Triangularity and dipole asymmetry in relativistic heavy ion collisions

We introduce a cumulant expansion to parametrize possible initial conditions in relativistic heavy ion collisions. We show that the cumulant expansion converges and that it can systematically reproduce the results of Glauber type initial conditions. At third order in the gradient expansion the cumulants characterize the triangularity r³cos3(φ-ψ3,3) and the dipole asymmetry r³cos(φ-ψ1,3) of the initial entropy distribution. We show that for midperipheral collisions the orientation angle of the dipole asymmetry ψ1,3 has a $20%$ preference out of plane. This leads to a small net v₁ out of plane. In peripheral and midcentral collisions the orientation angles ψ1,3 and ψ3,3 are strongly correlated, but this correlation disappears towards central collisions. We study the ideal hydrodynamic response to these cumulants and determine the associated v₁/ε₁ and v₃/ε₃ for a massless ideal gas equation of state. The space time development of v₁ and v₃ is clarified with figures. These figures show that v₁ and v₃ develop toward the edge of the nucleus, and consequently the final spectra are more sensitive to the viscous dynamics of freezeout. The hydrodynamic calculations for v₃ are provisionally compared to Alver and Roland fit of STAR inclusive two-particle correlation functions. Finally, we propose to measure the v₁ associated with the dipole asymmetry and the correlations between ψ1,3 and ψ3,3 by measuring a two-particle correlation with respect to the participant plane cos(φ_α-3φ_β+2ΨPP). The hydrodynamic prediction for this correlation function is several times larger than a correlation currently measured by the STAR collaboration cos(φ_α+φ_β-2ΨPP). This experimental measurement would provide convincing evidence for the hydrodynamic and geometric interpretation of two-particle correlations at RHIC.