  We calculate the triviality bound on the Higgs mass in scalar field theory
models whose global symmetry group $SU(2)_L \times SU(2)_{\rm custodial}
\approx O(4)$ has been replaced by $O(N)$ and $N$ has been taken to infinity.
Limits on observable cutoff effects at four percent in several regularized
models with tunable couplings in the bare action yield triviality bounds
displaying a large degree of universality. Extrapolating from $N=\infty$ to
$N=4$ we conservatively estimate that a Higgs particle with mass up to
$0.750~TeV$ and width up to $0.290~TeV$ is realizable without large cutoff
effects, indicating that strong scalar self interactions in the standard model
are not ruled out. We also present preliminary numerical results of the
physical $N=4$ case for the $F_4$ lattice that are in agreement with the large
$N$ expectations. Note: The full ps file is also available via anonymous ftp to
ftp.scri.fsu.edu. To get the ps file, ftp to this address and use for username
"anonymous" and for password your name. The file is in the directory pub/vranas
(to go to that directory type: cd pub/vranas) and is called lat92_proc.ps (to
get it type: get lat92_proc.ps)
