Optimisation of the smoothing of $b$-jet identification efficiency and mistag rate simulation-to-data scale factors in ATLAS

The performance of the $b$-jet identification algorithms, in use by the ATLAS Collaboration for the analysis of data from Run 2 of the Large Hadron Collider, is evaluated in the simulation and the efficiency with which these algorithms identify $b$-jets, or misidentify $c$- and light-flavour jets, is measured in collision data. Simulation-to-data SFs, corresponding to the ratio of the measured $b$-jet tagging efficiency to the $b$-jet tagging efficiency evaluated from the simulation, are derived in bins of the jet transverse momentum and applied in physics analyses to jets in the simulation. Before SFs are incorporated into physics analyses, they are smoothed using the method of local polynomial kernel regression. Smoothing is applied since it is assumed that the SFs are smoothly varying functions of the jet kinematics. The parameter associated with the smoothing method is optimised based on minimising Stein's unbiased risk estimator. This note presents the smoothing procedure as applied to the $b$-tagging efficiency and mistag rate SFs extracted from proton-proton collisions collected at a centre-of-mass energy $\sqrt{s}=13\ \textrm{TeV}$, corresponding to an integrated luminosity of $139\ \textrm{fb}^{-1}$.