Triangularity and dipole asymmetry in relativistic heavy ion collisions

Abstract

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 $\ensuremath{\langle}{r}^{3}\mathrm{cos}3(\ensuremath{\phi}\ensuremath{-}{\ensuremath{\psi}}_{3,3})\ensuremath{\rangle}$ and the dipole asymmetry $\ensuremath{\langle}{r}^{3}\mathrm{cos}(\ensuremath{\phi}\ensuremath{-}{\ensuremath{\psi}}_{1,3})\ensuremath{\rangle}$ of the initial entropy distribution. We show that for midperipheral collisions the orientation angle of the dipole asymmetry ${\ensuremath{\psi}}_{1,3}$ has a $20%$ preference out of plane. This leads to a small net ${v}_{1}$ out of plane. In peripheral and midcentral collisions the orientation angles ${\ensuremath{\psi}}_{1,3}$ and ${\ensuremath{\psi}}_{3,3}$ are strongly correlated, but this correlation disappears towards central collisions. We study the ideal hydrodynamic response to these cumulants and determine the associated ${v}_{1}/{\ensuremath{\epsilon}}_{1}$ and ${v}_{3}/{\ensuremath{\epsilon}}_{3}$ for a massless ideal gas equation of state. The space time development of ${v}_{1}$ and ${v}_{3}$ is clarified with figures. These figures show that ${v}_{1}$ and ${v}_{3}$ 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 ${v}_{3}$ are provisionally compared to Alver and Roland fit of STAR inclusive two-particle correlation functions. Finally, we propose to measure the ${v}_{1}$ associated with the dipole asymmetry and the correlations between ${\ensuremath{\psi}}_{1,3}$ and ${\ensuremath{\psi}}_{3,3}$ by measuring a two-particle correlation with respect to the participant plane $\ensuremath{\langle}\mathrm{cos}({\ensuremath{\phi}}_{\ensuremath{\alpha}}\ensuremath{-}3{\ensuremath{\phi}}_{\ensuremath{\beta}}+2{\ensuremath{\Psi}}_{\mathit{PP}})\ensuremath{\rangle}$. The hydrodynamic prediction for this correlation function is several times larger than a correlation currently measured by the STAR collaboration $\ensuremath{\langle}\mathrm{cos}({\ensuremath{\phi}}_{\ensuremath{\alpha}}+{\ensuremath{\phi}}_{\ensuremath{\beta}}\ensuremath{-}2{\ensuremath{\Psi}}_{\mathit{PP}})\ensuremath{\rangle}$. This experimental measurement would provide convincing evidence for the hydrodynamic and geometric interpretation of two-particle correlations at RHIC.