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

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 mechanica l model. Weyl ordering the corresponding Hamiltonian in phase space, an ext ra two-loop counterterm. 1/8(R + g(ij)Gamma (il)(k)Gamma (k)(lj)) is produc ed in the action. Applying a recursive method we evaluate the components of the metric tensor in Riemann normal coordinates in six dimensions and cons truct the interaction lagrangian density by employing the background field method. The calculation of the anomaly is based on the end-point scalar pro pagator and not on the string inspired center-of-mass propagator which give s incorrect results for the local trace anomaly. The manipulation of the Fe ymnan diagrams is partly relied on the factorization of four-dimensional su bdiagrams and partly on a brute force computer algebra program developed to serve this specific purpose. The computer program enables one to perform i ndex contractions of twelve quantum fields (10 395 in the present case) a t ask which cannot be accomplished otherwise. We observe that the contributio n of the disconnected diagrams is no longer proportional to the two-dimensi onal trace anomaly (which vanishes in four dimensions). The integrated trac e anomaly is finally expressed in terms of the 17 linearly independent scal ar monomials constructed out of covariant derivatives and Riemann tensors. (C) 2001 Elsevier Science B.V. All rights reserved.