R-Symmetry, Gauge Mediation and Decaying Dark Matter by Santiago José De Lope Amigo a Thesis Submitted in Conformity with the R

R-symmetry, Gauge Mediation and Decaying Dark Matter

Santiago Jos´eDe Lope Amigo

A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Physics University of Toronto

Abstract

R-symmetry, Gauge Mediation and Decaying Dark Matter

Santiago Jos´eDe Lope Amigo

Doctor of Philosophy

Graduate Department of Physics

University of Toronto

2011

Different aspects of specific models in supersymmetry as well as constraints on decaying dark matter are analysed in this thesis. In chapter 1 we give a general introduction to supersymmetry, and briefly discuss some of the concepts that are used throughout the thesis.

In chapter 2 we present a version of Gauge Mediated Supersymmetry Breaking which

preserves an R-symmetry—the gauginos are Dirac particles, the A-terms are zero, and

there are four Higgs doublets. This oﬀers an alternative way for gauginos to acquire mass in the supersymmetry-breaking models of Intriligator, Seiberg, and Shih [1] . Addition-

ally, we investigate the possibility of using R-symmetric gauge mediation to realise the

spectrum and large sfermion mixing of the model of Kribs, Poppitz, and Weiner [2].

In chapter 3 we investigate the Higgs sector of the R-symmetric model presented in

chapter 2. Furthermore, a scan of the parameter space and sample spectra are provided.

Other attributes like the tuning of the model are discussed.

In chapter 4 we present a complete analysis of the cosmological constraints on decay-

ing dark matter. In order to do this, we have updated and extended previous analyses to include Lyman-α forest, large scale structure, and weak lensing observations. As-

trophysical constraints are not considered in this thesis. The bounds on the lifetime

of decaying dark matter are dominated by either the late-time integrated Sachs-Wolfe

eﬀect for the scenario with weak reionization, or CMB polarisation observations when

ii there is signiﬁcant reionization. For the respective scenarios, the lifetimes for decaying

1 1 8 dark matter are Γ− & 100 Gyr and (fΓ)− & 5.3 10 Gyr (at 95.4% confidence level), × where the phenomenological parameter f is the fraction of the decay energy deposited in baryonic gas. This allows us to constrain particle physics models with dark matter candidates through investigation of dark matter decays into Standard Model particles via effective operators. For decaying dark matter of 100 GeV mass, we found that the ∼ size of the coupling constant in the effective dimension-4 operators responsible for dark

22 matter decay has to generically be . 10− .

iii Dedication

A Iaia

Gracias por tu entusiasmo, tu alegr´ıa y tu apoyo, siempre te extra˜naremos.

iv Acknowledgements

I would like to express my sincere gratitude to all the people that have accompanied and

assisted me through this intellectual journey. I am particularly thankful to my parents

Jaime and Ana Mar´ıa and and to my brother Alfonso for being a source of encouragement.

Special thanks to my grandmother, to whom I dedicate this thesis, she passed away while I was writing it, and to my aunt Martha for supporting me and encouraging me in my

studies.

Academically, I would like to thank in particular my supervisor, Erich Poppitz for guiding me and for being a mentor during all this time. I would also like to thank my closest collaborators: Andrew E. Blechman, Patrick Fox (Paddy), William Man-Yin

Cheung, Siew-Phang Ng, and Zhiqi Huang. The members of my committee, Michael Luke and Pierre Savard, provided guidance and insight every year. Additionally, my professors and mentors in Mexico were of great support while I was applying for the programme and when I had to apply for funding. Alejandro Pomposo, Edmundo Palacios, Jorge

Cervantes, Gustavo Soto, Alfredo Sandoval, Dr. Garc´ıa Colin, Dr. Mariano Bauer, and

Jos´eJob Flores, thank you for all your letters of recommendation.

The people that I met everyday in the oﬃce kept me sane and contributed to a supportive envirionment. Simon, Catalina, Siavash, Saba, Brian and Donfang, thank you all. Moreover, the staﬀ at the university was always helpful, in particular Krystyna and Teresa.

The community at Knox College, where I lived several years, was always there for me. The list is endless but I want to thank in particular to Iv´an (chavo de oro), Poppy,

Alex, Nik, Nicol´as, Charles, Foad, Pierre, Tianna, Ernest, Jude, Ralph, Daniela, and many others with whom I shared conversations and good times.

I am particularly thankful to Gio for supporting me and giving me advice when I needed it most. Also to all of your family who became a family away from home for me:

Anna & Dave, Tina & Anthony, Guy, Neelu, Santina & Umberto.

v Many other friends and acquaintances from the department of physics and other departments also helped me to shape my experience as a graduate student. Fazel, Peter,

Behi, V´ıctor, Khash, Claude, Marilyn, and Olivera, thank you.

I would like to acknowledge the ﬁnancial support of the Natural Sciences and Engi-

neering Research Council of Canada (NSREC) and the Direcci´on General de Relaciones

Internacionales de la Secretar´ıa de Educaci´on P´ublica (DGRI-SEP) of Mexico.

Last but not least, I am particularly grateful to the Canadian society as a whole for their support, friendship, and for the opportunity to live and study here.

vi Contents

1 Introduction 1

1.1 Supersymmetry ...... 4

1.1.1 Supersymmetryformalism ...... 6

1.1.2 Minimal Supersymmetric Standard Model ...... 8

1.2 Gauge Mediation as a solution to the Supersymmetric Flavour Problem . 13

1.3 R-symmetryinSupersymmetry ...... 18

1.4 Minimal R-symmetricStandardModel ...... 23

1.5 DarkMatter...... 29

2 R-Symmetric Gauge Mediation 34

2.1 Motivation...... 34

2.2 ISS and R-symmetric direct gauge mediation ...... 35

2.2.1 The supersymmetry-breaking/mediation sector ...... 37

2.2.2 Scales of supersymmetry breaking and mediation ...... 39

2.2.3 Dirac gaugino masses, C-parity,and the extra adjoints ...... 43

2.3 Softtermsinthevisiblesector...... 45

2.3.1 Estimating the UV contributions ...... 46

2.3.2 Calculating the IR contributions ...... 50

2.4 Numerics ...... 55

2.4.1 HowhighcanΛbe? ...... 55

vii 2.4.2 SampleSpectra ...... 57 2.4.3 Estimationoftuning ...... 59

2.4.4 Lifetimeofthefalsevacuum ...... 61

3 Phenomenology of R-Symmetric Gauge Mediation 63

3.1 HiggsPotential ...... 63 3.2 Scan for λ =0...... 70

3.3 Turning on the λ terms...... 74

3.4 SampleSpectrum ...... 75

3.5 UniﬁcationoftheYukawas...... 79

3.6 Tuning...... 80

3.7 RelevantBetaFunctions ...... 81 3.8 FinalRemarks...... 82

4 Cosmological Constraints on Decaying Dark Matter 84

4.1 Motivation...... 84 4.2 DecayingColdDarkMatterCosmology ...... 86

4.3 Markov Chain Monte Carlo Results and Discussion ...... 92

4.4 Implications for Particle Physics Models with Decaying Cold Dark Matter 98

4.4.1 Spin-0DarkMatter...... 100

4.4.2 Spin-1/2DarkMatter ...... 103

4.4.3 Spin-1DarkMatter...... 109 4.4.4 General Dimensional Considerations ...... 113

4.5 CompendiumofDecayRates ...... 115

5 Conclusion 118

Bibliography 124

viii Chapter 1

Introduction

The Standard Model of particle has proven to be a successful description of Nature. In

this description, the matter and energy constituents are quarks and leptons (spin 1/2

particles described by fermion fields), and gauge fields that characterise the interaction among quarks and leptons (spin 1 particles described by vector fields). The strong

interactions are described by quantum chromodynamics, better known as QCD which

is an SU(3)c gauge theory, while weak and electromagnetic interactions are described in a single electroweak framework, an SU(2) U(1) gauge theory. Given the matter × Y content, the Standard Model Lagrangian can be realised by writing the most general renormalizable Lagrangian invariant under SU(3) SU(2) U(1) . These symmetries c × × Y are local, and hence the necessity of the gauge bosons to ensure gauge invariance. With these two principles -gauge invariance and renormalizability-, given the matter content, we are only allowed to write the kinetic terms of the Standard Model ﬁelds using covariant derivatives. Any mass term for the gauge bosons or for the fermions is forbidden by the symmetries.

However, since we know that fermionic and bosonic particles are massive (except for the photon and the gluons), we must have a mechanism to provide these particles a mass.

One proposal, the Glashow-Salam-Weinberg model, introduces a new scalar doublet in

1 Chapter 1. Introduction 2

the theory charged under SU(2) and U(1)Y . The introduction of this new scalar doublet Φ, known as the Higgs boson, must come with the corresponding gauge invariant kinetic term, a renormalizable potential, and since it is charged under SU(2) U(1) , we should × Y be able to couple it to the Standard Model fermions in a gauge invariant way. The main idea behind the model is that, through the Higgs potential, the Higgs will develop a nonzero vacuum expectation value v (we will see how this happens below), and thus break the electroweak symmetry down to U(1)em. As a result, gauge bosons, like the W and the Z will acquire a mass, and so do fermions. This is what we understand by electroweak symmetry breaking (EWSB). This procedure has proven to be quite successful, since it is consistent with measurements of the W and Z masses.

There are many aspects, besides the one mentioned above, of the electroweak theory that have been tested experimentally. Furthermore, hadron collisions have tested QCD in the perturbative regime. Therefore, overall, we can say that the Standard Model with the inclusion of the Higgs boson, is relatively successful as a description of Nature.

The Standard Model as it is, although successful, is clearly unaesthetic and incom- plete. One of the reasons for being dissatisﬁed with this model is that many of the parameters that the Standard Model requires need to be introduced “by hand” i.e. ﬁxed by experiment. Within the context of the Standard Model, there is no successful explanation that tells us how these parameters come about. For example, electroweak symmetry breaking allows the quarks and leptons to have a mass but it does not give us a concrete prediction for these masses. The “Yukawa couplings” of the quarks and leptons to the

Higgs boson have to be ﬁxed by measurements of the mass of these particles. Further- more, although the Higgs boson is a pleasant idea that explains how particles acquire a mass, we have not yet discovered such a particle. Indeed, one of the tasks of the Large Hadron Collider (LHC) is to unravel the mysteries of the electroweak scale and determine whether or not this fundamental particle exists.

Besides these aesthetic aspects of the Standard Model, we note that it is also incom- Chapter 1. Introduction 3

plete. For example, solar neutrino data strongly suggests that neutrinos have a mass, and the structure of the Standard Model does not allow us to give neutrinos a mass.

Furthermore, astrophysical and cosmological observations imply that there are unknown

types of energy and matter. The so-called “dark energy” is alluded by the observations of

type Ia supernovae. Also, measurements of the cosmic microwave background and of the

velocity of stars in the outer limits of galaxies suggest that there is a new form of matter

known as “dark matter”. Combined, dark energy and dark matter constitute 96% of the energy content of the universe (dark energy constitutes 74% while dark matter consti-

tutes around 22%). The nature of these forms of energy and matter is clearly far beyond

the reach of the Standard Model of particle physics. There are reasons to believe that

dark matter could be around the TeV scale. This makes the prospects for the LHC very

interesting since it is precisely this energy scale that it will be probing. Lastly, the Stan-

dard Model, as it is, does not incorporate gravity, which we know couples to all matter. Incorporating gravity into the particle physics framework has indeed proven to be one of

the most challenging intellectual endeavours of our time, perhaps mainly because of the

lack of experimental data at high energies, where quantum gravity becomes relevant.

Aside from all these problems that the Standard Model has, the introduction of a new fundamental scalar particle has theoretical challenges of its own. We will see in the discussion below that the corrections to the Higgs mass are quadratically divergent.

If the cutoﬀ of the theory is well above the electroweak scale, this posses a serious problem since the corrections to the physical mass of the Higgs would be well beyond the electroweak scale. This is where supersymmetry seems to give an appealing solution to this problem. Supersymmetry relates fermions and bosons in such a way that these quadratic corrections to the Higgs mass cancel exactly.

Besides being an extremely aesthetic theory, supersymmetry provides a solution to the aforementioned “naturalness” problem and many models of supersymmetry have a natural particle candidate for dark matter. Since the Minimal Supersymmetric Standard Chapter 1. Introduction 4

Model was constructed in the early 1980s there has been an enormous amount of papers and articles describing how supersymmetry cures certain problems, gives rise to other problems, and attempts to be a phenomenologically viable theory of Nature.

We have mentioned that there are, among others, two main goals for the LHC, namely, the possible discovery (or not) of the Higgs boson, and the possibility to investigate the nature of dark matter. A third endeavour is indeed to ﬁgure out whether supersymmetry exists. This could be determined by the discovery of supersymmetric partners and by the myriad of hypothetical supersymmetric signals that have been analysed even before the LHC was ready to start running. If supersymmetry is discovered, it will imply that, at least under our perception, Nature is more elegant than what we thought. By no means will the problems be over, but at least we would have taken another step, another glimpse into the understanding of our universe. We would then be left with the philosophical question, among many others, as of to why are symmetries indeed a good guiding principle for the description of Nature.

1.1 Supersymmetry

Supersymmetry is perhaps one of the most popular theories beyond the Standard Model that provides an elegant solution to the gauge hierarchy problem. It is well known that, upon accepting the hypothesis that there is a fundamental Higgs boson, the quantum corrections to the Higgs mass diverge. These corrections arise from calculating one loop diagrams that are quadratically divergent, i.e. proportional to Λ2, where Λ is the momentum cut-oﬀ of the theory.

To better illustrate the hierarchy problem, let us consider the scalar potential of the Higgs doublet

2 2 VHiggs = mH HH† + λ(HH†) . (1.1.1) Chapter 1. Introduction 5

When the electroweak symmetry of the Standard Model is broken, the Higgs acquires a vacuum expectation value (VEV) v = m2 /2λ, and the bare mass term of the − H 2 p potential, mHu , must be negative. Since this VEV ultimately provides a mass to the W and Z bosons, which can be experimentally measured, we know that the negative mass term must be of order m2 (100GeV)2. As mentioned before, the Higgs mass receives H ∼ − one loop corrections that are quadratically divergent. In particular, there are corrections containing Standard Model fermions in the loop. These corrections are as follows

c m2 m2 + Λ2 , (1.1.2) Hphys ∼ H 4π

2 where mHphys is the physical mass of the Higgs boson, c is just a constant that depends on the diﬀerent couplings of the Standard Model (the top Yukawa yt gives the largest contribution), and Λ is some large scale. If we assume that the fundamental scale of the theory is the Planck scale or some other uniﬁcation scale, then the corrections to the

2 Higgs mass are extremely large, much more than the aforementioned value for mH . This

2 is the essence of the hierarchy problem. It could very well be that mH is more negative, in which case we would still have a problem since there would be an enormous amount

2 2 of tuning between mH and Λ . Furthermore, unitarity arguments allow us to constrain the physical mass of the Higgs to less than 1TeV. Of course one can argue that the Higgs boson does not exist or even that the fundamental scale is much lower than the Planck scale or the uniﬁcation scale. However, for the purposes of illustration of supersymmetry, we will not concentrate on these possibilities.

To cure this hierarchy problem, one could imagine a scenario where the one loop quadratically divergent contributions to the Higgs mass cancel each other out due to the presence of new matter content. It turns out that fermionic loop contributions have a diﬀerent sign than those arising from bosonic loop contributions. What is desired is a symmetry relating fermions and bosons in such a way that the quadratic contributions cancel. This symmetry relating fermions and bosons is known as supersymmetry. In this Chapter 1. Introduction 6

way, supersymmetry can be thought of as a solution to the gauge hierarchy problem, by naturally cancelling large contributions to the Higgs boson even if the fundamental scale

is as high as the Planck scale.

