A kind of prediction from string phenomenology: extra matter at low
  energy
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Modern Physics Letters A
c© World Scientific Publishing Company
A KIND OF PREDICTION FROM STRING PHENOMENOLOGY:
EXTRA MATTER AT LOW ENERGY
CARLOS MUÑOZ
Departamento de F́ısica Teórica C-XI and Instituto de F́ısica Teórica C-XVI,
Universidad Autónoma de Madrid, Cantoblanco, E-28049 Madrid, Spain
carlos.munnoz@uam.es
We review the possibility that the Supersymmetric Standard Model arises from orbifold
constructions of the E8×E8 Heterotic Superstring, and the phenomenological properties
that such a model should have. In particular, trying to solve the discrepancy between the
unification scale predicted by the Heterotic Superstring (≈ gGUT × 5.27 · 10
17 GeV) and
the value deduced from LEP experiments (≈ 2 · 1016 GeV), we will predict the presence
at low energies of three families of Higgses and vector-like colour triplets. Our approach
relies on the Fayet-Iliopoulos breaking, and this is also a crucial ingredient, together with
having three Higgs families, to obtain in these models an interesting pattern of fermion
masses and mixing angles at the renormalizable lebel. Namely, after the gauge breaking
some physical particles appear combined with other states, and the Yukawa couplings
are modified in a well controlled way. On the other hand, dangerous flavour-changing
neutral currents may appear when fermions of a given charge receive their mass through
couplings with several Higgs doublets. We will address this potential problem, finding
that viable scenarios can be obtained for a reasonable light Higgs spectrum.
Keywords: Strings; Orbifolds; Phenomenology.
PACS Nos.: 11.25.Wx, 11.25.Mj, 12.60.Jv, 12.60.Fr
1. Introduction
In SuperString Theory the elementary particles are not point-like objects but ex-
tended, string-like objects. It is surprising that this apparently small change allows
us to answer fundamental questions that in the context of the quantum field theory
of point-like particles cannot even be posed. For example: Why is the Standard
Model gauge group SU(3) × SU(2)L × U(1)Y ? Why are there three families of
particles? Why is the mass of the electron me = 0.5 MeV? Why is the fine struc-
ture constant α = 1/137? In addition, only SuperString Theory has the potential
to unify all gauge interactions with gravity in a consistent way. In this sense, it
is a crucial step in the construction of the fundamental theory of particle physics
to find a consistent SuperString model in four dimensions accommodating the ob-
served Standard Model (SM), i.e. we need to find the SuperString Standard Model
(SSSM). Actually, this is the main task of what we call String Phenomenology.
In the late eighties, the compactification of the E8 × E8 Heterotic String on
http://arxiv.org/abs/0704.0987v4
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2 Carlos Muñoz
six-dimensional orbifolds proved to be an interesting method to carry out this taska
(for a brief historical account see the Introduction in Ref. 1 and references therein).
For example, it was shown that the use of two Wilson lines on the torus defining the
symmetric Z3 orbifold can give rise to four-dimensional supersymmetric models with
gauge group SU(3)×SU(2)×U(1)5×Ghidden and, automatically, three generations
of chiral particles2. In addition, it was also shown that the Fayet–Iliopoulos (FI)
D-term, which appears because of the presence of an anomalous U(1), can give rise
to the breaking of the extra U(1)’s. In this way it was possible to construct3,4,5
supersymmetric models with gauge group SU(3)×SU(2)×U(1)Y , three generations
of particles in the observable sector, and absence of dangerous baryon- and lepton-
number-violating operatorsb.
Unfortunately, we cannot claim that one of these Z3 orbifold models is the SSSM,
since several problems are always present. For example, the initially large number
of extra particles, which are generically present in these constructions, is highly
reduced through the FI mechanism, since many of them get a high mass (≈ 1016−17
GeV). However, in general, some extra SU(3) triplets, SU(2) doublets and SU(3)×
SU(2) singlets still remain at low energy. On the other hand, given the predicted
value for the unification scale in the Heterotic String12, MGUT ≈ gGUT ×5.27 ·1017
GeV, the values of the gauge couplings deduced from LEP experiments cannot13 be
obtainedc. It was also not possible to obtain in these models the necessary Yukawa
couplings reproducing the observed fermion masses5,4,14.
At this point, it is fair to say that almost 20 years have gone by since String
Phenomenology started, and the SSSM has not been found yetd As acquittal on the
charge we should remark that there are thousands of models (vacua) that can be
built. Some of them have the gauge group of the SM or GUT groups, three families
of particles, and other interesting properties, but many others have a number of
families different from three, no appropriate gauge groups, no appropriate matter,
etc. A perfect way of solving this problem would be to use a dynamical mechanism to
select the correct model (vacuum). Such a mechanism should be able to determine
a point in the parameter space of the Heterotic String determining the correct
compactification producing the gauge group SU(3)c×SU(2)L×U(1)Y , three families
of the known particles, the correct Yukawa couplings, etc. The problem is that such
a mechanism has not been discovered yet.
So, for the moment, the best we can do is keep trying, i.e. to use the experimental
aOther interesting attempts at model building used Calabi–Yau spaces and fermionic constructions.
bRecently, other interesting models in the context of the Z3 orbifold
6, as well as in the context of
the Z6 orbifold
7,8,9, and Z12 orbifold
10,11, have been analysed.
cRecall that this is only possible in the context of the Minimal Supersymmetric Standard Model
(MSSM) for MGUT ≈ 2× 10
16 GeV.
dAnd this sentence can also be applied to any of the interesting models constructed in more recent
years using D-brane technology15. Actually, the probability of obtaining an MSSM like gauge
group with three generation in the context of intersecting D-branes in an orientifold background
seems to be extremelly small16,17, of about 10−9.
