Download Shear viscosity of the quark gluon plasma

The Phase
Diagram of
Nuclear Matter
By Ryan Richards
Outline
1. Introduction
2. Calculating the QGP viscosity
a) Hydrodynamics (Macroscopic Approach)
b) Kinetic Theory (Microscopic Approach)
3. Hydrodynamic Models
a) Monte Carlo Glauber (MC)
b) Monte Carlo Kharzeev-Levin-Nardi (MC-KLN)
c) IP-Glasma
4. Extracting the Viscosity from Experimental data
a) Relativistic Heavy Ion Collider (RHIC)
b) Large Hadron Collider (LHC)
Introduction
• Natural Units
• Overview
Natural Units
• Natural units are used in both nuclear and particle physics.
• The fundamental constants c, k, and ħ are set to one.
• As a result, all measurements have units of energy in electron
volts (eV).
• Examples
a) Length, time ~ 1/eV
b) Mass, Temperature ~ eV
You would also see time measured in femtometers(fm),
1 𝑓𝑚−1 = 197 MeV
The proton size is about 1 fm.
Overview
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It has been known that for over half a century that high energy collisions between
hadrons and atomic nuclei can be described using relativistic fluid dynamics.
The idea of using this macroscopic approach to describe such phenomena became
more relevant in the late 1970s to mid 1990s.
One of the striking features of the collisions were the formation of dense matter that
expanded in the transverse directions to the beam (i.e. the x and y directions for
beam in the z direction).
However the emission was not isotropic and the dense matter emitted elliptically
around the beam. As a result, people began trying to solve the relativistic fluid
dynamics equations in both 1D and 2D transverse directions (hydrodynamics).
Beginning in 2000, RHIC began operating at ten times larger than the center of
mass energy, in which the QGP was mass produced.
It has only been recently that the experimental data has agreed with theoretical
predictions.
The Phase Diagram of
Nuclear Matter
Reference: http://www.bnl.gov/rhic/news/040808/story3.asp
Hadronization of the QGP
• The hadronization of the QGP is characterized by a first order
transition in the temperature-quark density plane which ends at the
critical point.
• In nucleus-nucleus collisions energy is transferred to the internal
structure of nucleons that liberate quarks and gluons for a small
period of time.
• An ultrarelativistic quark gas is formed which reaches thermal
equilibrium at temperature T, produced by the collision energy.
• At high energies, we can neglect the effects of strong interaction and
produce the quark gluon plasma, which consists of asymptotically
free quarks and gluons.
• At high energies the plasma expands - volume and number of
nucleons increase. The quark density decreases and the quarks in the
outer shell hadronize [1].
Qualitative Understanding
of the First Order Transition
• The first order transition is similar to the van der Waals liquidgas transition.
• Recall the equation of state
(p +an^2)(1 – bn) = nT
• Solve for the temperature as a function of quark density gives
T = bn^2 + a
• Require that
𝜕𝑇
𝜕𝑛
< 0, gives n =
𝐴
2𝐵
and T =
𝐴^2
4𝐵
• Now we want to study how T scales with the quark density [1].
Temperature as a function
of Quark Density
• Consider Nn nucleons in volume ~ Ro^3 and the nucleus is subjected to a
high energy collision. The energy per nucleon is 1 TeV for this example.
We will neglect the up and down quarks because the masses of these quarks
are small compared to the energy.
• The energy is mostly dominated by gluons except for the QGP, where the
quark gas in equilibrium with the gluons.
• The energy goes as
VT^4 /(ħc)^3
• The mean number of quarks goes as
VT^3/(ħc)^3
• Note that this the energy of both quarks and gluons. It is an approximation.
The energy of the system is dominated by the gluons and the quarks give a
small contribution to the energy. Also, the energy and mean number is
given by that of blackbody radiation aside from numerical factors.
T(n) continued
• Now we are interested in how these variables evolve in time.
• The temperature, mean number and volume scale as
T = To (1 + ct/Ro)^-3/4
N = No (1 + ct/Ro) ^3/4
V = Vo (1 + ct/Ro)^3
• Therefore the mean density of quarks is proportional to (1 +
ct/Ro)^9/4.
