Variations of the Hidden Sector in a Realistic Intersecting Brane Model
ACT-02-07, MIFP-07-11
Variations of the Hidden Sector in a Realistic
Intersecting Brane Model
Ching-Ming Chen,1 Tianjun Li,1, 2 V. E. Mayes,1 and Dimitri V. Nanopoulos1, 3, 4
1George P. and Cynthia W. Mitchell Institute for Fundamental Physics,
Texas A&M University, College Station, TX 77843, USA
2 Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100080, China
3Astroparticle Physics Group, Houston Advanced Research Center (HARC),
Mitchell Campus, Woodlands, TX 77381, USA
4Academy of Athens, Division of Natural Sciences,
28 Panepistimiou Avenue, Athens 10679, Greece
(Dated: November 1, 2018)
Abstract
Recently, we discussed the first example of a phenomenologically realistic intersecting D6-brane
model. In this model, the gauge symmetry in the hidden sector is USp(2)1×USp(2)2×USp(2)3×
USp(2)4. However, we find that the USp(2)1 × USp(2)2 gauge symmetry can be replaced by an
U(2)12 gauge symmetry, and/or the USp(2)3 × USp(2)4 gauge symmetry can be replaced by an
U(2)34 gauge symmetry since the USp(2)
2 stacks of D6-branes contribute to the same Ramond-
Ramond tadpoles as those of the U(2) stacks. Thus, there are three non-equivalent variations
of the hidden sector, and the corresponding gauge symmetries are U(2)12 × USp(2)3 × USp(2)4,
U(2)34 × USp(2)1 × USp(2)2, and U(2)12 × U(2)34, respectively. Moreover, we study the hidden
sector gauge symmetry breaking, discuss how to decouple the additional exotic particles, and briefly
comment on the phenomenological consequences.
PACS numbers: 11.10.Kk, 11.25.Mj, 11.25.-w, 12.60.Jv
http://arxiv.org/abs/0704.1855v1
I. INTRODUCTION
The goal of string phenomenology is to construct realistic standard-like string models with
all moduli stabilized. In the early days, string model building was mainly concentrated on the
weakly coupled heterotic string theory. After the second string revolution, consistent four-
dimensional chiral models with non-Abelian gauge symmetry on Type II orientifolds were
able to be constructed due to the advent of D-branes [1]. In particular, Type II orientifolds
with intersecting D-branes, where the chiral fermions arise from the intersections of D-branes
in the internal space [2] with T-dual description in terms of magnetized D-branes [3], have
played an important role in string model building during the last few years.
On Type IIA orientifolds with intersecting D6-branes, many non-supersymmetric three-
family standard-like models and Grand Unified Theories (GUTs) were constructed [4, 5, 6].
Although these models were globally consistent, there generically existed uncancelled Neveu-
Schwarz-Neveu-Schwarz (NSNS) tadpoles as well as the gauge hierarchy problem. To solve
these two problems, semi-realistic supersymmetric standard-like models, Pati-Salam models,
SU(5) models as well as flipped SU(5) models have been constructed in Type IIA theory
on T6/(Z2 × Z2) [7, 8, 9, 10, 11, 12, 13, 14] and T6/(Z2 × Z′2) [15, 16] orientifolds with
intersecting D6-branes, and some of their phenomenological consequences have been stud-
ied [17, 18]. Moreover, the supersymmetric constructions in Type IIA theory on other orien-
tifolds were also discussed [19]. There are two main constraints on supersymmetric D6-brane
model building: RR tadpole cancellation conditions and four-dimensional N = 1 supersym-
metric D6-brane configurations. Also, K-theory conditions provide minor constraints. In
addition, to stabilize the closed-string moduli via supergravity fluxes, the flux models on
Type II orientifolds have also been constructed [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30].
It is well known that there are two serious problems in almost all the supersymmetric
D-brane models: no gauge coupling unification at the string scale, and the rank one problem
in the Standard Model (SM) fermion Yukawa matrices. Although these problems can be
solved in the flux models of Ref. [29] where the RR tadpole cancellation conditions are
relaxed, these models are in the AdS vacua and the question of how to lift these AdS vacua
to the Minkowski vacua or dS vacua correctly is still a big challenge. Recently, we found that
there is one and only one intersecting D6-brane model on Type IIA T6/(Z2 × Z2) orientifold
where the above problems can be solved [11, 29]. Moreover, this model may has a realistic
low energy phenomenology [31]. Although its observable sector has unique phenomological
properties, it is possible to have different stacks of the D6-branes in the hidden sector.
In this paper, we discuss three non-equivalent variations of the hidden sector where the
RR tadpoles are cancelled, the four-dimensional N = 1 supersymmetry is perserved, and the
K-theory conditions are satisfied. These three variations seem to be the only possibilities.
