Beyond the Standard Model: The Pragmatic Approach to the...

Beyond the Standard Model: The Pragmatic Approach to

the Gauge Hierarchy Problem

A dissertation presented

by

Rakhi Mahbubani

to

The Department of Physics

in partial fulfillment of the requirements

for the degree of

Doctor of Philosophy

in the subject of

Physics

Harvard University

Cambridge, Massachusetts

May 2006

Thesis advisor Author

Nima Arkani-Hamed Rakhi Mahbubani

Beyond the Standard Model:

The Pragmatic Approach to the Gauge Hierarchy Problem

Abstract

The current favorite solution to the gauge hierarchy problem, the Minimal Su-

persymmetric Standard Model (MSSM), is looking increasingly fine tuned as recent results

from LEP-II have pushed it to regions of its parameter space where a light higgs seems

unnatural. Given this fact it seems sensible to explore other approaches to this problem;

we study three alternatives here.

The first is a Little Higgs theory, in which the Higgs particle is realized as the

pseudo-Goldstone boson of an approximate global chiral symmetry and so is naturally light.

We analyze precision electroweak observables in the Minimal Moose model, one example of

such a theory, and look for regions in its parameter space that are consistent with current

limits on these.

It is also possible to find a solution within a supersymmetric framework by adding

to the MSSM superpotential a λSHuHd term and UV completing with new strong dynamics

under which S is a composite before λ becomes non-perturbative. This allows us to increase

the MSSM tree level higgs mass bound to a value that alleviates the supersymmetric fine-

tuning problem with elementary higgs fields, maintaining gauge coupling unification in a

natural way

Finally we try an entirely different tack, in which we do not attempt to solve the

hierarchy problem, but rather assume that the tuning of the higgs can be explained in some

unnatural way, from environmental considerations for instance. With this philosophy in

mind we study in detail the low-energy phenomenology of the minimal extension to the

Standard Model with a dark matter candidate and gauge coupling unification, consisting

of additional fermions with the quantum numbers of SUSY higgsinos, and a singlet.

iii

Contents

Title Page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

Citations to Previously Published Work . . . . . . . . . . . . . . . . . . . . . . . vi

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

1 Introduction and summary 1

2 Precision Electroweak Observables in the Minimal Moose 4

2.1 The Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 The Gauge Boson Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Non-linear Sigma Model Sector . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4 Plaquette Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.5 Electroweak Symmetry Breaking . . . . . . . . . . . . . . . . . . . . . . . . 12

2.6 Fermion Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.7 Higgs Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.8 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.9 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3 The New Fat Higgs: Slimmer and More Attractive 22

3.1 Constructing a Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.1.1 Details of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.1.2 Conformality and Confinement . . . . . . . . . . . . . . . . . . . . . 26

3.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2.1 λ and the Higgs Mass Bound . . . . . . . . . . . . . . . . . . . . . . 28

3.2.2 Gauge Coupling Unification . . . . . . . . . . . . . . . . . . . . . . . 31

3.2.3 Phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.3 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4 The Minimal Model for Dark Matter and Unification 37

4.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.2 Relic Abundance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.2.1 Higgsino Dark Matter . . . . . . . . . . . . . . . . . . . . . . . . . . 43

iv

Contents v

4.2.2 Bino Dark Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.3 Direct Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.4 Electric Dipole Moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.5 Gauge Coupling Unification . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.5.1 Running and matching . . . . . . . . . . . . . . . . . . . . . . . . . . 514.6 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5 Conclusion 59

Bibliography 60

A The Workings of a Top Seesaw 68

B The Neutralino Mass Matrix 69

C Two-Loop Beta Functions for Gauge Couplings 70

Citations to Previously Published Work

Chapters 2-4 are lifted almost entirely from the following papers:

“Precision electroweak observables in the minimal moose little Higgs model”, C.Kilic and R. Mahbubani, JHEP 0407, 013 (2004),[arXiv:hep-ph/0312053].

“The new fat Higgs: Slimmer and more attractive”, S. Chang, C. Kilic and R.Mahbubani, Phys. Rev. D 71, 015003 (2005),[arXiv:hep-ph/0405267].

“The minimal model for dark matter and unification”, R. Mahbubani and L.Senatore, Phys. Rev. D 73, 043510 (2006),[arXiv:hep-ph/0510064].

Electronic preprints are available on the Internet at the URL

http://arXiv.org

vi

Acknowledgments

Thanks first and foremost to Nima Arkani-Hamed who enriched my years as a

graduate student with little nuggets of physics wisdom and endless amusing anecdotes

about physicists past and present. I feel enormously privileged to have been his student.

I also owe an huge debt of gratitude to Jay Wacker, without whose canny combination of

cajolery and threats I would probably still be advisor-less.

Thanks to my husband Will for his constant love and support and for bolstering my

confidence when I most needed it. Thanks to my friends for keeping me sane; to Can, whose

Daily Bad Joke will be sorely missed; and especially to Shiyamala, who taught me how to

keep being me, and imparted to my daily routine some much-needed feminine frivolity.

vii

To my parents and teachers

Who made me believe that nothing was impossible

viii

Chapter 1

Introduction and summary

Over the last few decades the search for physics beyond the Standard Model (SM)

has largely been driven by the principle of naturalness, according to which the parameters

of a low energy effective field theory should not be much smaller than the contributions

that come from running them up to the cutoff. This principle can be used to constrain

the couplings of the effective theory with positive mass dimension, which have a strong

dependence on UV physics. Requiring no fine tuning between bare parameters and the

corrections they receive from renormalization means that the theory must have a low cutoff.

New physics can enter at this scale to literally cut off the high-energy contributions from

renormalization.

Applying this principle to the SM means that despite spectacular agreement with

current experimental data, this theory is widely held to be incomplete due to an instability

in its Higgs sector; radiative corrections to the Higgs mass suffer from one-loop quadratic

divergences leading to an undesirable level of fine-tuning between the bare mass and quan-

tum corrections. This suggests the emergence of new physics at energy scales around a

TeV, which will be investigated in the near future with direct accelerator searches. The

electroweak sector of the SM has been probed to better than the 1% level by precision

experiments at low energies as well as at the Z-pole by LEP and SLC. The data obtained

can also severely constrain possible extensions of the SM at TeV energies [1, 2, 3, 4].

Supersymmetry (SUSY) provides arguably the most attractive solution for this hi-

erarchy, since it comes with gauge coupling unification as an automatic consequence. How-

ever its simplest implementation, the Minimal Supersymmetric Standard Model (MSSM),

is looking increasingly fine-tuned as recent results from LEP-II have pushed it to regions of

1

2 Chapter 1: Introduction and summary

parameter space where a light higgs seems unnatural.1 This is problematic for the MSSM

since SUSY relates the quartic coupling of the higgs to the electroweak gauge couplings,

which at tree level bounds the mass of the lightest higgs to be less than that of the Z.

Radiative corrections can help increase this bound, with the largest contribution coming

from the top yukawa, giving

m2h0 ≈ m2

Z +3

8π2h4t v

2 logm2t

m2t

(1.1)

for large tan β. Since this effect is only logarithmic in the stop mass however, consistency

with the LEP-II mass bound requires the stops to be pushed up to at least 500 GeV. At

the same time radiative corrections to m2Hu

are quadratic in the stop mass

δm2Hu≈ − 3

4π2m2tlog

Λ

mt

(1.2)

There is therefore a conflict between our expectation that the stop is heavy enough to

significantly increase the higgs mass through radiative corrections and yet light enough to

cut off the quadratic divergence in a natural way.2 Requiring consistency with LEP-II results

therefore forces us to live with a fine tuning of a few percent. This ‘little hierarchy’ problem

[8, 9] raises some doubts about the plausibility of low energy SUSY as an explanation for

the smallness of the higgs mass.

In this dissertation we attempt to take an entirely open-minded approach to this

long-unresolved problem and study solutions that are distinct from the MSSM both in

philosophy and low-energy phenomenology. Each of the following three chapters comes at

the problem from a different direction, and can be read relatively independently of the

others. Their contents are summarized as follows:

• Investigation of precision electroweak observables in a particular solution to the hi-

erarchy problem via the Little Higgs mechanism, in which the higgs is taken to be a

pseudo-goldstone boson of some larger global symmetry, with a mass that is protected

from large radiative corrections.

• Study of the NMSSM with a composite scalar higgs as a way to alleviate the super-

symmetric ’little hierarchy’ problem.

1See references [5, 6] for further discussion.2A recent paper [7] attempted to resolve this conflict by suppressing the size of radiative corrections to

m2Hu

from the stop.

Chapter 1: Introduction and summary 3

• Comprehensive analysis of the low-energy phenomenology of the most minimal non-

supersymmetric theory that has a good dark matter candidate and gives rise to gauge

coupling unification. We do not assume that the gauge hierarchy is solved in this

model; rather we entertain the possibility that a small higgs mass is selected for in the

string theory landscape by reasoning similar to that used by Weinberg in his anthropic

bound for the cosmological constant.

More detailed introductions for the relevant subject matter can be found at the

start of each chapter. The chapters end with a short summary and discussion of the results

obtained.

Chapter 2

Precision Electroweak Observables

in the Minimal Moose

Recently, a new class of theories known as Little Higgs (LH) models [10, 11, 12,

13, 14, 15, 16] have been proposed to understand the lightness of the Higgs by making it

a pseudo-Goldstone boson. Approximate global symmetries ensure the cancellation of all

quadratic sensitivity to the cutoff at one loop in the gauge, Yukawa and Higgs sectors,

by partners of the same quantum statistics: heavy gauge bosons cancel the divergence of

the SM gauge loop; massive scalars do the same for the Higgs self-coupling, as do heavy

fermions for the top loop contributions. These partner particles have masses of the order of

the symetry breaking scale f , which we take to be a few TeV. At lower energies the presence

of these new particles can be felt only through virtual exchanges and and their effects on

precision electroweak oberservables (PEWOs).

We calculate corrections to PEWOs in the Minimal Moose (MM) [10], and in a

similar model with a slight variation in gauge structure (the Modified Minimal Moose, or

MMM) in an attempt to find regions of parameter space where these are small with a

tolerable level of fine tuning in the Higgs sector.

Both models have a simple product gauge structure GL × GR and reduce to the

SM with additional Higgs doublets at low energies. Above the symmetry breaking scale the

Higgs sector is a nonlinear sigma model which becomes strongly coupled at Λ ' 10 TeV and

requires UV completion at higher energies. The enhanced gauge sector can contribute to

precision observables through the interaction of the partners to SM gauge bosons, W ′ and

4

Chapter 2: Precision Electroweak Observables in the Minimal Moose 5

B′, with fermions and Higgs doublets via currents jµF and jµH respectively, generating low

energy operators of the form jF jF , jF jH and jHjH . We group these into oblique and non-

oblique corrections, where the former impact precision experiments only via their effects on

gauge boson propagators, and summarize their salient properties below.

Oblique corrections can originate from:

• Interactions jHjH

B′ exchange modifies the Z0 mass and hence introduces custodial SU(2) violating

effects to which the T parameter is sensitive. This is a cause for concern in the MM,

but is reduced considerably in the Modified Moose by gauging a different subset of

the global symmetries.

• Non-linear sigma model (nlsm) kinetic terms

At energies above the global symmetry breaking scale, the Higgs doublets form compo-

nents of nlsm fields with self-interactions. This gives rise to custodial SU(2) violating

operators in the low energy theory which become our most significant constraint.

• Higgs-heavy scalar interactions

The theories also contain a scalar potential in the form of plaquette terms to en-

sure that electroweak symmetry is broken appropriately. This contributes to the T

parameter through the exchange of heavy scalar modes, which effect we show to be

negligible.

• Fermion loops

The presence of a vector-like partner to the top quark is another source for T and S

parameter contributions. We calculate these and show that they are tolerably small

for a wide range of parameters of the theory.

• Higgs loops

Since the MM is a two Higgs doublet theory at low energies, corrections due Higgs

loops are similar to those of the Minimal Supersymmetric Standard Model.

The following are the non-oblique corrections of concern to us:

• Four-fermion operators jF jF

These modify GF and can be controlled in the MMM in the near-oblique limit (see

below), in which light SM fermions decouple from the W ′ and B′.

6 Chapter 2: Precision Electroweak Observables in the Minimal Moose

• Interactions jF jH

Operators of this form shift the coupling of the SM gauge bosons to the fermions

(most easily seen in unitary gauge). These are also minimized in the near-oblique

limit.

LH gauge sectors generically have a simple limit in which highly constrained non-

oblique corrections vanish (the near-oblique limit [12]) in tandem with oblique corrections

from the gauge boson sector. In the MM however, the SU(3)×SU(2)×U(1) gauge structure

is too tightly constrained to allow for a decrease in the large oblique B ′ correction by

variation of the gauge couplings. This issue is resolved in the MMM by replacing the SU(3)

gauge group by another SU(2) × U(1) and charging the light fermions equally under both

U(1)s, giving

jH , jFlight∝ tan θ′ − cot θ′

for tan θ′ = g1R/g1L, the ratio of the U(1) couplings at the sites. Setting these nearly equal

to each other rids us of large heavy gauge boson contributions to the T parameter as well as

undesirable light four-fermion operators arising from B ′ exchange. This method does not

work with third-generation fermions which are coupled to the Higgs in a slightly different

way. Possible non-oblique corrections involving these will not be discussed since they are

not yet unambiguously constrained by experiment. For additional discussion of this see

[12]. W ′- exchange operators are more easily handled since, provided we stay away from

the strong coupling regime, increasing one of the SU(2) gauge couplings with respect to the

other increases the mass of the W ′, effectively decoupling it from our theory.

We begin this chapter with a brief review of the MM, keeping as far as possible

to the conventions used in [10]. In Sections 2.2 to 2.4 we calculate tree-level corrections

to PEWOs from different sectors. We go on to discuss electroweak symmetry breaking

in the low energy theory (Section 2.5) and determine loop effects due to a new heavy

fermion (Section 2.6) and Higgs doublets (Section 2.7). In spite of the fact that the MM

is inconsistent with current constraints on PEWOs we show in Section 2.8 that there are

regions in the parameter space of the MMM where all except third generation non-oblique

corrections can be eliminated, with tolerably small oblique corrections. We will see that

the most unforgiving aspect of both models is the non-linear sigma model sector which has

no residual SU(2)c symmetry and hence gives rise to a T parameter contribution that can

only be decreased by adjusting f . This compels us to choose f greater than ∼ 2 TeV. We


display two sets of parameters, one that is well within the 1.5-σ S-T ellipse with a 17% fine

tuning in the SM Higgs mass and another that falls just outside the ellipse with a 3% fine

tuning. We show that there are regions of parameter space where one can do even better

than the first set, however this is only possible for a rather specific choice of parameters.

