Gauge symmetry breaking in ten-dimensional Yang-Mills theory dynamically compactified on S^6

Abstract

We study fluctuation modes in ten-dimensional Yang-Mills theory with a higher derivative term for the gauge field. We consider the ten-dimensional space-time to be a product of a four-dimensional space-time and six-dimensional sphere which exhibits dynamical compactification. Because of the isometry on S^6, there are flat directions corresponding to the Nambu-Goldstone zero modes in the effective theory on the solution. The zero modes are absorbed into gauge fields and form massive vector fields as a consequence of the Higgs-Kibble mechanism. The mass of the vector fields is proportional to the inverse of the radius of the sphere and larger than the mass scale set by the radius because of the higher derivative term. Comment: 18 pages

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arXiv:0912.3128v2 [hep-th] 21 Dec 2009
KIAS-P09057
TIT/HEP-601
Gauge symmetry breaking in ten-dimensional Yang-Mills theory
dynamically compactified on S6
Pravabati Chingangbam∗
Korea Institute for Advanced Study, 207-43 Cheongnyangni 2-dong,
Dongdaemun-gu, Seoul 130-722, Republic of Korea
Hironobu Kihara1,3†and Muneto Nitta2,3‡
1Faculty of Science and Interactive Research Center of Science,
Graduate School of Science and Engineering, Tokyo Institute of Technology,
2-12-1 Oh-okayama, Meguro, Tokyo 152-8551, Japan
2Department of Physics, and3Research and Education
Center for Natural Sciences, Keio University,
4-1-1 Hiyoshi, Yokohama, Kanagawa 223-8521, Japan
(Dated: December 16, 2009)
Abstract
We study fluctuation modes in ten-dimensional Yang-Mills theory with a higher derivative term
for the gauge field. We consider the ten-dimensional space-time to be a product of a four-
dimensional space-time and six-dimensional sphere which exhibits dynamical compactification.
Because of the isometry on S6, there are flat directions corresponding to the Nambu-Goldstone
zero modes in the effective theory on the solution. The zero modes are absorbed into gauge fields
and form massive vector fields as a consequence of the Higgs-Kibble mechanism. The mass of the
vector fields is proportional to the inverse of the radius of the sphere and larger than the mass
scale set by the radius because of the higher derivative term.
∗prava(at)kias.re.kr
†hkihara(at)th.phys.titech.ac.jp
‡nitta(at)phys-h.keio.ac.jp
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I.INTRODUCTION
The idea of extra dimensions and their compactification have been attracting people [1] for
several decades. Among many scenarios of compactification of extra dimensions, Cremmer
and Scherk suggested an interesting idea that compactification may occur if a topologically
non-trivial gauge configuration exists in compactified space [2]. An example is given by
’t Hooft-Polyakov monopole [3] on S2. Recently in [4] some of us have studied a scenario
of dynamical compactification and inflation in ten-dimensional Einstein-Yang-Mills theory
with SO(6) gauge group, using the Cremmer-Scherk gauge configuration on S6[5, 6]. We
have added a higher derivative coupling term,∗originally introduced by Tchrakian [7], in
order to ensure the non-existence of tachyonic modes on the Cremmer-Scherk configuration
because of the Bogomol’nyi equation.
In this paper we study the issue of the stability and fluctuations of the Cremmer-Scherk
configuration in this scenario. We will work on the space-time given by the product of a
four-dimensional space-time and a six-dimensional sphere, where the radius of the sphere
shrinks to a constant value in the limit t → +∞. We assume that the four-dimensional
space-time can be treated as {(t,Nt)}t∈R, where Ntis diffeomorphic to a three-dimensional
manifold N.†
We first show the absence of tachyonic modes in general background gauge fields, and
then study if there are massless modes or not. In general the Cremmer-Scherk configuration
is obtained by the identification of compact direction and internal (gauge) direction as an
extension of ’t Hooft-Polyakov monopole. By rotating this identification, with the rotation
depending on R1,3, we obtain massless fluctuation modes which can be regarded as Nambu-
Goldstone bosons [9]. Then we show that these massless modes are actually absorbed into
gauge fields to form massive vector (Proca) fields by the Higgs-Kibble mechanism [10]. Since
the number of Nambu-Goldstone modes, fifteen, coincides with the dimension of SO(6), we
find that the gauge symmetry SO(6) is completely broken. By scaling fields for normalization
of the coefficient of their kinetic terms, we obtain the mass proportional to the inverse of the
radius of the compact space. We thus conclude that there are neither tachyonic nor massless
∗Such a quartic term is known to appear in the low-energy effective theory of quantum electro dynamics
too [8].
†For instance, four-dimensional Friedmann-Lemaitre-Robertson-Walker space-time satisfies the condition.
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modes in the physical spectrum around the background and that the configuration is stable.
Similar mechanisms for generating masses in compactifications have been discussed in the
literature. Scherk and Schwarz introduced mass by a generalized dimensional reduction or
a twisted boundary condition of fields [11]. Horvath et. al. considered system coupled with
Higgs fields to obtain mass of gauge fields [12]. Manton showed that the components of gauge
fields along extra-dimensions provide Higgs fields in the four dimensional effective theory
without additional scalar fields [13, 14]. Hosotani found a mechanism for obtaining Higgs
