Selected topics in Heavy Ion Physics Primorsko 2014 Peter Hristov

Lecture 2  QCD phase diagram  Yields, spectra, thermodynamics 2

Phase diagram of quark-gluon plasma (QGP) 3 Temperature Baryon Density Neutron stars Early Universe Nuclei Nuclon gas Hadron gas “color superconductor” Quark-gluon plasma TcTc 00 Critical point? vacuum “Color-Flavor Locking” High temperature High baryon density Free quarks Restoration of chiral symmetry Increased collision energy

Yields, spectra, thermodynamics 4 Based on the lectures of R.Bellwied, A.Kalweit, J.Stachel and QM2014 overview of J.F. Grosse-Oetringhaus

Statistical hadronization models 5  Assume systems in thermal and chemical equilibrium  Chemical freeze-out: defines yields & ratios  inelastic interactions cease  particle abundances fixed (except maybe resonances)  Thermal freeze-out: defines the shapes of p T, m T spectra:  elastic interactions cease  particle dynamics fixed

Reminder: Boltzmann thermodynamics 6  The maximum entropy principle leads to the thermal most likely distribution for different particle species.  Macro-state is defined by given set of macroscopic variables (E, V, N)  Entropy S = k B ln Ω, where Ω is the number of micro-states compatible with the macro-state  Compatibility to a given macroscopic state can be realized exactly or only in the statistical mean. L. Boltzmann

Ensembles in statistical mechanics 7  Micro-canonical ensemble: isolated system with fixed number of particles ("N"), volume ("V"), and energy ("E"). Micro-states have the same energy and probability.  Statistical model for e+e - collisions.  Canonical ensemble: system with constant number of particles ("N") and the volume ("V”), and with well defined temperature ("T"), which specifies fluctuation of energy (system coupled with heat bath).  in small system, with small particle multiplicity, conservation laws must be implemented locally on event-by-event basis (Hagedorn 1971, Shuryak 1972, Rafelski/Danos 1980, Hagedorn/Redlich 1985) => severe phase space reduction for particle production “canonical suppression”  Examples: Strangeness conservation in peripheral HI collisions; low energy HI collisions (Cleymans/Redlich/Oeschler 1998/1999); high energy hh or e+e - collisions (Becattini/Heinz 1996/1997)  Grand canonical ensemble: system with fixed volume ("V") which is in thermal and chemical equilibrium with a reservoir. Both, energy ("T") and particles ("N") are allowed to fluctuate. To specify the ("N") fluctuation it introduces a chemical potential (“ μ ”)  in large system, with large number of produced particles, conservation of additive quantum numbers (B, S, I 3 ) can be implemented on average by use of chemical potential; asymptotic realization of exact canonical approach  Example: Central relativistic heavy-ion collisions

Example: barometric formula 8  Describes the density of the atmosphere at a fixed temperature  Probability to find a particle on a given energy level j:  Energy on a given level is simply the potential energy: E pot = mgh Boltzmann factor Partition function Z (Zustandssumme = “sum over states”)

Thermal model for heavy ion collisions 9  Grand-canonical partition function for an relativistic ideal quantum gas of hadrons of particle type i (i = π, K, p,…):  quantum gas: (+) for bosons, (-) for fermions  g i – spin degeneracy  – relativistic dispersion relation  – chemical potential representing each conserved quantity  Only two free parameters are needed (T, μ B ), since the conservation laws permit to calculate  Baryon number: V Σ n i B i = Z+N =>V  Strangeness and charm: V Σ n i S i = 0 => μ S ; V Σ n i C i = 0 => μ C  Charge: V Σ n i I 3i = (Z-N)/2 => μ I3 See the derivation i.e. in the textbook of R. Vogt

Thermal model for heavy ion collisions 10  From the partition function we can calculate all other thermodynamic quantities:  The model uses iterative minimization procedure:  For a given set of (T, μ B ) the other quantities are recalculated to ensure conservation laws;  Compare calculated particle yields/ratios with experimental results in a 2 - minimization in (T, μ B ) plane (thermal fit).

Implementations of statistical models 11  Original ideas go back to Pomeranchuk (1950s) and Hagedorn (1970s).  Precise implementations and also interpretations differ from group to group:  K. Redlich  P. Braun-Munzinger, J. Stachel, A. Andronic (GSI)  Eigen-volume correction: ideal gas → Van-der-Waals gas  emphasis on complete hadron list  F. Becattini  non-equilibrium parameter SN  J. Rafelski (SHARE)  non-equilibrium parameter SN and qN  J. Cleymans (THERMUS)  Allows also canonical suppression in sub-volumes of the fireball  W. Broniowski, W. Florkowski (THERMINATOR)  space time evolution, pT-spectra, HBT, fluctuations

Comparison to experimental data 12 A. Andronic et al., Phys.Lett.B673 (2009) 142Thermal model from: A.Andronic et al, Nucl. Phys.A 772 (2006) 167 Resonance ratios deviate  Rescattering & regeneration Short life times [fm/c]  medium effects K* <  *<  (1520) <  4 < 6 < 13 < 40

