Observation of the Critical Point in the Phase

Observation of the Critical Point in the Phase Diagram for Hot and Dense Nuclear Matter Roy A. Lacey Stony Brook University Roy. Lacey@Stonybrook. edu Outline Ø Introduction ü Phase Diagram & its ``landmarks” Ø Strategy for CEP search ü Anatomical & operational Ø Finite Size Scaling (FSS) ü Basics ü Scaling Examples Ø Results ü FSS functions for susceptibility ü FSS functions for fluctuations ü Dynamic FSS Ø Epilogue Essential points; ü Finite-Size Scaling provides an important “window” for locating and characterizing the CEP ü The existence of the CEP is not a moot point Roy A. Lacey, International School of Physic, 38 th Course, Erice, Sept. 16 -24, 2016 1

Characterizing the phases of matter Phase diagram for H 2 O 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 1 st order (discontinuous) transition The properties of each phase is also of fundamental interest Roy A. Lacey, International School of Physic, 38 th Course, Erice, Sept. 16 -24, 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)? All are required to fully characterize the CEP Focus: The use of Finite-Size/time Scaling to locate and characterize the CEP! Roy A. Lacey, International School of Physic, 38 th Course, Erice, Sept. 16 -24, 2016 3

The CEP “Landmark” Transitions are anomalous near the CEP Critical opalescence [CO 2] 3 D-Ising Universality Class The critical end point is characterized by singular behavior at T = Tc+/Roy A. Lacey, International School of Physic, 38 th Course, Erice, Sept. 16 -24, 2016 4

QCD Knowns & unknowns Scaling relations Experimental verification and characterization of the CEP is an imperative ü Three slow modes (NL) ü z. T ~ 3 [critical slowing down] ü zv ~ 2 ü zs ~ -0. 8 [critical speeding-up] Y. Minami - Phys. Rev. D 83 (2011) 094019 Roy A. Lacey, International School of Physic, 38 th Course, Erice, Sept. 16 -24, 2016 5

Operational Strategy( Near the CEP, anomalous transitions can influence reaction dynamics and thermodynamics experimental signatures for the CEP J. Cleymans et al. Phys. Rev. C 73, 034905 A. Andronic et al. Acta. Phys. Polon. B 40, 1005 (2009) Roy A. Lacey, International School of Physic, 38 th Course, Erice, Sept. 16 -24, 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 Familiar cumulant ratios **Fluctuations can be expressed as a ratio of susceptibilities** Roy A. Lacey, International School of Physic, 38 th Course, Erice, Sept. 16 -24, 2016 7

Anatomy of search strategy Roy A. Lacey, International School of Physic, 38 th Course, Erice, Sept. 16 -24, 2016 8

Anatomy of search strategy Inconvenient truths: Ø Finite-size and finite-time effects complicate the search and characterization of the CEP ü They impose non-negligible constraints on the magnitude of ξ. Ø The observation of non-monotonic signatures, while helpful, is neither necessary nor sufficient for identification and characterization of the CEP. ü The prevailing practice to associate the onset of non-monotonic signatures with the actual location of the CEP is a ``gimmick’’. A Convenient Fact: Ø The effects of finite size 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, International School of Physic, 38 th Course, Erice, Sept. 16 -24, 2016 9

Basics of Finite-Size Effects Illustration large L T > Tc small L L characterizes the system size T close to Tc 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, International School of Physic, 38 th Course, Erice, Sept. 16 -24, 2016 10

The curse of Finite-Size effects Finite-size effects on the sixth order cumulant -- 3 D Ising model Influence of FSE on the phase diagram Pan Xue et al ar. Xiv: 1604. 06858 E. Fraga et. al. J. Phys. G 38: 085101, 2011 Displacement of pseudo-firstorder transition lines and CEP due to finite-size FSE on temp dependence of minimum ~ L 2. 5 n ü A flawless measurement, sensitive to FSE, can Not locate and characterize the CEP directly ü One solution exploit FSE Roy A. Lacey, International School of Physic, 38 th Course, Erice, Sept. 16 -24, 2016 11

