Introduction to Lattice Gauge Theory and Some Applications Roman H¨ollwieser Derar Altarawneh, Falk Bruckmann, Michael Engelhardt, Manfried Faber, Martin Gal, Jeff Greensite, Urs M. Heller, Andrei Ivanov, Thomas Layer, Stefanˇ Olejnik, Mario Pitschmann, Hugo Reinhardt, Thomas Schweigler, Lorenz von Smekal, Wolfgang S¨oldner, Mithat Unsal, Markus Wellenzohn Outlook Quantum Chromo Dynamics Motivation & Introduction Formalism (! formulas ;-) Properties of QCD Vacuum Methods to explore QCD Experiments & Successes Lattice QCD Path Integral Formalism Euclidean Formulation Lattice Discretization QCD on the Lattice Center Vortices Vortex Picture of Quark Confinement Center Vortices and Chiral Symmetry Breaking Approaching full QCD from smeared Center Vortices Random 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 zusammenh¨alt. (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 1 + Lowest I π = -baryon-octet and lowest I π = 0−-meson-octet 2 0 Y = S + C + B + T + B = 2(Q − I3) ::: ) SU(3)-multipletts Quarks and Antiquarks 0 1 1 0 0 1 0 0 1 jui $ @ 0 A ; jdi $ @ 1 A ; jsi $ @ 0 A 0 0 1 quark-triplet anti-triplet Baryons ! jqqq >, e.g. p = juud >; n = judd > Mesons ! jqq >, e.g. π+ = jud > Problem: ∆++ = juuu > with parallel spins and vanishing orbital angular momentum ! baryon wave function symmetric ! Pauli exclusion principle ! color charge colored quarks + gluons ! colorless hadrons ) Confinement Fields Quarks: a(x) massive spin-1=2 fermions with color charge Dirac-fields in the fundamental representation 3 of SU(3) electric charge −1=3 or 2=3 and weak isospin baryon number 1=3, hypercharge and flavor µ Gluons: Aa (x) spin-1 bosons with color charge adjoint representation 8 of gauge group SU(3) no electric charge, no weak interaction, no flavor Dynamics Three basic interactions: 1) quark emits (absorbs) gluon 2) gluon emits (absorbs) gluon 3) gluon interacts with gluon Feynman-diagrams: Lagrangian...summarizes dynamics of the system (L=T-V) LQCD = LDirac + LGauge 1 = (iγµD − m) − F a F µν µ 4 µν a 1 = (iγµ(@ + igT Aa ) − m) − F a F µν µ a µ 4 µν a 1 = (iγµ@ − m) − g( γµT )Aa − F a F µν µ a µ 4 µν a 1 = iγµ@ − m − g γµT Aa − F a F µν µ a µ 4 µν a with the Gluon Field Strength Tensor a a a b c Fµν = @µAν − @νAµ − gfabc AµAν γµ :::Dirac matrices, m :::fermion mass, g :::coupling constant, Ta ::: generators of SU(3), fabc :::structure constant Lagrangian ¯ µ 2 LQCD = (i~cγµ@ − mc ) 1 − (@µAν − @νAµ)(@ A − @ A ) 4 a a µ ν; a ν ν; a ¯ µ − gc¯ γµFa Aa g¯ µ ν + fabc (@µAν; a)Ab (x)Ac (x) ~ 2 1 g¯ µ ν − fabc Aµ, b(x)Aν; c (x)fade Ad (x)Ae (x) 4 ~2 Properties Important Properties of QCD are Asymptotic Freedom in very high energy reactions (small distances), quarks and gluons interact very weakly Color Charge Gain anti screening of color charged gluons Quark-Gluon Plasma phase of (almost) free moving quarks and gluons Confinement force between quarks does not diminish as they are separated Chiral Symmetry Breaking left- and right-handed quarks transform differently Asymptotic Freedom in high-energy scattering quarks move within nucleons mostly as free non-interacting particles (QGP) first discovered in SU(2) gauge theory as a mathematical curiosity Screening and Anti screening screening by virtual charged particle-antiparticle pairs (vacuum polarization) anti screening by virtual gluons carrying color charge and anti-color magnetic moment beta-function describing the variation of theory's coupling α2 11N n β(α) = − + f π 6 3 negative beta-function ) Asymptotic Freedom Color Confinement hadrons are colorless color charged particles (quarks) cannot be isolated color flux lines are compressed to a flux tube (string) linear rising quark-antiquark potential π V (r) ≈ σr − + c 12r with string tension σ p ( σ ≈ 0:44GeV) color electric flux-tube quark-antiquark pair production Methods Perturbative QCD asymptotic freedom allows perturbation theory accurately in experiments performed at very high energies most precise tests of QCD to date Lattice QCD discrete set of space-time points (lattice) solve path integrals on discrete space-time insight into parts of theory, inaccessible by other means 1/N expansion starts from the premise that the number of colors is infinite