Hydrodynamic fluctuations and QCD critical point M. Stephanov with - - PowerPoint PPT Presentation

Hydrodynamic fluctuations and QCD critical point

• M. Stephanov

with Y. Yin, 1712.10305; with X. An, G. Basar and H.-U. Yee, 1902.09517, 1912.13456;

• M. Stephanov

Fluctuations and QCD Critical Point WWND 2020 1 / 17

Critical point: intriguing hints

Where on the QCD phase boundary is the CP?

H a d r

• n

G a s

200 400 600 800 1000 1200 1400 1600 50 100 150 200 250 300

Temperature (MeV) Baryon Chemical Potential μB(MeV)

Vacuum Nuclear Maer

14.5 19.6 11.5 9.1 7.7 200 √s = 62.4 GeV 27 39 2760

B E S I I

• Quark-Gluon Plasma

Color Superconductor

The Phases of QCD

1

s t

O r d e r P h a s e T r a n s i t i

• n

Critical Point

Equilibrium κ4 vs T and µB: “intriguing hint” (2015 LRPNS)

Motivation for phase II of BES at RHIC and BEST topical collaboration.

• M. Stephanov

Fluctuations and QCD Critical Point WWND 2020 2 / 17

Theory/experiment gap: predictions assume equilibrium, but Non-equilibrium physics is essential near the critical point. Challenge: develop hydrodynamics with fluctuations capable of describing non-equilibrium effects on critical-point signatures. Also notable: Fluctuations are the first step to extend hydro to smaller systems.

• M. Stephanov

Fluctuations and QCD Critical Point WWND 2020 3 / 17

Stochastic hydrodynamics

Hydrodynamic eqs. are conservation equations (∂µT µν = 0): ∂tψ = −∇ · Flux[ψ];

• M. Stephanov

Fluctuations and QCD Critical Point WWND 2020 4 / 17

Stochastic hydrodynamics

Hydrodynamic eqs. are conservation equations (∂µT µν = 0): ∂tψ = −∇ · Flux[ψ]; Stochastic variables ˘ ψ = ( ˘ T i0, ˘ J0 ) are local operators coarse-grained (over “cells” b: ℓmic ≪ b ≪ L):

more

∂t ˘ ψ = −∇ ·

• Flux[ ˘

ψ] + Noise

• (Landau-Lifshitz)
• M. Stephanov

Fluctuations and QCD Critical Point WWND 2020 4 / 17

Stochastic hydrodynamics

Hydrodynamic eqs. are conservation equations (∂µT µν = 0): ∂tψ = −∇ · Flux[ψ]; Stochastic variables ˘ ψ = ( ˘ T i0, ˘ J0 ) are local operators coarse-grained (over “cells” b: ℓmic ≪ b ≪ L):

more

∂t ˘ ψ = −∇ ·

• Flux[ ˘

ψ] + Noise

• (Landau-Lifshitz)

Linearized version has been considered and applied to heavy- ion collisions (Kapusta-Muller-MS, Kapusta-Torres-Rincon, . . . ) Non-linearities + point-like noise ⇒ UV divergences. In numerical simulations – cutoff dependence.

• M. Stephanov

Fluctuations and QCD Critical Point WWND 2020 4 / 17

Deterministic approach

Variables are one- and two-point functions: ψ = ˘ ψ and G = ˘ ψ ˘ ψ − ˘ ψ ˘ ψ – equal-time correlator Nonlinearities lead to dependence of flux on G. ∂tψ = −∇ · Flux[ψ, G];

(conservation)

∂tG = L[G; ψ].

(relaxation)

In Bjorken flow by Akamatsu et al, Martinez-Schaefer. For arbitrary relativistic flow – by An et al (this talk). Earlier, in nonrelativistic context, – by Andreev in 1970s.

• M. Stephanov

Fluctuations and QCD Critical Point WWND 2020 5 / 17

Deterministic approach

Variables are one- and two-point functions: ψ = ˘ ψ and G = ˘ ψ ˘ ψ − ˘ ψ ˘ ψ – equal-time correlator Nonlinearities lead to dependence of flux on G. ∂tψ = −∇ · Flux[ψ, G];

(conservation)

∂tG = L[G; ψ].

