Outline
 Introduction
 Phase Diagram
 Search strategy
 Theoretical guidance
 Femtoscopic probe
 Analysis
 Finite-Size-Scaling
 Dynamic Finite-Size-Scaling
 Epilogue
Essential message
This is the first *significant and quantitative*
indication (from the experimental point of view)
of the CEP’s existence.
Anonymous PRL reviewer
Roy A. Lacey
Phys.Rev.Lett. 114 (2015) 14, 142301
Roy A. Lacey, Stony Brook University; Quark Matter 2015, Sept. 28, 2015
1
The QCD Phase Diagram
A central goal of the worldwide program in relativistic heavy ion
collisions, is to chart the QCD phase diagram
Essential Question
What new insights do we have on:
The CEP “landmark”?
𝑐𝑒𝑝
 Location (𝑇 𝑐𝑒𝑝 , 𝜇𝐵 ) values?
 Static critical exponents - 𝜈, 𝛾?
 Static universality class?
 Order of the transition
 Dynamic critical exponent – z?
 Dynamic universality class?
All are required to fully characterize the CEP
& drives the ongoing search
(New) measurements, analysis techniques and theory efforts which
investigate a broad range of the (T, 𝛍𝐁 )-plane are currently underway
Roy A. Lacey, Stony Brook University; Quark Matter 2015, Sept. 28, 2015
2
Theoretical Guidance
Theory consensus on the static
universality class for the CEP
3D-Ising Z(2)
 𝜈 ~ 0.63
 𝛾 ~ 1.2
𝐜𝐞𝐩
The predicted location (𝐓 𝐜𝐞𝐩 , 𝛍𝐁 ) of
the CEP is even less clear!
M. A. Stephanov
Int. J. Mod. Phys. A 20, 4387 (2005)
Dynamic Universality class
for the CEP less clear
 One slow mode
 z ~ 3 - Model H
Son & Stephanov
Phys.Rev. D70 (2004) 056001
Moore & Saremi ,
JHEP 0809, 015 (2008)
 Three slow modes
 zT ~ 3
 zv ~ 2
 zs ~ -0.8
Y. Minami - Phys.Rev. D83
(2011) 094019
M. A. Stephanov
Int. J. Mod. Phys. A 20, 4387 (2005)
Experimental verification and characterization
of the CEP is a crucial ingredient
 We use femtoscopic measurements
to perform our search
Roy A. Lacey, Stony Brook University; Quark Matter 2015, Sept. 28, 2015
3
Femtoscopy as a susceptibility probe
S. Afanasiev et al. (PHENIX)
PRL 100 (2008) 232301
3D Koonin Pratt Eqn.
Hanbury Brown & Twist (HBT) radii
obtained from two-pion
correlation functions
dN 2 / dp1dp 2
C (q) 
(dN1 / dp1 )( dN1 / dp 2 )
The expansion of the emitting
source (𝑅𝐿 , 𝑅𝑇𝑜 , 𝑅𝑇𝑠 )
produced in HI collisions
is driven by cs
૏ of the order parameter
diverges at the CEP
r
r
r r r
2
R(q)  C (q)  1  4  drr K 0 (q , r )S (r ) (1)
Correlation
function
Encodes FSI
Source function
(Distribution of pair
separations)
Inversion of this integral
equation  Source Function
(𝑅𝐿 , 𝑅𝑇𝑜 , 𝑅𝑇𝑠 )
cs2 
1

Susceptibility (૏)
In the vicinity of a phase transition or the CEP, the divergence of 𝜿
leads to anomalies in the expansion dynamics
Strategy
Search for non-monotonic patterns for HBT radii
combinations that are sensitive to the divergence of 𝜿
Roy A. Lacey, Stony Brook University; Quark Matter 2015, Sept. 28, 2015
4
Interferometry Probe
Hung, Shuryak, PRL. 75,4003 (95)
Chapman, Scotto, Heinz, PRL.74.4400 (95)
Makhlin, Sinyukov, ZPC.39.69 (88)
The measured HBT radii encode
space-time information for
the reaction dynamics
2
R
geo
2
Rside

emission
mT 2
duration
1
T
T
1
2
c

2
s
R

geo
2
2
2
Rout 
 T ( )
m
The divergence of the susceptibility 𝜿
1  T T2
 “softens’’ the sound speed cs
T
 extends the emission duration
T 2
2
Rlong 

