  A new sine observable, $R_{\Psi_2}(\Delta S)$, has been proposed to measure
the chiral magnetic effect (CME) in heavy-ion collisions; $\Delta S = \left
\langle \sin \varphi_+ \right \rangle - \left \langle \sin \varphi_- \right
\rangle$, where $\varphi_\pm$ are azimuthal angles of positively and negatively
charged particles relative to the reaction plane and averages are event-wise,
and $R_{\Psi_2}(\Delta S)$ is a normalized event probability distribution.
Preliminary STAR data reveal concave $R_{\Psi_2}(\Delta S)$ distributions in
200 GeV Au+Au collisions. Studies with a multiphase transport (AMPT) and
anomalous-viscous Fluid Dynamics (AVFD) models show concave $R_{\Psi_2}(\Delta
S)$ distributions for CME signals and convex ones for typical resonance
backgrounds. A recent hydrodynamic study, however, indicates concave shapes for
backgrounds as well. To better understand these results, we report a systematic
study of the elliptic flow ($v_{2}$) and transverse momentum ($p_{T}$)
dependences of resonance backgrounds with toy-model simulations and central
limit theorem (CLT) calculations. It is found that the concavity or convexity
of $R_{\Psi_2}(\Delta S)$ depends sensitively on the resonance $v_2$ (which
yields different numbers of decay $\pi^+\pi^-$ pairs in the in-plane and
out-of-plane directions) and $p_T$ (which affects the opening angle of the
decay $\pi^+\pi^-$ pair). Qualitatively, low $p_{T}$ resonances decay into
large opening-angle pairs and result in more `back-to-back' pairs out-of-plane,
mimicking a CME signal, or a concave $R_{\Psi_2}(\Delta S)$. Supplemental
studies of $R_{\Psi_3}(\Delta S)$ in terms of the triangular flow ($v_3$),
where only backgrounds exist but any CME would average to zero, are also
presented.
