The thermal model in heavy-ion collisions [fits of

The thermal model (at low energies) and the significance

of hadron yields measurements at FAIR & NICA

A.Andronic – GSI Darmstadt

• The thermal model in heavy-ion collisions

[fits of abundances of light quark (u,d,s) hadrons]

• Relevance for the QCD phase diagram

• Charmonium in the statistical hadronization model

• Summary and outlook

AA, P. Braun-Munzinger, J. Stachel, PLB 673 (2009) 142

AA, P. Braun-Munzinger, J. Stachel, H. Stocker, PLB 697 (2011) 203

NICA-FAIR workshop, FIAS, April 2-4, 2012

Hadron yields (at midrapidity, in central collisions)

10-1

1

10

10 2

10 102

√sNN (GeV)

dN/d

y

Npart=350

AGS SPS RHIC

π+ π-

p p–

K+ K-

Λ Λ–

• lots of particles, mostly newlycreated (m = E/c2)

• a great variety of species:

π± (ud, du), m=140 MeVK± (us, us), m=494 MeVp (uud), m=938 MeVΛ (uds), m=1116 MeValso: Ξ(dss), Ω(sss)...

• mass hierarchy in production(u, d quarks: remnants fromthe incoming nuclei)

The thermal model

grand canonical partition function for specie i (~ = c = 1):

lnZi =V gi2π2

∫ ∞

0±p2dp ln[1 ± exp(−(Ei − µi)/T )]

gi = (2Ji + 1) spin degeneracy factor; T temperature;

Ei =√

p2 +m2i total energy; (+) for fermions (–) for bosons

µi = µbBi + µI3I3i + µSSi + µCCi chemical potentials

µ ensure conservation (on average) of quantum numbers:i) baryon number: V

∑

i niBi = NBii) isospin: V

∑

i niI3i = Itot3iii) strangeness: V

∑

i niSi = 0iv) charm: V

∑

i niCi = 0.

Widths of resonances taken into accountShort-range repulsive core modelled via excluded volume correction (Rischke)(leads to an overall 10-20% decrease of densities; no effect on T )

The thermal model

...is in a way the simplest model

the analysis of hadron yields within the thermal model provides a “snapshot” ofAA collision at chemical freeze-out

(the earliest in the collision timeline we can look with hadronic observables)

...but the devil is in the details ...one needs:

- a complete hadron spectrum- canonical approach at low energies (and smaller systems)- to understand the data well (control fractions from weak decays)

The hadron mass spectrum as of 2008

Particle Data Group, Phys. Lett. B 667 (2008) 1

relative increase of calculated densities with mass spectrum 2008/2005 ↓

Additions (compared to 2005):

Many new resonances up to 3 GeV+(86)4 (non)strange mesons+(36)30 (non)strange baryons

σ meson [f0(600)]:mσ=484±17 MeV,Γσ=510±20 MeVGarcıa-Martın, Pelaez, Yndurain, Phys. Rev. D 76

(2007) 074034

in total 485 hadron species, includingcomposites (t, He, hypernuclei)

vacuum masses

0.95

1

1.05

1.1

1.15

1.2

1.25

10 102

103

104

√sNN (GeV)

Rel

ativ

e pr

oduc

tion

π+

with high-mass resonances

with res. and σ(484,510)

50

100

150

200

250

300

350

400

450

T (

MeV

)

Temperature

Strangeness suppression factor, γs

...a non-thermal fit parameter, to check possible non-thermalproduction of strangeness

for a hadron carrying “absolute” strangeness s = |S − S|: ni → niγss

Examples: K± (us, us): nKγs, Λ (uds): nΛγs,

Ξ(dss): nΞγ2s, Ω(sss): nΩγ

3s, φ(ss): nφγ

2s

in principle, usage of γs is to be avoided if one tests the basic thermal model

even as some models employ it (⇒ γs = 0.6 − 0.8), all agree that it is notneeded at RHIC energies (for central collisions)

here (central AA collisions) we fix γs=1

The thermal fits

ni =gi

2π2

1

NBW

∫ ∞

Mthr

dm

∫ ∞

0

Γ2i

(m−mi)2 + Γ2i/4

· p2dp

exp[(Emi − µi)/T ] ± 1

Minimize: χ2 =∑

i(N

expi −N therm

i )2

σ2i

Ni: hadron yield (⇒ T , µb, V ) or yieldratio (no V )

