Form factors of descendant operators: Free field construction and reflection relations

TL;DR

This paper extends the free field representation of form factors in integrable models to descendant operators, verifying their structure through counting and reflection relations, and proposes a generalization to kink sectors.

Contribution

It introduces a modified free field construction for descendant operators' form factors and proves their reflection relations, expanding the understanding of operator structures in integrable models.

Findings

Counted operators at each level in chiral sectors.

Verified reflection relations for descendant operators.

Proposed generalization to kink sector.

Abstract

The free field representation for form factors in the sinh-Gordon model and the sine-Gordon model in the breather sector is modified to describe the form factors of descendant operators, which are obtained from the exponential ones, $\e^{\i\alpha\phi}$, by means of the action of the Heisenberg algebra associated to the field $\phi(x)$. As a check of the validity of the construction we count the numbers of operators defined by the form factors at each level in each chiral sector. Another check is related to the so called reflection relations, which identify in the breather sector the descendants of the exponential fields $\e^{\i\alpha\phi}$ and $\e^{\i(2\alpha_0-\alpha)\phi}$ for generic values of $\alpha$. We prove the operators defined by the obtained families of form factors to satisfy such reflection relations. A generalization of the construction for form factors to the kink sector is also proposed.