turbulent spectra in non-abelian gauge theories sebastian scheffler, tu darmstadt, 30 january 2009,...

Turbulent spectra in non-Abelian gauge theories

Sebastian Scheffler, TU Darmstadt,

30 January 2009,¢(2009) Heidelberg

Journal references: • J. Berges, S. Scheffler, D. Sexty, PRD 77, 034504 (2008) , arXiv:0712.3514 [hep-ph]• J. Berges, S. Scheffler, D. Sexty, arXiv:0811.4293 [hep-ph], submitted to Elsevier• J. Berges, D. Gelfand, S. Scheffler, D. Sexty, arXiv 0812.3859 [hep-ph], submitted to Elsevier

Sebastian Scheffler 2

Outline of the talk

30.01.2009

1. Motivation

2. Formalism & setup

3. Results: Fast vs. slow dynamics

4. Conclusions & outlook



Motivation, part 1: Heavy-ion collisions

30.01.2009


Result from RHIC: • hydrodynamics works well starting at ¿0 ' 1 fm/c ,

(Luzum/Romatschke, PRC 78)

-> Rapid isotropisation essential (Arnold et al., PRL 94):• How is this achieved? • Need to understand what happens before ¿0; plasma instabilities?

Numerical approaches:1. Soft classical gauge fields coupled to hard classical particles2. Classical-statistical gauge field evolution

Introduction


Motivation, part 2: Non-equilibrium QFT

30.01.2009

There are still many open questions in non-equilibrium QFT - in

particular regarding gauge theories. An (incomplete) to-do list:

• Develop, test, and benchmark different approximation schemes

• Analyse and exploit analogies between various fields of non-

equilibrium physics (e. g. early universe, heavy-ion collisions,

cold atomic gases )

• Transport coefficients

• Non-thermal fixed points? Universality far from equilibrium?

Turbulent spectra in non-Abelian gauge theories Introduction


Reminder: Classical-statistical field theory

30.01.2009

Why use the classical approximation?

feasibility

good to study early times if occupation numbers are high

highly successful for scalar fields

can serve to test other methods (e. g. 2PI)

Turbulent spectra in non-Abelian gauge theories Formalism & setup


Setup

30.01.2009

• classical-statistical limit of pure SU(2) gauge theory: Evolve an initial

ensemble using the classical field equations

• discretize everything on a lattice

• use a static geometry

• pure gauge theory, i. e. no fermions

• anisotropic initial conditions (-> heavy-ion collisions)

• no separation of scales assumed

Formalism & setupTurbulent spectra in non-Abelian gauge theories


Setup

30.01.2009

Usecommon latticediscretization scheme:

Link variables: Ux;¹ := ei gaA ¹ (x)

Plaquette variables: Ux;¹ º := Ux;¹ U(x+¹̂ );º U¡ 1(x+º̂ );¹ U ¡ 1

x;º

Dynamics fromWilson- latticeaction in Minkowski- spacetime:

S = ¯ s

X

x

X

i ; ji < j

½1

2Tr1Tr

¡Ux ; i j + U y

x ; i j

¢¡ 1

¾¡ ¯ 0

X

x

X

i

½1

2Tr1Tr

¡Ux ;0i + U y

x ;0i

¢¡ 1

¾

where¯ 0 :=

2°Tr1

g20

; ¯ s :=2Tr1

°g2s

; ° :=as

at

Weuse temporal axial gaugeA0 ´ 0 and g0 = gs = 1.Variation w. r. t. spatial links ) Equations of motion

Turbulent spectra in non-Abelian gauge theories Formalism & setup


Sampling from the initial ensemble

30.01.2009

hA(t;x)A(t0;y) i =R

DA(t0) D _A(t0) P [A(t0); _A(t0)]A(t;x)A(t0;y)

Compute e. g. a correlation function according to

where the initial density function is characterised by

• ¢x À ¢z , distribution ±( pz ) – like on the lattice

• ( A/ t ) (t=0) = 0 => Gauss- constraint fulfilled

• Amplitude C dialed to give a fixed energy

• Convert to physical units via

hAaj (0;p)Ab

k(0;¡ q) i » C±ab±j k±p;q exp©¡

p2x +p2

y

2¢ 2x

¡ p2z

2¢ 2z

ª±( _A(t0))

" = "̂ ¢a¡ 4s

Formalism & setupTurbulent spectra in non-Abelian gauge theories


Instabilities: A brief reminder of ¢(2007)

30.01.2009

Turbulent spectra in non-Abelian gauge theories Results: Fast dynamics

Some general facts about instabilities:

• Gauge field possesses unstable (i. e. exponetially growing) modes if

distribution of charge carriers is anisotropic (Mrówczyńsky,

Romatschke/Strickland, ... )

• Bottom-up scenario by Baier et al. modified

• Can instabilities resolve the thermalization/isotropization puzzle?

(Arnold et al.)

Sebastian Scheffler 1030.01.2009

Instabilities: A brief reminder of ¢(2007)

30.01.2009 Sebastian Scheffler 10


Brief summary:

• instabilities occur using anisotropic init. cond.

