Position-Space Renormalisation of the Energy-Momentum Tensor

TL;DR

This paper introduces a novel position-space renormalisation method for the energy-momentum tensor in three-dimensional quantum field theories, addressing divergence issues and enabling continuum limit analysis relevant to holographic cosmology.

Contribution

It presents a new filtering technique using smooth bump functions to remove contact term divergences in the lattice calculation of the EMT two-point function.

Findings

Successfully removes contact divergences in EMT correlators

Enables extraction of continuum limit behaviour of the renormalised EMT

Applicable to theories relevant in holographic cosmology

Abstract

There is increasing interest in the study of nonperturbative aspects of three-dimensional quantum field theories (QFT). They appear as holographic dual to theories of (strongly coupled) gravity. For instance, in Holographic Cosmology, the two-point function of the Energy-Momentum Tensor (EMT) of a particular class of three-dimensional QFTs can be mapped into the power spectrum of the Cosmic Microwave Background in the gravitational theory. However, the presence of divergent contact terms poses challenges in extracting a renormalised EMT two-point function on the lattice. Using a $\phi^4$ theory of adjoint scalars valued in the $\mathfrak{su}(N)$ Lie Algebra as a proof-of-concept motivated by Holographic Cosmology, we apply a novel method for filtering out such contact terms by making use of infinitely differentiable "bump" functions which enforce a smooth window that excludes contributions at zero spatial separation. The process effectively removes the local contact terms and allows us to extract the continuum limit behaviour of the renormalised EMT two-point function.