research level - Proof for the Mass gap of sine-Gordon action with $g cos(beta Phi)$

This is the sine-Gordon action:

$$\frac{1}{4\pi} \int_{ \mathcal{M}^2} dt \; dx \; k\, \partial_t \Phi \partial_x \Phi - v \,\partial_x \Phi \partial_x \Phi + g \cos(\beta_{}^{} \cdot\Phi_{})$$ Here $\mathcal{M}^2$ is a 1+1 dimensional spacetime manifold, 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 well-known that at large coupling $g$ of the sine-Gordon 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 $g$?

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

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NOTE: I find S Coleman has discussion in

(1)"Aspects of Symmetry: Selected Erice Lectures" by Sidney Coleman

and this paper

(2)Quantum sine-Gordon 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?