Another motivation for supersymmetry is that in the supersymmetric extension of the Standard Model, known as the Minimal Supersymmetric Standard Model (MSSM), the gauge couplings seem to unify at a scale M 1016 GeV. This certainly does not U ∼ happen in the Standard Model. The MSSM seems to have the right particle content to

provide gauge coupling uniﬁcation.

One last motivation for supersymmetry comes from string theory. In string theory

there are many supersymmetric solutions. It is hard to have a reasonable ground state in

a non-supersymmetric vacuum. Consequently, if string theory is a fundamental theory of Nature it is very likely that low energy supersymmetry might be a possibility.

1.1.1 Supersymmetry formalism

Now that we have motivated supersymmetry as a possible theory of Nature, let us describe

some of the technical aspects related to this theory. One practical way to work with supersymmetric field theories is to make use of the superfield formalism. Superfields are

particularly useful since they contain, in a very compact notation, information about the

corresponding bosonic and fermionic states. In this discussion we will be concerned with

two types of superﬁelds, chiral and vector superﬁelds. Without going into much detail,

a chiral superﬁeld has the following form

2 Φ= φ(x)+ √2θψΦ(x)+ θ FΦ , (1.1.3)

where φ is the scalar component of the superﬁeld, ψΦ is the fermionic component, θ is a Grassmann variable, and FΦ is an auxiliary scalar term that has no kinetic term Chapter 1. Introduction 7

and allows the symmetry algebra to close oﬀ-shell1. If we perform a supersymmetric transformation on this chiral superﬁeld, its components transform as

δφ = √2ζψΦ , (1.1.4)

m δψΦ = √2ζFΦ + √2iσ ζ∂¯ mφ , (1.1.5)

δF = √2i∂ ψ σmζ¯ , (1.1.6) Φ − m Φ where ζ is an anticommuting parameter. It can be shown [3] that the product of two chiral superfields is a chiral superfield, and that the θ2 component of a chiral superfield is supersymmetric. It can also be shown, that

2 2 the θ θ¯ component of Φ†Φ contains the bosonic and fermionic supersymmetric kinetic terms. So, for example, a general renormalizable supersymmetric Lagrangian has the form

4 2 2 3 = d θΦ†Φ+ d θ λΦ+ mΦ + gΦ , (1.1.7) L Z Z where we have made use of the compact notation that extracts the desired terms from the superﬁelds, i.e. integrating over d2θ gives the θ2 component of the superﬁeld in question.

The ﬁrst term in 1.1.7 is usually referred as the K¨ahler potential and the other terms are referred to as the superpotential.

The other kind of superfield we are concerned with is the vector superfield. A vector superfield satisfies the condition V = V †. If one uses a particular gauge (the Wess-Zumino gauge) one can simplify the general expression for V as follows

1 V = θσmθv¯ iθ¯θθλ¯ + iθθθ¯λ¯ + θθθ¯θ¯(D + i∂ vm) , (1.1.8) − m − 2 m

where vm is the gauge ﬁeld, λ is its supersymmetric partner, namely the gaugino, and D is another auxiliary ﬁeld analogous to F . From V , using supercovariant derivatives, one

1Terms with higher order in θ vanish due to the Grassmann algebra. Chapter 1. Introduction 8

can construct a chiral gauge invariant superﬁeld W , which has the following component form

i ˙ W = iλ + θ D (σmσ¯nθ) F + θθσm ∂ λ¯β. (1.1.9) α − α α − 2 α mn αβ˙ m From this we can construct the supersymmetric generalisation of the Lagrangian for a

free vector ﬁeld. It has the form [3]

1 1 α˙ = dθ2W αW + dθ¯2W¯ W . (1.1.10) L 4 α 4 α˙ Z Z After expanding we have

1 1 = D2 vmnv iλσm∂ λ¯ , (1.1.11) L 2 − 4 mn − m where v is the usual gauge invariant term for a kinetic field v = ∂ v ∂ v . Finally, mn mn m n − n m we notice that the Kähler term in the Lagrangian in (1.1.7) is not invariant under local transformations. For it to be invariant, we need to introduce the vector superfield V

4 gV = d θΦ†e Φ. (1.1.12) L Z

1.1.2 Minimal Supersymmetric Standard Model

With these tools in place, we can now proceed to supersymmetrize the Standard Model.

We start by simply promoting the Standard Model ﬁelds to superﬁelds. All the matter

fields of the Standard Model, including the Higgs, are going to be chiral superfields and the usual vector fields are going to be described by vector superfields. The names of scalar partners of the quarks and leptons are going to be preceded by an “s”, for example selectron, stop, etc. The spin 1/2 partners of the spin 1 particles, like the gluon and the gauge bosons, are going to be referred as gluino, gaugino, etc. It is important to note that we will work with Weyl fermions unless otherwise specified. Chapter 1. Introduction 9

Promoting the Higgs field to a superfield must be done carefully. We are forced to introduce another Higgs field in the theory for two reasons. The first reason is that in

the Standard Model (SM) the Higgs ﬁeld that gives a mass to the up-type fermions is a

conjugate field of the form iσ2H†. In supersymmetry however, there is a property that demands that the terms in the superpotential must be holomorphic (or analytic) in the chiral superfields. This forbids the presence of conjugate superfields in the superpotential.

The second reason has to do with gauge anomalies. When promoting the Higgs ﬁeld to a superﬁeld, the corresponding fermionic components i.e., the Higgsinos, destroy the anomaly cancellation that exists in the SM. The anomaly cancellation demands that

3 2 Tr(Y ) and Tr(T3 Y ) vanish identically, where Y is the hypercharge and T3 is the third component of weak isospin. The Higgsinos would now have a hypercharge of Y = 1/2 which gives a non-zero contribution to the triangle anomalies. This can be solved if another Higgs supermultiplet with hypercharge Y = 1/2 is introduced in the theory. − So, in either case, the structure of the supersymmetric theories and the cancellation

of anomalies requires the introduction of a new Higgs supermultiplet. We will call the

Higgs supermultiplets Hu since it couples to up-type quarks and Hd since it couples to down-type quarks.

Without further discussion, the superpotential of the Minimal Supersymmetric Stan- dard Model (MSSM) is as follows

W = yuUQH ydDQH yeELH + µH H . (1.1.13) MSSM u − d − d u d

In this compact notation we have suppressed all the colour and weak isospin indices.

α α β Thus, for example, µHuHd should be understood as µHu Hdα = ǫαβHu Hd . Table 1.1 shows the diﬀerent ﬁelds of the MSSM with their charge assignments. Notice that the

last term in equation (1.1.13) is allowed by the symmetries. If supersymmetry were an exact theory of nature, then all the SM particles and

their superpartners would have the same mass. We have certainly not discovered any Chapter 1. Introduction 10

Superﬁelds SU(3) SU(2) U(1)Y

uL Q = 3 2 1   6 dL U  3¯ 1 2 − 3 ¯ 1 D 3 1 3

νeL L = 1 2 1   − 2 eL E  1 1 1

+ Hu 1 Hu = 1 2  0  2 Hu  H0  H = d 1 2¯ 1 d   − 2 Hd−   Table 1.1: Superﬁelds and their corresponding charges in the MSSM. U, D, and E refer to right handed ﬁelds. The R subscript is suppressed for simplicity. superpartners, so supersymmetry must be broken. Moreover, we want supersymmetry to be broken softly. That is, we want to break supersymmetry without re-introducing the dangerous quadratic divergences. Supersymmetry breaking terms that are added to the theory and do not re-introduce quadratic divergences are called soft breaking terms. These soft terms must be consistent with the symmetries of the theory (the MSSM in our case), and must introduce corrections to the Higgs mass that are at most logarithmic.

The soft supersymmetry breaking terms that can be present in the MSSM are gaugino mass terms for all the gauginos

1 = M λ λ , (1.1.14) Lgaugino −2 i i i where i = 1, 2, 3, one for each gauge group; non-holomorphic masses for all the scalar

ﬁelds including the Higgs ﬁelds Chapter 1. Introduction 11

2 2 = m H†H m H†H (1.1.15) Lscalar − Hu u u − Hd d d 2 i j 2 i j (m ) Q˜†Q˜ (m ) L˜†L˜ (1.1.16) − Q j i − L j i 2 i j 2 i j 2 i j (m ) U˜ †U˜ (m ) D˜ †D˜ (m ) E˜†E˜ . (1.1.17) − U j i − D j i − E j i

Notice that Hu and Hd in this case represent the scalar components of the superﬁelds

2 only ; the so-called Bµ term or holomorphic mass term

= B H H + h.c. ; (1.1.18) LBµ − µ u d and lastly, trilinear terms (also called A terms) that are consistent with the symmetries

= (A ) Q˜iH U˜ j (A ) Q˜iH D˜ j (A ) L˜iH E˜j h.c.. (1.1.19) LA − u ij u − d ij d − e ij d −

As one can see, the addition of these soft breaking terms introduces a lot of new arbitrary parameters in the MSSM Lagrangian. There are new sources of ﬂavour violation, new phases, and mixing angles that cannot be rotated away. Most of these problems are not present in the Standard Model.

For example, in the Standard Model the Flavour Changing Neutral Currents (FCNC) are forbidden at tree level but there are small contributions at one loop. To illustrate this, one can examine the Kaon system. The mass diﬀerence of KL and KS, ∆mK , can be computed from box diagrams like the one in Figure 1.1. As a matter of fact, the computation of the mass diﬀerence was used to predict the mass of the charm quark.

12 The experimental value of ∆m is extremely small, ∆m 3.5 10− MeV, that is K K ∼ × the mixing or interconversion of K0 and K¯ 0 is highly suppressed. This is consistent with

SM calculations. However, in supersymmetry there can be additional contributions to

K K¯ mixing that arise from box diagrams as well but with squarks and gluinos − 2The notation with the tilde on top of the ﬁeld refers to the scalar component of the superﬁeld. Chapter 1. Introduction 12

W − s¯ s

u, c, t u, c, t

d d¯ W +

Figure 1.1: Box diagram that contributes to FCNC.

˜∗ ˜∗ sR dR s¯ s

g˜ g˜

d d¯ ˜ dR s˜R

Figure 1.2: Box diagram that contributes to FCNC in the MSSM. in the loop. One of such diagrams is in Figure 1.2. If the oﬀ-diagonal terms of the mass matrices in (1.1.15) are of order one, then one can have an excessive mixing due to these kinds of contributions. In particular, one concludes that, for ∆mK to be consistent with experiment, one needs the term in the squark mass matrix that mixess ˜ and d˜ to be

2 inexplicably small, of order 10− for general 500 GeV squarks [4]. It is clear from this ∼ example that the soft parameters that were added to the MSSM introduce new sources of flavour violation that do not concur with experimental results. This is the so-called supersymmetric flavour problem. Additionally, there are very strong constraints on CP violation and on lepton flavour violation that further restrict the size of other off-diagonal terms in the soft Lagrangian. For example, large off-diagonal terms in the slepton sector can result in large decay rates for µ eγ. Also, measurements on the electron and → neutron dipole moment further constrain any CP violating parameters which are clearly present in the soft Lagrangian.

Thus, we see that soft breaking terms cannot be arbitrary for the MSSM to be con- Chapter 1. Introduction 13

sistent with experiment. One way to circumvent this problem is by assuming that all squark matrices are proportional to the identity matrix, that the A trilinear matrices are proportional to the usual yukawa matrices of the SM, and that all sources of CP violation are turned to zero (except for the usual CP-violating phase in the SM). This universality hypothesis solves at once many of the problems that emerge in the usual soft supersymmetry breaking story. Usually one can assume that this hypothesis is valid at some scale ΛUV and that the parameters are run down to the electroweak scale. Ideally, the troubling parameters would be kept reasonably small, like the oﬀ-diagonal terms in the squark mass matrices. Surely, this universality hypothesis must be explained somehow. The purpose of gauge mediation is to try give at least a partial explanation in which ﬂavour violating parameters are kept reasonably small.

1.2 Gauge Mediation as a solution to the Supersym-

metric Flavour Problem

Before presenting the main ideas of gauge mediated supersymmetry breaking (GSB) let us describe some generalities of spontaneous supersymmetry breaking (SSB) that will be useful. Ultimately, we want supersymmetry to be broken spontaneously. That is, we want the Lagrangian of the theory to be supersymmetric but not in a vacuum state.

There are two general types of SSB, F -type and D-type supersymmetry breaking. If supersymmetry is not broken in the vacuum state, then the Hamiltonian and hence the scalar potential of the theory will be zero at the vacuum, i.e. 0 = 0. However, if H| i supersymmetry is broken, the vacuum state will not be supersymmetric and the scalar potential V will have to be greater than zero. Therefore, the condition for supersymmetry breaking translates into a scalar potential V that is not zero at the vacuum state. In order to ﬁnd the scalar potential of the theory we must ﬁnd the corresponding F and D terms. The scalar potential of a supersymmetric theory is given by Chapter 1. Introduction 14

i 1 V = F F † + D D , (1.2.1) i 2 a a where the F terms and the D terms can be found as follows

∂W F i = † , (1.2.2) ∂Φi†

Da = gaφ†Taφ. (1.2.3)

Here Φ is a chiral superﬁeld, g a gauge coupling, and φ the scalar component of the

superﬁeld Φ. We see trivially, that if either the F terms or the D terms are not zero

at the vacuum, the potential will not be zero, and hence supersymmetry will be broken.

In the MSSM there is no gauge singlet candidate that can somehow develop a vacuum

expectation value (VEV) to break supersymmetry spontaneously. Therefore, we need to

extend the MSSM by breaking supersymmetry in some hidden sector that has little or

no interaction with the observable sector and communicate it down to the visible sector. Eventually we want to generate the soft terms that were discussed in the previous section.

One way to achieve this goal is through gravity mediated supersymmetry breaking.

Gravity is a natural candidate to communicate the supersymmetry breaking since it is a

universal interaction and it couples to the visible sector as well as to the hidden sector.

In this case, the soft supersymmetry breaking masses msoft are suppressed by the Planck scale as

Fhidden msoft h i , (1.2.4) ∼ Mp where F is a supersymmetry breaking VEV. To give a sense of the scales, if we h hiddeni take m 1TeV, then F (1011GeV)2 . However, one of the main problems soft ∼ h hiddeni ∼ with this approach is that since we do not have a knowledge of the theory beyond the

Planck scale, the Lagrangian is rather arbitrary. In particular, there is no reason to Chapter 1. Introduction 15

believe that there are ﬂavour symmetries at the Planck scale and thus, we have a serious conﬂict with experimental data. Besides this description requires a better understanding

of gravity than what we currently have.

An alternative way of communicating the supersymmetry breaking to the visible sec-

tor is gauge mediated supersymmetry breaking. In this scenario, new chiral superﬁelds, the messengers, have to be introduced. These messengers couple to the hidden sector

and to the visible sector through the usual gauge couplings of the MSSM. Since gauge

interactions are ﬂavour blind, we expect that the soft terms generated by this mechanism

will be ﬂavour preserving. To understand better how GMSB works let us show a simple

example of this framework [5].

We consider two chiral superﬁelds which will play the role of the messengers, Φ and

Φ,¯ that transform under the fundamental and antifundamental representation of SU(5)

respectively. The MSSM is considered to be embedded in SU(5).3 We also consider an

additional chiral superﬁeld S, whose scalar component s, and its auxiliary component FS acquire a VEV. Notice that there is no assumption on how these VEVs are generated, it is simply assumed that this is the case. The superpotential for this minimal model of

gauge mediation is

Wgm = λgSΦΦ¯ , (1.2.5)

2 where λg is some coupling constant. Integrating over d θ and adding the scalar potential, the Lagrangian looks like

= F φφ¯ + h.c. + λ s 2 φ 2 + φ¯ 2 + λ s ψ ψ¯ + ... (1.2.6) L h Si | gh i| | | | | gh i Φ Φ The fermionic masses of the messengers can readily be seen to be λ s . The scalar gh i masses squared are λ s 2 λ F . | gh i| ±| g S| 3In the usual MSSM, the gauge couplings seem to unify at a large scale. By having messengers that are complete multiplets of a uniﬁed gauge group SU(5) we do not spoil this gauge coupling uniﬁcation. Chapter 1. Introduction 16

λ ψφ λ

Figure 1.3: In Gauge Mediation gaugino masses are generated at one loop level. The ﬁelds in the loop are the messengers.

g g φ

q˜ φ q˜

Figure 1.4: Scalar masses are generated at two loops. Except for the gauge bosons, the ﬁelds in the loop are the messengers.

