Six-loop renormalization group analysis of the $\ddot{I}{•}^4+\ddot{I}{•}^6$ model

Abstract

We investigate the $λ\ph^4+g\ph^6$ model using the renormalization group method and the $\ep$ expansion. This model is used in a situation where the coefficients $\hat{I}»$, $g$ and the coefficient $\ddot{I}„$ of the term $τ\ph^2$ depend on two parameters $T$ and $P$, and there is a point ($T_c,P_c$) at which $\ddot{I}„$ and $\hat{I}»$ are zero. This point is named the tricritical point. The description of a system depends on a trajectory that leads to the tricritical point on the plane ($T,P$). In the trajectories, when $\hat{I}»$ goes to zero fast enough, the description is defined by the $\ph^6$ interaction and then the $\ph^4$ term can be considered as a composite operator. In this case, the logarithmic dimension is $d=3$, and the $\ep$ expansion is carried out in the dimension $d=3-2\ep$. The main exponents of the \textit{tricritical} model have been calculated in the third order of the $\ep$ expansion. Taking into account the $\ph^4$ interaction, we were able to calculate the value of the parameter that determines the required decrease rate in $\hat{I}»$ to implement the tricritical behavior. The tricritical dimensions of the composite operators $\ph^k$ for $k=1,2,4,6$ have been computed. The resulting values are compared to those known from a conformal field theory and non-perturbative renormalization group.

11 pages