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Anomalous U(1) gauge symmetry in superstring inspired low energy effective theories

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Anomalous U(1) gauge symmetry in superstring inspired low energy effective theories
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Irges, Nikolaos, 1965-
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vi, 149 leaves : ; 29 cm.

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Integers ( jstor )
Leptons ( jstor )
Matrices ( jstor )
Modeling ( jstor )
Neutrino masses ( jstor )
Neutrinos ( jstor )
Protons ( jstor )
Quarks ( jstor )
Supersymmetry breaking ( jstor )
Symmetry ( jstor )
Dissertations, Academic -- Physics -- UF ( lcsh )
Physics thesis, Ph. D ( lcsh )
Standard model (Nuclear physics) ( lcsh )
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Thesis:
Thesis (Ph. D.)--University of Florida, 1999.
Bibliography:
Includes bibliographical references (leaves 145-148).
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Nikolaos Irges.

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Full Text







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 Green-Schwarz Mechanism . . . . . . . . . . . . . . . . . . . . 9
1.3 The Anomalous U(1) and Yukawa Matrices . . . . . . . . . . . . . . 11

2 THE VACUUM . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.1 D-Flatness 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 Low-Energy 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 Inter-family 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 Vector-Like Matter . . . . . . . . . . .
4.6.1 Shift X. . . . . . . . . . . . . .
4.6.2 Discrete Symmetry . . . . . . .
4.6.3 Summary . . . . . . . . . . . .
4.7 The Hidden Sector . . . . . . . . . . .
4.8 R-Parity . . . . . . . . . . . . . . . . .
4.9 Proton Decay . . . . . . . . . . . . . .
4.10 Flat Direction Analysis . . . . . . . . .
4.10.1 Flat Directions with Vector-Like
4.11 Supersymmetry Breaking . . . . . . . .
4.11.1 Supersymmetry Breaking with U 4.11.2 Soft Parameters . . . . . . . . .

5 CONCLUSION . . . . . . . . . . . .


REFERENCES.............

BIOGRAPHICAL SKETCH . . . .


iv


Matter
.M.. s .


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. . .
. . .
. . .
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. . .
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. . .
. . .


. . . . . . . 145

. . . . . . . 149


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) space-time 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 non-Abelian 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 Green-Schwarz mechanism [27]. In these models the dilaton gets a vacuum expectation value, generating a non-zero Fayet-Iliopoulos 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, bottom-up, 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 Green-Schwarz 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 "superstring-inspired" 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 small-angle MSW effect

for the solar neutrino deficit, and the large angle mixing necessary for the

atmospheric neutrino effect.

" Natural R-parity 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 non-adiabatic 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 "bottom-up" 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 2-form B., and the dilaton S. In four dimensions, the antisymmetric two form appears through the field strength


H[p-p] =,,B-p + ... = 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 Kac-Moody levels. These are integers for non-Abelian 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 non-anomalous, 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 Yang-Mills 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 Fayet-Iliopoulos Dterm ( 2), which breaks supersymmetry. To restore supersymmetry, which should not break at such a high scale, a compensating term in the D-term of the anomalous U(1) appears. The corrected form of the anomalous D-term is then


Dx = -g2[D0 - 2].


(1.1.15)








9


1.2 The Green-Schwarz 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 Kac-Moody levels of the gauge factors. The anomalies of the Xsymmetry are compensated at the cut-off, as long as the ratio Ck/kk is universal. This is the four-dimensional equivalent of the Green-Schwarz 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 Green-Schwarz mechanism is that the Weinberg angle at cut-off 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 X-charges 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 Kac-Moody levels of the non-Abelian 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 non-zero 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 cut-off [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 supersymmetry-preserving vacuum is determined by the vanishing of the








13


N + 1 D-terms

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 family-dependent 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 D-term 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 D-Flatness 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 D-term Dx is of the form:


Dx = -qiloil2 _ (2.1.1)


where qi is the X-charge 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 right-handed neutrinos and we will denote them by N. Typically, for these fields to be interpreted as right-handed 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 X-charges 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 Dx-flatness (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. D-flatness 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 D-term 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 X-charge 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 X-charge of 6, x > 0 and that of N, xV < 0. With these fields the X D-term is:


(2.1.6)


Dx = X|0|2 + 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 F-terms forbid < 0 > and < 0, N >. Thus we lose the possibility of a Dx-flat 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 right-handed neutrino N, once 9 is allowed a vev. In this case, n = 2k + 1 and there must be the following relation between the X-charges:


XY 2k+1 (2.1.7)
x 2

Consider now another simple model, with 9 (X-charge x > 0) and I (X-charge x, < 0). With one field and one invariant, we have


Dx = x1912 + xI|vI12 _ 2 (2.1.8)


As previously, Dx-flatness kills the direction < I > and allows < 0 >. The main difference is that I being a composite field, I = 11_n 4 n, the F-terms corresponding to the invariant It9U are F = tnj(Hi , # 0) I"-10" and F = uIt6u-1; 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 + xI|vI12 _ 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 R-Parity

We now show that the constraints discussed above on the X quantum numbers of the low-energy fields may naturally lead to conserved R-parity. 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 i-vacuum, the powers no must be odd integers, or equivalently the X-charges X7, of the fields Ni must be, in units of x, half-odd 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 R-parity of the right-handed neutrino superfields to be negative. Let us study the X-charges 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 X-charges 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 R-parity. 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 R-parity 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 half-odd integers and there is no R-parity 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 Green-Schwarz 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 half-odd 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 non-integer 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 Green-Schwarz 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 family-independent symmetry). If XHI = 0 or more generally if XI"I is proportional to Nf (X[] = Nfz,), the charge in (2.1.21) is half-odd integer; it can only be compensated by odd powers of N: invariance under X means conservation of R-parity. For instance, the above allows the interaction:


iidN [nd+nu+zo(Nf-1)] (2.1.24)


This term allows baryon number violation, but preserves both B - L and R-parity. 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 R-parity violating operators, whatever their dimensions: through the righthanded neutrinos for example, R-parity is linked to half-odd integer charges, so that X charge invariance results in R-parity invariance. Thus none of the operators that violate R-parity can appear in holomorphic invariants: even after breaking of the anomalous X symmetry, the remaining interactions all respect R-parity, 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 non-anomalous 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 g-singlets by Xi. As stressed in the previous section, some of these fields must acquire non-vanishing vacuum expectation values through the Dine-Seiberg-Witten (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 F-term equations that breaks only the Abelian factors. The D-term equations:


Dx = . aXa|0)|2 _ 2 =- 0

(2.2.25)


Dy) = Z0 y 9)(O,)12










28


where the g-singlets with non-vanishing vevs are denoted by 0,, have in general several solutions, due to the large number of g-singlets 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 low-energy mass hierarchies (in particular fermion masses), which are

generated by the small parameters (90)/M, are completely determined by the high-energy 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 G-singlets (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 F-terms. 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 G-singlet 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 right-handed neutrino, this constraint leads to an automatic conservation of R-parity 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 non-renormalizable 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 D-flat 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 D-flat 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, g-invariants 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 g-invariant I = O... 0 (resp. g-singlet 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 non-anomalous 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 D-Flatness



Before characterizing the D-flat 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 D-terms can be classified in terms of the holomorphic gauge invariants. More precisely, a set of vevs (1, . . . , 0n) is a solution of the D-term constraints if and only [61] if there exists a G-invariant 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 D-flat 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 G-invariant I = ... /P', (resp. G-singlet x),








32


corresponds a D-flat 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 D-term constraints associated with I (resp. x). Note that this is not, in general, a flat direction of the scalar potential, because the F-term constraints F = 0 are not necessarily satisfied. Let us prove this first in the case of a g-singlet 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 D-term 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 D-flat 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 non-anomalous 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 F-term 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 g-invariant is straightforward. Applying (2.2.29) to I = 1. . ..0P, we find that the D-terms 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 D-terms:


(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 D-flat direction by (I, W) to stress the fact that the vevs of the fields in I are aligned. However, generic D-flat directions are not associated with a single basis g-invariant, but rather with a polynomial in the basis g-invariants. More precisely, flat directions involving a given set {0} of fields










34


charged under g are parameterized by the vevs of the g-invariants 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 D-term 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 F-term 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 F-Flatness



In this section, we examine under which conditions a D-flat direction is lifted by F-terms. We first assume no compensation between different contributions to the F-terms, so that each individual contribution has to vanish for a D-flat direction to be preserved. We shall come back to this point later.

We first restrict our attention to D-flat directions (I, W) that are associated with a single g-invariant 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 D-flat direction can be associated with some

holomorphic invariant Im 0'01 . .. 9N of the whole gauge group. This

invariant contributes to the F-terms 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 D-flat 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 non-holomorphic, e.g. Q~if1H, 0-101 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 F-terms that are likely to be non-vanishing along the D-flat 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 F-terms vanish only for v, = 0, and the D-flat direction


(2.2.41)


(Q1, i, Hu, 0, 1)








37


breaks down to the DSW vacuum (00, 01)Dsw. If no > 0, the F-term 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 D-flat 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 D-flat directions involving only g-singlets. 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 D-term equations do not allow for any other vacuum of singlets than the vacuum (2.2.27). The other solutions of (2.2.25) are D-flat 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 F-terms, 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 D-term equations








38


allow for vacua of singlets in which (0,) = 0, while some of the x fields have non-vanishing 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 g-charged fields as well as g-singlets. The relevant numbers here are {n a; n }, where the {n } are associated with the basis g-invariants {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 g-invariants 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 F-terms. The effect of such cancelations is to reduce the dimensionality of a D-flat 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 non-trivial and independent from each other. When higher order operators are added in the superpotential, the number of independent F-term 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 g-singlets is more subtle, and needs a separate discussion. Due to condition (2.2.26), the F-terms of the 6 fields are not independent from the other F-terms:


( 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 g-singlets are constrained by exactly as many equations as fields, and those (non-linear) 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 F-terms. 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 Fayet-Iliopoulos 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 D-term and F-term 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 Low-Energy 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 low-energy phenomenology, it is indeed essential to ensure that the flat direction analysis is relevant to the determination of the low-energy vacuum. The scalar potential of the low-energy 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. #"/Mpni-2 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 D-terms and F-terms 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 F-terms, which means that all contributions to the F-terms 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 F-terms. As a result, the low-energy 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 F-terms, 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 non-holomorphic invariant Qli1Hu 0-10'. 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 g-invariants involving only # fields;

add to this basis all g-singlets Xi, ..., Xq

2. for each element of the basis, compute the set of numbers (no, ..., nN)

defined by (2.2.28). The corresponding D-flat direction is (I; 0) in the case of a 9-invariant, and (X ; ) in the case of a g-singlet; 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 D-flat 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 F-terms. Two cases must be

distinguished: - flat directions for which all relevant {n,; n'} are positive are lifted down to the DSW vacuum by F-terms. - 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 F-terms, 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 see-saw mechanism via right handed neutrinos and if there are no supersymmetric zeros it is guaranteed that all the "non-DSW" 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 vector-like 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 F-term 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 F-terms 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 = -A-1Y1, (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 non-anomalous U(1) (a =1, ..., N):








50


Here, we have already used the assumption that the non-anomalous 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 1-q -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 (non-anomalous) U(1) charges (inverting 3.1.6-3.1.8):

(,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, vector-like, 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 inter-family 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 Inter-family 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 inter-family 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 inter-family hierarchy


m 4-6 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 left-handed 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 right-handed 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 right-handed neutrinos to the standard model, on the other hand, is written Y') = H L-N , where HLj is a vector independent of the right-handed 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 non-adiabatic 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 tree-level top quark mass, achieving Y and Y of the same order requires that the YF charge of the p-term, HUHd, be Y1] = --6. But avoiding anomalies such as Tr[YYYF] and Tr[SU(2)SU(2)YF forces the YF charge to be vector-like on the Higgs doublets, so that YAj' = 0. Hence Yb - Y requires YF to be anomalous (The Green-Schwarz mechanism cannot be invoked since Tr[SU(3)SU(3)YF] = 0). Furthermore, we shall soon see that a family-traceless YF cannot reproduce neutrino phenomenology.








60


To proceed, we need to assign YF charges to the right-handed 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 family-traceless, non-anomalous YF symmetry must be extended by adding a family-independent 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 family-independent piece X. The total flavor symmetry is now



x = X + YF (3.2.38)








61


To consider the implications of anomalies involving our family-independent 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, Li-eHd, 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 Green-Schwarz anomaly cancelation mechanism. We take the family-independent 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 vector-like, 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 Green-Schwarz 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 non-zero anomaly (YYxYx), which can be canceled by three families of standard model vector-like representations 5 + 5 of SU(5). With this addition made, the remaining anomaly structure is consistent with the Green-Schwarz 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 right-handed neutrinos:




Y(O) ~ MA-2X[7-2 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 X-charge 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 _ ~ 10-5eV2, consistent with the non-adiabatic 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
R|PS| 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)








68


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










69


-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








70


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 family-independent. 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 broken-up representations of E6, resulting in the cancelation of many anomalies. This also implies the presence of both matter that is vector-like with respect to standard model charges, and right-handed 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 Fayet-Iliopoulos 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


71








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, + 10-2 + 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 = 5-3 + 101 + 15 10 = 52 + 5-2


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








73


The two U(1)'s in SO(1O), can also be identified with baryon number minus lepton number and right-handed 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 vector-like matter in the 10 of SO(10) it cannot be interpreted as their baryon number minus their lepton number. We postulate the two non-anomalous 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:








74


" 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 right-handed neutrinos Ni, three vector-like 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 standard-model vector-like 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 standard-model gauge groups GsM, with coefficients

(GsmGsMGsm), which cancel for each chiral family and for vector-like 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 family-traceless Y(i) insures their vanishing over the three families of fermions with standard-model. Hence they must vanish on the Higgs fields: with Gsm = SU(2), it implies the Higgs pair is vector-like 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 standard-model vector-like 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 vector-like

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


75








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 vector-like on the Higgs pair. It follows that the standard-model 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 p-term, 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 Green-Schwarz relations we find the Kac-Moody

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 cut-off

tan2 6 - , (4.2.17)
Cweak 3

not surprisingly the same value as in SU(5) theories.

- The coefficients (XYC1)Y(2)). Since standard-model 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 right-handed 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 standard-model charges, but contributions are expected from other

sectors of the theory.

The vanishing of these anomaly coefficients is highly non-trivial, 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 cut-off. 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








78


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 non-zero values in the DSW vacuum, we would have undetermined flat directions and hierarchies, and Nambu-Goldstone 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 61|2 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









79


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 right-handed 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 F-term breaks supersymmetry at the DSW scale. The case p = 2 is more desirable since it translates into a Majorana mass for the right-handed 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 half-odd integer. Since right-handed neutrinos couple to the standard model invariants LjH, it implies that XL,;H is also a half-odd 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 X-charges 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








80


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 non-anomalous 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)









81


Since X is family-independent, it follows that the standard-model invariant operators QiiH., where i, j are family indices, have zero X-charge. 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








82


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 dimension-three terms, so that its X-charge, 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] , A3xM-6 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 Green-Schwarz 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)









83


while the quark intrafamily hierarchy is given by


Mb = cot 3A-3x d6 . (4.4.36)
mt

implying the relative suppression of the bottom to top quark masses, without large tan f. These quark-sector 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 dimension-three 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 inter-family 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









84


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 cut-off, 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 intra-family 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 right-handed neutrinos Ni acquire a Majorana mass in the DSW vacuum. The flat direction analysis then indicates that their X-charges must be negative half-odd integers, that is X7 = -1/2, - 3/2,.... Their standard-model invariant masses are generated by terms of the form


(-0) ( 02 (2)
0 P 1 02
MNiNj(M )) (4.5.41)








85


where M is the cut-off 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 right-handed neutrinos with masses


my,~ ACs myv, ~- y MAC (4.5.44)








86


By definition, right-handed neutrinos are those that couple to the standard-model 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
LiH-N () (-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 right-handed neutrinos. Hence the structure of the neutrino mass matrix depends only on the charges of the invariants LiHu, already fixed by phenomenology and








87


anomaly cancelation. In the few models with two non-anomalous 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




U-MNS A 1 1 , (4-5.49)

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 non-adiabatic 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 Super-Kamiokande [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 cut-off 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









88


(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 = df-e2 andy = d-L
d+f) d+f'

m 1-- 1-4x y2i22~~1-_2 = ;x sin2 20,, = 1 - . (4.5.52)
M13 1 + 1-4x 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








89


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: non-adiabatic MSW v, -+ v,,, transitions require [71]


4 x 10-6 eV2 < Am2 < 10-' eV2 (best fit: 5 x 10-6 eV2) , (4.5.53)


while an oscillation solution to the atmospheric neutrino anomaly requires [57]


5 x 104 eV2 < Am2 < 5 x 10-3 eV2 (best fit: 10-3 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 fine-tuning 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 fine-tuning (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








90


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 cut-off M. In scenario 1, the MSW effect requires mo - 10-3 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 cut-off, 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 X-ff favored by the flat direction analysis would yield precisely the neutrino mass scale needed to explain the solar neutrino deficit, mo ~ 10-3 eV. Other values of XV would give mass scales irrelevant to the data: XV = -1/2 corresponds to no 10-7 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 fine-tuned 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 order-one factors that appear in the neutrino mass matrices. The cut-off M, which is related to the neutrino mass scale, is determined to be close to the








91


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 Vector-Like Matter

To cancel anomalies involving hypercharge, vector-like matter with standardmodel charges must be present. Its nature is not fixed by phenomenology, but by a variety of theoretical requirements: vector-like 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 E6-inspired model, with vector-like 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 vector-like 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 X-charges of the standard model invariant mass terms are the same


X(DiDj) = X(EiEj) = 2a - 4y = -nVL. (4.6.57)








92


Its value determines the X-charge, 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 vector-like masses.

