Searches for lepton-flavour violating decays of the Higgs boson into $e \tau$ and $\mu \tau$ in $\sqrt{s}=13~\mathrm{TeV}$ $pp$ collisions with the ATLAS detector

Direct searches for lepton-flavour-violating decays of the Higgs boson into $e \tau$ and $\mu \tau$ final states with the ATLAS detector at the LHC with Run 2 data are presented. Both leptonically and hadronically decaying $\tau$-leptons are included and two different background estimation techniques are employed: a MC-template method, based on data-corrected simulation samples, and a data-driven method, based on exploiting the symmetry between electrons and muons in the Standard Model backgrounds. Observed (Expected) upper limits are set on the branching ratios at 95 \% confidence level, $\mathcal{B}(H \rightarrow e \tau)<0.20 \%~(0.12 \%)$ and $\mathcal{B}(H \rightarrow \mu \tau)<0.18\%~(0.09 \%)$, and a best-fit branching ratio difference, $\mathcal{B}(H \rightarrow \mu \tau) - \mathcal{B}(H \rightarrow e \tau)$, of $0.25 \pm 0.10 \%$ is found in the channel where the $\tau$-lepton decays to leptons, compatible with a value of zero within $2.5 \sigma$.