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Kenji Morita
CPOD2016@Wroclaw
Fluctuations of Charges at the
Phase Boundary
Kenji Morita
(Yukawa Institute for Theoretical Physics, Kyoto University)
1.
Fluctuations of Conserved Charges in Thermal Equilibrium
2.
Critical Behavior : Baryon number fluctuations
Consequence from O(4) and smearing by finite volume and quark mass
3.
Electric Charge Fluctuations in π gas and Hadron Resonance
Gas
Importance of “reference” distribution and role of quantum statistics
30 May, 2016
1
Kenji Morita (YITP, Kyoto)
CPOD2016@Wroclaw
“Thermodynamics” in Heavy Ion
Collisions~10fm/c, V=π(5fm)2×(10-100fm)
Conserved Charges in QCD
Baryon B (← Stopping)
Electric Charge Q (← Baryon)
Strangeness S (=0, pair creation)
Globally conserved:
Au+Au Collisions@RHIC
B=197×2
Q=79×2
S=0
Dynamical system
in local equilibrium
(Hydrodynamics)
T(x), μ(x), uμ(x)
Cooper-Frye
Exp. acceptance
Δy, Δpt, Δφ
Small Δy, Low pt
Subsystem in Grand-Canonical Ensemble Z(T,V,μB, μQ, μS)
30 May, 2016
2
Kenji Morita (YITP, Kyoto)
CPOD2016@Wroclaw
Fluctuation Studies in Heavy Ion
Collisions~10fm/c, V=π(5fm)2×(10-100fm)
Conserved Charges in QCD
Baryon B (← Stopping)
Electric Charge Q (← Baryon)
Strangeness S (=0, pair creation)
Counting # of particles
on e-by-e basis
STAR, net-proton, PRL112 (’14)
Globally conserved:
Au+Au Collisions@RHIC
B=197×2
Q=79×2
S=0
Exp. acceptance Δy, Δpt, Δφ
Small Δy, Low pt
Fluctuations of B, Q, S in
GC Subsystem Z(T,V,μB, μQ, μS)
30 May, 2016
3
Kenji Morita (YITP, Kyoto)
CPOD2016@Wroclaw
From Fluctuations to QCD Critical
Property
Hadronic observables - information
Extracted (T,μ) coincides with the
at freeze-out : Determination of T and
μ of a subsystem of the hot matter
from particle yields / fluctuations
Borsanyi, QM15
30 May, 2016
Crossover transition region in
QCD
Up to μB~ 400MeV
(Remnant) critical property of QCD
from fluctuation of conserved
charges
4
Kenji Morita (YITP, Kyoto)
CPOD2016@Wroclaw
Characterizing Fluctuations
10
Cumulants
0
Uncorrected Data 19.6GeV, 0-5%
Gaussian P(N)
Skellam
10-2
(=7.61)
(=9.16)
(=7.23)
P(N)
10-4
10-6
-8
10
10
-3
10
-4
10
-5
10
-6
30 May, 2016
-10
Skellam
(Data)
Gauss
18 19 20 21 22 23 24
-10
10
(=7.36)
0
10
20
N=N p-Npbar
30
Ratios
characterize the
shape of P(N)
5
Kenji Morita (YITP, Kyoto)
CPOD2016@Wroclaw
QCD Phase Transition / NB Fluctuations
Chiral Limit
Phys. quark mass T
Pisarski-Wilczek ‘84
2nd Order, 3d O(4)
TCP?
Crossover – Feel O(4)?
CP?
χ2 diverges
χ3,4 change the sign
mphys
mq
30 May, 2016
μ
Hatta-Ikeda ’03, Asakawa et al.,’09,
Skokov et al., ’11, Stephanov, ‘11
χ6 (μ=0), χ3,4(μ>>0)
General property from O(4) scaling function
changes the sign
(Engels and Karsch ’11, Friman et al., ’11)
across crossover
6
Kenji Morita (YITP, Kyoto)
CPOD2016@Wroclaw
Sign Change of Cumulants
Near TCP, Chiral Limit, Finite V
(Lattice QCD in Strong Coupling Limit, Ichihara, KM, Ohnishi, ‘15)
Divergence replaced by
sign changes due to finite
volume
30 May, 2016
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Kenji Morita (YITP, Kyoto)
CPOD2016@Wroclaw
Sign Change of Cumulants
Crossover near O(4) (PQM+FRG)
Skokov, Friman, Redlich, PRC’11
Weaker “Critical behavior”
No peak in χ2, no negative χ3
Smeared by finite mq and
fluctuations via FRG
30 May, 2016
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Kenji Morita (YITP, Kyoto)
CPOD2016@Wroclaw
Importance of Reference Distribution
Hadron Gas
Divergent : in χ limit or CP
Finite : cause sign changes in crossover
Suppression factor (μq/T)n
Obscured when
Remnant of the divergence may only appear as
deviation from the regular part contribution
 Net-Baryon : N, Δ, Hyperon, etc…
• m >> T,μ – Boltzmann Gas －Skellam Distribution
 Net-electric charge : π, K, ρ, p, ….
