Doctoral Thesis

Nonperturbative Dynamics
in Supersymmetric Gauge Theories
N. Maru∗
Department of Physics, Nagoya University,
Nagoya 464-8602, Japan∗ E-mail address maru@eken.phys.nagoya-u.ac.jp

Abstract
Recently, there has been a remarkable progress in the study of strong coupling dynamics of the gauge theories. Especially, in supersymmetric case, a certain sector
can be determined exactly. In the former part of this thesis, we discuss the phasestructure of N = 1 supersymmetric gauge theories, especially focusing on the confining phase and the non Abelian Coulomb phase (duality). In the latter part, we
discuss its phenomenological applications, especially focusing on the gauge mediatedsupersymmetry breaking.

Contents
1 Introduction

12 Phases of Supersymmetric QCD

6

2.1 Basics of SUSY QCD . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

2.2 Nf < Nc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92.3 Nf = Nc (quantum deformation) . . . . . . . . . . . . . . . . . . . . 11
2.4 Nf = Nc + 1 (s-confinement) . . . . . . . . . . . . . . . . . . . . . . . 13
2.5 Nf ≥ Nc + 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3 Negative Dimensional Group and Duality

26

3.1 Similarities in SO and SP duality . . . . . . . . . . . . . . . . . . . . 27
3.2 Negative dimensional group . . . . . . . . . . . . . . . . . . . . . . . 29
3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4 Confining Phase in SUSY SO(12) Gauge Theory

33

4.1 A Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.2 Low energy superpotentials . . . . . . . . . . . . . . . . . . . . . . . 39
4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5 Gauge Mediated SUSY Breaking

44

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.2 DNNS Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
6 Effective Messenger Sector Model50

6.1 Messenger sector as the effective theory of DSB sector . . . . . . . . . 50

i

CONTENTS

ii

6.2 The observable sector and estimation of the scales . . . . . . . . . . . 55
6.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
7 A DGM Model with an Affine QMS

59

7.1 A Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
7.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
8 Conclusion

69

A Superspace formalism

74

Chapter 1
Introduction
The Standard Model (SM) describes the various experimental measurements very
well, but none of particle physists believe the standard model as the fundamental
theory. One of the problems in the standard model is the hierarchy problem, whichappears in the Higgs sector, i.e. Higgs boson, which is introduced to trigger aspontaneous breaking of symmetry SU (2)L × U (1)Y → U (1)em . Taking into accountradiative corrections to the Higgs boson mass squared δm2H , one can easily see that
δm2H is quadratically divergent i.e. δm2H ∼ O(αΛ2 ), where α represents the gauge
coupling factor, and Λ is a cut-off scale (nothing but a parameter in the standard
model). The relation between the bare mass squared m2o and the renormalized one
m2 is written as follows, m2 = m2o + δm2H . In order to give the electroweak scale,m2 has to be O((100GeV)2 ). This needs a fine tuning of m2o if the cut-off scaleis assumed to be larger than the electroweak scale. For instance, if we assume thecut-off scale to be the Planck scale (∼ 1019GeV), then the above Higgs mass relation
implies O((100GeV)2 ) ' O((1019GeV)2 ) − O((1019GeV)2 ) for O(α) ∼ O(1). This isquite an accidental result! Once assuming the cut-off scale to be much larger thanthe electroweak scale, we cannot help accepting a terrible fine tuning of the Higgs
boson mass. In principle, though one cannot rule out this possibility, it appears to
be implausible that the nature behaves like this. We have to explain how the small
scale (the electroweak scale) arises from the fundamental scale of the theory (thePlanck scale, for instance). This is called “the hierarchy problem”.

1

CHAPTER 1. INTRODUCTION

2

One of the most attractive approaches to solve the hierarchy problem is “supersymmetry”(SUSY). In short, SUSY is a symmetry which transforms bosons into
fermions and vice versa. The origin of the hierarchy problem is that there is nosymmetry for scalars to guarantee the masslessness. Combining supersymmetry andchiral symmetry which guarantees the masslessness of the fermions, we can easily realize that scalars have to be massless from the requirement of symmetry. Indeed, thequadratic divergence of the Higgs scalar mass is milded by the cancellation between
boson contributions and fermion ones.
In 1994, there was a remarkable breakthrough in supersymmetric gauge theories.
First, Seiberg and Witten found the exact solutions in N = 2 four dimensional
SUSY gauge theories [1]. Second, Seiberg [2] conjectured the dual description of
N = 1 four dimensional SUSY gauge theories, which describes the same physics asin the original theory in the infrared limit. The key ingredient of these findings is“holomorphy”. Thanks to this holomorphy, certain functions (the prepotential in
N = 2 case or the superpotential in N = 1 case) are determined exactly. Therefore,
we can investigate the strong coupling dynamics in more detail compared to non
SUSY theories.
In this thesis, we focus on the issues on N = 1 SUSY gauge theories. They
are mainly separated into two parts. One is a theoretical aspect of N = 1 SUSY
gauge theories, the phase structure of the theory. Especially, the confining phase
dynamics and the duality will be discussed. One will be able to understand throughthis discussion how powerful the holomorphy is. The other is a phenomenologicalaspect of N =1 SUSY gauge theories, especially, the gauge mediated SUSY breaking.
This issue is inevitable from the viewpoint of the phenomenological applications since
SUSY must be broken at low energy. As the strong coupling dynamics of various
theories have been revealed, a large number of SUSY breaking models have also
been constructed. Therefore, it is worth while investigating the properties of eachmodel applied to phenomenology.
This thesis is organized as follows. In chapter 2, the phase structure of SUSY
SU (Nc ) gauge theory with Nf flavors is reviewed. It is instructive to review thistheory because it includes the dynamical features which are common to those inCHAPTER 1. INTRODUCTION

3

other theories. Various phases appear as a function of Nf (No vacuum, confinementwith chiral symmetry breaking, confinement without chiral symmetry breaking, afree magnetic phase, a non-Abelian Coulomb phase, a free electric phase.). Of thesephases, the non-Abelian Coulomb phase is remarkable because Seiberg [2] found adual pair which describe an equivalent physics in the infrared limit. Furthermore,
a strong coupling dynamics of one theory can be described by a weak coupling
dynamics of another theory and vice versa.
By using the duality, we can investigate the strong dynamics analytically. So, it
is interesting to ask whether this duality is generic or not. Unfortunately, althoughmany dualities have been discovered so far, we have no systematic approach to find aduality. In chapter 3 which is based on the work by myself and S. Kitakado, we foundthat the SO gauge group duality and the Sp gauge group duality are interrelated
through the negative dimensional group. This negative dimensional group technique
is useful and powerful in finding a new duality because if we know the SO(Sp) groupduality, we can easily obtain a new Sp(SO) duality by a simple manipulation.
Another approach to search for a duality is to extract its informations from theconfining phase dynamics. Since the duality must recover the result of the confining
phase, the informations are quite valuable. The advantage of this approach is thatthe confining phase can be investigated systematically by symmetry and holomorphy.
In chapter 4 which is based on the work by myself, I have studied the confining phaseof SUSY SO(12) gauge theory with one spinor and several vectors. The motivation
to study this particular model is that I would like to clarify the structure of the
duality in SUSY SO(Nc ) gauge theory with Nf vectors and NQ spinors. The known
dualities in this class have remarkable features which are not included in the duality
Seiberg found. From this point, I think that we might be able to understand N = 1
duality by clarifying this class of dualities. Along this line of thought, I constructed
the exact superpotentials systematically, and conjectured the subgroup of the dual
gauge group. Unfortunately, it turned out that we need more insight to find theabove mentioned duality.
In the remainder of the thesis, we focus on the phenomenological aspects. In
chapter 5, the concept of gauge mediated SUSY breaking is introduced. This is a

CHAPTER 1. INTRODUCTION

4

scenario that SUSY breaking effects are communicated to the low energy by gauge interactions, which is an alternative to the supergravity (SUGRA) mediation scenario.
Gauge mediated SUSY breaking has phenomenological advantages compared to thegravity mediation scenario, namely, the degeneracy of the sfermion masses is automatically achieved. Therefore, there is no flavor changing neutral currents (FCNC)
problem. Next, the typical model by Dine, Nelson, Nir and Shirman (DNNS model)
is reviewed. This model consists of three sectors, namely, the dynamical SUSY
breaking (DSB) sector, the messenger sector and the observable sector. The messenger sector makes the mechanism to communicate SUSY breaking effects to theobservable sector quite complicated. To make matters worse, in the true vacuum of
the DNNS model, the color and the electroweak symmetries are broken at the scale
which is higher than the weak scale. From these facts, the DNNS model became
the driving force to simplifying, modifying the model or finding new models. In thenext two sections, our new models are presented.
Chapter 6 is based on the work by N. Haba, T. Matsuoka and myself. We
constructed the model in which the messenger sector is the effective theory of the
DSB sector. More precisely, we identified the fields in the messenger sector with themassless composite fields in the DSB sector in contrast with the DNNS model inwhich the fields in the messenger sector are put by hand. This model simplifies thecomplicated structure of the DNNS model naturally. Furthermore, we have shownthat in the true vacuum, the color and the electroweak symmetries are not broken
at the scale higher than the weak scale. Then, we estimated various mass scales in
the model, which are phenomenologically viable.
From the viewpoint of the simplicity of the model, it is natural to ask whether or
not SUSY breaking effects can be communicated to the observable sector directly.
This framework is referred to as “Direct Gauge Mediation (DGM)”. In chapter 7
which is based on the work by myself, I constructed a new and a simple model ofthe DGM with an Affine quantum moduli space. The advantages of this model are
as follows. First, the effective Kähler potential is calculable and simple because on
Affine quantum moduli space, we can easily identify the massless composite fields
by ’t Hooft anomaly matching conditions. Second, SUSY breaking mechanism is

CHAPTER 1. INTRODUCTION

5

simple. I have shown that this model avoids some problems (the SUGRA mediation
dominance over the gauge mediation, the negative sfermion mass squared problem)in earlier DGM models. Furthermore, I have estimated various mass scales and it
turned out that these are phenomenologically viable. One can find conclusion inchapter 8. In Appendix A, the superspace formalism, which is used throughout thethesis, is summarized.

Chapter 2
Phases of Supersymmetric QCD
As a warm up and to make our discussion clear, we review here the phases of SU (Nc )supersymmetric QCD∗ .

2.1

Basics of SUSY QCD

The Lagrangian of SUSY SU (Nc ) QCD† is
L =
τ ≡

Z
4

† V

d θ(Q e Q + Q̄e−V1Q̄ ) +Im
16π†

Z

d2 θτW aα Wαa + h.c.,

θY M
4πi
+ 2,
2π
g

(2.1)
(2.2)

where Q and Q̄ are chiral superfields, V is a vector superfield, τ denotes a complex
gauge coupling (which is a useful combination in SUSY gauge theories), Waα is afield strength tensor superfield (a is a gauge index and α is a spinor index.), θ in the
Lagrangian represents Grassmannian coordinate, and θY M is the Yang-Mills theta
parameter. The first term in (2.1) includes kinetic terms for quarks and squarks andthe second term includes kinetic terms for gauge fields and gauginos.
This Lagrangian has the following classical symmetries:
SU (Nc ) × [SU (Nf )L × SU (Nf )R × U (1)V × U (1)A × U (1)R ]∗For further details, see [3, 4, 5].

†

We consider here the massless case.

6

(2.3)

CHAPTER 2. PHASES OF SUPERSYMMETRIC QCD

7

SU (Nc ) is a gauge symmetry, and symmetries in the bracket are flavor symmetries.
Transformation properties of superfields are‡
Qiα

(

;

Q̄αi

(

; 1,

Wα

(Adj; 1, 1, 0, 0, 1),, 1, 1, 1, R(Q)),(2.4), −1, 1, R(Q̄)),

(2.5)
(2.6)

where α = 1, · · · , Nc and i = 1, · · · , Nf . At the quantum level, U (1)A and U (1)R are
broken by anomalies but we can always make an anomaly free R symmetry by taking
a linear combination of U (1)A and U (1)R . In other words, we can choose the charges
so that only U (1)A is anomalous in the quantum theory. In fact, U (1)A , U (1)R
anomalies are∂µ jAµ = 2Nf

g2
F F̃ ,
32π 2∂µ jRµ = {(R(Q) + R(Q̄) − 2)Nf + 2Nc}

(2.7)
g2
F F̃ ,
32π 2

(2.8)

where ∂µ jAµ , ∂µ jRµ are the currents of U (1)A and U (1)R , respectively.
If we define the current of anomaly free U (1)R as follows,(jRµ )AF = jRµ −(R(Q) + R(Q̄) − 2)Nf + 2Nc µ
jA ,
2Nf

(2.9)

where “AF ” means anomaly free, then ∂µ (jRµ )AF = 0. So, quantum symmetries andthe charges are§
SU (Nc ) × [SU (Nf )L × SU (Nf )R × U (1)V × U (1)AF
R ]
−
N
N
f
c
( ; , 1, 1,
),
Qiα
Nf
Nf − NcQ̄αi( ; 1, , −1,),NfWα(Adj; 1, 1, 0, 1).

(2.10)(2.11)(2.12)(2.13)In order to study the vacuum structure in SUSY gauge theories, it is necessary
to investigate “the flat direction”, which means the field space where the scalar‡

In general, U (1)R charges of Q and Q̄ are arbitrary at the classical level.

§

We take R(Q) = R(Q̄), for simplicity.

CHAPTER 2. PHASES OF SUPERSYMMETRIC QCD

8potential vanishes. (the flat direction is also called “moduli space”. We will also use
this terminology in this thesis.)
Since the scalar potential in the present case is
1 X
1 X
V = g2 (Da )2 = g2 (Q† T a Q − Q̄T a Q̄† )2 ,22aa

(2.14)

the vanishing scalar potential condition is
0 = Q† T a Q − Q̄T a Q̄† = tr[(T a )βα ((Q† Q)αβ − (Q̄Q̄† )αβ )]⇒ (Q† Q)αβ − (Q̄Q̄† )αβ ∝ δβα ,(α, β = 1, · · · , Nc ),(2.15)(2.16)

where δβα is Nc by Nc matrix. The classical moduli space is parametrized by thevacuum expectation value (VEV) of Q, Q̄ modulo gauge and flavor rotations. For
Nf < Nc ,





q
 1



hQi = hQ̄i = 









.




q2
...

(2.17)

qNf
And for Nf ≥ Nc ,


q
 1



hQi = 





...










qNc 











q̄
 1
,

hQ̄i =



...





,
q̄Nc 


(2.18)

where the quantity |qi |2 −|q̄i |2 is independent of i. For Nf < Nc , the gauge symmetry
SU (Nc ) is broken to SU (Nc − Nf ), so we can describe the classical moduli space interms of VEVs of “mesons” Mji = QiQ̄j (which match massless degrees of freedom
correctly). For Nf ≥ Nc , the gauge symmetry is completely broken, the classicalmoduli space is described by VEVs of mesons and “baryons” which are defined asB ≡ α1 ···αNc i1 ···iNf Qiα11 · · · QiαNNcc ,(2.19)

B̄ ≡ α1 ···αNc i1 ···iNf Q̄αi11 · · · Q̄iNNcc .

(2.20)αNote that mesons and baryons are, in general, subject to the constraints as seen
later.

CHAPTER 2. PHASES OF SUPERSYMMETRIC QCD

2.2

9

Nf < Nc

As mentioned in the previous section, massless gauge invariant composites aremesons Mji = QiQ̄j (i, j = 1, · · · , Nf ). Baryons vanish identically due to the Bosestatistics of superfields. Since the effective theory should be described by gauge
invariant operators, the effective superpotential is a function of Mji . Taking into
account holomorphy, symmetries, dimensional analysis, we can fix the form of theeffective superpotential [7] (referred to as “ADS superpotential”) except for thecoefficient asWeff = CNc ,Nf 1 N −N

3N −N

cfΛNc ,Nfc



detQQ̄

f.

(2.21)

Here ΛNc ,Nf is the strong coupling scale of SUSY QCD and CNc ,Nf is a constantdepending on Nc , Nf . A coefficient in (2.21) is determined as follows. If we consider
the limit qNf → ∞, SU (Nc ) gauge group will be broken to SU (Nc − 1) by Higgs
mechanism, and the number of flavors is also reduced to Nf − 1. Matching thegauge coupling constant between the ultraviolet theory and the infrared theory at
the scale qNf through the 1-loop RGE, we obtain3(N −1)−(N −1) 2
3Nc −Nf
qNf = ΛNc ,N.
fcfΛNc −1,N
f −1(2.22)

Using this relation, the superpotential (2.21) can be rewritten as follows,3(Nc −1)−(Nf −1)  (Nc −1)−(Nf −1)
ΛNc −1,N
f −1Weff = CNc ,Nf ,
0det QQ̄
1(2.23)

where det’ is the determinant of (Nf −1) dimensional flavor space. This corresponds
to the superpotential for SU (Nc − 1) gauge theory with Nf − 1 flavors, so CNc ,Nf =CNc −1,Nf −1 . Iterating the same arguments for q1 · · · qNf −1 , we obtain
CNc ,Nf = CNc −Nf ,0 .

(2.24)

Next, if we add a mass term δW = mQ̄Nf QNf to (2.21) and integrate out the massivemode, the superpotential for SU (Nc ) gauge theory with Nf − 1 flavors is obtained.
In fact, if we add the mass term

Weff = CNc ,Nf 

3N −N

c
f
ΛNc ,Nf

detQQ̄

1
 N −Ncf+ mQ̄Nf QNf ,(2.25)

CHAPTER 2. PHASES OF SUPERSYMMETRIC QCD

10

by the equation of motion for Q̄Nf QNf , we find(Q̄Nf Q

Nf)

1
Nc −Nf

3N −NcfΛNc ,Nf1
 N −Nc
CNc ,Nf=m(Nc − Nf ) detQ̄Qf.

