Indications for a critical end point in the phase diagram for hot and dense nuclear matter Roy A. Lacey Stony Brook University 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: All are required to fully characterize the CEP & drives the ongoing search Roy A. Lacey, Stony Brook University; Quark Matter 2015, Sept. 28, 2015 2
Theoretical Guidance 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 z T ~ 3 z v ~ 2 z s ~ -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 Hanbury Brown & Twist (HBT) radii obtained from two-pion correlation functions dn2 / dp1dp2 C ( q) = ( dn / dp )( dn / dp 1 1 1 2 ) r r 2 r r R(q) = C(q) 1 = 4 π drr K0( q, r ) S( ) ( 1) Correlation function 3D Koonin Pratt Eqn. Encodes FSI S. Afanasiev et al. (PHENIX) PRL 100 (2008) 232301 Source function (Distribution of pair separations) 2 1 c s = ρκ 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) R R R emission lifetime 2 side R = m 1+ T R 2 geo T 2 T 2 2 geo 2 2 out = + βt ( Δτ ) mt 2 1+ βt T T τ 2 2 long mt β emission duration The measured HBT radii encode space-time information for the reaction dynamics (R side - R init )/ R long sensitive to c s 2 1 c s = Specific non-monotonic patterns expected as a function of s NN A maximum for (R 2 out - R2 side ) A minimum for (R side - R initial )/R long ρκ Roy A. Lacey, Stony Brook University; Quark Matter 2015, Sept. 28, 2015 5
Lacey QM2014. Adare et. al. (PHENIX) arxiv:1410.2559 Rlong τ ( ) out side R R Δτ 2 2 2 ( R R )/ R u R side i long initial = 2R à 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! 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 T > T c large L L characterizes the system size small L T close to T c ξ v ~ T -T c 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 7
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 à 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 8
The Blessings of Finite-Size ( ) χ T L = L P tl t = T T T γ / ν 1/ ν (, ) χ ( ) c / M. Suzuki, Prog. Theor. Phys. 58, 1142, 1977 c Finite-size effects have a specific L dependence L scales the volume ξ ~ T -T c v L 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
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
R R R = kr 2 2 side geom = kr + β ( Δτ ) 2 2 2 2 out geom T T τ 2 2 long mt c 2 1 s = ρκ S characteristic patterns signal the effects of finite-size III. Perform Finite-Size Scaling analysis with length scale L = R Roy A. Lacey, Stony Brook University; Quark Matter 2015, Sept. 28, 2015 11
Length Scale for Finite Size Scaling R is a characteristic length scale of the initial-state transverse size, σ x & σ y! RMS widths of density distribution Rout, Rside, Rlong R R R scales the full RHIC and LHC data sets Roy A. Lacey, Stony Brook University; Quark Matter 2015, Sept. 28, 2015 12
From slope ν ~ 0.66 1 ν s ( V ) = s ( ) k R NN NN 2 2 ( ) max out side R R R γ ν γ ~ 1.2 From intercept ~ 47.5 GeV s CEP The critical exponents validate the 3D Ising model (static) universality class 2 nd order phase transition for CEP cep cep T ~ 165 MeV, µ B ~ 95 MeV Roy A. Lacey, Stony Brook University; Quark Matter 2015, Sept. 28, 2015 13
2 nd order phase transition 3D Ising Model (static) universality class for CEP ν ~ 0.66 γ ~ 1.2 cep cep T ~ 165 MeV, µ B ~ 95 MeV χ = / 1/ ( T, L) L γ ν ν Pχ ( tl ) M. Suzuki, Prog. Theor. Phys. 58, 1142, 1977 **A further validation of the location of the CEP and the (static) critical exponents** Roy A. Lacey, Stony Brook University; Quark Matter 2015, Sept. 28, 2015 14
