A search for the leptonflavour violating decays $B^0_s \to e^{\pm} \mu^{\mp}$ and $B^0 \to e^{\pm} \mu^{\mp}$ is performed with a data sample, corresponding to an integrated luminosity of 1.0 fb$^{1}$ of $pp$ collisions at $\sqrt{s} = 7$ TeV, collected by the LHCb experiment. The observed number of $B^0_s \to e^{\pm} \mu^{\mp}$ and $B^0 \to e^{\pm} \mu^{\mp}$ candidates is consistent with background expectations. Upper limits on the branching fractions of both decays are determined to be $BR(B^0_s \to e^{\pm} \mu^{\mp}) < 1.1 (1.4) \times 10^{8}$ and $BR (B^0 \to e^{\pm} \mu^{\mp}) < 2.8 (3.7) \times 10^{9}$ at 90 (95) confidence level (C.L.). These limits are a factor of twenty lower than those set by previous experiments. Lower bounds on the PatiSalam leptoquark masses are also calculated, $M_{\rm LQ} (B^0_s \to e^{\pm} \mu^{\mp}) > 107$ TeV/c$^2$ and $M_{\rm LQ} (B^0 \to e^{\pm} \mu^{\mp}) > 126$ TeV/c$^2$ at 95 C.L., and are a factor of two higher than the previous bounds.
$\textrm{CL}_{\textrm{s}}$ as a function of the assumed branching fraction for (left) $ B^0_s\to e^{\pm}\mu^{\mp}$ and (right) $ B^0\to e^{\pm}\mu^{\mp}$ decays. The dashed lines are the medians of the expected $\textrm{CL}_{\textrm{s}}$ distributions if background only was observed. The yellow (green) area covers, at a given branching fraction, 34%(47.5%) of the expected $\textrm{CL}_{\textrm{s}}$ distribution on each side of its median. The solid black curves are the observed $\textrm{CL}_{\textrm{s}}$ . The upper limits at 90 % (95 %) C.L. are indicated by the dotted (solid) vertical lines in blue for the expectation and in red for the observation. 
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Animated gif made out of all figures. 
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Expected background (bkg) from the fit to the data sidebands, and expected $ B^0_{(s)}\to h^+h^{'}\to e^+ \mu^$ events, compared to the number of observed events in the mass signal region, in bins of BDT response. 
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Expected (background only) and observed limits on the $ B^{0}_{(s)}\to e^{\pm} \mu^{\mp}$ branching fractions. 
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Created on 20 February 2021.