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arXiv:hep-th/0205147v2 2 Oct 2002
FTUAM-02-10
IFT-UAM/CSIC-02-09
February 1, 2008
New Standard Model Vacua from Intersecting Branes
C. Kokorelis
Dep/to de F´ısica Te´orica C-XI and Instituto de F´ısica Te´orica C-XVI ,
Universidad Aut´onoma de Madrid, Cantoblanco, 28049, Madrid, Spain
ABSTRACT
We construct new D6-brane model vacua (non-supersymmetric) that have at low
energy exactly the standard model spectrum (with right handed neutrinos). The
minimal version of these models requires five stacks of branes. and the construc-
tion is based on D6-branes intersecting at angles in D= 4 type toroidal orien-
tifolds of type I strings. Three U(1)’s become massive through their couplings to
RR couplings and from the two surviving anomaly free U(1)’s, one is the standard
model hypercharge generator while the extra anomaly free U(1) could be broken
from its non-zero couplings to RR fields and also by triggering a vev to previously
massive particles. We suggest that extra massless U(1)’s should be broken by re-
quiring some intersection to respect N= 1 supersymmetry thus supporting the
appearance of massless charged singlets at the supersymmetric intersection. Pro-
ton is stable as baryon number is gauged and its anomalies are cancelled through
a generalized Green-Schwarz mechanism. Neutrinos are of Dirac type with small
masses, as in the four stack standard models of hep-th/0105155, as a result of the
existence of a similar PQ like-symmetry. The models are unique in the sense that
they predict the existence of only one supersymmetric particle, the superpartner
of νR.
1 Introduction
Over the years, one of the most difficult questions that string theory is facing is the
selection of the particular vacuum that includes at low energy all the necessary in-
gredients of the observable standard model spectrum a low energies. In the absence
of an underlying principle of picking up a particular vacuum, one can search for the
model with the correct low energy particle content. Such attempts have by far been
explored in the context of heterotic string theory as well to branes at singularities [1, 2].
The main characteristics of the models involved include the three generation massless
spectrum of the standard model (SM) accompanied by the presence of exotic matter
and /or gauge group factors. However, recently there has been some progress in the
study of string models as it has been possible in [3] to derive, at low energy, just the
SM spectrum together with right handed neutrinos. The models were studied in the
context of intersecting branes and have some satisfactory properties including proton
stability and small neutrino (of Dirac type) masses.
The purpose of this paper is to extend the four stack construction of type I four
dimensional toroidal orientifolded six-torus construction of [3], to a different structure
that involves one extra U(1) at the string scale and produces just the standard model
(SM) at low energies. Note that in [3] one starts with a U(3)a⊗U(2)b⊗U(1)c⊗U(1)d
open gauge group structure at string scale energies. Note that in our construction, as
in [3], right handed neutrinos are present along with the SM particle context at low
energies. Additional non-supersymmetric models along the same Type I backgrounds,
which give at low energy the SM structure, have been constructed in [4]. In the latter
case one starts with a four dimensional type I background on an orientifolded six
dimensional factorized torus, which at the string scale includes 1a Pati-Salam gauge
group. The models incorporate a number of interesting properties, like proton stability
and small neutrino masses.
In this work, one starts with five stacks of branes, namely with an U(3)a⊗U(2)b⊗
U(1)c⊗U(1)d⊗U(1)eat the string scale and get at low energy only the SM with right
handed neutrinos. The models also allow different generations of leptons-neutrinos to
be placed at different intersections, that could have interesting implications for the
phenomenology of the models. We also note that from the five string scale U(1)’s,
four couple to RR fields and one survives massless at low energies. The latter U(1)
corresponds to the hypercharge of the SM.
1The string scale gauge group structure, that includes four stacks of branes, that is an U(4)c⊗
U(2)L⊗U(2)R⊗U(1).
1
The models are non-supersymmetric and are build on a background of D6-branes in-
tersecting each other at non-trivial angles [5], in an orientifolded factorized six-torus,
while O6orientifold planes are on top of D6-branes [3]. Note that the latter picture
is just the T-dual of type I backgrounds of [6], [7] which make use of D9 branes with
background fluxes. Also, we note that the studied backgrounds are T-dual to models
with magnetic deformations [8].
Additional models on backgrounds, in the context of intersecting branes, have been
discussed in [9, 10, 11, 12, 13, 14, 15]. For an other proposal for realistic SM D-brane
model building, based not on a particular string construction, see [16].
The proposed models have 2the following distinctive features :
•The model starts with a gauge group at the string scale U(3) ×U(2) ×U(1) ×
U(1) ×U(1). The use of Green-Schwarz mechanism in the model renders three
of the U(1)’s massive while a combination of the other two remaining anomaly
free U(1)’s one makes the standard model hypercharge, while the other one 3gets
broken by its non-zero coupling to RR fields and by turning on N= 1 SUSY at
an intersection, while keeping the rest of the intersections non-supersymmetric.
•Neutrinos get a Dirac mass, as lepton number L is a gauged symmetry, of the right
order 4in consistency with the LSND neutrino oscillation experiments [17] as a
consequence of the existence of a PQ-like symmetry related to chiral symmetry
breaking.
•Proton is stable due to the fact that baryon number B is an unbroken gauged
global symmetry surviving at low energies whose anomalies cancel through a
generalized Green-Schwarz mechanism.
The paper is organized as follows. In section two we describe the general character-
istics of the new standard model vacua with particular emphasis on how to calculate
the fermionic spectrum from intersecting branes as well providing the classification
of multi-parameter solutions to the RR tadpole cancellation conditions. In section 3
we examine the cancellation of U(1) anomalies via a generalized Green-Schwarz (GS)
2The different classes of models of this work maintain essential characteristics of the classes of
models discussed in [3], including proton stability, and sizes of neutrino masses within experimental
limits.
3In an orthogonal basis, this U(1) will be broken only by the vev of a scalar created by turning
N=1 SYSY on an inbtersection.
4The same mechanism was employed in [3] for the four stack D6 orientifold counterpart of the
present SM’s.
2
mechanism [18, 19, 20] examining the conditions under which the hypercharge gener-
ator remains light at low energies. In section 4 we study the Higgs sector providing
for the tachyonic scalars that are used for the electroweak symmetry breaking of the
model. We also discuss the breaking of the extra anomaly free, other than hypercharge,
U(1), by turning on N= 1 supersymmetry at an intersection. In section 5, we examine
the Yukawa couplings and the smallness of neutrino masses. Section 6 contains our
conclusions.
2 New SM vacua from intersecting branes
In the present work, we are going to describe new type I compactification vacua, that
have as their low energy limit just the observable standard model interactions. The
proposed three generation non-supersymmetric standard models make use of five stacks
of branes at the string scale. They will originate from D6-branes wrapping on 3-cycles
of toroidal orientifolds of type IIA in four dimensions. Important characteristic of all
vacua coming from these type I constructions is the replication, at each intersection,
of the massless fermion spectrum by an equal number of massive particles in the same
representations and with the same quantum numbers.