Gaugino masses, Ma, arise radiatively at one loop order. To see how this happens let us take a look at the guage invariant term of the K¨ahler potential. Expanding the gauge invariant K¨ahler potential we have [3]

4 gV n 1 n i i dθ Φ¯e Φ gv ψ¯ σ¯ ψ + φ∗∂ φ ∂ φ∗φ (1.2.7) ⊃ 2 Φ Φ 2 n − 2 n Z i g φλ¯ψ¯Φ φ∗λψΦ , (1.2.8) −√2 − where λ is the gaugino and V is the vector superﬁeld. We must take into consideration a similar part in the MSSM with the same gauge interaction but with the corresponding matter ﬁelds. From this Lagrangian, it is straightforward to construct the diagram that gives gauginos a mass. This diagram is shown in Figure 1.3. The resulting gaugino mass Chapter 1. Introduction 17

is approximately given by

α F M a h Si. (1.2.9) a ∼ 4π s h i

The scalar masses of the MSSM arise at two loop order. There are a total of eight diagrams that contribute. Figure 1.4 shows an example of such diagrams. To a good approximation the scalar masses are given by

F 2 α 2 m˜ 2 h Si 2C a , (1.2.10) ∼ s a 4π a h i X

where Ca is the quadratic Casimir invariant of the gauge group. We observe from these equations that the gaugino mass and the scalar masses are approximately of the same order of magnitude since the expression in (1.2.10) is for the square of the scalar masses.

The trilinear terms also arise at two loop level, but since they are suppressed by an

αa additional factor of 4π , we can, to a ﬁrst approximation, neglect them. All these soft supersymmetry breaking parameters arise at the messenger scale and they have to be run down to the electroweak scale. Once that is done the Bµ and µ terms can be computed by demanding that the potential of the theory has a true minimum.

In summary, gauge mediation allows us to break supersymmetry in a hidden sector and communicate it to the MSSM by gauge interactions. Consequently, we get a natural solution to the flavour problem in supersymmetry since the soft terms produced are flavour diagonal. Phenomenologically, the gaugino and the scalar masses are of the same order of magnitude, and the A terms are relatively small. As a final note, if we want the superpartner masses to be consistent with experiment, i.e. heavier than 100GeV, the ratio F / s must be bigger than 1 TeV. h Si h i Chapter 1. Introduction 18

1.3 R-symmetry in Supersymmetry

The symmetries of the MSSM ﬁelds outlined in Table 1.1 allow us to write more terms in the superpotential than the ones mentioned in equation (1.1.13). In particular we could, in principle, add the following terms to the superpotential

W = λLQDLQD + λLLDLLE + µLH LHu + λUDDUDD. (1.3.1)

The ﬁrst three terms clearly violate lepton number and the last one violates baryon number. Furthermore, the ﬁrst and the last terms are particularly troublesome since if

λLQD and λUDD are of order one, we could have a process that leads to proton decay of the form p+ e+π0. The decay time would be extremely small compared with the → experimental constraint of 1032 years. Moreover, these baryon and lepton symmetries were accidental symmetries of the Standard Model. It seems that in the MSSM we would have to impose those symmetries for experimental consistency. One way around this issue is to enforce a new symmetry called R-parity or matter parity under which the lepton and quark superﬁelds are odd, and the Higgs, gauge bosons, and gaugino are even. This R-parity can be deﬁned by

3(B L)+2s R =( 1) − , (1.3.2) − where s is the spin of the ﬁeld. Under this parity, all terms from equation (1.3.1) are forbidden and the usual terms of the MSSM superpotential are still allowed. Additionally, the R-parity for all SM particles is even and odd for their corresponding superpartners.

This immediately tells us that when superpartners are produced in collider experiments, they must be produced in even numbers, otherwise there would be a violation of R-parity.

Moreover, the lightest supersymmetric partner (LSP), arises naturally because it has an odd charge and cannot decay to any SM particles. This could be an interesting candidate for dark matter. Of course, it may be the case that R-parity is not exact and it is slightly Chapter 1. Introduction 19 broken, which could lead to dark matter decay. We analyse this possibility in chapter 4.

R-parity is nothing more than discrete Z2 subgroup of a continuous global symmetry

U(1)R. Under this symmetry the chiral superﬁelds transform as [6]

iR α iα RΦ(θ)= e Φ Φ(θe− ) , (1.3.3) and the vector superﬁeld as

iα RV (θ)= V (θ− ). (1.3.4)

Since we must integrate over d2θ to obtain the Lagrangian from the superpotential, all superpotential terms must have an R-charge of +2.

Supersymmetry breaking and R-symmetry are connected in a particular way. N.

Seiberg and A. Nelson showed in [7] that generically, a spontaneously broken R-symmetry is a suﬃcient condition for supersymmetry breaking. They also showed that generically, if supersymmetry is broken, then the superpotential must have an R-symmetry. To illustrate this point, we follow the lines of [7,8]. Consider a superpotential with R charge

+2, and suppose that the ﬁeld φn acquires an expectation value and has an R charge qn. We can write the superpotential as

2/qn φi W = φ f(Xi), Xi = , (1.3.5) n qi/qn φn where f is a function of n 1 variables X . If we want supersymmetry to be unbroken, − i then the F terms must be zero

∂f =0, f =0. (1.3.6) ∂Xi

These are n equations for n 1 variables and they cannot be satisﬁed for a generic f, − thus supersymmetry is broken. Now let us prove that the existence of an R-symmetry is Chapter 1. Introduction 20

necessary for supersymmetry breaking. If there is no R-symmetry then the F terms of the superpotential

∂W =0 , (1.3.7) ∂φi

represent n conditions for n variables φi, and thus, generically there is a solution to these conditions, therefore supersymmetry is unbroken. This argument applies even if we have a continuous non-R-symmetry [7, 8]. The relation between R-symmetry breaking and

supersymmetry breaking points to a potential problem in supersymmetry for the following

reason. If the U(1)R symmetry is broken spontaneously, it implies the existence of a massless Goldstone boson known as the R-axion. This is troublesome phenomenologically

since there are strong constraints on massless particles. One then would be inclined to

believe that the R-symmetry should be explicitly broken. But since we have mentioned that the existence of an R-symmetry is a necessary condition for supersymmetry breaking, then supersymmetry would not be broken in Nature. One can obviously get around these arguments by postulating that the superpotential is not generic or that the underlying theory does not possess an R-symmetry, which emerges as an accidental symmetry in the eﬀective Lagrangian [8]. However, these ideas restrict the type of models we can work with.

This dilemma can be addressed through the approach of dynamical supersymmetry breaking in meta-stable vacua. If we assume that there is a fundamental theory of Na- ture describing Planck-scale physics, we would still have to explain the appearance of much smaller terms in the soft Lagrangian compared to the Planck scale. In dynamical supersymmetry breaking, the dynamics at some high scale are responsible for breaking supersymmetry. In this case supersymmetry breaking occurs through nonperturbative eﬀects and the resulting scale (the soft terms in the Lagrangian) are exponentially suppressed compared to the Planck scale Mp through dimensional transmutation. In other words, the fact that m M can be successfully explained in this context. soft ≪ p Chapter 1. Introduction 21

Due to reasons that will not be discussed here, until recently, it was believed that dynamical supersymmetry breaking was rather non-generic. Many of the theories that

break supersymmetry dynamically were complicated and they did not lead to phenomeno-

logically viable models of supersymmetry. In 2006, Intriligator, Seiberg, and Shih [1]

abandoned the requirement that models of dynamical supersymmetry breaking have no

supersymmetric vacua. Instead they proposed a long lived, false, meta-stable vacuum.

This leads to many, much simpler, phenomenologically viable models of supersymmetry breaking. We will refer to these models from now on as ISS models (Intrilligator, Seiberg,

and Shih).

We discussed above that there was an intimate relation between supersymmetry breaking and a spontaneously broken R-symmetry. We can explore the consequences of this relation in light of meta-stable theories of supersymmetry breaking. As pointed out earlier, to construct a viable model of supersymmetry breaking, the gauginos must have a nonzero mass, and therefore the R-symmetry must be broken. One possibility is that the

U(1)R symmetry is broken spontaneously, however in this case we would have a massless R-axion which is ruled out experimentally. To solve this conﬂict we can imagine a theory

with an approximate R-symmetry whose breaking is controlled by a small parameter ǫ.

In this context, if ǫ = 0, the R-symmetry is restored and supersymmetry is broken. If ǫ is nonzero but small, the theory is slightly deformed, and the non-supersymmetric states are still present. However, because now we do not have an exact R-symmetry, according to the previous discussion, there are supersymmetric ground states. These supersymmetric ground states are separated from the non-supersymmetric states by a barrier of width

1/ǫ in ﬁeld space, and in general it is not diﬃcult to guarantee that the lifetime of the latter is longer than the age of the universe. In this sense we say that supersymmetry is broken in a meta-stable vacuum.

By having a meta-stable vacuum in the theory we can incorporate the two phenomenological requirements in our models, namely that supersymmetry is broken without an Chapter 1. Introduction 22 exact R-symmetry. We must point out that if our universe is in a meta-stable vacuum, in the usual ISS models, we must still have suﬃciently strong R-symmetry breaking in the meta-stable state in order to give gauginos a large enough Majorana mass. Nor- mally, models have to be expanded to assure strong enough breaking of the R-symmetry, usually by adding additional singlets to the theory further complicating their structure.

This is one reason why it makes sense to construct a phenomenological model where the R-symmetry is unbroken in the meta-stable supersymmetry-breaking vacuum, up to (ǫ) terms. We will discuss this model in chapter 2. O

The main example of [1] consists of supersymmetric SU(Nc) QCD. The small parameter ǫ is given by

m ǫ , (1.3.8) ∼ Λ r where Λ is the strong-coupling scale and m is the scale of the quark masses. Consider an

O’Raifeartaigh model (which is an eﬀective “magnetic” description of supersymmetric

3 QCD for Nf = Nc + 1 and it is gauged for Nc +1 < Nf < 2 Nc), with the following superpotential

W = Trϕi Φ ϕ˜jc µ2TrΦ , (1.3.9) c ij − ij where ϕ andϕ ˜ are in the fundamental and antifundamental representations of SU(Nf ), Φ is in the adjoint representation, and µ2 mΛ . Notice that i runs through the number ∼ of ﬂavours Nf , and c = 1...Nc for Nc < Nf . We can readily compute the F term of the Φ chiral superﬁeld

F = ϕi ϕ˜jc µ2δij. (1.3.10) Φ c −

Not all of these terms can simultaneously vanish because of the rank condition, since the

2 term proportional to µ has rank Nf and the ﬁrst term has rank Nc < Nf . Therefore, Chapter 1. Introduction 23

supersymmetry is spontaneously broken. Some of the ﬁelds in (1.3.9) are Goldstone bosons of broken global symmetries and they are exactly massless. There are also massless

pseudo-moduli that receive quantum corrections through the one loop eﬀective potential.

It is in this spirit that a model that incorporates the ideas of ISS and gauge mediation

will be presented in chapter 2. Essentially it is a model of direct gauge mediation where

SU(6) is spontaneously broken to SU(5). The concepts of ISS and gauge mediation will be used to generate the ultraviolet completion of the Minimal R-symmetric Standard Model (MRSSM) that we discuss in the following section.

1.4 Minimal R-symmetric Standard Model

It was only recently realised [2] that a new universality class of supersymmetric particle physics models, characterised by an extra R-symmetry is not only phenomenologically viable, but also helps to signiﬁcantly alleviate the supersymmetric ﬂavour problem and has novel signatures at the TeV scale. A model with an exact R-symmetry, called the

“Minimal R-symmetric Supersymmetric Standard Model” (MRSSM), was constructed in

[2]. It was unexpectedly shown, that with the imposition of the new symmetry signiﬁcant

flavour violation in the sfermion sector is allowed by the current data. This is even the case for squarks and sleptons with mass of a few hundred GeV, provided the Dirac gauginos are sufficiently heavy. Stronger bounds on the allowed flavour violation, obtained by including the leading-log QCD corrections, were subsequently given in [9]. The Dirac nature of gauginos and Higgsinos and the possibility of large sfermion flavour violation in the MRSSM both present a departure from usual supersymmetric phenomenology.

One could wonder why is it that an unbroken R-symmetry has not been used as much in the construction of supersymmetric models. One reason is that the gaugino masses as well as the Higgsino masses in the presence of a Bµ term, require the R- symmetry to be broken. This is mainly because the gaugino masses are Majorana, Chapter 1. Introduction 24

a problem that can be eliminated if the gaugino masses were Dirac. Another reason has already been discussed in the previous section. Namely, that in many models of

dynamical supersymmetry breaking, the R-symmetry is also broken. However, we have

already argued in the previous section, that certain models of supersymmetry breaking

can preserve the R-symmetry. Finally, in the supergravity framework one adds a constant term to the superpotential in order to tune the cosmological constant to zero. This constant explicitly breaks the R-symmetry and guarantees that the R-axion that arises from a spontaneously broken R-symmetry is given a mass. Although these are acceptable reasons for not using an unbroken R-symmetry in supersymmetric models, we can modify the content of the MSSM to make it R-symmetric and explore the consequences.

In the context of the MRSSM both F -type and D-type supersymmetry breaking is

2 allowed. We can write these terms as spurions X = θ F and Wα′ = θαD, where the R-

charge assignments +2 and +1 respectively. The W ′ can be considered a hidden sector

U(1)′ that acquires a D-term. We further assume that the quark and lepton superﬁelds

have R-charge +1, the Higgs R-charge 0, and the gauge superﬁelds Wi have R-charge +1. We notice that by imposing this symmetry the MRSSM has the following properties [2]:

Majorana gaugino masses are forbidden. •

The µ-term, and hence Higgsino mass, is forbidden. •

A-terms are forbidden. •

Left-right squark and slepton mass mixing is absent since the A-terms and the µ • term are forbidden.

The operators in (1.3.1) that violate lepton and baryon number by one unit are • forbidden.

Proton decay through dimension-ﬁve operators, QQQL and UUDE, is forbidden. •

Majorana neutrino mass, H H LL, is allowed. • u u Chapter 1. Introduction 25

Gaugino masses can be Dirac instead of Majorana. This is achieved by adding an adjoint

chiral superﬁeld Φi to the theory (one for each gauge group). The operator that leads to a gaugino mass is [10,11]: W d2θ α′ W αΦ . (1.4.1) M i i Z susy This operator produces a Dirac mass of the form mλψ, where m D/M . ∼ susy Since the µ term is forbidden, we have to extend the Higgs sector to give mass to the

Higgs and the Higgsinos. Two chiral multiplets Ru and Rd that transform in the same

way as Hd and Hu respectively, are added to the theory. The only diﬀerence is that these new superﬁelds have R-charge +2. We can then write the appropriate mass terms

Wµ = µuHuRu + µdHdRd . (1.4.2)

It is important to mention that unlike the usual Higgses, these new chiral superﬁelds

do not acquire expectation values that break the electroweak symmetry. This is crucial since we want to preserve the R-symmetry.

Other terms that involve the Higgs ﬁelds and the R-Higgses, that are consistent with

the symmetries are

i i WΦ = λuHuΦiRu + λdRdΦiHd , (1.4.3) ˜ ˜ i=XB,W where i = B,˜ W˜ refer to the couplings of the U(1)Y or SU(2)L adjoints, respectively.

