Introduction to Lattice Gauge Theories I
M. M¨uller-Preussker Humboldt-University Berlin, Department of Physics
Dubna International Advanced School of Theoretical Physics - Helmholtz International School ”Lattice QCD, Hadron Structure and Hadronic Matter” September 5 to 17, 2011 Outline:
1. Introductory remarks: Why lattice field theory?
2. Path integrals in quantum (field) theory (a) Path integral and Euclidean correlation functions (b) Path integral quantization of scalar fields
3. Discretizing gauge fields (a) QCD: a short introduction (b) Gauge fields on a lattice (c) Wilson and Polyakov loops (d) Fixing the QCD scale (e) Renormalization group and continuum limit
4. Simulating gauge fields (a) How does Monte Carlo work? (b) Metropolis method
5. Fermions on the lattice (a) Naive discretization and fermion doubling (b) Wilson fermions and improvements (c) Staggered fermions (d) How to compute typical QCD observables Literature, text books: Path integrals:
• R.P. Feynman, A.R. Hibbs, Quantum mechanics and path integrals, Mc Graw Hill, 1965
• L.D. Faddeev, in Methods in field theory, Les Houches School, 1975
• V. Popov, Kontinualniye integraly v kvantovoi teorii polya i statisticheskoi fizikye, Moskva Atomizdat, 1976
• H. Kleinert, Path integrals in quantum mechanics, statistics, polymer physics, World Scientific
• G. Roepstorf, Path integral approach to quantum physics, Springer
• J. Zinn-Justin, Path integrals in quantum mechanics, Oxford University Press (2005), translated into russian
Lattice field theory:
• K. G. Wilson, Confinement of Quarks, Phys. Rev. D10 (1974) 2445
• J. Kogut, L. Susskind, Hamiltonian Formulation of Wilson’s Lattice Gauge Theories., Phys. Rev. D11 (1975) 395
• M. Creutz, Quarks, gluons and lattices, Cambrigde Univ. Press (1983), translated into russian
• H. Rothe, Lattice gauge theories - An introduction, World Scientific
• I. Montvay, G. M¨unster, Quantum fields on a lattice, Cambrigde Univ. Press
• J. Smit, Introduction to quantum fields on a lattice: A robust mate, Cambridge Lect. Notes Phys. 15 (2002) 1-271
• C. Davies, Lattice QCD, Lecture notes, hep-ph/0205181 (2002)
• A. Kronfeld, Progress in lattice QCD, Lecture notes, hep-ph/0209231 (2002)
• T. DeGrand, C. E. DeTar, Lattice methods for quantum chromodynamics, World Scientific (2006)
• C. Gattringer, C.B. Lang, Quantum Chromodynamics on the Lattice - An introductory presentation, Springer (2010)
More general on non-perturbative methods:
• Yu. Makeenko, Methods of contemporary gauge theory, Cambrigde Univ. Press (2002) 1. Introductory remarks: Why lattice field theory?
Allows to understand non-perturbative phenomena and to carry out ab initio computations in strong interactions (and beyond):
Spontaneous breaking of chiral symmetry: • QCD Lagrangian for 3 zero-mass flavours - among other symmetries -
symmetric with respect to SU(3)A:
ψ′ = eiξaγ5λa ψ, ψ =(u,d,s)
but < 0 ψψ¯ (x) 0 >=0 quark condensate | | = octet mesons (π’s, K’s, η) massless (Goldstone bosons). ⇒ Quark confinement: • Gluon field flux tube between static quarks: 4 αqq¯(r) r→∞ in pure gauge theory = Fqq¯(r)= σ string tension ⇒ 3 r2 −→ Scenarios for confinement made ‘visible’ in lattice QCD: - flux tube structure visualized
From Abelian projection of SU(2) LGT [Bali, Schilling, Schlichter, ’97] - condensation of U(1) monopoles: dual superconductor
’t Hooft; Mandelstam; Schierholz,....; Bali, Bornyakov, M.-P., Schilling;
Di Giacomo,...; ... - center vortices
’t Hooft; Mack; Greensite, Faber, Olejnik; Reinhardt,...; Polikarpov,...; ... - semiclassical approach via instantons, calorons, dyons ′ solving e.g. the problem of large η -mass (“UA(1)-problem”) better understood on the lattice, but responsibility for confinement ??? but cf. lecture by V. Zakharov Hadron masses, hadronic matrix elements, ...: • (cf. lectures by R. Sommer, M. Peardon, M. G¨ockeler, M. Polyakov)
Not calculable within perturbation theory of continuum QCD, need non-perturbative (model) assumptions ⊲ condensates < ψψ¯ >,< trG ν G ν >, = QCD sum rules ⇒ Novikov, Shifman, Vainshtein, Zakharov; ... ⊲ vacuum state scenarios: = most popular instanton liquid model ⇒ Callan, Dashen, Gross; Shuryak; Ilgenfritz, M.-P.; Dyakonov, Petrov; ... ⊲ Dyson-Schwinger and functional renormalization group equations
or alternatively effective field theories or quark models ...
