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.