dynamical instability of holographic qcd at finite density shoichi kawamoto 23 april 2010 at...
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23 April 2010 at NTUShoichi Kawamoto3 massless QCD [Rajagopal-Wilczek hep-ph/ ] 1 st orderTRANSCRIPT
Dynamical Instability of Holographic QCD at Finite Density
Shoichi Kawamoto
23 April 2010 at National Taiwan University
Based on arXiv:1004.0162 in collaboration withWu-Yen Chuang (Rutgers), Shou-Huang Dai, Feng-Li Lin (NTNU)and Chen-Pin Yeh (NTU)
National Taiwan Normal University
23 April 2010 at NTU Shoichi Kawamoto 2
Phase diagram of “real” QCD[hep-ph/0503184]
23 April 2010 at NTU Shoichi Kawamoto 3
massless QCD[Rajagopal-Wilczek hep-ph/0011333]
1st order
23 April 2010 at NTU Shoichi Kawamoto 4
Large N QCD and chiral density wave
Quark “Cooper pair” are not color singlet and then it is suppressed in large N limit.
CN
CT e
Instead, in large N limit, another spatially modulated phase will be favored.
)()()( )( yxFeyx i yxp C
eF
)0(
[Deryagin-Grigoriev-Rubakov]
In the large N limit, there will appear clear confining/deconfinement transition.
No color superconductivity (or CFL) in large N limit.
For high density, low temperature
“chiral density wave” phase
23 April 2010 at NTU Shoichi Kawamoto 5
large N QCD phase diagram???
CDW?
Another confinement/deconfinement transition??
quarkyonic? [McLerran, Pisarski, …]
23 April 2010 at NTU Shoichi Kawamoto 6
Phase diagram of holographic QCD
23 April 2010 at NTU Shoichi Kawamoto 7
Holographic Realization of Pure YM (1)Nc D4-brane compactified on S1 with SUSY breaking spin structure (Scherk-Schwarz circle)
4x
95 ,, xx
0 1 2 3 4 5 6 7 8 9
Nc D4 o o o o o
Fermions : tree level massive (anti-periodic boundary condition)5 scalars : 1-loop massive (no supersymmetry)
3+1D pure Yang-Mills theory (with KK modes)low energy theory on D4
23 April 2010 at NTU Shoichi Kawamoto 8
Holographic Realization of Pure YM (2)
4x
2
42
223
24
2223
2
)()( dU
UfdU
URdxUfdxdt
RUds i
3
3
1)(U
UUf KK
24
32
34
xRU KK
33scs lNgR
KKU U
4x
UTU
2
42
223
24
2223
2
)()( dU
UfdU
URdxdxdtUf
RUds i
3
3
1)(UUUf T
confining geometry
deconfined geometry
[Witten]
23 April 2010 at NTU Shoichi Kawamoto 9
Confinement/Deconfiment transitionCompactify on a thermal circle, we compare thermodynamic free energy.
tE tEx4 x4
At a critical temperature, we need to switch these two geometries (phase transition)
quark potentiallinear screened
[Aharony-Sonnenschein-Yankieowicz]
Confinement Deconfined
23 April 2010 at NTU Shoichi Kawamoto 10
Phase diagram (1)T
deconfined
confining
(This phase transition is leading and will not be changed by introducing flavors)
23 April 2010 at NTU Shoichi Kawamoto 11
Adding Quarks (Sakai-Sugimoto model)
4x
95 ,, xx
0 1 2 3 4 5 6 7 8 9
Nc D4 o o o o o o o o o o o o o o
L
Symmetry:
To add the quark degrees of freedom, we introduce Nf probe D8-branes. [Sakai-Sugimoto]
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
)5()()()( SONUNUNU RfLfc
In the gravity dual, this symmetry is broken down to the diagonal U(Nf).
RLA ,,
23 April 2010 at NTU Shoichi Kawamoto 12
Chiral symmetry breaking in SS modelKKU U
UTU
In this cigar geometry, D8 and anti-D8 need to connect.
diag)()()( fRfLf NUNUNU
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.
In the deconfined geometry, there will be two configurations for the same boundary condition of D8.
A
B
L
The one (A) breaks the chiral symmetry,while for the other configuration ending on the horizon (B)the chiral symmetry is restored.
23 April 2010 at NTU Shoichi Kawamoto 13
Chiral symmetry restoration
[Aharony etal. hep-th/0604161]
The restoration depends on the position of UT (the Hawking temperature) andthe asymptotic separation L.
separation L
temperature T
We will consider a fixed L. There is a critical temperature T.
Chiral symmetry restored
Chiral symmetry breaking
23 April 2010 at NTU Shoichi Kawamoto 14
Phase diagram (2)T
?
