Type II and heterotic one loop string effective actions in four dimensions

We analyze the reduction to four dimensions of the R4 terms which are part of the ten-dimensional string effective actions, both at tree level and one loop. We show that there are two independent combinations of R4 present, at one loop, in the type IIA four dimensional effective action, which means they both have their origin in M-theory. The d = 4 heterotic effective action also has such terms. This contradicts the common belief that there is only one R4 term in four-dimensional supergravity theories, given by the square of the Bel-Robinson tensor. We show that the supersymmetric completion of this new R4 term in 4 ≤ N ≤ 8 extended supergravity cannot be achieved at the linearized level without introducing nonlinear α corrections to the solutions of the Bianchi identities. In pure N = 1 supergravity this new R4 combination cannot be directly supersymmetrized, but we show that, when coupled to a scalar chiral multiplet (violating the U(1) R-symmetry), it emerges in the action after elimination of the auxiliary fields.