Next, we describe the construction of the standard model. It is based on type I
string with D9-branes compactified on a six-dimensional orientifolded torus T6, where
internal background gauge fluxes on the branes are turned on. If we perform a T-duality
transformation on the x4,x5,x6, directions the D9-branes with fluxes are translated into
D6-branes intersecting at angles. Note that the branes are not parallel to the orientifold
planes. Furthermore, we assume that the D6a-branes are wrapping 1-cycles (ni
a, mi
a)
along each of the ith-T2torus of the factorized T6torus, namely T6=T2×T2×T2.
That means that we allow our torus to wrap factorized 3-cycles, that can unwrap into
products of three 1-cycles, one for each T2. We define the homology of the 3-cycles as
[Πa] =
3
Y
i=1
(ni
a[ai] + mi
a[bi]) (2.1)
while we define the 3-cycle for the orientifold images as
[Πa⋆] =
3
Y
i=1
(ni
a[ai]−mi
a[bi]) (2.2)
In order to build the SM model structure a low energies, we consider five stacks of
D6-branes giving rise to their world-volume to an initial gauge group U(3) ×U(2) ×
3
U(1) ×U(1) ×U(1) or SU(3) ×SU(2) ×U(1)a×U(1)b×U(1)c×U(1)d×U(1)eat
the string scale. Also, we consider the addition of NS B-flux [21], such that the tori
involved are not orthogonal, thus avoiding an even number of families [6], and leading
to effective tilted wrapping numbers as,
(ni, m = ˜mi+ni/2); n, ˜m∈Z. (2.3)
In this way we allow semi-integer values for the m-wrapping numbers.
Because of the ΩRsymmetry, where Ω is the worldvolume parity and Ris the reflection
on the T-dualized coordinates,
T(ΩR)T−1= ΩR,(2.4)
each D6a-brane 1-cycle, must have its ΩRimage partner (ni
a,−mi
a).
Chiral fermions gets localized at the intersections between branes, by stretched open
strings between intersecting D6-branes [5]. Subsequently, the chiral spectrum of the
model may be obtained by solving simultaneously the intersection constraints coming
from the existence of the different sectors and the RR tadpole cancellation conditions.
Note that in the models we examine in this work, there are a number of different
sectors, which should be taken into account when computing the chiral spectrum. We
denote the action of ΩRon a sector a, b, by a⋆, b⋆, respectively. The possible sectors
are:
•The ab +ba sector: involves open strings stretching between the D6aand D6b
branes. Under the ΩRsymmetry this sector is mapped to its image, a⋆b⋆+b⋆a⋆
sector. The number, Iab, of chiral fermions in this sector, transforms in the
bifundamental representation (Na,¯
Na) of U(Na)×U(Nb), and reads
Iab = [Πa]·[Πb] = (n1
am1
b−m1
an1
b)(n2
am2
b−m2
an2
b)(n3
am3
b−m3
an3
b),(2.5)
where Iab is the intersection number of the wrapped cycles. Note that with the
sign of Iab intersection, we denote the chirality of the fermions, where Iab >0 de-
notes left handed fermions. Also negative multiplicity denotes opposite chirality.
•The ab⋆+b⋆asector : It involves chiral fermions transforming into the (Na, Nb)
representation with multiplicity given by
Iab⋆= [Πa]·[Πb⋆] = −(n1
am1
b+m1
an1
b)(n2
am2
b+m2
an2
b)(n3
am3
b+m3
an3
b).(2.6)
Under the ΩRsymmetry it transforms to itself.
4
•the aa⋆sector : under the ΩRsymmetry it transforms to itself. From this sec-
tor the invariant intersections will give 8m1
am2
am3
afermions in the antisymmetric
representation and the non-invariant intersections that come in pairs provide us
with 4m1
am2
am3
a(n1
an2
an3
a−1) additional fermions in the symmetric and antisym-
metric representation of the U(Na) gauge group. However as we explain later,
these sectors will be absent from our models.
Any vacuum derived from the previous intersection constraints is subject addition-
ally to constraints coming from RR tadpole cancellation conditions [6]. That demands
cancellation of D6-branes charges 5, wrapping on three cycles with homology [Πa] and
O6-plane 7-form charges wrapping on 3-cycles with homology [ΠO6]. Note that the RR
tadpole cancellation conditions in terms of cancellations of RR charges in homology
are
X
a
Na[Πa] + X
α′
Nα′[Πα′]−32[ΠO6] = 0.(2.7)
In explicit form, the RR tadpole conditions read
X
a
Nan1
an2
an3
a= 16,
X
a
Nam1
am2
an3
a= 0,
X
a
Nam1
an2
am3
a= 0,
X
a
Nan1
am2
am3
a= 0.(2.8)
That guarantees absence of non-abelian gauge anomalies. In D-brane model building,
by considering astacks of D-brane configurations with Na, a = 1,···, N , parallel branes
one gets the gauge group U(N1)×U(N2)× · · · × U(Na). Each U(Ni) factor will give
rise to an SU(Ni) charged under the associated U(1i) gauge group factor that appears
in the decomposition SU(Na)×U(1a). For the five stack model that we examine in this
work, the complete accommodation, where all other intersections are vanishing, of the
fermion structure can be seen in table (1). We note a number of interesting comments
:
a) There are various gauged low energy symmetries in the models. They are defined
in terms of the U(1) symmetries Qa,Qb,Qc,Qd,Qe. The baryon number B is equal to
Qa= 3B, the lepton number is L=Qd+Qewhile Qa−3Qd−3Qe= 3(B−L). Also
note that Qc= 2I3R,I3Rbeing the third component of weak isospin. Also, 3(B−L) and
5Taken together with their orientifold images (ni
a,−mi
a) wrapping on three cycles of homology
class [Πα′].
5
Matter Fields Intersection QaQbQcQdQeY
QL(3,2) Iab = 1 1 −1 0 0 0 1/6
qL2(3,2) Iab∗= 2 110001/6
UR3(¯
3,1) Iac =−3−1 0 1 0 0 −2/3
DR3(¯
3,1) Iac∗=−3−1 0 −1 0 0 1/3
L2(1,2) Ibd =−2 0 −1 0 1 0 −1/2
lL(1,2) Ibe =−1 0 −1 0 0 1 −1/2
NR2(1,1) Icd = 2 0 0 1 −1 0 0
ER2(1,1) Icd∗=−20 0 −1−1 0 1
νR(1,1) Ice = 1 0 0 1 0 −1 0
eR(1,1) Ice∗=−10 0 −1 0 −1 1
Table 1: Low energy fermionic spectrum of the five stack string scale SU(3)C⊗SU (2)L⊗
U(1)a⊗U(1)b⊗U(1)c⊗U(1)d⊗U(1)e, type I D6-brane model together with its U(1) charges.
Note that at low energies only the SM gauge group SU(3) ⊗SU(2)L⊗U(1)Ysurvives.
Qcare free of triangle anomalies. The U(1)bsymmetry plays the role of a Peccei-Quinn
symmetry in the sence of having mixed SU(3) anomalies. From the study of Green-
Schwarz mechanism, see later chapters, we deduce that Baryon and Lepton number
are unbroken gauged symmetries and thus proton should be stable, while Majorana
masses for right handed neutrinos are not allowed. That means that mass terms for
neutrinos should be of Dirac type.