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A kind of prediction from string phenomenology: extra matter at low energy 3
results available (such as the SM gauge group, three families, fermion masses, mixing
angles, etc.), to discard models. Although the model space is in principle huge, a
detailed analysis can reduce this to a reasonable size. For example, within the Z3
orbifold with two Wilson lines, one can construct in principle a number of order
50000 of three-generation models with the SU(3) × SU(2) × U(1)5 gauge group
associated to the first E8 of the Heterotic String. However, a study implied that
most of them are equivalent18, and in fact, at the end of the day, only 192 different
models were found19,18. This reduction is remarkable, but we should keep in mind
that the analysis of each one of these models is really complicated.
Nevertheless, a certain degree of optimism is important when working in String
Phenomenology, and one can argue that if the SM arises from SuperString Theory
there must exist one model with the right properties. In the present review we will
adopt this viewpoint, and will assume that the SM arises from orbifolds construc-
tions. Instead of the painful work of searching for the correct orbifold model, we
will try to deduce the phenomenological properties that such a model must have
in order to solve the crucial problems mentioned above, with the hope that this
analysis will allow us to make predictions that can be tested at the LHC.
In fact, all those problems, extra matter, gauge coupling unification, and cor-
rect Yukawa couplings, are closely related. The first two because the evolution of
the gauge couplings from high to low energy through the renormalization group
equations (RGEs), depends on the existing matter20,6. In Section 2 we will dis-
cuss a solution to the gauge coupling unification problem implying the prediction of
three generations of supersymmetric Higgses and vector-like colour triplets at low
energies1. In this solution the FI scale plays an important role.
Concerning the third problem, how to obtain the observed structure of fermion
masses and mixing angles, this is in our opinion the most difficult task in String
Phenomenology. For example, the right model must reproduce also the correct mass
hierarchy for quarks and leptons, mt
∼ 105, mτ
∼ 103, etc., and this is not a trivial
task, although it is true that one can find interesting results in the literature. In
particular, orbifold spaces have a beautiful mechanism to generate a mass hierar-
chy at the renormalizable level. Namely, Yukawa couplings between twisted matter
can be explicitly computed and they get suppression factors, which depend on the
distance between the fixed points to which the relevant fields are attached21−26.
The couplings can be schematically written as λ ∼ e−
Ti , with Re Ti ∼ R2i ,
and the Ti are the moduli fields associated to the size and shape of the orbifold.
The distances can be varied by giving different vacuum expectation values (VEVs)
to these moduli, implying that one can span in principle five orders of magnitude
the Yukawa couplings24−26. Unfortunately, this is not the end of the story, since
Nature tells us that a weak coupling matrix exists with weird magnitudes for the
entries, and that therefore we must arrange our up-and down-quark Yukawa cou-
plings in order to have specific off diagonal elementse. In Section 3 we will see that
e Needless to say, the recent experimental confirmation of neutrino masses makes the task even
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4 Carlos Muñoz
to obtain this at the renormalizable level is possible if three Higgs families and the FI
breaking are present26,27. Thus we have a common solution for the three problems
mentioned above.
On the other hand, it is well known that dangerous flavour-changing neutral
currents (FCNCs) may appear when fermions of a given charge receive their mass
through couplings with several Higgs doublets28,29. This situation might be present
here since we have three generations of supersymmetric Higgses. In Section 4 we
will address this potential problem, finding that viable scenarios can be obtained27.
2. Predictions from gauge coupling unification
Since we are interested in the analysis of gauge couplings, we need to first clarify
which is the relevant scale for the running between the supersymmetric scaleMS and
the unification point. Let us recall that in heterotic compactifications some scalars
singlets Ci develop vacuum expectation values (VEVs) in order to cancel the FI
D-term, without breaking the SM gauge group. An estimate about their VEVs
can be done with the average result 〈Ci〉 ∼ 1016−17 GeV (see e.g. Ref. 6). After
the breaking, many particles, say ξ, acquire a high mass because of the generation
of effective mass terms. These come for example from operators of the type Ciξξ.
In this way extra vector-like triplets and doublets and also singlets become very
heavy. We will use the above value as our relevant scale, the so-called FI scale
MFI ≈ 1016−17 GeV.
As discussed in the Introduction, we are interested in the unification of the gauge
couplings at MGUT ≈ gGUT ×5.27 ·1017 GeV. This is not a simple issue, and various
approaches towards understanding it have been proposed in the literature30. Some
of these proposals consist of using string GUT models, extra matter at intermedi-
ate scales, heavy string threshold corrections, non-standard hypercharge normaliza-
tions, etc. In our case, we will try to obtain this value by using first the existence of
extra matter at the scale MS. We will see that this is not sufficient and, as a conse-
quence, the FI scale must be included. Let us concentrate for the moment on α3 and
α2. Recalling that three generations appear automatically for all the matter in Z3
orbifold scenarios with two Wilson lines, the most natural possibility is to assume
the presence of three light generations of supersymmetric Higgses. This implies that
we have four extra Higgs doublets, n2 = 4, with respect to the case of the MSSM.
Unfortunately, this goes wrong. Whereas α−13 remains unchanged, since the number
of extra triplets is n3 = 0, the line for α
2 is pushed down with respect to the case
of the MSSM. As a consequence, the two couplings cross at a very low scale (≈ 1012
GeV). We could try to improve this situation by assuming the presence of extra
triplets in addition to the four extra doublets. Then the line for α−13 is also pushed
down and therefore the crossing might be obtained for larger scales. However, even
more involved. We have to explain also the weak coupling matrix with the charged leptons. Besides,
in addition to the hierarchies shown above, we have to explain others such as me
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A kind of prediction from string phenomenology: extra matter at low energy 5
Fig. 1. Unification of the gauge couplings at MGUT ≈ gGUT × 5.27 · 10
17 GeV with three light
generations of supersymmetric Higgses and vector-like colour triplets. In this example we show
one of the four possible patterns of heavy matter in eq. (2), in particular that with a) nFI3 = 0.