We measure fine tuning by (m/δm)2, where δm is the top loop correction to the mass of

the Higgs doublet, and m is the physical Higgs mass.

2.1 The Theory

G2G1 GRGL

Figure 2.1: The Minimal Moose

The Minimal Moose is a two-site four-link model with gauge symmetry GR = SU(3) at

one site and GL = SU(2) × U(1) at the other. The standard model fermions are charged

under GL, with their usual quantum numbers while the link fields Xj = exp(2ixj/f) are 3

by 3 nonlinear sigma model fields transforming as bifundamentals under GL ×GR where a

fundamental of GL is 21/6 ⊕ 1−1/3. These fields get strongly coupled at a scale Λ = 4πf ,

beneath which the theory is described by the Lagrangian

L = LG + Lχ + Lt + Lψ (2.1)

LG includes all kinetic terms and gauge interactions, while Lχ contains plaquette couplings

between the Xj:

Lχ =

(f

2

)4 (Tr

[A1X1X

†2X3X

†4

]+ Tr

[A2X2X

†3X4X

†1

]

+Tr[A3X3X

†4X1X

†2

]+ Tr

[A4X4X

†1X2X

†3

])+ h.c. (2.2)

with Ai = κi + εiT8 for ε ∼ κ/10. This is a natural relation since any radiative corrections

to ε require spurions from both the gauge and plaquette sectors and so can only arise at

two loops. The ε terms give the little Higgses a mass (see Equation 2.18) and are required

to stabilize electroweak symmetry breaking (EWSB).


The third generation quark doublet is coupled to a pair of colored Weyl fermions

U ,U c via Yukawa terms in Lt

Lt = λf(

0 0 uc′

3

)X1X

†4

q3

U

+ λ

′

fUU c + h.c (2.3)

and Lψ contains the remaining Yukawa couplings. These take the same form as above for

the light up-type quarks, but with U and U c removed, while the down and charged lepton

sectors look like

Lψ ⊃ λD(q 0

)X1X

†4

0

0

dc

+ λE

(l 0

)X1X

†4

0

0

ec

+ h.c (2.4)

We also impose a Z4 symmetry which cyclically permutes the link fields and hence

requires equality of all the decay constants (fi = f) and plaquette couplings (κi = κ, εi = ε).

The only Z4-breaking terms arise in the fermion sector and are small.

2.2 The Gauge Boson Sector

The link fields Higgs the GL×GR gauge groups down to the diagonal SU(2)×U(1)

subgroup, leaving one set each of massive and massless gauge bosons. This can be seen

explicitly by considering the link field covariant derivatives:

DµXj = ∂µXj − ig3XjAA3,RµT

A + ig2Aa2,LµT

aXj + iqg1A1,LµT8Xj (2.5)

where the Ts for A = 1, ..., 8 and a = 1, 2, 3 are SU(3) generators normalized such that

Tr[TATB] = δAB

2 (similarly for a,b indices); and q = 1/√

3 to ensure that the Higgs doublet

eventually has the correct SM hypercharge. Expanding out the fields (Xj = exp(2ixj/f))

in the kinetic term

f2

4Tr

4∑

j=1

(DµXj)(DµXj)

†

(2.6)

shows that the eaten Goldstone boson, w, is proportional to x1 + x2 + x3 + x4. Orthogonal

combinations x,y and z can be defined as follows:

w

z

x

y

=1

2

+1 +1 +1 +1

+1 −1 +1 −1

−1 −1 +1 +1

−1 +1 +1 −1

x1

x2

x3

x4

(2.7)


where each of the above fields decomposes under SU(2)× U(1) as 30 (φ) + 10 (η) + 2±1/2

(h and h†).

x =

φx + ηx√

12hx√

2h†x√

2

−ηx√3

(2.8)

The x and y contain two Higgs doublets in the more familiar form

h1 =hx + ihy√

2(2.9)

h2 =hx − ihy√

2(2.10)

and plaquette terms give z a large tree level mass, so it can be integrated out of the theory

at a TeV.

Going to unitary gauge results in a mass matrix for heavy gauge bosons W ′µ, B

′µ

and A3µ with eigenvalues 2gf/ sin 2θ, 2qg′f/ sin 2θ′ and fg/ sin θ respectively. The W ′ s

and B′ s are admixtures of A3,R, A2,L and A1,L:

W aµ = cos θAa2,Lµ + sin θAa3,Rµ

W ′aµ = − sin θAa2,Lµ + cos θAa3,Rµ

Bµ = cos θ′A1,Lµ + sin θ′A83,Rµ (2.11)

B′µ = − sin θ′A1,Lµ + cos θ′A8

3,Rµ

with mixing angles defined as follows:

g =g2g3√g22 + g2

3

sin θ =g

g3

g′ =g1g3√

(qg1)2 + g2

3

sin θ′ =qg′

g3

The Higgses couple to heavy gauge bosons via the following currents:

jaW ′µ

= − ig

2 tan 2θ

[h†1σ

a←→Dµh1 + h†2σa←→Dµh2

]

jB′µ = −

√3iqg

′

2 tan 2θ′

[h†1←→Dµh1 + h†2

←→Dµh2

](2.12)

where Dµ is a Standard Model covariant derivative and σs are Pauli matrices.

Explicitly integrating out the heavy gauge bosons results in the following SU(2)c


violating terms:

3

16f2cos 22θ′

[(h†1Dµh1

)2+

(h†2Dµh2

)2+ 2

(h†1Dµh1

)(h†2D

µh2

)]

+1

8f2cos 22θ

[(h†1Dµh2

)(h†2D

µh1

)−

(h†1Dµh1

)(h†2D

µh2

)]+ h.c. (2.13)

It seems surprising that there is a contribution from the heavy W at all (cos2 2θ term)

since its coupling is custodial SU(2)-symmetric! The relevant operators appear with a

relative minus sign, however, and cancel when we break electroweak symmetry, giving a

total contribution to the T parameter of 1.6(

1TeVf

)2cos2 2θ′. This mechanism is responsible

for some more fortuitous cancellation in the next section.

At first glance it seems like we can minimize the B ′ contribution to precision

measurements by varying θ′. However we are constrained to sin θ ′ . 1/3 by the relation

tan θW =1

q

sin θ′

sin θ

which gives us an unacceptably high T parameter as well as large corrections to GF from

B′ exchange. To overcome this problem the MM can be modified by replacing the SU(3)×SU(2)×U(1) gauge symmetry by [SU(2)×U(1)]2 whose generators can be embedded into

SU(3) as T1,2,3 and T8. This sidesteps the constraint, since we now have enough freedom

to vary θ′ independently of θ. If we charge the fermions under GL as before, we will still

have to tolerate large non-oblique corrections. Altering the fermion couplings, however, by

charging them under both U(1)s, gives a B ′- fermion coupling of:

ig′∑

i

f i

( qLtan θ′

− qR tan θ′)σµB′

µfi (2.14)

where qL and qR are the fermion charges under each of the U(1)s. We can set qL = qR =

qSM/2 for the light fermions to eliminate this coupling at θ ′ ' π/4, provided we adjust the

light yukawa couplings to account for the new gauge structure:

Lup =

λU

(0 0 uc

)X1X

†4

q

0

[X33]

− 34 + h.c (2.15)

Ldown =

λD

(q 0

)X1X

†4

0

0

dc

+ λE

(l 0

)X1X

†4

0

0

ec

[X33]

34 + h.c

where X33 is the 33 component of any of the link fields.


Gauging an SU(2) × U(1) at both sites gives rise to an extra Higgs doublet, hw,

which is no longer eaten by gauge bosons. Its mass is zero at tree level, but its one-loop

effective potential contains a logarithmically divergent contribution that is of the same order

as that of h1 and h2. Since it is not coupled to the fermion sector or the Little Higgses,

though, it does not pick up a vev. We can therefore avoid the complications of working

with three Higgs doublets in favor of just two.

2.3 Non-linear Sigma Model Sector

SU(2)c violating operators are also contained in the link field kinetic terms. It is

straightforward to show that these are generated with the following coefficients:

1

16f2

[(h†1Dµh1

)2+

(h†2Dµh2

)2+ 2

(h†1Dµh1

)(h†2D

µh2

)]

+1

16f2

[2(h†1Dµh2

)(h†2D

µh1

)−

(h†1Dµh2

)2−

(h†2Dµh1

)2]

+ h.c. (2.16)

Like the operators that originate from integrating out W ′, the terms in the second bracket

will not give any contribution to the T parameter after EWSB. The contribution from the

first bracket is 0.53(

1TeVf

)2.

2.4 Plaquette Terms

For an analysis of the plaquette terms we use the Baker-Campbell-Hausdorff pre-

scription to expand them to quartic order in the light Higgs fields. The Z4 symmetry of the

theory simplifies things greatly: it gets rid of the z tadpole, for example, leaving a z mass:

M2z = 4f2<(κ) +O(ε) (2.17)

a tree level mass for the Higgses which stabilizes the flat direction in the potential and

triggers electroweak symmetry breaking;

√3f2

4=(ε)

(h†1h1 − h†2h2

)(2.18)

a z-Higgs coupling of the form, jaza, with

ja = −f2=(ε)Tr

(Ta[x, [x,T8]]−Ta[y, [y,T8]]

)+ ... (2.19)


and the leading quartic Higgs interaction

<(κ)Tr [x, y]2 (2.20)

We will neglect the T contribution from integrating out the heavy z since this is O(ε2/κ2)

and so is suppressed by a factor of 100 in relation to the other terms considered.

2.5 Electroweak Symmetry Breaking

The leading order terms in the Higgs potential (in manifestly CP invariant form)

are

V ≈ m21h

†1h1 +m2

2h†2h2 +m2

12

(h†1h2 + h†2h1

)(2.21)

+ λh

[(h†1h1

)2+

(h†2h2

)2−

(h†1h1

)(h†2h2

)−

(h†1h2

)(h†2h1

)]

where the couplings include radiative corrections as well as the tree level terms detailed in

the previous section. We are unable to say anything more precise since two loop radiative

corrections to the Higgs mass terms, for example, are parametrically of the same order as

one loop corrections. We can, however, place some constraints on the relative values of

these by imposing that the potential go to positive infinity far from the origin. The quartic

terms will dominate in this limit, but there is a flat direction, namely h1 = eiϕh2 in which

we demand that the quadratic part of the potential be positive definite. This gives us the

constraint

m21 +m2

2 ≥ 2|m212| (2.22)

Further requiring that the mass matrix for h1 and h2 have one negative eigenvalue at the

origin tells us that

m21m

22 < m4

12 (2.23)

The potential (Equation 2.21) is minimized for vevs of the form

h1 =1√2

0

v cos β

h2 =1√2

0

v sinβ

(2.24)


where

v2 =1

λh

[−m2

1 −m22 +|m2

1 −m22|

cos 2β

]

sin 2β = − 2m212

m21 +m2

2

(2.25)

An examination of the solution shows that it is consistent with the constraints (2.22) and

(2.23).

The masses of the physical states in the two-doublet sector satisfy the relations

4m2H± = m2

h0 +m2H0 + 3m2

A0 (2.26)

m2H± = m2

A0 + λhv2

2.6 Fermion Sector

Armed with this information we can now calculate the T and S parameters from

the fermion sector. We look directly at corrections to the W and Z masses from vacuum

polarization diagrams containing fermion loops.

The Higgses give rise to a small mixing term for the top and heavy fermion in our

theory so we need to find the fermion mass eigenstates. Diagonalizing the Yukawa coupling

in two stages: to zeroth order in v to start with, we get Lt in terms of the new eigenstates:

Lt = f√λ2 + λ′2

[U cU + sin2 ξ

(0 0 U c

)(X1X

†4 − 1

) q3

U

(2.27)

+ sin ξ cos ξ(

0 0 uc3

)(X1X

†4 − 1

) q3

U

]

where

sin ξ =λ√

λ2 + λ′2

U c = cos ξU c + sin ξuc′

3 (2.28)

uc3 = − sin ξU c + cos ξuc′

3

Expanding the link fields to first order in v/f, a convenient phase rotation gives us the


following terms in the t− U mass matrix:

mtt =√λ2 + λ′2 sin ξ cos ξ

v√2(sinβ + cos β)

mtU =√λ2 + λ′2 sin2 ξ

v√2(sinβ + cos β) (2.29)

mUU = f√λ2 + λ′2

Using mtt,mtU << mUU , mUt = 0 we approximate the results in Appendix A to obtain

m2t ≈ m2

tt

m2U ≈ m2

UU

(1 +

m2tU

m2UU

)(2.30)

cos θL ≈ 1− m2tU

2m2UU

Now we can fix the top Yukawa coupling to its value λt in the SM, which for a given value

of tanβ relates λ to λ′ in the following way:

λ′

λ=

λt√

1 + tan2 β√λ2(1 + tan β)2 − λ2

t (1 + tan2 β)(2.31)

with λ constrained by

λ2 >1 + tan2 β

(1 + tanβ)2λ2t (2.32)

Having determined the fermion mass eigenvalues, we use [17] to find:

Tf =3

16π sin2 θW cos2 θW

[sin2 θLΘ+

(m2U

m2Z

,m2b

m2Z

)− sin2 θLΘ+

(m2t

m2Z

,m2b

m2Z

)(2.33)

− sin2 θL cos2 θLΘ+

(m2U

m2Z

,m2t

m2Z

)]

Sf =3

2π

[sin2 θLΨ+

(m2U

m2Z

,m2b

m2Z

)− sin2 θLΨ+

(m2t

m2Z

,m2b

m2Z

)− sin2 θL cos2 θLχ+

(m2U

m2Z

,m2t

m2Z

)]

for

Θ+(y1, y2) = y1 + y2 −2y1y2

y1 − y2lny1

y2Ψ+(y1, y2) =

1

3− 1

9lny1

y2

χ+(y1, y2) =5(y2

1 + y22)− 22y1y2

9(y1 − y2)2+

3y1y2(y1 + y2)− y31 − y3

2

3(y1 − y2)3lny1

y2(2.34)


2.7 Higgs Sector

There is a contribution to the vacuum polarization diagrams from additional phys-

ical Higgs states running around the loop. This is a standard calculation (see [12, 18]) which

yields

Th =1

16π sin2 θWm2W

(F (m2

A0 ,m2H±) + cos2(α− β)

(F (m2

H± ,m2h0)− F (m2

A0 ,m2h0)

)

+sin2(α− β)(F (m2

H± ,m2H0)− F (m2

A0 ,m2H0)

))

Sh =1

12π

(cos2(β − α) log

m2H0

m2h0

− 11

6+ sin2(β − α)G(m2

H0 ,m2A0 ,m

2H±) (2.35)

+ cos2(β − α)G(m2h0 ,m

2A0 ,m

2H±)

)

where

F (x, y) =1

2(x+ y)− xy

x− y logx

y

G(x, y, z) =x2 + y2

(x− y)2 +(x− 3y)x2 log x

z − (y − 3x)y2 log yz

(x− y)3

The A, H, h are Higgs mass eigenstates and α is the mixing angle between H 0 and h0, as

detailed in [18].