(2.26)

Substituting this for Q̄Q in (2.25), we obtain
Weff = (Nc − Nf + 1)

CNc ,Nf
Nc − Nf1! Nc −Nf  3Nc −Nf +1  Nc −N
f +1
Nc −Nf +1
ΛNc ,Nf −1.

detQ̄Q(2.27)

Here one-loop matching relation for the gauge coupling constant mΛ3Nc −Nf = Λ3Nc −Nf +1
is used. Since Eq. (2.27) corresponds to the superpotential for SU (Nc ) gauge theory
with Nf − 1 flavors,
CNc ,Nf(Nc − Nf + 1)
Nc − Nf! Nc −Nf

Nc −Nf +1= CNc ,Nf −1 .

(2.28)

Combining this and Eq. (2.24), one can derive1

CNc ,Nf = (Nc − Nf )C Nc −Nf ,(2.29)

where C is a numerical constant. Direct instanton calculation [7] shows that thisnumerical constant is equal to one. Therefore, the final expression of the effective
superpotential is

Weff = (Nc − Nf ) 

3N −N

c
f
ΛNc ,Nf

detQQ̄

1
 N −Ncf.

(2.30)

Note that the superpotential (2.30) is nonperturbative because non renormalization
theorem [6] forbids to induce the superpotential perturbatively. Note that this typeof superpotential cannot appear for Nf ≥ Nc because for Nf = Nc , the power issingular and for Nf > Nc , it is also singular in the weak coupling limit Λ → 0, QQ̄ →∞.
While Weff is generated by one instanton for Nf = Nc − 1, for Nf < Nc − 1, it isgenerated by gaugino condensation in the unbroken SU (Nc − Nf ) gauge group [7].
As for the vacuum, the superpotential (2.30) pushes the minimum away to infinity, so the vacuum is not well-defined.

CHAPTER 2. PHASES OF SUPERSYMMETRIC QCD

2.3

11

Nf = Nc (quantum deformation)

The superpotential (2.30) is singular at Nf = Nc . Furthermore, since R-chargesof the matter fields Q, Q̄ vanish, it is impossible to construct any superpotentials.
This leads to the fact that the classical flat direction cannot be lifted quantum mechanically. Recalling here that the gauge symmetry is completely broken, massless
degrees of freedom (Nf2 + 1) and the degrees of freedom of the gauge invariant composites (Nf2 + 2) differ by one. This implies that M, B, B̄ are not independent andone classical constraint
detM − B B̄ = 0(2.31)

should exist. This constraint follows from the Bose statistics of superfields Q, Q̄.
Seiberg argued [8] that this constraint is modified quantum mechanically as
detM − B B̄ = Λ2Nc .

(2.32)

This modification is due to one instanton effect because the right hand side of(2.32) is proportional to one instanton action. One may write this constraint using
Lagrange multiplier superfield X as
W = X(detM − B B̄ − Λ2Nc ).

(2.33)

The consistency of (2.33) is verified as follows. Adding the mass term for theNlast flavor W = mMNff to (2.33) and integrating out massive mode using F-flatness
conditions,
0 =

∂W
N
∂MNff

N

= X(cofM )Nff + m,∂W
= detM − B B̄ − Λ2Nc ,
∂X
∂W
0 =
= X B̄,
∂B
∂W
= XB,
0 =
∂ B̄
∂W
Nf
0 =
(i = 1, · · · , Nf − 1),
Nf = X(cofM )i∂Mi
0 =(2.34)(2.35)(2.36)(2.37)(2.38)

CHAPTER 2. PHASES OF SUPERSYMMETRIC QCD

12

where (cofM )ij is a cofactor of a Nf × Nf matrix defined as(cofM )ij =

jN
1
∂detM
ii2 ···iNf jj2 ···jNf Mij22 · · · MiN f =.
f(Nf − 1)!
∂Mij

(2.39)The third and fourth equations lead to B = B̄ = 0. The last equation can restrict
the form of M as follows,


 M̃



N
MNff


,

(2.40)

where M̃ denotes (Nf − 1) × (Nf − 1) matrix. Using (2.32) and the above results,
one obtainsmΛ2Nc
Λ̃2Nc +1=.
(2.41)
detM̃
detM̃
In the second equality, we used 1-loop matching relation of the gauge coupling
Weff =

constant mΛ2Nc = Λ̃2Nc +1 , where Λ̃ is the strong coupling scale of SU (Nc ) gauge
theory with Nc − 1 flavors. This is nothing but ADS superpotential for Nf = Nc − 1,which is a consistent result.
The origin of the moduli space hM i = hBi = hB̄i = 0 is not on the quantum moduli space (2.32). Therefore, chiral symmetries SU (Nf )L ×SU (Nf )R ×U (1)B ×U (1)Rare necessarily broken. It is straightforward to check that fundamental massless
fermions and composite fermions satisfy ’t Hooft anomaly matching conditions forunbroken global symmetries [8].
The first case we consider is where hBi = hB̄i = 0 and hMji i = δji . In this case,
SU (Nf )L × SU (Nf )R is broken to their diagonal SU (Nf )D , and U (1)B , U (1)R are
unbroken. The explicit calculations of global anomalies are
U (1)R SU (Nf )2 E (−1) × Nc × µ(

) + (−1) × Nc × µ(

C (−1) × µ(Adj) = −2Nf µ() = −2Ncµ(),

U (1)R E (−1) × Nc Nf + (−1) × Nc Nf + 1 × (Nc2 − 1)
= −2NcNf + Nc2 − 1,
C (−1) × (Nf2 − 1) + (−1) × 1 + (−1) × 1 = −Nf2 − 1,
U (1)3R E (−1)3 × Nc Nf + (−1)3 × Nc Nf + 1 × (Nc2 − 1)
= −2NcNf + Nc2 − 1,),

CHAPTER 2. PHASES OF SUPERSYMMETRIC QCD

13

C (−1)3 × (Nf2 − 1) + (−1)3 × 1 + (−1)3 × 1 = −Nf2 − 1,
U (1)R U (1)B E (−1) × 1 × Nf Nc + (−1) × 1 × Nf Nc = −2Nc2,
C (−1) × Nc2 + (−1) × Nc2 = −2Nc2 ,(2.42)

where E, C means “elementary” ,“composite”, respectively and µ(R) is a Dynkinindex for the representation R.
The second case is where hBi = −hB̄i = ΛNf , hM i = 0. In this case, SU (Nf )L ×
SU (Nf )R is unbroken, and U (1)B is broken. The explicit global anomalies are
SU (Nf )3L,R E Nc A()

C Nf A(),

U (1)R SU (Nf )2L,R E (−1) × Nc × µ(),

C (−1) × Nf × µ(),

U (1)R E (−1) × Nc Nf + (−1) × Nc Nf + 1 × (Nc2 − 1)
= −2NcNf + Nc2 − 1,
C (−1) × (Nf2 − 1) + (−1) × 1 + (−1) × 1 = −Nf2 − 1,
U (1)3R E (−1)3 × Nc Nf + (−1)3 × Nc Nf + 1 × (Nc2 − 1)
= −2NcNf + Nc2 − 1,
C (−1)3 × (Nf2 − 1) + (−1)3 × 1 + (−1)3 × 1= −Nf2 − 1,

(2.43)

where A(R) denotes anomaly coefficient for the representation R.

2.4

Nf = Nc + 1 (s-confinement)

In this case, symmetry properties of M, B, B̄ are following.
SU (Nf ) SU (Nf )
M
B
B̄

1
1

U (1)BU (1)R0

2/Nf

Nf − 1

Nc /Nf

−Nf + 1

Nc /NfCHAPTER 2. PHASES OF SUPERSYMMETRIC QCD

14

We can easily derive the effective superpotential from symmetry and holomorphy[8]:
Weff =

1
Λ2Nf −3(BM B̄ − detM ).

(2.44)

It is important that the relative sign is opposite because the equations of motion
derived from (2.44) reproduce the classical constraints∂Weff= 0 ⇒ Bi B̄ j − (cofM )ji = 0,
∂Mji
∂Weff
= 0 ⇒ Mji B̄ j = 0,
∂Bi
∂Weff
= 0 ⇒ Bj Mij = 0.
i
∂ B̄(2.45)(2.46)(2.47)

This is consistent with the fact that the classical moduli space cannot be lifted at
the quantum level for Nf ≥ Nc as mentioned earlier.
N

We can also check that by adding the mass term W = mMNff to (2.44), andintegrating out the massive mode, the quantum constraint (2.32) is reproduced.
Indeed, if we add the mass term explicitly
W =

1

N

Λ2Nf −3(BM B̄ − detM ) + mMNff ,(2.48)and integrating out the massive modes, the quantum constraint (2.32) is obtained
0 =

∂W
N
∂MNff

=

1
Λ2Nf −3(B Nf B̄Nf − (M −1 )Nff detM ) + m
N⇔ B Nf B̄Nf − detM + Λ2Nf −2 = 0.

(2.49)(2.50)

Note that detM in the last equation is a determinant of (Nf − 1) dimensional flavor
space and the scale is also written by the scale of Nf − 1 flavors. This last equationis the constraint of the quantum deformation.
In contrast to Nf = Nc case, the origin hM i = hBi = hB̄i = 0 is on thequantum moduli space (2.44). Therefore, ’t Hooft matching conditions between thefundamental massless fermions and the composite massless fermions are satisfied
everywhere on the moduli space. Explicit calculations go as follows,
SU (Nf )3L,R E Nc A()

CHAPTER 2. PHASES OF SUPERSYMMETRIC QCD

U (1)R SU (Nf )2L,R

U (1)B SU (Nf )2L,R

U (1)RU (1)R U (1)2B

U (1)2R U (1)BC Nf A( ) + A( ) = (Nf − 1)A( ),!
1
N2E− 1 Nc × µ( ) = − c µ( ),
Nf
Nf
!
Nc
2
C (
− 1)Nf × µ( ) +− 1 × µ(Nf
Nf
(Nf − 1)2
=−
µ( ),
Nf
E 1 × Nc × µ( ),)C (Nf − 1) × µ( ),
!
!
1
1
E
− 1 × Nc Nf +− 1 × Nc Nf + 1 × (Nc2 − 1)
Nf
Nf
= −Nc2 − 1,
!
!
!
2
Nc
Nc
2
C
− 1 × Nf +− 1 × Nf +− 1 × Nf
Nf
Nf
Nf
= −Nc2 − 1,
!
!
1
1
E 1×
− 1 × Nf Nc + 1 ×− 1 × Nf Nc = −2Nc2 ,
Nf
Nf
!
Nc
C
− 1 × (Nf − 1)2 × 2Nf = 2(Nc − Nf )Nc2 ,
Nf
1
E 1×
−1
Nf
C

Nc
−1
Nf
Nc
−1
Nf

U (1)3R

15

E

1
−1
Nf

!2

!2

1
× Nf Nc − 1 ×−1Nf

× Nf Nc = 0,

× (Nf − 1) × Nf +!2× (−Nf + 1) × Nf = 0,
!3

× 2Nc Nf + 1 × (Nc2 − 1)

= −Nf2 + 6Nf − 12 +
2
−1
Nf

!2

!3

8
2
− 2,
Nf
Nf
!3

Nc
C
− 1 × 2Nf
Nf
8
2
= −Nf2 + 6Nf − 12 +
− 2.
Nf
Nf
× Nf2 +

(2.51)

Finally, we comment that the models with above features are referred to as “sconfinement”. The s-confinement models based on simple groups are completely

CHAPTER 2. PHASES OF SUPERSYMMETRIC QCD

16classified by C. Csáki, M. Schmaltz and W. Skiba [9].

2.5Nf ≥ Nc + 2

As in the previous sections, we can infer the form of the effective superpotential asj
Weff ∼ Bij···k MliMm
Mnk B̄ lmn − detM.

(2.52)

However, this observation is not correct in the following respects. First, R-chargeof (2.52) is not two. Second, M, B, B̄ do not satisfy ’t Hooft anomaly matchingconditions without violating the previous results. This implies a signal for thetermination of the confining phase.
Seiberg found a solution to this problem [2]. In the range 32 Nc ≤ Nf ≤ 3Nc ,there are two equivalent descriptions which flow to the same infrared fixed point.
At this fixed point, the gauge coupling is finite.
The exact beta function¶ in SUSY QCD is [10, 11]β(g) = −g3 3Nc − Nf + γ(g2 )
,
g2
16π 2
1 − 8π2 Nc

g2 Nc2 − 1γ(g ) = − 2+ O(g4 ),8π
Nc
2(2.53)(2.54)

where γ(g2 ) is the anomalous dimension.
N

If we take Nc , Nf to infinity holding Nc g2 and 3Nfc = 1 − (0 <   1) fixed, we
can readβ0 = −3Nc + Nf = −3Nc < 0 ⇔ Nf < 3Nc ,Nf (Nc2 − 1)
β1 = −Nc (3Nc − Nf ) +Nc
3' Nc (−3Nc + 2Nf ) > 0 ⇔ Nc < Nf .
2

(2.55)(2.56)

There is a range where 1-loop beta function β0 is negative, but 2-loop beta function
β1 is positive. This might be a non-trivial fixed point [13]. The coupling constantat the fixed point is determined from β(g∗ ) = 0, that is, Nc g∗2 = 8π 2  + O(2 ).
¶The gauge coupling constant appearing in the beta function is the “canonical” gauge coupling

constant not the “holomorphic” gauge coupling constant. Related issues are discussed in Ref. [12].

CHAPTER 2. PHASES OF SUPERSYMMETRIC QCD

17

The infrared theory is a non-trivial four dimensional superconformal field theory. The elementary quarks and gluons appear as interacting massless particles.
Therefore, this phase is referred to as “the non-Abelian Coulomb phase”.
Given that such a fixed point exists, we can use the superconformal algebra toderive some results about the theory. This algebra includes an R symmetry. The
requirement that a representation of the superconformal algebra be unitary puts
restriction on the scaling dimension D and the R-charge. In particular,
3
D ≥ |R|2(2.57)and this inequality is saturated for chiral D = 32 R or antichiral D = − 32 R superfields.
For the gauge invariant operators QQ̄,3
3(Nf − Nc )D(QQ̄) = R =.
2
NfThis also follows from (2.53), D = γ∗ + 2 =(2.58)

3(Nf −Nc )Ncbecause of γ∗ = −3 N
+ 1.
Nf
f

The bound 32 Nc < Nf is also consistent with the unitarity bound D ≥ 1 forthe scalar fields. D(QQ̄) ≥ 1 is rewritten as 32 Nc < Nf . Note that the bound issaturated for free fields.
Let us discuss these equivalent theories in more detail. We list below the matter
content and its charge of symmetries and their superpotentials.
Original theory
SU (Nc )SU (Nf )L

SU (Nf )R

U (1)BU (1)R

1

1(Nf − Nc )/Nf−1

(Nf − Nc )/Nf

SU (Nf )R

U (1)BU (1)R1

Nc /(Nf − Nc )Nc /NfNc /(Nf − Nc )Nc /Nf02(Nf − Nc )/Nf

Q
1

Q̄

W =0
Dual theory
SU (Nc )SU (Nf )L

q
q̄
M

1
1

CHAPTER 2. PHASES OF SUPERSYMMETRIC QCD

18

W = µ1 qM q̄
Original theory is nothing but SUSY QCD discussed so far. The “dual” theory isequivalent to SUSY QCD in the infrared limit. (The meaning of “dual” will be
explained.) The remarkable thing between two theories is that the gauge symmetry
is not the same but the global symmetries are same. This is not so surprisingbecause searching for dual description heavily depends on ’t Hooft anomaly matchingconditions. U (1)B,R charges of dual quarks q, q̄ are determined through the mappingof baryons
B = QNc ↔ b =B̄ = Q̄Nc↔ b̄ =q−(−µ)Nc −Nf Λ3Nc −Nf i1 ···iNf q Nf −Nc ,q−(−µ)Nc −Nf Λ3Nc −Nf i1 ···iNf q̄ Nf −Nc .

(2.59)The normalization constant is determined by symmetry and dimensional analysis.
Let us suppose that the normalization constant is C. The mass dimension of C is
2Nc −Nf since the mass dimension of B, b is Nc , Nf −Nc respectively. Mass scales inthe theory are Λ3Nc −Nf and µ.k Therefore, we can write C = (Λ3Nc −Nf )a µb without
loss of generality, here a, b are constants. It leads to (3Nc − Nf )a + b = 2Nc − Nffrom the argument of mass dimension. As will be discussed later, the scale Λ3Nc −Nf2Nfand µ have charges 2Nf and Nf −N
under anomalous U (1). Taking into account thatcboth baryons B and b have charge Nc , one finds that the normalization constant2N

fshould be neutral under anomalous U (1), namely, 2Nf a + Nf −N
b = 0. These twocconsiderations fix the normalization constant except for the sign. It is easy to seethat the sign dependence is correct from Nf = Nc + 2 argument below.
We have to emphasize the following. First, M in the dual theory is an elementarysuperfield though the charges of M corresponds to those of meson QQ̄ in the original
theory. Note that QQ̄ ↔ M identification is valid only in the infrared limit. Second,
the superpotential in the dual theory is not only consistent with symmetries but
also indispensable for the dual theory to possess the same U (1) symmetries. If thissuperpotential does not exist, extra U (1) symmetries appear. Mass scale µ in thek

There is also a scale of the theory Λ̃, but this scale is related to Λ3Nc −Nf and µ through (2.60).

