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 Summer School ”Lattice QCD, Hadron Structure and Hadronic Matter” August 25 - September 5, 2014 Outline: • 1st Lecture 1. Introductory remarks: Why lattice field theory? 2. Path integrals in quantum (field) theory 2.1. Path integral and Euclidean correlation functions 2.2. Path integral quantization of scalar fields
• 2nd Lecture 3. Discretizing gauge fields 3.1. QCD: a short introduction 3.2. Gauge fields on a lattice 3.3. Wilson and Polyakov loops 3.4. Fixing the QCD scale 3.5. Renormalization group and continuum limit
• 3rd Lecture 4. Simulating gauge fields 4.1. How does Monte Carlo work? 4.2. Metropolis method 5. Fermions on the lattice 5.1. Naive discretization and fermion doubling 5.2. Wilson fermions and improvements 5.3. Staggeblu fermions 5.4. How to compute typical QCD observables
• 4th Lecture 6. Topology, cooling and the Wilson (gradient) flow (has not been presented during the lectures given) 6.1. Topology of gauge fields 6.2. Measuring topology on the lattice 7. Conclusions, outlook 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 (4th ed. 2012)
• 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) 1st Lecture
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. Bornyakov Hadron masses, hadronic matrix elements, ...: • (cf. lectures by C. Hoelbling, M. G¨ockeler)
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 O. Philipsen, G. Endrodi, P. Buividovich 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 ⇒ K. Jansen,...; J. Kuti,...; ... , – (broken) SUSY cf. lecture notes HISS school 2011 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):
ˆ Z = n e−βH n = e−βEn e−βE0 1+ O(e−β(E1−E0)) for β . | | ∝ →∞ n 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. ⇒ 2nd Lecture
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 Homework I: Prove the continuum limit in the (non-) Abelian case: 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) - Useful test for programmers: 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, new extended edition ’12]: 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 →∞ (3) ′ (3) ′ G ′ ′ δ ( x x ) δ ( 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) ∝ ⇐⇒ ∝ −
Simulation: V (R) E(R) (Sommer scale: r0 0.5 fm, see below). ≡ ≃
3 β = 6.0 β = 6.2 2 β = 6.4 Cornell 1 0 )] r
0 0
-1 [V(r)-V(r -2
-3
-4 0.5 1 1.5 2 2.5 3 r/r [G. Bali, Phys. Reports, ’01] 0 Polyakov loops and non-zero temperature QCD [cf. lecture by O. Philipsen]
[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 – order parameter for (de)confinement only if mq , ⇒ →∞ quark condensate ψψ¯ – strict order parameter ⇒ for chiral symmetry breaking / restoration only if mq 0. → 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
β − 1 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¯. 3rd Lecture
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
Homework II: Write heat bath code for 4D Ising model tutorial −→ 4.2. Creutz’ heat bath method
Assume statistical weight w[U] e−SG[U] with plaquette action ∼ 1 S [U]= β¯ 1 Re tr Un, ν . G − n, <ν Nc Select a single link variable: Un , U0. There are 6 plaquettes containing 0 0 ≡ this link and contributing to SG:
β¯ 6 SG[U0; U U0] = C[ U U0] Re tr (U0US ) { }\ { }\ − Nc S =1 β¯ 6 = C Re tr (U0A), A = US − Nc S =1 Call open plaquette US = “staple”. Assume A be fixed (“heat bath”) = link variable U0 to be determined with ⇒ probability β¯ w(U0) [dU0] exp Re tr (U0A) [dU0] ∼ N c . Metropolis: ′ - apply random shifts U0 U = GU0, with G Uǫ(1) SU(Nc), → 0 ∈ ⊂ - carry out Metropolis acceptance steps.
