The Pragmatic Approach to the Gauge Hierarchy Problem

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 c 2006 - 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 ...... 44 4.3 Direct Detection ...... 46 4.4 Electric Dipole Moment ...... 46 4.5 Gauge Coupling Uniﬁcation ...... 49 4.5.1 Running and matching ...... 51 4.6 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 quantum 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 hierarchy, 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

2 2 2 3 4 2 mt˜ mh0 mZ + 2 ht v log 2 (1.1) ≈ 8π mt 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 2 the same time radiative corrections to mHu are quadratic in the stop mass

2 3 2 Λ δmHu 2 mt˜ log (1.2) ≈ −4π mt˜ 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 2 mHu 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 G G and reduce to the L × R 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 0 and

4 Chapter 2: Precision Electroweak Observables in the Minimal Moose 5

µ µ B0, with fermions and Higgs doublets via currents jF and jH respectively, generating low energy operators of the form jF jF , jF jH and jH jH . 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 j j • H H B0 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 components 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 ensure 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 j j • F F 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 0 and B0. 6 Chapter 2: Precision Electroweak Observables in the Minimal Moose

Interactions j j • F H 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 0 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

j , j tan θ0 cot θ0 H Flight ∝ − for tan θ0 = 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 0 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 0- 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 0, 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 ∼ Chapter 2: Precision Electroweak Observables in the Minimal Moose 7 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

GG G 1L GR2

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 G = SU(2) U(1) at the other. The standard model fermions are charged L × 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 G G where a L × R 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

= + + + (2.1) L LG Lχ Lt Lψ

includes all kinetic terms and gauge interactions, while contains plaquette couplings LG Lχ between the Xj:

f 4 = Tr A X X†X X† + Tr A X X†X X† Lχ 2 1 1 2 3 4 2 2 3 4 1 h i h i +Tr A3X3X4† X1X2† + Tr A4X4X1† X2X3† + h.c. (2.2) h i h i with A = κ + T 8 for κ/10. This is a natural relation since any radiative corrections i i i ∼ to require spurions from both the gauge and plaquette sectors and so can only arise at two loops. The terms give the little Higgses a mass (see Equation 2.18) and are required to stabilize electroweak symmetry breaking (EWSB). 8 Chapter 2: Precision Electroweak Observables in the Minimal Moose

The third generation quark doublet is coupled to a pair of colored Weyl fermions U,U c via Yukawa terms in Lt

q 0 c0 3 c t = λf 0 0 u3 X1X4† + λ fUU + h.c (2.3) L  U  and contains the remaining Yukawa couplings. These take the same form as above for Lψ the light up-type quarks, but with U and U c removed, while the down and charged lepton sectors look like 0 0

λD q 0 X1X†  0  + λE l 0 X1X†  0  + h.c (2.4) Lψ ⊃ 4 4  dc   ec          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, i = ).

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

2.2 The Gauge Boson Sector

The link ﬁelds Higgs the G G gauge groups down to the diagonal SU(2) U(1) L × R × subgroup, leaving one set each of massive and massless gauge bosons. This can be seen explicitly by considering the link ﬁeld covariant derivatives:

A A a a 8 D X = ∂ X ig X A3 R T + ig A2 L T X + iqg A1 L T X (2.5) µ j µ j − 3 j , µ 2 , µ j 1 , µ j where the Tes for A = 1, ..., 8 and a = 1, 2, 3 are SU(3) generators normalized such that A B δAB √ Tr[T T ] = 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 2 4 f µ Tr (D X )(D X )† (2.6) 4  µ j j  Xj=1 shows that the eaten Goldstone boson, w, iseproportionale to x1 + x2 + x3 + x4. Orthogonal combinations x,y and z can be deﬁned as follows:

w +1 +1 +1 +1 x1

 z  1  +1 1 +1 1   x2  = − − (2.7)  x  2  1 1 +1 +1   x     − −   3         y   1 +1 +1 1   x4     − −          Chapter 2: Precision Electroweak Observables in the Minimal Moose 9

where each of the above ﬁelds decomposes under SU(2) U(1) as 30 (φ) + 10 (η) + 2 1/2 × (h and h†). φ + ηx hx x √12 √2 x = † (2.8) h ηx  x −  e √2 √3   The x and y contain two Higgs doublets in the more familiar form

hx + ihy h1 = (2.9) √2 hx ihy h2 = − (2.10) √2 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 Wµ0 , Bµ0 and A3µ with eigenvalues 2gf/ sin 2θ, 2qg0f/ sin 2θ0 and fg/ sin θ respectively. The W 0 s and B0 s are admixtures of A3,R, A2,L and A1,L:

