  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.
