lattice qcd and qgp

Table of contents

Aspects of QCD

Lattice formulation Wick rotation

Bosonic part

Wilson loop

Fermionic part

Monte Carlo simulations

Problems and constraints Sign problem

Thermodynamics of QGP Phase diagram

Equation of state

Non-zero density

Short summary

17.12.09 | Daniil Gelfand | Seminar: „Relativistische Schwerionenphysik" | 2

Aspects of QCD

• Theory of quarks and gluons

• QCD Lagrangian

• Generating functional


Aspects of QCD

• Expectation values

• Asymptotic freedom

• Perturbative description at high

energies

• Length scales ≤ 1 fm


S. Bethke, Prog. Part. Nucl. Phys. 58 (2007) 351 .

Aspects of QCD

• Non-perturbative approach needed for:

• Length scales ≥ 1 fm

• Energies of ≈ 1 GeV

• Confinement

• Hadrons

• Glueballs

• K. Wilson (1974) proposed to discretize the theory


Lattice QCD

Lattice formulation

Wick rotation

• Leaving Minkowski space for Euclidian

• Equivalent to a 4D statistical theory

• Monte Carlo algorithms applicable

• Hypercubic lattice with spacings and

• Discrete imaginary time Matsubara frequencies


Bosons

• Periodic boundary conditions

• Gauge-invariant discretization of gauge fields

• Link variables

• Using

• Every Link connects two neighbouring points on a lattice

• Plaquette is a two-dimensional loop over 4 lattice points

Lattice formulation


Lattice formulation

Bosons


Ux,μν

Lattice formulation

Bosons

• Expanding gives:

• Bosonic part of the action can be written as

• Links can also be used to define an effektive potential

• Wilson loop


Lattice formulation

Wilson loop

• Product of link variables on the

path

• May be used to define quark-

antiquark potential for a

rectangular path


G.S

. Bali a

nd K

. Schilling, Phys. Rev. D

47 (

1993)

661.

Lattice formulation


Fermions

• Represented by Grassmann numbers

• Hard to simulate on computers

• But Gaussian integration possible

• Using

• Antiperiodic boundary conditions in (imaginary) time direction

and

Lattice formulation

Fermions

• Covariant derivative on the lattice?

• Central difference approximation

• Links to preserve gauge invariance


Lattice formulation

Fermions

• Momentum space propagator

• With unphysical poles at

• „Naive“ method leads to degenerate fermions

• Fermion doubling problem

• Nielsen-Ninomiya Theorem


Where is no lattice fermion action which is:

− real − translation invariant − doubler-free

− bilinear − local − chirally symmetric

Lattice formulation

Fermions

• Is there a way to get rid of doublers in practice?

• Wilson fermions

• Adding a new term to the action

• Propagator changes to

• fermions get a mass of the order of the lattice UV cutoff


YES!

Lattice formulation

Fermions

• But: Wilson term breaks chiral symmetry

• Staggered fermions (Kogut, Susskind, 1977)

• Reinterpretation of doublers as spinor components

• 4 flavors instead of one

• Solution: take fourth root of the fermion determinant

• Pro: Input consists of only one spinor component

• Lower computational effort


Lattice formulation


Fermions

• Contra: Fourth root trick is an approximation

• No proof of validity exists

• Other approaches:

• Domain-wall fermions (Furman, Shamir, 1995)

• Overlap fermions (Narayanan, Neuberger, 1995)

• Mostly used:

• Different improved staggered formulations (asqtad, p4fat3 etc.)

• Variations of Wilson fermions (clover-improved etc.)

Lattice formulation

Monte Carlo

• How to actually perform a simulation in Lattice QCD?

