Non-critical strings at high energy

TL;DR

This paper analyzes high-energy scattering amplitudes in non-critical string theory, proposing an analytic continuation in the matter central charge and revealing a saddle point dominated by electrostatic configurations, with explicit solutions for 3- and 4-point functions.

Contribution

It introduces an analytic continuation approach for amplitudes in non-critical string theory and solves the electrostatic problem explicitly for certain cases, advancing understanding of high-energy behavior.

Findings

Amplitudes are complex analytic in the matter central charge c.

High energy limit dominated by electrostatic saddle point.

Explicit solutions for 3- and 4-point functions using hyper-elliptic integrals.

Abstract

We consider scattering amplitudes in non-critical string theory of $N$ external states in the limit where the energy of all external states is large compared to the string tension. We argue that the amplitudes are naturally complex analytic in the matter central charge $c$ and we propose to define the amplitudes for arbitrary value of $c$ by analytic continuation. We show that the high energy limit is dominated by a saddle point that can be mapped onto an equilibrium electro-static energy configuration of an assembly of $N$ pointlike (Minkowskian) charges, together with a density of charges arising from the Liouville field. We argue that the Liouville charges accumulate on segments of curves, and produce quadratic branch cuts on the worldsheet. The electro-statics problem is solved for string tree level in terms of hyper-elliptic integrals and is given explicitly for 3- and 4-point functions. We show that the high energy limit should behave in a string-like fashion with exponential dependence on the energy scale for generic values of $c$.