First we note that nVL must be a non-negative integer. The reason is that the power of 01 is nVL, the X-charge of the invariant and by holomorphy, it must be zero or a positive integer. Thus if nVL is negative, all vector-like matter is massless, which is not acceptable. The exponents for the heavy quark matrix are given by the integer nVL Plus


(0,)-3,7-3) (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.








93


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


(4-6.61)


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 vector-like 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








94


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 one-loop analysis using self-consistently M = MU in the mass of the vector-like 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 vector-like matter becomes too light. This can easily spoil gauge coupling unification by two loop effects [73] and cause




Full Text

PAGE 1

ANOMALOUS f/(l) 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

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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

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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 Green-Schwarz Mechanism 9 1.3 The Anomalous U{1) and Yukawa Matrices 11 2 THE VACUUM 18 2.1 D-Flatness and Holomorphy 18 2.1.1 Models with 2 Fields 20 2.1.2 Models with 3 Fields 22 2.1.3 /^-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 Summary 44 2.2.5 The Ideal Vacuum 45 3 MASS MATRICES 48 3.1 Quark Mass Matrices from L''(l)'s: Hierarchies 48 3.2 A Model with a Single L''(l) Family Symmetry 55 3.2.1 Inter-family Hierarchy 55 3.2.2 Intrafamily Hierarchy 59 3.3 Squark Mass Matrices from f/(l)'s: FCNC 64 3.4 Summary 69 4 MODEL BUILDING: A REALISTIC MODEL 71 4.1 General Structure 71 4.2 Anomalies 74 iii

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4.3 The DSW Vacuum 77 4.4 Quark and Charged Lepton Masses 80 4.5 Neutrino Masses 84 4.6 Vector-Like Matter 91 4.6.1 Shift X 97 4.6.2 Discrete Symmetry 101 4.6.3 Summary 102 4.7 The Hidden Sector 103 4.8 /^-Parity 107 4.9 Proton Decay 109 4.10 Flat Direction Analysis 114 4.10.1 Flat Directions with Vector-Like Matter 119 4.11 Supersymmetry Breaking 121 4.11.1 Supersymmetry Breaking with C/(l)'s 126 4.11.2 Soft Parameters 136 5 CONCLUSION 143 REFERENCES 145 BIOGRAPHICAL SKETCH 149 iv

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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

PAGE 6

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

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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 = 0, SU{3Y x SU{2)w x U{1)y gauge theory, where A'' 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], [3], 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 ~ 10^^ GeV in 10 (11) space-time dimensions and the real world, at ~ 10^ 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

PAGE 8

2 space (space of possible vacua), by imposing phenomenological constraints [15], [16], [17], [18], [19], [20], [21], [22], [23], [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 non-Abelian gauge group that contains the standard model. In many cases, in addition, one of these U{l)s is anomalous with its anomalies canceled by what is known as the Green-Schwarz mechanism [27]. In these models the dilaton gets a vacuum expectation value, generating a non-zero Fayet-Iliopoulos 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, bottom-up, 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 C/(l). 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 Green-Schwarz 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

PAGE 9

3 analysis. In other words, a consistent "superstring-inspired" 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 small-angle MSW effect for the solar neutrino deficit, and the large angle mixing necessary for the atmospheric neutrino effect. • Natural i?-parity conservation. • Proton decay into + 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 non-adiabatic MSW eflfect [52], [53], [54], [55], [56] and the atmospheric neutrino anomaly [57] and a

PAGE 10

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{1) 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 "bottom-up" 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 2-form and the dilaton S. In four dimensions, the antisymmetric two form appears through the field strength -^^[/ji/p] = duB^p + ... = e^^pad^a + ... (l-l-l) which is clearly invariant under the shift a a + rj. The field a lies in a supermultiplet, called the dilaton supermultiplet. From the effective action we will see that

PAGE 11

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 eff'ective field theory: =yo = I (1.1.3) Here, gstr{Mstr) is the coupling at the scale MstrWe will denote gstr{Mstr) = 9 (1-1-4) and assume from now on for simplicity that M,tr = 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 ^eff = ^h f d^e [S W" W] + c.c (1.1.6) k with kk being the Kac-Moody levels. These are integers for non-Abelian groups and rational numbers for Abelian factors depending on the normalization of the U{1). The summation over A; is over all gauge groups. The gauge supermultiplet

PAGE 12

6 is W'' = X'' + ... (1.1.7) with A*^ being the gaugino. The bosonic part of the gauge kinetic term in component form is given by W''-W'' = F'^^F^"'' + F''^,F^"'K (1.1.8) The effective Lagrangian in component form for the gauge kinetic term then becomes ^e// = E ^FlF'"' + « E hF'^uF'''". (1.1.9) 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 F(i) X X ... X Ghid X Z)W (1.1.10) Here, G is a nonabelian gauge group that contains SU {ZY x SU (2)w x t/(l)y of the standard model. X and F^"^ are (anomalous and non-anomalous, respectively) Abelian gauge factors present at low energies. Ghid is the gauge group of the hidden sector (we define Q = Gx Ghid) and Z)''^ 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.

PAGE 13

7 • Specify the number of low energy supersymmetries. In the following we will be interested only in N = 1 supersymmetric models. • Specify a Kahler potential. This will be important when we talk about supersymmetry breaking. In general, we will assume minimal Kahler 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 fmnGiven the above, we can write a unique low energy effective action; it is just = 1 supergravity (which is fixed by Kahler potential K, the superpotential W and the gauge kinetic functions fmn), coupled to A'^ = 1 super Yang-Mills 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 where is a polynomial consisting of the superfields invariant under Q. The superscript refers to the type of the field and the subscript is a family index. Xi ^^re Q singlet fields and /(M) = ^, (1.1.12) 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

PAGE 14

8 superfields (if it belongs to the Kahler 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 i7(l)'s since we know that these are not present at low energy. A gauged U (1) factor will be called anomalous if rr[X] ^ 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: 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 cutoflf M which is an energy scale below the scale of the string theory, generating an anomalous Fayet-Iliopoulos Dterm (^^), which breaks supersymmetry. To restore supersymmetry, which should not break at such a high scale, a compensating term in the D-term of the anomalous U{1) appears. The corrected form of the anomalous D-term is then Tr[Uil)x] = Tr[X] / 0. (1.1.13) Tr[X] = Cgrav ^ 0. (1.1.14) (1.1.15)

PAGE 15

9 1.2 The Green-Schwarz 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 Under a gauge transformation of the vector potential, A^^, shifts as A^-^A^^ + d^A (1.2.17) and the axion as a-^a + IGtt^SgsK (1.2.18) where ^ TrjX) 1927r2 The Lagrangian then changes by (1.2.19) = + 167r%5A:fc)F;,F^„ (1.2.20) with kk the Kac-Moody levels of the gauge factors. The anomalies of the Xsymmetry are compensated at the cut-off, as long as the ratio C^/kk is universal. This is the four-dimensional equivalent of the Green-Schwarz anomaly cancelation mechanism [27]. Consistency requires all other anomaly coefficients to vanish. CcoioD ^we&k the mixed anomalies between the X current and the

PAGE 16

10 standard model gauge currents, (XG^G^) = <5^^Q„,„r ; (Xiy^W^) = (5"^C,eak ; (XYY) = Cy , (1.2.21) where are the QCD currents, and the weak isospin currents. We must have '''Color "-weak "'Y and (xrWrW) = s'^c^^ . (1.2.23) All the other anomaly coefficients must vanish by themselves: (y(a)y('')y(c)) _ (y(a)y(i))y^ _ (Y^°'^G^G^) — {Y^'^^W°'W^) = (F^"^yy) — 0 . (1.2.24) as well as: {XYY^"^) = (XXY) = {XXY^"^) = (yWrr) = o . (1.2.25) A consequence of the Green-Schwarz mechanism is that the Weinberg angle at cut-ofT can be understood [30] as a ratio of anomaly coefficients: tan^ 9^ = -^ = ^ = ^ . (1.2.26)

PAGE 17

11 1.3 The Anomalous U{1) and Yukawa Matrices The nonzero anomaly coefficients can be computed from the X-charges 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 Q.UjH, ; QidjH, L.e^H, H^H, , (1.3.27) where i,j are the family indices, with X charges X\f, Xlf, X^, XM, (1.3.28) respectively; a simple computation yields <^oior = Eixlf + X^) SXM , (1.3.29) i + ^weak ^Qoior = 2 f:(x|f x|f ) + 2^^ . (1.3.30) Since the Kac-Moody levels of the non-Abelian factors are the same, the GreenSchwarz condition requires ^weak = ^color ' (1.3.31)

PAGE 18

12 from which we deduce Cy = til^i:^ + 2X1?) . (1.3.32) Similar equations hold for the mixed anomalies of the F^"^ currents; their vanishing imposes constraints on the F^"^ charges of the standard model invariants. The further constraint that the Weinberg angle be at its canonical SU{5) value, sin^^u; = 3/8, that is 3Cy = 5Cweak> yields the relations = j^ixlf Xjf) . (1.3.33) t Qoior = E [4^' 2Xjfl + 3Xjfl] , (1.3.34) i as well as C;=oior = ^ E [4"' 2X1?' + 3X^1] . (1.3.35) Since Ccoior does not vanish, these equations imply that some standard model invariants have non-zero 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 cut-off [58]; this is the origin of Yukawa hierarchies and mixings. A theory with A'^ -|1 extra Abelian gauged symmetries X, Y^^\ . . . , F^'^) will contain (as we will explain later), A'-t1 standard model singlet chiral superfields 9o,...9r^,to serve as their order parameters. The anomalyinduced supersymmetry-preserving vacuum is determined by the vanishing of the

PAGE 19

13 + 1 D-terms Y^xM' = (1-3-36) Q = 0 "tvi^W = 0, a = l,2,...,iV. (1.3.37) Q=0 These equations can be solved as long as the (A^ + 1) x (A'^ + 1) matrix A, with rows equal to the A'' + 1 vectors x = {xo,Xi, ...,xn), y^°'^ = (?/o"^yl"^ •••jJ/jv') 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) Q^djH,Jli^-^j , (1.3.38) where holomorphy requires the n-"^ to be zero or positive integers. Invariance under the A^ + 1 symmetries yields Xlf + Y^x,n^^ = 0, (1.3.39) a ^.$"^^'^ + E2/i"^«l?-0. (1.3.40) These involve the same matrix A, and here a solution also requires that det A ^ 0, linking hierarchy to vacuum structure. Evaluated at the vacuum values of the 9a fields, the terms shown above can produce a family-dependent 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,

PAGE 20

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{\) breaks just below the cutoff when a standard model singlet aquires a vacuum expectation value via the DSW mechanism. We will call this field 9. If there are additional U{\) factors, those have to be broken also at a similar scale by vevs of fields 9aWe 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: • We can always absorb the anomalies into only one of the C/(l)'s (which we will called X). We assume that the resulting anomalous symmetry is family independent, except when there is only one such U{\) in which case it has

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15 to be anomalous and family dependent. After X breaking, the effective term in W will be We notice that an effective suppression factor Aq =< 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) 6 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 C/(l)'s is determined by solving the D-term equations in the supersymmetric vacuum. In fact, we will assume that in a realistic low energy model all the f/(l)'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 C/(l)'s is unknown. We will take a minimalistic approach. We will try to construct a model, with the smallest possible number of C/(l)'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 Ac. Then, we require that the model reproduces the following:

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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: Q3U3HU. (1.3.42) 2. The ratios of quark masses extrapolated to the scale M via the renoramization group equations are rUt TUt TUb (1.3.43) 3. The quark mixing (CKM) matrix in this (Wolfenstein) parametrization is given by 4. The corresponding relations in the charged lepton sector are (1.3.44) ~ A , ~ A rrir m-r (1.3.45) 5. Unification of the down quark and r Yukawa coupligs at M m-r ~ nih (1.3.46)

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6. If the MSSM parameter tan/3 ~ 1, the interfamily relation 17 ^ ~ (1.3.47) rrit 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{\) context. • 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.

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CHAPTER 2 THE VACUUM 2.1 D-Flatness 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 f/(l), called X. Later, we will leave the number of f/(l)'s arbitrary. The anomalous D-term Dx is of the form: Dx = T,QM'-^' (2.1-1) i where qi is the X-charge of a generic scalar field 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 9. • fields charged under 5C/(3)^ x SU{2)w x t/(l)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

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19 in the superpotential. Typically for the superfields in the MSSM: HdHy^ , QidkHd ,LiHciek ,QiUkHu , LiHu , QidkLj ,LiLjek ,Uidjdk , (2.1.2) 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 /. • 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 right-handed neutrinos and we will denote them by N. Typically, for these fields to be interpreted as right-handed neutrinos, one needs terms in the superpotential to generate Majorana mass terms: Wm ~ MN^ {e/MY (2.1.3) and terms to generate Dirac mass terms for the neutrinos: WD-^IN{d/MY (2.1.4) where / is the invariant / = 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 < 0i, 02, • • • , the direction in scalar field parameter space where the fields (t)u (t)2, • An acquire a common vacuum expectation value of order ^. Our basic requirement is to choose the X-charges and the superpotential

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20 so as to forbid all solutions to the vacuum equations except the one corresponding to < 9 >. 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, = dW/d(l)i = 0. (2.1.5) For instance, the assumption of D^-Aatness {Dx = 0 in (2.1.1)) automatically takes care of the directions < n > where Xj < 0 for z G {1, • • , n}. Finally, there is necessarily some gauge symmetry other than the anomalous X, for example the symmetries of the standard model. D-flatness for these symmetries plays an important role for / invariants: it tends to align the fields present in /. We will give the general theorem later. Now we just demonstrate it with a simple example. Take / = (;Ai=< 02 >= Vj. The contribution to Dx from these fields is xi\vj\'^, where X[ is the total X-charge of /. Hence a positive x/ will allow a vacuum with the flat direction < 0i, 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 6 field and one N field. Take the X-charge of 6, x > 0 and that of N, x-^ < 0. With these fields the X D-term is: Dx = xiei-" + xj^\N\'' e (2.1.6)

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21 There are three different flat directions to consider: the desired < 6 >, < N > and < 6,N > which we wish to avoid. The direction < ^ > is favored by choosing X > 0, and < TV > is forbidden if x;^ < 0. The third direction is allowed by = 0. However, since x xj^ < 0, we can form a holomorphic invariant involving A'' and 6. The simplest possible invariant in the superpotential is A^^" but the corresponding F-terms forbid < 6 > and < ^, >. Thus we lose the possibility of a Dx-flat direction with < 6 >~ ^. Since all possible flat directions are lifted, supersymmetry is spontaneously broken. We must therefore require the presence of an invariant A^^" with p > 2 and n ^ 0 mod{p) to forbid only the direction < A'^,^ >. The case p = 2 corresponds precisely to a Majorana mass term for the right-handed neutrino A'^, once 9 is allowed a vev. In this case, n = 2k + 1 and there must be the following relation between the X-charges: -f = (2.1.7) Consider now another simple model, with 9 (X-charge x > 0) and / (X-charge X/ < 0). With one field and one invariant, we have Dx=x\9\' + xr\vt\'-e (2.1.8) As previously, D^-flatness kills the direction < / > and allows <9 >. The main diff'erence is that / being a composite fleld, I = U7=i (}>l\ the F-terms corresponding to the invariant are Fj = tnj{X[ii.j(t>T), even for ^ = 1. One is therefore left with a vev of order ^ along the single direction < 9 >. It is certainly en-

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22 couraging that linear terms in / can appear in the superpotential. Terms such as QiUjHu{0/MY are needed to implement hierarchies among the Yukawa couplings. Conversely, requiring that with n integer ^ 0, is sufficient to insure the linear appearance of the invariant /. In this case, the vacuum structure is related to the Yukawa hierarchies. However if xi = 0, there is no danger associated with /, and the above discussion does not apply. 2.1.2 Models with 3 Fields. First, consider a model with ^ (x > 0), iV {xj^) and / (x/). Let us take 3^7vi ^/ < 0. The vanishing of the Dx term forbids the directions < iV >, < / > and < iV,/ >, but allows the directions < 9,N >, < e,I > and < e,NJ >. We saw earlier that an invariant Nd forbids the desired direction < 6 >. We must require the presence of an invariant ]V'^" with 9 > 2 and n / 0 mod{q), to disallow the directions < 9,N > and < e,N, I > {q = 2 generates masses for N). The last direction < 7,^ > is disposed of by adding an invariant of the form /'(9", which is also allowed given the signs of the charges. (2.1.9) Dx=x\9\' + xj^\N\^ + xj\vr\^-e (2.1.10)

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23 2.1.3 i^-Parity We now show that the constraints discussed above on the X quantum numbers of the low-energy fields may naturally lead to conserved i?-parity. 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: iViiV.K + LiiV,//„0"o (2.1.11) We saw that, in order not to spoil the ^-vacuum, the powers n% must be odd integers, or equivalently the X-charges Xj^. of the fields Ni must be, in units of x, half-odd integers: X-^. 2ki + 1 ^ = -^ (2.1.12) where ki is an integer. Henceforth we set x = 1. The last term in (2.1.11) determines the i2-parity of the right-handed neutrino superfields to be negative. Let us study the ^-charges of possible standard model invariant operators made up of the basic fields Qi, Ui, di, Li, Ei (i being a family index) and of the Higgs fields Hu, and H^. The cubic standard model invariants that respect baryon and lepton numbers are, in presence of the gauge singlets Fj, QidjHd , Q.UjHu , LiCjEd , LiNjH^ , (2.1.13) with charges Xlf, xlf, xjf and xlf respectively. To avoid undesirable flat directions, all must appear in the superpotential, restricting their X-charges to be of

