Baryon Chemical Potential in AdSCFT

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Baryon Chemical Potential in AdS/CFT

• Shin Nakamura ?? ?
• Hanyang Univ. and CQUeST
• (??????)

Refs. S.N.-Seo-Sin-Yogendran, hep-th/0611021
Kobayashi-Mateos-Matsuura-Myers-Thomson,
hep-th/0611099
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Motivation
AdS/CFT a useful tool for analysis of
strongly coupled YM theories.
Established
Yet to be completed
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AdS/CFT at finite temperature

• Imaginary-time formalism

• Phase transitions

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D7
mq
quark
D3
5
The system we consider 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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AdS/CFT at finite R-charge chemical potential
SO(5) on the S5
R-symmetry
10 dim.
angular momentum on the S5
R-charge
From the 5d point of view
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
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How about finite baryon-number density?
D4-D8-D8 case
A0 on the flavor brane at the boudary
U(1)B chemical potential?
?
Kim-Sin-Zahed,2006/8 Horigome-Tanii,2006/8
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How about gauge invariance?
S.N.-Seo-Sin-Yogendran,2006/11
Kobayashi-Mateos-Matsuura-
Myers-Thomson,2006/11
We should use
boundary
D7
?
AdS-BH
?
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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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
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Thermodynamics as classical electromagnetism
DBI action of the flavor D7-branes with F?0
?-derivative
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Legendre transformation
Hamiltonian is interpreted as the Helmholtz
free energy in the canonical ensemble.
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A problem
Kobayashi-Mateos-Matsuura-Myers-Thomson (KMMMT)
claims the Minkowski branch is
unphysical.
Our (S.N.-Seo-Sin-Yogendran) treatment with the
Minkowski branch.
(Analysis canonical ensemble in both papers)
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KMMMTs claim
charged source
Gauss-law constraint
Black-hole branch
Minkowski branch
D7
E
E
F1
AdS-BH
1st order
horizon
D7 falls into the BH and no Minkowski branch.
1st order in canonical ensemble
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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, black-hole
branch has only high-temperature region. (Full
temperature region cannot be covered within
KMMMTs proposal.)
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Brane configurations
y
Minkowski branch (y0 / yH gt1)
D7
y0
yH
Black-hole branch (y0 / yH lt1)
y0
BH
?
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If black-hole branch only,
1/T
No low-temp. region in the theory??
µconst.
y0/yH
BH branch
Minkowski branch
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Conclusion

• Attempt to introduce U(1)B-chemical potential has
  been started last year.
• The KMMMTs claim itself is reasonable, but we
  found that their proposal produces another
  serious problem.
• AdS/CFT with U(1)B-chemical potential is still
  under construction in the above sense, and still
  under debate.