The mixed anomalies Aij of the four surplus U(1)’s with the non-abelian gauge
groups SU (Na) of the theory cancel through a generalized GS mechanism [18, 19], in-
volving close string modes couplings to worldsheet gauge fields. Two combinations of
the U(1)’s are anomalous and become massive, their orthogonal non-anomalous com-
binations survive, combining to a single U(1) that remains massless, the hypercharge.
b) In order to cancel the appearance of exotic representations in the model appearing
from the aa⋆sector, in antisymmetric and symmetric representations of the U(Na)
group, we will impose the condition
Π3
i=1mi= 0.(2.9)
6
The solutions satisfying simultaneously the intersection constraints and the cancellation
of the RR crosscap tadpole constraints are parametric. They are given in table (2).
The solutions represent the most general solution of the RR tadpoles and they depend
on five integer parameters n2
a,n2
d,n2
e,n1
b,n1
c, the phase parameters ǫ=±1, ˜ǫ=±1, and
the NS-background parameter βi= 1 −bi, which is associated to the presence of the
NS B-field by bi= 0,1/2. Note that the different solutions to the tadpole constraints
represent deformations of the D6 brane RR charges within the same homology class. In
the rest of the paper we will be discussing, for simplicity the case with ǫ= ˜ǫ= 1. The
multiparameter tadpole solutions of table (2) represent deformations of the D6-brane
intersection spectrum of table (1), within the same homology class of the factorizable
three-cycles. By using the tadpole solutions of table (2) in (2.8) all tadpole equations
but the first are automatically satisfied, the 6latter yielding 7:
Ni(n1
i, m1
i) (n2
i, m2
i) (n3
i, m3
i)
Na= 3 (1/β1,0) (n2
a, ǫβ2) (3,˜ǫ/2)
Nb= 2 (n1
b,−ǫβ1) (1/β2,0) (˜ǫ, 1/2)
Nc= 1 (n1
c, ǫβ1) (1/β2,0) (0,1)
Nd= 1 (1/β1,0) (n2
d,2ǫβ2) (1,−˜ǫ/2)
Ne= 1 (1/β1,0) (n2
e, ǫβ2) (1,−˜ǫ/2)
Table 2: Tadpole solutions of D6-branes wrapping numbers giving rise to the standard model
gauge group and spectrum at low energies. The solutions depend on five integer parameters,
n2
a,n2
d,n2
e,n1
b,n1
c, the NS-background βiand the phase parameters ǫ=±1, ˜ǫ=±1.
6We have added an arbitrary number of NDbranes which don’t contribute to the rest of the
tadpoles and intersection number constraints.
7We have set for simplicity ˜ǫ= 1.
7
9n2
a
β1+ 2n1
b
β2+n2
d
β1+n2
e
β1+ND
2
β1β2= 16.(2.10)
Note that we had added the presence of extra NDbranes (hidden branes). Their
contribution to the RR tadpole conditions is best described by placing them in the
three-factorizable cycle
ND(1/β1,0)(1/β2,0)(2, m3
D) (2.11)
and we have set m3
D= 0. To see clearly the cancellation of tadpoles, we have to choose
a consistent numerical set of wrappings, e.g.
n2
a= 1, n1
b= 1, n1
c∈Z, n2
d=−1, ne= 1, β1= 1/2, β2= 1, ǫ = 1.(2.12)
With the above choices, all tadpole conditions but the first are satisfied, the latter is
satisfied when we add one ¯
D6 brane, e.g. ND=−1. Thus the tadpole structure 8
becomes
Na= 3 (2,0)(1,1)(3,1/2)
Nb= 2 (1,−1/2)(1,0)(1,1/2)
Nc= 1 (n1
c,1/2)(1,0)(0,1)
Nd= 1 (2,0)(−1,2)(1,−1/2)
Ne= 1 (2,0)(1,1)(1,−1/2).(2.13)
Actually, the satisfaction of the tadpole conditions is independent of n1
c. Thus, when
all other parameters are fixed, n1
cis a global parameter that can vary. However, finally
it will be fixed in terms of the remaining parameters once we specify, the tadpole
subclass that corresponds to the massless spectrum with the hypercharge embedding
of the standard model.
Note that there are always choices of wrapping numbers of wrapping numbers that
satisfy the RR tadpole constraints without the need of adding extra parallel branes,
e.g. the following choice satisfies all RR tadpoles
n2
a= 1, n1
b= 3, n1
c∈Z, n2
d=−3, ne=−1, β1= 1/2, β2= 1, ǫ = 1.(2.14)
with cycle wrapping numbers
8Note that the parameter n1
cshould be defined such that its choice is consistent with a tilted tori,
e.g. n1
c= 1.
8
Na= 3 (2,0)(1,1)(3,1/2)
Nb= 2 (3,−1/2)(1,0)(1,1/2)
Nc= 1 (n1
c,1/2)(1,0)(0,1)
Nd= 1 (2,0)(−3,2)(1,−1/2)
Ne= 1 (2,0)(−1,1)(1,−1/2).(2.15)
Another alternative choice, satisfied by all RR tadpoles will be
n2
a= 1, n1
b= 1, n1
c∈Z, n2
d= 2, ne= 1, β1= 1, β2= 1/2, ǫ = 1.(2.16)
with cycle wrapping numbers
Na= 3 (1,0)(1,1/2)(3,1/2)
Nb= 2 (1,−1)(2,0)(1,1/2)
Nc= 1 (n1
c,1/2)(2,0)(0,1)
Nd= 1 (1,0)(2,1)(1,−1/2)
Ne= 1 (1,0)(1,1/2)(1,−1/2).(2.17)
f) the hypercharge operator in the model is defined as a linear combination of the
three generators of the SU(3), U(1)c,U(1)d,U(1)egauge groups:
Y=1
6U(1)a−1
2U(1)c−1
2U(1)d−1
2U(1)e.(2.18)
3 Cancellation of U(1) Anomalies
In general the mixed anomalies Aij of the four U(1)’s with the non-Abelian gauge
groups are given by
Aij =1
2(Iij −Iij⋆)Ni.(3.1)
Moreover, analyzing the mixed anomalies of the extra U(1)’s with the non-abelian
gauge groups SU(3)c,SU(2)b, we can conclude that there are two anomaly free combi-
nations Qc,Qa−3Qd−3Qe. Also, note that the gravitational anomalies cancel since
D6-branes never intersect O6-planes. In the orientifolded type I torus models gauge
anomaly cancellation [18] [20] is guaranteed through a generalized GS mechanism [3]
9
that uses the 10-dimensional RR gauge fields C2and C6and gives at four dimensions
the following couplings to gauge fields
Nam1
am2
am3
aZM4
Bo
2∧Fa;n1
bn2
bn3
bZM4
Co∧Fb∧Fb,(3.2)
NanJnKmIZM4
BI
2∧Fa;nI
bmJ
bmK
bZM4
CI∧Fb∧Fb,(3.3)
where C2≡Bo
2and BI
2≡R(T2)J×(T2)KC6with I= 1,2,3 and I6=J6=K. Notice the
four dimensional duals of Bo
2, BI
2:
Co≡Z(T2)1×(T2)2×(T2)3C6;CI≡R(T2)IC2,(3.4)
where dCo=−⋆dBo
2, dCI=−⋆dBI
2.
