1. Heavy Quark Potential at Finite-T in AdS/CFT Yuri Kovchegov The Ohio State University work done with J. Albacete and A. Taliotis, arXiv:0807.4747 [hep-th]

2. Outline AdS/CFT techniques Heavy quark potential at finite-T at strong coupling for N =4 SYM Heavy quark potential at weak coupling at finite-T: and what we really know there Conclusions

3. AdS/CFT techniques

4. AdS/CFT Approach z z=0 Our 4d world 5d (super) gravity lives here in the AdS space AdS 5 space – a 5-dim space with a cosmological constant  = -6/L 2. (L is the radius of the AdS space.) 5 th dimension

5. AdS/CFT Correspondence (Gauge-Gravity Duality) Large-N c, large  g 2 N c N=4 SYM theory in our 4 space-time dimensions Weakly coupled supergravity in 5d anti-de Sitter space!  Can solve Einstein equations of supergravity in 5d to learn about energy-momentum tensor in our 4d world in the limit of strong coupling!  Can calculate Wilson loops by extremizing string configurations.  Can calculate e.v.’s of operators, correlators, etc.

6. Calculating Wilson loops using AdS/CFT correspondence Our 4d world Wilson loop String stretching into the 5 th dimension of AdS 5. To calculate a Wilson loop, need to i.find the extremal (classical) string configuration ii.find the corresponding Nambu-Goto action S NG iii.then the expectation value of the Wilson loop is given by Maldacena, ‘98 z

7. Energy-momentum tensor is dual to the metric in AdS. Using Fefferman-Graham coordinates one can write the metric as with z the 5 th dimension variable and the 4d metric.  Expand near the boundary of the AdS space:  For Minkowski world and with Holographic renormalization de Haro, Skenderis, Solodukhin ‘00

8. Finite-T medium in AdS Finite-T static medium in AdS is represented by a black hole solution: with z h =1/ (  T) the black hole horizon, T – the temperature of gauge theory. z black hole horizon zhzh 0our world

9. Finite-T medium in AdS The black hole metric can be recast into Fefferman-Graham coordinates: Here Using holographic renormalization prescription we read off the energy-momentum tensor of the thermal medium:

10. Heavy Quark Potential at Finite-T at Strong Coupling

11. Heavy Quark Potential At finite temperature the heavy quark potential is defined using a correlator of two Polyakov loops (in Euclidean time formalism) Quark and anti-quark can be in the singlet or adjoint (octet) color state. Hence there are two potentials: We want to find the singlet potential V 1 (r). singletadjoint (octet)

12. N C Counting for Potentials At LO the Wilson loop correlator is zero: One gluon exchange = o(beta) term. Equating that to zero we obtain: The adjoint gluon potential is repulsive and N C -suppressed. Wilson loop Wilson loop =0

13. Heavy Quark Potential Using the rules of AdS/CFT to find the correlator of Polyakov loops we connect the strings in all possible ways, obtaining the following configurations: Each configuration is a saddle point in the integral over string coordinates. Summing over the saddle point contributions yields: black hole horizon connect to N c D3-branes Bak, Karch, Yaffe ‘07

14. N C Counting in AdS Space There are N C 2 configurations of two straight strings, as each of them can connect to one of the N C different branes: In black hole metric strings still “remember” which D3 they connect to through Chan-Paton indices. N C D3-branes horizon

15. The Potentials Compare the two formulas: We conclude that the hanging string configuration yields a singlet potential V 1 (r). The two straight strings configuration gives the repulsive adjoint potential V adj (r). As V adj (r)=o(1/N C 2 ) one should expect S straight ren =0 at LO in N C : this is exactly what happens! field th’y AdS

16. Calculating Singlet Potential Our 4d world Polyakov loops String stretching into the 5 th dimension of AdS 5. To calculate singlet potential V 1 (r), need to i.find the extremal (classical) hanging string configuration ii.find the corresponding Nambu-Goto action S NG iii.then the potential will be given by z r r    0 Euclidean time

17. Hanging String Configuration We extremize the Nambu-Goto action to obtain the classical string configuration: We label the string by its maximum extent in the 5 th dimension z max. The solution for the string is (Rey et al, hep-th/9803135; Branhuber et al, hep-th/9803137): Maximum extent z max is determined by the following equation:

18. Complex saddle point There are several solutions for z max, most of them complex! Two more relevant complex roots for z max are shown here. The solutions become complex starting from some minimum separation r=r c.

