quark-gluon plasma physicsreygers/lectures/...a. andronic, arxiv:1407.5003v1 qgp physics ss2017 | k....

Quark-Gluon Plasma Physics  5. Statistical Model and Strangeness  

Prof. Dr. Klaus Reygers Heidelberg University SS 2017

QGP physics SS2017 | K. Reygers | 5. Statistical Model and Strangeness

Strangeness production in hadronic interactions

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⇤ = (uds), ⌃ = (qqs), ⌅ = (qss), ⌦� = (sss)

Particles with strange quarks:

Creation in collisions of hadrons:

Example 1:

associated production of strangeness

Example 2: p + p ! p + p + ⇤+ ⇤̄, Q = 2m⇤ ⇡ 2230MeV

"hidden strangeness"

p + p ! p + K+ + ⇤, Q = m⇤ +mK+ �mp ⇡ 670MeV

K+ = (us̄), K� = (ūs), K 0 = (ds̄), K̄ 0 = (d̄s), � = (ss̄),


Strangeness production in the QGP

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QQGP ⇡ 2ms ⇡ 200MeV

Q value in the QGP significantly lower than in hadronic interactions

This reflects the difference between the current quarks mass (QGP) and the constituent quark mass (chiral symmetry breaking)


Strangeness enhancement:  One of the earliest proposed QGP signals

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J. Rafelski, B. Müller, PRL 48 (1982) 1066

assumed typical lifetime of the plasma

strangeness  per baryon

Strangeness equilibration was expected to be sufficiently fast


Quark composition of the ideal QGP

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� = eµ/T

"Boltzmann approximation"  (neglect "±1"): first term of the sum

Quarks: fermions ("upper sign"),  mu = 2.2 MeV, md = 4.7 MeV,  ms = 96 MeV,

In a QGP with μ = 0 and 150 < T < 300 MeV:

[But s-quarks eventually have to materialize as  constituent quarks …]

Particle densities for a non-interacting massive gas of fermions (upper sign)/bosons (lower sign):

0.15 0.20 0.25 0.30

0.93

0.94

0.95

0.96

0.97

0.98

T (GeV)

2n s

n u+n d

2(ns + ns̄)

nu + nū + nd + nd̄⇡ 0.92–0.98

upper sign: fermions, lower sign: bosons

ni = gi4⇡

(2⇡)3

Z 1

0

p2 dp

exp

✓pp2+m2�µ

T

◆± 1

=gi2⇡2

m2T1X

k=1

(⌥1)k+1

k�kK2

✓km

T

◆


Fraction of strange quarks: A+A vs. e+e–, πp, and pp

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√ s (GeV)

λ S

RHIC

SPSPbPb

AGSAuAu

K+p collisionsπ+p collisionspp collisionspp

– collisions

e+e- collisions

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

1 10 102

103

�s =2hss̄i

huūi+ hdd̄i

ratio of newly created valence quark pairs before strong decays (ρ, Δ, …)

arXiv:0907.1031

“Wròblewski factor”,  Acta Phys. Pol. B16 (1985) 379

Strangeness indeed enhanced in nucleus-nucleus collisions relative to e+e–, πp, and pp collisions


Strangeness Enhancement in Pb-Pb relative to p-Pb at √sNN = 17.3 GeV

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1

10

Strangeness |S|

Enha

ncem

ent E

h- K0S Λ Λ–

Ξ Ξ–

Ω+Ω–

0 1 12 2 3

Figure 4: Strange particle enhancement versus strangeness content.

