1. Multiple parton interactions in heavy-ion collisions
Cyrille Marquet
Columbia University

2. High-energy heavy-ion collisions
relativistic heavy ion collisions
 multiple partonic interactions (MPI)
what can be described with QCD at weak coupling (1st principle calculation)?
maybe the early stages of the collision
space-time picture of heavy-ion collisions
5. Individual hadrons
freeze out
4. Hadron gas
cooling with expansion
3. Quark Gluon Plasma
thermalization, expansion
2. Pre-equilibrium state
collision
1. Nuclei (initial condition)
 Glasma
 CGC
MPI at weak coupling
 the saturation scale

3. Outline
The saturation scale in the Color Glass Condensate the nuclear wave function at small-x the saturation phenomenon
The saturation scale in the Glasma multiple partonic interactions particle production in AA collisions
The saturation scale in the quark-gluon plasma medium-induced energy loss and pT broadening weakly-coupled vs strongly-coupled plasma

4. The saturation scale in the nuclear wave function
Color Glass
Condensate

5. The hadron wave function in QCD
one can distinguish three regimes
S (kT ) << 1
perturbative regime,
dilute system of partons:
hard QCD (leading-twist
approximation)
weakly-coupled regime,
dense system of partons (gluons)
non linear QCD
the saturation regime
non-perturbative
regime: soft QCD
relevant for instance for
the total cross-section in
hadron-hadron collisions
relevant for instance for
top quark production
not relevant to experiments
until the mid 90’s
with HERA and RHIC: recent gain of interest for saturation physics

6. The dilute regime hadron = a dilute system of partons
transverse view of the hadron
hadron = a dilute system of partons
1/kT ~ parton transverse size
evolution: as kT increases,
the hadron gets more dilute
leading-twist regime
Dokshitzer Gribov
Lipatov Altarelli Parisi
the partons interact incoherently
for instance, the total cross-section in DIS
partonic cross-section
parton density
higher twist
not valid if x is too small when the hadron becomes a dense system of partons

7. The saturation regime the saturation scale Qs(x)
the separation between the dilute and dense
regimes is caracterized by a momentum scale:
the saturation scale Qs(x)
in the saturation regime,
higher-twists are important:
Balitsky Fadin Kuraev Lipatov
hadron = a dense system of partons
the saturation regime of QCD: - non-linear yet weakly-coupled - describes the collective behavior of partons in the nuclear wave function
evolution: as x decreases,
the hadron gets more dense
in the saturation regime,
the evolution becomes non linear
the partons interact coherently

8. The saturation scale and MPI
gluon recombination in the hadronic wave function
gluon density per unit area
it grows with decreasing x
recombination cross-section
recombinations important when
gluon kinematics
the saturation regime: for with
occupation numbers are large
multiple scattering
parton saturation is as important as MPI, to be consistent, both should be included
so far I only discussed the nuclear
wave function, multiple scatterings
occur during the collision, when

9. The Color Glass Condensate
the idea in the CGC is to take into account saturation via strong classical fields
the CGC: an effective theory to describe the saturation regime
McLerran and Venugopalan (1994)
lifetime of the fluctuations
in the wave function ~
high-x partons ≡ static sources
low-x partons ≡ dynamical fields  
the solution gives
small x gluons
as radiation field
valence partons
as static random
color source
separation between
the long-lived high-x partons
and the short-lived low-x gluons
CGC wave function
the evolution of with x is a
renormalization group equation
classical Yang-Mills equations
Jalilian-Marian, Iancu, McLerran,
Weigert, Leonidov, Kovner ( )

10. The saturation scale in the Glasma

11. Collision of two CGCs
the initial condition for the time evolution in heavy-ion collisions
before the collision:
the distributions of ρ contain the small-x
evolution of the nuclear wave function r1 r2
after the collision
compute the gluon field in the forward light-cone
express cross-sections in terms of the field
the fields decay, once they are not strong (classical)
anymore, a particle description is again appropriate
this is where multiple partonic
interactions should be included

12. Particle production in the Glasma
single scattering
factorization theorems in
pQCD answer this question
multiple scatterings
this is what happens in heavy-ion collisions
one needs to reformulate quantum field theories to address the problem
Gelis, Lappi and Venugopalan (2006-now)
one needs to understand how strong color fields decay into particles
the time scale is
possible application to pp collisions: a first principle calculation of the underlying event

13. Total multiplicity in AA
predictions including NLO evolution
Albacete (2007)
the extrapolation from RHIC to LHC
is driven by the small-x evolution
yellow band: uncertainties due to the
starting point of the small-x evolution
and the initial saturation scale
caveat: kT factorization is assumed,
meaning MPI are not treated properly
the focus was the correct NLO evolution,
however numbers are similar with an exact
numerical treatment of MPI (but an
approximate treatment of the evolution)
Lappi (2008)
day-1 measurement
~ 1400 charged particles is a robust prediction,
if one gets <1000 or >2500, this rules out the CGC

14. The saturation scale in the QCD plasma
Quark-Gluon Plasma

15. 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 are suppressed
 absence of radiation in a forward cone
Dokshitzer and Kharzeev (2001)

16. Medium induced gluon radiation
Baier, Dokshitzer, Mueller,
Peigne, Schiff (1997)
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
only property of the medium needed
mean free path
only the fluctuations which pick up enough transverse momentum are freed
this discussion is also valid for light quarks
the saturation scale of the pQCD plasma 

17. 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 relevant fluctuations in the wave function have a smaller energy
the case of finite extend matter of length
the maximum transverse momentum that gluons can pick-up is
the radiated gluons which dominate the energy loss have

18. 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 (2007)
PHENIX, PRL (2007)
suppression similar to light
hadron suppression at high pT

19. Energy-loss at strong coupling?
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
main result: one can also identify a momentum scale which
controls what fluctuations become emitted radiation

20. Energy loss and Qs results for energy loss infinite matter or
Dominguez, C.M., Mueller, Wu and Xiao (2008)
infinite matter or
finite matter with
QCD at weak coupling
SYM at strong coupling
heavy-quark energy loss
results for energy loss
coherence time
- 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

21. Conclusions
MPI in heavy-ion collisions - are not only important but crucial (HIC probe non-linear QCD) - are at the origin of the interesting phenomena
MPI at weak coupling - are characterized by the saturation scale Qs - at RHIC, Qs may be hard enough to justify using weak coupling - CGC & Glasma : first-principle description of the early stages - parton saturation & MPI equally important
MPI in the quark-gluon plasma - are responsible for the energy loss and pT-broadening of hard probes - the corresponding saturation scale is related to - at strong coupling, MPI can be also described in terms of a saturation scale