The Lattice Cutoff for $\lambda\phi^4_4$ and $\lambda\phi^6_3$

TL;DR

This paper investigates the relationship between continuum cutoff and lattice spacing in scalar field theories $mbda\u03c6^4_4$ and $mbda\u03c6^6_3$ using perturbative analysis and lattice data comparison.

Contribution

It provides a perturbative method to determine the lattice cutoff in relation to the continuum for specific scalar field theories, including theoretical predictions for the cutoff.

Findings

Quantified the relation between continuum cutoff and lattice spacing for mbdac6^4_4

Quantified the relation for mbdac6^6_3 in 3 dimensions

Presented two theoretical predictions for the cutoff mbda a

Abstract

We analyze the critical line of $\lambda\phi^4_4$ perturbatively in the bare coupling $\lambda_0$, by setting the daisy-improved renormalized mass to zero. By comparing to lattice data, we can then quantify the relation between the continuum cutoff and the lattice spacing; for the 4-dimensional hypercubic lattice we find $(\Lambda a)_{C4} = 4.893$. We perform a similar analysis for $\lambda\phi^6_3$, and find in 3 dimensions $(\Lambda a)_{C3} = 4.67$. We present two theoretical predictions for $(\Lambda a)$. For small $\lambda_0$, both the critical line and the renormalized mass near criticality are easily and accurately calculated from the lattice input parameters.