[email protected]
Outline
 Introduction
 Phase Diagram & its ``landmarks”
 Strategy for CEP search
 Anatomical & operational
 Finite-Size/Finite-Time Scaling (FSS/FTS)
 Basics
 Scaling Examples
 Results
 FSS functions for susceptibility
 Dynamic FSS
 FSS functions“A
forrich
fluctuations
life is lived
 Epilogue
Essential point;
 Finite-Size/Finite-Time Scaling provides an
important “window” in the desolate landscape of
from apossibilities
giving heartfor
not
a selfish
locating
andmind.”
characterizing the
― Rasheed
criticalOgunlaru
end point (CEP).
Roy A. Lacey, Zimányi School'16, Dec. 07, 2016
1
Characterizing the phases of matter
Phase diagram for H2O
The location of the critical
End point (CEP) and the
phase coexistence regions
are fundamental to the
phase diagram of any
substance !
Critical Point
End point of 1st order
(discontinuous) transition
The properties of each
phase is also of
fundamental interest
Roy A. Lacey, Zimányi School'16, Dec. 07, 2016
2
QCD Phase Diagram
Quantitative study of the QCD phase diagram is a current
focus of our field
Essential question:
What new insights do we have on
the CEP “landmark” from the RHIC
beam energy scan (BES-I)?
𝑐𝑒𝑝
 Its location (𝑇 𝑐𝑒𝑝 , 𝜇𝐵 )?
 Its static critical exponents - 𝜈, 𝛾?
 Static universality class?
 Order of the transition
 Dynamic critical exponent/s – z?
 Dynamic universality class?
Measurements spanning the full
(T, 𝛍𝐁 )-plane are ongoing/slated at
RHIC and other facilities
All are required to fully characterize
the CEP
Focus:
The use of Finite-Size-Finite-Time Scaling
to locate and characterize the CEP!
Roy A. Lacey, Zimányi School'16, Dec. 07, 2016
3
The CEP “Landmark”
 I) Large scale correlated fluctuations
of the order parameter [OP] develop
 The correlation length (𝜉 )“diverges”
Transitions are anomalous
near the CEP
T
Tc T  Tc T  Tc T
and renders microscopic details
irrelevant for critical behavior (𝜉 ≫ 𝑟).
 II) Only the structure of the
Critical
correlations matter – spatial dimension
opalescence
(d) tensor character (n) of the OP
[CO2]
 Diverse systems can be grouped into
universality classes (d, n) displaying the
same static/dynamic behavior 
characterized by critical exponents
Tc
3D-Ising Universality Class
v
Corr. Length
 (T )  T - Tc
Mag. Sucep.
 M (T )  T - Tc
Heat Cap. CV (T) =

1d E

 T - Tc
V dT
[OP] Magnetization
M(T)  T - Tc

The critical end point is characterized by singular
behavior at T = Tc+/Roy A. Lacey, Zimányi School'16, Dec. 07, 2016
4
QCD Knowns & unknowns

Scaling relations
2    2  
Known known
Theory consensus on the static
universality class for the CEP
 (1   )  2  
3  2  
2  
3D-Ising Z(2), 𝜈 ~ 0.63 , 𝛾 ~ 1.2
Summary - M. A. Stephanov Int. J. Mod. Phys. A 20, 4387 (2005)
Known unknowns
𝐜𝐞𝐩
 Location (𝐓 𝐜𝐞𝐩 , 𝛍𝐁 ) of the CEP?
Summary - M. A. Stephanov Int. J. Mod. Phys. A 20, 4387 (2005)
 Dynamic Universality class for the CEP?
 One slow mode (L),
z ~ 3 - Model H
Son & Stephanov, Phys.Rev. D70 (2004) 056001
Moore & Saremi, JHEP 0809, 015 (2008)
 Three slow modes (NL)
 zT ~ 3
[critical slowing down]
 zv ~ 2
 zs ~ -0.8 [critical speeding-up]
Y. Minami - Phys.Rev. D83 (2011) 094019