(relaxation)

In Bjorken flow by Akamatsu et al, Martinez-Schaefer. For arbitrary relativistic flow – by An et al (this talk). Earlier, in nonrelativistic context, – by Andreev in 1970s. Advantage: deterministic equations. “Infinite noise” causes UV renormalization of EOS and transport coefficients – can be taken care of analytically (1902.09517)

• M. Stephanov

Fluctuations and QCD Critical Point WWND 2020 5 / 17

Fluctuation dynamics near CP: Hydro+

Yin, MS, 1712.10305

Fluctuation dynamics near CP requires two main ingredients: Critical fluctuations (ξ → ∞) Slow relaxation mode with τrelax ∼ ξ3 (leading to ζ → ∞)

• M. Stephanov

Fluctuations and QCD Critical Point WWND 2020 6 / 17

Fluctuation dynamics near CP: Hydro+

Yin, MS, 1712.10305

Fluctuation dynamics near CP requires two main ingredients: Critical fluctuations (ξ → ∞) Slow relaxation mode with τrelax ∼ ξ3 (leading to ζ → ∞) Both described by the same object: the two-point function

• f the slowest hydrodynamic mode m ≡ (s/n),

i.e., δm(x1) δm(x2) . Without this mode, hydrodynamics would break down near CP when τexpansion ∼ τrelax ∼ ξ3.

• M. Stephanov

Fluctuations and QCD Critical Point WWND 2020 6 / 17

Additional variables in Hydro+

At the CP the slowest new variable is the 2-pt function δmδm

• f the slowest hydro variable:

φQ(x) =

• ∆x

δm (x+) δm (x−) eiQ·∆x where x = (x+ + x−)/2 and ∆x = x+ − x−. Wigner transformed b/c dependence on x (∼ L) is slow and relevant ∆x ≪ L. Scale separation similar to kinetic theory.

• M. Stephanov

Fluctuations and QCD Critical Point WWND 2020 7 / 17

Relaxation of fluctuations towards equilibrium

As usual, equilibration maximizes entropy S =

i pi log(1/pi):

s(+)(ǫ, n, φQ) = s(ǫ, n) + 1 2

• Q
• log φQ

¯ φQ − φQ ¯ φQ + 1

• M. Stephanov

Fluctuations and QCD Critical Point WWND 2020 8 / 17

Relaxation of fluctuations towards equilibrium

As usual, equilibration maximizes entropy S =

i pi log(1/pi):

s(+)(ǫ, n, φQ) = s(ǫ, n) + 1 2

• Q
• log φQ

¯ φQ − φQ ¯ φQ + 1

• Entropy = log # of states, which depends on

the width of P(mQ), i.e., φQ: Wider distribution – more microstates – more entropy: log(φ/¯ φ)1/2 ; vs Penalty for larger deviations from peak entropy (at δm = 0): −(1/2)φ/¯ φ.

• - - equilibrium (variance ¯

φ) —- actual (variance φ)

Maximum of s(+) is achieved at φ = ¯ φ.

• M. Stephanov

Fluctuations and QCD Critical Point WWND 2020 8 / 17

Hydro+ mode kinetics

The equation for φQ is a relaxation equation: (u · ∂)φQ = −γπ(Q)πQ, πQ = − ∂s(+) ∂φQ

• ǫ,n

γπ(Q) is known from mode-coupling calculation in ‘model H’. It is universal (Kawasaki function). γπ(Q) ∼ 2DQ2 for Q < ξ−1 and ∼ Q3 for Q > ξ−1.

more

Characteristic rate: Γ(Q) ∼ γπ(Q) ∼ ξ−3 at Q ∼ ξ−1. Slowness of this relaxation process is behind the divergence of ζ ∼ 1/Γ ∼ ξ3 and the breakdown of ordinary hydro near CP (frequency depedence of ζ at ω ∼ ξ−3).

• M. Stephanov

Fluctuations and QCD Critical Point WWND 2020 9 / 17

Towards a general deterministic formalism

An, Basar, Yee, MS, 1902.09517,1912.13456

To embed Hydro+ into a unified theory for critical as well as non- critical fluctuations we develop a general deterministic (hydro- kinetic) formalism. We expand hydrodynamic eqs. in {δm, δp, δuµ} ∼ φ and then average, using equal-time correlator G(x, y) ≡ φ(x + y/2) φ(x − y/2) . What is “equal-time” in relativistic hydro? Renormalization.

• M. Stephanov

Fluctuations and QCD Critical Point WWND 2020 10 / 17

Equal time

We use equal-time correlator G = φ(t, x+)φ(t, x−). But what does “equal time” mean? Needs a frame choice. The most natural choice is local u(x) (x = (x+ + x−)/2). Derivatives wrt x at “y-fixed” should take this into account: using Λ(∆x)u(x + ∆x) = u(x): ∆x · ¯ ∇G(x, y) ≡ G(x + ∆x, Λ(∆x)−1y) − G(x, y) . not G(x + ∆x, y) − G(x, y) .