In the original model [11, 29], the gauge symmetry in the hidden sector is USp(2)1 ×
USp(2)2 × USp(2)3 × USp(2)4. Interestingly, we can replace the USp(2)1 × USp(2)2 gauge
symmetry by an U(2)12 gauge symmetry, and/or the USp(2)3 × USp(2)4 gauge symmetry
by an U(2)34 gauge symmetry since the contributions to the RR tadpoles from the USp(2)
stacks of D6-branes are the same as those of the U(2) stacks. Thus, there are three non-
equivalent variations, and the corresponding gauge symmetries in the hidden sector are
U(2)12×USp(2)3×USp(2)4, U(2)34×USp(2)1×USp(2)2, and U(2)12×U(2)34, respectively.
Moreover, we discuss the hidden sector gauge symmetry breaking, and consider how to
decouple the additional exotic particles. Because the observable sector is the same, the
discussions on phenomenological consequences, for example, the gauge coupling unification,
supersymmetry breaking soft terms, low energy supersymmetric particle spectrum, dark
matter density, and the SM fermion masses and mixings, are the same as those in Ref. [31,
This paper is organized as follows. We briefly review the intersecting D6-brane model
building on Type IIA T6/(Z2 × Z2) orientifold in Section II and the realistic intersecting
D6-brane model in Section III. We study the three variations of the hidden sector in Section
IV. Discussion and conclusions are given in Section V.
II. INTERSECTING D6-BRANE MODEL BUILDING IN TYPE IIA THEORY
ON T6/(Z2 × Z2) ORIENTIFOLD
We briefly review the intersecting D6-brane model building in Type IIA theory on
T6/(Z2 × Z2) orientifold [7, 8]. We consider T6 to be a six torus factorized as T6 =
T2 × T2 ×T2 whose complex coordinates are zi, i = 1, 2, 3 for the i-th two torus, respec-
tively. The θ and ω generators for the orbifold group Z2×Z2 act on the complex coordinates
as following
θ : (z1, z2, z3) → (−z1,−z2, z3) ,
ω : (z1, z2, z3) → (z1,−z2,−z3) . (1)
We implement an orientifold projection ΩR, where Ω is the world-sheet parity, and R acts
on the complex coordinates as
R : (z1, z2, z3) → (z1, z2, z3) . (2)
So, there are four kinds of orientifold 6-planes (O6-planes) for the actions of ΩR, ΩRθ,
ΩRω, and ΩRθω, respectively. Also, we have two kinds of complex structures consistent
with orientifold projection for a two torus – rectangular and tilted [32]. If we denote the
TABLE I: General spectrum for intersecting D6-branes at generic angles, where Iaa′ =
−23−k
i=1(n
a), and IaO6 = 2
3−k(−l1al2al3a + l1an2an3a + n1al2an3a + n1an2al3a). Moreover, M is the
multiplicity, and aS and aA denote the symmetric and anti-symmetric representations of U(Na/2),
respectively.
Sector Representation
aa U(Na/2) vector multiplet and 3 adjoint chiral multiplets
ab+ ba M(Na
) = Iab = 2
i=1(n
ab′ + b′a M(Na
) = Iab′ = −2−k
i=1(n
aa′ + a′a M(aS) = 12(Iaa′ −
IaO6) ; M(aA) = 12(Iaa′ +
IaO6)
homology classes of the three cycles wrapped by the D6-brane stacks as nia[ai] +m
a[bi] and
nia[a
i] +m
a[bi] with [a
i] = [ai] +
[bi] for the rectangular and tilted tori respectively, we can
label a generic one cycle by (nia, l
a) in either case, where in terms of the wrapping numbers
lia ≡ mia for a rectangular two torus and lia ≡ 2m̃ia = 2mia+nia for a tilted two torus. So, the
homology three-cycles for stack a of Na D6-branes and its orientifold image a
′ take the form
[Πa] =
nia[ai] + 2
−βilia[bi]
, [Πa′ ] =
nia[ai]− 2
−βilia[bi]
, (3)
where βi = 0 if the i-th two torus is rectangular and βi = 1 if it is tilted. Also, we define
k ≡ β1 + β2 + β3.
For a stack of N D6-branes that do not lie on the top of any O6-plane, we obtain the
U(N/2) gauge symmetry with three adjoint chiral superfields due to the orbifold projections.
While for a stack of N D6-branes on the top of an O6-plane, we obtain the USp(N) gauge
symmetry with three anti-symmetric chiral superfields. The bifundamental chiral superfields
arise from the intersections of two different stacks of D6-branes or one stack of D6-branes
and its ΩR image [7, 8]. In short, the general spectrum for intersecting D6-branes at generic
angles, which is valid for both rectangular and tilted two tori, is given in Table I. Moreover,
a model may contain additional non-chiral (vector-like) multiplet pairs from ab+ba, ab′+b′a,
and aa′+ a′a sectors if two stacks of the corresponding D-branes are parallel and on the top
of each other on one two torus. The multiplicity of the non-chiral multiplet pairs is given
by the product of the intersection numbers on the other two two-tori.
