  We revisit the hadronic production of the four-lepton final state, e^- e^+
\mu^- \mu^+, through the fusion of initial state gluons. This process is
mediated by loops of quarks and we provide first full analytic results for
helicity amplitudes that account for both the effects of the quark mass in the
loop and off-shell vector bosons. The analytic results have been implemented in
the Monte Carlo program MCFM and are both fast, and numerically stable in the
region of low Z transverse momentum. We use our results to study the
interference between Higgs-mediated and continuum production of four-lepton
final states, which is necessary in order to obtain accurate theoretical
predictions outside the Higgs resonance region. We have confirmed and extended
a recent analysis of Caola and Melnikov that proposes to use a measurement of
the off-shell region to constrain the total width of the Higgs boson. Using a
simple cut-and-count method, existing LHC data should bound the width at the
level of 25-45 times the Standard Model expectation. We investigate the power
of using a matrix element method to construct a kinematic discriminant to
sharpen the constraint. In our analysis the bound on the Higgs width is
improved by a factor of about 1.6 using a simple cut on the MEM discriminant,
compared to an invariant mass cut m_{4l} > 300 GeV.