8

0-40% Pb-Pb at √sNN = 17.3 GeV

Strangeness enhancement increases with s quark contents  (up to factor 17 for the Ω baryon)

WA97,  PLB 449 (1999) 401, CERN-EP/99-29

E =

✓Y

Npart

◆

Pb�Pb/

✓Y

Npart

◆

p�Pb

p-Be reference instead of p-Pb: similar behavior (NA57)


Ξ/π and Ω/π enhancement in Pb-Pb at √sNN = 2.76 TeV

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-π++π-+K+K

-π++πpp+

0KΛ2

-π++π+Ξ+-Ξ

-π++π+Ω+-Ω d

dHe 3

-π++π HΛH+ Λ

-+K+Kφ

-+K+KK*+

0

0.2

0.4

1× 3× 0.5× 30× 250× 50× 100× 5 4 10× 2× 1×

ALICE Preliminary = 7 TeVspp = 5.02 TeVNNsp-Pb

V0A Multiplicity (Pb-Side) 0-5% = 2.76 TeV, 0-10%NNsPb-Pb

Extrapolated (Pb-Pb 0-10%)

Extrapolated (p-Pb 0-5%)

K/ p/ /K0 / / d/p He/d H/ /K K*/K3

Interestingly, φ/π very similar in pp, p-Pb, and Pb-Pb

3


Particle yields from the hadron resonance gas■ Idea: Freeze-out of the QGP creates an equilibrated hadron resonance gas ■ The HRG then freezes out with a characteristic temperature Tch close to Tc

which determines the yields of different particle species ■ What is the appropriate statistical ensemble for the theoretical treatment?

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system S T, V, N

heat bath T

canonical ensemble: N and V fixed, energy E of the system fluctuates  (Es + Eb = E, T is given)

grand-canonical ensemble: V fixed, energy E and particle number N fluctuate (T, μ given)

system S T, V, μ

heat bath T, μ

pp collisions, strangeness locally conserved

central A-A collisions, local  strangeness fluctuations possible,“there is a medium”

Braun-Munzinger, Redlich, Stachel, nucl-th/0304013v1


Grand canonical ensemble: Large volume limit of the canonical treatment

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Canonical suppression factor Fs:

nK : Density of particles with strangeness K = |S|,  S =-1, -2, -3

Modified Bessel function of the first kind

nCK = nGCK · FS

FS =IK (2nGCK V )

I0(2nGCK V )

In :

A. Tounsi, K. Redlich, hep-ph/0111159

Already at moderately central Pb-Pb collisions the grand canonical ansatz is justified


Statistical model  (hadron gas, grand canonical ensemble)Partition function  (particle species i):

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lnZi =Vgi2⇡2

Z 1

0±p2dp ln(1± exp(�(Ei � µi )/T ))

“-” for bosons, “+” for fermions

Particle densities: ni = N/V = �T

V

@ lnZi@µ

=gi2⇡2

Z 1

0

p2 dp

exp((Ei � µi )/T )± 1

For every conserved quantum number there is a chemical potential:

µi = µBBi + µSSi + µI3 I3,i

Use conservation laws to constrain V ,µs ,µI3strangeness:

charge:

baryon number:

X

i

niSi = 0 ! µs

VX

i

niBi = Z + N ! µB

VX

i

ni I3,i =Z � N

2! µI3

Only two parameters left (T, μB) Example:  → determine (T, μB) for different √sNN from fits to data

gi = (2 Ji +1) spin degeneracy factorEi2 = pi2 + mi2

n(p̄)/n(p) = exp(�2µB/T )

Boltzmann  approximation


χ2 fit of the statistical models to LHC data

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LHC, Pb–Pb, 0-10%9 [email protected]

y/d

NYi

eld

d

-410

-310

-210

-110

1

10

210

310

Data, ALICE, 0-10%=33.0/14)dfN/2χStatistical model fit (

3=5330 fmV=0 MeV, b

µ=156.0 MeV, T

=2.76 TeVNNsPb-Pb

+π -π +K -K s0K 0K* φ p p Λ-Ξ

+Ξ -Ω

+Ω d He3 HΛ

3 HΛ3

Dat

a/Fi

t

0.4

0.6

0.8

1

1.2

1.4

1.6=2.76 TeVNNsPb-Pb

Data, ALICE, 0-10%

σ(D

ata-

Fit)/

-3-2-1

012

33=5330 fmV=0 MeV,

bµ=156.0 MeV, TFit:

