Lecture 1: basics of lattice QCD Peter Petreczky • Lattice regularization and gauge symmetry : Wilson gauge action, fermion doubling • Different fermion formulations • Meson correlation function and Wilson loops • Scale setting, continuum limit and lines of constant physics (LCP) • Numerical simulations : path integral, quenched approximation • Improved actions and thermodynamics • The integral method and equation of state from lattice QCD 1 Finite Temperature QCD and its Lattice Formulation evolution operator in imaginary time Integral over functions integral with very large (but finite) Lattice dimension ( > 1000000 ) Costs : difficult to study real time properties: spectral functions, 2 transport coefficients Quarks and gluon fields on a lattice U (x) G†(x)U G(x + µ) µ ! µ fermion doubling ! 16 d.o.f ! 3 Wilson fermions Wilson (1975) chiral symmetry is broken even in the massless case ! additive mass renormalization Wilson Dirac operator is not bounded from below difficulties in numerical simulations 4 Discretization errors ~ a g2 , used for study of hadron properties, spectral functions Staggered fermions Kogut, Susskid (1975) different flavors, spin componets sit in different corners of the Brillouin zone or in hypercube 4-flavor theory 5 Chiral fermions on the lattice ? We would like the following properties for the lattice Dirac operator: Nielsen-Ninomiya no-go theorem : conditions one 1-4 cannot be satisfied simultaneously Nielsen, Ninomiya (1981) Wilson fermion formulation gives up 4) Staggered fermion formulation gives up 3) 6 Ginsparg-Wilson fermions Ginsparg, Wilson (1982) • anti-commutation properties are recovered in the continuum limit (a->0) • the r.h.s. of the Ginsparg-Wilson relation is zero for the solutions mildest way to break the chiral symmetry on the lattice : physical consequences of the chiral symmetry are mantained ( e.g. chiral perturbation theory ) 7 Generalized chiral symmetry and topology GW relation Luescher (1998) flavor singlet transformation : Luescher (1998) Hasenfratz, Laliena, Niedermeyer (1998) for flavor non-singlet transformation no anomaly ! 8 Constructing chiral fermion action I Overlap fermions : Neuberger (1998) using it can be shown that GW relation with R=1/2 9 Constructing chiral fermion action II Domain wall fermions : introduce the fictitious 5th dimension of extent : Shamir (1993) Extensively used in numerical simulations : (see P. Boyle, 2007 for review) 10 QCD at finite baryon density Hasenfratz, Karsch, PLB 125 (83) 308 S ( µa (x) −µa + (x) ) (x) ( (x) + (x) ) am = ψ x e U 0 ψ x+0 −ψ x e U 0 ψ x−0 + ∑ηi ψ xU i ψ x+i −ψ xU i ψ x−i + ∑ψ xψ x x,i x det M is complex => sign problem det M exp(-S) 11 cannot be a probability Meson correlators and Wilson loops Meson states are created by quark bilenear operators: J(x, y; ⌧)= ¯(⌧,x)Γ (x, y) (⌧,y) U Fixes the quantum number of of mesons, Γ is one Of the Dirac matrices Most often one considers point operators x=y and their correlation function: decay constant Consider static quarks : 12 Static meson correlation function functions after integrating out the static quark fields: τ x y Static quark anti-quark potential 0 x y R String tension n=2 and larger : hybrid potentials 13 