Semi-Analytical Solution of the $\phi^4$ Theory on an $F_4$ Lattice

TL;DR

This paper presents a semi-analytical non-perturbative solution to the $^4$ theory on an $F_4$ lattice, providing insights into the Higgs mass triviality bound and confirming recent Monte Carlo results.

Contribution

It introduces a semi-analytical approach using high temperature expansion on an $F_4$ lattice to study the $^4$ theory non-perturbatively, improving understanding of the Higgs mass bound.

Findings

Renormalized coupling remains below 2/3 of unitarity bound in the broken phase.

Upper bound for Higgs mass ratio $m_R/f_pi \u2264 2.46$ at $\u0394/m_R=2$.

Results agree with recent Monte Carlo data.

Abstract

Investigating the cutoff dependence of the Higgs mass triviality bound, the $\phi^4$ theory is formulated on an $F_4$ lattice which preserves Lorentz invariance to a higher degree than the commonly used hypercubic lattice. I solve this model non-perturbatively by evaluating the high temperature expansion through 13th order following the approach of L\"uscher and Weisz. The results are continued across the transition line into the broken phase by integrating the perturbative RG equations. In the broken phase, the renormalized coupling never exceeds 2/3 of the tree level unitarity bound when $\Lambda/m_R \geq 2$. The results confirm recent Monte Carlo data and I obtain as an upper bound for the Higgs mass $m_R/f_\pi \leq 2.46 \pm 0.02_{\rm HTE} \pm 0.08_{\rm PT}$ at $\Lambda/m_R=2$.