Outline
 Introduction
 Phase Diagram & HIC
 Search strategy for the CEP Theoretical guidance Guiding principles for search The probe Femtoscopic“susceptibility”
 Analysis
 Finite-Size-Scaling Dynamic Finite-Size-Scaling Summary Epilogue

Roy A. Lacey, Stony Brook University; TGSW workshop, Tsukuba Japan, Sept. 30, 2015

1

Characterizing the phases of matter

The location of the critical
End point and the phasecoexistence regions arefundamental to the phase
diagram of any substance !
The properties of eachphase is also offundamental interest

Roy A. Lacey, Stony Brook University; TGSW workshop, Tsukuba Japan, Sept. 30, 2015

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The QCD Phase Diagram
Known knowns
Spectacular achievement: Validation of the crossovertransition leading to the QGP
 Initial estimates for thetransport properties of the QGP
Known
unknownsp Location of the critical point (CEP)?
 Order of the phase transition?
 Value of the critical exponents?

 Location of phase coexistence regions?
 Detailed properties of each phase?
All are fundamental to charting the phase
diagram(New) measurements, analysis techniques and theory efforts which probea broad range of the (T, 𝛍𝐁 )-plane, are essential to fully unravel the unknowns!
Roy A. Lacey, Stony Brook University; TGSW workshop, Tsukuba Japan, Sept. 30, 2015

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RHICFirst collisions 2000
 p+p, d+Au, Cu+Cu,Cu+Au, Au+Au, U+U sNN ~ 7 – 200 GeV
LHCFirst collisions 2010
 p+p, Pb+Pb, p+Pb
 sNN =2.76 TeV
 (5.5 TeV in 2015-16)
The RHIC and LHC are two major experimental facilities currently
used to study the QCD phase diagram
Roy A. Lacey, Stony Brook University; TGSW workshop, Tsukuba Japan, Sept. 30, 20154

# participants
Npart

Heavy ion collisions are used to produce the hot and dense
matter used to probe the QCD phase diagram
Roy A. Lacey, Stony Brook University; TGSW workshop, Tsukuba Japan, Sept. 30, 2015

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

RHIC

A Typical Event

LHC

The particles produced in collision events are used tostudy the produced medium
Roy A. Lacey, Stony Brook University; TGSW workshop, Tsukuba Japan, Sept. 30, 2015

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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

Roy A. Lacey, Stony Brook University; TGSW workshop, Tsukuba Japan, Sept. 30, 2015

7

Theoretical Guidance
Theory consensus on the staticuniversality class for the CEP
3D-Ising Z(2) 𝜈 ~ 0.63
 𝛾 ~ 1.2

𝐜𝐞𝐩

The predicted location (𝐓 𝐜𝐞𝐩 , 𝛍𝐁 ) ofthe CEP is even less clear!

M. A. StephanovInt. J. Mod. Phys. A 20, 4387 (2005)

Dynamic Universality classfor the CEP less clear
 One slow mode z ~ 3 - Model H
Son & StephanovPhys.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. StephanovInt. J. Mod. Phys. A 20, 4387 (2005)

Experimental verification and characterizationof the CEP is a crucial ingredient

Roy A. Lacey, Stony Brook University; TGSW workshop, Tsukuba Japan, Sept. 30, 20158

Anatomy of search strategy
𝐇𝟐 O

Cp ~  T ~   ~ T - Tc𝜹𝒏v~ 𝝃𝟐  search for “critical”
fluctuations in HIC

Stephanov, Rajagopal, Shuryak, PRL.81, 4816 (98)The critical point is characterized by several (power law) divergences
Central idea  use beam energy scans to vary 𝛍𝐁 & T to search for theinfluence of such divergences!
 We use femtoscopic measurements to perform our search
Roy A. Lacey, Stony Brook University; TGSW workshop, Tsukuba Japan, Sept. 30, 2015

9Interferometry 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-particle
correlation functionsdN 2 / dp1dp 2C (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(q)  C (q)  1  4  drr 2 K0 (q , r )S (r ) (1)Correlation
function

Encodes FSISource function(Distribution of pairseparations)

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 𝜿leads 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; TGSW workshop, Tsukuba Japan, Sept. 30, 201510

Interferometry signalC (q) dN 2 / dp1dp 2(dN1 / dp1 )(dN1 / dp 2 )A. Adare et. al. (PHENIX)
arXiv:1410.2559

StrategySearch for non-monotonic patterns for HBT radii
combinations that are sensitive to the divergence of 𝜿Roy A. Lacey, Stony Brook University; TGSW workshop, Tsukuba Japan, Sept. 30, 201511

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

2Rgeo
2Rsideemission
mT 2duration
1TT12c2sRgeo
222Rout  T ( )mThe divergence of the susceptibility 𝜿1  T T2
 “softens’’ the sound speed csT extends the emission duration
T 22Rlong (R2out - R2side) sensitive to the 𝜿mTemission
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; TGSW workshop, Tsukuba Japan, Sept. 30, 201512√𝒔𝑵𝑵 dependence of interferometry signal

Lacey QM2014
Adare et. al. (PHENIX)
arXiv:1410.2559.

