Second-order and fluctuation-induced first-order phase transitions with functional renormalization group equations

We investigate phase transitions in scalar field theories using the functional renormalization group (RG) equation. We analyze a system with U(2) x U(2) symmetry, in which there is a parameter lambda(2) that controls the strength of the first-order phase transition driven by fluctuations. In the limit of lambda(2) -> 0, the U(2) x U(2) theory is reduced to an O(8) scalar theory that exhibits a second-order phase transition in three dimensions. We develop a new insight for the understanding of the fluctuation-induced first-order phase transition as a smooth continuation from the standard RG flow in the O(8) system. In our view from the RG flow diagram on coupling parameter space, the region that favors the first-order transition emerges from the unphysical region to the physical one as lambda(2) increases from zero. We give this interpretation based on the Taylor expansion of the functional RG equations up to the fourth order in terms of the field, which encompasses the epsilon-expansion results. We compare results from the expansion and from the full numerical calculation and find that the fourth-order expansion is only of qualitative use and that the sixth-order expansion improves the quantitative agreement.