QCD critical point, fluctuations and hydrodynamics

Transcription

1 QCD critical point, fluctuations and hydrodynamics M. Stephanov M. Stephanov QCD critical point, fluctuations and hydro Oxford / 32

2 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, fluctuations and hydro Oxford / 32

3 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, fluctuations and hydro Oxford / 32

4 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, fluctuations and hydro Oxford / 32

5 Critical opalescence M. Stephanov QCD critical point, fluctuations and hydro Oxford / 32

6 Critical point is a ubiquitous phenomenon M. Stephanov QCD critical point, fluctuations and hydro Oxford / 32

7 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? hadron gas nuclear matter Quarkyonic regime? CFL+ M. Stephanov QCD critical point, fluctuations and hydro Oxford / 32

8 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? hadron gas nuclear matter Quarkyonic regime? CFL+ Lattice QCD at µ B 2T a crossover. C.P. is ubiquitous in models (NJL, RM, Holog., Strong coupl. LQCD,... ) M. Stephanov QCD critical point, fluctuations and hydro Oxford / 32

9 Essentially two approaches to discovering the QCD critical point. Each with its own challenges. Lattice simulations. 200 The sign problem restricts reliable lattice calculations to µ B = 0. T, MeV 150 LTE04 LTE08 LTE03 LR04 LR01 Under different assumptions one can estimate the position of the critical point, assuming it exists, by extrapolation from µ = µb, MeV Heavy-ion collisions. M. Stephanov QCD critical point, fluctuations and hydro Oxford / 32

10 Essentially two approaches to discovering the QCD critical point. Each with its own challenges. Lattice simulations. The sign problem restricts reliable lattice calculations to µ B = 0. Under different assumptions one can estimate the position of the critical point, assuming it exists, by extrapolation from µ = LTE04 T, LTE08 LTE03 MeV 17 LR LR µb, MeV 2 Heavy-ion collisions. M. Stephanov QCD critical point, fluctuations and hydro Oxford / 32

11 Essentially two approaches to discovering the QCD critical point. Each with its own challenges. Lattice simulations. The sign problem restricts reliable lattice calculations to µ B = 0. Under different assumptions one can estimate the position of the critical point, assuming it exists, by extrapolation from µ = LTE04 T, LTE08 LTE03 MeV 17 LR RHIC scan 5 LR µb, MeV 2 Heavy-ion collisions. M. Stephanov QCD critical point, fluctuations and hydro Oxford / 32

12 Essentially two approaches to discovering the QCD critical point. Each with its own challenges. Lattice simulations. The sign problem restricts reliable lattice calculations to µ B = 0. Under different assumptions one can estimate the position of the critical point, assuming it exists, by extrapolation from µ = LTE04 T, LTE08 LTE03 MeV 17 LR RHIC scan 5 LR µb, MeV 2 Heavy-ion collisions. Non-equilibrium. M. Stephanov QCD critical point, fluctuations and hydro Oxford / 32

13 Outline Equilibrium Non-equilibrium M. Stephanov QCD critical point, fluctuations and hydro Oxford / 32

14 Why fluctuations are large at a critical point? The key equation: P (σ) e S(σ) (Einstein 1910) M. Stephanov QCD critical point, fluctuations and hydro Oxford / 32

15 Why fluctuations are large at a critical point? The key equation: P (σ) e S(σ) (Einstein 1910) M. Stephanov QCD critical point, fluctuations and hydro Oxford / 32

16 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, fluctuations and hydro Oxford / 32

17 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, fluctuations and hydro Oxford / 32

18 Fluctuations of order parameter and ξ Fluctuations at CP conformal field theory. Parameter-free universality. Only one scale ξ = m 1 σ <, Ω = P [σ] exp { Ω[σ]/T }, [ 1 d 3 x 2 ( σ)2 + m2 σ 2 σ2 + λ 3 3 σ3 + λ ] 4 4 σ Width/shape of P (σ 0 xσ) best expressed via cumulants: Higher cumulants (shape of P (σ 0 )) depend stronger on ξ. Universal: σ k 0 c V ξ p, p = k(3 [σ]) 3, [σ] = β/ν 1/2. E.g., p 2 for k = 2, but p 7 for k = 4. M. Stephanov QCD critical point, fluctuations and hydro Oxford / 32