(which will be discussed later).

CHAPTER 2. PHASES OF SUPERSYMMETRIC QCD

19dual superpotential is needed to compensate for the difference of mass dimension of
QQ̄ and M at the ultraviolet.
Here, we comment on the meaning of “dual”. Using anomalous U (1) symmetries,
we can derive the relation between the strong coupling scale of the electric theoryΛ and the dual’s Λ̃ as
Λ3Nc −Nf Λ̃2Nf −3Nc = (−1)Nf −Nc µNf .

(2.60)

This is derived as follows. Let us note that Q, Q̄ have charge +1 and q, q̄ havec(through the operator mapping QNc ↔ q Nf −Nc ) under anomalous
charge NfN−Nc

U (1). Moreover, the strong coupling scale of the original theory and the dual theory
2N N

2N

f c
f
have charge 2Nf and Nf −N, and µ has Nf −N. We can fix the relation except forcccoefficient asΛ3Nc −Nf Λ̃2Nf −3Nc = C(Nc , Nf )µNf ,(2.61)

where C(Nc , Nf ) is a dimensionless constant. Nc , Nf dependence is determinedby the flat direction deformation and the mass term deformation as we fixed thecoefficient of ADS superpotential before.
If we consider the weak coupling limit of the original theory (Λ → 0)∗∗ , thenΛ̃ → ∞, that is, the dual is in the strong coupling regime and vice versa. This is thegeneralization of Dirac quantization condition. The relation of two theories underconsideration is analogous to the electro-magnetic duality. Therefore, we use theterminology “dual” or we also use “electric” for the original theory and “magnetic”
for the dual theory.
Next we comment on the scale relation (2.60). If we perform another dualitytransformation (Nc → Ñc , Λ → Λ̃, µ → µ̃), Eq. (2.60) becomes Λ̃3Ñc −Nf Λ3Nc −Nf =(−1)Nc µ̃Nf , therefore µ̃ = −µ. This minus sign is crucial when we dualize again.
The dual of the dual theory is SU (Nc ) gauge theory with Nf flavors of quarks di , d¯j̄
and singlet mesons Mj̄i and Nij̄ = qi q j̄ , and the superpotential
W =1 j̄ i ¯
1
Ni d dj̄ + Mj̄i Nij̄
µ̃
µ

∗∗

In supersymmetric gauge theory, the gauge coupling constant and the strong coupling scale  b0Λ, where θ is theta angle, b0 is 1-loopΛ are related through 1-loop RGE, exp − 8π
g 2 + iθ =µbeta function coefficient, µ is renormalization scale.

CHAPTER 2. PHASES OF SUPERSYMMETRIC QCD

20

1 j̄
N (−di d¯j̄ + Mj̄i ).
µ i=(2.62)

M and N are massive and can be integrated out using equations of motion,∂W= Nij̄ ,
i
∂Mj̄
∂W
0 =
= −did¯j̄ + Mj̄i .
ī
∂Nj
0 =

(2.63)(2.64)The second equation M = dd¯ indicates that the quarks d, d¯ can be identified withthe original quarks Q, Q̄! Thus, the dual of the dual reproduces the original theory.
Although this Seiberg duality is not proved but only conjectured, there are many
evidences which support this duality. First of all, as mentioned earlier, ’t Hooft
anomaly matching conditions are satisfied.
SU (Nf )3L,R E Nc A(

U (1)R SU (Nf )2L,R

U (1)B SU (Nf )2L,R

U (1)RU (1)R U (1)2B)

D Nf A( ) + (Nf − Nc )A( ) = Nc A( ),
!
Nc
N2
E
−
Nc × µ( ) = − c µ( ),NfNfNc
Nf − 2Nc
D (− 1)(Nf − Nc ) × µ( ) +× Nf µ( )NfNfN2= − c µ( ),Nf
E 1 × Nc × µ( ),Nc
× (Nf − Nc )µ( ),D
Nf − Nc!
Nc
E−× 2NcNf + 1 × (Nc2 − 1) = −Nc2 − 1,
Nf!
!
2(Nf − Nc )Nc
2D
− 1 × Nf +− 1 × 2Nf (Nf − Nc )Nf
Nf+(Nf − Nc )2 − 1 = −Nc2 − 1,!
Nc
E 1× −
× 2Nf Nc = −2Nc2,
Nf!

U (1)3R

D

Nc
Nc
−1 ×
Nf
Nf − Nc

E

Nc
−
Nf

!2

× 2Nf (Nf − Nc ) = −2Nc2 ,!3

× 2Nc Nf + 1 × (Nc2 − 1) =−2Nc4
+ Nc2 − 1,
Nf2

CHAPTER 2. PHASES OF SUPERSYMMETRIC QCD
2(Nf − Nc )
−1
Nf

D

!3

× Nf2 +

Nc
−1
Nf

21
!3

× 2Nf (Nf − Nc )−2Nc4+(Nf − Nc ) − 1 =+ Nc2 − 1.
2
Nf2

(2.65)

Here D denotes “dual”. Second, all gauge invariant operators of the electric theory
are mapped to those of the magnetic theory.
QQ̄ ↔ M,

(2.66)

B = QNc ↔ b = q Nf −Nc ,(2.67)

B̄ = Q̄Nc ↔ b̄ = q̄ Nf −Nc .

(2.68)

This mapping is compatible with global symmetries. Note that the meson q q̄ in the
dual is redundant because of 0 = ∂W/∂M = q q̄.
Third, the duality is preserved under various deformations of the theory. For
example, adding the mass term for Nf -th quark δW = mQNf Q̄Nf and integratingout, we obtain the effective theory of SU (Nc ) gauge theory with (Nf − 1) flavors. InN

the dual theory, this mass term corresponds to δW = mMNff . Solving the equations
NNof motion for MNff , Mi f , MNi f (i = 1, · · · , Nf − 1), we find hq Nf q̄Nf i = −mµ andhqii = hq̄ i i = 0. This breaks the gauge symmetry SU (Nf − Nc) to SU (Nf − Nc − 1).
Then, the effective theory is SU (Nf − Nc − 1) gauge theory with (Nf − 1) flavors,gauge singlet meson M and the same form of the superpotential. The resulting pairof theories is nothing but theories which Nf − 1 is substituted for Nf . This is aconsistent result. The above discussion is incomplete for Nf = Nc + 2, where themass terms trigger complete magnetic gauge symmetry breaking. Then, the low
energy should include instanton effects in the broken group. In this case, there arefermion zero modes, one mode for each ψq , ψq̄ . The superpotential coupling φM ψq ψq̄
lifts zero modes, then the instanton generates the superpotential
Winst = Λ̃−Nc +2 det(µ−1 M̂ ) == −Λ̃4−Nfdet(µ−1 M̂ )hq Nf +2q̄Nf +2 i

detM̂
detM̂
Λ̃4−Nf
det
M̂
=
−
=
−
.
N
+2
2N
−2
mµ c
mΛ c
ΛL2Nc −1

(2.69)

Here M̂ is (Nf − 1) × (Nf − 1) matrix. We used the scale matching relation in+2
in the third
the second and the last equalities, the equation of motion for MNNcc+2

CHAPTER 2. PHASES OF SUPERSYMMETRIC QCD

22equality. The scale relation (2.60) is used in the fourth equality. One can see that
this superpotential is generated by 1-instanton effect in completely broken magnetic
gauge group. Following Ref. [14], the ’t Hooft vertex for 1-instanton in SU (2) gauge
theory with Nf (= Nc + 2) flavors is given byψq2Nf λ̃4 Λ̃−Nc −4 ,(2.70)

where λ̃ is the gaugino in the magnetic theory, and ψq is obviously the fermioniccomponents of q. First, inserting gaugino-fermion-scalar vertex ψq λ̃φ∗q four times,weconvert the ’t Hooft vertex toψq2(Nf −2) (φ∗q )4 Λ̃−Nc −4 .

(2.71)

Next, we use the superpotential coupling µ1 ψM φq ψq̄ twice to obtain
1 2(Nf −3) ∗ 4 2 2 −Nc −4ψ(φq ) ψM φq Λ̃.
µ2 q(2.72)

Finally, we use the superpotential coupling µ1 φM ψq ψq to obtain the term
1

N −3

µNf −1

φMf

2
ψM(φ∗q )4 φ2q Λ̃−Nc −4 .

(2.73)2ψMh(φ∗q )i4 hφq i2 Λ̃−Nc −4 .

(2.74)

Substituting VEV for φq , we obtain1

N −3µNf −1

φMf

The integration over the instanton size will result in the additional factor of hφ∗q i−4
because the dependence on hφ∗q i has to be cancelled in order for the superpotential
to be holomorphic. Thus, 1-instanton in the completely broken SU (2) group gives
a contribution to the superpotential of the formdet( µ1 M )hφq i2

Λ̃−Nc +4 .

(2.75)

Adding this to the tree level superpotential, we obtain the following superpotential
11
M q q̄ − 2Nc −1 detMµΛL
1=
(M B B̄ − detM ),2N
Λ c −1W =(2.76)

CHAPTER 2. PHASES OF SUPERSYMMETRIC QCD
qwhere we used q =µ/ΛL2Nc −1 B, q̄ =q

23µ/ΛL2Nc −1 B̄. This is the same superpotential

as that of s-confinement phase (2.44). In the electric description, Eq.(2.44) is due tothe strong coupling effects. In the magnetic description, it is rederived in a weakly
coupled framework.
One more example of the deformation is “flat direction deformation”. If VEVof Nf -th quark is non-zero and large compared to the dynamical scale, the effective
theory will become SU (Nc − 1) gauge theory with (Nf − 1) flavors. In the dualNtheory, this deformation corresponds to hMNff i =6 0. Since M is a gauge singlet, thegauge symmetry is unbroken. The effective theory is SU (Nf − Nc ) gauge theory
with (Nf − 1) flavors. The resulting pairs of theories are nothing but theories where
Nc − 1 and Nf − 1 are substituted for Nc and Nf .
Classically, the electric and magnetic theories have different moduli space. It isonly after taking non-perturbative effects into account that they are seen to be identical. For example, in the electric theory, there is a classical constraint rankhM i ≤ Nc .
In the dual theory, M is an independent field whose VEV is unconstrained to allorders of perturbation theory. The constraint arises in the dual theory by quantum
effects. The equation of motion for M sets hN i = 0. However, the magnetic theory
has Nf − rankhM i massless flavors, so if Nf − rankhM i < Nf − Nc , the superpotential, which is analogous to ADS superpotential, is generated dynamically and there
is no vacuum with hN i = 0. Therefore, the vacua of the dual theory also satisfies
rankhM i ≤ Nc as a result of quantum effects rather than as a classical constraint.
Similarly, for rankhM i = Nc , the magnetic theory has Nf − Nc = Ñc masslessflavors and a constraint similar to (2.32)
detN − bb̄ = Λ̃2LÑc(2.77)

appears, where Λ̃2LÑc = det0 hµ−1 M iΛ̃3Ñc −Nf , det0 hµ−1 M i means the product of the
Nc non zero eigenvalues of hµ−1 M i. Using the equation of motion for M , i,e, hN i = 0
and the mapping (2.59) and the scale relation (2.60), the constraint (2.77) becomeshB B̄i = det0 hM i.

(2.78)

This is a constraint which is classical in the electric theory but quantum in the dual

CHAPTER 2. PHASES OF SUPERSYMMETRIC QCD

24

theory. It is quite interesting that classical relations arise via quantum effects in the
dual theory.
Before closing this chapter, we comment on phases of other ranges of Nf which
are not touched upon. For Nf ≥ 3Nc , the original theory is not asymptotically
free. The low energy spectrum is elementary quarks and gluons. The phase in thisrange is referred to as “non-Abelian free electric phase”. For Nc + 2 ≤ Nf ≤ 32 Nc ,the magnetic theory is not asymptotically free. The low energy spectrum consistsof SU (Nf − Nc ) gauge fields and M, q, q̄. The phase in this range is referred to as“non-Abelian free magnetic phase”. The phases of SUSY QCD is summarized inFig. 2.1.

CHAPTER 2. PHASES OF SUPERSYMMETRIC QCD

25

Nf
Original
theory is
IR free
3Nc
IR fixed point

IR fixed point

strongercouplingof dualtheory3/2Nc
Dual theory is IR free
Nc +2

Nc +1
stronger Moduli spaceof vacua forcouplingof original
M,B,B
theoryNc

Nc -1
No vacuum

Figure 2.1: Phases of SUSY QCD

Chapter 3
Negative Dimensional Group andDuality
Duality proposed by Seiberg [2] gives a clue to analyze the strong coupling dynamics.
Many authors have proposed dualities which have various gauge groups and various
matter contents [15]-[45].
In this chapter we point out that the similarities of dualities in N = 1 supersymmetric SO, SP gauge theories can be explained by using the trick of negative
dimensional groups. In constructing the duality in N = 1 supersymmetric gauge
theories one of the key ideas is anomaly matching [48]. Anomaly matching conditions heavily depend on the representations and the charges of matter fields in themodel. Therefore we have to find the matter content of the model by trial and error. However if we know one SO(SP ) duality we can easily obtain another SP (SO)duality automatically satisfying anomaly matching conditions by using the trick ofnegative dimensional groups.
The idea of negative dimensional groups is not new and goes back to Penrose[49], who has constructed the SU (2) (= SP (2)) representations in terms of SO(−2).
Since then many relations have been observed among the expressions for the SU (N),
SO(N ) and SP (N) group invariants under the substitution N → −N [50]. On
the other hand, Parisi and Sourlas [51] have observed that a Grassmann spaceof dimension N can be interpreted as an ordinary space of dimension −N . In

26

CHAPTER 3. NEGATIVE DIMENSIONAL GROUP AND DUALITY

27

supersymmetric theories, the N → −N relations are, in a sense, built in and weare going to utilize this property. We also give a comment on the problem which
one encounters in applying the trick of negative dimensional group to Spin groupduality.
This chapter is based on the work in collaboration with S. Kitakado [47].

3.1

Similarities in SO and SP duality

In order to show the advantages of our trick of negative dimensional groups we
focus here on dualities with SO and SP gauge groups. ∗ It is well known that thesemodels strongly resemble each other in appearance. As an example, we shall takethe model proposed by Intriligator [27]. In Ref. [27], duality in supersymmetric
SO and SP gauge theories are discussed. The electric theory of SO dual model
is a N = 1 supersymmetric SO(2Nc)† gauge theory with 2Nf fields Qi in thefundamental representation and a symmetric traceless tensor X. The anomaly freeglobal symmetries are SU (2Nf ) × U (1)R with the fields transforming as
!

Q

i

X

2(Nc − k),1 −,(k + 1)Nf


2
1,
.
k+1

(3.1)

The superpotential is
W = gk TrX k+1 .

(3.2)

The magnetic theory is N = 1 supersymmetric SO(2Ñc) gauge theory, whereÑc ≡ k(Nf + 2) − Nc , with 2Nf fields q i in the fundamental representation, asymmetric traceless tensor Y and singlets Mj (j = 1, · · · , k). The anomaly freeglobal symmetries are SU (2Nf ) × U (1)R with the fields transforming as
!

q
∗

i

2(Ñc − k)
,1 −,(k + 1)Nf

As will be explained later, SU (N ) group is self-dual under N → −N. Therefore we don’t

discuss SU group here.
†
In order to see the relation with the SP groups, we are restricting our discussion to the evendimensional SO(2Nc ) groups leaving aside the SO(2Nc + 1) groups.

CHAPTER 3. NEGATIVE DIMENSIONAL GROUP AND DUALITY




2
1,
,
k+1
!
2(j + k) 4(Nc − k)−,.
(k + 1)(k + 1)Nf

Y
Mj

28
(3.3)

The superpotential is
W = TrY k+1 +kX

Mj qY k−j q.

(3.4)

j=1

On the other hand, the electric theory of SP dual model is a N = 1 supersymmetric SP (2Nc ) gauge theory with 2Nf fields Qi in the fundamental representation andan antisymmetric traceless tensor X ‡ . The global symmetries are SU (2Nf )×U (1)Rwith fields transforming as
!

Q

i

X

2(Nc + k),1 −,(k + 1)Nf


2
1,
.
k+1

(3.5)

The superpotential is
W = gk TrX k+1 .

(3.6)

The magnetic theory is N = 1 supersymmetric SP (2Ñc ) gauge theory, whereÑc ≡ k(Nf − 2) − Nc , with 2Nf fields q i in the fundamental representation, an antisymmetric traceless tensor Y and singlets Mj (j = 1, · · · , k). The global symmetries
are SU (2Nf ) × U (1)R with the fields transforming as
!

qi
Y
Mj

2(Ñc + k)
,1 −,(k + 1)Nf


2
1,
,
k+1
!
2(j + k) 4(Nc + k),.
−
(k + 1)(k + 1)Nf(3.7)

The superpotential is
W = TrY k+1 +kX

Mj qY k−j q.

(3.8)

j=1

It is easy to recognize that the representations and the charges of fields are quite
similar. Furthermore, it can be seen that we can obtain SP (SO) duality from the‡In this thesis, we denote the symplectic group as SP (2Nc ) whose fundamental representation

is 2Nc dimensional.

CHAPTER 3. NEGATIVE DIMENSIONAL GROUP AND DUALITY

29

SO(SP ) duality by changing the signs of Nc , Nf into −Nc , −Nf , and exchanging
a symmetric (an antisymmetric) tensor for an antisymmetric (a symmetric) tensor.
This feature is not specific to this model and is applicable to SO, SP dual models
discovered so far [17, 26, 22].