fields belonging to representations different from the adjoint representation and endowing
mass to fermions on orbifold [15].
This paper is organized as follows: in Sec. II we will study the general treatment of
the fluctuation of gauge fields about some classical solution. In the effective theory on the
classical background, the fluctuation of the ten-dimensional vector field splits into two parts,
a four-dimensional vector field and six scalar fields. We will see that the lowest Kaluza-Klein
mode of the four-dimensional vector field gets mass because of the gauge configuration. In
Sec. III we review the Cremmer-Scherk gauge configuration and the Tchrakian’s self-duality
equation. In Sec. IV we show the Higgs mechanism during the dynamical compactification.
By considering rotated identification between the internal (gauge) space and the compact
space (S6), we explicitly show that the Nambu-Goldstone bosons form the Proca fields with
vector fields. In Sec. V we summarize this article. In Appendix A, we prove the non-
existence of covariantly constant functions belonging to the adjoint representation on the
Cremmer-Scherk configuration.
II.FLUCTUATIONS AROUND SOLUTIONS OF BOGOMOL’NYI EQUATION
Let us consider the SO(6) Yang-Mills theory on the ten-dimensional curved space-time
which is a direct product of a four-dimensional space-time M and a six-dimensional sphere
S6, whose radius varies. We assume that the four-dimensional space-time M is the form of
M = {(t,Nt)}t∈R, where Ntis diffeomorphic to a three-dimensional space N, and that any
tangent vectors on the time slice Ntare space like. The metric on the ten-dimensional space
is given as
ds2= gµν(x)dxµdxν+ gij(x,y)dyidyj,gij(x,y) = R2(x)2
δij
(1 + |y|2/4)2.(1)
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The indices are µ,ν = 0,7,8,9 and i,j = 1,··· ,6. xµrepresent four-dimensional coordinates
along the four-dimensional space-time M and yirepresent six-dimensional coordinates along
the sphere. We denote XM≡ (xµ,yi) for the coordinates of total space-time and their
indices are represented by capital scripts. R2(x) is the radius of the six-dimensional sphere
and depends on the four-dimensional coordinates x. We consider only the case where R2(x)
converges to a nonzero constant value in the limit t → +∞. We start from the following
action,
S :=
1
16
?
Tr?−F ∧ ∗F + α2(F ∧ F) ∧ ∗(F ∧ F)?
, (2)
where F is the field strength two-form and α is the quartic coupling constant. The second
term quartic in F is the term introduced by Tchrakian [7] which we call the Tchrakian
term. This action is quadratic in the time derivative acting on the gauge field, ∂A/∂t.
We consider the gauge potential one-form A taking a value in so(6). In our notation the
generators of so(6) are represented with spinor indices, but one should not confuse them with
spinor fields. In order to define the product of field strength two-form, we have to indicate
the representation matrix of gauge fields. Let us use the Clifford algebra {γa,γb} = 2δab,
(a,b = 1,2,··· ,6) with respect to SO(6). Here γaare Hermitian 8×8 matrices. The internal
indices are represented by a,b,c,···. This is independent of the space indices along the six-
dimensional sphere. Commutators γab:= (1/2)[γa,γb] of γ-matrices satisfy the commutation
relation of so(6) and their normalization is shown as follows,
[γab,γcd] = 2(δbcγad− δbdγac− δacγbd+ δadγbc) ,
Trγabγcd= 8(δbcδad− δacδbd) .(3)
The anticommutation relation of these generators γabis given by
{γab,γcd} = 2γabcd+ 2(δbcδad− δacδbd) ,(4)
where we have used the following notation,
γa(1)···a(p):=1
p!
?
σ∈Sp
sgn(σ)γa(σ(1))γa(σ(2))···γa(σ(p)), (5)
where Spis the symmetric group of p characters. Further, we use the matrix γ7:= −iγ123456.
The gauge potential can then be written as
A :=1
2Aab
MγabdXM. (6)
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The field strength is defined as F := dA + gA ∧ A, with gauge coupling constant g. The
equation of motion of the Lagrangian (2) reads
D?∗F − α2(F ∧ ∗(F ∧ F) + (F ∧ F) ∧ F)?= 0 .
Let us suppose that A(0)is some solution of Eq. (7), which we fix to be our background
(7)
solution. We make the assumtions that A(0)has components only along the compact di-
rections and depends only on yi. Then it follows that the corresponding field strength,
F(0):= dA(0)+ gA(0)∧ A(0), has components only along the compact directions.
Let us consider fluctuations δA around the solution A(0), as
A = A(0)+ δA , δA = v + Φ ,v = vµdxµ,Φ = Φidyi
(8)
The fluctuation δA is divided into two parts, v and Φ. v is a one-form whose components are
nonzero only along the four-dimensional space-time, while Φ has components only along the
six-dimensional sphere. The coefficients depend on x and y, vµ:= vµ(x,y), Φi:= Φi(x,y).
Our objective is then to obtain the four-dimensional effective theory for these fluctuations.
The fluctuation v is a vector field and the fluctuation Φi,(i = 1,2,··· ,6), are six scalar fields
under the general coordinate transformation of the four-dimensional space-time. Each vµor
Φibelongs to the adjoint representation of the gauge group SO(6) and (Φab
1,Φab
2,··· ,Φab
6)
transform as vectors under the rotation of the six-dimensional space. Let us decompose the
field strengths in terms of v and Φ,
F = F(0)+ dδA + g?A(0)∧ δA + δA ∧ A(0)?+ gδA ∧ δA
= F(0)+ d(4)v + d(4)Φ + d(6)v + d(6)Φ
+ g?A(0)∧ v + v ∧ A(0)?+ g?A(0)∧ Φ + Φ ∧ A(0)?
+ g (v ∧ v + v ∧ Φ + Φ ∧ v + Φ ∧ Φ)(9)
where the exterior derivative d is decomposed into two parts, d = d(4)+d(6). These operators
are defined as
d(4):= dxµ∂
∂xµ,d(6):= dyi∂
∂yi. (10)
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