1. Baryon Chemical Potential in AdS/CFT Shin Nakamura CQUeST and Hanyang Univ. Refs. S.N.-Seo-Sin-Yogendran, hep-th/0611021 and arXiv:0708.2818(hep-th)

2. Motivation AdS/CFT may be alternative useful tool. Interesting phenomena in quark-hadron systems often lie in the strongly coupled region: (Example: RHIC quark-gluon plasma) Non-perturbative analysis is necessary. However, there is a technical difficulty in analysis of: Finite baryon density (chemical potential) systems. Time-dependent systems. ………….. Lattice QCD: a first-principle computation.

3. Kim-Sin-Zahed Horigome-Tanii S.N.-Seo-Sin-Yogendran (D3-D7) Kobayashi-Mateos-Matsuura-Myers-Thomson (KMMMT) (D3-D7) Baryon chemical Potential in AdS/CFT Attempts started last summer: There are much progress, but the complete framework is yet to be constructed. How much have been achieved? What is the problem? We’ll see in D3-D7 systems. (D4-D8-D8)

4. Introduction of flavors N=4 SYM theory does not have fundamental quarks (i.e. no hadron). If we introduce many D7’s: many flavors U(N f ) NfNf Introduction of quarks: Introduction of flavor-branes D3 D7 mqmq quark 4d SYM Dp-brane: (p+1)-dim. object

5. D3 D7 mqmq quark anti-quark AdS-BH D7 horizon gravity dual meson D7-brane’s fluctuation Mesons 4d SYM

6. The system we have considered: D3-D7 system YM theory: N=2 large-N c SYM with quarks Flavor branes: N f D7-branes Flavor symmetry: U(N f ) Quarks are massive (in general): m q Probe approximation (N c >>N f ) Free energy ～ Flavor-brane action No back reaction to the bulk gometry from the flavor branes. ( ～ quenched approx.)

7. AdS-BH D7 horizon Minkowski branch (mesons’ spectrum has a gap) Black-hole branch (Gap-less meson’s spectrum) 1 st order T<Tc Tc<T A phase transition of meson’s system Mateos, Myers, and Thomson, hep-th/0605046 Albash, Filev, Johnson and Kundu, hep-th/0605088, hep-th/0605175 Karch and O'Bannon, hep-th/0605120

8. How about finite baryon-number density? We need flavor branes （ D7-branes). U(1) B symmetry: This is a local (gauge) symmetry on the D7-branes. U(1) B charge:“electric charge” for the U(1) gauge field on the D7-brane. A 0 on the flavor brane at the boudary of the geometry The diagonal part of the flavor symmetry. U(1) B chemical potential Kim-Sin-Zahed,2006/8; Horigome-Tanii,2006/8 conjugate

9. How about gauge invariance? We should use A “physical” ? meaning: a work necessary to bring a single quark charge from the boundary to ρ min against the electric field. S.N.-Seo-Sin-Yogendran,2006/11,2007/8 ρ E D7 ρ boundary Kobayashi-Mateos-Matsuura- Myers-Thomson,2006/11 AdS-BH ρ-derivative ρ: radial coordinate

10. Thermodynamics as classical electromagnetism DBI action of the flavor D7-branes with F ρ0 : Gauss-law constraint: “electric charge” density A function of A 0 ’: grand potential in the grand canonical ensemble. =Ω quark number density

11. Legendre transformation “Hamiltonian” is interpreted as the Helmholtz free energy in the canonical ensemble. Thermodynamics in the YM side Electromagnetism in the gravity side

12. A problem pointed out by KMMMT AdS-BH D7 horizon Minkowski branch Black-hole branch 1 st order Gauss-law constraint: charged source D7 falls into the BH and no Minkowski branch. E E (Kobayashi-Mateos-Matsuura-Myers-Thomson, 2006) strings “ We should include the charged objects.”

13. However, If we use the black-hole branch only, we have other serious problems. (S.N.-Seo-Sin-Yogendran, to appear)

14. D7-brane solutions in the grand- canonical ensemble y 0 /y H 1/T We have flavor branes in all the temp. region. BH-branchMinkowski branch If we abandon the Minkowski-type solutions, the theory does not cover the low-temp. region. “Incompleteness of the theory”

15. Furthermore, in the canonical ensemble, thermodynamic instability Minkowski: ABCD Black-hole: DEFGHI The Minkowski branch provides a stable final state, otherwise the system is unstable. The model need to have the Minkowski branch. Q F QLQL QHQH F’

16. A possible interpretation What is the physical interpretation of the present setup with the Minkowski branch? Why does it look to be consistent? A possible interpretation: “A meson’s effective theory under the presence of an external source charged under U(1) B.”

17. sigma-omega model Baryon Scalar meson (sigma) Vector meson (omega)

18. What we are doing may be….. Q Source

19. (Cf. Bergman-Lifschytz-Lippert, arXiv.0708.0326 for D4-D8-D8.) Discussion We should introduce baryons (D5-branes on S5) instead of the quarks (F1’s) in the Minkowski branch. For a complete setup,

20. Conclusion D3-D7 systems at finite baryon-charge chemical potential with the Minkowski branch looks to be consistent. If we abandon the Minkowski branch, the theory becomes incomplete. For a complete framework for finite baryon density, perhaps we need to introduce homogeniously distributed dynamical quarks/baryons on the flavor brane. AdS/CFT with U(1) B -chemical potential is still under construction (but in progress).