Introduction to Lattice GaugeTheory and Some Applications

Roman Hollwieser

Derar Altarawneh, Falk Bruckmann, Michael Engelhardt, Manfried Faber,

Martin Gal, Jeff Greensite, Urs M. Heller, Andrei Ivanov, Thomas Layer,

Martin Luscher, Stefan Olejnik, Mario Pitschmann, Hugo Reinhardt,

Thomas Schweigler, Lorenz von Smekal, Wolfgang Soldner, Hideo

Suganuma, Mithat Unsal, Markus Wellenzohn, Bo Zhang

OutlookQuantum Chromo Dynamics

Motivation & IntroductionFormalism (→ formulas ;-)Properties of QCD VacuumMethods to explore QCDExperiments & Successes

Lattice QCD

Path Integral FormalismEuclidean FormulationLattice DiscretizationQCD on the Lattice

Center Vortices

Vortex Picture of Quark ConfinementCenter Vortices and Chiral Symmetry BreakingApproaching full QCD from smeared Center VorticesRandom Center Vortices in 3D/4D Space-Time Continuum

Electric Polarizabilities of the Neutron in Lattice QCD

Motivation & Introduction

Dr. Heinrich Faust in Johann Wolfgang von Goethes Faust I:Dass ich erkenne, was die Welt, im Innersten zusammenhalt.

(So that I may perceive whatever holds,the world together in its inmost folds.)

Theory of strong interactions between quarks and gluons.

The Eightfold Way

Lowest Iπ =1

2

+

-baryon-octet and lowest Iπ = 0−-meson-octet

Y = S + C + B ′ + T + B = 2(Q − I3) . . .⇒ SU(3)-multipletts

Quarks and Antiquarks

|u〉 ↔

100

, |d〉 ↔

010

, |s〉 ↔

001

quark-triplet anti-triplet

Baryons → |qqq >, e.g. p = |uud >, n = |udd >

Mesons → |qq >, e.g. π+ = |ud >

Problem: ∆++ = |uuu > with parallel spins and vanishing orbitalangular momentum → baryon wave function symmetric → Pauliexclusion principle → color charge

colored quarks + gluons → colorless hadrons ⇒ Confinement

Fields

Quarks: ψa(x)

massive spin-1/2 fermions with color chargeDirac-fields in the fundamental representation 3 of SU(3)electric charge −1/3 or 2/3 and weak isospinbaryon number 1/3, hypercharge and flavor

Gluons: Aµa (x)

spin-1 bosons with color chargeadjoint representation 8 of gauge group SU(3)no electric charge, no weak interaction, no flavor

Dynamics

Three basic interactions:1) quark emits (absorbs) gluon2) gluon emits (absorbs) gluon3) gluon interacts with gluon

Feynman-diagrams:

Lagrangian...summarizes dynamics of the system (L=T-V)

LQCD = LDirac + LGauge

= ψ(iγµDµ −m)ψ − 1

4F aµνFµν

a

= ψ(iγµ(∂µ + igTaAaµ)−m)ψ − 1

4F aµνFµν

a

= ψ(iγµ∂µ −m)ψ − g(ψγµTaψ)Aaµ −

1

4F aµνFµν

a

= ψiγµ∂µψ − ψmψ − gψγµTaψAaµ −

1

4F aµνFµν

a

with the Gluon Field Strength Tensor

F aµν = ∂µAa

ν − ∂νAaµ − gfabcAb

µAcν

γµ . . .Dirac matrices, m . . .fermion mass, g . . .coupling constant,Ta . . . generators of SU(3), fabc . . .structure constant

Lagrangian

LQCD = ψ(i~cγµ∂µ −mc2)ψ

− 1

4(∂µAνa − ∂νAµa )(∂µAν, a − ∂νAν, a)

− g cψγµFaψAµa

+g

~fabc(∂µAν, a)Aµb(x)Aνc (x)

− 1

4

g 2

~2fabcAµ, b(x)Aν, c(x)fadeAµd(x)Aνe (x)

Properties

Important Properties of QCD are

Asymptotic Freedomin very high energy reactions (small distances),quarks and gluons interact very weakly

Color Charge Gainanti screening of color charged gluons

Quark-Gluon Plasmaphase of (almost) free moving quarks and gluons

Confinementforce between quarks does not diminish as they are separated

Chiral Symmetry Breakingleft- and right-handed quarks transform differently

Color Confinement

hadrons are colorless

color charged particles (quarks)cannot be isolated

color flux lines are compressed to aflux tube (string)

linear rising quark-antiquarkpotential

V (r) ≈ σr − π

12r+ c

with string tension σ(√σ ≈ 0.44GeV)

color electric flux-tube

quark-antiquark pair production

MethodsPerturbative QCD

asymptotic freedom allows perturbation theoryaccurately in experiments performed at very high energiesmost precise tests of QCD to date

