Measuring CP Violation in $h \to \tau^+ \tau^-$ at Colliders

We investigate the LHC and Higgs Factory prospects for measuring the CP phase in the Higgs-tau-tau coupling. Currently this phase can be anywhere between 0 degrees (CP even) and 90 degrees (CP odd). A new, ideal observable is identified from an analytic calculation for the $\tau^\pm \to \rho^\pm\nu \to \pi^\pm \pi^0 \nu$ channel. It is demonstrated to have promising sensitivity at the LHC and superior sensitivity at the ILC compared to previous proposals. Our observable requires the reconstruction of the internal substructure of decaying taus but does not rely on measuring the impact parameter of tau decays. It is the first proposal for such a measurement at the LHC. For the 14 TeV LHC, we estimate that about 1 ab$^{-1}$ data can discriminate CP-even versus CP-odd at the $5\sigma$ level. With 3 ab$^{-1}$, the CP phase should be measurable to an accuracy of $\sim 11$ degrees. At an $e^+e^-$ Higgs Factory, we project that a 250 GeV run with 1 ab$^{-1}$ luminosity can measure the phase to $\sim 4.4$ degrees accuracy.