We explore the sensitivity of the Higgs decay to four leptons, the so-called golden channel, to higher dimensional loop-induced couplings of the Higgs boson to $ZZ$, $Z\gamma$, and $\gamma\gamma$, allowing for general CP mixtures. The larger standard model tree level coupling $hZ^\mu Z_\mu$ is the dominant "background" for the loop induced couplings. However this large background interferes with the smaller loop induced couplings, enhancing the sensitivity. We perform a maximum likelihood analysis based on analytic expressions of the fully differential decay width for $h\to 4\ell$ ($4\ell \equiv 2e2\mu, 4e, 4\mu$) including all interference effects. We find that the spectral shapes induced by Higgs couplings to photons are particularly different than the $hZ^\mu Z_\mu$ background leading to enhanced sensitivity to these couplings. We show that even if the $h\to\gamma\gamma$ and $h\to 4\ell$ rates agree with that predicted by the Standard Model, the golden channel has the potential to probe both the CP nature as well as the overall sign of the Higgs coupling to photons well before the end of high-luminosity LHC running ($\sim$3 ab$^{-1}$).
