Background: The LHCb Collaboration has studied a number of kinematic correlations between $B$-hadron pairs through their subsequent decays to $J/psi$ pairs at 7 and 8 TeV for four minimum values of the $J/psi$ $p_T$. Purpose: In this work, these measurements are compared to calculations of $b bar b$ pairs and their hadronization and inclusive decays to $J/psi J/psi$ are compared to the same observables. Potential cold matter effects on the $b bar b$ pair observables are discussed to determine which are most likely to provide insights about the system and why. Methods: The calculations, employing the exclusive HVQMNR code, assume the same intrinsic $k_T$-broadening and fragmentation as in [R. Vogt, Phys. Rev. C {bf 98} (2018) 034907]. The pair distributions presented by LHCb are calculated in this approach, both for the parent $b bar b$ and the $J/psi J/psi$ pairs produced in their decay. The sensitivity of the results to the intrinsic $k_T$ broadening is shown. The theoretical uncertainties due to the $b$ quark mass and scale variations on both the initial $b bar b$ pairs and the resulting $J/psi$ pairs are also shown. Possible effects due to the presence of the nucleus are studied by increasing the size of the $k_T$ broadening and modification of the fragmentation parameter. Results: Good agreement with the LHCb data is found for all observables. The parent $b bar b$ distributions are more sensitive to the $k_T$ broadening than are the final-state $J/psi$ pairs. Conclusions: Next-to-leading order calculations with $k_T$ broadening, as in [R. Vogt, Phys. Rev. C {bf 98} (2018) 034907], can describe all correlated observables. Multiple measurements of correlated observables are sensitive to different nuclear effects which can help distinguish between them.
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