On the dimensional reduction of quadratic higher-derivative gravitational terms

Gravitational Lagrangian theories, that are formulated initially in D(= 4+ N) dimensions, produce scalar moduli sA, sB upon reduction to four dimensions, via the metric decomposition <^> gAB(X c) = e -v 2.0sAgij(x k). e 2.0sB/v N g mu.(y.), where. 20 is the bare four-dimensional gravitational coupling, while gij(x k) and g mu.(y.) are the physical four-and internal-space N-metrics, respectively. After integration over the N-space, the four-Lagrangian resulting from the Einstein-Hilbert D-theory <^> L = -e -2f [<^> R + 4(<^>. f) 2]/2 <^>. 2 is L = -R/2. 2 +(. sA) 2 /2+(. sB) 2 /2, in which the kinetic-energy terms for sA, sB have canonical coefficients 1/2. These coefficients are modified, however, if <^> L contains in addition quadratic higher-derivative terms R <^> 2 = a <^> 1 <^> R 2 + <^> a2 <^> R AB <^> R AB + <^> a3 <^> R ABCD <^> R ABCD, due to the rescaling under the conformal transformation <^> gAB. e -v 2.0sAgij, which is typically of the form R <^> 2. e 2 v 2.0sA[R2 + R(. sA, B) 2 +(. sA, B) 4]. Previously, we analyzed the effect of the terms R(. sA, B) 2 quadratic in. sA, B, which in general lead to a mixing of sA and sB, and consequently instability at high energies. Here, we consider the quartic terms (. sA, B) 4, that also give rise to instabilities, both for arbitrary <^> an and in the specific case of the heterotic superstring theory, for which <^> a1 = <^> a3 = -<^> a2/4 =. 20 /2, and become significant if sA, sB behave as massless scalars.