Before further discussions, let us define the products of wrapping numbers
Aa ≡ −n1an2an3a, Ba ≡ n1al2al3a, Ca ≡ l1an2al3a, Da ≡ l1al2an3a,
Ãa ≡ −l1al2al3a, B̃a ≡ l1an2an3a, C̃a ≡ n1al2an3a, D̃a ≡ n1an2al3a.
TABLE II: Wrapping numbers of the four O6-planes.
Orientifold Action O6-Plane (n1, l1)× (n2, l2)× (n3, l3)
ΩR 1 (2β1 , 0)× (2β2 , 0) × (2β3 , 0)
ΩRω 2 (2β1 , 0)× (0,−2β2)× (0, 2β3)
ΩRθω 3 (0,−2β1)× (2β2 , 0)× (0, 2β3)
ΩRθ 4 (0,−2β1)× (0, 2β2)× (2β3 , 0)
The four-dimensional N = 1 supersymmetric models from Type IIA orientifolds with
intersecting D6-branes are mainly constrained by the RR tadpole cancellation conditions
and the four-dimensional N = 1 supersymmetric D6-brane configurations, and also
constrained by the K-theory conditions:
(1) RR Tadpole Cancellation Conditions
The total RR charges of D6-branes and O6-planes must vanish since the RR field flux
lines are conserved. And then we obtain the RR tadpole cancellation conditions as follows
− 2kN (1) +
NaAa = −2kN (2) +
NaBa =
−2kN (3) +
NaCa = −2kN (4) +
NaDa = −16, (5)
where 2N (i) are the number of D6-branes wrapping along the i-th O6-plane which is defined
in Table II.
(2) Four-Dimensional N = 1 Supersymmetric D6-Brane Configurations
The four-dimensional N = 1 supersymmetry can be preserved by the orientation projec-
tion if and only if the rotation angle of any D6-brane with respect to the O6-plane is an
element of SU(3) [2], or in other words, θ1+θ2+θ3 = 0 mod 2π, where θi is the angle between
the D6-brane and the O6-plane in the i-th two torus. This supersymmetry conditions can
be rewritten as [9]
xAÃa + xBB̃a + xCC̃a + xDD̃a = 0,
Aa/xA +Ba/xB + Ca/xC +Da/xD < 0, (6)
where xA = λ, xB = λ2
β2+β3/χ2χ3, xC = λ2
β1+β3/χ1χ3, xD = λ2
β1+β2/χ1χ2, and
χi = R
i are the complex structure parameters. The positive parameter λ has been
introduced to put all the variables A, B, C, and D on an equal footing.
(3) K-theory Conditions
The discrete D-brane RR charges classified by the Z2 K-theory groups in the presence of
orientifolds, which are subtle and invisible by the ordinary homology [22, 33], should also
be taken into account [21]. The K-theory conditions are
2−kÃa =
2−β1NaB̃a =
2−β2NaC̃a =
2−β3NaD̃a = 0 mod 4 . (7)
III. THE REALISTIC INTERSECTING D6-BRANE MODEL
There may be one and only one intersecting D6-brane model in Type IIA theory on
T6/(Z2 × Z2) orientifold with a realistic phenomenology [11, 29, 31]. Let us briefly review
it. We present the D6-brane configurations and intersection numbers in Table III, and its
spectrum in Table IV. We put the a′, b, and c stacks of D6-branes on the top of each other
on the third two torus, and then we have the additional vector-like particles from N = 2
subsectors.
We have shown that the gauge symmetry in the observable sector can be broken down
to the SM gauge symmetry via the Green-Schwarz mechanism, D6-brane splittings and su-
persymmtry preserving Higgs mechanism. The gauge couplings for SU(4)C , SU(2)L and
SU(2)R are unified at the string scale, and the additional exotic particles may be decoupled
around the string scale. Also, we calculated the supersymmetry breaking soft terms, and
the corresponding low energy supersymmetric particle spectrum that can be tested at the
Large Hadron Collider (LHC). The observed dark matter density can also be generated.
In addition, we can explain the SM quark masses and mixings, and the tau lepton mass.