• M. Stephanov

Fluctuations and QCD Critical Point WWND 2020 11 / 17

Confluent derivative, connection and correlator

Take out dependence of components of φ due to change of u(x): ∆x · ¯ ∇φ = Λ(∆x)φ(x + ∆x) − φ(x)

Confluent two-point correlator: ¯ G(x, y) = Λ(y/2) φ(x + y/2) φ(x − y/2) Λ(−y/2)T (boost to u(x) – rest frame at midpoint)

¯ ∇µ ¯ GAB = ∂µ ¯ GAB − ¯ ωC

µA ¯

GCB − ¯ ωC

µB ¯

GAC − ˚ ωb

µa ya ∂

∂yb ¯ GAB . Connection ¯ ω corresponds to the boost Λ. Connection ˚ ω makes sure derivative is independent of the choice of local space triad ea needed to express y ≡ x+ − x−. We then define the Wigner transform WAB(x, q) of ¯ GAB(x, y).

• M. Stephanov

Fluctuations and QCD Critical Point WWND 2020 12 / 17

Sound-sound correlation and phonon kinetic equation

Upon lots of algebra with many miraculous cancellations we ar- rive at “hydro-kinetic” equations for components of W. The longitudinal components, corresponding to p and uµ fluctua- tions at δ(s/n) = 0, obey the following eq. (NL ≡ WL/(wcs|q⊥|))

• (u + v) · ¯

∇ + f · ∂ ∂q

• NL
• L[NL] – Liouville op.

= −γLq2

• NL −

T cs|q⊥|

N (0)

L

• Kinetic eq. for phonons with E = cs(x)|q⊥| and v = csq⊥/|q⊥|

fµ = −E(aµ + 2vνωνµ)

• inertial + Coriolis

−q⊥ν∂⊥µuν

• “Hubble”

− ¯ ∇⊥µE . N(0)

L

is equilibrium Bose-distribution.

• M. Stephanov

Fluctuations and QCD Critical Point WWND 2020 13 / 17

Diffusive mode fluctuations

Fluctuations of m ≡ s/n and transverse components of uµ obey

(entropy-entropy)

L[Nmm] = −2Γλ

• Nmm − cp

n

• + . . .

(entropy-velosity)

L[Nmi] = −2(Γη + Γλ)Nmi + . . .

(velocity-velocity)

L[Nij] = −2Γη

• Nij − Tw

n

• + . . .

L is Liouville operator with v = f = 0, i.e., no propagation, but diffusion: ΓX = γXq2, where γλ = λ/cp and γη = η/w. “. . . ” are terms ∼ background grads, mixing Nmm ↔ Nmi ↔ Nij. Near critical point Γλ is smallest, γλ = λ/cp ∼ 1/ξ → 0. Nmm equation decouples and matches Hydro+ (φQ = nNmm). Very nontrivially!

• M. Stephanov

Fluctuations and QCD Critical Point WWND 2020 14 / 17

Beyond Hydro+

Hydro+ breaks down when hydro frequency/rate is of order ξ−2 due to next-to-slowest modes (Wmi and Wij). The formalism extends Hydro+ to higher frequencies, i.e., shorter hydrodynamic scales (all the way to ξ.) Fluctuations (Wmi) enhance conductivity for small ω.

Hydro Hydro+ Hydro++

• 3

ξ

• 2

ξ

• 1

ξ ω ζ λ

scaling regime (model H)

• M. Stephanov

Fluctuations and QCD Critical Point WWND 2020 15 / 17

Renormalization

Expansion of T µν contains φ(x)φ(x) = G(x, 0) =

• d3q

(2π)3 W(x, q).

This integral is divergent (equilibrium G(0)(x, y) ∼ δ3(y)).

• M. Stephanov

Fluctuations and QCD Critical Point WWND 2020 16 / 17

Renormalization

Expansion of T µν contains φ(x)φ(x) = G(x, 0) =

• d3q

(2π)3 W(x, q).

This integral is divergent (equilibrium G(0)(x, y) ∼ δ3(y)). W(x, q) ∼ W (0)

Tw

+ W (1)

∂u/q2

+

• W

(∼“OPE” or gradient expansion)

• M. Stephanov

Fluctuations and QCD Critical Point WWND 2020 16 / 17

Renormalization

Expansion of T µν contains φ(x)φ(x) = G(x, 0) =

• d3q

(2π)3 W(x, q).

This integral is divergent (equilibrium G(0)(x, y) ∼ δ3(y)). W(x, q) ∼ W (0)

Tw

+ W (1)

∂u/q2

+

• W

(∼“OPE” or gradient expansion) G(x, 0) ∼ Λ3

• ideal (EOS)

+ Λ ∂u

• visc. terms

+

• G
• finite “∂3/2”
• M. Stephanov

Fluctuations and QCD Critical Point WWND 2020 16 / 17

Renormalization

Expansion of T µν contains φ(x)φ(x) = G(x, 0) =

• d3q

(2π)3 W(x, q).

