Searching for QCD critical point theoretical overview M. Stephanov M. Stephanov QCD critical point search Fudan 2017 1 / 40
History Cagniard de la Tour (1822): discovered continuos transition from liquid to vapour by heating alcohol, water, etc. in a gun barrel, glass tubes. M. Stephanov QCD critical point search Fudan 2017 2 / 40
Name Faraday (1844) – liquefying gases: “Cagniard de la Tour made an experiment some years ago which gave me occasion to want a new word.” Mendeleev (1860) – measured vanishing of liquid-vapour surface tension: “Absolute boiling temperature”. Andrews (1869) – systematic studies of many substances established continuity of vapour-liquid phases. Coined the name “critical point”. M. Stephanov QCD critical point search Fudan 2017 3 / 40
Theory van der Waals (1879) – in “On the continuity of the gas and liquid state” (PhD thesis) wrote e.o.s. with a critical point. Smoluchowski, Einstein (1908,1910) – explained critical opalescence. Landau – classical theory of critical phenomena Fisher, Kadanoff, Wilson – scaling, full fluctuation theory based on RG. M. Stephanov QCD critical point search Fudan 2017 4 / 40
Critical point is a ubiquitous phenomenon M. Stephanov QCD critical point search Fudan 2017 5 / 40
Critical point between the QGP and hadron gas phases? QCD is a relativistic theory of a fundamental force. CP is a singularity of EOS, anchors the 1st order transition. QGP (liquid) critical point ? Quarkyonic regime hadron gas nuclear CFL+ ? matter M. Stephanov QCD critical point search Fudan 2017 6 / 40
Critical point between the QGP and hadron gas phases? QCD is a relativistic theory of a fundamental force. CP is a singularity of EOS, anchors the 1st order transition. QGP (liquid) critical point ? Quarkyonic regime hadron gas nuclear CFL+ ? matter Lattice QCD at µ B � 2 T – a crossover. C.P . is ubiquitous in models (NJL, RM, Holog., Strong coupl. LQCD, . . . ) M. Stephanov QCD critical point search Fudan 2017 6 / 40
Essentially two approaches to discovering the QCD critical point. Each with its own challenges. Lattice simulations. 200 LTE04 LTE03 T , LTE08 The sign problem restricts reliable lat- LR01 MeV LR04 150 tice calculations to µ B = 0 . 100 Under different assumptions one can estimate the position of the critical point, assuming it exists, by extrapo- 50 lation from µ = 0 . 0 0 200 400 600 800 µ B , MeV Heavy-ion collisions. M. Stephanov QCD critical point search Fudan 2017 7 / 40
Essentially two approaches to discovering the QCD critical point. Each with its own challenges. Lattice simulations. 200 130 LTE04 LTE03 T , LTE08 The sign problem restricts reliable lat- LR01 17 MeV LR04 150 9 tice calculations to µ B = 0 . 5 100 Under different assumptions one can estimate the position of the critical point, assuming it exists, by extrapo- 50 2 lation from µ = 0 . 0 0 200 400 600 800 µ B , MeV Heavy-ion collisions. M. Stephanov QCD critical point search Fudan 2017 7 / 40
Essentially two approaches to discovering the QCD critical point. Each with its own challenges. Lattice simulations. 200 130 LTE04 LTE03 T , LTE08 The sign problem restricts reliable lat- LR01 17 MeV LR04 150 R H I 9 C tice calculations to µ B = 0 . s c a n 5 100 Under different assumptions one can estimate the position of the critical point, assuming it exists, by extrapo- 50 2 lation from µ = 0 . 0 0 200 400 600 800 µ B , MeV Heavy-ion collisions. M. Stephanov QCD critical point search Fudan 2017 7 / 40
Essentially two approaches to discovering the QCD critical point. Each with its own challenges. Lattice simulations. 200 130 LTE04 LTE03 T , LTE08 The sign problem restricts reliable lat- LR01 17 MeV LR04 150 R H I 9 C tice calculations to µ B = 0 . s c a n 5 100 Under different assumptions one can estimate the position of the critical point, assuming it exists, by extrapo- 50 2 lation from µ = 0 . 0 0 200 400 600 800 µ B , MeV Heavy-ion collisions. Non-equilibrium. M. Stephanov QCD critical point search Fudan 2017 7 / 40
Outline Equilibrium Non-equilibrium Experimental hints M. Stephanov QCD critical point search Fudan 2017 8 / 40
Why fluctuations are large at a critical point? The key equation: P ( σ ) ∼ e S ( σ ) (Einstein 1910) M. Stephanov QCD critical point search Fudan 2017 9 / 40