23 April 2010 at NTU Shoichi Kawamoto 15
Introducing Baryon chemical potential
U(1) part of chiral U(Nf) symmetry: I
RLiI
RL e ,,
The conserved charge is the ordinary fermion number.
The corresponding gauge field will be turned on. Bcq NUAA )(:0
qUxA ),(0
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. 00
0 AnjA
The Baryon vertex
23 April 2010 at NTU Shoichi Kawamoto 16
Baryons in Sakai-Sugimoto model
4 040St cRR ANFA
D4-brane wrapping on S4 is a baryon vertex. [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 underflavor 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.
23 April 2010 at NTU Shoichi Kawamoto 17
Baryons as D4-instantons
8
25 '2
DFC
45
D
C
A nontrivial gauge field configurationon 4-submanifold in 8-brane That gauge fields configuration carries
D4-brane charge.
Codimension 4 solition (instanton) on D8 is identified with D4-brane inside D8.
D8-brane Wess-Zumino term includes the following coupling:
040
3
3 )(Tr)'2(Tr AnNFFANFC cc
Instanton number (D4-charges)density
Instantons are indeed a source for U(1) charge.
We consider a smeared instanton over 3+1D
23 April 2010 at NTU Shoichi Kawamoto 18
D8-brane profile with D4-instantonFor 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.
)()Tr(8 0
202 cb
cCS UanFaNS Then D8 WZ-term (Chern-Simons term) is
nb is proportional to instanton density.
D8 profile is the same as before except U=Uc (tip). The new configuration isdetermined by minimizing the total action with respect to Uc
Uc
),()()(8 cbCScDBIcD UnSUSUS
c
L
For given L and nb, Uc is uniquely fixed and the angle at the tip is
52
2
coscb
bc Un
n
23 April 2010 at NTU Shoichi Kawamoto 19
Chiral symmetry restoration due to nb
In the deconfinement geometry, chiral symmetry can be restored by having baryon density.
Large baryon number density (nb) is “heavy” due to the tension of D4, andis pilling the tip of D8 towards the horizon.
nb large
23 April 2010 at NTU Shoichi Kawamoto 20
Phase diagram (3)T
23 April 2010 at NTU Shoichi Kawamoto 21
Fluctuations on D8-brane
Dictionary of gauge/gravity correspondense
)(),( UBUAUUx bulk field
leading sub-leading
nonnormalizablemode
normalizablemode
AOboundary O
source term
Finally, we will consider the fluctuation on D8-brane and see that it suggests an instability.
23 April 2010 at NTU Shoichi Kawamoto 22
Dynamical instability
Assume that if normalizable solution (A=0) develops growing mode.texB )(
no source term and tachyonic mode of O
Ospontaneous symmetry breakingwith 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.
23 April 2010 at NTU Shoichi Kawamoto 23
FluctuationsU(1) gauge field: Ui aaaAA ,,000
D8-brane embedding: yxx 44
Take quadratic order in fluctuations
],;,[],[],;,[ 42404 xAyaLxALyaxAL
6 Linearlized equation of motion
)(),,( UgeUxt iti xk
)()()()(')()('')( 2 UgUgUCUgUBUgUA kkk
Using expansion:
23 April 2010 at NTU Shoichi Kawamoto 24
Boundary conditions
)()()()(')()('')( 2 UgUgUCUgUBUgUA kkk
(Coupled) euqations of motion take the form of 2nd order ordinary linear differential equations.
UmUg )(
With the boundary condition (m=0), this is an eigenvalue equation and a solution exists forspecific 2.
cU U
Need to specify the boundary condition for the other “end” U=Uc.
Ui aa , : Dirichlet or Neumann
y
0a: Dirichlet (fixing the position of the tip)
: Neumann (fixing the electric source)
23 April 2010 at NTU Shoichi Kawamoto 25
Instability from Chern-Simons termWe just look at 3 equations of motion.
From Chern-Simons term0, AEff Ujk
ijki 2 2k
Domokos-Harvey (and Nakamura-Ooguri-Park) found that with this Chern-Simons termwith electric field background the solution can develop unstable modes.
23 April 2010 at NTU Shoichi Kawamoto 26
“Shooting” to find solutionsFirst, look at the marginal case (2=0).
We tune k to find a normalizable solution (shooting method).
UmUg )( Solution starts to exist.
Large nb (instanton density) and low temperature tend to develop the instability.
23 April 2010 at NTU Shoichi Kawamoto 27
Result of the numerical analysis
k
-2The solution is confirmed to representactual unstable mode.
2=0 solution means onset of instability.
Only ai modes develop unstable modes.
ii Ja
vector current
xk itii eJ
unstable for nonzero k
Spatially modulated!
23 April 2010 at NTU Shoichi Kawamoto 28
Phase diagram of holographic QCD
23 April 2010 at NTU Shoichi Kawamoto 29
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.