The first observation of the decays $Lambda_b^0 to chi_{c1} p K^-$ and $Lambda_b^0 to chi_{c2} p K^-$ is reported using a data sample corresponding to an integrated luminosity of $3.0$ fb$^{-1}$, collected by the LHCb experiment in $pp$ collisions at centre-of-mass energies of 7 and 8 TeV. The following ratios of branching fractions are measured begin{eqnarray*} frac{{cal B}(Lambda_b^0 to chi_{c1} p K^-)}{{cal B}(Lambda_b^0 to J/psi p K^-)} = 0.242 pm 0.014 pm 0.013 pm 0.009, frac{{cal B}(Lambda_b^0 to chi_{c2} p K^-)}{{cal B}(Lambda_b^0 to J/psi p K^-)} = 0.248 pm 0.020 pm 0.014 pm 0.009, frac{{cal B}(Lambda_b^0 to chi_{c2} p K^-)}{{cal B}(Lambda_b^0 to chi_{c1} p K^-)} = 1.02 pm 0.10 pm 0.02 pm 0.05, end{eqnarray*} where the first uncertainty is statistical, the second systematic and the third due to the uncertainty on the branching fractions of the $chi_{c1}to J/psigamma$ and $chi_{c2} to J/psigamma$ decays. Using both decay modes, the mass of the $Lambda_b^0$ baryon is also measured to be $m_{Lambda_b^0} = 5619.44 pm 0.28 pm 0.26$ MeV/$c^2$, where the first and second uncertainties are statistical and systematic, respectively.
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