ANOMALOUS U(1) GAUGE SYMMETRY IN
SUPERSTRING INSPIRED LOW ENERGY EFFECTIVE THEORIES
BY
NIKOLAOS IRGES
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1999
ACKNOWLEDGMENTS
I would like to express my gratitude to my advisor, Dr. Pierre Ramond, who helped and supported me in numerous ways during the whole time of my work towards the completion of this thesis. I am also grateful to all the professors at the Physiscs Department of the University of Florida who taught me various topics of theoretical physics and especially to Dr. John Klauder, Dr. Pierre Sikivie and Dr. Charles Thorn. I would like to thank Dr. Pierre Binetruy, Dr. Sang Hyeon Chang, Dr. Claudio Coriano, Dr. John Elwood, Dr. Alon Faraggi, Dr. Richard Field, Dr. Youli Kanev and Dr. Stephane Lavignac for discussions and/or collaboration on topics related to supersymmetric particle physics. Finally, I would like to express my gratitude to my parents for their patience and support.
ii
TABLE OF CONTENTS
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . .ii
ABSTRACT ..... ... ... ............................ v
CHAPTERS
1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Superstring Inspired Low Energy Effective Theories and the Anomalous U (1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 The GreenSchwarz Mechanism . . . . . . . . . . . . . . . . . . . . 9
1.3 The Anomalous U(1) and Yukawa Matrices . . . . . . . . . . . . . . 11
2 THE VACUUM . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.1 DFlatness and Holomorphy . . . . . . . . . . . . . . . . . . . . . . 18
2.1.1 M odels with 2 Fields. . . . . . . . . . . . . . . . . . . . . . . 20
2.1.2 M odels with 3 Fields. . . . . . . . . . . . . . . . . . . . . . . 22
2.1.3 R Parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 The General Flat Direction Analysis . . . . . . . . . . . . . . . . . 27
2.2.1 D Flatness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2.2 F Flatness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.2.3 Supersymmetry Breaking and LowEnergy Vacuum . . . . . 41
2.2.4 Sum m ary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.2.5 The Ideal Vacuum . . . . . . . . . . . . . . . . . . . . . . . 45
3 MASS MATRICES . . . . . . . . . . . . . . . . . . . . . . . . 48
3.1 Quark Mass Matrices from U(1)'s: Hierarchies . . . . . . . . . . . . 48
3.2 A Model with a Single U(1) Family Symmetry . . . . . . . . . . . . 55
3.2.1 Interfamily Hierarchy . . . . . . . . . . . . . . . . . . . . . 55
3.2.2 Intrafamily Hierarchy . . . . . . . . . . . . . . . . . . . . . . 59
3.3 Squark Mass Matrices from U(1)'s: FCNC . . . . . . . . . . . . . . 64
3.4 Sum m ary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4 MODEL BUILDING: A REALISTIC MODEL . . . . . . . . . . . 71
4.1 General Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.2 A nom alies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
iii
4.3 The DSW Vacuum . 4.4 Quark and Charged L 4.5 Neutrino Masses . .
. . . . . . . . . .
epton Masses . . . . . . . . . . . .
4.6 VectorLike Matter . . . . . . . . . . .
4.6.1 Shift X. . . . . . . . . . . . . .
4.6.2 Discrete Symmetry . . . . . . .
4.6.3 Summary . . . . . . . . . . . .
4.7 The Hidden Sector . . . . . . . . . . .
4.8 RParity . . . . . . . . . . . . . . . . .
4.9 Proton Decay . . . . . . . . . . . . . .
4.10 Flat Direction Analysis . . . . . . . . .
4.10.1 Flat Directions with VectorLike
4.11 Supersymmetry Breaking . . . . . . . .
4.11.1 Supersymmetry Breaking with U 4.11.2 Soft Parameters . . . . . . . . .
5 CONCLUSION . . . . . . . . . . . .
REFERENCES.............
BIOGRAPHICAL SKETCH . . . .
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Matter
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77 80
84 91 97 101
102 103 107 109
114 119
121 126 136
143
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
ANOMALOUS U(1) GAUGE SYMMETRY IN
SUPERSTRING INSPIRED LOW ENERGY EFFECTIVE THEORIES By
Nikolaos Irges
May 1999
Chairman: Pierre Ramond
Major Department: Physics
The Standard Model (SM) of elementary particles is a theoretical model that describes quite accurately what seem to be the constituents of matter and the forces that govern their dymanics, with the exception of gravity. Our confidence in the validity of the SM lies in experimental results obtained in accelerator experiments that, up to now, have not contradicted it in a radical way. One of the shortcomings of the SM from a theoretical point of view is that it has many parameters input "by hand." These are parameters that are necessary for its consistency but their origin is unknown. However, what we would like to call the real model of nature is one where all the parameters are self determined dynamically rather than put by hand. In addition, theoretical investigations of its underlying mathematical structure, as well as attempts to extend the model so that it includes gravity, revealed certain inconsistencies at energy scales far above our current experimental capabilities and led to the conclusion that the SM is probably correct but not complete; it has to be complemented by additional structure. One of the most popular such extensions is a new symmetry, so called supersymmetry, that provides a theoretically promising
v
candidate that can solve many of these problems and it is consistent with the only consistent quantum gravity theory, M theory. The model in this thesis is, to our knowledge, the first which has these characteristics. First, it provides a scheme that can explain the origin of most of the arbitrary parameters of the SM, it is supersymmetric and it naturally predicts properties of elementary particles that will be tested very soon in experimental laboratories. Two of the most striking examples of such predictions are the masses and the mixing properties of neutrinos and the mass of the only particle that is believed to be elementary in the SM but it has not been experimentally detected yet: the Higgs particle. Second, it is a model that has many of the signature features of models that come directly from string theory (M theory) compactifications. We would like to stress the fact that since the model we are presenting here is not a direct descendant of a string theory, it can not be viewed as a fundamental theory but rather as a phenomenological extension of the SM that could come from a string theory. Given the fact that up this day we are not sure if string theory is the relevant mathematical description of the universe and that no viable "string compactification" has been constructed yet, this model not only proposes a simple link between the exotic string theories that live at huge energy scales and our experimentally reachable world, but also provides a possible guide to those who are hoping to prove that string theory is correct by showing that the SM naturally emerges after compactifying a string theory to four dimensions  a highly non trivial and non unique process.
vi
CHAPTER 1
INTRODUCTION
It is believed that the standard model of elementary particles (SM) can not be the ultimate theory of nature. There are questions and problems that can not be solved within the context of an N = 0, SU(3)C x SU(2)w x U(1)y gauge theory, where N is the number of supersymmetries [1]. For example, it is well known that in the standard model, scalar masses admit large quantum corrections and as a consequence their masses are driven to the Planck scale. A correction that is several orders of magnitude larger than the bare mass is not only excluded by experiment but also considered to be unnatural. Another related problem is the huge disparity of scales between the electroweak and Planck scales. Low energy supersymmetry [2],[31, is one of the most popular ways to evade these problems, even though at this point there is no experimental evidence for its existence. On the other hand, supersymmetric string theory [4], is the only theory that incorporates all the known fundamental forces of nature in a consistent and unified way. One problem that immediately arises is to make the connection between the string (M) theory that lives at a scale ~ 1017 GeV in 10 (11) spacetime dimensions and the real world, at ~ 103 GeV and 4 space time dimensions. One of the main difficulties of realistic superstring model building (for some recent works, see [5], [6], [7], [8], [9], [10], [11], [12], [13], [14]) is the fact that one has to "find the vacuum of our world," among many other possible equivalent vacua (flat directions). The choice of the vacuum is usually an arbitrary input of the model builder, who constraints the moduli
1
2
space (space of possible vacua), by imposing phenomenological constraints [15], [16], [17], [18], [19], [20], [21], [22], [231, [24], [25], [26]. However, there are certain features that are rather model and compactification independent. Such is the fact that compactification leaves us a number of horizontal U(1) gauge symmetries in addition to the nonAbelian gauge group that contains the standard model. In many cases, in addition, one of these U(1)s is anomalous with its anomalies canceled by what is known as the GreenSchwarz mechanism [27]. In these models the dilaton gets a vacuum expectation value, generating a nonzero FayetIliopoulos term that triggers the breaking of the anomalous U(1) at a scale just below the string scale [28], [29] (which will be assumed in this work to coincide with the scale at which the gauge couplings unify).
On the other hand, it was also realized that one could start directly from an effective quantum field theory that has many of the stable features of a compactified theory and draw conclusions about the low energy phenomenology. In this, bottomup, approach, one postulates a gauge group that in most cases contains an anomalous U(1) and follows the consequences of its breaking to low energies [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45]. The first attempts in this direction involved a single, flavor dependent, anomalous U(1). They clearly indicated that the presence of this symmetry can be very useful and not only the hierarchical structure of the fermion mass matrices could be explained but also the value of the Weinberg angle could be predicted, with the help of the GreenSchwarz mechanism. These early attempts, however, were never complete because they touched on certain features like quark and charged lepton masses and ignored others like for example neutrino masses or a proper vacuum
3
analysis. In other words, a consistent "superstringinspired" model has never been constructed. The purpose of this thesis is to show that such a model can indeed be constructed. We will demonstrate how a minimalistic model building point of view leads us to a model that is consistent with the features present in superstring compactifications and explain a considerably large part of the low energy data, especially the one related to particle masses and mixings. Let us summarize its main features. It explains/predicts:
" All quark and charged lepton Yukawa hierarchies and mixings, including the
bottom to top Yukawa suppression.
" The value of the Weinberg angle at unification.
" Three massive neutrinos with mixings that give the smallangle MSW effect
for the solar neutrino deficit, and the large angle mixing necessary for the
atmospheric neutrino effect.
" Natural Rparity conservation.
" Proton decay into K0 + p+ near the experimental limit.
* A hidden sector that contains strong gauge interactions.
" A susy breaking mechanism yielding squark masses compatible with bounds
on fcnc and a Higgs mass of 104 GeV.
Surprisingly, as we mention above, it predicts massive neutrinos [46], [47], [48], with masses [49], [50] and mixings [51] consistent with the nonadiabatic MSW effect [52], [53], [54], [55], [56] and the atmospheric neutrino anomaly [57] and a
4
Higgs of about 100 GeV mass. The predictions associated with the neutrino sector and the Higgs mass will be undoubtedly the most serious tests of the model.
1.1 Superstring Inspired Low Energy Effective Theories and the Anomalous U(l)
In this section, we will give the necessary ingredients in order to write a minimal low energy four dimensional superstring effective action. By "minimal" we mean that we will not construct a complete compactified to four dimensions string low energy effective action, but rather take a minimal point of view and write only the part of such an action which is universal to all compactifications and is only the part necessary to include the minimal supersymmetric standard model (MSSM). This is called a "bottomup" approach.
Superstring compactification from ten dimensions to four, yields generically a large number of massless modes. Realistic compactifications are considered those that contain in their gauge group the standard model gauge group and that have a field content at least that of the MSSM. There are, however, some massless modes that are in addition always present in the string ground state. Such is the antisymmetric 2form B., and the dilaton S. In four dimensions, the antisymmetric two form appears through the field strength
H[pp] =,,Bp + ... = EgpOa + ... (1.1.1)
which is clearly invariant under the shift a a +rq. The field a lies in a supermultiplet, called the dilaton supermultiplet. From the effective action we will see that
5
it has the couplings of an axion. The field content of the dilaton supermultiplet is
S = y + ia + fermionic superpartners (1.1.2)
The field y is the (the real part of the) dilaton field and it gives the name to the whole multiplet. The vacuum expectation value of the dilaton becomes the string coupling constant in the effective field theory:
1
< y >= yo = 2. (1.1.3)
Here, gt,(M,,.) is the coupling at the scale Mtr. We will denote
gstr(Mtr) g (1.1.4)
and assume from now on for simplicity that
Mtr = M ;> MGUT, (1.1.5)
i.e. that there is no mass scale available in the theory other than M.
The dilaton superfield couples to the regular matter through its coupling to the gauge fields
Leff kk / d20 [S Wk . Wk] +c.c (1.1.6)
k
with kk being the KacMoody levels. These are integers for nonAbelian groups and rational numbers for Abelian factors depending on the normalization of the U(1). The summation over k is over all gauge groups. The gauge supermultiplet
6
is
Wk =A k+..(11)
with Ak being the gaugino. The bosonic part of the gauge kinetic term in component form is given by
wk. W k = F kFttvk + F,# "t. (1.1.8)
The effective Lagrangian in component form for the gauge kinetic term then becomes
kFk Fk Vkkpvk (1.1.9
Le55 = F ,F M + a EkkF F'#9* (.
k 9 k
To write an effective action, we also have to specify the following:
" Specify the gauge group that survives just below the string scale. In general
the gauge group will be denoted as
G x X x Y(1) x y(2) x ... x Ghid x DIz] (1.1.10)
Here, G is a nonabelian gauge group that contains SU(3)C x SU(2)w x U(1)y of the standard model. X and y(a) are (anomalous and nonanomalous, respectively) Abelian gauge factors present at low energies. Ghid is the gauge group of the hidden sector (we define g G x Ghid) and D[i] are local discrete
symmetries.
" Specify the massless particle spectrum. Of course, any realistic low energy
model should include all the particles that have been observed in experiments.
7
" Specify the number of low energy supersymmetries. In the following we will
be interested only in N = 1 supersymmetric models.
" Specify a Kihler potential. This will be important when we talk about supersymmetry breaking. In general, we will assume minimal Kihler couplings for all fields except the dilaton, which we believe to be responsible for supersymmetry breaking. We will also assume, for simplicity, a minimal form for
the gauge kinetic function finn.
Given the above, we can write a unique low energy effective action; it is just N = 1 supergravity (which is fixed by Kahler potential K, the superpotential W and the gauge kinetic functions fmn), coupled to N = 1 super YangMills with matter.