+π -π +K -K s0K 0K* φ p p Λ -Ξ +Ξ -Ω +Ω d He3 HΛ

3 HΛ

3

T = 156.0± 1.5 MeV, µB = 0± 2 MeV, V = 5330± 410 fm3

π, K±, K0 from charm included (0.7%, 2.6%, 2.9% for the best fit)

[ no p,p̄ in fit: T = 158+1−2 MeV, V = 5170+450−210 fm

3, χ2/Ndf=11.5/12 ]

■ Overall good agreement with data ■ T = 156 ± 1.5 MeV, μB = 0 ± 2 MeV, V = 5330 ± 400 fm3

Andronic, Braun-Munzinger, Stachel arXiv:1106.632, arXiv:1210.7724, arXiv:1311.4662, talk A. Andronic Trento

3σ deviation for  protons and anti-protons

Statistical yields for primaries + feed-down from strong decay, e.g., ρ → π+π−, η → π+π−π0, φ → K+ K−

http://www.ectstar.eu/sites/www.ectstar.eu/files/talks/andronic_trento14.pdf


√sNN dependence of T and μB

■ Smooth evolution of T and μB with √sNN ■ T reaches limiting value of Tlim = 159 ± 2 MeV

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Energy dependence of T , µB (central collisions)4 [email protected]

40

60

80

100

120

140

160

180

1 10 102

103

√sNN (GeV)

T (M

eV)

fits of yields

dN/dy

parametrization4π

0

100

200

300

400

500

600

700

800

900

1 10 102

103

√sNN (GeV)

µB

(MeV

)

thermal fits exhibit a limiting temperature: Tlim = 159± 2 MeV

T = Tlim1

1+exp(2.60−ln(√sNN (GeV))/0.45)

, µB[MeV] =1307.5

1+0.288√sNN (GeV)

PLB 673 (2009) 142 ...with updates (Tlim was 164± 4 MeV)

update of PLB 673 (2009) 142


K/π ratio vs. √sNN

■ Maximum in K+/π+ (“the horn”) was discussed as a signal for the onset of deconfinement at √sNN ≈ a few GeV 

■ However, in the GC statistical model the structure can be reproduced with T, μB that vary smoothly with √sNN

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(GeV)NNs10 210 310

dN/d

y ra

tio

0

0.05

0.1

0.15

0.2

0.25

0.3

+π/+K-π/-K

symbols: data, lines: thermal model

A. Andronic, arXiv:1407.5003v1


Freeze-out points for √sNN ≳ 10 GeV from thermal model fits coincide with Tc from lattice calculations 

■ What is the origin of equilibrium particle yields? ‣ General property of the

QCD hadronization process (“particle born into equilibrium”)

‣ Or does the hadron gas thermalizes via particle scattering after the transition?

■ Possible mechanism for fast thermalization after the transition: multi-hadron scattering resulting from high particle densities

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Braun-Munzinger, Stachel, Wetterich,  PLB 596 (2004) 61


Strangeness enhancement already in small systems: Multiplicity dependence of Ω/π in pp, p-Pb, and Pb-Pb

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Significant increase in Ω/π with dNch/dη already in pp and p-Pb


Even yields in e+e– are not so far from chemical equilibrium

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Becattini, Castorina, Manninen, Satz, 0805.0964Statistical model + phenomenological factor γs < 1, reducing hadron yields by γsN where N is the number of strange quarks (or antiquarks)

T not so different from the one in central A+A


Summary/questions strangeness

■ Strangeness is enhanced in A-A collisions relative to e+e– and pp ■ LHC: Strangeness enhancement in high-multiplicity pp collisions approaches

the enhancement in Pb-Pb ■ Origin of the strangeness enhancement? ‣ Collisional equilibration? ‣ Or "born into equilibrium"? ‣ Strange quark coalescence ("recombination")? ‣ Or something else?

■ Strangeness provides important information and probably points to QGP formation ‣ But why does the statistical approach also work to some degree in e+e– where

no QGP is expected? ‣ Better understanding of the mechanisms of strangeness enhancement is

needed

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quark-gluon plasma physicsreygers/lectures/...a. andronic, arxiv:1407.5003v1 qgp physics ss2017 | k....

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