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 pQCDmedium induced gluon radiation and dead cone effectthe saturation scale of the pQCD plasma

• Heavy quark energy loss in SYM theorythe AdS/CFT correspondencethe trailing string picturethe saturation scale of the strongly coupled SYM plasma

• DIS off the SYM plasmathe structure functions and the saturation scale

• Quarkonium dissociation in the SYM plasmathe screening length and the saturation scale

Heavy quark energy loss in a weakly-coupled QCD plasma

The heavy quark wave function

its transverse momentum is denoted

• 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

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 effectcompared to massless quarks, the fluctuation with are suppressed

absence of radiation in a forward cone

Medium induced gluon radiation

only property of the medium needed

• 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

mean free path

average pT picked up

in each scattering

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

Heavy quark energy loss

for heavy quarks, the radiated gluons which dominate the energy loss have

• the case of infinite extend matter

and

this allows to express Qs in terms of T and E/M only

and

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

and the heavy quark energy loss is

Indications from RHIC data

suppression similar to light

hadron suppression at high pT

PHENIX, PRL 172301 (2007)PHENIX, PRL 172301 (2007)

STAR, PRL 192301 (2007)STAR, PRL 192301 (2007)

• light-quark energy loss

• heavy-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 in astrongly-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 lossby Herzog et al., and address the question of finite-extend matter

The AdS/CFT correspondence

strong coupling means ‘t Hooft limit in gauge theory:

• 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

classical gravity is a good approximation

• the equivalent string theory in AdS5 x S5 : weak coupling and small curvature

fifth dimension

curvature radius of AdS5

T = Hawking temperature of the black

hole = temperature of the SYM plasma

the SYM theory lives on the boundary at r = infinity

horizon

• the AdS5 black-hole metric

quantum fluctuations in the SYM theory are mapped onto the 5th dimension

• a heavy quark lives on a brane atwith a string attached to it, hanging down to the horizon

• points on the string can be identified to quantum fluctuationsin the quark wave function with virtuality ~ u

• the string dynamics is given by the Nambu-Goto action:

A heavy quark in the plasma

induced metric on the worldsheetarea of the string worldsheet

equation of motion:

rate at which energy flows down the string:

• parameterization:

The trailing string solutionassume the quark is being pulled at a constant velocity v:

solution (known as the trailing string) :

Herzog et al (2006)Gubser et al (2006)Liu et al (2006)

corresponding rate of energy flow down the string:

this is naturally understood after this key observation:the part of string above is genuinely part of heavy quarkthe part of string below is emitted radiation

limiting velocity: the picture is valid for meaning

one has, similarly to the weak-coupling result:

Energy loss in the partonic picture

• 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

the energy density is unchanged around the heavy quark up to distances ~ 1/Qs

one gets forGubser et al (2006), Chesler and Yaffe (2007)

the radiated partons in the wavefunction have transverse momentum and energy

giving the maximum (dominant) values and

and therefore a coherence time

• simple derivation of the energy loss:

thenthis does not give the overall coefficient

but it gets the right v and T dependences

The case of finite-extend matterwe 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 ?

describe the creation with a brief acceleration to the desired speed

then stopping the acceleration triggers the building of the wavefunctionour proposal:

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

a can be interpreted as the acceleration

of the quark solution

the equation of motion at zero temperature:

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

Xiao (2008)

when stopping the acceleration, this separation goes down as :the heavy quark is building its wavefunction

when the time it takes to build the fluctuations which dominate the energy loss in the infinite matter case), the separation crosses Qs, hence:

a

v

if , the result is as before

if , then softer fluctuations dominate:

with

for , only soft components contributeto the heavy quark

Summary

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

- one easily gets the infinite matter result which is non trivial to get with a direct calculation

About pT broadening

- same parametric form for the pT broadening in pQCD and SYM at strong coupling !

- in the finite matter case, (at weak-coupling: )

Gubser (2007), Solana and Teaney (2007)

results for pT broadening

- at strong coupling: no multiple scattering with local transfer of momentum no equivalent of

- again, similar to radiative pT broadening in pQCD

for infinite or finite length plasma

DIS off the SYM plasmaY. 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

the current-plasma interaction is described by the propagation

of a vector field which obeys Maxwell equations in AdS5

R-current, equivalent of EM

current for SYM theory

• properties of the current

it probes plasma fluctuations with energy fraction

assume high energy high virtuality:

coherence time of the current

The saturation scale

structure functions exponentially small, no large-x partons

for , the vector field is prevented to penetrate AdS space by a potential barrier

decreasing x at fixed Q2, the barrier disappears for

structure functions saturated, all the partons at small x

• a partonic picture

• 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 dissociationin 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

the transition between the two regimes defines the screening length

the quark and antiquark are screened from each other when the string breaks

for large L, there is no solution , and the minimum of the

action is obtained with two strings hanging down to the horizon

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

forone finds

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