This article presents differential measurements of the asymmetry between $Lambda_b^0$ and $overline{Lambda}_b^0$ baryon production rates in proton-proton collisions at centre-of-mass energies of $sqrt{s}=7$ and $8,textrm{TeV}$ collected with the LHCb experiment, corresponding to an integrated luminosity of $3,textrm{fb}^{-1}$. The $Lambda_b^0$ baryons are reconstructed through the inclusive semileptonic decay $Lambda_b^0rightarrowLambda_c^+mu^-overline{ u}_{mu}X$. The production asymmetry is measured both in intervals of rapidity in the range $2.15<y<4.10$ and transverse momentum in $2<p_T<27,textrm{GeV}/c$. The results are found to be incompatible with symmetric production with a significance of 5.8 standard deviations for both $sqrt{s}=7$ and $8,textrm{TeV}$ data, assuming no $C!P$ violation in the decay. There is evidence for a trend as a function of rapidity with a significance of 4 standard deviations. Comparisons to predictions from hadronisation models in PYTHIA and heavy-quark recombination are provided. This result constitutes the first observation of a particle-antiparticle asymmetry in $b$-hadron production at LHC energies.
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