Chiral Symmetry Breaking in Brane Models

0.41 MB·

by  Norio Horigome

STUPP–07–193
October, 2007
8 Chiral Symmetry Breaking in Brane Models
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n
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J Norio Horigome,a∗ Madoka Nishimurab and Yoshiaki Taniia†

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aDivision of Material Science
]
h Graduate School of Science and Engineering
t
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Saitama University, Saitama 338-8570, Japan
p
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h bDepartment of Community Service and Science
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Tohoku University of Community Service and Science
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v Iimoriyama 3-5-1, Sakata 998-8580, Japan
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Abstract
.
0
1
7 We discuss the chiral symmetry breaking in general intersecting
0 Dq/Dp brane models consisting of N Dq-branes and a single Dp-brane
: c
v with an s-dimensional intersection. There exists a QCD-like theory lo-
i
X calized at the intersection and the Dq/Dp model gives a holographic de-
r scription of it. The rotational symmetry of directions transverse to both
a
of the Dq and Dp-branes can be identified with a chiral symmetry, which
is non-Abelian for certain cases. The asymptotic distance between the
Dq-branes and the Dp-brane corresponds to a quark mass. By studying
the probe Dp-branedynamics in a Dq-branebackground in the near hori-
zon and large N limit we find that the chiral symmetry is spontaneously
c
broken and there appear (pseudo-)Nambu-Goldstone bosons. We also
discuss the models at finite temperature.
∗ e-mail: [email protected]
† e-mail: [email protected]
1. Introduction
The AdS/CFT correspondence [1, 2, 3] (see [4] for a review) provides a new non-
perturbative approach to strongly coupled gauge theories. This duality relates a
string theory in(d+1)-dimensional anti de Sitter spacetime (times a compact space)
to a d-dimensional conformal field theory. The AdS/CFT correspondence can be
extended to the string/gauge duality, which is a generalization to non-conformal
and non-supersymmetric theories. The string/gauge duality also provides us with a
useful tool for the analysis of low energy behaviors of QCD such as the confinement
and the spontaneous chiral symmetry breaking. This approach is often called the
holographic QCD [5, 6, 7, 8, 9, 10, 11, 12, 13, 14] (and references therein).
OneofthemostinterestingphenomenaofthelowenergyQCDisthespontaneous
breaking of the chiral symmetry. In the holographic approach the chiral symmetry
can be realized in two different ways. In this approach one introduces N color Dq-
c
branes and N flavor Dp-branes. The U(N ) gauge field on the Dq-branes represents
f c
a gluon field of a QCD-like theory. Open strings connecting the Dq-branes and the
Dp-branesrepresent quarksinthefundamentalrepresentationofU(N ). Whenthese
c
brane configurations have directions transverse to both of the Dq and Dp-branes, a
rotational symmetry in these directions can be understood as a chiral symmetry of
the dual gauge theories in certain cases [7, 8, 9, 10, 11, 12]. One can separate color
branes and flavor branes in such directions. The asymptotic distance between these
branes is identified with a quark mass. So one can study the spontaneous chiral
symmetry breaking starting from a theory with a non-vanishing quark mass and
taking the massless limit. So far only the Abelian chiral symmetry U(1) × U(1)
V A
is considered in this approach.
Alternatively, the chiral symmetry can be realized as a gauge symmetry on the
flavor branes. When Dp-Dp-brane pairs are used as flavor branes, one can obtain
a non-Abelian U(N ) × U(N ) chiral symmetry [13, 14, 15, 16, 17, 18, 19, 20].
f L f R
These configurations of physical interest often do not have directions transverse to
both of the color and flavor branes. Therefore, it is not obvious how to introduce a
quark mass in these models. For work toward an introduction of a quark mass in
this type of models and on related issues see refs. [21, 22, 23, 24, 25]. In both of
these two approaches, the spontaneous breaking of the chiral symmetry is closely
related to the configurations of the probe branes in the background geometry.
The chiral symmetry breaking was also discussed at finite temperature [7, 8, 26,
2
27, 28, 29, 30, 31] and at finite chemical potential [32, 33, 34, 35, 36, 37]. The
temperature T is related to a period δt of the S1 compactified Euclidean time
E
coordinate as T = 1/δt . The chemical potential µ is introduced as a non-vanishing
E
asymptotic value of the time component of the gauge field on the probe brane
A ∼ µ. One can study a chiral phase transition and obtain a phase diagram of the
0
QCD-like theories.
