1. Lattice QCD and the strongly interacting matter Péter Petreczky Physics Department Zimányi School 2012 and Ortvay Colloquium, December 6, 2012, ELTE, Budapest What Quantum Chromo-Dynamics (QCD) tells about the properties of nuclear matter at very high temperatures when solved numerically using discretized space-time ? Lattice QCD (LQCD) Matter that consists of hadrons cannot exist beyond some temperature and density What happens to it then ?

2. asymptotic freedom confinement Strong interactions and QCD Quarks and gluons cannot exist as free particles. Observed particles are color charge neutral Quantum Chromo Dynamics (QCD) : SU(3) non-Abelian gauge theory coupled to fermions Structure and Interaction of Hadrons gluon self interactions Mesons Baryons (includes proton and neutron) heavy Nobel Prize 2004 3-5MeV100 MeV quark masses make up only 2% of the mass of the proton or neutron (~940Mev) ! 98% of the visible mass in the Universe comes from QCD Dürr et al, Science 322 (2008) 1224 Gross, Politzer, Wilczek q qqqqqq

3. Global symmetries of QCD in the vacuum Chiral symmetry The vacuum (ground state) is not invariant under this symmetry (spontaneous symmetry breaking or Nambu-Goldstone realization of the symmetry) Nobel Prize 2008 hadrons with opposite parity have very different masses pions and kaons are light For vanishing quark masses QCD has a large symmetry group SU(3) V ×SU(3) A ×U(1) B ×U(1) A U(1) B Unbroken symmetries : SU(3) V SU(3) A Baryon (quark) number conservation Eightfold way, isospin symmetry Axial anomaly U(1) A symmetry is dynamically broken m(a 0 )=980 MeV >> m(π)=135 MeV The flavor singlet pseuo-scalar meson is heavy: m(η’)=958 MeV

4. Deconfinement at high temperature and density Hadron Gas Transition Quark Gluon Plasma (QGP) temperature and/or density Why this is interesting ? : Basic properties of strong interaction Compact stars ? Early Universe few microseconds after Big Bang LQCD Color screening

5. Relativistic Heavy Ion Collisions Monte-Carlo Simulations of Lattice QCD at T>0 Now PC and GPU clusters are more important: FNALEötvös University

6. Finite Temperature QCD and its Lattice Formulation Lattice integral with very large dimensions Monte-Carlo Methods sign problem Staggered fermion discretization (p4, asqtad, HISQ, stout ), part of the chiral symmetry, no flavor symmetry, 4-flavor in the continuum limit, det D q  (det D q ) 1/4 (rooting trick) relatively inexpensive computationally Wilson fermions : preserve flavor symmetry but not chiral symmetry, expensive (x10) Chiral Fermions (DWF or overlap) : preserves all the symmetries but very expensive (x100) Taylor expansion for not too large μ Computational costs ~ 1/a 7 ~ N τ 7

7. Equation of state rapid change in the number of degrees of freedom at T=160-200MeV: deconfinement deviation from ideal gas limit is about 10-20% at high T consistent with the perturbative result discrepancies between stout and p4 (asqtad) calculations (understood at low T ) energy density at the chiral transition temperature ε(T c =154MeV)=240 MeV/fm 3 : free gas of quarks and gluons = 18 quark+18 anti-quarks +16 gluons =52 mass-less d.o.f meson gas = 3 light d.o.f. Bazavov et al (HotQCD), PRD 80 (09) 14504 P.P. arXiv:1203.5320

8. Equation of state rapid change in the number of degrees of freedom at T=160-200MeV: deconfinement deviation from ideal gas limit is about 10-20% at high T consistent with the perturbative result discrepancies between stout and p4 (asqtad) calculations energy density at the chiral transition temperature ε(T c =154MeV)=240 MeV/fm 3 equation of state can be described as Hadron Resonance Gas at low T P.P. arXiv:1203.5320 Borsányi et al, JHEP 1009 (2010) 073 Huovinen P.P., NPA 837 (2010) 26

9. Symmetries of QCD at T>0 Chiral symmetry and U(1) A expected to be restored at very high T: relation to spin models Evidence for 2 nd order transition in the chiral limit  universal properties of QCD transition: Center (Z3) symmetry: invariance of the QCD partition function under global gauge transformation Exact symmetry for infinitely heavy quarks Polyakov loop : restored broken Center symmetry does not seem to play any role in QCD LQCD calculations with staggered quarks suggest crossover, e.g. Aoki et al, Nature 443 (2006) 675 if U A (1) restoration at significantly higher temperature SU(2) A is a relevant (good) symmetry SU(3) A restoration ? U(1) A restoration ?

10. Renormalized chiral condensate introduced by Budapest-Wuppertal collaboration with our choice : after extrapolation to the continuum limit and physical quark mass HISQ/tree calculation agree with stout results strange quark condensate does not show a rapid change at the chiral crossover => strange quark do not play a role in the chiral transition The temperature dependence of chiral condensate Borsányi et al, 2010, HotQCD : Phys. Rev. D85 (12) 054503; Bazavov, P.P, 2012 preliminary

11. O(N) scaling and the chiral transition temperature For sufficiently small m l and in the vicinity of the transition temperature: governed by universal O(4) scaling T c 0 is critical temperature in the mass-less limit, h 0 and t 0 are scale parameters Pseudo-critical temperatures for non-zero quark mass are defined as peaks in the response functions ( susceptibilities) : == = T c 0 in the zero quark mass limit universal scaling function has a peak at z=z p Caveat : staggered fermions O(2) m l →0, a > 0, proper limit a →0, before m l → 0