The neutrino masses and mixings may be generated via seesaw mechanism as well. Simi-
lar to the GUTs [34], we have roughly the wrong fermion mass relation me/mµ ≃ md/ms,
and the correct electron and muon masses can be generated via high-dimensional opera-
stackN (n1,l1)(n2,l2)(n3,l3) A S b b
′ c c′ O61O62O63O64
a 8 ( 0,-1) ( 1, 1)( 1, 1) 0 0 3 0(3) -3 0(3) 1 -1 0 0
b 4 ( 3, 1) ( 1, 0)( 1,-1) -2 2 - - 0(6)0(1) 0 1 0 -3
c 4 ( 3,-1) ( 0, 1)( 1,-1) 2 -2 - - - - -1 0 3 0
O61 2 ( 1, 0) ( 1, 0)( 2, 0) - - - - - - - - - -
O62 2 ( 1, 0) ( 0,-1)( 0, 2) - - - - - - - - - -
O63 2 ( 0, -1)( 1, 0)( 0, 2) - - - - - - - - - -
O64 2 ( 0, -1)( 0, 1)( 2, 0) - - - - - - - - - -
TABLE III: The D6-brane configurations and intersection numbers on Type IIA T6/Z2 × Z2
orientifold. The gauge symmetry is [U(4)C × U(2)L × U(2)R]Observable × [USp(2)1 × USp(2)2 ×
USp(2)3 × USp(2)4]Hidden, the SM fermions and Higgs fields arise from the intersections on the
first two torus, and the complex structure parameters are 2χ1 = 6χ2 = 3χ3 = 6. Also, the beta
functions for all USp(2)i gauge symmetries are −3.
tors [35]. Furthermore, all the USp(2)i gauge symmetries will become strong around the
string scale [35].
IV. THREE VARIATIONS OF THE HIDDEN SECTOR
In the realistic intersecting D6-brane model [11, 29], the observable sector is unique. Inter-
estingly, we find three non-equivalent variations of the hidden sector where we can cancel the
RR tadpoles, preserve the four-dimensional N = 1 supersymmetry, and satisfy the K-theory
conditions. And it seems to us that there is no other variation. In the original model [11, 29],
the gauge symmetry in the hidden sector is USp(2)1 × USp(2)2 × USp(2)3 × USp(2)4. We
notice that the USp(2)1 × USp(2)2 gauge symmetry can be replaced by an U(2)12 gauge
symmetry, and/or the USp(2)3 × USp(2)4 gauge symmetry by an U(2)34 gauge symmetry
because the contributions to the RR tadpoles from the USp(2)2 stacks of D6-branes are
the same as those of the U(2) stacks. Thus, there are three non-equivalent variations, and
the corresponding gauge symmetries in the hidden sector are U(2)12 × USp(2)3 × USp(2)4,
U(2)34×USp(2)1×USp(2)2, and U(2)12 ×U(2)34, respectively. Let us present them one by
one in the following subsections.
TABLE IV: The chiral and vector-like superfields, and their quantum numbers under the gauge
symmetry SU(4)C × SU(2)L × SU(2)R × USp(2)1 × USp(2)2 × USp(2)3 × USp(2)4.
Quantum Number Q4 Q2L Q2R Field
ab 3× (4, 2, 1, 1, 1, 1, 1) 1 -1 0 FL(QL, LL)
ac 3× (4, 1, 2, 1, 1, 1, 1) -1 0 1 FR(QR, LR)
a1 1× (4, 1, 1, 2, 1, 1, 1) 1 0 0 Xa1
a2 1× (4, 1, 1, 1, 2, 1, 1) -1 0 0 Xa2
b2 1× (1, 2, 1, 1, 2, 1, 1) 0 1 0 Xb2
b4 3× (1, 2, 1, 1, 1, 1, 2) 0 -1 0 X i
c1 1× (1, 1, 2, 2, 1, 1, 1) 0 0 -1 Xc1
c3 3× (1, 1, 2, 1, 1, 2, 1) 0 0 1 X i
bS 2× (1, 3, 1, 1, 1, 1, 1) 0 2 0 T iL
bA 2× (1, 1, 1, 1, 1, 1, 1) 0 -2 0 SiL
cS 2× (1, 1, 3, 1, 1, 1, 1) 0 0 -2 T iR
cA 2× (1, 1, 1, 1, 1, 1, 1) 0 0 2 SiR
ab′ 3× (4, 2, 1, 1, 1, 1, 1) 1 1 0
3× (4, 2, 1, 1, 1, 1, 1) -1 -1 0
ac′ 3× (4, 1, 2, 1, 1, 1, 1) 1 1 Φi
3× (4, 1, 2, 1, 1, 1, 1) -1 0 -1 Φi
bc 6× (1, 2, 2, 1, 1, 1, 1) 0 1 -1 Hi
6× (1, 2, 2, 1, 1, 1, 1) 0 -1 1
A. U(2)12 × USp(2)3 × USp(2)4 Hidden Sector
In the first variation of the hidden sector, we replace the USp(2)1 × USp(2)2 gauge
symmetry by an U(2)12 gauge symmetry. We present the D6-brane configurations and
intersection numbers in Table V. Moreover, the particle spectrum has two parts: (1) the
spectrum for old particles is given in Table IV by removing all the particles that are charged
under USp(2)1 × USp(2)2; (2) the spectrum for the new particles is given in Table VI.