2.8 Results

The graphs below give some idea of the size of oblique corrections from the Higgs

and fermion sectors. It can be seen in Figure 2.2 that the T parameter contribution from

fermions is rather small (S is negligible) for the most part, and decreases with increasing

tanβ. However the top partner also gets heavier in this limit, increasing the level of fine

tuning in the theory, since the quadratically divergent fermion loop diagram is cut off at a

higher energy.

The Higgs sector contribution to the T parameter is generically negative, although

there is no such restriction on the S parameter (see Figure 2.3). As for the fermions, though,

the latter is usually small and can be ignored . The biggest constraint in our models is the

large T parameter arising in the nonlinear sigma model sector. Keeping this at a manageable

level limits us to f & 2 TeV. At this breaking scale the remaining parameters can have a

range of values that do not take us beyond 1.5-σ in the S-T plane. To illustrate this we chose


0.5 1 1.5 2 2.5 3 3.5

0.01

0.02

0.03

0.04

0.05f= 1.9 TeV

f=2.6 TeV

Tf

βtan0.5 1 1.5 2 2.5 3 3.5

2.8

3.2

3.4

3.6

3.8

4

4.2 f=2.6 TeV

f=1.9 TeV

mU (TeV)

tanβ

Figure 2.2: Fermion sector contribution to T and mass of top partner as a function of tanβ.λ andλ′ were chosen to minimize mU with a fixed top quark mass.

0.5 1 1.5 2 2.5 3 3.5

-0.125

-0.1

-0.075

-0.05

-0.025

0

0.05

Ref. values 2

Ref. values 1

Thβtan

0.5 1 1.5 2 2.5 3 3.5

-0.002

0.002

0.004

0.006

0.008Ref. values 1

Ref. values 2

Sh

tanβ

Figure 2.3: Higgs sector contribution to PEWOs as a function of tanβ. Values of other variablestaken from Table 2.1

two representative sets of free parameters (Table 2.1) and plotted T against S, subtracting

out the SM T and S contributions. The first reference set (see Figure 2.4), which contains

a moderately heavy Higgs, has parameters which were chosen to obtain a sizable negative

T from the Higgs sector to partly cancels the nonlinear sigma model contribution, thus

allowing us to make f as low as 1.9 TeV without leaving the ellipse. We also plot the fine

tuning for different regions of parameter space within the ellipse in Figure 2.5 by varying

the Higgs quartic coupling and tanβ around this reference set. One can see that there are

viable regions with larger quartic coupling which can give even less fine tuning in the Higgs

mass, however these lie in a smaller band in parameter space and thus correspond to a more

specific choice of the parameters of the theory. In fact, the allowed region ends for large

values of the quartic coupling because it is driven out of the ellipse by a T contribution

from the Higgs sector that is too negative. One could imagine taking an even smaller value


for f and thus increasing the positive nonlinear sigma model contribution to T , expanding

the allowed region and decreasing the fine tuning further, since the Higgs mass is increased

as the top quark partner mass is decreased. However, as before, this occurs for more and

more specific choices of parameters where large T contributions from the nonlinear sigma

model and Higgs sectors are delicately cancelling out and we chose not to work with such

values. Our second set of parameters (see Figure 2.6), which was picked to contain a light

Higgs but is otherwise fairly random, takes us only slightly out of the S-T ellipse. It has

a small negative S and no cancellation between sectors, which forces us to choose a larger

value for f . We vary λh and tan β around this reference set in Figure 2.7. We see that

there is a large region of parameter space where the PEWOs are no further outside the 1.5σ

ellipse than the reference point we chose, and the fine tuning is even better. Since S and T

are not as sensitive to the other parameters one can conclude that the results we quote are

quite generic in the parameter space of the model. Note that although the theory seems to

favour a heavy Higgs, it is still possible to find acceptable data sets in which it is light.

More generically consistency with PEWOs constrains us to values of f greater

than 2 TeV. The increase of the heavy quark mass with f bounds the latter to be less than

2.5 TeV for the fine tuning to be any better than that of the SM. The acceptable region

in parameter space is larger for higher values of f , however this comes with the price of

increased fine tuning in the higgs mass.

Parameter Reference values 1 Reference values 2

f(TeV) 1.9 2.6

θ 40 25

θ′ 47 50

λ 0.9 1.1

λh 1.6 0.5

tanβ 1.1 2.0

mH±(GeV) 234 206

mH0(GeV) 381 206

mA0(GeV) 98 191

mh0(GeV) 220 134

mU (TeV) 2.77 3.89

m′W (TeV) 2.56 4.50

m′B(TeV) 0.78 1.08

fine tuning 17% 3%

Table 2.1: Two sets of reference parameters for the Modified Minimal Moose.


B’ and fermion

Higgs doublets

NLSM

REFERENCE VALUES 10.1

T

−0.025 0.0250S

−0.1

0

"!#%$'&&)(*+,-./$1023(*456,78 -$'&3(*

:9; 4

Figure 2.4: S and T values in the MMM for reference set 1 in Table 2.1 plotted on a 1.5-σ oval inthe S-T plane

2.9 Summary and Discussion

Little Higgs models, like the Minimal Moose, predict new heavy particles at the

TeV scale. Upon integrating out these particles the SM is recovered at low energies, with

possibly one or more extra Higgs doublets. At higher energies the Higgses form components

of nonlinear sigma model fields which become strongly coupled at around 10 TeV. At still

higher energies a UV completion of the theory is needed. This could be achieved with

strongly coupled dynamics [19], a linear sigma model or supersymmetry. In the latter case

the SUSY breaking scale is pushed to 10 TeV, alleviating the difficulties of flavor-changing


0.5 1 1.5 2 2.5 3 3.5tan Β

1

2

3

4

5

6

Λh

ref. set 1

30%

15%

20%

25%

35%

40%

Figure 2.5: Fine tuning plotted as a function of tan(β) and λh where all other parameters have beentaken from Table 2.1. We only plot regions of parameter space which lie within the S-T ellipse. Weindicate the position of reference set 1. Note that the fine tuning improves for larger values of λh asthe Higgs becomes heavier.

neutral currents associated with TeV-scale superpartners.

At energies below the masses of these new particles we rely on precision electroweak

data to gauge the feasibility of a particular model as a possible extension to the SM. In

the absence of new flavor physics (due to the introduction of a partner for the top quark

only), precision measurements can be divided into oblique and non-oblique corrections. We

analyzed these for two such models at around a TeV, translating the low energy theory

into effective operator language as far as possible. We saw that we ran into significant

problems in more than one sector when we considered the constrained gauge structure of

the MM. Gauging two copies of SU(2)×U(1) instead and charging the SM fermions equally

under both U(1)s, as in the MMM, does away with these issues as we can then go to the

near oblique limit without reintroducing large contributions to the T parameter from B ′

exchange.

This might be understood better in the context of other LH theories, the Littlest


REFERENCE VALUES 2

0.1

T

−0.025 0.0250S

−0.1

0

B’ and fermion

NLSM

Higgs doublets (negligible)

! "#$ &%'("*),+.-/!0123#)54.-/6718 ),+.-/

Figure 2.6: S and T values in the MMM for reference set 2 in Table 2.1 plotted on a 1.5-σ oval inthe S-T plane

Higgs [11] for example. The greatest contrast between this and the MM is the nonlinear

sigma model sector, where the Littlest Higgs has a built-in SU(2)c symmetry which protects

it from any T parameter contribution. This symmetry is explicitly broken in the top sector,

but only by a small amount. The gauge sectors of the theories are identical except for a B ′

mass in the Littlest Higgs which is lighter by a factor of 2 (since the theory only contains 1

link field), but heavier by√

5/3 to account for the different group structure involved. Aside

from this, the similarity in the general framework of the models implies that a lower cutoff

can be tolerated in the case of the Littlest Higgs, giving rise to lower masses for the heavy

particles, and a subsequent decrease in the level of fine tuning. The relative success of the

MM is rather surprising, however, given that it was designed for minimality rather than


1.5 2 2.5 3 3.5 4 4.5tan Β

0.5

1

1.5

2

2.5

3

Λh

ref. set 2

4%

6%

8%

10%

12%

14%

16%

18%

2%

Figure 2.7: Fine tuning plotted as a function of tanβ and λh where all other parameters have beentaken from Table 2.1. We plot regions of parameter space which are at least as close to the S-Tellipse as reference set 2.

freedom from precision electroweak constraints.

In summary we see that the MMM, which contains a gauged [SU(2)× U(1)]2 is a

viable candidate for TeV-scale physics. The heavy counterparts for SM particles give rise

to precision electroweak corrections that are within acceptable experimental bounds for a

large range of parameters of the theory. It leads to at least moderate improvements over

the SM in terms of the gauge hierarchy problem for generic regions of parameter space,

and very significant improvements for less generic regions, which are nevertheless plausibly

large.

Chapter 3

The New Fat Higgs: Slimmer and

More Attractive

One way to resolve the SUSY ’little hierarchy’ problem is by generation of a larger

tree level quartic coupling for the higgses. This can be accomplished through new F-terms

as in the Next To Minimal Supersymmetric Standard Model (NMSSM) [20, 21, 22, 23];

new D-terms by charging the higgs under a new gauge symmetry [24, 25, 26]; or by using

“hard” SUSY breaking at low scales [27, 28]. We choose to focus on the NMSSM, where

the addition of a gauge singlet S allows for the following term in the superpotential

W = λSHuHd (3.1)

and results in an additional quartic coupling for the higgses of the form |λ|2|HuHd|2. Un-

fortunately the requirement of perturbativity up to the GUT scale limits the size of λ at

the electroweak scale [29, 30] giving a maximum higgs mass bound of about 150 GeV. This

constraint was evaded in what is known as the Fat Higgs model [31] by allowing the coupling

to become nonperturbative at an energy lower than the GUT scale, where S,Hu and Hd

were seen to be composites of new strong dynamics. All couplings were asymptotically free

above this point and the higgs mass bound could be pushed up to 500 GeV. On the other

hand the composite nature of the higgs doublets gave rise to a different problem - gauge

coupling unification was not manifest and some ad hoc matter content had to be added to

the theory to preserve it. In addition, elementary higgs fields needed to be reintroduced in

order to generate the usual Standard Model yukawas at low energies.

We will argue that UV completion of the NMSSM does not require us to sacrifice

22

Chapter 3: The New Fat Higgs: Slimmer and More Attractive 23

the desirable natural unification properties of weak scale SUSY. We will keep the higgs

fields elementary, making unification manifest while permitting the usual Standard Model

yukawas to be written down. Like the Fat Higgs, we use a composite S but instead we

replace the λ coupling above the compositeness scale by asymptotically free yukawas. Since

we will no longer have to run λ, which grows in the UV, all the way to the GUT scale, we

can afford to start at a larger value at the electroweak scale. Unfortunately our scheme will

require us to compromise slightly on how heavy we can make the higgs, but this seems a

small price to pay for natural gauge coupling unification.

3.1 Constructing a Model

In SUSY models gauge contributions to anomalous dimensions are negative, tend-

ing to make yukawa couplings asymptotically free. The yukawas themselves, on the other

hand contribute positive anomalous dimensions. These competing effects, which are evident

in the Renormalization Group Equation (RGE) for the NMSSM λ coupling

dλ

dt=

λ

16π2

[4λ2 + 3h2

t − 3g22 −

3

5g21

]+ · · · (3.2)

result in an asymptotically free λ only when the gauge couplings involved are larger than

λ itself. Even when they do not dominate the running, maximizing the negative contri-

butions from the gauge sector by adding as many SU(5) 5 + 5 multiplets as are allowed

by perturbative unification gives an upper bound on the low energy λ coupling [29]. The

benefit is small here, however, since the electroweak gauge couplings remain quite weak for

the majority of the running and g3 only affects ht at one loop. This makes it difficult to

significantly increase the low energy value of λ.

One way to improve the situation is to introduce new gauge dynamics through the

following superpotential:

Wλ = λ1 φXHu + λ2 φcXcHd +MXXX

c +MXXXc. (3.3)

We have added the fields φ, φc, X,Xc, X, Xc, which are charged under a new strong gauge

symmetry, with the Xs also charged under the Standard Model as seen in Table 3.1. We

choose SU(n) to be our strong group as this permits our scheme to be most easily im-

plemented. Since the strong gauge coupling (gs) can now dominate the running, the λ1,2

yukawas can be asymptotically free for larger initial values and the resulting gain in λ will

24 Chapter 3: The New Fat Higgs: Slimmer and More Attractive

SU(3) × SU(2)L × U(1)Y SU(n)sφ (1, 1, 0) n

φc (1, 1, 0) n

X (1, 2,− 12 ) n

Xc (1, 2, 12 ) n

X (3, 1, 13) n

Xc (3, 1,− 13 ) n

Table 3.1: Preliminary charge assignments for the new particles

be more substantial. The two X fields have been given a supersymmetric mass, MX , and

are completed into (5,n)+(5, n) multiplets of SU(5)×SU(n)s by the Xs and thus maintain

gauge coupling unification. Note that this doesn’t require any MSSM particles to be gauged

under SU(n)s. The fields that are gauged under both the Standard Model and the new

group have large supersymmetric mass terms and thus decouple from low energy physics.