• inverse growth rates » 1 fm/c (for ² = 30

GeV/fm^3)

• low-momentum sector driven towards isotropy

Two disadvantages of the original setup:

• SU(2) instead of SU(3)

• Gauss constraint enforced by ( A/ t ) (t=0) = 0

Sebastian Scheffler 1130.01.200930.01.2009 Sebastian Scheffler 11

Instabilities: Some new results



B. Sc. theses of D. Gelfand and N. Balanešković

SU(3):

• different time scales, but can be accounted for

in terms of the number of colours

• see arXiv:0812.3859 [hep-ph]

Gauss- constraint:

• Can implement more general initial conditions

• no differences discernible


Fast vs. slow dynamics

30.01.2009

Turbulent spectra in non-Abelian gauge theories Results: Fast vs. slow dynamics

Early times: • dominated by fast processes (instabilities)

Late times:• governed by slow/stationary processes • fixed points / turbulence / scaling

solutions ?

Why is this interesting? -> Cf. early universe


UV- fixed points: Motivation from scalars

30.01.2009

Stationary power-law spectra reminiscient of Kolmogorov turbulence are commonly encountered in early-universe cosmology following a phase of parametric resonance:

Micha/Tkachev, PRD 70 Berges/Rothkopf/Schmidt, PRL 101

The spectral index 3/2 is derived in terms of Boltzmann- eqns. or 2PI- calculations, respectively.

Turbulent spectra in non-Abelian gauge theories Results: Slow dynamics


UV- fixed points: What about gauge theories?

30.01.2009

Yes!

1. Arnold & Moore (PRD 73) find particle number spectra with

spectral index κ = 2.

2. Müller et al. predict κ = 1 (thermal value) , NPB 760 .

3. This work: See next slides…


Are there analogous phenomena in gauge theories?


Search for UV- fixed points - Analytics (I)

30.01.2009

F (t;t;~p) :=R

d3xe¡ i~p¢~xhA(t;~x)A(t;0) i » (n(~p) + 12)=! (~p)

F (t;t;p) » j p j¡ (1+· )


Consider in the following

Search for solutions of the form

Are there solutions of this kind? If yes, what is the value of κ?


Search for UV- fixed points - Analytics (II)

30.01.2009

§ (½)¹ · (p)F · ¹ (p) ¡ § (F )

¹ · (p)½· ¹ (p) = 0

J. Berges / G. Hoffmeister, arXiv:0809.5208:

Stationary and translationally-invariant correlation functions fulfill the

identity

where

and denote the non-local contributions to the self-energy

of odd and even symmetry, respectively.

F¹ º (x;y) ´ 12hfA¹ (x);Aº (y)gi and ½¹ º (x;y) ´ ih[A¹ (x);Aº (y)]i

§ (½)¹ · § (F )

¹ ·



Search for UV- fixed points - Analytics (III)

30.01.2009

F (sp) = s¡ (2+®)F (p) ½(sp) = s¡ 2½(p)

Evaluate 1-loop contribution to the self-energy:

Assume scaling behaviour of the kind

and demand

Rd3p

n§ (½)

¹ · (p)F · ¹ (p) ¡ § (F )¹ · (p)½· ¹ (p)

o!= 0



Search for UV- fixed points - Analytics (IV)

30.01.2009

0= g2±ahZ

d3pZ

d4k(2¼)4

Zd4q

(2¼)4±(p+k +q)

n

F · ½(k)F¸¾(q)½° º (p)h:::

i h:::

i+½· ½(k)F¸¾(q)F° º (p)

h:::

i h:::

i

+F · ½(k)½̧¾(q)F° º (p)h:::

i h:::

i¡ ½· ½(k)½̧¾(q)½° º (p)

h:::

i h:::

i o:

0=Z

d3pZ

d4k(2¼)4

Zd4q

(2¼)4±(p+k+q)½° º (p)

h:::

i h:::

i np4¡ 2·0 +k4¡ 2·

0 +q4¡ 2·0

p(4¡ 2· )0

F · ½(k)F¸¾(q)¡ ½· ½(k)½̧¾(q)| {z }quant.

o:

First, this yields a rather unwieldy integral:3- vertex

Carrying out a Zakharov- transformation, this can be cast into the form

Classical limit: |F F | À | ½ ½ | ,

4¡ 2· = 1 , · =32


No fixed-point solution in the full quantum theory!solution for


Search for UV- fixed points - Numerics

30.01.2009

const:£ p¡ (1+· )

F (t;t;p)

Find that the equal-time correlators

converge to a stationary solution after the saturation of instabilities.

Computation on a 128^3- lattice in Coulomb gauge

Fit spectrum to



Universality far from equilibrium?

30.01.2009

Early-universe (scalars) Heavy-ion coll.(Yang-Mills)


parametric resonance, instabilities

fixed-point solutions, turbulence, power-law spectra


Fixed points: Obstacles on the way to equilibrium?


Wanted to reach equilibrium by

fast processes (instabilities) ….

… but seem to get stuck at a

fixed point instead!


However: No UV- fixed point in the

full quantum theory


Summary

30.01.2009

Turbulent spectra in non-Abelian gauge theories Conclusions & outlook

Instabilities:

• inverse growth rates of order 1 fm/c

• no qualitative difference for SU(3) and more general initial conditions

UV- fixed point:

• Find quasi-stationary power-law spectrum in Coulomb gauge

• characterised by spectral index κ = 3/2

• very similar to results for scalars – universality far from equilibrium?


Future projects

30.01.2009

1. Establish a description of the UV- fixed point in terms of gauge

invariant quantities

2. Investigate the IR- regime: Are there power-law solutions as in the

scalar field theory?

3. Couple the gauge fields to fermions

4. Compare classical-statistical simulations to 2PI- calculations

Turbulent spectra in non-Abelian gauge theories Conclusions & outlook


Thanks for your attention!

turbulent spectra in non-abelian gauge theories sebastian scheffler, tu darmstadt, 30 january 2009,...

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