• Generate a set of gauge configurations according to the

probability distribution

• Each configuration contains values for every link on the lattice

• Recipe:

1) Start with a random configuration

2) Change a link to create a new configuration

3) Accept U„ with a given probability and/or procede with 2)


Lattice formulation

Monte Carlo

• By choosing a proper (e.g. Metropolis algorithm) the

system equilibrates to the given probability distribution

• Then the gauge configurations are used to compute expectation

values of a given operator


Problems and constraints

• Statistical errors from Monte Carlo

• Finite amount of gauge configurations

• Typical values: 400 – 800 configurations

• Lattice artefacts

• Discretization errors due to finite lattice spacing

• Typically

• Finite size effects

• Finite volume and boundaries

• Solutions?



• Bigger lattices

• Finer resolution

• More gauge configurations

• But: computational limitations

• Many degrees of freedom

• Main difficulty: fermionic determinant

• has to be performed for matrices with up to

• lines/rows

• Inverse of these large matrices needed for fermionic expectation values


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• Possible solution is to keep thedeterminant constant

• Quenched QCD

• Means neglecting quark loops

• Vacuum polarization turned off

• Reduces computation time by a factor of about 103-105

• But quenched QCD is not a controlled approximation

• Usually one encounters errors of

the order of 10 - 20 %

• False long range behaviour


R. G

upta

, “I

ntr

oduction t

o L

att

ice Q

CD

”, a

rXiv

:hep-l

at/

9807028


• Another issue:

• Quark masses very small compared to QCD scale of

• Forced to simulate at higher quark masses to avoid numerical instability

Sign problem

• At least the Dirac operator is hermitian

• Eigenvalues and determinant are real

• Positive probability measure allows Monte Carlo sampling

• Introduction of a chemical potential changes everything



Sign problem

• Eigenvalues and determinant become complex

• Complex determinant prevents Monte Carlo sampling

• Physical meaning:

• Both time directions no longer equivalent since chemical potential induces

more quarks than anti-quarks

• Particles travel forwards in time and anti-particles backwards

• Sign problem even worse for staggered fermions

• Fourth root of a complex number has to be taken



Sign problem

• Same problem also relevant for quantities like transport

coefficients which depend on the spectral function

• Real-time properties crucial

• Translation of Euclidian observables into Minkowski space

• Possible ways how to avoid/solve the sign problem in order to

simulate at non-zero densities follow in the next part of this talk


withEuclidiancorrelator

Thermodynamics of QGP

Phase diagram

• How to find and

describe the phase

transition to QGP?

• Find an appropriate

order parameter

and calculate it„s

expectation values!


Simon Hands: »The Phase Diagram of QCD«, Phys. Rev. D 77 (2008) 034504.


• Chiral condensate

• Exact order parameter formassless up/down quarks

• Indicator of a phasetransition

• Should be nonzero belowand zero above criticaltemperature

• Defined for each quarkflavor


MIL

C C

ollabora

tion, C.

Bern

ard

et

al., Q

CD

therm

odynam

ics w

ith thre

e fla

vors

of

impro

ved

sta

ggere

d q

uark

s, Phys. Rev. D

71 (

2005)

034504,

[arX

iv:

hep-l

at/

0405029].


• Polyakov loop

• Order parameter for deconfining phase transition in the limit of

infinite quark masses

• Measures how introduction of a static quark would change the free

energy

• Should be zero below and nonzero above critical temperature



• is at low

temperatures the binding

energy of a light meson

• Inside of QGP is

the screening energy

• Should decrease abovethe phase transition


R. G

upta

et

al., The E

OS fro

m s

imula

tions o

n B

lueG

ene

L S

uperc

om

pute

r at

LLN

L a

nd N

YBlu

e, PoS

LAT2008 (

2008)

170.


• Chiral susceptibilities

• Measure fluctuations of chiralcondensate for different flavours

• In the region of a phase transitionorder parameters exhibit large fluctuations

• E.g. disconnected chiralsusceptibility


HotQ

CD

Collabora

tion, C.

DeTar

and R

. G

upta

, Tow

ard

a p

recis

e d

ete

rmin

ation o

f Tc

with

2+

1 fla

vors

of

quark

s, PoS

LAT2007 (

2007)

179,

[arX

iv:

0710.1

655].