(R2out - R2side) sensitive to the 𝜿
mT
emission
lifetime
(Rside - Rinit)/ Rlong sensitive to cs
Specific non-monotonic patterns expected as a function of √sNN
 A maximum for (R2out - R2side)
 A minimum for (Rside - Rinitial)/Rlong
Roy A. Lacey, Stony Brook University; Quark Matter 2015, Sept. 28, 2015
5
√𝒔𝑵𝑵 dependence of interferometry signal
Lacey QM2014
Adare et. al. (PHENIX)
arXiv:1410.2559
.
Rlong  
R
2
out
2
 Rside
   2
 Rside  Ri  / Rlong  u
Rinitial  2 R
The measurements validate the expected non-monotonic patterns!
 Reaction trajectories spend a fair amount of time near a
“soft point’’ in the EOS that coincides with the CEP!
** Note that 𝐑 𝐥𝐨𝐧𝐠 , 𝐑 𝐨𝐮𝐭 and 𝐑 𝐬𝐢𝐝𝐞 [all] increase with √𝐬𝐍𝐍 **
Finite-Size Scaling (FSS) is used for further
validation of the CEP, as well as to characterize
its static and dynamic properties
Roy A. Lacey, Stony Brook University; Quark Matter 2015, Sept. 28, 2015
6
Basis of Finite-Size Effects
Illustration
large L
T > Tc
small L
L characterizes the system size
T close to Tc
 ~ T - Tc
v
L
note change in peak heights,
positions & widths
 A curse of Finite-Size Effects (FSE)
 Only a pseudo-critical point is observed  shifted
from the genuine CEP
Roy A. Lacey, Stony Brook University; Quark Matter 2015, Sept. 28, 2015
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The curse of Finite-Size effects
E. Fraga et. al.
J. Phys.G 38:085101, 2011
Displacement of pseudo-first-order transition lines and CEP due to finite-size
Finite-size shifts both the pseudo-critical point
and the transition line
 A flawless measurement, sensitive to FSE, can not give
the precise location of the CEP
Roy A. Lacey, Stony Brook University; Quark Matter 2015, Sept. 28, 2015
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The Blessings of Finite-Size
 (T , L)  L / P (tL1/ )
t  T  Tc  / Tc
M. Suzuki,
Prog. Theor. Phys. 58, 1142, 1977
Finite-size effects have a specific
L dependence
 ~ T - Tc
v
L
L scales
the volume
 Finite-size effects have specific identifiable
dependencies on size (L)
 The scaling of these dependencies give access
to the CEP’s location and it’s critical exponents
Roy A. Lacey, Stony Brook University; Quark Matter 2015, Sept. 28, 2015
9
𝑺𝒊𝒛𝒆 𝒅𝒆𝒑𝒆𝒏𝒅𝒆𝒏𝒄𝒆 𝒐𝒇 𝑯𝑩𝑻 𝒆𝒙𝒄𝒊𝒕𝒂𝒕𝒊𝒐𝒏 𝒇𝒖𝒏𝒄𝒕𝒊𝒐𝒏𝒔
Roy A. Lacey
Phys.Rev.Lett. 114 (2015) 14, 142301
Data from L. Adamczyk et al. (STAR)
Phys.Rev. C92 (2015) 1, 014904
Large L
small L
The data validate the expected patterns for Finite-Size Effects
 Max values decrease with decreasing system size
 Peak positions shift with decreasing system size
 Widths increase with decreasing system size
Roy A. Lacey, Stony Brook University; Quark Matter 2015, Sept. 28, 2015
10
𝑺𝒊𝒛𝒆 𝒅𝒆𝒑𝒆𝒏𝒅𝒆𝒏𝒄𝒆 𝒐𝒇 𝑯𝑩𝑻 𝒆𝒙𝒄𝒊𝒕𝒂𝒕𝒊𝒐𝒏 𝒇𝒖𝒏𝒄𝒕𝒊𝒐𝒏𝒔
2
R 2 side  kRgeom
2
2
Rout
 kRgeom
 T2 ( ) 2
2
long
R
T 2


mT
cs2 
1
 S
I. Use (𝑹𝟐𝒐𝒖𝒕 -𝑹𝟐𝒔𝒊𝒅𝒆) as a proxy for the susceptibility
characteristic patterns signal
the effects of finite-size
II. Parameterize distance to the CEP by 𝒔𝑵𝑵
𝝉𝒔 = 𝒔 − 𝒔𝑪𝑬𝑷 / 𝒔𝑪𝑬𝑷
III. Perform Finite-Size Scaling analysis
with length scale L  R
Roy A. Lacey, Stony Brook University; Quark Matter 2015, Sept. 28, 2015
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Length Scale for Finite Size Scaling
R is a characteristic length scale of the
initial-state transverse size,
σx & σy  RMS widths of density distribution
Rout , Rside , Rlong  R
R scales
the volume
 R scales the full RHIC and LHC data sets
Roy A. Lacey, Stony Brook University; Quark Matter 2015, Sept. 28, 2015
12
𝑭𝒊𝒏𝒊𝒕𝒆 − 𝑺𝒊𝒛𝒆 𝑺𝒄𝒂𝒍𝒊𝒏𝒈
sNN (V )  sNN ()  k  R
R
2
out
R