Experimental verification
and characterization of
the CEP is an imperative
Ongoing beam energy scans
to probe a large (T,𝛍𝐁 )
domain
Roy A. Lacey, Zimányi School'16, Dec. 07, 2016
5
Operational Strategy(
Near the CEP, anomalous transitions can influence reaction dynamics
and thermodynamics  experimental signatures for the CEP
(𝛍𝐁 ,T) at chemical freeze-out
J. Cleymans et al.
Phys. Rev. C73, 034905
A. Andronic et al.
Acta. Phys.Polon. B40,
1005 (2009)
 LHC  access to high T and small 𝝁𝑩
 RHIC  access to different systems and
a broad domain of the (𝛍𝐁 ,T)-plane
RHICBES to LHC  ~360 𝒔𝑵𝑵 increase
Vary 𝐬𝐍𝐍 to explore the (𝐓, 𝛍𝐁 )-plane for
experimental signatures which could signal the CEP
Roy A. Lacey, Zimányi School'16, Dec. 07, 2016 6
Anatomy of search strategy
Anomalous transitions near the CEP lead to critical fluctuations of
the conserved charges (q = Q, B, S)
The moments of the distributions grow as powers of 
Cumulants of the distributions
 ~ T - Tc
v
Familiar cumulant ratios
Fluctuation moments
can be expressed as
a ratio of
susceptibilities
A pervasive idea  use beam energy scans to vary 𝛍𝐁 & T,
and search for enhanced/non-monotonic fluctuation patterns
Proviso  Finite size/time effects
Roy A. Lacey, Zimányi School'16, Dec. 07, 2016
7
Anatomy of search strategy
𝐇𝟐 O
Cp ~ 
T ~ 




 ~ T - Tc
v
Other thermodynamic quantities show (power law) divergences
Similarly use beam energy scans to vary 𝛍𝐁 & T, and search for the
influence of such divergences on reaction dynamics!
Important proviso:
Finite size/time effects can significantly dampen divergences
 weak non-monotonic response?
Roy A. Lacey, Zimányi School'16, Dec. 07, 2016
8
Anatomy of search strategy
Inconvenient truths:
 Finite-size and finite-time effects complicate the search and
characterization of the CEP
 They impose non-negligible constraints on the magnitude of ξ.
 The observation of non-monotonic signatures, while helpful, is
neither necessary nor sufficient for identification and
characterization of the CEP.
 The prevailing practice to associate the onset of non-monotonic
signatures with the actual location of the CEP is a ``gimmick’’ .
A Convenient Fact:
 The effects of finite size/time lead to specific volume dependencies
which can be leveraged, via scaling, to locate and characterize the CEP
 This constitutes the only credible approach to date!
Roy A. Lacey, Zimányi School'16, Dec. 07, 2016
9
Basics 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, Zimányi School'16, Dec. 07, 2016 10
The curse of Finite-Size effects
Finite-size effects on the sixth order cumulant
-- 3D Ising model
Influence of FSE on the phase diagram
Pan Xue et al arXiv:1604.06858
E. Fraga et. al.
FSE on temp
dependence
of minimum
~ L2.5n
J. Phys.G 38:085101, 2011
Displacement of pseudo-firstorder transition lines and
CEP due to finite-size
 A flawless measurement, sensitive to FSE, can Not locate
and characterize the CEP directly
 One solution  exploit FSE
Roy A. Lacey, Zimányi School'16, Dec. 07, 2016 11
The Blessings of Finite-Size
 and 
critical exponents
 (T , L)  L / P (tL1/ )
t  T  Tc  / Tc
M. Suzuki, Prog. Theor. Phys. 58, 1142, 1977
Finite-size effects have a specific
influence on the succeptibility
 ~ T - Tc
v
L
L scales
the volume
 The scaling of these dependencies give access
to the CEP’s location, critical exponents and
scaling function.
Roy A. Lacey, Zimányi School'16, Dec. 07, 2016 12
Model demonstration of Finite-Size Scaling
Locating the CEP and 𝜈 via
the susceptibility -- 2D-Ising model
 ( )
  1
Finite-Size Scaling of the peak position gives the location of
the deconfinement transition and the critical exponent 𝝂
Roy A. Lacey, Zimányi School'16, Dec. 07, 2016
13
Model demonstration of Finite-Size Scaling
Extracton of 𝛾 𝜈 via the susceptibility
-- 2D-Ising model

