Five loop renormalization of $\phi^3$ theory with applications to the Lee-Yang edge singularity and percolation theory

TL;DR

This paper computes five-loop renormalization group functions for $\,\phi^3$ theory using graphical functions, deriving critical exponents for Lee-Yang and percolation models, and providing improved estimates in multiple dimensions.

Contribution

It extends the graphical functions method to five loops in $\,\phi^3$ theory, enabling precise critical exponent calculations for related models.

Findings

Five-loop $eta$ and anomalous dimensions computed.

Derived $\,\varepsilon$ expansions up to $\,\varepsilon^5$ for critical exponents.

Resummation yields improved estimates in 3, 4, and 5 dimensions.

Abstract

We apply the method of graphical functions that was recently extended to six dimensions for scalar theories, to $\phi^3$ theory and compute the $\beta$ function, the wave function anomalous dimension as well as the mass anomalous dimension in the $\overline{\mbox{MS}}$ scheme to five loops. From the results we derive the corresponding renormalization group functions for the Lee-Yang edge singularity problem and percolation theory. After determining the $\varepsilon$ expansions of the respective critical exponents to $\mathcal{O}(\varepsilon^5)$ we apply recent resummation technology to obtain improved exponent estimates in 3, 4 and 5 dimensions. These compare favourably with estimates from fixed dimension numerical techniques and refine the four loop results. To assist with this comparison we collated a substantial amount of data from numerical techniques which are included in tables for each exponent.