Joe Rebholz

New Progress in Heavy Ion Collisions:What is hot in the QGP? October 06, 2015 Wuhan, China

On Wave Turbulence in Non-Abelian Plasmas

(In preparation)Yacine Mehtar-Tani

INT, University of Washington

Yacine Mehtar-Tani What is Hot in the QGP?

Wave Turbulence ( I )

• Similarity with fluid turbulence: inviscid transport of conserved quantities from large to small scales through the so-called transparency window (or inertial range)

2

• Out-of-equilibrium statistics of random non-linear waves

• Some examples:

Atmospheric Rossby waves

Water surface gravity and capillary waves

Waves in plasmas

Nonlinear Schrödinger equation (NL Optics, BEC)

Yacine Mehtar-Tani What is Hot in the QGP?

Wave Turbulence ( II )

• Waves are excited by external processes. Driven turbulence: Open system with source and sink → away from thermodynamical equilibrium

• Steady states characterized by constant fluxes P and Q rather than temperature and thermodynamical potentials

• Kolmogorov-Obukhov (KO41) theory relies on Locality of interactions: Only eddies (waves) with comparable sizes (wavelengths) interact. Steady state power spectra in momentum space depend on the fluxes and not on the pumping and dissipation scales

• Weak (Wave) Turbulence Theory: Kinetic description in the limit of weak nonlinearities (NB: no theory for strong turbulence)

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V. E. Zakharov, V. S. L’vov, G. Falkovich (Springer- Verlag, 1992)

Yacine Mehtar-Tani What is Hot in the QGP?

• Immediately after the collision the system is far from equilibrium. Anisotropic particle distribution in momentum space.

• Chromo-Weibel Instabilities : Momentum anisotropy induces exponential growth of soft modes - transverse magnetic and electric fields - (early stage: as in abelian plasmas) which turns into a linear growth due to nonlinear interactions inherent to non-abelian plasmas

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E. S. Weibel (1959) S. Mrowczynski (1993)

P. Arnold, J. Lenaghan, G. Moore, L. Yaffe (2005) A. Rebhan, P. Romatschke, and M. Strickland (2005)

D. Bödeker, K. Rummukainen (2005) P. Arnold, G. D. Moore (2005)

Turbulence in early stages of Heavy Ion Collisions

pz

p�

Yacine Mehtar-Tani What is Hot in the QGP?

• Hard-loop simulations (large scale separation between hard modes and soft excitations) : Nonlinear interactions develop a turbulent cascade in the UV with exponent 2

Turbulence in early stages of Heavy Ion Collisions

5

P.Arnold, G. D. Moore (2005) A. Ipp, A. Rebhan, M. Strickland (2011)

k�2

Yacine Mehtar-Tani What is Hot in the QGP?

Weak Turbulence in Kinetic Theory

• Can one understand this power spectrum from first principles?

• Note: from A. H. Mueller, A. I. Shoshi, S. M. H. Wong (2006):

Turbulence in QCD is nonlocal ⇒

• Some caveats (in this work):

Homogeneous and isotropic system of gluons

Forcing: Energy injection with constant rate at : Dispersion relation

Weak nonlinearities in the classical limit (high occupancy):

6

n(k) � k�1

�(k) � |k|

P kf � m

1 � n(k) � 1

g2

k�2

g2 � 1 and

Yacine Mehtar-Tani What is Hot in the QGP?

Elastic 2 to 2 process (4-waves interactions)

• Elastic gluon-gluon scattering

7

1 k

2 3

• Two constant of motion: particle number and energy ⇒ Two fluxes

P �E =

�d3k |k| n(k)Q �N =

�d3k n(k)

�

�tnk =

1

2

�

k1,k2,k3

1

2�(k)|M12�3k|2 �(

�

i

ki)�(�

i

�i) F[n]

M12�3k �

F[n] � [nk1nk2

nk + nk1nk2

nk3� nk1

nk3nk � nk2

nk3nk] � n3

Yacine Mehtar-Tani What is Hot in the QGP?

Elastic 2 to 2 process (4-waves interactions)

• Elastic gluon-gluon scattering

8

1 k

2 3

• H-theorem ⇒ Thermal fixed-point (vanishing fluxes)

�

�tnk =

1

2

�

k1,k2,k3

1

2�(k)|M12�3k|2 �(

�

i

ki)�(�

i

�i) F[n]

M12�3k �

F[n] � [nk1nk2

nk + nk1nk2

nk3� nk1

nk3nk � nk2

nk3nk] � n3

P = Q = 0

nk =T

�(k) � µ(Rayleigh-Jeans distribution)

Yacine Mehtar-Tani What is Hot in the QGP?