The triangle anomalies (3.1) cancel from the existence of the string amplitude in-
volved in the GS mechanism [19] in four dimensions [18]. The latter amplitude, where
the U(1)agauge field couples to one of the propagating B2fields, coupled to dual
scalars, that couple in turn to two SU (N) gauge bosons, is proportional [3] to
−Nam1
am2
am3
an1
bn2
bn3
b−NaX
I
nI
anJ
anK
bmI
amJ
bmK
b, I 6=J, K (3.5)
Taking into account the phenomenological requirements of eqn. (2.9) the RR cou-
plings BI
2of (3.3) then appear into three terms 9:
B1
2∧ −2ǫ˜ǫβ1
β2!Fb,
B2
2∧ ǫβ2
β1!(9Fa+ 2Fd+Fe),
B3
2∧ 3˜ǫn2
a
2β1Fa+n1
b
β2Fb+n1
c
β2Fc−˜ǫn2
d
2β1Fd−˜ǫn2
e
2β1Fe!.(3.6)
At this point we should list the couplings of the dual scalars CIof BI
2that required
to cancel the mixed anomalies of the five U(1)’s with the non-abelian gauge groups
SU (Na). They are given by
C1∧[ǫ˜ǫβ2
2β1(Fa∧Fa)−ǫ˜ǫβ2
β1(Fd∧Fd)−ǫ˜ǫβ2
2β1Fe∧Fe)],
C2∧[−ǫβ1
2β2(Fb∧Fb) + ǫβ1
β2(Fc∧Fc)],
Co∧ 3n2
a
β1(Fa∧Fa) + ˜ǫn1
b
β2(Fb∧Fb) + n2
d
β1(Fd∧Fd) + n2
e
β1(Fe∧Fe)!,(3.7)
9For convenience we have included the dependence on ǫ, ˜ǫparameters.
10
Notice that the RR scalar B0
2does not couple to any field Fias we have imposed
the condition (2.9) which excludes the appearance of any exotic matter.
Looking at (3.6) we can conclude that there are two anomalous U(1)’s that become
massive through their couplings to the RR fields. They are the model independent
fields, U(1)band the combination 9U(1)a+ 2U(1)d+U(1)e, which become massive
through their couplings to the RR 2-form fields B1
2, B2
2respectively. In addition, there
is a model dependent, non-anomalous and massive U(1) field coupled to B3
2RR field.
That means that the two massless and anomaly free combinations are U(1)cand U(1)a−
3U(1)d−3U(1)e. Also, note that the mixed anomalies Aij are cancelled by the GS
mechanism set by the couplings (3.6, 3.7).
The question we want to address at this point is how we can, from the general class
of models, associated with the generic SM’s of tables (1) and (2), pick up the subclass
that corresponds to the ones associated with just the observable SM at low energies.
Clearly, for this to happen we have to identify the subclass of tadpole solutions of table
(2) that corresponds to the hypercharge assignment (2.18) of the standard model 10
spectrum.
In general, the generalized Green-Schwarz mechanism that cancels non-abelian
anomalies of the U(1)’s to the non-abelian gauge fields involves couplings of closed
string modes to the U(1) field strengths 11 in the form
X
a
fi
aBa∧tr(Fi).(3.9)
Effectively, the mixture of couplings in the form
Ai k +X
a
fi
agk
a= 0 (3.10)
cancels the all non-abelian U(1) gauge anomalies. That means that if we want to
keep some U(1) massless we have to keep it decoupled from some closed string mode
couplings that can make it massive, that is
X
a
(1
6fα
a−1
2fc
a−1
2fd
a−1
2fe
a) = 0 .(3.11)
10At this point, we recall an argument that have appeared in [3].
11In addition, to the couplings of the Poincare dual scalars ηaof the fields Ba,
X
a
gk
aηatr(Fk∧Fk).(3.8)
11
In conclusion, the combination of the U(1)’s which remains light at low energies, is
(3n2
a+ 3n2
d+ 3n2
e)6= 0, Ql=n1
c(Qa−3Qd−3Qe)−3˜ǫβ2(n2
a+n2
d+n2
e)
2β1Qc.(3.12)
The subclass of tadpole solutions of (3.12) having the SM hypercharge assignment at
low energies is exactly the one which is proportional to (2.18). It satisfies the condition,
n1
c=˜ǫβ2
2β1(n2
a+n2
d+n2
e).(3.13)
We note that there is one extra anomaly free, model dependent U(1) beyond the
hypercharge combination, and orthogonal to the latter, which is 12
QN=3˜ǫβ2
2β1(Qa−3Qd−3Qe) + 19n1
cQc.(3.14)
Lets us summarize what we have found up to now. The tadpole solutions of table
(2), taking into account the condition (3.13), give at low energies classes of models that
have the low energy spectrum of the SM with the correct hypercharge assignments. At
this level the gauge group content of the model includes beyond SU(3)⊗SU(2)⊗U(1)Y
the additional U(1)Ngenerator and all SM particles gets charged under the additional
U(1)Nsymmetry. However, notice that QNhas a non-zero coupling to RR field B3
2.
That is it receives a mass of order of the string scale Msand disappears from the low
energy spectrum. Hence at low energy only the SM remains.
In the next sections we will see that in the present five stack constructions it is
possible to use an additional mechanism to break this extra U(1) symmetry, comple-
mentary to the one associated with the RR fields. In involves the usual mechanism
of giving a vev to a scalar field. In this case we will have to require that the inter-
section where the right handed neutrino is localized, respects N= 1 supersymmetry.
In the latter case the immediate effect on obtaining just the SM at low energies will
be one additional linear condition on the RR tadpole solutions of table (2). We note
that when n1
c= 0, it is possible to have massless in the low energy spectrum both the
U(1) generators, Qc, and the B-L generator (1/3)(Qa−3Qd−3Qe) as long as n1
c= 0,
n2
a=−n2
d−n2
e.
12We note that alternatively, in an orthogonal basis, the three U(1)’s are coupled to Bi
2’s, the 4th
U(1) is the hypercharge (3.12) and its condition (3.13), the fifth U(1) is U(1)(5) = (−3
29 +3
28 )Fa−
1
29 Fd+1
28 Fe, the latter surviving massless the Green-Schwarz mechanism when n2
a=−(28/9)n2
d. In
this case, U(1)(5) should be broken by the vev of sνRonly (see section 5).
12
4 Electroweak Higgs and symmetry breaking from
open string tachyons
The mechanism of electroweak symmetry breaking is a well understood effect at the
level of gauge theories with or without supersymmetry. At the string theory level
the mechanism is believed to take place either by using open string tachyonic modes
between parallel branes or following a recent suggestion using brane recombination
[13]. In the former mechanism, the mass of the Higgs field receives contributions
from the distance between the branes b,c(b,c⋆) which are parallel across the second
tori (see table 2). By varying the distance between the branes, across the 2nd tori,
the Higgs mass could become tachyonic signalling electroweak symmetry breaking. In
the latter mechanism, one of the two factorizable b-branes, making the SU(2)-stack,
recombine with the single U(1) c-brane into a single non-factorizable j-brane. After
the recombination the electroweak symmetry is broken, a result better seen from the
new intersection numbers produced. Note that the latter procedure is topological and
cannot be described using field theoretical methods. During the recombination process
instead of us working with our usual wrapping numbers (2.3), one must work with
cycles associated with a non-factorizable T6torus. Also one has to preserve the RR
charge in homology before and after the recombination process. In this work, we will
follow the former method.
