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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)
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
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
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
“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
