Heavy Quark Energy Loss due to
Three-body Scattering in a QuarkGluon Plasma
Wei Liu
Texas A&M University

Introduction

Heavy quark scattering in QGP

Heavy quark drag coefficients

Heavy quark momentum spectra

Nuclear modification factor for electron

Summary and discussions
Original work was done in collaboration with C. M. Ko.
QM2006, shanghai, china
Jet quenching
1. Light jet: light quark or gluon
I, vitev, hep-ph/0511237
Energy loss by radiation when
passing through the hot dense
medium
ΔE ~ q̂L2 , q̂  1 ~ 3 GeV/fm 2
WG model: static color scattering
center, fit data at dN/dy=1000
2. Heavy quark jet
S. Wicks et al, nucl-th/0512076
WG models fails by including
only radiative contribution
2
Dead cone:
m
 θ02 
dP  dP0 1  2  with θ0  Q
Ec
 θ 
Possible solutions?
a. Non-perturbative process (resonance)
b. First order radiation (dN/dy=3500)
c. Elastic plus radiation (dN/dy=1000)
Our approach
T  300 MeV, N g  N q  N q   5 / fm 3
 distance between a pair of partons L  0.3 fm
 L slightly smaller th an 1/m D  1 / gT  1 /( 600 MeV)  0.33 fm
Multiple partonic collisions may be
important !
Three-quark and gluon elastic scattering are
found to be important for quark and gluon
thermalization in the initial stage
X. M. Xu et al. Nucl.
Phys. A 744, 347
(2004); X. M. Xu et al.
Phys. Lett. B 629, 68
(2005)
Fokker-Planck equation: studying heavy quark momentum degradation
Lowest order QCD approach: Binary elastic + radiative + three-body elastic
2 - body elas : Qq  Qq, Qq  Qq, Qg  Qg,
radiative : Qq  Qqg, Qq  Qqg, Qg  Qgg,
3 - body elas : Qqq  Qqq, Qqq  Qqq, Qq q  Qq q,
Qqg  Qqg,Qqg  Qqg, Qgg  Qgg.
Fokker-Planck equation

f (p, t )
 


Bij (p) f ( p, t )
 Ai ( p) 
t
pi 
p j

Ai (p)  pi γ( p )

d pT
dt
γ( p )  1 
  γ(p T , T)p T
p  p'
p
2
...   ... f (p, t )dp
  
22
1 1
X ( p j1 ) 
2 Ei1  i1
 
23
 
32
gik d 3 pik
d 3 p jl
3
3
    33
  2  2 E   2  2 E  f ( p ) 1  f ( p
k  2 ,,m
ik l 1,,n
jl k  2 ,,m
2


 M i1im  j1 jn (2 )4  ( 4 )   pik   p jl  X ( p j1 )
l 1,,n
 k 1,,m

ik
l  2 ,,n
jl
)
Processes calculated exactly
1. 2-body elastic
2. Radiative
Qq  Qq, Qq  Qq, Qg  Qg
screening mass mD  gT
thermal mass mg  3mq 
coupling α s  g 2 /4π  0.3
mD
2
Qq  Qqg, Qq  Qqg, Qg  Qgg
3. 3-body elastic collisions
1. Qqq  Qqq, Qqq  Qqq, Qqq  Qq q
with different flavor (7 diagrams)
true three-body process
(p  m)/(p 2  m 2 )  Re[( p  m)/(p 2  m 2  ip 0 Γ)]
Γ : collision width
Processes calculated approximately
2. Qqq  Qqq, Qq q  Qq q with same flavor
6 extra diagrams are obtained by interchanging final two light quarks, and give same
contribution as that due to direct diagrams. Interference terms are found to be two
order of magnitude smaller and neglected.
3. Qqq  Qqq with same flavor
5 extra diagrams are obtained by exchanging a gluon between heavy quark, and
light quark, antiquark, or virtual gluon from quark and antiquark annihilation. The
contribution is also two order of magnitude smaller than that due to direct diagrams.
4. Qqg  Qqg and Qqg  Qqg
36 diagrams are obtained by attaching an extra gluon to all parton
lines and three-gluon vertex in Qq→Qqg. Only six diagrams with two
gluons attached to both heavy quark and light partons are evaluated.
5. Qgg  Qgg
123 diagrams are obtained from Qg→Qgg by attaching an extra
gluon. Again, only six diagrams with two gluons attached to both
heavy quark and light partons are calculated.
Collision width of heavy and light partons
Q ,q /   i M i
M i1im  j1 jn


