RG studies of scalar-field models of long-range interactions

In this work, we study the long-range interactions in nongravitational field theories and their behavior in the deep infrared. To model such effects, we consider a nonlocal scalar theory obtained by adding a ϕ □ − 1 ϕ term to the local action. Using the functional renormalization group, we analyze its infrared fixed-point structure. Within the local potential approximation (LPA), we show that nonlocality modifies phase-transition patterns and can induce symmetry breaking. Extending the LPA beyond polynomial truncations, we examine the convexity property of the effective potential as k → 0 and find that the flow becomes singular for λ 2 > 0 before reaching the deep infrared. Including the scale-dependent wave function renormalization, Z k ( LPA ′ ), we find that the infrared-stable fixed point is the nonlocal Gaussian fixed point. We then generalize the model to ϕ □ σ / 2 ϕ and analyze how the infrared properties depend on σ . With appropriate scaling choices, we show that the infrared behavior remains unchanged up to σ = d / 2 and follows Sak’s prediction up to σ = 2 . Finally, we study higher-derivative cases within the LPA, focusing on σ = 4 , which corresponds to isotropic Lifshitz criticality, and obtain results consistent with earlier work.