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

Dynamics of inhomogeneous chiral condensates Gastão Krein Instituto de Física Teórica, São Paulo

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

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 Ginzburg-Landau framework Z d 3 x ω ( T, µ ; φ ( x )) F [ T, µ ; φ ( x )] = α 2 2 φ ( x ) 2 + α 4 n φ ( x ) 4 + ⇤ 2 o ⇥ ω ( T, µ ; φ ( x )) = r φ ( x ) 4 ⇢ ⇤ 2 φ ( x ) 2 + 1 � α 6 φ ( x ) 6 + 5 ⇤ 2 r 2 φ ( x ) ⇥ ⇥ + r φ ( x ) 6 2 Condensate profile (1-dim) φ ( z ) = √ ν q sn( qz ; ν ) χ SR χ SB p 36 / 5 α 2 α 6 α 4 = − χ SB (hom . ) ν = 1 (inhom . ) φ ( z ) = q tanh( qz ) Nickel, PRL 103, 072301 (2009)

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 of highly (not in equilibrium) inhomogeneous states

Dynamics — Coarse-graining 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 — cut off short wave lengths A. Zee book

Dynamical equations First-principles derivation: — Schwinger-Keldysh effective action; real-time Phenomenological: — Ginzburg-Landau-Langevin equations Smallish deviations from equilibrium

At equilibrium At equilibrium, system described by a macroscopic free energy (Landau), functional of the order parameter: Example: F = F [ ϕ ( x )] Z κ ( r ϕ ) 2 + V ( ϕ ) d 3 x ⇥ ⇤ F [ ϕ ] = : order parameter ϕ ( x ) V ( ϕ ) = 1 2 m 2 ϕ 2 ( x ) + 1 V H j L 4 λϕ 4 ( x ) δ F [ ϕ ] Equilibrium: δϕ ( x ) = 0 j

At equilibrium At equilibrium, system described by a macroscopic free energy (Landau), functional of the order parameter: Example: F = F [ ϕ ( x )] Z κ ( r ϕ ) 2 + V ( ϕ ) d 3 x ⇥ ⇤ F [ ϕ ] = : order parameter ϕ ( x ) V ( ϕ ) = 1 2 m 2 ϕ 2 ( x ) + 1 V H j L 4 λϕ 4 ( x ) δ F [ ϕ ] Equilibrium: δϕ ( x ) = 0 j Mechanics: equilibrium, zero force, gradient of potential energy is zero Thermodynamics: gradient of F is zero, is the thermodynamic force δ F [ ϕ ] δϕ ( x )

Close to equilibrium F H j L ϕ ( x ) → ϕ ( x, t ) Equation of ∂ϕ ( x, t ) = − Γ δ F [ ϕ ] motion δϕ ( x, t ) ∂ t j ∂ϕ ( x, t ) = − Γ δ F [ ϕ ] δϕ ( x, t ) ∂ t

Close to equilibrium F H j L ϕ ( x ) → ϕ ( x, t ) Equation of ∂ϕ ( x, t ) = − Γ δ F [ ϕ ] motion δϕ ( x, t ) ∂ t j V ( ϕ ) ⇡ 1 ( r ϕ ) 2 ⇡ 0 , 2 m 2 ϕ 2 Near the minimum: ∂ϕ ( x, t ) = − Γ δ F [ ϕ ] ϕ ( x, t ) ≈ e − m 2 t/ Γ δϕ ( x, t ) ∂ t

Close to equilibrium F H j L ϕ ( x ) → ϕ ( x, t ) Equation of ∂ϕ ( x, t ) = − Γ δ F [ ϕ ] motion δϕ ( x, t ) ∂ t j V ( ϕ ) ⇡ 1 ( r ϕ ) 2 ⇡ 0 , 2 m 2 ϕ 2 Near the minimum: ∂ϕ ( x, t ) = − Γ δ F [ ϕ ] ϕ ( x, t ) ≈ e − m 2 t/ Γ δϕ ( x, t ) ∂ t Purely diffusive, FLUCTUATIONS are missing

Fluctuations — noise fields Example: white noise ∂ϕ ( x, t ) = − Γ δ F [ ϕ ] h ξ ( x, t ) ξ ( x 0 , t 0 ) i = 2 Γ T δ ( x � x 0 ) δ ( t � t 0 ) δϕ ( x, t ) + ξ ( x, t ) ∂ t

Fluctuations — noise fields Example: white noise ∂ϕ ( x, t ) = − Γ δ F [ ϕ ] h ξ ( x, t ) ξ ( x 0 , t 0 ) i = 2 Γ T δ ( x � x 0 ) δ ( t � t 0 ) δϕ ( x, t ) + ξ ( x, t ) ∂ t What is this T?