Phase transitions: • QCD: lattice predicts deconfinement or chiral transition: hadrons quark-gluon plasma ⇐⇒ under extreme conditions T, , B cf. lectures by P. Petreczky, F. Karsch, C. Schmidt, M. Polikarpov, O. Teryaev Standard model and beyond: • Non-perturbative lattice approach very useful also for
– strongly coupled QED Kogut,...; Schierholz,...; Mitrjushkin, M.-P.,...; ... – Higgs-Yukawa model = e.g. bounds for Higgs mass, 4th generation ? ⇒ K. Jansen,...; J. Kuti,...; ... , – (broken) SUSY cf. lectures by D. Kazakov, S. Catteral
– lattice studies (large Nc, Nf ) motivated by string theory and AdS/CFT correspondence M. Teper,...; J. Kuti,...; V. Zakharov,...; ... – ... 2. Path integrals in quantum (field) theory
2.1. Path integral and Euclidean correlation functions Quantum physics mostly starts from hermitean, time-independent Hamiltonian
Hˆ n = En n , n =0, 1, 2,... with m n = δm,n, n n = 1ˆ . | | | | | n Schr¨odinger Equation for time evolution:
d i − Hˆ (t−t0) i~ ψ(t) = Hˆ ψ(t) , ψ(t) = e ~ ψ(t0) . dt | | | | or i − Hˆ (t−t0) ψ(x,t) x ψ(t) = dx0 x e ~ x0 x0 ψ(t0) . ≡ | | | | Standard task: find En , in particular ground state energy E0. { } Useful quantity for this: quantum statistical partition function:
ˆ ˆ 1 Z = tr e−βH = dx x e−βH x = free energy F (V,T )= kT log Z, β | | ⇒ − ≡ kB T i 1 Note formal replacement: real time ~ t imaginary time β . ↔ ≡ kB T Extracting the ground state E0 (Feynman-Kac formula):
Z = e−βEn e−βE0 1+ O(e−β(E1−E0)) for β . ∝ →∞ n
Extracting the energy (or mass) gap E1 E0 : − two-point correlation function in (Euclidean) Heisenberg picture (~ = 1): operators: Xˆ(τ)= eHτˆ xe−Hτˆ , states: Ψ = eHτˆ ψ(τ) , | | quantum statistical mean values:
ˆ ... Z−1 tr (... e−βH ) . ≡
ˆ ˆ Xˆ(τ) = Z−1 dx x e−H(β−τ)xeˆ −H(τ−0) x | | = Z−1 n xˆ n e−Enβ Z−1 0 xˆ 0 e−E0β + ... | | ∝ | | n 0 xˆ 0 + ... for β . ∝ | | →∞ For two-point function assume β (τ2 τ1) 1: ≫ − ≫
−1 −Hˆ (β−τ2) −Hˆ (τ2−τ1) −Hˆ (τ1−0) T Xˆ(τ2)Xˆ(τ1) = Z dx x e xeˆ xeˆ x | | Z−1 0 xˆ n 2e−(En−E0)(τ2−τ1) e−E0β + ... ∝ | | | | n Z−1 0 xˆ 0 2 + 0 xˆ 1 2e−(E1−E0)(τ2−τ1) e−E0β + ... ∝ | | | | | | | | Then connected two-point correlator:
2 X(τ2)X(τ1) T Xˆ(τ2)Xˆ(τ1) ( Xˆ(τ) ) ≡ − e−(τ2−τ1)(E1−E0)(1 + ...) ∝ Higher order correlations can be defined.
Z and ... allow path integral representations ´ala Feynman. Path integral representation Subdivide β = Nǫ = fix with N , ǫ 0 . →∞ → Use (N 1) times completeness relation dx x x = 1. − | | N −Hǫˆ −Hǫˆ −Hǫˆ Z = dx x e x − x − e x − x1 e x0 i N | | N 1 N 1| | N 2 | | i =1 with x0 x , i.e. “periodic boundary condition”. ≡ N
For one degree of freedom: Hˆ = Tˆ + Vˆ =p ˆ2/2m + V (ˆx) , [ˆx, pˆ]= i
′ ′ ′ −Hǫˆ ǫ − 1 (x −x)2− ǫ [V (x)+V (x )] 3 Transfer matrix : x e x = e 2ǫ 2 + O(ǫ ) , | | 2π = Z Dx(τ) e−SE [x(τ)] ⇒ ∼ Proof applies Trotter product formula: N exp( βHˆ ) exp( (Tˆ + Vˆ )ǫ)N = lim exp( ǫTˆ) exp( ǫVˆ ) − ≡ − N→∞ − − N ǫ N/2 m dx(τ) 2 Dx(τ) lim dxi, SE [x(τ)] dτ + V (x(τ)) . ≡ N→∞ 2π ∼ 2 dτ i =1 Result: summation over fictitious paths in imaginary time
xn τn
τn−1
τ3 τ τ2 1 x1
Our main interest: correlation functions in Heisenberg picture:
Ω(τ ′,τ ′′,...) = Z−1 Dx(τ) Ω(τ ′,τ ′′,...) e−SE [x(τ)] . (Proof as for partition function Z.)
= no operators, but trajectories x(τ) and Euclidean action S [x(τ)], ⇒ E = correlators resemble classical statistical averages, ⇒ = we use β = Nǫ with N finite, but large, i.e. “lattice approximation”. ⇒ 2.2. Path integral quantization of scalar fields
Lattice formulation allows non-perturbative computation • from first principles. Axiomatic field theory starts from lattice formulation. •
Illustration: scalar field, Euclidean space-time.
1 1 1 S[φ]= d4x (∂ φ)2 + V (φ) , V (φ)= m2φ2 + g φ4. 2 2 0 4! 0 Z a
x xn −→ a 4
N φ(x) φn φ(xn) =
4 −→ ≡ L 1 ∂ φ(x) (φn+ˆ φn) −→ a −
L = NSa [“Lattice” taken from A. Kronfeld, hep-ph/0209231]
4 4 1 2 1 S = a (φ φn) + V (φn) Latt n+ˆ − 2 n 2 a =1 = no unique discretization, ⇒ = even more - systematic improvement scheme exists (Symanzik). ⇒ Boundary condition: φ(x + L ˆ)= φ(x) π 2π π Spectrum: k = n − a ≤ L ≤ a Quantization: Path integral for time ordered correlation functions and vacuum transition amplitude
−1 −S[φ] <φ(x1 φ(xN ) > = Z dφ(x) φ(x1) φ(xN ) e (1) { } x Z Y Z = dφ(x) e−S[φ] (2) x Z Y
lattice approximation and rescaling φ φ √g0 . −→
− 1 −1 g SLatt <φn φn > = Z dφn φn φn e 0 (3) 1 N Latt { 1 N } n Z Y − 1 g SLatt ZLatt = dφn e 0 (4) n Z Y 1 f − H(qi,pi) Compare with: (T,V )= dqi dpi e kT Z i=1 R Q Scalar Higgs model in some detail:
ϕ 6λ 2 1 2λ 8κ φ = √2κ , g0 = , m = − − . a κ2 0 a2κ
2 2 2 SLatt = 2κ ϕnϕn+ˆ + λ (ϕ 1) + ϕ − n − n n, n n X X X (κ ‘Hopping parameter’) No lattice spacing visible. How does the continuum limit occur? In practice need for determining correlation lengths
−1 L4 = N4a ζ m a ≫ ≡ ≫ Continuum limit corresponds to 2nd order phase transition:
ζ/a whereas m1/m2 const. →∞ → λ : κ =fixed = ϕn = 1 . →∞ ⇒ ±
ZLatt const. exp 2κ ϕnϕn+ˆ → − ϕ =±1 n, ! nX X = Ising model in 4D, ⇒ nd = 2 order phase transition at κc =0.0748. ⇒
λ< : find critical line κ = κc(λ). ∞
Comment: renormalized coupling λren 0 for κ κc → → = non-interacting theory in the continuum limit or ⇒ = effective theory at finite cutoff. ⇒ 3. Discretizing gauge fields
3.1. QCD: a short introduction The strong force of hadrons realized by quarks interacting with gluons Quarks (= fermionic matter field) occur with • – Nc = 3 “colour” states (gauge group SU(Nc)), – Nf = 6 (?) “flavour” states (up, down, strange, charm, bottom, top, ...), – 4 “spinor” states. Gluons (= bosonic gauge field) occur with • 2 – N 1 = 8 “colour” states (adjoint repr. of SU(Nc)), c − – 4 space-time components. Colour neutral bound states:
qqq: baryons – p, n, ..., qq¯: mesons – π, ρ,K, ....
