Dense Holographic QCD in the Wigner-Seitz Approximation

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by  Keun-Young Kim

Dense Holographic QCD in the Wigner-Seitz Approximation
Keun-Young Kima, Sang-Jin Sin a,b and Ismail Zaheda
a Department of Physics and Astronomy, SUNY Stony-Brook, NY 11794
b Department of Physics, BK21 division, Hanyang University, Seoul 133-791, Korea
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2 Abstract

n
a WeinvestigatecolddensematterinthecontextofSakaiandSugimoto’sholographic
J
model of QCD in the Wigner-Seitz approximation. In bulk, baryons are treated as
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1 instantons on S3 R1 in each Wigner-Seitz cell. In holographic QCD, Skyrmions
×
] are instanton holonomies along the conformal direction. The high density phase is
h
t identified with a crystal of holographic Skyrmions with restored chiral symmetry at
-
p
e about4M3 /π5. As theaverage density goes up,itapproaches touniformdistribution
KK
h
while the chiral condensate approaches to p-wave over a cell. The chiral symmetry is
[

2 effectively restoredinlongwavelength limitsincethechiralorderparameterisaveraged
v 5/3
to be zero over a cell. The energy density in dense medium varies as n , which is the
2 B
8 expected power for non-relativistic fermion. This shows that the Pauli exclusion effect
5
1 in boundary is encoded in the Coulomb repulsion in the bulk.
.
2
1
7
0
:
v
i
X
r
a
1 Introduction
The AdS/CFT approach [1] provides a powerful framework for discussing large N gauge
c
theories at strong coupling λ = g2N . The model suggested by Sakai and Sugimoto (SS) [2]
c
offers a specific holographic realization that includes N flavors and is chiral. For N N ,
f f c
≪
chiralQCDisobtainedasagravitydualtoN D8-D8branesembeddedintoaD4background
f
in10dimensions where supersymmetry isbroken by theKaluza-Klein(KK)mechanism. The
SSmodelyields a holographicdescription ofpions, vectors, axialsandbaryonsthat isingood
agreement with experiment [2, 3, 4]. The SS model at finite temperature has been discussed
in [5] and at finite baryon density in [6, 7, 8]. The finite baryon chemical potential problem
in D3/D7 model was discussed in [9] and Isospin chemical potential and glueball decay also
has been discussed in [10].
ColdanddensehadronicmatterinQCDisdifficult totrackfromfirstprinciplesincurrent
lattice simulations owing to the sign problem. In large N QCD baryons are solitons and a
c
densematterdescriptionusingSkyrme’schiralmodel[11]wasoriginallysuggestedbySkyrme
and others [12]. At large N and high density matter consisting of solitons crystallizes, as
c
the ratio of potential to kinetic energy Γ = V/K N2 is much larger than 1. QCD matter
≈ c
at large N was recently revisited in [14].
c
The many-soliton problem can be simplified in the crystal limit by first considering all
solitons to be the same and second by reducing the crystal to a single cell with boundary
conditions much like the Wigner-Seitz approximation in the theory of solids. A natural
way to describe the crystal topology is through T3 with periodic boundary conditions. In
so far, this problem can only be addressed numerically. A much simpler and analytically
tractable approximation consists of treating each Wigner-Seitz cell as S3 with no boundary
condition involved. The result is dense Skyrmion matter on S3 [13]. Interestingly enough,
the energetics of this phase is only few percent above the energetics of a more involved
numerical analysis based on T3. Skyrmions on S3 restore chiral symmetry on the average
above a critical density. While Skyrmions on S3 are unstable against T3, they still capture
the essentials of dense matter and chiral restoration in an analytically tractable framework.
Cold dense matter in holographic QCD is a crystal of instantons with Γ = √λ/v 1
F
≫
where v 1/N is the Fermi velocity. (In contrast hot holographic QCD has Γ = √λ 1).
