Dynamical Instability of Holographic QCD at Finite Density Shoichi Kawamoto 23 April 2010 at National Taiwan University Based on arXiv:1004.0162 in collaboration.

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]


massless QCD[Rajagopal-Wilczek hep-ph/0011333]

1st order


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


large N QCD phase diagram???

CDW?

Another confinement/deconfinement transition??

quarkyonic? [McLerran, Pisarski, …]


Phase diagram of holographic QCD


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


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]


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


Phase diagram (1)T

deconfined

confining

(This phase transition is leading and will not be changed by introducing flavors)


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 ,,


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.


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


Phase diagram (2)T

?


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


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.


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


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


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


Phase diagram (3)T


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.


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.


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:


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)


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.


“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.


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!


Phase diagram of holographic QCD


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.

Dynamical Instability of Holographic QCD at Finite Density Shoichi Kawamoto 23 April 2010 at National Taiwan University Based on arXiv:1004.0162 in collaboration.

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