A search for the leptonflavour violating decays $B^+ \to K^+ {\mu}^{\pm} e^{\mp}$ is performed using a sample of protonproton collision data, collected with the LHCb experiment at centreofmass energies of $7$ and $8 {\rm TeV}$ and corresponding to an integrated luminosity of 3$\rm fb^{1}$. No significant signal is observed, and upper limits on the branching fractions are set as $\mathcal{B}(B^+ \to K^+ {\mu}^ e^+) < 7.0 (9.5) \times 10^{9}$ and $\mathcal{B}(B^+ \to K^+ {\mu}^+ e^) < 6.4 (8.8) \times 10^{9}$ at 90 (95) confidence level. The results improve the current best limits on these decays by more than one order of magnitude.
Efficiency of (left) $ B ^+ \rightarrow K ^+ \mu ^ e ^+ $ and (right) $ B ^+ \rightarrow K ^+ \mu ^+ e ^ $ as function of the squared invariant masses $m^2_{K^+e^\pm}$ and $m^2_{K^+\mu^\mp}$. The variation of efficiency across the Dalitz plane is due to applied vetoes. The efficiencies are given in per mille. 
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Invariantmass distributions of the (left) $ B ^+ \rightarrow K ^+ \mu ^ e ^+ $ and (right) $ B ^+ \rightarrow K ^+ \mu ^+ e ^ $ candidates obtained on the combined data sets recorded in 2011 and 2012 with background only fit functions (blue continuous line) and the signal model normalised to 10 candidates (red dashed line) superimposed. The signal window is indicated with grey dotted lines. Difference between the two distributions arises from the effect of the $m(K^+\ell^)$ requirement. 
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Upper limits on the branching fractions of (left) $ B ^+ \rightarrow K ^+ \mu ^ e ^+ $ and (right) $ B ^+ \rightarrow K ^+ \mu ^+ e ^ $ decays obtained on the combined data sets recorded in 2011 and 2012. The red solid line (black solid line with data points) corresponds to the distribution of the expected (observed) upper limits, and the light blue (dark blue) band contains the $1\sigma$ ($2\sigma$) uncertainties. 
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Invariantmass distributions of the $ B ^+ \rightarrow K ^+ { J \mskip 3mu/\mskip 2mu\psi \mskip 2mu} ( \rightarrow \mu ^+ \mu ^ )$ normalisation channel for (left) 2011 data and (right) 2012 data. 
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Animated gif made out of all figures. 
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Normalisation factor $\alpha$ for $ B ^+ \rightarrow K ^+ \mu ^ e ^+ $ and $ B ^+ \rightarrow K ^+ \mu ^+ e ^ $ final states. The ratio $\alpha/\mathcal{B}( B ^+ \rightarrow K ^+ { J \mskip 3mu/\mskip 2mu\psi \mskip 2mu} ( \rightarrow \mu ^+ \mu ^ ) )$ is independent of external inputs. 
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Upper limit on the branching fraction of $ B ^+ \rightarrow K ^+ \mu ^ e ^+ $ and $ B ^+ \rightarrow K ^+ \mu ^+ e ^ $ decays obtained on the combined data sets recorded in 2011 and 2012 for confidence levels of $90\%$ and $95\%$. 
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Summary of systematic uncertainties. 
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Created on 20 February 2021.