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Introduction to Lattice GaugeTheory and Some Applications
Roman Höllwieser
Derar Altarawneh, Falk Bruckmann, Michael Engelhardt, Manfried Faber,
Martin Gal, Jeff Greensite, Urs M. Heller, Andrei Ivanov, Thomas Layer,
Štefan Olejnik, Mario Pitschmann, Hugo Reinhardt, Thomas Schweigler,
Lorenz von Smekal, Wolfgang Söldner, Mithat Unsal, Markus Wellenzohn
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 zusammenhält.
(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〉 ↔ 01
0
, |s〉 ↔ 00
1
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 aµνF
µνa
= ψ(iγµ(∂µ + igTaAaµ)−m)ψ −
1
4F aµνF
µνa
= ψ(iγµ∂µ −m)ψ − g(ψγµTaψ)Aaµ −1
4F aµνF
µνa
= ψiγµ∂µψ − ψmψ − gψγµTaψAaµ −1
4F aµνF
µνa
with the Gluon Field Strength Tensor
F aµν = ∂µAaν − ∂νAaµ − gfabcAbµAcν
γµ . . .Dirac matrices, m . . .fermion mass, g . . .coupling constant,Ta . . . generators of SU(3), fabc . . .structure constant
Lagrangian
LQCD = ψ̄(i~cγµ∂µ −mc2)ψ
− 14
(∂µAνa − ∂νAµa )(∂µAν, a − ∂νAν, a)
− ḡ cψ̄γµFaψAµa
+ḡ
~fabc(∂µAν, a)A
µb(x)A
νc (x)
− 14
ḡ 2
~2fabcAµ, b(x)Aν, c(x)fadeA
µd(x)A
νe (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
Asymptotic Freedom
in high-energy scattering quarks move within nucleons mostlyas free non-interacting particles (QGP)
first discovered in SU(2) gauge theory as a mathematicalcuriosity
Screening and Anti screening
screening by virtual charged particle-antiparticle pairs (vacuumpolarization)anti screening by virtual gluons carrying color charge andanti-color magnetic moment
beta-function describing the variation of theory’s coupling
β(α) =α2
π
(−11N
6+
nf3
)negative beta-function ⇒ Asymptotic Freedom
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 − π12r
+ 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′〉 statetime 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 integralsextend 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 mechanicsMinkovski-metric
ds2 = −dt2 + dx21 + dx22 + dx23
changes to Euclidean-metric
ds2 = dτ2 + dx21 + dx22 + dx
23
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 + aµ̂)− 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) = eiagAµ(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 + aν̂)Uν(x + aµ̂)Uµ(x)
Lattice Gauge Action
plaquette loop with the link elements
Uµν(x) =
exp {iag [(Aν(x + aµ̂)− Aν(x))− (Aµ(x + aν̂)− (Aµ(x))]}
discretization of the field strength
Fµν(x) = ∂µAν(x)− ∂νAµ(x) ⇐⇒
Fµν(x) =1
a[(Aν(x + aµ̂)− Aν(x))− (Aµ(x + aν̂)− (Aµ(x))]
in the continuum limit one identifies
Uµν(x) = eia2gFµν(x)
Lattice Gauge Action
possible and very common choice:
Wilson’s pure gauge action
SW = β∑x ,µ
Lattice Fermion Action
naive discretization of the Dirac operator
Kogut-Susskind term
ψ̄DKSψ =1
2aψ̄(x)
∑µ
γµ[Uµ(x)ψ(x + µ̂)−U†µ(x− µ̂)ψ(x− µ̂)]
higher terms may be included (asqtad improvement)
fermionic action:
ψ̄DKSψ + . . .+ mq∑x
ψ̄(x)ψ(x)
Simulation
Monte-Carlo method determines sequence of configurations Ci
representative set of states → expectation value of observable
〈O〉 ≈ 1N
N∑i=1
O(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
finite-size effects: a
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 → ConfinementCreutz 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, 1979Mack, 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 − 12β~B2
− i2γE1~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) = −12αE ~E
2 − 12γE1~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,
Štefan Olejnik, Mario Pitschmann, Hugo Reinhardt, Thomas Schweigler,
Lorenz von Smekal, Wolfgang Söldner, Mithat Unsal, Markus Wellenzohn
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