The product of the $\Lambda_b^0$ ($\overline{B}^0$) differential production crosssection and the branching fraction of the decay $\Lambda_b^0\rightarrow J/\psi pK^$ ($\overline{B}^0\rightarrow J/\psi\overline{K}^*(892)^0$) is measured as a function of the beauty hadron transverse momentum, $p_{\rm T}$, and rapidity, $y$. The kinematic region of the measurements is $p_{\rm T}<20 {\rm GeV}/c$ and $2.0<y<4.5$. The measurements use a data sample corresponding to an integrated luminosity of $3 {\rm fb}^{1}$ collected by the LHCb detector in $pp$ collisions at centreofmass energies $\sqrt{s}=7 {\rm TeV}$ in 2011 and $\sqrt{s}=8 {\rm TeV}$ in 2012. Based on previous LHCb results of the fragmentation fraction ratio, $f_{\Lambda_B^0}/f_d$, the branching fraction of the decay $\Lambda_b^0\rightarrow J/\psi pK^$ is measured to be \begin{equation*} \mathcal{B}(\Lambda_b^0\rightarrow J/\psi pK^)= (3.17\pm0.04\pm0.07\pm0.34^{+0.45}_{0.28})\times10^{4}, \end{equation*} where the first uncertainty is statistical, the second is systematic, the third is due to the uncertainty on the branching fraction of the decay $\overline{B}^0\rightarrow J/\psi\overline{K}^*(892)^0$, and the fourth is due to the knowledge of $f_{\Lambda_b^0}/f_d$. The sum of the asymmetries in the production and decay between $\Lambda_b^0$ and $\overline{\Lambda}_b^0$ is also measured as a function of $p_{\rm T}$ and $y$. The previously published branching fraction of $\Lambda_b^0\rightarrow J/\psi p\pi^$, relative to that of $\Lambda_b^0\rightarrow J/\psi pK^$, is updated. The branching fractions of $\Lambda_b^0\rightarrow P_c^+(\rightarrow J/\psi p)K^$ are determined.
Fit to the (left) $ { J \mskip 3mu/\mskip 2mu\psi \mskip 2mu} p K ^ $ and (right) $ { J \mskip 3mu/\mskip 2mu\psi \mskip 2mu} \overline{ K }{} {}^{*0} $ invariant mass distributions with $ p_{\rm T} \in[6,7] {\mathrm{\,GeV\!/}c} $ and $y\in[3.0,3.5]$ for the 2012 data sample. The hatched (red) area represents the signals, the filled (green) area $ B ^0_ s \rightarrow { J \mskip 3mu/\mskip 2mu\psi \mskip 2mu} \overline{ K }{} {}^{*0} $, and the dashed (magenta) lines the combinatorial background. The dotdashed (black) lines indicate the doubly misidentified background (left) and partially reconstructed background (right). The solid (blue) lines represent the sum of the above components and the points with error bars show the data. 
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Products of production crosssections and branching fractions as functions of $p_{\rm T}$ in $y$ bins for (left) $\Lambda ^0_ b \rightarrow { J \mskip 3mu/\mskip 2mu\psi \mskip 2mu} p K ^ $ and (right) $\overline{ B }{} {}^0 \rightarrow { J \mskip 3mu/\mskip 2mu\psi \mskip 2mu} \overline{ K }{} {}^{*0} $ . The top (bottom) plots represent the 2011 (2012) sample. The error bars represent the total uncertainties. 
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Products of production crosssections and branching fractions as functions of (top) $p_{\rm T}$ or (bottom) $ y$ . The left (right) plots represent $\Lambda ^0_ b $ ( $\overline{ B }{} {}^0$ ) hadrons. The error bars indicate the statistical uncertainties and the hatched areas represent the total uncertainties. 
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Fit to the $\Lambda ^0_ b $ distribution with the Tsallis function. 
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Production ratios of (left) $\Lambda ^0_ b $ and (right) $\overline{ B }{} {}^0 $ at $8\mathrm{\,TeV} $ and $7\mathrm{\,TeV} $ as functions of the (top) $ p_{\rm T} $ and (bottom) $y$ of the $b$ hadron. The blue lines are FONLL predictions. The error bars represent uncorrelated uncertainties, while the hatched areas show the total uncertainties. Linear fits are also shown. 
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Ratio $ R_{\Lambda ^0_ b /\overline{ B }{} {}^0 } $ as a function of (left) $ p_{\rm T} $ and (right) $y$ for the 2011 and 2012 samples, where the error bars indicate statistical uncertainties and the hatched areas the total uncertainties. The red solid (blue dashed) line in the left plot represents the fit to the ratio $ f_{\Lambda ^0_ b }/f_d ( p_{\rm T} )$ from Ref. [8] for the 2011 (2012) data sample. 
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Asymmetries $ a_{\rm p+d}$ between $\Lambda ^0_ b $ and $\overline{\Lambda} {}^0_ b $ as functions of (left) $ p_{\rm T} $ and (right) $y$. The error bars indicate statistical uncertainties, and the hatched areas the total uncertainties. 
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Animated gif made out of all figures. 
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Summary of the systematic uncertainties (%) for the production crosssections of $\Lambda ^0_ b $ and $\overline{ B }{} {}^0 $. The large uncertainties affect the bins with very few candidates. 
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Summary of the absolute systematic uncertainties (%) for the asymmetry of $\Lambda ^0_ b $ and $\overline{\Lambda} {}^0_ b $. The large uncertainties affect the bins with very few candidates. 
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Products of $\Lambda ^0_ b $ production crosssections ( $\rm pb$ ) and the branching fraction $\cal B (\Lambda ^0_ b \rightarrow { J \mskip 3mu/\mskip 2mu\psi \mskip 2mu} p K ^ )$ in bins of $ p_{\rm T} $ and $y$ in the 2011 data sample. The first uncertainties are statistical and the second are systematic. 
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Products of $\Lambda ^0_ b $ production crosssections ( $\rm pb$ ) and the branching fraction $\cal B (\Lambda ^0_ b \rightarrow { J \mskip 3mu/\mskip 2mu\psi \mskip 2mu} p K ^ )$ in bins of $p_{\rm T}$ and $ y$ in the 2012 data sample. The first uncertainties are statistical and the second are systematic. 
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Products of $\overline{ B }{} {}^0 $ production crosssections ( $\rm pb$ ) and $\cal B (\overline{ B }{} {}^0 \rightarrow { J \mskip 3mu/\mskip 2mu\psi \mskip 2mu} \overline{ K }{} {}^{*0} )$ in bins of $ p_{\rm T} $ and $y$ in the 2011 data sample. The first uncertainties are statistical and the second are systematic. 
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Products of $\overline{ B }{} {}^0 $ production crosssections ($\rm pb $) and $\cal B (\overline{ B }{} {}^0 \rightarrow { J \mskip 3mu/\mskip 2mu\psi \mskip 2mu} \overline{ K }{} {}^{*0} )$ in bins of $ p_{\rm T} $ and $y$ in the 2012 data sample. The first uncertainties are statistical and the second are systematic. 
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Asymmetries $ a_{\rm p+d}$ $(\%)$ of $\Lambda ^0_ b $ and $\overline{\Lambda} {}^0_ b $ in bins of $ p_{\rm T} $ and $y$ for the 2011 and 2012 samples. The first uncertainties are statistical and the second are systematic. 
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Created on 09 December 2018.Citation count from INSPIRE on 18 December 2018.