  U(2) lattice gauge theory with $\theta$-term in 2 space-time dimensions is
investigated. It has non-Abelian real action and Abelian( U(1) type) imaginary
action. The imaginary action is defined as the standard $\theta$-term. As the
effect of renormalization group (RG) transformation, non-Abelian imaginary
action is induced. After many steps of RG transformation, non-Abelian part will
die away. After several steps of RG transformations, renormalized action
approaches so called heat kernel action. Phase transition is found at $\theta
=\pi$ only.
