The d=6 trace anomaly from quantum field theory four-loop graphs in one dimension

TL;DR

This paper computes the six-dimensional trace anomaly for a scalar field using a path integral approach, recursive metric evaluation, and advanced computer algebra, revealing new insights into quantum field theory in curved space.

Contribution

It introduces a recursive method and a specialized computer algebra program to accurately evaluate the six-dimensional trace anomaly from four-loop graphs.

Findings

Calculated the integrated trace anomaly in six dimensions.

Developed a computer algebra tool for complex tensor contractions.

Found that disconnected diagrams contribute differently than in lower dimensions.

Abstract

We calculate the integrated trace anomaly for a real spin-0 scalar field in six dimensions in a torsionless curved space without a boundary. We use a path integral approach for a corresponding supersymmetric quantum mechanical model. Weyl ordering the corresponding Hamiltonian in phase space, an extra two-loop counterterm ${1/8}\bigg(R + g^{ij} \Gamma^{l}_{k i} \Gamma^{k}_{l j} \bigg)$ is produced in the action. Applying a recursive method we evaluate the components of the metric tensor in Riemann normal coordinates in six dimensions and construct the interaction Langrangian density by employing the background field method. The calculation of the anomaly is based on the end-point scalar propagator and not on the string inspired center-of-mass propagator which gives incorrect results for the local trace anomaly. The manipulation of the Feynman diagrams is partly relied on the factorization of four dimensional subdiagrams and partly on a brute force computer algebra program developed to serve this specific purpose. The computer program enables one to perform index contractions of twelve quantum fields (10395 in the present case) a task which cannot be accomplished otherwise. We observe that the contribution of the disconnected diagrams is no longer proportional to the two dimensional trace anomaly (which vanishes in four dimensions). The integrated trace anomaly is finally expressed in terms of the 17 linearly independent scalar monomials constructed out of covariant derivatives and Riemann tensors.