Tachyon-massless couplings at order $α'$ in bosonic string theory

RESUMO

Recently, it has been proposed that the classical effective action of bosonic string theory should contain only couplings with an even number of tachyon fields. We employ the T-duality procedure to derive such couplings at order \(\alpha'\) (four-derivative order) for the tachyon and massless fields. We first observe that, through higher-derivative field redefinitions of the tachyon potential term, one can choose a scheme in which, apart from the tachyon potential itself, all remaining couplings involve only covariant derivatives of the tachyon and the massless fields. We then construct a minimal basis of such couplings at four-derivative order, consisting of 14 independent terms. Imposing T-duality reduces this set to 6 nonzero couplings, expressed in terms of two unfixed parameters. One of these parameters is determined by matching to the known effective action in the zero-tachyon limit, while the other is fixed by comparison with the sphere-level S-matrix element of four tachyon vertex operators. This latter comparison also establishes that the coefficient of the \(T^4\) term in the tachyon potential is positive. Remarkably, the tachyon mass term and this quartic term are consistent with a potential of the form \((2/\alpha')(-1+\cos T)\), which implies that the closed string tachyon condenses to the minimum of the potential at \(T=\pi\), thereby generating an Anti-de Sitter spacetime with cosmological constant \(\Lambda =-4/\alpha'\).