 /   1.75
Finite-Size Scaling of the peak
heights give  / 
Recall; these critical exponents reflect
the static universality class and the
order of the phase transition
Roy A. Lacey, Zimányi School'16, Dec. 07, 2016 14
Model demonstration of Finite-Size Scaling
 (T , L)  L / P (tL1/ )
Extracton of the scaling function for
the susceptibility -- 2D-Ising model

t  T  Tc  / Tc
L /  vs. t L1/
M. Suzuki,
Prog. Theor. Phys. 58, 1142, 1977
Tc ,  ,
The susceptibility distribution
for different system sizes, can
be re-scaled to an identical
non-singular Scaling function
Roy A. Lacey, Zimányi School'16, Dec. 07, 2016 15
Model demonstration of Finite-Size Scaling
Extracton of the scaling function for
critical fluctuations -- 3D-Ising model
**Fluctuations 
susceptibility ratio**
tT  T  Tc  / Tc
 r (T , L)  L '/ P (tL1/ )
L '/ S vs. L1/ tT ,
1
Tc ,  ',
3D Ising Model
0
S
Values for the CEP
+
Critical exponents
 '/  = 0
-1
L = 2.5
L = 2.0
L = 1.5
L = 1.0
-2
-1.5
-1.0
1/
Corollary:
L
-0.5
0.0
1/
tT (fm
Scaling function can be used to extract critical
exponents and the location of the CEP
Roy A. Lacey, Zimányi School'16, Dec. 07, 2016 16
Model demonstration of Finite-Size Scaling
Extracton of the scaling function for
critical fluctuations -- 3D-Ising model
**Fluctuations 
susceptibility ratio**
 r (T , L)  L '/ P (tL1/ )
tT  T  Tc  / Tc
L '/  vs. L1/ tT ,
 '/  = 0
6
3D Ising Model
5
Values for the CEP
+
Critical exponents
Tc ,  ',
4
L = 2.5
L = 2.0
L = 1.5
L = 1.0
3
2
1
0
-1.5
Corollary:
-1.0
-0.5
0.0
L1/ tT (fm1/
Scaling function can be used to extract critical
exponents and the location of the CEP
Roy A. Lacey, Zimányi School'16, Dec. 07, 2016 17
A 𝑭𝑨𝑸
The Role of Finite-Time Effects (FTE)?
𝜒𝑜𝑝 diverges at the CEP
so relaxation of the order parameter could be anomalously slow
 ~
dynamic
critical exponent
z
Non-linear dynamics 
Multiple slow modes
zT ~ 3, zv ~ 2, zs ~ -0.8
zs < 0 - Critical speeding up
z > 0 - Critical slowing down
Y. Minami - Phys.Rev. D83 (2011) 094019
An important consequence
Significant signal attenuation for
short-lived processes
with zT ~ 3 or zv ~ 2
eg. 𝜹𝒏 ~ 𝝃𝟐 (without FTE)
𝜹𝒏 ~ 𝝉𝟏/𝒛 ≪ 𝝃𝟐 (with FTE)
**Note that observables driven by the sound mode
would NOT be similarly attenuated**
FTE could depend on
 The specific observable
 Associated dynamic critical exponent/s
FTE could also influence FSE
Roy A. Lacey, Zimányi School'16, Dec. 07, 2016
18
𝑭𝑺𝑺/𝑭𝑻𝑺
Finite-Size-Finite-Time Scaling form
FSS form
V=L3 ,  = T  T cep  / T cep
FTS form
r  z  1/ 
R  cooling rate Ti  T cep
ˆ  effective cor. length for FTE
If Finite-Size Effects Dominate
  ˆ  L
If Finite-Time effects dominate
  L  ˆ
Strategy:
Apply Finite-Size-Finite-Time Scaling to candidate
signals!
Roy A. Lacey, Zimányi School'16, Dec. 07, 2016
19
STAR Phys. Rev. Lett. 116 (2016) 112302
dv1/dy|y=0
𝑪𝒂𝒏𝒅𝒊𝒅𝒂𝒕𝒆 signals
Niseem Magdy