Below the scale MX , integrating out the Xs and Xs generates the nonrenormaliz-

able operator

Weff = −λ1λ2

MXφφcHuHd. (3.4)

There are two ways in which the NMSSM λ coupling can be recovered. One is to break

SU(n)s by giving a vev to φ; as long as this breaking takes place close to the MX scale, λ

can be satisfactorily large. A simpler approach, which we adopt here, is to use the fact that

below MX ,MX there are 5 fewer flavors of the strong group, making the gauge coupling

stronger at lower energies and forcing the φ fields to confine into an NMSSM singlet which

we will call S.

Building a realistic theory from this philosophy is simply a matter of deciding

what n will be. We use the fact that there is a restriction on the number of SU(5) flavors

that can be added to the Standard Model for gauge couplings to perturbatively unify given

that the added SU(5) fundamentals do not decouple until the TeV scale.1 This requires 4

flavors or less and hence n ≤ 4. Another important constraint is on the number of flavors

of SU(n)s that remain after the 5 flavors in X and X have been integrated out. We want

to avoid nf < n where there is an Affleck-Dine-Seiberg vacuum instability [33] and will

ignore the potentially interesting case nf = n, where the Quantum Modified Moduli Space

constraint might shed some light on the µ problem. Instead we will choose to start with

1The possibility of a model with accelerated unification [32] and a lowered unification scale will not beconsidered here.


SU(3)× SU(2)L × U(1)Y SU(4)s

φ (1, 1, 0) 4

φc (1, 1, 0) 4

ψi for i = 1, · · · , 4 (1, 1, 0) 4

ψci for i = 1, · · · , 4 (1, 1, 0) 4

X (1, 2,− 12 ) 4

Xc (1, 2, 12) 4

X (3, 1, 13 ) 4

Xc (3, 1,− 13 ) 4

Table 3.2: Final charge assignments for new particles

n + 6 flavors of SU(n)s, where integrating out the 5 flavors gives nf = n + 1, making the

theory s-confine. Now combining the requirement for asymptotic freedom (n+6 < 3n) with

the perturbative unitarity constraint (n ≤ 4) discussed earlier uniquely fixes n = 4.2

3.1.1 Details of the Model

We now summarize the content and interactions of the model. There is a strong

SU(4)s gauge group, with the particle content shown in Table 3.2. The superpotential

contains

W = Wλ +WS +Wd where (3.5)

WS = mφφc (3.6)

Wd = y(T i φψci + T c i ψiφc + T ij ψiψ

cj) +

y′

M2GUT

(εijkl TBi φψjψkψl + εijkl TBc

i φc ψcjψckψ

cl ). (3.7)

where we have introduced some singlets denoted by T . After confinement, WS gives a

linear term in S as in the Fat Higgs [31] while Wd decouples the extra mesons by giving

them mass terms with the singlets T i, T c i, T ij . Note that in the second line of Wd there

is a nonrenormalizable mass term for the baryons with the T Bs which is suppressed by the

GUT scale MGUT and thus gives rise to light baryon states. The constraints imposed by

2The case of SU(3) with 9 flavors might also be useful for our purpose. This model has been argued tohave a linear family of conformal fixed points in (λi, g) space [34] and would therefore be convenient whenwe discuss the possibility of having a new superconformal fixed point in Section 3.1.2. Alternatively if theXs required for unification were not also charged under the strong group, satisfying the resulting constraintswould be easier since we would have more room to maneuver. However this theory is not naturally unified,and so will not be pursued here.


these light states will be discussed in Section 3.2.3. Note that there is a non-anomalous

U(1)R symmetry (under which ψi, ψci are neutral and all other SU(4)s flavors have charge

1) that makes the given superpotential natural.

3.1.2 Conformality and Confinement

At high energies the strong group has 10 flavors and is within the conformal window

(32n < 10 < 3n) implying, in the absence of λ1,2, that the theory flows to an interacting fixed

point in the IR [33]. As discussed previously the strong gauge coupling gives large negative

contributions to the beta functions of the λ1,2 couplings making them asymptotically free

for gs λ1,2. Ignoring electroweak couplings and the top yukawa, near Seiberg’s fixed

point we have the RGE:

dλ1,2

dt=

7λ31,2

16π2+ γ∗λ1,2 + · · · (3.8)

The first term is the usual one loop term due to the yukawa couplings while the second term

contains contributions from all orders in the fixed point gauge coupling g∗. If the theory is

at the fixed point then we have very precise information on the value of γ∗ in the weak limit,

since this is related to the U(1)R charges of the fields by the superconformal algebra. Within

the conformal window for example, −1 < γ∗ < 0 which indicates that the λ1,2 couplings

are relevant; they grow in the IR. Our limited understanding of strongly coupled theories

prevents us from proceeding in full generality, so from now on we will restrict ourselves to

two plausible types of behavior.

The first possibility is the emergence of a new superconformal phase where both

the new yukawas and gauge couplings hit fixed points in the IR; in this case it is hard to

be quantitative about possible values of the NMSSM λ coupling. At best, we can specify a

range of fixed point values of λ1,2 which give interesting λ couplings, without being able to

justify if those values can be obtained. Still, the insensitivity of this scenario to UV initial

conditions is very attractive.

In the second possibility the yukawa couplings get strong and disrupt the confor-

mality, pushing the theory away from the fixed point. In this case a reasonable bound on

the sizes of λ1,2 can be given using their apparent fixed point values from Eq. 3.8. We will

refer to this as the weak limit bound. It is nontrivial that this bound on λ will be large

enough to be of interest to us. In fact, the naive estimate will be in the right range, but as

we will see in Section 3.2.1, there are many unknown order one factors that can change its


size. An undesirable aspect of this case is that the UV boundary conditions for λ1,2 have to

be tuned to small values in order for these couplings to be just below their one loop fixed

points at low energies. This tuning could be improved somewhat if the gauge coupling hits

its fixed point at some intermediate scale. It is also worth noting that this weak limit bound

could give us a rough estimate of the fixed point values of λ1,2 in the first scenario.

At energies around the mass of the Xs and their colored partners X, these 5 flavors

are integrated out of the theory. The terms in the RGEs for the supersymmetric masses

typically give an ordering m < MX < MX . Thus, the colored partners are integrated out

first which leaves 7 flavors of SU(4)s; this is still within the conformal window and, in the

electric description, has a stronger fixed point than the UV theory. This would take |γ∗|from 1/5 to 5/7 and also increase the weak limit bound on λ1,2 at the scale MX . For the

coupling to approach this fixed point the ratio MX/MX must be small. As discussed in

Section 3.2.2, there are no constraints on the size of this parameter from unification as long

as MX = MX at the GUT scale.

Below MX , the theory is a SU(4)s gauge theory with 5 flavors ΨI = (φ, ψi),ΨcI =

(φc, ψci ) for I = 0, · · · , 4. There is a dynamically generated superpotential

Wdyn =1

Λ7

[MIJB

IBc J − detM]

(3.9)

written in terms of gauge invariant mesons (MIJ ∼ ΨIΨcJ) and baryons (BI ∼ εIJKLMΨJ · · ·ΨM).

At the scale Λ . MX this theory confines and the superpotential should be written in terms

of the canonically normalized meson and baryon fields. Since the gauge coupling is strong

the sizes of the interactions after matching are in principle unknown. However estimating

their sizes by Naive Dimensional Analysis (NDA) [35, 36] gives:

W = Weff +WS +Wd +Wdyn where (3.10)

Weff →[√

nλ1λ2

4π

Λ

MX

]SHuHd (3.11)

WS → mΛ

4πS (3.12)

Wd → yΛ

4π(T iM0i + T c iMi0 + T ijMij) +

y′Λ3

4πM2GUT

(TBi Bi + TB

c

i Bc i) (3.13)

Wdyn →[(4π)MIJB

IBc J − (4π)3

Λ2detM

](3.14)

and we have defined M00 to be S. The first two terms give us an NMSSM-like model at

energy scales below Λ. In theWeff term we have done not only the normal NDA analysis, but


also the large n counting - notice that this partly compensates for the 4π NDA suppression.

Up to an unknown O(1) constant, this results in a value for λ at the confinement scale of:

λ =√nλ1λ2

4π

Λ

MX. (3.15)

WS contains a term linear in S that favors electroweak symmetry breaking and explicitly

breaks the Peccei-Quinn symmetry that would give rise to an undesirable light axion. Wd

marries up the superfluous baryons and mesons with singlet T partners as desired. In

addition, integrating out the heavy mesons will decouple their interactions in Wdyn. It is

also possible to add interactions that will give rise to the standard NMSSM S 3 coupling to

eliminate the new µ problem arising from the supersymmetric parameter m but we will not

address this or the µ problems of MX and MX here.

3.2 Analysis

3.2.1 λ and the Higgs Mass Bound

So far we have shown how our model approximately reduces to the NMSSM below

the confinement scale. Before analyzing this further it is important to determine what range

of λ will be most useful for our purposes. The value of the higgs quartic can be found by

running the λ coupling from the compositeness scale down to the electroweak scale (µ). We

can solve for λ in Eq. 3.2 by ignoring all except the λ3 term to obtain:

λ(µ)2 =

(1

λ(Λ)2+

1

2π2ln

Λ

µ

)−1

. (3.16)

We summarize the resulting running in Figure 3.1, in which the low energy value λ(µ)

is plotted as a function of the initial value λ(Λ), for Λ/µ of different orders of magnitude.

Notice that the value of λ at low energies is largely insensitive to its value at the confinement

scale for λ(Λ) & 3; it is this crucial feature that allows this model to compare favorably

with the Fat Higgs. Unlike the Fat Higgs however, we do not have to start in the limit

of strong coupling to get λ(µ) parameterically higher than the NMSSM bound of 0.8 [29].

In the analysis that follows we will arbitrarily choose as our region of interest λ(µ) & 1.5,

which translates to λ(Λ) & (1.8, 2.2, 3.3) for running over 1, 2, and 3 decades respectively.

Returning to the first scenario in which there is a new superconformal fixed point,

we can now relate the above values of λ to the fixed point values of λ1,2. Using Eq.


2 4 6 8 10 12λHΛL0.5

1

1.5

2

2.5

λHµLΛ/µ = 1000

Λ/µ = 10

Λ/µ = 100

Figure 3.1: The low energy values of the λ coupling after running from the compositeness scale Λdown to the scale µ.

3.15 and assuming comparable fixed points for the two yukawas, we see that we need

λ1,2 & (3.4, 3.7, 4.5) at MX . Unfortunately, we cannot say whether the actual fixed points

satisfy this condition, although these values are at least feasible since the flatness of the

RGE running of λ(µ) means that λ does not have to equal 4π at the confinement scale. It

would be interesting to do a more detailed study to determine whether this occurs.

It is possible to be more quantitative than this in the second case by relying on

our knowledge of the model in the weak limit. Using Eq. 3.8 we see that

λ(Λ) ∼√

4λ1λ2

4π

Λ

MX. − Λ

2πMX

16π2

7γ∗ ∼

8π

7γ∗ ∼ 3.6 γ∗. (3.17)

If we start with all 10 flavors of SU(4)s we have γ∗ = −1/5 and λ(Λ) . 0.7 which is too

low to be of interest. However, integrating out the Xs leaves us with 7 flavors, which at

the fixed point gives γ∗ = −5/7 and λ(Λ) . 2.6. We saw that this gives rise to a λ that is

in the interesting range for almost 3 decades of running between the confinement scale and

the electroweak scale, suggesting that there are regions of parameter space where the low

energy λ coupling is large enough to be of interest.

We can calculate the tree level bound on the higgs mass by assuming that we are

somewhere in the region 1.5 . λ(µ) . 2 and using the NMSSM equation:

m2h ≤ m2

Z cos2 2β + λ2v2 sin2 2β/2. (3.18)

to obtain

mh . 260− 350 GeV (3.19)


which is a substantial improvement over the MSSM bound of 90 GeV. Taking the largest

λ(µ) in Figure 3.1 pushes this bound up to 490 GeV, but this is probably less generic in

the parameter space. Radiative corrections from the top sector can increase this further

although these are no longer necessary to satisfy the LEP-II bound.

We emphasize that it is rather surprising to obtain interesting results in the weak

limit bound in spite of the NDA suppression factor of 4π. This is a direct consequence

of λ not having to start off at 4π; moderately large coupling is sufficient. However, the

robustness of our conclusions in the weak limit depends on a number of O(1) unknowns

which we ignored in the above analysis. These are listed below and discussed in turn.

• the value of the factor Λ/MX

• the running of the nonrenormalizable operator in Eq. 3.4 due to gauge coupling

contributions in the region Λ ≤ E ≤MX

• the coefficient in the NDA matching that was used in Eq. 3.15

• loop-level corrections from g∗ to the coefficient of λ31,2 in Eq. 3.8.

• restrictions due to the large top yukawa.

The first tends to suppress the value of λ at low energies. The strong dynamics after flavor

decoupling suggests that this factor is close to one, but it cannot be determined exactly

since we do not have detailed information on the fixed point value and exact running of

gs below MX . It might, however, be compensated for by the effect of the second which

enhances λ, hence we might be able to make a case for neglecting them both, especially

since this allows us to make a quantitative prediction. The O(1) coefficient in the third item

parametrizes our ignorance of the physics of strong coupling and unfortunately cannot be

eliminated. The fourth point is that we ignored higher order gauge corrections to the λ31,2

term in Eq. 3.8 at the gauge coupling fixed point. If the coefficient of this term decreases,

the upper bound on the λ coupling increases and vice versa. Notice however, that higher

loop λ1,2 corrections to the RGE are suppressed and have been rightfully ignored since the

loop suppression factor λ21,2/(16π

2) . −γ∗/7 ≤ 1/7 1. Finally, the fact that the top

yukawa is not neglible at low energies places some constraints on how large we can make λ1

without losing perturbativity for both these couplings to the GUT scale. Doing a simple

one loop analysis, for tanβ near 1 (where the gain in the tree level bound is greatest), the


λ1 fixed point is about half of the value in the above analysis which in turn halves the size of

λ(Λ). In general, we expect that there is some O(1) suppression from this effect, but there

is no comparable suppression in λ2 due to the smallness of the bottom yukawa. Although it

is unfortunate that these factors cannot be evaluated to determine a more specific bound,

that the naive answer is in the interesting range suggests that the actual value of λ can be

similarly large.