F c
≈ ≫
When the wigner-Seitz cell is approximated by S3, the pertinent instanton is defined on
S3 R. In this paper, we investigate cold QCD matter using instantons on S3 R in bulk.
× ×
As a result the initial D4 background is deformed to accomodate for the S3 which is just the
back reaction of the flavour crystal structure on the pure gauge theory. Holographic dense
1
matter can be organized in 1/λ at largeN . In our model the baryon density is uniform while
c
the chiral condensate is p-wave over a cell. However, the chiral condensation is averaged to
be zero over a cell so that the chiral symmetry is effectively restored in long wavelength limit.
We will show that as the average density goes up, it approaches to uniform distribution while
the chiral condensate approaches to p-wave over a cell. The energy density in dense medium
5/3
varies as n , which is the expected power for non-relativistic fermion. This shows that the
B
Pauli exclusion effect in boundary is encoded in the Coulomb repulsion in the bulk.
In section 2, we define this deformation and discuss the D8 brane embedding structure.
The instantons on S3 R in the flavour D8 brane is discussed in section 3. In section 4,5,6
×
we derive the equation of state of cold holographic matter using the small size instanton
expansion and in general. In section 7 we show how the holographic small instantons in bulk
transmute large size Skyrmions on the boundary. The comparison to other models of nuclear
matter is carried in section 8. Our conclusions are in section 9.
2 D8 brane action
We consider crystallized skyrmions at finite density in the Wigner Seitz approximation.
Spatial R3 is naturally converted to T3 with periodic boundary conditions. As a result the
D4 background geometry is deformed. The baryons are then instantons on T3 R. Most
×
solutions are only known numerically on the lattice. A simpler and analytically tractable
analysis that captures the essentials of dense matter is to substitute T3 by S3 in bulk with
no boundary conditions altogether. As a result, the D4 background dual to the crystal is
modified with the boundary special space as S3. Specifically, the 10 dimensional space is
that of (R1 S3) R1 S4. The ensuing metric on D4 is therefore
× × ×
U 3/2 R 3/2 dU2
ds2 = dt2 +R2dΩ2 +f(U)dτ2 + +U2dΩ2 , (1)
R − 3 U f(U) 4
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
(cid:0) (cid:1) U3
dΩ dψ2 +sin2ψ dθ2 +sin2ψsin2θ dφ2 , f(U) 1 KK , (2)
3 ≡ ≡ − U3
3/4
U 2πN
eφ = g , F dC = cǫ , (3)
s 4 3 4
R ≡ V
(cid:18) (cid:19) 4
While this compactified metric is not an exact solution to the general relativity (GR) equa-
tions for small size S3, it can be regarded as an approximate solution for large size S3.
Indeed, in this case, the GR equations are seen to be sourced by terms wich are down by the
size of S3. Here, (3) can be regarded as an approximation to the stable metric with T3 for
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a dense matter analysis. Clearly, the former is unstable against decay to the latter, which
will be reflected by the fact that the energy of dense matter on S3 is higher than that on T3.
As indicated in the introduction, the Skyrmion analysis shows that the energy on S3 is only
few percent that of T3. So we expect the current approximation to capture the essentials
of dense matter in holographic QCD. Specifically, the nature and strength of the attraction
and repulsion in dense matter. Indeed, this will be the case as we will detail below.
Now, consider N probe D8-branes in the N D4-branes background. With U(N ) gauge
f c f
field A on the D8-branes, the effective action consists of the DBI action and the Chern-
M
Simons action
S = S +S ,
D8 DBI CS
S = T d9x e−φ tr det(g +2πα′F ) , (4)
DBI 8 MN MN
− −
Z
1 p
S = C trF3 . (5)
CS 48π3 3
ZD8
where T = 1/((2π)8l9), the tension of the D8-brane, F = ∂ A ∂ A i[A ,A ]
8 s MN M N − N M − M N
(M,N = 0,1, ,8), and g is the induced metric on the D8-branes
MN
···
3/2 3/2
U R
ds2 = ( dt2 +R2dΩ2)+g dσ2 + U2dΩ2 , (6)
D8 R − 3 σσ U 4
(cid:18) (cid:19) (cid:18) (cid:19)
g G ∂ τ∂ τ +G ∂ U∂ U , (7)
σσ ττ σ σ UU σ σ
≡
where G refer to the background metric (1) and the profile of the D8 brane is parame-
MN
terized by U(σ) and τ(σ).
