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Search for the critical point in the phase diagram

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1 Search for the critical point in the phase diagram
for QCD matter Roy A. Lacey Stony Brook University Plan Introduction QCD phase diagram & it’s ``landmarks” Strategy for CEP search Anatomical & operational Finite-Size Scaling functions (FSS) Basics Proof of principle Experimental Scaling Functions Demonstrate FSS for 𝜿 𝑻 and 𝝌 𝑩 𝒏 ( 𝝉 𝝁 ,𝐋) 𝝌 𝑩 π’Ž ( 𝝉 𝝁 ,𝐋) Dynamic FSS to estimate z? Epilogue Takeaway messages; The critical end point (CEP) exists and is characterizable Finite-Size/Finite-time Scaling is essential to credible searches for the CEP Finite-size susceptibility scaling functions extracted from currently available data, give estimates for; the location ( 𝑻 𝒄𝒆𝒑 , 𝝁 𝑩 𝒄𝒆𝒑 ) of the CEP the critical exponents - 𝝂, 𝜸, 𝜷, 𝜹 𝒂𝒏𝒅 𝒛 Roy A. Lacey, Weihai-Shandong University, China

2 QCD Phase Diagram Quantitative study of the QCD phase diagram is a major focus of our field NICA π’Ž 𝒔 > π’Ž 𝒔 π’„π’“π’Šπ’• > π’Ž 𝒖,𝒅 CEP validation requires knowledge about; Its location ( 𝑇 𝑐𝑒𝑝 , πœ‡ 𝐡 𝑐𝑒𝑝 )? Its static critical exponents - Ξ²,𝜈, 𝛾, etc.? Static universality class? Order of the transition It’s dynamic critical exponent/s – z? Dynamic universality class? M. A. Halasz, et. al. Phys. Rev. DΒ 58, (1998) All are required to fully characterize the CEP! Roy A. Lacey, Weihai-Shandong University, China

3 3D-Ising Universality Class
Exploration strategy - Anatomical( The critical end point is characterized by singular behavior at/near 𝑻 𝒄𝒆𝒑 , 𝝁 𝑩 𝒄𝒆𝒑 J. Cleymans et al., Phys. Rev. C73, Andronic et al., Acta. Phys.Polon. B40, 1005 (2009) 3D-Ising Universality Class If reaction trajectories pass through the critical region, it can; Influence baryon number susceptibility and the associated critical fluctuations The freezeout and the critical region should not be too far apart Influence baryon number susceptibility and the associated expansion dynamics The freezeout and the critical region do not have to be similarly close Vary the beam energy ( 𝐬 𝐍𝐍 ) to explore the ( 𝐓, 𝛍 𝐁 )-plane for anomalous transitions which signal the CEP Roy A. Lacey, Weihai-Shandong University, China

4 Exploration strategy - Dynamics driven
Dirk Rischke and Miklos Gyulassy Nucl.Phys.A608: ,1996 Exploration strategy - Dynamics driven 𝑐 𝑠 2 = 1 𝜌 πœ… 𝑠 πœ… 𝑠 πœ… 𝑇 = 𝑐 𝑉 𝑐 𝑃 The radii of the β€œfireball” encode space-time information related to the expansion dynamics 𝐇 𝟐 O (R2out - R2side) sensitive to 𝜿 The divergence of 𝜿 at or near the CEP β€œsoftens’’ the sound speed cs extends the emission duration The compressibility, which influence the expansion dynamics, shows a (power law) divergence at/near the CEP Search for this influence with femtoscopy? Measure (R2out - R2side) as a function of 𝐬 𝐍𝐍 [ 𝛍 𝐁 , 𝑻], to search for non-monotonic signature signaling the CEP! Proviso οƒ  Finite size/time effects Roy A. Lacey, Weihai-Shandong University, China

