On the heterotic string loop expansion

The effective action for the bosonic sector of the heterotic superstring theory, considered as an expansion in string loops, is reduced to four dimensions, with particular regard to the effects of the one-loop, parity-even term involving the eight-form X8X8 introduced by Green et al. Formulae are given for the one-loop corrections to the ten-dimensional, tree-level quartic gravitational term Rˆ4→Rˆ′4, which determines the renormalization of the four-dimensional quadratic term R2R2 via (Rˆ4→R2R¯2)→(Rˆ′4→R′2R¯2). We also derive the formula for the four-dimensional, gauge kinetic function up to two loops, f=Ar+βBr+γAr−1Br4, where ArAr and BrBr are moduli and γ   is the unknown two-loop coefficient of the E8×E8′ string. In the theory of gaugino condensation in the hidden sector, based upon the no-scale supergravity theory, the parameter β   determines the ratio of the masses of the gaugino and gravitino to one loop, ξ≡m1/2/m3/2≈βBr/(Ar+βBr)ξ≡m1/2/m3/2≈βBr/(Ar+βBr). The torus approximation for the internal space g¯μν predicts, setting Ar≈2Ar≈2, Br≈3Br≈3, that this ratio is close to unity, ξ≈12/13ξ≈12/13, while the six-sphere would yield a somewhat smaller ratio, ξ≈1/15ξ≈1/15.