Zero Mode Divergence Problem in String Theory

TL;DR

This paper addresses the zero mode divergence in 2D string theory's perturbative S-matrices by proposing a wave packet-based scheme that yields finite, physically meaningful S-matrices and justifies formal amplitudes.

Contribution

It introduces a novel method using wave packets to obtain finite S-matrices in 2D string theory, clarifying the physical interpretation of formal procedures.

Findings

The scheme produces well-defined S-matrices for simple processes.

It justifies the shifted Virasoro-Shapiro amplitude.

Tree-level unitarity is demonstrated for these processes.

Abstract

For $2D$ string theory, the perturbative $S$-matrices are not well-defined due to a zero mode divergence. Although there exist formal procedures to make the integral convergent, their physical meanings are unclear. We describe a method to obtain finite $S$-matrices physically to justify the formal schemes. The scheme uses asymptotic states by wave packets which fall faster than exponentials. It is shown that the scheme gives well-defined $S$-matrices and justifies the formal shifted Virasoro-Shapiro amplitude for simple processes. The tree-level unitarity for these processes is also shown. We point out a problem in this scheme.