19. Complex saddle point In picking the correct solution/string configuration we demand that the resulting potential is physical. We demand that:  V 1 (r) becomes Coulomb-like as r→0. In fact it shout map onto T=0 potential calculated by Maldacena in ’98.  As we demand that Im[ V 1 (r) ] < 0. This leaves us with the root denoted by the solid line in the plots. What to do when string coordinates become complex? Let’s analytically continue into complex domain, similar things are done in the “method of complex trajectories” in quasi-classical QM.

20. Status of the field before this work Before us people did not identify the two string configurations as corresponding to two different potentials – singlet and adjoint. People would also stop the calculation if string coordinates became complex. In fact they stop the calculation even earlier, when the energy of the hanging string configuration becomes zero (equal to the energy of the two straight strings configuration). The resulting heavy quark potential was (Rey et al, hep-th/9803135; Branhuber et al, hep-th/9803137): Dashed line – their expectation of what the answer should look like; solid line – their answer. kink? zero potential? =r =V(r)

21. UV regularization and the answer If one substitutes the string coordinates into the Nambu-Goto action to find the classical NG action, one gets UV divergences due to infinite masses of the quarks. To renormalize the theory one has to subtract those divergences out. This procedure is defined up to a r-independent constant which may move the resulting V(r) up or down by an overall constant. We will use the following prescription: The answer is:

22. Heavy Quark Potential: Real Part N=4 SYM theory is not confining. At T=0 one only gets Coulomb-like (~ -1/r) potential. At finite-T we get: at small distances scales as Coulomb potential ~ -1/r at large distances it falls off as a power law, proportional to 1/r 4 ! No Debye-type screening!

23. Asymptotics of Re part By exploring the large-r and small-r limits of the full result we get the following asymptotics:  At small-r we recover Maldacena’s ’98 vacuum result:  At large-r we get an unexpected power-law falloff, instead of usually expected exponential falloff:  Does not contradict any fundamental principles (more later).

24. Heavy Quark Potential: Real Part The (real part of the) potential is well-approximated by with the “screening length”

25. Heavy Quark Potential: Imaginary Part The potential becomes absorptive at large separations: it develops an imaginary part. This is due to color singlet state easily decaying into a color-octet (adjoint) configuration. at large distances scales ~r cf. perturbative QCD calculations by M. Laine et al, hep-ph/0611300; N. Brambilla et al, arXiv:0804.0993.

26. Asymptotics of Im part At large-r the Im part of the potential is Makes sense: the wider part the quarks are the more likely the singlet state to decay into an octet (adjoint) state. Due to Im part the color-singlet state is a metastable state which will of course decay into an octet state with a high probability. One can think of Im part of V(r) as meson decay width in a thermal medium.

27. Adjoint potential To find the adjoint potential consider two straight strings: The LO part is r-independent and after renormalization gives V adj (r)=0, in agreement with expected 1/N C 2 suppression. NLO graviton exchange corrections are o(1/N C 2 ) and can be calculated (see Bak, Karch,Yaffe ‘07) giving V adj (r). graviton

28. Heavy Quark Potential at Finite-T at Weak Coupling

29. Perturbative Potential The standard perturbative calculation yields: Usually people approximate (exact for Abelian theories) and get

30. Perturbative Potential However, in non-Abelian theories the static self-energy is not a constant: Keeping the linear term in self-energy yields (C. Gale, J. Kapusta ‘87): Note that one gets the same power as at strong coupling in AdS!

31. Perturbative Potential A word of caution: the linear term becomes important when it is comparable to q 2 : This happens at and hence We are dealing with ultrasoft (magnetic) modes, which are non- perturbative. Hence we should not trust either exponential and power-law screening. It is curious though that is we take perturbative expression “as is” we get a power-law falloff.

32. Conclusions We have identified the hanging string and two straight strings configurations with the singlet and adjoint heavy quark potentials. We have used the complex-string worldsheet technique to find the singlet heavy quark potential in the strongly coupled SYM theory at finite temperature. The resulting potential is smooth, has no kinks, with a negative- definite Re part, and has a non-zero Im part giving the decay width for the meson states. The Re part of the potential at large separations falls off as 1/r 4 : an interesting alternative to exponential Debye screening. Apparently there is no consensus on what the screened potential is at weak coupling.