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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). Purpose of this talk. - PowerPoint PPT Presentation
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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.