4.1 The angle structure
We have up to now describe the appearance in the R-sector of open strings of Iab mass-
less chiral fermions in the D-brane intersections that transform under bifundamental
representations Na,¯
Nb. We should note that in backgrounds with intersecting branes,
besides the actual presence of massless fermions at each intersection, we have evident
the presence of an equal number of massive scalars (MS), in the NS-sector, in exactly
the same representations as the massless fermions [10]. The mass of the MS is of order
of the string scale. In some cases, it is possible that some of those MS may become
tachyonic, triggering a potential that looks like the Higgs potential of the SM, espe-
cially when their mass, that depends on the angles between the branes, is such that
is decreases the world volume of the 3-cycles involved in the recombination process of
joining the two branes into a single one [22].
The models examined in this work, are based on orientifolded six-tori on type I
strings. In those configurations the bulk has N= 4 SUSY. Lets us now describe
13
the open string sector of the model. In order to describe the open string spectrum
we introduce a four dimensional twist [9, 10] vector υθ, whose I-th entry is given by
ϑij , with ϑij the angle between the branes iand j-branes. After GSO projection the
states are labeled by a four dimensional twisted vector r+υθ, where PIrI=odd and
rI∈Z,Z+1
2for NS, R sectors respectively. The Lorentz quantum numbers are denoted
by the last entry. The mass operator for the states is provided by:
α′M2
ij =Y2
4π2α′+Nbos(ϑ) + (r+υ)2
2−1
2+Eij ,(4.1)
where Eij the contribution to the mass operator from bosonic oscillators, and Nosc (ϑ)
their number operator, with
Eij =X
I
1
2|ϑI|(1 − |ϑI|),(4.2)
and Ymeasures the minimum distance between branes for minimum winding states.
If we represent the twisted vector r+υ, by (ϑ1, ϑ2, ϑ3,0), in the NS open string
sector, the lowest lying states are given 13 by:
State Mass
(−1 + ϑ1, ϑ2, ϑ3,0) α′M2=1
2(−ϑ1+ϑ2+ϑ3)
(ϑ1,−1 + ϑ2, ϑ3,0) α′M2=1
2(ϑ1−ϑ2+ϑ3)
(ϑ1, ϑ2,−1 + ϑ3,0) α′M2=1
2(ϑ1+ϑ2−ϑ3)
(−1 + ϑ1,−1 + ϑ2,−1 + ϑ3,0) α′M2= 1 −1
2(ϑ1+ϑ2+ϑ3)
(4.3)
The angles at the ten different intersections can be expressed in terms of the tadpole
solutions parameters. Let us define the angles :
θ1=1
πtan−1β1R(1)
2
n1
bR(1)
1
, θ2=1
πtan−1β2R(2)
2
n2
aR(2)
1
, θ3=1
πtan−1R(3)
2
6R(3)
1
,
˜
θ1=1
πtan−1β1R(1)
2
n1
cR(1)
1
,˜
θ2=1
πtan−12β2R(2)
2
n2
dR(2)
1
,˜
θ3=1
πtan−1R(3)
2
2R(3)
1
,
¯
θ2=1
πtan−1β2R(2)
2
n2
eR(2)
1
,;¯
θ3=1
πtan−1R(3)
2
2R(3)
1
,(4.4)
where R(i)
1,2are the compactification radii for the three i= 1,2,3 tori, namely projections
of the radii onto the X(i)
1,2directions when the NS flux B field, bi, is turned on and we
have chosen for convenience ǫ= ˜ǫ= 1.
13we assumed 0 ≤ϑi≤1 .
14
2
(1)
X
a, a*,d,d* b,b*,c,c*
2
(2)
X
1
X(1)
θ1
∼
2
(2)
R
1
(2)
R1
(3)
R
2
(1)
R
c*
c
b
(1)
R1e, e*
b*
θ1
X(2)
1
a*
d
d*
θ2
∼θ2
a
θ2
e*
e
a
b,e*,d*
θ
∼
3
θ3
θ3
a*
d,b*
1
(3)
X
c
c* e,
(3)
R2
(3)
2
X
Figure 1: Assignment of angles between D6-branes on the five stack type I model giving rise
to the SM at low energies. The angles between branes are shown on a product of T2×T2×T2.
We have chosen β1=β2= 1, n1
b, n1
c, n2
a, n2
d>0, ǫ= ˜ǫ= 1.
At each of the ten non-trivial intersections we have the presence of four states
ti, i = 1,···,4, associated to the states (4.3). Hence we have a total of forty different
scalars in the model 14 .
The following mass relations hold between the different intersections of the classes
of models :
m2
ab(t2) + m2
ab(t3) = m2
ab⋆(t2) + m2
ab⋆(t3) = m2
bd(t2) + m2
bd(t3),
m2
ac(t2) + m2
ac(t3) = m2
ac⋆(t2) + m2
ac⋆(t3) = m2
cd(t2) + m2
cd(t3),
m2
ce⋆(t1) + m2
ce⋆(t2) = m2
cd⋆(t1) + m2
cd⋆(t2),(4.5)
or equivalently
m2
QL(t2) + m2
QL(t3) = m2
qL(t2) + m2
qL(t3) = m2
L(t2) + m2
L(t3),
m2
UR(t2) + m2
UR(t3) = m2
DR(t2) + m2
DR(t3) = m2
NR(t2) + m2
NR(t3),
m2
eR(t1) + m2
eR(t2) = m2
ER(t1) + m2
ER(t2),(4.6)
14In figure one, we can see the D6 branes angle setup in the present models.
15
We note that in this work, we will not discuss the stability conditions for absence of
tachyonic scalars such that the D-brane configurations discussed will be stable as this
will be discussed elsewhere. Similar conditions have been examined before in [3, 4].
4.2 Tachyon Higgs mechanism in detail
In this section, we will study the electroweak Higgs sector of the models. We note
that below the string scale the massless spectrum of the model is that of the SM with
all particles having the correct hypercharge assignments but with the gauge symmetry
being SU(3) ⊗SU(2) ⊗U(1) ⊗U(1)N. For the time being we will accept that the addi-
tional U(1)Ngenerator breaks to a scale higher than the scale of electroweak symmetry
breaking. The latter issue will be discussed in detail in the next section. Thus in the
following we will focus our attention to the Higgs sector of the theory.
In general, tachyonic scalars stretching between two different branes can be used
as Higgs scalars as they can become non-tachyonic by varying the distance between
parallel branes. This happens in the models under discussion as the complex scalars
h±,H±get localized between the b,cand between b,c∗branes respectively and can be
interpreted from the field theory point of view [3] as Higgs fields which are responsible
for the breaking the electroweak symmetry. We note that the intersection numbers of
the b, c and b, c⋆branes across the six-dimensional torus vanish as a result of the fact
that the b, c and b, c⋆branes are parallel across the second tori. The electroweak Higgs
fields, appearing as Hi(resp. hi), i= 1,2, in table (3), come from the NS sector, from
open strings stretching between the parallel b,c⋆(resp. c) branes along the second tori,
and from open strings stretching between intersecting branes along the first and third
tori.