5 Exploration strategy – Fluctuations driven
Anomalous transitions near the CEP lead to critical fluctuations of the conserved charges (q = Q, B, S) Cumulants often used to Characterize distribution 𝐢 1 π‘ž = 𝑁 𝐢 2 π‘ž = 𝛿𝑁 2 𝐢 3 π‘ž = 𝛿𝑁 3 𝐢 4 π‘ž = 𝛿𝑁 4 βˆ’ 𝛿𝑁 The moments of these distributions grow as powers of 𝛿𝑁= π‘βˆ’ 𝑁 A flat distribution has only one non-zero cumulant C1. 𝐢 𝑛 π‘ž =0, for n > 2, for a Gaussian distribution Higher order cumulants characterizes the non-Gaussianity of a distribution Higher order cumulants reach their maximum at/near the CEP 𝑀=𝐢 1 𝐡 𝜎 2 =𝐢 2 𝐡 𝑆= 𝐢 3 𝐡 𝜎 2 3/2 πœ…= 𝐢 4 𝐡 𝜎 2 2 **Fluctuations can be expressed as a ratio of susceptibilities** Measure cumulants of conserved charge distributions as a function of 𝐬 𝐍𝐍 ( 𝛍 𝐁 ,T), to search for critical fluctuations! Proviso οƒ  Finite size/time effects Roy A. Lacey, Weihai-Shandong University, China

6 Exploration strategy – Fluctations/Dynamics
Non-monotonic patterns for cumulants or susceptibility ratios could signal the CEP οƒ  strong dependence on model specifics reaction trajectory Holographic QCD phase diagram with critical point Consistency Check 1 𝑁 πœ• 𝑁 πœ•π›½πœ‡ = 𝑁 2 βˆ’ 𝑁 2 𝑁 πœ… 𝑇 ∝ 𝑁 2 βˆ’ 𝑁 2 𝑁 = 𝐢 2 𝐢 1 The singular contributions to baryon number susceptibility ratios are universal features for phase transitions and the existence of the CEP! Proviso οƒ  Finite size/time effects Roy A. Lacey, Weihai-Shandong University, China

7 Collision systems are small and short-lived!
Finite-Size effects strongly influence characterization of the CEP Collision systems are small and short-lived! Significant signal attenuation Shifts the CEP to a new pseudocritical point Finite-size effects on the sixth order cumulant -- 3D Ising model E. Fraga et. al. J. Phys.G 38:085101, 2011 Pan Xue et al arXiv: Displacement of pseudo-CEP & pseudo-first-order transition lines due to finite-size FSE on temp dependence of minimum ~ L2.5n A flawless measurement, sensitive to FSE, cannot locate and characterize the CEP directly Roy A. Lacey, Weihai-Shandong University, China

8 Basics of the Finite-Size Effect
Illustration large L small L T > Tc L characterizes the system size T close to Tc note change in peak heights, positions & widths οƒ  The curse of Finite-Size Effects (FSE) In addition to signal attenuation, only a pseudo-critical point (shifted from the genuine CEP) can be observed! Roy A. Lacey, Weihai-Shandong University, China

9 Basics of the Finite-Time Effect
πœ’ π‘œπ‘ diverges at the CEP so relaxation of the order parameter could be anomalously slow z is the dynamic critical exponent Consequence 𝑑 π‘’π‘ž ~ πœ‰ 𝑧 ≑ 𝜏 βˆ’πœˆπ‘§ πœ‰ ~ 𝑑 1/𝑧 Non-linear dynamics οƒ  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) Significant signal attenuation for short-lived processes with zT ~ 3 or zv ~ 2 eg. π‘ͺ 𝟐 ~ 𝝃 𝟐 (without FTE) π‘ͺ 𝟐 ~ 𝒕 𝟐/𝒛 β‰ͺ 𝝃 𝟐 (with FTE) **Note that observables driven by thermoacoustic coupling would not be similarly affected** FTE complicate the search for the CEP in short-lived systems. Roy A. Lacey, Weihai-Shandong University, China

