Responses of the chiral-magnetic-effect–sensitive sine observable to resonance backgrounds in heavy-ion collisions

A new sine observable, $R_{{\mathrm{\Psi }}_2}\left(\mathrm{\Delta }S\right)$, has been proposed to measure the chiral magnetic effect (CME) in heavy-ion collisions; $\mathrm{\Delta }S=〈sin{\phi }_{+}〉-〈sin{\phi }_{-}〉$, where ${\phi }_{\pm }$ are azimuthal angles of positively and negatively charged particles relative to the reaction plane and averages are event-wise, and $R_{{\mathrm{\Psi }}_2}\left(\mathrm{\Delta }S\right)$ is a normalized event probability distribution. Preliminary STAR data reveal concave $R_{{\mathrm{\Psi }}_2}\left(\mathrm{\Delta }S\right)$ distributions in 200 GeV Au+Au collisions. Studies with a multiphase transport (AMPT) and anomalous-viscous fluid dynamics (AVFD) models show concave $R_{{\mathrm{\Psi }}_2}\left(\mathrm{\Delta }S\right)$ 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 $R_{{\mathrm{\Psi }}_2}\left(\mathrm{\Delta }S\right)$ 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 $R_{{\mathrm{\Psi }}_2}\left(\mathrm{\Delta }S\right)$. Supplemental studies of $R_{{\mathrm{\Psi }}_3}\left(\mathrm{\Delta }S\right)$ in terms of the triangular flow ($v_3$), where only backgrounds exist but any CME would average to zero, are also presented.

Figure 1

Observable distributions for various values of $v_{2,\rho }$ (with $v_{2,\pi }$ fixed to its default distribution). Here, $v_{2,\rho }^{\mathrm{def}}$ is the default distribution of $v_{2,\rho }$ obtained from data [36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46].

Figure 2

$R_{{\mathrm{\Psi }}_2}\left(\mathrm{\Delta }S\right)$ for various values of $v_{2,\pi }$ (with $v_{2,\rho }$ fixed to 0). Here, $v_{2,\pi }^{\mathrm{def}}$ is the default distribution of $v_{2,\pi }$ obtained from data [36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46].

Figure 3

Observable distributions for various values of $p_{T,\rho }$ (with $v_{2,\rho }$ fixed to 0.06).

Figure 4

RMS of $C_{{\mathrm{\Psi }}_2}$ and $C_{{\mathrm{\Psi }}_2}^{\perp }$ depending on $v_{2,\rho }$ and $p_{T,\rho }$. (a) RMS of $C_{{\mathrm{\Psi }}_2}$ and $C_{{\mathrm{\Psi }}_2}^{\perp }$ (shown in Fig. 1) depending on $v_{2,\rho }$ (with $v_{2,\pi }$ fixed to its default distribution). (b) RMS of $C_{{\mathrm{\Psi }}_2}$ and $C_{{\mathrm{\Psi }}_2}^{\perp }$ (shown in Fig. 3) depending on $p_{T,\rho }$ (with $v_{2,\rho }$ fixed to 0.06 and $v_{2,\pi }$ fixed to its default distribution).

Figure 5

Definition A: observable distributions for various values of $v_{3,\rho }$ and $p_{T,\rho }$ (with $v_{3,\pi }$ fixed to 0). The $C_{{\mathrm{\Psi }}_3}$ and $C_{{\mathrm{\Psi }}_3}^{\perp }$ curves, with the same $p_{T,\rho }$ but various $v_{3,\rho }$, are very close to each other in panels (a) and (b) (concave dashed lines for low $p_{T,\rho }$ and convex solid lines for high $p_{T,\rho }$).

Figure 6

Definition B: The upper plots (a)–(c) show observable distributions for various values of $v_{3,\rho }$ (with $v_{3,\pi }$ fixed to 0). Here, $v_{3,\rho }^{\mathrm{def}}$ is the default distribution of $v_{3,\rho }$ approximated by $0.5v_{2,\rho }^{\mathrm{def}}$ [Eq. (11)]. The lower plots (d)–(f) show observable distributions for various values of $p_{T,\rho }$ (with $v_{3,\rho }$ fixed to 0.03 and $v_{3,\pi }$ fixed to 0).

Figure 7

$\text{Var}\left[\delta \right]$ in the default simulation and the scan of $p_{T,\rho }$. (a) $\text{Var}\left[\delta \right]$ depending on $p_{T,\rho }$. (b) $\delta$ distribution in the default simulation.

Figure 8

The choice of the azimuthal range affects the physical results using Definition B.