Numerial results on the potentials Static quark anti-quark potential Hybrid potentials Sommer scale 14 Scale setting in lattice QCD and continuum limit Hadron masses in lattice QCD are dimensionless: m=mphys a Continuum limit: Physics does not depend on the details on the regularization, e.g dimensionless ratios : Should be independent of the lattice spacing The gauge coupling constant depends on the lattice spacing: 15 Lattice QCD calculations Staggered fermions : we get 4nf flavors to get 1-flavor replace nf by ¼ (rooting trick) Monte-Carlo Methods, importance sampling: sign problem Costs : improved discretization schemes are needed: 16 p4, asqtad, stout, HISQ where W k,l denotes a symmetrized combination of k l Wilson loops in the (µ, ν)- µ,ν × plane of the lattice, 1 W k,l (x)=1 Re TrL(k) L(l) L(k)+ L(l)+ +(k l) . (2.2) µ,ν − 2N ! x,µ x+kµ,ˆ ν x+lνˆ,µ x,ν ↔ " (k) k−1 Here we have introduced the short hand notation for long links, Lx,µ = j=0 Ux+jµ,µˆ and x =(n ,n ,n ,n )denotesthesitesonanasymmetriclatticeofsize# N 3 N . 1 2 3 4 σ × τ With a suitable choice of the coefficients ak,l it can be achieved that the gener- alized Wilson actions reproduce the continuum Euclidean Yang-Mills Lagrangian, = 1 F F ,uptosomeorderO(a2n)[1].Thestandardone-plaquetteWilsonac- L − 2 µ,ν µ,ν tion, S(1,1),witha =1anda =0forall(k, l) =(1, 1) receives O(a2)corrections 1,1 k,l ̸ in the naive continuum limit. Expanding the link variables, Ux,µ =exp(igaAµ(x)), in powers of a one findsImproved gauge action 1 S(1,1)(x)=1 Re TrU U U + U + µ,ν − N x,µ x+ˆµ,ν x+ˆν,µ x,ν 1 1 = g2a4 F F + a2F (∂2 + ∂2)F −2N ! µ,ν µ,ν 12 µ,ν µ ν µ,ν +O(a4) . (2.3) " can be eliminated by adding larger loops By adding an additional Wilson loop to the action one can achieve that corrections start only at O(a4). In particular we will consider here actions obtained by adding aplanar6-linkand8-linkloop,respectively.Thenon-vanishing coefficients in these cases are, 5 1 I (1, 2) : a = ,a= ≡ 1,1 3 1,2 −6 4 1 I (2, 2) : a = ,a= 17 (2.4) ≡ 1,1 3 2,2 −48 The action S(1,2) is a specific choice of the 6-link improved actions originallyproposed by Symanzik [1]. The action S(2,2) has recently also been discussed in the context of NRQCD calculations [15, 16]. Its generalization to largerquadraticloopsgivesa straightforward procedure to eliminate also higher order cut-off effects. Already in O(a4)thereexisttwoindependentoperatorsofdimensioneight,which contribute to the finite cut-off effects. In general, one thus needs three independent loops of length six and eight in order to eliminate all cut-off effects proportional to a2 and a4,respectively.Inanactionconstructedonlyfromquadraticloopsthedifferent 4 2.1 Improvement of rotational symmetry at tree level 4 3.5 py=pz=0 E(p) 3 momentum2.5 space we find then 2 (0) ¯ SF = i ψ(p) γµhµ(p)+m ψ(p) , p " µ $ 1.5 ! # N 2 1 with S˜(0) = i c − ψ¯(p) γ h(2)(p) ψ(p) , F 1 µ µ 2Nc p µ AAppendix ! # (1) ¯ STANDARDˆ(1) 2 2 SF0.5 = i ψ(p) γµSµ (NAIKp, k) ψ(p k) , sF (p)= hµ(p)+m . p k µ − We give here explicit expressionsµ as well as some technical details of the perturbative ! ! # P4 p ! calculations. It is organized as follows: In part A.1 we definethefundamentalterms with(2)0 i ¯ with ˆ(2) SF 0 = 