Confronting fluctuations of conserved charges in
HIC at the LHC with lattice QCD
LHC
LQCD
T ^
Tc
A-A collisions
fixed s
Quark-Gluon
Plasma
Chiral
symmetry
restored
Hadronic matter
Fluctuations of conserved charges at the
LHC and LQCD results
Chiral symmetry
broken
P. Braun-Munzinger, A. Kalweit and J. Stachel
The influence of critical fluctuations on
the probability distribution of net baryon
number B. Friman & K. Morita
1st
x
B
>
principle calculations:
, T  QCD :  perturbation theory
pQCD >
, T  QCD :
q  T
:
LGT
Deconfinement and chiral symmetry restoration in QCD
L
H
C
Tc

Pisarski & Wilczek;
Critical
region
region
O. Kaczmarek et.al. Phys.Rev. D83, 014504 (2011)
TCP
CP
The QCD chiral transition is
crossover Y.Aoki, et al Nature (2006)
and appears in the O(4) critical
Asakawa-Yazaki
Rajagopal, Schuryak
Stephanov; Hatta, et al.

Chiral transition temperature
Tc  155 18 MeV
T. Bhattacharya et.al.
Phys. Rev. Lett. 113, 082001 (2014)

Deconfinement of quarks sets in at
the chiral crossover
A.Bazavov, Phys.Rev. D85 (2012) 054503

See also:
Y. Aoki, S. Borsanyi, S. Durr, Z. Fodor, S. D. Katz, et al.
JHEP, 0906 (2009)
The shift of Tc with chemical potential
Tc (B )  Tc (0)[1  0.0066  (B / Tc )2 ]
Ch. Schmidt Phys.Rev. D83 (2011) 014504
Excellent data of ALICE Collaboration for particle yields
ALICE Collaboration
A. Kalweit
ALICE Time Projection Chamber (TPC), Time of Flight Detector (TOF), High Momentum Particle
Identification Detector (HMPID) together with the Transition Radiation Detector (TRD) and the
Inner Tracking System (ITS) provide information on the flavour composition of the collision fireball,
vector meson resonances, as well as charm and beauty production through the measurement of
leptonic observables.
Consider fluctuations and correlations of
conserved charges

They are quantified by susceptibilities:
If P(T , B , Q , S ) denotes pressure, then
N
 2 ( P)

2
T
(  N )2
 NM
T2
 2 ( P)

 N  M
N  Nq  Nq , N , M  ( B, S , Q),

Susceptibility is connected with variance
N
T2

   / T, P  P / T 4

1
2
2
(

N



N

)
3
VT
If P( N ) probability distribution of N then
 N n   N n P( N )
N
Consider special case:
P. Braun-Munzinger,
B. Friman, F. Karsch,
V Skokov &K.R.
Phys .Rev. C84 (2011) 064911
Nucl. Phys. A880 (2012) 48)
 N q  N q
=>


Charge and anti-charge uncorrelated
and Poisson distributed, then
P( N ) the Skellam distribution
 Nq 
P( N )  

 N q 



N /2
I N (2 N  q N q ) exp[( N  q  N q )]
Then the susceptibility
N
1
T
2

VT
3
( N q    N  q  )
Consider special case: particles carrying
P. Braun-Munzinger,
B. Friman, F. Karsch,
V Skokov &K.R.
Phys .Rev. C84 (2011) 064911
Nucl. Phys. A880 (2012) 48)
Fluctuations

q  1, 2, 3
The probability distribution
 S q  S  q
q  1, 2, 3
Correlations
Variance at 200 GeV AA central coll. at RHIC
STAR Collaboration
P. Braun-Munzinger, et al.
Nucl. Phys. A880 (2012) 48)

Consistent with Skellam distribution
 p p


2
p p
 6.18  0.14 in 0.4  pt
2
 1.022  0.016
1
 1.076  0.035
3
The maxima of P( N ) have a very
similar values at RHIC and LHC
thus N p  N p  const., indeed
 0.8GeV
Momentum integrated:
p p
 7.67  1.86 in 0.0  pt   GeV
p p
RHIC  2  p    p  61.4  5.7
LHC  2  p    p  61.04  3.5
Probing O(4) chiral criticality with charge fluctuations