The purpose of the present paper is to study the chiral symmetry breaking in
general intersecting Dq/Dp brane systems consisting of N color Dq-branes and
c
a single probe Dp-brane with an s-dimensional intersection. They are holographic
duals of QCD-like theories in (s+1)-dimensional spacetime QCD in certain cases.
s+1
As in refs. [7, 8] these models can have directions transverse to both of the Dq
and Dp-branes. A rotational symmetry of these directions can be interpreted as a
chiral symmetry in certain cases. This symmetry can be non-Abelian in contrast
to the models in refs. [7, 8]. We can separate the Dq-branes and the Dp-brane in
these transverse directions and break the rotational symmetry. In the holographic
description this deformation makes quarks on the intersection massive and leads to
anexplicit chiralsymmetrybreaking. InthenearhorizonlimitandthelargeN limit
c
we can treat the Dq-branes as a background geometry and the Dp-brane as a probe
which does not affect this background. We discuss the chiral symmetry breaking by
analyzing the Dp-brane dynamics in the Dq-brane background geometry.
The organization of this paper is as follows. In sect. 2 we study the low energy
spectrum at an s-dimensional intersection of the Dq/Dp brane system. In general
dual theories are defect field theories [38, 39]. We are interested in field theories
without defects. There are systems corresponding to QCD-like theories at the inter-
section. In particular, the D2/D4 model with s = 1, the D3/D5 model with s = 2
and the D4/D6 model with s = 3 correspond to QCD , QCD and QCD , respec-
2 3 4
tively. For certain (q,p,s) the rotational symmetry of the transverse directions can
be understood as a chiral symmetry in the QCD-like theories. This chiral symmetry
is non-Abelian SU(2) × SU(2) for QCD .
L R 2
In sect. 3 we study the chiral symmetry breaking in the QCD-like theories by
using a supergravity analysis. The near horizon limit and the large N limit N ≫ 1
c c
allow us to study the probe Dp-brane dynamics in the Dq-brane background. We
find that a Dp-brane embedding breaks the rotational symmetry of the transverse
directions. Thiscorrespondstoachiralsymmetry breakingintheQCD-liketheories.
¯
The quark mass m and the quark condensate ψψ can be read from the asymp-
q
(cid:10) (cid:11)
3
totic behavior of the Dp-brane embedding. There is a non-zero value of the quark
¯
condensate ψψ even for the massless quark limit. This leads to a spontaneous
chiral symmetry breaking in the QCD-like theories.
(cid:10) (cid:11)
Insect.4weconsider fluctuationsoftheDp-branearoundthevacuumembedding
discussed in sect. 3. For m = 0 there appear (8−q−p+s) massless scalar bosons,
q
which can be understood as the Nambu-Goldstone (NG) bosons associated with
the spontaneous symmetry breaking. For a non-zero but small quark mass there
appear pseudo-NG bosons. We show that these pseudo-NG bosons satisfy the Gell-
Mann-Oakes-Renner (GMOR) relation [40]. The effective action of the fluctuations
is obtained at quartic order.
In sect. 5 we discuss the theories at finite temperature. We study the probe Dp-
brane dynamics in the Euclidean Dq background. The Dp-brane embedding breaks
the rotational symmetry as in the zero temperature case. Then the chiral symmetry
is also broken at finite temperature. We find that the quark condensate vanishes
and the chiral symmetry is restored only in the high temperature limit. We also
study the models with s = q. We conclude in sect. 6.
2. General setup
We consider an intersecting brane system consisting of N color Dq-branes and a
c
single probe Dp-brane
x0 ··· xs xs+1 ··· xq xq+1 ··· xq+p−s xq+p−s+1 ··· x9
N Dq ◦ ··· ◦ ◦ ··· ◦ − ··· − − ··· −
c
Dp ◦ ··· ◦ − ··· − ◦ ··· ◦ − ··· −
(2.1)
with xq being a coordinate of S1. It has an s-dimensional intersection in the di-
rections x1,··· ,xs. The configuration (2.1) is a T-dual of Ds′/D9 system with
s′ = 9−(q +p−2s) ≥ s. Following ref. [19] we call it a transverse intersection if
s′ = s (q +p −s = 9) and a non-transverse intersection if s′ > s (q +p −s < 9).
Non-transverse intersections have directions transverse to bothof the Dq-branesand
the Dp-brane, while transverse intersections do not.