SHM model comparison based on yields including multi- strange baryons Data: L.Milano for ALICE (QM 2012) Fit: R. Bellwied Either a bad fit with a common freeze-out… or a good fit with a flavor specific freeze-out => Potential evidence of flavor dependence in equilibrium freeze-out 13

Thermal Fits p-Pb  Statistical model fits  Freeze-out temperature  Volume  µ b  Equilibrium? (  S,  q )  High-multiplicity p-Pb  THERMUS 2.3   S = 0.96 ± 0.04  approximate thermal equilibrium   2 /ndf rather large though (about 30) (model – data) /  data  K K 0 K *  p    d THERMUS: CPC 180 (2009)

Physics origin? Non equilibrium thermal model Baryon annihilation Freeze-out temperature hierarchy Incomplete hadron spectrum Thermal Fits Pb-Pb  Equilibrium models yields T = MeV  But with  2 /ndf of about 2  Plenary: M. Floris Fits without the proton (and K*) –similar T, V but  2 /ndf drops from about 2 to about 1  proton anomaly? Thermus 2.3GSISHARE 3 (model – data) /  data THERMUS: CPC 180 (2009) 84 | GSI: PLB 673 (2009) 142 | SHARE: arXiv:

Statistical Hadronic Models: Misconceptions 16  Model says nothing about how system reaches chemical equilibrium  Model says nothing about when system reaches chemical equilibrium  Model makes no predictions of dynamical quantities  Some models use a strangeness suppression factor, others not  Model does not make assumptions about a partonic phase; However the model findings can complement other studies of the phase diagram (e.g. Lattice-QCD)

Experimental QCD phase diagram 17  Build accelerator capable to work at different energies and to use different nuclei  Build experiments with PID capabilities  Take and calibrate data  For all the particles: π, K, p, Λ, Ξ, Ω, Φ, K* 0, d, 3 He we have to:  Use PID detectors and topological selection to measure raw spectrum  Correct for efficiency, acceptance, feed-down, contamination  Fit the spectrum and extrapolate unmeasured region  Determine the integrated yield dN/dy  Do the thermal fit and extract (T, μ B )

Strangeness: Two historic QGP predictions  restoration of  symmetry  increased production of s  mass of strange quark in QGP expected to go back to current value (m S ~ 150 MeV ~ T c )  Pauli suppression for (u,d)  copious production of ss pairs, mostly by gg fusion [Rafelski: Phys. Rep. 88 (1982) 331] [Rafelski-Müller: P. R. Lett. 48 (1982) 1066]  deconfinement  stronger effect for multi-strange baryons  by using uncorrelated s quarks produced in independent partonic reactions, faster and more copious than in hadronic phase  strangeness enhancement increasing with strangeness content [Koch, Müller & Rafelski: Phys. Rep. 142 (1986) 167] 18

The SPS ‘discovery plot’ (WA97/NA57) Unusual strangeness enhancement N part 19

Strangeness enhancement: SPS, RHIC, LHC  enhancement still there at RHIC and LHC  effect decreases with increasing √s  strange/non-strange increases with √s in pp 20

Strangeness enhancement (in AA) or suppression (in pp)? 21  For smaller collision systems (pp, pPb, peripheral HI), the total number of produced strange quarks is small and strangeness conservation has to be explicitly taken into account => canonical ensemble => suppression in small systems  Since the enhancements are quoted relative to pp they are due to a canonical suppression of strangeness in pp. s s canonical grand-canonical s s s s s s s s TBTB TBTB μ s s P. Braun-Munzinger, K. Redlich, J. Stachel arXiv:nucl-th/

“Thermal” Spectra Invariant spectrum of particles radiated by a thermal source: where:m T = (m 2 +p T 2 ) ½ transverse mass (requires knowledge of mass)  = b  b + s  s grand canonical chem. potential (central AA) Ttemperature of source Neglect quantum statistics (small effect) and integrating over rapidity gives: R. Hagedorn, Supplemento al Nuovo Cimento Vol. III, No.2 (1965) At mid-rapidity E = m T cosh y = m T and hence: “Boltzmann” 22

“Thermal” spectra and radial expansion (flow) The “thermal” fit fails at low p T Different spectral shapes for particles of differing mass  strong collective radial flow Spectral shape is determined by more than a simple T  at a minimum T,  T mTmT 1/m T dN/dm T light heavy T purely thermal source explosive source T,  mTmT 1/m T dN/dm T light heavy 23

Thermal + Flow: “Traditional” Approach 1. Fit Data  T2. Plot T(m)  T th,  T  is the transverse expansion velocity. 2 nd term = KE term (½ m  2 )  common T th, . Assume common flow pattern and common temperature T th 24

Blast wave: a hydro inspired description of spectra R ss Ref. : Schnedermann, Sollfrank & Heinz, PRC48 (1993) 2462 Spectrum of longitudinal and transverse boosted thermal source: Static Freeze-out picture, No dynamical evolution to freeze-out 25

Momentum spectra for identified particles 26

Ratios:  K K * K 0 p     d 3 He 3 H pp: no significant energy dependence Strangeness enhancement Deuteron enhancement K* Suppression p ? pp 0.9 TeV 2.76 TeV 7 TeV pp p-Pb Pb-Pb