The soft terms can be generated through F -type supersymmetry breaking. For ex-

ample, the Bµ term can be generated through

X X d4θ † H H , (1.4.4) M 2 u d Z susy while the usual soft masses for squarks, sleptons, Higgses, R-Higgses and for the scalar chiral adjoints Φi are allowed through operators of the type Chapter 1. Introduction 26

Field (SU(3)c, SU(2)L)U(1)Y U(1)R

QL (3, 2)1/6 1

UR (3¯, 1) 2/3 1 −

DR (3¯, 1)1/3 1

LL (1, 2) 1/2 1 −

ER (1, 1)1 1

ΦB˜ (1, 1)0 0

ΦW˜ (1, 3)0 0

Φg˜ (8, 1)0 0

Hu (1, 2)1/2 0

Hd (1, 2¯) 1/2 0 −

Ru (1, 2¯) 1/2 2 −

Rd (1, 2)+1/2 2

Table 1.2: Matter and R-charges in the R-symmetric supersymmetric model.

4 X†X d θ Q†Q , (1.4.5) M 2 i j Z susy where only the soft terms for the squarks is shown but the operator is also applicable for the other ﬁelds.

Finally, holomorphic masses for the scalar adjoints are allowed through

β X X W ′ W ′ d4θ † tr Φ2 + d2θ β tr Φ2 . (1.4.6) M 2 i M 2 i Z susy Z susy Table 1.2 summarises the ﬁeld content of the MRSSM as well as the matter and R-charges. As we have stated in section 1.1, in the MSSM processes that contribute to K K¯ − and B B¯ mixing are enhanced due to box diagrams with gluinos and squark mixing in − the loops. Looking at Figure 1.2, we realise that if the oﬀ-diagonal terms of the squark

mass matrices are of order one, then a diagram like this can contribute excessively to the

process, well beyond experimental limits. For example, if the gluinos and the squarks Chapter 1. Introduction 27

are of the same order of magnitude 500GeV, then the ratio of the squark mass matrix ∼ δ =(m2) / m2 must be δ < .06 in the best case scenario that δ = 0 [2]. ij q˜ ij | q˜| LL RR,LR There are two eﬀects that contribute to the suppression of ﬂavour-violating processes

in the MRSSM. First, the squark masses receive ﬁnite loop corrections from Dirac gaug-

inos in the loop. This is not the case for Majorana gauginos where the corresponding

squark masses are logarithmically enhanced. Therefore, Dirac gauginos can be naturally

heavier than the squarks by a factor of 10. Second, the R-symmetry forbids dimension ﬁve operators that would contribute to K K¯ mixing after integrating out the gluinos −

1 ˜ ¯ dR∗ s˜L∗ dRsL. (1.4.7) Mg Instead the leading operator that respects the R-symmetry is dimension six, like the

following

1 ˜ ¯ µ 2 dL∂µs˜L∗ dLγ sL. (1.4.8) Mg 2 2 2 Diagrams formed with these kinds of operators lead to a suppression of m /m 10− , q˜ g˜ ∼ (we have mentioned that the gauginos can naturally be a factor of 10 heavier than the

squarks). These effects sufficiently suppress the box diagrams even if we have (1) O flavour-violating terms in the squark mass matrices. The same arguments apply for B meson mixing. Amusingly, since the flavour-violating terms could be of order one, these could lead to interesting phenomenology where, although flavour violation is suppressed in the known low-energy particle physics, there could still be significant flavour violation in the squark sector, which could be observable at colliders.

Other ﬂavour-violating observables like µ eγ and b sγ arise at one loop in the → → MSSM. Typically, these diagrams involve a Majorana gaugino in the loop with a helicity

flip. In the case of Dirac gauginos the diagram with helicity flip on the gaugino is absent since the opposite helicity state has no couplings to sleptons [11,12]. As a result there is a smaller diagram that contributes with the helicity flip in the internal line where there Chapter 1. Introduction 28

is gaugino-Higgsino mixing, or on the external lepton or quark line. This means that the calculations resulting from these diagrams are consistent with experimental constraints

even with large ﬂavour violation in the squark or slepton sector. Here we did not show

all the diagrams that contribute to these processes in the MSSM versus the MRSSM.

However, in general we can say that the constraints on these processes are weaker due to

the absence of the A terms, the heavier gauginos, and the Dirac nature of the gauginos.

Needless to say, there are also constraints on the imaginary parts of the ﬂavour violation in supersymmetry like ǫK , the CP violating observable ǫ′/ǫ, constraints on the electric dipole moments, and constraints coming from strong CP. Although these constraints can also be analysed within the framework of the MRSSM, they will not be mentioned here.

As a ﬁnal note, we point out that all the beneﬁts that arise from having a continuous

U(1)R symmetry arise also if we have a Z6 subgroup of this continuous R-symmetry.

These benefits are also present even if the subgroup is Z4, except that proton decay operators would be allowed in such case. In [2] the authors even consider the effects of a weak breaking of the R-symmetry to Z2, and they mention that to some extent the severity of the supersymmetric flavour problem is reduced in this context.

In summary, the MRSSM provides an elegant solution to the flavour problem in supersymmetry. This is achieved by imposing a continuous U(1)R symmetry and modifying the matter content of the MSSM. As we have seen, new fields have to be introduced to give gauginos a Dirac mass and to have a viable Higgs sector. This seems to be the price that one needs to pay to preserve the R-symmetry. As a consequence, one can have order one flavour-violating soft masses and still be consistent with experimental constraints.

Phenomenologically this could lead to interesting signals in the squark sector. Moreover, the heavy gauginos make this model quite distinct from other models of supersymmetry.

In chapter 2 we construct a UV model to realise the MRSSM in the framework of gauge Chapter 1. Introduction 29 mediated supersymmetry breaking. We explore the possibility that the R-symmetry is preserved within the context of ISS models by having gauginos that have Dirac masses instead of Majorana. In a few words, we constructed a model of R-symmetric Gauge

Mediated Supersymmetry Breaking. We considered two speciﬁc examples of the supersymmetry breaking sector: a version of ISS and a more general O’Raifeartaigh model.

We provided sample spectra in which the gauginos are sufficiently heavy. Additionally, since there are flavour non-diagonal UV operators whose size is unknown, we found out that, making a reasonable assumption on their size, the MRSSM can be realised only at the expense of fine tuning.

1.5 Dark Matter

After the discussion in supersymmetric theories, we turn our attention to another interesting problem in modern astrophysics: dark matter. The past decade and a half has seen tremendous progress in the ﬁeld of cosmology. The numerous experiments and the plethora of accumulated data have elevated cosmology into a precision science. With the knowledge we have gained, a remarkably consistent consensus known as the standard model of cosmology has emerged from these attempts to understand the nature of the universe. The picture that we currently have is that of a universe that is composed of

74% dark energy, 22% dark matter (DM) and 4% baryonic matter [13,14]. Despite its ex- traordinary success in explaining a variety of diverse observations, fundamental questions do remain. Arguably, the most vexatious is the question “What are these dark components of the universe?” Despite the fact that it makes up 96% of the universe, we have so far been unable to say deﬁnitively what they are. Perhaps the most compelling ideas to resolve this question arise from particle physics. Dark energy is commonly attributed to the vacuum energy while the identity of dark matter is hypothesised to be one of the new particles in theories that extend the Standard Model of particle physics. This conﬂuence Chapter 1. Introduction 30

and cross-fertilisation of ideas from two major ﬁelds of scientiﬁc endeavour promises to herald an exciting new era in the understanding of the universe. With the resumption of

operation of the Large Hadron Collider at CERN, we could be close to detecting, albeit

indirectly through missing energy signatures, dark matter particles.

There are many troubling questions that need to be addressed if we are to take the

idea seriously that some particle from an extension of the Standard Model is indeed the

ubiquitous dark matter of the universe. Most worryingly, to ensure the presence of a dark matter candidate in a number of beyond the Standard Model theories of particle physics,

it is often necessary to impose global symmetries. For instance, we have the T-parity in

Little Higgs [15] and the already discussed R-parity in supersymmetry [4]. In the limit

where the global symmetries are exact, the lightest particle carrying such a global charge

would be stable from decay to lighter particles that do not possess this charge. It is this

point that could potentially destroy this promising marriage of ideas from cosmology and particle physics, for it is well known that global symmetries are never exact.

The presence of anomalies, as in the case of T-parity [16], or R-parity violating terms

in supersymmetry [17] would often mean that the dark matter candidates arising from

these theories are neither stable nor long-lived in the cosmological sense. Even if this

had not been the case, the presence of gravity would necessarily induce the violation of global symmetries as was ﬁrst revealed in studies of black holes [18–21]. Therefore, the

lightest particle charged under a particular global symmetry would have, at best, a very

long lifetime. Indeed, it has even been conjectured that discrete global symmetries are

violated maximally by gravity [22,23].

Additional motivation can be found in numerical simulations of the universe (based on conventional cosmology) which predict an overabundance of substructures as compared to actual observations. Models with decaying dark matter [24,25] provide an extremely compelling and natural mechanism for suppressing the power spectrum at small scales, thus resolving the discrepancy. Chapter 1. Introduction 31

Continuing on the line of thought leading from particle physics to cosmology, the question that then naturally springs to mind is “What can cosmology say about decaying

dark matter and the particle physics theories that contain them?” It is this intriguing

prospect that we explore in chapter 4.

Diﬀerent observations of the universe can help us answer this question. The wealth of data that we have from these observations today is by no means negligible. The most useful source of information is the latest data from the cosmic microwave background.

In addition to this, data from Type Ia supernova, Lyman-α forest, large scale structure and weak lensing observations can help us further constrain the lifetime of dark matter.

All these data can help us determine the primordial curvature perturbation. The faster dark matter decays, the more dark matter we would need in the early universe to explain the current observations. This has an effect in the primordial curvature that can affect the evolution of the universe and that is precisely what we are trying to determine. The accuracy of our results is obviously increased if we have more observations. Furthermore, we will be assuming a ΛCDM cosmology, that is a Friedman-Robertson-Walker universe with dark matter and a cosmological constant Λ. In other words a homogenous and isotropic universe. Thus, these observations are used as well to cross check the predictions from this model. By doing this we can have further confidence in the ΛCDM model.

There are two eﬀects frequently referred to in astrophysics that will be mentioned when constraining the lifetime of decaying dark matter. We give a brief description of these eﬀects below.

Sachs-Wolfe eﬀect

Perturbations in the primordial curvature can produce a shift in the spectrum of radiation emitted. One of the eﬀects that contributes to this shift is the so-called Sachs-

Wolfe eﬀect. This is caused by a diﬀerence in the gravitational potential that causes a Chapter 1. Introduction 32

shift in the radiation due to gravitational redshift. If the gravitational potential changes with time, then the redshifting would be diﬀerent. This eﬀect is called the Integrated

Sachs-Wolfe eﬀect. In the usual models of ΛCDM, the Integrated Sachs-Wolfe eﬀect ap-

pears to have signiﬁcance before the matter-radiation equality, and at later times when

the vacuum energy dominates and the gravitational potential changes (the so-called late

Integrated Sachs-Wolfe eﬀect). Diﬀerent (and changing) amounts of dark matter in the

history of the universe will have an impact on the radiation, and thus we expect the Integrated Sachs-Wolfe eﬀect to be important when analysing decaying dark matter.

Reionization

After recombination took place, most of the hydrogen in the universe was neutral and

there were absorption lines for photons travelling through hydrogen clouds. However, the universe did not always remain neutral since observation of distant quasars indicates

that most of the hydrogen was completely ionised at a redshift of z 5. This can clearly ∼ be seen in the absence of absorption lines from intergalactic hydrogen. One possible explanation is that at that time star formation contributed signiﬁcantly to the reionization of the universe by overheating the hydrogen in the medium. Another possibility is that an additional source of reionization contributed to this process. A decaying dark matter candidate decaying to ectromagnetically charged particles can certainly contribute, to some degree, to the reionization of the universe. In the reionization lingo, an essential concept is that of the optical depth τ. The physical interpretation of this quantity can

be understood if we look at the fraction of photons that would not be inﬂuenced by the

τ(zreion) τ(zreion) scattering oﬀ electrons. This fraction is e− . The remaining fraction, 1 e− , − will be scattered oﬀ the electrons.

When analysing the constraints on the lifetime of dark matter, the concepts discussed

above will be used. We will see that decaying dark matter gives an enhancement to the Chapter 1. Introduction 33 late Integrated Sachs-Wolfe eﬀect. Furthermore, special attention will be put on electromagnetically charged decay products, since they aﬀect the reionization of the universe.

In chapter 4 we investigate these constraints on decaying dark matter by looking at the latest data from cosmology. We used all the available and most recent data sets of cosmology and we found that, if we assume that the dark matter decays into relativistic

1 particles, then the dark matter lifetime is Γ− & 100 Gyr (at 95% conﬁdence level). In the case where the decay products are electromagnetically charged, since they can inﬂuence the reionization history of the universe, we found that the decay rate is constrained by

1 8 (fΓ)− & 5.3 10 Gyr (at 95% conﬁdence level), where f is the fraction of the decay × energy that is deposited in the baryonic gas. If we want to have a particle physics description of a decaying dark matter candidate, given these constraints, it is reasonable to assume that the eﬀective operator that gives rise to dark matter decay must be of dimension-6. Chapter 2

R-Symmetric Gauge Mediation

The contents of this chapter were published in the Journal of High Energy Physics, JHEP

01 (2009) 018.

2.1 Motivation

The analysis of the Minimal R-symmetric Standard Model (MRSSM) in [2] was performed in the framework of an effective supersymmetric theory with the most general soft terms respecting the R-symmetry. The place of this model in a grander framework, including the breaking and mediation of supersymmetry, was not addressed in detail. The purpose of this chapter is to investigate a possible ultraviolet completion of the MRSSM in the framework of gauge-mediated supersymmetry breaking, with the hope that an ultraviolet completion will help narrow the choice of parameters of the effective field theory analysis.

The focus of this chapter on gauge mediation is motivated by several recent observations.

First of all, phenomenological studies [26] of the MRSSM have shown that Dirac charginos are typically the next-to-lightest supersymmetric particles (NLSPs) in the visible sector. This points toward a possible small scale of supersymmetry breaking, with the resulting light gravitino allowing a decay channel of the light charginos.

Secondly, it has been known [7] for a while that models with non-generic super-

34 Chapter 2. R-Symmetric Gauge Mediation 35

potentials can have both broken supersymmetry and unbroken R-symmetry. As was pointed out before, Intriligator, Seiberg, and Shih (ISS) [1] observed that metastable

supersymmetry-breaking and R-preserving vacua in supersymmetric gauge theories are,

in a colloquial sense, quite generic. Majorana gaugino masses require breaking of the

R-symmetry; instead we explore the possibility that the gauginos are Dirac and the

R-symmetry is unbroken. Combined with the fact that these vacua can preserve large

nonabelian ﬂavor symmetries, it makes sense to use ISS models to build R-symmetric models of direct mediation of supersymmetry breaking.

In this chapter, we present a model that uses direct gauge mediation and the metastable

solution of ISS to generate the MRSSM. In the next section, we will discuss the relevant

details of the ISS model and how it can be used to generate direct gauge mediation with an R symmetry. We will also introduce notation for computing masses that will be used throughout the thesis. In section 2.3 we will consider how to use the model presented in section 2.2 to generate soft terms in the visible sector. This section is divided into two parts: contributions from the cutoﬀ scale UV physics, and direct contributions from the messenger sector, that we call “IR contributions”. At this stage we will also discuss a generalization of the model where we identify the important features of the metastable ISS solution and consider how these essential features can be extracted in a general, phenomenologically viable way. Then, in section 2.4, we present some examples of qualitatively diﬀerent spectra, and discuss constraints such as perturbativity and tuning.