a a a Wµ = cos θA2,Lµ + sin θA3,Rµ a a a W 0 = sin θA2 L + cos θA3 R µ − , µ , µ 8 Bµ = cos θ0A1,Lµ + sin θ0A3,Rµ (2.11) 8 B0 = sin θ0A1 L + cos θ0A3 R µ − , µ , µ

with mixing angles deﬁned as follows:

g g g g = 2 3 sin θ = 2 2 g g2 + g3 3 g1g3 qg0 g0 = p sin θ0 = 2 2 g3 (qg1) + g3 q The Higgses couple to heavy gauge bosons via the following currents:

a ig a a j 0 = h† σ ←D→h + h†σ ←D→h W µ −2 tan 2θ 1 µ 1 2 µ 2 0 √3iqg h i j 0 = h†←D→h + h† ←D→h (2.12) B µ −2 tan 2θ0 1 µ 1 2 µ 2 h i 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 10 Chapter 2: Precision Electroweak Observables in the Minimal Moose violating terms:

2 2 3 2 µ cos 2θ0 h† D h + h† D h + 2 h† D h h† D h 16f 2 1 µ 1 2 µ 2 1 µ 1 2 2 1 2 µ µ + cos 2θ h† D h h† D h h† D h h† D h + h.c. (2.13) 8f 2 1 µ 2 2 1 − 1 µ 1 2 2 h i 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 2 1TeV 2 total contribution to the T parameter of 1.6 f cos 2θ0. This mechanism is responsible for some more fortuitous cancellation in thenext section.

At ﬁrst glance it seems like we can minimize the B 0 contribution to precision measurements by varying θ0. However we are constrained to sin θ0 . 1/3 by the relation 1 sin θ tan θ = 0 W q 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) 0 × 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 θ0 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 B0- fermion coupling of:

qL µ ig0 f i qR tan θ0 σ Bµ0 fi (2.14) tan θ0 − Xi where qL and qR are the fermion charges under each of the U(1)s. We can set qL = qR = q /2 for the light fermions to eliminate this coupling at θ π/4, provided we adjust the SM 0 ' light yukawa couplings to account for the new gauge structure:

q 3 c 4 up = λU 0 0 u X1X4† [X33]− + h.c (2.15) L   0     0 0 3 = λD q 0 X1X†  0  + λE l 0 X1X†  0  [X33] 4 + h.c Ldown 4 4   dc   ec            where X33 is the 33 component of any of the link ﬁelds. Chapter 2: Precision Electroweak Observables in the Minimal Moose 11

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

2 2 1 µ h†D h + h† D h + 2 h†D h h† D h 16f 2 1 µ 1 2 µ 2 1 µ 1 2 2 2 2 1 µ + 2 h† D h h† D h h† D h h†D h + h.c. (2.16) 16f 2 1 µ 2 2 1 − 1 µ 2 − 2 µ 1 Like the operators that originate from integrating out W 0, the terms in the second bracket will not give any contribution to the T parameter after EWSB. The contribution from the 2 1TeV ﬁrst bracket is 0.53 f . 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:

M 2 = 4f 2 (κ) + O() (2.17) z < a tree level mass for the Higgses which stabilizes the ﬂat direction in the potential and triggers electroweak symmetry breaking;

√3f 2 () h† h h† h (2.18) 4 = 1 1 − 2 2 a z-Higgs coupling of the form, jaza, with

f ja = ()Tr Ta[x, [x, T8]] Ta[y, [y, T8]] + ... (2.19) − 2 = − 12 Chapter 2: Precision Electroweak Observables in the Minimal Moose 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(2/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

2 2 2 V m h† h + m h† h + m h† h + h† h (2.21) ≈ 1 1 1 2 2 2 12 1 2 2 1 2 2 + λ h† h + h† h h† h h†h h† h h† h h 1 1 2 2 − 1 1 2 2 − 1 2 2 1 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 iϕ terms will dominate in this limit, but there is a flat direction, namely h1 = e h2 in which we demand that the quadratic part of the potential be positive definite. This gives us the constraint m2 + m2 2 m2 (2.22) 1 2 ≥ | 12|

Further requiring that the mass matrix for h1 and h2 have one negative eigenvalue at the origin tells us that 2 2 4 m1m2 < m12 (2.23)

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

1 0 h1 = √2  v cos β    1 0 h2 = (2.24) √2  v sin β    Chapter 2: Precision Electroweak Observables in the Minimal Moose 13 where

