Methods for the Study of Transverse Momentum Differential Correlations

Abstract

We introduce and compare three differential correlation functions for the study of transverse momentum correlation in $p+p$ and $A+A$ collisions. These consist of {\it inclusive}, {\it event-wise} and a differential version of the correlation measure $\tilde C$ introduced by Gavin \cite{Gavin} for experimental study of the viscosity per unit entropy of the matter produced in $A+A$ collisions. We study the quantitative difference between the three observables on the basis of PYTHIA simulations of $p+p$ collisions and $A+A$ collisions consisting of an arbitrary superposition of $p+p$ collision events at $\sqrt{s} = $200 GeV. We observe that {\it inclusive} and {\it event-wise} correlation functions are remarkably identical to each other where as the observable $\tilde C$ differs from the two. We study the robustness and efficiency dependencies of these observables based on truncated Taylor expansions in efficiency in $p+p$ collisions and on the basis of Monte Carlo simulation using an adhoc detector efficiency parameterization. We find that all the three observables are essentially independent of detector efficiency. We additionally study the scaling of the correlation measures and find all the observables exhibit an approximate $1/N$ dependence of the number of participants ({\it N}) in $A+A$ collisions. Finally, we study the impact of flow-like anisotropy on the {\it inclusive} correlation function and find flow imparts azimuthal modulations similar to those observed with two-particle densities.