Transcript Dynamical Instability of Holographic QCD at Finite Density

23 April 2010 at National Taiwan University
Dynamical Instability of Holographic
QCD at Finite Density
Shoichi Kawamoto
National Taiwan Normal University
Based on arXiv:1004.0162 in collaboration with
Wu-Yen Chuang (Rutgers), Shou-Huang Dai, Feng-Li Lin (NTNU)
and Chen-Pin Yeh (NTU)
Phase diagram of “real” QCD
[hep-ph/0503184]
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massless QCD
[Rajagopal-Wilczek hep-ph/0011333]
1st order
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Large N QCD and chiral density wave
In the large N limit, there will appear clear confining/deconfinement transition.
Quark “Cooper pair” are not color singlet and then it is suppressed in large N limit.
  e
T
C
NC

No color superconductivity (or CFL) in large N limit.
Instead, in large N limit, another spatially modulated phase will be favored.
 ( x) ( y)  eip(xy ) F ( x  y)
For high density, low temperature
F (0)  e

C

[Deryagin-Grigoriev-Rubakov]
“chiral density wave” phase
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large N QCD phase diagram???
CDW
?
quarkyonic? [McLerran, Pisarski, …]
Another confinement/deconfinement transition??
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Phase diagram of holographic QCD
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Holographic Realization of Pure YM (1)
Nc D4-brane compactified on S1 with SUSY breaking spin structure (Scherk-Schwarz circle)
x 5 , , x 9
0 1 2 3 4 5 6 7 8 9
Nc D4
x4
o o o o o
Fermions : tree level massive (anti-periodic boundary condition)
5 scalars : 1-loop massive (no supersymmetry)
low energy theory on D4
3+1D pure Yang-Mills theory (with KK modes)
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Holographic Realization of Pure YM (2)
U KK
confining geometry
U 
3
2


U 
R
ds     dt2  dxi2  f (U )dx42   
R
U 
2
3
2
 dU 2


 U 2 d 24 
 f (U )

x4
f (U )  1 
3
U KK
U3
R3   gs Nc ls3
3
 4  R
U KK  

2
 3  x 4
2
[Witten]
deconfined geometry
UT
U 
3
2
3
2
2

U 
 R   dU
2
2
2
ds     f (U )dt  dxi  dx4    
 U 2 d 24 
R
 U   f (U )

2


f (U )  1 
x4
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U T3
U3
Confinement/Deconfiment transition
Compactify on a thermal circle, we compare thermodynamic free energy.
[Aharony-Sonnenschein-Yankieowicz]
tE
tE
x4
linear
quark potential
Confinement
x4
screened
Deconfined
At a critical temperature, we need to switch these two geometries (phase transition)
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Phase diagram (1)
T
deconfined
confining

(This phase transition is leading and will not be changed by introducing flavors)
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Adding Quarks (Sakai-Sugimoto model)
[Sakai-Sugimoto]
To add the quark degrees of freedom, we introduce Nf probe D8-branes.
x 5 , , x 9
x4
0 1 2 3 4 5 6 7 8 9
L
Nc D4
o o o o o
o o o o
o o o o o
4-8 open strings give chiral (from D8) and anti-chiral (from anti-D8)
fermions in the fundamental representation.
Nf flavor massless U(Nc) QCD in 3+1D
Symmetry:
A , L , R
U ( Nc ) U ( N f ) L U ( N f ) R  SO(5)
In the gravity dual, this symmetry is broken down to the diagonal U(Nf).
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Chiral symmetry breaking in SS model
U KK
U 
In this cigar geometry, D8 and anti-D8 need to connect.
U ( N f ) L U ( N f ) R  U ( N f )diag
Geometrical realization of chiral symmetry breaking
U(1)B subgroup is counting the number of quarks.
Later we will introduce the chemical potential for
this conserved quantity.
UT
U 
In the deconfined geometry, there will be two configurations for
the same boundary condition of D8.
B
The one (A) breaks the chiral symmetry,
while for the other configuration ending on the horizon (B)
the chiral symmetry is restored.
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A
L
12
Chiral symmetry restoration
The restoration depends on the position of UT (the Hawking temperature) and
the asymptotic separation L.
temperature T
Chiral symmetry restored
Chiral symmetry breaking
[Aharony etal. hep-th/0604161]
separation L
There is a critical temperature T.
We will consider a fixed L.
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Phase diagram (2)
T
?

