Difference between signal and background of the chiral magnetic effect relative to spectator and participant planes in isobar collisions at $\sqrt{s_{_{\rm NN}}} = 200$ GeV

The search for the chiral magnetic effect (CME) in relativistic heavy-ion collisions helps us understand the $\mathcal{CP}$ symmetry breaking in strong interactions and the topological nature of the QCD vacuum. Since the background and signal of the CME have different correlations with the spectator and participant planes, a two-plane method has been proposed to extract the fraction of the CME signal inside the CME observable of $\Delta\gamma$ from the experimental measurements relative to the two planes. Using a multiphase transport model with different strengths of the CME, we reexamine the two-plane method in isobar collisions at $\sqrt{s_{_{\rm NN}}} = 200$ GeV. The ratios of the CME signals and the elliptic flow backgrounds relative to the two different planes are found to be different, which is inconsistent with the assumptions made in the current experimental measurements. This difference arises from the decorrelation effect of the chiral magnetic effect relative to the spectator and participant planes caused by final state interactions. Our finding suggests that the current experimental measurements may overestimate the fraction of the CME signal in the CME observable in the final state of relativistic heavy-ion collisions.