The gauge field A has nine components, A , A = A , A (= A ), and A (α =
M 0 i 1,2,3 σ 4 α
5,6,7,8, the coordinates on the S4). We assume
A = A (σ) U(1) , (8)
0 0
∈
(A = A (xi,σ), A = A (xi,σ)) SU(N ) , (9)
i i σ σ f
∈
A = 0 . (10)
α
3
Then the action becomes 5-dimensional:
8π2T R3
8
S = tr dtǫ dσ U
DBI 3
− 3g
s
Z
3/2 3
U U 1
g (2πα′)2(∂ A )2 + (2πα′)2F Fij
σσ σ 0 ij
R − R 2
" ( )( )
(cid:18) (cid:19) (cid:18) (cid:19)
1/2
3
U 1
+ (2πα′)2F F i + (2πα′)4(ǫ F F )(ǫ Fi Fjk) , (11)
R σi σ 4 ijk iσ jk ijk σ
#
(cid:18) (cid:19)
N
S = c tr A F F , (12)
CS 24π2 ∧ ∧
Z
where ǫ is the volume form of S3 space and the indices i,j,k( ψ,θ,φ ) are raised by the
3
∈ { }
metric g˜ij defined by
1 1 1
g˜ij = , , . (13)
R2 R2sin2ψ R2sin2ψsin2θ
(cid:18) (cid:19)
3 Instanton in S3 R1
×
OnlyA willbedetermineddynamicallyinthegiveninstantonbackgroundA ,A . Theexact
0 i σ
background instanton solution is unknown. Thus we start with an approximate solution
which is the SU(2) Yang-Mills instanton solution in the space with metric,
ds2 = dσ2 +R2dΩ . (14)
3
This metric is different from our metric in (6) and (11), where there are warping factors.
Furthermore our action is the nonlinear DBI action and not a Yang-Mills action. However
it can be shown that the Yang-Mills instanton in the space (14) is the leading order solution
of 1/λ expansion of the full metric and the DBI action as shown in [3] . So the solution can
be used in the leading order calculation.
We summarize here the (anti) self dual instanton solution obatained in [15]. Using the
ansatz ,
A = f(σ)U−1dU , U cosψ +iτ rˆa(θ,φ)sinψ , (15)
a
≡
we get the field strength, in terms of vielbein whose relation to the coordinate ψ,θ,φ, is
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specified in [15],
(∂ f)τ 1 2(f2 f)τ ǫd
F = σ a e0 ea + − d bc eb ec , (16)
R ∧ 2 R2 ∧
(cid:20) (cid:21)
where we used L = U−1∂ U = τ /R. If we require (anti) self-duality,
a a a
2(f2 f)
∂ f = − , (17)
σ R
±
then f is determined as
1
f , (18)
± R
≡ 1+e∓2(σ−σ0)/
so the field strength of one (anti) instanton solution is
τ 1
F± = (∂ f ) a(e0 ea ǫa eb ec) . (19)
σ ± R ∧ ± 2 bc ∧
4 D8 brane plus Instanton
Now that we have the background instanton configurations, the remaining dynamical vari-
ables are τ and A . However it can be shown that ∂ τ(σ) = 0 is always a solution of the
0 σ
Euler-Lagrange equation regardless of the gauge field. For simplicity we will work with this
specific configuration so that the only dynamical variable is A . Let us parameterize A by
0 0
Z defined as
U (U3 +U σ2)1/3 ,
