Second-order viscous corrections to the harmonic spectrum in heavy-ion collisions

We calculate the second-order viscous correction to the kinetic distribution, δf(2), and use this result in a hydrodynamic simulation of heavy-ion collisions to determine the complete second-order correction to the harmonic spectrum, vn. At leading order in a conformal fluid, the first viscous correction is determined by one scalar function, χ0p. One moment of this scalar function is constrained by the shear viscosity. At second order in a conformal fluid, we find that δf(p) can be characterized by two scalar functions of momentum, χ1p and χ2p. The momentum dependence of these functions is largely determined by the kinematics of the streaming operator. Again, one moment of these functions is constrained by the parameters of second-order hydrodynamics, τπ and λ1. The effect of δf(2) on the integrated flow is small (up to v4) but is quite important for the higher harmonics at modestly large pT. Generally, δf(2) increases the value of vn at a given pT and is most important in small systems.