Since we are dealing with an effective theory, we have to include in the action all possible gauge invariant terms. An arbitrary gauge invariant term in W will be of the form
T =f (M)I1(4), 4)...f,) X, n,' 11.
where I(4)1 ... , ) is a polynomial consisting of the superfields V , invariant under g. The superscript refers to the type of the field and the subscript is a family index. Xi are g singlet fields and
1
f (M)= ,,(..2
where r is such that T has superfield dimension 3 as it is appropriate for an effective tree level superpotential term. In addition, if T is a term that belongs to the superpotential W, it has to be an holomorphic function of the component
8
superfields (if it belongs to the Kihler potential K, it has to be a real function of the superfields but not necessarily an holomorphic function).
The next step is to give a scenario which describes the breaking of the additional U(1)'s since we know that these are not present at low energy. A gauged U(1) factor will be called anomalous if
Tr[U(1)x] = Tr[X] 5 0. (1.1.13)
Tr[X] 5 0 means in the quantum field theory language that for example the anomaly triangle graph XTT = Cgrav with an X gauge field and two gravitons does not vanish:
Tr[X] = Cgrav $ 0. (1.1.14)
In our convention, Cgrav will turn out to be negative. The framework in which this happens was explained by Dine, Seiberg and Witten (DSW), in [28]. In these models, the U(1) breaks at a scale close to the cutoff M which is an energy scale below the scale of the string theory, generating an anomalous FayetIliopoulos Dterm ( 2), which breaks supersymmetry. To restore supersymmetry, which should not break at such a high scale, a compensating term in the Dterm of the anomalous U(1) appears. The corrected form of the anomalous Dterm is then
Dx = g2[D0  2].
(1.1.15)
9
1.2 The GreenSchwarz Mechanism
Since any viable theory has to be anomaly free, there has to be a way to cancel the apparent anomaly. Indeed, since the divergence of the anomalous current is
2
jf = 2C F 2. (1.2.16)
Under a gauge transformation of the vector potential, A,, shifts as
AX a Ax +,9A (1.2.17)
and the axion as
a * a + 167r26GSA, (1.2.18)
where
GS Tr(X) (1.2.19)
1927r2
The Lagrangian then changes by
2
6C = A (Ck + 167r26Gskk)FF,",, (1.2.20)
r k
with kk the KacMoody levels of the gauge factors. The anomalies of the Xsymmetry are compensated at the cutoff, as long as the ratio Ck/kk is universal. This is the fourdimensional equivalent of the GreenSchwarz anomaly cancelation mechanism [27]. Consistency requires all other anomaly coefficients to vanish. Color, Cweak and Cy are the mixed anomalies between the X current and the
10
standard model gauge currents,
(X G^G B) _ 6AB color ; (XW"W3) = 6*"Cwek ; (XYY) =Cy , (1.2.21)
where GA are the QCD currents, and W" the weak isospin currents. We must have
167r20s _coior _ weak C 167r 26GS =Coo  kweak = CY: 0 ,(1.2.22) kcolor kweak ky
and
(Xy(a)y(b)) = 6jC() . (1.2.23)
All the other anomaly coefficients must vanish by themselves:
(y(a)y(b)y(c)) = (y(a)y(b)y) = (y(a)GAGB) (y(a)W"Wj6)  (y(a)yy) = 0.
(1.2.24)
as well as:
(XYY(a)) = (XXY) = (XXy(a)) = (y(a)TT) = 0 . (1.2.25)
A consequence of the GreenSchwarz mechanism is that the Weinberg angle at cutoff can be understood [30] as a ratio of anomaly coefficients:
tan26w = = 2 kweak _ Cweak (1.2.26)
weak ky CY
11
1.3 The Anomalous U(1) and Yukawa Matrices
The nonzero anomaly coefficients can be computed from the Xcharges of chiral fermions. Such fermions can come in two varieties, those from the three chiral families and those from standard model pairs with chiral X values. The anomaly coefficients from the three chiral families can be related to the X charges of the standard model invariants. The minimal supersymmetric standard model contains the invariants
QiuH; QidHd; Li Hd; H.Hd, (1.3.27)
where i, j are the family indices, with X charges
X , X , XX ,(1.3.28)
respectively; a simple computation yields
3
Ccoor = (X + X  3X 1P , (1.3.29)
Cy + Cek  8Ccoior 2 (X  X') + 2X1"9 . (1.3.30)
Since the KacMoody levels of the nonAbelian factors are the same, the GreenSchwarz condition requires
Cweak = Ccolor , (1.3.31)
12
from which we deduce
35
Cy  X  X1 + 2X1)  3XIAI . (1.3.32)
Similar equations hold for the mixed anomalies of the yCa) currents; their vanishing imposes constraints on the Y(a) charges of the standard model invariants. The further constraint that the Weinberg angle be at its canonical SU(5) value, sin20" = 3/8, that is 3Cy = 5Cweak, yields the relations
3
Xl1 = E(XIf  Xl ) . (1.3.33)
Ccoior = [X[']  2XI + 3Xl , (1.3.34)
as well as
Ccoior = 2  2X[ + 3X] . (1.3.35)
Since Ccolor does not vanish, these equations imply that some standard model invariants have nonzero X charges. In the framework of an effective field theory, it means that these invariants will appear in the superpotential multiplied by fields that balance the excess X charge. These higher dimension interactions are suppressed by inverse powers of the cutoff [58]; this is the origin of Yukawa hierarchies and mixings. A theory with N + 1 extra Abelian gauged symmetries X, Y), . . . , y(N) will contain (as we will explain later), N+ 1 standard model singlet chiral superfields 00, ... ON, tO serve as their order parameters. The anomalyinduced supersymmetrypreserving vacuum is determined by the vanishing of the
13
N + 1 Dterms
N
Z Xio2 2 , (1.3.36)
Q=0
y(a) 1.12 = 0, a = 1, 2, ..., N. (1.3.37)
a=O
These equations can be solved as long as the (N + 1) x (N + 1) matrix A, with rows equal to the N + 1 vectors x = (XO, x1, ..., XN), y(a) _ (a), (a), ) has an inverse with a positive first row. A typical term in the superpotential, invariant under these N + 1 symmetries will then be of the form
(a)
QidHd (1.3.38)
where holomorphy requires the nr') to be zero or positive integers. Invariance under the N + 1 symmetries yields
X1 + E n = 0 , (1.3.39)
y(a) [al + y na = 0. (1.3.40)
These involve the same matrix A, and here a solution also requires that det A 5 0, linking hierarchy to vacuum structure. Evaluated at the vacuum values of the 0, fields, the terms shown above can produce a familydependent Yukawa hierarchy. A successful model of this type is highly constrained: it must satisfy all anomaly conditions and reproduce the observed Yukawa hierarchies. In addition,
14
the breaking triggered by the anomalous U(l)x must preserve supersymmetry, as well as the standard model gauge symmetries.
To summarize, in our search for a realistic model we made a number of assumptions, most of which happen to be generic features of realistic superstring compactifications. We consider low energy effective field theories originating from a string compactification such that the broken gauge group contains the standard model gauge group and at least one additional, anomalous U(1). The superfields appearing in the theory are at least those that contain the fermions and bosons that are observed at low energy. We assume only three families of chiral fermions. Additional visible matter may be present but since we do not have any experimental information on its existence, we will be assuming the existence of any, only if its presence is required by anomaly cancelation. Hidden matter may also be present and its presence from our point of view is again dictated by anomaly cancelation. The anomalous U(1) breaks just below the cutoff when a standard model singlet aquires a vacuum expectation value via the DSW mechanism. We will call this field 0. If there are additional U(1) factors, those have to be broken also at a similar scale by vevs of fields 0,.
We now summarise all additional assumptions that we will make in the course of building a realistic model. In the following chapters we will justify each one of these in detail and we will see that they will be natural consequences of phenomenological constraints rather than arbitrary assumptions:
e We can always absorb the anomalies into only one of the U(1)'s (which we
will called X). We assume that the resulting anomalous symmetry is family independent, except when there is only one such U(1) in which case it has
15
to be anomalous and family dependent. After X breaking, the effective term in W will be
( )", >  I( .. .(1.3.41)
We notice that an effective suppression factor A, =< 6, > /M ~ 0.1 has been generated in front of the G invariant term in W. The remainig U(1) symmetries are anomaly free. They are family traceless over the three families of visible chiral fields. The latter is an assumption that is strongly favored by anomaly cancelations. We assume that they spontaneously break by a similar mechanism that X breaks. This is achieved by assuming the presence of one (and only one) 0 field for each U(1). The vacuum expectation value of each singlet breaks one (and only one) U(1). The relative breaking scales of all the U(1)'s is determined by solving the Dterm equations in the supersymmetric vacuum. In fact, we will assume that in a realistic low energy model all the U(1)'s break exactly at the same scale. As we will see this assumption simplifies enormously the phenomenological analysis of our models. The number of such U(1)'s is unknown. We will take a minimalistic approach.
We will try to construct a model, with the smallest possible number of U(1)'s.
* The mass matrices in the Yukawa sector have to be compatible with experimental data. They can be expressed in a convenient form by using as expansion parameter the Cabbibo angle A,. Then, we require that the model
reproduces the following:
16
1. The only Yukawa coupling term that appears at tree level in the superpotential is the one that gives mass to the top quark:
Q3U3H,.
(1.3.42)
2. The ratios of quark masses extrapolated to the scale M via the renoramization group equations are
mus
AC
rnt
CC 4 Cd m  ~
Mt Mb
SAC. Mb
(1.3.43)
3. The quark mixing (CKM) matrix in this (Wolfenstein) parametrization is given by
UCKM A,
A
1
(1.3.44)
A 3 Ac2
4. The corresponding relations in the charged lepton sector are
in,
m A A2. m,
(1.3.45)
5. Unification of the down quark and r Yukawa coupligs at M
(1.3.46)
M7 ~ Mb
17
6. If the MSSM parameter tan  1, the interfamily relation
M ~ A (1.3.47)
mt
holds with satisfactory accuracy. We will see that the high value regime for tan 3 does not allow for a phenomenologically consistent model in our U(1)
context.
e The number of invariants that are missing from the superpotential due to
holomorphicity, i.e. the number of supersymmetric zeros in W is the minimum possible.
CHAPTER 2
THE VACUUM
2.1 DFlatness and Holomorphy
One of the most important ingredients of supersymmetric models is the existence of flat directions. These are solutions to the supersymmetric vacuum equations, D = 0 and F = 0 which give to inequivalent physics. Most of the times there is a large number of possible solutions to these equations, some of which are completely unacceptable from the phenomenological point of view. To illustrate the problem, we consider here models with only one, anomalous U(1), called X. Later, we will leave the number of U(1)'s arbitrary.
The anomalous Dterm Dx is of the form:
Dx = qiloil2 _ (2.1.1)
where qi is the Xcharge of a generic scalar field #i and is the anomalous FayetIliopoulos term. We consider three types of fields:
" fields which we would like to acquire a vacuum expectation value of order (
when the X symmetry is broken: we denote them generically by 0.
" fields charged under SU(3)C x SU(2)w x U(1)y, typically the fields found in
the MSSM; these fields should not acquire vacuum expectation values. They appear in invariants which form the building blocks used to construct terms
18
19
in the superpotential. Typically for the superfields in the MSSM:
HdH. , QidkHd , LiHdk ,QinkH.
Li H. ,Qidk Lj Li Lge , Uidid , (.1.2)
where i, j, k are family indices. In general, we can also write higher order invariants like QQQL, uude, etc. We will denote an invariant generically by
I.
e scalar fields, singlets under the standard model gauge group, which do not
receive vacuum expectation values of order . These fields are natural candidates for the righthanded neutrinos and we will denote them by N. Typically, for these fields to be interpreted as righthanded neutrinos, one needs
terms in the superpotential to generate Majorana mass terms:
Wm ~ MN2 (0/M)" (2.1.3)
and terms to generate Dirac mass terms for the neutrinos:
WD ~ IN (6/M)P (2.1.4)
where I is the invariant I = LHa. The presence of both terms (2.1.3) and
(2.1.4) is necessary to implement the seesaw mechanism [49], [50].
Finally, we will denote by < 01, #2,   , On > the direction in scalar field parameter space where the fields 01, 02,   , On acquire a common vacuum expectation value of order . Our basic requirement is to choose the Xcharges and the superpotential
20
so as to forbid all solutions to the vacuum equations except the one corresponding to < 6 >. Since we work in the context of global supersymmetry unbroken at the scale , directions in the scalar field parameter space will be determined by the conditions
Dx = 0 and F = oW/aoi = 0. (2.1.5)
For instance, the assumption of Dxflatness (Dx = 0 in (2.1.1)) automatically takes care of the directions < 41, 02,   , On > where Xi < 0 for i E {1,   ,n}. Finally, there is necessarily some gauge symmetry other than the anomalous X, for example the symmetries of the standard model. Dflatness for these symmetries plays an important role for I invariants: it tends to align the fields present in I. We will give the general theorem later. Now we just demonstrate it with a simple example. Take I = 0102. Invariance under U(1)y implies that the hypercharges of 01 and 02 are opposite: Y = Y2. Then the corresponding Dterm reads: D ~ y1(112  10212) +  And Dy = 0 implies < 01 >=< 02 >= v1. The contribution to Dx from these fields is xIIvI2, where x, is the total Xcharge of I. Hence a positive x, will allow a vacuum with the flat direction < 01, 02 >We now give a few, simple, illustrative examples of the vacuum uniqueness problem, with an increasing number of fields.