2.2 ISS and R-symmetric direct gauge mediation

Direct gauge mediation postulates that the SM gauge group GSM is part of the global symmetry of the supersymmetry breaking sector, thus relaxing the need to have a sepa-

rate messenger sector of supersymmetry breaking. Dynamical models of direct mediation Chapter 2. R-Symmetric Gauge Mediation 36

have been considered in the past, see e.g. [27, 28]. The ISS models [1] of metastable supersymmetry breaking are attractive setups for constructing models of gauge mediation,

particularly in the R-symmetric setup. As we shall see in this chapter, using ISS as an illustrative example of an R-symmetric supersymmetry-breaking/mediation sector will teach us some general lessons on R-symmetric mediation; these open the way for the future study of more general models with diﬀerent phenomenology.

The “electric” (high-energy) ISS model is supersymmetric QCD with gauge group

SU(Nˆc) and Nf ﬂavors of quarks Q and Q¯, with a tree level superpotential:

Wel. = Tr m QQ.¯ (2.2.1)

The dual “magnetic” (low-energy) theory has gauge group SU(N ), N = N Nˆ , N c c f − c f

ﬂavors of magnetic quarks q, q¯, gauge-singlets M, transforming as (Nf , N¯f ) under the ﬂavor group, and superpotential:

W =q ¯ q + Tr m Λ + . . . (2.2.2) magn. M M where the dots denote nonperturbatively generated terms (that are not important in the metastable supersymmetry breaking vacuum) and Λ is the duality scale. As ISS show, there exists a metastable supersymmetry-breaking vacuum in this theory, since the equation of motion for following from (2.2.2):q ¯i q = Λmi (the dot denotes M · j j summation over the gauge indices) can not be satisfied, for Nf > Nc and a mass matrix of maximal rank Nf , due to the rank condition . The flavor symmetry preserved by the mass terms in (2.2.2) is broken in the supersymmetry breaking vacuum, while an R- symmetry, under which has R-charge 2 and q, q¯ have R-charge 0, remains unbroken. M That the R-symmetry is unbroken follows from the Coleman-Weinberg calculation of [1], which shows what while the dual quarks get expectation values, the trace of , which M is a classical flat direction, does not.

R-symmetry breaking is needed to obtain Majorana gaugino masses. Thus, a lot of the model building using ISS and other supersymmetry-breaking models has focused on Chapter 2. R-Symmetric Gauge Mediation 37

breaking the R symmetry, either explicitly or spontaneously; for example [29–41]. As described in the Introduction, in light of the recent observations of [2] on the interesting

phenomenological features of supersymmetric models with unbroken R symmetry, we

explore here the contrary possibility. We build R-symmetric models of direct gauge

mediation, where gauginos are Dirac, and study their phenomenological consequences.

2.2.1 The supersymmetry-breaking/mediation sector

To be more concrete, we consider a simple ISS model allowing for direct gauge mediation.1

For simplicity, we take Nc = 1, Nf =6(Nˆc = 5), as done by [32]. The “magnetic” dual theory is then an O’Raifertaigh model. The supersymmetry-breaking vacuum has

a reduced vectorlike global symmetry SU(6) SU(5) due to the vevs of the dual V → V squark fields q andq ¯. We will describe the model in terms of a set of fields with definite

quantum numbers under the unbroken SU(5)V , related to the ones in (2.2.2) as follows:

M N ϕ ϕ¯ =   , q =   , q¯ =   . (2.2.3) M N¯ X ψ ψ¯             In addition to the ﬁelds in (2.2.3), as we will shortly explain, our model will also

require the introduction of two other ﬁelds which transform as adjoints under SU(5)V

and carry vanishing R-charge. We will call these ﬁelds M ′ and Φ. In what follows, it

will only be necessary for Φ to be an adjoint under GSM rather than the full SU(5)V symmetry (Φ will be used to give Dirac masses to the gauginos). This avoids the need for “bachelor” ﬁelds of [11]; they can be added with minimal trouble, but in the spirit of

minimization of the model, we will leave them out. The charges of the various superﬁelds of the supersymmetry-breaking/mediation sec-

tor under the global SU(5)V symmetry of the ISS model, the U(1)R symmetry, and a

1Since our purpose here is more to emphasize the general features of R-symmetric gauge mediation rather than to construct a model with minimal ﬁne-tuning, in most of this chapter we consider this simple example where GSM SU(5)V . More general constructions are possible, perhaps even desirable, and will be discussed later. ⊂ Chapter 2. R-Symmetric Gauge Mediation 38

SU(5)V U(1) U(1)R M Adj+1 0 +2 X 1 0 +2 N 5 +6 +2 N¯ 5¯ –6 +2 ϕ 5 +1 0 ϕ¯ 5¯ –1 0 ψ 1 –5 0 ψ¯ 1 +5 0

Φ Adj′ 0 0

M ′ Adj 0 0

Table 2.1: Charges of superﬁelds of the supersymmetry breaking and mediation sector.

Note that the chiral superﬁelds Φ are only adjoints under GSM and not the full SU(5)V ,

denoted by Adj′. In addition to the continuous symmetries indicated, we impose a charge-conjugation symmetry (C) under which barred and unbarred fields are exchanged, the G SU(5) vector superfields change sign, as does Φ; the fields M, M ′,X are SM ⊂ V invariant.

residual U(1) global symmetry (which is spontaneously broken by the dual squark vevs) are given in Table 2.1.

The spontaneous breaking of SU(6) SU(5) in the ISS model will leave behind V → V a massless Nambu-Goldstone (NG) boson in the messenger sector. However, the gauging of GSM explicitly breaks the the full SU(6)V and the NG boson will acquire a mass. Since the symmetry is broken in this way we consider the more general case where we

“tilt” the couplings in the superpotential in eqn. (2.2.2) so that the SU(6)V symmetry is explicitly broken, keeping certain ratios of couplings ﬁxed as would be the case for the gauging of GSM, e.g. κ in Wmagn below. Finally, the most general nontrivial tilting of Chapter 2. R-Symmetric Gauge Mediation 39 the superpotential that is consistent with the remaining symmetries is

W = Wmagn + W1, (2.2.4) where:

2 W = λ ϕMϕ¯ + κ′ ψXψ¯ + κ ϕNψ¯ + κ ψ¯Nϕ¯ f (X + ω TrM), (2.2.5) magn − is the (tilted) ISS superpotential from Equation (2.2.2), while

W = y ϕ¯ΦN N¯Φϕ (2.2.6) 1 − are additional terms, which explicitly break the global U(1) of Table 2.1.2 The couplings in W1 are needed to generate Dirac gaugino mass. For now, we simply postulate a C- parity, deﬁned in the caption to Table 2.1, which explains the relative minus sign in

(2.2.6); we will come back to this point below. Notice that we can recover the SU(6)V limit by setting κ = κ′ = ω = 1. By rephasing ﬁelds it is possible to take all the parameters in (2.2.5) and (2.2.6) to be real, which we do in the following.

2.2.2 Scales of supersymmetry breaking and mediation

The F -term equations at the SUSY breaking metastable minimum of (2.2.5) give:3

f 2 ψψ¯ v2 = , (2.2.7) h i ≡ λκ′ F = ωf 2 . (2.2.8) h TrM i

2 In a complete SU(6)V description this term can be thought of originating from a termq ¯[Φˆ, ]q M in the magnetic superpotential (2.2.2), with Φˆ being an extension of Φ to SU(6)V , similar to the relation between M and in (2.2.3); this allows following Seiberg duality for the construction of the corresponding electric theory,M if such a thing is desired. However, for the purposes of this thesis we will simply treat these terms as additional terms allowed by symmetries, making no assumptions as to the origin of the y couplings. 3Notice that canonically normalizing Tr M would require a factor of √5 be introduced in Equation (2.2.8), as in [32]. This factor can be reabsorbed into our deﬁnition of ω and not doing so only serves to clutter the notation, so we will not include it here. Omitting this factor has no eﬀect on the low-energy phenomenology of the model. Chapter 2. R-Symmetric Gauge Mediation 40

We also ﬁnd ϕ = ϕ¯ = N = N¯ = X = 0, all with masses near f. The other h i h i h i h i h i ﬁelds are stabilized at higher order in the loop expansion, as we will see below.

At the minimum (2.2.7, 2.2.8) the scalar mass squared terms are:

λκ2 λω 0 0 ϕ κ′ −  λκ2    λω 0 0 ϕ¯ 2 − κ ∗ ϕ ϕ¯ N N¯ f  ′ 2  , (2.2.9) ∗ ∗  λκ     0 0 0   N   κ     ′ 2     λκ  N¯   0 0 0   ∗  κ′        and fermion masses are:

λκ2 0 eiξ ϕ¯ (ϕ N) f κ′ . (2.2.10)  2 r    λκ iξ N¯  e− 0  r κ′        Notice that all the masses can be scaled to depend on two variables:

x λω , ≡ ωκ z ′ , (2.2.11) ≡ κ2

and we can deﬁne an overall messenger mass scale:

x M 2 f 2 (2.2.12) mess ≡ z which is independent of SUSY breaking. From the above mass matrices, we see that the N, N¯ scalars as well as the two fermion messengers all have mass Mmess. The mass eigenstates of the upper 2 2 block of the scalar mass matrix are: ×

φ = 1 (ϕ +ϕ ¯ ) , + √2 ∗

1 φ = ( ϕ +ϕ ¯∗) , (2.2.13) − √2 − and have mass squareds:

2 2 m = (1 z)Mmess. (2.2.14) ± ± Chapter 2. R-Symmetric Gauge Mediation 41

Hence, to avoid tachyons, we require z 1. In fact, z = 1 is the SU(6) limit where there ≤ V is a massless messenger, as we can see explicitly from (2.2.14). Further in our analysis,

we will take z 0.9. We note, from (2.2.14), that there is a signiﬁcant mass hierarchy ∼

in the messenger sector for small breaking of SU(6)V . The F -term conditions (2.2.7) and (2.2.8) do not ﬁx the vacuum expectation values of M and ψ, ψ¯:

ψ = ve(η+iξ)/v (2.2.15)

(η+iξ)/v ψ¯ = ve− . (2.2.16)

At one loop, following the calculation of ISS, we ﬁnd M = η = 0 with masses one h i h i order of magnitude down from the messenger scale. This leaves the massless ﬁeld ξ, which is the NG boson of the spontaneously broken U(1) of Table 1. The Yukawa terms in W1 break the U(1) symmetry,4 hence this NG boson will get a mass starting at two loops.

To leading order there is only one diagram that generates this mass, shown in Figure 2.1 plus its complex conjugate. This diagram is finite. Expanding around the minimum, with vacuum expectation values from Equations (2.2.15) and (2.2.16) and η = 0, we h i find the effective potential for ξ is

2ξ V (ξ)= µ2v2 cos , (2.2.17) eﬀ − v where µ2 is the value of the loop in Figure 2.1:

2 2 (λκy) µ = [I(m+, m+)+ I(m , m ) 2I(m+, m )] , (2.2.18) 4 − − − − and the Euclidean loop integrals I(m1, m2) are computed in [5] and have the form

ddk ddq 1 1 1 . (2π)d (2π)d (k + q)2 k2 + m2 q2 + m2 Z Z 1 2

4Notice that if these operators come from the dimension four superpotential termq ¯[Φˆ, ]q as mentioned in a previous footnote, then these operators no longer explicitly break the symmetry,M and in fact the ξ ﬁeld remains a true massless NG boson. Chapter 2. R-Symmetric Gauge Mediation 42

ϕ ϕ

ψ Φ ψ ϕ ϕ

Figure 2.1: The leading diagram contributing to the ξ mass.

Thus, the minimum of the potential is at ξ = 0, leading to the conclusion that the h i C-symmetry of the model is not spontaneously broken, and a mass of ξ:

λκy 2 m2 = M 2 H(z) , (2.2.19) ξ 16π2 mess where

1+ z 2z H(z)=(1+ z) log2 2Li − +(z z) . (2.2.20) − 1 z − 2 1 z → − − − In particular: H(1) = 2π2/3, and vanishes for z = 0 (the SUSY limit).

Finally, we discuss the remaining messenger fermions. These come with a mass matrix

0 0 1 ψ     (ψ ψ¯ X) v 0 0 1 ψ¯ , (2.2.21)         1 1 0 X         where v M is given by (2.2.7) and we have set η = ξ = 0. This matrix can be ∼ mess h i h i ¯ diagonalized with the result that the X fermion has mass mX˜ = v and the ψ, ψ ﬁelds mix maximally, one getting a mass √2v and the other with vanishing mass. This spectrum is not a surprise: the X fermion, having R = +1, can only mix with the goldstino, the fermionic partner of TrM; one of these fermions marries the gravitino, while the other

has a mass M . The ψ, ψ¯ fermions each have R = 1 and can mix. That there is ∼ mess − a massless fermion is not surprising, since the ψ, ψ¯ sector contains the pseudo-NG boson

discussed above, and by supersymmetry this must come with a massless fermion (notice Chapter 2. R-Symmetric Gauge Mediation 43

that there is no supersymmetry breaking in these fermion masses). The ψ, ψ¯ superﬁelds can couple to the SM fermions starting at two loops, with gauge ﬁelds and messengers

in the loops; however, since these operators are generated by gauge bosons the operator

is ﬂavor diagonal. These can then generate four-fermion (ﬂavor conserving) operators

that, thanks to supersymmetry, are ﬁnite and small, with any divergent loop integrals

cut oﬀ by the ξ mass (2.2.19). Such massless fermions might have some interesting phenomenological or cosmological consequences; from the R symmetry they can only be pair-produced. We will not say any more about them here.

This completes the discussion of the spectrum in the messenger sector. We may now

discuss the phenomenology of the visible sector. Before doing so we comment on a few technical features of our model.

2.2.3 Dirac gaugino masses, C-parity, and the extra adjoints

Generating a Dirac gaugino mass requires a chirality ﬂip on a fermion line as explained in

Section 2.3.2. This can only come from a superpotential fermion mass term and requires

the sum of the R-charges of the ﬁelds involved to be 2. The mass of the scalar involved in the loop must be diﬀerent from the fermion—if not there is a cancellation and the

gaugino mass is zero. This SUSY breaking splitting must come from oﬀ-diagonal terms

in the scalar mass matrix, otherwise the supertrace is non-zero and there will logarithmic

divergences in the scalar masses [42]. Only scalar ﬁelds of zero R-charge can have these

oﬀ diagonal mass terms. Hence in order to generate a non-zero Dirac gaugino mass in

R-symmetric gauge mediation one needs ﬁelds with both R-charge 2 and R-charge 0. The model discussed here is of this general form: the ϕ andϕ ¯ have zero R-charge and acquire

oﬀ diagonal masses (2.2.9), while the ﬁelds N and N¯ have R-charge 2 and supply the

needed chirality ﬂip. We will discuss a more general version of this model, involving fewer

adjoints in Section 2.3.2. Chapter 2. R-Symmetric Gauge Mediation 44

Recall now that in two component notation, Dirac gaugino mass terms are

a a m1/2λ ψ , (2.2.22)

where λ is a Weyl fermion in the adjoint of the gauge group, part of the = 1 vector N multiplet, and ψ is the Weyl fermion component of a chiral supermultiplet Φ, also in the adjoint of the gauge group. In addition to preserving an R-symmetry, Dirac (2.2.22), as opposed to Majorana, gaugino masses are odd in the gaugino field λ. Hence, they change sign under C-parity if only λ is C-odd. Note that this already implies that Dirac gaugino masses can not be generated by coupling the adjoint field M from the supersymmetry breaking sector to the gauginos λ, even in modifications of the ISS model with broken R symmetry, as the field M is even under C (provided the ISS-modification does not break

C).

We chose to assign negative C-parity to the chiral adjoint Φ making the Dirac gaugino mass C-even. This also requires the relative minus sign between the two couplings in W1 in (2.2.6). Naively, one might think that with a diﬀerent ratio of the two couplings in

(2.2.6) the loop-induced Dirac gaugino mass might be reduced or even made to vanish.

