(ed.). Gravitational waves (IOP, 2001)(422s).

Appendix A. The string effective action 321Let me note, finally, that the expansion around x 0 can be continued to higherorders,g(x) = η + R ˆx ˆx + ∂ R ˆx ˆx ˆx + R 2 ˆx ˆx ˆx ˆx +···, (16.132)thus introducing higher curvature terms, and higher powers of α ′ , in the effectiveaction:S =− 12λ d−1 s∫d d+1 x √ |g|e −φ [R + (∇φ) 2 − α′4 R2 µναβ +···]. (16.133)At any given order, unfortunately, there is an intrinsic ambiguity in the action dueto the fact that, with an appropriate field redefinition of order α ′ ,g µν → g µν + α ′ (R µν + ∂ µ φ∂ ν φ +···)φ → φ + α ′ (R +∇ 2 φ +···), (16.134)we obtain a number of different actions, again of the same order in α ′ (see, forinstance, [84]). This ambiguity cannot be eliminated until we limit to an effectiveaction truncated to a given finite order.The higher curvature (or higher derivative) expansion of the effective actionis typical of string theory: it is controlled by the fundamental, minimal lengthparameter λ s = (2πα ′ ) 1/2 , in such a way that the higher order correctionsdisappear in the point-particle limit λ s → 0. At any given order in α ′ , however,there is also the more conventional expansion in power of the coupling constantg s (i.e. the loop expansion of quantum field theory: tree-level ∼ g −2 , one-loop∼g 0 , two-loop ∼g 2 ,...). The important observation is that, in a string theorycontext, the effective coupling constant is controlled by the dilaton. Consider,for instance, a process of graviton scattering, in four dimensions. Comparing theaction of (16.4) with the standard, gravitational Einstein action (16.1), it followsthat the effective coupling constant, to lowest order, is√8πG = λp = λ s e φ/2 (16.135)(G is the usual Newton constant). Each loop adds an integer power of the squareof the dimensionless coupling constant, which is controlled by the dilaton asg 2 s = (λ p/λ s ) 2 = (M s /M p ) 2 = e φ . (16.136)We may thus expect, for the loop expansion of the action, the following generalscheme:∫S =− e −φ√ −g(R +∇φ 2 + α ′ R 2 +···) tree level∫ √−g(R− +∇φ 2 + α ′ R 2 +···) one-loop∫− e +φ√ −g(R +∇φ 2 + α ′ R 2 +···) two-loop... (16.137)