Heat bath for SU(2): Normalize V = A/√det A SU(2), put VU0 U and use invariance of Haar ∈ ≡ measure β¯√det A w(U)[dU] exp(ρ tr U)[dU], ρ = ∼ 2 1 U B01 + i σB, [dU]= δ(B2 1) d4B, tr U =2B0 ≡ π2 − = w(U)[dU] exp 2ρB0 δ(B2 1) d4B ⇒ ∼ † − = determine U and U0 = V U. ⇒
Heat bath for SU(Nc) [Cabibbo, Marinari, ’82] Use SU(2) heat bath algorithm for various subgroup embeddings into SU(Nc). 5. Fermions on the lattice 5.1. Naive discretization and fermion doubling Path integral for fermions requires anticomm. variables – Grassmann algebra: ηi, η¯i,i =1, 2,...,N withη ¯i adjoint to ηi, η , η η η + η η { i j } ≡ i j j i = η¯ , η = η¯ , η¯ = ... =0, { i j } { i j } 2 ηi = 0, such that any function has representation
f(η)= f0 + fi ηi + fij ηiηj + ...f12...N η1η2 . . . ηN , i i =j correspondingly for f(η, η¯). E.g. g(η, η¯) = exp( η¯ A η )= N (1 η¯ A η ). − i,j i ij j ij=1 − i ij j Integration rules (same as differentiation):
dηi = dη¯i =0, dηiηi = dη¯iη¯i =1, dη ,dη = dη¯ ,dη = dη¯ ,dη¯ = dη ,η = ... =0. { i j } { i j } { i j } { i j } Most important for us:
N N I[A] = dη¯ dη exp η¯ A η = det A, l l − i ij j l =1 ij =1 N N Z[A; ρ, ρ¯] = dη¯ dη exp η¯ A η + (¯η ρ +ρ ¯ η ) , l l − i ij j i i i i l =1 ij =1 i N = det A exp ρ¯ A−1ρ , i ij j ij =1 DηDη¯ ηiη¯j exp( ηAη¯ ) −1 ηiη¯j = − = A . DηDη¯ exp( ηAη¯ ) ij − 0 i Dirac Propagator (with Euclidean: γ ,γν =2δ ν , γ4 γ ,γ iγ ) { } ≡ M i ≡− M ¯ ¯ ¯ ¯ DψDψ ψα(x)ψβ (y)exp SF [ψψ ) ψα(x)ψβ (y) = − , DψDψ¯ exp S [ψψ¯ ] − F 4 SF [ψψ¯ ] = d x ψ¯(x)(γ ∂ + m)ψ(x). Naive lattice discretization: 1 1 rescale m M/a, ψα(xn) ψˆα(n), ∂ ψα(xn) ∂ˆ ψˆα(n) → → a3/2 → a5/2
1 ∂ˆ ψˆα(n)= [ψˆα(n + ˆ) ψˆα(n ˆ)] 2 − −
S [ψψ¯ ] ψˆ (n)K (n,m)ψˆ (m) F ≃ α αβ β n,m ;α,β 1 with K (n,m)= (γ ) [δ δ ]+ M δmnδ αβ 2 αβ m,n+ˆ − m,n− ˆ αβ Free lattice Dirac propagator: π/a 4 [ i γ p˜ + m] ¯ d p αβ ip(x−y) ψα(x)ψβ (y) = lim − e a→0 (2π)4 p˜2 + m2 −π/a 1 ˜ 2 withp ˜ = a sin(p a) to be compared with scalar case k = a sin(k a/2). For M = 0 in momentum space we get poles at all 2d corners of the Brillouin zone [(0000), (π/a 000),...]. = “Doubling of fermion degrees of freedom”. ⇒
Homework III: Determine lattice propagator for free scalar field tutorial −→ Theorem by Nielsen, Ninomiya, ’81: Doubling problem can be avoided only by giving up at least one of: - reflexion positivity, - cubic symmetry, - locality, - chiral invariance in the zero mass case.
