Momentum dependent measures of correlations between mean transverse momentum and harmonic flow in heavy ion collisions

The correlation between the mean transverse momentum and the harmonic flow coefficients is an observable which is of great interest; it is sensitive to shape fluctuations in the initial state of a relativistic nuclear collision. The measurement of that correlation coefficient in central collisions allows one to infer about the intrinsic deformation of the colliding nuclei. We propose to study the momentum dependent covariance and correlation coefficient between the mean transverse momentum and the harmonic flow in a given transverse momentum bin. Two possible constructions of such observables are provided and predictions are obtained from a viscous hydrodynamic model. We find that such momentum dependent correlation coefficients between the mean transverse momentum and the harmonic flow show a strong and nontrivial momentum dependence. We also explore the effects of granularity (nucleon width) in the initial state, the nuclear deformation, and the shear viscosity on this momentum dependent correlation coefficient. The shape of the momentum dependence of the correlation coefficient for the triangular flow is found to be sensitive to the size of small scale fluctuations in the initial state. On the other hand, the shape of the momentum dependence of the covariance between the mean transverse momentum and the harmonic flow coefficients is found to be sensitive to the value of the shear viscosity and to the granularity of the initial state.