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

TL;DR

This paper investigates the infrared behavior of nonlocal scalar field theories with long-range interactions using the functional renormalisation group, revealing how nonlocality influences phase transitions, symmetry breaking, and fixed points.

Contribution

It introduces a nonlocal scalar model with a $\, ext{phi}\,ox^{-1} ext{phi}$ term, analyzes its fixed points, and explores the effects of nonlocality and higher derivatives on infrared properties.

Findings

Nonlocality modifies phase transition patterns and can induce symmetry breaking.

The infrared fixed point is the nonlocal Gaussian fixed point.

Infrared behavior remains consistent up to certain nonlocality parameters.

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 $\phi\Box^{-1}\phi$ 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.