Exact potential and scattering amplitudes from the tachyon non-linear $\beta$-function

TL;DR

This paper computes the non-linear tachyon $eta$-function in open bosonic string theory, constructs a universal effective action, and provides an exact tachyon potential useful for studying off-shell processes like tachyon condensation.

Contribution

It derives the all-order in derivatives and field expansion of the tachyon $eta$-function and constructs a simple, universal effective action aligned with string field theory results.

Findings

Exact tachyon potential obtained in a small derivative expansion.

Effective action matches cubic string field theory on-shell.

Coordinates with non-linear $eta$-function are suitable for studying D-brane solitons.

Abstract

We compute, on the disk, the non-linear tachyon $\beta$-function, $\beta^T$, of the open bosonic string theory. $\beta^T$ is determined both in an expansion to the third power of the field and to all orders in derivatives and in an expansion to any power of the tachyon field in the leading order in derivatives. We construct the Witten-Shatashvili (WS) space-time effective action $S$ and prove that it has a very simple universal form in terms of the renormalized tachyon field and $\beta^T$. The expression for $S$ is well suited to studying both processes that are far off-shell, such as tachyon condensation, and close to the mass-shell, such as perturbative on-shell amplitudes. We evaluate $S$ in a small derivative expansion, providing the exact tachyon potential. The normalization of $S$ is fixed by requiring that the field redefinition that maps $S$ into the tachyon effective action derived from the cubic string field theory is regular on-shell. The normalization factor is in precise agreement with the one required for verifying all the conjectures on tachyon condensation. The coordinates in the space of couplings in which the tachyon $\beta$-function is non linear are the most appropriate to study RG fixed points that can be interpreted as solitons of $S$, $i.e.$ D-branes.