Abstract :
The low-energy, ten-dimensional effective Lagrangian for the bosonic sector of the heterotic superstring theory, in the string metric ĝAB, is L̂=−12K̂−2e−22K0φ[R̂−18α′RE2+∑n=1∞(α′)2n+1R̂2n+2+…], where K̂−2 and K0−2 are the ten-dimensional and bare four-dimensional gravitational coupling parameters, respectively, and φ is the dilaton. Dimensional reduction yields the effective four-Lagrangian L=−12K−2R+…, where the renormalized coupling K−2 includes contributions from terms of the form R∑n=1∞ (R̄μνξ0)2n+1, where R̄μνξ0 is the Riemann–Christoffel tensor of the six-dimensional internal space. The lowest order correction, deriving from the terms R̂4 in L̂, has been obtained previously, but the effect of the higher-order terms for which n≥2 cannot be calculated precisely, because they are unknown. Here, we attempt to assess their importance, and by consideration of the effect in cosmology of the analogous renormalization of the coefficient of R2, argue that they produce no substantial change, so that K−2≈K0−2. This is of relevance to the difference between the superstring mass scale Ms≈5.3×1017g0 GeV and the grand unification scale Mx≈1016±0.3 GeV, which we further argue cannot be attributed to gravitational effects.
