Download Baryon Chemical Potential in AdS/CFT

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
• 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.)
Motivation
Hadron 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 enable
us to analyze strongly coupled QCD.
• Effective theories, Lattice QCD,…
• AdS/CFT
AdS/CFT
(Original, weak version)
Classical Supergravity on AdS5  S 5
=
conjecture
10 dim.
Maldacena ‘97
4dim. Large-Nc SU(Nc)
N=4 Super Yang-Mills
at the large ‘t Hooft coupling
Strongly interacting quantum YM !!
What is AdS/CFT?
Analogy: Euclidean
V ( )
A

3
theory
1 2 2 1 3
V ( )   m   
2
3!
B

B
m2
-m2
2 solutions:
A: Ф=0
“trivial” vacuum
B: Ф=ФB
“non-trivial” vacuum
Physics around the “non-trivial” vacuum
2 equivalent methods:
1. Perturbation theory around the
“non-trivial” vacuum.
  B  
dynamical
=
2. Perturbation theory around the
“trivial” vacuum (with source).
  0  ˆ,
Jˆ
dynamical
source term
Propagator around the non-trivial
vacuum
method 1: (around non-trivial)
method 2: (around trivial)

+
+
J
1 
p 2  m2  1 

1
1
p m2
2
1
p 2  m2
+…..
consistency

1

J  p 2  m2

=
 
1
m2
J
2m
4

(Comment after the seminar: we have to understand more about this.)
What we have learned
Same physics can be described in
two different ways:
1. non-trivial vacuum, without source
Single Feynmann diagram
=
2. trivial vacuum, with source
• Re-summation of infinitely many diagrams
• The source carries non-perturbative information
J
2m 4

Let us do the same thing in string theory
Type IIB Superstring Theory
Defined in 10d spacetime
Theory of closed strings (perturbatively)
Low energy: 10d type IIB supergravity
Many different vacua. Two of them:
1. A curved spacetime: black 3-brane solution
“non-trivial”
Asymptotically flat
Extremal black hole
“trivial”
2. Flat spacetime
“Source for closed strings”: D3-brane
3+1 dim. hypersurface, gauge theory on it
Superstring theory around black 3-brane geometry
asymptotically flat
Black hole
(3+1 dim. object)
The
near
limit :
We do
nothorizon
want here.
AdS5  S
5
=
?
SU(N
U(Nc)
c) 3+1 dim N=4 Super YM theory
at low energy on the D3-branes
Superstring theory around flat geometry
+ source (Nc D3-brane)
AdS/CFT
(Original, weak version)
Classical Supergravity on AdS5  S 5
=
conjecture
10 dim.
Maldacena ‘97
4dim. Large-Nc SU(Nc)
N=4 Super Yang-Mills
at the large ‘t Hooft coupling
Strongly interacting quantum YM !!
What we have learned
Same physics can be described in
two different ways:
1. non-trivial vacuum, without source
Single Feynmann diagram
=
2. trivial vacuum, with source
• Re-summation of infinitely many diagrams
• The source carries non-perturbative information
J
2m 4

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 Dbrane.)
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
string
q
The end of the string
is a quark or antiquark.
q
3+1 dim.
D3-brane
The quark-antiquark pair
is a single string coming
from the boundary of AdS.
AdS5
Introduction of dynamical quarks
flavor brane
Nf D7
quark
Nc D3
mq
gravity dual
AdS5 + flavor branes
Nf D7
meson
AdS5
AdS/CFT and statistical mechanics
AdS/CFT : a useful tool for analysis of
strongly coupled YM theories.
We need to describe systems with
finite temperature and finite density.
Finite temperature
Established
Yet to be completed
Finite baryon-number
density (chemical potential)
AdS/CFT at finite temperature
Classical Supergravity on AdS-BH×S5
=
conjecture
Hawking temp.
Witten ‘98
4dim. Large-Nc strongly coupled
SU(Nc) N=4 SYM at finite temperature
(in the deconfinement phase).
Phase transitions
“confinement” phase
Thermal AdS
“de-confinement” phase
AdS-BH
Hawking-Page transition
Transition of bulk geometry at the same β(=1/T).
Transition related to quark condensate
Transition of flavor-brane configuration,
on a common branch of bulk geometry
Phase transition related to quarks
flavor brane
Nf D7
mq
quark
Nc D3
gravity dual
Minkowski branch
Black-hole branch
D7
AdS-BH
T<Tc
1st order
horizon
Tc<T
Brane configurations
ds  d   d  dy  y d
2
6
y
2
2
2
3
2
2
2
1
Minkowski branch
D7
y0
y  mq  a
yH
y0
BH
Black-hole branch
qq