1 m2 m2 v2 = m2 m2 + | 1 − 2| λ − 1 − 2 cos 2β h 2 2m12 sin 2β = 2 2 (2.25) −m1 + m2

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

2 2 2 2 4mH = mh0 + mH0 + 3mA0 (2.26) 2 2 2 mH = mA0 + λhv

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 in terms of the new eigenstates: Lt

q 2 02 c 2 c 3 t = f λ + λ U U + sin ξ 0 0 U X1X4† 1 (2.27) L −  U  p e e   q c 3 + sin ξ cos ξ 0 0 u3 X1X4† 1 −  U    where

λ sin ξ = √λ2 + λ02 c c c0 U = cos ξU + sin ξu3 (2.28) 0 uc = sin ξU c + cos ξuc e3 − 3

Expanding the link ﬁelds to ﬁrst order in v/f, a convenient phase rotation gives us the 14 Chapter 2: Precision Electroweak Observables in the Minimal Moose following terms in the t U mass matrix: −

2 2 v mtt = λ + λ sin ξ cos ξ (sin β + cos β) 0 √2 p 2 2 2 v mtU = λ + λ0 sin ξ (sin β + cos β) (2.29) √2 p 2 2 mUU = f λ + λ0 p Using mtt, mtU << mUU , mUt = 0 we approximate the results in Appendix A to obtain

m2 m2 t ≈ tt m2 m2 m2 1 + tU (2.30) U ≈ UU m2 UU 2 mtU cos θL 1 2 ≈ − 2mUU

Now we can ﬁx the top Yukawa coupling to its value λt in the SM, which for a given value of tan β relates λ to λ0 in the following way:

λ λ 1 + tan2 β 0 = t (2.31) λ λ2(1 + tan β)2 λ2(1 + tan2 β) p − t p with λ constrained by 1 + tan2 β λ2 > λ2 (2.32) (1 + tan β)2 t

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

2 2 2 2 3 2 mU mb 2 mt mb Tf = sin θLΘ+ , sin θLΘ+ , (2.33) 16π sin2 θ cos2 θ m2 m2 − m2 m2 W W Z Z Z Z m2 m2 sin2 θ cos2 θ Θ U , t − L L + m2 m2 Z Z 3 m2 m2 m2 m2 m2 m2 S = sin2 θ Ψ U , b sin2 θ Ψ t , b sin2 θ cos2 θ χ U , t f 2π L + m2 m2 − L + m2 m2 − L L + m2 m2 Z Z Z Z Z Z for

2y y y 1 1 y Θ (y , y ) = y + y 1 2 ln 1 Ψ (y , y ) = ln 1 + 1 2 1 2 − y y y + 1 2 3 − 9 y 1 − 2 2 2 5(y2 + y2) 22y y 3y y (y + y ) y3 y3 y χ (y , y ) = 1 2 − 1 2 + 1 2 1 2 − 1 − 2 ln 1 (2.34) + 1 2 9(y y )2 3(y y )3 y 1 − 2 1 − 2 2 Chapter 2: Precision Electroweak Observables in the Minimal Moose 15

2.7 Higgs Sector

There is a contribution to the vacuum polarization diagrams from additional physical Higgs states running around the loop. This is a standard calculation (see [12, 18]) which yields

1 2 2 2 2 2 2 2 Th = 2 2 F (mA0 , mH ) + cos (α β) F (mH , mh0 ) F (mA0 , mh0 ) 16π sin θW m − − W 2 2 2 2 2 + sin (α β) F (m , m 0 ) F (m 0 , m 0 ) − H H − A H 2 1 2 mH0 11 2 2 2 2 Sh = cos (β α) log 2 + sin (β α)G(mH0 , mA0 , mH ) (2.35) 12π − m 0 − 6 − h 2 2 2 2 + cos (β α)G(m 0 , m 0 , m ) − h A H where

1 xy x F (x, y) = (x + y) log 2 − x y y − x2 + y2 (x 3y)x2 log x (y 3x)y2 log y G(x, y, z) = + − z − − z (x y)2 (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 tan β. 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 16 Chapter 2: Precision Electroweak Observables in the Minimal Moose

4.2 f=2.6 TeV 0.05 T mU (TeV) f f= 1.9 TeV 4 0.04 3.8

3.6 0.03 f=2.6 TeV 3.4 0.02 3.2 f=1.9 TeV 0.01 0.5 1 1.5 2 2.5 3 3.5 tanβ 2.8 tanβ 0.5 1 1.5 2 2.5 3 3.5

Figure 2.2: Fermion sector contribution to T and mass of top partner as a function of tan β.λ and 0 λ were chosen to minimize mU with a ﬁxed top quark mass.