Suggestive non-monotonic behavior at a ~common √s
Roy A. Lacey, Zimányi School'16, Dec. 07, 2016 20
Experimental Results
Strategy
Extract scaling functions for the susceptibility and
critical fluctuations (susceptibility ratios)?
The scaling functions for the susceptibility and critical
fluctuations (susceptibility ratios) should result from the same
values of the CEP and critical exponents!
Interferometry measurements
used as a susceptibility (𝜿) probe
Phys.Rev.Lett. 114 (2015) no.14, 142301
Fluctuations measurements
used as a susceptibility-ratio probe
arXiv:1606.08071
Roy A. Lacey, Zimányi School'16, Dec. 07, 2016
21
Interferometry Measurements
3D Koonin Pratt Eqn.
The space-time dimensions of the
Emitting source (𝑅𝐿 , 𝑅𝑇𝑜 and 𝑅𝑇𝑠 )
can be extracted from two-pion
correlation functions
R (q)  C2 (q)  1  4  drr 2 K 0 (q, r )S (r )
Correlation
function
Encodes FSI
Source function
(Distribution of pair
separations)
Two pion correlation function
C2 (q) 
dN 2 / dp1dp 2
(dN1 / dp1 )(dN1 / dp 2 )
q = p2 – p1
Inversion of this integral
equation  Source Function
(𝑅𝐿 , 𝑅𝑇𝑜 , 𝑅𝑇𝑠 )
S. Afanasiev et al.
PRL 100 (2008) 232301
Interferometry measurements give access to the space-time
dimensions of the emitting source produced in the collisions
Roy A. Lacey, Zimányi School'16, Dec. 07, 2016
22
Interferometry as a susceptibility probe
Hung, Shuryak, PRL. 75,4003 (95)
T. Csörgő. and B. Lörstad, PRC54 (1996) 1390-1403
Chapman, Scotto, Heinz, PRL.74.4400 (95)
Makhlin, Sinyukov, ZPC.39.69 (88)
The measured radii encode
space-time information for
the reaction dynamics
2
R
geo
2
emission
Rside

m
duration
1  T T2
T
1
cs2 
2

R
geo
2
2
2
Rout 
  T (  )
The divergence of the susceptibility 𝜿
mT 2
1
T
 “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, Zimányi School'16, Dec. 07, 2016
23
√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
Ri  2 R
The measurements validate the expected non-monotonic patterns!
** Note that 𝐑 𝐥𝐨𝐧𝐠 , 𝐑 𝐨𝐮𝐭 and 𝐑 𝐬𝐢𝐝𝐞 [all] increase with √𝐬𝐍𝐍 **
 Confirm the expected patterns for Finite-Size-Effects?
 Use Finite-Size Scaling (FSS) to locate and characterize
the CEP
Roy A. Lacey, Zimányi School'16, Dec. 07, 2016
24
𝑺𝒊𝒛𝒆 𝒅𝒆𝒑𝒆𝒏𝒅𝒆𝒏𝒄𝒆 𝒐𝒇 𝑯𝑩𝑻 𝒆𝒙𝒄𝒊𝒕𝒂𝒕𝒊𝒐𝒏 𝒇𝒖𝒏𝒄𝒕𝒊𝒐𝒏𝒔
Phys.Rev.Lett. 114 (2015) no.14, 142301
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
Perform Finite-Size Scaling analysis
with length scale L  R
Roy A. Lacey, Zimányi School'16, Dec. 07, 2016
25
Lacey QM2014
Adare et. al. (PHENIX)
arXiv:1410.2559
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, Zimányi School'16, Dec. 07, 2016
26
𝑬𝒙𝒕𝒓𝒂𝒄𝒕𝒊𝒐𝒏 𝒐𝒇 𝑪𝑬𝑷 & critical exponents
Phys.Rev.Lett. 114 (2015) no.14, 142301
sNN (V )  sNN ()  k  R  
1
From slope
 ~ 0.66
2
 R out2  R side