2.1.1 Models with 2 Fields.
Consider a model with one 0 field and one N field. Take the Xcharge of 6, x > 0 and that of N, xV < 0. With these fields the X Dterm is:
(2.1.6)
Dx = X02 + X 1V2 _ 2 ,
21
There are three different flat directions to consider: the desired < 0 >, < N > and < 9, N > which we wish to avoid. The direction < 9 > is favored by choosing x > 0, and < N > is forbidden if xy < 0. The third direction is allowed by Dx = 0. However, since x  xy < 0, we can form a holomorphic invariant involving N and 9. The simplest possible invariant in the superpotential is NO" but the corresponding Fterms forbid < 0 > and < 0, N >. Thus we lose the possibility of a Dxflat direction with < 0 >~ . Since all possible flat directions are lifted, supersymmetry is spontaneously broken. We must therefore require the presence of an invariant NT'6 with p > 2 and n $ 0 mod(p) to forbid only the direction < N, 9 >. The case p = 2 corresponds precisely to a Majorana mass term for the righthanded neutrino N, once 9 is allowed a vev. In this case, n = 2k + 1 and there must be the following relation between the Xcharges:
XY 2k+1 (2.1.7)
x 2
Consider now another simple model, with 9 (Xcharge x > 0) and I (Xcharge x, < 0). With one field and one invariant, we have
Dx = x1912 + xIvI12 _ 2 (2.1.8)
As previously, Dxflatness kills the direction < I > and allows < 0 >. The main difference is that I being a composite field, I = 11_n 4 n, the Fterms corresponding to the invariant It9U are F = tnj(Hi , # 0) I"10" and F = uIt6u1; they therefore only forbid the direction < I, 9 >, even for t = 1. One is therefore left with a vev of order along the single direction < 9 >. It is certainly en
couraging that linear terms in I can appear in the superpotential. Terms such as QjiUH (0/M)" are needed to implement hierarchies among the Yukawa couplings. Conversely, requiring that
=  n, (2.1.9)
x
with n integer $ 0, is sufficient to insure the linear appearance of the invariant I. In this case, the vacuum structure is related to the Yukawa hierarchies. However if x, = 0, there is no danger associated with I, and the above discussion does not apply.
2.1.2 Models with 3 Fields.
First, consider a model with 0 (x > 0), N (x7) and I (x1). Let us take xy, x, < 0. The vanishing of the Dx term
Dx = x1012 + X71N2 + xIvI12 _ 2 (2.1.10)
forbids the directions < N >, < I > and < N, I >, but allows the directions < 0, N >, < 0, 1 > and < 0, N, I >. We saw earlier that an invariant NO forbids the desired direction < 0 >. We must require the presence of an invariant N qO with q > 2 and n $ 0 mod(q), to disallow the directions < 0, N > and < 0, N, I > (q = 2 generates masses for N). The last direction < I, 0 > is disposed of by adding an invariant of the form IP', which is also allowed given the signs of the charges.
23
2.1.3 RParity
We now show that the constraints discussed above on the X quantum numbers of the lowenergy fields may naturally lead to conserved Rparity. The presence of standard model singlets Ni necessary to implement the seesaw mechanism plays in this respect a key role. We assume the seesaw mechanism requiring the presence of the invariants:
NiNon"ii + Li~NH,9O (2.1.11)
We saw that, in order not to spoil the ivacuum, the powers no must be odd integers, or equivalently the Xcharges X7, of the fields Ni must be, in units of x, halfodd integers:
X, 2k, + 1 (2.1.12)
X 2
where ki is an integer. Henceforth we set x = 1. The last term in (2.1.11) determines the Rparity of the righthanded neutrino superfields to be negative. Let us study the Xcharges of possible standard model invariant operators made up of the basic fields Qi, fLi, di, Li, ei (i being a family index) and of the Higgs fields H,, and Hd. The cubic standard model invariants that respect baryon and lepton numbers are, in presence of the gauge singlets Ni,
Qijj Hd , QiiijH. , Li, jHd , Li7V H. (2.1.13)
with charges X., Xu], Xi and X] respectively. To avoid undesirable flat directions, all must appear in the superpotential, restricting their Xcharges to be of
24
the form
X ev  u d,e,v (2.1.14)
where n' de'v are all positive integers or zero. We now turn to the invariants which break Rparity. We have already encountered the quadratic invariants LjHu whose charges are determined by the seesaw couplings (2.1.11) to be half odd integers
2 (kj  nv) 1 (2.1.15)
XLHU .112
2
Consider the cubic Rparity violating operators, LiLjEk, LjQjdk and uididk. The charges of the first two, which violate lepton number, satisfy the relations:
XLILk = X + X v] X  X] , (2.1.16)
XLiQ, Xk + i X  (2.1.17)
where the index 1 can be chosen arbitrarily. As a consequence, if X[43 is integer,
po, both charges are halfodd integers and there is no Rparity violation from these operators. However they can still appear as
L L3y n% ( +l po) LOQ >(n4+>* p0) ( 2.1.18)
in the superpotential. A similar conclusion is reached if X[I3 is a multiple of one third, in which case one needs to include also appropriate powers of No. To determine the charges of the operators iidjdk in terms of the charges of the parityconserving invariants, one must use the GreenSchwarz condition on mixed anoma
25
lies Cweak CcoIor which reads:
Z(XQ + XL,)  (Xui + X;,) + X[" = 0. (2.1.19)
One obtains:
Xq.dd PX] + X3 + X[  X  2X[d + X
+1  N XIAJ  (2.1.20)
NNm
true for any two family indices p, m, and where Nf is the number of families which we will take to be three. In a large class of models, the charge Xuj.4 thus obtained will be such as to forbid not only a term uididk in the superpotential but also any term obtained from it by multiplying by any powers of 0, Ni or No. Let us consider for illustrative purpose an anomalous symmetry which is family independent. Then (2.1.20) simplifies to:
Xiigz = X[d] + X["]  (I )X[A]  X . (..1
Nf W(..1
Remember that Xy is halfodd integer and Nf = 3. If XENI is integer not proportional to Nf = 3, then the charge Xua is such that no term iidO'Nn can be invariant. If X'] is noninteger and a multiple of one third, then similarly no term idd'NNPh can be made invariant. In the low energy theory, baryon number violation becomes negligible. If we restrict our attention to models which yield
26
sin2 0w = 3/8, the GreenSchwarz condition 5Cweak = 3Cy reads:
Z(7XQ, + XL)  Z(4Xu, + X& + 3Xj) + XIA1 = 0. (2.1.22)
One infers from (2.1.19) and (2.1.22) the following relation:
X[A =E (X]  Xi') , (2.1.23)
which tends to favor models with integer X[A1 (proportional to Nf in the case of a familyindependent symmetry). If XHI = 0 or more generally if XI"I is proportional to Nf (X[] = Nfz,), the charge in (2.1.21) is halfodd integer; it can only be compensated by odd powers of N: invariance under X means conservation of Rparity. For instance, the above allows the interaction:
iidN [nd+nu+zo(Nf1)] (2.1.24)
This term allows baryon number violation, but preserves both B  L and Rparity. A very similar discussion can obviously be given for the general case of a family dependent anomalous symmetry. To conclude, in a large class of models, there are no Rparity violating operators, whatever their dimensions: through the righthanded neutrinos for example, Rparity is linked to halfodd integer charges, so that X charge invariance results in Rparity invariance. Thus none of the operators that violate Rparity can appear in holomorphic invariants: even after breaking of the anomalous X symmetry, the remaining interactions all respect Rparity, leading to an absolutely stable superpartner.
2
2.2 The General Flat Direction Analysis
We now leave the number of the Abelian symmetries arbitrary and develop the formalism for the uniqueness of the DSW vacuum. We consider supersymmetric models with a gauge group G x X x Y(') x . . x y(N) x Ghid, where G contains either the Standard Model or a GUT group, Ghid is some hidden gauge group, and there is a set of Abelian (horizontal) factors connecting both sectors. We will assume in this analysis that there are no discrete symmetries. We denote the anomalous U(1) by X, and the nonanomalous ones by Y(), ..., y(N). In general, but not always, fields charged under G are singlets under Ghid and vice versa, both carrying X, YO), ...and y(N) charges. In the following, we call g = G x Ghid, and we denote generically the fields charged under 9 by #j, and the gsinglets by Xi. As stressed in the previous section, some of these fields must acquire nonvanishing vacuum expectation values through the DineSeibergWitten (DSW) mechanism in order for supersymmetry to be preserved. This in turn breaks X slightly below the string scale, possibly together with some other symmetries. Since the Standard Model symmetries must not be broken at that scale, we shall assume that there exists a solution of the D and Fterm equations that breaks only the Abelian factors. The Dterm equations:
Dx = . aXa0)2 _ 2 = 0
(2.2.25)
Dy) = Z0 y 9)(O,)12
28
where the gsinglets with nonvanishing vevs are denoted by 0,, have in general several solutions, due to the large number of gsinglets generally present in string models. We shall assume the existence of at least one solution {(O)} of (2.2.25) satisfying the following requirements:
" all Abelian symmetries connecting the hidden sector to the observable sector
are broken at the scale ; while probably too strong, this requirement enables the models to escape many phenomenological problems. The number of 0
fields must then be equal to the number of U(1)'s or greater.
" the lowenergy mass hierarchies (in particular fermion masses), which are
generated by the small parameters (90)/M, are completely determined by the highenergy theory. This means that there must be no more 0 fields than
U(1)'s, otherwise the (0n) would not be uniquely determined by (2.2.25).
We stress here the fact that the above two assumptions are crucial for the claim that we will make, namely that there is essentially one model with unique vacuum that reproduces the experimental data in the quark and the charged lepton sectors. In other words, we assume the existence of at least one (N + 1)plet of Gsinglets (0, ..., ON) such that: (a) the matrix of the 0 field charges is invertible, i.e. det A $ 0, where
XO X1 . .. XN
() (1) (1)
A =Y(2.2.26)
(N) (N) (N)
Yo Yi ... YN
29
and the first column of A only contains strictly positive entries. This ensures the existence of a vacuum (0, ... , ON) DSW with
(6)DSW
(1 A _1 (2.2.27)
(ON) DSW 0
which we shall refer to as the DSW vacuum. In addition, one must check that this vacuum is not spoiled by the Fterms. Since condition (2.2.26) ensures that there is no invariant of the form 0"0"..1 ..N, this can happen only if the superpotential contains a term linear in the X and 0 fields. Our second assumption is then: (b) there is no holomorphic invariant x 0'0" ... 6"N linear in x, where x is a Gsinglet other than the 0 fields. This amounts to a condition on the charges of x, namely at least one of the numbers no ..., nN defined by
no X
ni Y11
S  A X (2.2.28)
nN y[N]
has to be either fractional or negative. Note that if X is a righthanded neutrino, this constraint leads to an automatic conservation of Rparity as we have argued in the previous section. Since we are dealing with an effective field theory, we must put in the superpotential all possible interactions allowed by the symmetries of the theory, including nonrenormalizable terms suppressed by inverse powers of the
30
cut off scale (in the following, we set for a while M = 1). An important comment is in order here. Due to discrete symmetries and conformal selection rules, the superpotential of an effective string theory does not contain every term allowed by the (continuous) gauge symmetries. This may have important consequences, in particular some Dflat directions that one would naively expect to be lifted by Fterms could remain flat to all orders [59]. In order to keep our discussion as general as possible, we shall neglect this effect. Thus the criteria that we gave for a Dflat direction to be lifted should be regarded as necessary conditions only. Let us have a look at the generic form of superpotential terms. In general, ginvariants and qsinglets are not neutral under the Abelian symmetries, and must appear multiplied by powers of the 0 fields. Condition (a) allows us to assign, through Eq. (2.2.28), a set of numbers {} = (no, ..., nN) to each ginvariant I = O... 0 (resp. gsinglet x). If all n, are positive integers, then Io"n,"1 ... " N is an holomorphic invariant and can be present in the superpotential. It is quite remarkable that condition (2.2.26), which ensures the existence of the DSW vacuum, is at the same time the one that is required for invariants of the form I 9on"0n ... Q"N to exist. Those invariants are precisely the ones needed to generate mass hierarchies in the DSW vacuum, with I being Yukawa couplings. Note that this is not true in the nonanomalous case: the condition required for the 0 fields to develop a nonzero vev is det A = 0, which forbids the existence of the invariants I0 . .."N except for some very specific charge assignments for which the powers (no, ..., nN) are not uniquely determined. If all n, are positive, but some of them are fractional, the invariant appears at higher order: (I00j" N. . )."". Finally, if some n0 is negative, one can not form any holomorphic invariant out of I and
31
the 0 fields; we shall refer to this last situation by saying that I corresponds to a supersymmetric zero in the superpotential.
