Universal Effective Potential for Scalar Field Theory in Three Dimensions by Monte Carlo Computation

We study the low-energy effective action for the theory of a one-component real scalar field in three Euclidean dimensions (3D), in the symmetric phase, concentrating on its static part—effective potential $V_{\mathrm{eff}}\left(\varphi \right)$. We compute it from the probability distributions of the average magnetization in the 3D Ising model in an external field, obtained by Monte Carlo computation. We find that the ${\varphi }^6$ term in $V_{\mathrm{eff}}$ is important in 3D, and compute the values of the universal four-point and six-point couplings.