Heavy quark energy loss in
pQCD and SYM plasmas
Cyrille Marquet
Columbia University
based on F. Dominguez, C. Marquet, A. Mueller, B. Wu and B.-W. Xiao,
arXiv:0803.3234, Nucl. Phys. A811 (2008) 197
Outline
• Heavy quark energy loss in pQCD
medium induced gluon radiation and dead cone effect
the saturation scale of the pQCD plasma
• Heavy quark energy loss in SYM theory
the AdS/CFT correspondence
the trailing string picture
the saturation scale of the strongly coupled SYM plasma
• DIS off the SYM plasma
the structure functions and the saturation scale
• Quarkonium dissociation in the SYM plasma
the screening length and the saturation scale
Heavy quark energy loss in a
weakly-coupled QCD plasma
The heavy quark wave function
• consider a heavy quark of mass M and energy E
the heavy quark wave function at lowest order
the energy of the gluon is denoted
its transverse momentum is denoted
the virtuality of the fluctuations is measured by their lifetime or coherence time
short-lived fluctuations are highly virtual
the probability of this fluctuation is
Lorentz factor of the heavy quark
• the dead cone effect
compared to massless quarks, the fluctuation with
 absence of radiation in a forward cone
are suppressed
Medium induced gluon radiation
• multiple scattering of the radiated gluon
this is how the virtual gluon in the heavy quark wave function is put on shell
it becomes emitted radiation if it picks up enough transverse momentum
the accumulated transverse momentum picked up by a gluon of coherence time
average pT picked up
in each scattering
mean free path
only property of the medium needed
• the saturation scale of the pQCD plasma
only the fluctuations which pick up enough transverse momentum are freed

this discussion is also valid for light quarks
Heavy quark energy loss
• the case of infinite extend matter
for heavy quarks, the radiated gluons which dominate the energy loss have
and
this allows to express Qs in terms of T and E/M only

and
and the heavy quark energy loss is
• the case of finite extend matter of length
the relevant fluctuations in the wave function have a smaller energy
the maximum transverse momentum that gluons can pick-up is
the radiated gluons which dominate the energy loss have
Indications from RHIC data
• light-quark energy loss
comparisons between models and data indicate the need for
however, for a weakly-coupled pQCD plasma we expect
• heavy-quark energy loss
STAR, PRL 192301 (2007)
PHENIX, PRL 172301 (2007)
suppression similar to light
hadron suppression at high pT
Heavy quark energy loss in a
strongly-coupled SYM plasma
Motivations
• it is unclear if the perturbative QCD approach can describe the
suppression of high-pT particles in Au+Au collisions at RHIC, in
particular for heavy-quark energy loss:
high-pT electrons from c and b decays indicate similar suppression
for light and heavy quarks, while the dead-cone effect in pQCD
implies a weaker suppression for heavier quarks
 this motivates to think about a strongly-coupled plasma
• for the N=4 SYM theory, the AdS/CFT correspondence allows to
investigate the strong coupling regime
limited tools to address the QCD dynamics at strong coupling
 the results for SYM may provide insight on strongly-coupled
gauge theories, some aspects may be universal
in this work, we consider the trailing string picture of heavy-quark energy loss
by Herzog et al., and address the question of finite-extend matter
The AdS/CFT correspondence
•
the N=4 SYM theory: 1 gauge field, 4 fermions, 6 scalars, all adjoint
in the large Nc limit, the ‘t Hooft coupling λ controls the theory
strong coupling means ‘t Hooft limit in gauge theory:
•
the equivalent string theory in AdS5 x S5 : weak coupling and small curvature
classical gravity is a good approximation
•
the AdS5 black-hole metric
curvature radius of AdS5
fifth dimension
T = Hawking temperature of the black
hole = temperature of the SYM plasma
horizon
the SYM theory lives on the boundary at r = infinity
quantum fluctuations in the SYM theory are mapped onto the 5th dimension
A heavy quark in the plasma
•
a heavy quark lives on a brane at
with a string attached to it, hanging down to the horizon
•
points on the string can be identified to quantum fluctuations
in the quark wave function with virtuality ~ u
•
the string dynamics is given by the Nambu-Goto action:
area of the string worldsheet
•
parameterization:
equation of motion:
rate at which energy flows down the string:
induced metric on the worldsheet
The trailing string solution
assume the quark is being pulled at a constant velocity v:
solution (known as the trailing string) :
corresponding rate of energy flow down the string:
Herzog et al (2006)
Gubser et al (2006)
Liu et al (2006)
one has, similarly to the weak-coupling result:
this is naturally understood after this key observation:
the part of string above
is genuinely part of heavy quark
the part of string below
is emitted radiation
limiting velocity: the picture is valid for
meaning
Energy loss in the partonic picture
• simple derivation of the energy loss:
the radiated partons in the wavefunction have transverse momentum
and
energy
giving the maximum (dominant) values
and
and therefore a coherence time
this does not give the overall coefficient
then
but it gets the right v and T dependences
• this picture is obtained from several results
- the part of the string below Qs is not causally connected with the part of the string
above: Qs corresponds to a horizon in the rest frame of the string
- when computing the stress-tensor on the boundary:
the trailing string is a source of metric perturbations in the bulk which give
Gubser et al (2006), Chesler and Yaffe (2007)
one gets
for
the energy density is unchanged around the heavy quark up to distances ~ 1/Qs
The case of finite-extend matter
we would like to know the medium length L dependence of the energy loss
exact calculation difficult to set up, need another scale in the metric
using the partonic picture, we can get the L dependence
the heavy quark is bare when produced and then builds its wave function
while interacting with the medium, how to set this up in AdS ?
our proposal: describe the creation with a brief acceleration to the desired speed
then stopping the acceleration triggers the building of the wavefunction
key issue: the time it takes for the heavy quark to build the partonic
fluctuations which will be freed and control the energy loss
if the ones that dominate in the infinite matter case have time to build before
the heavy quark escapes the plasma, then the result is as before:
if not, the hardest fluctuations which could be build dominate, and one finds:
The accelerating string
the equation of motion at zero temperature:
a can be interpreted as the acceleration
of the quark
Xiao (2008)
solution
the acceleration acts like an effective temperature (Unruh effect): the part of string
below u =a is not causally connected with the part of the string above
at finite T, this separation is not affected, provided T << a
when stopping the acceleration, this separation goes down as
the heavy quark is building its wavefunction
v
a
:
when
the time it takes to build the
fluctuations which dominate the energy loss in the
infinite matter case), the separation crosses Qs, hence:
if
, the result is as before
if
, then softer fluctuations dominate:
for
, only soft components
to the heavy quark

