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

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

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

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

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

2nd Rio-Saclay Meeting - Rio de Janeiro, 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" the theory of the strong interaction" Child’s questions:

2nd Rio-Saclay Meeting - Rio de Janeiro, 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” Experiments: “squeeze”, “heat”, “break” Theory: in-medium quantum field theory, i.e. Theory: in-medium quantum field theory, i.e. - finite-temperature QCD (François’ lectures) - finite-temperature QCD (François’ lectures) - finite-density QCD (Lecture 3) - finite-density QCD (Lecture 3) - effective models (Lecture 2) - effective models (Lecture 2)

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

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

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

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

2nd Rio-Saclay Meeting - Rio de Janeiro, 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) In the lab - heavy-ion collisions

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

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

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

2nd Rio-Saclay Meeting - Rio de Janeiro, Some numbers: RHIC: v beam ≈ c T c ≈ 200 MeV ≈ 2 x K QFT at “high” temperature QFT at “high” temperature Compact stars: n 0 = 3 x g/cm 3 = 0.16 fm -3 n core ≈ ( ) n 0 [ ≈ 5.5 g/cm 3 ] M ≈ (1 -- 2) solar masses [M S ≈ 2 x g] R ≈ ( ) Km [R S ≈ 7 x 10 5 km] QFT at “high” density + General Relativity

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

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

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

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

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

2nd Rio-Saclay Meeting - Rio de Janeiro, 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 ~  s Asymptotic freedom: free quarks Asymptotic freedom: free quarks and gluons inside the color and gluons inside the color singlet bags singlet bags Confinement: vector current Confinement: vector current vanishes on the boundary vanishes on the boundary

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

2nd Rio-Saclay Meeting - Rio de Janeiro, Assuming a deconfining transition, the pressure in the QGP Assuming a deconfining transition, the pressure in the QGP phase within this model is given by (see François’ lectures!) 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:  = 3, b = 2 (N c 2 - 1) and f = 2 N c N f

2nd Rio-Saclay Meeting - Rio de Janeiro, For instance, for N c = 3, N f = 2 and B 1/4 = 200 MeV: and a 1 st order transition [not so bad as compared to lattice QCD results] to lattice QCD results] [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, [From Karsch, Lattice 2007]

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

2nd Rio-Saclay Meeting - Rio de Janeiro, 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(N c ) gauge theory, with gluons First, it is a non-abelian SU(N c ) gauge theory, with gluons living in the adjoint representation living in the adjoint representation Then, there are N f dynamical quarks (who live in the fund. rep.) Then, there are N f dynamical quarks (who live in the fund. rep.) On top of that, all these quarks have masses which are On top of that, all these quarks have masses which are all different! Very annoying from the point of view of all different! Very annoying from the point of view of symmetries! 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, The basic hierarchy is the following: pure glue SU(N): Z(N) symmetry (SSB) order parameter: Polyakov loop L order parameter: Polyakov loop L deconfining trans.: N=2 (2 nd order), N=3 (weakly 1 st order) deconfining trans.: N=2 (2 nd order), N=3 (weakly 1 st order) + massless quarks: chiral symmetry (SSB) order parameter: chiral condensate  order parameter: chiral condensate  Z(N) explicitly broken, but rise of L  deconf. Z(N) explicitly broken, but rise of L  deconf. chiral trans.: N=3,2 (N f =2) – 2 nd order chiral trans.: N=3,2 (N f =2) – 2 nd order + massive quarks: Z(N) and chiral explicitly broken Z(N) and chiral explicitly broken Yet vary remarkably and L   Yet vary remarkably and L  

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

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

2nd Rio-Saclay Meeting - Rio de Janeiro, At very high T, g ~ 0, and  -> 0, so that and we have a N-fold degenerate vacuum, signaling SSB of global Z(N c ). At T = 0, confinement implies that l 0 = 0. Then, l 0 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 to the propagator for a test quark?

2nd Rio-Saclay Meeting - Rio de Janeiro, 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: [Petrov, 2006] Remarkable variation of L around T c Remarkable variation of L around T c L plays the role of an approximate order parameter L plays the role of an approximate order parameter N.B.: Z(3) is broken at high, not low T! N.B.: Z(3) is broken at high, not low T! Exercise: compare this description to that of spin systems (Ising, Potts, etc.) Exercise: compare this description to that of spin systems (Ising, Potts, etc.)

2nd Rio-Saclay Meeting - Rio de Janeiro, Adding massless quarks (chiral symmetry) In the limit of massless quarks, QCD is invariant under global chiral rotations U(N f ) L x U(N f ) 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, In QCD, the remaining SU(N f ) L x SU(N f ) R is explicitly broken by a nonzero mass term. Take, for simplicity, N f =2: so that, for non-vanishing m u = m d, 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, 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  s ), one expects to restore chiral symmetry, melting the condensate that is a function of T and  and plays the role of an order parameter for the chiral transition in QCD.

2nd Rio-Saclay Meeting - Rio de Janeiro, 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: [MILC collab., 2003] Remarkable variation of the condensate around T c Remarkable variation of the condensate around T c The condensate plays the role of an approximate order parameter The condensate plays the role of an approximate order parameter

2nd Rio-Saclay Meeting - Rio de Janeiro, Summary Two relevant phase transitions in QCD associated with SSB mechanisms for different symmetries of the action Two relevant phase transitions in QCD associated with SSB mechanisms for different symmetries of the action Approximate Z(N c ) symmetry and deconfinement [exact for pure gauge SU(N c )]. Order parameter: Polyakov loop Approximate Z(N c ) symmetry and deconfinement [exact for pure gauge SU(N c )]. Order parameter: Polyakov loop Approximate chiral symmetry and chiral transition [exact for massless quarks]. Order parameter: chiral condensate. 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… 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 S QCD ) Going beyond: effective models (based on symmetries of S QCD )

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