3.2

Negative dimensional group

Group theoretically, these can be anticipated by considering the negative dimensional groups first proposed by Penrose [49]. This is a technique to calculate the algebraic invariants. Using this technique, we can find the peculiar relations for dimensions of the irreducible representations of the classical groups SU (N), SO(N), SP (N)[50]. If λs is a Young tableau with s boxes and if the dimensions of the corresponding
irreducible representations of SU (N), SO(N) and SP (N) are denoted by D{λs ; N },D[λs ; N ] and Dhλs ; N i, respectively, it was noticed by King [50] thatD{λs ; N } = (−1)s D{λ˜s ; −N },D[λs ; N ] = (−1)s Dhλ˜s ; −N i.

(3.9)

Here λ̃ stands for the “transposed” (rows and columns interchanged ) Young tableau.
The simplest example of the relations (3.9) is a two index tensor,
for SU (N), SP (2N )(SO(2N )) ↔

for SU (−N ), SO(−2N )(SP (−2N )).
(3.10)

Moreover, it is useful to give the relations among the generalized Casimirs of theclassical groups in totally symmetric and totally antisymmetric representations.
CpSU (N) (1, 1, · · · , 1) = (−1)p−1 CpSU (−N )(r, 0, · · · , 0),
CpSO(2N ) (1, 1, · · · , 1) = (−1)p−1 CpSP (−2N )(r, 0, · · · , 0),(3.11)CpSP (2N )(1, 1, · · · , 1) = (−1)p−1 CpSO(−2N )(r, 0, · · · , 0),
where Cp (1, 1, · · · , 1) and Cp (r, 0, · · · , 0) mean the p-th order generalized Casimir in
totally antisymmetric and totally symmetric rank-r tensor representations respectively. These relations Eqs. (3.9) and (3.11) are necessary to take anomaly matchingconditions into account.

CHAPTER 3. NEGATIVE DIMENSIONAL GROUP AND DUALITY30

We can express these results symbolically as follows [50],SU (−N ) ∼= SU (N),SO(−2N ) ∼= SP (2N ),(3.12)SP (−2N ) ∼= SO(2N ),where the overbar means symmetrization and antisymmetrization are interchanged.
Since supersymmetric theories are “invariant” under this interchange we can use
this technique as a useful method of obtaining a new dual theory from a known
dual theory through the extrapolation to the negative dimensional groups. The
procedures are :• Change the sign of the group dimension, Nc ↔ −Nc , Nf ↔ −Nf .

• Interchange the symmetrization and antisymmetrization of the representations.
We can actually convince ourselves that with these procedures Eq. (3.7) follows
from Eq. (3.3). This negative dimensional group technique is very powerful since
the anomaly matching is automatically satisfied.

3.3

Discussion

In this chapter, we have pointed out that the similarities of dualities in N = 1
supersymmetric SO, SP gauge theories can be explained by using the trick of negative dimensional groups. If we know the duality in supersymmetric SO(SP ) gauge
theory, we can easily obtain another duality in supersymmetric SP (SO) gauge theory automatically satisfying the anomaly matching conditions by extrapolating thegroups to the negative dimensions. By this trick we can also know the representations and the charges of the fields. On the other hand, when there is no known
duality in SO(SP ) gauge theory this trick is powerless.
Explicit application of this trick in finding new dualities is left for future studies.
It is interesting to see whether this trick is applicable to other groups. If we try

CHAPTER 3. NEGATIVE DIMENSIONAL GROUP AND DUALITY

31

to apply the trick of negative dimensional groups to the Spin group dualities [32][37], we immediately get into trouble because the spinors in negative dimensions(called “spinsters” in [52]) have an infinite dimensional representation. This can beseen as follows. Let us recall how Dirac constructed the spinor representation. He
considered the orthogonal norm as the squared of the linear form,(x1 )2 + (x2 )2 + · · · + (xn )2 = (γ 1 x1 + γ 2 x2 + · · · + γ nxn )2 ,(3.13)

where xi means components of n-dimensional vector. This requires the γ i to satisfy
the anticommutation relations which define a Clifford algebra Cn{γ i, γ j } = 2δ ij .

(3.14)

Note that there exists a 2[n/2] × 2[n/2] matrix representation of Cn. Now consider amatrix in the defining representation of the orthogonal group O(n) acting on the
Euclidean vector x as
xi → x0i = Oji xj .

(3.15)In order for Eq. (3.13) to be invariant, γ i is transformed as follows,
γ i → γ 0i = Oji γ j .

(3.16)

Since the γ i also satisfy the Clifford algebra (3.14) and the matrix representation is
unique, we must have
γ 0i = S(O)γS(O)−1,(3.17)

where S is a 2[n/2] × 2[n/2] matrix depending on the orthogonal matrix O. In fact, theS forms a double-valued unitary representation of O(n), which we call the spinor
representation.
Next, we consider the same procedure on a Grassmann tensor space since thenegative dimensional group is defined in terms of their action on a Grassmann spacein a sense. In this case, it is natural to consider the symplectic form Θ2 ≡ Θi Jij Θj ,where Jij is an invariant metric of Sp(n). Then, we look for algebraic quantities β isuch thatΘ2 ≡ ΘiJij Θj = 2(Θ1 Θ2 + Θ3 Θ4 + · · · + Θn−1 Θn )= (β 1 Θ2 − β 2 Θ1 + · · · + β n−1 Θn − β n Θn−1 )2 = (β i Jij Θj )2 .

(3.18)

CHAPTER 3. NEGATIVE DIMENSIONAL GROUP AND DUALITY

32

This leads to the commutation relation[β i, β j ] = 2J ij .

(3.19)

We can represent the β i by a unitary irreducible infinite dimensional representation.
The remaining argument is same except that O is a matrix in the defining representation of Sp(n) in this case, and the spinster is represented by S, which is an
infinite dimensional matrix depending on the symplectic matrix O.
It seems to be impossible to match the global anomalies from the above argument. One may think that the above problem can be avoided in the case of Spin(8)gauge theory [33] because vector, spinor and conjugate spinor representations areequivalent in this case. However, it can be shown that duality does not work in
the theory extrapolated from Spin(8) duality. For the present, we have no idea
how spinsters should be treated. It is also interesting to consider the case of exceptional groups. The relations among dimensions of the irreducible representations
and Casimirs are also known for exceptional groups [53]. We have described thesubstitution N → −N just as a useful trick for studying the duality structures ofsupersymmetric theories leaving aside the direct significance it might have in suchtheories. Thus it might be interesting to study the symmetries under N → −N
directly in the supersymmetric theories where duality is realized explicitly. We hope
to report elsewhere on these together with the related problems.

Chapter 4
Confining Phase in SUSY SO(12)Gauge Theory
Our understanding of non-perturbative nature in N = 1 supersymmetric gauge theories has much advanced since the pioneering works of Seiberg and his collaborators[8, 17]. Especially, the physics of confining phase in N = 1 supersymmetric gauge
theories was enriched (quantum deformed moduli space, “s-confinement”). These
works have been also extended to the theories with various types of gauge groups
and matter contents [9, 56, 57, 58]. Furthermore, these theories have recently been
applied to construction of models with dynamical supersymmetry breaking or SUSY
composite models.
In this chapter, we present the confining phase in N = 1 supersymmetric SO(12)gauge theory with Nf ≤ 7 vectors and one spinor. There are two motivations weare interested in this particular model. First, from the theoretical point of view,
it will provide useful informations for finding the dual to SO(Nc )(Nc > 10) withan arbitrary number of vectors and spinors. Although the duality of this classof models has only been generalized to SO(10) [37], the known dualities have thefollowing remarkable properties which are not contained in Seiberg’s duality [2]:(i). Chiral-Nonchiral duality [32, 33, 36].
(ii). Each gauge group belongs to different Cartan families [32]-[37].

33CHAPTER 4. CONFINING PHASE IN SUSY SO(12) GAUGE THEORY

34(iii). Reducibility to the exceptional group (G2 ) duality [32, 38, 39].
(iv). Simple and Semi-simple group duality∗ [37].
(v). Identification of massive spinors and Z2 monopoles under duality [59].
It is interesting to ask whether these properties persist in the duality for SO(Nc )(Nc >10) theory. However, from the result of Ref. [37] looking for this class of duals seems
to be highly non-trivial. Cho [56] has already investigated in detail the confining
phase of SO(11) gauge theory with Nf ≤ 6 vectors and a spinor and extracted
some clues in search for duals. It is interesting to pursue further following the lineof his argument in order to clarify the dual to SO(Nc )(Nc > 10) theory. Second,as mentioned in the above paragraph, the theory under consideration may provide phenomenologically viable models with dynamical supersymmetry breaking or
SUSY composite models.
This chapter is based on the work [54].

4.1

A ModelThe model we consider has the following symmetry groups
G = SO(12)gauge × [SU (Nf )V × U (1)V × U (1)Q × U (1)R ]global(4.1)

under which the superfields transform as†Vµi ∼ (12,, 1, 0, 0),Qα ∼ (32, 1, 0, 1, 0)

(4.2)
(4.3)

and no tree level superpotential. Note that since each of the U (1) symmetries in
Eq. (4.1) are anomalous, the action is transformed as
S → S − Cα
∗
†

Z

d4 x

g2
F F̃ ,
32π 2

(4.4)

We do not consider here the duality derived by using deconfinement technique [23].
We implicitly regard the 32 dimensional SO(12) spinor as a projection Q = P− Q64 where Q64

means the 64 dimensional spinor of SO(13) and P− = 12 (1 − Γ13 ).

CHAPTER 4. CONFINING PHASE IN SUSY SO(12) GAUGE THEORY

35

where C denotes the anomaly coefficient of the corresponding U (1)V SO(12)2, U (1)Q SO(12)2or U (1)R SO(12)2 anomalies and α is a transformation parameter. If the theta parameter in the Lagrangian is shifted under these anomalous U (1)’s as θ → θ + Cα,then anomalies can be cancelled. Recalling the relation
Λ
µ

!b0

!

8π 2
+ iθ ,= exp − 2
g (µ)

(4.5)

where b0 represents 1-loop beta function coefficient ‡X
1b0 = [3µ(Adj) −µ(R)] = 26 − Nf ,2matter(4.6)

and Λ is the strong coupling scale of the theory, the spurion superfield Λb0 is transformed asΛb0 ∼ (1, 1, 2Nf , 8, 12 − 2Nf ).

(4.7)

Using these symmetries and holomorphy, we can easily fix the form of the dynamically generated superpotential Wdyn for the small value of Nf so that U (1)Rcharge of Wdyn be 2 and U (1)V , U (1)Q charges vanish. The results are summarizedin Table 4.1. Nf = 6 case is special because R charge of Λb0 vanishes. This meansthat we cannot construct the dynamically generated superpotential. However, theclassical constraint among matter superfields is modified by non-perturbative effects
and this quantum constraint can be included in the superpotential by using the Lagrange multiplier superfield X [8]. Thus, Nf = 6 case is analogous to Nf = Nc SUSYQCD (quantum deformation of moduli space). Furthermore, we will immediately
find that Nf = 7 case is analogous to Nf = Nc +1 SUSY QCD (“s-confinement”) [8],and Nf ≤ 5 case is analogous to Nf ≤ Nc − 1 SUSY QCD (runaway superpotential)[7].
In order to describe the low energy effective theory, we need to find gauge invariant operators which behave as the moduli space coordinate § . It is in general‡µ denotes quadratic Dynkin index defined as Tr T a (R)T b (R) = µ(R)δ ab (T a : the generatorsof the group, R: representation which the superfield belongs to). We use the following values :µ(12) = 2, µ(32) = 8, µ(66) = 20.
§
Vacuum expectation values (VEVs) of these gauge invariant operators are in one to one correspondence to the solutions of D-flatness conditions [61].

CHAPTER 4. CONFINING PHASE IN SUSY SO(12) GAUGE THEORY

Nf

R(Λb0 )

Wdyn

0

12

(Λ26 /Q8 )1/6

1

10

(Λ25 /Q8V 2 )1/5

2

8

(Λ24 /Q8V 4 )1/4

3

6

(Λ23 /Q8V 6 )1/3

4

4

(Λ22 /Q8V 8 )1/2

5

2

Λ21 /Q8V 10

60

X(Q8 V 12 − Λ20 )7−2

Q8 V 14 /Λ19

36

Table 4.1: The form of dynamically generated superpotentials
troublesome to do this task. However, if the gauge symmetry breaking pattern is
known at generic points on the moduli space, one can easily identify these gauge
invariant operators. We illustrate below how it works in the present model. The
gauge symmetry breaking pattern we utilize is [62]
<32>

<12>

<12>

<12>

<12>

<12>

SO(12) −→ SU (6) −→ SU (5) −→ SU (4)−→ SU (3) −→ SU (2) −→ 1.

(4.8)

With this information in hand, counting degrees of freedom of gauge invariant operators is nothing but a group theoretical exercise. We display in Table 4.2 parton
degrees of freedom, unbroken subgroups, eaten degrees of freedom by Higgs mechanism and hadron degrees of freedom. We are now in a position to construct gauge
invariant operators explicitly. Before doing this, we need to notice that SO(12)spinor product decomposes into the following irreducible representations˜ ,32 × 32 = [0]A + [2]S + [4]A + [6]
S

(4.9)

where [n] represents rank-n antisymmetric tensor, subscripts “A” and “S” mean
antisymmetry and symmetry under spinor exchange, and the tilde of the last term
implies that the rank-6 tensor is self-dual. Since our model has only one spinor,
˜ in Eq. (4.9).
gauge invariant operators can include [2]S and [6]S

CHAPTER 4. CONFINING PHASE IN SUSY SO(12) GAUGE THEORY

Nf

PartonUnbroken

Eaten

HadronDOF

Subgroup

DOF

DOF

0

32

SU(6)

66 − 35 = 31

1

1

44

SU(5)

66 − 24 = 42

2

2

56

SU(4)

66 − 15 = 51

5

3

68

SU(3)

66 − 8 = 58

10

4

80

SU(2)

66 − 3 = 63

17

5

92

1

66

26

6

104

1

66

38

7

116

1

66

50

37

Table 4.2: Degrees of freedom of independent gauge invariants

Taking this into account, we can construct gauge invariant composites as follows¶L =M (ij) =N [ij] =P [ijklmn] =R[ijklmn] =1(QT Γ[µ Γν]CQ)(QT Γ[µ Γν]CQ) ∼ (1; 1; 4Nf ; 4R),
2!2!
(V iµ )T Vµj ∼ (1;; −8; 2R),
1 T i jQ /V /V CQ ∼ (1; ; 2Nf − 8; 4R),2
1 T [i j k l m n]
Q /V /V /V /V /V /V CQ ∼ (1; ; 2Nf − 24; 8R),
6!
1 µ1 ···µ12 T(Q Γµ1 Γν CQ)(QT Γν Γµ2 · · · Γµ6 CQ)Vµi7 Vµj8 Vµk9 Vµl10 Vµm11 Vµn12

6!

∼ (1;

; 4Nf − 24; 10R),(4.10)

where the square bracket means the antisymmetrization of the corresponding indices,/ i ≡ Vµi Γµ and we use here the following SO(12) Gamma matrices,V¶

The representations and the charge in Eq. (4.10) are those under Eq. (4.11).

CHAPTER 4. CONFINING PHASE IN SUSY SO(12) GAUGE THEORYΓ 1 = σ2 ⊗ σ 3 ⊗ σ3 ⊗ σ 3 ⊗ σ3 ⊗ σ 3 ,Γ2 = −σ1 ⊗ σ3 ⊗ σ3 ⊗ σ3 ⊗ σ3 ⊗ σ3 ,

Γ 3 = 1 ⊗ σ 2 ⊗ σ 3 ⊗ σ 3 ⊗ σ3 ⊗ σ 3 ,Γ4 = −1 ⊗ σ1 ⊗ σ3 ⊗ σ3 ⊗ σ3 ⊗ σ3 ,

Γ 5 = 1 ⊗ 1 ⊗ σ 2 ⊗ σ3 ⊗ σ 3 ⊗ σ3 ,

Γ6 = −1 ⊗ 1 ⊗ σ1 ⊗ σ3 ⊗ σ3 ⊗ σ3 ,

Γ 7 = 1 ⊗ 1 ⊗ 1 ⊗ σ 2 ⊗ σ 3 ⊗ σ3 ,

Γ8 = −1 ⊗ 1 ⊗ 1 ⊗ σ1 ⊗ σ3 ⊗ σ3 ,

Γ 9 = 1 ⊗ 1 ⊗ 1 ⊗ 1 ⊗ σ 2 ⊗ σ3 ,

Γ10 = −1 ⊗ 1 ⊗ 1 ⊗ 1 ⊗ σ1 ⊗ σ3 ,

Γ11 = 1 ⊗ 1 ⊗ 1 ⊗ 1 ⊗ 1 ⊗ σ2 ,

Γ12 = −1 ⊗ 1 ⊗ 1 ⊗ 1 ⊗ 1 ⊗ σ1 ,38

Γ13 = σ3 ⊗ σ3 ⊗ σ3 ⊗ σ3 ⊗ σ3 ⊗ σ3 ,and C is a charge conjugation matrix.
In order to see that these gauge invariant operators are in fact the coordinatesof the moduli space, one has to check whether the total degrees of freedom of thesegauge invariant operators in Eq. (4.10) coincide with hadronic degrees of freedom.
In Table 4.3, these degrees of freedom are listed as a function of Nf . For Nf ≤ 5,

Nf

Hadron DOF L

M

N

P

R Constraints

0

1

1

0

0

0

0

1

2

1

1

0

0

0

2

5

1

3

1

0

0

3

10

1

6

3

0

0

4

17

1

10

6

0

0

5

26

1

15

10

0

0

6

38

1

21

15

1

1

−1

7

50

1

28

21

7

7

−14

Table 4.3: Hadron degrees of freedom counthadronic degrees of freedom and that of L, M, N, P and R agree with each other. For
Nf = 6, degrees of freedom of L, M, N, P and R are larger than those of hadrons by
one. This implies that L, M, N, P and R are not independent and a single constraint
among them exists. This statement is also consistent with the previous argumentfor dynamically generated superpotentials. For Nf = 7, 14 constraints are expected
to come from the equations of motion as Nf = Nc + 1 SUSY QCD.