• Plugging in the temperature dependence, the temperature goes
as
T =ħcn^1/3
Hadronization diagram in
(n,T) plane
• This is the temperature dependence of the quark density. The
temperature dependence is proportional to n^1/3 [1].
Summary
• In the hadronization process, the quarks on the outer shells of
the plasma, which is small compared to the number of quarks
begin to feel the interaction.
• Due to the interaction, they decouple from the QGP and form a
quark gas (ultrarelativistic) which moves within the volume of
the QGP and is in thermal equilibrium with the plasma [2].
Shear Viscosity of the
QGP
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The shear viscosity of the QGP is of both theoretical and experimental interest.
In most papers, the shear viscosity is consider in a ratio, the shear viscosity to
entropy ratio, η/s.
The quantum time scale is given τq ~ ħ/kT.
The ratio η/s is on the order of τr/τq, where τr is the relaxation time. The relaxation
time of the particle is estimated from the diffusion coefficient, D.
The diffusion coefficient is defined as η/H, where H is the enthalpy. D is on the
order of v^2τr, where v is the velocity of a particle due to a sound wave propagating
[4].
Shear viscosity causes anisotropic deviations from equilibrium which are
characteristic from off-diagonal elements in the stress energy tensor.
Hydrodynamics
• Start with the stress tensor and its equation of motion.
• The equations of motion are relativistic:
De = - (e + P)𝛻μ uν
Duμ = - ∇μP/(e +P)
• The first equation says that the change in the energy density is due
to mechanical work, while the second says that the pressure gradient
causes acceleration.
Kinetic Theory
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Start with the spectrum of particles.
The shear viscosity affects the spectra in two ways.
It is increases the magnitude and decreases the anisotropies of the transverse
flow.
It also causes a deviation of the phase space distribution.
The phase space distribution goes from
f(x,p) = fo(x,p) + δf(x,p)
The second term is depends on microscopic interactions and it is only sensitive
to the shear viscosity.
To calculate how the shear viscosity is related to the initial distribution, you can
assume the small correction is proportional to strains. If the most important
strain is shear strain for example, one can show that the η is related to the
integral of the δf(x,p) and the strains over all momentum [3,4].
Hydrodynamic Models
• Viscous hydrodynamic models are created by using Monte Carlo
methods.
• There are various algorithms such as VISHNU is a hybrid code that
couples viscous hydrodynamics with microscopic dynamics.
• Some of the models include the Monte Carlo Glauber, Mote Carlo
Kharzeev-Levin-Nardi model and the IP Glasma model.
Measuring (η/s)
• Hydrodynamics converts spatial deformation into final state momentum
anisotropies.
• The momentum anisotropy, εp is related to the shear viscosity to entropy
ratio monotonically.
• The observable related to the momentum anisotropy is the total charged
hadron elliptic flow, v2. The elliptical flow depends on the pT spectra of
different hadronic species.
• The elliptical flow is can be expressed as a function of the transverse
momentum by expanding the particle differential in a Fourier series.
• The Fourier series would include higher level harmonic flow coefficients as
well.
• The average elliptical flow would give you the ratio of particles in the
transverse directions [3,4].
η/s of the QGP at RHIC
• This is a graph of the eccentricity scaled elliptical
flow as a function of particle differential between Tc
< T < 2Tc probed at RHIC [5].
η/s of the QGP at the LHC
• Data from ALICE at the LHC compared with VISHNU, a
theoretical algorithm with the initial conditions that η/s =0.2
[5].
References
1) M. Apostol, Phase Diagram in Quantum Chromodynamics,
23 Nov. 2013
2) M. Apostol, Hadronization of the Quark Gluon Plasma,
2007
3) U. Heinz and R. Snellings, Collective Flow and Viscosity in
Relativistic Heavy-Ion Collisions 13 Jan. 2103.
4) D. Teaney, Viscous Hydrodynamics and the Quark Gluon
Plasma, 14 May 2009.
5) U. Heinz, C. Shen, and H. Song, The Viscosity of the Quark
Gluon Plasma at RHIC and the LHC, 26 Aug. 2011.