Pure spinor indications of ultraviolet finiteness in D=4 maximal supergravity

The ultraviolet divergences of amplitude diagrams in maximal supergravity are characterised by a first possible divergence at seven loops for the 4-point amplitude (logarithmic) and, in its absence, at eight loops. We revisit the pure spinor superfield theory results of [arXiv:1412.5983], stating the absence of the divergence originating in the 4-point 7-loop amplitude as well as those of more than seven loops. The analysis, performed in terms of the one-particle irreducible loop structures giving rise to the divergences, is extended, especially with respect to the limits on the dimension for finiteness. The results correspond to those mentioned, known from other approaches, indicating an ultraviolet finiteness of maximal supergravity in D=4.Comment: 28 pages, v2: Footnote added: further analysis require