CHAPTER 4. CONFINING PHASE IN SUSY SO(12) GAUGE THEORY39

We can obtain more non-trivial support which convinces us that gauge invariant operators L, M, N, P and R are the moduli. One of the powerful methods tostudy the low energy spectrum is ’t Hooft anomaly matching [48]. To see thatanomalies between elementary fields and composite ones match, we take anomaly
free symmetry group instead of Eq. (4.1)
GAF = SO(12)gauge × [SU (Nf ) × U (1) × U (1)R ]global ,(4.11)

where new U (1) and U (1)R are linear combinations of the original U (1)’s in Eq.(4.1).
Matter superfields transform under these symmetries as
Vµi ∼ (12,, −4, R),

(4.12)

Qα ∼ (32, 1, Nf , R),(4.13)

N −6

where R = Nff +4 . We can calculate anomalies and see that anomalies match forµ( ), SU (7)2 U (1) : −48µ( ), U (1)R :Nf = 7; SU (7)3 : 12A( ), SU (7)2 U (1)R : − 120
11− 434, U (1) : −112, U (1)2RU (1) : 2128
, U (1)R U (1)2 : − 29120, U (1)3R : − 28154, U (1)3 :
11
121
11
1331
5600, where A( ) and µ( ) are cubic and quadratic Dynkin indices for fundamental
representation of SU (Nf ). Recalling that anomalies are saturated in Nf = Nc + 1
SUSY QCD, we are convinced that this coincidence for Nf = 7 is very natural andgives a strong support that the theory in this case is in “s-confinement” phase.
For Nf ≥ 8, it is impossible to satisfy anomaly matching conditions without
violating Nf ≤ 7 result even if other gauge invariant operators are added. Thisimplies that a confining phase of this model terminates at Nf = 7.

4.2

Low energy superpotentials

In this section, we determine explicitly the low energy superpotentials in terms of
L, M, N, P and R. Since we know which of gauge invariant operators are moduli inthe previous section, it is straightforward to work out which of operators should be in
the superpotential. Following Ref. [56], we first determine the quantum deformed
constraint in Nf = 6 theory. Dimensional analysis, symmetries and holomorphy

CHAPTER 4. CONFINING PHASE IN SUSY SO(12) GAUGE THEORY

40

restrict the superpotential as follows
WNf =6 = X(R2 + P 2 L + 2P PfN + L2 detM
(4.14)
1
i i i i i i j j j j j j LN i1 j1 N i2 j2 M i3 j3 M i4 j4 M i5 j5 M i6 j6+
2!4! 1 2 3 4 5 6 1 2 3 4 5 61+
i i i i i i j j j j j j N i1 j1 N i2 j2 N i3 j3 N i4 j4 M i5 j5 M i6 j6 − Λ20
6 ),4!2! 1 2 3 4 5 6 1 2 3 4 5 6
where Λ6 is the strong coupling scale of SO(12) gauge theory with 6 vector flavors
and a spinor. Coefficients of each terms are determined so that it reproduce thesuperpotential in SO(11) gauge theory with 5 flavors [56]. (If VEV < V126 >6= 0
is given, SO(12) gauge theory with 6 flavors under consideration here reduces toSO(11) gauge theory with 5 flavors)
One may also determine these coefficients by using the symmetry breaking along<32>

the spinor flat direction SO(12) −→ SU (6), explicitly < 32 >T = (0, a, 0, · · · , 0, a, 0).
Vµi decomposes into 6 + 6̄ under SU (6), which is explicitly

q1 + q̄1  i(q1 − q̄1 )  q + q̄



2
2




 i(q2 − q̄2 )  q + q̄

3
3






 i(q3 − q̄3 ) ,Vµ = 
 q4 + q̄4  i(q − q̄ ) 

4
4 




 q5 + q̄5  i(q5 − q̄5 ) 






 q6 + q̄6 (4.15)i(q6 − q̄6 )where qi , q̄i (i = 1, · · · , 6) mean SU (6) quarks, antiquarks, respectively. The reduced
theory is SU (6) gauge theory with 6(+), therefore it has a single quantum

constraint [8].
According to this decomposition rule, SO(12) gauge invariant operators are de-CHAPTER 4. CONFINING PHASE IN SUSY SO(12) GAUGE THEORY

41

composed into the following SU (6) meson mij , baryon b and anti-baryon b̄;
L → 12a4 ,M ij → 2(mij + mji),
N [ij] → −4ia2 (mij − mji),
P → 64ia2 (b + b̄) + 2ia2 ijklmn mij mkl mmn ,R → 64a4 (b − b̄).

(4.16)

Using this information, one can also determine coefficients so that detm − bb̄ = Λ12
be reproduced.
The superpotential of Nf = 7 case can be found in a similar way. In this case,the superpotential must have the following features. 1. It is smooth everywhere on
the moduli space. 2. Equations of motion give classical constraints among vectors
and a spinor. 3. Adding mass term for one vector flavor to this superpotential andintegrating out this massive vector, the superpotential (4.14) must be reproduced.
The result isk
WNf =7 =1(M ij Ri Rj − 2iN ij PiRj + LPi Pj M ij
(4.17)
Λ19
1+L2 detM +i ···i Pj M ji1 N i2 i3 N i4 i5 N i6 i7
3!22 1 7
1
i i i i i i i j j j j j j j LN i1 j1 N i2 j2 M i3 j3 M i4 j4 M i5 j5 M i6 j6 M i7 j7+
3!4! 1 2 3 4 5 6 7 1 2 3 4 5 6 71
i i i i i i i j j j j j j j N i1 j1 N i2 j2 N i3 j3 N i4 j4 M i5 j5 M i6 j6 M i7 j7 ).
+4!3! 1 2 3 4 5 6 7 1 2 3 4 5 6 7

By adding the mass terms for vector fields δW = mij M ij to the superpotential(4.14) and integrating out each massive vectors successively, we can readily derive
the superpotentials for Nf ≤ 5 systematically. As a matter of fact, we obtain
2
WNf =5 = Λ21
5 /(L detM +WNf =4k1
LN i1 j1 N i2 j2 M i3 j3 M i4j4 M i5 j5 i1 i2 i3 i4 i5 j1 j2 j3 j4 j5
2!3!

1+ N i1 j1 N i2 j2 N i3 j3 N i4 j4 M i5 j5 i1 i2 i3 i4 i5 j1 j2 j3 j4 j5 ),
4!
!1/2
Λ22
4
= 2
,
1
L2 detM + 2!2!
LN i1 j1 N i2 j2 M i3 j3 M i4 j4 i1 i2 i3 i4 j1 j2 j3 j4 + (PfN )2

Although this superpotential has already been derived in Ref. [9] by using the index argument,a P M N 3 term was missing. Without this term, the result of Ref. [56] cannot be correctly recovered.

CHAPTER 4. CONFINING PHASE IN SUSY SO(12) GAUGE THEORY

WNf =3

Λ23
3= 3
L2 detM + 2!1 LM i1 j1 N i2 j2 N i3 j3 i1 i2 i3 j1 j2 j3

WNf =2Λ24
2= 4
2
L detM + LN 2

WNf =1

Λ25
1
= 5
L2 M

WNf =0

Λ26
0
= 6
L2

42

!1/3

,

!1/4

,

!1/5

,

!1/6

,

(4.18)

where the strong coupling scales for each flavor are related to each other throughone-loop matching of gauge coupling as follows
24
23
Λ26
= m11 Λ25
0
1 = m11 m22 Λ2 = m11 m22 m33 Λ3
21= m11 m22m33 m44 Λ22
4 = m11 m22 m33 m44 m55 Λ5
19= m11 m22m33 m44 m55m66 Λ20
6 = m11 m22 m33 m44 m55 m66 m77 Λ7 . (4.19)

It is worth to note that one can confirm the above superpotentials (4.18) to recovercorrectly the superpotentials for Nf ≤ 4 in SO(11) theory [56], which is obtained
when one flavor vector field has non-vanishing VEV.
Before closing this section, we briefly discuss dynamical supersymmetry breaking.
For Nf = 6, if we take the tree level superpotential as
Wtree = λ1 S1 V 2 + λ2 S2 Q2V 6 + λ3 S3 Q4V 6 ,(4.20)

where S1,2,3 are singlet superfields, then equations of motion with respect to S1,2,3
and the quantum constraint (4.14) are incompatible. Therefore, supersymmetry is
dynamically broken [63, 64]. For Nf = 0, since we cannot add terms which lift
a classical flat direction (i.e.L) preserving U (1)R to the tree level superpotential,
supersymmetry remains unbroken. The same argument seems to be applicable for1 ≤ Nf ≤ 5 ∗∗ .

∗∗In Ref. [9], the authors construct a model with dynamical supersymmetry breaking for Nf = 1

by promoting a global U (1) to a local U (1) and adding singlets to cancel U (1) gauge anomaly.

CHAPTER 4. CONFINING PHASE IN SUSY SO(12) GAUGE THEORY

4.3

43

Discussion

In this chapter, we have studied the confining phase in N = 1 SO(12) SUSY gauge
theory with Nf ≤ 7 vectors and a spinor. Utilizing the gauge symmetry breaking
pattern at generic points on the moduli space, which plays a crucial role in our study,
we have identified gauge invariant operators which behave as the moduli coordinate.
Then we have derived explicitly low energy superpotentials for Nf ≤ 7.
Some clues in search for duals are obtained from the results of this work. Let us
suppose that the dual with the gauge group G̃ exists for Nf ≥ 8. The original SO(12)theory breaks down to SU (6)+Nf ( + ) along the spinor flat direction. On the other
hand, the gauge group of the dual theory is usually unbroken since in the dual theory
the gauge invariant operators develop VEV, in the present case, it corresponds to
L. Since SU (6) + Nf (+)(Nf ≥ 8) theory is dual to SU (Nf − 6) + Nf (+)[2], we can guess that at least G̃ must include SU (Nf − 6) as a subgroup to preserve
the duality along this direction. Furthermore, the superpotential in the dual theory
must recover Nf = 7 superpotential. We also note that Nf = 8 case in our model isknown to be self-dual [44].
Although it seems to be quite difficult to find a dual which is compatible withthe above requirements, we hope that this work will provide useful informations tosearch for the dual to SO(Nc )(Nc ≥ 11) theory.

Chapter 5
Gauge Mediated SUSY Breaking
5.1

Introduction

In the previous chapters, we have discussed phases of N = 1 supersymmetric gauge
theories. This is a theoretical aspect in a sense. Once we turn to the phenomenological application of SUSY theories, we are necessarily faced with the issue of supersymmetry breaking. If the real world is supersymmetric, then the mass of bosons
and fermions are degenerate. However, we have not yet observed, for instance, thescalar electron (selectron), which is the superpartner of the electron. Therefore, wecannot help concluding that supersymmetry is broken at low energy. Moreover, weemphasize that the mechanism of supersymmetry breaking plays an important rolein the spectrum of the soft terms. This provides crucial informations on experimental methods of the superparticle searches.
There are two questions for supersymmetry breaking. How is SUSY broken?
To answer this question, we have to recall first “the supertrace theorem”[66]. In a
global supersymmetric theory with a gauge group free from gravitational anomalies[67], the sum of the tree level squared mass, weighted by the corresponding numberof degrees of freedom, is equal in the bosonic and fermionic sectors, namely
StrM2 =

X(−1)2J (2J + 1)M2J = 0,

(5.1)

J

where MJ denotes the tree level mass of a particle with spin J. In general, this44CHAPTER 5. GAUGE MEDIATED SUSY BREAKING

45

theorem implies the existence of a superpartner lighter than its ordinary particle.
In other words, this theorem rules out the scenario that supersymmetry is broken
at the tree level and its breaking is communicated to the ordinary supermultipletsat the tree level.
Therefore, we have to consider the scenario in which supersymmetry is broken
at the quantum level not at the classical level. In supersymmetric theories, if supersymmetry is unbroken at the classical level, it is unbroken perturbatively due to the
non-renormalization theorem [6]. Supersymmetry can be broken by certain nonperturbative effects (instantons, gaugino condensation, confinement etc). This is a very
nice feature for the hierarchy problem because nonperturbative effects relate thesmall scale (weak scale) to the large scale (Planck scale or GUT scale) exponentially[68].
How is supersymmetry breaking communicated to the real world once SUSY isbroken? There are mainly two alternatives. One is supergravity (SUGRA) mediation, the other is gauge mediation. In this thesis, we focus on the latter case because
it has many advantages phenomenologically compared to the former case. In the
SUGRA mediation, there is no obvious reason to expect that sfermion masses aredegenerate. For instance, the flavor non symmetric terms in the Kähler potential
are not forbidden by any symmetry. These terms lead to flavor breaking effects inthe soft terms. Thus, the degeneracy of the soft terms is not guaranteed in this
scenario.
On the other hand, in the gauge mediation, since SUSY breaking is transmitted
by the SM gauge interactions, the sfermion masses are degenerate automatically due
to the flavor blindness of the gauge interaction. Although the flavor violation only
arises through Yukawa coupling, these are suppressed enough because their effects
can only appear at higher loop order due to super GIM mechanism. Furthermore,
the number of parameters other than the SM parameters in the soft terms can belargely reduced. This leads to relations among soft masses, and the theory becomes
highly predictive∗ .

∗For more details, see [65].

CHAPTER 5. GAUGE MEDIATED SUSY BREAKING

U(1) mDSB

46

GSM
Messenger

Observable

Figure 5.1: Schematic structure of DNNS model

5.2

DNNS ModelFirst realistic model in this context was constructed by Dine, Nelson, Nir and Shirman [73]. This work was the beginning to attract much attention and interest for
the gauge mediation of SUSY breaking, and to explore models along the various
directions. It is instructive to review this model briefly.
The model consists of three sectors, namely, dynamical SUSY breaking sector(DSB sector), messenger sector and the observable sector as shown in Fig. 5.1. In
DSB sector, SUSY is broken by non perturbative effects and its breaking effects
are transmitted to the messenger sector through the radiative corrections by U (1)mgauge interactions, which is different from the hypercharge U (1) in the SM gauge
group. Then, its effects are transmitted to the observable sector through the radiative corrections by the SM gauge interactions. Note that the DSB sector is neutral
under the SM gauge group and that the observable sector is neutral under U (1)m .
The DSB sector and the observable sector does not connect directly.
It is natural to ask why these two sectors are obliged to be separated, why SUSY
breaking effects should not be transmitted directly to the observable sector. Let us
digress for a while to explain the reason why they introduce the messenger sector. Of
course, such an attempt had been made before by Affleck, Dine and Seiberg [60]. In
order to embed the SM gauge group into the unbroken global symmetry group in the
DSB sector, the gauge group in the DSB sector has to be very large† . This implies a†

Since most of SUSY breaking models are chiral, the size of the gauge group and that of the

global symmetry group are interrelated through the gauge anomaly cancellation condition.

CHAPTER 5. GAUGE MEDIATED SUSY BREAKING

47

large number of extra fields (which play a role of the messenger fields‡ ) transforming
nontrivially under the SM gauge group. As a result, the SM gauge couplings blow
up slightly above the messenger scale, which is often below the GUT scale. For
example, consider the DSB model based on SU (N) (N:odd) with an antisymmetrictensor and N −4 antifundamentals. The smallest gauge group including usual SU (5)GUT group in the unbroken global symmetry group is SU (15), namely, there are 15
flavors messengers. Then the QCD coupling constant blows up at 106−7 GeV. If weembedded SU (2)L × U (1)Y instead of SU (3)C × SU (2)L × U (1)Y , the gauge coupling
unification will be lost and the gluino will be too light because it is generated at ahigher order.
Let us turn back to the model of Ref. [73]. Here we do not specify the DSB
model concretely because what is now of interest is how to communicate SUSY
breaking to the observable sector. Anyway, we assume that SUSY is broken in the
DSB sector.
The superpotential in the messenger sector is1
Wm = k1 Xφ+ φ− + k2 X 3 + k3 XΦΦ̄,
3

(5.2)

where X is a singlet, φ± have charge ±1 under U (1)m and Φ, Φ̄ are the messengers.
The scalar potential in the messenger sector can be obtained
Vm = k12 |X|2 (|φ+ |2 + |φ− |2 ) + |k1 φ+ φ− + k2 X 2 + k3 ΦΦ̄|2
g2+k32 |X|2 (|Φ|2 + |Φ̄|2 ) + m (|φ+ |2 − |φ− |2 )2
2
2
+ 2
− 2
+Mφ± (|φ | + |φ | ),(5.3)

where we denote the scalar components of the superfields by the same symbols as
superfields. The first three terms come from F-term of Wm , the fourth term is
U (1)m D-term and the last term gives the negative soft masses squared for φ± .
Minimizing Vm in a certain range of parameters, non-zero VEVs for φ± , X, FX are
obtained, while Φ, Φ̄ VEVs are zero. It is crucial for the gauge mediation to havehXi =6 0, hFX i =6 0 as follows.
‡

Messenger field is defined as the field transforming nontrivially both under the DSB gauge

group and the SM gauge group, and having non-zero SUSY mass and SUSY breaking mass.