• π and Δ++ cause substantial deviation from Skellam
 Net-strangeness : K, K*, Λ, Σ, Ξ, ….
• Heavy enough, but multi-charged (up to 3!)
30 May, 2016
9
Kenji Morita (YITP, Kyoto)
CPOD2016@Wroclaw
Baryon Gas : Skellam Distribution
# of baryons （Poisson)
# of antibaryon (Poisson)
 Statistical mechanics : Boltzmann
distribution
 2 parameters
Independent of momentum
integration range
Expectation (confirmed by lattice): χ1 and χ2 are well described by HRG
Deviation from Skellam in higher order χn can reflect the phase transition
30 May, 2016
10
Kenji Morita (YITP, Kyoto)
CPOD2016@Wroclaw
(Negative) Binomial Distribution?
NB (κσ2 >1), BD(κσ2<1)
can fit the critical χ2 and
χ4, but cannot reproduce
χ6
4/2, 6/2
1
0.5
0
Higher (n>4) cumulants
necessary to identify the
critical fluctuations
FRG, 4/
FRG, 6/
NBD/BD, 6/2
Skellam
-0.5
0.6
0.7
0.8
0.9
T/Tpc
1
1.1
KM, Friman, Redlich, PLB ’15
30 May, 2016
11
Kenji Morita (YITP, Kyoto)
CPOD2016@Wroclaw
Charge Fluctuations in π gas
Pion mπ~T : Bose statistics
P. Braun-Munzinger et al., NPA’12
= Multicomponent Boltzmann gas with mass km,
charge k, and degeneracy kn-4
 Leading order : Skellam
 all k >1 terms : positive
 p-integration range can change χn ratio
30 May, 2016
12
Kenji Morita (YITP, Kyoto)
CPOD2016@Wroclaw
Effect of Low pt Cut
Karsch, KM, Redlich, PRC ‘16
Electric Charge Cumulants of a free gas
tmin=0
1
Q
Q
/
1 1 |p
χ1 – Density vs Mass:
T=160MeV, m
mK
m
mK*
0.8
Heavier particles
less affected by pt
cut
Q=-1MeV
0.6
0.4
π contribution
suppressed by
pt cut
0.2
0
0
0.2
0.4
ptmin
30 May, 2016
0.6
[GeV/c]
0.8
1
13
Kenji Morita (YITP, Kyoto)
CPOD2016@Wroclaw
Effect of Low pt Cut
Karsch, KM, Redlich, PRC ‘16
Electric Charge Cumulants of a free gas
2.5
Low pt cut : closer χ4/χ2 to 1
Skellam
m/T=0.75
1.0
1.25
1.5
Q
Q
4 /2
2
For small μQ/T,
free  gas, ptmax=2GeV/c
1.5
Q = -1MeV, T=160MeV
1
0
0.2
0.4
ptmin
0.6
[GeV/c]
0.8
1
Substantial decrease from ptmin=0 to ptmin = 0.2 (STAR) or 0.3 (PHENIX)
GeV
30 May, 2016
14
Kenji Morita (YITP, Kyoto)
CPOD2016@Wroclaw
M/σ2 in HRG
M = (π, ρ, etc) + (strange mesons) + (baryons) +
(hyperons)
> 0 (μS > 0)
> 0 (μB > 0)
> 0 (μB-(1~3)μS > 0)
< 0 (μ < 0)
Q
Negative meson contribution
is reduced by pt cut
Q=-1MeV
0.6
0.4
0.8
0
ptmin
0.6
[GeV/c]
0.8
Single component
30 May, 2016
1
s1/2
NN=200GeV
1/2
ptcut in  only, s NN=200GeV
1/2
sNN=19.6GeV
ptcut in  only, s 1/2
NN=19.6GeV
0.4
0
0.4
All particles
0.6
0.2
0.2
pt cut π only
1
0.2
0
1.2
0
0.2
0.4
ptmin
0.6
[GeV/c]
M in HRG
1
2/2(ptmin=0)
tmin=0
Q
Q
1 /1 |p
0.8
1.2
1/1(ptmin=0)
T=160MeV, m
mK
m
mK*
1
Total M can be non-monotonic in ptmin
σ2 simply decreases
π only
0.8
0.6
All particles
s1/2
NN=200GeV
1/2
ptcut in  only, s NN=200GeV
1/2
sNN=19.6GeV
ptcut in  only, s 1/2
NN=19.6GeV
0.4
0.2
0
0.8
1
0
0.2
0.4
ptmin
0.6
[GeV/c]
0.8
1
σ2 in HRG
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Kenji Morita (YITP, Kyoto)
CPOD2016@Wroclaw
M/σ2 in HRG : STAR vs PHENIX
