XQCD 2011 San Carlos, July 2011 Effective theories for QCD at finite temperature and density from strong coupling Owe Philipsen Introduction to strong coupling expansions SCE for finite temperature: free energy, screening masses Constructing effective theories for QCD by SCE in collaboration with M. Fromm, S. Lottini (Frankfurt) and J. Langelage (Bielefeld)
Why strong coupling expansions? SCE produce convergent series, finite radius of convergence Complementary to weak coupling approach, Monte Carlo Only analytical approach from first principles for confined phase Study onset of finite T-effects Study finite density (straight Monte Carlo impossible) Finite T: establish connection between QCD and strong coupling limit
SCE at T=0: free energy density β = 2N g 2 SCE valid for large g(a), i.e. on coarse lattices; continuum physics??
SCE: calculational technology
The free energy density
The graphs to be calculated
Introducing a physical temperature Münster, Langelage, Philipsen 08 Consider two lattices, one with finite temporal extent, periodic b.c. T = 1 an t continuum limit a 0,N t Small β(a) small T Subtract vacuum contribution (renormalisation, cf. continuum) Physical free energy density: New class of diagrams, LO: N t
The hard part: corrections
The series for the free energy density
Free energy from free glueball gas
Quality check: comparison with Monte Carlo
Quality rapidly worse for finer lattices and SU(3)...
Temperature dependence of screening masses
Including heavy fermions K f = 1 2(aM + 4)
Temperature effects Now two character expansions, compute again
The free energy density HRG arises as strong coupling effective theory!
Effective theories Need to get to larger Nt for continuum physics Idea: derive effective theory analytically, solve on lattice Example: perturbative dimensional reduction for high T describes phase diagram of e.w. theory, QCD only at high T Dim. red. does not work for the QCD transition, breaks Z(3) symmetry of Yang-Mills theory Approach for finite density!
Here: effective lattice theory, general strategy
The effective theory for SU(2)
Generalisation to SU(3) L
Numerical evaluation of effective theories Monte Carlo simulation of scalar model, Metropolis update Search for criticality: Binder cumulant: Susceptibility: Finite size scaling:
Numerical results for SU(3) 1.5 1 λ 1 = 0.178 λ 1 = 0.202 Im(L) 0.5 0 Order-disorder transition -0.5-1 -1.5-1.5-1 -0.5 0 0.5 1 1.5 Re(L) 1.25 1.2 1.15 N s = 06 N s = 08 N s = 10 N s = 12 N s = 14 0.01 0.008 N s = 06 N s = 08 N s = 10 N s = 12 N s = 14 1.1 0.006 L 1.05 χ L 0.004 1 0.95 0.002 0.9 0.175 0.18 0.185 0.19 0.195 0.2 0.205 0 0.175 0.18 0.185 0.19 0.195 0.2 0.205 λ 1 λ 1
! c! <! =! c! >! c Abundance 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 L 0.67 0.665 B L minimum 0.66 0.655 0.65 0.645 0.64 Data Fit 2/3 Asymptotic value Histogram estimate 6 8 10 12 14 N s First order phase transition for SU(3) in the thermodynamic limit!
The influence of a second coupling Next-to-nearest neighbour, adjoint rep. loops λ 1 0.19 0.185 0.18 0.175 0.17 0.165 0.16 0.155 0.15 from χ L from B L second-order fit on B L N τ = 2 N τ = 3 N τ = 4 N τ = 6 N τ = 8 0 0.002 0.004 0.006 0.008 0.01 λ 1 0.19 0.185 0.18 0.175 0.17 0.165 0.16 0.155 0.15 0.145 0.14 from χ L from B L third-order fit on χ L N τ = 1 N τ = 2 N τ = 3 N τ = 4 N τ = 6 N τ = 10 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 λ 2 λ a...gets very small for large N τ
Numerical results for SU(2), one coupling Abundance! 1 = 0.1940! 1 = 0.1944! 1 = 0.1948! 1 = 0.1952! 1 = 0.1956! 1 = 0.1960! 1 = 0.1964! 1 = 0.1968! 1 = 0.1972! 1 = 0.1976! 1 = 0.1980 0.92 0.94 0.96 0.98 1 1.02 1.04 L 0.204 data second order fit ["=0.63002] TD limit 0.195374(42) 0.202! 1 0.2 0.198 0.196 10 15 20 25 N s Second order (3d Ising) phase transition for SU(2) in the thermodynamic limit!
Mapping back to 4d Inverting λ 1 (N τ, β) β c (λ 1,c,N τ )...points at reasonable convergence β c 7.2 7 6.8 6.6 6.4 6.2 6 5.8 5.6 5.4 5.2 5 2 4 6 8 10 12 14 16 N τ Order 10 Order 9 Order 8 SU(3)
Comparison with 4d Monte Carlo Relative accuracy for β c compared to the full theory SU(2) SU(3) 1 8 0 6-1 4 % deviation -2-3 -4 % deviation of β c 2 0-2 -5-4 -6-7 M=1 results M=infinity results 2 4 6 8 10 12 14 16-6 -8 From (1) From (1,2) From (1,a) 2 4 6 8 10 12 14 16 N! N τ
First principles estimate for continuum limit! 800 700 600 T c [MeV] 500 400 300 200 One-coupling effective theory 270 MeV Linear fit: T c = 250(14) MeV 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 2 1/N t -error bars: difference between last two orders in strong coupling exp. -non-perturbative beta-function (4d T=0 lattice) -all from one single 3d MC simulation!