The line corresponding to α1 is just one of the many possible examples.
for the minimum number of extra triplets that can be naturally obtained in our
scenario, 3 × {(3, 1) + (3̄, 1)}, i.e. n3 = 6, the “unification” scale turns out to be
too large (≈ 1021 GeV). One can check that other possibilities including more extra
doublets and/or triplets do not work1. Thus, using extra matter at MS we are not
able to obtain the Heterotic String unification scale, since α3 never crosses α2 at
MGUT ≈ gGUT ×5.27 ·1017 GeV. Fortunately, this is not the end of the story. As we
will show now, the FI scale MFI is going to play an important role in the analysis.
In order to determine whether or not the Heterotic String unification scale can
be obtained, we need to know the number of doublets nFI2 and triplets n
3 in our
construction with masses of order the FI scale MFI . It is possible to show that
within the Z3 orbifold with two Wilson lines, three-generation standard-like models
must fulfil the following relation for the extra matter: 2+n2+n
2 = n3+n
3 +12.
Then, it is now straightforward to check that only models with n2 = 4, n3 = 6, and
therefore nFI2 − nFI3 = 12, may give rise to the Heterotic String unification scale1
(other possibilities for n2, n3 do not even produce the crossing of α3 and α2). This
is shown in Fig. 1 for an example with nFI3 = 0, and assuming MS = 500 GeV.
There we are using MFI = 2 · 1016 GeV as will be discussed below.
Note that at low energy we then have (excluding singlets)
3× {(3, 2) + 2(3̄, 1) + (1, 2)}+ 3× {(3, 1) + (3̄, 1) + 2(1, 2)} , (1)
i.e. the matter content of the Supersymmetric SM with three generations of Higgses
and vector-like colour triplets.
Let us remark that in these constructions only the following patterns of matter
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6 Carlos Muñoz
with masses of order MFI are allowed:
a) nFI3 = 0 , n
2 = 12 → 3× {4(1, 2)} ,
b) nFI3 = 6 , n
2 = 18 → 3× {(3, 1) + (3̄, 1) + 6(1, 2)} ,
c) nFI3 = 12 , n
2 = 24 → 3× {2[(3, 1) + (3̄, 1)] + 8(1, 2)} ,
d) nFI3 = 18 , n
2 = 30 → 3× {3[(3, 1) + (3̄, 1)] + 10(1, 2)} . (2)
Thus for a given FI scale, MFI , each one of the four patterns in eq. (2) will give
rise to a different value for gGUT . Adjusting MFI appropriately, we can always get
MGUT ≈ gGUT ×5.27 ·1017 GeV. In particular this is so for MFI ≈ 2×1016 GeV as
shown in Fig. 1. It is remarkable that this number is within the allowed range for
the FI breaking scale as discussed above. For the pattern in Fig. 1 corresponding
to case a) we have gGUT ≈ 1.1, and therefore MGUT ≈ 5.8 · 1017 GeV.
Of course, we cannot claim to have obtained the Heterotic String unification scale
until we have shown that the coupling α1 joins the other two couplings at MGUT .
The analysis becomes more involved now and a detailed account of this issue can be
found in Ref. 1. Let us just mention that the fact that the normalization constant,
C, of the U(1)Y hypercharge generator is not fixed in these constructions as in the
case of GUTs (e.g., for SU(5), C2 = 3/5) is crucial in order to obtain the unification
with the other couplings.
Summarizing, the main characteristic of this scenario is the presence at low
energy of extra matter. In particular, we have obtained that three generations of
Higgses and vector-like colour triplets are necessary.
Since more Higgs particles than in the MSSM are present, there will be of course
a much richer phenomenology. Note for instance that the presence of six Higgs
doublets implies the existence of sixteen physical Higgs bosons, where eleven of
them are neutral and five charged.
Concerning the three generations of vector-like colour triplets, sayD andD, they
should acquire masses above the experimental limit O(200 GeV). This is possible,
in principle, through couplings with some of the extra singlets with vanishing hy-
percharge, say Ni, which are usually left at low energies, even after the FI breaking.
For example, in the model of Ref. 3, there are 13 of these singlets. Thus couplings
NiDD might be present. From the electroweak symmetry breaking, the fields Ni
a VEV might develop. Note in this sense that the Giudice–Masiero mechanism to
generate a µ term through the Kähler potential is not available in prime orbifolds
as Z3. Thus an interesting possibility to generate VEVs, given the large number of
singlets present in orbifold models, is to consider couplings of the type NiHuHd,
similarly to the Next-to-Minimal Supersymmetric Standard Model (NMSSM). It is
also worth noticing that some of these singlets might not have the necessary cou-
plings to develop VEVs and then their fermionic partners might be candidates for
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A kind of prediction from string phenomenology: extra matter at low energy 7
right-handed neutrinosf .
For the models studied in Refs. 4, 6 the extra colour triplets have non-standard
fractional electric charge, ±1/15 and ±1/6 respectively. In fact, the existence of this
kind of matter is a generic property of the massless spectrum of supersymmetric
models. This means that they have necessarily colour-neutral fractionally charged
states, since the triplets bind with the ordinary quarks. For example, the model
with triplets with electric charge ±1/6 will have mesons and baryons with charges
±1/2 and ±3/2. On the other hand, the model studied in Ref. 3 has ‘standard’ extra
triplets, i.e. with electric charges ∓1/3 and ±2/3; these will therefore give rise to
colour-neutral integrally charged states. For example, a d-like quark D forms states
of the type uD, uuD, etc.
Let us finally mention that a detailed discussion about the stability of these
charged states, how to solve possible conflicts with cosmological bounds, and their
production modes can be found in Ref. 1.
3. Quark and lepton masses and mixing angles
Crucial ingredients in the above analysis were that all three generations of super-
symmetric Higgses remain light (Hui , H
i ), i = 1, 2, 3, and the FI breaking. And,
precisely, both ingredients favour to obtain the correct Yukawa couplings at the
renormalizable levelg. Namely, having three families of Higgses introduces more
Yukawa couplings, and after the FI breaking some physical particles appear com-
bined with other states, and the Yukawa couplings are modified in a well controlled
way. This, of course introduces more flexibility in the computation of the mass
matrices.