Take, for example, the extreme case of a positive relative sign between the two terms

in W1. Then one might argue that the Dirac gaugino mass should vanish: indeed, in this case, we could modify our deﬁnition of C so that Φ was even, thus forbidding the

loop-induced Dirac gaugino mass term (2.2.22). However, in this case the diagram of

Figure 2.1 would generate a positive-cosine eﬀective potential for ξ, instead of (2.2.17),

leading to spontaneous C symmetry breaking, and giving rise to the same absolute value

of the loop-generated Dirac gaugino mass.5

However, a choice of C with even Φ, or the absence of any symmetry, would allow

for the generation of a tadpole for ΦY —the gauge singlet hypercharge “adjoint.” Such

5That a maximal absolute value gaugino mass is always generated is true for any value of the couplings in (2.2.6)—the theory simply wants to maximize the (negative) vacuum energy contribution from gauginos. Chapter 2. R-Symmetric Gauge Mediation 45

tadpoles are known to destabilize the hierarchy, see e.g. [43]. Having ΦY odd under an unbroken discrete symmetry eliminates this tadpole, at least the supersymmetry-

breaking/messenger sector contribution. Also, this parity can be used to forbid kinetic

mixing of the SUSY-breaking spurion with the hypercharge gauge ﬁeld strength, which

could lead to large tachyonic scalar masses. C-violation in the SM may introduce other

contributions that will involve loops of quarks and leptons and will be suppressed by

products of SM gauge and Yukawa couplings. In what follows we will assume that these contributions are small and can be ignored. This is similar to the standard “messenger

parity” that goes along with gauge mediation [44–47] except Φ is also charged under the

parity.

The introduction of yet another zero-R-charge adjoint, M ′, of even C-parity, is neces- sitated by the requirement to give the adjoint M a mass. This is because in the absence

of R symmetry breaking the fermionic components of M are forbidden from obtaining

masses due to loops, as is usually expected in a model where R is broken.

Finally, take note that G SU(5) , and therefore the appearance of these new SM ⊂ V messengers will have a strong eﬀect on the Standard Model running couplings. In particular, all the couplings lose asymptotic freedom and will develop Landau poles. For typical choices of parameters used below, these Landau poles occur relatively close to the messenger mass scale.

2.3 Soft terms in the visible sector

Now we proceed to the calculation of the soft terms in the visible sector. To begin, we note that there are two main sources of visible sector soft masses in our model:

1. Ultraviolet (UV) contributions due to higher-dimensional operators. Typical in

models with direct mediation of supersymmetry breaking, all couplings in the SM

lose asymptotic freedom. In our model, the scales of the SM Landau poles are not Chapter 2. R-Symmetric Gauge Mediation 46

too far above the messenger scale Mmess.

As usual, the UV contributions can not be calculated in the low-energy theory. We

estimate the scale suppressing the higher dimensional operators and their contri-

bution to the SM soft parameters in Section 2.3.1 using naive dimensional analysis

(NDA). The largest UV contributions are to soft scalar masses, which are expected

to be ﬂavor-nondiagonal, and to µ and Bµ terms. UV contributions to gaugino masses are suppressed, similar to the well-known pre-anomaly mediation gaugino

mass problem of supergravity hidden-sector models.

2. Infrared contributions to the soft parameters arise due to loops of the particles in

the direct-mediation sector and are calculable in the low-energy theory. Messenger loops generate Dirac gaugino masses and ﬂavor-diagonal soft scalar masses. The

IR contributions to the soft parameters are a loop factor below Mmess and are calculated in Section 2.3.2.

There is an interplay between these two types of soft masses in our model. As we discuss in Section 2.4.1, the scale suppressing the UV contributions to the soft parameters is about a loop factor above Mmess. Thus the loop-suppressed IR contributions are typically similar to that due to the higher-dimensional operators. This allows us, at the cost of moderate cancellations of the various contributions in the scalar sector (see Section 2.4.3) to realize the scenario proposed in [2], where Dirac gauginos heavier than the scalars suppress the

ﬂavor-changing neutral currents due to non-degenerate squarks.

2.3.1 Estimating the UV contributions

We begin by discussing the typical size of UV contributions. From eqn. (2.2.8), the

F -term supersymmetry breaking spurion of R-charge 2 is:

Ξ TrM = θ2ωf 2 . (2.3.1) ≡h i Chapter 2. R-Symmetric Gauge Mediation 47

Using this spurion, many R-symmetric higher-dimensional operators that communicate supersymmetry breaking to the SM can be written down. They are all suppressed by

some high scale Λ, the value of which will be discussed later, in Section 2.4.1. These

UV-operator induced soft mass contributions are of order MUV , deﬁned as:

ωf 2 z M M = = mess M , (2.3.2) UV Λ λ Λ mess 2 where for future use we chose to rewrite MUV in terms of the messenger scale Mmess and the dimensionless parameters of eqn. (2.2.11)-(2.2.12).

Λ is the scale at which these operators are generated and is a model-dependent parameter. However, before we study the operators that are generated at this scale, a few words can be said about its size.

One possibility is that Λ M : this is the usual expectation from gauge + gravity ∼ P mediation, where any “UV operators” are generated by new physics at the Planck scale and are irrelevant. It solves the flavor problem trivially, since all flavor-changing operators are Planck suppressed; however it assumes that all physics below the Planck scale is flavor- conserving, which is a strong assumption. As it does nothing to realize the features of the MRSSM, we do not consider this possibility further here.

The other extreme is that Λ is related to Λ3, the QCD Landau pole, where presumably there is a new dual description that takes over. It is quite reasonable to assume that there are new states in this dual theory that can generate ﬂavor-violating operators.

We will discuss more careful estimates of Λ below but as this turns out to be the most constraining possibility we will consider it throughout this chapter.

We now enumerate the R-symmetric higher-dimensional operators that can be written down. Dirac gaugino masses m1/2 can be generated by the “supersoft” operator [11]:

2 1 α 2 MUV d θ Tr(W Φ) D¯ D Ξ†Ξ m M . (2.3.3) Λ3 α → 1/2 ∼ UV Λ Z

Similarly, soft scalar masses, m0 ij, generically ﬂavor non-diagonal, for the SM ﬁelds (say, Chapter 2. R-Symmetric Gauge Mediation 48

quark superﬁelds Q) are given by:

4 cij d θ Ξ†Ξ Q†Q m M . (2.3.4) Λ2 i j → 0 ij ∼ UV Z where c is a naively ﬂavor-anarchic matrix with (1) entries. We note that unless ij O M /Λ = (1), Dirac gaugino masses due to higher dimensional operators are sup- UV O pressed6 compared to the soft scalar mass. In addition, the smallness of this operator means that we can ignore supersoft contributions to the scalar masses [11]. As we will

see below, the problem of too-small masses due to higher-dimensional operators will be

addressed by direct gauge mediation in this model, along with an estimate of the relevant

cutoﬀ scale.

Next, we recall that in the R-symmetric MSSM the usual µ-term is forbidden by R-

symmetry and that there are, instead, two µ-terms, µuHuRu and µdHdRd, where Ru,d are

two new R-charge 2 Higgs doublets. The µu,d terms, as opposed to the MSSM, preserve a

Peccei-Quinn (PQ) symmetry, which forbids Bµ but not µd, µu (Hd,u can be taken to have PQ charge +2, R charge 2, and the quark and lepton superﬁelds charge 1). This u,d − −

symmetry implies that, unlike the MSSM, µu/d and Bµ originate from diﬀerent operators.

The Bµ term Bµhuhd is, however, allowed by R symmetry. Bµ is generated by an R-preserving Giudice-Masiero type operator:

4 1 d θ (Ξ†Ξ) H H B M , (2.3.5) Λ2 u d → µ ∼ UV Z p which yields Bµ similar to the soft scalar mass (2.3.4). The µu,d-terms are instead generated by R-preserving operators7 of the form:

4 1 d θ Ξ† H R µ M . (2.3.6) Λ u(d) u(d) → u(d) ∼ UV Z In addition, there is an operator that is not forbidden by any symmetry allowing

6 For Λ MPl this is the well known pre-anomaly-mediation problem of gaugino masses in supergravity without∼ singlets. 7 Notice that C parity implies µu = µd, but this need not be required in general; considerations along these lines is delegated for future work. Chapter 2. R-Symmetric Gauge Mediation 49

(2.3.5), is renormalizable, naively expected to be unsuppressed and generating an unac-

ceptably large Bµ term:

d2θ ΞH H B M . (2.3.7) u d → µ ∼ mess Z p However, one can put forward arguments in defense of ignoring (2.3.7). The only diﬀer- ence between the desirable (2.3.5) and the undesirable (2.3.7) (as written), is that the

former vanishes as Λ while the latter does not. Now, the scale Λ is expected to → ∞ be proportional to the SM Landau pole. Thus all UV-suppressed operators coupling the

SM to the supersymmetry-breaking sector that we wrote so far—except (2.3.7)—vanish

as one takes the SM gauge couplings to zero, since the Landau pole scale goes to inﬁnity

in this limit. One might adopt a broad deﬁnition of “gauge mediation” by requiring that all couplings of SM to supersymmetry-breaking sector ﬁelds vanish as one takes the SM

gauge coupling to zero (and, in our model, the coupling y of W1, which may be related by a high-scale = 2 supersymmetry to the gauge coupling). Clearly, imposing this N criterion amounts to an assumption on the nature of the unknown UV theory: in par-

ticular, it should have an accidental PQ symmetry which forbids (2.3.7) but is broken

by higher-dimensional operators such as (2.3.5). In the absence of an explicit dual, it is hard to be more precise; in practical terms, in what follows we will set the coeﬃcient of

(2.3.7) to zero and appeal to technical naturalness in supersymmetry.

The scalars in the adjoint chiral multiplets Φ (of zero R-charge) will also obtain soft

masses of order MUV from K¨ahler potential terms, such as:

4 Ξ†Ξ 2 d θ TrΦ†Φ + TrΦ . (2.3.8) Λ2 Z We can also write a large superpotential “B term” for Φ but chose not to for the same reasons as avoiding (2.3.7). Finally, as explained in Section 2.2.3, to avoid massless

SM adjoint fermions, we introduced (see Table 1) another SU(5)V adjoint, M ′, of zero R-charge. The R-preserving operator:

4 1 M d θ Ξ† TrMM ′ m M , (2.3.9) Λ → 1/2 ∼ UV Z Chapter 2. R-Symmetric Gauge Mediation 50

gives rise to a Dirac mass for M and M ′ of the same order as the soft scalar mass (2.3.4).

2.3.2 Calculating the IR contributions

We now consider the calculable IR contributions to the soft mass parameters. There is a 1-loop contribution (similar graphs were considered in [48]) to the Dirac gaugino mass from graphs involving the ϕ (¯ϕ) and N¯ (N) messengers, shown in Figure 2.2, as well as the usual two-loop gauge mediated contributions to the scalar masses. We now proceed to calculate these soft masses.

Soft masses in ISS-models ϕ

λ Φ ϕ N

Figure 2.2: One of the diagrams contributing to the 1-loop gaugino mass. The other graphs are obtained by diﬀerent choices of ϕ,ϕ ¯ N, and N¯ running in the loop.

The diagram of Figure 2.2 involves an R preserving fermion mass insertion and a scalar with a SUSY-breaking mass and generates a Dirac gaugino mass. Using the values for our masses and couplings of Section 2.2.2, we ﬁnd that the loop-induced Dirac gaugino masses can be written as:

gy ξ m = M R(z) cos h i , (2.3.10) 1/2 16π2 mess v where: 1 R(z)= [(1 + z)log(1 + z) (1 z) log(1 z) 2z] , (2.3.11) z − − − − Chapter 2. R-Symmetric Gauge Mediation 51

where z is deﬁned in (2.2.11) and measures the oﬀ-diagonal supersymmetry breaking mass splitting in the scalar mass matrix (2.2.9). Notice the dependence of the gaugino

mass on cos( ξ /v). Since (see discussion in Section 2.2.3) ξ = 0 this factor is just 1. h i h i

In principle the SU(3), SU(2) and U(1) pieces of the SU(5)V may have diﬀerent κ and y coeﬃcients. However, for simplicity, we take the 3-2-1 pieces to all be the same; breaking

this would eﬀect the relative size of the gauginos and sfermions associated with each SM

group.

The sfermions acquire a gauge-mediated mass from loops involving the messengers

ϕ, ϕ¯, but not N, N¯ since they do not have supersymmetry-breaking masses. Following [5], this contribution can be calculated. Thus, the contribution from gauge group a to a

sfermion mass squared is:

α 2 m(IR)2 =2C(a) a M 2 F (z) , (2.3.12) 0 F 4π mess where C(a) =(N 2 1)/2N for SU(N) and 3 Y 2 for U(1) and: F − 5 Y z 1 2z F (z)=(1+ z) log(1 + z) 2Li + Li +(z z) . (2.3.13) − 2 1+ z 2 2 1+ z → −

We note that the contribution of the R-symmetric messenger sector to soft scalar masses (2.3.12) is the same as that of one messenger multiplet in usual gauge mediation.

The function F (z) from (2.3.12), with our parameter z identiﬁed with F/λS2 of usual

gauge mediation, is the same appearing in, e.g., [5]. The Dirac gaugino mass (2.3.10),

however, is governed by a diﬀerent function of z than in the case of Majorana mass.

This qualitative diﬀerence arises because the Dirac gaugino mass requires the presence

of an R-preserving chirality ﬂip in the loop. This R-symmetric chirality ﬂip does not appear in the two-loop diagrams generating the scalar mass, which are thus identical

to those in one-ﬂavor gauge mediation—the messenger scalars ϕ andϕ ¯, which have

a supersymmetry-breaking spectrum contribute to the scalar masses, while N and N¯,

which are supersymmetric, do not. Chapter 2. R-Symmetric Gauge Mediation 52

R z F z H L H L 0.7 0.6 0.5 0.4 0.3 0.2 0.1 z 0.2 0.4 0.6 0.8 1

Figure 2.3: The function of z entering the ratio (2.3.14) of gaugino to scalar mass. At z = 1 there is a branch point, with R(z)/ F (z) 0.732 as z 1 . → → −

In addition note that R(z 0) z2, unlike usual gauge mediation where m z. → → 1/2 ∼ This is easy to understand since due to R-charges the Dirac mass operator (2.3.3) needs

two insertions of the spurion, ω f 2, unlike a Majorana mass that needs just one insertion. This qualitative diﬀerence leads to the general fact that in R-symmetric gauge mediation

the gaugino mass is typically lighter than the scalar mass, in contrast to usual gauge

mediation, where the m1/2 : m0 ratio is larger than unity, see [5]. The ratio of gaugino to sfermion mass in R-symmetric gauge mediation is:

m 1 y R(z) 1/2 = (2.3.14) mIR √2C g 0 F F (z)! p The ratio R/√F , as Figure 2.3 shows, is strictly less than 1: for z =0.99, R/√F = .64. | | Thus, in order to solve the supersymmetry ﬂavor puzzle along the lines of [2], which requires large gaugino to squark mass ratios, within an ISS supersymmetry-breaking- cum-mediation sector, we must have a large Yukawa coupling y (near the boundary

allowed by perturbativity, as we will discuss in Section 2.4.3).

Finally, the scalar adjoint ﬁelds in the Φ supermultiplets also get real and holomorphic Chapter 2. R-Symmetric Gauge Mediation 53

masses from the messenger loops:8

2 1 2 2 V = m φ∗φ + B φ + φ∗ . eff φ 2 φ These are given by

y2 m2 = M 2 R (z), (2.3.15) φ 16π2 mess s

y2 B = M 2 R(z), (2.3.16) φ 16π2 mess

where

1 R (z)= (1 + z)2 log(1 + z) (1 z)2 log(1 z) 2z , (2.3.17) s z − − − − and the z dependence in (2.3.16) is the same as in the gaugino mass (2.3.11). These masses are the same order, but it can be seen that B < m2 for any value of z, so the | φ| φ gauge symmetry is protected. Also notice that Bφ is strictly negative, which means that the scalar will always be lighter than the pseudoscalar. This is the reverse of ordinary supersoft mediation [11]. Notice that since this is a one-loop scalar mass, it is enhanced compared to the gaugino mass:

m 4π φ , (2.3.18) m1/2 ∼ r α where α is the ﬁne structure constant of the relevant gauge group. Thus, we generally expect the adjoint scalars to be roughly an order of magnitude heavier than the gauginos, although there could be a sizable cancellation between the real and holomorphic mass.

