Negative Coupling $\phi^4$ on the Lattice

TL;DR

This paper explores the possibility of formulating negative coupling $\,\phi^4$ theory in four dimensions on the lattice, using contour deformation into the complex domain to avoid triviality and connect with $\,\mathcal{PT}$-symmetric quantum mechanics.

Contribution

The work investigates lattice formulation of negative coupling $\,\phi^4$ theory in 4D, extending methods from lower dimensions and complex contour deformation techniques.

Findings

Potential to avoid triviality in 4D $\,\phi^4$ theory

Connection with $\,\mathcal{PT}$-symmetric quantum mechanics

Initial steps towards lattice implementation of negative coupling theories

Abstract

Triviality of $\phi^4$ theory in four dimensions can be avoided if the bare coupling constant is negative in the UV. Theories with negative coupling can be put on the lattice if the integration domain for $\phi(x)$ is contour-deformed from the real to the complex domain. In 0+1d (quantum mechanics), one can recover results from $\mathcal{PT}$-symmetric quantum mechanics in this way. In this work, I report on an attempt to put negative coupling $\phi^4$ theory in 4 dimensions on the lattice.