Let us recall that the Z3 orbifold is constructed by dividing R
6 by the [SU(3)]3
root lattice modded by the point group (P ) with generator θ, where the action of
θ on the lattice basis is θei = ei+1, θei+1 = −(ei + ei+1), with i = 1, 3, 5. The
two-dimensional sublattices associated to [SU(3)]3 are shown in Fig. 2. In orbifold
constructions, twisted strings appear attached to fixed points under the point group.
In the case of the Z3 orbifold there are 27 fixed points under P , and therefore there
are 27 twisted sectors. We will denote the three fixed points of each two-dimensional
sublattice as shown in Fig. 2. Thus the three generations arise because in addition to
the overall factor of 3 coming from the right-moving part of the untwisted matter,
the twisted matter come in 9 sets with 3 equivalent sectors on each one. Let us
fLet us remark however, that right-handed neutrino superfields with R-parity breaking couplings
of the type NiHuHd have been proposed recently
31 to solve the µ problem.
gLet us recall that the major problem that one encounters when trying to obtain models with en-
tirely renormalizable Yukawas lies at the phenomenological level, and is deeply related to obtaining
the correct quark mixing. Summarizing the analyses of Refs. 24, 25, for prime orbifolds with the
minimal Higgs content the space selection rules and the need for a fermion hierarchy forces the
fermion mass matrices to be diagonal at the renormalizable level. Thus, in these cases the CKM
parameters must arise at the non-renormalizable level. For analyses of non-prime orbifolds see
Refs. 24, 25, 32.
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8 Carlos Muñoz
Fig. 2. Two dimensional sublattices (i = 1, 3, 5) of the Z3 orbifold. The fixed point components
are also shown.
suppose that the two Wilson lines correspond to the first and second sublattices.
The three generations correspond to move the third sublattice component (x · o) of
the fixed point keeping the other two fixed.
As mentioned in the Introduction, we must arrange our up-and down-quark
Yukawa couplings in order to have specific off diagonal elements,
HuūLαλ
u uRγ +Hdd̄Lαλ
dRγ . (3)
In principle this property arises naturally in the Z3 orbifold with two Wilson
lines23−26. For example, if the SU(2) doublet Hu corresponds to (o o o), the three
generations of (3,2) quarks to (o o (o, x, ·)) and the three generations of (3̄, 1) up-
quarks to (o o (o, x, ·)), then there are three couplings allowed from the space group
selection rule (the components of the three fixed points in each sublattice must be
either equal or different): λttHut̄LtR associated to (o o o)(o o o)(o o o) with λtt ∼
1, λcuHuc̄LuR associated to (o o o)(o o x)(o o ·) with λcu ∼ e−T5 , and λucHuūLcR
associated to (o o o)(o o ·)(o o x) with λuc ∼ e−T5 . In this simple example one
gets one diagonal Yukawa coupling without suppression factor and two off diagonal
degenerate ones ∼ e−T5 , but other more realistic examples producing the observed
structure of quark and lepton masses and mixing angles can be obtained using three
generations of Higgses26,27.
Let us first study the situation before taking into account the effect of the FI
breaking. Consider for example the following assignments of observable matter to
fixed point components in the first two sublattices;
Q o o uc o o dc x o
Hu o o Hd · o (4)
In this case the up- and down-quark mass matrices, assuming three different radii,
are given by
Mu = gNAu , Md = gNε1A
d , (5)
where g is the gauge coupling constant, N is proportional to the square root of
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A kind of prediction from string phenomenology: extra matter at low energy 9
volume of the unit cell for the Z3 lattice, and
vu1 v
3 ε5 v
vu3 ε5 v
vu2 ε5 v
1 ε5 v
 , Ad =
vd1 v
3ε5 v
vd3ε5 v
vd2ε5 v
1ε5 v
 . (6)
Here vui , v
i denote the VEVs of the HiggsesH
i , H
i respectively, and εi = 3 e
For example, for T5 ∼ 1.95 one has ǫ5 ∼ 0.05.
The elements in the above matrices can be obtained straightforwardly. For ex-
ample, if the Higgs Hu1 corresponds to (o,o,o), then since the three generations of
(3,2) quarks Q correspond to (o,o,(o,x,·)) and the three generations of (3̄,1) quarks
uc to (o,o,(o,x,·)), there are only three allowed couplings,
(o,o,o)(o,o,o)(o,o,o) ,
(o,o,o)(o,o,x)(o,o,·) ,
(o,o,o)(o,o,·)(o,o,x) .
The corresponding suppression factors are given by 1, ε5, ε5 respectively, and are
associated with the elements 11, 23, 32 in the matrix Mu.
These matrices clearly improve the result obtained with only one Higgs family.
However, it is possible to show that although the observed quark mass ratios and
Cabbibo angle can be reproduced correctly, the 13 and 23 elements of the CKM
matrix cannot be obtained26. Fortunately, this is not the end of the story because
the previous result is modified when one takes into account the FI breaking. In
particular, it will be possible to get the right spectrum and a CKM with the right
form26,27.
As discussed in the Introduction and Section 2, some scalars Ci develop large
VEVs in order to cancel the FI D-term generated by the anomalous U(1). Thus
many particles ξ are expected to acquire a high mass because of the generation of
effective mass terms, and in this way vector-like triplets and doublets and also sin-
glets become heavy and disappear from the low-energy spectrum. This is the type
of extra matter that typically appears in orbifold constructions. The remarkable
point is that the SM matter remain massless, surviving through certain combina-
tions with other states3,4,14. Let us consider the simplest example, a model with
the Yukawa couplings
C1ξ1f , C2ξ1ξ2 , (7)
where f denotes a SM field, ξ1,2 denote two extra matter fields (triplets, doublets
or singlets), and C1,2 are the fields developing large VEVs denoted by 〈C1,2〉 = c1,2.
It is worth noting here that f can be an uc, dc, L, νc or ec field, but not a Q field.
