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

In this work we studies the long-range interactions in non-gravitational field theories and their behaviour in the deep infrared. To model such effects, we consider a nonlocal scalar theory obtained by adding a $\phi\Box^{-1}\phi$ term to the local action. Using the functional renormalisation group, we analyse its infrared fixed-point structure. Within the 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 $k\rightarrow 0$ and find that the flow becomes singular for $\lambda^{2}>0$ before reaching the deep infrared. In the LPA$'$ framework, we find that the infrared-stable fixed point is the nonlocal Gaussian fixed point. We then generalise the model to $\phi\Box^{\sigma/2}\phi$ and analyse how the infrared properties depend on $\sigma$. With appropriate scaling choices, we show that the infrared behaviour 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.