max
 R


 ~ 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
( 𝐬𝐂𝐄𝐏 & chemical freeze-out
 2nd order phase transition for CEP
systematics)
cep
cep
T
~ 165 MeV,  B ~ 95 MeV
Roy A. Lacey, Zimányi School'16, Dec. 07, 2016
27
𝑬𝒙𝒕𝒓𝒂𝒄𝒕𝒊𝒐𝒏 𝒐𝒇 𝒕𝒉𝒆 𝜒 𝒔𝒄𝒂𝒍𝒊𝒏𝒈 𝒇𝒖𝒏𝒄𝒕𝒊𝒐𝒏
Phys.Rev.Lett. 114 (2015) no.14, 142301
 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
𝐓 𝐚𝐧𝐟 𝛍𝐁 are from √𝐬𝐍𝐍
**Scaling function validates
the location of the CEP and
the (static) critical exponents**
Roy A. Lacey, Zimányi School'16, Dec. 07, 2016
28
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, Zimányi School'16, Dec. 07, 2016
29
𝑭𝒍𝒖𝒄𝒕𝒂𝒕𝒊𝒐𝒏 𝑫𝒂𝒕𝒂
Cumulants of the distributions
(STAR) Phys.Rev.Lett. 112 (2014) 032302
**Fluctuations  susceptibility ratio**
Scaling functions are derived
from these data
Roy A. Lacey, Zimányi School'16, Dec. 07, 2016
30
Statistical Mechanics – Not serendipity
Compressibility
For an isothermal change
VdP  Nd
1
  
 P 
N

V




 N V ,T
 N V ,T  T
Stoki´c, Friman, and K. Redlich, Phys.Lett.B673:192-196,2009
From partition function one can show that
N2  N
1  N 