Data: 4π or dN/dy (at y=0)

fit only STAR data: T=162 MeV, µb=24 MeV, V=2400

fm3, χ2/Ndf=10.9/12

µI3=-0.8 MeV; µS=6.6 MeV; µC=-3.5 MeV

dN/d

y-110

1

10

210

Data

STAR

PHENIX

BRAHMS

=31.6/12df/N2χModel, 3= 24 MeV, V=1950 fmbµT=164 MeV,

=200 GeVNNs

+π -π +K-

K p p Λ Λ -Ξ+

Ξ Ω φ d d K* *Σ *Λ He3/He3

The hadron abundances are in agreement with a thermally equilibrated system

A thermal fit displayed differently

Mass (MeV)0 500 1000 1500 2000

dN/d

y/(2

*J+

1)

-210

-110

1

10

210

Data (STAR experiment)

)3Model, T=162 MeV (V=2400 fm

=200 GeVNNs

Au+Au, central

π

K

K*

p

φ

Λ

Ξ *Σ

*Λ

Ω

d

Energy dependence of T , µb

40

60

80

100

120

140

160

180

1 10 102

√sNN (GeV)

T (

MeV

)

new fits (yields)

dN/dy

parametrization

4π

0

100

200

300

400

500

600

700

800

900

1 10 102

√sNN (GeV)µ b

(MeV

)

ratios

2005 fits, dN/dy data

yields

thermal fits exhibit a limiting temperature: Tlim = 164 ± 4 MeV

T = Tlim1

1+exp(2.60−ln(√sNN (GeV))/0.45)

, µb[MeV] = 13031+0.286

√sNN (GeV)

Energy dependence of the thermal parameters

40

60

80

100

120

140

160

180

T (

MeV

)

Becattini et al. (4π)Letessier,Rafelski (4π)

dN/dyAndronic et al.4π

0

100

200

300

400

500

600

700

800

900

1 10 102

√sNN (GeV)

µ b (M

eV)

Cleymans et al. (4π)Kaneta,Xu (dN/dy)

Dumitru et al. (4π)

...a comparison of models

• Becattini et al.: +γs

Phys. Rev. C 73 (2006) 044905

Phys. Rev. C 78 (2008) 054901

• Rafelski et al.: +γs,q, λq,s,I3Eur. Phys. J. A 35 (2008) 221γS=0.18,0.36,1.72,1.64,...γq=0.33,0.48,1.74,1.49,1.39,1.47...

• Dumitru et al.: inhomogeneous freeze-out(δT , δµb)

Phys. Rev. C 73 (2006) 024902

• Kaneta, Xu, nucl-th/0405068

• Cleymans et al., Phys. Rev. C 57 (1998)3319

Volume in central collisions

10 2

10 3

10 102

103

104

√sNN (GeV)

dNch

/dy

Npart=350

148⋅√s0.30

(hep-ph/0402291)

AGS SPS RHIC LHC

E895, E877

NA49,NA44

NA50,NA60

PHOBOS,BRAHMS

ALICE

(GeV)NNs10 210 310

)3V

olum

e (f

m

0

1000

2000

3000

4000

5000 y=1)∆ (chemV

HBTV

central collisions

Vchem(∆y = 1) = dNch/dy|y=0/nthermch

VHBT = (2π)3/2R2sideRlong ...data from ALICE, PLB 696, 328 (2011)

General features of (relative) hadron production

(GeV)NNs10 210 310

dN/d

y ra

tio

-410

-310

-210

-110

1

10

+πp/-π/p

symbols: data, lines: thermal model

(GeV)NNs10 210 310

dN/d

y ra

tio0

0.05

0.1

0.15

0.2

0.25

0.3

+π/+K-π/-K

symbols: data, lines: thermal model

good agreement data-model

...but something special about protons (ALICE preliminary) at LHC?

