The boundary sine-Gordon theory: classical and semi-classical analysis

Abstract

We consider the sine-Gordon model on a half-line, with an additional potential term of the form $-M\cos{β\over 2}(φ-φ_0)$ at the boundary. We compute the classical time delay for general values of $M$, $β$ and $φ_0$ using $τ$-function methods and show that in the classical limit, the method of images still works, despite the non-linearity of the problem. We also perform a semi-classical analysis, and find agreement with the exact quantum S-matrix conjectured by Ghoshal and Zamolodchikov.