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Phase Transitions in QCD

Phase Transitions in QCD. Eduardo S. Fraga. Instituto de Física Universidade Federal do Rio de Janeiro. Outline. 1st Lecture Phase transitions in QCD Why? Where? How? A very simple framework: the bag model Symmetries in the QCD action: general picture

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Phase Transitions in QCD

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  1. Phase Transitions in QCD Eduardo S. Fraga Instituto de Física Universidade Federal do Rio de Janeiro

  2. Outline • 1st Lecture • Phase transitions in QCD Why? Where? How? • A very simple framework: the bag model • Symmetries in the QCD action: general picture • SU(Nc), Z(Nc) and Polyakov loops • Adding massless quarks (chiral symmetry) • Summary 2nd Rio-Saclay Meeting - Rio de Janeiro, 2007

  3. 2nd Lecture • Effective models: general idea • Effective theory for the deconfinement transition using the Polyakov loop • Effective theory for the chiral transition: the linear s model • Nonzero quark mass effects • Combining chiral and deconfinement transitions • Summary 2nd Rio-Saclay Meeting - Rio de Janeiro, 2007

  4. 3rd Lecture • Finite T x finite m: pQCD, lattice, sign problem, etc • Nuclear EoS: relativistic and non-relativistic (brief) • pQCD at nonzero T and m (brief) • Cold pQCD at high density for massless quarks • Nonzero mass effects • Compact stars and QCD at high density • Summary 2nd Rio-Saclay Meeting - Rio de Janeiro, 2007

  5. Phase transitions in QCD Why? 2nd Rio-Saclay Meeting - Rio de Janeiro, 2007

  6. Child’s questions: The behavior of QCD at high temperature is of obvious interest. It provides the answer to a childlike question: What happens if you keep making things hotter and hotter? The behavior of QCD at large net baryon density (and low temperature) is also of obvious interest. It answers yet another childlike question:What will happen when you keep squeezing things harder and harder? (Frank Wilczek, Phys. Today, August 2000) The Nobel Prize in Physics 2004: "for the discovery of asymptotic freedom in the theory of the strong interaction" 2nd Rio-Saclay Meeting - Rio de Janeiro, 2007

  7. Physicist’s question: What is the inner structure of matter and the nature of strong interactions under extreme conditions of temperature and density? • Experiments:“squeeze”, “heat”, “break” • Theory:in-medium quantum field theory, i.e. - finite-temperature QCD (François’ lectures) - finite-density QCD (Lecture 3) - effective models (Lecture 2) 2nd Rio-Saclay Meeting - Rio de Janeiro, 2007

  8. Drawing the QCD phase diagrams… 2nd Rio-Saclay Meeting - Rio de Janeiro, 2007

  9. Phase transitions in QCD Where? 2nd Rio-Saclay Meeting - Rio de Janeiro, 2007

  10. First time: the early universe (RHIC) 2nd Rio-Saclay Meeting - Rio de Janeiro, 2007

  11. Temperature x time after the Big Bang Temperature-driven transitions (very low m) (RHIC) Observables: relics from that time? 2nd Rio-Saclay Meeting - Rio de Janeiro, 2007

  12. In the lab - heavy-ion collisions RHIC makes its debut 14 June 2000 The first collisions have been detected at the Relativistic Heavy Ion Collider (RHIC) at the Brookhaven National Laboratory in the US. The STAR detector recorded the first collisions at 9pm local time on Monday, while the PHOBOS detector recorded its first events early on Tuesday. The first physics results from RHIC are expected at the beginning of next year.(physicsweb - IOP) 2nd Rio-Saclay Meeting - Rio de Janeiro, 2007

  13. RHIC in action - “Little Bang” LEP: e+ + e- -> q qbar (≈200 GeV) (RHIC) 2nd Rio-Saclay Meeting - Rio de Janeiro, 2007 (LEP)

  14. Some scales: • Electroweak transition ~ 100 GeV -> way too high… GUT: a lot higher… • Chiral & deconfinement transitions ~ 150 MeV !!! QCD phase transitions in the lab! (very low m, temperature-driven) 2nd Rio-Saclay Meeting - Rio de Janeiro, 2007

  15. Compact stars: New phases, condensates, color superconductivity, etc in the core of very dense stars (neutron stars, quark stars, strange stars) (Thorsett & Chakrabarty, 1999) (F. Weber, 2000) 2nd Rio-Saclay Meeting - Rio de Janeiro, 2007 (NASA)

  16. Some numbers: RHIC: vbeam ≈ 0.99995 c Tc ≈ 200 MeV ≈ 2 x 1012 K QFT at “high” temperature Compact stars: n0 = 3 x 1014 g/cm3 = 0.16 fm-3 ncore ≈ (4 -- 15) n0[<nEarth> ≈ 5.5 g/cm3] M ≈ (1 -- 2) solar masses [MS ≈ 2 x 1033 g] R ≈ (6 -- 16) Km [RS ≈ 7 x 105 km] QFT at “high” density + General Relativity 2nd Rio-Saclay Meeting - Rio de Janeiro, 2007