α QCD: quark fields ψ (x) (ψ (x)) , α =1,...,Nc, s =1,..., 4 f ≡{ f s } a minimally coupled to gluon fields A (x), =0,.., 3 , x =( x, t). Nf ¯ 4 1 ν ¯ S[ψ,ψ,A]= d x tr G G ν + ψf (iγ D mf )ψf − 2 − f =1 a a G ν = ∂ Aν ∂ν A + ig0[A ,Aν ] , D = ∂ 1 + ig0A , A = A T , − a 2 T (a =1,...,Nc 1) – generators of SU(Nc), a b abc−c a b a with [T ,T ]= if T , 2tr(T T )= δab, trT = 0. local gauge invariance – the main principle of the standard model • ′ ′ † i † ψ (x)= g(x)ψ(x), A (x)= g(x)A g (x) (∂ g(x))g (x), g(x) SU(Nc) − g0 ∈ leaves action S[ψ,ψ,A¯ ] invariant for any g(x). quantization with Euclidean path integral analogously to QM: • −1 −S [ψ,ψ,A¯ ] Ω(ψ,ψ,A¯ ) = Z DA DψDψ¯ Ω(ψ,ψ,A¯ ) e E DA = dA (x); Dψ = dψ(x) Grassmannian measure − x x = weak coupling (αs O(1)): perturbation theory, requires gauge fixing; ⇒ ≪ = stronger coupling (αs O(1)): lattice theory, Dyson-Schwinger eqs., ... ⇒ ≃ 3.2. Gauge fields on a lattice
K. Wilson ’74: Local SU(Nc) gauge symmetry kept exact by non-local prescription on a 4d Euclidean lattice = link variables ⇒ xn+ˆ a A (xn) = Un, P exp ig0 A (x)dx SU(Nc) ⇒ ≡ ∈ xn iag0A (xn) 2 e 1 + iag0A (xn)+ O(a ) ≃ ≃
Gauge transformation gn g(xn) SU(Nc) for a 0: ≡ ∈ → ′ † i † A (xn) gnA gn (gn+ˆ gn)gn ≃ − ag0 − ′ ′ 2 Un, = 1 + iag0A (xn)+ O(a ) † † 2 = iag0 gnA gn + gn+ˆ gn + O(a ), gn = gn+ˆ + O(a) 2 † = gn+ˆ (1 + iag0A (xn)+ O(a ))gn † = gn+ˆ Un, gn Transformation of a product of variables of connected links
′ ′ ′ † † U (n + ˆ +ˆν,n) U U = g U Un, g = g U(n + ˆ +ˆν,n)g ≡ n+ˆ ,ν n, n+ˆ +ˆν n+ˆ ,ν n n+ˆ +ˆν n can be generalized to arbitrary continuum path: “Schwinger line” (parallel transporter)
x2 U(x2,x1) = P exp ig0 A dx , x1 ′ † = U (x2,x1) = g(x2)U(x2,x1)g (x1). ⇒
= tr P exp ig0 A dx = invariant. ⇒ Combination with matter field: ψ′(x)= g(x)ψ(x), ψ′†(x)= ψ†(x)g†(x)
† = ψ (x2)U(x2,x1)ψ(x1) = invariant. ⇒ Lattice gauge action: from elementary closed (Wilson) loops (“plaquettes”)
† † Un, ν Un U U U , ≡ n+ˆ ,ν n+ˆν, n,ν W 1 2Nc S = β¯ 1 Re tr Un, ν , β¯ = G − 2 n, <ν Nc g0 1 4 ν 2 = a tr G G ν + O(a ), 2 n 1 4 ν d x tr G G ν . → 2 Proof: homework, tutorial. −→ Comments: W - Improvement of SG : suppression of O(a)-corrections by adding contributions from larger loops (e.g. various 6-link loops) or contributions of loops in higher group representations (e.g. adjoint representation) - Test: non-trivial vacuum field = pure gauge
