Exact Mass-Coupling Relation for the Homogeneous Sine-Gordon Model

We derive the exact mass-coupling relation of the simplest multiscale quantum integrable model, i.e., the homogeneous sine-Gordon model with two mass scales. The relation is obtained by comparing the perturbed conformal field theory description of the model valid at short distances to the large distance bootstrap description based on the model’s integrability. In particular, we find a differential equation for the relation by constructing conserved tensor currents, which satisfy a generalization of the $\mathrm{\Theta }$ sum rule Ward identity. The mass-coupling relation is written in terms of hypergeometric functions.

Figure 1

Plots of (${\mu }_1$, ${\mu }_2$) versus (${\lambda }_1$, ${\lambda }_2$). On the left, the red and blue surfaces represent ${\mu }_1({\lambda }_i)$ and ${\mu }_2({\lambda }_i)$ in (29) and (30), respectively. The red and blue points represent the numerical data $[{\lambda }_1({\mu }_a),{\lambda }_2({\mu }_a),{\mu }_b]$ $(b=1,2)$ from the TBA equations, which are solved for given ${\mu }_a={\overline{\mu }}_a$. Each sequence from the bottom to the top corresponds to $({\mu }_2{)}^{2/5}=1/2,1,3/2,2$, with ${\mu }_1$ varied. ${\lambda }_i$ are determined by comparing the TBA free energy with the CFT perturbation. On the right, the diamonds (⋄) represent the projections of the left points to the (${\lambda }_1$, ${\lambda }_2$) plane. The solid lines are the contours in the fundamental domain for $[{\mu }_2({\lambda }_i){]}^{2/5}=1/2,1,3/2,2$ from (30).