Kolmogorov-Zakharov (KZ) Spectra

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P � Q � n � n3 � n � P1/3 � Q1/3

n(k) �Q1/3

k4/3n(k) �

P1/3

k5/3

• Dimensional analysis determines uniquely the out-of-equilibrium steady state (KZ) power spectra if the interactions are local in momentum space

• From collision integral the flus scales as the cube of the the occupation numebr: nonlinear 4-wave interactions

particle cascade energy cascade

• Same exponents for scalar theories in the absence of condensation

Yacine Mehtar-Tani What is Hot in the QGP?

Dual cascade: Fjørthoft argument (1953)

• Q: Direction of fluxes? Injection of energy at and dissipating at

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kf

k� � kf � k+

• Reductio ad absurdum: If energy was dissipating at low momenta then particles would dissipate faster than the pumping rate! ⟹ Direct energy cascade

Q� �P

k�� P

kf� Q

Direct energy cascade

Inverse particle cascade

kfk� k+

n(k)

k�4/3

k�5/3

P � kfQ

If no damping: condensation

absurd

Are KZ spectra in QCD physically relevant?

Yacine Mehtar-Tani What is Hot in the QGP?

Inelastic scattering in the small angle approximation

• Small angle approx: Fokker-Plank equation: Diffusion and drag

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�

�tnk � q

4k2

�

�kk2

��

�knk +

n2k

T�

�

t � q2 � s

• KZ spectra are not stationary solutions of the collision integral (contrary to non-relativistic Coulomb scattering! A. V. Kats, V. M. Kontorovich, S. S. Moiseev, and V. E. Novikov

(1975) )

• Furthermore: diverges in the IR for and for in the UV

k � q � m• Coulomb interaction is singular at small momentum transfer

q n � k�5/3 n � k�4/3

Diffusion coefficient Screening mass Effective temperature

|Mk1�23|2 � �2 s2

t2

L. D. Landau (1937) B. Svetitski (1988)

m2 ��

�d3k

|k|nk T� �

q

�m2q � � �2

�d3kn2

k

Yacine Mehtar-Tani What is Hot in the QGP?

Steady state solutions

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• Thermal fixed point:

• Non-thermal fixed point (inverse particle cascade):

A � 1

2T�

�1 +

�

1 +16Q

qT�

�

�

�tnk � q

4k2

�

�kk2

��

�knk +

n2k

T�

�+F� D

Forcing Dumping

• No stationary solution for the energy cascade without a sink

n(k) �A

k>

T�k

• Warm cascade behavior:

2-D Optical turbulence: S. Dyachenko, A.C. Newell, A. Pushkarev, V.E. Zakharov (1992)

Boltzmann equation: D. Proment, S. Nazarenko, P. Asinari, and M. Onorato (2011)

T�k � µ

Yacine Mehtar-Tani What is Hot in the QGP?

0.01 0.1 1 10k

-0.00001

-8x10-6

-6x10-6

-4x10-6

-2x10-6

0

2x10-6

4x10-6

6x10-6

E an

d N flu

xes

Numerical simulation of FK equation with forcing

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0.01 0.1 1k

0.0001

0.001

0.01

0.1

1

10

100

1000

n(k)

kf

Q(k)

P(k)

• The occupation number (left) and, the energy and particle number fluxes (right) at late times

• Constant particle flux at k=0 ⇒ Bose-Einstein condensate

A

k

kf

Yacine Mehtar-Tani What is Hot in the QGP?

Contribution from inelastic processes?

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Naively we would expect inelastic processes to be suppressed by powers of the coupling constant

In non-abelian plasmas inelastic processes are enhanced due to collinear divergences and hence cannot be neglected compared to elastic processes

In what follows we shall proceed in the small angle approximation: 2 → 3 process reduces to an effective 1 → 2

Yacine Mehtar-Tani What is Hot in the QGP? 16

• LPM regime: many scatterings can cause a gluon to branch with the rate

• Bethe-Heitler regime for

kd�

dk�

�

tf(k)� �

�q

k

tf(k) �k

k2�

�k

qtf

tf(k) < �mfp � m2/q

k

k

R. Baier, Y. Dokshitzer, A. H. Mueller, S. Peigné, D. Schiff (1995) V. Zakharov (1996)

J. F. Gunion and G. Bertsch (1982)

kd�

dk�

�

�mfp

formation time:

Effective 3 waves interaction (1 to 2 scattering)

Yacine Mehtar-Tani What is Hot in the QGP?

Effective 3 waves interaction (1 to 2 scatterings)

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�

�tnk � 1

k3

���

0

dqK(k + q, q)F(k + q, q) �

�k

0

dqK(k, q)F(k, q)

�

F(k, q) � nk+qnk + (nk+q � nk)nq � n2

K(k, q) � ��

q(k + q)7/2

k1/2q3/2

nk �P1/2

q1/4 k7/4

• Direct energy cascade (if interactions are local!)

k + q

k

q

k � q

k

q

k + q

k

k

q q

k � q

- - +

R. Baier, Y. Dokshitzer, A. H. Mueller, D. Schiff, D. T. Son (2000) P. Arnold, G. D. Moore, L. G. Yaffe (2002)

P. Arnold, G. D. Moore (2005)

Yacine Mehtar-Tani What is Hot in the QGP?