In addition, there could be cancellations with the UV operators that we deﬁned in the previous section (2.3.1).

8We thank Markus Luty for pointing this out. Chapter 2. R-Symmetric Gauge Mediation 54

Generalized R-symmetric gauge mediation

In this section, we introduce a model of generalized R-symmetric gauge mediation. In-

spired by previous discussions of generalized gauge mediation, see e.g. [5], we implement supersymmetry breaking in terms of an R-charge 2 spurion Ξ θ2f 2, instead of a dy- ∼ namical supersymmetry-breaking sector.

From the ISS model considered in the previous sections, we learned that only the

ﬁelds ϕ, ϕ¯, and N, N¯ of ISS play a role in the mediation of supersymmetry breaking to

leading order in the loop expansion. Furthermore, as we explained in Section 2.2.3, this

is the minimal set of messenger ﬁelds required to achieve R-symmetric gauge mediation. Thus, in our generalized model, we will keep only these ﬁelds and consider a messenger

sector consisting of Nmess copies:

Nmess W = Ξϕ ¯iϕi + M ϕ¯iN i + M N¯ iϕi + y ϕ¯iΦN i y N¯ iΦϕi . (2.3.19) mess mess mess − i=1 X

Here Ξ is the supersymmetry breaking spurion (2.3.1), Mmess is a rigid messenger mass scale; the R-assignments of the multiple copies of messengers are the same as their

namesakes of Table 1, as is their C-parity.

The messenger sector (2.3.19) gives rise, through the same set of one and two loop

diagrams as the ones discussed in the previous section, to N the gaugino mass mess× contribution of (2.3.10) and N the scalar mass squared contribution of (2.3.12). mess× Thus the ratio of loop-induced gaugino to scalar mass of eqn. (2.3.14) is enhanced by

a factor of √Nmess. The enhancement of the Dirac gaugino mass by √Nmess in the generalized model relaxes (some of, see Section 2.4.3) the need of having a large Yukawa coupling y. In addition, the absence of the SM adjoints M, M ′ from (2.3.19) pushes the SM Landau pole up: we note that the αs beta function of the MRSSM vanishes already above the scale of the Dirac gaugino mass and thus adding any colored messenger inevitably leads to a Landau pole. We will have to say more about this below.

To end this section, we note that in light of its phenomenogical desirability, it would Chapter 2. R-Symmetric Gauge Mediation 55

be of some interest to have a UV completion of the generalized messenger model of (2.3.19), ideally including both the dynamics of supersymmetry breaking and generating

the messenger mass scale Mmess without introducing the extra adjoint baggage of the ISS model; we leave this for future work.9

2.4 Numerics

2.4.1 How high can Λ be?

It is well known that in order to avoid constraints from K K¯ mixing, the dimension − six operator Q QQ Q d4θ † † Λ2 Z Q must have a cutoﬀ Λ 103 TeV. Thus we need to chose parameters such that our cutoﬀ Q ∼ is no smaller than this.

To understand how large the scale suppressing the UV contributions (Λ) can be, we must consider the location of the Landau pole. Consider the beta functions of GSM: d 1 b = i . (2.4.1) d ln µ αi(µ) −2π The model (2.2.4) contains ﬁelds that transform under SU(3) SU(2) as: C × L

M, M ′ : (8, 1)+(1, 3)+(1, 1)+(3, 2)+(3¯, 2)

Φ : (8, 1)+(1, 3)+(1, 1)

ϕ, N : (3, 1)+(1, 2)

ϕ,¯ N¯ : (3¯, 1)+(1, 2) . (2.4.2)

Now we must consider how the spectrum behaves, since the running will be sensitive to the fermionic and bosonic mass thresholds of the various multiplets. We solve the

9One simple way to achieve this is to make Ξ dynamical and add a linear term f 2Ξ to (2.3.19). 2 The model with superpotential Wmess + f Ξ has an R-preserving supersymmetry-breaking (possibly metastable) vacuum at the origin of moduli space (Ξ = 0) as a consequence of the R-charge assignments [49]. One could further “retrofit” [50] the explicit mass scales. Chapter 2. R-Symmetric Gauge Mediation 56 one-loop renormalization group equations including the various contributions as we pass their mass threshold, but we do not include finite threshold effects.

The presence of a large number of ﬁelds charged under the SM means that the Landau pole of SU(3) typically occurs at a relatively low scale, resulting in potentially sizeable

UV-induced soft masses (2.3.2). However, the Dirac gaugino mass will still be too small if the UV-generated operator (2.3.3) is the only source of its mass. For the Yukawa couplings in (2.2.6) of order one the gauginos have phenomenologically viable masses but the gluino will still be somewhat lighter than the squarks, see (2.3.14). Without a large value for y it is not possible to realize the scenario of [2]. For larger values of y, suﬃcient to allow for large squark mixing and the interesting ﬂavor physics of the MRSSM, there will be a Landau pole for some Yukawas below the strong coupling scale of SU(3). The generalized model of Section 2.3.2 alleviates some of these issues by removing some of the adjoints, raising the Landau pole, and increasing the number of messenger families, which lowers the Landau pole but also raises the gaugino:scalar mass ratio.

Once the location of the SU(3) Landau pole Λ3 has been determined we may estimate the size of the UV contributions. If all gauge and Yukawa couplings became strong at the same scale one would expect that the scale Λ of Section 2.3.1 would be related to the strong coupling scale Λ by, Λ 4πΛ. However, not all couplings become strong 3 3 ∼ at the same scale and the operators involve Ξ, which is not charged under SU(3). Such operators should have a suppression from the perturbative coupling which is weak at that scale, weakening some of the constraints that we will ﬁnd below.

Of course, we know that while there should be a suppression, it is hard to estimate: above Λ3, in the absence of an explicit dual description, we have no idea how the other couplings run (as we have a duality cascade, where after dualizing SU(3), the other gauge content will change) and where the other Landau poles now are. For the purposes of estimating the UV contributions we will therefore make the simplifying but conservative assumption that Λ = Λ3/4π, which potentially overestimates the size of the UV Chapter 2. R-Symmetric Gauge Mediation 57

SU(3) mq˜ 1400 GeV mg˜ 880 GeV

SU(2) m˜l 360 GeV mW˜ 520 GeV

U(1) me˜c 160 GeV mB˜ 370 GeV

Messenger M, M ′, Φ˜ 15 TeV m 10 TeV −

sector Mmess 100 TeV mξ 3100 GeV

Table 2.2: Spectrum for y = 2, λ = 1 and all other Yukawas are (1). Here and in O Tables 2.3, 2.4, only the IR contributions to squark and slepton masses are shown.

contributions, especially in the electroweak sector.

2.4.2 Sample Spectra

In this section we will consider three examples of spectra: the full ISS model with pertur-

bative Yukawas, the full ISS model with large y (and consequently large gaugino masses)

and the generalised model. In all cases we consider z = 0.99. All squark and slepton

masses in the tables below are from the IR-direct gauge mediation contribution; we will

discuss the UV mass contributions in the next section.

Spectrum at small Yukawa

We consider a case where the Yukawas of (2.2.5) and (2.2.6) remain perturbative up to

the Landau pole of SU(3); so we consider here the case of y = 2, λ = 1 and all other Yukawas are (1). As discussed below (2.3.14), this results in too light a gluino mass O and this situation does not allow for large squark mixing. We will assume that the UV contributions to the scalar masses have small coeﬃcients so that the ﬂavor diagonal,

IR contributions (2.3.12) dominate. Solving the RGEs we ﬁnd the spectrum, at the messenger scale, shown in Table 2.2: the Landau pole occurs at Λ 8 103 TeV and 3 ∼ × α (M ) 0.12. 3 mess ∼ Chapter 2. R-Symmetric Gauge Mediation 58

SU(3) mq˜ 1300 GeV mg˜ 3500 GeV

SU(2) m˜l 350 GeV mW˜ 2100 GeV

U(1) me˜c 160 GeV mB˜ 1500 GeV

Messenger M, M ′, Φ˜ 13 TeV m 10 TeV −

sector Mmess 100 TeV mξ 13 TeV

Table 2.3: Spectrum for y = 8 and all other Yukawas are (1). O

Spectrum at large Yukawa

As discussed in Section 2.3.2, to get large gaugino masses and so allow large sflavor violation in the MRSSM [2] we need a large Yukawa; here we consider the case of y =8 and all other Yukawas are (1). For such a large Yukawa the Yukawa Landau pole is O close to the messenger scale. The squark masses are somewhat large, but below we will assume some cancellation between the UV (2.3.4) and IR (2.3.12) contributions, allowing for large squark mixing àla [2]: this will require some tuning and we will discuss this in the next section. In this case, we find the spectrum of Table 2.3 while α (M ) 0.11 3 mess ∼ and Λ 104 TeV. The Landau poles of the other SM gauge groups are significantly 3 ∼ higher, but as we mentioned above, “dualizing” color at Λ3 would necessarily change that estimate. We emphasize that we do not expect this spectrum to be an accurate sample of the parameter space with such a large Yukawa coupling; rather, we can see from this exercise that the only hope we have to realize an MRSSM scenario in the IR masses is to go to strong coupling, which would necessitate a more detailed analysis, including effects of higher loops.

Spectrum in the generalized model

In the models of generalized R-symmetric gauge mediation of (2.3.19) increasing the number of messenger families, Nmess, increases the ratio of the gaugino mass to the scalar Chapter 2. R-Symmetric Gauge Mediation 59

SU(3) mq˜ 1900 GeV mg˜ 5300 GeV

SU(2) m˜l 620 GeV mW˜ 3500 GeV

U(1) me˜c 290 GeV mB˜ 2600 GeV

Messenger sector Mmess 80 TeV

Table 2.4: Spectrum in the generalized model for y = 3 and Nmess = 6. mass. Furthermore, the SM Landau pole is postponed because of the absence of the

SU(5)V adjoints M, M ′, which allows us to take a lower messenger scale. Performing the same analysis as above, we ﬁnd that for y = 3, Nmess = 6 and Mmess = 80 TeV we have α (M )=0.08 and Λ = 5 104 TeV. The corresponding spectrum is shown in s mess 3 × Table 2.4. Because of the large number of messengers the Yukawa has a Landau pole below Λ3.

2.4.3 Estimation of tuning

Recall that there are two contributions to the soft squark masses: one from the direct mediation, which is ﬁxed by the calculation in Section 2.3.2, and the other from the

UV operators in (2.3.4). The latter comes with a coeﬃcient that we will call cD for the

flavor-diagonal terms, and cOD for the flavor-off-diagonal terms. Ideally we would like these coefficients to be (1), and to solve the flavor puzzle we would also want c c . O D ∼ OD This means that there are two potential sources of tuning: one coming from the UV-IR cancellation of the diagonal masses, and one coming from the smallness of the flavor- violating terms relative to the flavor-diagonal terms. We will discuss each of these in turn.

First of all, some general comments can be made about the ﬁrst kind of tuning between UV and IR contributions. Recall that we made the conservative assumption that the scale of the UV operators was proportional to the QCD Landau pole Λ3 i.e. Chapter 2. R-Symmetric Gauge Mediation 60

Λ=Λ /4π. This means that M M 2 /λΛ is typically quite large unless we wish 3 UV ∼ mess to make λ big, which would introduce another Landau pole. This mass scale is typically

(10) TeV in the ISS models, and smaller for the generalized models, as can be seen O

from the tables in the previous section. If the ﬁnal scalar mass is m0, we have

2 2 m0 mIR cD = −2 , (2.4.3) MUV

2 2 with m given by (2.3.12). If m < m 1 TeV, this means that c 10− in the IR 0 IR ∼ | D| ∼ ISS models, and c 1 in the generalized models. This is smaller than hoped for in the | D| ∼ ISS case, although it does very well in the generalized model; but it should be noted that it depended on the cutoﬀ being so low, and our hopes to avoid another Landau pole in λ. If we are willing to accept strong coupling, or the added assumption that the generation of ﬂavor-changing operators is postponed to a higher scale (the SU(2) Landau pole, for instance), then this tuning can be weakened.

To analyze the second form of tuning, if δ is the ratio of the ﬂavor-changing mass

2 squared term over m0, we have

m 2 c = δ 0 . (2.4.4) OD M UV Given Equations (2.4.3)-(2.4.4), we can immediately write down a formula that quantiﬁes the ﬂavor tuning: c δ t OD = . (2.4.5) ≡ c 1 (m /m )2 D IR 0 | − |

Notice that this expression is independent of MUV . Typical allowed values of δ are of order

0.1 or less [9], given m1/2/m0 of 5–10. We saw from (2.3.14) that mIR is typically larger or of order the gaugino mass, so we immediately see from (2.4.5) that this model will be

somewhat tuned. For example, if we demand a 10% tuning, we require m0 = mIR/√2, which is very hard to do while maintaining the gaugino:squark ratio. Lowering our

standard to a 1% tuning, we require m0 = mIR/√11 which is much easier to accomplish. So there is a trade oﬀ. Chapter 2. R-Symmetric Gauge Mediation 61

m0 δ t ISS with Large y 600 GeV 0.05 1.4% General Model 1 TeV 0.07 2.7%

Table 2.5: Size of the ﬂavor tuning for the MRSSM spectra considered above.

In Table 2.5 we give the ﬂavor tunings for the two models considered in Tables 2.3

and 2.4. The values of δ δ = δ are the maximum values for the given m and gluino ≡ L R 0 mass after QCD corrections to K K¯ mixing are taken into account [9]. −

2.4.4 Lifetime of the false vacuum

We have concentrated our attention on the physics around the SUSY breaking vacuum of ISS but this minimum of the potential is metastable. The true minimum of the

system, whose existence is due to the higher dimension non-perturbatively generated

term we ignored in (2.2.2), has unbroken supersymmetry. The additional operator is due

to instanton contributions, det W = M , (2.4.6) inst Λ3 where in this section Λ denotes the duality scale, the strong coupling scale of the gauge coupling in the microscopic theory. Once this additional term is included the rank condition can now be satisﬁed and there is a SUSY preserving minimum at,

Λ 3/5 M f 1I , q = q¯ =0 . (2.4.7) h i ∼ f h i h i Because the additional term is irrelevant this SUSY preserving minimum is far from the SUSY breaking minimum. It is this distance that results in the metastable vacuum being very long lived. Transitions from one vacuum to another are initiated by bubble formation, the rate for this process is determined by the 4 dimensional Euclidean bounce action,

Γ f 4 exp ( S ) . (2.4.8) ∼ − 4 Chapter 2. R-Symmetric Gauge Mediation 62

In general calculating the bounce action analytically is not possible and it must be determined numerically. For the case of ISS however the potential is well approximated

by a square potential for which there are known analytic solutions [51]. The bubble

action for our model is given [1,32] by

Λ 12/5 S . (2.4.9) 4 ∼ f Requiring that the false vacuum lives longer than the age of the universe results in the requirement [52], i.e. no transition to the supersymmetric minimum has occurred during the lifetime of the universe places a constraint on the theory:

Λ > 3 . (2.4.10) f ∼ As seen in Section 2.4.1 the SU(3) Landau pole, the upper bound on the duality scale, was approximately 100f, so (2.4.10) can be easily satisﬁed for the scales discussed earlier. Chapter 3

Phenomenology of R-Symmetric Gauge Mediation

3.1 Higgs Potential

We have seen in the introduction that the MRSSM was proposed as a solution to the

ﬂavour problem in supersymmetry. The purpose of this chapter is to analyse the Higgs sector of the MRSSM and extract the corresponding phenomenology within the context of R-symmetric gauge mediation. Before analysing the Higgs sector of this theory, let

us remind ourselves about the structure of the Higgs superpotential in the MRSSM. As

we have stated, since the R-symmetry is not broken, the introduction of new SU(2)

supermultiplets, Ru and Rd, is necessary to have a µ term. This can be realised as

Wµ = µuHuRu + µdHdRd. (3.1.1)

We will deﬁne

µ2 µ2 + µ2 , (3.1.2) ≡ u d synonymously with the usual “µ term” of the MSSM, and µ tan ν u . (3.1.3) ≡ µd

63 Chapter 3. Phenomenology of R-Symmetric Gauge Mediation 64

The symmetries of the MRSSM also allow the following term relevant to the Higgs sector to be written down (1.4.3)

i i WΦ = λuHuΦiRu + λdRdΦiHd , (3.1.4) ˜ ˜ i=XB,W where i = B,˜ W˜ refer to the couplings of the U(1)Y or SU(2)L adjoints, respectively, and Φ is the adjoint ﬁeld that is added in order to give gauginos a Dirac mass. From now on we will refer to these terms in the superpotential as λ terms.

