Extracting Muon Momentum Scale Corrections for Hadron Collider Experiments

We present a simple method for the extraction of corrections for bias in the measurement of the momentum of muons in hadron collider experiments. Such bias can originate from a variety of sources such as detector misalignment, software reconstruction bias, and uncertainties in the magnetic field. The two step method uses the mean $<1/p^\mu_T>$ for muons from $Z\to \mu\mu$ decays to determine the momentum scale corrections in bins of charge, $\eta$ and $\phi$. In the second step, the corrections are tuned by using the average invariant mass $<M^Z_{\mu\mu}>$ of $Z\to \mu\mu$ events in the same bins of charge $\eta$ and $\phi$. The forward-backward asymmetry of $Z/\gamma^{*} \to \mu\mu$ pairs as a function of $\mu^+\mu^-$ mass, and the $\phi$ distribution of $Z$ bosons in the Collins-Soper frame are used to ascertain that the corrections remove the bias in the momentum measurements for positive versus negatively charged muons. By taking the sum and difference of the momentum scale corrections for positive and negative muons, we isolate additive corrections to $1/p^\mu_T$ that may originate from misalignments and multiplicative corrections that may originate from mis-modeling of the magnetic field $(\int \vec{B} \cdot d\vec{L})$. This method has recently been used in the CDF experiment at Fermilab and in the CMS experiment at the Large Hadron Collider at CERN