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24 the form ^luAe,.] ^ _^u,d,e,u ^ ^2.1.14) where n^f'^'" are all positive integers or zero. We now turn to the invariants which break i?-parity. We have already encountered the quadratic invariants LiH^, whose charges are determined by the seesaw couplings (2.1.11) to be half odd integers 2 {k^ <,) + 1 , ^ Xl,h. = 2 • ^^-^-^^^ Consider the cubic i?-parity violating operators, LiLjik, LiQjdk and Uidjdk. The charges of the first two, which violate lepton number, satisfy the relations: Xl,l,,. = Xf^ + Xlr' Xj^^ (2.1.16) XL,Q,i. = Xjj + X^, (2.1.17) where the index / can be chosen arbitrarily. As a consequence, if X^ is integer, -po, both charges are half-odd integers and there is no i?-parity violation from these operators. However they can still appear as L.L.EkNi ^("'*+"r,-po) LiQjdkNi ^(";*+"r,-Po) (2.1.18) in the superpotential. A similar conclusion is reached if XM is a multiple of one third, in which case one needs to include also appropriate powers of iVo. To determine the charges of the operators Uidjdk in terms of the charges of the parityconserving invariants, one must use the Green-Schwarz condition on mixed anoma-

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25 lies Cweak = Ccoior which leads: YiiXQ, + X^J ^(X,. + X,) + XM = 0. (2.1.19) One obtains: l-^/^M-X^ , (2.1.20) 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 X^^j.^^ thus obtained will be such as to forbid not only a term Uidjdk in the superpotential but also any term obtained from it by multiplying by any powers of 9, Nt or iVoLet us consider for illustrative purpose an anomalous symmetry which is family independent. Then (2.1.20) simplifies to: Xnii = X^'^ + (1 -i-)XM . (2.1.21) IS f Remember that Xj^ is half-odd integer and Nf = 3. If XM is integer not proportional to Nf = 3, then the charge Xj^jj is such that no term udde"'Jt can be invariant. If X'''] is non-integer and a multiple of one third, then similarly no term uddO'^TPl^Q can be made invariant. In the low energy theory, baryon number violation becomes negligible. If we restrict our attention to models which yield

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26 sin^ 6w = 3/8, the Green-Schwarz condition 5Cweak = 3Cy reads: 53(7Xq. + Xl,) E(4^*i. + ^J. + 3X,J + = 0. (2.1.22) One infers from (2.1.19) and (2.1.22) the following relation: X^'^=^J:{x^-X^), . (2.1.23) i _ ' which tends to favor models with integer X^'^'^ (proportional to Nf in the case of a family-independent symmetry). If X^"] = 0 or more generally if X^'^l is proportional to Nf (XM = Nfz^), the charge in (2.1.21) is half-odd integer; it can only be compensated by odd powers of N: invariance under X means conservation of i?-parity. For instance, the above allows the interaction: uddN ^K+^^+^mC^/-!)] . (2.1.24) This term allows baryon number violation, but preserves both B-L and i?-parity. 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 i?-parity violating operators, whatever their dimensions: through the righthanded neutrinos for example, i?-parity is linked to half-odd integer charges, so that X charge invariance results in ii-parity invariance. Thus none of the operators that violate i?-parity can appear in holomorphic invariants: even after breaking of the anomalous X symmetry, the remaining interactions all respect i?-parity, leading to an absolutely stable superpartner.

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27 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 GxXx Y^^^ x . . . x y(^) 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 non-anomalous ones by y(^\ . . . , Y'^^K In general, but not always, fields charged under G are singlets under Ghid and vice versa, both carrying X, y(i), ... and r(^) charges. In the following, we call g = G x G^id, and we denote generically the fields charged under G hy and the ^-singlets by Xi As stressed in the previous section, some of these fields must acquire non-vanishing vacuum expectation values through the Dine-Seiberg-Witten (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 Dand F-term equations that breaks only the Abelian factors. The D-term equations: Dx = Ea^a KOP e = 0 (2.2.25) Dyii) = Ea 2/«|(^a)P = 0

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28 where the ^-singlets with nonvanishing vevs are denoted by 9a, have in general several solutions, due to the large number of ^-singlets generally present in string models. We shall assume the existence of at least one solution {(^q)} 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 6 fields must then be equal to the number of C/(l)'s or greater. • the low-energy mass hierarchies (in particular fermion masses), which are generated by the small parameters {9c)/M, are completely determined by the high-energy theory. This means that there must be no more 6 fields than f/(l)'s, otherwise the {Oa) 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 (A^ -Il)-plet of G-singlets (^0, • • , On) such that: (a) the matrix of the 6 field charges is invertible, i.e. det ^ / 0, where (1) (1) A = (2.2.26)

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29 and the first column of A ^ only contains strictly positive entries. This ensures the existence of a vacuum ^Qq^ . . . ,Qn)dsw with / I(^o)Idsw \ DSW \ I(^n)Idsw j ( C2\ (2.2.27) which we shall refer to as the DSW vacuum. In addition, one must check that this vacuum is not spoiled by the F-terms. Since condition (2.2.26) ensures that there is no invariant of the form 6q°9^^ . . . O^'^ , this can happen only if the superpotential contains a term linear in the x ^.nd 0 fields. Our second assumption is then: (b) there is no holomorphic invariant x ^o°^i ' • linear in x, where x is a G-singlet other than the 9 fields. This amounts to a condition on the charges of x, namely at least one of the numbers no, . . . , n^v defined by / \ no ni TIN ( = A-' X \ \^x ) (2.2.28) has to be either fractional or negative. Note that if x is a right-handed neutrino, this constraint leads to an automatic conservation of i?-parity 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 non-renormalizable terms suppressed by inverse powers of the

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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 D-Qat 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 D-flat 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, ^-invariants and Qsinglets are not neutral under the Abelian symmetries, and must appear multiplied by powers of the 9 fields. Condition (a) allows us to assign, through Eq. (2.2.28), a set of numbers {ria} = (no, . . . , un) to each ^-invariant I = 4l{\ . . (resp. ^-singlet x)If all are positive integers, then IOq^O'I' ...9'}f 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 I9q°6i^ •On" 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 non-anomalous case: the condition required for the 9 fields to develop a nonzero vev is det^ = 0, which forbids the existence of the invariants I9q°9i' except for some very specific charge assignments for which the powers (no, . . . , un) are not uniquely determined. If all are positive, but some of them are fractional, the invariant appears at higher order: (/^g"^"' ••^Jv')"'. Finally, if some Ua is negative, one can not form any holomorphic invariant out of / and

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31 the 9 fields; we shall refer to this last situation by saying that / corresponds to a supersymmetric zero in the superpotential. 2.2.1 £>-Flatness Before characterizing the JD-flat 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 Q, and no FayetIliopoulos term associated with the Abelian factors that Q may contain, the zeros of the D-terms can be classified in terms of the holomorphic gauge invariants. More precisely, a set of vevs . . . , „) is a solution of the D-tevm constraints if and only [61] if there exists a ^-invariant holomorphic polynomial ...,(/>„) such that: = C{(t>\) i = l...n (2.2.29) Mi) where C is a complex dimensional constant. A systematic way to study D-flat directions is then to find a finite basis of invariant monomials {/„} 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 G -invariant I = (f^^ . . . (ff^'' (resp. G -singlet x),

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32 corresponds a D-flat direction ((^i, ...(/>„; ^) (resp. {x;0)), with |(^a)P > milsw for > 0 (2.2.30) \{0a)\' < milsw for n« < 0 As we show below, this is the only solution of the D-term constraints associated with I (resp. x)Note that this is not, in general, a flat direction of the scalar potential, because the F-term constraints F, = 0 are not necessarily satisfied. Let us prove this first in the case of a ^-singlet xThe only input we need is the existence of the DSW vacuum (2.2.27). Then, using the definition (2.2.28) of the {ua}, the D-term constraints ( 0 (2.2.31) Y[N] can be rewritten as: ^ A|(^o)P ' no = Kx)r ni (2.2.32) where we have defined A | (6^) p = | {9^) p _ | {e^)\l^^. The sign of A | {9,) p is thus determined by the sign of n^. In particular, when the D-flat direction is associated

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with an holomorphic invariant of the whole gauge group Q xX x Y^^^ x . constraints (2.2.30) read: 33 |(^a)|' > \{Oa)\lsw « = 0,1... iV (2.2.33) This is a remarkable difference with the non-anomalous case, in which the vevs of the 6 fields are not bounded. The A |(^a)P depend on a single parameter, which may be fixed by the F-term 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, tiq = ui = . . . = = 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 ^-invariant is straightforward. Applying (2.2.29) to / = (/>?\ . . we find that the D-terms associated with G constrain the vevs of the 0 fields to be aligned. In this relation, stands for I where the are the components of the representation of Q spanned by (pi. One should keep in mind that (2.2.34) is a weaker constraint than the vanishing of the £)-terms: Pl Pn ^ ^ As a result, we end up with a relation similar to (2.2.32), with |(x)P replaced by v]. As before, we denote this D-flat direction by (7,^) to stress the fact that the vevs of the fields in I are aligned. However, generic D-flat directions are not associated with a single basis ^-invariant, but rather with a polynomial in the basis ^-invariants. More precisely, flat directions involving a given set {
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charged under Q are parameterized by the vevs of the ^-invariants can be formed out of those fields: 34 = n, that l(«^.>P = E ^IPi (2.2.35) a where vl is a vev associated with the invariant /„. In general, the parameters vj 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 D-term constraints is a set of vevs {Xi},0) with: 1 a ^ ( \ a n? i , A 1(^1% (2.2.36) Clearly the relation |(^,)p > \{e,)\lsw holds, for a given q, only when all powers K and nj, are positive. On the contrary, when one of these numbers is negative, |(^a)|2 can be smaller than K^a)lDswThis may lead to vacua in which {6^) vanishes after imposing the F-term 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.

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35 2.2.2 F-Flatness In this section, we examine under which conditions a D-flat direction is lifted by F-terms. We first assume no compensation between different contributions to the F-terms, so that each individual contribution has to vanish for a Z)-flat direction to be preserved. We shall come back to this point later. We first restrict our attention to D-flat directions (/, 9 ) that are associated with a single ^-invariant I = 4>{\ . . ^P^. Two cases must be distinguished, depending on the signs of the numbers = (n^, . . . , njy) associated with /: • all are positive, i.e. the D-flat direction can be associated with some holomorphic invariant I"" e^O"^' . . .9"^" of the whole gauge group. This invariant contributes to the F-terms as (F^.) = amC^'vf"'-'^ {cf>l) {9or° {9^r^ . . . {9^)"^^ + ... (2.2.37) where we made use of (2.2.29), a is a coupling constant, and vj is defined by (2.2.34). Since > \{9c,)\lsw along the flat direction, this contribution vanishes only if {(f)i) = 0, or equivalently vi = 0. As a result, the flat direction breaks down to the DSW vacuum. • some of the are negative, i.e. the D-flat 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 /' 9o°9i' ...9"^," (with /' a combination of basis ginvariants and x fields), where either one of the following two conditions is

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36 fulfilled: (i) /' contains no other field than the ones appearing in /, and n'o = 0 or 1 if tIq < 0 (with the additional constraint I3{a;n„e'l' (2.2.39) Using (2.2.29), we obtain: (F,,) = aC{u\) {6or'>{6,r^ (2.2.40) If no = 0, the F-terms vanish only for vj = 0, and the D-flat direction {Qi,uuHu,6o,ei) (2.2.41)

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37 breaks down to the DSW vacuum {9o,6i)dswIf > 0, the F-term constraints have two solutions: the DSW vacuum (which corresponds to u/ = 0), and a residual flat direction (Qi, ixi, //„, ^i) with v] = = \{ui)\^ = |(i/„)|2 = |(^o)lW and |(^i)p = 3|(^o)lW + Thus the initial Mat direction is only partially lifted, and the residual flat direction, along which (^o) = 0, can lead to another vacuum than the DSW vacuum. However, it can still be lifted by higher order operators. Indeed, the invariant (QiUi//u)"° (<3iM2^u) ^i"""*""' (which satisfies condition (ii)) contributes to (Fq^) as: PC^'o+l^jno ^^t^ ^^^^3no+n, (2.2.42) which obviously vanishes only in the DSW vacuum {v[ = 0). We consider now all possible D-flat directions involving only ^-singlets. Two cases must be distinguished, depending on the signs of the numbers nj, associated with each of the fields Xi• all n'^ are positive: in this case, > \{9a)\lsw along any flat direction of singlets. As a consequence, the D-term equations do not allow for any other vacuum of singlets than the vacuum (2.2.27). The other solutions of (2.2.25) are D-flat 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 F-terms, 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 1(9^)1^ can be smaller than in vacuum (2.2.27). As can be seen from (2.2.36), the D-term equations

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38 allow for vacua of singlets in which {6a) = 0, while some of the x fields have non-vanishing 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 {{(f>i}; {Xi};^) involving ^-charged fields as well as ^-singlets. The relevant numbers here are {n^ ; n^}, where the {n^} are associated with the basis ^-invariants {la} that contain the The general requirement for this flat direction to be lifted is that invariants of the form n' n' n' I' 9Q°9y^ . . .Off' be present in the superpotential (where /' is a combination of basis ^-invariants and x fields), where either one of the following two conditions is fulfilled: (i) /' contains no other field than the ones appearing in the flat direction, and = 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) /' 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 F-terms. The effect of such cancelations is to reduce the dimensionality of a D-flat direction, while one would naively expect it to be (at least partially) lifted. For instance, in the toy model of the previous subsection.

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39 (case no > 0), contributions (2.2.40) and (2.2.42) cancel against each other in (Fn^) if the following relation between vevs is satisfied: (2.2.43) (note that the case no = 0 does not suffer from this problem, since (2.2.40) is the only contribution to (Fuj)). Such compensations are possible because the Fterms, at least at low orders, are not all non-trivial and independent from each other. When higher order operators are added in the superpotential, the number of independent F-term 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 ^-singlets is more subtle, and needs a separate discussion. Due to condition (2.2.26), the F-terms of the 9 fields are not independent from the other F-terms: 9n Fg^ ^ A. \ XqFx^ J (2.2.44) where A^ 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 ^-singlets are constrained by exactly as many equations as fields, and those (non-linear) 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

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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 F-terms. 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 Fayet-Iliopoulos 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 {9)/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 9, 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: 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 1(^)1^5^' = and {xi)dsw {X2)dsw = 0. In the presence however of this term, there is an additional solution to the D-term and F-term constraints: W = crxle' + C2xle' + c,XiX29' (2.2.45) (2.2.46)

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41 where due to the positive powers in (2.2.45), |(^)| > |(^)|£)5H' = ^, 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 Low-Energy Vacuum The purpose of this section is to discuss how supersymmetry breaking affects the conclusions of the previous section. Since models with an anomalous C/(l) have numerous implications for low-energy phenomenology, it is indeed essential to ensure that the flat direction analysis is relevant to the determination of the low-energy vacuum. The scalar potential of the low-energy theory reads: ^ = In + E \{Fk)\' + Vsy^sy (2.2.47) c k Since supersymmetry has to be broken in a soft way, Vsi/sy is of the form mF^^) + m^V^'^\ (2.2.48) where rh ~ ITeV is the scale of supersymmetry breaking, and V^'^\ V'^^'i 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. (/»"/Mp,~^ 6 V^'^\ This has the obvious consequence that the actual minimum of the scalar potential is close to a flat direction of the supersymmetric theory;

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42 otherwise the D-terms and F-terms would give a positive contribution of order to V, while Vsy/sy would contribute at most as fn^^, with no possibility of compensation. We thus necessarily have (Fj) «: and {D") \{Oa)\Dsw still hold (provided that the necessary conditions are satisfied), and that fields charged under Q 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 F-terms, which means that all contributions to the F-terms must be much smaller than Let us first consider the flat directions for which > \{9a)\l,sw 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 m, or of an intermediate scale such as (mM)^/^ Clearly those cannot be the 6 fields. In addition, the x and 4> fields cannot have a vev of order ^, otherwise some invariant of the form /^o""^"' 07 or X'" erdT' OT would give a contribution of order e to the F-terms. As a result, the low-energy vacuum is a slight deviation from the DSW vacuum: m\' \{ea)\lsw {i), iXi) « e (2.2.49) This is perfectly consistent with the conclusions from the flat direction analysis: flat directions along which \{9a,)\^ > \{9a)\Dsw for all a are lifted down to the DSW vacuum by the F-terms, 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 > \{9a)\lsw does not hold for all a. Contrary to the previous type of flat directions, these may be

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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,Ui, H^, Oq, 6i) associated with the non-holomorphic invariant QiUiHu9q^9\. Since along this flat direction |(^o)P < |(^o)lzjsw> we can see from (2.2.39) that, in the case no > 0, F^^ can be small compared to while v] = |(<3i)P = = \{Hu)\^ is of order For example, in the case no = 1, the minimization of the scalar potential could yield (^o) ~ ^ (with KFu^)]^ ~ m^^^ being compensated for by e.g. mjj^ with m^^ 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 ~ ^ would give Fgj ~ 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 (j) 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.

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44 2.2.4 Summary We now summarize the generic procedure to analyze flat directions in anomalous C/(l) models that satisfy conditions (a) and (b): 1. find a basis /i, . . . , /p for holomorphic ^-invariants involving only (j) fields; add to this basis all ^-singlets Xi, • • • , Xq2. for each element of the basis, compute the set of numbers (no, . . . , un) defined by (2.2.28). The corresponding Mat direction is in the case of a ^-invariant, and (x ; ^) in the case of a ^-singlet; the vevs of the 6 fields are determined by (2.2.32), and they satisfy the constraints: m? > \{ea)\lsw > 0 (2.2.50) . m\' < miisw < 0 3. the most general D-flat direction involving a set of fields ({(/>i}; {x,}) is parameterized by the vevs of the Xi and the the latter being subject to constraints (2.2.35). The vevs of the 9 fields are determined by (2.2.36), which implies l(^a)|' > \{Oa)\lsw if K, < > 0 (2.2.51) for all relevant la and Xi-

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45 4. determine which flat directions are lifted by F-terms. Two cases must be distinguished: flat directions for which all relevant {n^ ; nj^} are positive are lifted down to the DSW vacuum by F-terms. other flat directions are only partially lifted, unless invariants of the form /'^q"^"'' Otf, where /' 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 F-terms, 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 see-saw mechanism via right handed neutrinos and if there are no supersymmetric zeros it is guaranteed that all the "non-DSW" flat directions are

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46 lifted except < anything, {9^} > (2.2.52) "Anything" in the above means standard model fields which form invariants that, as usual, we call generically by /f^ and/or fields with tree level mass terms such as fields vector-like with respect to the standard model gauge group, extra singlets X and hidden sector fields. We will call generically these latter by • (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 F-terms that come from the Yukawa sector lift the directions < /^^, {Oa} > . (2.2.54) Finally, a generic flat direction < {<^}, {I}, {Oa} > (2.2.55) that corresponds to an arbitrary polynomial of the basis invariants {V), is lifted by the invariant that is made of the product of all the different invariants that compose V. Supersymmetric zeros make the analysis of the vacuum much more complicated. For that, we strongly favor models with less number of supersymmetric zeros. As

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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.