19 Sign Higher moments also depend on which side of the CP we are κ 3 [σ] = 2V T 3/2 λ3 ξ 4.5 ; κ 4 [σ] = 6V T 2 [ 2( λ 3 ) 2 λ 4 ] ξ 7. This dependence is also universal. 2 relevant directions/parameters. Using Ising model variables: M. Stephanov QCD critical point, fluctuations and hydro Oxford / 32

20 Experiments do not measure σ. M. Stephanov QCD critical point, fluctuations and hydro Oxford / 32

21 Mapping to QCD and experimental observables Observed fluctuations are not the same as σ, but related: Think of a collective mode described by field σ such that m = m(σ): δn p = δn free p + n p σ δσ The cumulants of multiplicity M p n p: ( ) κ 4 [M] = M + κ }{{} 4 [σ] g , }{{} baseline M 4 g coupling of the critical mode (g = dm/dσ). M. Stephanov QCD critical point, fluctuations and hydro Oxford / 32

22 Mapping to QCD and experimental observables Observed fluctuations are not the same as σ, but related: Think of a collective mode described by field σ such that m = m(σ): δn p = δn free p + n p σ δσ The cumulants of multiplicity M p n p: ( ) κ 4 [M] = M + κ }{{} 4 [σ] g , }{{} baseline M 4 g coupling of the critical mode (g = dm/dσ). κ 4 [σ] < 0 means κ 4 [M] < baseline M. Stephanov QCD critical point, fluctuations and hydro Oxford / 32

23 Mapping to QCD and experimental observables Observed fluctuations are not the same as σ, but related: Think of a collective mode described by field σ such that m = m(σ): δn p = δn free p + n p σ δσ The cumulants of multiplicity M p n p: ( ) κ 4 [M] = M + κ }{{} 4 [σ] g , }{{} baseline M 4 g coupling of the critical mode (g = dm/dσ). κ 4 [σ] < 0 means κ 4 [M] < baseline NB: Sensitivity to M accepted : (κ 4 ) σ M 4 (number of 4-tets). M. Stephanov QCD critical point, fluctuations and hydro Oxford / 32

24 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, fluctuations and hydro Oxford / 32

25 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, fluctuations and hydro Oxford / 32

26 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, fluctuations and hydro Oxford / 32

27 Beam Energy Scan M. Stephanov QCD critical point, fluctuations and hydro Oxford / 32

28 Beam Energy Scan M. Stephanov QCD critical point, fluctuations and hydro Oxford / 32

29 Beam Energy Scan M. Stephanov QCD critical point, fluctuations and hydro Oxford / 32

30 Beam Energy Scan intriguing hint (2015 LRPNS) M. Stephanov QCD critical point, fluctuations and hydro Oxford / 32

31 QM2017 update: another intriguing hint Preliminary, but very interesting: Non-monotonous s dependence with max near 19 GeV. Charge/isospin blind. φ (in)dependence is as expected from critical correlations. Width η suggests soft thermal pions but p T dependence need to be checked. But: no signal in R 2 for K or p. M. Stephanov QCD critical point, fluctuations and hydro Oxford / 32

32 Non-equilibrium physics is essential near the critical point. The goal for M. Stephanov QCD critical point, fluctuations and hydro Oxford / 32

33 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, fluctuations and hydro Oxford / 32

34 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, fluctuations and hydro Oxford / 32

35 Lessons κ n ξ p and ξ max τ 1/z Therefore, the magnitude of fluctuation signals is determined by non-equilibrium physics. M. Stephanov QCD critical point, fluctuations and hydro Oxford / 32

36 Lessons κ n ξ p and ξ max τ 1/z 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, fluctuations and hydro Oxford / 32

37 Lessons κ n ξ p and ξ max τ 1/z 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 Can we get critical fluctuations from hydrodynamics directly? M. Stephanov QCD critical point, fluctuations and hydro Oxford / 32

38 Time evolution of cumulants (memory) Mukherjee-Venugopalan-Yin Relaxation to equilibrium dp (σ 0 ) = F[P (σ 0 )] dτ dκ n dτ = L[κ n, κ n 1,...] M. Stephanov QCD critical point, fluctuations and hydro Oxford / 32

