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.