5.2. Wilson fermions and improvements Wilson’s choice – break chiral symmetry even at m = 0:
W 1 ∗ ∗ DF = γ ∂ DLatt D = γ ( + ) ar , −→ ≡ F 2 ∇ ∇ − ∇ ∇ ∗ where ( ) forward (backward) gauge covariant derivatives ∇ ∇ 1 ψ(xn) = Un, ψ(n + ˆ) ψ(n) , ∇ a − ∗ 1 † ψ(xn) = ψ(n) Un,− ψ(n ˆ) , Un,− U ∇ a − − ≡ n− ˆ r – arbitrary real parameter, often chosen to be r = 1. Lattice free Wilson fermion action:
SW [ψψ¯ ] ψˆ (n)KW (n,m)ψˆ (m), F ≃ α αβ β n,m ;α,β W 1 K (n,m) = (M +4r)δmnδ (r γ ) δ +(r + γ ) δ αβ αβ − − αβ m,n+ˆ αβ m,n− ˆ 2 Propagator: π/a 4 [ i γ p˜ + m(p)] ¯ d p αβ ip(x−y) ψα(x)ψβ (y) = lim − e , a→0 (2π)4 p˜2 + m(p)2 −π/a 2r 2 p˜ as before, but m(p) = m + a sin (p a/2) For a 0 → π m(p) m for p = , → ± a π m(p) for p = . → ∞ ± a Problems:
Chiral SU(3) flavor symmetry explicitly broken. • A Eigenvalue value spectrum of DW strongly differs from continuum • F spectrum.
Discretization error δSW O(a) compared with δSW O(a2) • F ∼ G ∼ = improvement possible with clover term ⇒
clover W 5 i ˆ SF = SF + a cswψ(xn) σ ν F ν ψ(xn) . n 4 Sheikholeslami, Wohlert ’85 = alternative: twisted-mass fermions at maximal twist (cf. K. Jansen). ⇒ Chiral improvement: Ginsparg-Wilson fermions
Any lattice Dirac operator satisfying the Ginsparg-Wilson relation (GWR)
Ginsparg, Wilson ’82
γ5DLatt + DLattγ5 = aDLattγ5DLatt guaranties approximately local ( O(a)), but exact chiral symmetry. ∼ L¨uscher ’98 Strategies to solve GWR: Neuberger’s operator: exact solution of GWR • m † − 1 W D + m DN = 1+ (1 + A (A A) 2 ) , A =1+ s D → 2 − F Neuberger ’98 Properties: – (√A†A)−1 numerically involved: approximation by polynomials (Chebyshev approx.).
– Det DN hard to compute, but required for full fermionic simulation. – Discretization error still O(a2). Equivalent alternative domain wall fermions: extension to 5 dimensions • 1 ∗ ∗ W ρ DLatt = [γ5(∂ + ∂s) as∂ ∂s] + D , 0 <ρ< 2 2 s − s F − a with boundary condition
P+ψ( o,x)= P−ψ((Ns + 1) as,x)=0 , P± = (1 γ5)/2 ±
and limit Ns and as 0 →∞ → Kaplan ’92; Shamir ’93
Approximative methods: • – Renormalization group based perfect action approach
P. Hasenfratz, Niedermayer ’94; DeGrand, ... ’94
– generalized (less local) ansatzes for DLatt with parameters fixed from GWR
Gattringer ’01 5.3. Staggered fermions
Kogut, Susskind, ’75
Use naive discretization and diagonalize action w.r. to spinor degrees of • freedom.
Neglect three of four degenerate Dirac components. • Attribute the 16 fermionic degrees of freedom localized around one • elementary hypercube to four tastes with four Dirac indices each.
Chiral symmetry restored flavor symmetry broken. ⇐⇒ Naturally the mass-degenerated four-flavor case is described.
Rooting prescription:
for Nf = 2 + 1(+1) 4th-root of the fermionic determinant is taken. = Locality violated (??) ⇒ Improvement possible by ’smearing’ link variables. 5.4. How to compute typical QCD observables
Path integral quantization for Euclidean time = ’statistical averages’. ⇒ Fermions as anticommuting Grassmann variables = analytically integrated non-local effective action Seff (U). ⇒ ⇒ ’Partition function’ for f =1, ,N light quark flavors: f
G ¯ ¯ −S (U)+ f ψf Mf (U)ψf Z = [dU] [dψf ][dψf ] e P f −SG(U) = [dU] e DetMf (U) f eff = [dU] e−S (U), Seff (U) = SG(U) log(DetM (U)) − f f with M (U) D (U)+ m . f ≡ Latt f 3 To be simulated on a finite lattice Nt N , mostly with periodic boundary × s conditions for gluons (anti-periodic for quarks). Observables: mostly gauge invariant.