This is because in these orbifold models no extra (3,2) representations are present,
and therefore the Standard Model field Q cannot mix with other representations
through Yukawas.
Clearly the ‘old’ physical particle f will combine with ξ1,2. It is now straightfor-
ward to diagonalise the mass matrix arising from the mass terms in eq. (7) to find
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10 Carlos Muñoz
two very massive and one massless combination. The latter is given by
f ′ ≡ 1√
|c1|2 + |c2|2
(c∗2f − c∗1ξ2) . (8)
Notice for example that the mass terms (7) can be rewritten as
|c1|2 + |c2|2 ξ1ξ′2,
where ξ′2 ≡ 1√|c1|2+|c2|2 (c1f + c2ξ2). Indeed the unitary combination is the massless
field in eq. (8). The Yukawa couplings and hence mass matrices of the effective low
energy theory are modified accordingly. For example, consider a model where we
begin with a Yukawa coupling HQf . Since we have
|c1|2 + |c2|2
′ + c∗1ξ
2) , (9)
then the ‘new’ coupling (involving the light state) will beh
|c1|2 + |c2|2
HQf ′ .
The situation in realistic models is more involved since the fields appear in three
copies. All these effects modify the mass matrices of the low-energy effective theory
(see Eq. (5)), which, for the example studied in Ref. 26, are now given by
Mu = gNau
, Md = gNε1ad
, (10)
where
A B =
v1ε5β v3ε5 v2α
5β v2 v1α
5β v1ε5 v3α/ε5
 , (11)
and the parameters af , α, and β depend oni c1,2 ǫ1,3, and their possible values are
discussed in Ref. 26. As shown in Ref. 27, for natural values of those parameters
and the VEVs, one can find configurations that obey the electroweak symmetry
breaking conditions, and can account for the correct quark masses and mixings.
In addition to the magnitudes of the CKM matrix elements we also require a CP
violating phase. Although it has been shown that observable CP violation cannot
be obtained at the renormalizable level in odd order orbifolds34,33 for a minimal
Higgs sector, the above matrices having in addition to the ‘mixing’ of states three
families of Higgses avoid this problem35. Thus one possibility here (in addition to
the one already mentioned in footnote i) is to assume that the VEVs of the moduli
hWe should add that the coupling HQξ2, which would induce another contribution to HQf
′, is
not in fact allowed. For this to be the case the fields ξ2 and f would have had to have exactly the
same U(1)n charges. This is not possible since different particles all have different gauge quantum
numbers.
iNote that the ci are in general complex VEVs, and therefore they can give rise to a contribution to
the CP phase. This mechanism to generate the CP phase through the VEVs of the fields cancelling
the FI D-term was used first, in the context of non-renormalisable couplings, in Ref. 24. For a
recent analysis, see Ref. 33.
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A kind of prediction from string phenomenology: extra matter at low energy 11
(a) (b)
Fig. 3. Feynman diagrams contributing to ∆mK at tree-level. h
s(p) denote scalar (pseudoscalar)
Higgses.
have an imaginary phase, which can occur when the flat moduli directions are lifted
by supersymmetry breaking and find their minimum where the phases are non-
zero36,35,37. Such a phase feeds directly into ε5. It is easy to check that this phase
is physically observable, and leads to a non-zero δ phase for the CKM matrix which
is of order one.
Let us finally mention that the correct masses for charged leptons can be ob-
tained following a similar approach, as discussed in Ref. [26]. For neutrinos this
turns out to be not sufficient, but a see-saw mechanism arising in a natural way in
orbifolds might solve the problem [38].
4. Phenomenological viability of orbifold models with three Higgs
families
The most challenging implication of an extended Higgs sector is perhaps the oc-
currence of tree-level FCNCs mediated by the exchange of neutral Higgs states.
Clearly, having six Higgs doublets (and thus six quark Yukawa couplings) the trans-
formations diagonalising the fermion mass matrices do not diagonalise the Yukawa
interactions. Since experimental data is in good agreement with the SM predictions,
where such an effect is not present, the potentially large contributions arising from
the tree-level interactions must be suppressed in order to have a model which is
experimentally viable. In general, the most stringent limit on the flavour-changing
processes emerges from the small value of the KL −KS mass difference39.
A detailed discussion of FCNCs in multi-Higgs doublet models was presented in
Ref. 40 (see also the references therein). We summarise here some relevant points
and apply the method to the orbifold case27, focusing on the neutral kaon sector
and investigating the tree-level contributions to ∆mK . The latter is simply defined
as the mass difference between the long- and short-lived kaon masses,
∆mK = mKL −mKS ≃ 2
∣MK12
∣ = 2
∣H∆S=2eff
, (12)
where H∆S=2eff is the effective Hamiltonian for the diagrams in Fig. 3. Once all
September 15, 2021 7:34 WSPC/INSTRUCTION FILE mplareviewMunoz
12 Carlos Muñoz
the contributions to MK12 have been taken into account, the prediction of this
orbifold model regarding ∆mK should be compared with the experimental value,
(∆mK)exp ≃ 3.49× 10−12 MeV.
In Ref. 27 the numerical approach was divided in two steps. Firstly, one focus
on the string sector of the model, and for each point in the space generated by the
free parameters of the orbifold (ε5, α
f ), one derives the up- and down-quark mass
matrices and computes the CKMmatrix. Further imposing the conditions associated
with electroweak symmetry breaking, and fixing a value for tanβ, one can then
determine the values of g N and ε1. A secondary step requires specifying the several
Higgs parameters, which must obey the minimum criteria Finally, the last step
comprehends the analysis of how each of the Yukawa patterns constrains the Higgs
parameters in order to have compatibility with the FCNC data. In particular, we
want to investigate how heavy the scalar and pseudoscalar eigenstates are required
to be in order to accommodate the observed value of ∆mK .
Let us summarize the analysis of the orbifold parameter space by commenting on
the relative number of input parameters and number of observables fitted. Working
with the six Higgs VEVs (v
i ), and the orbifold parameters ε1, ε5, α
and αd
one can obtain the correct electroweak symmetry breaking (MZ), as well as the
correct quark masses and mixings (six masses and three mixing angles).
