Renormalization group theory of the generalized multi-vertex sine-Gordon model

We investigate the renormalization group theory of generalized multi-vertex sine-Gordon model by employing dimensional regularization method and also Wilson method. The vertex interaction is given $\cos(k_j\cdot \phi)$ where $k_j$ ($j=1,2,\cdots,M$) are momentum vectors $\phi$ an $N$-component scalar field. beta functions calculated for with multi cosine interactions. second-order correction in procedure two-point scattering amplitude tachyon scattering. show that new vector $k_{\ell}$ generated from two interactions $k_i$ when meet condition $k_{\ell}=k_i\pm k_j$ called triangle condition. Further $k_i\cdot k_j=\pm 1/2$ required within equations form a set closed $\{k_j\}$ equilateral $N=2$ or regular tetrahedron $N=3$. Wilsonian gives qualitatively same result functions.