Constraining Quark and Gluon Momentum Fractions Accessed by Observables in Polarized pp Collisions at PHENIX and sPHENIX

Observations of transverse single-spin asymmetries (TSSAs) in polarized proton-proton collisions at the Relativistic Heavy Ion Collider (RHIC) have proven useful for probing quark and gluon dynamics inside the proton. Such observations are made in proton collisions in the Pioneering High Energy Nuclear Interaction Experiment (PHENIX) and the upcoming sPHENIX experiment at RHIC. Individual quarks and gluons, collectively known as partons, carry some fraction of the proton momentum, denoted x. Parton distribution functions (PDFs), at leading order, describe the probability of finding a given quark or gluon at a given x. Analyses of TSSAs for direct photon, eta meson, and open heavy-flavor hadron production, as well as open heavy-flavor decays, provide access to spin-dependent quark and gluon PDFs. A Monte Carlo study of these events is done using the PYTHIA event generator. After the appropriate kinematic cuts are applied, final state particles are traced back to their quark or gluon ancestors, and their respective momentum fractions are plotted. This procedure therefore tells us what range of momentum fractions and mix of partonic flavors can be accessed by each existing or planned measurement.