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QCD critical point, fluctuations and hydrodynamics M. Stephanov M. Stephanov QCD CP , fluctuations and hydrodynamics Trento 2017 1 / 28 Critical point is a ubiquitous phenomenon M. Stephanov QCD CP , fluctuations and hydrodynamics Trento

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QCD critical point, fluctuations and hydrodynamics

  • M. Stephanov
  • M. Stephanov

QCD CP , fluctuations and hydrodynamics Trento 2017 1 / 28

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Critical point is a ubiquitous phenomenon

  • M. Stephanov

QCD CP , fluctuations and hydrodynamics Trento 2017 2 / 28

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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.

Quarkyonic regime

QGP (liquid)

critical point

nuclear matter

hadron gas ? CFL+ ?

  • M. Stephanov

QCD CP , fluctuations and hydrodynamics Trento 2017 3 / 28

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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.

Quarkyonic regime

QGP (liquid)

critical point

nuclear matter

hadron gas ? CFL+ ?

Lattice QCD at µB 2T – a crossover. C.P . is ubiquitous in models (NJL, RM, Holog., Strong coupl. LQCD, . . . )

  • M. Stephanov

QCD CP , fluctuations and hydrodynamics Trento 2017 3 / 28

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Essentially two approaches to discovering the QCD critical point. Each with its own challenges. Lattice simulations. The sign problem restricts reliable lat- tice calculations to µB = 0. Under different assumptions one can estimate the position of the critical point, assuming it exists, by extrapo- lation from µ = 0.

LTE03 LR01 LR04 LTE08 LTE04 50 100 150 200 400 800 600 200

T, MeV µB, MeV

Heavy-ion collisions.

  • M. Stephanov

QCD CP , fluctuations and hydrodynamics Trento 2017 4 / 28

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Essentially two approaches to discovering the QCD critical point. Each with its own challenges. Lattice simulations. The sign problem restricts reliable lat- tice calculations to µB = 0. Under different assumptions one can estimate the position of the critical point, assuming it exists, by extrapo- lation from µ = 0.

LTE03 LR01 LR04 LTE08 LTE04 130 9 5 2 17 50 100 150 200 400 800 600 200

T, MeV µB, MeV

Heavy-ion collisions.

  • M. Stephanov

QCD CP , fluctuations and hydrodynamics Trento 2017 4 / 28

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SLIDE 7

Essentially two approaches to discovering the QCD critical point. Each with its own challenges. Lattice simulations. The sign problem restricts reliable lat- tice calculations to µB = 0. Under different assumptions one can estimate the position of the critical point, assuming it exists, by extrapo- lation from µ = 0.

LTE03 LR01 LR04 LTE08 LTE04 130 9 5 2 17 50 100 150 200 400 800 600 200

R H I C s c a n T, MeV µB, MeV

Heavy-ion collisions.

  • M. Stephanov

QCD CP , fluctuations and hydrodynamics Trento 2017 4 / 28

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SLIDE 8

Essentially two approaches to discovering the QCD critical point. Each with its own challenges. Lattice simulations. The sign problem restricts reliable lat- tice calculations to µB = 0. Under different assumptions one can estimate the position of the critical point, assuming it exists, by extrapo- lation from µ = 0.

LTE03 LR01 LR04 LTE08 LTE04 130 9 5 2 17 50 100 150 200 400 800 600 200

R H I C s c a n T, MeV µB, MeV

Heavy-ion collisions. Non-equilibrium.

  • M. Stephanov

QCD CP , fluctuations and hydrodynamics Trento 2017 4 / 28

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Why fluctuations are large at a critical point?

The key equation: P(σ) ∼ eS(σ) (Einstein 1910)

  • M. Stephanov

QCD CP , fluctuations and hydrodynamics Trento 2017 5 / 28

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Why fluctuations are large at a critical point?

The key equation: P(σ) ∼ eS(σ) (Einstein 1910)

  • M. Stephanov

QCD CP , fluctuations and hydrodynamics Trento 2017 5 / 28

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Why fluctuations are large at a critical point?

The key equation: P(σ) ∼ eS(σ) (Einstein 1910) At the critical point S(σ) “flattens”. And χ ≡ σ2/V → ∞.

CLT?

  • M. Stephanov

QCD CP , fluctuations and hydrodynamics Trento 2017 5 / 28

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Why fluctuations are large at a critical point?

The key equation: P(σ) ∼ eS(σ) (Einstein 1910) At the critical point S(σ) “flattens”. And χ ≡ σ2/V → ∞.

