Six-loop renormalization group analysis of the $\phi^4 + \phi^6$ model

We investigate the $\lambda\ph^4+g\ph^6$ model using the renormalization group method and the $\ep$ expansion. This model is used in a situation where the coefficients $\lambda$, $g$ and the coefficient $\tau$ of the term $\tau \ph^2$ 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 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 $\lambda$ 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 $\lambda$ 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.