10 Inconvenient truths: Solution? Anatomy of search strategy - Summary
Finite-size and finite-time effects complicate the search for the CEP, as well as its characterization. They impose non-negligible constraints on the growth of ΞΎ. The observation of a non-monotonic experimental signature, while instructive, would not be sufficient to identify and characterize the CEP. One should not associate the onset of non-monotonic CEP-driven signatures with the actual location of the CEP. Solution? Leverage susceptibility scaling functions to locate and characterize the CEP To date, this constitutes the only credible experimental approach to characterize the CEP! Roy A. Lacey, Weihai-Shandong University, China

11 Finite-Size-Scaling (FSS) - Anatomical(
Generalized Finite-Size-Finite-Time Scaling form for πœ’ πœ’ 𝜏,𝐻, 𝑑,𝐿 = 𝐿 𝛾 𝜈 πœ’(𝜏 𝐿 1 𝜈 ,𝐻 𝐿 𝛽𝛿 𝜈 ,𝑑 𝐿 βˆ’π‘§ ) πœ‰= cor. length πœ‰ = eff. cor. length for FTE 𝐿= sys. size Ο„=π‘‡βˆ’ 𝑇 𝑐𝑒𝑝 H = field t = time Finite-Size Scaling For πœ‰> πœ‰ >𝐿 πœ’ 𝐻, 𝐿 = 𝐿 𝛾 𝜈 𝑓 1 𝑠 𝐻 𝐿 𝛽𝛿 𝜈 πœ’ 𝜏, 𝑑 = 𝐿 𝛾 𝜈 𝑓 2𝑑 𝑠 (𝜏 𝐿 1 𝜈 ,𝑑 𝐿 βˆ’π‘§ ) dynamic finite-size scaling form for the correlation time πœ’ 𝑇,𝐿 = 𝐿 𝛾 𝜈 𝑓 2 𝑠 (𝜏 𝐿 1 𝜈 ) 𝑑 𝐿 =𝐿 𝑧 𝑓 3𝑑 𝑠 (𝜏 𝐿 1 𝜈 ) An observation of Finite-Size Scaling of susceptibility measurements would give access to the CEP’s location and the critical exponents Roy A. Lacey, Weihai-Shandong University, China

12 Finite-Size-Scaling (FSS) - Operational(
Consider measurements of the susceptibility (Ο‡(𝑻)) for different lengths (𝐋= 𝐕 𝟏/πŸ‘ ) Scaling Function M. Suzuki, Prog. Theor. Phys. 58, 1142, 1977 Finite-size effects lead to characteristic dependencies of the peak heights, widths & positions on L, which scale with the critical exponents. The scaling of these dependencies give access to the CEP’s location, critical exponents and a non-singular scaling function. Roy A. Lacey, Weihai-Shandong University, China

13 Finite-Size Scaling Functions
πœ’(𝐻,𝐿)= 𝐿 𝛾/𝜈 𝑃 πœ’ ( 𝐻𝐿 𝛽𝛿/𝜈 ) πœ’ 𝐻𝐿 Ξ²Ξ΄/𝜈 πœ’πΏ βˆ’π›Ύ/𝜈 Finite-Size Scaling functions validate the CEP’s location and the associated critical exponents. Roy A. Lacey, Weihai-Shandong University, China

14 FSS - Proof of Principle Apply similar idea to data?
E. Fraga et. al. J. Phys.G 38:085101, 2011 FSS - Proof of Principle 𝜏 𝑇 = 𝑇 𝑐𝑒𝑝 𝑉 βˆ’ 𝑇 𝑐𝑒𝑝 ∞ ~ 𝐿 βˆ’1/𝜐 𝑇 𝑐𝑒𝑝 ∞ ΞΌ 𝑐𝑒𝑝 ∞ 𝜏 ΞΌ = ΞΌ 𝑐𝑒𝑝 𝑉 βˆ’ ΞΌ 𝑐𝑒𝑝 ∞ ~ 𝐿 βˆ’Ξ²Ξ΄/𝜐 Finite Size Scaling confirms the CEP’s infinite-volume location and it’s associated critical exponents. Apply similar idea to data? Roy A. Lacey, Weihai-Shandong University, China