MQ/2Q / MQ/2Q |p
tmin=0
1.4
1.3
STAR : set A
0.2 < pt < 2.0, |h| < 0.5 STAR : set B
0.3 < pt < 2.0, |h| < 0.35 STAR : set C
PHENIX : set A
PHENIX : set B
A : pt cut for π only
B : Same pt cut for all hadrons
C : ptmin = 0.4 GeV for protons (STAR)
10% larger M/σ2 in
PHENIX acceptance
1.2
1.1
0.3
rescaled
STAR
PHENIX
1
10
100
 sNN [GeV]

Data : PHENIX/STAR ~ 1.35
After rescaling, data become
closer
30 May, 2016
(MQ/ 2Q )/(MP/ 2P )
0.25
STAR
PHENIX
0.2
0.15
0.1
0.05
0
5
10
50
s


NN
100
5
10
50
100
s


NN
16
Kenji Morita (YITP, Kyoto)
CPOD2016@Wroclaw
κσ2 in HRG (prediction)
STAR : set A
STAR : set B
STAR : set C
PHENIX : set A
PHENIX : set B
Q 2Q /Q 2Q |p
tmin=0
0.94
0.92
0.9
0.88
0.86
6% difference btw.
STAR and PHENIX
expected
0.84
0.82
0.8
10
100
 sNN [GeV]

pt cut reduces κσ2 < 10% at
low energies, due to more
proton contribution
30 May, 2016
pt cut reduces κσ2 by 10-20%
at high energies
17
Kenji Morita (YITP, Kyoto)
CPOD2016@Wroclaw
Concluding Remarks
Higher-order fluctuations of net-B, Q around
phase boundary
Critical behavior : Smeared by finite V, nonzero mq
Lattice QCD in Strong Coupling Limit
Chiral model calculations with FRG
Deviation from the reference
net-B : Skellam distribution – momentum
independent
net-Q : Bose statistics – affected by low pt cut
30 May, 2016
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Kenji Morita (YITP, Kyoto)
CPOD2016@Wroclaw
Backup
30 May, 2016
19
Kenji Morita (YITP, Kyoto)
CPOD2016@Wroclaw
Effects of Interaction
Charge fluctuations in QM model (Nf=2)
μB=μQ=0
0.8
0.7
0.6
free  gas
QM+FRG,  only
QM+FRG, quark only
QM+FRG
Susceptibility:
quark dominant
(Caveat: deconfined –
quarks have Q=2/3, -1/3)
2
0.5
“π only”
: Qu,d=0
“quark only” : Qπ=0
0.4
0.3
0.2
0.1
0
0.05
0.1
0.15
0.2
0.25
T [GeV]
30 May, 2016
20
Kenji Morita (YITP, Kyoto)
CPOD2016@Wroclaw
Effects of Interaction
Charge fluctuations in QM model (Nf=2)
μB=μQ=0
0.8
0.7
0.6
free  gas
QM+FRG,  only
QM+FRG, quark only
QM+FRG
4th cumulant:
π dominant
(Caveat: deconfined –
quarks have Q=2/3, -1/3)
Quark contribution
suppressed by Fermi
statistics
4
0.5
0.4
0.3
“π only”
: Qu,d=0
“quark only” : Qπ=0
0.2
0.1
0
0.05
0.1
0.15
0.2
0.25
T [GeV]
30 May, 2016
21
Kenji Morita (YITP, Kyoto)
CPOD2016@Wroclaw
Effects of Interaction
Charge fluctuations in QM model (Nf=2)
μB=μQ=0
2
1.5
6th cumulant:
Negative χ6 from chiral
crossover
Obscured by π contribution
6
1
0.5
0
-0.5
free  gas
QM+FRG,  only
QM+FRG, quark only(x10)
QM+FRG
0.05
0.1
0.15
“π only”
: Qu,d=0
“quark only” : Qπ=0
0.2
0.25
T [GeV]
30 May, 2016
22
Kenji Morita (YITP, Kyoto)
CPOD2016@Wroclaw
Effect of Low pt Cut
Electric Charge Cumulants of a free gas
T=160MeV, m
T=160MeV, 3m
T=53MeV, m
Q
Q
1 /1 |p
tmin=0
1
χ1 – Density vs Mass:
0.8
Heavier particles
less affected by pt
cut
0.6
Q=-1MeV
0.4
0.2
0
0.2
0.4
ptmin
0.6
[GeV/c]
0.8
1
π contribution
suppressed by
pt cut
At ptmin=0, χn scale with m/T
Different pt cut effect for same m/T!
