Six-loop renormalization group analysis of the $φ^4 + φ^6$ model
L. Ts. Adzhemyan, M. V. Kompaniets, A. V. Trenogin
TL;DR
The paper performs a six-loop renormalization-group analysis of the φ^4+φ^6 model in the ε-expansion about d = 3 − 2ε, focusing on tricritical behavior at the point where τ and λ vanish. By renormalizing in the G-scheme and computing the renormalization constants Z_i and RG functions up to six loops, the authors determine the fixed point u_* and tricritical exponents η, ν, ω, along with the parameter b_0 and the tricritical dimensions of composite operators Δ_{φ^k}. The results are compared to conformal-field-theory and non-perturbative RG predictions, showing full agreement with several classic works but revealing discrepancies with Hager02 due to typos, which the authors address in follow-up work. A Padé resummation to d = 2 provides approximate values for the exponents and composite dimensions, illustrating reasonable agreement with exact CFT values and non-perturbative RG results, and enabling a detailed cross-check of perturbative data against alternative frameworks.
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 $λ$, $g$ and the coefficient $τ$ of the term $τ\ph^2$ depend on two parameters $T$ and $P$, and there is a point ($T_c,P_c$) at which $τ$ and $λ$ 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 $λ$ 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 $λ$ 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.