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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 ] - PowerPoint PPT Presentation
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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
Turbulent spectra in non-Abelian gauge theories
Sebastian Scheffler 3
Motivation, part 1: Heavy-ion collisions
30.01.2009
Turbulent spectra in non-Abelian gauge theories
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
Sebastian Scheffler 4
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
Sebastian Scheffler 5
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
Sebastian Scheffler 6
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
Sebastian Scheffler 7
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 ¡ 1x;º
Dynamics fromWilson- latticeaction in Minkowski- spacetime:
S = ¯ sX
x
X
i ; ji < j
½ 12Tr1 Tr ¡Ux ; i j + U y
x ; i j¢ ¡ 1
¾¡ ¯ 0
X
x
X
i
½ 12Tr1 Tr ¡Ux ;0i + U y
x ;0i¢ ¡ 1
¾
where¯ 0 := 2°Tr1
g20
; ¯ s := 2Tr1°g2s
; ° := asat
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
Sebastian Scheffler 8
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
y2¢ 2x
¡ p2z
2¢ 2z
ª ±( _A(t0))
" = "̂ ¢a¡ 4s
Formalism & setupTurbulent spectra in non-Abelian gauge theories
Sebastian Scheffler 9
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
Turbulent spectra in non-Abelian gauge theories Results: Fast dynamics
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
30.01.2009 Sebastian Scheffler 11
Turbulent spectra in non-Abelian gauge theories Results: Fast dynamics
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
Sebastian Scheffler 12
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
Sebastian Scheffler 13
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
Sebastian Scheffler 14
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…
Turbulent spectra in non-Abelian gauge theories Results: Slow dynamics
Are there analogous phenomena in gauge theories?
Sebastian Scheffler 15
Search for UV- fixed points - Analytics (I)
30.01.2009
F (t;t;~p) := Rd3xe¡ i~p¢~xhA(t;~x)A(t;0) i » (n(~p) + 12)=! (~p)
F (t;t;p) » j p j¡ (1+· )
Turbulent spectra in non-Abelian gauge theories Results: Slow dynamics
Consider in the following
Search for solutions of the form
Are there solutions of this kind? If yes, what is the value of κ?
Sebastian Scheffler 16
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 ·
Turbulent spectra in non-Abelian gauge theories Results: Slow dynamics
Sebastian Scheffler 17
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
R d3pn§ (½)
¹ · (p)F · ¹ (p) ¡ § (F )¹ · (p)½· ¹ (p)
o != 0
Turbulent spectra in non-Abelian gauge theories Results: Slow dynamics
Sebastian Scheffler 18
Search for UV- fixed points - Analytics (IV)
30.01.2009
0= g2±ahZ
d3pZ d4k
(2¼)4Z d4q
(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)4Z d4q
(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
Turbulent spectra in non-Abelian gauge theories Results: Slow dynamics
No fixed-point solution in the full quantum theory!solution for
Sebastian Scheffler 19
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
Turbulent spectra in non-Abelian gauge theories Results: Slow dynamics
Sebastian Scheffler 20
Universality far from equilibrium?
30.01.2009
Early-universe (scalars) Heavy-ion coll.(Yang-Mills)
Turbulent spectra in non-Abelian gauge theories Results: Slow dynamics
parametric resonance, instabilities
fixed-point solutions, turbulence, power-law spectra
Sebastian Scheffler 2130.01.2009
Fixed points: Obstacles on the way to equilibrium?
30.01.2009 Sebastian Scheffler 21
Wanted to reach equilibrium by
fast processes (instabilities) ….
… but seem to get stuck at a
fixed point instead!
Turbulent spectra in non-Abelian gauge theories Results: Slow dynamics
However: No UV- fixed point in the
full quantum theory
Sebastian Scheffler 22
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?
Sebastian Scheffler 23
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
Sebastian Scheffler 2430.01.2009
Thanks for your attention!
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