Finite-Size Scaling of Net-Proton Cumulants in Heavy-Ion Collisions: Remarks on the Interpretation of a Recent Analysis

Abstract: Finite-size scaling (FSS) provides a framework for investigating the possible presence of a critical end point (CEP) in the QCD phase diagram using fluctuation observables measured in relativistic heavy-ion collisions. A recent analysis reported a FSS collapse of a susceptibility constructed from net-proton cumulants, which was interpreted as evidence for a CEP near $μ_B \approx 625$ MeV \cite{FSSpaper}. This note examines several aspects of that scaling construction. These include identifying the pseudorapidity acceptance window with the physical system size used in the finite-size scaling relations, assessing the influence of acceptance-driven multiplicity scaling on the susceptibility used for the scaling analysis, and treating thermodynamic scaling fields in the scaling variable. These considerations clarify several issues relevant for the consistent implementation and interpretation of finite-size scaling analyses in heavy-ion collision experiments.