CLT? σ is not a sum of ∞ many uncorrelated contributions: ξ → ∞

  • M. Stephanov

QCD CP , fluctuations and hydrodynamics Trento 2017 5 / 28

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Higher order cumulants

Higher cumulants (shape of P(σ)) depend stronger on ξ. E.g., σ2 ∼ V ξ2 while σ4c ∼ V ξ7

[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 CP , fluctuations and hydrodynamics Trento 2017 6 / 28

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Mapping Ising to QCD phase diagram

T vs µB: In QCD (t, H) → (µ − µCP, T − TCP)

  • M. Stephanov

QCD CP , fluctuations and hydrodynamics Trento 2017 7 / 28

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Mapping Ising to QCD phase diagram

T vs µB: In QCD (t, H) → (µ − µCP, T − TCP)

  • M. Stephanov

QCD CP , fluctuations and hydrodynamics Trento 2017 7 / 28

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Mapping Ising to QCD phase diagram

T vs µB: In QCD (t, H) → (µ − µCP, T − TCP) κn(N) = N + O(κn(σ))

  • M. Stephanov

QCD CP , fluctuations and hydrodynamics Trento 2017 7 / 28

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Beam Energy Scan

  • M. Stephanov

QCD CP , fluctuations and hydrodynamics Trento 2017 8 / 28

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Beam Energy Scan

  • M. Stephanov

QCD CP , fluctuations and hydrodynamics Trento 2017 8 / 28

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Beam Energy Scan

  • M. Stephanov

QCD CP , fluctuations and hydrodynamics Trento 2017 8 / 28

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Beam Energy Scan

“intriguing hint” (2015 LRPNS)

  • M. Stephanov

QCD CP , fluctuations and hydrodynamics Trento 2017 8 / 28

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QM/CPOD2017: two-point correlations

Preliminary, but very interesting:

Rapidity Correlations Click to edit Master subtitle style W.J. Llope for STAR, CPOD2017, Aug. 8-11, 2017, Stony Brook, NY 21

R2(Δy,Δφ) for LS pions vs. √sNN, 0-5% central, convolution

✩Preliminary

7.7 GeV 11.5 GeV 14.5 GeV 19.6 GeV 27 GeV 39 GeV 62.4 GeV 200 GeV

  • W. Llope

Why this is interesting: Non-monotonous √s dependence with max near 19 GeV. Charge/isospin blind. ∆φ (in)dependence is as expected from critical correlations. C2 ∼ f(φ1)f(φ2). Width ∆η suggests soft pions – but pT dependence need to be checked. Why no signal in R2 for K or p?

  • M. Stephanov

QCD CP , fluctuations and hydrodynamics Trento 2017 9 / 28

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Non-equilibrium physics is essential near the critical point. The goal for

  • M. Stephanov

QCD CP , fluctuations and hydrodynamics Trento 2017 10 / 28

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Why ξ is finite

System expands and is out of equilibrium Kibble-Zurek mechanism: Critical slowing down means τrelax ∼ ξz. Given τrelax τ (expansion time scale): ξ τ 1/z, z ≈ 3 (universal).

  • M. Stephanov

QCD CP , fluctuations and hydrodynamics Trento 2017 11 / 28

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Why ξ is finite

System expands and is out of equilibrium Kibble-Zurek mechanism: Critical slowing down means τrelax ∼ ξz. Given τrelax 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 CP , fluctuations and hydrodynamics Trento 2017 11 / 28

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Magnitude of observables and ξ

κn ∼ ξp and ξmax ∼ τ 1/z Therefore, the magnitude of fluctuation signals is determined by non-equilibrium physics.

  • M. Stephanov

QCD CP , fluctuations and hydrodynamics Trento 2017 12 / 28

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Magnitude of observables and ξ

κn ∼ ξp and ξmax ∼ 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 CP , fluctuations and hydrodynamics Trento 2017 12 / 28

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Magnitude of observables and ξ

κn ∼ ξp and ξmax ∼ 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 CP , fluctuations and hydrodynamics Trento 2017 12 / 28

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Hydrodynamics breaks down at CP

Hydrodynamics relies on gradient expansion: T µν = ǫuµuν + p∆µν + ˜ T µν

visc

˜ T µν

visc = − ζ∆µν(∇ · u)

  • O(ζ k)≪1

+ . . .

