Six-loop renormalization group analysis of the \(\varphi^4 + \varphi^6\) model

Abstract

We investigate the \(\lambda\varphi^4+g\varphi^6\) model using the renormalization group method and the \(\varepsilon\) expansion. This model is used in a situation where the coefficients \(\lambda\), \(g\), and the coefficient \(\tau\) of the term \(\tau \varphi^2\) depend on two parameters \(T\) and \(P\), and there is a point (\(T_c,P_c\)) at which \(\tau\) and \(\lambda\) are zero. This point is called the tricritical point. The description of the system depends on the trajectory leading to the tricritical point on the plane (\(T,P\)). Along trajectories where \(\lambda\) approaches zero sufficiently fast, the asymptotic description is defined by the \(\varphi^6\) interaction and thus the \(\varphi^4\) term can be considered as a composite operator. In this case, the logarithmic dimension is \(d=3\), and the \(\varepsilon\) expansion is carried out in the dimension \(d=3-2\varepsilon\). The main exponents of the tricritical model have been calculated in the third order of the \(\varepsilon\) expansion. Taking into account the \(\varphi^4\) interaction, we were able to calculate the parameter that determines the required decay rate of \(\lambda\) to implement the tricritical behavior. The tricritical dimensions of the composite operators \(\varphi^k\) for \(k=1, 2, 4, 6\) have been computed. The resulting values are compared to those known from a conformal field theory and nonperturbative renormalization group.