2
 .......
ρ
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-MyersThomson, 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)
No back reaction to the bulk gometry from
the flavor branes. (～quenched approx.)
• Free energy～Flavor-brane action
AdS/CFT at finite R-charge
chemical potential
R-symmetry:
10 dim.
R-charge:
From the AdS5
point of view
SO(6) on the S5
angular momentum
on the S5
electric charge of the BH
Electric potential A0 at the boundary
is interpreted as a chemical potential
Chamblin-Emparan-Johnson-Myers,1999
Cvetic-Gubser,1999
First law in charged black hole
Entropy from the area of the horizon
Mass
dM  TdS  dQ
Hawking temperature
Charge
Electromagnetic potential
plays as a chemical potential
How about finite baryon-number density?
• We need flavor branes（D8,D7,….)
• U(1)B symmetry:
D4-D8-D8 case
U ( N f ) L  U ( N f ) R  U (1) B  U (1) A  SU ( N f ) L  SU ( N f ) R
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
How about gauge invariance?
We should use


 min
D7
S.N.-Seo-Sin-Yogendran,2006/11
Kobayashi-Mateos-MatsuuraMyers-Thomson,2006/11
d F 0  A0 ()  A0 (  min )
boundary
E
ρ
AdS-BH
A “physical” ? meaning: ρ
a work necessary to bring a single quark
charge from the boundary to ρmin against
the electric field.
More standard AdS/CFT language
(Nc D3-Nf D7 case)
U(1) part of the U(Nf) gauge symmetry: Aμ
Aμ couples the U(1)B current (density):
the boundary value of A0 corresponds to
the source for the U(1)B number density op.
μ

A0 (  )  A0 (  min )    a
q q

2
 ......
Thermodynamics as classical
electromagnetism
DBI action of the flavor D7-branes with Fρ0:
S /( V3 )  

 min
L   d3
d L( y, y; A0 )
det(G  2 F )
=Ω
ρ-derivative
A function of A0’:
grand potential in the grand canonical ensemble.
Gauss-law constraint:
“electric charge”
L
 Q density
A0
quark number
density

 Q
 T
Legendre transformation
L
H  L  A0
A0
F    Q
“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
charged source
Gauss-law constraint:
L
 Q (  )
  (A0 )
L
 Q
A0
Black-hole branch
Minkowski branch
D7
E
E
F1
1st order
AdS-BH
horizon
D7 falls into the BH and
no Minkowski branch.
1st order in canonical
ensemble
However,
(S.N.-Seo-Sin-Yogendran, to appear)
However, if we use only the black-hole branch,
we have another serious problem.
In the grand canonical ensemble, KMMM has
only high-temperature region.
(Full temperature region cannot be covered
within their framework.)
Brane configurations
ds  d   d  dy  y d
2
6
y
2
2
2
3
2
2
2
1
Minkowski branch (y0 / yH >1)
D7
y0
yH
y0
BH
Black-hole branch (y0 / yH <1)
ρ
If black-hole branch only,
1/T
1/T
2
1.8
Q=const.
No flavor brane!
1.6
No low-temp. region
in the theory??
1.4
1.2
d 32
1
0.8
μ=const.
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
y0
BH branch
Minkowski branch
2
y0/yH
Conclusion
• Basic ideas of AdS/CFT have been
reviewed in this talk.
• Attempts to introduce U(1)B-chemical
potential 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.