Baryon Chemical Potential in AdS/CFT

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Baryon 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)
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Motivation
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.
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Baryon 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.
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Introduction of flavors
N4 SYM theory does not have fundamental quarks
(i.e. no hadron).
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D7
mq
quark
D3
4d SYM
anti-quark
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The 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.)
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A 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
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How about finite baryon-number density?

• We need flavor branes(D7-branes).

• U(1)B symmetry

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
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How 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.
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Thermodynamics as classical electromagnetism
DBI action of the flavor D7-branes with F?0
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Legendre transformation
Hamiltonian is interpreted as the Helmholtz
free energy in the canonical ensemble.
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A 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.
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However,
(S.N.-Seo-Sin-Yogendran, to appear)
If we use the black-hole branch only, we have
other serious problems.
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D7-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
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Furthermore, in the canonical ensemble,
thermodynamic instability
F
The model need to have the Minkowski branch.
F
Q
QL
QH
Minkowski ABCD Black-hole DEFGHI
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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 mesons effective
theory under the presence of an external source
charged under U(1)B.
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sigma-omega model
Baryon
Scalar meson (sigma)
Vector meson (omega)
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What we are doing may be..
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Discussion
(Cf. Bergman-Lifschytz-Lippert, arXiv.0708.0326
for D4-D8-D8.)
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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).