The anomalies from the global U(1) of U(2)12 are cancelled by the Green-Schwarz mech-
anism, and its gauge field obtains mass via the linear B ∧ F couplings. Then, the effective
gauge symmetry is SU(2)12. The SU(2)12 gauge symmetry can be broken down to U(1)12
via D6-brane splitting. Interestingly, we do not have any additional chiral exotic particles
that are charged under SU(4)C . The simple way to give masses to the extra exotic particles
stackN (n1,l1)(n2,l2)(n3,l3) A S b b
′ c c′ d d′ O63O64
a 8 ( 0,-1) ( 1, 1) ( 1, 1) 0 0 3 0(3) -3 0(3) 0(2)0(1) 0 0
b 4 ( 3, 1) ( 1, 0) ( 1,-1) -2 2 - - 0(6)0(1) 1 0(1) 0 -3
c 4 ( 3,-1) ( 0, 1) ( 1,-1) 2 -2 - - - - -1 0(1) 3 0
d 4 ( 1, 0) ( 1,-1)( 1, 1) 0 0 - - - - - - -1 1
O63 2 ( 0, -1)( 1, 0) ( 0, 2) - - - - - - - - - -
O64 2 ( 0, -1)( 0, 1) ( 2, 0) - - - - - - - - - -
TABLE V: The D6-brane configurations and intersection numbers on Type IIA T6/Z2 ×Z2 orien-
tifold. The complete gauge symmetry is [U(4)C × U(2)L × U(2)R]Observable × [U(2)12 ×USp(2)3 ×
USp(2)4]Hidden.
TABLE VI: The new chiral superfields and their quantum numbers under the gauge symmetry
SU(4)C × SU(2)L × SU(2)R × U(2)12 × USp(2)3 × USp(2)4.
Representation Q4 Q2L Q2R Q12 Field
bd 1× (1, 2, 1, 2, 1, 1) 0 1 0 -1 Xbd
cd 1× (1, 1, 2, 2, 1, 1) 0 0 -1 1 Xcd
d3 1× (1, 1, 1, 2, 2, 1) 0 0 0 -1 Xd3
d4 1× (1, 1, 1, 2, 1, 2) 0 0 0 1 Xd4
Xbd and Xcd is instanton effects [36, 37, 38, 39]. However, we do not have the suitable three-
cycles wrapped by E2 instantons 1, and thus the instanton effects are not available. Similar
results hold for the next two subsections. In addition, the USp(2)3 and USp(2)4 will become
strong at about the string scale [35], and then we will have some composite particles in the
U(2)12 anti-symmetric and symmetric representations, S
d and T
d from Xd3, and S
d and T
from Xd4, respectively. So we can break the U(1)12 by giving suitable string-scale vacuum
expectation values (VEVs) to T
d and T
d, and we can give the string-scale VEVs to S
d and
S ′d. Note that we give the TeV-scale VEVs to S
L and the string-scale VEVs to S
R [31], we
can give the GUT-scale masses to X ic3 and Xcd and the TeV-scale masses to the X
b4 and
1 Note that the E2 branes must also wrap rigid cycles.
Xbd via the high-dimensional operators [35]. Furthermore, if we could give the string-scale
masses to the three U(2)12 adjoint chiral superfields and we do not break the SU(2)12 via
D6-brane splitting, the SU(2)12 gauge symmetry will become strong around the string scale.
Then we can have the singlet composite field S ′L in the U(2)L anti-symmetric representation
with charge +2 under U(1)L from Xbd. And we can give the string-scale VEVs to S
L and S
while keeping the D-flatness of U(1)L. Therefore, we may also give the GUT-scale masses
to the X ib4 and Xbd via the high-dimensional operators [35].
B. U(2)34 × USp(2)1 × USp(2)2 Hidden Sector
stackN (n1,l1)(n2,l2)(n3,l3) A S b b
′ c c′ e e′ O61O62
a 8 ( 0,-1)( 1, 1) ( 1, 1) 0 0 3 0(3) -3 0(3) 0(2)0(0) 1 -1
b 4 ( 3, 1) ( 1, 0) ( 1,-1) -2 2 - - 0(6)0(1) 0(3) -3 0 1
c 4 ( 3,-1)( 0, 1) ( 1,-1) 2 -2 - - - - 0(3) 3 -1 0
e 4 ( 0, 1) (-1, 1) (-1, 1) 0 0 - - - - - - -1 1
O61 2 ( 1, 0) ( 1, 0) ( 2, 0) - - - - - - - - - -
O62 2 ( 1, 0) ( 0,-1)( 0, 2) - - - - - - - - - -
TABLE VII: The D6-brane configurations and intersection numbers on Type IIA T6/Z2 × Z2
orientifold. The gauge symmetry is [U(4)C × U(2)L × U(2)R]Observable × [U(2)34 × USp(2)1 ×
USp(2)2]Hidden.