CHAPTER 5. GAUGE MEDIATED SUSY BREAKING48

Recall that SUSY breaking effects in the messenger sector are transmitted tothe observable sector through loops by the SM gauge interactions. At one loop, we
obtain gaugino masses
mλi = ci

αi hFX i,
4π hXi(5.4)

where c1 = 5/3, c2 = c3 = 1. Sfermion masses are generated at two loops as
hFX im̃ = 2
hXi
2

!2 "



C3

α3
4π

2



+ C2

α2
4π

2



5 Y
+
3 2

2 

α1
4π

2 #

.

(5.5)

Here, C3 = 4/3 for color triplets and zero for singlets, C2 = 3/4 for weak doubletsand zero for singlets, and Y is the hypercharge.
Although this model is a first phenomenologically viable model, there are someunsatisfactory points. First of all, many fields other than fields in the DSB sector
and those in the observable sector are introduced in the messenger sector. This
makes the model complicated. Second, SUSY breaking effects are not consideredcorrectly. To be more precise, the superpotentials in the DSB sector, which has adynamically generated superpotential, are not included in the superpotential of themessenger sector. It seems to be unnatural. To make matters worse, the vacuum
discussed above turned out to be the only local minimum [74]. In other words, inthe true vacuum, the SM gauge group is broken at the messenger scale, and SUSY
breaking is not transmitted to the observable sector. Because of the above problems,
the model of Ref. [73] became the driving force to simplifing, modifing the modelor finding new models.
Summarizing the structure of the gauge mediated SUSY breaking, the following
coupling is needed,
W = λXΦΦ̄,(5.6)

where X is a moduli field in DSB sector and Φ, Φ̄ are messengers. What is nontrivial
when one constructs the model is that both hXi and hFX i must be non-zero. If thisis realized, SUSY breaking is communicated to the observable sector, and soft SUSY
breaking masses are generated as
mλi = ciN

αi hFX i,
4π hXi(5.7)

CHAPTER 5. GAUGE MEDIATED SUSY BREAKING2m̃ = 2N

hFX i
hXi

!2 "



C3

α3
4π

2



+ C2

α2
4π

2



5 Y
+
3 2

where N is the flavor number of the messenger fields.

49
2 

α1
4π

2 #

,

(5.8)

Chapter 6
Effective Messenger Sector ModelIn the previous chapter, we pointed out some problems in the model of Dine, Nelson,
Nir and Shirman [73]. In this chapter, we present the model whose messenger
sector is the effective theory of DSB sector. It is also shown that SUSY breaking iscommunicated to the observable sector without breaking the SM gauge group.
This chapter is based on the work in collaboration with N. Haba and T. Matsuoka
[69].

6.1

Messenger sector as the effective theory of
DSB sector

Our model is based on the SU (3) × SU (2) model [60], which breaks supersymmetry dynamically. This model has the SU (3) × SU (2) gauge group, a global U (1)mand a non-anomalous global R-symmetry. In our model this U (1)m is gauged fromthe beginning. The representations and charges of the matter fields are summarized in Table 6.1. Note that the singlet superfield Ē is included to cancel U (1)manomaly, and that vector-like superfields f ≡ (q, l), f¯ ≡ (q̄, ¯l) are involved. Here
we introduce the parameter r which specifies the U (1)R -charges for each superfields
and we assume that the sum of R-charge of f and f¯ is 0. Under the SM gauge group
GSM ≡ SU (3)C × SU (2)L × U (1)Y , Q, Ū, D̄, L and Ē are neutral, and q, q̄, l and ¯l50

CHAPTER 6. EFFECTIVE MESSENGER SECTOR MODEL

ParticleSU (3) × SU (2)U (1)mU (1)R

Q

(3,2)

1/6

r

Ū

( 3̄ , 1 )−2/3−4 − 4rD̄( 3̄ , 1 )1/3

2 + 2r

L

(1,2)

−1/2

−3r

Ē

(1,1)

1

8 + 6r

f
f¯

(1,1)

0

rf

(1,1)

0

rf¯

51

Table 6.1: The representations and charges of matter fields in supersymmetry breaking sector
are transformed as
q : (3, 1, −2/3),q̄ : (3̄, 1, 2/3),

l : (1, 2, 1),l̄ : (1, 2, −1),

(6.1)

respectively.
The tree level superpotential consistent with the symmetries is
Wtree = λ1 QD̄L +0κ0¯ + λ2 (QŪ L)Ē(detQQ̄),
(Q
D̄L)f
f
M2
M5

(6.2)

where Q̄ = (Ū , D̄). In Eq. (6.2), the first term is a renormalizable term which has
been treated in Ref. [60]. The second and the third terms are non-renormalizableterms which will be at most cubic terms of the SU (3) × SU (2) gauge invariant
operators∗ . The coupling constants λ02 and κ0 are taken to be of order O(1). λ1 isassumed to be very small so that the theory becomes weakly coupled.
To analyze the model we first consider the case in which the superpotential isabsent. Under the condition that g3  g2  g1 , where g3 , g2 and g1 represent
SU (3), SU (2) and U (1)m gauge coupling constants, respectively, there exist the D-∗

Other non-renormalizable terms are highly suppressed by the scale M , which will be fixed to

the Planck scale later.

CHAPTER 6. EFFECTIVE MESSENGER SECTOR MODEL

52

flat directions for SU (3) and SU (2) [60]












 a 0 
 a 
 0 
√






2
2





hQi =  0 b  , hŪ i =  0  , hD̄i =  b  , hLi = (0, a − b ).




0 0





0





(6.3)

0

Here a and b are taken as real and positive parameters with a ≥ b. Gauge symmetries
are completely broken along these flat directions and the one-instanton effect induces
the non-perturbative superpotential [60]Wdyn =

Λ73,
detQQ̄

(6.4)

where Λ3 is the scale where SU (3) gauge coupling blows up. Note that we consider
the case Λ3  Λ2 , where Λ2 is the scale where SU (2) gauge coupling diverges † .
If we turn on the tree level superpotential, the flat directions are lifted. In our
1/7

case VEVs of the fields are close to those of Ref. [60], v ∼ Λ3 /λ1 Λ3 . The

vacuum energy is V ∼ λ21 v4 > 0 and then supersymmetry is broken. The moduli
space is described in terms of the SU (3) × SU (2) gauge invariant operators [61]listed in Table 6.2. From Eqs. (6.2) and (6.4) we have the effective superpotential
below the scale Λ3
Λ73
κ0
λ0
+ 2 X1 f f¯ + 25 X2 ĒX3 ,
X3 M
M
4
λ1 v
= λ1 v 2 Y ++ κY f f¯ + λ2 P N X,
X

Weff = λ1 X1 +(6.5)

where in the second equality we rescale the gauge invariant operators as X1 =v2 Y, X2 = v2 N, X3 = v3 X and Ē = P . Here we introduce the notations as κ ≡κ0 (v/M )2 and λ2 ≡ λ02 (v/M )5 . Equation (6.5) represents the effective theory of
the supersymmetry breaking sector. Since N and P have U (1)m charges −1 and+1, respectively, these superfields correspond to φ− and φ+ in the model of Ref.
[73]. Contrary to Ref. [73] we add no superfields to the effective superpotential.
In the present model we have hFY i ∼ λ1 v2 . Then, the supersymmetry breaking iscommunicated to the messenger fields f and f¯ through κY f f¯ term. It is worth†

If we consider the case Λ2  Λ3 , supersymmetry is broken due to the quantum deformationof the moduli space [64].

CHAPTER 6. EFFECTIVE MESSENGER SECTOR MODEL

U (1)mU (1)R

X1 = QD̄L

0

2

X2 = QŪ L−1−4 − 6r

X3 = detQQ̄

0

−2

Ē

1

8 + 6r

f f¯

0

0

53

Table 6.2: The gauge invariant operators which describe the moduli spacenoting that the U (1)m gauge interaction and λ2 P N X term do not play an essential
role in communicating the supersymmetry breaking to the messenger fields.
In order to analyze the scalar potential in our model, it is necessary to calculate
the effective Kähler potential. Under the condition λ1  1, we can calculate theeffective Kähler potential using the procedure given by Poppitz and Randall [76]because the theory is weakly coupled and the gauge symmetries are completely
broken. The effective Kähler potential is given byB
K =3 t+
,
t

(6.6)

√
√
t ≡ (A + A2 − B 3 )1/3 + (A − A2 − B 3 )1/3 ,
1
1 †
A ≡
(X1 X1 + X2† X2 ) = v4 (Y † Y + N † N ),
2
2
1 †
1
B ≡
(X X3 )1/2 = v3 (X † X)1/2 .
3 3
3

(6.7)

where

The inverse of the effective Kähler metric is
 

K

j ∗i



=




t 2
+ 2t Y † Y
2
v
2 †
N Y
t
2 †
X Y
t

2 †
Y N
 2t
t
+ 2t N † N
v2
2
X †N
t

2 †
Y X
t
2 †
N X
t
2t
|X| + 2t X † X
v3








(6.8)

for i, j = Y, N and X. Since P (= Ē), f and f¯ have no SU (3) charge, their components of the effective Kähler potential are assumed to be of canonical form. FromCHAPTER 6. EFFECTIVE MESSENGER SECTOR MODEL

54

Eqs. (6.5) and (6.8) the scalar potential of the effective theory is given byV

g12(|P |2 − |N|2 )2 + (MP2 |P |2 + MN2 |N|2 ),22
2
λ1 v 42¯=
λ1 v Y + κY f f + 2λ2 P N X −tX∗= Wj ∗ K j i Wi +t 2 2 2 22t
λ1 v 4+ 2
λ2 |X| |P | + |λ1 v2 + κf f¯|2 + 3 |X| λ2 P N −v
v
X2+λ22 |N|2 |X|2 + κ2 |Y |2 (|f|2 + |f¯|2 )g2+ 1 (|P |2 − |N|2 )2 + (MP2 |P |2 + MN2 |N|2 ),
2

2

(6.9)

where the second last term is U (1)m D-term, because U (1)m is gauged and U (1)mD-flatness condition is not imposed. The last term represents two-loop generated
soft supersymmetry
breaking
mass term. Note that MP2 and MN2 are negative andof order O



2
g12
λ21 v2
2
16π

[73].

By minimizing the scalar potential (6.9) under the conditions‡κ  λ1 ,g12
λ1  λ2 ,16π 2(6.10)

we obtain VEVs which are in the vicinity of the SU (3), SU (2) D-flat direction [60],namelyX = vX + x,|x|  vX ,Y = vY + y,|y|  vY ,(6.11)|f|, |f¯|, |P |, |N|  v,
√where vX ≡ a2 b2 /v 3 , vY ≡ a2 a2 − b2/v 2 and x, y represent the fluctuation aroundvX , vY , respectively. In the minimization it is important that the effective Kähler
potential has of the non-canonical form. We can easily derive
hf i = hf¯i = 0(6.12)

from the stationary conditions for f and f¯. This shows that the SM gauge group is
not broken at the minimum. By solving the minimization conditions with respect‡g2

κ  λ1 is a condition to ensure that the mass squared of the messengers is positive. 16π1 2 λ1 

λ2 is a condition to ensure that U (1)m is broken.

CHAPTER 6. EFFECTIVE MESSENGER SECTOR MODEL

55

to N and P we obtain
1q
g1−MP2 ∼
λ1 v,
g1
16π 2
λ2
g1
|hN i| '|hP i| ∼
λ2 v
λ1
16π 2
|hP i| '(6.13)(6.14)The order of VEVs of the fluctuation x and y isg12|hxi| ∼ |hyi| ∼ O 
16π 2

!2



λ22 v  vX , vY .

(6.15)As a consequence, our analysis is found to be self-consistent.

6.2

The observable sector and estimation of thescalesSupersymmetry breaking in the messenger sector is transmitted to gauginos,
squarks and sleptons in the observable sector radiatively through the interaction
κY f f¯ in Eq.(6.5). At one-loop we can obtain the masses for SU (3)C , SU (2)L andU (1)Y gauginos
mλi ∼

gi2
gi2 hFY i∼
λ1 v,
16π 2 hY i16π 2(6.16)

where gi stands for the corresponding gauge coupling of the standard model. Taking
mλi = 102.5±0.5 GeV, we obtain


λ1

v



MPlanck∼ 10−13.3±0.5 ,(6.17)

g2

where 16πi 2 ' 10−2.5 is used. On the other hand, the soft supersymmetry breaking
masses for squarks and sleptons are induced at two-loop. They are given by
m2φi ∼

gi2 hFY i
16π 2 hY i

!2

∼

gi2
λ1 v
16π 2

!2

.

(6.18)

Here we focus on the estimation of the various scales. In the supersymmetry
breaking sector we have a scale M which suppresses the non-renormalizable interactions in Eq.(6.2). It is natural to take this scale M as MPlanck. This implies that weCHAPTER 6. EFFECTIVE MESSENGER SECTOR MODEL

56

have to take the effects of the gravitational interaction into account in calculating
the masses for the gauginos, the squarks, and the sleptons. The scalar mass terms
which are induced by gravity come from the D-term
Z

!

Q† Q
dθ
+ · · · Φ†i Φi,2MPlanck
4(6.19)

where Φi are superfields of the standard model and i denotes flavor index. Therefore,
the gravity-induced scalar masses are
hF i,
MPlanck
2v∼ λ1MPlanck  102.5±0.5 GeV,
MPlanckmφi (grav) ∼(6.20)

where hF i is VEV of the F-term in the supersymmetry breaking sector. The lastinequality implies that we consider only the case in which the gravitational effects
is negligible. From this inequality we obtain


λ1

2

v
MPlanck

 10−15.8±0.5 ,(6.21)

where MPlanck ' 1018.3 GeV is used.
On the other hand, gaugino mass terms which are induced by gravity arise via
the term



Z



Z
2
v
Y
X1
α
2
W
W
=
d
θ
W α Wα ,(6.22)α3MPlanckMPlanck MPlanck
where W α is a field strength superfield. Thus, the gravity-induced gaugino masses

d2 θ

becomemλi (grav) ∼ λ1



v

4

MPlanck.
MPlanck
From Eq. (6.17) and the inequality (6.21) we find thatmλi  mλi (grav).

(6.23)(6.24)

Namely, the gauge-mediated contribution to the gaugino mass is dominant comparedwith the gravity-mediated contribution.
Taking the condition λ1  κ ∼vMPlanck2

into account together with Eqs. (6.17)

and (6.21), we obtain the allowed range of parameters
10−4.4±0.2 v

 10−2.5,
MPlanck
10−10.8±0.5  λ1  10−8.9±0.3.

(6.25)(6.26)

CHAPTER 6. EFFECTIVE MESSENGER SECTOR MODEL

57

If we take λ1 ∼ 10−9.3 as an example, various scales in the model are determined as,v ∼ 1014.3 GeV,√Λ ∼ 1013.0 GeV,F ∼q

λ1 v2 ∼ 109.6GeV,(6.27)m ∼ κv ∼ 106.3GeV,
mλi ∼ 102.5GeV,
where m means the mass of f and f¯. Equation (6.20) also represents the order ofthe gravitino mass. In the present example the gravitino mass is ∼ 101.5 GeV. We√
note that supersymmetry breaking scale F turns out to be the intermediate scale
between the GUT scale and the soft supersymmetry breaking scale.

6.3

Discussion

We have shown that the messenger sector can be considered as the effective
theory of supersymmetry breaking sector. No matter superfields and interactions areadded in the messenger sector. All interactions in our messenger sector are derived
from the supersymmetry breaking sector. Using this effective theory we have also
shown that supersymmetry breaking can be communicated to the observable sector
without breaking QCD color.
In the present framework, the essential role of communicating the supersymmetry breaking to the observable sector is played by the κY f f¯ term in the effective
superpotential. We do not need to rely on the U (1)m gauge interaction and also on
the P N X term in the superpotential. This situation is in sharp contrast to that inRef. [73]. In fact, when we do not introduce the field P (= Ē) and the U (1)m gauge
symmetry, we can make the model simpler. Even in that case the main point of ourresults remains unchanged.
There appear various scales in the present model. We have also estimated thegravitational effects in calculating masses for gauginos, squarks, and sleptons. It
was found that gauge-mediated contributions are dominant compared with gravity-

CHAPTER 6. EFFECTIVE MESSENGER SECTOR MODEL

58

mediated contributions. As a typical example, we found that gauge symmetry breaking scale (v) in supersymmetry breaking sector is 1014.3 GeV, the scale of SU (3)dynamics (Λ3 ) in SU (3) × SU (2) model is 1013.0 GeV, supersymmetry breaking scale√( F ) is 109.6 GeV and the mass of messenger fields (κv) is 106.3 GeV.
In the observable sector of our model, as discussed in Ref. [73], the masses for
gauginos are induced at one-loop through the standard model gauge interaction. On
the other hand, the masses for squarks, sleptons are induced at two-loop throughthe standard model gauge interaction. Therefore, FCNC are naturally suppressed
because these masses are proportional to the square of the flavor-blind standard
model gauge coupling constants.

