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

Shin Nakamura Shin Nakamura 中村　真中村　真Hanyang Univ. and CQUeSTHanyang Univ. and CQUeST

（韓国・漢陽大学（韓国・漢陽大学 ))

Ref. S.N.-Seo-Sin-Yogendran, hep-th/0611021 ( Kobayashi-Mateos-Matsuura-Myers, hep-th/0611099)

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.)

MotivationHadron 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 enableus to analyze strongly coupled QCD.

• Effective theories, Lattice QCD,…• AdS/CFT

AdS/CFT(Original, weak version)

Classical Supergravity on

4dim. Large-Nc SU(Nc) N=4 Super Yang-Mills at the large ‘t Hooft coupling

55 SAdS

conjecture

=

Maldacena ‘97

Strongly interacting quantum YM !!

10 dim.

What is AdS/CFT?Analogy: Euclidean

3 theory

B

)(V

A

322

!31

21)( mV

2 solutions:B

A: Ф=0 “trivial” vacuum

B: Ф=ФB “non-trivial” vacuum-m2

m2

Physics around the “non-trivial” vacuum2 equivalent methods:

dynamical B

=

1. Perturbation theory around the “non-trivial” vacuum.

source term

2. Perturbation theory around the “trivial” vacuum (with source).

ˆ ,ˆ0 Jdynamical

Propagator around the non-trivial vacuum

method 1: (around non-trivial) 22

1mp

method 2: (around trivial)

221122

11

11222 mpJmp

mmp

+ + +…..

=consistency

42mJ

J

(Comment after the seminar: we have to understand more about this.)

What we have learnedSame physics can be described in two different ways:

1. non-trivial vacuum, without source

2. trivial vacuum, with source• Re-summation of infinitely many diagrams• The source carries non-perturbative information

Single Feynmann diagram

42mJ

=

Let us do the same thing in string theoryType IIB Superstring Theory

Low energy: 10d type IIB supergravity

Many different vacua. Two of them:

1. A curved spacetime: black 3-brane solution

2. Flat spacetime

Asymptotically flat Extremal black hole

“Source for closed strings”: D3-brane

Theory of closed strings (perturbatively)

3+1 dim. hypersurface, gauge theory on it

Defined in 10d spacetime

“non-trivial”“trivial”

?

=

U(Nc) 3+1 dim N=4 Super YM theoryat low energy on the D3-branes

Superstring theory around black 3-brane geometry

Superstring theory around flat geometry

asymptotically flat

+ source (Nc D3-brane)

Black hole(3+1 dim. object)

The near horizon limit : 55 SAdS We do not want here.

SU(Nc)

AdS/CFT(Original, weak version)

Classical Supergravity on

4dim. Large-Nc SU(Nc) N=4 Super Yang-Mills at the large ‘t Hooft coupling

55 SAdS

conjecture

=

Maldacena ‘97

Strongly interacting quantum YM !!

10 dim.

What we have learnedSame physics can be described in two different ways:

1. non-trivial vacuum, without source

2. trivial vacuum, with source• Re-summation of infinitely many diagrams• The source carries non-perturbative information

Single Feynmann diagram

42mJ

=

Construction of gauge/gravity duality

1. Construct a D-brane configuration on which the gague theory you want is realized.

2. Find the supergravity solution which corresponds to the D-brane configuration. (Here, we have a curved spacetie, but no D-brane.)

3. Take near-horizon limit to make the unwanted modes (like gravity in the YM side) decoupled.

4. Take appropriate limits to make the supergravity approximation valid, if necessary.

Introduction of quark/antiquarks

D3-brane3+1 dim.

AdS5

string

q

q

The quark-antiquark pairis a single string coming from the boundary of AdS.

The end of the stringis a quark or antiquark.

Nc D3

Nf D7mqquark

flavor braneIntroduction of dynamical quarks

gravity dual

AdS5

Nf D7meson

AdS5 + flavor branes

AdS/CFT and statistical mechanics

AdS/CFT : a useful tool for analysis of strongly coupled YM theories.

Finite temperature

Finite baryon-number density (chemical potential)

Established

Yet to be completed

We need to describe systems withfinite temperature and finite density.

AdS/CFT at finite temperature

Classical Supergravity on AdS-BH×S5

4dim. Large-Nc strongly coupledSU(Nc) N=4 SYM at finite temperature(in the deconfinement phase).

conjecture

=

Witten ‘98

Hawking temp.

Phase transitions

Transition of bulk geometry at the same β(=1/T).

Thermal AdS AdS-BH

“confinement” phase “de-confinement” phase

Hawking-Page transition

Transition related to quark condensate

Transition of flavor-brane configuration, on a common branch of bulk geometry

gravity dual

AdS-BH

D7

horizon

Minkowski branch Black-hole branch

1st order

T<Tc Tc<T

Nc D3

Nf D7mqquark

flavor branePhase transition related to quarks

Brane configurations

Minkowski branch

Black-hole branchBH

y0

y

ρ

yH

D7

y0

21

2223

2226 dydyddds

.......2 qq

amy q

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-Thom

son, 2006/11

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 fromthe flavor branes. ( ～ quenched approx.)

AdS/CFT at finite R-charge chemical potential

R-symmetry: SO(6) on the S5

R-charge: angular momentumon the S5

electric charge of the BHFrom the AdS5 point of view

10 dim.

Electric potential A0 at the boundaryis interpreted as a chemical potential

Chamblin-Emparan-Johnson-Myers,1999Cvetic-Gubser,1999

First law in charged black hole

dQTdSdM Mass

Hawking temperature

Entropy from the area of the horizon

Electromagnetic potential

Charge

plays as a chemical potential

How about finite baryon-number density?

RfLfABRfLf NSUNSUUUNUNU )()()1()1()()(

• 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

A0 on the flavor brane at the boudary

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

D4-D8-D8 case

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

ρED7

ρ

boundary

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

)()( min000min

AAFd

AdS-BH

More standard AdS/CFT language

U(1) part of the U(Nf) gauge symmetry: Aμ

Aμ couples the U(1)B current (density):the boundary value of A0 corresponds tothe source for the U(1)B number density op.

μ

......)()( 2min00

qqaAA

(Nc D3-Nf D7 case)

Thermodynamics as classical electromagnetism

DBI action of the flavor D7-branes with Fρ0:

)2det(

);,()/(

3

03min

FGdL

AyyLdVS

Gauss-law constraint:

QAL

0

“electric charge” density

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

=Ω

QT

quark number density

ρ-derivative

Legendre transformation

00 ALALH

QF

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

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)

KMMM’s claim

AdS-BH

D7

horizon

Minkowski branch Black-hole branch

1st order

Gauss-law constraint:

QAL

0

)()( 0

QAL

charged source

F1

D7 falls into the BH andno Minkowski branch.

1st order in canonical ensemble

EE

However,

However, if we use only the black-hole branch, we have another serious problem.

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

In the grand canonical ensemble, KMMM hasonly high-temperature region. (Full temperature region cannot be covered within their framework.)

Brane configurations

Minkowski branch (y0 / yH >1)

Black-hole branch (y0 / yH <1) BH

y0

y

ρ

yH

D7

y0

21

2223

2226 dydyddds

If black-hole branch only,

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

y0

1/ T

No flavor brane!

μ=const.

BH branch Minkowski branch

No low-temp. regionin the theory??

Q=const.

y0/yH

1/T

23d

Conclusion• Basic ideas of AdS/CFT have been review

ed in this talk.• Attempts to introduce U(1)B-chemical pote

ntial 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.