Six-loop renormalization group analysis of the $ϕ^4 + ϕ^6$ model
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2026
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| _version_ | 1866918401048313856 |
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| author | Adzhemyan, L. Ts. Kompaniets, M. V. Trenogin, A. V. |
| author_facet | Adzhemyan, L. Ts. Kompaniets, M. V. Trenogin, A. V. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_21515 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Six-loop renormalization group analysis of the $ϕ^4 + ϕ^6$ model Adzhemyan, L. Ts. Kompaniets, M. V. Trenogin, A. V. Statistical Mechanics High Energy Physics - Theory Chaotic Dynamics 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. |
| title | Six-loop renormalization group analysis of the $ϕ^4 + ϕ^6$ model |
| topic | Statistical Mechanics High Energy Physics - Theory Chaotic Dynamics |
| url | https://arxiv.org/abs/2601.21515 |