15 π‘Ίπ’Šπ’›π’† 𝒅𝒆𝒑𝒆𝒏𝒅𝒆𝒏𝒄𝒆 𝒐𝒇 𝑯𝑩𝑻 π’†π’™π’„π’Šπ’•π’‚π’•π’Šπ’π’ π’‡π’–π’π’„π’•π’Šπ’π’π’”
The divergence of the susceptibility 𝛋 β€œsoftens’’ the sound speed cs extends the emission duration I. Use (𝑹 𝒐𝒖𝒕 𝟐 - 𝑹 π’”π’Šπ’…π’† 𝟐 ) as a proxy for the susceptibility II. Parameterize distance to the CEP by 𝒔 𝑡𝑡 𝝉 𝒔 = 𝒔 βˆ’ 𝒔 π‘ͺ𝑬𝑷 / 𝒔 π‘ͺ𝑬𝑷 The measurements indicate non-monotonic behavior! characteristic patterns, signal the effects of finite-size Perform Finite-Size Scaling analysis with length scale Roy A. Lacey, Weihai-Shandong University, China

16 scales the volume Length Scale for Finite Size Scaling
Lacey QM2014 Adare et. al. (PHENIX) . arXiv: Length Scale for Finite Size Scaling 𝑅 ≑𝐿 is a characteristic length scale of the initial-state transverse size, Οƒx & Οƒy οƒ  RMS widths of density distribution 𝑅 ≑𝐿 scales the volume scales the full RHIC and LHC data sets 𝑅 ≑𝐿 Roy A. Lacey, Weihai-Shandong University, China

17 π‘­π’Šπ’π’Šπ’•π’† π‘Ίπ’Šπ’›π’† π‘Ίπ’„π’‚π’π’Šπ’π’ˆ 𝒐𝒇 𝒕𝒉𝒆 π‘Ίπ’–π’„π’„π’†π’‘π’•π’Šπ’ƒπ’π’Šπ’•π’š π’‘π’“π’π’™π’š
Phys.Rev.Lett. 114 (2015) no.14, R out 2 βˆ’ R side 2 max ∝ 𝐿 𝛾 𝜈 𝑠 𝑁𝑁 𝑉 = 𝑠 𝑁𝑁 ∞ βˆ’π‘˜Γ— 𝐿 1 𝜈 𝑅 ≑𝐿 𝑠 𝑁𝑁 ∞ ~ GeV 𝑇 𝑐𝑒𝑝 ∞ 𝜈 ~ 0 .63 𝑠 𝑁𝑁 𝐢𝐸𝑃 ~ GeV The extracted critical exponents (𝝂 and 𝜸) validate the 3D Ising model (static) universality class 2nd order phase transition for the CEP Roy A. Lacey, Weihai-Shandong University, China

18 π‘Ίπ’–π’„π’„π’†π’‘π’•π’Šπ’ƒπ’π’Šπ’•π’š π‘Ίπ’„π’‚π’π’Šπ’π’ˆ π‘­π’–π’π’„π’•π’Šπ’π’
𝐿 βˆ’π›Ύ 𝜈 πœ’ 𝑠,𝐿 = 𝑓 2 𝑠 (𝑠 𝐿 1 𝜈 ) 𝑅 ≑𝐿 s= 𝒔 βˆ’ 𝒔 π‘ͺ𝑬𝑷 / 𝒔 π‘ͺ𝑬𝑷 Data collapse onto a single curve, confirms the expected non-singular scaling function. Roy A. Lacey, Weihai-Shandong University, China

