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

Abstract

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 $\varphi {\square }^{-1}\varphi$ 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 $k\to 0$ and find that the flow becomes singular for ${\lambda }^2>0$ before reaching the deep infrared. In the LPA${}^{\prime }$ framework, we find that the infrared-stable fixed point is the nonlocal Gaussian fixed point. We then generalise the model to $\varphi {\square }^{\sigma /2}\varphi$ 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.

30 pages, 10 figures

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