0.5 1 ψ(p) 1.5 γµSµ (2p, k1,k22.5) ψ(p k13 k2) , and functions we refer to in the following parts. Part A.2 and part A.3 contain some −2 p k1 k2 µ − − ! ! ! Improved# staggered fermion actionsdetails of the determination of the tree-level and 1-loop coefficients respectively. Figure 1: Dispersion relation E2= E(p⃗)with2 ⃗p =(s (p,p)=0, 0) for theh standard2 (p)+m2 staggered.A.2 Tree level coefficients for improved rotational symme- withsF (p)= h (p)+m . F µ 2 In part A.4 we give the explicit results of the 1-loop calculation of the fermionic Standard staggeredµ action has discretization errorsµ ~ a Heller, Karsch, Sturm, fermionwhere action (blue), the Naik-actionµ (green) and the p4-action! (red) comparedcontribution to to the free energy density. ! PRD60try (1999) 114502 the continuumEliminate dispersion those relation usingE higher= p.Solidanddashedlinesdenoterealand order difference scheme 2 2 s(1)F (p)= h (p)+m . complex polesˆ of the propagatorµ respectively. 4 Sµ (p, k)=µ Kµ;ρ(p,A.2 k) Aρ( Treek) , level coefficientsAn for expansion improved of the rotational free inverse symme- fermion propagator up to order p yields A.2 Tree level! coeρ fficients for improved rotationalA.1 symme- General 1-loop results and definitions # try The inverseˆ(2) free fermion propagator of the action 2.1 in momentum space takes Sµ (p,try k1,k2)= Lµ;ρ,σ(p, k1,k2) Aρ(k1)Aσ(k2) . the form ρ,σ ForD the(0)( trigonometricp)= h ( functionsp)h (p)= we useAp the2 shortA A hand+2 notatiB p2 on:+2B p2 + (p6) , A.2 Tree level coe# fficientsAn expansion for improved of the free rotationalinverse fermion symme- propagator up toµ orderµp4 yields µ ⎛ 1 µ 2 ν⎞ µ µ ν̸ µ O An expansion of the free inverse fermion propagator up to order p4 yields ! ! != (0)−1 tryi µ γµhµ(p)+m i µ γµhµ(p)+m ⎝ ⎠ Here∆ we(p use)= the− following definitions= − also referred to in appe, ndices A.2 -(2.6) A.4:sµ(p) sin(pµ) , F Free quark 2propagator:2 (0) (0) 2 2 2≡ 2 6 !ρ hρ(p)+m !D D(p)+(p)=m hµ(p)hµ(p)= ApwithA coeA +2fficientsB p +2B p + (p ) , µ ⎛ cµ(p) 1 µ cos(2 pµ) ν.⎞ (0) 2 µ 2 2 µ 4 6 ≡ ν≠ µ O AnD expansion(p)=! ofh theµ(p free)hµ(p inverse)= fermionAp(0)µA propagatorA(0)+2B!1pµ +2 upB to2 orderp(0)ν!p +yields(p ) , ! hµ(p)=2sµ(p) c , +2⎛ c , cν (2p) +2c , ⎞sµ(3p) ,⎝ (A.2) ⎠ µ µ 1 0 1 2 ν≠ µ 3 0 O where the functions hµ!(p)aregiveninappendixA.1(Eq.A.2).!% with coefficientsν≠ µ & ! The inverse gluon(0) propagator(0) of the(0) 1 2actionwascalculatedin[2].Forthe ⎝ # ⎠ A =2c1,0 +12c1,2 +6c3,0× , In order to obtain(0) a(2) rotational invariant free(2) 2 propagator(2) we2 should(2) achieve2 standard that6 plaquette action as for the 1 2actionthepropagatorcanbewrittenas with coeD ffi(cientshp)=µ (p)=2hµ(p)hµ(ps)=µ(p) c1Ap,0 +2µA cA1,2+2Bc1νp(2µ p+2) +2B2 c3,0psνµ(3+p) ,(p ) , 1 × D(0)(p)= h (p)h (p)isafunctionofp2 only.⎛ For the standard one-link⎞ prop-O (0) (0) (0) µ µ µ µ %µ ν≠ µ (0) & (0)ν≠ µ (0) B1 = c1,0 2c1,2 9c3,0 , ! ! A =2# c1,0 +12c1!,2 +6c3,0 , −3 − − agator of the staggered fermion action(0) fat this requirement⎝ (0) for(3,0)arotationalinvariant(0)⎠ (1,2) ! K (p, k)=(0) c(0) (0)(ω; p, k)+c 1 (0)(p, k)+(0)c (0) (p,∆ k)Gµ,, ν (k)=(0)DGµ(k)δµ,ν EGµ,ν(k)+ξ gµ(k)gν(k) , µ;ρ 4 1,0 µ;ρ (0) i.e.3,0 µ;ρ 1,2 µ;ρ B = 8c .