0.05Th S β 0.008 h tan Ref. values 1 0.5 1 1.5 2 2.5 3 3.5 0 0.006 Ref. values 2 -0.025 0.004 -0.05 Ref. values 1 0.002 -0.075 β -0.1 tan 0.5 1 1.5 2 2.5 3 3.5 -0.125 Ref. values 2 -0.002

Figure 2.3: Higgs sector contribution to PEWOs as a function of tan β. Values of other variables taken 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 tan β 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 Chapter 2: Precision Electroweak Observables in the Minimal Moose 17 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◦ θ0 47◦ 50◦ λ 0.9 1.1 λh 1.6 0.5 tan β 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 mW0 (TeV) 2.56 4.50 mB0 (TeV) 0.78 1.08 ﬁne tuning 17% 3%

Table 2.1: Two sets of reference parameters for the Modiﬁed Minimal Moose. 18 Chapter 2: Precision Electroweak Observables in the Minimal Moose

0.1 REFERENCE VALUES 1

NLSM B’ and fermion T 0

Higgs doublets

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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 Chapter 2: Precision Electroweak Observables in the Minimal Moose 19

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Figure 2.5: Fine tuning plotted as a function of tan(β) and λh where all other parameters have been taken from Table 2.1. We only plot regions of parameter space which lie within the S-T ellipse. We indicate the position of reference set 1. Note that the ﬁne tuning improves for larger values of λh as the 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 0 exchange. This might be understood better in the context of other LH theories, the Littlest 20 Chapter 2: Precision Electroweak Observables in the Minimal Moose

REFERENCE VALUES 2 0.1 B’ and fermion

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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 0 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 inpthe 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 Chapter 2: Precision Electroweak Observables in the Minimal Moose 21

3 18%

2.5 16% 14% 2 12%

h 10% Λ 1.5 8%

1 6%

¡ 0.5 ref. set 2 2%

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Figure 2.7: Fine tuning plotted as a function of tan β and λh where all other parameters have been taken from Table 2.1. We plot regions of parameter space which are at least as close to the S-T ellipse 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 H H 2. Un- | | | u d| 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 λ 3 = 4λ2 + 3h2 3g2 g2 + (3.2) dt 16π2 t − 2 − 5 1 · · · 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 contributions 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:

c c c ˜ ˜ c Wλ = λ1 φXHu + λ2 φ X Hd + MX XX + MX˜ XX . (3.3)

We have added the ﬁelds φ, φc, X, Xc, X˜ , X˜ c, 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) U(1) SU(n) × L × Y s φ (1, 1, 0) n φc (1, 1, 0) n¯ 1 X (1, 2, 2 ) n¯ c −1 X (1, 2, 2 ) n ˜ ¯ 1 X (3, 1, 3 ) n¯ X˜ c (3, 1, 1 ) n − 3 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) by the X˜ s and thus maintain × s 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 X˜s generates the nonrenormalizable operator λ1λ2 c Weﬀ = φφ HuHd. (3.4) − MX 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 be considered here. Chapter 3: The New Fat Higgs: Slimmer and More Attractive 25

SU(3) SU(2) U(1) SU(4) × L × Y s φ (1, 1, 0) 4 φc (1, 1, 0) 4¯ ψi for i = 1, , 4 (1, 1, 0) 4 c · · · ¯ ψi for i = 1, , 4 (1, 1, 0) 4 · · · 1 ¯ X (1, 2, 2 ) 4 c −1 X (1, 2, 2 ) 4 ˜ ¯ 1 ¯ X (3, 1, 3 ) 4 X˜ c (3, 1, 1 ) 4 − 3 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) c WS = m φφ (3.6) i c c i c ij c Wd = y(T φψi + T ψiφ + T ψiψj ) +

y0 ijkl B ijkl Bc c c c c 2 ( Ti φ ψjψkψl + Ti φ ψj ψkψl ). (3.7) MGUT

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 i c i ij them mass terms with the singlets T , T , T . 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 to have a linear family of conformal fixed points in (λi, g) space [34] and would therefore be convenient when we discuss the possibility of having a new superconformal fixed point in Section 3.1.2. Alternatively if the X˜s required for unification were not also charged under the strong group, satisfying the resulting constraints would be easier since we would have more room to maneuver. However this theory is not naturally unified, and so will not be pursued here. 26 Chapter 3: The New Fat Higgs: Slimmer and More Attractive these light states will be discussed in Section 3.2.3. Note that there is a non-anomalous c U(1)R symmetry (under which ψi, ψi 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 3 ( 2 n < 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 g λ . Ignoring electroweak couplings and the top yukawa, near Seiberg’s fixed s 1,2 point we have the RGE: 3 dλ1,2 7λ1,2 = + γ λ1,2 + (3.8) dt 16π2 ∗ · · · 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 conformality, 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 Chapter 3: The New Fat Higgs: Slimmer and More Attractive 27