Since we were motivated to explore this model by concerns of naturalness, we will

now discuss how this scenario helps the fine tuning. First of all, the higgs mass bound has

increased so it is no longer necessary for the top squarks to be made heavy to evade the

LEP-II bound. In fact, it is now possible for all the MSSM scalars including the higgs to

have masses that are of the same order. Thus, from a bottom-up perspective, there are

no unnatural hierarchies in these masses.3 On the other hand there is a new fine tuning

introduced in the weak limit (the second scenario), since the UV initial conditions have to

be precisely tuned to avoid breaking conformality. However, these parameters are at least

technically natural and so could still have the right size. There is no such fine tuning in the

new superconformal phase since the attractive IR fixed points reduce the sensitivity to UV

initial conditions. For further discussion of how a larger higgs quartic coupling helps the

fine tuning issue see [37] and Casas et. al. in [27].

3.2.2 Gauge Coupling Unification

In both the Fat Higgs and the New Fat Higgs SUSY guarantees that running the

SM gauge couplings through the strong coupling regions does not give corrections larger than

typical threshold effects. We will recount the argument here for completeness. Matching

holomorphic couplings of a high energy theory containing a massive field with those of a low

energy theory with the field integrated out, is constrained by holomorphy. In particular,

the matching depends only on the bare mass of the field and thus is not affected by strong

dynamics [38, 39, 40, 41, 42]. For instance, taking MX = MX = M at the cutoff MGUT,

the high and low energy SM gauge couplings (with and without the X, X respectively) are

3It could be argued that the top-down approach is still problematic since starting with universal scalar andgaugino masses (m0 and m 1

2

) at the unification scale, for example, force the top squarks to be heavy given

observational lower bounds on chargino and slepton masses. This is a property of current SUSY breakingscenarios, however, and it is possible to imagine alternatives with more random boundary conditions at theGUT scale that result in realistic particle spectra with light top squarks.


matched at the bare mass M :

gsm, le(M) = gsm, he(M) (3.20)

where the high energy gauge couplings have their unified value at MGUT. At other energies

these holomorphic couplings are determined by their one loop running (with beta functions

bi, le = bi,MSSM and bi, he = bi, le + 4). However, during this running the coefficients of the

matter kinetic terms (Z) can change. Thus to reach a more “physical” coupling, one should

go to canonical normalization for the matter fields. This rescaling is anomalous and relates

the couplings by

8π2

g2le, phys

=8π2

g2le

−∑

i

T i lnZi (3.21)

where i only runs over the matter fields in the low energy theory and the Tis are their

Dynkin indices. All potential strong coupling effects are contained within the Zis of the

low energy fields. As a matter of fact, there is actually no effect due to the RGE splitting

MX < MX , since the matching in Eq. 3.20 of the low energy couplings occurs at M , giving

no restriction on the ratio of these masses from unification. An order one lnZi gives a

contribution of the order of a typical theshold correction; thus it takes exponentially large

Zi to adversely affect unification. In this model, such large Zi can only occur for the higgses

when the λ1,2 couplings are strong for an exponentially large region. Thus, the weak limit

case is generically safe whereas in the new superconformal phase, the conformal region

for λ1,2 cannot be exponentially large without affecting unification. Note that a similar

constraint applies to the conformal region in the Fat Higgs model.

Aside from this potential constraint, gauge coupling unification occurs naturally

in this theory since the additional matter is charged under the SM in complete SU(5) multi-

plets and because the higgses are elementary (hence the beta functions of the SM couplings

are equivalent to those of the MSSM up to SU(5) symmetric terms as detailed earlier). In

comparison, the Fat Higgs model had elementary preons which correctly reproduced the

running of the higgs doublets above the compositeness scale, but also contained additional

fields which were put into both split GUT and non-GUT multiplets in order to restore unifi-

cation. In that model, explaining why unification is natural requires a setup that generates

the additional matter content as well as the required mass spectrum.


3.2.3 Phenomenology

Much of the phenomenology in this model is similar to the Fat Higgs. In both

theories the physics at the TeV scale is NMSSM-like with a linear term in S but no cubic.

The low energy λ coupling is large and gets strong before the GUT scale, but some asymp-

totically free dynamics takes over to UV complete the theory. They both have similar higgs

spectra which are in concordance with precision electroweak constraints. Also, the analysis

in [43] which concludes that UV insensitive Anomaly Mediation works in the Fat Higgs

should also apply to this model.

One notable difference between the two models is the additional baryon physics

in our model. The B0 and Bc 0 in this theory get a large supersymmetric mass from the S

vev and are not problematic. However we also have light baryon states, the four B is and

Bc is that are married to the TBi s and TBc

i s, with supersymmetric masses of order

MB ∼Λ3

4πM2GUT

∼ 10−13 − 10−7 eV (3.22)

for Λ ∼ 5 − 500 TeV. The scalar components of these chiral superfields get TeV sized

soft masses from SUSY breaking and it is possible to determine these from the masses of

the elementary fields using the the techniques in [44, 45]. The fermionic components are

more worrying since they remain light and thus give rise to some stringent cosmological

constraints. For instance, they decouple at a Tdec ∼ 10GeV, requiring Treheat . Tdec in

order to be consistent with Big Bang Nucleosynthesis. Whether the LSP in this theory

is a good Dark Matter candidate given such a low reheat temperature is also not clear.

However, the reason these fermions decouple late is due to the MIJBIBc J coupling in

Wdyn. Thus, if the scenario with the Quantum Modified Moduli Space could be made to

work, the light baryons would not couple to the Standard Model and there would no longer

be any cosmological problems.

It is also possible to circumvent the issues raised by these light fermions by making

models without baryons, with an Sp(2) ≡ SO(5) theory, for example, starting with 18

fundamentals of the Sp(2). Integrating out the X, X will reduce to the s-confining case

with 8 fundamentals. At high energies, this has a vanishing one loop beta function but

is not asymptotically free at two loops. With all 18 fundamentals and their yukawas, the

analysis in [34] suggests that there is a superconformal fixed point for the yukawa and gauge

couplings. Specifically there is a linear family of fixed points which run through the free


fixed point (g = 0, λi = 0) (see Footnote 2) and it needs to be determined whether the fixed

point values of λ1,2 are large enough to be in the interesting range. We can also work in

a limit analogous to our weak limit of the previous section, integrating out the Xs first;

this leaves the group in the conformal window with 12 fundamentals. Thus if MX/MX is

small enough the theory can run to Seiberg’s strong conformal fixed point before the Xs

are integrated out. In this case, the weak limit bound gives λ(Λ) . 1.8, so we would need

MX near the weak scale or some help from the unknown order one contributions detailed

above. However, since there are no baryons in Sp(n) theories we only have to to decouple

the extra mesons. From this reasoning we see that the physics associated with the baryons

does not appear generic to all implementations of our mechanism and thus cannot be used

to rule out all models of this type.


Supersymmetry does extremely well in solving the hierarchy problem, but as more

precise measurements have told us, the minimal implementation of weak scale supersym-

metry (the MSSM) is becoming fine tuned at about the percent level. Approaches that

attempt to alleviate this problem have been many and varied, all of which have their own

advantages and disadvantages. Led by the positive aspects of the MSSM, we analyzed a UV

complete NMSSM model which justifies the presence of a large λ at low energies, resulting

in a similarly large higgs quartic coupling. We did this by splitting the λ coupling into

two asymptotically free yukawa couplings, allowing the theory to be continued above the

apparent strong coupling scale. The simple model pursued in this chapter is similar in spirit

to the Fat Higgs model: we started at the electroweak scale with a large λ coupling which

grows with increasing energy scale. Rather than waiting for it to hit 4π before UV complet-

ing, we did this at a lower scale, leaving a theory with a composite S only (see Figure 3.2).

There was no need for a dynamically generated superpotential because the induced λ cou-

pling never became non-perturbative; instead moderately strong coupling was sufficient to

achieve a large tree level higgs mass bound without making the higgs fields composite. This

resulted in a higgs that was not as fat as in the Fat Higgs, but gauge coupling unification,

arguably the best evidence for weak scale SUSY, was naturally maintained.

We did not study in depth the potentially interesting scenario where the theory

hit a superconformal fixed point, since it was tricky to make any definitive statements


1 2 3 4

2

4

6

8

10

12

log Λ/µ

λ

Fat Higgs:

Composite S,

Hu, H

d

Composite S

New Fat Higgs:

Elementary H du, H

Figure 3.2: A comparison of UV completion scales in the Fat Higgs and the New Fat Higgs

about the fixed point values of λ1,2.4 The strong coupling dynamics also made it difficult

to give exact results in the second case we considered, but we were able to set a reasonable

upper bound on λ at low energies, up to some unknown order one coefficients, using the

properties of Seiberg’s fixed point and superconformality in the weak limit. That this bound

turned out to give large enough λ is comforting, since it suggests the possibility of realizing

our mechanism for a generic parameter space with similar results. However, to say any

more requires a detailed understanding of both the RGE equations at strong coupling and

matching at the confinement scale.

Finally, we discussed some of the implications of our model. We saw that the

fine tuning issue was indeed ameliorated, at least from a bottom-up perspective and that

unification was not affected by the strong coupling. We also discussed the equivalence of

the phenomenology to that of the Fat Higgs Model in that there was little difference in their

higgs spectra or compatibility with precision electroweak constraints. One notable difference

was the presence of light fermionic baryons in our theory. It would be interesting to analyze

the new baryon physics in more detail, especially since there are interesting cosmological

constraints. However, the existence of models which do not have baryons suggest that these

light states are not generic to this framework.

Using naturalness as a guideline, it already seems that the simplest SUSY models

4We are currently looking into a potential AdS dual to this theory which would allow us to do this.


are fine-tuned, which motivates us to attempt to generalize them. With this intuition we

have analyzed a theory which improves the naturalness of weak scale SUSY in a simple way

without losing the natural unification of the MSSM.

Chapter 4

The Minimal Model for Dark

Matter and Unification

The effective lagrangian of the SM contains two relevant parameters: the higgs

mass and the cosmological constant (c.c), both of which give rise to problems concerning

the interpretation of the low energy theory. Any discussion of large discrepancies between

expectation and observation must begin with what is known as the c.c. problem. This

relates to our failure to find a well-motivated dynamical explanation for the factor of 10120

between the observed c.c and the naive contribution to it from renormalization which is

proportional to Λ4, where Λ is the cutoff of the theory, usually taken to be equal to the

Planck scale. Until very recently there was still hope in the high energy physics community

that the c.c. might be set equal to zero by some mysterious symmetry of quantum gravity.

This possibility has become increasingly unlikely with time since the observation that our

universe is accelerating strongly suggests the presence of a non-zero cosmological constant

[46, 47, 48].

Both this issue and the gauge hierarchy problem can be understood from a different

perspective: the fact that the c.c. and the higgs mass are relevant parameters means that

they dominate low energy physics, allowing them to determine very gross properties of the

effective theory. We might therefore be able to put limits on them by requiring that this

theory satisfy the environmental conditions necessary for the universe not to be empty. This

approach was first used by Weinberg [49] to deduce an upper bound on the cosmological

constant from structure formation, and was later employed to solve the hierarchy problem

37

38 Chapter 4: The Minimal Model for Dark Matter and Unification

in an analogous way by invoking the atomic principle [50].

Potential motivation for this class of argument can be found in the string theory

landscape. At low energies some regions of the landscape can be thought of as a field theory

with many vacua, each having different physical properties. It is possible to imagine that

all these vacua might have been equally populated in the early universe, but observers can

evolve only in the few where the low energy conditions are conducive to life. The number

of vacua with this property can be such a small proportion of the total as to dwarf even

the tuning involved in the c.c. problem; resolving the hierarchy problem similarly needs

no further assumptions. This mechanism for dealing with both issues simultaneously by

scanning all relevant parameters of the low energy theory within a landscape was recently

proposed in [51, 52].

From this point of view there seems to be no fundamental inconsistency with

having the SM be the complete theory of our world up to the Planck scale; nevertheless

this scenario presents various problems. Firstly there is increasing evidence for dark matter

(DM) in the universe, and current cosmological observations fit well with the presence of

a stable weakly interacting particle at around the TeV scale. The SM contains no such

particle. Secondly, from a more aesthetic viewpoint gauge couplings do not quite unify

at high energies in the SM alone; adding weakly interacting particles changes the running

so unification works better. A well-motivated example of a model that does this is Split

Supersymmetry [51], which is however not the simplest possible theory of this type. In

light of this we study the minimal model with a finely-tuned higgs and a good thermal

dark matter candidate proposed in [52], which also allows for gauge coupling unification.

Although a systematic analysis of the complete set of such models was carried out in [53], the

simplest one we study here was missed because the authors did not consider the possibility

of having large UV threshold corrections that fix unification, as well as a GUT mechanism

suppressing proton decay.

Adding just two ‘higgsino’ doublets1 to the SM improves unification significantly.

This model is highly constrained since it contains only one new parameter, a Dirac mass

term for the doublets (‘µ’), the neutral components of which make ideal DM candidates

for 990 GeV. µ . 1150 GeV (see [53] for details). However a model with pure higgsino

dark matter is excluded by direct detection experiments since the degenerate neutralinos

1Here ‘higgsino’ is just a mnemonic for their quantum numbers, as these particles have nothing to dowith the SUSY partners of the higgs.

Chapter 4: The Minimal Model for Dark Matter and Unification 39

have unsuppressed vector-like couplings to the Z boson, giving rise to a spin-independent

direct detection cross-section that is 2-3 orders of magnitude above current limits2 [54, 55].

To circumvent this problem, it suffices to include a singlet (‘bino’) at some relatively high

energy (. 109 GeV), with yukawa couplings with the higgsinos and higgs, to lift the mass

degeneracy between the ‘LSP’ and ‘NLSP’3 by order 100 keV [56], as explained in Appendix

B. The instability of such a large mass splitting between the higgsinos and bino to radiative

corrections, which tend to make the higgsinos as heavy as the bino, leads us to consider

these masses to be separated by at most two orders of magnitude, which is technically

natural. We will see that the yukawa interactions allow the DM candidate to be as heavy

as 2.2 TeV. There is also a single reparametrization invariant CP violating phase which

gives rise to a two-loop contribution to the electron EDM that is well within the reach of

next-generation experiments.