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CHAPTER 3 MASS MATRICES 3.1 Quark Mass Matrices from C/(l)'s: Hierarchies Assume now a generic model with N + 1 extra f/(l)'s. A gauge invariant term in the superpotential has the form: We have displayed in the standard model invariant / and the exponents the family indices explicitly. The invariance of this term under the whole gauge group and in particular under the [/(l)'s allows us to compute the powers n]^'' to be n = -A-'Y^'\ (3.1.2) where we have introduced the following matrix notation: n is an (A'^+1) x 1 column vector with the powers of the 9 fields n — (3.1.3) 48

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49 yt^^ is an (A'^+l) x 1 column vector with the charges of the standard model invariant -^111213. ..1/ under the + 1 f/(l)'s. We denote the anomalous U{1) by X and the anomaly free ones by F . We also assume that X contains all of the trace of the N + 1 C/(l)'s. Then, y(i)a. . . .) (3.1.4) \li),i2ii...ii) } A is the matrix we defined in the previous chapter. Its inverse is assumed to be /I an aiN ^ 1 u Notice that all the elements of the first column of the above matrix being 1 means that all the Q fields (A'^ + 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 non-anomalous L''(l) (a = 1, ...,N):

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50 1st family 2nd family 3rd family Q [a] [a] Q2 [a] [a] -Qi 92 u [a] [a] d d? Here, we have already used the assumption that the non-anomalous factors are traceless. It is useful to introduce the quantities QS^29!''l+gr and Q^}^2q^^^ + q[ [2] C/g = 2uS"l + 4^1 and U^^ = 2u^^^ + u? 24"! + 4'] and = 24"! + 4']. ^12 (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 Xc{=<6> /M) to the power A''^ plus (M + K P + K 0 + K\ M+L P+L 0+L VM + 0 P + 0 0 + 0 J and in the down sector we get the powers N^"^^ plus fR + K T + K 0 + K\ R+L T+L 0+L \ R + 0 T + 0 0 + 0 J (3.1.9) (3.1.10)

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51 j\l[u,d\ jg (jefined as the total power appearing at the 33 position of the mass matrix Also, fK\ L Lc M M« = E P a Pa R Ra [tJ [Taj where La Ma rj[a] Pa Ra Ta ) (3.1.11) (3.1.12) As usual, a = 0, ...,N, a = I, ...,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/3 of order of one (we will justify the latter soon), amount to iVM = 0 and A^'"^ = 3 (3.1.13) respectively. The elements in the CKM matrix above the diagonal, require K = 3 and L = 2, (3.1.14)

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52 with Ka,La > 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 M c (3.1.15) 1 / in the up sector. This matrix has the (order of magnitude) eigenvalues 1 and and A.^+^ (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 = l and T = 0. (3.1.18) The above, also fix 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 f/(l)'s are unique in the up and down quark sectors. Also they give a unique CKM mixing matrix.

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53 We can therefore narrow down our search to matrices of the form: and An \7 \ A? 1 / (K A^ A^ A^ A^ 1 1 ) (3.1.19) (3.1.20) where =< i/„,d >. We now give an algorithmic procedure that one should follow when searching for models generating the above forms: • Find a set of arbitrary integers K^, L^,, M^, Pa, Ra, Sa that satisfy (3.1.17) and (3.1.18), (including A^M = 0 and N^'^ = 3). • Assume a matrix A, that satisfies (3.1.5). Calculate from the integers of the previous step, the quantities \q[Vj -A ( \ ~A \Ln) (3.1.21)

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and and /Mo\ Ml = -A \RnJ ( \ [1] 21 Pi \Pn} (R^\ = -A \RnJ (3.1.22) (3.1.23) • Calculate the actual (non-anomalous) [7(1) charges (inverting 3.1.6-3.1.8): 2 3 3 3 2 3 (3.1.24) and and Mi 2 3 -1 3 [a]. 12 ^21 12 -^21 (3.1.25) (3.1.26)

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55 • Check if the previously found charges can be supplemented by a lepton (and perhaps neutrino, vector-like, 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 i7(l), 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 inter-family 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 ^ to be 1, which gives us .4 = 1. Then, combining (3.1.14), (3.1.17), (3.1.18), (3.1.21), (3.1.22) and (3.1.23), we obtain: (3.2.27)

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56 2 -1 3 3 -1 2 3 3 2 -1 3 3 -1 2 3 3 A. We summarize the traceless part of the quark sector of our U{1) in the following table: -5^ 1= (-S/3 -V { 1/3 :) f-2/3 K 1/3 (3.2.28) (3.2.29) Q u d /-4/3\ /-8/3\ /-2/3\ -1/3 1/3 1/3 [ 5/3 ) \ 7/3 I \ 1/3 I 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;(l, 0, -1), (3.2.30) where 77 = 1 for both Q and u and 77 = 0 for d. That both Q and u possess the same T] charge is reminiscent of the SU (5) charge patterns, where the chiral fermions are split into 5 = {L,d) and 10 = {Q,u,e). This suggests we flesh out the multiplets by assigning the e singlet a value 77 = 1 and the L doublet r] = 0. We generalize the factor B appearing in Eq. (3.2.30) to its 50(10) analog (B-L). Note that t], on the other hand, is outside of 50(10). The quark and lepton charges may then be succinctly written: yi. = (J5-L)(2,-l,-,l)-27?(l,0,-l) (3.2.31)

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57 The inter-family exponents of the Yukawa matrix associated with the operator LiejHd follow: ei 62 63 4 5 3 L2 1 2 0 Lz 1 2 0 Its diagonalization yields the lepton inter-family hierarchy Hh. ^ x^-^ , ^ ~ , (3.2.32) nir rn-r fully consistent with phenomenology, as well as the contribution to the lepton mixing matrix from the rotation of the left-handed lepton doublet: 1 A3 A3 \ A3 1 1 V A3 1 1 / (3.2.33) As Yp contains B — L, it is natural to introduce three families of right handed neutrinos Ni. Before assigning them Yp charges, we note that certain predictions associated with neutrino phenomenology are completely independent of the charges of the A'^s. 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 right-handed neutrino Majorana mass matrix is symmetric, it may be written Y^p — NiNj,

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58 where A^, and Nj are vectors depending only on the antineutrino charges. The matrix coupHng right-handed neutrinos to the standard model, on the other hand, is written Y^[^ = H^LiNj, where H^Li is a vector independent of the right-handed charges. Taking £7° to be the matrix that diagonalizes Y^°\ Y(^) = U'D\uy , (3.2.34) with D° a diagonal matrix, the effective neutrino mixing matrix after the seesaw is given by: (3.2.35) Because of the form of Eq. (3.2.35), a cancelation of Ni charges results, and one discovers that (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). MNS 1 A3 ^ A3 1 1 A3 1 1 (3.2.37)

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59 This implies a small (order A^) mixing of the electron neutrino with the // and r species, and mixing between the n and r neutrinos of order one [65], [66], [67]. Remarkably enough, this mixing pattern is precisely the one suggested by the non-adiabatic MSW [52][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/rrit, is of order Since both cot^ and the Yukawa entries conspire to produce this suppression, there are two extreme possibilities. • The first possibility is that and Yt are of the same order, with cot p responsible for the suppression. With a tree-level top quark mass, achieving and Yt of the same order requires that the Yp charge of the //-term, HuHa, be Y^/^ = -6. But avoiding anomalies such as TrlYYYp] and Tr[SU{2)SU{2)YF] forces the Yp charge to be vector-like on the Higgs doublets, so that rj."^ = 0. Hence n ~ Yt requires Yp to be anomalous (The Green-Schwarz mechanism cannot be invoked since Tr[SU{Z)SU{2>)Yp] = 0). Furthermore, we shall soon see that a family-traceless Yp cannot reproduce neutrino phenomenology.

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60 To proceed, we need to assign y> charges to the right-handed neutrinos N. Since r] is contained in Ee, we give the N fields their Eq value, 77 = 2, which yields ypiNi) = (-2, -1,3). One obtains an NjNj Majorana mass matrix with familydependence N2 4 3 SZ N2 3 2 SZ Ns 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 vlX^M and vlX^^/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 10^^ Q^y^ suppressed by four orders of magnitude with respect to MqutThere is no mechanism in our model to effect such a suppression. We conclude that the family-traceless, non-anomalous Yp symmetry must be extended by adding a family-independent piece, hereafter called X. • We turn now to the alternate possibility, n ~ X^Yt and cot P of order 1, where the suppression comes from the family-independent piece X. The total flavor symmetry is now Yx = X + Yf (3.2.38)

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61 To consider the implications of anomalies involving our family-independent symmetry, we define the mixed anomaly coefficients of Yx with the Standard Model gauge fields by Cg, = Tr[GiGiYx] . (3.2.39) The satisfy the following relations: Cy + Cweak ^Ccolor = 6(JCM X^'^) + 2J^M , (3.2.40) Ccolor — 3(A:["^ -IX^'^) SXt"] , (3.2.41) where ^["•-'''H are the X charges of the operators QiUjH^, UejH^, Q.djHa, and HuHd, respectively. It is precisely these charges X^, XM, and XM that determine the intrafamily hierarchies nib /nit and nir/mb. Let us set — ~ cot^A^" , -I ~ A^^" . (3.2.42) iTT't nib Then one finds that Eqs. (3.2.40) and (3.2.41) above may be rewritten as Cy + Cweak 2Ccolor = -2(P(,t + 3Prb + 6) , (3.2.43) where we have used the fact that the top quark Yukawa coupling appears at tree level, and therefore that = o. The data suggest = 3 and P^b = 0, which through Eq. (3.2.43) tells us that our new symmetry Yx must be anomalous.

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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 Green-Schwarz anomaly cancelation mechanism. We take the family-independent X acting on the chiral fields to be a linear combination of a universal piece and of the two charges, V, V, defined through D 50(10) X U{l)v' ; 50(10) D SU{5) x U{l)v (3.2.44) Across the Higgs doublets, the X symmetry is taken to be vector-like, a necessary condition if the three C/(l) symmetries comprising Yx are gauged separately. These choices yield X^'^ = X^, and = X{LHjl) = 0. The Green-Schwarz structure has the added benefit of producing the correct value of the Weinberg angle at cutoflf [30][32]: C 5 tan^^^ = — ^ = . (3.2.45) ^weak " There still remains the non-zero anomaly (YYxYx), which can be canceled by three families of standard model vector-like representations 5 -I5 of SU{5). With this addition made, the remaining anomaly structure is consistent with the Green-Schwarz cancelation mechanism. We get ^ ~ cot /?A-(c^co,o.+i8)/3 ^ a^lr^i (3 2 46) mt rrib and agreement with the data is achieved for Ccoior = -27.

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63 We can now specify the form of the matrices involving right-handed neutrinos: A" 1 A 1 A-4 ^ A8 A^ A^ ^ A^ A" 1 A^ A^ 1 , (3.2.47) where Y^°^ is the WN Majorana mass matrix and the matrix coupling Li to Nj. Note that, to appear in the superpotential only as holomorphic quadratic mass terms, the X-charge of the Ns must be negative half odd integers. After the seesaw, we have the actual neutrino mass matrix t;^A^^'"^'+6 M ^ \' A3 A3 ^ A3 1 1 A3 1 1 (3.2.48) which produces light neutrinos with masses M M (3.2.49) The mass splitting between i/g and the other two neutrinos is Am^ ~ lO'^eV^ consistent with the non-adiabatic MSW solution to the solar neutrino problem if Xl^l = -9/2 and M ~ MqutTo check agreement with the atmospheric neutrino data, we must know the mass splitting between i/^ and v^, but this can be predicted only with a theory for the prefactors. Interestingly, prefactors of order 1 produce Am2^_^^ ~ O.OTeV^, so that the atmospheric data may be explained by the same solution that accommodates the solar neutrino data without any fine

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64 tuning. Moreover, this solution requires M to be of order Mqut 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 YxThe analysis proceeds much as that above, but in this case = -3 instead of X^"^ = -9. Also, as will be shown in the next chapter, the extra anomaly conditions fix all the charges of the fx term to zero, and the analysis of the vacuum in which all three symmetries are broken at the same scale favors X^^'^ = -3/2. Remarkably, it is precisely this charge assignment, corresponding to Xt^5 = -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 ~ Mqut3.3 Squark Mass Matrices from t/(l)'s: FCNC Every realistic supersymmeric model has to account for the large flavor changing neutral currents (fcnc) 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

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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'-'^^ that has supersymmetric zeros. The quark matrices are diagonalized by = V^^"^M"F^, = vfM'V^. (3.3.50) Similarly for the squarks we have mass matrices associated with the soft terms YlL' = Vt'' KiK'' (3.3.51) Yrr = K'm^rVR (3.3.52) yL = VfMUV^. (3.3.53) The phenomenological limits on the entries of these matrices (coming primarily from the neutral meson mixing experiments) are: (^l)i2 = {VtvfU = ~ \l, where = mzn[(y/)i2, {Vf),2. {vf)n + {vf)^] (3.3.54) = {V^V^U = ^ Xl, where = mzn[(F/)i2, (V/)i2, (V^'hs + {V^),s] (3.3.55)

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66 < >= ^J{Kt)x2{Ki)u = Al^") ~ (3.3.56) (^l)i2 = {VlVl'),2 = A^~ A„ where mf2" = min[{V^'),2, {Vf)n. (Fl"')i3 + (Vi"')^ (3.3.57) (^r)i2 = {V^V^')i2 = X<" ~ A„ where = m2n[(F/)i2, (F/)i2, (1^h"')i3 + (V/)i3] (3.3.58) < I= ^{KDuiKDn = aI^"^'"^ ^ ~ A^ (3.3.59) (KDu = {V^V^'),, = A^'a" ^ A„ where mfg" = mm[(V2'^)i3, (Vf),,, (y2'')i3 + (Vi"')^] (3.3.60) (^h)i3 = (V^K"V^fi')i3 = A-fa" ~ A„ where mf3" = mm[(V'/)i3, (l^«"')i3, (V/)i3 + (V'/)i3] (3.3.61) < \/(^E)i3(i^^)i3 = Al^""" ^""^ ) ~ A^. (3.3.62)

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67 If there are no supersymmetric zeros in the mass matrices, we can diagonalize them very easily. We obtain and / 1 {Ka La) Ka\ Kl = [Ka La) 1 1 La r 1 ^ 1 f 1 {Ma Pa) Ma\ {Ma Pa) 1 Pa K Ma Pa 1 / ( 1 \Ka La\ \Ka\\ \Ka-La\ 1 \ \Ka\ \La\ 1 / 1 \Ma-Pa\ \Ma\\ \Ma Pa\ 1 \Pa\ \ . \Ma\ 1 / / 1 \Ra-Sa\ \Ra\\ \Pa-Ta\ 1 \Ta\ \ \Pa\ \Ta\ 1 ) Furthermore, we can compute for example (3.3.63) (3.3.64) (3.3.65) (3.3.66) (3.3.67) m = mmE Ra Ta, ^ \Ra Tal Ra + |TJ] = = T.{Ra~Ta) = Y:R--Y:Ta = l-0=l (3.3.68)

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68 A similar relation holds for i? L. Clearly the fcnc constraints are not satisfied. From the general form of the constraints it is easy to see that to satisfy them, •^2^ and Y^i^ have to be supersymmetric zeros. But if these are zeros, then also Y^^ and Fg^f^ 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 ^23'^ can not be zeros if Y^*^^ should give the desired mass ratios and mixings. We have therefore proved that there is a unique F^"^) compatible with the "alignment scenario" of suppressing fcnc. On the other hand, F^") is fixed except the elements (21), (31) and (32). These are either supersymmmetric zeros or not. We write: (3.3.69) 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 9 fields have to simoultaneously satisfy the following: Q
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Also for every a: 69 -50 (3.3.77) Ta + La + N^>0 (3.3.78) Ma + Ka + N^>0 (3.3.79) Pa + La + N^>0 (3.3.80) Pa + ii:a + iV^>0 .... (3.3.81) and at the same time for some a: T,+N^ < 0 orR^+N^ < 0 or T^+K^+N^ < o or R^+L, + N^ < o. (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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70 reproduces (assuming no susy zeros and compatibility with neutrino data) the quark mass hierarchies and mixings is: Yx^X^ \{2Y + y)(2, -1, -1) ^(V + 3y')(l, 0, -1). (3.4.83) We have now several choices. We either gauge this U{\) 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 restwe can not have a suppression in the ratio nib/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{iys are separated. This happens because in the 3 f/(l) case, there is an anomaly, XY^^^Y^'^^ 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.

