Selected topics in Heavy Ion PhysicsPrimorsko 2014

Peter Hristov

2

Lecture 2 QCD phase diagram Yields, spectra, thermodynamics

3

Phase diagram of quark-gluon plasma (QGP)Te

mpe

ratu

re

Baryon DensityNeutron stars

Early Universe

NucleiNuclon gas

Hadron gas“color superconductor”

Quark-gluon plasmaTc

r0

Critical point?

vacuum

“Color-FlavorLocking”

High temperature

High baryon density

Free quarksRestoration of chiral symmetry

Increased collision energy

4

Yields, spectra, thermodynamics

Based on the lectures ofR.Bellwied, A.Kalweit, J.Stachel and QM2014 overviewof J.F. Grosse-Oetringhaus

5

Statistical hadronization models 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 pT, mT spectra: elastic interactions cease particle dynamics fixed

6

Reminder: Boltzmann thermodynamics 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 = kB lnΩ, 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

7

Ensembles in statistical mechanics 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, I3) can be implemented on average by use of chemical potential; asymptotic realization of exact canonical approach

Example: Central relativistic heavy-ion collisions

8

Example: barometric formula 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: Epot= mgh

€

Pj =e

−E j

kBT

Z

Boltzmann factor

Partition function Z (Zustandssumme = “sum over states”)

€

n(h1)n(h2)

=N P(h1)N P(h2)

= e−

ΔE pot

kBT = e−

mgkBT

Δh

9

Thermal model for heavy ion collisions 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

gi – spin degeneracy – relativistic dispersion relation – chemical potential representing each

conserved quantity Only two free parameters are needed (T,μB), since the

conservation laws permit to calculate Baryon number: VΣniBi = Z+N =>V Strangeness and charm: VΣniSi = 0 => μS; VΣniCi = 0 => μC Charge: VΣniI3i = (Z-N)/2 =>μI3

€

lnZ i = ±giV

2π 2h3 dpp2 ln 1± exp −E i( p) − μ i( )

kBT

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

0

∞

∫

€

E i( p) = p2 + mi2

€

μi = μB Bi + μSSi + μ I 3I3i + μCCi

See thederivationi.e. in thetextbook ofR. Vogt

10

Thermal model for heavy ion collisions From the partition function we can calculate all

other thermodynamic quantities:

The model uses iterative minimization procedure: For a given set of (T, μB) the other quantities are

recalculated to ensure conservation laws; Compare calculated particle yields/ratios with

experimental results in a 𝝌2- minimization in (T, μB) plane (thermal fit).

€

ni = N i /V =1V

∂ T lnZi( )∂μ i

=gi

2π 2h3

p2dpe E i ( p )−μ i( ) /(kBT ) ±10

∞

∫ ;

Pi =∂ T lnZ i( )

∂V; si =

1V

∂ T lnZi( )∂T

11

Implementations of statistical models 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 𝜸SN

J. Rafelski (SHARE) non-equilibrium parameter 𝜸SN and 𝜸qN

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

12

Comparison to experimental data

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 & regenerationShort life times [fm/c] medium effects K* < *< (1520) < 4 < 6 < 13 < 40

13

SHM model comparison based on yields including multi-strange baryons

Data: L.Milano for ALICE (QM 2012)Fit: R. Bellwied

Either a bad fitwith a commonfreeze-out…..

148

164

160

154

152

..or a good fit witha flavor specificfreeze-out

=> Potential evidence of flavor dependence in equilibrium freeze-out

14

Thermal Fits p-Pb Statistical model

fits Freeze-out

temperature Volume µb

Equilibrium? (gS, gq)

High-multiplicity p-Pb THERMUS 2.3 gS = 0.96 ± 0.04

approximate thermal equilibrium

c2/ndf rather large though (about 30)

(model – data) / sdata

p K K0 K* f p L X W dTHERMUS: CPC 180 (2009) 84

JFGO@QM2014

15

Physics origin?• Non equilibrium thermal model• Baryon annihilation• Freeze-out temperature hierarchy• Incomplete hadron spectrum

Thermal Fits Pb-Pb Equilibrium models yields T = 156-157 MeV

But with c2/ndf of about 2

Plenary: M. Floris

• Fits without the proton (and K*)– similar T, V but c2/ndf drops

from about 2 to about 1 proton anomaly?

Thermus 2.3 GSI SHARE 3

(model – data) / sdata

THERMUS: CPC 180 (2009) 84 | GSI: PLB 673 (2009) 142 | SHARE: arXiv:1310.5108

JFGO@QM2014

16

Statistical Hadronic Models: Misconceptions 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)

17

Experimental QCD phase diagram 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, 3He 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,μB)

18

Strangeness: Two historic QGP predictions restoration of c symmetry increased production of s

mass of strange quark in QGP expected to go back to current value (mS ~ 150 MeV ~ Tc)

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]

q q s s g g s s

p N L K

K p L N

Ethres 2ms 300 MeV

Ethres 530 MeVEthres 1420 MeV

19

The SPS ‘discovery plot’ (WA97/NA57)

Unusual strangeness enhancement

Npart Npart

20

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

21

Strangeness enhancement (in AA) or suppression (in pp)? 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

canonical

grand-canonicals

ssss

s

s

s

TB

TB

μs

s

P. Braun-Munzinger, K. Redlich, J. StachelarXiv:nucl-th/0304013

22

“Thermal” Spectra

pdNdE

3

3

TE

TT

Eeddmmdy

dNdp

NdE /)(3

3

μ

f

Invariant spectrum of particles radiated by a thermal source:

where: mT= (m2+pT2)½ transverse mass (requires

knowledge of mass)μ = b μb + s μsgrand canonical chem. potential

(central AA)T temperature of

source

Neglect quantum statistics (small effect) and integrating over rapidity gives:

TmT

TmTT

TT

TT emTmKmdmm

dN /1 )/(

R. Hagedorn, Supplemento al Nuovo Cimento Vol. III, No.2 (1965)

TmT

TT

Temdmm

dN /

At mid-rapidity E = mT cosh y = mT and hence:“Boltzmann”

23

“Thermal” spectra and radial expansion (flow)• The “thermal” fit fails at low pT

• 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, bT

mT

1/m

T dN/

dmT light

heavyT

purely thermalsource

explosivesource

T,b

mT1/

mT d

N/dm

T light

heavy

24

Thermal + Flow: “Traditional” Approach

shift) (blue for

11

for 2

mpT

mpmT

TT

T

Tth

TTth

measured

bb

b

1. Fit Data T 2. Plot T(m) Tth, bT

b is the transverse expansion velocity. 2nd term = KE term (½ m b2) common Tth, b.

Assume common flow pattern and commontemperature Tth

25

Blast wave: a hydro inspired description of spectra

R

bs

Ref. : Schnedermann, Sollfrank & Heinz,PRC48 (1993) 2462

Spectrum of longitudinal and transverse boosted thermal source:

r

n

sr

TTT

TT

Rrr

TmK

TpImdrr

dmmdN

br

bb

rr

tanh rapidity)(boost angleboost and

)(on distributi velocity transverse

with

cosh sinh

1

R

0 10

Static Freeze-out picture,No dynamical evolution to freeze-out

26

Momentum spectra for identified particles

Ratios: p K K* K0 p f L X W d 3He 3H

pp: no significant energy dependence

Strangeness enhancementDeuteron enhancement

K* Suppressionp ?

pp0.9 TeV2.76 TeV7 TeV pp

p-PbPb-Pb

JFGO@QM2014