Lattice QCD

discrete set of space-time points (lattice)solve path integrals on discrete space-timeinsight into parts of theory, inaccessible by other means

1/N expansion

starts from the premise that the number of colors is infiniteseries of corrections to account for the fact that it is notmodern variants include the AdS/CFT approach

Effective theories (special theories for specific problems)

chiral perturbation th. (expansion around light quark masses)heavy quark eff. theory (expansion around heavy quark masses)soft-colinear eff. th. (exp. around large ratios of energy scales)Nambu-Jona-Lasinio model, Effective Infrared Vortex Models

Experimentsfirst evidence for quarks in deep inelastic scattering at SLAC(Standford Linear Accelerator)

first evidence of gluons in three jet events at PETRA(Positron-Electron Tandem Ring Accelerator)

good quantitative tests of perturbative QCD

running of coupling as deduced from many observationsscaling violation in un-/polarized deep inelastic scatteringvector boson production at collidersjet cross section in collidersevent shape observables at the LEPheavy-quark production in colliders

best quantitative test of non-perturbative regime is therunning of the coupling as probed through lattice calculations

Lattice QCD

Path Integral in quantum mechanicsclassical mechanics: exact path of particle

quantum mechanics: quantum amplitude 〈q′t′|qt〉between initial |qt〉 and final |q′t′〉 state

time development of states described by Hamiltonian H

|q′t′〉 = e iH(t−t′)|qt〉

⇒ 〈q′t′|qt〉 = 〈q′t′|e−iH(t′−t)|qt〉

take n time-steps ∆t ⇒ insert n − 1 eigen-states

〈qtn |qt〉 =

∫. . .

∫dqt1 . . . dqtn−1

〈qtn |e−iH∆t |qtn−1〉 . . . 〈qt1 |e−iH∆t |qt0〉

integration over all possible paths

limn→∞

∫. . .

∫dqt1 . . . dqtn . . .→

∫Dx

Path Integral in quantum mechanics

〈qtn |qt0〉 =

∫Dxe iS

Eucliden Continuation

imaginary exponent e iS ⇒ non-converging integrals

extend real time t to imaginary

t → −iτ (τ > 0)

〈q′|e−iHt |q〉 → 〈q′|e−Hτ |q〉 =

∫Dxe−SE

every path contributes to quantum amplitude with e−SE

paths with high action are suppressed ⇒ classical mechanics

Minkovski-metric

ds2 = −dt2 + dx21 + dx2

2 + dx23

changes to Euclidean-metric

ds2 = dτ2 + dx21 + dx2

2 + dx23

Discretization on the lattice

path integral: time discretization and lim∆t→0

field theory: space-time discretization ⇒ lattice

xµ = anµ, a . . . lattice constant, nµ ∈ Z, µ = 0, 1, 2, 3

discrete derivatives and integrals (sums)

∂µφ(x) → ∆µφ(x) =1

a[φ(x + aµ)− φ(x)]∫

d4x → a4∑x

continuum limit: lattice spacing a→ 0 and volume →∞

QCD on the Lattice

matter field ψ(x) defined only on lattice sites xµ

gauge field Aµ(x) (gluons) defined on “links” (edges)

Uµ(x) = e iagAµ(x)

with lattice spacing a and renormalized coupling g

→ parallel transporter

Lattice Gauge Action

gauge invariant terms → closed loops of linkssimplest form is the “plaquette”

Uµν(x) = U†ν(x)U†µ(x + aν)Uν(x + aµ)Uµ(x)

Lattice Gauge Action

plaquette loop with the link elements

Uµν(x) =

exp {iag [(Aν(x + aµ)− Aν(x))− (Aµ(x + aν)− (Aµ(x))]}

discretization of the field strength

Fµν(x) = ∂µAν(x)− ∂νAµ(x) ⇐⇒

Fµν(x) =1

a[(Aν(x + aµ)− Aν(x))− (Aµ(x + aν)− (Aµ(x))]

in the continuum limit one identifies

Uµν(x) = e ia2gFµν(x)

Lattice Gauge Action

possible and very common choice:

Wilson’s pure gauge action

SW = β∑x ,µ<ν

a4

[1− 1

2

(Uµν(x) + U†µν(x)

)]=

1

4

∑x ,µ,ν

a4F 2µν

with lattice spacing a and (inverse) coupling β = 1/g 2

tends in continuum limit to 14

∫dx4F 2

µν

action can be improved by including bigger (Wilson) loopswith perturbatively determined couplings (Luscher-Weiszaction)

mean field approximations to make action more“continuum-like” (tadpole improvement)