series of corrections to account for the fact that it is not modern 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 Experiments first 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 observations scaling violation in un-/polarized deep inelastic scattering vector boson production at colliders jet cross section in colliders event shape observables at the LEP heavy-quark production in colliders best quantitative test of non-perturbative regime is the running of the coupling as probed through lattice calculations Lattice QCD Path Integral in quantum mechanics classical mechanics: exact path of particle quantum mechanics: quantum amplitude hq0t0jqt i between initial jqt i and final jq0t0i state time development of states described by Hamiltonian H iH(t−t0) jq0t0i = e jqt i −iH(t0−t) ) hq0t0jqt i = hq0t0je jqt i take n time-steps ∆t ) insert n − 1 eigen-states Z Z hqtn jqt i = ::: dqt1 ::: dqtn−1 −iH∆t −iH∆t hqtn je jqtn−1 i ::: hqt1 je jqt0 i integration over all possible paths Z Z Z limn!1 ::: dqt1 ::: dqtn ::: ! Dx Path Integral in quantum mechanics Z iS hqtn jqt0 i = Dxe Eucliden Continuation imaginary exponent eiS ) non-converging integrals extend real time t to imaginary t ! −iτ (τ > 0) Z hq0je−iHt jqi ! hq0je−Hτ jqi = Dxe−SE every path contributes to quantum amplitude with e−SE paths with high action are suppressed ) classical mechanics Minkovski-metric 2 2 2 2 2 ds = −dt + dx1 + dx2 + dx3 changes to Euclidean-metric 2 2 2 2 2 ds = dτ + dx1 + dx2 + dx3 Discretization on the lattice path integral: time discretization and lim∆t!0 field theory: space-time discretization ) lattice xµ = anµ; a ::: lattice constant; nµ 2 Z; µ = 0; 1; 2; 3 discrete derivatives and integrals (sums) 1 @ φ(x) ! ∆ φ(x) = [φ(x + aµ^) − φ(x)] µ µ a Z X d4x ! a4 x continuum limit: lattice spacing a ! 0 and volume ! 1 QCD on the Lattice matter field (x) defined only on lattice sites xµ gauge field Aµ(x) (gluons) defined on \links" (edges) iagAµ(x) Uµ(x) = e with lattice spacing a and renormalized coupling g ! parallel transporter Lattice Gauge Action gauge invariant terms ! closed loops of links simplest form is the \plaquette" y y Uµν(x) = Uν(x)Uµ(x + aν^)Uν(x + aµ^)Uµ(x) Lattice Gauge Action plaquette loop with the link elements Uµν(x) = exp fiag[(Aν(x + aµ^) − Aν(x)) − (Aµ(x + aν^) − (Aµ(x))]g discretization of the field strength Fµν(x) = @µAν(x) − @νAµ(x) () 1 F (x) = [(A (x + aµ^) − A (x)) − (A (x + aν^) − (A (x))] µν a ν ν µ µ in the continuum limit one identifies 2 ia gFµν (x) Uµν(x) = e Lattice Gauge Action possible and very common choice: Wilson's pure gauge action X 1 1 X S = β a4 1 − U (x) + Uy (x) = a4F 2 W 2 µν µν 4 µν x,µ<ν x,µ,ν with lattice spacing a and (inverse) coupling β = 1=g 2 1 R 4 2 tends in continuum limit to 4 dx Fµν action can be improved by including bigger (Wilson) loops with perturbatively determined couplings (Luscher-Weisz¨ action) mean field approximations to make action more \continuum-like" (tadpole improvement) Lattice Fermion Action naive discretization of the Dirac operator Kogut-Susskind term 1 X ¯D = ¯(x) γ [U (x) (x +µ ^)−Uy(x −µ^) (x −µ^)] KS 2a µ µ µ µ higher terms may be included (asqtad improvement) fermionic action: X ¯DKS + ::: + mq ¯(x) (x) x Simulation Monte-Carlo method determines sequence of configurations Ci representative set of states ! expectation value of observable N 1 X hOi ≈ O(C ) N i i=1 \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 2 [0; 1] Heat bath algorithm: e−∆S acceptance probability P(Cnew ) = e−∆S +e+∆S finite-size effects: a << ξ << L a :::latt. spacing, ξ : : :coherence length, L :::latt. extent Wilson loop closed loops around rectangular (R × T ), planar contour C Y −σRT W (R; T ) = h Uµ(x)i ! e x2C quark-antiquark \test-pair" heavy quark potential in limit T ! 1 1 V (R) = lim lnhW (R; T )i ! −σR T !1 T perimeter/area law ! Confinement Creutz ratio ! σ : : :string tension W (R + 1; T + 1)W (R; T ) χ = ! e−σ ) σ = −ln(χ) W (R + 1; T )W (R; T + 1) Confinement due to Magnetic Monopoles type II superconductor dual superconductor magnetic fluxoid quantization electric fluxoid quantization Center Vortices Center Vortices 35 years of vortices § '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 ..