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 $\varphi {\square }^{-1}\varphi$ 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 $k\to 0$ and find that the flow becomes singular for ${\lambda }^2>0$ before reaching the deep infrared. Including the scale-dependent wave function renormalization, $Z_k$ (${\mathrm{LPA}}^{\prime }$), we find that the infrared-stable fixed point is the nonlocal Gaussian fixed point. We then generalize the model to $\varphi {\square }^{\sigma /2}\varphi$ and analyze how the infrared properties depend on $\sigma$. With appropriate scaling choices, we show that the infrared behavior remains unchanged up to $\sigma =d/2$ and follows Sak’s prediction up to $\sigma =2$. Finally, we study higher-derivative cases within the LPA, focusing on $\sigma =4$, which corresponds to isotropic Lifshitz criticality, and obtain results consistent with earlier work.

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• Critical phenomena
• Effective field theory
• Functional renormalization group
• Nonperturbative effects in field theory