Unique Trajectory Method in Migdal Renormalization Group Approach and Crossover Phenomena

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Abstract
Migdal renormalization group approach, combined with Wilson-Kogut topoligical argument, is applied to four dimensional lattice gauge theory of finite subgroup Ĩ(120) of SU(2). i) A slight (compared with the Monte Carlo results) but clear crossover from strong coupling regime to weak coupling regime is observed for the Wilson action. ii) For mixed action, of the fundamental and the adjoint representation, a clearer stepwise transition, suggesting first order phase transition, is found at 1≲βab≲3 (where βab denotes the bare inverse coupling constant of the adjoint representation). This stepwise transition changes into crossover for smaller βab. iii) There are four critical lines in (βfb,βab) plane starting form a quadruple point (βfb∼0.75,βab∼3.2) where βfb denotes the bare inverse coupling constant of the fundamental representation; 1) SO(3) critical line, 2) Z(2) critical line, 3) a critical line due to the discreteness of Ĩ(120), 4) a critical line related to crossover. In this investigation, the unique trajectory of renormalization group is very important and plays a powerful role in finding crossover and stepwise transition.