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

在前人对U（1）规范理论研究的基础上，我们利用重整化群研究了二维带rheta项的U（2）格点规范理论，选择U（2）群的原因是我们对非Abel部分的作用感兴趣。其中最简单的规范群是U（2），因此我们开始了对U（2）规范群的研究。作用量由二维非阿贝尔真实的部分和阿贝尔虚部分给出。与四维理论相反，我们不能构造非阿贝尔虚部。作为<munu><munu>裸作用量，我们采用1）真实的作用量betal_1l_2=beta_<11>（l_1= 4 q，l_2= 2 I）* 0，（q表示U（1）电荷，I表示SU（2）同位素自旋）; 2）虚作用量标准rheta作用量i（rheta/2 pi）Treppal_<munu>F_<munu>。这是由U（1）部分定义的。经过重整化变换后，虚作用量中出现了非阿贝尔部分，但经过多次重整化群变换后，虚作用量收敛为零，相变只在reta =pi且在SU（2）中平凡的不可约表示中发生，即对于（（l_1，l_2）=（2，0），即q=1，I=0的表示），但对于非平凡SU（2）表示（（l_1，l_2）=（1，1），即q=1/2，I=1/2），则不成立.这是由于SU（2）禁闭机制在reta =pi时也不允许解除禁闭跃迁，真实的作用量通过重整化群变换接近“热核”型。我们还进行了1）4维Z_N理论的rheta项，这是有趣的，因为它与“对偶性”和“斜禁闭”有关（Imachi，Liu和Yoneyama），2）CP_<N-1>N大于2的数值研究（Imachi，Kanou和Yoneyama）。

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).