(vac) † U g g , for arbitrary gn SU(Nc) n ≡ n+ˆ n ∈ W Re tr Un, ν = Nc = S =0. ⇒ G Path integral quantization: replacement dA (x) [dU]n, , [dU] “Haar measure”. x, −→ n, Properties: U, V ∈ G = SU(Nc) in fundamental representation, then • G[dU] = 1 normalization, R • G f(U)[dU] = G f(VU)[dU] = G f(UV )[dU], [d(UV )] = [d(VU)] = [dU], R R R • −1 G f(U)[dU] = G f(U )[dU]. R R Examples: iϕ ⇒ 1 ≡ 1 2π U(1) : U = e = [dU] = 2π dϕ, f(U)[dU] 2π 0 dϕ f(U(ϕ)); R R SU(2) : U = B0 + i σ · B, (Bi)2 = 1 =⇒ [dU] = 1 δ(B2 − 1) d4B, i π2 P σ – Pauli matrices.
Expectation values, correlation functions in pure gauge theory:
1 −S [U] −S [U] W = ( [dUn ]) W [U] e G , Z = ( [dUn ]) e G . Z n, n, Gauge invariance:
′ † ′ ′ U = g Un g = dU = dUn , S [U ]= S [U] n n+ˆ n ⇒ n G G Then immediately 1 ′ W ′ = ( [dU ′ ]) W [U ′] e−SG[U ] ′ n Z n, 1 ′ −S [U] = ( [dUn ]) W [U ] e G Z n, = invariant, if W [U] itself is gauge invariant. ⇒
Fortunately, holds in most of the applications of lattice gauge theory !
Exceptions: Quark, gluon, ghost propagators or various vertex functions. Gauge fixing is required, e.g. Landau/Lorenz gauge, as in perturbation theory. However, gauge condition not uniquely solvable - “Gribov problem”. Lattice discretized path integral allows to study analytically:
β¯ , g0 0: (lattice) perturbation theory by expanding all link • →∞ → iag A (x ) variables Un = e 0 n in powers of g0. Inverse lattice spacing a−1 acts as UV cutoff.
β¯ 0, g0 : strong coupling expansion by expanding in powers of β¯ • → →∞
1 exp β¯ 1 Re tr Un, ν . − − n, <ν Nc Mostly we have to rely on g0 O(1) = Monte Carlo simulation. ≃ ⇒ 3.3. Wilson and Polyakov loops Closed Wilson loops
W [U] tr P exp ig0 A dx ≡ play important rˆole for explaining quark confinement.
Sketch of the proof [cf. book H. Rothe, ’92]: Consider (infinitely) heavy QQ¯-pair. Gauge-invariant state at fixed t′:
ψ¯(Q)( x, t′) U( x, t′; y, t′) ψ(Q)( y, t′) Ω α β | with y ′ ′ ′ U( x, t ; y, t )= P exp ig0 A ( z, t ) dz . x Compute 2-pt. function for QQ¯-pair - propagation from 0 to t: ′ ′ Gα′,β′;α,β ( x , y ,t; x, y, 0)
(Q) ′ ′ ′ (Q) ′ (Q) (Q) = Ω T ψ¯ ′ ( y ,t)U( y ,t; x ,t)ψ ′ ( x ,t) ψ¯ ( x, 0)U( x, 0; y, 0)ψ ( y, 0) Ω . | { β α α β }| Euclidean time t = iτ; for τ , M expect − →∞ Q →∞ ′ ′ G ′ ′ δ(3)( x x ) δ(3)( y y ) C ′ ′ ( x, y) exp( E(R)τ) α ,β ;α,β ∝ − − α ,β ;α,β − E(R) = energy of the lowest state (above vacuum), i.e. QQ¯-potential.