2.2.1 DFlatness
Before characterizing the Dflat directions of the models defined above, let us recall a very useful theorem [60] which we shall use throughout this section. In a globally supersymmetric theory with a compact gauge group g, and no FayetIliopoulos term associated with the Abelian factors that G may contain, the zeros of the Dterms can be classified in terms of the holomorphic gauge invariants. More precisely, a set of vevs (1, . . . , 0n) is a solution of the Dterm constraints if and only [61] if there exists a Ginvariant holomorphic polynomial I(,. . ,) such that:
01 = C () i = 1... n (2.2.29)
0i oi=(0.)
where C is a complex dimensional constant. A systematic way to study Dflat directions is then to find a finite basis of invariant monomials {Ia} over which any holomorphic invariant polynomial can be decomposed. Such a basis is characteristic of the gauge group and the field content of the theory. As an example, a basis of the MSSM invariants can be found in Ref. [62]. We are now ready to make the following statement: to each basis Ginvariant I = ... /P', (resp. Gsinglet x),
32
corresponds a Dflat direction (,... 0, ; W) (resp. (X ; W)), with
for nc, > 0
(2.2.30)
(9.)12 > 1(.)12Dsw
(0')12 < I .) 1sw
for n0 < 0
As we show below, this is the only solution of the Dterm constraints associated with I (resp. x). Note that this is not, in general, a flat direction of the scalar potential, because the Fterm constraints F = 0 are not necessarily satisfied. Let us prove this first in the case of a gsinglet x. The only input we need is the existence of the DSW vacuum (2.2.27). Then, using the definition (2.2.28) of the {na}, the Dterm constraints
/
Xx
Y111
yEN]
X
can be rewritten as:
(X)12 + A
I
r
A 1(00)12 A 1(61) 2 AI(N) 12
t
\(0)12 I(01)12 I(ON)12
I
/
no nN
r
02
0
(2.2.31)
(2.2.32)
/
where we have defined A I (a)12 = I(a)12 _ \(0a) >SW. The sign of A 1(0")12 is thus determined by the sign of n,. In particular, when the Dflat direction is associated
33
with an holomorphic invariant of the whole gauge group g x X x Y(l) x ... X y(N), constraints (2.2.30) read:
1(0.)12 ()2DSW  0,1 ... N (2.2.33)
This is a remarkable difference with the nonanomalous case, in which the vevs of the 0 fields are not bounded. The A (")12 depend on a single parameter, I(x)12, which may be fixed by the Fterm constraints, or by supersymmetry breaking. Notice that this is no longer true when x is a singlet of the whole gauge group. In this case, no = ni = ... = nN = 0 and (2.2.32) is nothing but the DSW vacuum, whatever (X) may be. In the particular case where (x) = 0, one recovers the DSW vacuum. The generalization of (2.2.32) to the case of a basis ginvariant is straightforward. Applying (2.2.29) to I = 1. . ..0P, we find that the Dterms associated with g constrain the vevs of the # fields to be aligned. In this relation, I(0j) 2 stands for La I(#?)2, where the # are the components of the representation of g spanned by 0j. One should keep in mind that (2.2.34) is a weaker constraint than the vanishing of the Dterms:
(01   = (2.2.34)
Pi Pn
As a result, we end up with a relation similar to (2.2.32), with i(x)12 replaced by vo. As before, we denote this Dflat direction by (I, W) to stress the fact that the vevs of the fields in I are aligned. However, generic Dflat directions are not associated with a single basis ginvariant, but rather with a polynomial in the basis ginvariants. More precisely, flat directions involving a given set {0} of fields
34
charged under g are parameterized by the vevs of the ginvariants Ia = li q, that can be formed out of those fields:
1(0,)2 v2 pa (2.2.35)
a
where v 2is a vev associated with the invariant Ia. In general, the parameters v2
a a
are complex but shall assume that they can always be chosen to be positive real numbers. While we do not have a general proof for this, it turns out to be the case in numerous explicit examples. Then the most general solution of the complete set of Dterm constraints is a set of vevs ({#i}, {xi}, ) with:
A L(00) 2
n a(1) ni
aV(2 += v + 2 (2.2.36)
a
A I(ON)j12 na
Clearly the relation 1(0")12 ;> 1(0a)I2sw holds, for a given a, only when all powers n. and n' are positive. On the contrary, when one of these numbers is negative, (6)I2 can be smaller than 1(0")1Dsw. This may lead to vacua in which (0") vanishes after imposing the Fterm constraints, or after supersymmetry breaking. We shall see in the following that formulae (2.2.32) and (2.2.36) considerably simplify the analysis of flat directions in anomalous U(1) models.
35
2.2.2 FFlatness
In this section, we examine under which conditions a Dflat direction is lifted by Fterms. We first assume no compensation between different contributions to the Fterms, so that each individual contribution has to vanish for a Dflat direction to be preserved. We shall come back to this point later.
We first restrict our attention to Dflat directions (I, W) that are associated with a single ginvariant I = 0 ... #,. Two cases must be distinguished, depending on the signs of the numbers n' = (n',. . ., n ) associated with I:
" all n. are positive, i.e. the Dflat direction can be associated with some
holomorphic invariant Im 0'01 . .. 9N of the whole gauge group. This
invariant contributes to the Fterms as
(FA ) = cm m C' v2" (01) (0)m'o (1)m ... (ON (2.2.37)
where we made use of (2.2.29), a is a coupling constant, and v, is defined by (2.2.34). Since 1(0")12 ;> 1(0")1DSW along the flat direction, this contribution vanishes only if (0j) = 0, or equivalently v, = 0. As a result, the flat direction
breaks down to the DSW vacuum.
" some of the n' are negative, i.e. the Dflat direction cannot be associated
with any holomorphic invariant of the whole gauge group. Such flat directions are in general not completely lifted, unless the superpotential contains an invariant of the form I' ...1 1 N (with I' a combination of basis ginvariants and X fields), where either one of the following two conditions is
36
fulfilled: (i) I' contains no other field than the ones appearing in I, and n' = 0 or 1 if n, < 0 (with the additional constraint < {a.gO} n' < 1); (ii)
I' contains only one field that does not appear in I, and n' = 0 if n" < 0. To illustrate the last point, let us consider a toy model with G = SU(3)c x SU(2)w x U(1)y, two U(1)'s, and the following field content: Q1, 'ii, u2, H,. We assume that the invariant associated with I = Q1jii1H, is nonholomorphic, e.g. Q~if1H, 0101 Then the superpotential contains only one term:
W = a Q12H, o" ;1 (2.2.38)
provided that no and ni are positive integers. Note that I' Qi 2H" contains only one field, ii2, that does not appear in I; therefore, it satisfies condition (ii) provided that no = 0. The only Fterms that are likely to be nonvanishing along the Dflat direction (Q, iii, Hu, 0o, 01) associated with I are:
Fu2 =a QiHu O0 "1 (2.2.39)
Using (2.2.29), we obtain:
(FIf) = a C (ti) (10, and (2.2.40)
If no = 0, the Fterms vanish only for v, = 0, and the Dflat direction
(2.2.41)
(Q1, i, Hu, 0, 1)
37
breaks down to the DSW vacuum (00, 01)Dsw. If no > 0, the Fterm constraints have two solutions: the DSW vacuum (which corresponds to v, = 0), and a residual flat direction (Qi, i, H, 1) with vI = (Q1) () (H)I2 = I(O)20sw
and 1(01)12 = 3 1(00)12SW + 1(1)1Dsw. Thus the initial Dflat direction is only partially lifted, and the residual flat direction, along which (0o) = 0, can lead to another vacuum than the DSW vacuum. However, it can still be lifted by higher order operators. Indeed, the invariant (Q1iL1H.)" (Qi2H.) 12o+fi (which satisfies condition (ii)) contributes to (FU2) as:
!3Cno+l Vflo (i) (61)3no+n(
which obviously vanishes only in the DSW vacuum (v1 = 0).
We consider now all possible Dflat directions involving only gsinglets. Two cases must be distinguished, depending on the signs of the numbers n' associated with each of the fields Xi:
* all n' are positive: in this case, 1(0)12 > 1(O)12Sw along any flat direction
of singlets. As a consequence, the Dterm equations do not allow for any other vacuum of singlets than the vacuum (2.2.27). The other solutions of (2.2.25) are Dflat directions parameterized by the vevs of the x fields.
Since these flat directions correspond to holomorphic invariants of the whole gauge group, they are lifted by Fterms, leaving only the vacuum (2.2.27).
Therefore, in this case, the DSW vacuum is unique.
" some n. are negative: in this case, some of the (9,)2 can be smaller than
in vacuum (2.2.27). As can be seen from (2.2.36)., the Dterm equations
38
allow for vacua of singlets in which (0,) = 0, while some of the x fields have nonvanishing vevs. Those vacua correspond to particular points along Dflat directions that are in general not completely lifted, unless the required holomorphic invariants are present in the superpotential. If this is the case,
one recovers the uniqueness of the DSW vacuum.
Consider now a generic flat direction ({Mi}; {xi}; W) involving gcharged fields as well as gsinglets. The relevant numbers here are {n a; n }, where the {n } are associated with the basis ginvariants {Ia} that contain the {qj}. The general requirement for this flat direction to be lifted is that invariants of the form I' 0 6 ...0 1 N be present in the superpotential (where I' is a combination of basis ginvariants and x fields), where either one of the following two conditions is fulfilled: (i) I' contains no other field than the ones appearing in the flat direction, and n' = 0 or 1 if one of the powers {n,; n'} is negative (with the additional constraint that no more than one such n' should be equal to 1); (ii) I' contains only one field that does not appear in the flat direction, and n' = 0 if one of the powers {n,.; n'} is negative. Several invariants are in general necessary to lift completely the flat direction. Clearly those conditions are automatically satisfied when all relevant {n,; n'} are positive. In all other cases, one has to check explicitly that the invariants required are present in the superpotential, even if they appear at high orders.
So far we did not consider the possibility of compensations between different contributions to the Fterms. The effect of such cancelations is to reduce the dimensionality of a Dflat direction, while one would naively expect it to be (at least partially) lifted. For instance, in the toy model of the previous subsection,
39
(case no > 0), contributions (2.2.40) and (2.2.42) cancel against each other in (Fu2) if the following relation between vevs is satisfied:
a(0)"o + /3(CV2)""(61)3o 0 (2.2.43)
(note that the case no = 0 does not suffer from this problem, since (2.2.40) is the only contribution to (Ff2)). Such compensations are possible because the Fterms, at least at low orders, are not all nontrivial and independent from each other. When higher order operators are added in the superpotential, the number of independent Fterm constraints generally increases, and cancelations become less likely. We will neglect them here, but in the flat direction analysis of an explicit model, they have to be taken into account. The case of flat directions involving only gsinglets is more subtle, and needs a separate discussion. Due to condition (2.2.26), the Fterms of the 6 fields are not independent from the other Fterms:
( o Fo. x1 Fxj
A : = AX (2.2.44)
ON FN Xq Fx
where AX is the matrix of the charges of the X fields, defined in an analogous way to the matrix A. As a result, flat directions of gsinglets are constrained by exactly as many equations as fields, and those (nonlinear) equations have in general several solutions. Thus the theory possesses, at any order, vacua of singlets that may compete with the DSW vacuum. However, while the DSW vacuum is welldefined and stable against the addition of higher order terms in the superpotential
40
(as implied by condition (b)), this is obviously not the case for the other vacua of singlets, which depend on the explicit form of the Fterms. This would not be a problem if all vevs were small compared with the mass scale by which nonrenormalizable operators are suppressed. But due to the anomalous FayetIliopoulos term, the singlet vevs are generally very close to the Planck scale and they are not expected to converge to any fixed value when higher order invariants are added in the superpotential (below we illustrate this point with a simple example). Such a situation obviously does not make sense in the context of an effective field theory, and for this reason we shall consider the DSW solution (in which (O)/M is typically of order 0.01  0.1) as the only plausible vacuum of singlets. To illustrate this point, consider a toy model with three fields Xi, X2 and 0, charged under the gauge group X with charges 5/3, 4/3 and 1. At order 8, the superpotential consists of the following three terms:
W = cx065 + C2X364 + C3X1X203 (2.2.45)
where ci, c2 and c3 are numerical coefficients of order one. If the last term in W were not present, there would be a unique DSW vacuum, with ()DSW and (x1)DSw = (X2)DSW = 0. In the presence however of this term, there is an additional solution to the Dterm and Fterm constraints:
(X )c ()) (46
M 27C2)
3X)c 1/3 ) 4/3
(X2 C27c(c) (2.2.46)
41
where due to the positive powers in (2.2.45), j(O) ;> I(O)IDsw = , and ~ (0.1  0.01) M. For coefficients ci, c2 and c3 of order one, this solution gives
(xi)/M of order one, which is unacceptable in the context of an effective field theory. In addition, when higher order terms are added in (2.2.45), the vacuum (2.2.46) changes.
2.2.3 Supersymmetry Breaking and LowEnergy Vacuum
The purpose of this section is to discuss how supersymmetry breaking affects the conclusions of the previous section. Since models with an anomalous U(1) have numerous implications for lowenergy phenomenology, it is indeed essential to ensure that the flat direction analysis is relevant to the determination of the lowenergy vacuum. The scalar potential of the lowenergy theory reads:
V = D + F + (2.2.47)
Since supersymmetry has to be broken in a soft way, Vsl/sy is of the form
in V( + mn2 V2, (2.2.48)
where ii ~ 1 TeV is the scale of supersymmetry breaking, and V , V(3 are functions of the scalar fields with dimensions 2 and 3, respectively. This definition allows for higher order terms suppressed by negative powers of the Planck mass, e.g. #"/Mpni2 E V(2). This has the obvious consequence that the actual minimum of the scalar potential is close to a flat direction of the supersymmetric theory;
42
otherwise the Dterms and Fterms would give a positive contribution of order 4 to V, while Vsy/sy would contribute at most as f5 3, with no possibility of compensation. We thus necessarily have (F) < 2 and (D') < p2, which implies in particular that the relations (0,)12 > (0,)1'sw still hold (provided that the necessary conditions are satisfied), and that fields charged under g with vevs of order should be aligned in the sense of Eq. (2.2.35). In addition, we shall assume that there are no compensations inside the Fterms, which means that all contributions to the Fterms must be much smaller than 2. Let us first consider the flat directions for which 1(0")12 ;> 1(0")12sw hold for all a. The minimization procedure amounts to adjusting the field vevs around this flat direction so as to obtain the lowest value for the scalar potential; as a result, some fields acquire vevs of the order of the supersymmetry breaking scale Fi, or of an intermediate scale such as (Fi M)1/2. Clearly those cannot be the 9 fields. In addition, the X and 0 fields cannot have a vev of order , otherwise some invariant of the form 10J" 0" ... 0"ON or Xm OW0 011 . N would give a contribution of order 2 to the Fterms. As a result, the lowenergy vacuum is a slight deviation from the DSW vacuum:
() ) (x) (2.2.49)
This is perfectly consistent with the conclusions from the flat direction analysis: flat directions along which 1(0")12 ;> 1(0")2SW for all a are lifted down to the DSW vacuum by the Fterms, and the only effect of supersymmetry breaking is to give a small vev to the x and 0 fields. Note that the symmetries of g are not broken at the scale . Consider now the flat directions for which 1(0")2 ;> ( 2sw does not hold for all a. Contrary to the previous type of flat directions, these may be
43
only partially lifted at the supersymmetric level, and the effect of supersymmetry breaking is to lift them completely, leading possibly to undesired vacua, as we illustrate bellow. For this purpose, we go back to our simple example and consider the flat direction (Qi, 1, Hu, 0o, 1) associated with the nonholomorphic invariant Qli1Hu 010'. Since along this flat direction 1(9o)I2 < SW, we can see from (2.2.39) that, in the case no > 0, Fa, can be small compared to 2 while vo= I(Q)1)2 = (i)2 = (Hu)12 is of order 2. For example, in the case no = 1, the minimization of the scalar potential could yield (0o) ~ in (with I(F2)I2 "'2 2 being compensated for by e.g. in2 2 (Hu)12, with fi2 negative). In this case, the Standard Model symmetries would be broken at the scale . We conclude that such flat directions are potentially dangerous and should be lifted at the supersymmetric level, in the way that has been discussed in the previous section. For instance, the problem disappears in the no = 0 case, since v, ~ would give F2 ~ 2, which as stressed before cannot be the case at the minimum of the scalar potential. From this qualitative discussion, we conclude that supersymmetry breaking does not change radically the conclusions from the flat direction analysis. Its two main effects are to modify slightly the DSW vacuum by giving small or intermediate vevs to the x and < fields, and to lift the flat directions that are present in the supersymmetric theory. It is therefore essential to check, in a specific model, that the flat directions that may lead to undesired vacua are completely lifted already in the supersymmetric limit.