The Blessings of Finite-Size critical exponents M. Suzuki, Prog. Theor. Phys. 58, 1142, 1977 Finite-size effects have a specific influence on the succeptibility L scales the volume ü The scaling of these dependencies give access to the CEP’s location, critical exponents and scaling function. Roy A. Lacey, International School of Physic, 38 th Course, Erice, Sept. 16 -24, 2016 12

Model demonstration of Finite-Size Scaling Roy A. Lacey, International School of Physic, 38 th Course, Erice, Sept. 16 -24, 2016 13

Model demonstration of 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, International School of Physic, 38 th Course, Erice, Sept. 16 -24, 2016 14

Model demonstration of Finite-Size Scaling Extracton of the scaling function for the susceptibility -- 2 D-Ising model M. Suzuki, Prog. Theor. Phys. 58, 1142, 1977 The susceptibility observed for different system sizes, can be re-scaled to an identical Scaling function Roy A. Lacey, International School of Physic, 38 th Course, Erice, Sept. 16 -24, 2016 15

Model demonstration of Finite-Size Scaling **Fluctuations susceptibility ratio** Extracton of the scaling function for critical fluctuations -- 3 D-Ising model Values for the CEP + Critical exponents Corollary: Scaling function can be used to extract critical exponents and the location of the CEP Roy A. Lacey, International School of Physic, 38 th Course, Erice, Sept. 16 -24, 2016 16

Model demonstration of Finite-Size Scaling **Fluctuations susceptibility ratio** Extracton of the scaling function for critical fluctuations -- 3 D-Ising model Values for the CEP + Critical exponents Corollary: Scaling function can be used to extract critical exponents and the location of the CEP Roy A. Lacey, International School of Physic, 38 th Course, Erice, Sept. 16 -24, 2016 17

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! Fluctuations measurements used as a susceptibility-ratio probe Phys. Rev. Lett. 114 (2015) no. 14, 142301 ar. Xiv: 1606. 08071 Roy A. Lacey, International School of Physic, 38 th Course, Erice, Sept. 16 -24, 2016 18

Statistical Mechanics – Not serendipity Compressibility For an isothermal change Stoki´c, Friman, and K. Redlich, Phys. Lett. B 673: 192 -196, 2009 From partition function one can show that The compressibility diverges at the CEP At the CEP the inverse compressibility 0 Scaling function provides an independent measure! Roy A. Lacey, International School of Physic, 38 th Course, Erice, Sept. 16 -24, 2016 19

Interferometry Measurements 3 D Koonin Pratt Eqn. Correlation Encodes FSI function Source function (Distribution of pair separations) Two pion correlation function q = p 2 – p 1 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, International School of Physic, 38 th Course, Erice, Sept. 16 -24, 2016 20

Interferometry as a susceptibility probe Hung, Shuryak, PRL. 75, 4003 (95) T. Csörgő. and B. Lörstad, PRC 54 (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 emission duration emission lifetime (Rside - Rinit)/ Rlong sensitive to cs Specific non-monotonic patterns expected as a function of √s. NN Ø A maximum for (R 2 out - R 2 side) Ø A minimum for (Rside - Rinitial)/Rlong Roy A. Lacey, International School of Physic, 38 th Course, Erice, Sept. 16 -24, 2016 21

Lacey QM 2014 Adare et. al. (PHENIX) ar. Xiv: 1410. 2559. The measurements validate the expected non-monotonic patterns! ü Confirm the expected patterns for Finite-Size-Effects? ü Use Finite-Size Scaling (FSS) to locate and characterize the CEP Roy A. Lacey, International School of Physic, 38 th Course, Erice, Sept. 16 -24, 2016 22

Phys. Rev. Lett. 114 (2015) no. 14, 142301 The data validate the expected patterns for Finite-Size Effects ü Max values decrease with decreasing system size ü Peak positions shift with decreasing system size ü Widths increase with decreasing system size Perform Finite-Size Scaling analysis with length scale Roy A. Lacey, International School of Physic, 38 th Course, Erice, Sept. 16 -24, 2016 23

