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Mach Conesin a Perturbative Quark-Gluon Plasma

Berndt Mueller – Duke University

Quark Matter 2008Jaipur, India, 2 - 10 February 2008

Credits to: M. Asakawa R.B. Neufeld C. Nonaka J. Ruppert

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An interesting question

Is a Mach cone created when a supersonic parton propagates through the quark gluon plasma?

A Mach cone is formed when an object moves faster than the speed of sound in the medium.

What is the energy and momentum perturbation of the medium due to a fast parton?

What happens here ?!

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In real life….

What happens here ?!

g2 %Q2 . Projectile color charge

mD2 . Screening mass

cs2 =

∂p∂ε

. Spεεd of sound

Γs =4η3sT

. Attεnuation lεngtη

Relevant dynamical quantities:

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FormalismCalculate the interaction of the color field of the supersonic parton with the medium by means of semi-classical transport theory:

pμ

E∂∂xμ + gfabcQ

aAμb ∂∂Qc

⎛⎝⎜

⎞⎠⎟ + gQa(Ea +v×Ba)

F1 24 44 34 4 4 ⋅∇p

⎡

⎣⎢⎢

⎤

⎦⎥⎥ f x, p,Q( )=0

If the medium is color neutral, to lowest order:

pμ

E∂∂xμ −∇p ⋅D(x, p)⋅∇p

⎡⎣⎢

⎤⎦⎥f x, p( )=0

witη D ij(x, p)= dt'Firx,t( )Fj

rx + rv(t'−t),t'( )

−∞

t

∫ .

medium

%Qa , uz

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Hydrodynamics

In the macrocopic limit this yields hydrodynamic equations with source terms:

∂∂xμ

T μν = J ν with T μν = ε + p( )uμuν − pgμν +Tdiss

μν

J ν = dp∫ pν ∇ p ⋅D(x, p) ⋅∇ p f (x, p)

⎧⎨⎪

⎩⎪

Momentum space integrals yield term ~ mD2:

J rx,t( )=iμ D

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(2p)8d 4k d 4k'εi(k+k')⋅x dv

Ea(k)+ v× Ba(k)( )4p w '−v⋅k'+ iε( )∫∫ v⋅ Ea(k')+ v× Ba(k')( )⎡⎣ ⎤⎦

J0 rx,t( )=iμ D

2

(2p)8d 4k d 4k'εi(k+k')⋅x dv

v⋅Ea(k)+ v× Ba(k)( )4p w '−v⋅k'+ iε( )∫∫ v⋅ Ea(k')+ v× Ba(k')( )⎡⎣ ⎤⎦

witη Ea(k)=i ε(k)wuz−k( )2pg %Qad w −kzu( )

ε(k) k2 −ε(k)w 2( ), Ba(k)=

i k × u( )2pg %Qad w −kzu( )k2 −ε(k)w 2( )

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Unscreened source

For an unscreened color charge, analytical result in u1 limit:

J 0 (r,z,t)= f(r,z,t)gu2 12−gu(z−ut)

2r⎛⎝⎜

⎞⎠⎟

Jz(r,z,t)=u J0(r,z,t)−f(r,z,t)u2

2+

1g 2

⎛⎝⎜

⎞⎠⎟z−utr

Jx/y(r,z,t)= −f(r,z,t)u2

2+

1g 2

⎛⎝⎜

⎞⎠⎟x / yr

⎫

⎬

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

witη

f(r,z,t)=g2 %Q2μ D

2

16p 2 r2 +g 2(z−ut)2⎡⎣ ⎤⎦3/2

Applying infrared (screening) and ultraviolet (quantum) cuts on the r-integral gives the standard expression for collisional energy loss:

−dEdx

= d 3x J 0 (x)∫ =g2 %Q2mD

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8πlnρmax

ρmin

=12α s %Q2mD

2 ln4ETmD

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With screening

Use HTL di-electric functions for w = ukz :

Expressions for Jn(x) can be reduced to sums of products of two-dimensional Fourier integrals, which can be performed numerically after contour rotation in the complex plane.

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Energy density

J0(r,z) unscreened J0(r,z) screened

(z - ut) r (z - ut) r

u = 0.99

(GeV)4 (GeV)4

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z-Momentum density

Jz(r,z) unscreened Jz(r,z) screened

(z - ut) r (z - ut)

r

u = 0.99

(GeV)4 (GeV)4

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x-Momentum density

Jx(r,z) unscreened Jx(r,z) screened

(z - ut)

r (z - ut)

r

u = 0.99

(GeV)4 (GeV)4

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More comparisons

Jz(z-ut) at r = 2 GeV-

1

Jx(z-ut) at r = 1 GeV-

1

UnscreenedScreened u = 0.99

(GeV)4

(GeV)4

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Linearized hydro

Linearize hydro eqs. for a weak source: T00 ε0 + dε, T0i gi .

∂∂tδε +∇ ⋅ rg = J 0 ∂

∂trg + cs

2∇δε +η

ε 0 + p0

43∇ ∇ ⋅ rg( ) =

rJ

Solve in Fourier space for longitudinal sound:

dε =iω + iΓ sk

2( )J 0 + kJL

ω 2 − cs2k2 + iΓ sωk

2 gL = ics

2kJ 0 +ωJL

ω 2 − cs2k2 + iΓ sωk

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… and dissipative transverse perturbation:gT =iJT

w + 34 iΓsk

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See: J. Casalderrey-Solana, E.V. Shuryak and D. Teaney, arXiv:hep-ph/0602183

Use: u =0.99955c, cs2 =

13, Γs =

13pT

for T =350 MεV .

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The Mach cone (at last!)

(z - ut)

r

Energy density

Momentum density

rx dε(rx)μ D

2 T

rx gz (rx)

mD2 T

gT gL

Unscreened source with rmin/max cutoff

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Contour plots

dε

z

r

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pQCD vs. N=4 SYM

(z - ut)

r

rx dε(rx)μ D

2 T

rx gz (rx)

mD2 T

Chesler & Yaffe

arXiv:0712.0050

R.B. Neufeld (preliminary)

u = 0.99955 c

u = 0.75 c

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Conclusion

Summary:We have calculated the energy and momentum density deposited into a perturbative, thermal QCD plasma by the color field of a fast moving parton. When treated as a source in linearized dissipative hydrodynamics, the perturbation induces a sonar Mach cone and a diffusive wake. Apart from logarithmic effects, the effect has a well defined relativistic limit.The emerging picture closely resembles that found in the N = 4 super-symmetric gauge theory at strong coupling.An attempt to explore the effects of the (screened) source term in a 3D relativistic, ideal hydro code in progress.