Search for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>C</mml:mi><mml:mi>P</mml:mi></mml:math> violation in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msubsup><mml:mi mathvariant="normal">Ξ</mml:mi><mml:mi>b</mml:mi><mml:mo>−</mml:mo></mml:msubsup><mml:mo stretchy="false">→</mml:mo><mml:mi>p</mml:mi><mml:msup><mml:mi>K</mml:mi><mml:mo>−</mml:mo></mml:msup><mml:msup><mml:mi>K</mml:mi><mml:mo>−</mml:mo></mml:msup></mml:math> decays

A search for $CP$ violation in charmless three-body ${\mathrm{\ensuremath{\Xi}}}_{b}^{\ensuremath{-}}\ensuremath{\rightarrow}p{K}^{\ensuremath{-}}{K}^{\ensuremath{-}}$ decays is performed using $pp$ collision data recorded with the LHCb detector, corresponding to integrated luminosities of $1\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ at a center-of-mass energy $\sqrt{s}=7\text{ }\text{ }\mathrm{TeV}$, $2\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ at $\sqrt{s}=8\text{ }\text{ }\mathrm{TeV}$ and $2\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ at $\sqrt{s}=13\text{ }\text{ }\mathrm{TeV}$. A good description of the phase-space distribution is obtained with an amplitude model containing contributions from $\mathrm{\ensuremath{\Sigma}}(1385)$, $\mathrm{\ensuremath{\Lambda}}(1405)$, $\mathrm{\ensuremath{\Lambda}}(1520)$, $\mathrm{\ensuremath{\Lambda}}(1670)$, $\mathrm{\ensuremath{\Sigma}}(1775)$ and $\mathrm{\ensuremath{\Sigma}}(1915)$ resonances. The model allows for $CP$ -violation effects, which are found to be consistent with zero. The branching fractions of ${\mathrm{\ensuremath{\Xi}}}_{b}^{\ensuremath{-}}\ensuremath{\rightarrow}\mathrm{\ensuremath{\Sigma}}(1385){K}^{\ensuremath{-}}$, ${\mathrm{\ensuremath{\Xi}}}_{b}^{\ensuremath{-}}\ensuremath{\rightarrow}\mathrm{\ensuremath{\Lambda}}(1405){K}^{\ensuremath{-}}$, ${\mathrm{\ensuremath{\Xi}}}_{b}^{\ensuremath{-}}\ensuremath{\rightarrow}\mathrm{\ensuremath{\Lambda}}(1520){K}^{\ensuremath{-}}$, ${\mathrm{\ensuremath{\Xi}}}_{b}^{\ensuremath{-}}\ensuremath{\rightarrow}\mathrm{\ensuremath{\Lambda}}(1670){K}^{\ensuremath{-}}$, ${\mathrm{\ensuremath{\Xi}}}_{b}^{\ensuremath{-}}\ensuremath{\rightarrow}\mathrm{\ensuremath{\Sigma}}(1775){K}^{\ensuremath{-}}$ and ${\mathrm{\ensuremath{\Xi}}}_{b}^{\ensuremath{-}}\ensuremath{\rightarrow}\mathrm{\ensuremath{\Sigma}}(1915){K}^{\ensuremath{-}}$ decays are also reported. In addition, an upper limit is placed on the product of ratios of ${\mathrm{\ensuremath{\Omega}}}_{b}^{\ensuremath{-}}$ and ${\mathrm{\ensuremath{\Xi}}}_{b}^{\ensuremath{-}}$ fragmentation fractions and the ${\mathrm{\ensuremath{\Omega}}}_{b}^{\ensuremath{-}}\ensuremath{\rightarrow}p{K}^{\ensuremath{-}}{K}^{\ensuremath{-}}$ and ${\mathrm{\ensuremath{\Xi}}}_{b}^{\ensuremath{-}}\ensuremath{\rightarrow}p{K}^{\ensuremath{-}}{K}^{\ensuremath{-}}$ branching fractions.