Properties of the є-expansion, Lagrange inversion and associahedra and the O (1) model
2020
We discuss properties of the $\epsilon$-expansion in $d=4-\epsilon$ dimensions. Using Lagrange inversion we write down an exact expression for the value of the Wilson-Fisher fixed point coupling order by order in $\epsilon$ in terms of the beta function coefficients. The $\epsilon$-expansion is combinatoric in the sense that the Wilson-Fisher fixed point coupling at each order depends on the beta function coefficients via Bell polynomials. Using certain properties of Lagrange inversion we then argue that the $\epsilon$-expansion of the Wilson-Fisher fixed point coupling equally well can be viewed as a geometric expansion which is controlled by the facial structure of associahedra. We then write down an exact expression for the value of anomalous dimensions at the Wilson-Fisher fixed point order by order in $\epsilon$ in terms of the coefficients of the beta function and anomalous dimensions.
We finally use our general results to compute the values for the Wilson-fisher fixed point coupling and critical exponents for the scalar $O(1)$ symmetric model to $O(\epsilon^7)$.
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