  The Higgs decay $h\to 4\ell$ has played an important role in discovering the
Higgs and measuring its mass thanks to low background and excellent resolution.
Current cuts in this channel have been optimized for Higgs discovery via the
dominant tree level $ZZ$ contribution arising from electroweak symmetry
breaking. Going forward, one of the primary objectives of this sensitive
channel will be to probe other Higgs couplings and search for new physics on
top of the tree level $ZZ$ `background'. Thanks to interference between these
small couplings and the large tree level contribution to $ZZ$, the $h\to 4\ell$
decay is uniquely capable of probing the magnitude \emph{and} CP phases of the
Higgs couplings to $\gamma\gamma$ and $Z\gamma$ as well as, to a lesser extent,
$ZZ$ couplings arising from higher dimensional operators. With this in mind we
examine how much relaxing current cuts can enhance the sensitivity while also
accounting for the dominant non-Higgs continuum $q\bar{q}\to4\ell$ background.
We find the largest enhancement in sensitivity for the $hZ\gamma$ couplings
($\gtrsim 100\%$) followed by $h\gamma\gamma$ ($\gtrsim 40\%$) and less so for
the higher dimensional $hZZ$ couplings (a few percent). With these
enhancements, we show that couplings of order Standard Model values for
$h\gamma\gamma$ may optimistically be probed by end of Run-II at the LHC while
for $hZ\gamma$ perhaps towards the end of a high luminosity LHC. Thus an
appropriately optimized $h\to4\ell$ analysis can complement direct decays of
the Higgs to on-shell $\gamma\gamma$ and $Z\gamma$ pairs giving a unique
opportunity to directly access the CP properties of these couplings.

High Energy Physics - Phenomenology;High Energy Physics - Experiment