This chapter is organized as follows: in Section 4.1 we briefly introduce the model,

in Section 4.2 we study the DM relic density in different regions of our parameter space with

a view to constraining these parameters; we look more closely at the experimental implica-

tions of this model in the context of dark matter direct detection and EDM experiments in

Sections 4.3 and 4.4. Next we study gauge coupling unification at two loops. We find that

this is consistent modulo unknown UV threshold corrections, however the unification scale

is too low to embed this model in a simple 4D GUT. This is not necessarily a disadvantage

since 4D GUTs have problems of their own, in splitting the higgs doublet and triplet for ex-

ample. A particularly appealing way to solve all these problems is by embedding our model

in a 5D orbifold GUT, in which we can calculate all large threshold corrections and achieve

unification. We also find a particular model with b-τ unification and a proton lifetime just

above current bounds. We discuss our results in Section 4.6.

4.1 The Model

As mentioned above, the model we study consists of the SM with the addition of

two fermion doublets with the quantum numbers of SUSY higgsinos, plus a singlet bino,

2A model obtained adding a single higgsino doublet, although more minimal, is anomalous and hence isnot considered here.

3From here on we will refer to these particles and couplings by their SUSY equivalents without thequotation marks for simplicity.


with the following renormalizable interaction terms:

µΨuΨd +1

2M1ΨsΨs + λuΨuhΨs + λdΨdh

†Ψs (4.1)

where Ψs is the bino, Ψu,d are the higgsinos, h is the finely-tuned higgs.

We forbid all other renormalizable couplings to SM fields by imposing a parity

symmetry under which our additional particles are odd whereas all SM fields are even. As

in SUSY conservation of this parity symmetry implies that our LSP is stable.

The size of the yukawa couplings between the new fermions and the higgs are lim-

ited by requiring perturbativity to the cutoff. For equal yukawas this constrains λu(MZ) =

λd(MZ) ≤ 0.88, while if we take one of the couplings to be small, say λd(MZ) = 0.1 then

λu(MZ) can be as large as 1.38.

The above couplings allow for the CP violating phase θ = Arg(µM1λ∗uλ

∗d), giving

5 free parameters in total. In spite of its similarity to the MSSM (and Split SUSY) weak-

ino sector, there are a number of important differences which have a qualitative effect

on the phenomenology of the model, especially from the perspective of the relic density.

Firstly a bino-like LSP, which usually mixes with the wino, will generically annihilate less

effectively in this model since the wino is absent. Secondly the new yukawa couplings are

free parameters so they can get much larger than in Split SUSY, where the usual relation

to gauge couplings is imposed at the high SUSY breaking scale. This will play a crucial role

in the relic density calculation since larger yukawas means greater mixing in the neutralino

sector as well as more efficient annihilation, especially for the bino which is a gauge singlet.

Our 3×3 neutralino mass matrix is shown below:

MN =

M1 λuv λdv

λuv 0 −µeiθ

λdv −µeiθ 0

for v = 174 GeV, where we have chosen to put the CP violating phase in the µ term. The

chargino is the charged component of the higgsino with tree level mass µ.

It is possible to get a feel for the behavior of this matrix by diagonalizing it

perturbatively for small off-diagonal terms, this is done in Appendix B.


4.2 Relic Abundance

In this section we study the regions of parameter space in which the DM abundance

is in accordance the post-WMAP 2σ region 0.094 < Ωdmh2 < 0.129 [47], where Ωdm is the

fraction of today’s critical density in DM, and h = 0.72 ± 0.05 is the Hubble constant in

units of 100 km/(s Mpc).

As in Split SUSY, the absence of sleptons in our model greatly decreases the

number of decay channels available to the LSP [57, 58, 59]. Also similar to Split SUSY is

the fact that our higgs can be heavier than in the MSSM (in our case the higgs mass is

actually a free parameter), hence new decay channels will be available to it, resulting in a

large enhancement of its width especially near the WW and ZZ thresholds. This in turn

makes accessible neutralino annihilation via a resonant higgs, decreasing the relic density

in regions of the parameter space where this channel is accessible. For a very bino-like LSP

this is easily the dominant annihilation channel, allowing the bino density to decrease to an

acceptable level. We use a modified version of the DarkSUSY [60, 61, 62] code for our relic

abundance calculations, explicitly adding the resonant decay of the heavy higgs to W and

Z pairs.

As mentioned in the previous section there are also some differences between our

model and Split SUSY that are relevant to this discussion: the first is that the Minimal

Model contains no wino equivalent (this feature also distinguishes this model from that in

[63], which contains a similar dark matter analysis). The second difference concerns the size

of the yukawa couplings which govern this mixing, as well as the annihilation cross-section

to higgses. Rather than being tied to the gauge couplings at the SUSY breaking scale, these

couplings are limited only by the constraint of perturbativity to the cutoff. This means that

the yukawas can be much larger in our model, helping a bino-like LSP to both mix more

and annihilate more efficiently. These effects are evident in our results and will be discussed

in more detail below.

We will restrict our study of DM relic abundance and direct detection in this model

to the case with no CP violating phase (θ = 0, π); we briefly comment on the general case in

Section 4.4. Our results for different values of the yukawa couplings are shown in Figure 4.1

below, in which we highlight the points in the µ-M1 plane that give rise to a relic density

within the cosmological bound. The higgs is relatively heavy (Mhiggs = 160 GeV) in this

plot in order to access processes with resonant annihilation through an s-channel higgs. As


we will explain below the only effect this has is to allow a low mass region for a bino-like

LSP with M1 ∼ Mhiggs/2. Notice that the relic abundance seems to be consistent with a

0 500 1000 1500 2000 2500 30000

1000

2000

3000

sgn(mu)=+, mh=159.5, lu=0.1, ld=0.1sgn(mu)=-, mh=159.5, lu=0.1, ld=0.1sgn(mu)=+1,mh=159.5,lu=0.88,ld=0.88sgn(mu)=-, mh=159.5, lu=0.88,ld=0.88sgn(mu)=+, mh=159.5,lu=0.1,ld=1.38sgn(mu)=-, mh=159.5, lu=1.39, ld=0

M1

(GeV

)

µ (GeV)

annihilates too much annihilates too little

Higgsino LSP

Bino LSP

θ=0, λu=1.38, λd=0.1

θ=π, λu=0.1, λd=0.1

θ=0, λu=0.1, λd=0.1

θ=π, λu=1.38, λd=0.1

θ=π, λu=0.88, λd=0.88

θ=0, λu=0.88, λd=0.88

Figure 4.1: Graph showing regions of parameter space consistent with WMAP.

dark matter mass as large as 2.2 TeV. Although a detailed analysis of the LHC signature

of this model is not within the scope of this paper, it is clear that a large part of this

parameter space will be inaccessible at LHC. The pure higgsino region for example, will

clearly be hard to explore since the higgsinos are heavy and also very degenerate. There is

more hope in the bino LSP region for a light enough spectrum.

While analyzing these results we must keep in mind that Ωdm ∼ 10−9GeV−2/〈σ〉eff ,

where 〈σ〉eff is an effective annihilation cross section for the LSP at the freeze out temper-

ature, which takes into consideration all coannihilation channels as well as the thermal

average [64]. It will be useful to approximate this quantity as the cross-section for the dom-

inant annihilation channel. Although rough, this approximation will help us build some

intuition on the behavior of the relic density in different parts of the parameter space. We


1M

(G

eV)

µ (GeV)

0.998

0.90.8

1000 2000 30000

−810

~ 0

3000

2000

1000

0.970.98

0.99

0.994

0.996

0.997

(a) θ=01

M

(GeV

)

µ (GeV)

0.03

0.01

0.1

0.50.8

0.9

0.97

3000200010000

3000

2000

1000

0.99

(b) θ=π

Figure 4.2: Gaugino fraction contours for λu = λd = 0.88 and θ=0, π

will not discuss the region close to the origin where the interpretation of the results become

more involved due to large mixing and coannihilation.

4.2.1 Higgsino Dark Matter

In order to get a feeling for the structure of Figure 4.1, it is useful to begin

by looking at the regions in which the physics is most simple. This can be achieved by

diminishing the the number of annihilation channels that are available to the LSP by taking

the limit of small yukawa couplings.

For θ = 0, mixing occurs only on the diagonal M1 = µ to a very good approxima-

tion (see Appendix B and Figure 4.2), hence the region above the diagonal corresponds to

a pure higgsino LSP with mass µ. For λu = λd = 0.1 the yukawa interactions are irrelevant

and the LSP dominantly annihilates by t-channel neutral (charged) higgsino exchange to

ZZ (WW ) pairs. Charginos, which have a tree-level mass µ and are almost degenerate

with the LSP, coannihilate with it, decreasing the relic density by a factor of 3. This fixes

the LSP mass to be around µ = 1 TeV, giving rise to the wide vertical band that can be

seen in the figure; for smaller µ the LSP over-annihilates, for larger µ it does not annihilate


enough.

Increasing the yukawa couplings increases the importance of t-channel bino ex-

change to higgs pairs. Notice that taking the limit M1 µ makes this new interaction

irrelevant, therefore the allowed region converges to the one in which only gauge interac-

tions are effective. Taking this as our starting point, as we approach the diagonal the mass

of the bino decreases, causing the t-channel bino exchange process to become less suppressed

and increasing the total annihilation cross-section. This explains the shift to higher masses,

which is more pronounced for larger yukawas as expected and peaks along the diagonal

where the higgsino and bino are degenerate and the bino mass suppression is minimal. The

increased coannihilation between higgsinos and binos close to the diagonal does not play a

large part here since both particles have access to a similar t-channel diagram.

Taking θ = π makes little qualitative difference when either of the yukawas is small

compared to M1 or µ, since in this limit the angle is unphysical and can be rotated away by

a redefinition of the higgsino fields. However we can see in Figure 4.2 that for large yukawas

the region above the diagonal M1 = µ changes to a mixed state, rather than being pure

higgsino as before. Starting again with the large M1 limit and decreasing M1 decreases the

mass suppression of the t-channel bino exchange diagram like in the θ = 0 case, but the

LSP also starts to mix more with the bino, an effect that acts in the opposite direction and

decreases 〈σ〉eff . This effect happens to outweigh the former, forcing the LSP to shift to

lower masses in order to annihilate enough.

With θ = π and yukawas large enough, there is an additional allowed region for

µ < MW . In this region the higgsino LSP is too light to annihilate to on-shell gauge bosons,

so the dominant annihilation channels are phase-space suppressed. Furthermore if the

splitting between the chargino and the LSP is large enough, the effect of coannihilation with

the chargino into photon and on-shell W is Boltzmann suppressed, substantially decreasing

the effective cross-section, and giving the right relic abundance even with such a light

higgsino LSP. Although acceptable from a cosmological standpoint, this region is excluded

by direct searches since it corresponds to a chargino that is too light.

4.2.2 Bino Dark Matter

The region below the diagonal M1 = µ corresponds to a bino-like LSP. Recall that

in the absence of yukawa couplings pure binos in this model do not couple to anything and


hence cannot annihilate at all. Turning on the yukawas allows them to mix with higgsinos

which have access to gauge annihilation channels. For λu = λd = 0.1 this effect is only

large enough when M1 and µ are comparable (in fact when they are equal, the neutralino

states are maximally mixed for arbitrarily small off-diagonal terms), explaining the stripe

near the diagonal in Figure 4.1. Once µ gets larger than ∼ 1 TeV even pure higgsinos are

too heavy to annihilate efficiently; this means that mixing is no longer sufficient to decrease

the dark matter relic density to acceptable values and the stripe ends.

Increasing the yukawas beyond a certain value (λu = λd = 0.88, which is slightly

larger than their values in Split SUSY, is enough), makes t-channel annihilation to higgses

become large enough that a bino LSP does not need to mix at all in order to have the

correct annihilation cross-section. This gives rise to an allowed region which is in the

shape of a stripe, where for fixed M1 the correct annihilation cross-section is achieved

only for the small range of µ that gives the right t-channel suppression. As M1 increases

the stripe converges towards the diagonal in order to compensate for the increase in LSP

mass by increasing the cross-section. Once the diagonal is reached this channel cannot be

enhanced any further, and there is no allowed region for heavier LSPs. In addition the cross-

section for annihilation through an s-channel resonant higgs, even though CP suppressed

(see Section 4.4 for details), becomes large enough to allow even LSPs that are very pure

bino to annihilate in this way. The annihilation rate for this process is not very sensitive to

the mixing, explaining the apparent horizontal line at M1 = 12Mhiggs ∼ 80 GeV. This line

ends when µ grows to the point where the mixing is too small.

As in the higgsino case, taking θ = π changes the shape of the contours of constant

gaugino fraction and spreads them out in the plane (see Figure 4.2), making mixing with

higgsinos relevant throughout the region. For small M1, the allowed region starts where the

mixing term is small enough for the combination of gauge and higgs channels not to cause

over-annihilation. Increasing M1 again makes the region move towards the diagonal, where

the increase in LSP mass is countered by increasing the cross-section for the gauge channel

from mixing more.

For either yukawa very large (λu = 1.38, λd = 0.1), annihilation to higgses via t-

channel higgsinos is so efficient that this process alone is sufficient to give bino-like LSPs the

correct abundance. As M1 increases the allowed region again moves towards the diagonal

in such a way as to keep the effective cross-section constant by decreasing the higgsino mass

suppression, thus compensating for the increase in LSP mass. As we remarked earlier since


λd is effectively zero in this case, the angle θ is unphysical and can be rotated away by a

redefinition of the higgsino fields.

4.3 Direct Detection

Dark matter is also detectable through elastic scatterings off ordinary matter. The

direct detection cross-section for this process can be divided up into a spin-dependent and

a spin-independent part; we will concentrate on the former since it is usually dominant.

As before we restrict to θ = 0 and π, we expect the result not to change significantly for

intermediate values.

The spin-independent interaction takes place through higgs exchange, via the

yukawa couplings which mix higgsinos and binos. Since the only χ01χ

01h term in our model

involves the product of the gaugino and higgsino fractions, the more mixed our dark mat-

ter is the more visible it will be to direct detection experiments. This effect can be seen

in Fig 4.3 below. Although it seems like we cannot currently use this measure as a con-

straint, the major proportion of our parameter space will be accessible at next-generation

experiments. Since higgsino LSPs are generally more pure than bino-type ones, the former

will escape detection as long as there is an order 100 keV splitting between its two neutral

components. This is is necessary in order to avoid the limit from spin-independent direct

detection measurements [56].

Also visible in the graph are the interesting discontinuities mentioned in [57], cor-

responding to the opening up of new annihilation channels at MLSP = 1/2Mhiggs through

an s-channel higgs. We also notice a similar discontinuity at the top threshold from anni-

hilation to tt; this effect becomes more pronounced as the new yukawa couplings increase.