x2
x1
¯ ψu(x1) U(x1,x2) Γ ψd(x2) Ω(U)=tr l Ul Non-localfermioniccurrent Wilsonloop [Taken from C. Davies, hep-ph/0205181]
Pure gauge observables:
1 G ¯ ¯ −S (U)+ f ψf M(U)ψf Ω = [dU] [dψf ][dψf ] Ω(U) e Z P f 1 eff = [dU] Ω(U) e−S (U) Z Fermionic observables through correlators, ¯a a e.g. for local (u,d)-meson current H(x)= ψu(x)Γ ψd (x)
1 G ¯ † ¯ † −S + f ψf M(U)ψf H (x)H(y) = [dU] [dψf ][dψf ] H (x)H(y) e Z P f 1 eff = [dU] (M(U)−1(x,y))ab Γ (M(U)−1(y,x))ba Γ e−S (U) Z u d −1 propagator (M(U) (x,y))f , f = u,d computed with conj. gradient method.
Quenched approximation: put Det M(U) 1 , i.e. pure gauge field simulation. ≡ Det M(U) = 1 can be taken into account, but time consuming = Hybrid Monte Carlo, multibosonic algorithms,... ⇒ = massively parallel supercomputers required. ⇒ Typical lattice size for correlation functions in d=4 dimensions: Ns = 32,Nt = 64 Gauge field to be simulated: 2 3 3 7 N d N Nt = 9 4 32 64 7.5 10 degrees of freedom. c × × s × × × × ≃ Sparse Wilson-Dirac matrix to be inverted: 3 3 7 7 (4NcN Nt) (4NcN Nt) 2.5 10 2.5 10 s × s ≃ ×
Three critical limites: - Continuum limit: correlation lengths in lattice units a diverge, 3 - Box size L Lt (Ls Nsa,Lt Nta ) be sufficiently large, s × ≡ ≡ - Realistically small u-, d-quark masses zero modes of Dirac matrix, ⇒ ≈
zm z za 20 MeV Ls L 0.1 fm CPU C ≃ m 3 fm a quark 2007: empirical values for HMC algorithm for improved Wilson fermions (ETM collaboration)
C 0.3 Tflops years, zm 1, z 5, za 6 ≃ × ≃ L ≃ ≃ 0.00 Tflops · years Ukawa Urbach et al. 1
0 0.5 1 m PS /m V 0 “Berlin Wall” = CPU time versus fixed ratio mπ/mρ A. Ukawa, Lattice ’01, HU Berlin; updated C. Urbach, Lattice ’07, U. Regensburg At present: Simulations at the physical point even with improved Wilson fermions are becoming realistic !! Our aim: Computation of hadronic masses and matrix elements from various 2-point or 3-point functions 2-point functions:
vac O n n O vac T →∞ O (T )O (0) O O = | f | | i| e−MnT e−M0T f i − f i ∼ n 2Mn O H†, O H = extract masses of hadrons with quantum • f ≡ i ≡ ⇒ numbers related to non-local current H.
O J, O H = extract vacuum-to-hadron matrix elements • f ≡ i ≡ ⇒ (decay constants) with local current J.
J=A 0 0 T 0 T
2pt function for spectrum 2pt function for decay constant 3-point functions: vac H† m m J n n H vac H′†(T )J(t)H(0) = | | | | | | e−Mnte−Mm(T −t) n m 2Mn2Mm allow to extract experimentally relevant hadron-to-hadron matrix elements • for decay constants, moments of structure functions and form factors of hadrons. W J = V 0 , V i
0 t T 3pt function for SL decay
[As previous diagrams taken from C. Davies, hep-ph/0205181] 4th Lecture
6. Topology, Cooling and the Wilson (gradient) flow
6.1. Topology of gauge fields [Belavin, Polyakov, Schwarz, Tyupkin, ’75; ’t Hooft,
’76; Callan, Dashen, Gross, ’78 -’79]
1 4 Euclidean Yang-Mills action: S[A]= d x tr(G ν G ν ) − 2g2 Topological charge:
4 1 1 Qt[A] d x ρt(x), ρt(x)= tr(G ν G˜ ν (x)), G˜ ν ǫ νρσGρσ. ≡ − 16π2 ≡ 2 q Qt[A] w Z , ≡ i ∈ i=1 w “windings” of continuous mappings S(3) SU(2) (homotopy classes), i → invariant w.r. to continuous deformations (but not on the lattice !!) Example for topologically non-trivial field - “instanton”:
[Belavin, Polyakov, Schwarz, Tyupkin, ’75]
2 4 2 8π d x tr[(G ν G˜ ν ) ] 0 = S[A] Qt[A] − ± ≥ ⇒ ≥ g2 | | 8π2 iff S[A] = Qt[A] , then G ν = G˜ ν (anti) selfduality . g2 | | ±
Qt = 1: BPST one-(anti)instanton solution (singular gauge) for SU(2): | | 2 (±) aα (±) 2 ρ (x z)ν a, (x z,ρ,R)= R ηα ν − , A − (x z)2 ((x z)2 + ρ2) − − For SU(Nc) embedding of SU(2) solutions required.