TABLE VIII: The new chiral superfields and their quantum numbers under the gauge symmetry
SU(4)C × SU(2)L × SU(2)R × U(2)34 × USp(2)1 × USp(2)2.
Representation Q4 Q2L Q2R Q34 Field
be′ 3× (1, 2, 1, 2, 1, 1) 0 -1 0 -1 X i
ce′ 3× (1, 1, 2, 2, 1, 1) 0 0 1 1 X i
e1 1× (1, 1, 1, 2, 2, 1) 0 0 0 -1 Xe1
e2 1× (1, 1, 1, 2, 1, 2) 0 0 0 1 Xe2
In the second variation of the hidden sector, we replace the USp(2)3 × USp(2)4 gauge
symmetry by an U(2)34 gauge symmetry. We present the D6-brane configurations and
intersection numbers in Table VII. The particle spectrum also has two parts: (1) the
spectrum for old particles is given in Table IV by removing all the particles that are charged
under USp(2)3 × USp(2)4; (2) the spectrum for the new particles is given in Table VIII.
Note that the wrapping numbers for the d stack of D6-branes are equivalent to those of
the a stack by T duality and orientifold action, we can think that we have an U(6) gauge
symmetry in the begining. Only the global U(1) of U(6) is anomalous U(1) symmetry, and
its gauge field obtains mass via the linear B ∧ F couplings. After we put four D6-branes
on the place with equivalent wrapping numbers (just like the D6-brane splittings), we break
the SU(6) down to the SU(4)C × SU(2)34 × U(1)′ where the U(1)′ generator in SU(6) is
TU(1)′ ≡
diag (1, 1, 1, 1,−2,−2) . (8)
Thus, the left-handed and right-handed SM fermions have U(1)′ charges +1/2
6 and
6, respectively. In order to keep the gauge coupling unification, we have to break
the U(1)′ so that it will not become part of the U(1)Y . In short, we have to break U(2)34
completely.
Because the USp(2)1 and USp(2)2 will become strong at about the string scale [35], we
will have some composite particles in the U(2)34 anti-symmetric and symmetric represen-
tations, S
e and T
e from Xe1, and S
e and T
e from Xe2, respectively. So we can break the
U(2)12 completely by giving suitable string-scale VEVs to S
e, and T
e. Moreover,
we can have the singlet composite particle S ′L in the U(2)L anti-symmetric representation
with charge +2 under U(1)L from Xb2. And then we can give the string-scale VEVs to S
and S ′L while keeping the D-flatness of U(1)L. Note that S
R also have string-scale VEVs,
we may give the GUT-scale masses to Xb2, Xc1, X
be′, and X
ce′ via the high-dimensional
operators [35]. Moreover, Xa1 and Xa2 may form the vector-like particles if we break the
USp(2)1 and USp(2)2 down to the diagonal USp(2)D12 [31].
C. U(2)12 × U(2)34 Hidden Sector
In the third variation of the hidden sector, we replace the USp(2)1 × USp(2)2 gauge
symmetry by U(2)12, and replace the USp(2)3 × USp(2)4 gauge symmetry by U(2)34. We
present the D6-brane configurations and intersection numbers in Table IX. The particle
spectrum also has two parts: (1) the spectrum for old particles is given in Table IV by
removing all the particles that are charged under USp(2)1×USp(2)2 ×USp(2)3 ×USp(2)4;
(2) the spectrum for the new particles is given in Table X.
stackN (n1,l1)(n2,l2)(n3,l3) A S b b
′ c c′ d d′ e e′
a 8 ( 0,-1)( 1, 1)( 1, 1) 0 0 3 0(3) -3 0(3) 0(2)0(1)0(2)0(0)
b 4 ( 3, 1) ( 1, 0)( 1,-1) -2 2 - - 0(6)0(1) 1 0(1)0(3) -3
c 4 ( 3,-1)( 0, 1)( 1,-1) 2 -2 - - - - -1 0(1)0(3) 3
d 4 ( 1, 0) ( 1,-1)( 1, 1) 0 0 - - - - - - 0(1)0(2)
e 4 ( 0, 1) (-1, 1)(-1, 1) 0 0 - - - - - - - -
TABLE IX: The D6-brane configurations and intersection numbers on Type IIA T6/Z2×Z2 orien-
tifold. The complete gauge symmetry is [U(4)C×U(2)L×U(2)R]Observable× [U(2)12×U(2)34]Hidden.
TABLE X: The chiral and vector-like superfields, and their quantum numbers under the gauge
symmetry SU(4)C × SU(2)L × SU(2)R × U(2)12 × U(2)34.
