Baryon Chemical Potential in AdS/CFT

1. Baryon Chemical Potential in AdS/CFT Shin Nakamura 中村　真 Hanyang Univ. and CQUeST （韓国・漢陽大学) Ref. S.N.-Seo-Sin-Yogendran, hep-th/0611021 ( Kobayashi-Mateos-Matsuura-Myers, hep-th/0611099)

2. Purpose of this talk • I would like to present an overview of AdS/CFT. (Incomplete, but “intuitive” hopefully.) • I will report the present status on construction of finite-density AdS/CFT. (What we know and what we do not know.)

3. Motivation Hadron physics is very interesting research area both theoretically and experimentally. • RHIC, LHC • Nuetron (quark) stars We encounter strongly coupled systems. We need theoretical frameworks which enable us to analyze strongly coupled QCD. • Effective theories, Lattice QCD,… • AdS/CFT

4. = conjecture AdS/CFT (Original, weak version) Classical Supergravity on 10 dim. Maldacena ‘97 4dim. Large-Nc SU(Nc) N=4 Super Yang-Mills at the large ‘t Hooft coupling Strongly interactingquantum YM !!

5. What is AdS/CFT? Analogy: Euclidean theory A: Ф=0 “trivial” vacuum B: Ф=ФB“non-trivial” vacuum m2 -m2 A 2 solutions: B

6. Physics around the “non-trivial” vacuum 2 equivalent methods: • Perturbation theory around the • “non-trivial” vacuum. dynamical = 2. Perturbation theory around the “trivial” vacuum (with source). source term dynamical

7. + + +….. Propagator around the non-trivial vacuum method 1: (around non-trivial) method 2: (around trivial) consistency = (Comment after the seminar: we have to understand more about this.)

8. What we have learned Same physics can be described in two different ways: 1. non-trivial vacuum, without source Single Feynmann diagram = 2. trivial vacuum, withsource • Re-summation of infinitely many diagrams • The source carries non-perturbative information

9. Let us do the same thing in string theory Type IIB Superstring Theory Defined in 10d spacetime Theory of closed strings (perturbatively) Low energy: 10d type IIB supergravity Many different vacua. Two of them: 1. A curved spacetime: black 3-brane solution “non-trivial” Asymptotically flat Extremal black hole “trivial” 2. Flat spacetime “Source for closed strings”: D3-brane 3+1 dim. hypersurface, gauge theory on it

10. Superstring theory around black 3-brane geometry asymptotically flat Black hole (3+1 dim. object) = ? The near horizon limit : We do not want here. U(Nc) 3+1 dim N=4 Super YM theory at low energy on the D3-branes Superstring theory around flat geometry + source (Nc D3-brane) SU(Nc)

11. = conjecture AdS/CFT (Original, weak version) Classical Supergravity on 10 dim. Maldacena ‘97 4dim. Large-Nc SU(Nc) N=4 Super Yang-Mills at the large ‘t Hooft coupling Strongly interactingquantum YM !!

12. What we have learned Same physics can be described in two different ways: 1. non-trivial vacuum, without source Single Feynmann diagram = 2. trivial vacuum, withsource • Re-summation of infinitely many diagrams • The source carries non-perturbative information

13. Construction of gauge/gravity duality • Construct a D-brane configuration on which the gague theory you want is realized. • Find the supergravity solution which corresponds to the D-brane configuration. (Here, we have a curved spacetie, but no D-brane.) • Take near-horizon limit to make the unwanted modes (like gravity in the YM side) decoupled. • Take appropriate limits to make the supergravity approximation valid, if necessary.

14. Introduction of quark/antiquarks string The end of the string is a quark or antiquark. D3-brane 3+1 dim. The quark-antiquark pair is a single string coming from the boundary of AdS. AdS5

15. gravity dual AdS5 + flavor branes Nf D7 meson AdS5 Introduction of dynamical quarks flavor brane Nf D7 mq quark Nc D3

16. Finite temperature Finite baryon-number density (chemical potential) AdS/CFT and statistical mechanics AdS/CFT : a useful tool for analysis of stronglycoupled YM theories. We need to describe systems with finite temperature and finite density. Established Yet to be completed

17. = conjecture AdS/CFT at finite temperature Classical Supergravity on AdS-BH×S5 Hawking temp. Witten ‘98 4dim. Large-Nc strongly coupled SU(Nc) N=4 SYM at finite temperature (in the deconfinement phase).