Dilute instanton gas (DIG) instanton liquid (IL): −→ path integral “approximated” by superpositions of (anti-) instantons and represented as partition function in the modular space of instanton parameters.
[Callan, Dashen, Gross, ’78 -’79; Ilgenfritz, M.-P., ’81; Shuryak, ’81 - ’82; Diakonov, Petrov, ’84]
= may explain chiral symmetry breaking, but fails to explain confinement. ⇒ Case T > 0: x4-periodic instantons - “calorons”
Semiclassical treatment of the partition function [Gross, Pisarski, Yaffe, ’81] with “caloron” solution x4-periodic instanton chain ( 1/T = b) ≡ [Harrington, Shepard, ’77]
aHS (±) A (x) = ηa ν ∂ν log(Φ(x)) ρ2 πρ2 sinh 2π | x − z| Φ(x)−1 = = b ( x − z)2 +(x − z − kb)2 b| x − z| cosh 2π | x − z| − cos 2π(x − z ) kX∈Z 4 4 b b 4 4
1 b 3 - Qt = dx4 d x ρt(x)= 1 . − 16π2 0 ± - (as for instantons) it exhibits trivial holonomy, i.e. Polyakov loop behaves as:
b 1 | x|→∞ tr P exp i A4( x, t) dt = 1 2 ⇒ ± 0 Kraan - van Baal solutions = (anti-) selfdual caloron solutions with non-trivial holonomy
[K. Lee, Lu, ’98, Kraan, van Baal, ’98 - ’99, Garcia-Perez et al. ’99]
b=1/T | x|→∞ P ( x)= P exp i A4( x, t) dt = ∞ / Z(Nc) ⇒ P ∈ 0
Action density of a single (but dissolved) SU(3) caloron with Qt = 1 (van Baal, ’99) =⇒ not a simple SU(2) embedding into SU(3) !!
Dissociation into caloron constituents (BPS monopoles or “dyons”) gives hope for modelling confinement for T [Gerhold, Ilgenfritz, M.-P., ’07; Diakonov, Petrov, et al., ’07 - ’12; Bruckmann, Dinter, Ilgenfritz, Maier, M.-P., Wagner, ’12; Shuryak, Sulejmanpasic, ’12-’13; Faccioli, Shuryak, ’13] Systematic development of the semiclassical approach + perturbation theory “resurgent trans-series expansions” ... [Dunne, Unsal¨ and collaborators, ’12-’14] Axial anomaly [Adler, ’69; Bell, Jackiw, ’69; Bardeen, ’74] 5 ∂ j (x)= D(x)+2Nf ρt(x) Nf Nf 5 ¯ 5 ¯ 5 with j (x)= ψf (x)γ γ ψf (x), D(x)=2im ψf (x)γ ψf (x) f f =1 ρt = 0 due to non-trivial topology = solution of the U (1) problem: ⇒ A ′ η -meson (pseudoscalar singlet) for m 0 not a Goldstone boson, m ′ mπ. → η ≫ Related Ward identity: 2 4 4 4N d x ρt(x)ρt(0) = 2iN 2mψ¯ ψ + d x D(x)D(0) f f − f f 2 2 2 = 2iNf mπFπ + O(m ) 1 2 i 2 2 4 χt Q = m F + O(m ) 0 for mπ 0. ≡ V t 2N π π π → → Nf f 1/Nc-expansion, i.e. fermion loops suppressed (“quenched approximation”) [Witten, ’79, Veneziano ’79] q 1 2 1 2 2 2 2 4 χ = Q = F [m ′ + m 2m ] (180MeV) . t V t 2N π η η − K ≃ Nf =0 f Integrating axial anomaly we get Atiyah-Singer index theorem Qt[A]= n+ n− Z − ∈ n± number of zero modes fr(x) of Dirac operator iγ [A] D with chirality γ5fr = fr. ± = For lattice computations employ a chiral operator iγ . ⇒ D = Not free of lattice artifacts, use improved gauge action. ⇒ Topology becomes unique only for lattice fields smooth enough. Sufficient (!) bound to plaquette values can be given. [L¨uscher, ’82]. 