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Fluctuations of harmonic and radial flow in heavy ion collisions with principal components

We analyze the spectrum of harmonic flow, $v_n(p_T)$ for $n=0\text{--}5$, in event-by-event hydrodynamic simulations of Pb+Pb collisions at the CERN Large Hadron Collider ($\sqrt{s_{NN}}=2.76\,{\text{TeV}}$) with principal component analysis (PCA). The PCA procedure finds two dominant contributions to the two-particle correlation function. The leading component is identified with the event plane $v_n(p_T)$, while the subleading component is responsible for factorization breaking in hydrodynamics. For $v_0$, $v_1$, and $v_3$ the subleading flow is a response to the radial excitation of the corresponding eccentricity. By contrast, for $v_2$ the subleading flow in \emph{peripheral collisions} is dominated by the nonlinear mixing between the leading elliptic flow and radial flow fluctuations. In the $v_2$ case, the sub-sub-leading mode more closely reflects the response to the radial excitation of $\varepsilon_2$. A consequence of this picture is that the elliptic flow fluctuations and factorization breaking change rapidly with centrality, and in central collisions (where the leading $v_2$ is small and nonlinear effects can be neglected) the subsub-leading mode becomes important. Radial flow fluctuations and nonlinear mixing also play a significant role in the factorization breaking of $v_4$ and $v_5$. We construct good geometric predictors for the orientation and magnitudes of the leading and subleading flows based on a linear response to the geometry, and a quadratic mixing between the leading principal components. Finally, we suggest a set of measurements involving three point correlations which can experimentally corroborate the nonlinear mixing of radial and elliptic flow and its important contribution to factorization breaking as a function of centrality.