1
2Ei1
2
charm
2
g i k d 3pi k
d 3p jl
3
3
  2π  2E   2π  2E
k  2,,m
i k l 1,,n
 f(p )  1  f(p
k  2,,m
ik
l  2,,n
jl
) M i1im  j1 jn
jl
2


 (2π24 δ(4)   pi k   p jl 
l 1,,n
 k 1,,m

 Widths are mainly due to 2-body
elastic scattering.
 Width of gluon is about twice of
that of light quark.
 Width of bottom quark is two
thirds of that of charm quark.
Light quark
Drag coefficient
M. Djordjevic and M. Gyulassy,
Nucl. Phys. A733 265 (2004)
At high pT, radiation dominates for
charm; contributions from threebody elastic collisions is 80% of
those from two-body elastic
collisions.
QGP fireball dynamics
Vτ   πR(τ) 2 τ
where R τ   R 0  a2 (τ  τ 0 )2
R 0  7 fm, τ 0  0.6 fm, a  0.1c 2 /fm
Initial temp Ti  350 MeV
This model gives a total
transverse energy comparable
to that measured in
experiments, and the time
dependence of temperature is
obtained from entropy
conservation.
Critical temp Tc  175 MeV at  c  5 fm
Initial heavy quark spectra

 
pT 2
6
dN c
dN
19.2 1 
 N coll 2 
2
d pT
d pT 1  pT /3.7 12 1  exp(0.9  2pT )
pp
c

dN b
dN pp
b

N
 0.0025 1 
coll 2
2
d pT
d pT
  exp 
pT 5
16
N coll  960 in Au  Au @ sNN  200 GeV
pT
1.495

Charm quarks: spectrum determined from fitting simultaneously measured
transverse spectrum of charm mesons from d+Au collisions and of
electrons from heavy meson decays in p+p collisions.
Bottom quarks: spectrum taken from the pQCD prediction.
Heavy quark spectrum and electron spectrum
R 02  R 2
number of collisions 
R 02
Petersen fragmentat ion
1
D( z ) 
z[1  1 / z   /(1  z )]2
 D  0.02;  B  0.002
R0
Electron RAA for charm and bottom
Au  Au @ s NN  200 GeV
Charm: Radiation dominates at high transverse momentum.
Bottom: 3-body elastic scattering is comparable to 2-body
and radiative scattering.
Combination of contributions from charm and bottom are still above
experimental data.
Strongly coupled QGP ?
O. Kaczmarek, F. Karsch, and F. Zantow, Phys. Rev. D 70, 074505
(2004); O. Kaczmarek, F. Zantow,Phys. Rev. D 71, 114510 (2005)
From lattice calculation
αs (T)  g2 (T)/4 π  2.1αpert (T)
Using two loop pQCD running coupling
 
ln 2 ln  
-2
gpert
(T )  811
ln
2

51
88 2
2T
 MS
with Tc /  MS  1.14.
2T
 MS
Screening mass is given by
mD
T

 A 1
Nf
6
g
1
2
with A  1.417
pert(T)
Summary and discussions
 We have calculated the drag coefficient for heavy
quark in quark gluon plasma, and found that threebody elastic collision is important for the heavy
quark momentum degradation.
 More reliable calculation is needed for the most
important process Qqg→Qqg (36 Feynman
diagrams).
 New experimental data seem to favor a strongly
coupled quark gluon plasma.
 Multiple partonic processes involving more than 6
partons need to be considerd (a theoretical
challenge).