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