This is the sineGordon action:
$$
\frac{1}{4\pi} \int_{ \mathcal{M}^2} dt \; dx \; k \partial_t \Phi \partial_x \Phi  v \partial_x \Phi \partial_x \Phi
+ g \cos(\ell_{}^{} \cdot\Phi_{})
$$
Here $\mathcal{M}^2$ is a 1+1 dimensional spacetime, where 1D space is a $S^1$ circle of length $L$.
At $g=0$ : it is a chiral boson theory with zero mass, gapless scalar boson $\Phi$.
At large $g$ : It seems to be wellknown that at large coupling $g$ of the sineGordon equation, the scalar boson $\Phi$ will have a mass gap.
Q1: What is the original Ref which states and proves this statement about the nonzero (or large) mass gap for large $g$?

Q2: What does the mass gap $m$ scale like in terms of other quantities (like $L$, $g$, etc)?

NOTE: I find S Coleman has discussion in
(1)"Aspects of Symmetry: Selected Erice Lectures" by Sidney Coleman
and this paper
(2)Quantum sineGordon equation as the massive Thirring model 
Phys. Rev. D 11, 2088 by Sidney Coleman
But I am not convinced that Coleman shows it explicitly. I read these, but could someone point out explicitly and explain it, how does he(or someone else) rigorously prove this mass gap?
Here Eq.(17) of this reference does a quadratic expansion to show the mass gap $m \simeq \sqrt{\Delta^2+\#(\frac{g}{L})^2}$ with $\Delta \simeq \sqrt{ \# g k^2 v}/(\# k)$, perhaps there are even more mathematical rigorous way to prove the mass gap with a full cosine term?
This post imported from StackExchange Physics at 20140604 08:00 (UCT), posted by SEuser Idear