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. c Below MX , the theory is a SU(4)s gauge theory with 5 flavors ΨI = (φ, ψi), ΨI = (φc, ψc) for I = 0, , 4. There is a dynamically generated superpotential i · · · 1 W = M BI Bc J det M (3.9) dyn Λ7 IJ − written in terms of gauge invariant mesons (M Ψ Ψc ) and baryons (BI IJKLM Ψ Ψ ). IJ ∼ I J ∼ J · · · M 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) λ λ Λ W √n 1 2 SH H (3.11) eff → 4π M u d X mΛ W S (3.12) S → 4π 3 yΛ i c i ij y0Λ B i Bc c i Wd (T M0i + T Mi0 + T Mij) + 2 (Ti B + Ti B ) (3.13) → 4π 4πMGUT (4π)3 W (4π)M BIBc J det M (3.14) dyn → IJ − Λ2 and we have defined M00 to be S. The first two terms give us an NMSSM-like model at energy scales below Λ. In the Weff term we have done not only the normal NDA analysis, but 28 Chapter 3: The New Fat Higgs: Slimmer and More Attractive 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 λ Λ λ = √n 1 2 . (3.15) 4π MX

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:

1 1 1 Λ − λ(µ)2 = + ln . (3.16) λ(Λ)2 2π2 µ 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. Chapter 3: The New Fat Higgs: Slimmer and More Attractive 29

λ µ Λ/µ = 10 2.5 H L 2 Λ/µ = 100 Λ/µ = 1000 1.5 1 0.5 λ Λ 2 4 6 8 10 12

Figure 3.1: The low energy values of the λ coupling after running fromH Lthe compositeness scale Λ down 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 Λ Λ 16π2 8π λ(Λ) √4 1 2 . γ γ 3.6 γ . (3.17) ∼ 4π MX −2πMX 7 ∗ ∼ 7 ∗ ∼ ∗

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 X˜s 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:

m2 m2 cos2 2β + λ2v2 sin2 2β/2. (3.18) h ≤ Z to obtain m . 260 350 GeV (3.19) h − 30 Chapter 3: The New Fat Higgs: Slimmer and More Attractive 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/M • X the running of the nonrenormalizable operator in Eq. 3.4 due to gauge coupling • contributions in the region Λ E M ≤ ≤ X the coefficient in the NDA matching that was used in Eq. 3.15 • 3 loop-level corrections from g to the coefficient of λ1,2 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 3 eliminated. The fourth point is that we ignored higher order gauge corrections to the λ1,2 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 2 2 loop suppression factor λ1,2/(16π ) . 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 tan β near 1 (where the gain in the tree level bound is greatest), the Chapter 3: The New Fat Higgs: Slimmer and More Attractive 31

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 and gaugino masses (m0 and m 1 ) at the uniﬁcation scale, for example, force the top squarks to be heavy given 2 observational lower bounds on chargino and slepton masses. This is a property of current SUSY breaking scenarios, however, and it is possible to imagine alternatives with more random boundary conditions at the GUT scale that result in realistic particle spectra with light top squarks. 32 Chapter 3: The New Fat Higgs: Slimmer and More Attractive 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

2 2 8π 8π i 2 = 2 T ln Zi (3.21) gle, phys gle − Xi 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 ln Zi 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) multiplets 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 unification. In that model, explaining why unification is natural requires a setup that generates the additional matter content as well as the required mass spectrum. Chapter 3: The New Fat Higgs: Slimmer and More Attractive 33

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 asymptotically 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 c i B Bc B s that are married to the Ti s and Ti s, with supersymmetric masses of order

3 Λ 13 7 MB 2 10− 10− eV (3.22) ∼ 4πMGUT ∼ − 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 T 10GeV, requiring T . T in dec ∼ reheat dec 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. I c J However, the reason these fermions decouple late is due to the MIJ B B 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 34 Chapter 3: The New Fat Higgs: Slimmer and More Attractive

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 X˜s 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.