This integral is divergent (equilibrium G(0)(x, y) ∼ δ3(y)). W(x, q) ∼ W (0)

Tw

+ W (1)

∂u/q2

+

• W

(∼“OPE” or gradient expansion) G(x, 0) ∼ Λ3

• ideal (EOS)

+ Λ ∂u

• visc. terms

+

• G
• finite “∂3/2”

T µν(x) = ǫuµuν + p(ǫ, n)∆µν + Πµν +

• G(x, 0)
• = ǫRuµ

Ruν + pR(ǫR, nR)∆µν R + Πµν R +

• ˜

G(x, 0)

• .
• M. Stephanov

Fluctuations and QCD Critical Point WWND 2020 16 / 17

Work in progress and outlook

Add higher-order correlators for non-gaussian fluctuations. Connect fluctuating hydro with freezeout kinetics and implement in full hydrodynamic code and event generator. Compare with experiment. First-order transition in fluctuating hydrodynamics? Connection to action principle (SK) formulation.

• M. Stephanov

Fluctuations and QCD Critical Point WWND 2020 17 / 17

More

• M. Stephanov

Fluctuations and QCD Critical Point WWND 2020 18 / 17

Separation of scales

G(x, y) = φ(x + y/2) φ(x − y/2) depends on x slowly (L), but on y – fast (ℓeq ∼ √ L ≪ L). Similar to separation of scales in QFT in kinetic regime. (q ≫ k)

• M. Stephanov

Fluctuations and QCD Critical Point WWND 2020 19 / 17

Scales

Hydro cell size b: coarse-grain quantum operators over scale b ≫ ℓmic to leave only slow modes for which quantum fluctuatu- ations are negligible compared to thermal, i.e., ω ≪ kT. ℓmic ∼ ℓmfp, cs/T. ˘ ψ = ( ˘ T i0, ˘ J0 ) are classical stochastic variables. Hydrodynamic gradients scale L: must be L ≫ b.

back

Size of local equlibrium cell ℓeq ≡ ℓ∗: diffusion length in evolution time scale, typically τev ∼ L/cs ℓ∗ ∼ √γτev ∼

• γL/cs.

b ≪ L implies the hierarchy: ℓmic ≪ b < ℓ∗ ≪ L

• r

T/cs ≫ Λ > q∗ ≫ k (γq2

∗ = csk)

• M. Stephanov

Fluctuations and QCD Critical Point WWND 2020 20 / 17

Critical fluctuations

Near CP there is parametric separation of relaxation time scales. The slowest and thus most out-of-equilibrium mode is charge diffusion at const p: δ(s/n) ≡ m.

back

• M. Stephanov

Fluctuations and QCD Critical Point WWND 2020 21 / 17

Critical fluctuations

Near CP there is parametric separation of relaxation time scales. The slowest and thus most out-of-equilibrium mode is charge diffusion at const p: δ(s/n) ≡ m.

back

Rate of m at scale k ∼ ξ−1, Γ ∼ Dξ−2 ∼ ξ−3, is of order of that for sound at much smaller k ∼ ξ−3.

• M. Stephanov

Fluctuations and QCD Critical Point WWND 2020 21 / 17

Critical fluctuations

Near CP there is parametric separation of relaxation time scales. The slowest and thus most out-of-equilibrium mode is charge diffusion at const p: δ(s/n) ≡ m.

back

Rate of m at scale k ∼ ξ−1, Γ ∼ Dξ−2 ∼ ξ−3, is of order of that for sound at much smaller k ∼ ξ−3. The effect of m fluctuations, 1/ √ V , is (kξ)3/2 = O(1)!

• M. Stephanov

Fluctuations and QCD Critical Point WWND 2020 21 / 17

Critical fluctuations

Near CP there is parametric separation of relaxation time scales. The slowest and thus most out-of-equilibrium mode is charge diffusion at const p: δ(s/n) ≡ m.

back

Rate of m at scale k ∼ ξ−1, Γ ∼ Dξ−2 ∼ ξ−3, is of order of that for sound at much smaller k ∼ ξ−3. The effect of m fluctuations, 1/ √ V , is (kξ)3/2 = O(1)! Thus we need mm as the independent variable(s) in hydro+ equations.

• M. Stephanov

Fluctuations and QCD Critical Point WWND 2020 21 / 17

Hydro+ vs Hydro: real-time bulk response

Hydrodynamics breaks down for processes faster than Γξ ∼ ξ−3 → Hydro+

Stiffness of eos (sound speed) is underestimated in hydro (- - -): cs → 0 at CP , but

• nly modes with ω ≪ Γξ are

critically soft. Dissipation during expansion is

• verestimated in hydro (- - -):

ζ → ∞ at CP , but

• nly modes with ω ≪ Γξ

experience large ζ.

• M. Stephanov

Fluctuations and QCD Critical Point WWND 2020 22 / 17