Complications in the interpretation of the charge asymmetry dependent $pi$ flow for the chiral magnetic wave

The charge asymmetry ($A_{rm ch}$) dependence of the $pi^{-}$ and $pi^{+}$ elliptic flow difference, $Delta v_{2}(A_{rm ch})$, has been regarded as a sensitive observable for the possible chiral magnetic wave (CMW) in relativistic heavy ion collisions. In this work, we first demonstrate that, due to non-flow backgrounds, the flow measurements by the Q-cumulant method using all charged particles as reference introduce a trivial linear term to $Delta v_{2}(A_{rm ch})$. The trivial slope can be negative in the triangle flow difference $Delta v_{3}(A_{rm ch})$ if the non-flow is dominated by back-to-back pairs. After eliminating the trivial term, we find that the non-flow between like-sign pairs gives rise to an additional positive slope to $Delta v_{2}(A_{rm ch})$ because of the larger dilution effect to $pi^{+}$ ($pi^{-}$) at positive (negative) $A_{rm ch}$. We further find that the competition between different $pi$ sources can introduce another non-trivial linear-$A_{rm ch}$ term due to their different multiplicity fluctuations and anisotropic flows. We then study the effect of neutral cluster (resonance) decays as a mechanism for local charge conservation on the slope parameter of $Delta v_{2}(A_{rm ch})$. We find that the slope parameter is sensitive to the kinematics of those neutral clusters. Light resonances give positive slopes while heavy resonances give negative slopes. Local charge conservation from continuum cluster mass distribution can give a positive slope parameter comparable to experimental data. Our studies indicate that many non-CMW physics mechanisms can give rise to a $A_{rm ch}$-dependent $Delta v_{2}(A_{rm ch})$ and the interpretation of $Delta v_{2}(A_{rm ch})$ in terms of the CMW is delicate.