Fluctuations — noise fields Example: white noise ∂ϕ ( x, t ) = − Γ δ F [ ϕ ] h ξ ( x, t ) ξ ( x 0 , t 0 ) i = 2 Γ T δ ( x � x 0 ) δ ( t � t 0 ) δϕ ( x, t ) + ξ ( x, t ) ∂ t What is this T? V H j L System is forced V H j L to change phase j j T > T c T < T c

What is the input? Γ ∂ϕ ( x, t ) = − δ F [ ϕ ] δϕ ( x, t ) + ξ ( x, t ) ∂ t ) h ξ ( x, t ) ξ ( x 0 , t 0 ) i = 2 Γ T δ ( x � x 0 ) δ ( t � t 0 ) Need from elsewhere: and Γ κ Z κ ( r ϕ ) 2 + V ( ϕ ) d 3 x ⇥ ⇤ F [ ϕ ] = Presently, rough estimates only : use equilibrium free energy V ( ϕ )

Example Two order parameters — PNJL model — Chiral condensate: σ — Polyakov loop: φ , ¯ φ

Example Two order parameters — PNJL model — Chiral condensate: σ — Polyakov loop: φ , ¯ φ T. Peixoto

Domain formation Time increases

Domain formation Positive order parameter Time increases Negative order parameter

NJL - inspired Z d 3 x ω ( T, µ ; φ ( x )) F [ T, µ ; φ ( x )] = α 2 2 φ ( x ) 2 + α 4 n φ ( x ) 4 + ⇤ 2 o ⇥ ω ( T, µ ; φ ( x )) = r φ ( x ) 4 ⇢ � ⇤ 2 φ ( x ) 2 + 1 α 6 φ ( x ) 6 + 5 ⇤ 2 r 2 φ ( x ) ⇥ ⇥ + r φ ( x ) 6 2 Condensate profile (1-dim) φ ( z ) = √ ν q sn( qz ; ν ) χ SR p χ SB χ SB 36 / 5 α 2 α 6 α 4 = − (hom . ) (inhom . ) ν = 1 φ ( z ) = q tanh( qz ) Nickel, PRL 103, 072301 (2009)

Dynamics p 36 / 5 α 2 α 6 — static medium α 4 = − Low T, noise has small effect Typical value (low temperatures) Γ ' 1 3 fm Nahrgang, Leupold & Bleicher, PLB 711, 109 (2008) φ ( z ) = q tanh( qz )

Dynamics — Bjorken expansion ds 2 = d τ 2 − τ 2 d η 2 = d τ 2 − ( τ / τ 0 ) 2 d ( τ 0 η ) 2 : expansion rate 1 / τ 0 { ∂φ ( η , τ ) = − Γ δ F [ φ ] δφ ( η , τ ) + ξ ( η , τ ) ∂τ

Dynamics — Bjorken expansion Homogeneous phase Volume average Z L φ ( τ ) = 1 dz φ ( η , τ ) L 0

Dynamics p 36 / 5 α 2 α 6 — Bjorken expansion α 4 = − Inhomogeneous phase Slow expansion Fast expansion ∆ τ / Γ ∼ 30 Qualitative same behavior for different combinations of parameters

Nonlocal NJL — interaction has a range  ( x ) 6 @ ( x ) � G � Z � i ¯ d 4 x S E = 2 j a ( x ) j a ( x ) Z d 4 z G ( z ) ¯ j a ( x ) = ( x + z/ 2) Γ a ( � z/ 2) = (1 , i � 5 ~ ⌧ ) Γ a : form factor 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