The configuration (2.1) has the following symmetries. The gauge symmetry
of this system is U(N ) × U(1). The U(1) gauge symmetry on the Dp-brane is
c
4
regarded as a global symmetry (baryon number symmetry) of an (s+1)-dimensional
field theory at the intersection. The ten-dimensional Lorentz symmetry SO(1,9) is
broken to its subgroup by the configuration (2.1). Therefore the global symmetry
preserved at the intersection contains
SO(1,s)×SO(9−q −p+s)×U(1), (2.2)
where SO(1,s) is the Lorentz symmetry at the intersection and SO(9−q−p+s) is
the rotational symmetry in the directions xq+p−s+1,··· ,x9.
The spectrum of the theory localized at the intersection is as follows. Massless
fields generated by q-q strings (open strings having both ends on the Dq-branes) are
a gauge field A (µ = 0,1,··· ,s), scalar fields Φi (i = s +1,··· ,9) and fermionic
µ
fields S. Imposing the periodic boundary condition for the bosonic fields and the
anti-periodiconeforthefermionicfieldsalongthecompactxq direction, thefermions
become massive at zero mode and supersymmetry is explicitly broken at low energy.
Then the scalars acquire mass at one-loop level. Thus only the gauge field A is
µ
massless at low energy. This gives a pure U(N ) gauge theory.
c
To study the lowest modes generated by q-p strings (open strings connecting the
Dq-branes and the Dp-brane), we note the zero-point energy in the R sector and the
NS sector [41]
#ND−4
aR = 0, aNS = , (2.3)
8
where #ND = q + p − 2s = 9 − s′ is the number of spatial coordinates of open
strings which have the Neumann boundary condition for one end and the Dirichlet
one for the other end. The lowest modes generated by q-p strings in the NS sector
are massive for #ND > 4 (#ND = 6, 8), massless for #ND = 4 and tachyonic for
#ND < 4 (#ND = 0, 2). We do not consider the tachyonic case #ND < 4. When
#ND ≥ 4, the lowest modes from the NS sector are massive (by loop effects for
#ND = 4) and are decoupled at low energy. There are only massless fermions from
the R sector. They belong to representations of the Clifford algebra for the NN and
DD directions. These fermions belong to the fundamental representation of U(N )
c
and are called “quarks”.
In general the Dq/Dp configuration (2.1) is dual to a defect field theory [38, 39].
We only consider the case s+1 = q, which corresponds to a theory without defects.
We are interested in non-transverse intersections satisfying s′ > s, which implies
s < 9−#ND. Possible cases are s = 1, 2, 3, 4 for #ND = 4 (p = q+2) and s = 1,
5
0 1 2 3 4 5 6 7 8 9
color D2 ◦ ◦ ◦ − − − − − − −
probe D4 ◦ ◦ − ◦ ◦ ◦ − − − −
color D3 ◦ ◦ ◦ ◦ − − − − − −
probe D5 ◦ ◦ ◦ − ◦ ◦ ◦ − − −
color D4 ◦ ◦ ◦ ◦ ◦ − − − − −
probe D6 ◦ ◦ ◦ ◦ − ◦ ◦ ◦ − −
Table 1: The Dq/D(q+2) brane configurations with #ND = 4. From top to bottom
these are dual to QCD , QCD and QCD , respectively.
2 3 4
2 for #ND = 6 (p = q + 4). The configurations with #ND = 6 do not preserve
supersymmetry and are most likely unstable. We further restrict ourselves to the
cases s = 1, 2, 3 since we are especially interested in theories in four and lower
dimensions. To summarize, we consider the Dq/D(q+2) configurations for q = 2, 3,
4 compactified on xq shown in Table 1. The effective theory on the intersection at
low energy is an (s+1)-dimensional non-supersymmetric U(N ) gauge theory with
c
quarks in the fundamental representation. We call this theory “QCD ” for the
s+1
sake of convenience.
Since these configurations are non-transverse intersections, there are directions
transverse to both of the Dq-branes and the Dp-brane. In refs. [7, 8] the rotational
symmetry SO(9−q −p+s) of such directions is interpreted as a chiral symmetry
in the dual gauge theory for certain sets of (q,p,s). When the Dq-branes and the
Dp-brane are separated along these directions, quarks on the intersection become
massive and the chiral symmetry is explicitly broken. As we will see in sect. 3, only
when aNS = 0 (#ND = 4) andp−s−2 > 0, which aresatisfied forthe configurations
in Table 1, an equation for probe brane embeddings derived from the DBI action
has a solution for which the distance between the color branes and the probe brane
asymptotically approaches a constant value. This distance is interpreted as a quark
mass.
