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Introduction to Lattice Gauge Theories I

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Introduction to Lattice Gauge Theories I 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¯ >,< trGµν Gµν >, ··· = 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,µ, 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 µ 2 0 4! 0 Z a x xn −→ a 4 N φ(x) φn φ(xn) = 4 −→ ≡ L 1 ∂µφ(x) (φn+ˆµ 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+ˆµ − 2 n 2 a µ=1 = no unique discretization, ⇒ = even more - systematic improvement scheme exists (Symanzik). ⇒ Boundary condition: φ(x + Lµ µˆ)= φ(x) · π 2π π Spectrum: kµ = nµ − a ≤ Lµ ≤ 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+ˆµ + λ (ϕ 1) + ϕ − n − n n,µ 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+ˆµ → · − ϕ =±1 n,µ ! nX X = Ising model in 4D, ⇒ nd = 2 order phase transition at κc =0.0748.
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