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The constraints imposed by maximal supersymmetry on multi-loop contributionsto the scattering of four open superstrings in the U(N) theory are examined byuse of the pure spinor formalism. The double-trace term k^2 t_8(tr F^2)^2(where k represents an external momentum and F the Yang--Mills field strength)only receives contributions from L<=2 (where L is the loop number) while thesingle-trace term k^2 t_8(tr F^4) receives contributions from all L. Weverified these statements up to L=5, but arguments based on supersymmetrysuggest they extend to all L. This explains why the single-trace contributionsto low energy maximally supersymmetric Yang--Mills field theory are moredivergent in the ultraviolet than the double-trace contributions. We alsocomment further on the constraints on closed string amplitudes and theirimplications for ultraviolet divergences in N=8 supergravity.

Non-renormalization conditions for four-gluon scattering in supersymmetric string and field theory