Dynamical Thermalization in the Quark-Meson Model Linda Shen Institute for Theoretical Physics Masterthesis with J. Berges, J. Pawlowski, A. Rothkopf

Dynamical Thermalization in the Quark-Meson Model

Linda Shen

Institute for Theoretical Physics

Masterthesis with J. Berges, J. Pawlowski, A. Rothkopf

From heavy ion collisions towards the QCD phase diagram:

an equilibration process

Figuresfromhttp://wl33.web.rice.edu/images/HI-cartoon.png, http://images.slideplayer.com/25/7893277/slides/slide_3.jpg

o QCD phase diagram: an equilibrium concept o deconfinement + chiral phase transition

baryon chemical potential

te mp er atu re

From heavy ion collisions towards the QCD phase diagram:

an equilibration process

guresfromhttp://wl33.web.rice.edu/images/HI-cartoon.png, tp://images.slideplayer.com/25/7893277/slides/slide_3.jpg

o QCD phase diagram: an equilibrium concept o Need of out-of-equilibrium dynamics to

describe initial stages of heavy ion collision.

baryon chemical potential

te mp er atu re

From heavy ion collisions towards the QCD phase diagram:

an equilibration process

Figuresfromhttp://wl33.web.rice.edu/images/HI-cartoon.png, http://images.slideplayer.com/25/7893277/slides/slide_3.jpg

non-perturbative regime of QCD

o QCD phase diagram: an equilibrium concept o Need of out-of-equilibrium dynamics to

describe initial stages of heavy ion collision.

baryon chemical potential

te mp er atu re

We can investigate this equilibration using effective field theories.

DIJSBM TZNNFUSJD

= 0

DIJSBM CSPLFO

= 0 DSJUJDBM QPJOU

µ

T

OPOFRVJMJC SJVN

o low-energy QCD

o non-equilibrium dynamics o approach of thermal state

initial state

thermal

state

The quark-meson model provides

a successful formulation of QCD below scales ~1 GeV.

up and down quark

σ-meson and pions

Yukawa coupling

scalar potential

o model for low-energy QCD o chiral symmetry breaking

o phase diagram with 1 ^st and 2 ^nd order transition

The 2PI effective action is a practical tool to study thermalization.

DMBTTJDBM BDUJPO S[ ¯, , , ]

non-equilibrium ü non-perturbative approximations

ü non-secular and universal non-equilibrium FWPMVUJPO FRVBUJPOT

GPS BOE G

1* FݏFDUJWF BDUJPO

[ , G]

double Legendre transf.

3F t t

Z[J, R] = e ^{iW [J,R]} = D e ^{iS[ ]+iJ · + 2 i ·R·}

with

OPOFRVJMJCSJVN HFOFSBUJOH GVODUJPOBM GPS ^(BVTT (t ₀ )

DMBTTJDBM BDUJPO S[ ¯, , , ]

1* FݏFDUJWF BDUJPO [ , G]

FWPMVUJPO FRVBUJPOT

GPS BOE G

double Legendre transf.

with

= S[ ] + MPPQ RVBOUVN

DPSSFDUJPOT + 1* EJBHSBNT

-0 EJBHSBNT

N

+ /-0 EJBHSBNT

N

+ . . . + GFSNJPOCPTPOMPPQ

large N expansion

stationary conditions

DMBTTJDBM BDUJPO S[ ¯, , , ]

1* FݏFDUJWF BDUJPO [ , G]

FWPMVUJPO FRVBUJPOT

GPS BOE G

double Legendre transf.

with

t 2 + M ² (x; ) (t) = GFSNJPO CBDLSFBDUJPO  1* DPSSFDUJPOT [ _x + M ȷ(x; )] G(x, y) = i

z

[ (x, z; , G)] G(z, y) i (x y)

effective mass self-energy large N expansion

stationary conditions

DMBTTJDBM BDUJPO S[ ¯, , , ]

