  The correlation of on- and off-shell Higgs boson production at the LHC in $gg
\to h^* \to ZZ$ to bound the Higgs width, under specific model-dependent
assumptions, has recently received a lot of attention. As off-shell cross
section measurements in this channel suffer from a small signal yield, large
backgrounds, and theoretical uncertainties, we propose an alternative
complementary constraint which is only possible through the combination of LEP
and LHC measurements. Previous precision electroweak measurements at LEP allow
for the determination of indirect constraints on Higgs couplings to vector
bosons by considering one-loop processes involving virtual Higgs exchange. As
the Higgs is off-shell in these diagrams we venture that LEP can be interpreted
as an off-shell `Higgs Factory'. By combining these LEP constraints with
current LHC 8 TeV Higgs measurements a stronger limit on the Higgs width can be
achieved than with LHC data alone for models with rescaled Higgs couplings.
Looking to the future, avoiding ambiguities arising due to new physics
modifications of the $hGG$ coupling, a theoretically more robust constraint can
be achieved by correlating LEP measurements with WBF Higgs production followed
by Higgs decays to $WW$ and $ZZ$. This method for indirectly constraining the
Higgs width is very effective for specific BSM scenarios and is highly
complementary to other proposed methods. The limits we obtain particularly
highlight the power of a concrete LEP+LHC combination, not only limited to
Higgs width measurements.