44
2.2.4 Summary
We now summarize the generic procedure to analyze flat directions in anomalous U(1) models that satisfy conditions (a) and (b):
1. find a basis I1, ..., I for holomorphic ginvariants involving only # fields;
add to this basis all gsinglets Xi, ..., Xq
2. for each element of the basis, compute the set of numbers (no, ..., nN)
defined by (2.2.28). The corresponding Dflat direction is (I; 0) in the case of a 9invariant, and (X ; ) in the case of a gsinglet; the vevs of the 0 fields
are determined by (2.2.32), and they satisfy the constraints:
(0.)I2 > 1(9) 1DSW na > 0
(2.2.50)
(0.)12 < 1(.2Dsw n, < 0
3. the most general Dflat direction involving a set of fields ({}; {xi}) is
parameterized by the vevs of the Xi and the Oi, the latter being subject to constraints (2.2.35). The vevs of the 0 fields are determined by (2.2.36),
which implies
I(0 )I2 > ()2SW if n a, ni > 0 (2.2.51)
for all relevant I, and Xi.
45
4. determine which flat directions are lifted by Fterms. Two cases must be
distinguished:  flat directions for which all relevant {n,; n'} are positive are lifted down to the DSW vacuum by Fterms.  other flat directions are only partially lifted, unless invariants of the form I'0 10 ... "', where I' contains no more than one field that does not have a vev, are present in the superpotential. For this to happen, the n' must satisfy the conditions we have previously specified. Several such invariants are in general necessary to
lift completely the flat direction.
5. once the presence of the invariants required to lift a given flat direction has
been checked, a more careful analysis should take into account the possibility of cancelations inside the Fterms, and show that the flat direction is indeed lifted at some order. Also, one should check that cancelations do not allow other stable vacua of singlets than the DSW solution, even though such a
possibility seems to be very unlikely.
2.2.5 The Ideal Vacuum
We now present the ideal scenario for the vacuum structure of a model. This is when there are absolutely no supersymetric zeros in the superpotential. Let us assume so and ask what additional conditions we must meet to have a really unique vacuum in the sense that even flat directions associated with an arbitrary polynomial of the basis is lifted down to the DSW vacuum. First of all we know that if we have made sure the existence of the DSW vacuum and made the connection to the seesaw mechanism via right handed neutrinos and if there are no supersymmetric zeros it is guaranteed that all the "nonDSW" flat directions are
46
lifted except
< anything, {"} > (2.2.52)
"Anything" in the above means standard model fields which form invariants that, as usual, we call generically by ISM and/or fields with tree level mass terms such as fields vectorlike with respect to the standard model gauge group, extra singlets x and hidden sector fields. We will call generically these latter by Ob. Then, the Fterm coming from the mass terms will lift the directions
< Ob, {0} > . (2.2.53)
For the standard model fields that appear at lowest order as Yukawa couplings, we have shown that the vevs of the fields that make up an invariant align, so that the Fterms that come from the Yukawa sector lift the directions
< Ism, {0} > . (2.2.54)
Finally, a generic flat direction
< {}, {I}, {0} > (2.2.55)
that corresponds to an arbitrary polynomial of the basis invariants (P), is lifted by the invariant that is made of the product of all the different invariants that compose P.
Supersymmetric zeros make the analysis of the vacuum much more complicated. For that, we strongly favor models with less number of supersymmetric zeros. As
47
we will see, we will not be able to find a model with no supersymmetric zeros. We will end up however with a model with very few.
CHAPTER 3
MASS MATRICES
3.1 Quark Mass Matrices from U(1)'s: Hierarchies
Assume now a generic model with N + 1 extra U(1)'s. A gauge invariant term in the superpotential has the form:
1il1213* .. 1 ,i 2 i3 I .. 1121 .3' .
(< 00 > )"0 < 01 > < < ON > ) N iii ..l
M M M. ..
We have displayed in the standard model invariant I and the exponents the family indices explicitly. The invariance of this term under the whole gauge group and in particular under the U(1)'s allows us to compute the powers n"" to be n = A1Y1, (3.1.2)
where we have introduced the following matrix notation: n is an (N+1) x 1 column vector with the powers of the 0 fields
=2% (....ZI
n (3.1.3)
48
49
Y['I is an (N+ 1) x 1 column vector with the charges of the standard model invariant IiIi2i3... i under the N + 1 U(1)'s. We denote the anomalous U(1) by X and the anomaly free ones by Y(a). We also assume that X contains all of the trace of the N + 1 U(1)'s. Then,
/ X (Iiii2i3...T)
Y(M)(Isii2i3 ...ifr)
Y['] = (3.1.4)
y(N)(.)
A is the matrix we defined in the previous chapter. Its inverse is assumed to be
1 al . . . aIN
1 a21 . . . a2N
A (3.1.5)
1 a(N+1)1 .   a(N+1)N Notice that all the elements of the first column of the above matrix being 1 means that all the 0 fields (N + 1 of them) take vacuum expectation values at the same scale. This is an assumption that we will keep until the end of this work, because it makes the discussion on mass matrices more lucid. We will later point out the possibility that such an assumption could be relaxed.
We also set the notation for the Abelian charges of the observed quarks under the a'th nonanomalous U(1) (a =1, ..., N):
50
Here, we have already used the assumption that the nonanomalous factors are traceless. It is useful to introduce the quantities
Q[a 2qi"] + q2 and Q[a] 2q ] + q[2] UI a 2[a] + u[2] and U a] [2]
D 2d, + d1 and D a] 2d'] +
(3.1.6)
(3.1.7) (3.1.8)
Assuming no supersymmetric zeros, we can easily compute the Yukawa matrices in the up and down sectors. In the up sector we get the Yukawa matrix with elements A,(=< 0 > /M) to the power N[u] plus
M+K M+L
M+0
P+K P+L P+0
0+K\ 0+L
0+0 J
(3.1.9)
and in the down sector we get the powers N[d] plus
R+K R+L R+0
T+K T+L T+0
0+K O+L
0+0J
(3.1.10)
1st family 2nd family 3rd family Q q[a] q[a] q[a] q[a]
Q 1q q
d d[a] d[a] d[a]  [a]
U 1 2 U1 U2
d_ _ _ _ [a] a] [4a] [a
51
NE'4d is defined as the total power appearing at the 33 position of the mass matrix which we always pull in front. Also, K Ka
L L,
M M.
= Pa (3.1.11)
P P.
R R,
T Ta
where
SKa\ La Q1
Ma)U
= aaa U . (3.1.12)
RaDa STa D
As usual, a 0, ..., N, a = 1, ..., N and summation over a is implied. The assumption that the top quark acquires mass at tree level and the fact that we would like to have a tan 0 of order of one (we will justify the latter soon), amount to
Nf"u = 0 and Na = 3 (3.1.13)
respectively. The elements in the CKM matrix above the diagonal, require
K = 3 and L = 2, (3.1.14)
52
with K0, L0 > 0. Finally, since the ratios of the eigenvalues of the up and down matrices should obey the geometric hierarchy stated in the introduction, we have to diagonalize the order of magnitude matrix
LAv
A P+3
C
A P+2
C
A2 3 CI 1,
(3.1.15)
in the up sector. This matrix has the (order of magnitude) eigenvalues
1 and AP+2 and AM+3
(3.1.16)
The unique choice that gives the correct phenomenology is
M = 5 and P = 2.
(3.1.17)
Identical arguments give in the down sector
R = 1 and T = 0.
(3.1.18)
The above, also fix A, to be a number close to the Cabbibo angle, which justifies the notation. We have therefore proven that the mass matrices with no supersymmetric zeros that are capable of reproducing the low energy data with U(1)'s are unique in the up and down quark sectors. Also they give a unique CKM mixing matrix.
We can therefore narrow down our search to matrices of the form:
M = vuyu = vI AC Ac
Md =Vd yd VdA 3A
and
A25
C
C'
1J
A 3
C
1
where Vu,d =< Hu,d >.
We now give an algorithmic procedure that one should for models generating the above forms:
follow when searching
* Find a set of arbitrary integers Ka, L,, M,, P,, R,, S, that satisfy (3.1.17)
and (3.1.18), (including Nu] = 0 and Na = 3).
* Assume a matrix A, that satisfies previous step, the quantities
Q12 Q112
KN)
(3.1.5). Calculate from the integers of the
Q \21
Qif \Q1112
A
/Lo \ L,
(3.1.21)
\\LN)
53
(3.1.19) (3.1.20)
Ko Ki
and 0T0
U1'2 MO U2' PO
= A = A (3.1.22)
U MN /N UN/ \N
and D01
/D12 \ /Ro \ /D[\ / Ro\
D[12 R, D[1 R,
A A . (3.1.23)
D RN / \D RN /
* Calculate the actual (nonanomalous) U(1) charges (inverting 3.1.63.1.8):
(,a ~ 1)(QQi)
3 312(3.1.24)
and
[ 3 )(3.1.25)
U2 U2"1
and
da D [2
\ d" \/3 D 12 (3.1.26)
55
* Check if the previously found charges can be supplemented by a lepton (and
perhaps neutrino, vectorlike, hidden) sector that gives rise to an anomaly
free system.
We would like to stress here the fact that the above procedure, in the case of only one, family dependent, anomalous U(1), fixes uniquely the traceless part of it (over the visible sector). The trace however, which will later play the role of the anomalous U(1), we will see that it is forced on us by phenomenology but it is by no means uniquely fixed by it.
3.2 A Model with a Single U(1) Family Symmetry
Before we continue, we will try to elucidate all the above with an example that will turn out to be extremely important for model building. Let us try to construct a model with a single U(1). We can begin by assuming that our U(1) is traceless and anomaly free to start out, in which case the interfamily structure of the mass matrices should fix its form over the visible sector, as we saw in the previous section. Then, we will impose the intrafamily structure and for that we will have to add a trace to it, so that it becomes anomalous.
3.2.1 Interfamily Hierarchy
We normalize the charge of 0 to be 1, which gives us A = 1. Then, combining (3.1.14), (3.1.17), (3.1.18), (3.1.21), (3.1.22) and (3.1.23), we obtain:
(q ) (  3 4/3
3 3(3.2.27)
q2) 2 1/3
31 j j 5 8/3
3 3/(3.2.28) =2 1  2 1/3
U2 3 ( 1) 2/
3 3 (3.2.29)
d2 =I 0 1/3
We summarize the traceless part of the quark sector of our U(1) in the following
table:
At first sight this looks unlikely that this symmetry can be made anomaly free. However, looking at it closer, we discover that it can be written as
YF = B(2, 1, 1)  2r(1, 0, 1), (3.2.30)
where r7 = 1 for both Q and U and r7 = 0 for a. That both Q and U possess the same rq charge is reminiscent of the SU(5) charge patterns, where the chiral fermions are split into 5 = (L, a) and 10 = (Q, U, E). This suggests we flesh out the multiplets by assigning the E singlet a value q = 1 and the L doublet r7 = 0. We generalize the factor B appearing in Eq. (3.2.30) to its SO(10) analog (B  L). Note that rq, on the other hand, is outside of SO(10). The quark and lepton charges may then be succinctly written:
YF = (B  L)(2, 1, ,1)  277(1, 0, 1) .(
__ I__ U __4/3 8/3 \ /2/3 Yl) 1/3 1/3 1/3
\ 5/3 7/3 1/3
I
(3.2.31)
57
The interfamily exponents of the Yukawa matrix associated with the operator L,EjHd follow:
i1 I Z2 1 53
Li 4 5 3
L2 1 2 0
L3 1 2 0
Its diagonalization yields the lepton interfamily hierarchy
m 46 A2, (3.2.32)
m, m,
fully consistent with phenomenology, as well as the contribution to the lepton mixing matrix from the rotation of the lefthanded lepton doublet:
1 A3
A3 1 1 (3.2.33)
A3 1 1
As YF contains B  L, it is natural to introduce three families of right handed neutrinos Ni. Before assigning them YF charges, we note that certain predictions associated with neutrino phenomenology are completely independent of the charges of the Ns. The neutrino mixing matrix, for example, is uniquely determined by the charges of the MSSM fields [63], [64]. This is a result of its seesaw [49], [50] origin, as can be seen via the following simple argument. Since the righthanded neutrino Majorana mass matrix is symmetric, it may be written Y = NiNg,
58
where Ni and N, are vectors depending only on the antineutrino charges. The matrix coupling righthanded neutrinos to the standard model, on the other hand, is written Y') = H LN , where HLj is a vector independent of the righthanded charges. Taking UO to be the matrix that diagonalizes Y01,
Y(O) = UODO (UO)T , (3.2.34)
with D0 a diagonal matrix, the effective neutrino mixing matrix after the seesaw is given by:
y(v) = .(Y(v)Uo)(Do)(Y()Uo)T . (3.2.35)
Because of the form of Eq. (3.2.35), a cancelation of Nj charges results, and one discovers that
H2LiL . (3.2.36)
The MNS neutrino mixing matrix [51] therefore depends only on the mixing of the Li. Thus, both the neutrino mass matrix and the MNS mixing matrix appearing in the leptonic charged current are determined by the Li charges, and the MNS mixing matrix will be of the form given in Eq. (3.2.33).