p,p data of STAR (62, 130, 200 GeV) ad-hoc “corrected” by -25% for feed-down (STAR BES prelim. data)

...and the state of other “horns”

0

0.05

0.1

0.15

0.2

Λ /

π-

E895 E896NA49

STAR (BES prel.)NA57,NA44

thermal model

0

0.05

0.1

0.15

0.2

10 x

Ξ /

π-

0

0.05

0.1

0.15

0.2

10 102

103

100

x Ω

/ π-

√sNN (GeV)

overall agreement model-data

Acta Phys. Pol. 40 (2009) 1005

STAR BES prelim., arXiv:1203.5183

...and more exotic “horns”

AA, PBM, K.Redlich, NPA 765 (2006) 211

...if only such composites would exist Akaishi, Yamazaki, PRC 65 (2002) 044005

Extending the range: (hyper-)nuclei predictions

(GeV)NNs10 210 310

eve

nts

6Y

ield

(dN

/dy)

for

10

-510

-410

-310

-210

-110

1

10

210

310

410

510

610 He3He, 3

He4He, 4

H3Λ

H5ΛΛ

He6ΛΛ

He7ΞΛΛ

STAR, Science 328 (2010) 58.

Ratio Exp. (STAR) Model3He/3He 0.45±0.02±0.04 0.42±0.033ΛH/3

ΛH 0.49±0.18±0.07 0.45±0.033ΛH/

3He 0.82±0.16±0.12 0.35±0.0033ΛH/3He 0.89±0.28±0.13 0.37±0.003

RHIC (√sNN=200 GeV):

T=164 MeV, µb = 24 ± 2 MeV

...discrepancy for 3ΛH/

3He?

could be resolved if an excited state of3ΛH exists

Phys.Lett.B697,203(2011)

a fertile ground for discoveries at FAIR - NICA

The phase diagram of QCD 1

as T → Tlim, is chemical freeze-out a determination of the phase boundary?

0

20

40

60

80

100

120

140

160

180

200

0 200 400 600 800 1000

µb (MeV)

T (

MeV

)

E/N=1.08 GeVs/T3=7percolation

Cleymans et al.

Becattini et al.

Andronic et al.

if yes, how is thermalizationachieved?

• for low µb (<∼300 MeV):

driven by the deconfinement tran-sition

PBM, Stachel, Wetterich, PLB 596 (2004) 61

• for high µb:

is the quarkyonic phase transitionthe “thermalizer”?McLerran, Pisarski, NPA 796 (2007) 83

AA et al., NPA 837 (2010) 65

...or maybe not?

Floerchinger, Wetterich, arXiv:1202.1671

The phase diagram of QCD 2

! "#"$

%&'!"# "$

K. Fukushima, PLB 695, 387(2011)

[PNJL, “forced” to agree withstat. model (for large µb)]

..but quarkyonic (confinedand chirally-symmetric) maynot exist

(justifies vacuum masses)

result predicted/confirmed by (RG) QCD approach;B.J. Schaefer, J.Pawlowski private comm.

The chemical vs. kinetic freeze-out

0

20

40

60

80

100

120

140

160

180

T (

MeV

)

central collisions (Au,Pb)

chemical freeze-out

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

10 102

103

√sNN (GeV)

<β>

FOPIEOSE802NA49PHENIXSTARALICE preliminary

not quite the right ordering atlow energies

begs for a clarification (atSIS100)[together with a host of flowfeatures]

-0.1

-0.05

0

0.05

0.1

10 102

103

Collision energy (GeV)

v 2 at

mid

rapi

dity

out-of-plane

in-plane

semi-central collisions

FOPIE895,E877NA49STARALICE

log(√s)