  17. Phase transitions in QCD How? 2nd Rio-Saclay Meeting - Rio de Janeiro, 2007

  18. So, we want to compute the EoS for QCD… Then, all we need is… Dynamical fields: Nf flavors of quarks in Nc colors and (Nc2-1) gluons; gauge symmetry given by SU(Nc), etc. and compute the thermodynamic potential, blah blah… Well, it’s not so simple… 2nd Rio-Saclay Meeting - Rio de Janeiro, 2007

  19. Asymptotic freedom & the vacuum of QCD • Matter becomes simpler at very high temperatures and densities (T and m as energy scales in a plasma), but very complicated in the opposite limit… • T and m are not high enough in the interesting cases… • Finite T pQCD is very sick… (François’ lectures) 2nd Rio-Saclay Meeting - Rio de Janeiro, 2007

  20. Equation of state - naïve field map hadrons ………………………………………………… quarks low temp. ??? high temp. and density T~Tc ; n~nc and density pQCD at T>0 & m>0 asymptotic freedom hadronic models, nuclear field theory where all the things that matter happen… there is no appropriate formalism yet! 2nd Rio-Saclay Meeting - Rio de Janeiro, 2007

  21. Ok, let’s not desperate… there are many ways out! • Some popular examples: • Very intelligent and sophisticated brute force: lattice QCD • Intensive use of symmetries: effective field theory models • Redefining degrees of freedom: quasiparticle models • “Moving down” from high-energy pQCD • “Moving up” from hadronic low-energy (nuclear) models We can also combine a bunch of them! 2nd Rio-Saclay Meeting - Rio de Janeiro, 2007

  22. But, before all that, let’s do something REALLY simple… The MIT bag model (70’s) • Asymptotic freedom + confinement • in the simplest and crudest • fashion: bubbles (bags) of • perturbative vacuum in a • confining medium. • + eventual corrections ~as • Asymptotic freedom: free quarks and gluons inside the color singlet bags • Confinement: vector current vanishes on the boundary 2nd Rio-Saclay Meeting - Rio de Janeiro, 2007

  23. Confinement achieved by assuming a constant energy density for the vacuum (negative pressure) -> bag constant (B) • B: phenomenological parameter, extracted from fits to masses (difference in energy density between the QCD and the pert. vacua) • Hadron mass (spherical bag): Eh = “vacuum” + kinetic ~ • Hadron pressure: (at equilibrium)

  24. Assuming a deconfining transition, the pressure in the QGP phase within this model is given by (see François’ lectures!) whereas the pressure in the hadronic phase (pion gas) is neglecting masses for simplicity. Here, we have the following numbers of d.o.f.’s: np = 3, nb = 2 (Nc2 - 1) and nf = 2 NcNf 2nd Rio-Saclay Meeting - Rio de Janeiro, 2007

  25. For instance, for Nc = 3 , Nf = 2 and B1/4 = 200 MeV: [not so bad as compared to lattice QCD results] and a 1st order transition [differs from lattice QCD results] Exercise:follow the same procedure in the case of finite density and T=0 to estimate the critical chemical potential for the quark-hadron transition. 2nd Rio-Saclay Meeting - Rio de Janeiro, 2007

  26. [From Karsch, Lattice 2007] 2nd Rio-Saclay Meeting - Rio de Janeiro, 2007

  27. Trace anomaly (“interaction measure”) - Nf = 2+1 [Bernard et al, 2006] 2nd Rio-Saclay Meeting - Rio de Janeiro, 2007

  28. To go beyond in our study of the phases of QCD, we need to • know its symmetries, and how they are broken spontaneously • or explicitly. But QCD is very complicated: • First, it is a non-abelian SU(Nc) gauge theory, with gluons living in the adjoint representation • Then, there are Nf dynamical quarks (who live in the fund. rep.) • On top of that, all these quarks have masses which are all different! Very annoying from the point of view of symmetries! So, in studying the phases of QCD, we do it by parts, and consider many “cousin theories” which are very similar to QCD but simpler (more symmetric). We also study the dependence of physics on parameters which are fixed in nature. 2nd Rio-Saclay Meeting - Rio de Janeiro, 2007

  29. The basic hierarchy is the following: • pure glue SU(N): • Z(N) symmetry (SSB) • order parameter: Polyakov loop L • deconfining trans.: N=2 (2nd order), N=3 (weakly 1st order) • + massless quarks: • chiral symmetry (SSB) • order parameter: chiral condensates • Z(N) explicitly broken, but rise of L deconf. • chiral trans.: N=3,2 (Nf=2) – 2nd order + massive quarks: Z(N) and chiral explicitly broken Yet vary remarkably and Ls 2nd Rio-Saclay Meeting - Rio de Janeiro, 2007