Including heavy quarks: LO hopping expansion Z(λ, a, b) = [dl]e V x ( [1+2λReL i L j ] )( det [ (1 + aw x ) 2 (1 + bw x) 2] N f ) <ij> x }{{} Q x Q x L x = TrW x a = N f (2κe +µ ) N τ, b = N f (2κe µ ) N τ, det(1 + aw ) = 1 + atrw + a 2 TrW + a 3 = 1 + al + a 2 L + a 3 Start with zero density: µ = 0 : a = b { } [ ] 2 2 Q x = (1 + a 3 )+(a + a 2 ) ReL x + [(a a 2 ] 2 )Im L x
a QCD: first order deconf. transition region m s m s tric 2nd order O(4)? phys. point N = 2 f 2nd order Z(2) N = 3 crossover f 1st Pure Gauge N = 1 f deconfinement p.t.: breaking of global Z(3) symmetry explicitly broken by quark masses transition weakens 1st 2nd order Z(2) 0 0 m, m u d Phase diagram in eff. theory: lambda I II deconfined confined a c cr.
Phase boundary in two-coupling space pseudo-critical! 0.1868 0.1866 0.1864 0.1862 " L "l " E B L #B L #B l #B E Individual fit results Final! c! 0.188 0.1875 0.187 0.1865 Final points Linear fit 0.186 0.186 0.1858 8 10 12 14 16 18 20 22 N s 0.1855 0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 a λ 1,c (a) =0.187971(37) + 0.187971(37)a
Observable to identify order of p.t.: δb Q = B 4 (δq) = (δq)4 (δq) 2 2 B 4 (x) = 1.604 + bl 1/ν (x x c )+... 1.85 1.8 1.75 1.7 N s = 16 N s = 18 N s = 20 N s = 22 N s = 24 Scaling function 2 1.9 1.8 N s = 12 N s = 14 N s = 16 N s = 18 N s = 20 N s = 22 N s = 24!B Q 1.65 1.6!B Q 1.7 1.55 1.6 1.5 1.5 1.45 1.4-0.02-0.015-0.01-0.005 0 0.005 0.01 0.015 0.02 rescaled x 1.4 0.0005 0.0006 0.0007 0.0008 0.0009 0.001 0.0011 0.0012 0.0013 a
Critical point (λ c,a c )= ( ) 0.186687(38), 0.0007505(58) Mapping back to QCD: K 1 2 exp( am) N t =4,N f =1:K c 0.083 N f =1: N f =2: N f =3: M c T =7.19(58) M c T =7.88(63) M c T =8.29(66) Comparison with 4d lattice QCD, Nt=4 Alexandrou et al. 1999 N t =4,N f =1:K c 0.08 Inclusion of finite density in progress, µ q /T 3 feasible!
Light quarks: staggered QCD in the chiral limit Chiral symmetry restoration at finite temperature and density, phase transition! Non-perturbative determination by worm Fromm, de Forcrand 10 at =! 2 / N t 1.5 1 "##$ % 0 2nd order "##$ = 0 TCP Observable: correlator of chiral condensate χχ(x) χχ(y) = 1 D χdχ z(x, µ) Z x,µ 0.5 1st order 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 aµ =! 2 a t µ What about gauge observables, deconfinement?
Gauge observables in the strong coupling limit Defined before the gauge-integration: O = D χdχ DU O(U)e S F [U, χ,χ] Example Polyakov loop: Z J = D χdχ DUe S F [U, χ,χ]+jp +J P D χdχz F ( JP U + J P U ) := Z J P = d dj ln Z J J=0
Polyakov loop across chiral transition 0.18 0.16 0.16 0.14 0.15 0.12 0.14 < P > 0.1 0.08 < P > 0.13 0.06 0.12 0.04 8 3 x4 0.02 12 16 0 0 0.5 1 1.5 2 2.5 T / T c 4 4 0.11 4 4 8 3 x4 12 16 0.1 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 T / T c P continuous across chiral transition: P exp (F Q F 0 )/T = Z Q Z Apparent weak volume dependence: lattice very coarse!
Derivatives: 0.3 0.25 4 4 8 3 x4 12 16 5.5 5 4 4 8 3 x4 12 16 0.2 4.5 d / dt < P > 0.15! E 4 3.5 0.1 3 0.05 2.5 0 0 0.5 1 1.5 2 2.5 2 0 0.5 1 1.5 2 2.5 T / T c T / T c specific heat : d/dt of pion density; diverges with α 0 ( O(2) ) Polyakov loop sensitive to chiral p.t.!
Computing LO gauge corrections 19 structure!
Conclusions SCE at finite T useful! Exponential smallness of pressure and near T-independence of screening masses are genuine strong coupling effects HRG emerges from the fully interacting theory in the strong coupling limit Deconfinement phase transition for finite chemical potential straightforward, chiral transition at finite beta? Hope for finite density QCD!