19 π‘­π’Šπ’π’Šπ’•π’† π‘Ίπ’Šπ’›π’† π‘Ίπ’„π’‚π’π’Šπ’π’ˆ 𝒐𝒇 𝒕𝒉𝒆 π‘Ίπ’–π’„π’„π’†π’‘π’•π’Šπ’ƒπ’π’Šπ’•π’š π’‘π’“π’π’™π’š ( 𝝁 𝑩 )
Karsch et. al 𝜏 ΞΌ = ΞΌ 𝑐𝑒𝑝 𝑉 βˆ’ ΞΌ 𝑐𝑒𝑝 ∞ ~ 𝐿 βˆ’Ξ²Ξ΄/𝜐 Strong Vol. dependence ΞΌ 𝑐𝑒𝑝 ∞ Mapping 𝑠 𝑁𝑁 β†’(𝑇, πœ‡ 𝐡 ) 𝑠 𝑁𝑁 𝐢𝐸𝑃 ~ GeV πœ‡ 𝐡 𝑐𝑒𝑝 ~ 90 MeV (Vβ†’βˆž) 𝑇 𝑐𝑒𝑝 ~ 155 MeV, πœ‡ 𝐡 𝑐𝑒𝑝 ~ 90 MeV (Vβ†’βˆž) Ξ΄ ~ 4.8 Ξ² ~ 0 .33 Further validation of the CEP and the critical exponents οƒ  3D Ising universality class Roy A. Lacey, Weihai-Shandong University, China

20 Use 𝒔 as a proxy for πœ‡ 𝐡 οƒ  πœ‡ 𝐡 𝐢𝐸𝑃 ~ 90 𝑀𝑒𝑉
π‘Ίπ’–π’„π’„π’†π’‘π’•π’Šπ’ƒπ’π’Šπ’•π’š π‘Ίπ’„π’‚π’π’Šπ’π’ˆ π‘­π’–π’π’„π’•π’Šπ’π’ πœ’( πœ‡ 𝑠 , L)= 𝐿 𝛽𝛿 𝜈 𝑓 1 πœ‡ πœ‡ 𝑠 𝐿 𝛽𝛿 𝜈 𝑅 ≑𝐿 Use 𝒔 as a proxy for πœ‡ 𝐡 οƒ  πœ‡ 𝐡 𝐢𝐸𝑃 ~ 90 𝑀𝑒𝑉 Data collapse onto a single curve, confirms the expected non-singular scaling function. Roy A. Lacey, Weihai-Shandong University, China

21 πœ’( πœ‡ 𝑠 , L)= 𝐿 βˆ’π›½π›Ώ 𝜈 𝑓 1 πœ‡ πœ‡ 𝑠 𝐿 𝛽𝛿 𝜈
π‘Ίπ’„π’‚π’π’Šπ’π’ˆ π‘­π’–π’π’„π’•π’Šπ’π’ 𝒇𝒐𝒓 𝑡𝒆𝒕 π’ƒπ’‚π’“π’šπ’π’ π‘Ίπ’–π’„π’„π’†π’‘π’•π’Šπ’ƒπ’π’Šπ’•π’š π’“π’‚π’•π’Šπ’ πœ’( πœ‡ 𝑠 , L)= 𝐿 βˆ’π›½π›Ώ 𝜈 𝑓 1 πœ‡ πœ‡ 𝑠 𝐿 𝛽𝛿 𝜈 πœ’( πœ‡ 𝑠 , L)= 𝐿 𝛽𝛿 𝜈 𝑓 1 πœ‡ πœ‡ 𝑠 𝐿 𝛽𝛿 𝜈 π‘ͺ 𝒏 πš«π‘΅ 𝒑 𝒆𝒙𝒕𝒓𝒂𝒄𝒕𝒆𝒅 from π’…π’Šπ’”π’•π’“π’Šπ’ƒπ’–π’•π’Šπ’π’π’” πœ… 𝑇 ∝ π‘ͺ 𝟐 πš«π‘΅ 𝒑 βˆ’ π‘ͺ 𝟐 πš«π‘΅ 𝒑 Δ𝑁 𝑝 = π‘ͺ 𝟐 πš«π‘΅ 𝒑 π‘ͺ 𝟏 πš«π‘΅ 𝒑 Data collapse onto a single curve, confirms the expected non-singular scaling function. Roy A. Lacey, Weihai-Shandong University, China

