Bounds for the cutoff, Euclidean $:{\Phi }^4:$ perturbation expansion

We study the unrenormalized perturbation expansion of the Euclidean, massive $\frac{\lambda :{\Phi }^4:}{4!}$ field theory in $d\ge 1$ space-time dimensions, with a volume cutoff, and with the free propagator regulated by an $\alpha$-parameter cutoff in case $d\ge 2$. In the formal expansion of the Schwinger $n$-point function, $S(x_1,\dots ,x_n)=\Sigma {l=0\mathrm{}}^{\infty }{(-\lambda )}^l{S}_l(x_1,\dots ,x_n)$, we show that $0\le S_l(x_1,\dots ,x_n) \le A^l{\mathrm{}(l!)\mathrm{}}^{-1\mathrm{}}\Pi {i=1\mathrm{}}^l[{[i+\frac{\mathrm{}(n-5\mathrm{})\mathrm{}}{4}]}^2\times c{S}_0(x_1,\dots ,x_n\left)\right]$. The constant $A$ diverges as the volume cutoff is removed, and, in $d\ge 2$ dimensions, as the ultraviolet cutoff is removed. We also give finite bounds for no volume cutoff, at the expense of lowering the mass in the free field and multiplying by an extra factor $l!$. We give analogous bounds for the connected $n$-point function in terms of the tree approximation. The method is combinatoric, once we establish $x$-space bounds on two basic diagrams. These follow from some properties of Bessel potentials of $\alpha$ cutoffs, which we believe to be new.