Chapter 7
A DGM Model with an Affine
QMS
In the previous chapter, we discussed the effective messenger sector model, whichsimplify the model of Ref. [73] quite naturally. There has also been much progress
to simplify the structure of the model along various lines [78]-[81].
The simplest idea of gauge mediated SUSY breaking is “Direct Gauge Mediation”(DGM). In this framework, the SM gauge group is coupled to the DSB sector
directly, in other words, the SM gauge group is embedded into the flavor symmetry of the DSB sector, and the fields transforming both under the gauge groupin the DSB sector and the SM gauge group communicate SUSY breaking to ourworld. As mentioned in the chapter 5, main problem in this scenario is that the SM
gauge group, especially QCD color gauge group becomes asymptotically non-freeand blows up below the GUT scale. Since the model which breaks SUSY dynamically has been increased, it is worth reconsidering this problem. Indeed, there aresome DGM models which solve the problem [82]-[91].
However, there arises other problems in these DGM models.
• SUGRA contributions to soft masses dominate over or are comparable to thegauge mediation contributions because SUSY breaking scale is relatively high(∼ 1010GeV) [82].
• If there are massive gauge multiplets charged under the SM gauge group (which

59

CHAPTER 7. A DGM MODEL WITH AN AFFINE QMS

60

is referred to as “gauge messenger”), these give negative contributions to soft
scalar masses squared [84, 85].
• If light scalars (∼ 104 GeV) charged under the SM gauge group exist, these
also give negative contributions to soft scalar masses squared through 2-loop
RGE [82, 83].
In this chapter, we present a new, simple model of direct gauge mediation of SUSY
breaking with an Affine quantum moduli space which avoids the above problems. ∗This chapter is based on the work [77].

7.1

A ModelThe symmetries of our model are
SU (6)1 × SU (6)2 × [SU (6)],(7.1)

where the first two SU (6)1,2 are gauge groups and SU (6) in the bracket is a global
symmetry. We will later identify the subgroup of this SU (6) with the SM gauge
groups.
The field content of our model is as follows.
X ∼ (, 1; 1),

Σ ∼ (

,

; 1),

Q ∼ (

, 1;

),

Q̄ ∼ (1,

;

).

(7.2)

We note that this field content consists of two sectors. The first sector includes
X only. This model is SU (6) +, which has been originally discussed by Csáki,

Schmaltz and Skiba [9] and also discussed recently in Ref. [92, 93]. According
to Ref. [9], the low energy effective theory of this model has two branches, one
is Wdyn = 0 (“an Affine quantum moduli space”) and another is Wdyn 6= 0. The∗

Our model has similar dynamics to the model in Ref. [90].

CHAPTER 7. A DGM MODEL WITH AN AFFINE QMS

61

definition of Affine quantum moduli space is that the moduli space is given by gauge
invariant polynomials with no relations among them and Wdyn = 0. Moreover, ’t
Hooft anomalies between fundamental fields and gauge invariant composites matchat all points in the moduli space (including the origin). Therefore, if the appropriate
gauge invariant operators to lift the flat directions are added to the superpotential,
then we can expect the model to break SUSY due to confinement such as the model
proposed by Intriligator, Seiberg and Shenker (ISS) [94]. We refer to this sector as
SUSY breaking sector or DSB sector throughout this paper. We will utilize thisbranch to construct the model of direct gauge mediation. On the other hand, we
will discuss later the case of Wdyn 6= 0.
The second sector contains Σ, Q, Q̄. These superfields are necessary to communicate SUSY breaking effects to the observable sector. The field content for eachgauge group is SU (6)1,2 + 6(+), which is the special case of the model in Ref.

[95].
Note also that this model is completely chiral, in other words, we cannot add
mass terms for any field to the superpotential† .
Here we take the following tree level superpotential
W = λ1 ΣQQ̄ +λ2 4
λ3
X + 3 detΣ,
MP
MP

(7.3)

where λ1,2,3 are the couplings of order unity and MP is the reduced Planck scale.
Although it is possible to add other nonrenormalizable terms to the superpotential,
we forbid them by imposing additional symmetries. Note that explicit contractionof indices in X 4 are X 4 ≡ Xa1 b1 c1 Xd1 e1 f1 Xa2 b2 c2 Xd2 e2 f2 a1 b1 f1 a2 b2 f2 d1 e1 c1 d2 e2 c2 .
In the presence of the first two terms in the superpotential, there exist someclassical flat directions: v6 ≡ detΣ, B ≡ Q6 , B ≡ Q̄6 , M ≡ XQ3 .
Let us first discuss the direction hΣi =6 0 (VEVs of other fields are zero) we areinterested in, which corresponds to the direction detΣ 6= 0;hΣi = diag(v, v, v, v, v, v).

(7.4)

Along this direction, the gauge group SU (6)1 × SU (6)2 is broken to their diagonal†

X 2 vanish identically due to the Bose statistics of the superfield.

CHAPTER 7. A DGM MODEL WITH AN AFFINE QMS

62

SU (6)D and Q, Q̄ become massive since the superpotential includes the mass term
λ1 hΣiQQ̄. For large v, the low energy effective theory is SU (6)D ++ one singlet

v. ‡ As mentioned earlier, the low energy dynamics of this model has already been
discussed [28] and this model is one of the theories with an Affine quantum moduli
space [93]. SUSY gauge theories based on simple groups which have an Affine quantum moduli space are completely classified by Dotti and Manohar [93] as in Table.
7.1. Here notations are as follows, G is the gauge group, ρ the matter representation

G

ρ

dM

µ

µadj

G∗

T1

SU (2N )+N +1

2N − 22N(SU(2))NT2

SU (6)1

3

6

SU (3) × SU (3)T3Sp(2N ), N ≥ 2

N −1

N −1

N +1

(SU(2))N

T4

SO(N ), N ≥ 5(N − 4)

1(N − 4)(N − 3)
2

N −4

N −2

SU (2) × SU (2)T5

SO(12)

2S

7

8

10

(SU(2))3

T6

SO(14)

S

1

8

12

G2 × G2

T7

SU (2)

1

5

2

Z3

T8

SU (8)

710

8(Z2 )6

T9

Sp(8)

6

7

5

(Z2 )6

T10

SO(N ), N ≥ 5

N −1

N +2

N −2

(Z2 )kT11

SO(16)

S

8

16

14

(Z2 )8S1

SU (5)+02

5

SU (5)

S2

SO(10)S

0

2

8

SO(10)

Table 7.1: All theories with an Affine quantum moduli space(S denotes the spinor representation), dM the number of gauge invariant composites, µ, µadj Dynkin index of matter representations and adjoint, for respectively,G∗ the unbroken gauge group at D-flat directions where G is maximally broken.
T1-T6 have µ < µadj , T7-T11 have µ > µadj . S1 and S2 have no gauge invariant
composites.
‡

We use the same notation hΣi for a singlet superfield.

CHAPTER 7. A DGM MODEL WITH AN AFFINE QMS

63

In the present case, U (1)R anomaly of the fermionic component of X, gaugino
and that of the fermionic component of X 4 are saturated. This implies that thetheory is in the confining phase below the scale ΛL (the strong coupling scale of
SU (6) +) and the low energy effective theory should be described in terms ofthe composite X 4 . The form of the dynamically generated superpotential due togaugino condensation from the subgroup SU (3) × SU (3) in SU (6) is
Wdyn = (ωr − ωs )Λ33 ,(7.5)

where ω represents the cube root of unity (r, s = 1, 2, 3) and Λ3 means the strong
coupling scale of pure SU (3) super Yang-Mills theory. Naively, one expects that thesuperpotential is the sum W = Λ3L (ωr + ωs ), so the relative sign in Eq. (7.5) is nontrivial and essential for Wdyn = 0.
We would like to digress here for a moment and outline the argument of Dotti
and Manohar that how Eq. (7.5) is derived. The VEV hX[123] i = v1 , hX[456] i = v2(|v1 | = |v2 | for D-flatness) breaks SU (6) → SU (3)1 × SU (3)2 . One can easily seethat the effective theory is pure SU (3) × SU (3) gauge theory (with singlet X 4 ) bycounting degrees of freedom through Higgs mechanism (20 − (35 − 8 − 8) = 1). The
1-loop matching of the gauge coupling between the high energy and the low energy
is Λ3Li = Λ5SU (6) /vi2 . One can interchange the two SU (3) groups by acting with the
SU (6) matrix





 0

 0



 0
U =

 i


 0



0 0 i 0 0 

0 0 0 i 0 

0 0 0 0 i 
0 0 0 0i 0 0 0

,

0 


0 



(7.6)

0 0 i 0 0 0which maps v1 → −iv2 , v2 → −iv1 . Under this U , one finds that Λ3L1,2 → −Λ3L2,1 ,therefore, the difference of each SU (3) superpotential is SU (6) invariant§ .
§Also, for other theories with µ < µadj , these theories have multi branches, and Wdyn = 0

is obtained from the cancellation of gaugino condensations for each unbroken subgroup. On
the other hand, in the theories with µ > µadj , Wdyn must be zero because the general

CHAPTER 7. A DGM MODEL WITH AN AFFINE QMS

64

Let us turn back to the model of the direct gauge mediation. What is important here is that this model has Wdyn = 0 branch. In this situation, the effective
superpotential becomes
λ2
u,
MP

Wef f =u ≡ X 4.

(7.7)

Taking into account that the effective Kähler potential is
Kef f ∼ u† u/|ΛL |6 ,(7.8)

we obtain the following scalar potential.
∂ 2 Kef f∂u† ∂uVef f =!−1

∂Wef f
∂u

2

λ22 6
Λ .
MP2 L

=

(7.9)

At the first glance, one may think that this model is the plateau model [84, 85, 91]
since the scalar potential is flat. However, we have to recall that ΛL depends on thescale v through 1-loop matching of the gauge couplings:


Λ1
v

9 

Λ2
v

12



ΛL
=
λ1 v

15

,

(7.10)

where Λ1,2 denotes the dynamical scale of SU (6)1 + +6( + ) and SU (6)2 +6( + ),respectively. Then, the scalar potential (7.9) becomesλ2
Vef f = 22MP

λ51 Λ31 Λ42
v2

!6
5

λ2
= 22
MP

λ51 Λ7v2!65,(7.11)

where Λ is defined as Λ7 ≡ Λ31 Λ42 for simplicity. This results in runaway behavior
but the term detΣ in Eq. (7.3) stabilizes the scalar potential.
Indeed, by minimizing the scalar potential
λ2
Vef f = 22MP

λ51 Λ7v2!6
5+

36λ23 10v ,MP6

(7.12)

form of the superpotential consistent with the R symmetry is a sum of the term like W =2µ /(µ−µadj )Λ(µ−3µadj )/(µ−µadj ) Πi φi i

, where φi are elementary fields with index µi . The product offields φi must be gauge and flavor invariant. For asymptotically free theories, µ < 3µadj , the powerof Λ is negative if µ > µadj . In this case, W diverges in the weak coupling limit φi → ∞, Λ → 0.
Therefore, W must vanish.

CHAPTER 7. A DGM MODEL WITH AN AFFINE QMS

65

we obtain131v ∼ (Λ21 MP10 ) ,

Fv ∼

Λ105
MP43

!1

31

,

Λ210
MP86

V0 ∼

!1

31

,

Fv
∼
v

Λ84
MP53

!1

31

. (7.13)Since the vacuum energy is non-zero, SUSY is certainly broken. This breaking effectis communicated to the observable sector as follows. Upon identifying the SU (5)subgroup of the flavor group SU (6) with usual GUT gauge group which includes theSM gauge groups, Q and Q̄ behave as 5 + 5̄ messenger fields of which SUSY mass is
λ1 v and SUSY breaking (mass)2 is λ1 Fv . They communicate SUSY breaking to thesoft masses through loop diagrams of Q, Q̄ in the usual way [72, 73]. Note also thatsince the original matter content is completely chiral, these vectorlike messengers
are dynamically generated as a result of symmetry breaking. Furthermore, since Σ
is a singlet for SU (5), there is no gauge messengers and no light scalars charged
under the SM gauge group, which gives a negative contribution to the soft (mass)2 .
Requiring Fv /v ∼ 104 GeV to obtain the soft masses of order 102∼3 GeV, we findΛ ∼ 7 × 1012GeV,ΛL ∼ 1 × 1012GeV,v ∼ 3 × 10 GeV,

Fv ∼ 1 × 109GeV,

14

q(7.14)

where we used MP = 2 × 1018 GeV.
We give some comments on the above scales in order. First of all, since Fv < v2 ,the determinant of the mass squared matrix for messenger fields is positive, so the
SM gauge groups are not broken at the minimum. The problem in Ref. [73] isavoided. Second, the messenger scale v is close to the GUT scale, so it is possible topreserve the perturbative unification in spite of six flavors of messengers. Third, one
may worry about that the SUGRA contribution to the soft masses is comparable toor dominates over the gauge mediation contribution because SUSY breaking scale√
Fv is relatively large. If we require that the gravitino mass m3/2 , which is the
typical scale of the SUGRA contribution, is less than 10 percent of the gluino mass
(such that RGE induced squark mass squareds have degeneracy at 1 percent level),
m3/2 = √

Fv
αs Fv
,
< 0.1 × 6 ×
4π v
3MP

(7.15)

CHAPTER 7. A DGM MODEL WITH AN AFFINE QMS

66

then we findαs √3MP ∼ 2 × 1016GeV.
(7.16)
4π
In our model, the above requirement is clearly satisfied, so the SUGRA contributionv < 0.1 × 6 ×

is suppressed enough.
Recall here that there still remain classical flat directions B, B̄ and M . We haveto argue whether or not these directions are lifted quantum mechanically.
Along the B direction, SU (6)1 and SU (6) are completely broken. Then Σ andQ̄ become massive, hence the low energy effective theory is SU (6)2 + singlets. The
dynamical superpotential is
Wdyn = Λ3L = Λ22 B 1/6 ,(7.17)

where we use the scale matching Λ3L = Λ22 B 1/6 . This leads to non-zero constantvacuum energy. At one loop, the correction to Kähler potential makes the scalarpotential stabilized near the origin [84, 85, 96].
Along the B̄ direction, SU (6)2 and SU (6) are completely broken. Then Σ andQ become massive, hence the low energy effective theory is SU (6)1 ++ a singlet.

The effective superpotential becomes ¶Wef f =λ2uMP

(7.18)

Using the canonical Kähler potential for u and the scale matching, we obtain thescalar potential
Vef f =

Λ6L
λ22 18/5 2/5= 2 Λ1 B̄ .
MP2
MP(7.19)

Clearly, this stabilizes B̄ direction.
Along the M direction, SU (6)1 and SU (6) are completely broken. The low
energy effective theory is SU (6)2 + singlets. This is the same effective theory along
the B direction. Therefore, the M direction is also lifted.
We now briefly discuss the drawback of our model. In the case with Wdyn 6= 0,
this Wdyn becomes a runaway potential for u, v. Using the matching condition


¶

Λ1
x

9 

Λ2
x

9



=

ΛL
x

30

,

We also assume here that the theory is in the Wdyn = 0 branch.

(7.20)

CHAPTER 7. A DGM MODEL WITH AN AFFINE QMS

67

where x means a VEV of X, we obtain
Λ5L
λ51 Λ31 Λ42∼.
x2
u1/2 v2(7.21)

Kef f ∼ u† u/|ΛL |6 + v† v,(7.22)

Wdyn ∼
Since the effective Kähler potential isthe effective scalar potential becomes
Vef f ∼ Λ

6

λ2
λ51 Λ7
− 3/2
MP
2u v2

!2

6λ3 v5 2λ51Λ7+− 1/2 3
MP3
u v

!2

.

(7.23)

Then one can easily see that SUSY vacuum exists (u ∼ (MP Λ21 )2/11 , v ∼ (MP8 Λ14 )1/22 ).
Therefore, our model does not work in this case. The same situation occurs if we
apply the model with µ < µadj in Ref. [93] instead of SU (6)+model. On the other

hand, if we apply the model with µ > µadj in Ref. [93], the situation is different
because Wdyn 6= 0 branch does not exist as mentioned beforek . For instance, it isan interesting challenge to construct a model with direct gauge mediation by using
ISS model [94] as SUSY breaking sector.

7.2

Discussion

In this chapter, we have presented a new, simple model of direct gauge mediationof SUSY breaking with an Affine quantum moduli space. SUSY breaking is due toconfinement, and is communicated to the observable sector by the SM gauge interactions. This model has no gauge messengers and no light scalars charged under
the SM gauge group. These give a negative contribution to soft masses. Large VEVat the minimum is obtained by balancing the runaway potential and dimension six
nonrenormalizable term. This makes it easy to preserve the perturbative unification. Enough suppression of the SUGRA contribution to soft masses can be shown.
Moreover, we have seen that phenomenologically viable results are obtained. ThekShirman [90] uses the model that has no flat directions as SUSY breaking sector in the firstexample. This model is also the case in which Wdyn necessarily vanishes.

CHAPTER 7. A DGM MODEL WITH AN AFFINE QMS

68

drawback is that if the SUSY breaking model has Wdyn 6= 0 branch, this dynamicallygenerated superpotential becomes runaway potential, then SUSY is unbroken. This
problem is not so serious because this is model dependent.