SIS AGS SPS RHIC LHC

Fits at AGS: 6 and 10.5 AGeVdN

/dy

1

10

210

Data

=0.8/3df/N2χModel, 3=610 MeV, V=5600 fm

bµT= 86 MeV,

=3.8 GeVNNs

+π −π +K−

K p ΛdN

/dy

−210

−110

1

10

210

Data

=3.9/7df/N2χModel, 3=550 MeV, V=1020 fm

bµT=126 MeV,

=4.9 GeVNNs

+π −π +K−

K p p Λ Λ φ d

AGS, 2-8 AGeV: a rather small set of hadron yields measured

this may lead to a downward bias on T

Importance of including interactions4

/Tε

2

4

6

8

10

12

14

R=0

0.05 fm±R=0.3

43P

/T

1

2

3

4

5

6

7

8

LQCD, Borsanyi et al.

T (MeV)

100 120 140 160 180 200 220 240

3s/

T

2

4

6

8

10

12

14

16

• need to go beyond Dashen, Ma andBernstein (dilute limit) when densitieslarge enough (even below Tc)

• it is not only a “technical” correc-tion (thermodynamic consistency ful-filled, free of acausal behavior)

reference for Lattice QCD (for T < Tc)Borsanyi et al., arXiv:1011.4229

• NB: Tc is not THagedorn...but rather Tc ≃ Tlim...THagedorn is no more

The end?

...not quite... we need something more

we need to look at hadrons for whichthe statistical model doesn’t work inelementary (pp and e+e−) collisions

...because it works (albeit less welland with extra parameters than inAA) for the large majority of hadrons(and gives similar T )

e+e− collisions (LEP) −→V - volume of 1 jetmix of qq jets (EW) from data

(full canonical calc.)

Mul

tiplic

ities

−310

−210

−110

1

10

Data

=494/28df/N2χModel,

=0.66s

γ, 3T=170 MeV, V=16 fm

=91 GeVs

+ π 0 π +K

0K η ’η 0f ± 0a 0 ρ ± ρ ω

±K

* 0K

* φ(1

285)

1f(1

420)

1f2f ’ 2f

0 * 2K

p Λ 0 Σ − Σ + Σ − Ξ ++

∆−

(138

5)Σ

+(1

385)

Σ0

(153

0)Ξ

−Ω (152

0)Λ

PLB 675 (2009) 312, 678 (2009) 350

Statistical hadronization of heavy quarks: assumptions

P.Braun-Munzinger, J.Stachel, PLB 490 (2000) 196

• all charm quarks are produced in primary hard collisions (tcc ∼ 1/2mc ≃ 0.1 fm/c)

• survive and thermalize in QGP (thermal, but not chemical equilibrium)

• charmed hadrons are formed at chemical freeze-out together with all hadronsstatistical laws, quantum no. conservation; stat. hadronization 6= coalescence

is freeze-out at(/the?) phase boundary? ...we believe yes

• focus on J/ψ: ...can it survive above Tc? ...not settled yet (LQCD)

Asakawa, Hatsuda, PRL 92 (2004) 012001; Mocsy, Petreczky, PRL 99 (2007) 211602

we assume no J/ψ survival in QGP (full screening)

in our model the full yield is from generation at the phase boundary

if this supported by data, J/ψ looses status as “thermometer” of QGP...and gains status as a powerful observable for the phase boundary

Statistical hadronization of charm: method and inputs

• Thermal model calculation (grand canonical) T ,µB: → nthX

• Ndircc = 1

2gcV (∑

i nthDi

+ nthΛi) + g2

cV (∑

i nthψi

+ nthχi)

• Ncc << 1 →Canonical (J.Cleymans, K.Redlich, E.Suhonen, Z. Phys. C51 (1991) 137):

Ndircc = 1

2gcNthocI1(gcN

thoc )

I0(gcN thoc )

+ g2cN

thcc → gc (charm fugacity)

Outcome: ND = gcV nthDI1/I0 NJ/ψ = g2

cV nthJ/ψ

Inputs: T , µB, V∆y=1(= (dNexpch /dy)/nthch), Ndir

cc (pQCD or exp.)

