A search for the leptonflavour violating decays $B^0_s \rightarrow e^\pm \mu^\mp$ and $B^0 \rightarrow e^\pm \mu^\mp$ is performed based on a sample of protonproton collision data corresponding to an integrated luminosity of 3 fb$^{1}$, collected with the LHCb experiment at centreofmass energies of 7 and 8 TeV. The observed yields are consistent with the backgroundonly hypothesis. Upper limits on the branching fractions are determined to be $\mathcal{B}(B^0_s \rightarrow e^\pm \mu^\mp) < 5.4 (6.3) \times 10^{9}$ and $\mathcal{B}(B^0 \rightarrow e^\pm \mu^\mp) < 1.0 (1.3) \times 10^{9}$ at $90\% (95\%)$ confidence level, which are the strongest limits on these decays to date.
Expected distribution of the BDT response for $ B ^0_{( s )} \rightarrow e^\pm\mu^\mp$ decays with recovered bremsstrahlung photons obtained from the $ B ^0 \rightarrow K ^+ \pi ^ $ control channel. The total uncertainty is shown as a light grey band. Each bin is normalised to its width. 
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Invariant mass distributions of the two normalisation channels with fit functions superimposed: (left) $ B ^0 \rightarrow K ^+ \pi ^ $ and (right) $ B ^+ \rightarrow { J \mskip 3mu/\mskip 2mu\psi \mskip 2mu} K ^+ $ . Pull distributions are shown below each plot. 
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Distribution of the $ m_{e^{\pm}\mu^{\mp}}$ invariant mass of simulated $ B ^0_ s $ candidates with no (left) and one or more (right) recovered bremsstrahlung photons. The overlaid fit function is a modified Crystal Ball function with two tails on opposite sides. 
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Distributions of the invariant mass of the $ B ^0_{( s )} \rightarrow e^\pm\mu^\mp$ candidates, $ m_{e^{\pm}\mu^{\mp}}$ , divided into bins of BDT response and two bremsstrahlung categories (left) without and (right) with bremsstrahlung photons recovered. The result of the fit is overlaid and the different components are detailed. The edges of the range that was examined only after finalising the selection and fit procedure are delimited by gray dashed vertical lines. This region includes 90% of the potential signal candidates. Given the result obtained from the fit, the $ B ^0 \rightarrow e^\pm\mu^\mp$ component is not visible in the plots. 
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Results of the $\mathrm{CL_s}$ scan used to obtain the limit on (left) $\mathcal{B} ( B ^0 \rightarrow e^\pm\mu^\mp )$ and (right) $\mathcal{B} ( B ^0_ s \rightarrow e^\pm\mu^\mp )$. The backgroundonly expectation is shown by the dashed line and the $1\sigma$ and $2\sigma$ bands are shown as dark (green) and light (yellow) bands respectively. The observed limit is shown as the solid black line. 
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Animated gif made out of all figures. 
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Yields of normalisation channels obtained from fits to data. 
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Expected (assuming no signal) and observed upper limits for $\mathcal{B} ( B ^0_ s \rightarrow e^\pm\mu^\mp )$ and $\mathcal{B} ( B ^0 \rightarrow e^\pm\mu^\mp )$ at 95% (90%) CL. The upper limit on the $\mathcal{B} ( B ^0_ s \rightarrow e^\pm\mu^\mp )$ is evaluated under the assumption of pure heavy eigenstate contribution on the decay amplitude. 
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Created on 21 October 2019.