Phase transition

• Order of the phase

transition at zero

baryon density for

different quark masses


E. Laermann and O. Philipsen, Status of lattice QCD at finite temperature, Ann. Rev. Nucl. Part. Sci. 53 (2003) 163–198, [arXiv: hep-ph/0303042].


Equation of state

• Energy density

• Defined as

• Since the common way is to variate

• Dependence of coupling and masses on is crucial but hard to compute

• Pressure

• Defined as

• Same problems but for



Equation of state

• Interaction measure

• Defined as

• To get physical results one has to substract values at zero temperature

• Assuming that is proportional to volume one gets

• Other quantities, like entropy or energy density



Equation of state


E H

otQ

CD

Collabora

tion, A.

Bazavov

et

al., Equation o

f sta

te a

nd Q

CD

tra

nsitio

n a

t finite

tem

pera

ture

, arX

iv:

0903.4

379.


Non-zero density

• Some methods to simulate at non-zero baryon density

• Reweighting

• Simulating with the absolute value of the determinant

• Reweighting results with

• Breaks down for large volumes and

• Taylor expansion

• Expand observables like pressure or susceptibilities in a Taylor

series in starting at



Non-zero density

• Coefficients can be evaluated with standard Monte Carlo

• But valid only for small and for the region with

• Imaginary chemical potential

• Analytic continuation to real

• Analytic form as a function of chemical potential needed

• But simulations can only produce a discrete amount of data points

• Using a Taylor series as ansatz previous method

• How severe is this problem at all?



Non-zero density


K. Splittorff and J. J. M. Verbaarschot, Phys. Rev. D 77, 014514 (2008) [arXiv:0709.2218 [hep-lat]].


Non-zero density

• Application of imaginary , current results

favor the scenario on the right-hand side


P. de Forcrand and O. Philipsen, The chiral critical line of N(f) = 2+1 QCD at zero and non-zero baryon density, JHEP 01 (2007) 077, [arXiv: hep-lat/0607017].


Non-zero density

• Stochastic quantization

• Here the strategy is to generate the equilibrium ensemble without

Monte Carlo integration

• Starting with Langevin equation

• Here is a scalar field and is a fifth coordinate, called

„Langevin time“

• No QCD applications yet


Short summary

Lattice QCD

Is a powerful tool to examine non-perturbative effects in QCD

Uses Monte Carlo methods to compute observables

Is computationally involved

Already accessible quantities/properties of QGP are

Equation of state (energy, entropy, pressure)

Chiral condensate and susceptibilities

Confinement

Challenges are

Fermions with exact chiral symmetry

Sign problem and non-zero density


Sources

C. DeTar and U. M. Heller, “QCD Thermodynamics from the Lattice”, Eur. Phys. J. A 41 405 (2009).

S. Bethke, Prog. Part. Nucl. Phys. 58 (2007) 351.

G.S. Bali and K. Schilling, Phys. Rev. D47 (1993) 661.

R. Gupta, “Introduction to Lattice QCD”, arXiv:hep-lat/9807028

K. Yagi, T. Hatsuda and Y. Miake, “Quark-Gluon Plasma”.

Simon Hands, Phys. Rev. D 77 (2008) 034504.

MILC Collaboration, C. Bernard et al., Phys. Rev. D71 (2005) 034504.

R. Gupta et al., PoS LAT2008 (2008) 170.

HotQCD Collaboration, C. DeTar and R. Gupta, PoS LAT2007 (2007) 179.

E. Laermann and O. Philipsen, Ann. Rev. Nucl. Part. Sci. 53 (2003) 163–198.

E HotQCD Collaboration, A. Bazavov et al., Equation of state and QCD transition at finite temperature, arXiv: 0903.4379.

K. Splittorff and J. J. M. Verbaarschot, Phys. Rev. D 77, 014514 (2008).

P. de Forcrand and O. Philipsen, JHEP 01 (2007) 077.


The End

Thank you for your attention!

Questions?


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