N   
N
N
C1
 T1  2

2
C2
N  N
2
The compressibility diverges at the CEP
At the CEP the inverse compressibility  0
Scaling function provides an independent measure!
Roy A. Lacey, Zimányi School'16, Dec. 07, 2016 31
arXiv:1606.08071
Compressibility
1.5
00-05%
05-10%
10-20%
20-30%
30-40%
40-50%
50-60%
Au + Au @ RHIC
 Fluctuation measurements
T cep
+
~ 165 MeV,  Bcep ~ 95 MeV
G (T , L)  L '/ P (tL1/ )
C1/C2
 ~ 0.66
1.5
1.0
1.0
0.5
0.5
C1/C2
𝑭𝒍𝒖𝒄𝒕𝒂𝒕𝒊𝒐𝒏 𝑺𝒄𝒂𝒍𝒊𝒏𝒈 𝑭𝒖𝒏𝒄𝒕𝒊𝒐𝒏𝒔
L '/ G (T , L) vs. L1/ tT ,  '/  = 0
M. Suzuki,
Prog. Theor. Phys. 58, 1142, 1977
0.0
0.0
-0.4
The scaling functions for the
critical fluctuations should
result from the same values of
the CEP and critical exponents
L1/ tT (fm1/
𝐜𝐞𝐩
𝐓 𝐜𝐞𝐩 , 𝛍𝐁 ,
𝐓 𝐚𝐧𝐟 𝛍𝐁 are from √𝐬𝐍𝐍
-0.2
0.0
-5
0
L1/ t
5
10
** The compressibility diverges
𝐚𝐧𝐝 𝛎
at the CEP **
Use
to obtain Scaling
Function 𝐏𝛘
Validation of the location of
the CEP and the (static)
critical exponents
Roy A. Lacey, Zimányi School'16, Dec. 07, 2016
32
𝑭𝒍𝒖𝒄𝒕𝒂𝒕𝒊𝒐𝒏 𝑺𝒄𝒂𝒍𝒊𝒏𝒈 𝑭𝒖𝒏𝒄𝒕𝒊𝒐𝒏𝒔
arXiv:1606.08071
 Fluctuation measurements
 ~ 0.66
T
cep
+
~ 165 MeV,  Bcep ~ 95 MeV
G (T , L)  L '/ P (tL1/ )
L '/ G (T , L) vs. L1/ tT ,  '/  = 0
M. Suzuki,
Prog. Theor. Phys. 58, 1142, 1977
The scaling functions for the
critical fluctuations should
result from the same values of
the CEP and critical exponents
1
𝐓 𝐜𝐞𝐩 ,
0
𝐜𝐞𝐩
𝛍𝐁 ,
S
Use
𝐚𝐧𝐝 𝛎
to obtain Scaling
Function 𝐏𝛘
-1
** Validation of the location of
the CEP and the (static)
critical exponents **
3D Ising Model
L = 2.5
L = 2.0
L = 1.5
L = 1.0
-2
-1.5
-1.0
1/
L
-0.5
0.0
1/
tT (fm
Roy A. Lacey, Zimányi School'16, Dec. 07, 2016
33
arXiv:1606.08071
𝑭𝒍𝒖𝒄𝒕𝒂𝒕𝒊𝒐𝒏 𝑺𝒄𝒂𝒍𝒊𝒏𝒈 𝑭𝒖𝒏𝒄𝒕𝒊𝒐𝒏𝒔
 Fluctuation measurements
 ~ 0.66
T cep
+
~ 165 MeV,  Bcep ~ 95 MeV
G (T , L)  L '/ P (tL1/ )
L '/ G (T , L) vs. L1/ tT ,  '/  = 0
M. Suzuki,
Prog. Theor. Phys. 58, 1142, 1977
6
3D Ising Model
5
𝐜𝐞𝐩
Use 𝐓 𝐜𝐞𝐩 , 𝛍𝐁 , 𝐚𝐧𝐝 𝛎
to obtain Scaling
Function 𝐏𝛘
4
L = 2.5
L = 2.0
L = 1.5
L = 1.0
3
2
**further validation of
the location of the CEP and
the (static) critical exponents**
1
0
-1.5
-1.0
-0.5
0.0
L1/ tT (fm1/
Roy A. Lacey, Zimányi School'16, Dec. 07, 2016
34
arXiv:1606.08071
𝑺𝒄𝒂𝒍𝒊𝒏𝒈 𝑭𝒖𝒏𝒄𝒕𝒊𝒐𝒏 𝑻𝒆𝒔𝒕𝒔
 Fluctuation measurements
 ~ 0.66
T
cep
+
~ 165 MeV,  Bcep ~ 95 MeV
G (T , L)  L '/ P (tL1/ )
L '/ G (T , L) vs. L1/ tT ,  '/  = 0
M. Suzuki,
Prog. Theor. Phys. 58, 1142, 1977
𝐜𝐞𝐩
Use incorrect values for 𝐓 𝐜𝐞𝐩 , 𝛍𝐁
obtain Scaling Function 𝐏𝛘
sNN ~ 20 GeV
T cep ~ 160 MeV,  Bcep ~ 205 MeV
~ Value for various maxima/minima
to
** Invalidates incorrect
CEP location **
FAQ
What about apparent limit obtained
from LQCD cep
 B  200 MeV
Calculations performed for L ~ 4 -5 fm
so they set a limit for the pseudo-critical point
cep
 Shifted to larger  B !!
Roy A. Lacey, Zimányi School'16, Dec. 07, 2016
35
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
 ~ 0.66
 ~ 1.2
~ 95 MeV z ~ 0.87

New Data from RHIC (BES-II)
together with theoretical
modeling, can provide
crucial validation tests for
the coexistence regions, as
well as to firm-up
characterization of the CEP!
 Landmark validated
 Crossover validated
 Deconfinement
validated
 (Static) Universality
class validated
 Model H dynamic
Universality class
invalidated?
 Other implications!
Much additional
work required to
get to “the end of
the line”
Roy A. Lacey, Zimányi School'16, Dec. 07, 2016
36
End
Roy A. Lacey, Zimányi School'16, Dec. 07, 2016
37