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This article presents the first measurement of the ratio of branching fractions $\mathcal{B}({\ensuremath{\Lambda}}_{b}^{0}\ensuremath{\rightarrow}{\ensuremath{\Lambda}}_{c}^{+}{\ensuremath{\mu}}^{\ensuremath{-}}{\overline{\ensuremath{\nu}}}_{\ensuremath{\mu}})/\mathcal{B}({\ensuremath{\Lambda}}_{b}^{0}\ensuremath{\rightarrow}{\ensuremath{\Lambda}}_{c}^{+}{\ensuremath{\pi}}^{\ensuremath{-}})$. Measurements in two control samples using the same technique $\mathcal{B}({\overline{B}}^{0}\ensuremath{\rightarrow}{D}^{+}{\ensuremath{\mu}}^{\ensuremath{-}}{\overline{\ensuremath{\nu}}}_{\ensuremath{\mu}})/\mathcal{B}({\overline{B}}^{0}\ensuremath{\rightarrow}{D}^{+}{\ensuremath{\pi}}^{\ensuremath{-}})$ and $\mathcal{B}({\overline{B}}^{0}\ensuremath{\rightarrow}{D}^{*}(2010{)}^{+}{\ensuremath{\mu}}^{\ensuremath{-}}{\overline{\ensuremath{\nu}}}_{\ensuremath{\mu}})/\mathcal{B}({\overline{B}}^{0}\ensuremath{\rightarrow}{D}^{*}(2010{)}^{+}{\ensuremath{\pi}}^{\ensuremath{-}})$ are also reported. The analysis uses data from an integrated luminosity of approximately $172\text{ }\text{ }{\mathrm{pb}}^{\ensuremath{-}1}$ of $p\overline{p}$ collisions at $\sqrt{s}=1.96\text{ }\text{ }\mathrm{TeV}$, collected with the CDF II detector at the Fermilab Tevatron. The relative branching fractions are measured to be $\frac{\mathcal{B}({\ensuremath{\Lambda}}_{b}^{0}\ensuremath{\rightarrow}{\ensuremath{\Lambda}}_{c}^{+}{\ensuremath{\mu}}^{\ensuremath{-}}{\overline{\ensuremath{\nu}}}_{\ensuremath{\mu}})}{\mathcal{B}({\ensuremath{\Lambda}}_{b}^{0}\ensuremath{\rightarrow}{\ensuremath{\Lambda}}_{c}^{+}{\ensuremath{\pi}}^{\ensuremath{-}})}=16.6\ifmmode\pm\else\textpm\fi{}3.0(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}1.0(\mathrm{syst})\genfrac{}{}{0}{}{+2.6}{\ensuremath{-}3.4}(\mathrm{PDG})\ifmmode\pm\else\textpm\fi{}0.3(\mathrm{EBR})$, $\frac{\mathcal{B}({\overline{B}}^{0}\ensuremath{\rightarrow}{D}^{+}{\ensuremath{\mu}}^{\ensuremath{-}}{\overline{\ensuremath{\nu}}}_{\ensuremath{\mu}})}{\mathcal{B}({\overline{B}}^{0}\ensuremath{\rightarrow}{D}^{+}{\ensuremath{\pi}}^{\ensuremath{-}})}=\phantom{\rule{0ex}{0ex}}9.9\ifmmode\pm\else\textpm\fi{}1.0(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.6(\mathrm{syst})\ifmmode\pm\else\textpm\fi{}0.4(\mathrm{PDG})\ifmmode\pm\else\textpm\fi{}0.5(\mathrm{EBR})$, and $\frac{\mathcal{B}({\overline{B}}^{0}\ensuremath{\rightarrow}{D}^{*}(2010{)}^{+}{\ensuremath{\mu}}^{\ensuremath{-}}{\overline{\ensuremath{\nu}}}_{\ensuremath{\mu}})}{\mathcal{B}({\overline{B}}^{0}\ensuremath{\rightarrow}{D}^{*}(2010{)}^{+}{\ensuremath{\pi}}^{\ensuremath{-}})}=16.5\ifmmode\pm\else\textpm\fi{}2.3(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}\phantom{\rule{0ex}{0ex}}0.6(\mathrm{syst})\ifmmode\pm\else\textpm\fi{}0.5(\mathrm{PDG})\ifmmode\pm\else\textpm\fi{}0.8(\mathrm{EBR})$. The uncertainties are from statistics (stat), internal systematics (syst), world averages of measurements published by the Particle Data Group or subsidiary measurements in this analysis (PDG), and unmeasured branching fractions estimated from theory (EBR), respectively. This article also presents measurements of the branching fractions of four new ${\ensuremath{\Lambda}}_{b}^{0}$ semileptonic decays: ${\ensuremath{\Lambda}}_{b}^{0}\ensuremath{\rightarrow}{\ensuremath{\Lambda}}_{c}(2595{)}^{+}{\ensuremath{\mu}}^{\ensuremath{-}}{\overline{\ensuremath{\nu}}}_{\ensuremath{\mu}}$, ${\ensuremath{\Lambda}}_{b}^{0}\ensuremath{\rightarrow}{\ensuremath{\Lambda}}_{c}(2625{)}^{+}{\ensuremath{\mu}}^{\ensuremath{-}}{\overline{\ensuremath{\nu}}}_{\ensuremath{\mu}}$, ${\ensuremath{\Lambda}}_{b}^{0}\ensuremath{\rightarrow}{\ensuremath{\Sigma}}_{c}(2455{)}^{0}{\ensuremath{\pi}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}}{\overline{\ensuremath{\nu}}}_{\ensuremath{\mu}}$, and ${\ensuremath{\Lambda}}_{b}^{0}\ensuremath{\rightarrow}{\ensuremath{\Sigma}}_{c}(2455{)}^{++}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\mu}}^{\ensuremath{-}}{\overline{\ensuremath{\nu}}}_{\ensuremath{\mu}}$, relative to the branching fraction of the ${\ensuremath{\Lambda}}_{b}^{0}\ensuremath{\rightarrow}{\ensuremath{\Lambda}}_{c}^{+}{\ensuremath{\mu}}^{\ensuremath{-}}{\overline{\ensuremath{\nu}}}_{\ensuremath{\mu}}$ decay. Finally, the transverse-momentum distribution of ${\ensuremath{\Lambda}}_{b}^{0}$ baryons produced in $p\overline{p}$ collisions is measured and found to be significantly different from that of ${\overline{B}}^{0}$ mesons, which results in a modification in the production cross-section ratio ${\ensuremath{\sigma}}_{{\ensuremath{\Lambda}}_{b}^{0}}/{\ensuremath{\sigma}}_{{\overline{B}}^{0}}$ with respect to the CDF I measurement.

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