Baryon Chemical Potential in Ad SCFT Shin Nakamura

Baryon Chemical Potential in Ad. S/CFT Shin Nakamura CQUe. ST and Hanyang Univ. Refs. S. N. -Seo-Sin-Yogendran, hep-th/0611021 and ar. Xiv: 0708. 2818(hep-th)

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

Baryon chemical Potential in Ad. S/CFT Attempts started last summer: • Kim-Sin-Zahed (D 4 -D 8) • Horigome-Tanii (D 4 -D 8) • S. N. -Seo-Sin-Yogendran (D 3 -D 7) • Kobayashi-Mateos-Matsuura-Myers-Thomson (KMMMT) (D 3 -D 7) 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 D 3 -D 7 systems.

Introduction of flavors N=4 SYM theory does not have fundamental quarks (i. e. no hadron). Dp-brane: (p+1)-dim. object Introduction of quarks: Introduction of flavor-branes Nf D 7 D 3 quark mq 4 d SYM If we introduce many D 7’s: many flavors U(Nf)

D 7 D 3 quark mq 4 d SYM anti-quark gravity dual D 7 meson Ad. S-BH horizon D 7 -brane’s fluctuation Mesons

The system we have considered: D 3 -D 7 system • • • YM theory: N=2 large-Nc SYM with quarks Flavor branes: Nf D 7 -branes Flavor symmetry: U(Nf) Quarks are massive (in general): mq Probe approximation (Nc>>Nf) No back reaction to the bulk gometry from the flavor branes. (～quenched approx. ) • Free energy～Flavor-brane action

A phase transition of meson’s system Minkowski branch Black-hole branch (mesons’ spectrum has a gap) (Gap-less meson’s spectrum) D 7 1 st order Ad. S-BH horizon T<Tc Tc<T 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

How about finite baryon-number density? • We need flavor branes（D 7 -branes). • U(1)B symmetry: The diagonal part of the flavor symmetry. This is a local (gauge) symmetry on the D 7 -branes. U(1)B charge: “electric charge” for the U(1) gauge field on the D 7 -brane. conjugate A 0 on the flavor brane at the boudary of the geometry U(1)B chemical potential Kim-Sin-Zahed, 2006/8; Horigome-Tanii, 2006/8

How about gauge invariance? We should use boundary D 7 S. N. -Seo-Sin-Yogendran, 2006/11, 2007/8 Kobayashi-Mateos-Matsuura. Myers-Thomson, 2006/11 ρ-derivative E ρ Ad. S-BH ρ ρ: radial coordinate A “physical” ? meaning: a work necessary to bring a single quark charge from the boundary to ρmin against the electric field.

Thermodynamics as classical electromagnetism DBI action of the flavor D 7 -branes with Fρ0: =Ω A function of A 0’: grand potential in the grand canonical ensemble. Gauss-law constraint: “electric charge” density quark number density

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

A problem pointed out by KMMMT (Kobayashi-Mateos-Matsuura-Myers-Thomson, 2006) Gauss-law constraint: charged source “ We should include the charged objects. ” Black-hole branch Minkowski branch D 7 E Ad. S-BH strings E 1 st order horizon D 7 falls into the BH and no Minkowski branch.

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

D 7 -brane solutions in the grand 1/T canonical ensemble If we abandon the Minkowski-type solutions, theory does not cover the low-temp. region. We have branes “Incompleteness of flavor theory” in all the temp. region. y 0/y. H BH-branch Minkowski branch

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

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. ”

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

What we are doing may be…. . Q Source

Discussion For a complete setup, • We should introduce baryons (D 5 -branes on S 5) instead of the quarks (F 1’s) in the Minkowski branch. (Cf. Bergman-Lifschytz-Lippert, ar. Xiv. 0708. 0326 for D 4 -D 8. )

Conclusion • D 3 -D 7 systems at finite baryon-charge chemical potential with the Minkowski branch looks to be consistent. • If we abandon the Minkowski branch, 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. • Ad. S/CFT with U(1)B-chemical potential is still under construction (but in progress).