In order to discuss now the tree-level FCNCs, let us remark that the present
orbifold model does not include a specific prediction regarding the Higgs sector.
For instance, we have no hint regarding the value of the several bilinear terms, nor
towards their origin. Concerning the soft breaking terms, the situation is similar.
In the absence of further information, we merely assume that the structure of the
soft breaking terms is the usual one (see Ref. 27 for further details), taking the
Higgs soft breaking masses and the Bµ-terms as free parameters (provided that the
electroweak symmetry breaking and minimisation conditions are verified).
In the absence of orbifold predictions for the Higgs sector parameters, and mo-
tivated by an argument of simplicity, we begin our analysis by considering textures
for the soft parameters as simple as possible. In particular, we arrive to four repre-
sentative cases with the following associated scalar and pseudoscalar Higgs spectra
(a) ms = {82.5, 190.6, 493.9, 515.9, 744.4, 760.2} GeV ;
mp = {186.8, 493.9, 515.9, 744.4, 760.2} GeV .
(b) ms = {84.6, 213.9, 387.4, 560.8, 785.9, 879.1} GeV ;
mp = {215.2, 387.3, 560.5, 785.9, 878.9} GeV .
(c) ms = {83.6, 292.9, 733.6, 785.9, 987.6, 1057.0} GeV ;
mp = {291.1, 733.6, 785.9, 987.6, 1057.0} GeV .
(d) ms = {79.4, 121.5, 296.9, 354.3, 794.6, 808.8} GeV ;
mp = {114.8, 296.9, 353.7, 794.6, 808.8} GeV .
In Fig. 4 we plot the ratio ∆mK/(∆mK)exp versus ε5, for cases (a)-(d), and
tanβ = 5. All the points displayed comply with the bounds from the CKM matrix.
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A kind of prediction from string phenomenology: extra matter at low energy 13
 0.013  0.014  0.015  0.016  0.017
 0.013  0.014  0.015  0.016  0.017
Fig. 4. ∆mK/(∆mK)exp as a function of ε5 for tanβ = 5. The Higgs parameters correspond to
textures (a)-(d).
From Fig. 4 it is clear that it is quite easy for the orbifold model to accommodate
the current experimental values for ∆mK . Even though the model presents the
possibility of important tree-level contributions to the kaon mass difference, all
the textures considered give rise to contributions very close to the experimental
value. Although (a) and (b) are not in agreement with the measured value of ∆mK ,
their contribution is within order of magnitude of (∆mK)exp. As seen from Fig. 4,
with a considerably light Higgs spectrum (i.e. mh0
< 1 TeV), one is safely below
the experimental bound, as exhibited by cases (c) and (d). This is not entirely
unexpected given the strongly hierarchical structure of the Yukawa couplings (notice
from Eq. (11) that λd21 is suppressed by ε
Let us finally mention that the analysis for other neutral meson systems, Bd,
Bs and D
0, can be carried out in an analogous way27.
Additionally, and given the existence of flavour violating neutral Higgs couplings,
and the possibility of having complex Yukawa couplings, it is natural to have tree-
level contributions to CP violation. In the kaon sector, indirect CP violation is
parameterised by εK . From experiment one has εK = (2.284 ± 0.014) × 10−3. A
comparison of this quantity with the theoretical result in orbifold models can be
found in Ref. 27.
5. Conclusions
We have attacked the problem of the unification of gauge couplings in Heterotic
String constructions. In particular, we have obtained that due to the Fayet-
Iliopoulos scale, α3 and α2 cross at the right scale when a certain type of extra
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14 Carlos Muñoz
matter is present. In this sense three families of supersymmetric Higgses and vector-
like colour triplets might be observed in forthcoming experiments. The unification
with α1 is obtained if the model has the appropriate normalization factor of the
hypercharge. Let us recall that although we have been working with explicit orb-
ifold examples, our arguments are quite general and can be used for other schemes
where the Standard Model gauge group with three generations of particles is ob-
tained, since extra matter and anomalous U(1)’s are generically present in string
compactifications.
Another advantage of these models is that they naturally predict three gener-
ations, and also that the three generations of Higgs fields give enough freedom to
allow an entirely geometric explanation of masses and mixings. The Fayet-Iliopoulos
mechanism plays also an important role here. Namely, after the gauge breaking some
physical particles appear combined with other states, and the Yukawa couplings are
modified in a well controlled way.
On the other hand, the presence of six Higgs doublets poses the potential prob-
lem of having tree-level FCNCs. By assuming simple textures for the Higgs free
parameters, we have verified for example that the experimental data on the neutral
kaon mass difference can be easily accommodated for a quite light Higgs spectra,
namely mh0
. 1 TeV.
The presence of a fairly light Higgs spectrum, composed by a total of 21 physical
states, may provide abundant experimental signatures at future colliders, like the
Tevatron or the LHC. In fact, flavour violating decays of the form hi → qq̄, or
hi → l+l− may provide the first clear evidence of this class of models.
Acknowledgments
This work was supported in part by the Spanish DGI of the MEC under Proyec-
tos Nacionales FPA2006-05423 and FPA2006-01105; by the Comunidad de Madrid
under Proyecto HEPHACOS, Ayudas de I+D S-0505/ESP-0346; and also by the
European Union under the RTN program MRTN-CT-2004-503369.
References
1. C. Muñoz, ‘A kind of prediction from superstring model building’, JHEP 12 (2001)
015 [arXiv:hep-ph/0110381].
2. L.E. Ibáñez, J.E. Kim, H.P. Nilles and F. Quevedo, ‘Orbifolds compactifications with
three families of SU(3) × SU(2)× U(1)n’, Phys. Lett. B191 (1987) 3.
3. J.A. Casas and C. Muñoz, ‘Three generation SU(3) × SU(2) × U(1)Y models from
orbifolds’, Phys. Lett. B214 (1988) 63.