6.2. Measuring topology on the lattice: Gauge field approaches: - Field theoretic with loop discretization of G ν Latt 1 Latt Latt ρ (xn) tr(G (xn)G˜ (xn)) t ≡− 16π2 ν ν with GLatt in the simplest case from plaquette expansion Latt 4 Un, ν = 1 + iG ν (xn)+ O(a ) thus using plaquette loops around site xn in a symmetrized way ǫ˜ νρσ = ǫ νρσ for ,ν,ρ,σ > 0, ǫ˜ νρσ =˜ǫ(− )νρσ, ... Latt Latt Latt 1 Q = ρ (xn) , ρ (xn)= ǫ˜ νρσ tr Un ν Unρσ t t t − 9 2 n 2 π νρσ= ±1,...,±4 [Fabricius, Di Vecchia, G.C. Rossi, Veneziano, ’81; Makhaldiani, M.-P., ’83] Problems: - QLatt integer-valued only for sufficiently ‘smooth’ lattice gauge fields. t ≃ - Top. susceptibility χLatt = 1 (QLatt)2 has be renormalized. t V t - Geometric definitions providing always integer Qt have been invented [L¨uscher, ’82; Woit, ’83; Phillips, Stone, ’86], (used with and without smoothing). Fermionic approaches: - Index of Ginsparg-Wilson fermion operators: Qt = n+ n− − [Hasenfratz, Laliena, Niedermayer, ’98; Neuberger, ’01;... ] - From corresponding spectral representation of ρt N 1 λn † ρt(x)=tr γ5( Dx,x 1) = ( 1)ψ (x)γ5ψn(x) 2 − 2 − n n =1 - Fermionic representation: [Smit, Vink, ’87] m γ5 Nf Qt = κ Tr , κ renorm. factor D + m - Topological susceptibility from higher moments and spectral projectors [L¨uscher, ’04; Giusti, L¨uscher, ’08; L¨uscher, Palombi, ’10; Cichy, Garcia Ramos, Jansen, ’13-’14 - ..... Stripping off quantum fluctuations: “Cooling”: Old days lattice search for multi-instanton solutions [Berg,’81; Iwasaki, et al., ’83; Teper, ’85; Ilgenfritz, Laursen, M.-P., Schierholz, ’86], more recently, for non-trivial holonomy KvBLL calorons [Garcia Perez, Gonzalez-Arroyo, Montero, van Baal, ’99; Ilgenfritz, Martemyanov, M.-P., Shcheredin, Veselov, ’02; Ilgenfritz, M.-P., Peschka, ’05] - Solve the lattice field equation locally (for a given link variable), - replace old by new link variable, - step through the lattice (order not unique), - find plateau values for the topological charge and action. Homework IV: Try to formulate and to solve the lattice field equation for a single link variable Un , in the SU(2) case tutorial 0 0 −→ - (Over-) improved cooling and improved G ν = for T ’96; Bruckmann, Ilgenfritz, Martemyanov, van Baal, ’04] Typical examples of gluodynamics for T > 0 (Qt, S). T =0.88 Tc T =1.12 Tc 12 5 10 4 8 3 6 2 |Q|, S/S_inst |Q|, S/S_inst 4 1 2 0 0 0 200 400 600 800 1000 1200 1400 0 200 400 600 800 1000 1200 1400 n_cool n_cool [Bornyakov, Ilgenfritz, Martemyanov, M.