A note on tachyon actions in string theory

A number of spacetime fields in string theory—notably the metric, dilaton, bulk (bosonic and type 0) closed string tachyon, and bosonic open string tachyon—have the property that, 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 that encodes those beta functions. We show that this constraint alone—and without needing actually to compute any 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.