3.3 Summary and Discussion

Supersymmetry does extremely well in solving the hierarchy problem, but as more precise measurements have told us, the minimal implementation of weak scale supersymmetry (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 completing, 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 coupling 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 Chapter 3: The New Fat Higgs: Slimmer and More Attractive 35

¡ ¡ ¡ ¡

¡ ¡ ¡

¡ Fat Higgs:

¡ ¡ ¡

10 ¡ Composite S,

¡ ¡ ¡ ¡

¡ ¡ ¡ ¡ Hu, Hd 8

4 New Fat Higgs: 2 Composite S Elementary Hu, Hd

1 2 3 4 log Λ/µ

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

4 about the fixed point values of λ1,2. 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. 36 Chapter 3: The New Fat Higgs: Slimmer and More Attractive 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 do with 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 implications 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 example. 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 is not considered here. 3From here on we will refer to these particles and couplings by their SUSY equivalents without the quotation marks for simplicity. 40 Chapter 4: The Minimal Model for Dark Matter and Uniﬁcation with the following renormalizable interaction terms:

1 µΨ Ψ + M Ψ Ψ + λ Ψ hΨ + λ Ψ h†Ψ (4.1) u d 2 1 s s u u s d d s

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 limited by requiring perturbativity to the cutoff. For equal yukawas this constrains λu(MZ ) = λ (M ) 0.88, while if we take one of the couplings to be small, say λ (M ) = 0.1 then d Z ≤ d Z λu(MZ ) can be as large as 1.38.

The above couplings allow for the CP violating phase θ = Arg(µM1λu∗ λd∗), 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: ×

M1 λuv λdv iθ MN =  λ v 0 µe  u −  λ v µeiθ 0   d −    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. Chapter 4: The Minimal Model for Dark Matter and Uniﬁcation 41

4.2 Relic Abundance

In this section we study the regions of parameter space in which the DM abundance 2 is in accordance the post-WMAP 2σ region 0.094 < Ωdmh < 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 W W 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 42 Chapter 4: The Minimal Model for Dark Matter and Unification we will explain below the only effect this has is to allow a low mass region for a bino-like LSP with M M /2. Notice that the relic abundance seems to be consistent with a 1 ∼ higgs

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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 Ω 10 9GeV 2/ σ , dm ∼ − − h ieff where σ is an effective annihilation cross section for the LSP at the freeze out temper- h ieff 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 dominant 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 Chapter 4: The Minimal Model for Dark Matter and Unification 43

3000 3000

0.01 0.998 2000 2000 0.03 ~ 0 0.997 1 1 0.1 M (GeV) 0.996 M (GeV) 0.994 0.5 0.8 1000 0.99 1000 0.9

0.98 0.97 0.97 10−8 0.9 0.99 0.8 0 1000 2000 3000 0 1000 2000 3000 µ (GeV) µ (GeV)

(a) θ=0 (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 approximation (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 (W W ) 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 44 Chapter 4: The Minimal Model for Dark Matter and Unification enough. Increasing the yukawa couplings increases the importance of t-channel bino exchange to higgs pairs. Notice that taking the limit M µ makes this new interaction 1 irrelevant, therefore the allowed region converges to the one in which only gauge interactions 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 . This effect happens to outweigh the former, forcing the LSP to shift to h ieff 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 Chapter 4: The Minimal Model for Dark Matter and Uniﬁcation 45 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 M = 1 M 80 GeV. This line 1 2 higgs ∼ 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 46 Chapter 4: The Minimal Model for Dark Matter and Unification

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 0 0 yukawa couplings which mix higgsinos and binos. Since the only χ1χ1h term in our model involves the product of the gaugino and higgsino fractions, the more mixed our dark matter 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 constraint, 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], corresponding 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 annihilation 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 (1) θ. The dominant diagram O 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 Chapter 4: The Minimal Model for Dark Matter and Unification 47

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[65]): dW 2 3 f α mf mχi µ L R 0 = Im (O O ∗) r , r (4.2) e 8π2s4 M 2 M 2 i i G i W W i=1 W X where

1 1 0 ∞ dγ y z (y + z/2) r , r = dz dy G i γ (z + y)3(z + K ) Z0 Z0 Z0 i 1 dγ 1 (y 3K )y + 2(K + y)y K (K 2y) K = dy y − i i + i i − ln i γ 4y(K y)2 2(K y)3 y Z0 Z0 i − i − and

0 2 2 ri r µ 0 mχi K = + , r , r , i 1 γ γ ≡ M 2 i ≡ M 2 − W W R iθ L O = √2N ∗ exp− , O = N i 2i i − 3i 48 Chapter 4: The Minimal Model for Dark Matter and Uniﬁcation

w j + χi W 0 W

f' f

Figure 4.4: The 2-loop contribution to the EDM of a fermion f.

