Title: Baryon Chemical Potential in AdS/CFT
1Baryon Chemical Potential in AdS/CFT
- Shin Nakamura
- CQUeST and Hanyang Univ.
Refs. S.N.-Seo-Sin-Yogendran, hep-th/0611021 and
arXiv0708.2818(hep-th)
2Motivation
Interesting phenomena in quark-hadron
systems often lie in the strongly coupled region
(Example RHIC quark-gluon plasma)
Non-perturbative analysis is necessary.
AdS/CFT may be alternative useful tool.
3Baryon chemical Potential in AdS/CFT
Attempts started last summer
- Kim-Sin-Zahed
- Horigome-Tanii
- S.N.-Seo-Sin-Yogendran (D3-D7)
- Kobayashi-Mateos-Matsuura-Myers-Thomson
- (KMMMT) (D3-D7)
There are much progress, but the
complete framework is yet to be constructed.
How much have been achieved? What is the problem?
Well see in D3-D7 systems.
4Introduction of flavors
N4 SYM theory does not have fundamental quarks
(i.e. no hadron).
5D7
mq
quark
D3
4d SYM
anti-quark
6The system we have considered D3-D7 system
- YM theory N2 large-Nc SYM with quarks
- Flavor branes Nf D7-branes
- Flavor symmetry U(Nf)
- Quarks are massive (in general) mq
- Probe approximation (NcgtgtNf)
- Free energyFlavor-brane action
No back reaction to the bulk gometry from the
flavor branes. (quenched approx.)
7A phase transition of mesons system
Minkowski branch (mesons spectrum has a gap)
Black-hole branch (Gap-less mesons spectrum)
D7
AdS-BH
1st order
horizon
TcltT
TltTc
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
8How about finite baryon-number density?
- We need flavor branes(D7-branes).
The diagonal part of the flavor symmetry.
This is a local (gauge) symmetry on the D7-branes.
A0 on the flavor brane at the boudary of the
geometry
9How about gauge invariance?
S.N.-Seo-Sin-Yogendran,2006/11,2007/8
Kobayashi-Mateos-Matsuura-
Myers-Thomson,2006/11
We should use
boundary
D7
?-derivative
?
AdS-BH
?
? radial coordinate
A physical ? meaning a work necessary to bring
a single quark charge from the boundary to ?min
against the electric field.
10Thermodynamics as classical electromagnetism
DBI action of the flavor D7-branes with F?0
11Legendre transformation
Hamiltonian is interpreted as the Helmholtz
free energy in the canonical ensemble.
12A problem pointed out by KMMMT
(Kobayashi-Mateos-Matsuura-Myers-Thomson, 2006)
Gauss-law constraint
charged source
Black-hole branch
Minkowski branch
D7
E
E
AdS-BH
1st order
horizon
D7 falls into the BH and no Minkowski branch.
13However,
(S.N.-Seo-Sin-Yogendran, to appear)
If we use the black-hole branch only, we have
other serious problems.
14D7-brane solutions in the grand-canonical ensemble
1/T
If we abandon the Minkowski-type solutions, the
theory does not cover the low-temp. region.
Incompleteness of the theory
We have flavor branes in all the temp. region.
y0/yH
BH-branch
Minkowski branch
15Furthermore, in the canonical ensemble,
thermodynamic instability
F
The model need to have the Minkowski branch.
F
Q
QL
QH
Minkowski ABCD Black-hole DEFGHI
16A 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 mesons effective
theory under the presence of an external source
charged under U(1)B.
17sigma-omega model
Baryon
Scalar meson (sigma)
Vector meson (omega)
18What we are doing may be..
19Discussion
(Cf. Bergman-Lifschytz-Lippert, arXiv.0708.0326
for D4-D8-D8.)
20Conclusion
- 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).