Baryon Chemical Potential in ABaryon Chemical Potential in AdS/CFTdS/CFT

Shin Nakamura Shin Nakamura CQUeST and Hanyang Univ. CQUeST and Hanyang Univ.

Refs. S.N.-Seo-Sin-Yogendran, hep-th/0611021 and arXiv:0708.2818(hep-th)

Motivation

AdS/CFT may be alternative useful tool.

Interesting phenomena in quark-hadron systemsoften 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.

• 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 completeframework is yet to be constructed.

How much have been achieved?What is the problem?

We’ll see inD3-D7 systems.

(D4-D8-D8)(D4-D8-D8)

Introduction of flavors

N=4 SYM theory does not have fundamentalquarks (i.e. no hadron).

If we introduce many D7’s: many flavors U(Nf)

Nf

Introduction of quarks: Introduction of flavor-branes

D3

D7mqquark

4d SYM

Dp-brane: (p+1)-dim. object

D3

D7mqquark

anti-quark

AdS-BH

D7

horizon

gravity dual

meson D7-brane’sfluctuation

Mesons

4d SYM

The system we have considered: 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 fromthe flavor branes. ( ～ quenched approx.)

AdS-BH

D7

horizon

Minkowski branch(mesons’ spectrum has a gap)

Black-hole branch(Gap-less meson’s spectrum)

1st order

T<Tc Tc<T

A phase transition of meson’s system

Mateos, Myers, and Thomson, hep-th/0605046Albash, Filev, Johnson and Kundu, hep-th/0605088, hep-th/0605175Karch and O'Bannon, hep-th/0605120

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.

A0 on the flavor brane at the boudary of the geometry

)()1()( fBf NSUUNU

The diagonal part of the flavor symmetry.

U(1)B chemical potentialKim-Sin-Zahed,2006/8; Horigome-Tanii,2006/8

conjugate

How about gauge invariance?

We should use

A “physical” ? meaning:a work necessary to bring a single quark charge from the boundary to ρmin againstthe electric field.

S.N.-Seo-Sin-Yogendran,2006/11,2007/8

ρED7

ρ

boundary

Kobayashi-Mateos-Matsuura- Myers-Thomson,2006/11

minmin

0min000 )()( AdAAFd

AdS-BH

ρ-derivative

ρ: radial coordinate

Thermodynamics as classical electromagnetism

DBI action of the flavor D7-branes with Fρ0:

)2det(

);,()/(

3

03min

FGdL

AyyLdVS

Gauss-law constraint:

QA

L

0

“electric charge” density

A function of A0’: grand potential in the grand canonical ensemble.

=Ω

QT

quark number density

Legendre transformation

00 A

LALH

QF

“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

AdS-BH

D7

horizon

Minkowski branch Black-hole branch

1st order

Gauss-law constraint:

)(0

QA

L

d

d

charged source

D7 falls into the BH andno Minkowski branch.

EE

(Kobayashi-Mateos-Matsuura-Myers-Thomson, 2006)

strings

“ We should includethe charged objects.”

However,

If we use the black-hole branch only,we have other serious problems.

(S.N.-Seo-Sin-Yogendran, to appear)

0

1

2

3

4

5

6

7

0 1 2 3 4 5 6 7

D7-brane solutions in the grand-canonical ensemble

y0/yH

1/T

We have flavor branesin all the temp. region.

BH-branch Minkowski branch

If we abandon the Minkowski-typesolutions, the theory does not coverthe low-temp. region. “Incompleteness of the theory”

Furthermore, in the canonical ensemble,

02

2

Q

F

Q

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 theMinkowski branch. Q

F

QL QH

F’

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

VVmFF

m

gmVgiL

v

s

sv

2

222

2

1

4

12

1

)()(

Baryon

Scalar meson (sigma)

Vector meson(omega)

What we are doing may be…..

VVmFF

m

VgL

v

s

v

2

222

00

2

1

4

12

1

Q Source

(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,

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).