Lepton Flavour Violation in Tau Decays: Results and Prospects at the LHC

In this report, searches for the lepton flavour violating decays $\tau^-\to\mu^-\mu^+\mu^-$, $Z^0\to\tau^{\pm}\mu^{\mp}$ and $D^0\to e^{\pm}\mu^{\mp}$ are presented, using proton--proton collision data recorded by the LHC experiments during 2011 and 2012. The ATLAS experiment puts an upper limit on $\mathcal{B}(\tau^-\to\mu^-\mu^+\mu^-) < 3.76 \times 10^{-7}$ at $90\%$ confidence level, and on $\mathcal{B}(Z^0\to\tau^{\pm}\mu^{\mp}) < 1.69 \times 10^{-5}$ at $95\%$ confidence level, based on a data sample corresponding to an integrated luminosity of $20.3\:fb^{-1}$. The LHCb experiment puts an upper limit on $\mathcal{B}(\tau^-\to\mu^-\mu^+\mu^-) < 4.6 \times 10^{-8}$ and on $\mathcal{B}(D^0\to e^{\pm}\mu^{\mp}) < 1.3 \times 10^{-8}$, both at $90\%$ confidence level, based on a data sample corresponding to an integrated luminosity of $3.0\:fb^{-1}$.