The Fixed Points of RG Flow with a Tachyon

TL;DR

This paper investigates the fixed points of first-order RG flow in non-linear sigma models with background fields, revealing conditions for non-zero tachyon fixed points on compact spaces and analyzing perturbative corrections on non-compact spaces.

Contribution

It establishes the link between tachyon fixed points and the second derivative of the potential, and computes second-order corrections to black hole solutions with a tachyon.

Findings

Fixed points with non-zero tachyon depend on the potential's second derivative.

Second-order perturbative corrections to black hole solutions are well-behaved.

Tachyon 'hair' persists in corrected solutions.

Abstract

We examine the fixed points to first-order RG flow of a non-linear sigma model with background metric, dilaton and tachyon fields. We show that on compact target spaces, the existence of fixed points with non-zero tachyon is linked to the sign of the second derivative of the tachyon potential $V''(T)$ (this is the analogue of a result of Bourguignon for the zero-tachyon case). For a tachyon potential with only the leading term, such fixed points are possible. On non-compact target spaces, we introduce a small non-zero tachyon and compute the correction to the Euclidean 2d black hole (cigar) solution at second order in perturbation theory with a tachyon potential containing a cubic term as well. The corrections to the metric, tachyon and dilaton are well-behaved at this order and tachyon `hair' persists. We also briefly discuss solutions to the RG flow equations in the presence of a tachyon that suggest a comparison to dynamical fixed point solutions obtained by Yang and Zwiebach.