Effective viscosities in a hydrodynamically expanding boost-invariant QCD plasma

Background: The near-equilibrium properties of a QCD plasma can be encoded into transport coefficients such as bulk and shear viscosity. In QCD, the ratio of these transport coefficients to entropy density, $\ensuremath{\zeta}/s$ and $\ensuremath{\eta}/s$, depends nontrivially on the plasma's temperature. This is unlike in conformal systems where they take constant values such as $\ensuremath{\eta}/s=1/(4\ensuremath{\pi})$.Purpose: In this work, we show that in a $(0+1)\mathrm{D}$ boost-invariant fluid with no transverse expansion, a temperature-dependent $\ensuremath{\zeta}/s(T)$ or $\ensuremath{\eta}/s(T)$ can be described by an equivalent effective viscosity ${\ensuremath{\langle}\ensuremath{\zeta}/s\ensuremath{\rangle}}_{\text{eff}}$ or ${\ensuremath{\langle}\ensuremath{\eta}/s\ensuremath{\rangle}}_{\text{eff}}$. This effective viscosity combines the actual temperature-dependent $\ensuremath{\zeta}/s(T)$ or $\ensuremath{\eta}/s(T)$ with the temperature profile of the fluid. We extend the concept of effective viscosity in systems with transverse expansion and discuss how effective viscosities can be used to identify families of $\ensuremath{\zeta}/s(T)$ and $\ensuremath{\eta}/s(T)$ that lead to similar hydrodynamic evolution.Methods: The Navier-Stokes relativistic hydrodynamic equations are used to provide a first definition of effective viscosity, in $(0+1)\mathrm{D}$ and $(1+1)\mathrm{D}$. In the $(0+1)\mathrm{D}$ case, the analysis is extended to Israel-Stewart-type second-order hydrodynamics to clarify the effect of higher-order hydrodynamics corrections on the effective viscosity.Results: In a boost-invariant fluid with no transverse expansion [$(0+1)\mathrm{D}$], the effective viscosity is expressed as a simple integral of $\ensuremath{\zeta}/s(T)$ or $\ensuremath{\eta}/s(T)$ over temperature, with a weight determined by the speed of sound of the fluid. The result is general for any equation of state with a moderate temperature dependence of the speed of sound, including the QCD equation of state. This definition of effective viscosity can be used to identify infinite families of $\ensuremath{\zeta}/s(T)$ or $\ensuremath{\eta}/s(T)$ that produce essentially indistinguishable temperature profiles. In a boost-invariant cylindrical system [$(1+1)\mathrm{D}$], a similar definition of effective viscosity is obtained in terms of characteristic trajectories in time and transverse direction. This leads to an infinite number of constraints on an infinite functional space for $\ensuremath{\zeta}/s(T)$ and $\ensuremath{\eta}/s(T)$. Realistic examples are presented by using a finite number of constraints on a finite functional space.Conclusions: The definition of effective viscosity in a $(0+1)\mathrm{D}$ system clarifies how infinite families of $\ensuremath{\zeta}/s(T)$ and $\ensuremath{\eta}/s(T)$ can result in nearly identical hydrodynamic temperature profiles. By extending the study to a boost-invariant cylindrical [$(1+1)\mathrm{D}$] fluid, we identify an approximate but more general definition of effective viscosity that highlights the potential and limits of the concept of effective viscosity in fluids with limited symmetries.