Due to expected O(4) scaling in QCD the free energy:
P  PR (T , q ,  I )  b PS (b
1



t (  ), b  / h)
Generalized susceptibilities of net baryon number
cB ( n ) 

(2  ) 1
 (P / T )

n
( B / T )
n
At   0 only
At   0 only
4
cR ( n)  cS ( n) with
cs(n) | 0  d h(2 n/2)/  f( n) ( z)
(2 n)/ 
c | d h
( n)
s  0
cB( n ) with n  6 receive contribution
cB( n ) with n  3 receive contribution
( n)

(n)
from S
(n)
from S
f ( z)
c
c
cBn2  B / T 2 Generalized susceptibilities of the net baryon
number never critical with respect to ch. sym.
8
Constructing net charge fluctuations and correlation
from ALICE data

B
T2

Net baryon number susceptibility
1



0

(
p

N














 par )
0
3
VT

Net strangeness
S
1

0



0


(
K

K









4


4


9

 par
S
0
2
3
T
VT
( K    K    K 0   K 0 )  )
S
QS
T2


L
Charge-strangeness correlation
1



(
K

2


3

 par
3
VT
(  K     K  )   ( K *  K    K *  K  ) K 0* )
0
0
B , S , QS





constructed from ALICE particle yields
s  25 GeV
use also 0 /   0.278 from pBe at
B
Net baryon fluctuations
T2
S
Net strangeness fluctuations
T2
QS
Charge-Strangeness corr.
T2

1
(203.7  11.4)
3
VT

1
(504.2  16.8)
3
VT

1
(191  12)
3
VT
Ratios is volume independent
B
 0.404  0.026
S
and
B
 1.066  0.09
QS
Compare the ratio with LQCD data:
A. Bazavov, H.-T. Ding, P. Hegde, O. Kaczmarek, F. Karsch, E. Laermann, Y. Maezawa and S. Mukherjee
Phys.Rev.Lett. 113 (2014) and HotQCD Coll. A. Bazavov et al. Phys.Rev. D86 (2012) 034509

Is there a temperature where calculated ratios from
ALICE data agree with LQCD?
Baryon number strangeness and Q-S correlations
Compare at chiral crossover


There is a very good
agreement, within
systematic uncertainties,
between extracted
susceptibilities from
ALICE data and LQCD at
the chiral crossover
How unique is the
determination of the
temperature at which such
agreement holds?
Consider T-dependent LQCD ratios and compare
with ALICE data



The LQCD susceptibilities ratios are weakly T-dependent for T  Tc
We can reject T  0.15 GeV for saturation of B , S and QS at LHC
and fixed to be in the range 0.15  T  0.21 GeV , however
LQCD => for T  0.163 GeV thermodynamics cannot be anymore
described by the hadronic degrees of freedom
Extract the volume by comparing data with LQCD
Since
thus

VB (T ) 
203.7  11.4
T 3 (  B / T 2 ) LQCD
VQS (T ) 

(  N / T ) LQCD
2
( N 2    N 2 ) LHC

VN T 3
VS (T ) 
504.2  24.2
T 3 (  B / T 2 ) LQCD
191  12
T 3 (  B / T 2 ) LQCD
All volumes, should be equal at a
given temperature if originating
from the same source
Particle density and percolation theory

Density of particles at a
exp
N
given volume n(T )  total
V (T )


Total number of particles in
HIC at LHC, ALICE
3
Percolation theory: 3-dim system of objects of volume V0  4 / 3 R0
1.22
p
3
T
n

0.57
[
fm
]
R

0.8
fm
nc 
take 0
=> c
=> c  154 [MeV ]
V0 P. Castorina, H. Satz &K.R. Eur.Phys.J. C59 (2009)
Constraining the volume from HBT and percolation theory
Some limitation on volume from
Hanbury-Brown–Twiss: HBT
volume
Take ALICE data from pion
interferometry VHBT  4800  640 fm3
If the system would decouple at
the chiral crossover, then V  VHBT

From these results: variance extracted from LHC data consistent with LQCD
at T  154  2 MeV where the fireball volume V  4500 fm3
Excellent description of the QCD Equation of States by
Hadron Resonance Gas
A. Bazavov et al. HotQCD Coll. July 2014
F. Karsch et al. HotQCD Coll.
2 (P / T 4 )

| 0
B
B2

“Uncorrelated” Hadron Gas provides an
excellent description of the QCD equation of
states in confined phase

