The Sine-Gordon field theory model at \(\alpha ^ 2=8\pi\), the non- superrenormalizable theory (Q1118272)

We study the Sine-Gordon field theory at \(\alpha^ 2=8\pi\). We prove that the theory is renormalizable but not superrenormalizable and we show how the perturbative renormalization procedure works in this case where the interaction is not polynomial. To go beyond the perturbative results we investigate the \(\beta\)-functional equation for this theory and discuss in what sense at \(\alpha^ 2=8\pi\) the theory is lacking the asymptotic freedom and how it is asymptotic free for \(\alpha^ 2<8\pi\) in an appropriate region of the coupling constants.