Why fluctuations are large at a critical point? The key equation: P ( σ ) ∼ e S ( σ ) (Einstein 1910) M. Stephanov QCD critical point search Fudan 2017 9 / 40
Why fluctuations are large at a critical point? The key equation: P ( σ ) ∼ e S ( σ ) (Einstein 1910) At the critical point S ( σ ) “flattens”. And χ ≡ � σ 2 � /V → ∞ . CLT? M. Stephanov QCD critical point search Fudan 2017 9 / 40
Why fluctuations are large at a critical point? The key equation: P ( σ ) ∼ e S ( σ ) (Einstein 1910) At the critical point S ( σ ) “flattens”. And χ ≡ � σ 2 � /V → ∞ . CLT? σ is not a sum of ∞ many uncorrelated contributions: ξ → ∞ M. Stephanov QCD critical point search Fudan 2017 9 / 40
Higher order cumulants Higher cumulants (shape of P ( σ ) ) depend stronger on ξ . E.g., � σ 2 � ∼ V ξ 2 while � σ 4 � c ∼ V ξ 7 [PRL102(2009)032301] Higher moment sign depends on which side of the CP we are. This dependence is also universal. [PRL107(2011)052301] Using Ising model variables: M. Stephanov QCD critical point search Fudan 2017 10 / 40
Experiments do not measure σ . M. Stephanov QCD critical point search Fudan 2017 11 / 40
Critical fluctuations and experimental observables Observed fluctuations are related to fluctuations of σ . [MS-Rajagopal-Shuryak PRD60(1999)114028; MS PRL102(2009)032301] Think of a collective mode described by field σ such that m = m ( σ ) : + ∂ � n p � δn p = δn free × δσ p ∂σ � The cumulants of multiplicity M ≡ p n p : ( M P ∼ n B × ∆ y ) + κ 4 [ σ ] × g 4 � � 4 κ 4 [ M ] = � M � + . . . , ���� � �� � baseline ∼ M 4 � �� � κ 4 ( a.k.a .C Bzdak-Koch this is ˆ ) 4 M. Stephanov QCD critical point search Fudan 2017 12 / 40
Mapping Ising to QCD phase diagram T vs µ B : In QCD ( t, H ) → ( µ − µ CP , T − T CP ) M. Stephanov QCD critical point search Fudan 2017 13 / 40
Mapping Ising to QCD phase diagram T vs µ B : In QCD ( t, H ) → ( µ − µ CP , T − T CP ) M. Stephanov QCD critical point search Fudan 2017 13 / 40
Mapping Ising to QCD phase diagram T vs µ B : In QCD ( t, H ) → ( µ − µ CP , T − T CP ) κ n ( N ) = N + O ( κ n ( σ )) M. Stephanov QCD critical point search Fudan 2017 13 / 40
Beam Energy Scan M. Stephanov QCD critical point search Fudan 2017 14 / 40
Beam Energy Scan M. Stephanov QCD critical point search Fudan 2017 14 / 40
Beam Energy Scan M. Stephanov QCD critical point search Fudan 2017 14 / 40
Beam Energy Scan “intriguing hint” (2015 LRPNS) M. Stephanov QCD critical point search Fudan 2017 14 / 40
Large µ B : n 4 B vs ξ 7 � The cumulants of multiplicity M ≡ p n p : ( M P ∼ n B × ∆ y ) + κ 4 [ σ ] × g 4 � � 4 κ 4 [ M ] = � M � + . . . , ���� � �� � baseline ∼ M 4 κ 4 [ M ] ≈ g 4 κ 4 [ σ ] M 4 ∼ ξ 7 × n 4 (∆ y ) 4 . ˆ × B [Athanasiou-Rajagopal-MS] � �� � compete at large µ B The ratio ˆ κ 4 [ M ] or ˆ κ 4 [ M ] ∼ κ 4 [ σ ] ∼ ξ 7 . n 4 M 4 B M. Stephanov QCD critical point search Fudan 2017 15 / 40
κ 4 /M 4 ˆ Bzdak Speculative, but: qualitatively (signs of) deviations from the baseline seem in agreement with C.P . expectations. M. Stephanov QCD critical point search Fudan 2017 16 / 40
Non-equilibrium physics is essential near the critical point. The goal for M. Stephanov QCD critical point search Fudan 2017 17 / 40
Why ξ is finite System expands and is out of equilibrium Kibble-Zurek mechanism: Critical slowing down means τ relax ∼ ξ z . Given τ relax � τ (expansion time scale): ξ � τ 1 /z , z ≈ 3 (universal). M. Stephanov QCD critical point search Fudan 2017 18 / 40
Why ξ is finite System expands and is out of equilibrium Kibble-Zurek mechanism: Critical slowing down means τ relax ∼ ξ z . Given τ relax � τ (expansion time scale): ξ � τ 1 /z , z ≈ 3 (universal). Estimates: ξ ∼ 2 − 3 fm (Berdnikov-Rajagopal) KZ scaling for ξ ( t ) and cumulants (Mukherjee-Venugopalan-Yin) M. Stephanov QCD critical point search Fudan 2017 18 / 40
Lessons κ n ∼ ξ p ξ max ∼ τ 1 /z and Therefore, the magnitude of fluctuation signals is determined by non-equilibrium physics. M. Stephanov QCD critical point search Fudan 2017 19 / 40
Lessons κ n ∼ ξ p ξ max ∼ τ 1 /z and Therefore, the magnitude of fluctuation signals is determined by non-equilibrium physics. Logic so far: Equilibrium fluctuations + a non-equilibrium effect (finite ξ ) − → Observable critical fluctuations M. Stephanov QCD critical point search Fudan 2017 19 / 40
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