59
This implies a small (order A') mixing of the electron neutrino with the Iy and r species, and mixing between the M and T neutrinos of order one [65], [66], [67]. Remarkably enough, this mixing pattern is precisely the one suggested by the nonadiabatic MSW [521[56] explanation of the solar neutrino deficit and by the oscillation interpretation of the reported anomaly in atmospheric neutrino fluxes [46], [47]. It is important to stress that this mixing matrix is a generic prediction of such models, and depends only on standard model charges already fixed by phenomenology. The neutrino masses, on the other hand, depend on the origin of the intrafamily hierarchy.
3.2.2 Intrafamily Hierarchy
The intrafamily hierarchy in the quark sector suggests that a family independent symmetry is not the end of the story. Recall that the ratio of third family quarks, mb/mt, is of order A3. Since both cot 3 and the Yukawa entries conspire to produce this suppression, there are two extreme possibilities.
* The first possibility is that Y and Y are of the same order, with cot # responsible for the suppression. With a treelevel top quark mass, achieving Y and Y of the same order requires that the YF charge of the pterm, HUHd, be Y1] = 6. But avoiding anomalies such as Tr[YYYF] and Tr[SU(2)SU(2)YF forces the YF charge to be vectorlike on the Higgs doublets, so that YAj' = 0. Hence Yb  Y requires YF to be anomalous (The GreenSchwarz mechanism cannot be invoked since Tr[SU(3)SU(3)YF] = 0). Furthermore, we shall soon see that a familytraceless YF cannot reproduce neutrino phenomenology.
60
To proceed, we need to assign YF charges to the righthanded neutrinos N. Since r7 is contained in E6, we give the N fields their E6 value, q = 2, which yields YF(NZ) (2, 1, 3). One obtains an NiNj Majorana mass matrix with family dependence
NVI1 N2 113
N1 4 3 SZ
N2 3 2 SZ
N3 SZ SZ SZ
where 'the SZ' stand for 'supersymmetric zeros' due to negative charges. With a null row, this matrix has a zero eigenvalue, and the third family neutrino drops out of the seesaw mechanism. We are then left with two light species of neutrinos, with masses voA /M and vUA12/M. This situation is inconsistent with the combined set of atmospheric and solar neutrino data. The predictions can be made to fit any one experiment, however, but only if M is of order 1012 GeV, suppressed by four orders of magnitude with respect to MGUT. There is no mechanism in our model to effect such a suppression. We conclude that the familytraceless, nonanomalous YF symmetry must be extended by adding a familyindependent piece, hereafter called X.
e We turn now to the alternate possibility, Yb  A3Y and cot 0 of order 1, where the suppression A' comes from the familyindependent piece X. The total flavor symmetry is now
x = X + YF (3.2.38)
61
To consider the implications of anomalies involving our familyindependent symmetry, we define the mixed anomaly coefficients of Yx with the Standard Model gauge fields by
CGi = Tr[GjGjYx] (3.2.39)
The CG, satisfy the following relations:
Cy + Cweak  8Ccoior = 6(X[e  X[) + 2XI'1 , (3.2.40)
3
CcoIor = 3(Xlul + X  3X[') , (3.2.41)
where X[ued'"I are the X charges of the operators QiujHu, LieHd, QidjHd, and HuHd, respectively. It is precisely these charges X[e], Xal, and X(Il that determine the intrafamily hierarchies mb/mt and mr/mb. Let us set
 ~ cot fAPb l A Pb (3.2.42)
mnt mb
Then one finds that Eqs. (3.2.40) and (3.2.41) above may be rewritten as
Cy + Cweak  2Ccoior = 2(Pt + 3Pb + 6) , (3.2.43)
where we have used the fact that the top quark Yukawa coupling appears at tree level, and therefore that Yu] = 0. The data suggest Pt = 3 and Pb = 0, which through Eq. (3.2.43) tells us that our new symmetry Yx must be anomalous.
62
The only consistent way to build a model with such an anomalous U(1) is the use of the four dimensional version of the GreenSchwarz anomaly cancelation mechanism. We take the familyindependent X acting on the chiral fields to be a linear combination of a universal piece and of the two E6 charges, V, V', defined through
E6 > SO(10) x U(1)v, ; SO(10) D SU(5) x U(1)v. (3.2.44)
Across the Higgs doublets, the X symmetry is taken to be vectorlike, a necessary condition if the three U(1) symmetries comprising Yx are gauged separately. These choices yield Xk'I = X[e], and XI"] = X(LHN) = 0. The GreenSchwarz structure has the added benefit of producing the correct value of the Weinberg angle at cutoff [30][32]:
tan2 . Cy  5 (3.2.45)
Cweak 3
There still remains the nonzero anomaly (YYxYx), which can be canceled by three families of standard model vectorlike representations 5 + 5 of SU(5). With this addition made, the remaining anomaly structure is consistent with the GreenSchwarz cancelation mechanism. We get
~ cot A (CcoIor+18)/3 I 1 , (3.2.46)
mnt mb
and agreement with the data is achieved for Ccoior = 27.
63
We can now specify the form of the matrices involving righthanded neutrinos:
Y(O) ~ MA2X[72 A5 A 4A1 y(') ~ v. ' A4 4 , (3.2.47)
SA I A 4 A5 A 4 1
where Y(O) is the NN Majorana mass matrix and YH"] the matrix coupling Li to Nj. Note that, to appear in the superpotential only as holomorphic quadratic mass terms, the Xcharge of the Ns must be negative half odd integers.
After the seesaw, we have the actual neutrino mass matrix
A 6 A3 A3
Y (") ~ A' J , (3.2.48)
which produces light neutrinos with masses
V A2xN]+12 ; 2 A 2X[N] 6
m M m M (3.2.49)
The mass splitting between v, and the other two neutrinos is Am _ ~ 105eV2, consistent with the nonadiabatic MSW solution to the solar neutrino problem if X[N = 9/2 and M ~ MGUT. To check agreement with the atmospheric neutrino data, we must know the mass splitting between v. and v,, but this can be predicted only with a theory for the prefactors. Interestingly, prefactors of order 1 produce Am,_ ~ 0.07eV2, so that the atmospheric data may be explained by the same solution that accommodates the solar neutrino data without any fine
64
tuning. Moreover, this solution requires M to be of order MGUT as well, and drives the mixing angle to maximal, in agreement with recent experimental results [46], [47].
As mentioned before, it is possible to gauge separately the three symmetries that make up Yx. The analysis proceeds much as that above, but in this case X[d = 3 instead of X[dl = 9. Also, as will be shown in the next chapter, the extra anomaly conditions fix all the charges of the IL term to zero, and the analysis of the vacuum in which all three symmetries are broken at the same scale favors X[N]= 3/2. Remarkably, it is precisely this charge assignment, corresponding to X[N] = 9/2 when the three symmetries are combined into a single gauged symmetry Yx, that leads to a fit of the neutrino data with M  MGUT3.3 Squark Mass Matrices from U(1)'s: FCNC
Every realistic supersymmeric model has to account for the large flavor changing neutral currents (fenc) that supersymmetric particles contribute. We consider two of the existing scenarios for suppression of the fcnc. The first is a supersymmetry breaking mechanism that yields (nearly) degenerate squark masses. This clearly requires detailed knowledge of the agent of supersymmetry breaking which in turn requires a complete model of supersymmetry breaking. Since we do not have in our hands such a model yet, we postpone the discussion of this topic for later.
The second mechanism, does not require detailed knowledge of the supersymmetry breaking mechanism. When the quark and squark matrices are "aligned" the fcnc are suppressed even if supersymmetry breaking does not yield degenerate
65
squark masses. We will now show that alignment is possible in the context of anomalous U(1) models only through supersymmetric zeros in the mass matrices, that as we saw, create dangerous flat directions. This will lead us to the conclusion that degenerate squarks are needed to suppress fcnc.
Let us now show the fact that alignment is possible only with a y(d) that has supersymmetric zeros. The quark matrices are diagonalized by
Y"_ VL"MUV, Y y = VL M RVJ. (3.3.50)
Similarly for the squarks we have mass matrices associated with the soft terms i 2j j:
;i = PV;' I 417u' (3.3.51)
YR  MRURVRU (3.3.52)
 =fM . (3.3.53)
The phenomenological limits on the entries of these matrices (coming primarily from the neutral meson mixing experiments) are:
((VLiff)i2 = A" ~ A, where
mLd min[(jf~ ) 2, (Vf )12, (Vf )13 + (Vdt)13] (3.3.54)
(K)12 = (V' )12 = A'12 A where
12 =m [(Pd ) 12, (V')12, (VL')13 + (V )131 (3.3.55)
2>= \(Kd)12(Ki)1 = A(+m2 5
(K1) =(VLVU) =A"K l ~ where m(L = min[(VL)1 (V2)1, (V= A )13 + ( ')13]
(KA)12 = (Va )_ = A" ' ~ Ac, where M1 = min[(u )12, (V) 12, (V )13 + (4u')13] < Kf2 >= \(Kfl12(K)2 = A2Cmi2~ 2l
(Ku)13 = (VVu')1 = A"12  Ac, where m12  min[(jR )13, (V2 )13, (VR7j)13 + (q? )13]
(Ku)13 = (VVu )13 = A" ~ Ac, where M1 = min[(VQ )13, (VLt)13, (Vaut)13 + (V, t)13] = = A 2 +3 1 ~ 2
13 R c c.
66
(3.3.56)
(3.3.57)
(3.3.58) (3.3.59)
(3.3.60)
(3.3.61)
(3.3.62)
67
If there are no supersymmetric zeros in the mass matrices, we can diagonalize them very easily. We obtain
Vr" = (K. L)
\K.
V u KR (M P)
( 1
M dt PT01
1 I
(K.  L.) K.\
1 L
(M. P) M\ 1P.
P 1
K.  L. K.\
1 La
L. 1 J IM1  PI M1
1 P1
1P01 I
RPS 10 R.  S. R.J
1 1T 1
T 1)
Furthermore, we can compute for example
Rd =
M1= min[E R.  T.,EJR.  T, R, +T=
=(R,,  T') =R,,  ET,, = 1  0 = 1
and
(3.3.63)
(3.3.64) (3.3.65) (3.3.66) (3.3.67)
(3.3.68)
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A similar relation holds for R * L. Clearly the fcnc constraints are not satisfied. From the general form of the constraints it is easy to see that to satisfy them, Y() and Y7) have to be supersymmetric zeros. But if these are zeros, then also Y3s) and Y(d) have to be supersymmetric zeros as well, as the sum rules indicate. This is the minimum number of supersymmetric zeros in the down sector and it is also the maximum since the diagonal elements, Y) and Y can not be zeros if Y(d) should give the desired mass ratios and mixings. We have therefore proved that there is a unique y(d) compatible with the "alignment scenario" of suppressing fcnc. On the other hand, Y(") is fixed except the elements (21), (31) and (32). These are either supersymmmetric zeros or not. We write:
/A8 A5 A\ /A4 0 A \
Y) cement= ? A4 Al and y(d) 0 A2 A . (3.3.69)
? ? 1/ 0 0 1J
Even though we will not present a model that has such s structure, we mention for completeness the conditions in order to generate such matrices in the U(1) scheme. The individual exponents of the 0 fields have to simoultaneously satisfy the following:
0 < K_ 5 3, and Ka 3 (3.3.70)
0
3
2< P <5, and EP =2 (3.3.73)
a
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5
a
6
a
O
Also for every a:
Ra + Ka + N > 0 (3.3.77)
Ta + La + Nd > 0 (3.3.78)
Ma + Ka + N >0 (3.3.79)
P, + L, + Nd 0 (3.3.80)
P, + K + N > 0 (3.3.81)
and at the same time for some a:
Ta+Nd < 0 orR0+Nd < 0 or Ta+Ka+Nd < 0 or Ra+La+Nd < 0. (3.3.82)
3.4 Summary
To summarize, we saw that alignment implies supersymmetric zeros in the mass matrices, which is an unwanted situation since they create serious problems of vacuum instability. We therefore exclude this class of models and from now on we search for models with no (if possible) supersymmetric zeros. This simplifies the tusk in hand but requires an explanation to the problem of supersymmetry breaking and squark degeneracy. The symmetry that we showed that uniquely
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reproduces (assuming no susy zeros and compatibility with neutrino data) the quark mass hierarchies and mixings is:
1 1
Yx = X + I (2Y + V)(2, 1, 1) (V + WV) (1, 0,  1). (3.4.83)
5 2
We have now several choices. We either gauge this U(1) as it is, or we can break it into several pieces in which case, we will assume that we loose no generality in separating the trace part X from the traceless part (which then we can break into other pieces). It is straightforward to check that if we try to break Yx into 2 pieces X and the rest we can not have a AC suppression in the ratio mb/mt. The next simplest choice is to break it into 3 pieces: X, the (2, 1, 1) part and the (1, 0, 1) part. This choice, will be the topic of the rest of this thesis. The motivation for doing this separation is that in the case of a single U(1), the anomaly structure does not imply the existence of a hidden sector, unlike the one where the U(1)'s are separated. This happens because in the 3 U(1) case, there is an anomaly, XYU1)Y(2) not present in the single U(1) case that has to be canceled. The cancelation of this (and only this) anomaly can come about if we attach to the model a hidden sector with a specific gauge and matter structure, as we will explain in a separate section. We therefore now precede and give the detailed analysis of the model with those 3 (additional to the SM) U(1) gauged symmetries.