Chapter 8
Conclusion
We have discussed the nonperturbative dynamics in N = 1 supersymmetric gauge
theories. In the former part of this thesis, we focused on their theoretical aspects.
Supersymmetric gauge theory is tractable compared to non SUSY theory thanks
to holomorphy. The form of the superpotential can be determined analytically andexactly by symmetry and holomorphy. Since the superpotential is crucial to analyze
the low energy behavior of the theory, we can extract informations about the strong
coupling dynamics. Especially, the confining phase can be examined systematically,and exhibits rich phenomena for example, gaugino condensation, chiral symmetry
breaking and confinement, which seems to be generic in N = 1 SUSY gauge theories.
In this thesis, I have studied t he confining phase of SUSY SO(12) gauge theory with
one spinor and Nf ≤ 7 vectors, which is motivated by the search for Spin group duality. Spin group duality has interesting features which are not included in Seiberg
duality. From this point, I think that clarifying spin group duality may provide aclue to understand N = 1 duality. I have completely determined the low energy
superpotentials, and it turned out that the confining phase structure is analogous to
SUSY SU (Nc ) QCD, that is, no vacuum (ADS-type superpotentials), confinementwith chiral symmetry breaking (quantum deformation of the moduli space), confinement without chiral symmetry breaking (s-confinement). Moreover, from thisanalysis I have conjectured that the dual gauge group must include SU (Nf − 6) asa subgroup. Although my analysis certainly gives a hint to find spin group duality,69

CHAPTER 8. CONCLUSION70

we need more insight and informations.
As another remarkable feature, a large number of N = 1 SUSY gauge theories
with dual descriptions have been conjectured, which are equivalent to the original
theory in the infrared limit, more precisely, which flow to the same infrared fixed
point of the original theory. This also opens up an avenue to study the strong
coupling physics because the strong coupling dynamics of one theory is described by
the weak coupling dynamics of another theory. However, there are many uncertain
points in the duality relations. For instance, we cannot in practice calculate theGreen function of the strong coupling theory by calculating the Green function of the
weak coupling theory because the explicit transformations of fields in both theories,
such as two dimensional theories, are unknown, we know only the correspondence of
their fields. Moreover, there is no systematic method to search for duality. In otherwords, it can be said that we know little about general properties of duality. Thus,
there are many problems in duality, and we hope to clarify the origin of duality infuture. In this thesis, for the second point mentioned above, I and S. Kitakado havefound that a certain class of dualities (Sp(2Nc ) gauge theory with arbitrary matters
and SO(2Nc ) gauge theory with vectors and its higher rank representations.) are
interrelated through the negative dimensional group trick. This trick is useful andpowerful in finding a new duality because if we know the SO(Sp) group duality, we
can easily obtain a new Sp(SO) duality by a simple manipulation. This gives some
informations on general structure of N = 1 SUSY duality.
In the latter part, phenomenological aspects of N = 1 SUSY gauge theories havebeen discussed, especially focusing on the gauge mediated SUSY breaking.
As our understanding of the low energy effective theory has advanced, a large
number of models which breaks SUSY has been proposed, and its phenomenologicalapplications have attracted much interest. As mentioned earlier, gauge mediated
SUSY breaking has phenomenological advantages, which solve supersymmetric flavor problems naturally, compared to SUGRA mediated SUSY breaking. In thiscontext, Dine et al. constructed the first interesting model, but this model does
not seem to describe nature because of the complexity of the model and the color
breaking minimum problem.

CHAPTER 8. CONCLUSION71

We have presented two models in this thesis. One is the model which considers the messenger sector as the effective theory of the DSB sector, which has been
constructed by N. Haba, T. Matsuoka and myself. This model simplified the complicated structure of the DNNS model naturally. We have shown that in the true
vacuum, the color and electroweak symmetries are not broken at the scale higher
than the weak scale. Moreover, we have estimated the various mass scales, which arephenomenologically desirable. This model has provided one direction to constructphenomenological viable models.
I have constructed a new and simple model of the DGM with an Affine quantum
moduli space. This model has avoided some phenomenologically dangerous problems (the SUGRA mediation dominance over the gauge mediation, the negativesfermion mass squared problem) in earlier DGM models. Furthermore, the desirable mass scales have been obtained. Although it turned out that the minimum weare interested in is the only local minimum, it is not serious because this is a model
dependent problem.
Although many models of gauge mediation have been proposed, it seems that
we do not know well what type of models lead to what pattern of sparticle spectrum. In order to see which of the models describes the nature well, we have toinvestigate sparticle spectroscopy. Therefore, it will be interesting theoretically andphenomenologically if we can specify the model (the gauge group, matter content,
messenger matter content etc.) from the sparticle analysis.
Finally, we comment on future directions. It will be most challenging to apply
the methods and knowledge developed in N = 1 supersymmetric gauge theories to
non SUSY theories, especially to QCD. Further insights for color confinement may
be gained through supersymmetry.
Instead of insisting on field theory, it will be interesting to study field theory dynamics using string theories or more generally using branes. Possibly, the information on the field theoretical dynamics may be interpreted in terms of the geometricalinformations in their framework, “We may be able to see the dynamics”.
Anyway, thanks to recent developments in supersymmetric gauge theories, we
can now attack the strong coupling dynamics without hesitation. It seems (at leastCHAPTER 8. CONCLUSION
to me) that SUSY is indispensable to understand nature!

72

Acknowledgements
The author wishes to express his sincere gratitude to S. Kitakado and T. Matsuoka
for stimulating discussions, various suggestions and careful reading of this thesis,
and to N. Haba, K. Ito and J. Nishimura, who are collabolators, for many fruitful
discussions. He is also grateful to H. Ikemori and K. Ito for valuable comments andcareful reading of this thesis. Without their heartly encouragement, this work would
not have been completed.
Finally, he expresses his gratitude for all members of the laboratory of the elementary particle physics (E-ken) in Nagoya University for their hospitality.

73Appendix A
Superspace formalism
In this appendix, we would like to summarize the superspace formalism since it
is used throughout this thesis, and for completeness of our discussion. For moredetails, see Refs. [97]-[101].
N = 1 SUSY algebra is generated by a momentum operator Pµ which are thegenerator of translation, and Weyl spinor operators Qα , Q̄α̇ which are the generatorof SUSY∗ .
[Qα , Pµ ] =where [ ,], { ,hiQ̄α̇ , Pµ = [Pµ , Pν ] = 0,(A.1)

{Qα, Qβ } = {Q̄α̇ , Q̄β̇ } = 0,(A.2){Qα, Q̄β̇ } = 2σαµβ̇ Pµ ,

(A.3)

µ
} represent commutator, anticommutator, respectively. σαβ=(1, σ), here 1 is two by two unit matrix, and σ’s are Pauli matrices.
The corresponding supertransformation group isG(x, θ, θ̄) = exp i(θQ + θ̄Q̄ − xµ Pµ ) ,(A.4)θα , θ̄α̇ are Grassmannian parameters.
Using Hausdorff formula eA eB = exp(A + B + 12 [A, B] + · · ·), and calculating the¯ we findgroup product G(x, θ, θ̄)G(aµ , ξ, ξ),¯ = G(xµ + aµ − iξσµ θ̄ + iθσµ ξ̄, θ + ξ, θ̄ + ξ).
¯G(x, θ, θ̄)G(aµ , ξ, ξ)∗In this appendix, we use gµν = (1, −1, −1, −1).

74

(A.5)

APPENDIX A. SUPERSPACE FORMALISM

75

We can read off the following transformations in parameter space.
¯ θ + ξ, θ̄ + ξ̄).
(x, θ, θ̄) → (xµ + aµ − iξσµ θ̄ + iθσµ ξ,(A.6)

Then, the transformations of the function F (x, θ, θ̄) (referred to as “superfield”) of(x, θ, θ̄) (referred to as “superspace”),

F (x, θ, θ̄) → exp i(ξQ + ξ¯Q̄ − aµ Pµ ) F (x, θ, θ̄)(A.7)

are generated by the following differential operators,
Pµ = i∂µ ,∂− iσαµα̇ θ̄α̇ ∂µ ,
iQα =
∂θα
∂
iQ̄α̇ = − α̇ + iθα σαµα̇ ∂µ .
∂ θ̄

(A.8)

Moreover, since the supercovariant derivatives on the superspace∂+ iσαµα̇ θ̄α̇ ∂µ ,∂θα∂
= − α̇ − iθα σαµα̇ ∂µ
∂ θ̄

Dα =

(A.9)

D̄α̇

(A.10)

generate the following SUSY algebra{Dα , D̄α̇ } = 2iσαµα̇ ∂µ ,(A.11){Dα , Dβ } = {D̄α̇ , D̄β̇ } = 0,

(A.12){Dα , Qβ̇ } = {Dα , Q̄β̇ } = {D̄α̇ , Qβ̇ } = {D̄α̇ , Q̄β̇ } = 0,

(A.13)

and anticommute with Q, Q̄. We obtain irreducible representations of SUSY algebra
by imposing the constraint on the superfield using the supercovariant derivatives.
Chiral superfield, which is one of the irreducible representations, is defined by
the conditionD̄α̇ Φ(x, θ, θ̄) = 0.

(A.14)

Using the variable y µ = xµ + iθσµ θ̄ we can expand the superfield Φ in θ, θ̄ as follows.
√Φ = φ(y) + 2θψ(y) + θ2 F (y)(A.15)√i= φ(x) + 2θψ(x) + θ2 F (x) + i∂µ φ(x)θσµ θ̄ − √ θ2 ∂µ ψ(x)σµ θ̄2
1(A.16)− ∂µ ∂ µ φ(x)θ2 θ̄2 ,4

APPENDIX A. SUPERSPACE FORMALISM

76

where φ, F are complex scalar fields, ψ is a left-handed Weyl spinor field.
If we make an infinitesimal supertransformationδΦ = i(ξQ + ξ̄ Q̄)Φ,(A.17)

from Eq. (A.8) we have
!

!

∂
∂
δΦ = ξ α− iσαµα̇ θ̄α̇ ∂µ Φ + ξ̄ α̇ − α̇ + iθα σαµα̇ ∂µ Φ
α
∂θ
∂ θ̄
√
i
2ξψ + 2ξθF + 2i∂µ φθσµ ξ̄ + √ θ2 ∂µ ψσµ ξ¯ + · · · ,=2and from Eq. (A.15)
δΦ = δφ +√2θδψ + θ2 δF + · · · .

(A.18)(A.19)(A.20)

Comparing the above two expressions, the transformation laws for the scalar, fermion,and the auxiliary field (referred to as F-term) are obtained, and these become irreducible representations.
√
2ξψ,
√
√
¯
2ξF + i 2∂µ φσµ ξ,δψ =iδF = √ ∂µ ψσµ ξ̄.
2δφ =(A.21)(A.22)(A.23)

Since supertransformation of F-term leads to the total derivative, this is invariant
under SUSY. It is also clear by definition that the product of the chiral superfields
is the chiral superfield.
On the other hand, the superfield satisfying
Dα Φ† = 0

(A.24)

is referred to as antichiral superfield, which is complex conjugate of the chiral superfield.
Next, the vector superfield is defined by
V = V †.

(A.25)

APPENDIX A. SUPERSPACE FORMALISM

77

Explicit expansion in θ, θ̄ is1
V (x, θ, θ̄) = C(x) + iθχ(x) − iθ̄χ̄(x) + iθ2 [M (x) + iN (x)]2
1 2
− iθ̄ [M (x) − iN (x)] + θσµ θ̄Vµ (x)2iiiθ2 θ̄[λ̄(x) + σ̄ µ ∂µ χ(x)] − iθ̄2 θ[λ(x) + σ µ ∂µ χ̄(x)]
2
2
1 2 2
1
θ θ̄ [D − ∂µ ∂ µ C],2
2(A.26)

where C, M, N, D are real scalar fields, χ, λ are Weyl spinor field, and Vµ is a vector
field. If we consider particularly as a vector superfield√i(Λ − Λ† ) = i(φ − φ† ) + i 2(θψ − θ̄ψ̄) + iθ2 F − iθ̄2 F †1
1−θσ µ θ̄∂µ (φ + φ† ) − √ θ2 θ̄σ̄ µ ∂µ ψ + √ θ̄2 θσµ ∂µ ψ̄22i− θ2 θ̄2 ∂µ ∂ µ (φ − φ† ),4(A.27)

where Λ is a chiral superfield, and Λ† is an antichiral superfield. Comparing with
Eq. (A.26), we can derive
C = i(φ − φ† ),
√
2ψ,
χ =
1
(M + iN ) = F,2Vµ = −∂µ (φ + φ† ),(A.28)

λ = 0,
D = 0.
It turns out that Eq. (A.27) corresponds to U (1) gauge transformation from thefourth equality in Eq. (A.28). In fact, the vector superfield is transformed underU (1) gauge transformation as follows,
V (x, θ, θ̄) → V 0 (x, θ, θ̄) = V (x, θ, θ̄) + i[Λ(x, θ, θ̄) − Λ† (x, θ, θ̄)].

(A.29)C, M, N, χ are not physical degrees of freedom because these can be eliminated by
the degrees of freedom of the gauge transformation. Wess-Zumino gauge, which

APPENDIX A. SUPERSPACE FORMALISM

78

eliminates them, is often used.
1
VW Z (x, θ, θ̄) = θσµ θ̄Vµ (x) + iθ2 θ̄λ̄(x) − iθ̄2 θλ(x) + θ2 θ̄2 D(x),21
VW2 Z (x, θ, θ̄) = −(θσµ θ̄)(θ̄σ̄ ν θ)Vµ Vν = θ2 θ̄2 V µ Vµ ,2
3
VW Z (x, θ, θ̄) = 0.

(A.30)

Supertransforming the vector superfield,δξ V = i(ξQ + ξ¯Q̄)V.

(A.31)The transformation laws of components are from Eqs. (A.8, A.30, A.31),1δξ λα = −iD(x)ξα − (σµ σ̄ ν )βα ξβ Fµν (x),2µµ¯ − ∂ µ (ξχ(x) + ξ¯χ̄(x)),= i(ξσ λ̄(x) − λ(x)σµ ξ)δξ V(A.33)¯δξ D = ∂µ (−ξσµ λ̄(x) + λ(x)σµ ξ),(A.34)(A.32)

where Fµν ≡ ∂µ Vν − ∂ν Vµ . Since¯δξ F µν = i∂ µ (ξσν λ̄ − λσ ν ξ) − i∂ ν (ξσµ λ̄ − λσ µ ξ),(A.35)λ, λ̄, Vµν , D make irreducible representation. In the vector superfield, as the coefficient (D(x), referred to as D-term) of θ2 θ̄2 becomes the total derivative, D-term issupersymmetric.
The field strength superfield is defined as follows,
Wα ≡

1 2
D̄ Dα V
4

(A.36)i= iλα (y) − [δαβ D(y) + (σµ σ̄ ν )βα Fµν (y)]θβ + θ2 σαµα̇ ∂µ λ̄α̇ (y).
2(A.37)

One can easily derive D̄Wα = 0 because D̄ 3 = 0 due to the Grassmannian propertyof D̄. In other words, Wα is a chiral superfield. Note that Wα is a spinor valued
field. The gauge kinetic term can be written as
Z

1
1
1
1
d2 θ W α Wα + h.c. = − Fµν F µν + iλσµ ∂µ λ̄ − F µν (∗ Fµν ) + D 2 ,
4
4
4
2
∗

i
Fµν ≡ µνρσ F ρσ ,2

(A.38)(A.39)

APPENDIX A. SUPERSPACE FORMALISM

79

where the first and the second terms are the kinetic term for the gauge field andgaugino, respectively. The third one is the anomaly term, and the last one is theterm for the auxiliary fields. ∗ Fµν is a dual of Fµν .
Charged matter fields under U (1) gauge group are transformed as
Φ → e−iΛ Φ,†Φ† → Φ† eiΛ ,(A.40)

so it is clear that the combination Φ† eV Φ is gauge invariant. Since this is a vector
superfield, D-term is supersymmetric, that is,Zd2 θd2 θ̄Φ† eV Φ

(A.41)

is both supersymmetric and gauge invariant.
Extension to non Abelian gauge group is straightforward. Matter field is transformed under non Abelian gauge group transformation similar to Eq. (A.40), butΛ ≡ Λa T a , V ≡ V a T a , Λa is a chiral superfield, V a is a vector superfield, T a is
the generators of the representation that Φ belongs to. If the vector superfield istransformed as†

eV → e−iΛ eV eiΛ ,(A.42)

Eq. (A.41) is both supersymmetric and gauge invariant under non Abelian gaugetransformation as well.
In this case, the field strength superfield is defined asi
Wα ≡ D̄ 2 e−V Dα eV ,4

(A.43)

Wα → e−iΛ Wα eiΛ ,(A.44)

1d2 θ Tr(W α Wα ) + h.c.
4(A.45)and is transformed asZ

is supersymmetric and gauge invariant.
Summing up the above content, the most general supersymmetric and gauge
invariant Lagrangian is written as follows,
L =

Z
4

† V

d θΦ e Φ +

Z







Z
1
d θ Tr(Wα Wα ) + h.c. +d2 θW (Φi ) + h.c. , (A.46)
42

APPENDIX A. SUPERSPACE FORMALISM

80

i
1 a 2
i
= (Dµ φi )† (Dµ φi ) + ( ψ̄i σ̄ µ Dµ ψi + h.c.) − (Fµν
) + ( λ̄a σ̄ µ Dµ λa + h.c.)
2
4!
2
2
a
√
1 ∂ W (φ)∂D−ψi ψj + i 2
ψi λa + h.c. − V.
(A.47)
2 ∂φi ∂φj∂φiHere we use Wess-Zumino gauge. W is referred to as superpotential, which is aholomorphic function of the chiral superfields. The covariant derivative, the fieldstrength are given as
Dµ φi = (∂µ − igT a Vµa )φi ,(A.48)

Dµ ψi = (∂µ − igT a Vµa )ψi ,(A.49)

Dµ λa = ∂µ λa + gf abc Vµb λc ,(A.50)a= ∂µ Vνa − ∂ν Vµa + gf abc Vµb Vνc .
Fµν(A.51)

V is the scalar potential, and is expressed as
V=

X|Fi |2 +i=

X ∂W 2i∂φi

1X a 2(D )2 a+

1X aX † a 2(g
φi T φi ) ,2 a
i(A.52)(A.53)

where i denotes the flavor index, and the first term is F-term contribution, and thesecond term is D-term contribution.

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