Scaling Properties of the $\Delta\gamma$ Correlator: Constraints on Background and CME-Sensitive Charge Separation in Heavy-Ion Collisions

The scaling properties of the $\Delta\gamma$ correlator, guided by calculations from the Anomalous Viscous Fluid Dynamics (AVFD) model, are used to investigate charge separation in $p$+Au, $d$+Au, Ru+Ru, Zr+Zr, and Au+Au collisions at $\sqrt{s_{\mathrm{NN}}}=200$~GeV, and in $p$+Pb and Pb+Pb collisions at $\sqrt{s_{\mathrm{NN}}}=5.02$ and $2.76$~TeV. The ratio $\Delta\gamma/v_2$, where $v_2$ is the elliptic-flow coefficient, exhibits a common approximate $1/N_{\rm ch}$ scaling behavior for $p$+Au, $d$+Au, $p$+Pb, and Pb+Pb collisions, establishing a common multiplicity-dilution baseline for background-driven charge correlations. In contrast, Ru+Ru, Zr+Zr, and Au+Au collisions show significant deviations from this scaling trend. These violations are qualitatively similar to those obtained in AVFD calculations with an input chiral magnetic effect (CME) signal and point to an additional charge-separation component beyond that expected from the observed background scaling. Quantitative estimates indicate that, in mid-central collisions, the corresponding CME-sensitive fraction of the measured $\Delta\gamma/v_2$ is approximately 27\% for Au+Au collisions and roughly a factor of two smaller for Ru+Ru and Zr+Zr collisions, which exhibit similar magnitudes within uncertainties. The extracted magnitudes imply an expected Ru+Ru--Zr+Zr signal difference of only $\sim1.3\%$, suggesting limited sensitivity of the $\Delta\gamma$ correlator to the small difference expected between the isobar signals.