Sphere free energy of scalar field theories with cubic interactions

Abstract

The dimensional continuation approach to calculating the free energy of $d$-dimensional Euclidean CFT on the round sphere $S^d$ has been used to develop its $4-ε$ expansion for a number of well-known non-supersymmetric theories, such as the $O(N)$ model. The resulting estimate of the sphere free energy $F$ in the 3D Ising model has turned out to be in good agreement with the numerical value obtained using the fuzzy sphere regularization. In this paper, we develop the $6-ε$ expansions for CFTs on $S^d$ described by scalar field theory with cubic interactions and use their resummations to estimate the values of $F$. In particular, we study the theories with purely imaginary coupling constants, which describe non-unitary universality classes arising when certain conformal minimal models are continued above two dimensions. The Yang-Lee model $M(2,5)$ is described by a field theory with one scalar field, while the $D$-series $M(3,8)$ model is described by two scalar fields. We also study the $OSp(1|2)$ symmetric cubic theory of one commuting and two anti-commuting scalar fields, which appears to describe the critical behavior of random spanning forests. In the course of our work, we revisit the calculations of beta functions of marginal operators containing the curvature. We also use another method for approximating $F$, which relies on perturbation theory around the bilocal action near the long-range/short-range crossover. The numerical values it gives for $F$ tend to be in good agreement with other available methods.

Figures (9)

• Figure 2.1: One and two-loop corrections to the propagator to order ${\cal O}(g_1^{n_1}g_2^{n_2})$ with $n_1+n_2=4$.
• Figure 2.2: Diagrams up to order ${\cal O}(g_1^{n_1}g_2^{n_2})$ with $n_1+n_2=5$ that contribute to the one-point function of $\sigma$. The external leg is amputated.
• Figure 2.3: Leading contributions of curvature terms to the one-point function. The empty dot represents an insertion of $\kappa$ and the cross represents an $\eta$ vertex.
• Figure 2.4: Leading contributions of curvature terms to the two-point function. The empty dot represents an insertion of $\kappa$ and the cross represents an $\eta$ vertex.
• Figure 5.1: Free energy in the $N=1$ quartic theory between two and four dimensions. The blue line is the $\epsilon$-expansion result $\widetilde{F}^{\text{Pad\'e}}_{\text{Ising}}$Fei:2015oha based on a two-sided Padé approximation, the red dots are the results of the long-range approach $\widetilde{F}^{\rm LRA}$, the blue cross is the fuzzy sphere result, and the green stars are the exact results in two $(\frac{\pi}{12})$ and four $(\frac{\pi}{180})$ dimensions.