We present three-particle mixed-harmonic correlations $langle cos (mphi_1
+ nphi_2 - (m+n) phi_3)
angle$ for charged particles in $sqrt{s_{NN}}=$200
GeV Au+Au collisions at RHIC. These measurements provide information on the
three-dimensional structure of the initial collision zone and are important for
constraining models of a subsequent low-viscosity quark-gluon plasma expansion
phase. We find evidence for correlations between the first, second and third
harmonics predicted as a consequence of fluctuations in the initial state. The
dependence of the correlations on the pseudorapidity separation between
particles is consistent with a breaking of longitudinal invariance. While the
correlator for $m,n=2$ is well described by a hydrodynamical model with a
temperature dependent shear viscosity-to-entropy ratio, the higher harmonic
correlations are significantly under-predicted. The data suggest that models of
heavy-ion collisions need to include the longitudinal dependence of the initial
state and stronger correlations produced in the final state.
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