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: , 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 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 effect compared 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 p T 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 p T PHENIX, PRL (2007) STAR, PRL (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 a strongly-coupled SYM plasma

Motivations it is unclear if the perturbative QCD approach can describe the suppression of high-p T particles in Au+Au collisions at RHIC, in particular for heavy-quark energy loss: high-p T 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 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 AdS 5 x S 5 : weak coupling and small curvature fifth dimension curvature radius of AdS 5 T = Hawking temperature of the black hole = temperature of the SYM plasma the SYM theory lives on the boundary at r = infinity horizon the AdS 5 black-hole metric quantum fluctuations in the SYM theory are mapped onto the 5th dimension

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: 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 solution assume 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 quark the 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 for Gubser 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: then this does not give the overall coefficient but it gets the right v and T dependences

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 ? describe the creation with a brief acceleration to the desired speed then stopping the acceleration triggers the building of the wavefunction our 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 contribute to the heavy quark 

Summary infinite matter or finite matter with QCD at weak couplingSYM 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 p T broadening - same parametric form for the p T broadening in pQCD and SYM at strong coupling ! - in the finite matter case, (at weak-coupling: ) Gubser (2007), Solana and Teaney (2007) results for p T broadening - at strong coupling: no multiple scattering with local transfer of momentum  no equivalent of - again, similar to radiative p T broadening in pQCD for infinite or finite length plasma

DIS off the SYM plasma Y. Hatta, E. Iancu and A. Mueller, arXiv: , 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 AdS 5 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 Q 2, 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 dissociation in the SYM plasma H. Liu, K. Rajagopal and U.A. Wiedemann, hep-ph/ , 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 S 0 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 p T 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 p T broadening Qs appears in other calculations, deep inelastic scattering and quarkonium dissociation