Path integral representation:
1 (Q) (Q) (Q) ′ (Q) (Q) (Q) ′ ′ ¯ ¯ − ¯ Gα ,β ;α,β = DA Dψ Dψ ψ ′ ( y ,τ) ...ψ ( y, 0) exp( S[ψ ,ψ ,A]) Z Z Z β β Carry out (Grassmanian) fermion integration product of Green’s functions for Q, Q¯ and Det const for M . −→ → Q →∞ 4 Static limit – neglect spatial derivatives (iγ D ) (iγ D4) → Green’s function: −→ 4 ′ (4) ′ [iγ D4(A) M ] K(z,z ; A)= δ (z z ). − Q − Solution: τ ′ ′ ′ (3) ′ K(z,z ; A) P exp ig0 A0( z, t ) dt δ ( z z ) K ∼ − 0 1 4 with K containing projectors P± = (1 γ ). 2 ± Combine time-like phase factors at z = x, y with space-like ones U( x, 0; y, 0), U( y, τ; x, τ) . Result:
G ′ ′ W tr P exp ig0 A dx α ,β ;α,β ∼ ≡ with closed rectangular path of extension R τ. ×
For τ : W exp( E(R) τ). →∞ ∝ − Confinement: E(R) σR for large R W exp( σ (R τ)) (area law) ∝ ⇐⇒ ∝ − Polyakov loops and non-zero temperature QCD [cf. lecture by P. Petreczky]
[Gross, Pisarski, Yaffe, Rev.Mod.Phys. 53 (1981) 43;
Svetitsky, Yaffe, Nucl.Phys. B210 (1982) 423; Svetitsky, Phys.Rept. 132 (1986) 1]
Partition function:
Z = Trexp( βHˆ ), β =1/k T, − B = x (exp( ǫHˆ ))N x , Nǫ = β, | − | x β m 2 = C Dx exp( SE [x(t)]), SE = dt [ x˙(t) + V (x(t))] − 2 0 with x(0) = x(β), i.e. periodicity. For Yang-Mills theory analogous proof of path integral representation is non-trivial:
a – Hamiltonian approach within gauge A0 = 0, a – trace integration produces integration over auxiliary field A4 , = full Euclidean Yang-Mills action recovered. ⇒ Result: β 1 −SG[A] 3 ZG = C DAe , SG[A]= dt d x tr(G ν G ν ) 2 0 a a a a with A ( x, 0) = A ( x, β), and G ν = G ν T Euclidean.
Applied to lattice pure gauge theory: 1 −SG[U] ZG [dU]e , with U ( x, 0) = U ( x, β), β = aN4 = . ≡ k T B We are interested in the thermodynamic limit: V (3) . →∞ In practice, “aspect ratio” Ns/N4 1 and periodic b.c.’s in spatial directions. ≫ Extension to full QCD: time-antiperiodic boundary conditions for fermionic fields from trace of statist. operator: ψ( x, 0) = ψ( x, β), ψ( x, 0) = ψ( x, β). − − Polyakov loop:
N 1 4 L( x) tr U4( x, x4), (a = 1), ≡ Nc x 4=1 invariant w.r. to time-periodic gauge transformations g( x, 1) = g( x, N4 + 1). Physical interpretation:
L( x) = exp( βF ), − Q with FQ free energy of an isolated infinitely heavy quark. Proof goes analogously as for Wilson loop expectation value.
= F , i.e. L( x) 0 within the confinement phase. ⇒ Q →∞ → = L( x) order parameter for the deconfinement transition. ⇒
Spontaneous breaking of ZN center symmetry: 2πi ν zν = e N 1 Z SU(N), ν =0, 1,...,N 1 ∈ N ⊂ − commute with all elements of SU(N). For SU(2): zν = 1. ± Global ZN -transformation:
U4( x, x4) z U4( x, x4) for all x and fixed x4 → = L( x) zL( x) not invariant. ⇒ → = Plaquette values U zz⋆U = U , i.e. S invariant. ⇒ n,i4 → n,i4 n,i4 G SU(2)-case: L( x) L( x) . Both states have same statistical weight. →− = L( x) =0. ⇒ = Order parameter for the deconfinement transition: ⇒ 1 0 confinement L L( x) | | ≡ | V (3) | ∼ 1 deconfinement x analogously to spin magnetization for 3d Ising model.