“Uncorrelated” Hadron Gas provides also an
excellent description of net baryon number
fluctuations
Thermal origin of particle yields with respect to HRG
Rolf Hagedorn => the Hadron Resonace Gas (HRG):
“uncorrelated” gas of hadrons and resonances
 Ni  V [nith (T ,  )   K i nithRe s. (T ,  )]
K
A. Andronic, Peter Braun-Munzinger, &
Johanna Stachel,
et al.
Particle yields with no resonance decay
contributions:
1 dN
 V (m / T ) 2 K 2 (m / T )
2 j  1 dy

Measured yields are reproduced with HRG at T  156 MeV
What is the influence of O(4) criticality on P(N)?

For the net baryon number use the
Skellam distribution (HRG baseline)
N /2

B
P( N )    I N (2 B B ) exp[( B  B)]
B
as the reference for the non-critical
behavior
Calculate P(N) in an effective chiral
model which exhibits O(4) scaling and
compare to the Skellam distribtuion
The influence of O(4) criticality on P(N) for   0

Take the ratio of P FRG ( N ) which contains O(4) dynamics to Skellam
distribution with the same Mean and Variance at different T / Tpc
K. Morita, B. Friman &K.R. (PQM model within renormalization group FRG)
 0

Ratios less than unity
near the chiral
crossover, indicating
the contribution of
the O(4) criticality to
the thermodynamic
pressure
Conclusions:
From a direct comparison of fluctuations constructed with
ALICE data to LQCD:

there is thermalization in heavy ion collisions at the LHC
and the 2nd order charge fluctuations and correlations are
saturated at the chiral crossover temperature
Skellam distribution, and its generalization, is a good
approximation of the net charge probability distribution
P(N) for small N. The chiral criticality sets in at larger N and
results in the shrinking of P(N) relative to the Skellam function .
Moments obtained from probability
distributions

Moments obtained from probability
distribution
 N k   N k P( N )
N

Probability quantified by all cumulants
2
1
P( N ) 
dy exp[iyN   (iy)]

2 0
k

(
y
)


V
[
p
(
T
,
y


)

p
(
T
,

)
]


y
Cumulants generating function:
k k
 In statistical physics
N
ZC ( N ) T
P( N ) 
e
ZGC
The influence of O(4) criticality on P(N) for   0

Take the ratio of P FRG ( N ) which contains O(4) dynamics to Skellam
distribution with the same Mean and Variance at different T / Tpc
K. Morita, B. Friman &K.R. (QM model within renormalization group FRG)
 0
Ratio < 1 at larger |N|
if c6/c2 < 1
Modelling O(4) transtion: effective Lagrangian and FRG
Effective potential is obtained by solving the exact flow equation (Wetterich
eq.) with the approximations resulting in the O(4) critical exponents
B.J. Schaefer & J. Wambach,; B. Stokic, B. Friman & K.R.
q
q
Full propagators with k < q < L
GL=S classical
Integrating from k=L to k=0 gives a full quantum effective potential
Put Wk=0(smin) into the integral formula for P(N)
Higher moments of baryon number fluctuations
B. Friman, K. Morita, V. Skokov & K.R.

If freeze-out in heavy ion
collisions occurs from a
thermalized system close
to the chiral crossover
temperature, this will lead
to a negative sixth and
eighth order moments of
net baryon number
fluctuations.
These properties are
universal and should be
observed in HIC
experiments at LHC and
RHIC
Figures: results of the PNJL model
obtained within the Functional
Renormalisation Group method 25
The influence of O(4) criticality on P(N) for   0

Take the ratio of P FRG ( N ) which contains O(4) dynamics to Skellam
distribution with the same Mean and Variance at different T / Tpc
K. Morita, B. Friman &K.R. (QM model within renormalization group FRG)
 0

Ratios less than unity
near the chiral
crossover, indicating
the contribution of
the O(4) criticality to
the thermodynamic
pressure
The influence of O(4) criticality on P(N) at   0

Take the ratio of P FRG ( N ) which contains O(4) dynamics to Skellam
distribution with the same Mean and Variance near Tpc ( )
K. Morita, B. Friman et al.
 0


 0
Asymmetric P(N) N   N 
Near Tpc ( ) the ratios less
than unity for