CHAPTER 4
MODEL BUILDING: A REALISTIC MODEL
4.1 General Structure
In the visible sector, the gauge structure is that of the standard model (= Gsm), augmented by three Abelian symmetries:
SU(3)" x SU(2)w x U(1)y x U(1)x x U(1)y( x U ) . (4.1.1)
One of the extra symmetries, which we call X, is anomalous in the sense of GreenSchwarz; Its charges are assumed to be familyindependent. The other two symmetries, Y(1) and y(2), are not anomalous, but have specific dependence on the three chiral families, designed to reproduce the Yukawa hierarchies. This theory is inspired by models generated from the E8 x E8 heterotic string and its chiral matter lies in brokenup representations of E6, resulting in the cancelation of many anomalies. This also implies the presence of both matter that is vectorlike with respect to standard model charges, and righthanded neutrinos, which trigger neutrino masses through the seesaw mechanism. The three symmetries, X, y(1,2) are spontaneously broken at a high scale by the FayetIliopoulos term generated by the dilaton vacuum. This (DSW) vacuum is required to preserve both supersymmetry and the standard model symmetries. Below its scale, our model displays only the standard model gauge symmetries. To set our notation, and explain our charge
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72
assignments, let us recall some basic E6 [68]. It contains two Abelian symmetries outside of the standard model: The first U(1), which we call V', appears in the embedding
E6 c SO(10) x U(1)v,
(4.1.2)
27 = 16, + 102 + 14 ,
(4.1.3)
where the U(1) value appears as a subscript. The second U(1), called V, appears in
SO(10) C SU(5) x U(l)v,
(4.1.4)
corresponding to
16 = 53 + 101 + 15 10 = 52 + 52
The familiar hypercharge, Y, appears in SU(5) C SU(2) x SU(3) x U(1)y with the representation content
5 = (2, 1r)1 + (1, jC)2/3 10 = (1, 1)2 + (2, 3C)1/3 + (1, Y)4/3
(4.1.5)
(4.1.6)
(4.1.7) (4.1.8)
with
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The two U(1)'s in SO(1O), can also be identified with baryon number minus lepton number and righthanded isospin as
1 1
B  L = (2Y + V) I3R (3Y  V) (4.1.9)
5 10
The first combination is B  L only on the standard model chiral families in the 16; on the vectorlike matter in the 10 of SO(10) it cannot be interpreted as their baryon number minus their lepton number. We postulate the two nonanomalous symmetries to be
Yl) = (2Y + V) 2, 1, 1 (4.1.10)
y(2) = (V 3V') 1, 0, 1 , (4.1.11)
The family matrices run over the three chiral families, so that y(1,2) are familytraceless.
We further assume that the X charges on the three chiral families in the 27 are of the form
X = (a +,3V + yV') (1, 1, 1 ,(4.1.12)
where a, 3, y are undetermined parameters. Since Tr(YY(i)) = Tr(YX) = 0, there is no appreciable kinetic mixing between the hypercharge and the three gauged symmetries. The matter content of this model is the smallest that reproduces the observed quark and charged lepton hierarchy, cancels the anomalies associated with the extra gauge symmetries, and produces a unique vacuum structure:
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" Three chiral families each with the quantum numbers of a 27 of E6. This
means three chiral families of the standard model, Qi, Ui, di, Li, and ei, together with three righthanded neutrinos Ni, three vectorlike pairs denoted by E, + D, and E, + Di, with the quantum numbers of the 5 + 5 of SU(5).
Our model does not contain the singlets that make up the rest of the 27.
With our charges, they are not required by anomaly cancelation, and their
presence would create unwanted flat directions in the vacuum.
" One standardmodel vectorlike pair of Higgs weak doublets.
" Chiral fields that are needed to break the three extra U(1) symmetries in
the DSW vacuum. We denote these fields by 0,. In our minimal model with three symmetries that break through the FI term, we just take a = 0, 1, 2.
The 0 sector is necessarily anomalous.
* Hidden sector gauge interactions and their matter, together with singlet
fields, needed to cancel the remaining anomalies.
4.2 Anomalies
In terms of the standard model, the vanishing anomalies are of the following types:
* The first involve only standardmodel gauge groups GsM, with coefficients
(GsmGsMGsm), which cancel for each chiral family and for vectorlike matter.
Also the hypercharge mixed gravitational anomaly (YTT) vanishes.
" The second type is where the new symmetries appear linearly, of the type
(Y(t)GSMGsM). The choice of familytraceless Y(i) insures their vanishing over the three families of fermions with standardmodel. Hence they must vanish on the Higgs fields: with Gsm = SU(2), it implies the Higgs pair is vectorlike with respect to the Y(). It follows that the mixed gravitational anomalies (Y()TT) are zero over the fields with standard model quantum numbers. They must therefore vanish as well over all other fermions in the
theory.
* The third type involve anomalies of the form (GsMY(1)Y(i)). These vanish
automatically except for those of the form (YY(i)Y(i)). Two types of fermions
contribute: the three chiral families and standardmodel vectorlike pairs
0 = (yy(i)y() = (YY()Y())chiral + (yy()y(j))reai  (4.2.13)
By choosing y(1,2) in E6, overall cancelation is assured, but the vectorlike
matter is necessary to cancel one of the anomaly coefficient, since we have
(YY(1)Y() )chiral = _(yy(1)y(2))reai = 12 . (4.2.14)
" The fourth type are the anomalies of the form (y(i)y(i)y(k)). Since standardmodel singlet fermions can contribute, it is not clear without a full theory, to determine how the cancelations come about. We know that over the fermions in an E6 representation, they vanish, but, as we shall see, the 0 sector is necessarily anomalous. In the following we will present a scenario
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76
for these cancelations, but it is the least motivated sector of the theory since it involves the addition of fields whose only purpose is to cancel anomalies.
* The remaining vanishing anomalies involve the anomalous charge X.
 Since both X and Y are family independent, and Y() are family traceless, the vanishing of the (XYY(1')) coefficients over the three families is assured, so they must vanish over the Higgs pair. This means that X is vectorlike on the Higgs pair. It follows that the standardmodel invariant H.Hd (the p term) has zero X and Y() charges; it can appear by itself in the superpotential, but we are dealing with a string theory, where mass terms do not appear in the superpotential: it can appear only in the Kdhler potential. This results, after supersymmetrybreaking in an induced pterm, of weak strength, as suggested by Giudice and Masiero [69]. Since the Higgs do not contribute to anomaly coefficients, we can compute the standard model anomaly coefficients.
We find
Color = 18a ; Cweak = 18a ; Cy = 30a . (4.2.15)
Applying these to the GreenSchwarz relations we find the KacMoody
levels for the color and weak groups to be the same
kcolor = kweak ,)
(4.2.16)
and through the Ibfiez relation [301, the value of the Weinberg angle
at the cutoff
tan2 6  , (4.2.17)
Cweak 3
not surprisingly the same value as in SU(5) theories.
 The coefficients (XYC1)Y(2)). Since standardmodel singlets can contribute, we expect its cancelation to come about through a combination of hidden sector and singlet fields. Its contribution over the chiral
fermions (including the righthanded neutrinos) is found to be
(XY(1NY(2))chiral + real =l8a . (4.2.18)
 The coefficient (XXY). With our choice for X, it is zero.
 The coefficients (XXY(z)) vanish over the three families of fermions
with standardmodel charges, but contributions are expected from other
sectors of the theory.
The vanishing of these anomaly coefficients is highly nontrivial, and it can be viewed as an alternative motivation for our choices of X, and Y().
4.3 The DSW Vacuum
The X, Y(;) and Y) Abelian symmetries are spontaneously broken below the cutoff. Phenomenological considerations require that neither supersymmetry nor any of the standard model symmetries be broken at that scale. Since three symmetries are to be broken, we assume that three fields, O, acquire a vacuum
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value as a result of the FI term. They are singlets under the standard model symmetries, but not under X and Y(,'2). If more fields than broken symmetries assume nonzero values in the DSW vacuum, we would have undetermined flat directions and hierarchies, and NambuGoldstone bosons associated with the extra symmetries. We express their charges in terms of our 3 x 3 matrix A, whose rows are the X, Yl) and Y) charges of the three 0 fields, respectively. Assuming the existence of a supersymmetric vacuum where only the 0 fields have vacuum values, implies from the vanishing of the three D terms
A 612 0 . (4.3.19)
10212 0
We have found no fundamental principle that fixes the charges of the 0 fields. However, by requiring that they all get the same vacuum value and reproduce the quark hierarchies (according to our assumptions in the introduction), we arrive at the simple assignment
1 0 0
A = o 1 1 . (4.3.20)
1 1 0
so that its inverse
1 0 0
A' 1 0 1 ,(4.3.21)
1 1 1
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is of the desired form. We see that all three 0 fields have the same vacuum expectation value
< 00 > =< 01 > < 02 >= (4.3.22)
The presence of other fields that do not get values in the DSW vacuum severely restricts the form of the superpotential. In particular, when the extra fields are righthanded neutrinos, the uniqueness of the DSW vacuum is attained only after adding to the superpotential terms of the form NTP(O), where p is an integer > 2, and P is a holomorphic polynomial in the 9 fields. If p = 1, its Fterm breaks supersymmetry at the DSW scale. The case p = 2 is more desirable since it translates into a Majorana mass for the righthanded neutrino, while the cases p > 3 leave the N massless in the DSW vacuum. To single out p = 2 we simply choose the X charge of the Ni to be a negative halfodd integer. Since righthanded neutrinos couple to the standard model invariants LjH, it implies that XL,;H is also a halfodd integer. The same analysis can be applied to the invariants of the MSSM. Since they must be present in the superpotential to give quarks and leptons their masses, their Xcharges must be negative integers. Remarkably, these are the very same conditions necessary to avoid flat directions along which these invariants do not vanish: with negative charge, these invariants cannot be the only contributors to DX in the DSW vacuum. The presence of a holomorphic invariant, linear in the MSSM invariant multiplied by a polynomial in the 9 fields, is necessary to avoid a flat direction where both the invariant and the 0 fields would get DSW vacuum values. The full analysis of the DSW vacuum in our model is rather involved, but it is greatly simplified by using the general methods introduced in chapter 2. We will discuss the question of the uniqueness of the vacuum in a later
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section. Finally, we note a curious connection between the DSW vacuum and the anomalies carried by the 9 fields. Assume that the 0 sector does not contribute to the mixed gravitational anomalies
(Y(2)TT)o = 0 . (4.3.23)
This means that the charges Y(i) are traceless over the 9 sector. They are therefore generators of the global SU(3) under which the three 9 fields form the 3 representation. However, SU(3) is anomalous, and it contains only one nonanomalous U(1) that resides in its SU(2) subgroup. Thus to avoid anomalies, the two charges y(1,2) need to be aligned over the 9 fields, but this would imply det A = 0, in contradiction with the necessary condition for the DSW vacuum. It follows that the vacuum structure requires the 9 sector to be anomalous. Indeed we find that, over the 0 fields,
(Y(YY(2) y1y(2)y(2))O = 1 . (4.3.24)
In a later section we discuss how these anomalies might be compensated.
4.4 Quark and Charged Lepton Masses
To account for the top quark mass, we assume that the superpotential contains the invariant
Q3U3H. (
(4.4.25)
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Since X is familyindependent, it follows that the standardmodel invariant operators QiiH., where i, j are family indices, have zero Xcharge. Together with the anomaly conditions, this fixes the Higgs charges
XH. = XHd= XQ  X , (4.4.26)
and
Yl = YHl = , Y =  2 (4.4.27)
X(QjiijHu)  X U 0 . (4.4.28)
The superpotential contains terms of higher dimensions. In the charge 2/3 sector, they are
'Q IM 0, 1) 0(2)
()jHu( ) (M) ' (4.4.29)
in which the exponents must be positive integers or zero. Invariance under the three charges yields
) = 0 , n) yY,) _y() [u]+ Y 2) [ , (4.4.30)
where Y() [u], and y(2) [u are the charges of Q iyHu, respectively. They are determined by our choice for the charges y(1,2). A straightforward computation yields the orders of magnitude in the charge 2/3 Yukawa matrix
A8 A5 3
Y(U) A A 4A , (4.4.31)
A 5 A 2 1
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where as usual A =< 10QI > /M is the expansion parameter. A similar computation is now applied to the charge 1/3 Yukawa standard model invariants QidjHd. The difference is the absence of dimensionthree terms, so that its Xcharge, which we denote by X[d need not vanish. We find that if X[d] > 3, one exponent in the
(33) position is negative, resulting in a supersymmetric zero and spoiling the quark hierarchy. Hence, as long as X[d] < 3, we deduce the charge 1/3 Yukawa matrix
(A4 /3 A3
Y[d] , A3xM6 3 2 2 , (4.4.32)
A 1 1
and diagonalization of the two Yukawa matrices yields the CKM matrix
tICKM A A (4.4.33)
A3 A2 1
This shows the expansion parameter to be of the same order of magnitude as the Cabbibo angle A,. For definiteness in what follows we take them to be equal, although as we show later, the GreenSchwarz evaluation of A gives a slightly higher value. The eigenvalues of these matrices reproduce the geometric interfamily hierarchy for quarks of both charges
~ A ,8 _~ A 4 (4.4.34)
mb mb
Md 4ms
~b c ,b ~ C (4.4.35)
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while the quark intrafamily hierarchy is given by
Mb = cot 3A3x d6 . (4.4.36)
mt
implying the relative suppression of the bottom to top quark masses, without large tan f. These quarksector results are the same as in the previous section, but our present model is different in the lepton sector. The analysis is much the same as for the down quark sector. No dimensionthree term appears and the standard model invariant LiejHd have charges X[el, Yi,2) e]. The pattern of eigenvalues depends on the X[e]: if X[e] > 3, we find a supersymmetric zero in the (33) position, and the wrong hierarchy for lepton masses; if X[el = 3, there are supersymmetric zeros in the (21) and (31) position, yielding
Yle) ~3A 0 A2 AC . (4.4.37)
0 A2
We could have avoided these supersymmetric zeros by relaxing the assumption that the 0 fields break at the same scale. We will not examine this possibility here. Notice also that in the single U(1) case, there is again no zeros at these positions. Diagonalization of the above matrix, yields the lepton interfamily hierarchy
me 4_6 mA ~ A2 (4.4.38)
The issue of the eigenvalues of this matrix is rather saddle. To make a definite statement, one should know the exact coefficients of order of one in the Yukawa
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matrices. We do not discuss this point any further. We refer the reader to [70] for a more detailed analysis. Our choice of X insures that XM = X[eIl, which guarantees through the anomaly conditions the correct value of the Weinberg angle at cutoff, since
sin2 + + XM] = X[e] ; (4.439)
8
it sets Xa = 3, so that
mb 1; ~ cot3A . (4.4.40)
M, Mt
It is a remarkable feature of this type of model that both inter and intrafamily hierarchies are linked not only with one another but with the value of the Weinberg angle as well. In addition, the model predicts a natural suppression of mb/mr, which suggests that tan # is of order one.