Representation Q4 Q2L Q2R Q12 Q34 Field
bd 1× (1, 2, 1, 2, 1) 0 1 0 -1 0 Xbd
be′ 3× (1, 2, 1, 1, 2) 0 -1 0 0 -1 X i
cd 1× (1, 1, 2, 2, 1) 0 0 -1 1 0 Xcd
ce′ 3× (1, 1, 2, 1, 2) 0 0 1 0 1 X i
de 1× (1, 1, 1, 2, 2) 0 0 0 1 -1 Xde
1× (1, 1, 1, 2, 2) 0 0 0 -1 1 Xde
de′ 2× (1, 1, 1, 2, 2) 0 0 0 1 1 X i
2× (1, 1, 1, 2, 2) 0 0 0 -1 -1 Xi
As discussed in above subsections, we can break the U(2)12 down to the U(1)12 gauge
symmetry via Green-Schwarz mechanism and D6-brane splitting, and we have to break
the U(2)34 gauge symmetry completely. In order to break the U(1)12 and U(2)34 gauge
symmetries, we put the d and e stacks of D6-branes on the top of each other on the second
two torus, and put the d and e′ stacks on the top of each other on the third two torus. Then,
we have additional vector-like particles Xde, Xde, X
de′, and X
de′ , as given in Table X. And
there exist the following Yukawa couplings
W ⊃ yAijXbdX
+ yBijXcdX
de′ , (9)
where yAij and y
ij are Yukawa couplings. If we give the diagonal string-scale VEVs to X
and X
de′ , we break the U(2)12×U(2)34 down to the diagonal U(2)D. Moreover, the Xbd and
one linear combination of X ibe′, and the Xcd and one linear combination of X
ce′ can have
vector-like masses close to the string scale. Note that we can give the TeV-scale VEVs to SiL
and the string-scale VEVs to SiR [31], we can give the GUT-scale masses to Xcd and the other
two linear combinaions of X ice′, and the TeV-scale masses to Xbd and the other two linear
combinations of X ibe′ via the high-dimensional operators [35]. Similar to the discussions in
the above subsection A, if we can give the string-scale masses to the three U(2)12 adjoint
chiral superfields and do not break the SU(2)12 gauge symmetry via D6-brane splitting, the
SU(2)12 gauge symmetry will become strong around the string scale. Then we can have
the singlet composite field S ′L in the U(2)L anti-symmetric representation with charge +2
under U(1)L from Xbd, and we can give the string-scale VEVs to S
L and S
L while keeping
the D-flatness of U(1)L. Therefore, we may also give the GUT-scale masses to Xbd and the
other two linear combinations of X ibe′ via the high-dimensional operators [35].
V. DISCUSSION AND CONCLUSIONS
At present, there is only one known example of an intersecting D6-brane model with a
realistic observable sector. Interestingly, there are three non-equivalent variations of the
hidden sector in which the theoretical constraints on model building can be satisfied. There
does not seem to be any other possible variation in the original model [11, 29], and the
gauge symmetry in the hidden sector is USp(2)1 × USp(2)2 × USp(2)3 × USp(2)4. We
noticed that the USp(2)1 × USp(2)2 gauge symmetry can be replaced by an U(2)12 gauge
symmetry, and/or the USp(2)3 × USp(2)4 gauge symmetry can be replaced by an U(2)34
gauge symmetry because the USp(2)2 stacks of D6-branes contribute to the same RR tad-
poles as those of the U(2) stacks. Thus, we obtained three non-equivalent variations, and
the corresponding gauge symmetries in the hidden sector are U(2)12 × USp(2)3 × USp(2)4,
U(2)34 ×USp(2)1 ×USp(2)2, and U(2)12 ×U(2)34, respectively. In addition, we studied the
hidden sector gauge symmetry breaking, and discussed how to decouple the additional exotic
particles. Because the observable sector is the same, the phenomenological discussions in
the observable sector are the same as those in Ref. [31, 35].
Acknowledgments
This research was supported in part by the Mitchell-Heep Chair in High Energy Physics
(CMC), by the Cambridge-Mitchell Collaboration in Theoretical Cosmology (TL), and by
the DOE grant DE-FG03-95-Er-40917 (DVN).
[1] J. Polchinski and E. Witten, Nucl. Phys. B 460, 525 (1996).
[2] M. Berkooz, M. R. Douglas and R. G. Leigh, Nucl. Phys. B 480 (1996) 265.
[3] C. Bachas, arXiv:hep-th/9503030.
[4] R. Blumenhagen, L. Goerlich, B. Kors and D. Lust, JHEP 0010, 006 (2000).
[5] C. Angelantonj, I. Antoniadis, E. Dudas and A. Sagnotti, Phys. Lett. B 489, 223 (2000).
[6] R. Blumenhagen, M. Cvetic, P. Langacker and G. Shiu, Ann. Rev. Nucl. Part. Sci. 55, 71
(2005), and the references therein.
[7] M. Cvetič, G. Shiu and A. M. Uranga, Phys. Rev. Lett. 87, 201801 (2001).