Initially, the Higgses of table (3), are part of the massive spectrum of fields localized
in the intersections bc, bc⋆. However, we emphasize that the Higgses Hi,hibecome
massless by varying the distance along the second tori between the b, c⋆,b, c branes
respectively. In fact, a similar set of Higgs fields appear in the four stack models of [3],
but obviously with different geometrical data. We should note that the representations
of Higgs fields Hi,hiis the maximum allowed by quantization. Their number is model
dependent.
The number of complex scalar doublets present in the models is equal to the non-
zero intersection number product between the bc,bc⋆branes in the first and third
complex planes. Thus
nH±=Ibc⋆=|ǫβ1(n1
b−n1
c)|, nh±=Ibc =|ǫβ1(n1
b+n1
c)|.(4.7)
16
Intersection EW breaking Higgs QbQcY
bc h11−1 1/2
bc h2−1 1 -1/2
bc⋆H1−1−1 1/2
bc⋆H21 1 -1/2
Table 3: Higgs fields responsible for electroweak symmetry breaking.
The precise geometrical data for the scalar doublets are
State Mass2
(−1 + ϑ1,0, ϑ3,0) α′(Mass)2
Y=Z2
4π2+1
2(ϑ3−ϑ1)
(ϑ1,0,−1 + ϑ3, , 0) α′(Mass)2
X=Z2
4π2+1
2(ϑ1−ϑ3)(4.8)
where X={H+
bc⋆, h+
bc},Y={H−
bc⋆, h−
bc}and Z2is the distance2in transverse space
along the second torus, ϑ1,ϑ3are the (relative) angles between the b-, c⋆(for H±) (or
b,cfor h±) branes in the first and third complex planes.
Also we note the presence of two ”Higgsino masses” at each of the bc or bc⋆inter-
sections, with the same quantum numbers and representations as the Higgs fields and
masses corresponding to
State Mass2
(−1/2 + ϑ1,∓1/2,−1/2 + ϑ3,±1/2) (Mass)2=Z2
4π2α′.(4.9)
We note that in this picture while the Higgs fields can be made massless by varying the
distance between the branes, the Higginos are not massless and are part of the N= 2
massive spectrum accompanying the “massless” Higgs fields at the intersections bc,bc⋆.
As we noted the presence of scalar doublets H±, h±, can be seen as coming from
the field theory mass matrix
(H∗
1H2)M2
H1
H∗
2
+ (h∗
1h2)m2
h1
h∗
2
+h.c. (4.10)
where
M2=M2
s
Z(bc∗)
21
2|ϑ(bc∗)
1−ϑ(bc∗)
3|
1
2|ϑ(bc∗)
1−ϑ(bc∗)
3|Z(bc∗)
2
,(4.11)
m2=M2
s
Z(bc)
21
2|ϑ(bc)
1−ϑ(bc)
3|
1
2|ϑ(bc)
1−ϑ(bc)
3|Z(bc)
2
(4.12)
17
The fields Hiand hiare thus defined as
H±=1
2(H∗
1±H2); h±=1
2(h∗
1±h2).(4.13)
As a result the effective potential which corresponds to the spectrum of Higgs scalars
is given by
VHiggs =m2
H(|H1|2+|H2|2) + m2
h(|h1|2+|h2|2)
+m2
BH1H2+m2
Bh1h2+h.c., (4.14)
where
mh2=Z(bc)
2
4π2α′;mH2=Z(bc∗)
2
4π2α′
m2
b=1
2α′|ϑ(bc)
1−ϑ(bc)
3|;m2
B=1
2α′|ϑ(bc∗)
1−ϑ(bc∗)
3|(4.15)
We note that the Z2is a free parameter, a moduli, and can become very small in
relation to the Planck scale. However, the m2
Bmass can be expressed in terms of the
scalar masses of the particles present at the different intersections. Going one step
further, we can express the “angle” part of the Higgs masses in terms of the angles
defined in (4.4) and in figure 1. Explicitly, we find 15 :
m2
B=1
2α′|˜
ϑ1−ϑ1|;m2
b=1
2α′|˜
ϑ1+ϑ1+ϑ3−1
2|
m2
h=1
2α′(χb−χc)2;m2
H=1
2α′(χb+χc)2,(4.16)
where χb,χcthe distances from the orientifold plane of the branes b, c. Making use of
the scalar mass relations at the intersections of the model we can reexpress the mass
relations (4.16), in terms of (4.5). The values of m2
B,m2
bare given in appendix A.
5 SUSY at intersections and intermediate scale
In the present classes of models the U(1) symmetry gets broken and the associated
gauge boson receives a mass, as there is a non-zero coupling of (3.14) to RR two form
field B3
2. However, for special values of the RR tadpole solution parameters is is possible
that the usual mechanism, of giving a vev to a scalar, contributes additionally to the
mass of the massive gauge boson associated with (3.14).
15We have chosen a configuration with ǫ= ˜ǫ= 1, nb, nc, nd, ne>0.
18
Thus our aim in this section is to provide us with a complementary mechanism
to break the additional generator U(1)N. That may happen by demanding that the
sector ce preserves N= 1 SUSY. That will have as an effect the appearance of Ice
massless scalars in the intersection with the same quantum numbers as the massless
Ice fermions. Because Ice = 1, and the massless fermion localized in the intersection is
νR, the massive partner of νRwhich become massless will be a sνR. Consequently, by
giving a vev to sνR, the sνRgets charged and thus breaks U(1)N, leaving only the SM
gauge group SU(3) ⊗SU(2) ⊗U(1)Yat low energies. Lets us describe the procedure
in more detail.
We want the particles localized on the intersection ce to respect some amount of
SUSY, in our case N= 1. That means that the relative angle between branes c,e,
should obey the SUSY preserving condition
±˜
θ1±¯
θ2±(π
2+θ3) = 0 (5.1)
In this case, a massless scalar field appear in the intersection ce, the superpartner of
the νR, the sνRfield. It is charged under the additional U(1)Nsymmetry, thus breaks
U(1)Nby receiving a vev. In this case the surviving gauge symmetry is of SM. The
scale of the additional breaking, MN, is set from the vev of sνRand in principle, can
be anywhere between MZand the string scale.
The following choice :
tan−1β1U(1)
n1
c
+tan−1β2U(2)
n2
e
−tan−1(U(3)
6)−π
2= 0,(5.2)
with
n2
e= 0,β1U(1)
n1
c
=U(3)
6, α =β1U(1)
n1
c
, U(i)=R(i)
2
R(i)
1
.(5.3)
solves (5.1). In particular,
n2
e= 0 ⇒β2= 1 (5.4)
thus the second tori is not tilted. The angle content of the branes, c,e, when the gauge
symmetry breaks to the SM is given in table (4).
