The success of ideal Relativistic Fluid Dynamics (RFD) in describing hadron spectra and elliptic flow at RHIC has led to a strong interest in the transport coefficients of QCD, in particular the shear- and bulk-viscosity as well as the shear-viscosity over entropy-density ratio η/s. Of course ideal RFD assumes η to be zero, which is unphysical, but its success has hinted at a very small value for η/s. The purpose of this post is to review what we currently know about η/s of QCD matter and put the different approaches used to study this quantity into context.
The development of a consistent causal theory of relativistic viscous fluid dynamics has allowed for a phenomenological analysis to determine QCD transport coefficients such as η/s: it has been shown that in particular the elliptic flow coefficient v2 is sensitive to the value of η/s used in the calculation. Current calculations show that a value of η/s between 0.1 and 0.4 is required to describe the elliptic flow data measured at RHIC:
- Conformal Relativistic Viscous Hydrodynamics: Applications to RHIC results at s(NN)**(1/2) = 200-GeV
- Extracting the QGP viscosity from RHIC data — a status report from viscous hydrodynamics
- What do elliptic flow measurements tell us about the matter created in the little bang at RHIC?
One should note, however, that all phenomenological analyses relying on the fit of viscous fluid dynamics to elliptic flow data suffer from large systematic uncertainties due to the choice of initial conditions and the treatment of the late dissipative hadronic phase:
Ideally one should be able to determine η or η/s from lattice QCD calculations. However these calculations are technically very difficult and currently still suffer from large error-bars:
Effective theories have been able to provide additional guidance, albeit come with their own drawbacks:
- It has been argued that the N=4 supersymmetric Yang-Mills theory in (3+1)-dimensional space-time represents a good model for QCD in the strong-coupling limit. This N=4 SYM theory can be related to a 10-dimensional superstring theory in AdS5 via the AdS/CFT correspondence. Using this correspondence, a lower bound of η/s = \hbar/4π has been derived (the so-called KSS bound). Using the same techniques, many other quantities of interest to the relativistic heavy-ion community (e.g. drag coefficients and the jet energy-loss transport coefficient q-hat) have been calculated as well. One should note, however, that N=4 SYM is not QCD and that it is by no means clear how closely related these quantities in QCD are to those in N=4 SYM. Most importantly for the discussion at hand, the lower bound for η/s postulated does not provide us with any insight on the microscopic mechanism (i.e. the nature of the interaction and the degrees of freedom) of how the low value of η/s comes about.
- There is currently still disagreement among Parton Cascade Model calculations on whether the elliptic flow seen at RHIC can be generated with purely perturbative cross sections or how big the value of η/s for a system of partons interacting with pQCD cross sections actually is. A study by Molnar and Huovinen has shown that when one assumes the QGP to be quasi-particulate in nature and only allows for binary 2-2 scattering among the partons, that unphysically large cross sections (incompatible with pQCD) are required to reproduce the elliptic flow seen by the RHIC experiments. The Frankfurt Group have reported that the inclusion of radiative corrections (i.e. 2-3 and 3-2 processes) have a large effect and lead to very small values of η/s while using pQCD cross sections. However, this result hinges strongly on the implementation of the LPM effect in the calculation, which is still a matter of debate in the community.
- One should note that large cross sections and/or multi-particle processes may not necessarily be required to generate a small value for η/s: fluid dynamics does not care about the origin of the viscosity, since the viscosity appears only in the form of an expansion coefficient of the stress-energy tensor. Field degrees of freedom (e.g. turbulent color fields, which are thought to drive the early thermalization at RHIC) may induce an anomalous viscosity without the need for large scattering cross sections and could provide a mechanism for generating a small viscosity before the system fully thermalizes (which is a pre-requisite for the lattice-QCD calculations).
Coming back to the viscous fluid dynamical analysis of elliptic flow data at RHIC: one of the major shortcomings of the current analyses is that they all assume a fixed temperature independent value for η/s. However, η/s of real gases, such as Nitrogen and Helium, exhibits a strong temperature dependence with a pronounced minimum at the critical temperature. Correlating this observation with η/s calculations for a gas of chiral pions at low temperature and a HTL calculation for a QGP in the high-temperature limit, Csernai, Kapusta and McLerran have argued that η/s of QCD matter should exhibit a similar behavior. Having η/s exhibit a minimum at the critical temperature would be in line with the observed low value of η/s at RHIC, since the produced matter is thought to spend a significant fraction of its life-time between 2 Tc and 0.6 Tc. Unfortunately, until very recently no reliable calculations of η/s for QCD matter have existed in that temperature range. For the deconfined phase, this is due to the strongly interacting non-perturbative regime of QCD which is only accessible via lattice QCD calculations (see above), and for the hadron gas phase it has been the large number of hadronic states which has made any attempt at an analytical calculation of η/s beyond a gas of chiral pions or a binary pion+kaon mixture not feasible.
Over the past few years, however, there has been considerable progress in numerical calculations of the shear viscosity as well as η/s for a hadron gas, utilizing microscopic transport models in the infinite matter limit (i.e. confining the calculation to a box with periodic boundary conditions) and extracting the shear viscosity coefficient via the Kubo formalism:
- Shear viscosity coefficient from microscopic models
- A Calculation of the viscosity over entropy ratio of a hadronic gas
However, the first of the two calculations was only done for the shear viscosity coefficient of a meson-gas (w/o calculating its entropy-density) and suffered from low statistics for the correlators, whereas the second one contained only a limited set of hadronic species and a rather simple ansatz for the calculation of the entropy, neglecting the difference in specific entropies between fermions and bosons as well as the mass-dependence of that quantity. A more systematic study, containing a full hadron gas and a rigorous calculation of the entropy density was just concluded recently:
- Shear-Viscosity to Entropy Density Ratio of a Relativistic Hadron Gas at RHIC: Approaching the AdS/CFT bound?
The key results of this calculation are that η/s of a full hadron gas in chemical equilibrium falls as a function of increasing temperature, and levels off at a value of approximately 1.0 near Tc. However, the inclusion of non-zero fugacities, which are bound to arise due to the separation of chemical and kinetic freeze-out during the evolution of the collision, will reduce η/s to approximately 0.5. This value is most likely still too large to ensure the successful application of the viscous fluid dynamics calculations to the full reaction evolution at RHIC.
Putting all the constraints currently known for η/s together, one is therefore bound to speculate that the collision system initially forms a QGP with a value of η/s significantly lower than 0.5 at an initial temperature on the order of twice Tc. The QGP then expands and cools to the transition temperature while retaining this small value of η/s. At hadronization, η/s rises sharply to the value computed for the hadron gas in chemical equilibrium (potentially this rise could be delayed by a phase containing hadronic Hagedorn resonance states), but then decreases again with the rise of non-unit value fugacities after the chemical decoupling of the system. At temperatures close to kinetic freeze-out around 110 MeV, η/s will rise sharply beyond a value of 1.0 . Note, that lattice calculations have shown a sharp rise in the entropy density at Tc. It is therefore not unexpected for η/s to exhibit a sharp change in value, since it would be quite surprising that the rise in s would be exactly compensated by a likewise rise in the shear viscosity coefficient to yield a flat behavior of η/s at Tc. Of course a rigorous calculation of η/s in the range between Tc and 2-3 Tc remains a challenge for the future.