contribute
with
Summary
results for energy loss
QCD at weak coupling SYM at strong coupling
heavy-quark energy loss
coherence time
infinite matter or
finite matter with
- same parametric form for the energy loss in pQCD and SYM at strong coupling !
- first estimate of the plasma length dependence of heavy quark energy loss
About pT broadening
results for pT broadening
-
again, similar to radiative pT broadening in pQCD
for infinite or finite length plasma
- one easily gets the infinite matter result
which is non trivial to get
with a direct calculation Gubser (2007), Solana and Teaney (2007)
- in the finite matter case,
(at weak-coupling:
)
- same parametric form for the pT broadening in pQCD and SYM at strong coupling !
- at strong coupling: no multiple scattering with local transfer of momentum
 no equivalent of
DIS off the SYM plasma
Y. Hatta, E. Iancu and A. Mueller, arXiv:0710.5297, JHEP 0801 (2008) 063
DIS off the SYM plasma
• the retarded current-current correlator
its imaginary part gives the plasma structure functions
R-current, equivalent of EM
current for SYM theory
the current-plasma interaction is described by the propagation
of a vector field which obeys Maxwell equations in AdS5
• properties of the current
assume high energy high virtuality:
coherence time of the current
it probes plasma fluctuations with energy fraction
The saturation scale
• a partonic picture
for
, the vector field is prevented to penetrate AdS space by a potential barrier
 structure functions exponentially small, no large-x partons
decreasing x at fixed Q2, the barrier disappears for
 structure functions saturated, all the partons at small x
• the saturation scale
consistent with what we found in the energy loss case
• the energy density
dominates
for a given
, all partons in the plasma have
Quarkonium dissociation
in the SYM plasma
H. Liu, K. Rajagopal and U.A. Wiedemann, hep-ph/0612168, JHEP 0703 (2007) 066
The quark-antiquark potential
•
the quark and antiquark live on a brane at
each hooked to the end of a string hanging down in the fifth dimension
•
the string dynamics is given by the Nambu-Goto action
parameterization:
for small L, there is string connecting the
pair, hanging down in the fifth dimension
quark-antiquark potential
substraction of S0 so that
at small L
obtained from implicit equation
The screening length
for large L, there is no solution
, and the minimum of the
action is obtained with two strings hanging down to the horizon
the quark and antiquark are screened from each other when the string breaks
the transition between the two regimes defines the screening length
one finds
for
consistent with what we found in the energy loss case
up a to a distance ~ 1/Qs away from the quark, the plasma is not felt
in fact before the string breaks, it doesn’t tilt in the direction of motion of the pair
Conclusions
• same parametric form for the heavy quark energy loss and pT
broadening when written in terms of the saturation scale Qs
• only the saturation scale differs between pQCD and SYM
theories
• the plasma length L dependence is stronger in SYM compared
to pQCD, for both the energy loss and pT broadening
• Qs appears in other calculations, deep inelastic scattering and
quarkonium dissociation