  • M. Stephanov

QCD CP , fluctuations and hydrodynamics Trento 2017 13 / 28

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Hydrodynamics breaks down at CP

Hydrodynamics relies on gradient expansion: T µν = ǫuµuν + p∆µν + ˜ T µν

visc

˜ T µν

visc = − ζ∆µν(∇ · u)

  • O(ζ k)≪1

+ . . . Near CP: ζ ∼ ξ3 → ∞ (z − α/ν ≈ 3).

  • M. Stephanov

QCD CP , fluctuations and hydrodynamics Trento 2017 13 / 28

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Hydrodynamics breaks down at CP

Hydrodynamics relies on gradient expansion: T µν = ǫuµuν + p∆µν + ˜ T µν

visc

˜ T µν

visc = − ζ∆µν(∇ · u)

  • O(ζ k)≪1

+ . . . Near CP: ζ ∼ ξ3 → ∞ (z − α/ν ≈ 3). [Units: T = 1] When k ∼ ξ−3 hydrodynamics breaks down, while k ∼ ξ−3 ≪ ξ−1 still. Why not at k ∼ 1 or at least k ∼ ξ−1?

  • M. Stephanov

QCD CP , fluctuations and hydrodynamics Trento 2017 13 / 28

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Critical slowing down and bulk viscosity

Bulk viscosity is the effect of system taking time to adjust to local equilibrium (Mandel’shtam-Leontovich, Khalatnikov-Landau). phydro = pequilibrium − ζ ∇ · v ∇ · v – expansion rate ζ ∼ τrelaxation ∼ ξ3

  • M. Stephanov

QCD CP , fluctuations and hydrodynamics Trento 2017 14 / 28

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Critical slowing down and bulk viscosity

Bulk viscosity is the effect of system taking time to adjust to local equilibrium (Mandel’shtam-Leontovich, Khalatnikov-Landau). phydro = pequilibrium − ζ ∇ · v ∇ · v – expansion rate ζ ∼ τrelaxation ∼ ξ3 Hydrodynamics breaks down because of large relaxation time. Similar to breakdown of an effective theory due to a low-energy mode which should not have been integrated out.

  • M. Stephanov

QCD CP , fluctuations and hydrodynamics Trento 2017 14 / 28

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Critical slowing down and Hydro+

There is a critically slow mode φ with relaxation time τφ ∼ ξ3.

  • M. Stephanov

QCD CP , fluctuations and hydrodynamics Trento 2017 15 / 28

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Critical slowing down and Hydro+

There is a critically slow mode φ with relaxation time τφ ∼ ξ3. To extend the range of hydro – extend hydro by the slow mode. (MS-Yin 1704.07396, to appear)

  • M. Stephanov

QCD CP , fluctuations and hydrodynamics Trento 2017 15 / 28

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Critical slowing down and Hydro+

There is a critically slow mode φ with relaxation time τφ ∼ ξ3. To extend the range of hydro – extend hydro by the slow mode. (MS-Yin 1704.07396, to appear) “Hydro+” has two competing time scales τφ → ∞ and τhydro ∼ (csk)−1 → ∞.

  • M. Stephanov

QCD CP , fluctuations and hydrodynamics Trento 2017 15 / 28

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Critical slowing down and Hydro+

There is a critically slow mode φ with relaxation time τφ ∼ ξ3. To extend the range of hydro – extend hydro by the slow mode. (MS-Yin 1704.07396, to appear) “Hydro+” has two competing time scales τφ → ∞ and τhydro ∼ (csk)−1 → ∞. Regime I: τhydro ≫ τφ ≫ τmicro – ordinary hydro (with ζ ∼ τφ).

  • M. Stephanov

QCD CP , fluctuations and hydrodynamics Trento 2017 15 / 28

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Critical slowing down and Hydro+

There is a critically slow mode φ with relaxation time τφ ∼ ξ3. To extend the range of hydro – extend hydro by the slow mode. (MS-Yin 1704.07396, to appear) “Hydro+” has two competing time scales τφ → ∞ and τhydro ∼ (csk)−1 → ∞. Regime I: τhydro ≫ τφ ≫ τmicro – ordinary hydro (with ζ ∼ τφ). Crossover occurs when τhydro ∼ τφ, or k ∼ 1/τφ. Regime II: τφ ≫ τhydro ≫ τmicro – “Hydro+” regime.