-P., Mitrjushkin, ’13] Stability (decay) of plateaus for T (clover-impr. N =2, mπ 1 GeV) f ≃ 0.25 1 0.2 0.8 4 0.6 0.15 c 4 c /T χ /T χ 0.1 0.4 0.05 0.2 0 0 0.9 1 1.1 1.2 0.9 1 1.1 1.2 T /Tc T /Tc - χt smoother in full QCD (crossover) than in gluodynamics (1st order). Should show up also in the UA(1) restoration. - What about chiral limit ? Wilson or gradient flow: Proposed and thoroughly investigated by M. L¨uscher (perturbatively with P. Weisz) since 2009 (cf. talks at LATTICE 2010 and 2013). Flow time evolution uniquely defined for arbitrary lattice field U (x) by { } 2 V˙ (x,τ)= g ∂x, S(V (τ)) V (x,τ), V (x, 0) = U (x) . − 0 2 - Diffusion process continuously minimizing action, scale λs √8t, t = a τ. ≃ - Allows efficient scale-setting (t0,t1) 2 1 2 by demanding e.g. t tr G ν G ν t=t ,t =0.3, . − 2 | 0 1 3 - Emergence of topological sectors at sufficient large length scale becomes clear. - Renormalization becomes simple (in particular in the fermionic sector). = Easy to handle, theoretically sound prescription !! ⇒ How does it work? [Bonati, D’Elia, arXiv:1401.2441 [hep-lat]] Wilson plaquette action: 2Nc 1 † S = 1 tr(U ν (x)+ U (x)) , 2 − ν g0 x, <ν 2Nc “Staples” = (in general non unitary) matrices W (x), defined by † † W (x)= Uν (x)U (x +ˆν)U (x + ˆ)+ U (x νˆ)U (x νˆ)U (x νˆ + ˆ) ν ν − − − ν = Part of action involving a given link variable U (x): 2 † 2 2 † (2/g )Retr[U (x)W (x)] (2/g )Retr[Ω (x)] = (1/g ) tr[Ω (x)+Ω (x)] − 0 ≡− 0 − 0 Link derivatives (with Hermitean generators T a): (a) d isT a ∂x, f(U (x)) = f(e U (x)) ds s=0 (a) 2 a ∂x, S(U)= Im tr T Ω g2 0 2 a a 1 † 1 † g ∂x, S(U) 2i T Im tr T Ω = Ω Ω tr Ω Ω . 0 ≡ − − − a 2 2Nc Comparison gradient flow with cooling: [C. Bonati, M. D’Elia, 1401.2441] Pure gluodynamics: - For given number of cooling sweeps nc find gradient flow time τ yielding same Wilson plaquette action value. - Perturbation theory: τ = nc/3 1 4 τ/nc scaling χt vs. λs 10 gradient flow β=5.95 β=5.95 200 gradient flow β=6.07 β=6.07 8 gradient flow β=6.2 β=6.2 cooling β=5.95 198 cooling β=6.07 6 β ) cooling =6.2 c (n [MeV] τ 196 4 1/4 χ 194 2 192 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 5 10 15 20 25 30 λ nc S [fm] - Lattice spacing dependence at fixed λs clearly visible. - Moreover: cooling and gradient flow show same spatial topological structure. Holds also for ρt(x) filtered with adjusted # ferm. (overlap) modes [Solbrig et al., ’07; Ilgenfritz et al., ’08]. 7. Conclusions, outlook Have discussed the basics of lattice gauge theories. • Main aim was the introduction of lattice QCD. • As main features were discussed: • – the path integral method, in particular for fermion matter fields; – Wilson’s representation of gauge fields on the lattice, such that gauge invariance is maintained; – the continuum limit on the basis of the renormalization group and ‘asymptotic freedom’. Basic Monte Carlo methods for the pure gauge case were introduced. • The basic approach to compute hadron correlation functions was • presented. QCD at non-zero temperature was introduced. • Methods to reveal the topological structure of QCD were presented. •