Summarizing the SM exists between MZand the mass scale MNof the additional
U(1)N. A set of SM wrappings exists only if we consider the hypercharge (3.13) and
the gauge symmetry breaking condition (5.3) when defining numerically the tadpole
solutions of table (2). Taking into account both conditions a consistent set, for the
observable SM to exist, wrapping numbers is given by
n2
e= 0, β2= 1, β1= 1/2, nb=−1, n2
d=−1, n2
a= 2, n1
c= 1 (5.5)
19
Brane θ1
aθ2
aθ3
a
c tan−1α0π
2
e0π
2tan−1(α)
Table 4: Angle content for branes participating in the gauge symmetry breaking to the SM.
Imposing N= 1 SUSY on the open sector ce breaks the surplus U(1)Nby a vev of sνR.
or
Na= 3 (2,0)(2,1)(3,1/2)
Nb= 2 (−1,−1/2)(1,0)(1,1/2)
Nc= 1 (1,1/2)(1,0)(0,1)
Nd= 1 (2,0)(−1,2)(1,−1/2)
Ne= 1 (2,0)(0,1)(1,−1/2).(5.6)
It satisfies all tadpole conditions but the first, the latter is satisfied with the addition
of four ¯
D6located at (2,0)(1,0)(2,0).
The number of electroweak Higgs present in the model can be investigated further.
The most interesting cases that have a minimal Higgs content follow :
•The Higgs system of MSSM
For (4.7), we can see that the minimal set of Higgs in the models is obtained for
either nH= 0, nh= 1, or nh= 1, nH= 0. The two cases, are studied in table
(5). We found two families of models that depend on a single integer n2
d. We
also list the number of necessary NDbranes required to cancel the first tadpole
condition. We have taken into account the conditions (3.13), (5.4) necessary to
obtain the observable SM at low energies.
The case nH= 1, nh= 0 appears to be the most interesting as this appears
to give a plausible explanation for the existence of small and different neutrino
masses to the different generations. These issues are examined in more detail in
the next section.
•Double Higgs system The next to minimal set of Higgses is obtained when nH=
1, nh= 1. In this case, quarks and leptons get their mass from the start.
20
Higgs fields β1β2nbncn2
aND
nH= 1, nh= 0 1/2 1 1 −1−1−n2
d8 + 4n2
d
nH= 1, nh= 0 1/2 1 −1 1 1 −n2
d4n2
d
nH= 0, nh= 1 1/2 1 1 1 1 −n2
d−1 + 4n2
d
nh= 0, nh= 1 1/2 1 −1−1−1−n2
d9 + 4n2
d
nH= 1, nh= 1 1 1 1 0 −n2
d7 + 4n2
d
nH= 1, nh= 1 1 1 −1 0 −n2
d9 + 4n2
d
nH= 1, nh= 1 1 1 0 1 2 −n2
d−1 + 4n2
d
nH= 1, nh= 1 1 1 0 −1−2−n2
d16 + 4n2
d
Table 5: Families of models with with minimal Higgs structure. They depend on a single
integer, nd. The surplus gauge symmetry breaking condition (5.4) has been taken into account.
6 Neutrino couplings and masses
The Yukawa couplings in this model follow the usual pattern that appears in inter-
secting branes [10]. The couplings between the two fermion states Fi
L,¯
Fj
Rand the
Higgs fields Hk, arise from the stretching of the worldsheet between the three D6-
branes which cross at those intersections. For a six dimensional torus they can take
the following form in the leading order [10],
Yklm =e−˜
Aklm ,(6.1)
where ˜
Aklm is the worldsheet area connecting the three vertices in the six dimensional
space. The areas of each of the two dimensional torus involved in this interaction is
typically of order one in string units. For the models discussed in table (1), the Yukawa
interactions for the chiral spectrum of the SM’s yield:
YU
jQLUj
Rh1+YD
jQLDj
RH2+
Yu
ij qi
LUj
RH1+Yd
ij qi
LDj
Rh2+
Yl
hlh
Lνh
Rh1+Ye
hlh
Leh
RH2+
YN
ij LiNj
Rh1+YE
ij LiEj
RH2+h.c
(6.2)
21
where i= 1,2, j= 1,2,3, h= 1.
The nature of Yukawa couplings is such that the lepton and neutrino sector of the
models distinguish between different generations, e.g. the ”first” from the other two
generations, as one generation of neutrinos (resp. leptons) is placed on a different
intersection from the other two one’s. For example looking at the charged leptons
of table (1) we see that one generation of charged leptons lLgets localized on be-
intersection, while the other two generations Lget localized in the bd-intersection.
There are a number of scalar doublets in the model present that are interpreted
in terms of the low energy theory as Higgs doublets. The most interesting ones were
mentioned briefly in the last section. There are two possibilities to be discussed. The
minimal case when nH= 1, nh= 0 and the next to minimal case, nH= 1, nh= 1.
•Minimal Higgs presence
Without loss of generality we choose nH= 1, nh= 0, only the H1,H2fields are
present. The mechanism that will give masses to the charged lepton/quark sector
is similar to what happended in the four stack models of [3]. At tree level two
U-quarks and one D-quark as well the charge leptons gain masses. Identifying
the massive quarks with t, c, b, the rest of the quarks as well as neutrinos remain
massless at tree level. The rest of the quarks are expected to receive masses from
strong interaction effects, that create effective couplings of the form QLUj
RH1,
qi
LDj
RH2creating masses for the u-, d-, s-quarks of less than equal of ΛQCD . Note
that the latter couplings are not allowed at tree level since otherwise the U(1)b
global symmetry would have been violated.
The neutrino sector is slightly different from the four stack counterparts [3] of
the SM’s discussed in the present work. Contrary to the models of [3] where
all neutrinos come from the same intersection, in this work the SM’s are build
such that one generation of neutrinos (and leptons) are placed at a different
intersection from the remaining two generations 16. Thus the structure of Yukawa
couplings in the neutrino sector suggests that they is a distinction between the
neutrinos of the one (e.g. first) generation and the other two one’s. That could
be used in principle to discuss neutrino mass textures in the context of recent
results 17 of SuperKamiokande [24], which suggest that at least two generations
of neutrinos may be massive.
16The latter though are placed both at the same intersection.
17see for example ref. [23].
22
Because Lepton number is a gauged symmetry, there are only Dirac masses al-
lowed. Thus a see-saw mechanism is excluded. The origin of small neutrino
masses originates from dimension 6 operators in the form, breaking the U(1)b
PQ like symmetry through chiral symmetry breaking,
α′(LNR) (QLUR)⋆, α′(lνR)(qLUR).(6.3)
Hence, the smallness of neutrino masses is related to the existence of the dominant
u-quark chiral condensate, < uRuR>. For values 18 of the u-chiral condensate,
and assuming that all generations of neutrino species receive the same value for
u-condensate, of order (240MeV )3, the neutrinos get a mass of order
< uRuL>
M2
s
=(240MeV )3
M2
s
(6.4)
Hence neutrino masses of order 0.1 - 10 eV are easily achieved in consistency
with LSND oscillation experiments [17]. In this case, a generation mixing among
neutrino species is generated from the leading order coupling behavior (6.1).
