Triangularity and Dipole Asymmetry in Heavy Ion Collisions
We introduce a cumulant expansion to parameterize 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 $<r^3 \cos3(φ - ψ_3,3)>$ and the dipole asymmetry $<r^3 \cos(φ- ψ_1,3)>$ of the initial entropy distribution. We show that for mid-peripheral collisions the orientation angle of the dipole asymmetry $ψ_1,3$ has a $20%$ preference out of plane. This leads to a small net $v_1$ out of plane. In peripheral and mid-central collisions the orientation angles $ψ_1,3$ and $ψ_3,3$ are strongly correlated. We study the ideal hydrodynamic response to these cumulants and determine the associated $v_1/ε_1$ and $v_3/ε_3$ for a massless ideal gas equation of state. $v_1$ and $v_3$ develop towards the edge of the nucleus, and consequently the final spectra are more sensitive to the viscous dynamics of freezeout. The hydrodynamic calculations for $v_3$ are compared to Alver and Roland fit of two particle correlation functions. Finally, we propose to measure the $v_1$ associated with the dipole asymmetry and the correlations between $ψ_1,3$ and $ψ_3,3$ by measuring a two particle correlation with respect to the participant plane, $<\cos(φ_a - 3φ_b + 2Ψ_PP)>$. The hydrodynamic prediction for this correlation function is several times larger than a correlation currently measured by the STAR collaboration, $<\cos(φ_a + φ_\b - 2Ψ_PP)>$.