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CHAPTER 4 MODEL BUILDING: A REALISTIC MODEL 4.1 General Structure In the visible sector, the gauge structure is that of the standard model {Q = Gsm), augmented by three Abelian symmetries: SU{3y X SU{2)w X U{1)y x U{1)x x U{1)yw x U{l)ym . (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 family-independent. The other two symmetries, y(^) and Y^'^\ 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 Eg x Es heterotic string and its chiral matter lies in broken-up representations of £^6, resulting in the cancelation of many anomalies. This also implies the presence of both matter that is vector-like with respect to standard model charges, and right-handed neutrinos, which trigger neutrino masses through the seesaw mechanism. The three symmetries, X, ^j.^ spontaneously broken at a high scale by the Fayet-Iliopoulos 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 71

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72 assignments, let us recall some basic Eq [68]. It contains two Abelian symmetries outside of the standard model: The first U{1), which we call V, appears in the embedding Eg C 50(10) X U{l)v' (4.1.2) with 27 = 16i + 10_2 + U , (4.1.3) where the U{1) value appears as a subscript. The second U{1), called V, appears in 50(10) C SU{5) X U{l)v , (4.1.4) corresponding to 16 = 5_3 + lOi + I5 ; 10 = 52 + 5_2 . (4.1.5) The familiar hypercharge, Y, appears in SU{5) c SU{2) X SU{3) X U{1)y , (4.1.6) with the representation content 5 = (2,r)_i + (l,r)2/3 , (4.1.7) 10 = (1, ^2 + (2, 3^^)1/3 + (1, r)_4/3 . (4.1.8)

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73 The two C/(l)'s in S0{1Q), can also be identified with baryon number minus lepton number and right-handed isospin as B-L=^{2Y + V)/3^ = 1(3F-F). (4.1.9) The first combination is B — L only on the standard model chiral families in the 16; on the vector-like matter in the 10 of S'O(IO) it cannot be interpreted as their baryon number minus their lepton number. We postulate the two non-anomalous symmetries to be r(i) = l(2r + y)(2, -1, -i) (4.1.10) r(^) = J(V + 3n( 1, 0, -1 ) , (4.1.11) The family matrices run over the three chiral families, so that F^^'^^ are familytraceless. We further assume that the X charges on the three chiral families in the 27 are of the form X = {a + pV + yV')(^ 1, 1, 1 ) , (4.1.12) where a,/?, 7 are undetermined parameters. Since Tr {YY'^'^) = Tv{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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74 • Three chiral families each with the quantum numbers of a 27 of Eq. This means three chiral families of the standard model, Qi, Uj, di, Li, and ej, together with three right-handed neutrinos Ni, three vector-like pairs denoted by Ei -\Di and Ei -f Di, with the quantum numbers of the 5 -I5 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 standard-model vector-like 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 OaIn our minimal model with three symmetries that break through the FI term, we just take a = 0, 1,2. The 9 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 es: • The first involve only standard-model gauge groups Gsm, with coefficients (GsuGsuGsm), which cancel for each chiral family and for vector-like matter. Also the hypercharge mixed gravitational anomaly {YTT) vanishes.

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75 The second type is where the new symmetries appear linearly, of the type (F^'^GsmGsm)The choice of family-traceless F^'^ insures their vanishing over the three families of fermions with standard-model. Hence they must vanish on the Higgs fields: with Gsm = SU{2), it implies the Higgs pair is vector-like 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 (Gsm^^^'^^^-'^)These vanish automatically except for those of the form (YY^'^^Y^^^). Two types of fermions contribute: the three chiral families and standard-model vector-like pairs 0 = (rywyo)) = (rF(^)y(^))ehirai + {yy^'^y^^^,^ . (4.2.13) By choosing F^^'^) in Ee, overall cancelation is assured, but the vector-like matter is necessary to cancel one of the anomaly coefficient, since we have {YY^'^Y^'%,,^ = -(Fy(i)F(2)),eai = 12 . (4.2.14) The fourth type are the anomalies of the form (y(OyO)y(*)). 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 representation, they vanish, but, as we shall see, the 9 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 F^'^ are family traceless, the vanishing of the {XYY^^'"^^) coefficients over the three families is assured, so they must vanish over the Higgs pair. This means that X is vector-like on the Higgs pair. It follows that the standard-model invariant HuH^ (the // term) has zero X and F''^ 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 Kahler potential. This results, after supersymmetrybreaking in an induced /^-term, 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 coeflRcients. We find Ccoior = 18a ; Cweak = 18^ ; Cy = 30a . (4.2.15) Applying these to the Green-Schwarz relations we find the Kac-Moody levels for the color and weak groups to be the same ^color — ^weak > (4.2.16)

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77 and through the Ibaiiez relation [30] , the value of the Weinberg angle at the cut-off tan^^, = -^^ = 1 , (4.2.17) not surprisingly the same value as in SU (5) theories. The coefficients {XY^^^Y^'^^). Since standard-model singlets can contribute, we expect its cancelation to come about through a combination of hidden sector and singlet fields. Its contribution over the chiral ferniions (including the right-handed neutrinos) is found to be (Xy(l)F(2)),hi,al + real = 18a . (4.2.18) The coefficient (XXY). With our choice for X, it is zero. The coefficients (XXY^^^) vanish over the three families of fermions with standard-model charges, but contributions are expected from other sectors of the theory. The vanishing of these anomaly coefficients is highly non-trivial, and it can be viewed as an alternative motivation for our choices of X, and Y^K 4.3 The DSW Vacuum The X, y(^) and F^^) Abelian symmetries are spontaneously broken below the cut-off. 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, 9^, acquire a vacuum

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78 value as a result of the FI term. They are singlets under the standard model symmetries, but not under X and Y^^''^\ If more fields than broken symmetries assume non-zero values in the DSW vacuum, we would have undetermined flat directions and hierarchies, and Nambu-Goldstone bosons associated with the extra symmetries. We express their charges in terms of our 3x3 matrix A, whose rows are the X, Y^^^ and F^^^ charges of the three 9 fields, respectively. Assuming the existence of a supersymmetric vacuum where only the 9 fields have vacuum values, implies from the vanishing of the three D terms 0 (4.3.19) We have found no fundamental principle that fixes the charges of the 9 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 so that its inverse ^ 0 0^ 0 -1 1 1 -1 0 (4.3.20) A-' = 1 0 -1 1 1 -1 (4.3.21)

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79 is of the desired form. We see that all three d fields have the same vacuum expectation value |<^o>| = |<^i>|-|<^2>|=e(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 right-handed neutrinos, the uniqueness of the DSW vacuum is attained only after adding to the superpotential terms of the form WV{6), where p is an integer > 2, and "P is a holomorphic polynomial in the 9 fields. If p = 1, its F-term breaks supersymmetry at the DSW scale. The case p = 2 is more desirable since it translates into a Majorana mass for the right-handed neutrino, while the cases p > 3 leave the massless in the DSW vacuum. To single out p — 2 we simply choose the X charge of the Ni to be a negative half-odd integer. Since right-handed neutrinos couple to the standard model invariants LiH^, it implies that XiiHu is also a half-odd 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 X-charges 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 9 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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80 section. Finally, we note a curious connection between the DSW vacuum and the anomalies carried by the 9 fields. Assume that the 9 sector does not contribute to the mixed gravitational anomalies (r«Tr), = 0 . (4.3.23) This means that the charges are traceless over the 9 sector. They are therefore generators of the global SU{Z) under which the three 9 fields form the 3 representation. However, 5f/(3) is anomalous, and it contains only one non-anomalous U{\) that resides in its SU{2) subgroup. Thus to avoid anomalies, the two charges F(^'2) need to be aligned over the 9 fields, but this would imply det^ = 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 9 fields, (y(i)y(i)y(2))^ = (r(i)r(2)y(2))^ ^ _1 ^^ ^^^^ 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 Q3U3HU . (4.4.25)

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81 Since X is family-independent, it follows that the standard-model invariant operators QiUjH^, where i,j are family indices, have zero X-charge. Together with the anomaly conditions, this fixes the Higgs charges (4.4.26) and y(i) = _f(i) = 0 y(2) _ _y{2) _ X{Q,uM = XH = 0 . (4.4.27) (4.4.28) The superpotential contains terms of higher dimensions. In the charge 2/3 sector, they are (4.4.29) in which the exponents must be positive integers or zero. Invariance under the three charges yields (4.4.30) where Y^^ and are the charges oiQfijH^, respectively. They are determined by our choice for the charges F^'^). a straightforward computation yields the orders of magnitude in the charge 2/3 Yukawa matrix y(u) ^ A« A^ A^ A2 A^ A2 1 (4.4.31)

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82 where as usual A =< > /M is the expansion parameter. A similar computation is now applied to the charge -1/3 Yukawa standard model invariants QidjH^. The difference is the absence of dimension-three terms, so that its X-charge, which we denote by X^'^ need not vanish. We find that if X^''^ > -3, one exponent in the (33) position is negative, resulting in a supersymmetric zero and spoiling the quark hierarchy. Hence, as long as < -3, we deduce the charge -1/3 Yukawa matrix A^ A^ A3 A2 A2 A 1 1 (4.4.32) and diagonalization of the two Yukawa matrices yields the CKM matrix ^ A A3 ^ U CKM A 1 A2 A3 A2 1 (4.4.33) This shows the expansion parameter to be of the same order of magnitude as the Cabbibo angle Ac. For definiteness in what follows we take them to be equal, although as we show later, the Green-Schwarz evaluation of A gives a slightly higher value. The eigenvalues of these matrices reproduce the geometric interfamily hierarchy for quarks of both charges rut A? rric 4 ~ A. rrit (4.4.34) (4.4.35)

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while the quark intrafamily hierarchy is given by 83 mt (4.4.36) implying the relative suppression of the bottom to top quark masses, without large tan p. These quark-sector 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 dimension-three term appears and the standard model invariant LiejHd have charges X^^\ y}^'^^ '"l The pattern of eigenvalues depends on the X'^l: if Xl^l > -3, we find a supersymmetric zero in the (33) position, and the wrong hierarchy for lepton masses; if = _3, there are supersymmetric zeros in the (21) and (31) position, yielding K A? 0 1 0 1 (4.4.37) ion We could have avoided these supersymmetric zeros by relaxing the assumpt that the 9 fields break at the same scale. We will not examine this possibility here. Notice also that in the single U{\) case, there is again no zeros at these positions. Diagonalization of the above matrix, yields the lepton inter-family hierarchy rur ~ Ac TTIt (4.4.38) The issue of the eigenvalues of this matrix is rather saddle. To make a definite statement, one should know the exact coeflicients of order of one in the Yukawa

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84 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 X^'^ = JsT^, which guarantees through the anomaly conditions the correct value of the Weinberg angle at cut-off, since smH^ = ^ ^ Xt'^=X[«]; (4.4.39) it sets X^'^ = -3, so that -~1; --cot/?A3. (4.4.40) It is a remarkable feature of this type of model that both interand intra-family 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 mfc/m^, which suggests that tan ^ is of order one. 4,5 Neutrino Masses Our model, based on E^, has all the features of 50(10); in particular, neutrino masses are naturally generated by the seesaw mechanism if the three right-handed neutrinos Ni acquire a Majorana mass in the DSW vacuum. The flat direction analysis then indicates that their J^-charges must be negative half-odd integers, that is X^ = -1/2, 3/2, . . .. Their standard-model invariant masses are generated by terms of the form '^m' W 'm' ' (4.5.41)

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85 where M is the cut-off of the theory. In the {ij) matrix element, the exponents are computed to be equal to -2Xj^ plus (4.5.42) (0,4,0) (0,2,1) (0,0,-1) (0,2,1) (0,0,2) (0,-2,0) \^ (0,0,-1) (0,-2,0) (0,-4,-2) j If Xjf = -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 (•3) ^ 0; such flat directions are dangerous because they can lead to vacua other than the DSW vacuum. If Xjj < -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 = -3/2, which still yields one harmless supersymmetric zero in the Majorana mass matrix, now of the form MXl Ac A^ 1 1 0 (4.5.43) Its diagonalization yields three massive right-handed neutrinos with masses MX 13 (4.5.44)

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86 By definition, right-handed neutrinos are those that couple to the standard-model invariant LiHu, and serve as Dirac partners to the chiral neutrinos. In our model, X{LiH^Nj) = ^ 0 . The superpotential contains the terms (4.5.45) (0) (1) (2) (4.5.46) resulting, after electroweak symmetry breaking, in the orders of magnitude (we note V, — /uo TO) AJ A^ A^ 1 Ac^ A^ 1 (4.5.47) 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 MA3 A^ A? Ac^ 1 1 A? (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 right-handed 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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87 anomaly cancelation. In the few models with two non-anomalous horizontal symmetries based on Ee 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 f 1 A? Wmns = 1 1 [ac^ 1 (4.5.49) so that the mixing of the electron neutrino is small, of the order of A^, while the mixing between the /i and r neutrinos is of order one. Remarkably enough, this mixing pattern is precisely the one suggested by the non-adiabatic 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 Super-Kamiokande [46] and Soudan [47] collaborations). A naive order of magnitude diagonalization gives a fi and r neutrinos of comparable masses, and a much lighter electron neutrino: m^, ~ TUoXl; m^^ ~ mo ; mo = . (4.5.50) The overall neutrino mass scale mo depends on the cut-oflF 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 m^,. However, the estimates

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88 (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 Ac in (4.5.48) can spoil the naive determination of the mass eigenvalues. In order to take this elTect into account, we rewrite the neutrino mass matrix, expressed in the basis of charged lepton mass eigenstates, as: / mo bXl cXl hXl d e cXl e f \ (4.5.51) where the prefactors a, b, c, d, e and /, unconstrained by any symmetry, are assumed to be of order one, say 0.5 < a, . . . / < 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 = ^-dy=^: m^^ 1 VI 4a; m^3 1 + v'l 4a; ' sin^2^^^ = 1 1 4a; (4.5.52) Scenario 1 corresponds to both regimes 4a; ~ 1 and (-4a;) > 1, while scenario 2 requires |a;| < 1. Let us stress that small values of \x\ are very generic when d and / have same sign, provided that df ~ e^. 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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89 point of view. We have in this limit, m^^ < m^^. The two distinct oscillation frequencies Arnfj and Amj^ ~ Amjj can explain both the solar and atmospheric neutrino data: non-adiabatic MSW u^^r transitions require [71] 4 X 10-^ eV^ < Am^ < 10"' eV^ (best fit: 5 x 10"' eV^) , (4.5.53) while an oscillation solution to the atmospheric neutrino anomaly requires [57] 5 X 10-'' eF^ < Am' < 5 x 10"^ eF^ (best fit: 10"' eF^) . (4.5.54) To accommodate both, we need 0.03 < ^ ~ a: < 0.15 (with x = 0.06 for the best fits), which can be achieved without any fine-tuning in our model. Interestingly enough, such small values of x generically push sin' 29 towards its maximum, as can be seen from (4.5.52). Indeed, since d and / have the same sign and are both of order one, y' 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 sin' 29 = 1. To be more quantitative, let us fix x and try to adjust y to make sin' 29f,r as close to 1 as possible. With x = 0.06, one obtains sin' 29^^ = 0.9 for y ~ 0.3, sin' 29^r = 0.95 for y ~ 0.2 and sin' 29^^ = 0.98 for i/ ~ 0.1. This shows that very large values of sin' 29^r can be obtained without any fine-tuning (note that y = 1/3 already for d/f = 2). Thus, in the regime a; < 1, i/^ o z/^ 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 /i and r neutrinos, with parameters Am' = Amf^ and sin' 26' = 4m'A^ To match the mixing angle with experimental data, one

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90 needs m ~ 3 — 5 ; we note that such moderate values of u are favored by the fact that df ~ e^. In both scenarios, the scale of the neutrino masses measures the cut-off M. In scenario 1, the MSW effect requires mo ~ 10"^ eF, which gives M ~ 10^^ GeV. In scenario 2, the best fit to the atmospheric neutrino data gives mo {d + f) — + mt,3 ~ 0.03 eV, which corresponds to a slightly lower cut-off, 10^^ GeV < M < 4 X 10^^ GeV (assuming 0.2 < d + / < 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 Xjf, since = I XT*'^\ (4.5.55) but the value Xj^ = -3/2 is precisely that favored by the flat direction analysis. As a comparison, Xj^ -1/2 would give M ~ lO^^ GeV, and Xj^ < -5/2 corresponds to M < 10^^ GeV. Turning the argument the other way, had we set M = Mu ab initio, the value of Xjf favored by the flat direction analysis would yield precisely the neutrino mass scale needed to explain the solar neutrino deficit, mo ~ 10"^ eV. Other values of Xj; would give mass scales irrelevant to the data: X^ = -1/2 corresponds to mo ~ lO"*" eV, which is not interesting for neutrino phenomenology, and Xj^ < -5/2 to mo > 10 eV, which, given the large mixing between /x and r neutrinos (and assuming no fine-tuned 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 order-one factors that appear in the neutrino mass matrices. The cut-off M , which is related to the neutrino mass scale, is determined to be close to the

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91 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 Vector-Like Matter To cancel anomalies involving hypercharge, vector-like matter with standardmodel charges must be present. Its nature is not fixed by phenomenology, but by a variety of theoretical requirements: vector-like 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 ^Jg-inspired model, with vector-like 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 vector-like matter are determined through the same procedure, namely operators of the form The JC-charges of the standard model invariant mass terms are the same X{DiDj) = X{EiE,) = 2a-4y= -uvl(4.6.57)

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92 Its value determines the X-charge, since X^'^ = -3 and Xjj. = -3/2 already fix P = -3/20 and a + j = -3/4. It also fixes the orders of magnitude of the vector-like masses. First we note that nyi must be a non-negative integer. The reason is that the power of 6i is hvl, the X-charge of the invariant and by holomorphy, it must be zero or a positive integer. Thus if uvl is negative, all vector-like matter is massless, which is not acceptable. The exponents for the heavy quark matrix are given by the integer Uvl plus ^ (0,-3,-3) (0,-1,-3) (0,1,-1) ^ (0,-2,0) (0,0,0) (0,2,2) \^ (0,-1,1) (0,1,1) (0,3,3) Those of the heavy leptons, by uvl plus ^ (0,-3,-3) (0,-2,-2) (0,-1,-1) ^ (0,-1,-1) (0,0,0) (0,1,1) (0,1,1) (0,2,2) (0,3,3) DiDj (4.6.58) (4.6.59) 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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93 • i^vL = 0. We obtain the mass matrices MDD M /O 0 0 \ 0 1 VO XlJ I 0 0 0 1 0 \ 2 x6 (4.6.60) Diagonalization of these matrices yields one zero eigenvalue for both matrices and nonzero (order of magnitude) eigenvalues M and AjM for M-^j^ and M and A^M for Mg^;. 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 ny^ = 0. • J^KL = 1The mass matrices are MDD M 0 A3 A^ A^; Mee^M /O 0 \c\ Ac Ag Ag (4.6.61) The eigenvalues for M;^^ are A^M, A^M and A^M and for M-^^ X^M, X^M and XlM. The splitting between the members of the third family vector-like fields is too large and as a consequence, gauge coupling unification is spoiled. • ^VL = 2. The mass matrices are: / 0 0 XI \ 10 A^ A A« Xl^J /O A^ MEE M A^ A^ A? VA 10 (4.6.62) The eigenvalues are now A^M, X^M, X^M and A^M, A^M, Aj^M, respectively. There is again splitting between the families of the doublet and the triplet and

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94 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. • = 3. We obtain the mass matrices A! A? AP VA n Mee M ( A? A^ Aj \ Aj A^ A^^ AJ^ \'^} (4.6.63) with eigenvalues: and Me = {XlM, XlM, X'J'M} , (4.6.64) (4.6.65) respectively. The unification of couplings in this case is preserved. For uvl > 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 one-loop analysis using self-consistently M — My in the mass of the vector-like particles and for the unification scale, yields unified gauge couplings at the unification scale, Mu 19 ' M[/ ~ 3 X lO^^GeV (4.6.66) For nvL large, other problems arise as the vector-like matter becomes too light. This can easily spoil gauge coupling unification by two loop effects [73] and cause

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95 Table 4.1: Operators that mix MSSM fields with vector-like matter with /? = -3/20. Place 1 (ylass 6 XT' Tj / 3 nv r \ EHu — / ^ \ uDD (-nuL f ) (-f ^) Dd (-^) (-nvL f ) gc^L) (-5 — ^) QDHd (-3 ^) EEe {-nvL f ) tieHcL { — 6 QuE (-1 ^) EQd (-1 ^) LQD (-1 ^) LEe (-1 2^) Due (-f ^) significant deviations from precision measurements of standard model parameters [74], [73]. Thus the unification of the gauge couplings favors Uvl = 3. The value of nyi also determines the mixing between the chiral and vector-like matter. Indeed, the quantum numbers of the vector-like matter allow for mixing with the chiral families, since {Ei, Lj, Ha), (Ei with and (A with 5^) have the same standard model quantum numbers. This generates new standard model invariants. In table 4.1, we give a set of mixed operators up to superfield dimension 3. Next to the operator we show its X-charge.