22 Epilogue Strong experimental indication for the CEP and its location
New Data from RHIC (BES-II) together with theoretical modeling, are essential for further validation tests for characterization of the CEP and the coexistence regions of the phase diagram (Dynamic) Finite-Size Scalig analysis 3D Ising Model (static) universality class for CEP 2nd order phase transition Ξ²~ 0 .33 𝑇 𝑐𝑒𝑝 ~ 155 MeV, πœ‡ 𝐡 𝑐𝑒𝑝 ~ 90 MeV z~ 0 .87 Landmark validated Crossover validated Deconfinement validated (Static) Universality class validated Model H dynamic Universality class? Other implications! Excitation functions for more systems are especially important Roy A. Lacey, Weihai-Shandong University, China

23 End Roy A. Lacey, Weihai-Shandong University, China

24 Note that 𝝁 𝑩 𝒇 , 𝑻 𝒇 is not strongly dependent on L
π‘΄π’‚π’‘π’‘π’Šπ’π’ˆ 𝒔 𝑡𝑡 𝒕𝒐 ( 𝑻, 𝝁 𝑩 ) Karsch et. al Use Lattice EOS to map 𝑠 𝑁𝑁 β†’(𝑇, πœ‡ 𝐡 ) Note that 𝝁 𝑩 𝒇 , 𝑻 𝒇 is not strongly dependent on L Roy A. Lacey, Weihai-Shandong University, China

25 Experimental estimate of one of the dynamic critical exponents
π‘«π’šπ’π’‚π’Žπ’Šπ’„ π‘­π’Šπ’π’Šπ’•π’†βˆ’π‘Ίπ’Šπ’›π’† π‘Ίπ’„π’‚π’π’Šπ’π’ˆ 2nd order phase transition 𝜈 ~ 0 .63 𝛾 ~ 1 .2 𝑇 𝑐𝑒𝑝 ~ 155 MeV, πœ‡ 𝐡 𝑐𝑒𝑝 ~ 90 MeV (Vβ†’βˆž) DFSS ansatz at time t when T is near Tcep πœ’ 𝐿,𝑇,𝑑 = 𝐿 𝛾/𝜈 𝑓 𝜏𝐿 1/𝜈 ,𝑑 𝐿 βˆ’π‘§ For T = Tc πœ’ 𝐿, 𝑇 𝑐 ,𝑑 = 𝐿 𝛾/𝜈 𝑓 𝑑 𝐿 βˆ’π‘§ Experimental estimate of one of the dynamic critical exponents M. Suzuki, Prog. Theor. Phys. 58, 1142, 1977 Roy A. Lacey, Weihai-Shandong University, China

26 Large theoretical uncertainties Additional experimental
CEP Knowns & unknowns S. Datta, R. V. Gavai and S. Gupta, QM 2012 Scaling relations Known knowns Theory consensus on the static universality class for the CEP 3D-Ising Z(2), 𝜈 ~ 0.63 , 𝛾 ~ 1.2 Summary - M. A. Stephanov Int. J. Mod. Phys. A 20, 4387 (2005) Experimental verification - RL, PRL 114, (2015) Frithjof Karsch CPOD 2016 Known knowns/unknowns Location ( 𝐓 𝐜𝐞𝐩 , 𝛍 𝐁 𝐜𝐞𝐩 ) of the CEP? Experimental estimate - RL, PRL 114, (2015) Dynamic Universality class for the CEP? One slow mode (L), z ~ 3 - Model H Son & Stephanov, Phys.Rev. D70 (2004) Moore & Saremi, JHEP 0809, 015 (2008) Three slow modes (NL) zT ~ 3 zv ~ 2 zs ~ [critical speeding-up] Y. Minami - Phys.Rev. D83 (2011) Large theoretical uncertainties [critical slowing down] Additional experimental study to further characterize the CEP is an imperative Roy A. Lacey, Weihai-Shandong University, China

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