Dynamics of inhomogeneous chiral condensates Gasto Krein Instituto - - PowerPoint PPT Presentation

Dynamics of inhomogeneous chiral condensates

Gastão Krein Instituto de Física Teórica, São Paulo

— Daniel Kroff (IFT, École Polytechnique Paris) — Juan Pablo Carlomagno (Univ. La Plata, IFT) — Thiago Peixoto (IFT)

In collaboration with

Motivation

QCD phase diagram at low temperature and finite baryon density might be more interesting than initially thought

Model calculations, large Nc arguments, predict that different kinds of inhomogeneous phases in QCD matter might exist at finite density

Chiral condensate might be inhomogeneous

NJL model: Nakano & Tatsumi, Phys. Rev. D 71, 114006 (2005) Basar, Dunne & Thies, PRD 79, 105012 (2009)

• D. Nickel, Phys. Rev. Lett. 103, 072301 (2009)

NJL - inspired

Nickel, PRL 103, 072301 (2009)

χSR

χSB

χSB

(hom.)

(inhom.)

Condensate profile (1-dim)

φ(z) = √ν q sn(qz; ν)

α4 = − p 36/5 α2α6

ν = 1

φ(z) = q tanh(qz)

F[T, µ; φ(x)] = Z d3x ω(T, µ; φ(x)) ω(T, µ; φ(x)) = α2 2 φ(x)2 + α4 4 n φ(x)4 + ⇥ rφ(x) ⇤2o + α6 6 ⇢ φ(x)6 + 5 ⇥ rφ(x) ⇤2φ(x)2 + 1 2 ⇥ r2φ(x) ⇤2

• Ginzburg-Landau framework

Observable?

Compact stars:

— neutrino emissivity, larger than standard Urca cooling

— EOS supports stars with 2 M⦿

Tatsumi & Muto, PRD 89, 103005 (2014), Carignano, Ferrer, Incera & Paulucci, PRD 92, 105018 (2015) Buballa & Carignano, EPJA 52, 57 (2016)

Observable?

Heavy-ion collisions:

— seem not yet fully explored — inhomogeneous phase, different momentum distribution, eccentricities (geometric information) — CBM, NICA, fragments of cold matter with inhomogeneous phase — inhomogeneous phase in nuclear matter*

Important here is time evolution of condensate, formation of inhomogeneous condensate

*Heinz, Giacosa & Rischke, NPA 933, 34 (2014)

Phase Change

— time dependence Typical situation:

— A system is forced to change from a thermodynamic equilibrium phase to another, out of equilibrium phase — Evolution to new equilibrium through spatial fluctuations that take the system (initially homogeneous) through a sequence

• f highly (not in equilibrium) inhomogeneous states

Dynamics

Rational:

• 1. It is hopeless to obtain a macroscopic description with microscopic

d.o.f.

• 2. Focus on a small number of semi-macroscopic variables; the order

parameters

• 3. Dynamics of the order parameters is slow in comparison to that of the

(remaining) microscopic degrees of freedom

— Coarse-graining

• A. Zee book

Coarse-graining

— cut off short wave lengths

Dynamical equations

First-principles derivation:

— Schwinger-Keldysh effective action; real-time

Phenomenological:

— Ginzburg-Landau-Langevin equations

Smallish deviations from equilibrium

j VHjL

At equilibrium

F = F[ϕ(x)]

ϕ(x)

At equilibrium, system described by a macroscopic free energy (Landau), functional of the order parameter:

: order parameter

Equilibrium:

δF[ϕ] δϕ(x) = 0

Example:

F[ϕ] = Z d3x ⇥ κ(rϕ)2 + V (ϕ) ⇤

V (ϕ) = 1 2m2ϕ2(x) + 1 4λϕ4(x)

j VHjL

At equilibrium

F = F[ϕ(x)]

ϕ(x)

At equilibrium, system described by a macroscopic free energy (Landau), functional of the order parameter:

: order parameter

Equilibrium:

Mechanics: equilibrium, zero force, gradient of potential energy is zero Thermodynamics: gradient of F is zero, is the thermodynamic force

δF[ϕ] δϕ(x)

δF[ϕ] δϕ(x) = 0

Example:

F[ϕ] = Z d3x ⇥ κ(rϕ)2 + V (ϕ) ⇤

V (ϕ) = 1 2m2ϕ2(x) + 1 4λϕ4(x)

Close to equilibrium

j FHjL

ϕ(x) → ϕ(x, t)

Equation of motion

∂ϕ(x, t) ∂t = −Γ δF[ϕ] δϕ(x, t) ∂ϕ(x, t) ∂t = −Γ δF[ϕ] δϕ(x, t)

Close to equilibrium

j FHjL

Near the minimum:

ϕ(x) → ϕ(x, t)

Equation of motion

(rϕ)2 ⇡ 0, V (ϕ) ⇡ 1 2m2ϕ2

ϕ(x, t) ≈ e−m2t/Γ

∂ϕ(x, t) ∂t = −Γ δF[ϕ] δϕ(x, t) ∂ϕ(x, t) ∂t = −Γ δF[ϕ] δϕ(x, t)

