Research of Lattice Field Theory with rheta-Term by Renormalization Group

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On the basis of the previous research on U (1) gauge theory, we investigated U (2) lattice gauge theory with rheta-term in 2 dimensions by renormalization group.The reason to choose the group U(2) is that we are interested in the role of non Abelian part. The simplest among such group is U (2), so we began the study on this gauge group. The action is given by non Abelian real part and Abelian imaginary part in 2 dimensions. In contrast to 4 dimensional theory, we can not construct non Abelian imaginary part. This is because the topological term is given by iTrepsilon_<munu>F_<munu> in 2 dimensions, and it gives zero when we choose SU (2) (non Abelian) part.As a bare action, we adopt 1) real action ; defined by couplings, betal_1l_2=beta_<11> (l_1=4q, l_2=2I) * 0, (q means U (1) charge, and I means SU (2) isotopic spin), 2) imaginary action ; standard rheta action (i (rheta/2pi) Trepsilon_<munu>F_<munu>. This is defined by U (1) part.).After renomalization transformations, there appears non Abelian part in imaginary action, it, however, converges to zero after many renomalization group transformations.Phase transition occurs only when rheta=pi and in the irreducible representation which is trivial in SU (2), i.e., for ( (l_1, l_2) = (2,0), namely, the representation with q=1, I=0), but not in non trivial SU (2) representation ( (l_1, l_2) = (1,1), namely, q=1/2, I=1/2). This is due to the SU (2) confinement mechanism which forbids deconfinement transition even at rheta=pi.Real action approaches "heat kernel" type by renormalization group transformations. We are performing also 1) 4 dimensional Z_N theory with rheta-term, which is interesting because it is related with "duality" and "oblique confinement" (Imachi, Liu and Yoneyama), 2) numerical study of CP^<N-1> with N lager than 2 (Imachi, Kanou and Yoneyama).