We investigate the LHC and Higgs factory prospects for measuring the $CP$ phase in the Higgs-$\tau \mathrm{\text{−}}\tau$ coupling. Currently this phase can be anywhere between 0° ($CP$ even) and 90° ($CP$ 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 $1{\mathrm{ab}}^{-1}$ data can discriminate $CP$ even versus $CP$ odd at the $5\sigma$ level. With $3{\mathrm{ab}}^{-1}$, the $CP$ phase should be measurable to an accuracy of $\sim 11°$. At an $e^{+}{e}^{-}$ Higgs factory, we project that a 250 GeV run with $1{\mathrm{ab}}^{-1}$ luminosity can measure the phase to $\sim 4.4°$ accuracy.

• Received 20 August 2013

DOI:https://doi.org/10.1103/PhysRevD.88.076009