max
2
side
 R

 1

From slope
 ~ 0.66
 ~ 1.2
From intercept
sCEP ~ 47.5 GeV
** Same 𝝂 value from analysis of the widths **
 The critical exponents validate
 the 3D Ising model (static) universality class
 2nd order phase transition for CEP
T cep ~ 165 MeV,  Bcep ~ 95 MeV ( 𝐬𝐂𝐄𝐏 & chemical freeze-out
systematics)
Roy A. Lacey, Stony Brook University; Quark Matter 2015, Sept. 28, 2015
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𝑪𝒍𝒐𝒔𝒖𝒓𝒆𝒓 𝒕𝒆𝒔𝒕 𝒇𝒐𝒓 𝑭𝑺𝑺
 2nd order phase transition
 3D Ising Model (static)
universality class for CEP
 ~ 0.66
 ~ 1.2
T cep ~ 165 MeV,  Bcep ~ 95 MeV
 (T , L)  L / P (tL1/ )
M. Suzuki,
Prog. Theor. Phys. 58, 1142, 1977
𝐜𝐞𝐩
Use 𝐓 𝐜𝐞𝐩 , 𝛍𝐁 , 𝛎 and 𝛄
to obtain Scaling
Function 𝐏𝛘
**A further validation of
𝐓 𝐚𝐧𝐟 𝛍𝐁 are from √𝐬𝐍𝐍
the location of the CEP and
the (static) critical exponents**
Roy A. Lacey, Stony Brook University; Quark Matter 2015, Sept. 28, 2015
14
A 𝑭𝑨𝑸
What about Finite-Time Effects (FTE)?
𝜒𝑜𝑝 diverges at the CEP
so relaxation of the order parameter could be anomalously slow
 ~
z
dynamic
critical exponent
z > 0 - Critical slowing down
Multiple slow modes?
zT ~ 3, zv ~ 2, zs ~ -0.8
z < 0 - Critical speeding up
Y. Minami - Xiv:1201.6408
An important consequence
Significant signal attenuation for
short-lived processes
with zT ~ 3 or zv ~ 2
eg.
𝜹𝒏 ~ 𝝃𝟐 (without FTE)
𝜹𝒏 ≪ 𝝃𝟐 (with FTE)
The value of the dynamic critical exponent/s is crucial for HIC
Dynamic Finite-Size Scaling (DFSS) is used to
estimate the dynamic critical exponent z
Roy A. Lacey, Stony Brook University; Quark Matter 2015, Sept. 28, 2015
15
𝑫𝒚𝒏𝒂𝒎𝒊𝒄 𝑭𝒊𝒏𝒊𝒕𝒆 − 𝑺𝒊𝒛𝒆 𝑺𝒄𝒂𝒍𝒊𝒏𝒈
**Experimental estimate of the
dynamic critical exponent**
 ~ 1.2
5
(%)
z ~ 0.87
00-05
05-10
10-20
20-30
30-40
40-50
2-
2
10
4
(fm
12
2
-R
2
-
6
out
2
3
8
-R
out
2
  L, T ,   L / f  L1/ tT , L z 
(R
at time 𝜏 when T is near Tcep
side)
fm
DFSS ansatz
2
T cep ~ 165 MeV,  Bcep ~ 95 MeV
)(R
 ~ 0.66
side)
order phase transition
For
T = Tc
  L, Tc ,   L / f  L z 
M. Suzuki,
Prog. Theor. Phys. 58, 1142, 1977
1
4
2
3
4
5
Rlong fm
Rlong  
6
7
2.5
3.0
3.5
(R

2nd
0
4.0
Rlong L-z (fm1-z)
The magnitude of z is similar to
the predicted value for zs but
the sign is opposite
Roy A. Lacey, Stony Brook University; Quark Matter 2015, Sept. 28, 2015
16
Epilogue
Strong experimental indication for the CEP
and its location
(Dynamic) Finite-Size Scalig analysis
 3D Ising Model (static)
universality class for CEP
 2nd order phase transition
T
cep
~ 165 MeV, 
cep
B
 Landmark validated
 Crossover validated
 Deconfinement
validated
 (Static) Universality
class validated
 Model H Universality
class invalidated?
 Other implications!
 ~ 0.66
 ~ 1.2
~ 95 MeV z ~ 0.87