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96 One notices that the operators fall into three classes. For nyi odd only the operators of the first class can appear in the superpotential and for tivl even only operators of the second class appear. The third class is excluded for any integer value oinvLLet us examine these two possibilities in more detail. nvL = 2,4,6,... Only operators of the second class are allowed in W and D mixes with d. The mixing is computed by diagonalizing the down type quark mass matrices. To see this, we give a one family example where the operators DD, QdHd, QDHd, and Dd are all present in the superpotential. After electroweak breaking the masses of the down type quark fields come from diagonaUzing the matrix The extra quark fields aflPect the down quark mass matrices of section 5 and modify our previous order of magnitude estimates. The same type of mixing happens in the lepton sector due to the operators EE, LE and EeHdIf allowed, this type of mixing produces phenomenologically unacceptable mass patterns for quarks and charged leptons. nvL = 3,5,... Operators of the first class are allowed since their X charges are all negative integers. Due to the mixing of the heavy leptons with the Higgs doublets, we have to diagonalize the following mass matrix (we give again a simple one family example) : (4.6.67) (4.6.68)

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97 The 11 entry is the fx term generated by the Giudice-Masiero mechanism and is naturally of order of a TeV. The Higgs eigenstates will be modified to K^H^ + Y^dl-E, (4.6.69) i and K = H, + Y^c^-Ei . • (4.6.70) t where c"''' are mixing angles to be obtained upon diagonalization. With both off diagonal entries present, this matrix has two large eigenvalues and consequently the Higgs mass is driven to the Planck scale. If one of the off diagonal entries is missing, then the matrix has one small and one large eigenvalue and the mixing is harmless as long as the angles c"'"^ are small (see later). There are several ways to evade these problems. One is to relax the simple but very restrictive assumption that X is the same for both the MSSM and the vector-like fields and another is to assume the existence of a discrete symmetry that prohibits the dangerous operators. We now procede to examine each of these possibilities in detail. 4.6.1 Shift X. The vector-like matter could come from a different 27 than the MSSM fields so that the X-charges of the vector-like fields are shifted relative to the fields in the 16: XvL ^a + pV + jV. (4.6.71)

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98 Table 4.2: Operators with vector-like matter with Xy^ = a + /3V + jV. Class 1 Class 2 Class 3 EHJ2P ^ + ^) LE(-2B+^ + ^) uDDiAd nvr —) a\ !^ 2 10/ Dd(-2d ^ ^) EOD(43 nvr —) QQD(-2p-^ m ^ "V. " 2 in/ j-ij^^y-zy "'V L If)/ udD{2P ? ^ lALL 1 EeHa{2'^-^-^^) i — ; ^ HI / QuE{-2p ^ '^,) EQd{2'^ LQD{2P ^ II) LEe{2P '4) Due{-2P ^ '4) In table 4.2 we show the different operators with their X-charges. It is interesting to notice that the X charges of these operators depend only on /? and nvL = -2a -I47. We have again two possibilities. • No MSSMVector Like mixing We can choose /? in such a way that none of the X charges of the operators appearing in table 4.2 is an integer for any integer uvlNone of them will appear in W and therefore we avoid the mixing problem. Then, the lightest of the vector-like fields will be stable. To avoid cosmological problems, this

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99 requires a reheating temperature lower than the lowest vector-like mass in order to dilute their abundance during inflation. Recall that the mass of the lightest pair of D and E for uvl = 3 is Aj^M ~ \Qi^-''GeV, and therefore a reheating temperature of at most this order of magnitude is required, uvl = 4 or higher result in lower eigenvalues and thus lower reheating temperatures. We therefore favor in this case ny^, = 3. Similar arguments apply to any other scenario with stable heavy vector-like states. • Partial MSSMVector Like mixing Let us take uvl = 3 which avoids the dangerous d-^ and L-'E mixing. The X charges of the operators that could give rise to mixing are X{EHu) = 3/10 and X(EHd) = -2P 33/10. We can choose p in a way that X{EH^) is positive and X{EHa) is negative and so prohibit EHu from appearing but allow EHd. This yields the mass matrix (7.10) with its 21 element being zero. As we mentioned before, the mixing is harmless if the angles c"''' are small which is indeed the case. We still have to check if the proton decays due to mixed operators slowly enough to avoid conflict with experimental data. Proton decay due to operators consisting only of MSSM fields will be discussed in a separate section, since it is independent of the choice of the charges of the vector-like matter. We find that the dominant proton decay channels come from the operators LQD and udD (4.6.72)

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100 and QQD and Due (4.6.73) via an intermediate heavy quark. They appear after DSW breaking as KjkUQjDk + XijkuidjDk (4.6.74) and pijkQiQjDk + PijkDiUjek (4.6.75) where is the suppression factor in the DSW vacuum in front of the corresponding operator with flavor indices i,j,k. Similar expressions hold for Ai^^, pijk and PijjtThe experimental constraint on these is [75] KjkKjk < Ml 10-32 {A.^.n) and similarly P^3kP^jk < Ml 10-32 GeV~\ (4.6.78) We computed the suppression factors of these operators in the DSW vacuum that the model gives for = 7/20 and we found that the above constraints are not easily satisfied, which excludes this possibility.

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101 4.6.2 Discrete Symmetry It is known that superstring models usually contain discrete symmetries. If present, they could forbid the dangerous mixed operators, leaving the mass terms for the vector-like matter intact. As an example, consider the discrete symmetry where E^-E, E^-E, -£>, D -D. (4.6.79) This additional symmetry indeed completely decouples the MSSM fields from the vector-like matter. No operator with an odd number of vector-like fields is allowed for any value of uvlSpecifically, all operators that mix MSSM fields and vectorlike matter and that can cause proton decay are also prohibited. Such, are the dimension-3 operators LQD and udD (4.6.80) that belong to class 1 and the dimension-4 operators QQQE, TPuDe. (4.6.81) As a consequence of this discrete symmetry the vector-like matter has no available decay channels. This can have undesired cosmological implications except if inflation takes place at a temperature lower than the lightest of the vector-like particles. For this reason we strongly favor the value uvl = 3. Also in this case we can keep the simple universal X charge assignment X = a + l3V + yV' for both the MSSM and the vector-like fields which makes the flat direction analysis particularly sim-

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102 pie because the superpotential has a very small number of supersymmetric zeros corresponding to standard model invariants with vector-like fields. 4.6.3 Summary To summarize, we have given three alternative ways to fix the X charges of the vector-like fields. The solution of section 7.1.1 is viable fornvL = 3. A reheating temperature 10^~^ GeV is required. Lower reheating temperatures are required as uvl increases so in this case nyi = 3 is clearly favored. The solution of section 7.1.2 = 7/20) is not viable even if the mixing angles c"''' are small. The proton decays rather fast. The vector-like particles can decay. The solution of section 7.2 involves a discrete symmetry. Stable heavy quarks and leptons require a reheating temperature ~ 10^~^ GeV for uvl = 3 and lower temperatures for higher values of uvl, so tivl = 3 is again favored. In this case the flat direction analysis is particularly simple. We do not have any physical motivation that can tell us which of the above proposed mechanisms is the correct one. The simplest is the scenario with the discrete symmetry and from now on we will continue our discussion on the hidden sector, flat directions and proton decay in this context.

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103 4.7 The Hidden Sector So far we have described the matter necessary to satisfy the anomaly conditions that involve standard model quantum numbers, the breaking of the extra gauge symmetries, and phenomenology. These are the three chiral families, the three right-handed neutrinos, the three vector-like families just described, and three 9 fields necessary to produce the DSW vacuum. We refer to this as visible matter. By fixing the value oiX(EE) = X(DD) = -nyi, the X charge is totally determined. Since gauge unification favors uvl = 3, the weak and color anomalies are fixed, Ccoior = C'weak = "18. This enables us to "predict" the value of the Cabbibo angle through the relation where it is assumed for simplicity that the Kahler potential takes its tree level weak coupling form: (4.7.82) K^-\og{S + S) = -\og{2y). (4.7.83) Using the Green-Schwarz relation gray 'weak weak 12 (4.7.84) and the identification (4.7.85)

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104 we relate the Cabbibo angle to the gauge couplings at the cut-ofF a(M), using only visible matter contributions For nvL = 3, the couplings unify with a ~ 1/19, which yields A = 0.28, clearly of the same order of magnitude as the Cabbibo angle! Given the many uncertainties in this type of theory, the consistency of these results with Nature is remarkable. We note that the numerical value of the expansion parameter clearly depends on the contribution of the vector-like matter to Cweak, about which we have no direct experimental information. In addition, the values of the mixed gravitational anomaly is also determined through the relation For integer ^^eak and uvl = 3, this implies that Cgrav = -216, -108, -72, ... for ^weak = 1, 2, 3 • • •, to be compared with the visible matter contribution to Cgrav = -80. Thus additional fields are required, and fc^eak < 2, to avoid fields with positive X-charges that spoil the DSW vacuum. Another argument for new fields is that not all anomalies are canceled, since we have from the d sector (4.7.86) Cg = 12 weak (4.7.87) weak y(i)y^(i)y(2) ^ y(i)y(2)y(2) ^ _j (4.7.88)

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105 and from all visible matter j!^y(i)y(2) ^ (4.7.89) The construction of a hidden sector theory that cancels these anomalies, and provides the requisite Cgrav is rather arbitrary, since we have few guidelines: anomaly cancelation, and the absence of flat directions which indicates that the X charges of the hidden matter should be negative. In particular, the authors of [76], [77] considered a hidden gauge group Ghid with a pair of matter fields with the same Xcharge, but vector-like with respect to all other symmetries, causing supersymmetry breaking. This theory contributes to few anomalies, only in Cgrav, (XY'-^^Y^^^) and the anomaly associated with the hidden gauge group Ghid, related by the Green-Schwarz relation Cg = -18t^ (4.7.90) "-weak where ka is the Kac-Moody integer level {ka integer heavily constrains possible theories of this type). It must be augmented by other fields, since it does not cancel the remaining anomalies {XXY^^^), (r(i)r(i)r(2)) and (y(i)y(2)r(2)) jhese will be accounted for by singlet fields. There is a simple set of four singlet fields, Ea which absorb many of the remaining anomalies, without creating unwanted flat directions. Their charges are given in the following table:

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106 Si S2 S3 E4 X -1/2 -1/2 0 0 0 0 1/2 -1/2 y(2) -9/4 -7/4 9/4 7/4 They cancel the anomalies from the 9 sector, since over the E fields XF(l)y(2) = 0 , XXF(2) = _i ^ y(l)y(l)y(2) ^ y(l)y(2)y(2) ^ j ^ ^ gj^ as well as Cgrav = -1The remaining anomalies can be accounted for by a sim,ple gauge theory based on Ghid = A^d three families of vector like hidden fields transforming under the fundamental (anti-fundamental) representation (Nc) of SU{Nc) with charges under the C/(l)'s as Qi Qi X -3 -3 2/iVc -2/7Vc y(2) 1/2 -1/2 where z = 1, 2, 3 since Nf = 3. The rest of the anomalies is carried by a set of singlet fields {Tj) which have no charges under the non-anomalous C/(l)'s and their charge under X is -3. This last set of fields is given for completeness, since their only purpose here is, to adjust the gravitational anomaly to be compatible with the Green-Schwarz mechanism. For a hidden sector with kh 1, we need (4.7.92)

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107 The singlet fields have little effect on low energy phenomenology. Computation of the powers of the 9 fields in the mass invariants H^a^b, yield in the DSW vacuum the mass matrix of the E fields before SUSY breaking /O 0 0 0 0 0 0 0 \ 0 0 0 MA? (4.7.93) \0 0 MXl 0 / it has two zero eigenvalues. The Giudice-Masiero mechanism can fill in the 12 (and 21) entries after SUSY breaking, yielding: / 0 mXl mXl 0 0 0 0 \ 0 MA? (4.7.94) 0 0 0 V 0 0 MXl 0 j where m is of order of the SUSY breaking scale. The above matrix has now two large (~ lO^i GeV) and two small (1 100 MeV) eigenvalues. The two heavy states get diluted during inflation. The two light states are stable since their lowest order coupling to the light fields is quartic, dominated by terms like lliHiHuHd. Although stable, and undiluted by inflation, their contribution to the energy density of the universe is negligible. 4.8 ii-Parity The invariants of the minimal standard model and their associated flat directions have been analyzed in detail in the literature [62]. In models with an

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108 anomalous U{1), these invariants carry in general X-charges, which, as we have seen, determines their suppression in the effective Lagrangian. Just as there is a basis of invariants, the charges of these invariants are not all independent; they can in fact be expressed in terms of the charges of the lowest order invariants built out of the fields of the minimal standard model, and some anomaly coefficients. The X-charges of the three types of cubic standard model invariants that violate i?-parity as well as baryon and/or lepton numbers can be expressed in terms of the X-charges of the MSSM invariants and the /^-parity violating invariant X^^ = X{LH^) , (4.8.95) through the relations ^LQd = X^"^ ^f'^' + , (4.8.96) Xllb = X^'^ XM + . (4.8.97) X^id = X^'^ + + i (Ccolor Cweak) " jx^"^ (4.8.98) Although they vanish in our model, we still display Xt"l and X^f"^ = 0, since these sum rules are more general. In the analysis of the flat directions, we have seen how the seesaw mechanism forces the X-charge of N to be half-odd integer. Also, the Froggatt-Nielsen [58] suppression of the minimal standard model invariants, and the holomorphy of the superpotential require x!"''^-^] to be zero or negative integers, and the equality of the Kac-Moody levels of SU{2) and SU{3) forces Ccoior = C'weak, through the Green-Schwarz mechanism. Thus we conclude that the X-charges of these operators are half-odd integers, and thus they cannot appear

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109 in the superpotential unless multiplied by at least one iV. This reasoning can be applied to the higher-order ^ operators since their charges are given by ^QQQH, = ^'"1 + X^'^ ixl'^] , (4.8.99) XdddLL = 2X['^ + 3Xt^ , (4.8.100) Xqqqqu = 2XM + Xl'^-Ix['']-X[/^ , (4.8.101) Xuuuee = Xt'^ + 2X[^1 X^^ , (4.8.102) It follows that there are no i2-parity violating operators, whatever their dimensions : through the right-handed neutrinos, i?-parity is linked to half-odd integer charges, so that charge invariance results in il-parity invariance. Thus none of the operators that violate i?-parity can appear in holomorphic invariants: even after breaking of the anomalous X symmetry, the remaining interactions all respect i?-parity, leading to an absolutely stable superpartner. This is a general result deduced from the uniqueness of the DSW vacuum, the Green-Schwarz anomaly cancelations, and the seesaw mechanisms. 4.9 Proton Decay In the presence of the extra discrete symmetry we introduced before, the operators that mix MSSM fields and vector-like matter and trigger proton decay are excluded. Since /^-parity is exactly conserved, the dangerous dimension 3 operators LQd and udd that usually induce fast proton decay are also excluded. This leaves for the dominant sources of proton decay the dimension 5 operators that

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110 appear in the effective Lagrangian as W = —[Kn2iQ\QiQ2Li + KijkiUiUjdkei] (4.9.103) where for the first operator the flavor index i = 1, 2 if there is a charged lepton in the final state and z = 1,2,3 if there is a neutrino and j = 2,3, k,l = 1,2. We have denoted the suppression factors in the DSW vacuum in front of the operators by k. and k. These operators could for example give rise the proton decay modes p n'^Vi and p -> n^lf or to p -> K'^Ui and p K^lf . In [75], the phenomenological limits on these suppression factors were computed to be: «112i ^ A U c (4.9.104) and (4.9.105) where K'^n^. = V^V^. Vr are the matrices that diagonalize on the right the quark and the squark matrices respectively. We can easily calculate it in this model: / 1 ^RRlj 1 XI c Ac (4.9.106) 1 / In table 4.3, we give in the first column a list of the dangerous operators QQQL (mde) and in the second column the suppression Kijki (KijuK^R) that we computed in our model.