  • M. Stephanov

QCD CP , fluctuations and hydrodynamics Trento 2017 15 / 28

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Advantages/motivation of Hydro+

Extends the range of validity of “vanilla” hydro to shorter time scales (ω ≫ 1/τφ) than ordinary hydro. And to shorter length scales – near CP to scales k ≫ ξ−3.

  • M. Stephanov

QCD CP , fluctuations and hydrodynamics Trento 2017 16 / 28

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Advantages/motivation of Hydro+

Extends the range of validity of “vanilla” hydro to shorter time scales (ω ≫ 1/τφ) than ordinary hydro. And to shorter length scales – near CP to scales k ≫ ξ−3. No large kinetic coefficients. Large ζ generated “dynamically”.

  • M. Stephanov

QCD CP , fluctuations and hydrodynamics Trento 2017 16 / 28

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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 CP , fluctuations and hydrodynamics Trento 2017 17 / 28

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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 = −γππ − Aφ(∂ · u), where π = −∂s(+) ∂φ φ relaxes to equilibrium (π = 0) at a rate 1/τφ ≡ Γ = γπ(∂π/∂φ).

  • M. Stephanov

QCD CP , fluctuations and hydrodynamics Trento 2017 17 / 28

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Linearized Hydro+

4 longitudinal modes (sound×2 + density + φ). In addition to cs, D, etc. Hydro+ has two more parameters ∆c2 = c2

(+) − c2 s

and Γ = 1/τφ.

  • M. Stephanov

QCD CP , fluctuations and hydrodynamics Trento 2017 18 / 28

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Linearized Hydro+

4 longitudinal modes (sound×2 + density + φ). In addition to cs, D, etc. Hydro+ has two more parameters ∆c2 = c2

(+) − c2 s

and Γ = 1/τφ. The sound velocities, i.e., eos stiffness, are different in Regime I (csk ≪ Γ) and Regime II: c2

s =

∂p ∂ε

  • s/n,π=0

and c2

(+) =

∂p(+) ∂ε

  • s/n,φ
  • M. Stephanov

QCD CP , fluctuations and hydrodynamics Trento 2017 18 / 28

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Linearized Hydro+

4 longitudinal modes (sound×2 + density + φ). In addition to cs, D, etc. Hydro+ has two more parameters ∆c2 = c2

(+) − c2 s

and Γ = 1/τφ. The sound velocities, i.e., eos stiffness, are different in Regime I (csk ≪ Γ) and Regime II: c2

s =

∂p ∂ε

  • s/n,π=0

and c2

(+) =

∂p(+) ∂ε

  • s/n,φ

In Regime I bulk viscosity receives large contribution from the slow mode ∆ζ = w∆c2/Γ In Regime II bulk viscosity drops.

  • M. Stephanov

QCD CP , fluctuations and hydrodynamics Trento 2017 18 / 28

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Modes

  • M. Stephanov

QCD CP , fluctuations and hydrodynamics Trento 2017 19 / 28

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Modes

  • M. Stephanov

QCD CP , fluctuations and hydrodynamics Trento 2017 19 / 28

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Modes

  • M. Stephanov

QCD CP , fluctuations and hydrodynamics Trento 2017 19 / 28

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What is the slow mode? Nonequilibrium fluctuations.

An equilibrium thermodynamic state is completely characterized by ¯ ε, ¯ n, . . .. Fluctuations of ε, n are given by eos, i.e., 2nd derivs. of s(¯ ε, ¯ n).

  • M. Stephanov

QCD CP , fluctuations and hydrodynamics Trento 2017 20 / 28

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What is the slow mode? Nonequilibrium fluctuations.

An equilibrium thermodynamic state is completely characterized by ¯ ε, ¯ n, . . .. Fluctuations of ε, n are given by eos, i.e., 2nd derivs. of s(¯ ε, ¯ n). Hydrodynamics describes quasiequilibrium states, i.e., equilibrium is only local, because it takes time. Fluctuations in such states are not necessarily in equilibrium, depending on what spatial scale we look at and what are the relaxation rates of fluctuations at that scale.

  • M. Stephanov

QCD CP , fluctuations and hydrodynamics Trento 2017 20 / 28

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What is the slow mode? Nonequilibrium fluctuations.

An equilibrium thermodynamic state is completely characterized by ¯ ε, ¯ n, . . .. Fluctuations of ε, n are given by eos, i.e., 2nd derivs. of s(¯ ε, ¯ n). Hydrodynamics describes quasiequilibrium states, i.e., equilibrium is only local, because it takes time. Fluctuations in such states are not necessarily in equilibrium, depending on what spatial scale we look at and what are the relaxation rates of fluctuations at that scale. Therefore, magnitudes of fluctuations are additional parameters, independent of ¯ ε, ¯ n, needed to characterize the state.