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Introducing Baryon chemical potential
U(1) part of chiral U(Nf) symmetry:
 LI ,R  ei LI ,R
The conserved charge is the ordinary fermion number.
The corresponding gauge field will be turned on.
A0 : A(U  )  q  Nc B
 A ( x,U  )    
0
q
Temporal component of the gauge field is electric: we need to have a source.
Then we will introduce the source for the gauge field on D8-brane.
A
0
j 0  n  A0
The Baryon vertex
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Baryons in Sakai-Sugimoto model
D4-brane wrapping on S4 is a baryon vertex.

tS
4
A0  F4RR  N c  A0
[Witten]
electric charge on a compact space
To cancel charge, need to attach Nc strings
Nc quark bound state (baryon)
With dynamical quarks (D8-brane), baryons are charged under
flavor symmetry as well
Strings are ending on D8 and being a source for a0
However, this configuration is unstable.
[Callan-Guijosa-Savvidy-Tafjord]
D4 brane is attracted to D8 and becomes an instanton on it.
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Baryons as D4-instantons
A nontrivial gauge field configuration
on 4-submanifold in 8-brane

D8
C5  2 ' F 
2
C
5
That gauge fields configuration carries
D4-brane charge.
D4
Codimension 4 solition (instanton) on D8 is identified with D4-brane inside D8.
D8-brane Wess-Zumino term includes the following coupling:
C
3
Instanton number (D4-charges)
density
3
 T r (2 ' F )  N c  A0  T r( F  F )  N c n4  A0
Instantons are indeed a source for U(1) charge.
We consider a smeared instanton over 3+1D
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D8-brane profile with D4-instanton
For single instanton, an explicit profile is known (Hata-Sakai-Sugimoto-Yamato) and has a finite size.
However, the profile for multi-instanton is difficult to determine in general.
Consider a small instanton (zero-size) localized at the tip of D8.
Then D8 WZ-term (Chern-Simons term) is
SCS 
Nc
a  Tr( F 2 )  nb a0 (U c )
2  0
8
nb is proportional to
instanton density.
D8 profile is the same as before except U=Uc (tip). The new configuration is
determined by minimizing the total action with respect to Uc
SD8 (Uc )  SDBI (U c )  SCS (nb ,Uc )
L
For given L and nb, Uc is uniquely fixed and the angle at the tip is
qc
Uc
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nb2
cosqc  2
nb  U c5
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Chiral symmetry restoration due to nb
In the deconfinement geometry, chiral symmetry can be restored by having
baryon density.
nb large
Large baryon number density (nb) is “heavy” due to the tension of D4, and
is pilling the tip of D8 towards the horizon.
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Phase diagram (3)
T

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Fluctuations on D8-brane
Finally, we will consider the fluctuation on D8-brane and see that it suggests an
instability.
Dictionary of gauge/gravity correspondense
leading
bulk field
sub-leading

 ( x,U )  AU  BU   (U  )
nonnormalizable
mode
boundary

 AO
normalizable
mode
O
source term
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Dynamical instability
Assume that if normalizable solution (A=0) develops growing mode.
no source term and tachyonic mode of
spontaneous symmetry breaking
B( x)  et
O
O
with order parameter <O>
In the bulk side, normalizable modes correspond to small perturbations around the solution.
instability of the solution
We then look for normalizable tachyonic (growing in time) solution in the bulk.
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Fluctuations
A0  A0  a0 , ai , aU
U(1) gauge field:
D8-brane embedding:
x4  x4  y
Take quadratic order in fluctuations
L[ A, x4 ; a , y]  L0[ A, x4 ]  L2[a , y; A, x4 ]  
6 Linearlized equation of motion
Using expansion:
 (t , x,U )  eit ikx g (U )
Ak (U ) g ' ' (U )  Bk (U ) g ' (U )  Ck (U ) g (U )   2 g (U )
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Boundary conditions
(Coupled) euqations of motion take the form of 2nd order ordinary linear differential equations.
Ak (U ) g ' ' (U )  Bk (U ) g ' (U )  Ck (U ) g (U )   2 g (U )
g (U )  m U   
With the boundary condition (m=0), this is an eigenvalue equation and a solution exists for
specific 2.
U 
Uc

Need to specify the boundary condition for the other “end” U=Uc.
ai , aU : Dirichlet or Neumann
y
a0
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: Dirichlet (fixing the position of the tip)
: Neumann (fixing the electric source)
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Instability from Chern-Simons term
We just look at 3 equations of motion.
fi   ijk f jk , E  U A0
2
 k2
From Chern-Simons term
Domokos-Harvey (and Nakamura-Ooguri-Park) found that with this Chern-Simons term
with electric field background the solution can develop unstable modes.
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“Shooting” to find solutions
First, look at the marginal case (2=0).
We tune k to find a normalizable solution (shooting method).
g (U )  m U   
Solution starts to exist.
Large nb (instanton density) and low temperature tend to develop the instability.
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Result of the numerical analysis
The solution is confirmed to represent
actual unstable mode.
-2
2=0 solution means onset of instability.
k
Only ai modes develop unstable modes.
a J
J i  e it ikx
i
i
unstable for nonzero k
vector current
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Spatially modulated!
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Phase diagram of holographic QCD
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Conclusion



In holographic QCD (Sakai-Sugimoto model),
we draw a phase diagram including a spatially
modulated phase.
The onset of phase transition is signaled by
appearance of unstable mode in the presence
of Chern-Simons term.
CS term here is given directly by background
baryon density.
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