We can explicitly write down the symmetry (2.2) for the configurations in Table
1. In the D2/D4 model, which has a one-dimensional intersection and is dual to
QCD , we can identify the SO(4) rotational symmetry in the x6, x7, x8, x9
2 6789
directions with an SU(2) × SU(2) chiral symmetry of quarks. Indeed, the GSO
L R
projection in the R sector of open strings requires that the chiralities of SO(1,1)
01
and SO(4) are correlated. Left-handed (right-handed) quarks of SO(1,1) have
6789 01
6
the positive (negative) chirality of SO(4) and transform as (2,1) ((1,2)) under
6789
SU(2) ×SU(2) . The gauge symmetry U(1) on the probe brane acts on quarks as
L R
a baryon number symmetry U(1) . Therefore the global symmetry (apart from the
V
Lorentz symmetry) of QCD at the intersection is
2
SO(4) ×U(1) ∼ SU(2) ×SU(2) ×U(1) . (2.4)
6789 L R V
Thus we can realize a non-Abelian chiral symmetry in a holographic model of this
type, although spacetime is two-dimensional.
In the D3/D5 model, which has a two-dimensional intersection and is dual to
QCD , we can identify the SO(3) rotational symmetry in the x7, x8, x9 directions
3 789
with an SU(2) symmetry of QCD . Then the global symmetry of QCD at the
3 3
intersection is
SO(3) ×U(1) ∼ SU(2)×U(1). (2.5)
789
Quarks transform as 2 under SU(2). Note that there is no chirality in QCD and
3
therefore the symmetry (2.5) is not a chiral symmetry.
Finally, in the D4/D6 model, which has a three-dimensional intersection and
is dual to QCD , we can identify the SO(2) rotational symmetry in the x8, x9
4 89
directions with an axial U(1) symmetry of QCD as discussed in refs. [7, 8]. The
A 4
global symmetry of QCD at the intersection is
4
SO(2) ×U(1) ∼ U(1) ×U(1) . (2.6)
89 A V
3. Chiral symmetry breaking from supergravity analysis
The dynamics of a strongly coupled large N gauge theory can be analyzed by
c
supergravity. We study the chiral symmetry breaking in this section. The near
horizon limit and the large N limit N ≫ 1 allow us to treat the Dq-branes as
c c
a background geometry and the Dp-brane as a probe which does not affect this
background. We will find that the Dp-brane embedding breaks the SO(9−q −p+
s) rotational symmetry in the directions transverse to both of the branes. This
can be interpreted as the chiral symmetry breaking in QCD and QCD . We will
2 4
calculate the quark condensate and find a non-zero value even in the massless quark
limit. Although we are most interested in the configurations in Table 1, we will give
formulae for the configuration (2.1) with general q, p, s.
7
3.1 Dq-brane background
The near horizon geometry of S1 compactified N Dq-branes is [42]
c
7−q q−1 7−q
ds2 = U 2 −dt2 + dxi 2 +f(U)(dxq)2 + R 2 dU2 +U2dΩ2 ,
R U f(U) 8−q
!
(cid:18) (cid:19) i=1 (cid:18) (cid:19) (cid:18) (cid:19)
X(cid:0) (cid:1)
7−q
f(U) = 1− UUKK , R7−q = (4π)5−2q Γ 7−2q gsNc ℓs7−q, (3.1)
(cid:18) (cid:19)
(cid:0) (cid:1)
where dΩ2 , g and ℓ are the metric of a unit S8−q, the string coupling and the
8−q s s
string length, respectively. xq is a coordinate of S1 and its period is denoted as
δxq = 2π/M . M is a compactification scale. To avoid a conical singularity at
KK KK
U = U in the U-xq plane the period must be related to a constant U as
KK KK
7−q
4π R
δxq = 2 . (3.2)
5−q
(7−q) U 2
KK
The dilaton field and the Ramond-Ramond (RR) flux are given by
(7−q)(3−q)
R 4 N
eφ = g , F = c ǫ , (3.3)
s 8−q 8−q
U V
(cid:18) (cid:19) 8−q
where ǫ and V are the volume form and the volume of a unit S8−q.