1* FݏFDUJWF BDUJPO [ , G]

FWPMVUJPO FRVBUJPOT

GPS BOE G

double Legendre transf.

with

t 2 + M ² (x; ) (t) = GFSNJPO CBDLSFBDUJPO  1* DPSSFDUJPOT [ _x + M ȷ(x; )] G(x, y) = i

z

[ (x, z; , G)] G(z, y) i (x y)

= S[ ] + MPPQ RVBOUVN

DPSSFDUJPOT + 1* EJBHSBNT

-0 EJBHSBNT

N

+ /-0 EJBHSBNT

N

+ . . . + GFSNJPOCPTPOMPPQ

large N expansion

stationary conditions

FWPMVUJPO FRVBUJPOT GPS BOE G

add fermions

add e.o.m. for 4 fermion propagator

components

DMBTTJDBM BDUJPO S[ ¯, , , ]

1* FݏFDUJWF BDUJPO

[ , G]

Numerical solution of the equations of motion

o Symmetries: spatial homogeneity & isotropy

o Propagator decomposition:

o real-time evolution:

• specify initial conditions as

free-field propagators + vanishing field

• iterative numerical computation of the time-evolution in two temporal directions

t

t G(x, y) = F (x, y) + i

2 (x, y) THO(x ⁰ y ⁰ )

statistical function spectral function

0 20 40 60 80 100 t

0.0 0.2 0.4 0.6 0.8

(t )

= 90.0, g = 5.0, m ² = 0.008, m = 0.18 Real-time evolution of the macroscopic field

When does this stationary

state become thermal?

(1) Thermal equilibrium is a time-translation invariant state.

o time-translation invariance implies

o temporal Wigner transformation:

G(t, t , |T|) G( , |T|) POMZ t t OP t + t

(t, t , |T|) (X ⁰ , , |T|)

The two-point functions become time-translation invariant.

0.0 0.5 1.0 1.5 2.0

! 2.5

0.0 2.5 5.0 7.5 10.0

i⇢ ⇡ (X 0 ,! ,| p| )

|p| = 0.016

X ⁰ = 25.0

X ⁰ = 37.5

X ⁰ = 50.0

X ⁰ = 62.5

X ⁰ = 125.0

X ⁰ = 250.0

DPOTUBOU JO X ⁰ t + t GPS X

150

(2) Thermal eq. as state with thermal particle distributions.

o thermal initial density matrix implies fluctuation-dissipation relation:

o effective particle number:

o in thermal equilibrium:

F _FR ( , |T|) = i 1

2 + n _UI ( ) _FR ( , |T|)

n( , |T|) = i F ( , |T|) ( , |T|)

1

2 n _UI ( )?

n( , |T|) n _"1/6. ( ) = 1

e 1 XJUI = 1/T

Determination of the thermalization temperature using the Bose-Einstein and Fermi-Dirac distribution

0 1 2 3 4

! 0

2 4

log [n 1 b/f (! ,| p| )± 1] T ^fit _b = 1.022849

T _f ^fit = 1.022852

boson (pion) fermion

T m Rjy J2o

Particle masses from spectral functions by dispersion relation

0 1 2 3

|p|

1.0 1.2 1.4 1.6

m ⇡ (| p| )

m f it = 1.024 data

m( |T|) = _T2F ² ( |T| ² ) |T| ² fit

o particle mass from peak position:

o physical mass at zero momentum

0 1 2 3 4

! 0

5 10

⇢ ⇡ (X 0 ,! ,| p| ) X ⁰ = 250.0 X ⁰ = 250.0 X ⁰ = 250.0

|p| = 0.016

|p| = 1.571

|p| = 3.142

A step forward

in describing the thermalizing of the QGP in a heavy ion collision We were able to

o include non-equilibrium dynamics

o observe the approach of thermal equilibrium o determine the physical mass spectrum

Next steps:

o non-zero baryon-chemical potential o expanding box size

o scaling behavior around critical point