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A search for $B^{0}_{s}\to\tau^{\pm}\mu^{\mp}$ and$B^{0}\to\tau^{\pm}\mu^{\mp}$ decays is performed using data corresponding toan integrated luminosity of 3 fb$^{-1}$ of proton-proton collisions, recordedwith the LHCb detector in 2011 and 2012. For this search, the $\tau$ lepton isreconstructed in the $\tau^{-}\to\pi^{-}\pi^{+}\pi^{-}\nu_{\tau}$ channel. Nosignificant signal is observed. Assuming no contribution from$B^{0}\to\tau^{\pm}\mu^{\mp}$ decays, an upper limit is set on the$B^{0}_{s}\to\tau^{\pm}\mu^{\mp}$ branching fraction of $\mathcal{B}\left(B^{0}_{s}\to\tau^{\pm}\mu^{\mp}\right) < 4.2\times 10^{-5}$ at $95\%$confidence level. If instead no contribution from$B^{0}_{s}\to\tau^{\pm}\mu^{\mp}$ decays is assumed, a limit of$\mathcal{B}\left( B^{0}\to\tau^{\pm}\mu^{\mp}\right) < 1.4\times 10^{-5}$ isobtained at $95\%$ confidence level. These are the first limit on$\mathcal{B}\left( B^{0}_{s}\to\tau^{\pm}\mu^{\mp}\right)$ and the world's bestlimit on $\mathcal{B}\left( B^{0}\to\tau^{\pm}\mu^{\mp}\right)$.

Search for the lepton-flavour-violating decays $B^{0}_{s}\to\tau^{\pm}\mu^{\mp}$ and $B^{0}\to\tau^{\pm}\mu^{\mp}$