Sigma model approach to string theory effective actions with tachyons

Motivated by recent discussions of actions for tachyon and vector fields related to tachyon condensation in open string theory we review and clarify some aspects of their derivation within sigma model approach. In particular, we demonstrate that the renormalized partition function $Z(T,A)$ of boundary sigma model gives the effective actionfor massless vectors which is consistent with string S-matrix and beta-function resolving an old problem with this suggestion in bosonic string case at the level of the leading $F^2 (dF)^2$ derivative corrections to Born-Infeld action. We give manifestly gauge invariant definition of $Z(T,A)$ in non-abelian NSR open string theory case and demonstrate that for a particular scheme choice its derivative over the tachyon field reproduces the tachyon beta function. We also discuss derivation of similar actions for tachyon + massless modes from sigma model partition functions in closed bosonic and NSR (or type 0) string theories. In the bosonic case the tachyon potential is $T^2 e^{-T} $ unless D=2, but it vanishes in the NSR string case.