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Ill Table 4.3: Operators inducing proton decay and their suppression. Operator Supression Q1Q1Q2L1 QlQlQ2L2,3 u\U2d\ei uiU2die2 UiU2d2e\ UlU2d2e2 UiU3die2 XI' uiU3d2ei xi^ Uxuzd2e2

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112 Even though all operators in table 4.3 seem naively sufficiently suppressed so that proton decay is within the experimental bound, it is interesting to examine them more closely from the phenomenological point of view. Consider the operator QiQiQ2L2This operator can lead to proton decay via a wino, gluino, zino, photino or Higgsino exchange. The contribution via gluino exchange could be the dominant due to the strong coupling of the gluino. Here let us recall that experimental data strongly suggests a near degeneracy between squark masses in order to avoid large contributions to flavor changing neutral currents (fcnc). One mechanism that has been suggested [78] is where alignment between quarks and squarks takes place and therefore fcnc are suppressed irrespectively of the SUSY breaking mechanism. One can calculate in the model the extent of such an alignment. We find that there is no sufficient quark-squark alignment and therefore fcnc are not sufficiently suppressed. If squarks are approximately degenerate on the other hand, the contribution due to gluino exchange is negligible. Generically, a careful calculation of a proton decay process not only involves uncertainties due to our ignorance of superpartner masses but also due to large uncertainties in hadronic matrix elements. Assuming nearly degenerate squarks, the dominant decay mode is via wino exchange and the decay rate for the process p ->• is given by [79]: r{p ^ /^V^) = ( d^-^^ Lmlf^ |0-7/cn22/(m^,m,)P (4.9.107) were here b = (0.003 0.03) GeV^ is an unknown strong matrix element, ^2 = a/ sin^ dw and from our earlier estimates of the cut-off, M ~ 3 x 10^^ GeV. We

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113 have two regimes to consider « rUg : f{m^,mq) = —^, (4.9.108) and 1 TTt^» TUq : /{ma,, ms) = In — ^ . (4.9.109) The experimental bound on the decay p ->• + )Lt+, which is the dominant one in our model is: Tip -> K^n+) < 10^2 years-^ . (4.9.110) For wino mass much larger than squark masses, this decay rate is several orders of magnitude lower than the experimental limit. For wino masses much lower than squark masses, the rate is near the experimental limit. For example, with ma, ~ 100 GeV, ~ SOOGeV, and b = .003, we get the lifetime ~ 10^^ years, near the experimental bound. Unfortunately our model cannot be more precise, because of the unknown prefactors of order one terms in the effective interactions; Still it predicts that the proton decays preferentially into a neutral K and an antimuon with a lifetime at or near the present experimental limit. Finally we note that if we use the expansion parameter determined through the Green-Schwarz relation, and not the Cabbibo angle, our estimates get worse and our model implies a proton lifetime slightly shorter than the experimental bound. As we remarked earlier, this value of the expansion parameter depends on the contribution of the vector-like matter to Cweak-

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114 4.10 Flat Direction Analysis In the top-bottom approach of constructing models with an anomalous U{1), one typically obtains upon compactification of a superstring a certain set of fields that here we called "^-singlet". Then this set has to be divided into a subset that takes a vev at a scale ^ ("^ set") and a subset that does not ("x set"). Unfortunately, in realistic model building this separation has turned out to be far from unique. From the vacuum point of view therefore the minimalistic, bottomup approach is advantageous, in the sense that one can ensure that the ^-set is unique. The simplest way to achieve this in a model with an anomalous, flavor blind X, is to allow vevs only to those ^-singlets that have nonnegative X charges. The fact that det^ ^ 0 and that the first column of is (1,1,1), ensures the existence of the vacuum configuration (^o, ^i, ^2)Then, we have to extend the basis of MSSM holomorphic invariants of ref. [62] to take into account the presence of vector-like matter in the model, and compute the powers of the 6 fields for each basis invariant, as well as for each Standard Model singlet. In addition we must check that the superpotential does not contain a term linear in a x field (such a term would be for example iVi^o°^r^r with no,ni,n2 nonnegative integers). It turns out that, for a convenient choice of the X-charge, all the supersymmetric zeroes of this model reside in the sector that contains the fields lying in the 16 of 50(10). Consider first the flat directions involving only standard model singlets. Assuming for simplicity that only one x field acquires a vev, we must distinguish between two cases:

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115 • all Ua are positive. Then |(^a)P > for a = 0, 1, 2 , whatever (x) may be. In addition, the superpotential contains an invariant of the form x"* O^^O^^O^^, with rua = mua (as discussed in Section 4.10, m > 2 is required in order not to spoil the DSW vacuum). The F-term constraints then impose (x) = 0: the flat direction is lifted down to the DSW vacuum. • some of the are negative. The relations |(^q)P > no longer hold, and the low-energy vacuum may be different from the DSW vacuum. In our model, this happens only for N3, for which rzi, ^2) = (3/2, -1/2, 1/2). One can then see from (2.2.31) that the vacuum (N3, 60,62) with |(]V3)P = 2^2^ \{6o)\^ = 4e and |(^2)P = 2e is perfectly allowed by Dterm constraints. This is a rather unwelcome feature, because most Yukawa couplings vanish in this vacuum. Fortunately, the superpotential contains an invariant 6^6^, with no power of di, which lifts the undesired vacuum. This discussion can be generalized to flat directions involving several x fields; we conclude that the model does not possess any other stable vacuum of singlets than the DSW vacuum. Thus, the low-energy mass hierarchies are completely determined by the symmetries at high energy. Remarkably enough, those conditions are always fulfilled in our model, despite the great number of standard model invariants. A list of all the supersymmetric zeroes of the model can be found in table 4.4 and 4.5. It is remarkable that among the total number of invariants which is of order of thousands we find only a few zeroes due to negative powers. The first column of

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116 Table 4.4: Flat Directions (FD) of MSSM and N fields. Basis Invariant (ni,n2,n3) FD FD lifted by Ns (3/2,-1/2,1/2) < N3,ei,93 > N2N3 0^,0i L\Hu (-3/2,1/2,5/2) < Li, Hu, 02, 63 > J. u u ^ L2HU (-3/2,1/2,-1/2) < L2, Hu, 02, O3 > L2N3HU (-3/2,1/2,-1/2) < -^3, Hu, 02, 03 > L3N3HU (3,2,-1) < L2,ei, Hd,9i,92 > L2e3Hid\ (3,2,-1) < L3,ei,Ha,ei,02 > L3e3Ha0\ -£'2^361 (3/2,5/2,-3/2) < L2, L3,ei, 61,62 > L2L3e,Ni0l0l £'2-^>363 (3/2,1/2,-1/2) < L2, L3,e3,6i,62 > L2L3e3N36\ (3/2,1/2,-1/2) < J^2, Q3, d2, 01, 62 > L2Q3d2Ns6l (3/2,1/2,-1/2) < L3,Q3,d2,6i,62 > LzQzd2Nz6\ L2Qzdi (3/2,1/2,-1/2) < L2,Q3,d3,6i,62 > L2Qzd3Nz6\ (3/2,1/2,-1/2) < L3, Q3, 6,3, 01,02 > LzQ3d3Nz6\

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Table 4.5: Continued: Flat Directions (FD) of MSSM and fields. Invariant (ni,n2,n3) FD FD lifted by (3/2,1/2,-1/2) U3d2d3N39\ QsUseiHd (9/2,3/2,-1/2) < Q3,U3,ei,Hd,9i,92 > Q3U3HU (9/2,-1/2,1/2) < Q3,U3,e3,Hd,9i,93 > (3,2,-1) < Q3,U3,L2,ei,ei,92 > Q3U3HU (3,2,-1) < Q3,U3,L3,ei,9i,92 > Q3U3HU (9/2,3/2,-1/2) < Q3,U3,ei,9i,92 > Q3U3HU (9/2,-1/2,1/2) < Q3,U3,e3,9i,93 > Q3U3HU

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118 that table contains the basis invariant. As an example, consider the invariant: didid^LiU. (4.10.111) In the second column, by solving (2.2.28), we compute the powers of the 9 fields corresponding to that basis invariant. For the above invariant, these are computed to be: (4.10.112) The third column shows the potentially dangerous flat direction associated with the invariant of the first column. For our example this is the direction: < di,d2,d3,L2,L3,9i,92 > . (4.10.113) The fourth column contains an holomorphic invariant that lifts this flat direction. In our case such an invariant would be: did2diL2Lj^z9i^92. (4.10.114) Finally, we would like to stress the fact that in this example we checked flat directions that correspond only to basis invariants. However, as mentioned in the general discussion, in a complete analysis one has to look at all possible invariant polynomials containing negative powers and their associated "composite" flat directions, and check that the invariants present in the superpotential actually lift them.

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119 4.10.1 Flat Directions with Vector-Like Matter The case of flat directions involving vector-like matter is slightly diflterent. Since we have assumed the existence of a discrete symmetry that prevents numerous invariants from appearing in the superpotential, there could be flat directions associated with these invariants. But this is not the case, as long as the vector-like fields are massive. Their F-terms take indeed the following form (gauge indices are not shown, and powers of the 9 fields have been absorbed in the mass matrices for simplicity): F-E, = M-^,EjEj + ... F-^^ = M-^^i,.Dj + ... (4.10.115) Fe, = M^^^.Ei + ... Fd, = M^,DjDi + (4.10.116) where the dots stand for possible higher order contributions. Since the matrices ^EE ^DD invertible, one concludes that the vanishing of (4.10.115) and (4.10.116) forbids any flat direction involving vector-like fields, provided that it is associated with an invariant for which all are positive. That this is true also for invariants with one or several negative is less obvious. It is due to the following features of the model: the (1,1) entry of the vector-like mass matrices are generated from the superpotential terms EiEi 9^ and DjDi 9^, and all invariants that have one or several negative both satisfy no > 0 and contain at least one vector-like field of the first family. Therefore, condition (ii) is always fulfilled. This can be checked in table 4.6 (where only operators up to superfield dimension 4 have been displayed).

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120 Table 4.6: Flat Directions (FD) involving vector-like matter (up to quartic operators) in the discrete symmetry scenario. Basis invariant (ni,n2,n3) FD FD lifted by (3,-1,-1) EiEi 91 (3/2,-1/2,3/2) < Di,di,ei,93 > DiDi9\ (3/2,-1/2,1/2) < Dud2,3,9i,93 > DiDiOf (3/2,-1/2,3/2) < Li, El, 61,93 > EiEi 91 (3/2,-1/2,-3/2) < -f'2,3. El, 9i,92 > EiEi9\ (9/2,1/2,-1/2) DiDi6\ Q3U3E1 (3,-1,-1) < Q3,U3,Ei,9i > EiEi9\ L2,zQiDi (3,1,-1) < L2,3,Q3,Di,9i,92 > DiDi9\ (3,-1,1) < Di,u3,e3,9i,93 > DiDi9l uzd2,zDi (3,1,-1) < U3,d2,3,Di,9i,92 > DiDi6\ QsUsQsDi (9/2,1/2,-1/2) < Q3,U3,Q3,Di,ei,92 > DiDi9\ DiU2U3ei (9/2,7/2,-1/2) < Di,U2,U3,ei,9i,92 > DiDi 9\ Q3D1D2E1 (9/2,-1/2,1/2) DxDi 91

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121 We have thus checked that the superpotential contains terms that lift all flat directions associated with a single standard model invariant. This is not suflScient, however, to ensure that the standard model symmetries are not broken at the scale ^. Other invariants than those of tables 4.4, 4.5 and 4.6 are in general necessary to lift completely the flat directions associated with several standard model invariants and singlets. While we did not perform a complete analysis which would be rather tedious, it is clear that most, if not all, flat directions are forbidden by the F-term constraints. We conclude that the vacuum structure of our model is satisfactory: the only stable vacuum of singlets allowed by Dand F-term constraints is the DSW vacuum, and flat directions associated with a single SU{3YxSU{2)wxU{1)y invariant are lifted by the F-terms. The only expected effects of supersymmetry breaking are to lift the possible remaining flat directions, and to shift slightly the DSW vacuum by giving a small or intermediate vev to other singlets or to fields with standard model quantum numbers. 4.11 Supersymmetry Breaking We saw before that our hidden sector is capable of breaking supersymmetry. We know examine in detail if such a mechanism applied to our model can produce a low energy picture, consistent with existing experimental bounds. Starting, we remind that the Green-Schwarz anomaly cancelation mechanism occurs if the non zero anomaly coefficients Ck and the corresponding Kac-Moody levels kk, satisfy Y = IStt^^gs, for all k, (4.11.117)

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122 with 'Jgs = j^, C,=Tr{X) (4.11.118) where the vacuum expectation value of the dilaton generates a Fayet-Iliopoulos term that triggers the breaking, generating a scale ^, slightly below the string scale. Some time ago, a class of models in which supersymmetry breaking is communicated to the low energy world by an anomalous U{1) was proposed, in the context of global supersymmetry [76], [77], [80]. The soft masses, in the context of these models, have two types of contributions: ml = ml + mf = ^K2\\'' Qli, (4.11.119) a=x,y('),... where mo is the (family independent) contribution from the dilaton F-term, rui the (generically family dependent) contribution from D-terms and K2 is the second derivative of the Kahler potential K, with respect to the real part of the dilaton field 5: 2/ = (l/2)(5+5). K is an unknown function of y. We will also assume that the non-universal couplings of the dilaton are suppressed. We will try to answer the following question: Is it possible to construct a model in which the presence of the family dependent t/(l)'s is not disastrous for flavor physics? We distinguish two possible phenomenologically viable scenarios. • ml « mf « m\i2. The first is a "no scale" type of scenario, where mo is very close to zero and the non-universal rrii are larger but still small enough, so that when extrapolated to the MSSM scale, they do not give dangerous contributions to fcnc because the running of the soft masses to low scale is dominated

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123 by (large) gaugino masses. Such a boundary condition is obtained when the dilaton is stabilized at a very small value of K2This can be achieved by assuming a weakly coupled form for K. An example of such a Kahler potential was proposed in [80]: which as long as 6 > 0 and 6 < l/^^, has a minimum near = sq {1/q) and the values of the derivatives at the minimum are given by: Indeed, \Ki\ < 1 and K2 « 1, since for reasonable values of the parameters, g is a number ~ 20 80 (see below) and therefore Sq ~ 3/2. This type of models however, as we will see, are plagued by charge/color breaking minima because of the absence of large contribution to mo from the dilaton, that tends to stabilize the vacuum. In order, therefore, to construct viable models of this type, we will have to assume the existence of additional family independent F-term contributions from other moduli that stabilize the low energy vacuum. • mf « ml ~ The second is a "minimal sugra" type of scenario, where in order to suppress fcnc, we require that all the D-term contributions are very small. We will now argue that in the extreme case where these exactly vanish, we can make

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124 predictions for the soft parameters. Following [80], upon integrating out the heavy gauge field associated with X and taking the D term part of its equation of motion at the minimum, we obtain a relation between the vacuum expectation values of the anomalous D-term Dx and that of the dilaton F-term F5: < >= -i| < Fs > P[| (4,1,22) In the presence of additional non-anomalous U{\) factors, we can similarly integrate out their heavy gauge fields and equation (4.11.122) still holds. We assume that there is no appreciable kinetic mixing between the i7(l)'s. The scale of the FI term can be evaluated from ~j = (4.11.123) For a confining gauge group in the hidden sector, there is a non perturbative contribution to the superpotential that is of the form Winp) ^ Q^-qS^ (4.11.124) where 9 is a model dependent (group theoretical) number and the prefactor B has units of mass cubed. The contribution of the dilaton to the scalar potential then becomes: = \kM\^ = ji^^n? = VB^I^, (4,1,25)

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125 where we have used (4.11.124) and that -4 In a dilaton dominated scenario (where V ^ V^), the above term dominates the minimization condition Vi = 0 and therefore at the minimum we get the condition ^ = -2q. (4.11.127) Substituting (4.11.127) into (4.11.122), we deduce that in order the D-term contribution to the soft masses to vanish, the following has to be satisfied at the minimum: ^ = -2q. (4.11.128) This implies that in order to have degenerate squarks after supersymmetry breaking, the form of the Kahler potential (at the minimum) has to be of the form: K = ce-^'fo. (4.11.129) The constant c can be fixed from (4.11.123). Doing so, we obtain for K2 at the minimum: Knowing K2, allows us to compute the soft masses from (4.11.119), in a model with known superpotential.

