Mach Cones in a Perturbative Quark-Gluon Plasma
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Transcript of Mach Cones in a Perturbative Quark-Gluon Plasma
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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
2
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.