  • M. Stephanov

QCD CP , fluctuations and hydrodynamics Trento 2017 20 / 28

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Near CP there is parametric separation of relaxation time scales. The slowest and thus most out-of-equilibrium mode is m = s/n. Thus we need to consider the correlator δmδm as an indepen- dent variable φ in hydro equations. What is s(+)(ε, n, φ)?

  • M. Stephanov

QCD CP , fluctuations and hydrodynamics Trento 2017 21 / 28

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Entropy of fluctuations

The entropy depends on φQ (Wigner transform ofδmδm ): s(+)(ε, n, φQ) = s(ε, n) + 1 2

  • Q
  • 1 − φQ/¯

φQ + log φQ/¯ φQ

  • Two competing effects:

log(φ/¯ φ)1/2 reflects increase of the # of micro states in ensemble due to larger than equilibrium fluctuations; −(1/2)φ/¯ φ – penalty for deviating from maximum entropy (at m = ¯ m). Balance is achieved (max. entropy) at φ = ¯ φ.

  • M. Stephanov

QCD CP , fluctuations and hydrodynamics Trento 2017 22 / 28

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Hydro+ entropy is similar to 2PI action in QFT. The mode distribution function φQ is similar to particle distribu- tion function in kinetic theory.

  • M. Stephanov

QCD CP , fluctuations and hydrodynamics Trento 2017 23 / 28

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Hydro+ entropy is similar to 2PI action in QFT. The mode distribution function φQ is similar to particle distribu- tion function in kinetic theory. The equation for φQ is a relaxation equation: (u · ∂)φQ = −γπ(Q)πQ, πQ = − ∂s(+) ∂φQ

  • ε,n
  • M. Stephanov

QCD CP , fluctuations and hydrodynamics Trento 2017 23 / 28

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Hydro+ entropy is similar to 2PI action in QFT. The mode distribution function φQ is similar to particle distribu- tion function in kinetic theory. The equation for φQ is a relaxation equation: (u · ∂)φQ = −γπ(Q)πQ, πQ = − ∂s(+) ∂φQ

  • ε,n

Near the critical point the relevant scale of Q is ξ−1. Characteristic rate Γξ = Dξ−2 ∼ ξ−3.

  • M. Stephanov

QCD CP , fluctuations and hydrodynamics Trento 2017 23 / 28

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SLIDE 56

Hydro+ vs Hydro: real-time response

Dissipation during expansion is

  • verestimated in hydro (dashed):

Stiffness of eos (sound speed) is underestimated.

  • M. Stephanov

QCD CP , fluctuations and hydrodynamics Trento 2017 24 / 28

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SLIDE 57

Summary

A fundamental question for Heavy-Ion collision experiments: Is there a critical point on the boundary between QGP and hadron gas phases? Intriguing results from experiments (BES-I). More to come (BES-II, FAIR/CBM, NICA, J-PARC). Quantitative theoretical framework is needed ⇒ . Large (non-gaussian) fluctuations – universal signature of a crit- ical point. In H.I.C., the magnitude of the signatures is controlled by dy- namical non-equilibrium effects. The physics of the interplay of critical and dynamical phenomena can be captured in Hydro+.

  • M. Stephanov

QCD CP , fluctuations and hydrodynamics Trento 2017 25 / 28

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More

  • M. Stephanov

QCD CP , fluctuations and hydrodynamics Trento 2017 26 / 28

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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): δnp = δnfree

p

+ ∂np ∂σ × δσ The cumulants of multiplicity M ≡

  • p np:

κ4[M] = M

  • baseline

+ κ4[σ] × g4 4

  • ∼M4
  • this is ˆ

κ4(a.k.a.CBzdak-Koch

4

)

+ . . . , =

  • p

np γp ← acceptance dependent

  • M. Stephanov

QCD CP , fluctuations and hydrodynamics Trento 2017 27 / 28

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Spinodal region

[An-Mesterhazy-MS]

Fonseca-Zamolodchikov conjecture: spinodal point is off the real axis

  • f H.

Spinodal singularity is an artefact of the mean field approximation. No thermodynamics in the metastabe/unstable region. Question: What is the meaning of EOS?

  • M. Stephanov

QCD CP , fluctuations and hydrodynamics Trento 2017 28 / 28