39 Time evolution of cumulants (memory) Mukherjee-Venugopalan-Yin Relaxation to equilibrium dp (σ 0 ) = F[P (σ 0 )] dτ dκ n dτ = L[κ n, κ n 1,...] κ 3 κ 4 Signs of cumulants also depend on off-equilibrium dynamics. M. Stephanov QCD critical point, fluctuations and hydro Oxford / 32

40 Time evolution of cumulants (memory) Mukherjee-Venugopalan-Yin Relaxation to equilibrium dp (σ 0 ) = F[P (σ 0 )] dτ dκ n dτ = L[κ n, κ n 1,...] κ 3 κ 4 Signs of cumulants also depend on off-equilibrium dynamics. M. Stephanov QCD critical point, fluctuations and hydro Oxford / 32

41 Time evolution of cumulants (memory) Mukherjee-Venugopalan-Yin Relaxation to equilibrium dp (σ 0 ) = F[P (σ 0 )] dτ dκ n dτ = L[κ n, κ n 1,...] κ 3 κ 4 Signs of cumulants also depend on off-equilibrium dynamics. M. Stephanov QCD critical point, fluctuations and hydro Oxford / 32

42 Hydrodynamics breaks down at CP T µν = ɛu µ u ν + p µν + T µν visc M. Stephanov QCD critical point, fluctuations and hydro Oxford / 32

43 Hydrodynamics breaks down at CP T µν = ɛu µ u ν + p µν + T µν visc T µν visc = ζ µν ( u) +... Near CP gradient terms are dominated by ζ ξ 3 (z α/ν 3). M. Stephanov QCD critical point, fluctuations and hydro Oxford / 32

44 Hydrodynamics breaks down at CP T µν = ɛu µ u ν + p µν + T µν visc T µν visc = ζ µν ( u) +... Near CP gradient terms are dominated by ζ ξ 3 (z α/ν 3). When k ξ 3 hydrodynamics breaks down, i.e., while k ξ 1 still. (For simplicity, measure dim-ful quantities in units of T, i.e., k T (T ξ) 3.) Why does hydro break at so small k? M. Stephanov QCD critical point, fluctuations and hydro Oxford / 32

45 Critical slowing down and bulk viscosity Bulk viscosity is the effect of system taking time to adjust to local equilibrium (Khalatnikov-Landau). p hydro = p equilibrium ζ v v expansion rate ζ τ relaxation ξ 3 M. Stephanov QCD critical point, fluctuations and hydro Oxford / 32

46 Critical slowing down and bulk viscosity Bulk viscosity is the effect of system taking time to adjust to local equilibrium (Khalatnikov-Landau). p hydro = p equilibrium ζ v v expansion rate ζ τ relaxation ξ 3 Hydrodynamics breaks down because of large relaxation time (critical slowing down). Similar to breakdown of an effective theory due to a low-energy mode which should not have been integrated out. M. Stephanov QCD critical point, fluctuations and hydro Oxford / 32

47 Critical slowing down and Hydro+ There is a critically slow mode φ with relaxation time τ φ ξ 3. M. Stephanov QCD critical point, fluctuations and hydro Oxford / 32

48 Critical slowing down and Hydro+ There is a critically slow mode φ with relaxation time τ φ ξ 3. To extend the range of hydro extend hydro by the slow mode. (MS-Yin , in preparation) M. Stephanov QCD critical point, fluctuations and hydro Oxford / 32

49 Critical slowing down and Hydro+ There is a critically slow mode φ with relaxation time τ φ ξ 3. To extend the range of hydro extend hydro by the slow mode. (MS-Yin , in preparation) Hydro+ has two competing limits, k 0 and ξ ; or competing rates Γ φ ξ 3 0 and Γ hydro k 0. M. Stephanov QCD critical point, fluctuations and hydro Oxford / 32

50 Critical slowing down and Hydro+ There is a critically slow mode φ with relaxation time τ φ ξ 3. To extend the range of hydro extend hydro by the slow mode. (MS-Yin , in preparation) Hydro+ has two competing limits, k 0 and ξ ; or competing rates Γ φ ξ 3 0 and Γ hydro k 0. Regime I: Γ φ Γ hydro ordinary hydro (ζ ξ 3 at CP). M. Stephanov QCD critical point, fluctuations and hydro Oxford / 32