Non-locality of interactions in momentum space

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• Assume a power spectrum and require the energy flux to be independent of

n � k�x

• We obtain and

• The above integral diverges at the KZ spectrum

x = 7/4

k

P =��

q

�1

0

dz(1 � z)x + zx � 1

zx+1/2(1 � z)x+3/2ln

1

z

⇒ Effective 3 waves Interaction is nonlocal in momentum space

and the KZ spectrum cannot be realized

Yacine Mehtar-Tani What is Hot in the QGP?

Thermalization of the soft sector ( k ≪ kf )

• Nonlocallity ⇒ The collision integral is dominated by strongly asymmetric

branchings

• In the regime: To the left of the source

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k � kf

�

�tnk � 1

k3

���

0

dqK(k + q, q)F(k + q, q)

�a � �

�q

k7/2[T� � kn(k)]

k + q

k

q q

k + q

k

-

• Late times (steady state) solution is thermal (no fluxes) :

n(k) � T�k

kf

n(k)

• Therefore, to the left of the forcing the system thermalizes rapidly

Yacine Mehtar-Tani What is Hot in the QGP?

Nonlocal energy cascade in the UV ( k ≫ kf )

• In the regime: To the right of the source. We perform a gradient expansion around

• We obtain a diffusion equation in “4-D”

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k � kf

k � q

�

�tn(k) � qinel

4k3

�

�kk3 �

�kn(k)

qinel =��

q

��

0

dq�

qn(q)

• with the inelastic diffusion coefficient (in the LPM regime)

kf

n(k)

• Straightforward generalization including the BH regime

Yacine Mehtar-Tani What is Hot in the QGP? 21

�

�tn(k) � qinel

4k3

�

�kk3 �

�kn(k)

• Recall that 3-D diffusion conserves number of particles:

Its fixed point (direct particle cascade):

N �

�dkk2 n(k)

E �

�dkk3 n(k)

n(k) �1

k

n(k) �1

k2

• 4-D diffusion conserves energy:

Its fixed point (direct energy cascade):

Nonlocal energy cascade in the UV ( k ≫ kf )

Yacine Mehtar-Tani What is Hot in the QGP?

0.1 1k

1x10-5

1x10-4

1x10-3

1x10-2

1x10-1

1

n(k)

Numerical simulation with forcing

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kf

nKZ � k�7/4

n � k�2

Parametric estimate:

n(k) �P

qk2�

P1/3 k1/3f

k2

q � k3fn2

⇓

kmax(t) ��

qinelt

increasing time

• Wave front moving towards the UV leaving in its wake the predicted nonlocal steady state spectrum: hard gluons in the inertial range interact dominantly with gluons at the forcing scale (energy gain)

wave front evolution:

Yacine Mehtar-Tani What is Hot in the QGP?

Interplay between elastic and inelastic processes ( I )

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• For a spectrum falling faster than 1/k one can neglect the drag term in the elastic part. Then, the collision integral in the UV reads

�

�tn(k) � qinel

4k3

�

�kk3 �

�kn(k) +

qel

4k2

�

�kk2 �

�kn(k)

• Steady state solution: n(k) �1

k2��

� B

4k3��

�

�kk3�� �

�kn(k)

B = qinel + qelwhere

0 < � < 1

� =2

1 + qinel/qeland

Yacine Mehtar-Tani What is Hot in the QGP?

Interplay between elastic and inelastic processes ( II )

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0.1 1k

0.01

0.1

1

10

n(k)

n � k�2

kf

n � k�2+�

A. Ipp, A. Rebhan, M. Strickland (2011)

Exponent is computed self-consistently

• Elastic processes reduce slightly the exponent at asymptotically late times. At intermediate times k -2 spectrum is observed: balance between the drag and diffusion terms?

� � 0.24

Yacine Mehtar-Tani What is Hot in the QGP?

Summary

Wave turbulence in QCD is different from scalar theories. It is characterized by nonlocal interactions in momentum space: Kolmogorov-Zakharov spectra are not physically relevant

Inelastic processes dominates the dynamics with a direct energy cascade

To the left of the forcing scale the system is in thermal equilibrium: warm cascade

To the right of the forcing scale: kinetic theory predicts a steady state spectrum (in the LPM and BH regimes) in agreement with Hard Loop simulations

Outlook: mass corrections, anisotropic fluxes, strong turbulence in the presence of strong fields (on the lattice): different exponents?

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� k�2