Close to equilibrium

j FHjL

Near the minimum:

ϕ(x) → ϕ(x, t)

Equation of motion

(rϕ)2 ⇡ 0, V (ϕ) ⇡ 1 2m2ϕ2

ϕ(x, t) ≈ e−m2t/Γ

Purely diffusive, FLUCTUATIONS are missing

∂ϕ(x, t) ∂t = −Γ δF[ϕ] δϕ(x, t) ∂ϕ(x, t) ∂t = −Γ δF[ϕ] δϕ(x, t)

Fluctuations

— noise fields

hξ(x, t)ξ(x0, t0)i = 2 Γ T δ(x x0)δ(t t0)

Example: white noise

∂ϕ(x, t) ∂t = −Γ δF[ϕ] δϕ(x, t) + ξ(x, t)

Fluctuations

— noise fields

hξ(x, t)ξ(x0, t0)i = 2 Γ T δ(x x0)δ(t t0)

Example: white noise

What is this T? ∂ϕ(x, t) ∂t = −Γ δF[ϕ] δϕ(x, t) + ξ(x, t)

Fluctuations

— noise fields

hξ(x, t)ξ(x0, t0)i = 2 Γ T δ(x x0)δ(t t0)

Example: white noise

j VHjL

j VHjL

T > Tc T < Tc

System is forced to change phase What is this T? ∂ϕ(x, t) ∂t = −Γ δF[ϕ] δϕ(x, t) + ξ(x, t)

F[ϕ] = Z d3x ⇥ κ(rϕ)2 + V (ϕ) ⇤

What is the input?

Γ∂ϕ(x, t) ∂t = − δF[ϕ] δϕ(x, t) + ξ(x, t)

hξ(x, t)ξ(x0, t0)i = 2 Γ T δ(x x0)δ(t t0)

V (ϕ)

)

Need from elsewhere: : use equilibrium free energy

Γ

κ

and

Presently, rough estimates only

Example

— PNJL model

Two order parameters

— Chiral condensate: — Polyakov loop:

σ

φ, ¯ φ

Example

— PNJL model

Two order parameters

— Chiral condensate: — Polyakov loop:

σ

φ, ¯ φ

• T. Peixoto

Domain formation

Time increases

Domain formation

Time increases Negative

• rder parameter

Positive

• rder parameter

NJL - inspired

Nickel, PRL 103, 072301 (2009)

χSR

χSB

χSB

(hom.)

(inhom.)

Condensate profile (1-dim)

φ(z) = √ν q sn(qz; ν)

α4 = − p 36/5 α2α6

ν = 1

φ(z) = q tanh(qz)

F[T, µ; φ(x)] = Z d3x ω(T, µ; φ(x)) ω(T, µ; φ(x)) = α2 2 φ(x)2 + α4 4 n φ(x)4 + ⇥ rφ(x) ⇤2o + α6 6 ⇢ φ(x)6 + 5 ⇥ rφ(x) ⇤2φ(x)2 + 1 2 ⇥ r2φ(x) ⇤2

Dynamics

— static medium

α4 = − p 36/5 α2α6

Typical value

(low temperatures)

Γ ' 1 3 fm

Nahrgang, Leupold & Bleicher, PLB 711, 109 (2008)

Low T, noise has small effect φ(z) = q tanh(qz)

Dynamics

— Bjorken expansion

∂φ(η, τ) ∂τ = −Γ δF[φ] δφ(η, τ) + ξ(η, τ)

{

ds2 = dτ 2 − τ 2dη2 = dτ 2 − (τ/τ0)2d(τ0η)2

: expansion rate

1/τ0

Dynamics

— Bjorken expansion

Homogeneous phase

φ(τ) = 1 L Z L dz φ(η, τ)

Volume average

Dynamics

— Bjorken expansion

α4 = − p 36/5 α2α6

Inhomogeneous phase

Slow expansion Fast expansion

∆τ/Γ ∼ 30

Qualitative same behavior for different combinations

• f parameters

Nonlocal NJL

— interaction has a range

: form factor

SE = Z d4x  i ¯ (x)6@ (x) G 2 ja(x) ja(x)

• ja(x)

= Z d4z G(z) ¯ (x + z/2) Γa (z/2) Γa = (1, i5~ ⌧) G(z)

Diakonov & Petrov, JETP 62 (1985) 204; NPB 245, 259 (1989) Bowler & Birse, NPA 582, 655 (1995) Gomez Dumm & Scoccola, PRD D65, 074021 … Carlomagno, Gomez Dumm & Scoccola, PLB 745, 1 (2015)

Free energy

Coefficients depend on temperature and baryon chemical potential

Coefficients

— expressions

Coefficients

— T & μ dependences

Profiles

— homogeneous states reached after long time

Reach homogeneous state

— passing through inhomogeneous states

Lifetime of inhomogeneous states

— can be increased, “right'' initial conditions

σ only π = 0 σ + π

Perspectives

— Results are qualitative, not quantitative — Evolution probes inhomogeneous configurations — Observable signatures in heavy-ion collision — Need go to three dimensions — More realistic expansion — Derive GLL equations from microscopic model

Funding