4. A. Font, L.E. Ibáñez, H.P. Nilles and F. Quevedo, ‘Yukawa couplings in degenerate
orbifolds: towards a realistic SU(3) × SU(2) × U(1) superstring’, Phys. Lett. B210
(1988) 101.
5. J.A. Casas, E.K. Katehou and C. Muñoz, ‘U(1) charges in orbifolds: anomaly cancel-
lation and phenomenological consequences’, Nucl. Phys. B317 (1989) 171.
6. J. Giedt, ‘Spectra in standard-like Z3 orbifold models’, Annals Phys. 297 (2002) 67
[arXiv:hep-th/0108244].
http://arxiv.org/abs/hep-ph/0110381
http://arxiv.org/abs/hep-th/0108244
September 15, 2021 7:34 WSPC/INSTRUCTION FILE mplareviewMunoz
A kind of prediction from string phenomenology: extra matter at low energy 15
7. T. Kobayashi, S. Raby, R.-J. Zhang ‘Searching for realistic 4d string models with a Pati-
Salam symmetry: Orbifold grand unified theories from heterotic string compactification
on a Z6 orbifold’ Nucl. Phys. B704 (2005) 3 [arXiv:hep-ph/0409098].
8. W. Buchmuller, K. Hamaguchi, O. Lebedev, M. Ratz, ‘Supersymmetric standard model
from the heterotic string’, Phys. Rev. Lett. 96 (2006) 121602 [arXiv:hep-ph/0511035];
‘Supersymmetric standard model from the heterotic string (II)’, arXiv:hep-th/0606187.
9. O. Lebedev, H.P. Nilles, S. Raby, S. Ramos-Sánchez, M. Ratz, P.K. Vaudrevange
and A. Wingerter, ‘A mini-landscape of exact MSSM spectra in heterotic orbifolds’,
arXiv:hep-th/0611095.
10. J.E. Kim and B. Kyae, ‘String MSSM through flipped SU(5) from Z12 orb-
ifold’, arXiv:hep-th/0608085; ‘Flipped SU(5) from Z12−I orbifold with Wilson line’,
arXiv:hep-th/0608086.
11. J.E. Kim, J.-H. Kim and B. Kyae, ‘Superstring standard model from Z12−I
orbifold compactification with and without exotics, and effective R-parity’,
arXiv:hep-ph/0702278.
12. V.S. Kaplunovsky, ‘One loop threshold effects in string unification’, Nucl. Phys. B307
(1988) 145, Erratum, ibid. B382 (1992) 436 [arXiv:hep-th/9205070].
13. J.A. Casas and C. Muñoz, ‘Restrictions on realistic superstring models from renor-
malization group equations’, Phys. Lett. B214 (1988) 543.
14. J.A. Casas and C. Muñoz, ‘Yukawa couplings in SU(3) × SU(2) × U(1)Y orbifolds
models’, Phys. Lett. B212 (1988) 343.
15. For a review, see for instance L.E. Ibáñez, ‘Standard Model engineering with inter-
secting branes’, hep-ph/0109082, and references therein.
16. F. Gmeiner, R. Blumenhagen, G. Honecker, D. Lüst and T. Weigand, ‘One in a billion:
MSSM-like D-brane statistics’, JHEP 01 (2006) 004 [arXiv:hep-th/0510170].
17. M.R. Douglas and W. Taylor, ‘The landscape of intersecting brane models’, JHEP 01
(2007) 031 [arXiv:hep-th/0606109].
18. J.A. Casas, M. Mondragon and C. Muñoz, ‘Reducing the number of candidates to
standard model in the Z3 orbifold’, Phys. Lett. B230 (1989) 63.
19. J. Giedt, ‘Completion of standard-like embeddings’, Ann. Phys. B289 (2001) 251
[hep-th/0009104].
20. For previous works using this mechanism, see I. Antoniadis, J. Ellis, S. Kelley and
D.V. Nanopoulos, ‘The price of deriving the standard model from string’, Phys. Lett.
B272 (1991) 31;
I. Antoniadis, G.K. Leontaris and N.D. Tracas, ‘Grand unified string models and low
energy couplings’, Phys. Lett. B279 (1992) 58;
D. Bailin and A. Love, ‘String unification, grand unification and string loop threshold
corrections’, Phys. Lett. B292 (1992) 315;
M.K. Gaillard and R. Xiu, ‘Analysis of running coupling constant unification in string
theory’, Phys. Lett. B296 (1992) 71 [hep-ph/9206206];
A.E. Faraggi, ‘Gauge coupling unification in superstring derived standard-like models’,
Phys. Lett. B302 (1993) 202 [hep-ph/9301268];
K.R. Dienes, A.E. Faraggi, ‘Making ends meet: String unification and low-energy data’,
Phys. Rev. Lett. 75 (1995) 2646 [hep-th/9505018];
K.R. Dienes, A.E. Faraggi, ‘Gauge coupling unification in realistic free fermionic string
models’, Nucl. Phys. B457 (1995) 409 [hep-th/9505046];
C. Bachas, C. Fabre and T. Yanagida, ‘Natural gauge-coupling unification at the string
scale’ Phys. Lett. B370 (1996) 49 [hep-th/9510094];
D. Emmanuel-Costa, P. Fileviez-Perez and R. Gonzalez Felipe, ‘Natural gauge and grav-
itational coupling unification and the superpartner masses’ Phys. Lett. B648 (2007) 60
http://arxiv.org/abs/hep-ph/0409098
http://arxiv.org/abs/hep-ph/0511035
http://arxiv.org/abs/hep-th/0606187
http://arxiv.org/abs/hep-th/0611095
http://arxiv.org/abs/hep-th/0608085
http://arxiv.org/abs/hep-th/0608086
http://arxiv.org/abs/hep-ph/0702278
http://arxiv.org/abs/hep-th/9205070
http://arxiv.org/abs/hep-ph/0109082
http://arxiv.org/abs/hep-th/0510170
http://arxiv.org/abs/hep-th/0606109
http://arxiv.org/abs/hep-th/0009104
http://arxiv.org/abs/hep-ph/9206206
http://arxiv.org/abs/hep-ph/9301268
http://arxiv.org/abs/hep-th/9505018
http://arxiv.org/abs/hep-th/9505046
http://arxiv.org/abs/hep-th/9510094
September 15, 2021 7:34 WSPC/INSTRUCTION FILE mplareviewMunoz
16 Carlos Muñoz
[hep-ph/0610178];
