Exact quantum S-matrices for solitons in simply-laced affine Toda field theories

TL;DR

This paper derives exact quantum S-matrices for solitons in simply-laced affine Toda field theories using a q-deformation approach, ensuring consistency with classical limits and known theoretical constraints.

Contribution

It introduces a novel q-deformation method to obtain exact S-matrices, aligning quantum results with classical time delays and known particle S-matrices in affine Toda theories.

Findings

Derived exact quantum S-matrices for solitons

Confirmed S-matrices satisfy crossing, unitarity, bootstrap

Matched breather S-matrices with known results

Abstract

Exact solutions to the quantum S-matrices for solitons in simply-laced affine Toda field theories are obtained, except for certain factors of simple type which remain undetermined in some cases. These are found by postulating solutions which are consistent with the semi-classical limit, $\hbar\rightarrow 0$, and the known time delays for a classical two soliton interaction. This is done by a `$q$-deformation' procedure, to move from the classical time delay to the exact S-matrix, by inserting a special function called the `regularised' quantum dilogarithm, which only holds when $|q|=1$. It is then checked that the solutions satisfy the crossing, unitarity and bootstrap constraints of S-matrix theory. These properties essentially follow from analogous properties satisfied by the classical time delay. Furthermore, the lowest mass breather S-matrices are computed by the bootstrap, and it is shown that these agree with the particle S-matrices known already in the affine Toda field theories, in all simply-laced cases.