Searches for the lepton flavour violating decay $\tau^\rightarrow \mu^\mu^+\mu^$ and the lepton flavour and baryon number violating decays $\tau^\rightarrow \bar{p}\mu^+\mu^$ and $\tau^\rightarrow p\mu^\mu^$ have been carried out using protonproton collision data, corresponding to an integrated luminosity of 1.0 fb$^{1}$, taken by the LHCb experiment at $\sqrt{s} = 7$ TeV. No evidence has been found for any signal, and limits have been set at 90% confidence level on the branching fractions: $\cal B(\tau^\rightarrow \mu^\mu^+\mu^) < 8.0 \times 10^{8}$, $\cal B(\tau^\rightarrow \bar{p}\mu^+\mu^) < 3.3 \times 10^{7}$ and $\cal B(\tau^\rightarrow p\mu^\mu^) < 4.4 \times 10^{7}$. The results for the $\tau^\rightarrow \bar{p}\mu^+\mu^$ and $\tau^\rightarrow p\mu^\mu^$ decay modes represent the first direct experimental limits on these channels.
Distribution of (a) $ {\rm \mathcal{M}_{3body}}$ and (b) $ {\rm \mathcal{M}_{PID}}$ for $\tau^\rightarrow \mu^\mu^+\mu^$ where the binning corresponds to that used in the limit calculation. The short dashed (red) lines show the response of the data sidebands, whilst the long dashed (blue) and solid (black) lines show the response of simulated signal events before and after calibration. Note that in both cases the lowest likelihood bin is later excluded from the analysis. 
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Invariant mass distribution of $\phi(\mu^+\mu^)\pi^$ after (a) the $\tau^\rightarrow \mu^\mu^+\mu^$ selection and (b) the $\tau\rightarrow p\mu\mu$ selection and PID cuts. The solid (blue) lines show the overall fits, the long dashed (green) and short dashed (red) lines show the two Gaussian components of the signal and the dot dashed (black) lines show the backgrounds. 
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Invariant mass distributions and fits to the mass sidebands in data for (a) $\mu^+\mu^\mu^$ candidates in the four merged bins that contain the highest signal probabilities, (b) ${\bar p}\mu^+\mu^$ candidates in the two merged bins with the highest signal probabilities, and (c) $p\mu^\mu^$ candidates in the two merged bins with the highest signal probabilities. 
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Distribution of $\textrm{CL}_{\textrm{s}}$ values as functions of the assumed branching fractions, under the hypothesis to observe background events only, for (a) $\tau^\rightarrow \mu^\mu^+\mu^$ , (b) $\tau^\rightarrow \bar{p}\mu^+\mu^$ and (c) $\tau^\rightarrow p\mu^\mu^$ . The dashed lines indicate the expected curves and the solid lines the observed ones. The light (yellow) and dark (green) bands cover the regions of 68% and 95% confidence for the expected limits. 
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Animated gif made out of all figures. 
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Terms entering in the normalisation factor $\alpha$ for $\tau^\rightarrow \mu^\mu^+\mu^$ , $\tau^\rightarrow \bar{p}\mu^+\mu^$ and $\tau^\rightarrow p\mu^\mu^$ , and their combined statistical and systematic uncertainties. 
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Expected background candidate yields, with their systematic uncertainties, and observed candidate yields within the $\tau$ signal window in the different likelihood bins for the $\tau^\rightarrow \mu^\mu^+\mu^$ analysis. The likelihood values for $ {\rm \mathcal{M}_{PID}}$ range from $0$ (most backgroundlike) to $+1$ (most signallike), while those for $ {\rm \mathcal{M}_{3body}}$ range from $1$ (most backgroundlike) to $+1$ (most signallike). The lowest likelihood bins have been excluded from the analysis. 
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Expected background candidate yields, with their systematic uncertainties, and observed candidate yields within the $\tau$ mass window in the different likelihood bins for the $\tau\rightarrow p\mu\mu$ analysis. The likelihood values for $ {\rm \mathcal{M}_{3body}}$ range from $1$ (most backgroundlike) to $+1$ (most signallike). The lowest likelihood bin has been excluded from the analysis. 
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Supplementary material full pdf 
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This zip file contains supplementary material for the publication LHCbPAPER2013014. The files included are: Supplementary.pdf: The appendix from the paper, including the figures with captions and some brief explanation. *pdf, *eps, *png: The figures 
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Created on 17 August 2019.Citation count from INSPIRE on 20 August 2019.