Minimal volume for QGP: V minQGP=400 fm3

The “null hypothesis”

0

0.05

0.1

0.15

0.2

0.25

10 102

103

104

√sNN (GeV)

σ ψ, /

σ J/ψ

Statistical model

pApp(p

–)Data

PbPb

dataaverage

charmonium in pp(A) collisions

...is far from thermalized(model is for AA)

...while a thermal value isreached in central PbPb(NA50, SPS)

Charmonium in the thermal model

partN0 50 100 150 200 250 300 350 400

ψ J

/A

AR

0

0.2

0.4

0.6

0.8

1

1.2

1.4 =2.76 TeVNN

sALICE (2.5<y<4.0 ),

=0.2 TeVNN

sPHENIX (1.2<y<2.2),

/dy=0.030 mb)cc

σ=0.2 TeV (dNN

sModel,

=2.76 TeVNN

sModel,

forward y

/dy=0.25 mbcc

σd

/dy=0.15 mbcc

σd

• ”suppression” at RHIC

• ”enhancement” at LHC

NJ/ψ ∼ (Ndircc )2

“generic” predictions

validated by data

What is so different at LHC?

(compared to RHIC)

σcc: ∼10x, Volume: 2.2-3x

A.Andronic et al., arXiv:1106.6321

...an ultimate observable to measure the phase boundary (how low in√sNN?)

...after melting and with charm quarks equilibrating in the deconfined stage

Summary and outlook

the thermal model provides one clear way to obtain “experimental” pointson the QCD phase diagram(with fits of hadron yields; higher moments to follow)

• thermal fits work remarkably well (AGS-RHIC) ⇒ (T ,µb,V )

• limiting temperature ⇒ phase boundary (Tlim ≃ Tc)

→ for the skeptics... LHC case decissive ...smaller exp. errors expected

...would we be able (do we need?) to constrain also δT ?

• indications (bad fits) around the critical point? ...maybe, at SPS...

...but not a strong case due to disagreements between experiments

→ RHIC low-energy run (and CBM?) will clarify this

Needed: a better freeze-out line (or phase boundary?) for µb >500 MeV

• Good agreement with J/ψ (and ψ′) data ...is scrutinized further (LHC)

A.Andronic@GSI.de

NICA - FAIR Workshop - FIAS, 04.04.2012

Backup slides

Canonical correction (“canonical suppression”)

needed whenever the abundance of hadrons with a given quantum number isvery small ...so that one needs to enforce exact QN conservation

in AA collisions:strangeness at low energies

nCi,S = nGCi,S · Is(NS)I0(NS)

NS = V ·∑

S · ni,S,total amount of strangeness-carryinghadrons (part.+antipart.)

nCK,Λ = nGCK,Λ · I1(NS)I0(NS)

,

nCφ = nGCφ

...negligible for√sNN >5 GeV

1

10

10 2

10

√sNN (GeV)

supp

ress

ion

fact

or

I0/I1

I0/I2

I0/I3

VC=1000 fm3

A.Andronic@GSI.de

Fraction of yields from weak decays

...is not always well controlled in measurements

0

0.1

0.2

0.3

0.4

0.5

0.6

10 102

103

√sNN (GeV)

frac

tion

from

wea

k de

cays

π

p

Λ

A.Andronic@GSI.de

Fits at SPS: 30 and 158 GeVdN

/dy

−110

1

10

210

Data (NA49)=16.3/8

df/N2χModel,

3=380 MeV, V=1380 fmb

µT=138 MeV,

=7.6 GeVNNs

+π −π +K−

K p p Λ Λ −Ξ+

Ξ φdN

/dy

−110

1

10

210

Data

NA49

NA44+NA57

=24.4/10df

/N2χModel, 3=230 MeV, V=1240 fm

bµT=154 MeV,

=17.3 GeVNNs

+π −π +K−

K p p Λ Λ −Ξ+

Ξ −Ω+

Ω φ

only NA49 data: T=148 MeV, µb=215 MeV, V=1660 fm3, χ2/Ndf=36/10

only NA44+NA57: T=172 MeV, µb=245 MeV, V=700 fm3, χ2/Ndf=30/10

A.Andronic@GSI.de

“The horn”(s) as of 2009

0

0.05

0.1

0.15

0.2

0.25

K+ /

π+

E866,E895E802,E866NA49STAR

NA44PHENIX

thermal model

60

80

100

120

140

160

T (

Me

V)