4.4 Electric Dipole Moment

Since our model does not contain any sleptons it induces an electron EDM only

at two loops, proportional to sin(θ) for θ as defined above. This is a two-loop effect, we

therefore expect it to be close to the experimental bound for O(1) θ. The dominant diagram

responsible for the EDM is generated by charginos and neutralinos in a loop and can be seen

in Figure 4.4 below. This diagram is also present in Split SUSY where it gives a comparable

contribution to the one with only charginos in the loop [65, 66]. The induced EDM is (see


0 500 1000 1500 2000 2500 30001e-54

1e-51

1e-48

1e-45

1e-42

blahblahblahblah

Proposed bound from

Current bound

next−generation experiments

Mχ (GeV)

σ(χ

-nucl

eon)

(cm

2)

θ=0, λu=0.88, λd=0.88θ=π, λu=0.88, λd=0.88

θ=0, λu=1.38, λd=0.1

θ=π, λu=0.1, λd=0.1

Figure 4.3: Spin-independent part of dark matter direct detection cross-section

[65]):

dWfe

= ± α2mf

8π2s4WM2W

3∑

i=1

mχiµ

M2W

Im (OLi O

R∗i )G

(r0i , r

±)(4.2)

where

G(r0i , r

±)=

∫ ∞

0dz

∫ 1

0

dγ

γ

∫ 1

0dy

y z (y + z/2)

(z + y)3(z +Ki)

=

∫ 1

0

dγ

γ

∫ 1

0dy y

[(y − 3Ki)y + 2(Ki + y)y

4y(Ki − y)2+Ki(Ki − 2y)

2(Ki − y)3lnKi

y

]

and

Ki =r0i

1− γ +r±

γ, r± ≡ µ2

M2W

, r0i ≡m2χi

M2W

,

ORi =√

2N∗2i exp−iθ, OLi = −N3i


ff’

W W

w j+

iχ0

Figure 4.4: The 2-loop contribution to the EDM of a fermion f.

NTMNN = diag(mχ1,mχ2

,mχ3) with real and positive diagonal elements. The sign on the

right-hand side of equation (4.2) corresponds to the fermion f with weak isospin ± 12 and

f ′ is its electroweak partner.

In principle it should be possible to cross-correlate the region of our parameter

space which is consistent with relic abundance measurements, with that consistent with

electron EDM measurements in order to further constrain our parameters. However since

the current release of DarkSUSY does not support CP violating phases and a version in-

cluding CP violations seems almost ready for public release4 we leave an accurate study of

the consequences of non-zero CP phase in relic abundance and direct detection calculations

for a future work. We can still draw some interesting conclusions by estimating the effect of

non-zero CP phase. Because there is no reason for these new contributions to be suppressed

with respect to the CP-conserving ones (for θ of O(1)), we might naively expect their in-

clusion to enhance the annihilation cross-section by around a factor of 2, increasing the

acceptable LSP masses by ∼√

2 for constant relic abundance. This is discussed in greater

detail in [67] (and [68] for direct detection) in which we see that this observation holds for

most of the parameter space. We must note, however, that in particular small regions of

the space the enhancement to the annihilation cross-section and the suppression to the elas-

tic cross section can be much larger, justifying further investigation of this point in future

work. With this assumption in mind we see in Figure 4.5 that although the majority of our

allowed region is below the current experimental limit of de < 1.7× 10−27e cm at 95% C.L.

[69], most of it will be accessible to next generation EDM experiments. These propose to

improve the precision of the electron EDM measurement by 4 orders of magnitude in the

next 5 years, and maybe even up to 8 orders of magnitude, funding permitting [70, 71, 72].

4Private communication with one of the authors of DarkSUSY.


We also see in this figure that CP violation is enhanced on the diagonal where the mixing

is largest. This is as expected since the yukawas that govern the mixing are necessary for

there to be any CP violating phase at all. For the same reason, decoupling either particle

sends the EDM to zero.

0 500 1000 1500 2000 2500 30000

1000

2000

3000sgn(mu)=+1,mh=159.5,lu=0.88,ld=0.88sgn(mu)=-, mh=159.5, lu=0.88,ld=0.88

µ (GeV)

M1

(GeV

)

1 × 10−28

1 × 10−27

1.7 × 10−27

2.2 × 10−27

3 × 10−27

θ=π, λu=0.88, λd=0.88

θ=0, λu=0.88, λd=0.88

Figure 4.5: Electron edm contours for maximal θ. The excluded region is bounded by the blackcontours

4.5 Gauge Coupling Unification

In this section we study the running of gauge couplings in our model at two loops.

The addition of higgsinos largely improves unification as compared to the SM case, but their

effect is still not large enough and the model predicts a value for αs(Mz) around 9σ lower

than the experimental value of αs(Mz) = 0.119 ± 0.002 [1]. Moreover the scale at which

the couplings unify is very low, around 1014 GeV, making proton decay occur much too

quickly to embed in a simple GUT theory5. These problems can be avoided by adding the

5It is possible to evade the constraint from proton decay by setting some of the relevant mixing parametersto zero [73]. However we are not aware of any GUT model in which such an assertion is justified by symmetry


Split SUSY particle content at higher energies, as in [74], at the cost of losing minimality;

instead we choose to solve this problem by embedding our minimal model in an extra

dimensional theory. This decision is well motivated: even though normal 4D GUTs have

had some successes, explaining the quark-lepton mass relations for example, and charge

quantization in the SM, there are many reasons why these simple theories are not ideal. In

spite of the fact that the matter content of the SM falls naturally into representations of

SU(5), there are some components that seem particularly resistant to this. This is especially

true of the higgs sector, the unification of which gives rise to the doublet-triplet splitting

problem. Even in the matter sector, although b-τ unification works reasonably well the

same cannot be said for unification of the first two generations. In other words, it seems

like gauge couplings want to unify while the matter content of the SM does not, at least

not to the same extent. This dilemma is easily addressed in an extra dimensional model

with a GUT symmetry in the bulk, broken down to the SM gauge group on a brane by

boundary conditions [75, 76, 77, 78, 79, 80, 81] since we can now choose where we put fields

based on whether they unify consistently or not. Unified matter can be placed in the bulk

whereas non-unified matter can be placed on the GUT-breaking brane. The low energy

theory will then contain the zero modes of the 5D bulk fields as well as the brane matter.

While solving many of the problems of standard 4D GUTs these extra dimensional theories

have the drawback of having a large number of discrete choices for the location of the matter

fields, as we shall see later.

We will consider a model with one flat extra dimension compactified on a circle of

radius R, with orbifolds S1/(Z2 × Z ′2), whose action is obtained from the identifications

Z2 : y ∼ 2π R− y, Z ′2 : y ∼ πR− y, y ε [0, 2π] (4.3)

where y is the coordinate of the fifth dimension. There are two fixed points under this

action, at (0, πR) and (πR/2, 3πR/2), at which are located two branes. We impose an

SU(5) symmetry in the bulk and on the y = 0 brane; this symmetry is broken down to the

SM SU(3)× SU(2)×U(1) on the other brane by a suitable choice of boundary conditions.

All fields need to have definite transformation properties under the orbifold action - we

choose the action on the fundamental to be φ → ±Pφ and on the adjoint, ±[P,A], for

projection operators PZ = (+,+,+,+,+) and PZ′ = (+,+,+,−,−). This gives SM gauge

fields and their corresponding KK towers Aaµ for a = 1, ..., 12 (+,+) parity; and the towers

arguments.


Aaµ for a = 13, ..., 24 (+,−) parity, achieving the required symmetry-breaking pattern. By

gauge invariance the unphysical fifth component of the gauge field, which is eaten in unitary

gauge, gets opposite boundary conditions.6 We still have the freedom to choose the location

of the matter fields. In this model SU(5)-symmetric matter fields in the bulk will get split

by the action of the Z ′ orbifold: the SM 5 for instance will either contain a massless dc or

a massless l, with the other component only having massive modes. Matter fields in the

bulk must therefore come in pairs with opposite eigenvalues under the orbifold projections,

so for each SM generation in the bulk we will need two copies of 10 + 5. This provides us

with a simple mechanism to forbid proton decay from X- and Y -exchange and also to split

the color triplet higgs field from the doublet. To summarize, unification of SM matter fields

in complete multiplets of SU(5) cannot be achieved in the bulk but on the SU(5) brane,

while matter on the SM brane is not unified into complete GUT representations.

4.5.1 Running and matching

We run the gauge couplings from the weak scale to the cutoff Λ by treating our

model as a succession of effective field theories (EFTs) characterized by the differing particle

content at different energies. The influence of the yukawa couplings between the higgsinos

and the singlet on the two-loop running is negligible, hence it is fine to assume that the

singlet is degenerate with the higgsinos so there is only one threshold from Mtop to the

compactification scale 1/R, at which we will need to match with the full 5D theory.

The SU(5)-symmetric bulk gauge coupling g5 can be matched on to the low energy

couplings at the renormalization scale M via the equation

1

g2i (M)

=2πR

g25

+ ∆i(M) + λi(MR) (4.4)

The first term on the right represents a tree level contribution from the 5d kinetic term,

∆i are similar contributions from brane-localized kinetic terms and λi encode radiative

contributions from KK modes. The latter come from renormalization of the 4D brane

kinetic terms which run logarithmically as usual.

6From an effective field theory point of view an orbifold is not absolutely necessary, our theory cansimply be thought of as a theory with a compact extra dimension on an interval, with two branes on theboundaries. Because of the presence of the boundaries we are free to impose either Dirichlet or Neumannboundary conditions for the bulk field on each of the branes breaking the SU(5) to the SM gauge grouppurely by choice of boundary conditions and similarly splitting the multiplets accordingly. Our orbifoldprojection is therefore nothing more than a further restriction to the set of all possible choices we can make.


To understand this in more detail let us consider radiative corrections to a U(1)

gauge coupling in an extra dimension compactified on a circle with no orbifolds, due to a

5D massless scalar field [82]. Since 1/g25 has mass dimension 1, by dimensional analysis

we might expect corrections to it to go like Λ +m log Λ where m is some mass parameter

in the theory. The linearly divergent term is UV sensitive and can be reabsorbed into the

definition of g5, whereas the log term cannot exist since there is no mass parameter in

the theory. Hence the 5D gauge coupling does not run, and neither does the 4D gauge

coupling. This can also be interpreted from a 4D point of view, where the KK partners of

the scalar cut off the divergences of the zero mode. Since there is no distinction between

the wavefunctions for even (cosine) and odd (sine) KK modes in the absence of an orbifold,

and we know that the sum of their contributions must cancel the log divergence of the 4D

massless scalar, each of these must give a contribution equal to −1/2 times that of the 4D

massless scalar.

When we impose a Z2 orbifold projection and add two 3-branes at the orbifold

fixed points, the scalar field must now transform as an eigenstate of this orbifold action

and can either be even ((+,+), with Neuman boudary conditions on the branes), or odd

((−,−), with Dirichlet boundary conditions on the branes). This restricts us to a subset

of the original modes and the cancellation of the log divergence no longer works. Since

this running can only be due to 4D gauge kinetic terms localized on the branes, where the

gauge coupling is dimensionless and can therefore receive logarithmic corrections, locality

implies that the contribution from a tower of states on a particular brane can only be due

to its boundary condition on that brane, with the total running equal to the sum of the

contributions on each brane. In fact, it is only in the vicinity of the brane that imposing a

particular boundary condition has any effect. As argued above, a (+,+) tower (excluding

the zero mode) and a (−,−) tower must each give a total contribution equal to −1/2 times

that of the zero mode, which corresponds to a coefficient of −1/4 to the running of each

brane-localized kinetic term. Taking into account the contribution of the zero mode we can

say that a tower of modes with + boundary conditions on a brane contributes +1/4 times

the corresponding 4D coefficient, while a − boundary condition contributes −1/4 times the

same quantity. This argument makes it explicit that the orbifold projection can be seen as

a prescription on the boundary conditions of the fields in the extra dimension, which only

affect the physics near each brane.

Adding another orbifold projection as we are doing in this case also allows for


towers with (+,−) and (−,+) boundary conditions which, from the above argument, both

give a contribution of ±1/4 ∓ 1/4 = 0. The contribution of the (+,+) and (−,−) towers

clearly remains unchanged.

Explicitly integrating out the KK modes at one loop at the compactification scale

allows us to verify this fact, and also compute the constant parts of the threshold corrections,

which are scheme-dependent. In DR 7 we obtain [82]:

λi(MR) =1

96π2

((bSi − 21bGi + 8bFi

)Fe(MR) +

(bSi − 21bGi + 8bFi

)F0

)(4.5)

with

Fe(µR) = I − 1− log(π)− log(MR), F0 = − log(2) (4.6)

I =1

2

∫ +∞

1dt

(t−1 + t−1/2

)(θ3(it)− 1) ' 0.02, θ3(it) =

+∞∑

n=−∞e−πtn

2

where bS,G,Fi (bS,G,Fi ) are the Casimirs of the KK modes of real scalars (not including

goldstone bosons), massive vector bosons and Dirac fermions respectively with even (odd)

masses 2n/R ((2n+1)/R). As explained above, the logarithmic part of the above expression

is equal to exactly −1/2 times the contribution of the same fields in 4D [84, 85]. Since the

compactification scale 1/R will always be relatively close to the unification scale Λ (so our

5D theory remains perturbative), it will be sufficient for us to use one loop matching in our

two loop analysis as long as the matching is done at a scale M close to the compactification

scale.

As an aside, from equation (4.4) we can get:

d

dt∆i =

bi − bMMi

8π2(4.7)

where bi is shorthand for the combination (bSi −21bGi +8bFi )/12 and bMMi are the coefficients

of the renormalization group equations below the compactification scale (see Appendix C

for details). It is clear from this equation that it is unnatural to require ∆i(1/R) 1/(8π2).

The most natural assumption ∆i(Λ) ∼ 1/(8π2) gives a one-loop contribution comparable

to the tree level term, implying the presence of some strong dynamics in the brane gauge

sector at the scale Λ. We know that the 5D gauge theory becomes strong at the scale

24π3/g25 , so from naive dimensional analysis (NDA) (see for example [86]) we find that it is

quite natural for Λ to coincide with the strong coupling scale for the bulk gauge group.

7We use this renormalization scheme even though our theory is non-supersymmetric since 4D thresholdcorrections in this scheme contain no constant part [83].