T N MN N = 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 1 and 2 f 0 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 including 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 (1)), we might naively expect their in- O 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 elastic 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 d < 1.7 10 27e cm at 95% C.L. e × − [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. Chapter 4: The Minimal Model for Dark Matter and Uniﬁcation 49

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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 α (M ) = 0.119 0.002 [1]. Moreover the scale at which s z 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 parameters to zero [73]. However we are not aware of any GUT model in which such an assertion is justiﬁed by symmetry 50 Chapter 4: The Minimal Model for Dark Matter and Uniﬁcation

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/(Z Z ), whose action is obtained from the identifications 2 × 20

Z : y 2π R y, Z0 : y πR y, y [0, 2π] (4.3) 2 ∼ − 2 ∼ − 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 P φ and on the adjoint, [P, A], for → projection operators P = (+, +, +, +, +) and P 0 = (+, +, +, , ). This gives SM gauge Z Z − − fields and their corresponding KK towers Aa for a = 1, ..., 12 (+, +) parity; and the towers µ { } arguments. Chapter 4: The Minimal Model for Dark Matter and Unification 51

Aaˆ for aˆ = 13, ..., 24 (+, ) parity, achieving the required symmetry-breaking pattern. By µ { } − 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 c by the action of the Z0 orbifold: the SM 5 for instance will either contain a massless d 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 2πR 2 = 2 + ∆i(M) + λi(MR) (4.4) gi (M) g5 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 can simply be thought of as a theory with a compact extra dimension on an interval, with two branes on the boundaries. Because of the presence of the boundaries we are free to impose either Dirichlet or Neumann boundary conditions for the bulk field on each of the branes breaking the SU(5) to the SM gauge group purely by choice of boundary conditions and similarly splitting the multiplets accordingly. Our orbifold projection is therefore nothing more than a further restriction to the set of all possible choices we can make. 52 Chapter 4: The Minimal Model for Dark Matter and Unification

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 2 5D massless scalar field [82]. Since 1/g5 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 Chapter 4: The Minimal Model for Dark Matter and Unification 53 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]: 1 λ (MR) = bS 21bG + 8bF F (MR) + ˜bS 21˜bG + 8˜bF F (4.5) i 96π2 i − i i e i − i i 0 with

F (µR) = 1 log(π) log(MR), F = log(2) (4.6) e I − − − 0 − + + 1 ∞ 1 1/2 ∞ πtn2 = dt t− + t− (θ (it) 1) 0.02, θ (it) = e− I 2 3 − ' 3 1 n= Z X−∞ S,G,F ˜S,G,F where bi (bi ) 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 b bMM ∆ = i − i (4.7) dt i 8π2 where b is shorthand for the combination (bS 21bG +8bF )/12 and bMM are the coefficients i i − i i i 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 ∆ (1/R) 1/(8π2). i The most natural assumption ∆ (Λ) 1/(8π2) gives a one-loop contribution comparable i ∼ 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 3 2 24π /g5 , 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 threshold corrections in this scheme contain no constant part [83]. 54 Chapter 4: The Minimal Model for Dark Matter and Uniﬁcation

Running equation (4.7) to the compactification scale we obtain b bMM ∆ (1/R) = ∆ (Λ) + i − i log(ΛR) (4.8) i i 8π2 For ΛR 1 the unknown bare parameter is negligible compared to the log-enhanced part, 2 and can be ignored, leaving us with a calculable correction. Using gGUT = 2πR/g5 we expect ΛR 8π2/g2 100. Keeping this in mind, we shall check whether unification is possible ∼ GUT ∼ 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 4π bMM Λ bSM bMM µ = +4πλ (ΛR)+λconv(ΛR)+ i log + i − i log (4.9) α (M ) g2 i i 2π M 2π M i z GUT z z SM bi are the SM beta function coeﬃcients (see Appendix C), µ is the scale of the higgsinos and singlet, λconv = ( 3 , 2 , 0) are conversion factors from MS, in which the low- − 12π − 12π energy experimental values for the gauge couplings are deﬁned, to DR [87]. Taking the 1 1 1 linear combination (9/14)α− (23/14)α− + α− allows us to eliminate the Λ dependence 1 − 2 3 as well as all SU(5)-symmetric terms, leaving

1 9/14 23/14 bSM bMM µ = + + 4πλ(ΛR) + λconv + − log (4.10) α (M ) −α (M ) α (M ) 2π M 3 z 1 z 2 z z where X = (9/14)X (23/14)X + X for any quantity X. Recall that the leading 1 − 2 3 threshold correction from the 5D GUT is proportional to log(ΛR). The low-energy value of

