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

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.