≡ KK KK
σ
Z , K 1+Z2 . (20)
≡ U ≡
KK
Then the field strength is expressed in terms of Z and the dimensionless radius R R/U ,
KK
≡
F = 1 f′ τa , b
Za
U R
KK
1 ǫ cτ
F = f′ ab c , (21)
ab U2 bR
KK
b
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where f′ ∂ f. The instanton configuration is
Z
≡
1
f = . (22)
± Rc
1+e∓2(Z−Z0)/
The DBI action reads
8π2T R3
S = 8 tr dtǫ dZK1/3
DBI 3
− 3g
s
Z
3
4 U 1
U2 K−1/3 (2πα′)2(A′)2 K KK + (2πα′)2F2
9 KK − 0 R 2 ab
" ( )
(cid:26)(cid:18) (cid:19) (cid:27) (cid:18) (cid:19)
1/2
3
U 1
+ K KK (2πα′)2F2 + (2πα′)4(ǫ F F )2 , (23)
R Za 4 abc aZ bc
#
(cid:18) (cid:19)
where A′ ∂ A . We are using the same vielbein coordinates as (16). Since the instanton
0 ≡ Z 0
size (R) is of order O(λ−1/2) we define a new dimensionless parameter R, which is order of
(λ0), as
e
R
R √λR = √λ . (24)
≡ U
KK
e b
Furthermore we rescale the coordinate and the instanton field strength for a systematic 1/λ
expansion
xa λ−1/2xa , Z λ−1/2Z , t t ,
→ → →
F λF , F λF , A A ,
ab ab aZ aZ 0 0
→ → →
1
K = (1+Z2) 1+ Z2 K , (25)
λ
→ λ ≡
(cid:18) (cid:19)
so all coordinates and gauge fields become of order of O(λ0) and we
By using the instanton solution (22) we get
N λ
c 1/3
S = tr dtǫ dZK
DBI −39π5U M−3 3 λ
KK KK Z
37π2 f′2 1 36π2
1+ K1/3 K1/3(A′)2
4M2 U2 λ R2 − λ4M2 λ 0
(cid:20) (cid:26) KK KK KK (cid:27)
37π2 f′2 1/2
M2 U2 K4/3 + K1/3 (26)
KK KK λ 4Me2 U2 λ R2
(cid:26) KK KK (cid:27) (cid:21)
e
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If we let U = M−1 for simplicity, then the DBI action yields
KK KK
S = dN λ dtǫ dZ 1+K1/3F2 1K1/3(A′)2 K3/4 +K1/3F2
DBI − c 3 s λ − λ λ 0 λ λ
Z (cid:26) (cid:27)n o
3Z2 e Z2 e1 ′ e
= dN λ dtǫ dZ 1+ +F2 + F2 (A )2 +O((1/λ)2) , (27)
− c 3 8λ 3λ − 2λ 0
Z (cid:20) (cid:21)
e e e
where
2M4 33π 37π2
d KK , A A , F2 J ,
≡ 39π5 0 ≡ 2M 0 ≡ 4
KK
f′2 sech4(Z/R) 1
e e
J = δ(Z) ,
≡ R2 4R4 ∼ 3R3
e
1 1 1
∂ K ∂ tanh(Z/R) 1+ sech2(Z/R) sgn(Z) (28)
Z e≡ Z6R3 (cid:20)e (cid:18)e 2 (cid:19)(cid:21) ∼ 6R3
e e
The Chern-Simons action does not change by the recaling (25) , and it is order of λ0.
e e
With the instanton solution (22) the Chern-Simons action reduces to
N N 1 f′2
S = c tr A F F = c tr dtǫ dZA 24M3
CS 24π2 ∧ ∧ 8π2 3 02 KKR2
Z Z (cid:18) (cid:19)
= cN dtǫ dZA F2 , (29)
c 3 0
e
Z
e e
where
4M4
c KK . (30)
≡ 39π5
It also can be written as
3
S = 3N R dZA ∂ K N , for A = 1 , (31)
CS c 0 Z c 0
→
Z
e
which confirms that the field configuration (21), and (22) describe the single (anti) self dual
instanton since S corresponds to N the Pontryagin index when A = 1.
