Shear viscosity in the quark-gluon plasma

The shear viscosity for QCD, η(T), is revisited with particular focus on the relevant scale(s) that determine α, the strong-coupling. Perturbation theory was previously deemed incompatible with the inference from heavy ion collisions (which indicated "perfect fluid" characteristics) for reasons we explain in 1704.06284. Also in the aforementioned preprint, we give the revised temperature dependence for T≈T_c, which is compatible with other estimates from lattice QCD and hydrodynamic simulations.

Background / kinetic theory

The QCD Boltzmann equation (in the quenched case) was first linearised by Baym et al. who used hard thermal loop (HTL) insertions to screen the collision term. Their calculation fixed the overall leading-log prefactor, and has been extended to include n_f>0 flavours of massless quarks [plus other gauge theories, besides SU(N)] due to Arnold, Moore & Jaffe (AMY) [Part I]. AMY also showed how to go beyond logarithmic accuracy by incorporating inelastic processes in [Part II]. This settled the LO result for η, as an expansion in α and allowed the next-to-leading-log coefficients to be calculated.

Data

fixed-α calculation

Here we present results for the LO viscosity, which are accurate to 1% (we only dropped inelastic contributions and used a single-function ansatz for the off equilibrium solution). Data is saved under out/data/, where file names indicate etaT3 (for η/t³). Firstly, the files tagged 'fixed' give the parametric dependence on the coupling parameter g=(4πα)^1/2. Some useful links are given below:

• etaT3_HTL_nf0_fixed.dat
• etaT3_kappa0.50_nf0_fixed.dat
• etaT3_kappa0.25_nf0_fixed.dat
• etaT3_HTL_nf2_fixed.dat
• etaT3_kappa0.50_nf2_fixed.dat
• etaT3_kappa0.25_nf2_fixed.dat
• etaT3_HTL_nf3_fixed.dat
• etaT3_kappa0.50_nf3_fixed.dat
• etaT3_kappa0.25_nf3_fixed.dat

With Running

Here we implement the scheme devised by AMY (minus the inelastic processes), with a coupling that depends on the virtuality Q²=ω²-q². An effective version of the one-loop running is used, to account for both s-channel (Q²>0) and t- or u-channel (Q²<0) processes. To account for the singularity at the Landau pole, a maximal value α≤[1...10] which turns out to be of little importance.

The files tagged as 'running' give the temperature dependence dependence (T in units of Λ_QCD).

• etaT3_HTL_nf0_running.dat
• etaT3_kappa0.50_nf0_running.dat
• etaT3_HTL_nf2_running.dat
• etaT3_kappa0.50_nf2_running.dat
• etaT3_HTL_nf3_running.dat
• etaT3_kappa0.50_nf3_running.dat

Usage

I have designed this piece of code for my personal use, simply to calculate η (for QCD). However, this has become a central quantity in heavy-ion physics and these results may be of wider interest.

Visualisation

Use gnuplot to plot data (scripts are under out/plotter). For example, to show the temperature dependent viscosity for 3-flavour QCD:

make 'NF=3' temperature

Details on 5-dimensional integration

Use the makefile to compile the binary, execute with

./bin/eta NF

where NF={0,1,...} is the number of light quark flavours. (The default value is Nf=0.)

• eval_g(gmin, gmax) evaluates η(g)/T³
• eval_T(Tmin, Tmax) evaluates η(T/Λ)/T³
• rate_E(Emin, Emax, Temp) evaluates absorbtion rate R(E)/T

Using HTL function (set HTL=1) OR eff. mass mu^2 = kappa*mD^2 (set HTL=0)

Integrator using gsl implementation of (pick one). The absolute value of (int) calls indicates how many times the integrand is sampled (the sign determines which codes below are used).

• VEGAS, calls > 0
• Cubature, calls < 0 ~ recommended

Note that ±1e5 is just stable for n_f=0, and should be higher by a factor ~10³ with quarks.