[8] M. Cvetič, G. Shiu and A. M. Uranga, Nucl. Phys. B 615, 3 (2001).
[9] M. Cvetič, I. Papadimitriou and G. Shiu, Nucl. Phys. B 659, 193 (2003) [Erratum-ibid. B
696, 298 (2004)].
[10] M. Cvetič and I. Papadimitriou, Phys. Rev. D 67, 126006 (2003).
[11] M. Cvetič, T. Li and T. Liu, Nucl. Phys. B 698, 163 (2004).
[12] M. Cvetič, P. Langacker, T. Li and T. Liu, Nucl. Phys. B 709, 241 (2005).
[13] C.-M. Chen, G. V. Kraniotis, V. E. Mayes, D. V. Nanopoulos and J. W. Walker, Phys. Lett.
B 611, 156 (2005); Phys. Lett. B 625, 96 (2005).
[14] C.-M. Chen, T. Li and D. V. Nanopoulos, Nucl. Phys. B 732, 224 (2006).
[15] R. Blumenhagen, M. Cvetic, F. Marchesano and G. Shiu, JHEP 0503, 050 (2005).
[16] C.-M. Chen, V. E. Mayes and D. V. Nanopoulos, arXiv:hep-th/0612087.
[17] M. Cvetič, P. Langacker and G. Shiu, Phys. Rev. D 66, 066004 (2002); Nucl. Phys. B 642,
139 (2002).
[18] M. Cvetič, P. Langacker and J. Wang, Phys. Rev. D 68, 046002 (2003).
[19] R. Blumenhagen, L. Görlich and T. Ott, JHEP 0301, 021 (2003); G. Honecker, Nucl. Phys.
B666, 175 (2003); G. Honecker and T. Ott, Phys. Rev. D 70, 126010 (2004) [Erratum-ibid.
D 71, 069902 (2005)].
[20] R. Blumenhagen, D. Lust and T. R. Taylor, Nucl. Phys. B 663, 319 (2003).
[21] J. F. G. Cascales and A. M. Uranga, JHEP 0305, 011 (2003).
[22] F. Marchesano and G. Shiu, Phys. Rev. D 71, 011701 (2005); JHEP 0411, 041 (2004).
[23] M. Cvetič and T. Liu, Phys. Lett. B 610, 122 (2005).
[24] M. Cvetič, T. Li and T. Liu, Phys. Rev. D 71, 106008 (2005).
http://arxiv.org/abs/hep-th/9503030
http://arxiv.org/abs/hep-th/0612087
[25] J. Kumar and J. D. Wells, JHEP 0509, 067 (2005).
[26] C.-M. Chen, V. E. Mayes and D. V. Nanopoulos, Phys. Lett. B 633, 618 (2006).
[27] G. Villadoro and F. Zwirner, JHEP 0506, 047 (2005).
[28] P. G. Camara, A. Font and L. E. Ibanez, JHEP 0509, 013 (2005).
[29] C.-M. Chen, T. Li and D. V. Nanopoulos, Nucl. Phys. B 740, 79 (2006).
[30] C.-M. Chen, T. Li and D. V. Nanopoulos, Nucl. Phys. B 751, 260 (2006).
[31] C.-M. Chen, T. Li, V. E. Mayes and D. V. Nanopoulos, arXiv:hep-th/0703280.
[32] R. Blumenhagen, B. Körs and D. Lüst, JHEP 0102 (2001) 030.
[33] E. Witten, JHEP 9812, 019 (1998).
[34] J.R. Ellis and M.K. Gaillard, Phys. Lett. B 88, 315 (1979); D.V. Nanopoulos and M. Sred-
nicki, Phys. Lett. B 124, 37 (1983).
[35] C.-M. Chen, T. Li, V.E. Mayes and D.V. Nanopoulos, in preparation.
[36] R. Blumenhagen, M. Cvetic and T. Weigand, arXiv:hep-th/0609191.
[37] L. E. Ibanez and A. M. Uranga, JHEP 0703, 052 (2007).
[38] M. Cvetic, R. Richter and T. Weigand, arXiv:hep-th/0703028.
[39] L. E. Ibanez, A. N. Schellekens and A. M. Uranga, arXiv:0704.1079 [hep-th].
http://arxiv.org/abs/hep-th/0703280
http://arxiv.org/abs/hep-th/0609191
http://arxiv.org/abs/hep-th/0703028
http://arxiv.org/abs/0704.1079
	Introduction
	Intersecting D6-Brane Model Building in Type IIA Theory on T6/(Z2Z2) Orientifold
	The Realistic Intersecting D6-Brane Model
	Three variations of the Hidden Sector
	U(2)12USp(2)3 USp(2)4 Hidden Sector
	U(2)34USp(2)1 USp(2)2 Hidden Sector
	U(2)12U(2)34 Hidden Sector
	Discussion and Conclusions
	Acknowledgments
	References