51 Critical slowing down and Hydro+ There is a critically slow mode φ with relaxation time τ φ ξ 3. To extend the range of hydro extend hydro by the slow mode. (MS-Yin , in preparation) Hydro+ has two competing limits, k 0 and ξ ; or competing rates Γ φ ξ 3 0 and Γ hydro k 0. Regime I: Γ φ Γ hydro ordinary hydro (ζ ξ 3 at CP). Crossover occurs when Γ hydro Γ φ, or k ξ 3. Regime II: k > ξ 3 Hydro+ regime. M. Stephanov QCD critical point, fluctuations and hydro Oxford / 32

52 Advantages/motivation of Hydro+ Extends the range of validity of vanilla hydro near CP to length/time scales shorter than O(ξ 3 ). M. Stephanov QCD critical point, fluctuations and hydro Oxford / 32

53 Advantages/motivation of Hydro+ Extends the range of validity of vanilla hydro near CP to length/time scales shorter than O(ξ 3 ). No kinetic coefficients diverging as ξ 3. (Since noise ζ, also the noise is not large.) M. Stephanov QCD critical point, fluctuations and hydro Oxford / 32

54 Ingredients of Hydro+ Nonequilibrium entropy, or quasistatic EOS: s (ε, n, φ) Equilibrium entropy is the maximum of s : s(ε, n) = max φ s (ε, n, φ) M. Stephanov QCD critical point, fluctuations and hydro Oxford / 32

55 Ingredients of Hydro+ Nonequilibrium entropy, or quasistatic EOS: s (ε, n, φ) Equilibrium entropy is the maximum of s : s(ε, n) = max φ s (ε, n, φ) The 6th equation (constrained by 2nd law): (u )φ = γ φ π G φ ( u), where π = s φ Another example: relaxation of axial charge. M. Stephanov QCD critical point, fluctuations and hydro Oxford / 32

56 Linearized Hydro+ Linearized Hydro+ has 4 longitudinal modes (sound 2 + density + φ). In addition to the usual c s, D, etc. Hydro+ has two more parameters c 2 = c 2 c 2 s and Γ = Γ φ. The sound velocities are different in Regime I (c s k Γ) and II: c 2 s = ( ) p ε s/n,π=0 ( ) p and c 2 = ε s/n,φ The bulk viscosity receives large contribution from the slow mode given by Landau-Khalatnikov formula ζ = w c 2 /Γ M. Stephanov QCD critical point, fluctuations and hydro Oxford / 32

57 Modes M. Stephanov QCD critical point, fluctuations and hydro Oxford / 32

58 Modes M. Stephanov QCD critical point, fluctuations and hydro Oxford / 32

59 Modes M. Stephanov QCD critical point, fluctuations and hydro Oxford / 32

60 Microscopic origins of Hydro+ Understanding the microscopic origin of the slow mode: The fluctuations around equilibrium are controlled by the entropy functional P e S. Near the critical point convenient to rotate the basis of variables to Ising -like critical variables E and M. M s/n (s/n) CP. [ 1 δs[δe, δm] = 2 a M (δm) ] 2 a E (δe) 2 + b δe δm Since a M a E fluctuations of M are large and are slow to equilibrate. Their magnitude is related to the slow relaxation mode φ. M. Stephanov QCD critical point, fluctuations and hydro Oxford / 32

61 Hydro + mode distribution Separate hard k > ξ 1 and soft k ξ 1 modes. The new variable, mode distribution function : φ(t, x, Q) = δm(t, x + y/2) δm(t, x y/2) e iq y y The additional mode distribution function relaxation equation: (u )φ(t, x, Q) = 2Γ M (Q) [ a 1 M φ(t, x, Q)] where Γ M (Q) is known from model H (Kawasaki). M. Stephanov QCD critical point, fluctuations and hydro Oxford / 32

62 Relaxation of slow mode(s). M. Stephanov QCD critical point, fluctuations and hydro Oxford / 32

63 Summary A fundamental question for Heavy-Ion collision experiments: Is there a critical point on the boundary between QGP and hadron gas phases? Theoretical framework is needed the goal for. Large (non-gaussian) fluctuations universal signature of a critical point. In H.I.C., the magnitude of the signatures is controlled by dynamical non-equilibrium effects. The physics of the interplay of critical and dynamical phenomena can be captured by hydrodynamics with a critically slow mode(s) Hydro+. M. Stephanov QCD critical point, fluctuations and hydro Oxford / 32