New Data from RHIC (BES-II)
together with theoretical
modeling, will provide crucial
validation tests for the
coexistence regions, as well
as to firm-up characterization
of the CEP!
Much additional
work required to
get to “the end of
the line”
Roy A. Lacey, Stony Brook University; Quark Matter 2015, Sept. 28, 2015
17
End
Roy A. Lacey, Stony Brook University; Quark Matter 2015, Sept. 28, 2015
18
𝑭𝒊𝒏𝒊𝒕𝒆 − 𝑺𝒊𝒛𝒆 𝑺𝒄𝒂𝒍𝒊𝒏𝒈 Analysis
(only two exponents
are independent )


f
f
Note that  B , T is not strongly dependent on V
Roy A. Lacey, Stony Brook University; Quark Matter 2015, Sept. 28, 2015
19
Phys.Rev.Lett.100:232301,2008)
Phys.Lett. B685 (2010) 41-46
   0  a
Space-time correlation parameter
Roy A. Lacey, Stony Brook University; Quark Matter 2015, Sept. 28, 2015
20
Non Monotonic behavior of the viscous coefficient
 Initial experimental
indication for η/s variation in
the 𝑻, 𝝁𝑩 -plane
R. Lacey et al., Phys. Rev. Lett. 112, 082302 (2014)
 CEP?
RHIC
Roy A. Lacey, Stony Brook University; Quark Matter 2015, Sept. 28, 2015
21
√𝒔𝑵𝑵 dependence of various signals

Systematic study of
various quantities
as function of √s:

dv1/dy|y=0
◦ Source radii
◦ Directed flow v1
◦ Scaled kurtosis
(baryon fluctuations)
Suggestive non-monotonic
behavior at a ~common √s
 Compelling need for
RHIC Beam Energy Scan II
◦ Greatly increase statistical precision
◦ Additional √s points
Roy A. Lacey, Stony Brook University; Quark Matter 2015, Sept. 28, 2015
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Finite size scaling and the Crossover Transition
Finite size scaling played an essential role for identification
of the crossover transition!
Y. Aoki, et. Al.,Nature ,
443, 675(2006).
Reminder
Crossover: size independent.
1st-order: finite-size scaling function, and scaling exponent is
determined by spatial dimension (integer).
 /
1/
2nd-order: finite-size scaling function  (T , L)  L P (tL )
Roy A. Lacey, Stony Brook University; Quark Matter 2015, Sept. 28, 2015 23
Interferometry as a susceptibility probe
Dirk Rischke and Miklos Gyulassy
Nucl.Phys.A608:479-512,1996
In the vicinity of a phase transition or
the CEP, the sound speed is expected
to soften considerably.
1
cs2 
 S
Divergence of the compressibility (𝜿)
 non-monotonic excitation function
for (R2out - R2side) due to an enhanced
emission duration
Roy A. Lacey, Stony Brook University; Quark Matter 2015, Sept. 28, 2015
24
Interferometry signal
C (q) 
dN 2 / dp1dp 2
(dN1 / dp1 )( dN1 / dp 2 )
A. Adare et. al. (PHENIX)
arXiv:1410.2559
Roy A. Lacey, Stony Brook University; Quark Matter 2015, Sept. 28, 2015
25
HBT Measurements
STAR - 1403.4972
ALICE -PoSWPCF2011
These data
are used to search
for non-monotonic
patterns as a function
of 𝑠𝑁𝑁
Exquisite data set for study of the HBT excitation function!
Roy A. Lacey, Stony Brook University; Quark Matter 2015, Sept. 28, 2015 26
Scaling properties of HBT
Viscous Hydrodynamics – B. Schenke
5
Freezeout size reflects;
Pb+Pb
 Initial size
+ expanded size
tR
20-25%
40-45%
65-70%
4
4
3
Rf (fm)
R(t) (fm)
Characteristic acoustic
Scaling validated in
Viscous hydrodynamics
5
3
2
2
1
1
0
0
0.0
E. Shuryak, I. Zahed
Phys.Rev. D89 (2014) 094001
0
2
4
6
8
10
12
t (fm/c)
Freeze-out time varies
with initial size
0.5
1.0
1.5
2.0
Ri (fm)
Freeze-out transverse
size proportional to initial
transverse size
Roy A. Lacey, Stony Brook University; Quark Matter 2015, Sept. 28, 2015 27
Acoustic Scaling of HBT Radii
2.76 TeV Pb+Pb
200 GeV Au+Au
200 GeV Au+Au
200 GeV d+Au
Rside (fm)
8
 Larger expansion
rate at the LHC
6
arXiv:1404.5291
- ALICE
- PHENIX
- STAR
- PHENIX
Pb+Pb 2.76 TeV
p+Pb 5.02 TeV
0.2 <kT<0.3 GeV/c
ALICE
16
12
kT~ 0.4 GeV/c
4
8
2
4
(b)
(a)
0
0.0
Rinv (fm)
Femtoscopic measurements
0
0.5
1.0
1.5
R (fm)
2.0
2.5
1
2
R (fm)
t  R exquisitely demonstrated via HBT
measurement for several systems
Roy A. Lacey, Stony Brook University; Quark Matter 2015, Sept. 28, 2015
28