Renormalization effects in the superstring moduli from the higher-derivative terms

The dimensional reduction of the effective ten-action (S) over cap[(g) over cap (AB)] = 1/2<(<kappa>)over cap>(-2) integral d(10)X root-(g) over cap e(-2 phi) [- (R) over cap - 4(<(<del>)over cap>phi)(2) + 1/8 alpha '(R) over cap (2)(E) + . . .] of the heterotic superstring theory to the physical four-action S[g(ij)] results in the appearance of three moduli B(a), whose real parts B-r drop e root (2/3 kappa0 sigmaB), set equal for simplicity, define the radius and shape of the compact internal six-space (g) over bar (mu nu), in addition to the dilaton A(r) drop e root (2 kappa0 sigmaA), the ten-interval being d (s) over cap (2) drop (g) over cap (AB)dX(A) dX(B) = A(r)(-1) g(ij) dx(i) dx(j) + B-r(g) over bar mu nu dy(mu) dy(nu). These scalar fields are massless at tree level and can be put into canonical form with coefficients 1/2 for the positive kinetic-energy terms, when higher-derivative terms are ignored, as found by Witten, so that S = integral d(4)x root -g[-R/2 kappa (2) + 1/2(del sigma (A))(2) + 1/2(del sigma (B))(2) + . . .]. Previously, we have shown that sigma (A) and sigma (B) acquire a potential from the higher-derivative terms (R) over cap (2)(E) and (R) over cap (4), which becomes large close to the Planck era. Here, we discuss the renormalization of the kinetic-energy terms due to (R) over cap (2)(E) which, after diagonalization, results in a mixing of del sigma (B) with del sigma (A), while the remaining coefficient of (del sigma (B))(2) vanishes at t(c) approximate to t(P)/12 in a radiation-dominated Universe, corresponding to a temperature T-c approximate to 5 x 10(17) GeV, where the four-theory is still classical. At earlier times, the energy is unbounded from below, signalling that the four-theory has become unphysical, and that the string must still be in its uncompactified form with one dilaton phi, whose canonical kinetic energy is positive in the Einstein metric. This mechanism depends upon the equation of state of the source for the Friedmann expansion assumed, and is only effective for values of the adiabatic index in the range 1.14 < gamma < 2.63, which thus includes radiation (gamma = 4/3) and the Zel'dovich equation of state (gamma = 2).