Nonlinear coupling of flow harmonics: Hexagonal flow and beyond

Higher Fourier harmonics of anisotropic flow ($v_4$ and beyond) get large contributions induced by elliptic and triangular flow through nonlinear response. We present a general framework of nonlinear hydrodynamic response which encompasses the existing one, and allows to take into account the mutual correlation between the nonlinear couplings affecting Fourier harmonics of any order. Using Large Hadron Collider data on Pb+Pb collisions at ~$\sqrt[]{s}=2.76$ TeV, we perform an application of our formalism to hexagonal flow, $v_6$, a coefficient affected by several nonlinear contributions which are of the same order of magnitude. We obtain the first experimental measure of the coefficient $\chi_{624}$, which couples $v_6$ to $v_2$ and $v_4$. This is achieved by putting together the information from several analyses: event-plane correlations, symmetric cumulants, as well as new higher-order moments recently analyzed by the ALICE collaboration. The value of $\chi_{624}$ extracted from data is in fair agreement with hydrodynamic calculations, although with large error bars, which would be dramatically reduced by a dedicated analysis. We argue that within our formalism the nonlinear structure of a given higher harmonic can be determined more accurately than the harmonic itself, and we emphasize potential applications to future measurements of $v_7$ and $v_8$.