18. “confinement” phase “de-confinement” phase Thermal AdS AdS-BH Hawking-Page transition Transition of bulk geometryat the same β(=1/T). Transition related to quark condensate Transition of flavor-brane configuration, on a common branch of bulk geometry Phase transitions

19. flavor brane Nf D7 mq quark Nc D3 gravity dual Minkowski branch Black-hole branch D7 AdS-BH 1st order horizon T<Tc Tc<T Phase transition related to quarks

20. Brane configurations y Minkowski branch D7 y0 yH y0 Black-hole branch BH ρ

21. How to introduce finite density(or chemical potential)? • Kim-Sin-Zahed, 2006/8 • Horigome-Tanii, 2006/8 • S.N.-Seo-Sin-Yogendran, 2006/11 • Kobayashi-Mateos-Matsuura-Myers-Thomson, 2006/11

22. The system we consider: D3-D7 system • YM theory: N=2 large-Nc SYM with quarks • Flavor branes: Nf D7-branes • Flavor symmetry: U(Nf) • Quarks are massive (in general): mq • Probe approximation (Nc>>Nf) • Free energy～Flavor-brane action No back reaction to the bulk gometry from the flavor branes. (～quenched approx.)

23. AdS/CFT at finite R-charge chemical potential SO(6) on the S5 R-symmetry: 10 dim. angular momentum on the S5 R-charge: From the AdS5 point of view electric charge of the BH Electric potentialA0at the boundary is interpreted as a chemical potential Chamblin-Emparan-Johnson-Myers,1999 Cvetic-Gubser,1999

24. First law in charged black hole Entropy from the area of the horizon Mass Charge Hawking temperature Electromagnetic potential plays as a chemical potential

25. We need flavor branes（D8,D7,….) • U(1)B symmetry: Local (gauge) symmetry on the flavor branes U(1)B charge: “electric charge” for the U(1) gauge field on the flavor brane How about finite baryon-number density? D4-D8-D8 case A0 on the flavor brane at the boudary ? U(1)B chemical potential? Kim-Sin-Zahed,2006/8; Horigome-Tanii,2006/8

26. E How about gauge invariance? S.N.-Seo-Sin-Yogendran,2006/11 Kobayashi-Mateos-Matsuura- Myers-Thomson,2006/11 We should use boundary D7 ρ AdS-BH ρ A “physical” ? meaning: a work necessary to bring a single quark charge from the boundary to ρmin against the electric field.

27. More standard AdS/CFT language (Nc D3-Nf D7 case) U(1) part of the U(Nf) gauge symmetry: Aμ Aμ couples the U(1)B current (density): the boundary value of A0 corresponds to the source for the U(1)B number density op. μ

28. =Ω A function of A0’: grand potential in the grand canonical ensemble. Gauss-law constraint: quark number density “electric charge” density Thermodynamics as classical electromagnetism DBI action of the flavor D7-branes with Fρ0: ρ-derivative

29. Legendre transformation “Hamiltonian” is interpreted as the Helmholtz free energy in the canonical ensemble.

30. A problem Kobayashi-Mateos-Matsuura-Myers (KMMM) claims: “the Minkowski branch is unphysical.” Our (S.N.-Seo-Sin-Yogendran) treatment: with the Minkowski branch. (Analysis: canonical ensemble in both papers)

31. KMMM’s claim charged source Gauss-law constraint: Black-hole branch Minkowski branch D7 E E F1 AdS-BH 1st order horizon D7 falls into the BH and no Minkowski branch. 1st order in canonical ensemble

32. However, (S.N.-Seo-Sin-Yogendran, to appear) However, if we use only the black-hole branch, we have another serious problem. In the grand canonical ensemble, KMMM has only high-temperature region. (Full temperature region cannot be covered within their framework.)

33. Brane configurations y Minkowski branch (y0 / yH >1) D7 y0 yH y0 Black-hole branch (y0 / yH <1) BH ρ

34. No flavor brane! Q=const. If black-hole branch only, 1/T No low-temp. region in the theory?? μ=const. y0/yH BH branch Minkowski branch

35. Conclusion • Basic ideas of AdS/CFT have been reviewed in this talk. • Attempts to introduce U(1)B-chemical potential have been started last year. • The KMMMT’s claim looks reasonable, but we found that their proposal produces another serious problem. • AdS/CFT with U(1)B-chemical potential is still under construction.