8−q 8−q
The relations between the parameters in the gauge theory and those in the string
theory are
5−q
7−q U 2
g2 = (2π)q−2g ℓq−3, M = KK , (3.4)
q+1 s s KK 2(4π)5−4q Γ(7−2q)21 (gsNc)12 ℓs7−2q
where g is the (q + 1)-dimensional gauge coupling. The (q + 1)-dimensional
q+1
’t Hooft coupling is defined as
g2 N
q+1 c
λ = . (3.5)
q+1 (2π)q−2
Note that the supergravity description is valid for [42, 19]
q−3
U 4
1 ≪ λ KK ≪ N7−q. (3.6)
q+1 ℓ2 c
(cid:18) s (cid:19)
8
We introduce isotropic coordinates in the directions (U,Ω ) to simplify the
8−q
following analysis. Introducing a new radial coordinate ρ defined by
2
U = ρ7−2q + UK77−−Kqq 7−q , ρ2 = 9 (xα)2 (3.7)
4ρ 2 ! α=q+1
X
the metric for the transverse space (U,Ω ) in eq. (3.1) can be written as
8−q
7−q
R 2 dU2
+U2dΩ2 = K(ρ) dρ2 +ρ2dΩ2
U f(U) 8−q 8−q
(cid:18) (cid:19) (cid:18) (cid:19)
(cid:0) (cid:1)
9
= K(ρ) (dxα)2, (3.8)
α=q+1
X
where
7−q q−3
R U
2 2
K(ρ) = . (3.9)
ρ2
We divide the coordinates xq+1,··· ,x9 into two parts and introduce spherical co-
ordinates (λ,Ω ) for the xq+1,··· ,xq+p−s directions and (r,Ω ) for the
p−s−1 8−q−p+s
xq+p−s+1,··· ,x9 directions. Then the Dq background becomes
7−q q−1
U 2
ds2 = −dt2 + (dxi)2 +f(U)(dxq)2
R
!
(cid:18) (cid:19) i=1
X
+K(ρ) dλ2 +λ2dΩ2 +dr2 +r2dΩ2 , (3.10)
p−s−1 8−q−p+s
where ρ2 = λ2 + r2. We wil(cid:0)l wrap the probe Dp-brane around Sp(cid:1)−s−1 in the next
subsection.
3.2 Dp-brane embeddings
We study the dynamics of a Dp-brane in the Dq background. In the limit N ≫ 1
c
the Dp-braneis introduced into the Dq background as a probe, which does not affect
the background geometry. The dynamics of the probe Dp-brane in the background
(3.10) is described by the Dirac-Born-Infeld (DBI) action
S = −T dp+1xe−φ −detg , (3.11)
Dp p MN
Z
p
where g (M,N = 0,1,··· ,p) is the induced metric on the world-volume and T
MN p
is the tension of the Dp-brane. For simplicity we have ignored the gauge field on
the probe Dp-brane.
9
We use a physical gauge for Dp-brane world-volume reparametrizations and use
the spacetime coordinates xµ (µ = 0,1,··· ,s), λ, Ω as the world-volume co-
p−s−1
ordinates. Then the configurations of the Dp-brane are determined by xi (i =
s+1,··· ,q), r and Ω as a function of those world-volume coordinates. We
8−q−p+s
make an ansatz
xs+1,··· ,xq = constant, r = r(λ), θa = constant, (3.12)
where θa (a = 1,2,··· ,8−q −p+s) are coordinates of S8−q−p+s.
With this ansatz, the induced metric on the Dp-brane is
7−q
U 2
ds2 = η dxµdxν +K(ρ) 1+(r′)2 dλ2 +λ2dΩ2 , (3.13)
R µν p−s−1
(cid:18) (cid:19)
(cid:2)(cid:0) (cid:1) (cid:3)
where r′ = dr. Then the DBI action of the Dp-brane becomes
dλ
U7−q β
S = −T˜V ds+1x dλ ρα 1+ KK λp−s−1 1+(r′)2, (3.14)
Dp p p−s−1 4ρ7−q
Z Z (cid:18) (cid:19)
p
where T˜ ≡ g−1T R−α and V is the volume of Sp−s−1. The parameters α and β
p s p p−s−1
are defined as
1 1 2(p−s)
α = (7−q)(4+2s−q −p) , β = (4+2s−q −p)+ . (3.15)
4 2 7−q
The action (3.14) leads to the equation of motion for r(λ)
d U7−q β λp−s−1 r′ ∂ U7−q β
ρα 1+ KK = ρα 1+ KK λp−s−1 1+(r′)2.
dλ " 4ρ7−q 1+(r′)2# ∂r " 4ρ7−q #
(cid:18) (cid:19) (cid:18) (cid:19)
p
(3.16)
p
As in refs. [7, 8] we are interested in the situation in which the asymptotic
distance between the Dq-branes and the Dp-brane is a finite constant r . This
∞
constant is proportional to the quark mass. Therefore we impose the boundary
conditions for λ → ∞
r(λ)| = r , r′(λ)| = 0. (3.17)
λ→∞ ∞ λ→∞
Then, eq.(3.16) can be linearized at large λ as
d
λα+p−s−1r′ = α λα+p−s−3r, (3.18)
dλ
(cid:0) (cid:1)
10