Lacey QM 2014 Adare et. al. (PHENIX) ar. Xiv: 1410. 2559 Length Scale for Finite Size Scaling . is a characteristic length scale of the initial-state transverse size, σx & σy RMS widths of density distribution Ø scales the full RHIC and LHC data sets Roy A. Lacey, International School of Physic, 38 th Course, Erice, Sept. 16 -24, 2016 24

Phys. Rev. Lett. 114 (2015) no. 14, 142301 From slope From intercept Ø The critical exponents validate ü the 3 D Ising model (static) universality class ü 2 nd order phase transition for CEP Roy A. Lacey, International School of Physic, 38 th Course, Erice, Sept. 16 -24, 2016 25

Phys. Rev. Lett. 114 (2015) no. 14, 142301 Ø 2 nd order phase transition Ø 3 D Ising Model (static) universality class for CEP M. Suzuki, Prog. Theor. Phys. 58, 1142, 1977 **Scaling function validates the location of the CEP and the (static) critical exponents** Roy A. Lacey, International School of Physic, 38 th Course, Erice, Sept. 16 -24, 2016 26

Cumulants of the distributions (STAR) Phys. Rev. Lett. 112 (2014) 032302 **Fluctuations susceptibility ratio** Scaling functions are derived from these data Roy A. Lacey, International School of Physic, 38 th Course, Erice, Sept. 16 -24, 2016 27

ar. Xiv: 1606. 08071 Compressibility Ø Fluctuation measurements + 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 ** The compressibility diverges at the CEP ** Validation of the location of the CEP and the (static) critical exponents Roy A. Lacey, International School of Physic, 38 th Course, Erice, Sept. 16 -24, 2016 28

ar. Xiv: 1606. 08071 Ø Fluctuation measurements + 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 ** Validation of the location of the CEP and the (static) critical exponents ** Roy A. Lacey, International School of Physic, 38 th Course, Erice, Sept. 16 -24, 2016 29

ar. Xiv: 1606. 08071 Ø Fluctuation measurements + M. Suzuki, Prog. Theor. Phys. 58, 1142, 1977 **further validation of the location of the CEP and the (static) critical exponents** Roy A. Lacey, International School of Physic, 38 th Course, Erice, Sept. 16 -24, 2016 30

ar. Xiv: 1606. 08071 Ø Fluctuation measurements + M. Suzuki, Prog. Theor. Phys. 58, 1142, 1977 ** Invalidate incorrect CEP location ** Roy A. Lacey, International School of Physic, 38 th Course, Erice, Sept. 16 -24, 2016 31

What about Finite-Time Effects (FTE)? dynamic critical exponent Non-linear dynamics Multiple slow modes z. T ~ 3, zv ~ 2, zs ~ -0. 8 zs < 0 - Critical speeding up z > 0 - Critical slowing down An important consequence Significant signal attenuation for short-lived processes with z. T ~ 3 or zv ~ 2 Y. Minami - Phys. Rev. D 83 (2011) 094019 **Note that observables driven by the sound mode would NOT be similarly attenuated** The value of the dynamic critical exponent/s is crucial for HIC Dynamic Finite-Size Scaling (DFSS) can be used to estimate the dynamic critical exponent z employed in this study Roy A. Lacey, International School of Physic, 38 th Course, Erice, Sept. 16 -24, 2016 32

Ø 2 nd order phase transition **Experimental estimate of the dynamic critical exponent** DFSS ansatz For T = Tc M. Suzuki, Prog. Theor. Phys. 58, 1142, 1977 The magnitude of z is similar to the predicted value for zs but the sign is opposite Roy A. Lacey, International School of Physic, 38 th Course, Erice, Sept. 16 -24, 2016 33

Epilogue Strong experimental indication for the CEP and its location (Dynamic) Finite-Size Scalig analysis Ø 3 D Ising Model (static) universality class for CEP Ø 2 nd order phase transition 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! ü Landmark validated ü Crossover validated ü Deconfinement validated ü (Static) Universality class validated ü Model H dynamic Universality class invalidated? ü Other implications! Much additional work required to get to “the end of the line” Roy A. Lacey, International School of Physic, 38 th Course, Erice, Sept. 16 -24, 2016 34

End Roy A. Lacey, International School of Physic, 38 th Course, Erice, Sept. 16 -24, 2016 35