Viscosity Estimate with Staggered Fermions

**Principal Investigator:**

Szabolcs Borsanyi

**Affiliation:**

University of Wuppertal (Germany)

**Local Project ID:**

hwu30

**HPC Platform used:**

JUQUEEN of JSC

**Date published:**

**Lattice QCD has played a crucial role in the determination of the true equation of state. It is a numerical approach to solve the theory of strong interactions, quantum chromodynamics (QCD), which provides first principles access to the physics of QGP. Lattice QCD assumes thermal equilibrium, where also the equation of state is defined. Researchers calculated this input to hydrodynamics in a succession of projects with the goal to investigate, whether such first principle determination of the (shear) viscosity parameter is possible from lattice QCD. **

Viscosity is the resistence of a fluid to flow. It describes the forces that dissipates energy while a liquid is in motion. Its value varies greatly for every day liquids ranging from water (1 mPa s), through vegetable oil (50 mPa s) to honey (>2000 mPa s). Gases can be seen as viscous fluids, too. Air, for example has a viscosity of about 20 µPa s.

Perhaps somewhat surprisingly, quark gluon plasma (QGP) also has a viscosity parameter. It actually takes a quite notable place among the characteristics of QGP, since its value can be addressed not only in theory, but also in those very experiments where QGP is produced.

Quark gluon plasma is a relatively recently discovered phase of matter. When the Universe was just a few microseconds old, our whole world was in the QGP phase. At those high temperatures of about two trillion Kelvin, electrons were not bound to nuclei to form atoms, even the protons and neutrons of the nuclei could not stay together. Actually, at that temperature even protons and neutrons could not form. Their constituents, quarks and gluons were floating around, building a form of matter that we call quark gluon plasma.

One of the great achievements of experimental high energy physics in the past decades is that QGP can be produced by man in heavy ion collision experiments, for example at the LHC in CERN, or at RHIC in Brookhaven, USA. The plasma then develops from very strong fields that are left behind by the colliding nuclei. This happens at a very small scale. Artificial QGP has a volume of a few thousand cubic femtometers. It is short lived, such that light, or energetic particles, can travel just ten femtometers during its lifetime.

How well the hydrodynamic description works for this tiny droplet of matter was a surprise to many when the RHIC accellerator went into production in 2001. Since then, the flow equations were equipped with the non-ideal componets, like viscosity and relaxation times. Assuming various values for these parameters hydrodynamics makes post-dictions on the momentum distribution of the outgoing charged particles. A matching result indicates that that a likely value was found for the input parameters.

The energy — pressure relation, or, in other words, the equation of state — is a principal ingredient of the flow equations, whether viscous effects are included or not. Initially, ad-hoc assumptions were made for this equation of state, which has led to the false early conclusion that the QGP fluid was ideal, that is, viscosity was zero.

Lattice QCD has played a crucial role in the determination of the true equation of state. Lattice QCD is a numerical approach to solve the theory of strong interactions, quantum chromodynamics (QCD), which provides first principles access to the physics of QGP. It assumes thermal equilibrium, where also the equation of state is defined. Our group calculated this input to hydrodynamics in a succession of projects on the Blue Gene/Q computer at FZ-Jülich.

The goal of our present study was to investigate, whether such first principle determination of the (shear) viscosity parameter is possible from lattice QCD. There are two great challenges that prohibit this.

Frist, we face the noise problem. Viscosity is defined through a correlator of the energy momentum tensor, which is a well defined observable on the lattice. The diagonal components of this energy momentum tensor was used earlier to calculate the equtaion of state. To calculate a correlator is far more difficult, however. The reason for this is that the correlation between two points in space-time is very small, and it is established though the exchange of gluon particles. Gluons are defined as random variables in lattice QCD, and their small correlation is completely dominated by stochastical noise.

The second problem is, that not only the viscous effects contribute to the small value of the correlator. Actually, most of the correlation is coming from high momentum gluons. If we substract these, we get an even smaller number, that has to be determined accurately.

For the first problem there is an algorithm for quark-less systems, the so called multi-level algorithm. Because of the way quarks are introduced into lattice QCD a generalization of this method to QGP is very difficult. We found a simpler method by removing the noisy, ultraviolet contribution coming from the gluons. We also calibrated this method in the quark-less case and found the ideal parameters where we only filter out the irrelevant, noisy part, whereas the physics remanis unchanged.

For the second problem we quantified how accurate the propagator needs to be in order the separation of the high momentum gluons becomes possible. This is in the promille region. We were able to get to this precision in our lattice simulations.

An important point in our study was continuum extrapolation. This means that the simulations are repeated on finder and even finer lattices, until the correlator can be extrapolated to zero lattice spacing. During this process precision is lost. Nevertheless, this is the first lattice study, where a continuum extrapolation of the energy momentum tensor correlator is actually performed.

We also studied several technicalities, for example, whether the temperature direction and the space direction should have the same discretization. This so called anisotropic discretization was very popular in earlier related works. We conclude that there is no benefit coming from such anisotropic treatment.

Our viscosity estimate is given in units of its entropy density, its value is between 0.15 and 0.20, which is compatible with the current estimates from the matching of the hydrodinamic evolution codes to experimental results.

How could future studies increase the precision of the viscosity estimate? In our procedure the errors were mostly coming from two sources. As already mentioned, precision is lost in the continuum extrapolation. This could be improved by a more sophisticated discretization scheme. The larger uncertainty, however, is coming from the estimate of the high momentum gluons. To calculate these the gluon interactions are either ignored, or approximated to the lowest order. Neither of these limiting factors can be easily improved just by increasing the simulation statistics beyond what we have achieved.

The next step to make the estimate will be to reduce the perturbative contribution or to calculate higher orders in perturbation theory. These tasks as well as to generalize the simulations for dynamical quantities are one of the greatest challenges for lattice QCD.

As illustration for the difficulty of the problem we show two equally well motivated theoretical correlators. In the representation of the top panel, viscosity is proportional to the value at ⍵/T=0. In a lattice simulation we observe only the Fourier transform. The ratio of the lattice correlators is shown in the bottom panel. Observe the narrow scale of the Y-axis. The viscious behaviour is encoded in the fourth digit of this quantity.

**References:**

• Ulrich Heinz (Ohio State U.), Raimond Snellings (Utrecht U.) Collective flow and viscosity in relativistic heavy-ion collisions

Ann.Rev.Nucl.Part.Sci. 63 (2013) 123-151

• Szabolcs Borsanyi (Wuppertal U.) et al. Full result for the QCD equation of state with 2+1 flavors

Phys.Lett. B730 (2014) 99-104

• Szabolcs Borsanyi (Wuppertal U.) et al. High statistics lattice study of stress tensor correlators in pure *SU*(3) gauge theory

Phys.Rev. D98 (2018) no.1, 014512

**Scientific Contact: **

Szabolcs Borsányi

Theoretische Physik, Fachbereich C - Bergische Universität Wuppertal

D-42097 Wuppertal/Germany

e-mail: borsanyi@uni-wuppertal.de

*December 2018*

*Project ID: hwu30*