We compute an array of Standard Model Higgs boson (\hsm) signals and
backgrounds for a possible upgrade of the Tevatron to E_{\rm cm}=4\tev.
Taking \mt\geq 140\gev, and assuming a total accumulated luminosity of
L=30\fbi, we find that a Standard Model Higgs boson with \mhsm\lsim 110\gev
could almost certainly be detected using the \wpm\hsm\rta l\nu b\anti b mode.
A Higgs boson with mass between \sim 120\gev and \sim 140\gev or above
\sim 230-250\gev almost certainly would not be seen. A Higgs boson with
\mhsm\sim 150\gev or 200\lsim\mhsm\lsim 230-250\gev has a decent chance of
being detected in the ZZ\rta 4l mode. There would also be some possibility of
discovering the \hsm in the WW\rta l\nu jj mode for 150\lsim\mhsm\lsim
200\gev. Finally, hints of an event excess in the WW\rta ll \nu\nu mode due
to the \hsm might emerge for 140\lsim\mhsm\lsim 180\gev. Given the
difficult nature of the Higgs boson signals for \mhsm values beyond the reach
of LEP-200, and the discontinuous \mhsm range that could potentially be
probed, justification of an upgrade of the Tevatron to 4\tev on the basis of
its potential for Standard Model Higgs boson discovery would seem
inappropriate.Comment: 21 pages; requires phyzzx.tex and tables.tex; full postscript file
including embedded figures available via anonymous ftp at ucdhep.ucdavis.edu
as [anonymous.gunion]4tev.ps, preprint UCD-94-1