T

0

0.025

0.05

0.075

0.1

0.125

0.15

0.175

0.2

10 102

103

Λ /

π-

E895 E896NA49

STARNA57,NA44

thermal model

0

100

200

300

400

500

600

700

800

µ b (

Me

V)

µb

√sNN (GeV)

rather well explained by the model

...as due to detailed features of thehadron mass spectrum...which leads to a limiting temperature(“Hagedorn”, T < TH)...and contains the QCD phase transition

the horn’s sensitivity to the phase bound-ary is determined (via strangeness neu-trality condition) by the Λ abundance(determined by both T and µb)

PLB 673 (2009) 142

Thermodynamical quantities at chemical freeze-out

2 3

)3 (

MeV

/fmε

0

50100

150200

250

300350

400450

2 3

)3P

(M

eV/fm

0

10

20

30

40

50

60

70

80

)-3

s (f

m

0

0.5

1

1.5

2

2.5

3

3.5

(GeV)NNs10

210 310

)-3

n (f

m

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

)bbaryons (b+mesons

...follow T dependence on√sNN

Timescales for charm(onium) production

Karsch & Petronzio, PLB 193 (1987) 105, Blaizot & Ollitrault, PRD 39 (1989) 232

• QGP formation time, tQGP

– SPS (FAIR): tQGP ≃ 1 fm/c ∼ tJ/ψ– RHIC, LHC: tQGP . 0.1 fm/c ∼ tcc

survival of initially-produced J/ψ at SPS/FAIR energies? (Td ∼ Tc)

• collision time, tcoll = 2R/γcm

– SPS (FAIR): tcoll & tJ/ψ– RHIC: tcoll < tJ/ψ, LHC: tcoll << tJ/ψ

cold nuclear suppression (breakup by initial nucleons) important at SPS/FAIRenergies but not at RHIC and LHC

shadowing is yet another (cold nuclear) effect - important at LHC (RHIC?)

NB: the only way to distinguish: measure σcc in pA and AA

J/ψ at RHIC: effect of shadowing

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

-2 -1 0 1 2

σcc: pQCD FONLLσcc: PHENIX+shadowing(dAu)

Au+Au 0-20% (Npart=280)

RA

AJ/ψ

-2 -1 0 1 2

Au+Au 20-40% (Npart=140)

y

model describes data with PHENIX σcc (lower error plotted)J. Phys. G 35 (2008) 104155

J/ψ production relative to charm

0

0.2

0.4

0.6

0.8

1

1.2

50 100 150 200 250 300 350

Npart

100

x (d

NJ/

ψ /d

y) /

(dN

cc /d

y)

SPS (dσcc/dy=5.7 µb)

RHIC (dσcc/dy=63.1 µb)

LHC (dσcc/dy=639 µb)

pp, PHENIX data • ...the most ”solid” observable

...with similar features as RAA

• similar values at RHIC and SPS

...with differences in fine details

...determined by canonical sup-pression of open charm

• enhancement-like at LHC

can. suppr. lifted, quadratic termdominant

Nucl. Phys. A 789 (2007) 334

J/ψ at LHC

0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

50 100 150 200 250 300 350

Npart

100

x (d

NJ/

ψ /d

y) /

(dN

cc /d

y)

dσcc/dy (mb)

0.32

0.43

0.64

0.85

1.28

J. Phys. G 35 (2008) 104155

solid expectations for LHC

...providing we know well (from mea-surements) the charm productioncross section in Pb-Pb

agreement that (re)generation is thegame at LHC?

Liu, Qu, Xu, Zhuang, arXiv:0907.2723Song, Park, Lee, arXiv:1002.1884

“2-component” (kinetic, coales-cence) models

...as Grandchamp, Rapp, PLB 523 (2001) 60, NPA

709 (2002) 415