Lattice QCD, Hadron Structure and Hadronic Matter Dubna, August/September 2014 Lecture II: Owe Philipsen The ideal gas on the lattice QCD in the static and chiral limit The strong coupling expansion at finite temperature The equation of state 1
The ideal gas on the lattice Starting point: propagator of a free scalar field Lattice momenta: Matsubara sum by analytic continuation, use: 2
Substitution: Expand in small lattice spacing about continuum limit: a 0 0 So we may put a=0 in the integration limits! Now expand the dispersion relation 3
breaks rotation invariance The bosonic dispersion relation has leading O(a 2 ) cut-off effects Improvement: subtracting these, the dispersion relation is p4-improved Expansion of the pressure now simple: expand down e aen τ, then expand log Use dimensionless variables: Free boson gas has leading cut-off effects! O(a 2 ) 4
Free fermion gas on the lattice Analogous calculation, massless case starts also at O(a 2 ) O.P., Zeidlewicz, PRD 81 (2010) For Wilson fermions with finite mass, the leading correction is O(a), staggered O(a 2 ) Note: N τ 10 required for leading cut-off effects to dominate! 5
Quenched limit of QCD and Z(N) symmetry Infinite quark masses (omitting flavour index) m Static quark propagator: ψ a α(τ, x) ψ b β(0, x) = δ αβ e mτ ( Te i R τ 0 dτa 0(τ,x) ) ab On the finite T lattice: Polyakov loop Static QCD: (one flavour) S static [U] =S g [U]+ x m S g [U] ( e mn τ TrL(x)+e mn τ TrL (x) ) Gauge transformations: Periodic b.c.: Action gauge invariant: 6
Topologically non-trivial gauge transformations: Modified b.c. for trafo matrix: global twist needs to be periodic for correct finite T physics! Centre of SU(N) S g [U g ]=S g [U] invariant: centre symmetry of pure gauge action Note: this is not a symmetry of H, but of H z! Requires compact time direction with periodic b.c. ; finite T! 7
Polyakov loop picks up a phase under centre transformations Partition function in the presence of one static quark: gives free energy difference of thermal YM-system with and without a static quark Small T: because of confinement Large T: Thus Polyakov loop is non-analytic function of T phase transition! Deconfinement phase transition in YM: spontaneous breaking of Z(N) symmetry 8
Now add dynamical quarks: needs to be anti-periodic for correct finite T physics! h = 1 only Centre symmetry explicitly broken by dynamical quarks! for all T! Confined and deconfined region analytically connected (only one phase!) No need for a phase transition! 9
II.The chiral limit of QCD Massless QCD and chiral symmetry (continuum) massless quarks: S f invariant under global chiral transformations U A (1) SU(N f ) L SU(N f ) R spontaneous symmetry breaking: SU(N f ) L SU(N f ) R SU(N f ) V N 2 f 1 massless Goldstone bosons, pions order parameter: chiral condensate ψψ = 1 L 3 N t m q ln Z ψψ { > 0 symmetry broken phase, T < Tc = 0 symmetric phase, T > T c chiral transition: spontaneous restoration of global SU(N f ) L SU(N f ) R at high T Chiral But: chiral symmetry limit cannot explicitly be simulated!! broken by dynamical quarks, no need for phase transition! 10
Physical QCD...breaks both chiral and Z(3) symmetry explicitly...but displays confinement and very light pions no order parameter if there is a p.t.: no phase transition necessary! are there two distinct transitions? T deconf c < T chiral c if there is just one p.t.: is it related to chiral or Z(3) dynamics? if there is no phase transition: how do the properties of matter change? 11
Strong coupling expansion (pure gauge) Wilson action: Plaquette action Character of rep. r: group element representation matrix of group element Character expansion: dimension of rep. matrix Expansion coefficients: combinations of modified Bessel fcns. for SU(N) all others can be expressed by fundamental one 12
Corrections:... Remarkable result: Glueball masses in SCE: (plaquette correlators)......... At strong coupling the QCD partition function is that of a free hadron resonance gas! 13
Equation of state: ideal (non-interacting) gases partition fcn. for one relativistic bosonic/fermionic d.o.f.: ln Z = V d 3 p ( ) (2π) 3 ln 1 ± e (E(p) µ)/t ±1, E(p) = p2 + m 2 equation of state for g d.o.f., two relevant limits: Stefan-Boltzmann Relativistic Boson, m T (Fermion) Non-relativistic, m T p r = g π2 90 T ( 4 7 ) 8 p nr = gt ( ) 3 mt 2 2π exp( m/t ) ɛ r = g π2 30 T ( 4 7 ) 8 ɛ nr = m T p nr p nr p r = ɛ r /3, p nr 0 14
The QCD equation of state Task: compute free energy density or pressure density f = T V ln Z(T, V ) all bulk thermodynamic properties follow: p = f, ɛ 3p T 4 = T d dt ( p T 4 ), s T 3 = ɛ + p T 4, c2 s = dp dɛ Technical problem: partition function in Monte Carlo normalized to 1. not directly calculable, only Z, p, f O = Z 1 Tr(ρO) Integral method: f T 4 T T o = 1 V T dx x 3 ln Z(x, V ) T o x 15
modify for lattice action: Integration along line of constant physics! f T 4 β = N 3 τ β L3 o β β o dβ ( ln Z β ln Z β T =0 ) N.B.: lower integration constant not rigorously defined, but exponentially suppressed f T 4 (β 0) e mglueball/t Hadron 0 cut-off effects in the high temperature, ideal gas limit: momenta T 1 a p T 4 = p Nτ T 4 + c N 2 τ + O(N 4 τ ) (staggered) 16
Quantities to be calculated: For the numerical integration along lines of constant physics, need beta-functions! Directly accessible before integration: trace anomaly 17
Numerical results, pure gauge Boyd et al., NPB 469 (1996) Ideal gas behaviour at high and low T Continuum extrapolation using Nt=6,8 18
Flavour Numerical dependence results on of the equation of of state staggered p4-improved, N τ = 4 Bielefeld Karsch et al., PLB 478 (2000) 5 4 p/t 4 p SB /T 4 compare with ideal gas: 3 Ideales Gas: Pions 2 1 0 3 flavour 2+1 flavour 2 flavour pure gauge T [MeV] 100 200 300 400 500 600 ɛ SB T 4 = 3p SB T 4 = { 3 π2 30 (16 + 21 2 N f ) π2 30 Gluons and Quarks, T < T c, T > T c T > T c : more degrees of freedom, but significant interaction! sqgp or `almost ideal gas...? so far qualitative, lattices too coarse, quark masses too heavy... 19
Deconfinement: 20
Beware of cut-off effects! Different versions of improved staggered actions: Taste splittings of staggered actions give different contributions to pressure 21
Equation of state for physical quark masses, continuum Karsch et al., PLB 478 (2000) Hadron resonance gas model Budapest-Marseille-Wuppertal Symanzik-improved gauge action, staggered quarks with stout links 22
Summary Lecture II Perturbation theory allows assessment of cut-off effects, but only at high T In the strong coupling limit QCD reduces to hadron resonance gas Equation of state accessible at physical masses in the continuum limit 23