Exploring CP violation in $H \to \tau^+ \tau^- \gamma$

We propose a method of measuring the CP-odd part of the Yukawa interaction of Higgs boson and $\tau$ leptons by observing the forward-backward asymmetry in the decay $H \to \tau^+ \, \tau^- \, \gamma$. The source of such asymmetry is the interference of the CP-even loop-level contribution coming from $H \to Z \, \gamma \to \tau^+ \, \tau^- \, \gamma$ decay channel with the contribution from tree-level CP-odd Yukawa interaction. We find that the CP violating effect is maximum when the invariant mass of the $\tau^+ \, \tau^-$ pair is equal to the mass of the $Z$ boson. We propose and utilise various Dalitz plot asymmetries to quantify the maximal size of the asymmetry and perform Monte Carlo simulations to study the feasibility of measuring it in the high luminosity phase of the Large Hadron Collider (HL-LHC).