CS c 0
×
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1
5 Equation of State in /λ
The equation of state of cold holographic matter is the energy following from the action
functional. The total action up to order of λ0 is
S dtǫ dZ(L +L )
3 DBI CS
≡
Z
Z2 1 ′
= dN dtǫ dZ λF2 + F2 (A )2 +cN d4xdZA F2 . (32)
− c 3 3 − 2 0 c 0
Z (cid:20) (cid:21) Z
e e e e e
whereA isanauxillaryfieldwithnotime-dependencethatcanbeeliminatedbytheequation
0
of motion or Gauss law,
Π′ = cN F2 , (33)
c
e
with
∂L ′
Π = dN A , (34)
≡ ∂A′ c 0
0
e
The integral of the equation of motion weith F2 in (28) is
e 1
Π(Z) = Π( ) tanh(Z/R) 1+ sech2(Z/R) ,
∞ 2
(cid:20) (cid:18) (cid:19)(cid:21)
M4 N
Π( ) = KK c , e e (35)
∞ 54π3 R3
where we have set Π(0) = 0.
e
The energy of one cell is
E = ǫ dZ(L +L )
cell 3 DBI CS
−
Z
Z2 1 Π2
= dN ǫ dZ λF2 + F2 ǫ dZA Π′ ,
c 3 3 − 2(dN )2 − 3 0
Z (cid:20) c (cid:21) Z
Z2 1 Π2 ∞
= dN ǫ dZ λeF2 + eF2 + ǫ Ae(Z)Π(Z) . (36)
c Z 3 (cid:20) 3 2(dNc)2(cid:21)−Z 3 0 (cid:12)−∞
(cid:12)
e e e
(cid:12)
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Thus the energy density (ε) of the crystalline structure is
NE E
cell cell
ε
≡ V ≈ ǫ
3
Z2 1 Π2 ∞
= dN dZR λF2 + F2 + A (Z)Π(Z) , (37)
cZ (cid:20) 3 2(dNc)2(cid:21)− 0 (cid:12)−∞
(cid:12)
e e e
(cid:12)
where N is the total number of baryons(cells) and V is the total volume which is approx-
imated by N ǫ . Interestingly the second term in (37) is equal to µ n since the density
3 B B
and the baryon chemical potential is given by
R
1 1 1
n = = = , µ N A ( ) . (38)
B ǫ3 2π2(UKKR)3 2π2(√λR)3 B ≡ c 0 ∞
R
respectively and because e
∞ 1
A (Z)Π(Z) = 2A ( )Π( ) = N A ( ) = µ n , (39)
0 0 c 0 B B
−∞ ∞ ∞ ∞ 2π2(U R)3
KK
(cid:12)
(cid:12)
e e
(cid:12)
Notice that R is of order of (λ)0 from (24) so the baryon densitey is of order (N λ)0. Since
c
the action is finite and concentrated in a finite size, we can restrict the integral to the region
e
Z Z and expand the action in 1/λ.
c
≤
Zc Z2 1 Π2
ε = dN dZ λF2 + F2 + (40)
c 3 2(dN )2
Z0 (cid:20) c (cid:21)
a b
= M n + n1/3e+ Z ne2 , (41)
0 B λ B λ c B
(cid:20) (cid:21)
where
λN
M 8π2κM , κ c ,
0 ≡ KK ≡ 216π3
(π2 6)M2 36π4
a − KK , b , (42)
≡ 36(2π2)2/3 ≡ 2M3
KK
A few remarks are in order.
1. Z isintroducedasanarbitrarycut-offwhichisbiggerthantheinstantonsize. However
c
in section 8, we will argue that Z should be identified as baryon size by explicitly
c
contructing the Skyrmion out of the instanton Therefore it is not an arbitrary number.
2. Even in the case the instanton size is small, the baryon size on the boundary is not.
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