Alternatively, if we assume that the chiral condensate generates the genera-
tion mixing, receiving different values for different neutrino species the neutrino
masses will depend weakly on the precise form of the couplings (6.1).
•Next to minimal Higgs presence
In this case, all couplings to quarks and leptons are realized from the start. All
particles get a mass. The hierarchy of masses depend both on the Higgs fields
and the leading order Yukawa behavior (6.1).
7 Conclusions and future directions
In this work, we have presented the first examples of string models, not based on a
GUT group at the string scale, that have at low energies only the standard model
and are derived from five stacks of ( D6 ) branes at the string scale 19. We note the
following :
•Baryon number is a gauged symmetry and proton is stable. If proton was not
stable in the models then we should have push the string scale higher than 1016 GeV
18See for example ref. [25].
19The first examples of models giving rise to the SM at low energies have been considered in [3],
using four stacks of branes and the same type I orientifolded T6constructions.
23
to suppress dimension six operators that could potentially contribute to proton decay.
However, in the present class of models, this is not necessary as proton is stable.
•The additional U(1)Nsymmetry may break from its non-zero coupling to the RR
two form fields.
However, as we have noted an additional mechanism is available for certain values of
the tadpole and complex structure parameters, and thus complementary to the Abelian
gauge field mass term generation induced by their non-zero couplings to the RR two
form fields Bi
2,i= 1,2,3. Crucial for this new mechanism, thus contributing to the
breaking of the additional anomaly free U(1) generator and getting only the SM at
low energies, was the novel partial imposition of N= 1 supersymmetry at only one
intersection, e.g. ce. That had as an immediate effect to pull out from the massive
modes the superpartner of νR. It would be interesting to investigate in detail the
symmetry breaking patterns that follow from having a SUSY intersection, at an open
string sector, of the non-SUSY SM’s examined in this work.
•We emphasize that the breaking of the U(1)Nsymmetry implies the existence of
an extra Zoboson above the electroweak scale. Bounds on additional gauge bosons
exist [26] placing them in the range between 500-800 GeV. Thus we conclude that the
string scale should be at least equal to MNor higher. Improved bounds of the string
scale for the present models would require a generalization of the four stack D6 model
[3] analysis of [27], for the masses of the extra U(1) gauge bosons made massive by the
Green-Schwarz mechanism. It would be interesting to extend the analysis of [27] to
the present SM’s that predict an additional intermediate scale between MZand MS.
•A natural extension of the five-stack D6-brane SM’s of this work is to examine
how we can construct D6-brane models that respect some supersymmetry at every
intersection as was detailed 20 recently.
•The models have vanishing RR tadpoles but some NSNS tadpoles remain, leaving
a open issue the full stability of the configurations. It is then an open question if the
backgrounds can be cured using Fischler-Susskind mechanism [28] in redefining the
background [29] as in [30].
Concluding this work, it is very interesting that the present class of models predicts
not only the existence of a non-supersymmetric standard model at low energies but in
addition other classes of models predicting the unique existence of a SUSY partner of
the right handed neutrino, the sνR.
20see the first two references of [13] for a similar construction for the 4-stack SM’s of [3].
24
8 Acknowledgments
I am grateful to D. Cremades, L. Ib´a˜nez and A. Uranga for usuful discussions.
25
9 Appendix A
In this appendix, we list the values of the mass parameters, of section 4, involved in
the mass of the set of four Higgses taking part in the process of electroweak symmetry
breaking. As we remark in the main body of the paper, the quadratic parts of the
Higgs mixing mass terms in the effective field theory potential are exactly calculable
at tree level in the D6-brane models.
m2
B=1
2|m2
QL(t2) + m2
QL(t3)−m2
UR(t2)−m2
UR(t3)|
=1
2|m2
QL(t2) + m2
QL(t3)−m2
DR(t2)−m2
DR(t3)|
=1
2|m2
QL(t2) + m2
QL(t3)−m2
NR(t2)−m2
NR(t3)|
=1
2|m2
qL(t2) + m2
qL(t3)−m2
UR(t2)−m2
UR(t3)|
=1
2|m2
qL(t2) + m2
qL(t3)−m2
DR(t2)−m2
DR(t3)|
=1
2|m2
qL(t2) + m2
qL(t3)−m2
NR(t2)−m2
NR(t3)|
=1
2|m2
L(t2) + m2
L(t3)−m2
UR(t2)−m2
UR(t3)|
=1
2|m2
L(t2) + m2
L(t3)−m2
DR(t2)−m2
DR(t3)|
=1
2|m2
L(t2) + m2
L(t3)−m2
NR(t2)−m2
NR(t3)|
(9.1)
m2
b=1
2|m2
QL(t2) + m2
QL(t3) + m2
UR(t2) + m2
UR(t3)−m2
eR(t1)−m2
eR(t2)|
=1
2|m2
QL(t2) + m2
QL(t3) + m2
UR(t2) + m2
UR(t3)−m2
ER(t1)−m2
ER(t2)|
=1
2|m2
QL(t2) + m2
QL(t3) + m2
NR(t2) + m2
NR(t3)−m2
eR(t1)−m2
eR(t2)|
=1
2|m2
QL(t2) + m2
QL(t3) + m2
NR(t2) + m2
NR(t3)−m2
ER(t1)−m2
ER(t2)|
=1
2|m2
qL(t2) + m2
qL(t3) + m2
UR(t2) + m2
UR(t3)−m2
eR(t1)−m2
eR(t2)|
=1
2|m2
qL(t2) + m2
qL(t3) + m2
UR(t2) + m2
UR(t3)−m2
ER(t1)−m2
ER(t2)|
=1
2|m2
qL(t2) + m2
qL(t3) + m2
DR(t2) + m2
DR(t3)−m2
eR(t1)−m2
eR(t2)|
26
=1
2|m2
qL(t2) + m2
qL(t3) + m2
DR(t2) + m2
DR(t3)−m2
ER(t1)−m2
ER(t2)|
=1
2|m2
qL(t2) + m2
qL(t3) + m2
NR(t2) + m2
NR(t3)−m2
eR(t1)−m2
eR(t2)|
=1
2|m2
qL(t2) + m2
qL(t3) + m2
NR(t2) + m2
NR(t3)−m2
ER(t1)−m2
ER(t2)|
=1
2|m2
L(t2) + m2
L(t3) + m2
UR(t2) + m2
UR(t3)−m2
eR(t1)−m2
eR(t2)|
=1
2|m2
L(t2) + m2
L(t3) + m2
UR(t2) + m2
UR(t3)−m2
ER(t1)−m2
ER(t2)|
=1
2|m2
L(t2) + m2
L(t3) + m2
DR(t2) + m2
DR(t3)−m2
eR(t1)−m2
eR(t2)|
=1
2|m2
L(t2) + m2
L(t3) + m2
DR(t2) + m2
DR(t3)−m2
ER(t1)−m2
ER(t2)|
=1
2|m2
L(t2) + m2
L(t3) + m2
NR(t2) + m2
NR(t3)−m2
eR(t1)−m2
eR(t2)|
=1
2|m2
L(t2) + m2
L(t3) + m2
NR(t2) + m2
NR(t3)−m2
ER(t1)−m2
ER(t2)|
(9.2)
27
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