α3 is therefore changed by b δα (M ) = α (M )2 log(ΛR) (4.11) 3 z 3 z 2π 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). − Chapter 4: The Minimal Model for Dark Matter and Unification 55

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

8 As explained in [88] the two yukawa couplings λb and λτ run differently only below the compactification scale. Because of locality, the fields living on the SU(5) brane do not feel the SU(5) breaking until energies below the compactification scale; hence if they are unified at some high energy they keep being unified until this scale. 56 Chapter 4: The Minimal Model for Dark Matter and Unification

SU(5) SU(3)xSU(2)xU(1) SU(5) SU(3)xSU(2)xU(1) q 3 uc 10 3 c 3 _ q d3 5 2 3 q1 l2 l3 c l1 e3 H ψ ψ u ψ H u ψ d ψ d ψ _ s _ s 102 52 _ 102 10'2 52 5'2 _ _ _ 101 10'1 5 5'1 1 101 51

π R/2 πR/2

(a) Higgs on the broken brane (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 2 2 2 g4 mkaon 2 was computed in [89] and is proportional to 2 1 2 (Rd† )23(Ru† )13(Ld)31 (1/R) − mproton | | where Rd,u and Ld are the rotation matricesof the right-handed do wn-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 Chapter 4: The Minimal Model for Dark Matter and Unification 57

λb(MZ) 0.028

0.027

0.026

0.025 4σ ΛR=100 ΛR=10 interval 0.024 4D 0.023

0.022

1000 2000 3000 4000 5000 µ (GeV)

Figure 4.7: Low energy prediction for λb(MZ ), as a function of the higgsino mass µ, for the model with 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

1/R 4 τ (p K+ν¯ ) 6.6 1038 years 4 1035 years (4.12) p → τ ' × × 1014GeV ' × 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.

4.6 Summary and Discussion

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 CKM matrix. 58 Chapter 4: The Minimal Model for Dark Matter and Uniﬁcation

α3(MZ) 0.125

0.12 13 4σ 1/R = 1.7 × 10 GeV interval 0.115 1/R = 1.6 × 1013 GeV ΛR=100

0.11 1/R = 4.1 × 1013 GeV

0.105 1/R = 3.8 × 1013 GeV ΛR=10

1000 2000 3000 4000 5000 µ (GeV) 0.095 4D

Figure 4.8: Low energy prediction for α3(Mz), as a function of the higgsino mass µ, for the model with 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.

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The Workings of a Top Seesaw

Starting with a top sector mass term of the form

c mtt mtU t t U (A.1) −  m m   U c  Ut UU     we can diagonalize the M †M matrix as in [100] giving mass eigenvalues of 1 m2 = m2 + m2 + m2 + m2 (m2 + m2 + m2 + m2 )2 4(m m m m )2 t,U 2 UU tt Ut tU UU tt Ut tU − UU tt − tU Ut q (A.2) with diﬀerent mixing angles on the right and left

t0 cL sL t = −  U 0   sL cL   U        tc0 cR sR tc = − (A.3)  U 0   s c   U  c − R R c       and cL given by 1 1 m2 m2 + m2 m2 2 cos θ = 1 + UU − tt Ut − tU (A.4) L √2 m2 m2 U − t

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 perturbativity 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 M µ). The approximate eigenvalues and eigenvectors are shown in the Table B.1 1 below: for cos θ = 1. M 2 Gaugino fraction 2 2 2 2 2 M1 λ v M + 4λ v 1 2 2 1 M1 µ (M1 µ) − µ2 0 2 2 2 µ λ2v2 µ 4λ v 2 2 M1 µ (M1 µ) 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

3 d 1 MM 1 MM 2 2 2 ( 2π) α− = b + 4πB α d λ d0 (λ + λ ) − dt i i (4π)2  ij j − i t − i u d  Xj with β-function coeﬃcients  

104 18 44 25 5 5 MM 9 15 MM 6 17 3 3 1 b = , , 7 B =  14 12  d = , , 2 d0 = , , 0 2 − 6 − 5 10 2 20 4  11 9 26   10 2 −    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). 3 2 9 2 1 2 2 (4π) λt = λt 3 4πciαi + λt + (λu + λd) "− 2 2 # Xi=1 17 3 8 with c = 60 , 4 , 3 . The two-lo op 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

3 MM 1 1 1 MM Λ 1 Bij 1 MM Λ 1 2 Λ αi− (M) = αG− + b ln + MM ln 1 + bj αG(Λ) ln 3 diλt ln 2π M 4π bj 4π M −32π M Xj=1