A Lower Bound on the Renormalized Nelson Model

We provide explicit lower bounds for the ground-state energy of the renormalized Nelson model in terms of the coupling constant $\alpha$ and the number of particles $N$, uniform in the meson mass and valid even in the massless case. In particular, for any number of particles $N$ and large enough $\alpha$ we provide a bound of the form $-C\alpha^2 N^3\log^2(\alpha N)$, where $C$ is an explicit positive numerical constant; and if $\alpha$ is sufficiently small, we give one of the form $-C\alpha^2 N^3\log^2 N$ for $N \geq 2$, and $-C\alpha^2$ for $N = 1$. Whereas it is known that the renormalized Hamiltonian of the Nelson model is bounded below (as realized by E. Nelson) and implicit lower bounds have been given elsewhere (as in a recent work by Gubinelli, Hiroshima, and L\"{o}rinczi), ours seem to be the first fully explicit lower bounds with a reasonable dependence on $\alpha$ and $N$. We emphasize that the logarithmic term in the bounds above is probably an artifact in our calculations, since one would expect that the ground-state energy should behave as $-C\alpha^2 N^3$ for large $N$ or $\alpha$, as in the polaron model of H. Fr\"{o}hlich.