Running equation (4.7) to the compactification scale we obtain

∆i(1/R) = ∆i(Λ) +bi − bMM

i

8π2log(ΛR) (4.8)

For ΛR 1 the unknown bare parameter is negligible compared to the log-enhanced part,

and can be ignored, leaving us with a calculable correction. Using gGUT = 2πR/g25 we expect

ΛR ∼ 8π2/g2GUT ∼ 100. Keeping this in mind, we shall check whether unification is possible

in our model with ΛR in the regime where the bare brane gauge coupling is negligible. To

this purpose we will impose the matching equation (4.4) at the scale Λ assuming ∆i(Λ) = 0;

we will then check whether the value of ΛR found justifies this approximation.

In order to develop some intuition for the direction that these thresholds go in, we

can analyze the one-loop expression (with one-loop thresholds) for the gauge couplings at

Mz:

1

αi(Mz)=

4π

g2GUT

+4πλi(ΛR)+λconvi (ΛR)+

bMMi

2πlog

(Λ

Mz

)+

(bSMi − bMM

i

)

2πlog

(µ

Mz

)(4.9)

bSMi are the SM beta function coefficients (see Appendix C), µ is the scale of the higgsinos

and singlet, λconv = (− 312π ,− 2

12π , 0) are conversion factors from MS, in which the low-

energy experimental values for the gauge couplings are defined, to DR [87]. Taking the

linear combination (9/14)α−11 − (23/14)α−1

2 +α−13 allows us to eliminate the Λ dependence

as well as all SU(5)-symmetric terms, leaving

1

α3(Mz)= − 9/14

α1(Mz)+

23/14

α2(Mz)+ 4πλ(ΛR) + λconv +

(bSM − bMM

)

2πlog

(µ

Mz

)(4.10)

where X = (9/14)X1 − (23/14)X2 + X3 for any quantity X. Recall that the leading

threshold correction from the 5D GUT is proportional to log(ΛR). The low-energy value of

α3 is therefore changed by

δα3(Mz) = α3(Mz)2 b

2πlog(ΛR) (4.11)

We still have the freedom to choose the positions of the various matter fields. In order to

determine the best setup for gauge coupling unification we need to keep in mind two facts:

the first is that adding SU(5) multiplets in the bulk does not have any effect on α3(MZ);

and the second is that b contains only contributions from (+,+) modes (in unitary gauge

none of our bulk modes have (−,−) boundary conditions; our SU(5) bulk multiplets are

split into (+,+) and (+,−) modes).


As stated at the beginning of this section our 4D prediction for α3(Mz) is too low.

Since fermions have a larger effect on running than scalars, this problem is most efficiently

tackled by splitting up the fermion content of the SM into non-SU(5) symmetric parts in

order to make b as positive as possible. Examining the particular linear combination that

eliminated the dependence on Λ at one loop we find that one or more SU(5)-incomplete

colored multiplets are needed in the bulk, or equivalently the weakly-interacting part of the

same multiplet has to be on one of the branes. Since matter in the bulk is naturally split

by the orbifold projections, this just involves separating the pair of multiplets whose zero

modes make up one SM family.

With this in mind we find that for fixed ΛR and µ, since separating different

numbers of SM generations allows us to vary the low energy value of α3 anywhere from

its experimental value to several σs off, gauge coupling unification really does work in

this model for some fraction of all available configurations. Although this may seem a

little unsatisfactory from the point of view of predictivity, the situation can be somewhat

ameliorated by further refining our requirements. For example, we can go some way towards

explaining the hierarchy between the SM fermion masses by placing the first generation in

the bulk, the second generation split between the bulk and a brane and the third generation

entirely on a brane. This way, in addition to breaking the approximate flavor symmetry

in the fermion sector we also obtain helpful factors of order 1/√

ΛR between the masses

of the different generations. The location of the higgs does not have a very large effect on

unification, the simplest choice would be to put it, as well as the higgsinos and singlet, on

the SU(5)-breaking brane, where there is no need to introduce corresponding color triplet

fields. This also helps to explain the hierarchy. In this model, which can be seen on the

left-hand side of Figure 4.6, we also need to put our third generation and split second

generation on the same brane in order for them to interact with the higgs.

On the right is another model which capitalizes on every shred of evidence we

have about GUT physics: we put the higgs in the bulk in this case (recall that the orbifold

naturally gives rise to doublet-triplet splitting) so that we can switch the third generation

to the SU(5)-preserving brane and obtain b-τ unification (see Figure 4.7) without having

analogous relationships for the other two generations8. We also need to flip the positions

8As explained in [88] the two yukawa couplings λb and λτ run differently only below the compactificationscale. Because of locality, the fields living on the SU(5) brane do not feel the SU(5) breaking until energiesbelow the compactification scale; hence if they are unified at some high energy they keep being unified untilthis scale.


2

d

u

sψψ

ψH

q l

q

10 5

10 10’ 5 5’

_

_ _

3cel3

3

cd

uc3

2

3

π R/2

SU(3)xSU(2)xU(1)SU(5)

1 1 1 1

22

(a) Higgs on the broken brane

_

Hd

u

s

105 q l

ψψ ψ

3 1 1

π R/2

SU(5)

510_

10 10’ 5’_

5_

SU(3)xSU(2)xU(1)

1 1

2222

3

(b) Higgs in the bulk and third generation on

the SU(5) brane

Figure 4.6: Matter content of the two orbifold GUT models we propose.

of the first and second generations if we want to keep the suppression of the mass of the

first generation with respect to the second. The low-energy values for α3 as a function of

µ in these two models can be seen in Figure 4.8 for different ΛR. Note that unification can

be acheived in the regime where ΛR 1, justifying our initial assumption that the brane

kinetic terms could be neglected. We see that although the dependence on µ is very slight,

small µ seems to be preferred. However we cannot use this observation to put a firm upper

limit on µ because of the uncertainties associated with ignoring the bare kinetic terms on

the branes. The second configuration, Figure 4.6(b), also gives proton decay through the

mixing of the third generation with the first two. From our knowledge of the CKM matrix

we infer that all mixing matrices will be close to the unit matrix, proton decay will therefore

be suppressed by off-diagonal elements. To minimize this suppression it is best to have an

anti-neutrino and a strange quark in the final state. The proton decay rate for this process

was computed in [89] and is proportional to(

g24(1/R)2

)2 (1− m2

kaon

m2proton

)|(R†

d)23(R†u)13(Ld)31|2

where Rd,u and Ld are the rotation matrices of the right-handed down-type and up-type

quarks, and the left-handed down-type quarks, which are unknown. We assume that the

2-3 and the 1-3 mixing elements are 0.05 and 0.01 respectively, similar to the corresponding


0.022

0.023

0.024

0.025

0.026

0.027

0.028

interval4σ

1000 2000 3000 4000 5000µ (GeV)

ΛR=100ΛR=104D

λb(MZ)

Figure 4.7: Low energy prediction for λb(MZ), as a function of the higgsino mass µ, for the modelwith the higgs in the bulk, for ΛR = 10, 100 and for the 4D model. 4σ interval taken from [1].

CKM matrix elements, giving a proton lifetime of

τp(p→ K+ντ ) ' 6.6× 1038 years ×(

1/R

1014GeV

)4

' 4× 1035 years (4.12)

for 1/R = 1.6 × 1013 GeV. This is above the current limit from Super-Kamiokande of

1.9× 1033 years at 90% C.L. [90, 91], although there are multi-megaton experiments in the

planning stages that are expected to reach a sensitivity of up to 6× 1034 years [91] . Given

our lack of information about the mixing matrices involved9, we see that there might be

some possibility that proton decay in this model will be seen in the not-too-distant future.


The identification of a TeV-scale weakly-interacting particle as a good dark matter

candidate, and the unification of the gauge couplings are usually taken as indications of the

presence of low-energy SUSY. However this might not necessarily be the case.

If we assume that the tuning of the higgs mass can be explained in some other

unnatural way, through environmental reasoning for instance, then new possibilities open

up for physics beyond the SM. In this chapter we studied the minimal model consistent

with current experimental limits, that has both a good thermal dark matter candidate and

gauge coupling unification. To this end we added to the SM two higgsino-like particles and

9Experiments have only constrained the particular combination that appears in the SM as the CKMmatrix.


1000 2000 3000 4000 5000

0.095

0.105

0.11

0.115

0.12

0.125

interval4σ

ΛR=100

ΛR=10

4D

α3(MZ)

µ (GeV)

1/R = 1.7 × 1013 GeV

1/R = 4.1 × 1013 GeV

1/R = 1.6 × 1013 GeV

1/R = 3.8 × 1013 GeV

Figure 4.8: Low energy prediction for α3(Mz), as a function of the higgsino mass µ, for the modelwith the higgs on the brane (black line), and for the model with the higgs in the bulk (green line),for ΛR = 10, 100, and for the 4D case (dashed line). Some typical values of 1/R are shown.

a singlet, with a singlet majorana mass of . 100 TeV in order to split the two neutralinos

and so avoid direct detection constraints. Making the singlet light allowed for a new region

of dark matter with mixed states as heavy as ∼ 2.2 TeV, well beyond the reach of the LHC

and the generic expectation for a weakly interacting particle. Nevertheless we do have some

handles on this model: firstly via the 2-loop induced electron EDM contribution which is just

beyond present limits for CP angle of order 1, and secondly by the spin-independent direct

detection cross section, both of which should be accessible at next-generation experiments.

Turning to gauge coupling unification we saw that this was much improved at

two loops by the presence of the higgsinos. A full 4D GUT model is nevertheless excluded

by the smallness of the GUT scale ∼ 1014 GeV, which induces too fast proton decay. We

embedded the model in a 5D orbifold GUT in which the threshold corrections were calculable

and pushed α3 in the right direction for unification (for a suitable matter configuration).

It is very gratifying that such a model can help explain the pattern in the fermion mass

hierarchy, give b-τ unification, and predict a rate for proton decay that can be tested in the

future.

Chapter 5

Conclusion

Much of recent research in BSM physics has been centred on the problematic

hierarchies that abound in higgs physics. As the simplest addition to the Standard Model

that can satisfactorily explain the origin of mass through spontaneous symmetry breaking,

the higgs is in the privileged position of being the only fundamental scalar in the pantheon of

Standard Model particles. This brings with it a whole host of issues to do with stabilization

of its mass scale relative to the cutoff of the theory: quantum corrections to the masses of

scalars are generally large and tend to outweigh their tree-level values, pushing them up

to the cutoff. This problem, known as the gauge hierarchy problem, was first discussed

by Wilson in 1974 and has occupied a large fraction of all available phenomenological and

model-building resources since then.

Unfortunately our failure to find the higgs at colliders like LEP makes the most

simple and elegant implementation of low-energy SUSY (a firm favorite among theorists),

which was built to make the higgs sector more natural, look increasingly finely tuned itself,

bringing into question its viability as a solution to the gauge hierarchy problem.

In this dissertation we have analysed three alternative approaches. In preparation

for the advent of the Large Hadron Collider at CERN in August 2007 it is important to

continue to take a pragmatic and even-handed approach and investigate both natural and

non-natural theories. It remains for the LHC to confirm whether there is a role for any of

these particular theories in what lies beyond the SM.

59

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Appendix A

The Workings of a Top Seesaw

Starting with a top sector mass term of the form

−(t U

) mtt mtU

mUt mUU

tc

U c

(A.1)

we can diagonalize the M †M matrix as in [100] giving mass eigenvalues of

m2t,U =

1

2

[m2UU +m2

tt +m2Ut +m2

tU ±√

(m2UU +m2

tt +m2Ut +m2

tU )2 − 4(mUUmtt −mtUmUt)2]

(A.2)

with different mixing angles on the right and left

t′

U ′

=

cL −sL

sL cL

t

U

tc

′

Uc′

=

cR sR

−sR cR

−tc

Uc

(A.3)

and cL given by

cos θL =1√2

[1 +

m2UU −m2

tt +m2Ut −m2

tU

m2U −m2

t

] 12

(A.4)

68

Appendix B

The Neutralino Mass Matrix

We have a 3×3 neutralino mass matrix in which the mixing terms (see equation

(4.1)) are unrelated to gauge couplings and are limited only by the requirement of perturba-

tivity to the cutoff. It is possible to get a feel for the behavior of this matrix by finding the

approximate eigenvalues and eigenvectors in the limit of equal and small off-diagonal terms

(λv M1 ± µ). The approximate eigenvalues and eigenvectors are shown in the Table B.1

below: for cos θ = ±1.

M2 Gaugino fraction

M21 + 4λ2v2 M1

M1±µ 1− 2 λ2v2

(M1±µ)2

µ2 0

µ2 ± 4λ2v2 µM1±µ 2 λ2v2

(M1±µ)2

Table B.1: Approximate eigenvalues and eigenvectors of neutralino mass matrix

If M1 << µ the first eigenstate will be the LSP. In the opposite limit the com-

position of the LSP is dependent upon the sign of cos(θ). For cos(θ) positive the second

eigenstate, which is pure higgsino, is the LSP, while for cos(θ) negative, the mixed third

eigenstate becomes the LSP.

69

Appendix C

Two-Loop Beta Functions for

Gauge Couplings

The two-loop RGE for the gauge couplings in the Minimal Model is

(−2π)d

dtα−1i = bMM

i +1

(4π)2

3∑

j

4πBMMij αj − diλ2

t − d′i(λ2u + λ2

d)

with β-function coefficients

bMM =

(9

2,−15

6,−7

)BMM =

10425

185

445

65 14 12

1110

92 −26

d =

(17

10,3

2, 2

)d′ =

(3

20,1

4, 0

)

The running of the yukawa couplings is the same as in [53] but we will reproduce

their RGEs here for convenience - we ignore all except the top yukawa coupling (we found

that our two new yukawas do not have a significant effect).

(4π)2λt = λt

[−3

3∑

i=1

4πciαi +9

2λ2t +

1

2(λ2u + λ2

d)

]

with c =(

1760 ,

34 ,

83

).

The two-loop coupled RGEs can be solved analytically if we approximate the top

yukawa coupling as a constant over the entire range of integration (see [87] for a study on

the validity of this approximation). The solution is

α−1i (M) = α−1

G +1

2πbMM ln

Λ

M+

1

4π

3∑

j=1

BMMij

bMMj

ln

(1 +

1

4πbMMj αG(Λ) ln

Λ

M

)− 1

32π3diλ

2t ln

Λ

M

70

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