Heavy quark energy loss in pQCD and SYM plasmas

1. 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

2. 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

3. Heavy quark energy loss in a weakly-coupled QCD plasma

4. 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 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

5. the accumulated transverse momentum picked up by a gluon of coherence time • 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 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 average pT picked up in each scattering only property of the medium needed mean free path

6. and • 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 Heavy quark energy loss • the case of infinite extend matter for heavy quarks, the radiated gluons which dominate the energy loss have this allows to express Qs in terms of T and E/M only  and and the heavy quark energy loss is

7. 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

8. Heavy quark energy loss in astrongly-coupled SYM plasma

9. 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 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

10. the AdS5 black-hole metric T = Hawking temperature of the black hole = temperature of the SYM plasma curvature radius of AdS5 fifth dimension horizon the SYM theory lives on the boundary at r = infinity quantum fluctuations in the SYM theory are mapped onto the 5th dimension 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

11. parameterization: equation of motion: rate at which energy flows down the string: A heavy quark in the plasma • 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: area of the string worldsheet induced metric on the worldsheet

12. 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 quarkthe part of string below is emitted radiation limiting velocity: the picture is valid for meaning 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)

13. 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 Gubser et al (2006),Chesler and Yaffe (2007) one gets for 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 but it gets the right v and T dependences then

14. 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 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

15. when stopping the acceleration, this separation goes down as : the heavy quark is building its wavefunction v 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 if , the result is as before if , then softer fluctuations dominate: for , only soft components contribute to the heavy quark  with The accelerating string Xiao (2008) the equation of motion at zero temperature: a can be interpreted as the acceleration of the quark 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

16. results for energy loss QCD at weak coupling SYM at strong coupling heavy-quark energy loss coherence time infinite matter or finite matter with Summary - 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

17. 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

18. DIS off the SYM plasma Y. Hatta, E. Iancu and A. Mueller, arXiv:0710.5297, JHEP0801 (2008) 063

19. properties of the current assume high energy high virtuality: coherence time of the current it probes plasma fluctuations with energy fraction 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

20. the energy density dominates for a given , all partons in the plasma have 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

21. Quarkonium dissociationin the SYM plasma H. Liu, K. Rajagopal and U.A. Wiedemann, hep-ph/0612168, JHEP0703 (2007) 066

22. for small L, there is string connecting the pair, hanging down in the fifth dimension quark-antiquark potential substraction of S0 so that at smallL obtained from implicit equation The quark-antiquark potential • the quark and antiquark live on a brane ateach hooked to the end of a string hanging down in the fifth dimension • the string dynamics is given by the Nambu-Goto action parameterization:

23. 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 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

24. 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