SU(3)-case: L complex-valued.
Confinement L 0, → ≃ Deconfinement L zν , ν =0, 1, 2. → ≃
Notice: fermions break center symmetry. Then Polyakov loop – no order parameter, but still useful indicator. ⇒ Condensate ψψ¯ – order parameter for chiral symmetry restoration. ⇒ 3.4. Fixing the QCD scale Expectation values W (of Wilson loops etc.) in pure gauge theory only 2 depend on parameter β¯ 2Nc/g (and linear lattice extensions N ). ≡ 0 View some Monte Carlo results in gluodynamics.
V(r) [GeV] 0.5
0
perturbation theory −0.5
0 0.2 0.4 0.6 0.8 r [fm]
Static QQ¯ potential V (r) ( E(R)) at small distances from Wilson loops. ≡ Necco, Sommer, ’02
Alternative method: Polyakov loop correlator † L( x1L( x2 ) exp( βV (r))(1 + ...),r x1 x2 ∝ − ≡| − | 1.5 1.0 fm 0.5 fm V // (r) [GeV/fm] 1.4
1.3
1.2
1.1 σ 1 0 2 4 6 ( r [fm] ) -2
Static QQ¯ force F (r) dV/dr from Polyakov loop correlators at T = 0. ≡ = in perfect agreement with string model prediction ⇒ π V (r) σr + + O(r−2). ∼ − 12 r
L¨uscher, Weisz, ’02
F (r) allows to fix the scale by comparing with phenomenologically knowncc ¯ - or ¯bb-potential [R. Sommer, ’94]:
2 F (r0) r = 1.65 r0 0.5 fm 0 ↔ ≃
If Sommer scale r0 is determined in lattice units a for certain β¯, spacing a(β¯) can be fixed in physical units. 3.5. Renormalization group and continuum limit
Assume for a physical observable Ω (e.g. string tension σ, critical temperature Tc, glueball mass Mg,...) in the continuum limit
lim Ω(g0(a),a)=Ωc a→0 Then renormalization group Eq. d Ω ∂ ∂ =0 = β(g0) Ω(g0,a)=0 d(ln a) ⇒ ∂(ln a) − ∂g 0 ∂g0 with β(g0) known from perturbation theory ≡− ∂(ln a) 3 2 4 β(g)/g = β0 β1g + O(g ). − −
For SU(Nc) and Nf massless fermions, independent on renormalization scheme: 1 11 2 β0 = Nc Nf , (4π)2 3 − 3 2 1 34 2 10 Nc 1 β1 = N NcNf − Nf . (4π)4 3 c − 3 − N c For pure gluodynamics: Nc =3, N =0 β0 > 0. f ⇒ Solution yields continuum limit
− β1 1 2 2β2 1 2 a(g0)= (β0g ) 0 exp (1 + O(g )). Λ 0 − 2β g2 0 Latt 0 0
= 1/a for g0 0 (or β¯ ), asymptotic freedom. ⇒ →∞ → →∞
Corresponds to second order phase transition.
In practice, tune g0,N4 for getting correlation lengths ξ, such that:
N4 ξ/a(g0) 1 . ≫ ≫
At non-zero T =1/L4 =1/(N4a(β¯)) :
T can be varied by changing β¯ at fixed N4 or N4 at fixed β¯. 4. Simulating gauge fields 4.1. How does Monte Carlo work?
Realization in quantum mechanics:
M. Creutz, B. Freedman, A stat. approach to quantum mechanics, Annals Phys. 132(1981)427
Here consider n-dim. integral: f = dnxf(x)w(x) with 0 w(x) 1, w(x) dnx =1. Ω ≤ ≤ Ω w(x) 1 f = dnxf(x) dη = dnx dη f(x)Θ(w(x) η) − Ω 0 Ω 0 Importance sampling by selecting x in acc. with w(x):
(a) choose randomly: x Ω and η [0, 1], ∈ ∈ (b) acceptance check: accept x, if η satisfies η