4.5 Neutrino Masses
Our model, based on E6, has all the features of SO(10); in particular, neutrino masses are naturally generated by the seesaw mechanism if the three righthanded neutrinos Ni acquire a Majorana mass in the DSW vacuum. The flat direction analysis then indicates that their Xcharges must be negative halfodd integers, that is X7 = 1/2,  3/2,.... Their standardmodel invariant masses are generated by terms of the form
(0) ( 02 (2)
0 P 1 02
MNiNj(M )) (4.5.41)
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where M is the cutoff of the theory. In the (ij) matrix element, the exponents are computed to be equal to 2Xy plus
(0,4,0) (0,2,1) (0,0,1)
(0,2,1) (0,0,2) (0,2,0) , (4.5.42)
(0,0, 1) (0, 2, 0) (0,4,2)
If Xy = 1/2, this matrix has supersymmetric zeros in the (23), (32) and (33) elements. While this does not result in a zero eigenvalue, the absence of these invariants from the superpotential creates flat directions along which (N3) 0 0; such flat directions are dangerous because they can lead to vacua other than the DSW vacuum. If XV <; 5/2, none of the entries of the Majorana mass matrix vanishes; but then the vacuum analysis indicates that flat directions are allowed which involve MSSM fields. For those reasons, we choose XV = 3/2, which still yields one harmless supersymmetric zero in the Majorana mass matrix, now of the form
MAC A5 A (4.5.43)
Ac 1 0
Its diagonalization yields three massive righthanded neutrinos with masses
my,~ ACs myv, ~ y MAC (4.5.44)
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By definition, righthanded neutrinos are those that couple to the standardmodel invariant LiH,, and serve as Dirac partners to the chiral neutrinos. In our model,
X(LiH.7N,) = X  = 0 . (4.5.45)
The superpotential contains the terms
(0) (1) (2)
00 01 q' 2
LiHN () (1 ) ( (4.5.46)
resulting, after electroweak symmetry breaking, in the orders of magnitude (we note v, =(H0))
VA 4 1 (4.5.47)
\C C
for the neutrino Dirac mass matrix. The actual neutrino mass matrix is generated by the seesaw mechanism. A careful calculation yields the orders of magnitude
v2
MA4 A 1 . (4.5.48)
A characteristic of the seesaw mechanism is that the charges of the Ni do not enter in the determination of these orders of magnitude as long as there are no massless righthanded neutrinos. Hence the structure of the neutrino mass matrix depends only on the charges of the invariants LiHu, already fixed by phenomenology and
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anomaly cancelation. In the few models with two nonanomalous horizontal symmetries based on E6 that reproduce the observed quark and charged lepton masses and mixings, the neutrino mass spectrum exhibits the same hierarchical structure: the matrix (4.5.48) is a very stable prediction of our model. Its diagonalization yields the neutrino mixing matrix
UMNS A 1 1 , (45.49)
CA' 1 1)
so that the mixing of the electron neutrino is small, of the order of AC, while the mixing between the M and r neutrinos is of order one. Remarkably enough, this mixing pattern is precisely the one suggested by the nonadiabatic MSW [52], [53] explanation of the solar neutrino deficit and by the oscillation interpretation of the reported anomaly in atmospheric neutrino fluxes (which has been recently confirmed by the SuperKamiokande [46] and Soudan [47] collaborations). A naive order of magnitude diagonalization gives a [y and T neutrinos of comparable masses, and a much lighter electron neutrino:
U2
me ~ m 6 A ; m, , ~ mo ; mV . (4.5.50)
The overall neutrino mass scale mo depends on the cutoff M. Thus the neutrino sector allows us, in principle, to measure it. At first sight, this spectrum is not compatible with a simultaneous explanation of the solar and atmospheric neutrino problems, which requires a hierarchy between m, and mw,. However, the estimates
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(4.5.50) are too crude: since the (2,2), (2,3) and (3,3) entries of the mass matrix all have the same order of magnitude, the prefactors that multiply the powers of A, in (4.5.48) can spoil the naive determination of the mass eigenvalues. In order to take this effect into account, we rewrite the neutrino mass matrix, expressed in the basis of charged lepton mass eigenstates, as:
aA' bA3 cA3
mO bA3 d e , (4.5.51)
cA' e f
where the prefactors a, b, c, d, e and f, unconstrained by any symmetry, are assumed to be of order one, say 0.5 < a,... f < 2. Depending on their values, the two heaviest neutrinos may be either approximately degenerate (scenario 1) or well separated in mass (scenario 2). It will prove convenient in the following discussion to express their mass ratio and mixing angle in terms of the two parameters x = dfe2 andy = dL
d+f) d+f'
m 1 14x y2i22~~1_2 = ;x sin2 20,, = 1  . (4.5.52)
M13 1 + 14x 1  4x
Scenario 1 corresponds to both regimes 4x 1 and (4x) > 1, while scenario 2 requires Ijx < 1. Let us stress that small values of x are very generic when d and f have same sign, provided that df ~ e2. Since this condition is very often satisfied by arbitrary numbers of order one, a mass hierarchy is not less natural, given the structure (4.5.48), than an approximate degeneracy. We examine scenario 2 only in detail, since it will turn out to be the most promising from the phenomenological
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point of view. We have in this limit, m,, < m,,. The two distinct oscillation frequencies A n2 and Am3 ~ Am23 can explain both the solar and atmospheric neutrino data: nonadiabatic MSW v, + v,,, transitions require [71]
4 x 106 eV2 < Am2 < 10' eV2 (best fit: 5 x 106 eV2) , (4.5.53)
while an oscillation solution to the atmospheric neutrino anomaly requires [57]
5 x 104 eV2 < Am2 < 5 x 103 eV2 (best fit: 103 eV2) . (4.5.54)
To accommodate both, we need 0.03 < 2  x < 0.15 (with x = 0.06 for the best
fits), which can be achieved without any finetuning in our model. Interestingly enough, such small values of x generically push sin2 20, towards its maximum, as can be seen from (4.5.52). Indeed, since d and f have the same sign and are both of order one, y2 is naturally small compared with (1 4x). This is certainly a welcome feature, since the best fit to the atmospheric neutrino data is obtained precisely for sin2 20 = 1. To be more quantitative, let us fix x and try to adjust y to make sin2 20., as close to 1 as possible. With x = 0.06, one obtains sin2 20,, = 0.9 for y ~ 0.3, sin2 2AT = 0.95 for y ~ 0.2 and sin2 20,r = 0.98 for y ~ 0.1. This shows that very large values of sin2 20, can be obtained without any finetuning (note that y = 1/3 already for d/f = 2). Thus, in the regime x < 1, v, ++ v, oscillations provide a natural explanation for the observed atmospheric neutrino anomaly. As for the solar neutrino deficit, it can be accounted for by MSW transitions from the electron neutrinos to both M and r neutrinos, with parameters Am2 = Am12 and sin2 20 = 4 u2A. To match the mixing angle with experimental data, one
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needs u ~ 3  5; we note that such moderate values of u are favored by the fact that df ~ e2. In both scenarios, the scale of the neutrino masses measures the cutoff M. In scenario 1, the MSW effect requires mo  103 eV, which gives M ~ 1018 GeV. In scenario 2, the best fit to the atmospheric neutrino data gives mo (d + f) = m, + m, ~ 0.03 eV, which corresponds to a slightly lower cutoff, 1016 GeV < M < 4 x 10"7 GeV (assuming 0.2 < d + f K 5). It is remarkable that those values are so close to the unification scale obtained by running the standard model gauge couplings. This result depends of course on our choice for XY, since
mo = A (+x) (4.5.55)
=M
but the value XT = 3/2 is precisely that favored by the flat direction analysis. As a comparison, X = 1/2 would give M ~ 1022 GeV, and Xy < 5/2 corresponds to M < 10" GeV. Turning the argument the other way, had we set M = MU ab initio, the value of Xff favored by the flat direction analysis would yield precisely the neutrino mass scale needed to explain the solar neutrino deficit, mo ~ 103 eV. Other values of XV would give mass scales irrelevant to the data: XV = 1/2 corresponds to no 107 eV, which is not interesting for neutrino phenomenology, and Xy7 5/2 to mo > 10 eV, which, given the large mixing between p and r neutrinos (and assuming no finetuned degeneracy between them), is excluded by oscillation experiments. To conclude, our model can explain both the solar neutrino deficit and the atmospheric neutrino anomaly, depending on the values of the orderone factors that appear in the neutrino mass matrices. The cutoff M, which is related to the neutrino mass scale, is determined to be close to the
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unification scale. Finally, the model predicts neither a neutrino mass in the few eV range, which could account for the hot component of the dark matter needed to understand structure formation, nor the LSND result [72]. The upcoming flood of experimental data on neutrinos will severely test the model.
4.6 VectorLike Matter
To cancel anomalies involving hypercharge, vectorlike matter with standardmodel charges must be present. Its nature is not fixed by phenomenology, but by a variety of theoretical requirements: vectorlike matter must not affect the unification of gauge couplings, must cancel anomalies, must yield the value of the Cabbibo angle, must not create unwanted flat directions in the MSW vacuum, and of course must be sufficiently massive to have avoided detection. As we shall see below, our E6inspired model, with vectorlike matter in 5  5 combinations, comes close to satisfying these requirements, except that it produces a high value for the expansion parameter. The masses of the three families of standard model vectorlike matter are determined through the same procedure, namely operators of the form
(t 1 2 (0) ( 1) (2)
23 t
 00 s 01 s 02 )s  0 0 02 )'
MDD, (M M M) + MEjE,() M . (4.6.56)
The Xcharges of the standard model invariant mass terms are the same
X(DiDj) = X(EiEj) = 2a  4y = nVL. (4.6.57)
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Its value determines the Xcharge, since X[d] = 3 and X7 = 3/2 already fix 0 = 3/20 and a + 7 = 3/4. It also fixes the orders of magnitude of the vectorlike masses.
First we note that nVL must be a nonnegative integer. The reason is that the power of 01 is nVL, the Xcharge of the invariant and by holomorphy, it must be zero or a positive integer. Thus if nVL is negative, all vectorlike matter is massless, which is not acceptable. The exponents for the heavy quark matrix are given by the integer nVL Plus
(0,)3,73) (0, l,3) (0, 1, l)
(0, 2, 0) (0, 0, 0) (0, 2, 2) DiD. (4.6.58)
(0, 1, 1) (0,1,1) (0,3,3)
Those of the heavy leptons, by nVL plus
(0, 3, 3) (0, 2, 2) (0, 1, 1)
(0,1,1) (0,0,0) (0,1,1) : EE. (4.6.59)
(0,1,1) (0,2,2) (0,3,3)
Since these particles carry standard model quantum numbers, they can affect gauge coupling unification. As these states fall into complete SU(5) representations, the gauge couplings unify at one loop like in the MSSM, provided that the mass splitting between the doublet and the triplet is not too large.
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e nVL = 0. We obtain the mass matrices
0
1
/0
M_5D = M 0
0 Cl A 6J
/0
,MEE = M 0
AAl
EEM(0
0
1 A4
0 A 2
(4.6.60)
Diagonalization of these matrices yields one zero eigenvalue for both matrices and nonzero (order of magnitude) eigenvalues M and A'M for MTD and M and A2M for M E. The pair of zero eigenvalues is clearly undesirable and furthermore the mass splitting between the second family E and D destroys gauge coupling unification. This excludes nVL 00 nVL = 1. The mass matrices are
/ 0
MD D = M 0
0 A 3 A5
A 3 A 7
C
A C
/E0
MEE = M Ac,
0
C
A, A C A9J
(46.61)
The eigenvalues for MDD are A3M, AIM and A3M and for MTE ACM, ACM and A'M. The splitting between the members of the third family vectorlike fields is too large and as a consequence, gauge coupling unification is spoiled.
* nVL = 2. The mass matrices are:
0
M_5D = M AC
0
A6
C
A\
C
A10C
AJ1
0
,MEE = M A
A 2
C
A]
C
A12
C
(4.6.62)
The eigenvalues are now A'M, A6M, A'M and A2M, A4M, A 2M, respectively. There is again splitting between the families of the doublet and the triplet and
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therefore the gauge couplings do not unify at one loop. The splitting in this case is not too big and a two loop analysis may actually prove this case viable from the gauge coupling unification point of view.
* nVL 3. We obtain the mass matrices
A3A5 A9 A A 5 A7
MD7 A9 A3 , M = M (4.6.63)
: : EEM( C C?{i
A Al A 15Al A13A5
with eigenvalues:
MD {A3M, A9M, A15M} (4.6.64)
and
ME = {A3M, A9M, A 5M} , (4.6.65)
respectively. The unification of couplings in this case is preserved. For nVL : 3, there are no supersymmetric zeros in the mass matrices and the mass eigenvalues are just the diagonal entries, so there is no splitting between masses of the same family of D and E. A simple oneloop analysis using selfconsistently M = MU in the mass of the vectorlike particles and for the unification scale, yields unified gauge couplings at the unification scale, MU
1
a(Mu) ~ ; Mu ~~ 3 x 1O'6GeV . (4.6.66)
For nVL large, other problems arise as the vectorlike matter becomes too light. This can easily spoil gauge coupling unification by two loop effects [73] and cause