Lattice Fermion Action

naive discretization of the Dirac operator

Kogut-Susskind term

ψDKSψ =1

2aψ(x)

∑µ

γµ[Uµ(x)ψ(x + µ)−U†µ(x− µ)ψ(x− µ)]

higher terms may be included (asqtad improvement)

fermionic action:

ψDKSψ + . . .+ mq

∑x

ψ(x)ψ(x)

Simulation

Monte-Carlo method determines sequence of configurations Ci

“Markov-chain” Ci , i = 1,N → “Markov process”

start with random (hot) or ordered (cold) configuration

different update algorithms to reach balance, p.ex.:

Metropolis algorithm:e−[S(Cnew )−S(Cold )] < ρ, random number ρ ∈ [0, 1]Heat bath algorithm:

acceptance probability P(Cnew ) = e−∆S

e−∆S+e+∆S

representative set of states → expectation value of observable

〈O〉 ≈ 1

N

N∑i=1

O(Ci )

Wilson loop

closed loops around rectangular (R × T ), planar contour C

W (R,T ) = 〈∏x∈C

Uµ(x)〉 → e−σRT

quark-antiquark “test-pair”

heavy quark potential in limit T →∞

V (R) = limT→∞1

Tln〈W (R,T )〉 → −σR

perimeter/area law → Confinement

Creutz ratio → σ . . .string tension

χ =W (R + 1,T + 1)W (R,T )

W (R + 1,T )W (R,T + 1)→ e−σ ⇒ σ = −ln(χ)

Confinement due to Magnetic Monopoles

type II superconductor dual superconductor

magnetic fluxoid quantization electric fluxoid quantization

Center Vortices

Center Vortices

35 years of vortices

Ü ’t Hooft 1979, Nielsen, Ambjorn, Olesen, Cornwall, 1979

Mack, 1980; Feynman, 1981

QCD vacuum is a condensate of closed magnetic flux-lines,they have topology of tubes (3D) or surfaces (4D),

magnetic flux corresponds to the center of the group,

Vortex model may explain ...

Confinement → piercing of Wilson loop ≡ crossing of staticelectric flux tube and moving closed magnetic fluxTopological charge: vortices carry topological charge atintersection points and writhing points + color structureSpontaneous chiral symmetry breaking: alsocenter-projected configurations show χSB

Center Projection and Vortex Removal

A plaquette is pierced by a P-vortex, if the product of its centerprojected links gives -1.

Structure of P-Vortices

In 4D they form closed 2D-surfaces in Dual Space,Random Structure

3-dimensional cut through the dual of a 124-lattice.

Area law for center projected loops

denote f the probability that a plaquette has the value -1

〈W (A)〉 = [f · (−1) + (1− f ) · 1]A = exp[ln(1− 2f )︸ ︷︷ ︸−σ

A],=

= exp[−σR × T ], σ ≡ − ln(1− 2f ) ≈ 2f

Center Dominance

Creutz ratios: χ(I , J) = W (I ,J) W (I−1,J−1)W (I−1,J) W (I ,J−1) → σ

Continuous 3D Center Vortex Line Model

String Tension

OutlookExtend the Continuous Vortex Model to 4D...

Approaching full QCD by Vortex Smearing

Topological Susceptibility

Neutron Electric Polarizabilities

Interaction with weak electromagnetic field:

LQM = ψ(x , t)[∂

∂t+ (−i ~∇− q~A)2 − µ~σ · ~B +

1

2αE~E 2 − 1

2β~B2

− i

2γE1~σ · ~E × ~E −

i

2γM1~σ · ~B × ~B + . . .]ψ(x , t)

~E = −∂~A

∂t− ~∇A4, ~B = ~∇× ~A, ~E =

∂~E

∂t, ~B =

∂~B

∂t

Lattice → Energy/Mass shift:

∆m = m(~E )−m(0) = −1

2αE~E 2 − 1

2γE1~σ ·(~E× ~E ) − 1

2αEν

~E 2 +. . .

Thank you for your attention!

Questions?

Thank You &

Derar Altarawneh, Falk Bruckmann, Michael Engelhardt, Manfried Faber,

Martin Gal, Jeff Greensite, Urs M. Heller, Andrei Ivanov, Thomas Layer,

Martin Luscher, Stefan Olejnik, Mario Pitschmann, Hugo Reinhardt,

Thomas Schweigler, Lorenz von Smekal, Wolfgang Soldner, Hideo

Suganuma, Mithat Unsal, Markus Wellenzohn, Bo Zhang