

TL;DR

This paper constructs a finite, conserved energy-momentum tensor for the six-dimensional scalar $$ theory using renormalisation group techniques, demonstrating conformal invariance at the fixed point.

Contribution

It provides a method to obtain a finite, conserved energy-momentum tensor in six-dimensional $$ theory, clarifying its conformal properties and RG behavior.

Findings

Finite, conserved energy-momentum tensor constructed.

Trace vanishes at the fixed point, indicating conformal invariance.

Energy tensor can be made RG-invariant everywhere with additional transverse terms.

Abstract

Following Brown[1], we construct composite operators for the scalar $\phi^3$ theory in six dimensions using renormalisation group methods with dimensional regularisation. We express bare scalar operators in terms of renormalised composite operators of low dimension, then do this with traceless tensor operators. We then express the bare energy momentum tensor in terms of the renormalised composite operators, with some terms having divergent coefficients. We subtract these away and obtain a manifestly finite energy tensor. The subtracted terms are transverse, so this does not affect the conservation of the energy momentum tensor. The trace of this finite improved energy momentum tensor vanishes at the fixed point indicating conformal invariance. Interestingly it is not RG-invariant except at the fixed point, but can be made RG invariant everywhere by further addition of transverse terms, whose coefficients vanish at the fixed point.