12. O(N) scaling and the transition temperature The notion of the transition temperature is only useful if it can be related to the critical temperature in the chiral limit : fit the lattice data on the chiral condensate with scaling form + simple Ansatz for the regular part 6 parameter fit : T c 0, t 0, h 0, a 1, a 2, b 1

13. Meson correlators and chiral symmetry 1.1T c 1.2T c 1.3T c The restoration of the chiral symmetry manifests itself in the degeneracy of vector and axial-vector for T>T c The flavor non-singlet pseudo-scalar and scalar correlators become degenerate only at 1.3T c => U A (1) is still broken at 1.2T c Cheng et al, Eur. Phys. J. C71 (2011) 1564

14. Domain wall Fermions and U A (1) symmetry restoration chiral:axial: Domain Wall Fermions, Bazavov et al (HotQCD), arXiv:1205.3535 axial symmetry is till broken at T=200 MeV ! Peak position roughly agrees with previous staggered results

15. Deconfinement and color screening infinite in the pure glue theory or large in the “hadronic” phase ~600MeV Decreases in the deconfined phase free energy of static quark anti-quark pair shows Debye screening at high temperatures melting of bound states of heavy quarks => quarkonium suppression at RHIC: Pure glue ≠ QCD !

16. Polyakov and gas of static-light hadrons Z QQ (T) / Z(T) ≈ exp( -E n /T ) Σ n Energies of static-light mesons E n =M n -m Q Megias, Arriola, Salcedo, PRL 109 (12) 151601 Bazavov, PP, work in progress Ground state and first excited states are from lattice QCD Michael, Shindler, Wagner, arXiv1004.4235 Higher excited state energies are estimated from potential model Wagner, Wiese, JHEP 1107 016,2011 Gas of static-light mesons Can reasonably well described the T-dependence of the Polyakov loop T < 200 MeV

17. Staggered versus Wilson and Overlap Fermioms Comparison with Wilson Fermion calculations, m π ≈ 500 MeV, Borsányi et al, arXiv:1205.0440 Chiral condensate Strangeness susceptibility Polyakov loop Comparison with overlap Fermion calculations, m π ≈ 350 MeV Borsányi et al, PLB713 (12) 342

18. QCD thermodynamics at non-zero chemical potential Taylor expansion : hadronic quark Taylor expansion coefficients give the fluctuations and correlations of conserved charges, e.g. Computation of Taylor expansion coefficients reduces to calculating the product of inverse fermion matrix with different source vectors => can be done effectively on GPUs

19. Deconfinement : fluctuations of conserved charges baryon number electric charge strangeness Ideal gas of massless quarks : conserved charges are carried by massive hadrons conserved charges carried by light quarks HotQCD: Bazavov et al arXiv:1203.0784 Budapest-Wuppertal: Borsányi et al, JHEP 1201 (12) 138

20. Deconfinement : fluctuations of conserved charges baryon number electric charge strangeness Ideal gas of massless quarks : conserved charges are carried by massive hadrons conserved charges carried by light quarks Quark Matter ‘12, Confinement’ 12 BNL-Bielefeld : C. Schmidt Budapest-Wuppertal: Borsányi

21. Correlations of conserved charges Correlations between strange and light quarks at low T are due to the fact that strange hadrons contain both strange and light quarks but very small at high T (>250 MeV) => weakly interacting quark gas For baryon-strangeness correlations HISQ results are close to the physical HRG result, at T>250 MeV these correlations are very close to the ideal gas value The transition region where degrees of freedom change from hadronic to quark-like is broad ~ 50 MeV P.P. arXiv:1203.5320

22. Quark number fluctuations at high T At high temperatures quark number fluctuations can be described by weak coupling approach due to asymptotic freedom of QCD Lattice results converge as the continuum limit is approached Good agreement between lattice and the weak coupling approach for 2 nd order quark number fluctuations For 4 th order the weak coupling result is above the continuum estimate from lattice 2 nd order quark number fluctuations 4 th order quark number fluctuations Andersen, Mogliacci, Su, Vuorinen, arXiv:1210.0912 BW

23. Transition temperature and equation of state at non-zero baryon density from Taylor expansion Continuum result with stout action: Borsányi, arXiv:1204.6710Endrödi, JHEP 1104 (2011) 001 The dependence of the transition temperature on the baryon density is very small, Similar conclusions with p4 action with N τ =6,8 Kaczmarek, PRD83 (11) 014504 The Taylor series truncated at order μ 2 for the pressure reproduces HRG result

24. Summary Lattice QCD show that at high temperatures strongly interacting matter undergoes a transition to a new state QGP characterized by deconfinement and chiral symmetry restoration We see evidence that provide evidence that the relevant degrees of freedom are quarks and gluons; lattice results agree well with perturbative calculations, while at low T thermodynamics can be understood in terms of hadron resonance gas The deconfinement transition can understood as transition from hadron resonance gas to quark gluon gas it is gradual and has no T c The chiral aspects of the transition are very similar to the transition in spin system in external magnetic fields: it is governed by universal scaling. Different calculations with improved staggered actions agree in the continuum limit resulting in a chiral transition temperature T c = ( 154 ± 9 ) MeV The effective restoration of U A (1) symmetry happens for T>200MeV and thus does not effect the chiral transition Color screening can be seen at temperatures T>200 MeV Calculations with Wilson, Domain-Wall and overlap fermion formulations seem to confirm the staggered fermion results

25. Lattice results on trace anomaly Bazavov, Lattice 2011, arXiv:1201.5345v1PP, Lattice 2012, Bazavov Quark Matter 2012 No significant disagreement at high (T>300 MeV) and low (T<180 MeV) temperatures Subtleties in the continuum extrapolations for HISQ

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