  A number of spacetime fields in string theory (notably the metric, dilaton,
bosonic and type 0 bulk closed-string tachyon, and bosonic open-string tachyon)
have the following property: whenever the spacetime field configuration
factorizes in an appropriate sense, the matter sector of the world-sheet theory
factorizes into a tensor product of two decoupled theories. Since the beta
functions for such a product theory necessarily also factorize, this property
strongly constrains the form of the spacetime action encoding those beta
functions. We show that this constraint alone--without needing actually to
compute any of the beta functions--is sufficient to fix the form of the
two-derivative action for the metric-dilaton system, as well as the potential
for the bosonic open-string tachyon. We also show that no action consistent
with this constraint exists for the closed-string tachyon coupled to the metric
and dilaton.