What about Finite-Time Effects (FTE)? τ ~ ξ z dynamic critical exponent An important consequence z > 0 - Critical slowing down Multiple slow modes? z T ~ 3, z v ~ 2, z s ~ -0.8 z < 0 - Critical speeding up Y. Minami - Xiv:1201.6408 Significant signal attenuation for short-lived processes with z T ~ 3 or z v ~ 2 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
2 nd order phase transition ν ~ 0.66 γ ~ 1.2 cep cep T ~ 165 MeV, µ B ~ 95 MeV DFSS ansatz / ( ) ( 1/ z L, T, L γ ν ν = f L t, ) T L χ τ τ For T = T c γ / ν (,, ) ( z L T ) c = L f L χ τ τ M. Suzuki, Prog. Theor. Phys. 58, 1142, 1977 (R 2 out - R2 side ) fm2 Rlong (%) 12 00-05 05-10 10-20 20-30 30-40 40-50 10 8 6 4 2 3 4 5 6 7 R long fm τ **Experimental estimate of the dynamic critical exponent** z ~ 0.87 0 2.5 3.0 3.5 4.0 R long L -z (fm 1-z ) The magnitude of z is similar to the predicted value for z s but the sign is opposite 5 4 3 2 1 (R -γ/ν )(R 2 out - R2 side ) (fm2-γ/ν ) 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) } ν ~ 0.66 universality class for CEP 2 nd order phase transition γ ~ 1.2 cep cep T ~ 165 MeV, µ B ~ 95 MeV z ~ 0.87 Landmark validated Crossover validated Deconfinement validated (Static) Universality class validated Model H Universality class invalidated? Other implications! 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
(only two exponents are independent ) Note that ( µ f f, T B ) 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 τ = τ 0 + aρ Space-time correlation parameter Roy A. Lacey, Stony Brook University; Quark Matter 2015, Sept. 28, 2015 20
Non Monotonic behavior of the viscous coefficient R. Lacey et al., Phys. Rev. Lett. 112, 082302 (2014) RHIC Roy A. Lacey, Stony Brook University; Quark Matter 2015, Sept. 28, 2015 21
! Systematic study of various quantities as function of s: Source radii Directed flow v 1 Scaled kurtosis (baryon fluctuations) dv 1 /dy y=0! 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 22
Finite size scaling and the Crossover Transition Y. Aoki, et. Al.,Nature, 443, 675(2006). Finite size scaling played an essential role for identification of the crossover transition! Reminder Crossover: size independent. 1 st -order: finite-size scaling function, and scaling exponent is determined by spatial dimension (integer). 2 nd / 1/ -order: finite-size scaling function ( T, L) L γ ν ν χ = P ( tl ) χ 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. 2 1 cs = ρκ S Roy A. Lacey, Stony Brook University; Quark Matter 2015, Sept. 28, 2015 24
Interferometry signal C ( q) = dn2 / dp1dp2 ( dn / dp )( dn / dp 1 1 1 2 ) A. Adare et. al. (PHENIX) arxiv:1410.2559 Roy A. Lacey, Stony Brook University; Quark Matter 2015, Sept. 28, 2015 25
HBT Measurements ALICE -PoSWPCF2011 STAR - 1403.4972 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 Freezeout size reflects; Initial size + expanded size 5 4 Characteristic acoustic Scaling validated in Viscous hydrodynamics R f (fm) 3 2 t R E. Shuryak, I. Zahed Phys.Rev. D89 (2014) 094001 1 Freeze-out time varies with initial size 0 0.0 0.5 1.0 1.5 2.0 R i (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 R side (fm) Larger expansion rate at the LHC 8 6 4 2 Femtoscopic measurements 2.76 TeV Pb+Pb - ALICE 200 GeV Au+Au - PHENIX 200 GeV Au+Au - STAR 200 GeV d+au - PHENIX k T ~ 0.4 GeV/c 0 0.0 0.5 1.0 1.5 2.0 2.5 R (fm) (a) arxiv:1404.5291 Pb+Pb 2.76 TeV p+pb 5.02 TeV 0.2 <k T <0.3 GeV/c ALICE 1 2 R (fm) (b) 16 12 8 4 0 R inv (fm) t R exquisitely demonstrated via HBT measurement for several systems Roy A. Lacey, Stony Brook University; Quark Matter 2015, Sept. 28, 2015 28