  30. SU(Nc), Z(Nc) and the Polyakov loop For the QCD Lagrangian (massless quarks) we have invariance under local SU(Nc). In particular, we have invariance under elements of the center Z(Nc) At finite temperature, one has also to impose the following boundary conditions (François’ lectures): Any gauge transf. that is periodic in t will do it. However, ’t Hooft noticed that the class of possible tranfs. is more general! 2nd Rio-Saclay Meeting - Rio de Janeiro, 2007

  31. They are such that keeping the gauge fields invariant but not the quarks! For pure glue, this Z(Nc) symmetry is exact, and we can define an order parameter - the Polyakov loop: 2nd Rio-Saclay Meeting - Rio de Janeiro, 2007

  32. At very high T, g ~ 0, and b -> 0, so that and we have a N-fold degenerate vacuum, signaling SSB of global Z(Nc). At T = 0, confinement implies that l0 = 0. Then, l0 can be used as an order parameter for the deconfining transition: Usually the Polyakov loop is related to the free energy of an infinitely heavy test quark via (confinement: no free quark) Exercise:do you see any possible problem in the equation above? If so, could we, instead, relate < l > to the propagator for a test quark? 2nd Rio-Saclay Meeting - Rio de Janeiro, 2007

  33. The analysis above is valid only for pure glue (no dynamical quarks). However, we can still ask:is Z(3) an approximate symmetry in QCD? On the lattice (full QCD), one sees: • Remarkable variation of L around Tc • L plays the role of an approximate order parameter • N.B.: Z(3) is broken at high, not low T! • Exercise: compare this description to that of spin systems (Ising, Potts, etc.) [Petrov, 2006] 2nd Rio-Saclay Meeting - Rio de Janeiro, 2007

  34. Adding massless quarks (chiral symmetry) In the limit of massless quarks, QCD is invariant under global chiral rotations U(Nf)L x U(Nf)R of the quark fields. One can rewrite this symmetry in terms of vector (V = R + L) and axial (A = R - L) rotations As U(N) ~ SU(N) x U(1), one finds where we see the U(1)V from quark number conservation and the U(1)A broken by instantons. Exercise: can we restore U(1)A in a hot and dense medium? Think of possible consequences and observables. 2nd Rio-Saclay Meeting - Rio de Janeiro, 2007

  35. In QCD, the remaining SU(Nf)L x SU(Nf)R is explicitly broken by a nonzero mass term. Take, for simplicity, Nf=2: so that, for non-vanishing mu= md, the only symmetry that remains is the vector isospin SU(2)V. In the light quark sector of QCD, the chiral symmetry is just approximate. Then, for massless QCD, one should find parity doublets in the vacuum, which is not confirmed in the hadronic spectrum. Thus, chiral symmetry must be broken in the vacuum by the presence of a quark chiral condensate, so that and the broken generators allow for the existence of pions, kaons, … 2nd Rio-Saclay Meeting - Rio de Janeiro, 2007

  36. Hence, for massless QCD, we can define an order parameter for the SSB of chiral symmetry in the vacuum - the chiral condensate: so that this vacuum expectation value couples together the L & R sectors, unless in the case it vanishes. For very high temperatures or densities (low as), one expects to restore chiral symmetry, melting the condensate that is a function of T and m and plays the role of an order parameter for the chiral transition in QCD. 2nd Rio-Saclay Meeting - Rio de Janeiro, 2007

  37. Again: the analysis above is valid only for massless quarks. However, we can still ask:is QCD approximately chiral in the light quark sector? On the lattice (full massive QCD), one sees: • Remarkable variation of the condensate around Tc • The condensate plays the role of an approximate order parameter [MILC collab., 2003] 2nd Rio-Saclay Meeting - Rio de Janeiro, 2007

  38. Summary • Two relevant phase transitions in QCD associated with SSB mechanisms for different symmetries of the action • Approximate Z(Nc) symmetry and deconfinement [exact for pure gauge SU(Nc)]. Order parameter: Polyakov loop • Approximate chiral symmetry and chiral transition [exact for massless quarks]. Order parameter: chiral condensate. • Some good estimates within a very simple framework: the bag model. Very crude, disagrees with lattice QCD on the nature of the transition, but still used in hydro… • Going beyond: effective models (based on symmetries of SQCD) 2nd Rio-Saclay Meeting - Rio de Janeiro, 2007

  39. 2nd Lecture: • Effective models: general idea • Effective theory for the deconfinement transition using the Polyakov loop • Effective theory for the chiral transition: the linear s model • Nonzero quark mass effects • Combining chiral and deconfinement transitions • Summary 2nd Rio-Saclay Meeting - Rio de Janeiro, 2007

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