A search for the lepton-flavour 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 Pati-Salam 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.
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