Rlong   R  R   2out2side2

 Rside  Ri  / Rlong  uRinitial  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 furthervalidation of the CEP, as well as to characterizeits static and dynamic properties
Roy A. Lacey, Stony Brook University; TGSW workshop, Tsukuba Japan, Sept. 30,201513

Basis of Finite-Size Effects
Illustration

large LT > Tcsmall LL characterizes the system size
T close to Tc

 ~ T - Tc

v

Lnote change in peak heights,
positions & widths
 A curse of Finite-Size Effects (FSE) Only a pseudo-critical point is observed  shiftedfrom the genuine CEPRoy A. Lacey, Stony Brook University; TGSW workshop, Tsukuba Japan, Sept. 30, 2015 14

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 givethe precise location of the CEPRoy A. Lacey, Stony Brook University; TGSW workshop, Tsukuba Japan, Sept. 30, 2015 15

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

LL scales
the volume

 Finite-size effects have specific identifiable
dependencies on size (L) The scaling of these dependencies give accessto the CEP’s location and it’s critical exponents
Roy A. Lacey, Stony Brook University; TGSW workshop, Tsukuba Japan, Sept. 30, 2015 16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 Lsmall LThe 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; TGSW workshop, Tsukuba Japan, Sept. 30,
2015

17

𝑺𝒊𝒛𝒆 𝒅𝒆𝒑𝒆𝒏𝒅𝒆𝒏𝒄𝒆 𝒐𝒇 𝑯𝑩𝑻 𝒆𝒙𝒄𝒊𝒕𝒂𝒕𝒊𝒐𝒏 𝒇𝒖𝒏𝒄𝒕𝒊𝒐𝒏𝒔2
R 2 side  kRgeom
22Rout kRgeom T2 ( )22longRT 2
mTcs2 1

 SI. 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; TGSW workshop, Tsukuba Japan,
Sept. 30, 201518

Length Scale for Finite Size Scaling

R is a characteristic length scale of theinitial-state transverse size,
σx & σy  RMS widths of density distribution

Rout , Rside , Rlong  R

R scalesthe volume

 R scales the full RHIC and LHC data sets
Roy A. Lacey, Stony Brook University; TGSW workshop, Tsukuba Japan, Sept. 30, 2015

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𝑭𝒊𝒏𝒊𝒕𝒆 − 𝑺𝒊𝒛𝒆 𝑺𝒄𝒂𝒍𝒊𝒏𝒈sNN (V )  sNN ()  k  RR  R 2outmax
2side 1 R

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-outsystematics)

Roy A. Lacey, Stony Brook University; TGSW workshop, Tsukuba Japan, Sept. 30,
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 of𝐓 𝐚𝐧𝐟 𝛍𝐁 are from √𝐬𝐍𝐍

the location of the CEP andthe (static) critical exponents**
Roy A. Lacey, Stony Brook University; TGSW workshop, Tsukuba Japan, Sept.
30, 2015

21

A 𝑭𝑨𝑸

What about Finite-Time Effects (FTE)?
𝜒𝑜𝑝 diverges at the CEP
so relaxation of the order parameter could be anomalously slow

 ~zdynamic
critical exponentz > 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 ~ 2eg.

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 toestimate the dynamic critical exponent z
Roy A. Lacey, Stony Brook University; TGSW workshop, Tsukuba Japan, Sept.
30, 2015

22

𝑫𝒚𝒏𝒂𝒎𝒊𝒄 𝑭𝒊𝒏𝒊𝒕𝒆 − 𝑺𝒊𝒛𝒆 𝑺𝒄𝒂𝒍𝒊𝒏𝒈**Experimental estimate of thedynamic critical exponent**

 ~ 1.25(%)

z ~ 0.87

00-05
05-10
10-20
20-30
30-40
40-50

12

3

2

8
2

6For
T = Tc  L, Tc ,   L / f  L z M. Suzuki,Prog. Theor. Phys. 58, 1142, 1977

142
3

4

5Rlong fmRlong  67

2.5

3.0

3.5

(R-  L, T ,   L / f  L1/ tT , L z 

2at time 𝜏 when T is near Tcep

2DFSS ansatz(R out - R side) fm

2

1042-T cep ~ 165 MeV, Bcep ~ 95 MeV2

 ~ 0.66)(R out - R side) (fm 2nd order phase transition

04.0

Rlong L-z (fm1-z)

The magnitude of z is similar tothe predicted value for zs but
the sign is opposite

Roy A. Lacey, Stony Brook University; TGSW workshop, Tsukuba Japan, Sept.
30, 201523

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

Tcep~ 165 MeV, cepB

 Landmark validated Crossover validated
 Deconfinementvalidated (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 theoreticalmodeling, will provide crucial
validation tests for thecoexistence regions, as wellas to firm-up characterizationof the CEP!

Much additional
work required to
get to “the end ofthe line”
Roy A. Lacey, Stony Brook University; TGSW workshop, Tsukuba Japan, Sept. 30, 201524

EndRoy A. Lacey, Stony Brook University; TGSW workshop, Tsukuba Japan, Sept. 30, 2015

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𝑭𝒊𝒏𝒊𝒕𝒆 − 𝑺𝒊𝒛𝒆 𝑺𝒄𝒂𝒍𝒊𝒏𝒈 Analysis(only two exponents
are independent )ff
Note that B , T is not strongly dependent on V

Locate (𝑻, 𝛍𝐁 ) position ofdeconfinement transition andextract critical exponents

Determine Universality Class

Determine order of the phasetransition to identify CEP

Roy A. Lacey, Stony Brook University; TGSW workshop, Tsukuba Japan, Sept. 30,
201526