  This paper provides a more detailed background to the results of
arXiv:1004.2692 concerning properties of multi-loop amplitudes in a pure spinor
formulation of field theories with maximal supersymmetry. This involves the
development of a first quantised field theory version of the non-minimal pure
spinor formalism originally designed for describing superstring amplitudes. In
addition to superspace world-line fields, the formalism involves a set of
world-line ghost fields that are required for implementing BRST invariance with
a composite b ghost. In particular, we show that BRST invariance requires the
presence of certain contact terms. For four-point amplitudes, these are
important beyond two loops. We also present an alternative proof of the
"no-triangle hypothesis" and the vanishing of amplitudes with fewer than four
external particles.