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126 We emphasize that the two types of limits are quite different. In the first "no scale" limit, we have to assume a form for the Kahler potential which stabilizes the dilaton at a very small value of K2 [80]. In the "minimal sugra" limit, we assume a form for K, such that the value of K2 at the minimum is ~ 1, and we impose that the D-term contribution to the soft masses vanishes. The function K in this case may contain both perturbative and non-perturbative contributions. For example, the potential K{y) = -ln{2y) + + c, + c, j'e-'^^'-^^^' dt, (4.11.131) stabilizes the dilaton at Vq — Sqcq (for cq small) and satisfies the fcnc constraint K^z{yf))Ki{yQ) = Ksiyo^ with ^'2(2/0) = 1 with q being appropriate constants. The natural value for yo in our model will be the value of 1/5^ at the unification point, 1.43. 4.11.1 Supersymmetry Breaking with ?7(l)'s In this section we extend the supersymmetric breaking mechanism of [76], [77] and [80] for the case of one anomalous and an arbitrary number of non-anomalous C/(l)'s. The U{iys break slightly below the string scale by the vevs of a set of singlet fields that we call OaThe number of these singlets is equal to the number

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127 of the additional f/(l)'s, so that their charges form a nonsingular square matrix: X2 (1) (1) (1) 2/0 Vi ?/2 yo^^ y? (4.11.132) \ 7 where the first row contains the charges of da with respect to the anomalous symmetry X, the second row the charges with respect to the non-anomalous F^^^ and so forth. The supersymmetric vacuum is defined to be the solution of the equations (4.11.133) Denoting the vevs < > by t;^, the L>-flatness condition 0 0 V • / V • / and the gauge invariance condition for the mass term of a field t (4.11.134) 2 ^m) ^m) ^m) (4.11.135)

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128 in the superpotential, A fPo\ Pi P2 0 0 (4.11.136) V • / V-/ give the supersymmetric vacuum constraint (p^\ vl Pi vl 1 P2 p \ ) [ J (4.11.137) where is the Fayet-Iliopoulos term generated by the breaking of the anomalous t/(l), and p = -n/^^ where n is the X charge of the field t^. M is the cut-off scale of our theory ~ W^'^'^GeV. From (4.11.137), we can see that the ratio Pa/vl = const. We will be looking for a supersymmetry breaking vacuum in the vicinity of this vacuum. In the following, we assume for simplicity that Ghid is a semi-simple, compact, non-Abelian gauge group and that there is only one type of hidden condensates. If there are other hidden fields besides those forming the condensates, they are singlets of ChidWe also assume that the number of hidden colors Nc, is greater than the number of hidden families A^;, in which case the non-perturbative super-

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129 potential is particularly simple. Gaugino condensation occurs at a scale where the hidden sector beta function blows up. This scale, is calculated from the renormalization group equation to be A = Me-^'''*=''(2^)/''° = Me~S^, (4.11.138) where h is the Kac Moody level of the hidden group G'' and 60 is the one loop beta function of the hidden sector. Below this scale, condensates of the hidden "quark" fields qi will be formed: ti = {2qiq,Y'\ (4.11.139) where the index i counts the number of hidden families Nj. In the following we will always assume that it is possible to diagonalize the condensate's mass matrix and in addition that all the condensates have the same mass. In this case t becomes a diagonal matrix with equal entries along the diagonal so we can simplify the calculation by minimizing the scalar potential for a single t and keeping in mind that it is multiplied by an Nj x Nj unit matrix. We are ready now to write down the scalar potential to be minimized. It's general form is V = F0 + V^^ (4.11.140) where ^'' = E|^|+|^ + E (4.11.141)

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130 the D-terms in the above are Dx = -9l[xo\eo\^ + + x^m' + ... + in|t|2 + e]+ • (4.11.142) ^y(a) = -gliyi'W + 2/i"Vir + yi'^W + ] + • , (4.11.143) where a runs over only the non-anomalous C/(l)'s. The superpotential is given by (4.11.144) where dr is the Dynkin index of the representation r of the hidden gauge group (r = o is the adjoint). Using (4.11.138) and (4.11.144), we can express the model dependent constant q in terms of group theoretical numbers: g= A. M (4.11.145) Consider now the minimization conditions 9^^ = {po l)\Fe,\^+Po\Fe,\^+Po\Fe,\''+Po-^F;W^^ -\9o\\xoDx + yi'^Dy, + Dy, + •••) = 0 (4.11.146) dV \F,\' da 2dr -Nf)'

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131 -InfDx + t\V^t\ = 0 (4.11.147) 2 where \Fg\'' = \Fg,\'' + \Fg,\'' + jF^.p + • • • and -2 dV^ tV\ = where V^t = (4.11.148) t-Nf ' dt Defining Nc=^, (which for fields transforming in the fundamental of SU{Nc) is just the usual color Nc), the F-terms entering the above equations are and We are looking for a minimum in the vicinity of the DSW vacuum: < Ft >~ 0, < ^ >~ 0 and < 6^ >~ C (4.11.151) The first of the above conditions, implies that W^(P) = (4.11.152) ^ f in the vacuum. Then, (4.11.147) becomes ^F/W^f") = -ft\V't\ f\Fg\^, (4.11.153)

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where we have introduced / = (4.11.146), we get 132 — Nf + 1). Substituting this into {po l)\Fef+po{l f){\Fe\' \Fe,\') pofW poft^^ = \eo\HxoDx + yi'^Dy,,, + yl'^Dym + • • •) (2) (4.11.154) which in the vacuum (where < >= t;^ and everything is evaluated at the minimum), becomes W^(P)V[(1 /)(P0 +Pl +P2 + ...) 1] Pft xo < Dyw > +yo' < Dyw >+•• , (4.11.155) ,(2) where we have used (4.11.137). Notice that the left hand side of the above equation does not depend on the index "0", so all the minimization conditions with respect to all 6a can be obtained from this, by interchanging the subscripts of the right hand side by "a". This in turn implies that we can solve these equations for the vevs of the D-terms: f < DyW > < Dym > 1\T /1\ 1 1 (4.11.156) V-/

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133 where N, + 1-Nf 1 (P0+Pl+P2 + -")-l]-P Nc + l-Nf 2 , (4.11.157) which after some algebra, can be written in the more convenient form: C = (e^m^n^ )[( iVe + 1 iV^ 1 (P0+Pl+P2 + -)-l) ( {Nc-Nf + l)nK2^M^ ;j(4.11.158) Before defining the new quantities that appear in this formula, we can make a useful remark. The above relation together with (4.11.156), imply that the values of the D-terms are proportional: where Ax is the sum of the entries of the first column of Ai is the sum of the entries of the second column of A'^, etc. This shows that in general, the £)-terms contribute to supersymmetry breaking, but also that for detA / 0, if < > vanishes, then all other < Dy^a) > vanish as well. In addition, using (4.11.126), (4.11.138) and (4.11.144), we find that the vacuum value of the F-term associated with the dilaton is < Dy(a) >^ < Dx > (4.11.159) (4.11.160) and we have defined, as usual, the helpful variables m^MA^AfAf... with A« =< ^ M >, (4.11.161)

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134 Nf-Nc-l l»e= ' = with A = Me-"'"'% (4.11.162) Here, yo =< y >= l/g{Mf where g{M) is the value of the gauge coupling at the unification scale M and we have assumed that the dilaton gets somehow stabilized to a reasonable value yoThe one loop beta function is given by 60 = 3c?a ^dr. (4.11.163) In our normalization of the indices, for SU{Nc) with Nj families of "quarks" and "antiquaries", the beta function is 60 = 2(3A^c Nj). We normalize the Dynkin indices so that Tr{T:Tl) = drSat (4.11.164) with being the generators of in the representation r. Having the expressions of the vevs of the D and F-terms, we can now calculate the soft masses. We can assign to each field -generically denoted by 4>i, with i being a family indexa set of numbers such that the term (4.11.165) is invariant under the t/'(l)'s. The soft masses then can be written as ml. =ml + mf = v.= + yinh + n\+ni + ...)c]\ (4.11.166)

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135 The gaugino masses are: mi/2 = mo. (4.11.167) The trilinear soft couplings are: 4;-''^' ~< Fs > yI;''''^^ = ^0^1/2^"'''^^ = aoF^-'^'^l, (4.11.168) where Aq is a constant of order of one and yi"''''^' is the corresponding Yukawa coupUng in the superpotential. We now consider the two different boundary conditions. • ml « mf « m\i2 First, notice that in (4.11.158), the second term inside the brackets dominates over the first for reasonable values of the parameters, so the first term can safely be neglected. Then, the ratios that are expected to be small in this limit are >(r.i M^i a.-r,i ...\(4.11.169) "I? ' n ' {n\ + + ni • • •) and 2 mi 2^(4.11.170) "^1/2 In order that both of these ratios be suppressed, we have to assume that it is possible to have a vanishing tree level cosmological constant and then we can write the second ratio as

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136 and clearly, in order to be satisfied, K2 has to be rather small. • m?/2 — rnl » mf We saw that if the conditions (4.11.127) and (4.11.128) are satisfied then the only contribution to the soft masses comes from the dilaton F-term: with K2 given by (4.11.130) and it is manifestly flavor and family universal for all fermions. The common gaugino mass and the trilinear couplings are 4.11.2 Soft Parameters As mentioned before, there are in addition three families of vector like hidden fields transforming under the fundamental (anti-fundamental) representation Nc (Nc) of SU{Nc). This implies that Po = Pi = P2 = 6. As it stands, this model is anomaly free. The Green-Schwarz relations are all satisfied with the Kac-Moody levels of the non-Abelian factors all equal to 1, including kh. Ck in the above are the non-zero anomaly coeflficients associated with the different gauge factors and kk the corresponding Kac-Moody levels. We (4.11.172) Co ~ mi/2 ~ mo. (4.11.173) — = constant, for all k kk (4.11.174)

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137 distinguish again the two limits and present for each case examples in the context of this model. • ml « m] « m\i2 For this, "no scale" case, the universal contribution to the soft masses is very small (it vanishes for all practical purposes: mo ~ 0) and the family dependent contribution from the D-terms is ^(<3,=,5,L,e) ^ ^ . sf^^)^ (4.11.175) where nf is the sum of the exponents defined in (4.11.165) for the field (f) and i is its family index. They are easily calculable in the model. The gaugino masses and the trilinear couplings are X = mi/2 ~ Co. (4.11.176) In table 4.7, we show some typical values that our model gives for the parameters VC and X with ^/M as a free parameter. Unfortunately, all these models have problems associated with charge and/or color breaking minima of the scalar potential and therefore can not be considered as viable [81] as they stand. This is a generic feature of the 'no scale' boundary conditions. There is however the possibility of other moduli Fterms contributing to mo, with vevs large enough to protect the vacuum In such a case, we could have a viable model of the "no scale" type. An example would be, if these contributions for the = 4 and ^/M = 0.24 case (see

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138 Table 4.7: Values for x and \/C at the high scale M, for different choices of Nc and ^/M, in the "no scale" regime. iVc — 0 C / A/f l;/M XyUeV ) FF^i „\ r\ vC (GeK j TV r A x[GeV) \fC[GeV) 0.10 60 4 0.22 60 1 0.12 435 22 0.24 190 3 0.125 680 33 0.26 565 8 0.13 1035 48 0.28 1533 20 table 4.11.2.), were ~ 200 GeV . We do not have a natural mechanism in which such a scenario could be realized in our model, so we will concentrate from now on on our favorite scenario of the minimal sugra type, which is more natural and predictive in our scheme. As an example, we take A^c = 5 and ^/M = 0.28. The choice of A^e = 5 is motivated by the fact that it is probably the only value that results in reasonable squark masses and the choice ^/M = 0.28 is the value of the expansion parameter that we obtain from 4.11.123, a number close to the Cabbibo angle and where the minimal tree level K = -log (5 + 5) was assumed for simplicity, even though we know that other (exponetial) terms should be present as well. It turns out that for our first numerical example this is a satisfactory approximation. For these values, the small

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139 expansion parameter is e ~ 0.86 • 10"^ and the condensation scale becomes A ~ 2-10^2 GeV. The dilaton is assumed to be stabiHzed at yo = 1.44, which is the value of at the unification point. From 4.11.130, we can see that it yields K2 ~ 1.3. Substituting these values into (4.11.172) and (4.11.167), we obtain mo ~ 200 GeV ; mi/2 ~ 200 GeV ; ao ~ 200 GeV. (4.11.177) Of course, these parameters, are predicted at the unification scale M, so we have to extrapolate their values to Mz to obtain the low energy spectrum. In table 4.8, we show the MSSM parameters corresponding to this particular model. It is an example of a phenomenologically viable model, with no charge/color breaking minima, consistent with EWSB and fcnc. Finally, in table 4.9, we demonstrate our predictions for the Higgs mass, for a class of models where we freeze the value of (/M to 0.222 as the visible sector and the CKM matrix in particular implied and K2 to its canonical value 1. The latter of course implies that the Kahler potential involves terms in addition to its usual weak coupling form. We proposed a form for K in 4.11.131 that can do that. Then, the only remaining free parameters which we can vary is M and tan Experimental signatures that this type of models could imply is for example a trileptonic signal in pp collisions: pp + N2+X h'^iNi + l2p2N2+X (three leptons and missing energy) [82], or at the LHC, the decay h'> 77 [83]. Also, e+eannihilation could pair produce the lowest mass charginos

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140 Table 4.8: The first column contains the name of the parameter. The second column contains the low energy value of the parameter for a "minimal sugra" type model corresponding to = 5, ^/M = 0.28 and yo = 1.48. The input to the RGB's at M, is mo = mi/2 = ao = 200 GeV, tan f3{Mz) = 4, syn(/i) = +1 and mt = 175 GeV. The values of {B, W,g) are their lowest order pole masses. Pa vfi m pf pr J. cLi allies UvJi VdlUe dl iW^ 11 a lUcLSSj {Mz,VHiggs) (90.4,174.1) imp 4 {B,W,~9) (62,122,367) (0.116,0.033,0.0165,0.232) V^33 ) ^33 ) ^33 ) (1,0.08,0.042) (125,233) {ul, ur, di, dn, cl, cr, Uei) (407, 399, 414, 394, 238, 216, 226) (407, 399, 414, 394, 238, 216, 226) {ti, tR, bi, bR, tl, tr, t/^i) (435, 283, 395, 364, 238, 215, 226) {h\H^,A°,H^) (104,346,343,352) (95, 270) (53, 99, 240, 272) LSP-^ (Ni) 53

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141 Table 4.9: Higgs mass versus tan for different values of M. The second column is for M ~ Mqut = 4 • 10^^ GeV, the third for M = 8 • 10^^ GeV and the fourth for M = 1.2 • 10^^ GeV. tan (5 /io (GeV) K (GeV) h (GeV) 2 79.0 90.0 95.0 3 80.2 92.0 97.0 4 91.5 103.0 107.0 5 99.6 109.5 112.0 and sfermions. The exact degeneracy of the squark masses is a result of the simultaneous conditions (4.11.127) and (4.11.128). If these are not exactly obeyed, then the squark masses split with the splittings proportional to the vevs of the D-terms times the U{1) charges of the quark fields. Soft masses with non-universal contributions from D-terms may be a more realistic scenario but in that case the mass differences, especially between the first two families should be small, in order to avoid conflict with experimental bounds on fcnc. The lesson from this model is clearly that the success of the visible sector of the model transfers to the hidden sector as well, since we predict supersymmetry breaking parameters and a Higgs mass consistent with experimental constraints. Furthermore, our predictions seem to be stable, at least as far as the Higgs mass prediction is concerned. The number of the free parameters of the model is very small, provided we look only for models that give sensible phenomenological predictions. We can essentially vary only A^, (but it has to be very close to 0.222), tan /? (but it has to be in its low value regime), K2 (but it has to be close to 1) and possibly M (which must not necessarily be Mqut but it could be slightly higher).

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142 If, as in table 4.9, we fix iiTz = 1 and Ac = 0.222, we have only two remaining free parameters which specify the model! In the next few years, where experiments testing supersymmetry will be carried out, our model will be tested in a very definite way.

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CHAPTER 5 CONCLUSION In this work, we presented a systematic approach to low energy supersymmetric model building, inspired mostly by compactifications of the EgxEs heterotic string. We showed how to analyze the vacuum and proposed a systematic way to build a model with a unique vacuum. We found a unique model that has a remarkably well behaved vacuum and at the same time reproduces all the known order of magnitude relations between the chiral fermion masses. It predicts neutrino masses that can give an answer to the puzzle in the solar and atmospheric neutrino fluxes. A remarkable feature of this model is the fact that it has only one mass scale, the one at which the gauge couplings unify. The gauge coupling at that scale together with the anomalies of the non-Abelian gauge group of the observable sector predict consistently the correct value for the expansion parameter which is used to express fermion masses. The expansion parameter is a number of the order of magnitude of the Cabbibo angle. Proton decay is within the experimental limits. We emphasized the fact that the source of all the instability in the observable sector in the model are the extra vector-like states that are necessarily present to cancel anomalies. We do not have any evidence for their existence so we assumed a minimal number of them and followed the consequences. We showed that vacuum uniqueness implied that squark-quark alignment is not possible in this type of models and therefore we relied on a dilaton dominated supersymmetry breaking mechanism to yield degenerate squarks in order not to contradict data on flavor changing neutral 143

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144 currents. Our hidden sector was indeed capable of breaking supersymmetry via a dynamical mechanism with gaugino condensation. The former approach turned out to be fruitful, since as the hidden sector is specified, we could immediately calculate the soft parameters. We proposed a hidden sector that was motivated by purely anomaly cancelation and we showed that it is possible to construct a model that yields a low energy supersymmetric parameter spectrum, consistent with experimental and theoretical bounds. Finally, we did not address in this work questions related to CP violation and Cosmology.

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BIOGRAPHICAL SKETCH Nikolaos Irges was born in Szeged, Hungary, on the 31"* of July, 1965. In 1976, he and his family moved to Greece. There, he took the (Greek) citizenship of his parents and since then he is a Greek citizen. He finished high school in Athens and in 1985 he was admitted to the physics department of the University of Athens. During his studies, he was involved in research in the areas of nuclear physics and health physics and developed Monte Carlo codes for measuring radiation dose in diagnostic radiology. In 1990, he graduated from the University of Athens and did his military obligation, until 1991. In 1991, he was granted admission with financial assistance to the graduate program in physics at the University of Florida. In August 1993, he enrolled in the Ph.D. program under the supervision of his advisor. Dr. Pierre Ramond. He is currently preparing his Ph.D. dissertation in the area of high energy physics in which he is investigating supersymmetric models inspired from superstrings, in particular models including an anomalous f/(l) family gauge symmetry. 149

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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Pjj»rr^namond, Chairman Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of P hiloso phy. shardTield Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Pierre Sikivie Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Charles B. Thorn Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Christopher W. Stark Associate Professor of Mathematics

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This dissertation was submitted to the Graduate Faculty of the Department of Physics in the College of Liberal Arts and Sciences and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. August 1999 Dean, Graduate School