V. Barger, J. Jiang, P. Langacker and T. Li, ‘String scale gauge coupling unification
with vector-like exotics and non-canonical U(1)Y normalization’, hep-ph/0612206
21. S. Hamidi and C. Vafa, ‘Interactions on orbifolds’, Nucl. Phys. B279 (1987) 465.
22. L.J. Dixon, D. Friedan, E. Martinec and S. Shenker, ‘The conformal field theory of
orbifolds’, Nucl. Phys. B282 (1987) 13.
23. L.E. Ibáñez, ‘Hierarchy of quark-lepton masses in orbifolds superstring compactifica-
tion’, Phys. Lett. B181 (1986) 269.
24. J.A. Casas and C. Muñoz, ‘Fermion masses and mixing angles: a test for string vacua’,
Nucl. Phys. B332 (1990) 189, Erratum-ibid. B340 (1990) 280.
25. J.A. Casas, F. Gómez and C. Muñoz, ‘Fitting the quark and lepton masses in string
theories’, Phys. Lett. B292 (1992) 42 [hep-th/9206083].
26. S. A. Abel and C. Muñoz, ‘Quark and lepton masses and mixing angles from super-
string constructions’, JHEP 02 (2003) 010 [arXiv:hep-ph/0212258].
27. N. Escudero, C. Muñoz and A.M. Teixeira, ‘Phenomenological viability of orbifold
models with three Higgs families’, JHEP 07 (2006) 041 [arXiv:hep-ph/0512301].
28. S.L. Glashow and S. Weinberg, ‘Natural conservation laws for neutral currents’, Phys.
Rev. D15 (1977) 1958.
29. E.A. Paschos, ‘Diagonal neutral currents’, Phys. Rev. D15 (1977) 1966.
30. For a review, see K.R. Dienes, ‘String theory and the path to unification: a review of
recent developments’, Phys. Rep. 287 (1997) 447 [hep-th/9602045].
31. D.E. Lopez-Fogliani and C. Muñoz, ‘Proposal for a supersymmetric standard model’,
Phys. Rev. Lett. 97 (2006) 041801 [arXiv:hep-ph/0508297].
32. P. Ko, T. Kobayashi and J.H. Park, ‘Quark masses and mixing angles in het-
erotic orbifold models’, Phys. Lett. B598 (2004) 263 [hep-ph/0406041]; ‘Lepton masses
and mixing angles from heterotic orbifold models’, Phys. Rev. D71 (2005) 095010
[hep-ph/0503029].
33. J. Giedt, “The KM phase in semi-realistic heterotic orbifold models”, Nucl. Phys.
B595 (2001) 3 [Erratum-ibid. B632 (2002) 397] [arXiv:hep-ph/0007193]; “CP viola-
tion and moduli stabilization in heterotic models”, Mod. Phys. Lett. A17 (2002) 1465
[arXiv:hep-ph/0204017].
34. O. Lebedev, ‘The CKM phase in heterotic orbifold models’, Phys. Lett. B521 (2001)
71 [hep-th/0108218].
35. T. Kobayashi and O. Lebedev, ‘Heterotic string backgrounds and CP violation’, Phys.
Lett. B565 (2003) 193 [hep-th/0304212].
36. B. Acharya, D. Bailin, A. Love, W.A. Sabra and S. Thomas, ‘Spontaneous breaking
of CP symmetry by orbifold moduli’, Phys. Lett. B357 (1995) 387 [hep-th/9506143];
D. Bailin, G.V. Kraniotis and A. Love, ‘CP violation by soft supersymmetry breaking
terms in orbifold compactifications’, Phys. Lett. B414 (1997) 269 [hep-th/970524]; ‘CP
violating phases in the CKM matrix in orbifold compactifications’, Phys. Lett. B435
(1998) 323 [hep-th/9805111].
37. O. Lebedev and S. Morris, ‘Towards a realistic picture of CP violation in heterotic
string models’, JHEP 08 (2002) 007 [hep-th/0203246].
38. S. A. Abel, N. Escudero, C. Muñoz and A. M. Teixeira, in preparation.
39. B. McWilliams and L. F. Li, ‘Virtual effects of Higgs particles’, Nucl. Phys. B179
(1981) 62; O. Shanker, ‘Flavor violation, scalar particles and leptoquarks’, Nucl. Phys.
B206 (1982) 253.
40. N. Escudero, C. Muñoz and A. M. Teixeira, ‘Flavor changing neutral currents
in supersymmetric multi-Higgs doublet models’, Phys. Rev. D73 (2006) 055015
[hep-ph/0512046].
http://arxiv.org/abs/hep-ph/0610178
http://arxiv.org/abs/hep-ph/0612206
http://arxiv.org/abs/hep-th/9206083
http://arxiv.org/abs/hep-ph/0212258
http://arxiv.org/abs/hep-ph/0512301
http://arxiv.org/abs/hep-th/9602045
http://arxiv.org/abs/hep-ph/0508297
http://arxiv.org/abs/hep-ph/0406041
http://arxiv.org/abs/hep-ph/0503029
http://arxiv.org/abs/hep-ph/0007193
http://arxiv.org/abs/hep-ph/0204017
http://arxiv.org/abs/hep-th/0108218
http://arxiv.org/abs/hep-th/0304212
http://arxiv.org/abs/hep-th/9506143
http://arxiv.org/abs/hep-th/9805111
http://arxiv.org/abs/hep-th/0203246
http://arxiv.org/abs/hep-ph/0512046