30 May, 2016
23
Kenji Morita (YITP, Kyoto)
CPOD2016@Wroclaw
Pseudorapidity Cut
2.2
1.8
2
1.6
Q = 0, T=160MeV
1.4
Skellam
0 < pT < 
0.2 < pT < 2.0 GeV
0.3 < pT < 2.0 GeV
1.8
Q
Q
4 /2
2
Q
Q
4 /2
2.2
Skellam
||=0.35
0.5
1
3
5
1.6
1.4
1.2
1.2
Static  gas, T=160MeV
1
1
0
0.2
0.4
0.6
ptmin [GeV/c]
0.8
1
No significant dependence on η cut
0
0.5
1
1.5
||=2max
2
Difference coming from lower pt cut
Lower cut induces effective mass heavier than m,
thus approaching Skellam distribution by
increasing ptmin.
Cuts in ηmax only affect high momentum particle
contribution.
30 May, 2016
24
Kenji Morita (YITP, Kyoto)
CPOD2016@Wroclaw
High p Cut Effects on Cumulant Ratios
2.4
Skellam
T=160MeV, || < 0.35
T=120MeV w/ Flow, | | < 0.35
2.2
Q
Q
/
4 2
2
1.8
free  gas, p tmin=0.3 GeV/c
1.6
1.4
1.2
1
0.4
0.6
0.8
1
1.2 1.4
ptmax [GeV/c]
1.6
1.8
ptmax < 1GeV : stronger influences from Bose
statistics
30 May, 2016
25
Kenji Morita (YITP, Kyoto)
CPOD2016@Wroclaw
Electric Charge Fluctuations
Recent measurements @RHIC
STAR(‘14), PHENIX(‘15)
Statistics problem (particularly STAR)
PHENIX: above Skellam ?
30 May, 2016
26
Kenji Morita (YITP, Kyoto)
CPOD2016@Wroclaw
Effects of expansion
Boost-invariant + Transverse Gaussian + Linear flow
10
10
3
2.4
STAR
T=160MeV, yT=0
T=120MeV, f=0.373
Skellam
Static, T=160MeV, |  | < 0.35
T=160MeV, yT =0, | | < 0.35
T=120MeV, f=0.373, |  | < 0.35
2.2
2
R=6fm
Q4 /Q2
d2N/(2dypTdpT) [(GeV/c)-2]
10
2
1.8
free  gas, p tmax=2GeV/c
1.6
1.4
1.2
-
STAR  Au+Au 0-5%, |y| < 0.1
1
1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
pT [GeV/c]
0
0.2
0.4
0.6
ptmin [GeV/c]
0.8
1
Similar χ4/χ2 in T=120MeV w/ Flow and
T=160MeV w/o flow for ptmin ~ 0.2GeV
HRG results found in P. Garg et al., ‘13
30 May, 2016
27
Kenji Morita (YITP, Kyoto)
CPOD2016@Wroclaw
Finite Size and Low pt cut
?
Pion gas in a finite (L3)
box
Periodic boundary
0.25
condition
0.6
ptmin = k 2/L
k=0.707,m /T=0.75
1
1.25
1.5
1.75
2
0.5
0.2
0.4
|η| < 0.35
k = 0.707
2
Q
4
Q
0.15
0.1
0.3
[|η| < 5 : k=0.5]
0.2
k=0.707,m /T=0.75
1
1.25
1.5
1.75
2
0.05
0
0
0.1
LQCD
30 May,
: 2016
0.2
0.3
1/ (Lm )
0.1
ptmin = k 2/L
0
0.4
0.5
0
0.1
0.2
0.3
1/ (Lm )
0.4
0.5
28
Kenji Morita (YITP, Kyoto)
CPOD2016@Wroclaw
Electric Charge Fluctuations
Complementary to net-baryon
Leading singularity from chiral transition
No “proton≠baryon” problem
Larger multiplicity (as π dominates)
than net-baryon
Less efficiency, however.
30 May, 2016
29