The explicit form of the Higgs doublets is as follows

H = + 0 H = 0 , (3.1.5) u Hu Hu d Hd Hd−

R = 0 R = + 0 . (3.1.6) u Ru Ru− d Rd Hd The ﬁeld ΦB is a singlet and ΦW is a triplet under SU(2) that is parametrized as

1 φ0 √2φ+ Φ = . (3.1.7) W 2   √2φ φ0  − −  Furthermore, we will explicitly rewrite the adjoint neutral complex ﬁelds into their real and imaginary parts as

ϕi + iηi φi = . (3.1.8) √2 Let us now expand the scalar potential, setting the λ terms to zero for now. The neutral potential is given by

V = VF + Vsoft + Vgauge. (3.1.9)

The ﬁrst part of the potential, VF , comes from the F terms in supersymmetry that have already been discussed in the introduction Chapter 3. Phenomenology of R-Symmetric Gauge Mediation 65

V = µ 2 H0 2 + R0 2 + µ 2 H0 2 + R0 2 . (3.1.10) F | u| u u | d| d d

The second term, Vsoft encaptures all the soft supersymmetry breaking terms of the scalar potential:

B B V = B H0H0 + c.c + φ ϕ2 η2 + φ ϕ2 η2 soft − µ u d 2 B − B 8 0 − 0 m2 m 2 + φ ϕ2 + η2 + φ ϕ2 + η2 2 B B 8 0 0 2 0 2 2 02 2 0 2 2 0 2 + mHu Hu + mHd Hd + mRu Ru + mRd Rd . (3.1.11)

This part of the potential contains the usual Bµ terms as well as the U(1) and SU(2) Bµ terms which we denote by Bφ. The corresponding non-holomorphic mass terms for Φi,

2 mφ, have also been included. The last contribution to the potential comes from the D terms. This contribution looks like

1 V = Tr (DaDa) . (3.1.12) gauge 2 i i

Following the notation of [11], we see that the UV operator that gives rise to the gaugino masses

α W ′ W Φj d2θ√2 α j , (3.1.13) M Z modiﬁes the D terms. For instance, for SU(2), the general form of the D term is now given by

a a a a(0) D = √2M Φ + Φ † + D , (3.1.14) 2 − 2 W W 2 with Chapter 3. Phenomenology of R-Symmetric Gauge Mediation 66

a a(0) σ D = g H∗ H . (3.1.15) 2 − 2 j 2 j j X Expanding we have

2 2 2 2 Vgauge = 2M1 ϕB +2M2 ϕ0

+ (M g ϕ M ϕ g ) H0 2 + R0 2 H0 2 R0 2 1 1 B − 2 0 2 u d − d − u 2 2 g + g 2 2 2 2 2 + 1 2 H0 + R0 H0 R0 . (3.1.16) 8 u d − d − u

In order to express the mass matrices of the diﬀerent ﬁelds, we can, analogously with the

Φ ﬁelds, expand the neutral Higgses in terms of their real and imaginary components as

h + ih H0 = uR uI u √2 h + ih H0 = dR dI , (3.1.17) d √2 where hiR and hiI are the real and imaginary parts of the respective Higgses. The real parts get a VEV equal to h = √2v and h = √2v . Furthermore, we deﬁne h uRi u h dRi d vu tan β as it is customary in the MSSM. We can write the conditions ∂V/∂huR = 0 ≡ vd

and ∂V/∂hdR = 0, under which the potential V has a minimum that satisﬁes

1 m2 + µ2 m2 cos(2β) B cot β + g M ϕ g M ϕ =0 , (3.1.18) Hu u − 2 Z − µ 1 1 B − 2 2 0

1 m2 + µ2 + m2 cos(2β) B tan β g M ϕ + g M ϕ =0 , (3.1.19) Hd d 2 Z − µ − 1 1 B 2 2 0

2 2 2 2 2 where we have used vu +vd =2mZ /(g1 +g2) to simplify the third term of these equations. These expressions are very similar to the usual ﬁrst derivative equations of the MSSM

potential except for the last two terms [4]. Additionally, taking the ﬁrst derivative of the Chapter 3. Phenomenology of R-Symmetric Gauge Mediation 67

potential with respect to ϕB and ϕ0 and equating to zero, we realise that these ﬁelds acquire a VEV given by

2 M1g1v cos(2β) ϕB = 2 2 , (3.1.20) h i B1 + m1 +4M1 M g v2 cos(2β) ϕ = 2 2 . (3.1.21) 0 m2 h i − B2 2 2 4 + 4 +4M2 We can now proceed to write down the mass matrices of the various ﬁelds in this

theory. The squared mass matrix of the real components of the neutral Higgs scalars

looks like

huR   h 1 2 dR ne = huR hdR ϕB ϕ0 ne   , (3.1.22) L 2 M    ϕB       ϕ0    with  

2 2 2 ′2 ′2 Bµct +M s Bµ M s c M s M s 0 β Z β − − Z β β 1 β − 2 β 1

B 2 2 2 ′2 ′2 C B Bµ M s c Bµt +M c M c M c C B − − Z β β β Z β − 1 β 2 β C 2 =B C , (3.1.23) Mne B C B C B M ′2s M ′2c 4M 2+m2 +B 0 C B 1 β 1 β 1 φ φ C B − C B C B C B M ′2s M ′2vs 0 4M 2+ 1 (m2 +B ) C B 2 β 1 β 2 4 φ φ C B − C @ A 2 where Mi′ = √2giMiv. From here onwards, we abbreviate sin β, cos β, tan β, and cot β as sβ, cβ, tβ, and ctβ respectively. The imaginary components of the neutral Higgs ﬁelds may be written as

ηuI   η 1 2 dI im = ηuI ηdI ηB η0 im   , (3.1.24) L 2 M   ηB       η0      Chapter 3. Phenomenology of R-Symmetric Gauge Mediation 68

with

Bµctβ Bµ 0 0   B B t 0 0 2 µ µ β im =   . (3.1.25) M  2   0 0 mφ Bφ 0   −   1 2   0 0 0 (mφ Bφ)   4 −    We notice that this matrix is essentially the same matrix of the imaginary components of the Higgs fields in the usual MSSM. The difference is that it is expanded to include the corresponding components of the adjoint Φ fields.

The Lagrangian that contains the charged ﬁelds has the form

+ Hu

+ 2   H = H ∗ H− φ− H−∗ , (3.1.26) L ± u d M± d    +   φ      with

1 2 2 2 1 2 2 ′2 g v c +2g M ϕ +Bµct g v s c +Bµ M s 0 2 2 β 2 2 0 β 4 2 β β 2 β 1

±=B 1 2 2 1 2 2 2 ′2 C. B g v s c +B g2 v s 2g2M2ϕ0+Bµt √2M c C (3.1.27) M B 4 2 β β β 2 β− β 2 β C B C B C B C B M ′2s M ′2c 1 B +8M 2+m 2 C B 2 β 2 β 2 ( φ 2 φ ) C B C @ A Finally, the matrices containing the R-Higgses completely decouple form the rest. This is an obvious result that arises from the unbroken R-symmetry. The neutral R-Higgs masses are as follows

0 1 Ru = 0 0 2 , (3.1.28) R0 Ru Rd R0 L 2 M  0 Rd     with Chapter 3. Phenomenology of R-Symmetric Gauge Mediation 69

1 2 2 MZ cos(2β)+ Mgϕ 0 = 2 (u,d) , (3.1.29) R0   M 0 1 M 2 cos(2β) M 2 2 Z gϕ(u,d)  − −    where

2 2 2 M = m + µ g1M1ϕB + g2M2ϕ0. (3.1.30) gϕ(u,d) R(u,d) (u,d) −

The charged R-Higgses masses have the following form

Ru−∗ = , (3.1.31) Ru− Rd+ R   L ∗ M ± + Rd     with

g2 v2 cos(2β) M 2 0 1 2 gϕ′ (u,d) R = − − , (3.1.32) M ±  2 2 2  0 g1 2v cos(2β)+ Mgϕ′ u,d  − − ( )    where

2 2 2 M ′ = m + µ g M ϕ + g M ϕ , (3.1.33) gϕ(u,d) R(u,d) (u,d) 1 1 B 2 2 0

2 1 2 2 g1 2 = g1 g2 . (3.1.34) − 4 − The neutralino and chargino mass matrices will be given when we present a sample spectrum. All this analysis was done setting the λ terms to zero. Because of the length of the matrices and the little insight that can be gained from them, we do not present the full matrices with non-zero λ terms here. We use instead the power of computing to rapidly perform calculations and scan the parameter space. Chapter 3. Phenomenology of R-Symmetric Gauge Mediation 70

3.2 Scan for λ =0

We would like to perform a scan in the “y z” parameter space by making the λ terms in − (3.1.4) zero to begin with. As we have seen in chapter 2, y is proportional to the gaugino mass and z is a parameter that tells us how much supersymmetry breaking there is. The

supersymmetric limit is when z = 0, and z = 1 corresponds to the SU(6)V limit of the theory presented in chapter 2. Before we perform this scan, we would like to know how

big the supersoft contributions are to see how precise our computations will be.

The model introduced in [2] is analogous to extending the gauge superpartners to

have N = 2 supersymmetry in the gauge sector (that is supersymmetry with two pairs

of supercharges). To allow for chirality, one can have only N = 1 supersymmetry for the matter ﬁelds. The UV operators introduced in chapter 2 were consistent with this

scenario. In this case, supersymmetry breaking introduces no new divergences. These

types of operators are known as “supersoft”. It was argued in [11] that all radiative cor-

rections to the scalar soft masses are ﬁnite. Even though we know that these corrections

are ﬁnite, we would still like to know how big they are. The hope is that they are less

than one at one loop order so that we do not have to worry about two loop contributions to the scalar masses, and as we see, this is certainly the case.

In Figure 3.1, we plot the supersoft/soft contributions to the stop masses for diﬀerent

values of z. We concentrate on the stop masses since these give the largest contributions.

Note that in the plot there are diﬀerent points for each z that correspond to diﬀerent values of y. Generically, the supersoft/soft ratio increases as y increases. We can justify

this plot by making rough estimate of this ratio. In chapter 2 we mentioned that the

gaugino masses are given by (at the messenger scale)

g y M = i M N R(z) , (3.2.1) i 16π2 mess mess

and the scalar masses are given by Chapter 3. Phenomenology of R-Symmetric Gauge Mediation 71

supersoftsoft 0.20

0.15

0.10

0.05

0.00 z 0.0 0.2 0.4 0.6 0.8 1.0

Figure 3.1: supersoft/soft vs z for diﬀerent values of y. The plot is for Nmess = 6,

Mmess = 100TeV, tan(β) = 5, and tan(ν) = 1. The red (light) end of the spectrum corresponds to small values of y, the blue (dark) end corresponds to large (y 3) values ∼ of y.

g2 2 m˜ 2 =2C i M 2 N F (z) , (3.2.2) 0 F 16π2 mess mess where R(z) and F (z) are given in chapter 2. From [11] we can compute a rough estimate of the supersoft/soft ratio as

2 2 mΦ 2 2 2y Nmess log 2 G(z) mQ3supersoft M3 2 2 . (3.2.3) mQ3soft ∼ 16π F(z)

For z = 0.95, y = 3, and Nmess = 6, we have that this value is around 0.8. In other words, the supersoft contributions seem to be under control and we need not to worry about performing two loop calculations to get more accurate results. We now proceed with the scan of the “y z” parameter space to exclude certain points − that are not phenomenologically viable. The idea is to take our messenger scale at some

high scale 102 TeV and to run all the relevant parameters down to the electroweak ∼ scale to give a notion of the phenomenology. The relevant parameters that run are the

2 2 2 top yukawa yt, µu, µd, Bµ, mHu , mQ3 , mu3 (these last two contribute to the left and right Chapter 3. Phenomenology of R-Symmetric Gauge Mediation 72

handed stops respectively), and the gaugino masses Mi. After the running is performed, we demand that the first derivative equations (3.1.18,3.1.19) are satisfied. Through this, we obtain a value for µ and Bµ. These values are compared with initial guess values of µ and Bµ. The process is then repeated until we obtain accurate results. Once the results are accurate enough, the different masses are computed. To exclude points

from the parameter space, we demand that ﬁrst, the points scanned allow for electroweak

symmetry breaking and that the points are in a stable vacuum of the potential. Secondly, we require that the lightest Higgs is bigger than m 115GeV, and that the chargino h ≥ mass is bigger than m ˜ 101GeV. Finally, we check if the points are consistent with C ≥ ρ parameter measurements. In the Standard Model the tree level contribution to ρ is

exactly 1

2 MW ρSM 2 2 =1. (3.2.4) ≡ MZ cos (θw) This parameter is sensitive to physics beyond the Standard Model. In particular, we are concerned with the contributions to this parameter coming from the triplets charged under SU(2) ΦW . All points scanned seem to be consistent with this measurement. The

reason for this being that the VEV of ϕ0 is extremely suppressed by the holomorphic and non-holomorphic mass terms of the scalar adjoints and by the mass of the gaugino as it can be seen from (3.1.20). If we take the denominator of the expression for ϕ0 to be of order 108GeV2 and the gaugino mass 1TeV we have that ∼

ϕ 0.1, (3.2.5) h 0i ∼ then the correction to the ρ parameter goes as

2 2 7 ∆ρ 2 ϕ /v 7 10− , (3.2.6) | | ≃ h 0i ∼ × which is consistent with experimental results. We present the results of diﬀerent scans of the parameter space in Figure 3.2 Chapter 3. Phenomenology of R-Symmetric Gauge Mediation 73

0.8 0.8

0.6 0.6 z z 0.4 0.4

0.2 0.2

0.0 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 y y

(a) (b)

0.8 0.8

0.6 0.6 z z 0.4 0.4

0.2 0.2

0.0 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 y y

0.8

0.6 z 0.4

0.2

0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 y

(e)

Figure 3.2: Excluded points in the y z parameter space for (a) N = 2, M = − mess mess 50TeV, tan(β) = 5, and tan(ν) = 1, (b) Nmess = 2, Mmess = 100TeV, tan(β) = 5, and

tan(ν) = 1, (c) Nmess = 6, Mmess = 50TeV, tan(β) = 5, and tan(ν) = 1, (d) Nmess = 6,

Mmess = 100TeV, tan(β) = 5, and tan(ν) = 1, (e) Nmess = 6, Mmess = 100TeV, tan(β) = 40, and tan(ν)=0.1. The dotted points are excluded by electroweak symmetry breaking. The dark blue region is excluded for a Higgs m 115GeV. Finally, the light h ≤ blue region is excluded for a chargino m ˜ 101GeV. C ≤ Chapter 3. Phenomenology of R-Symmetric Gauge Mediation 74

3.3 Turning on the λ terms

When the λ terms are turned on, the ﬁrst derivative equations with respect to huR and

hdR lead to the following conditions

2 1 2 2 m m cos(2β)+ g M ϕ g M ϕ B cot β + µ′ =0 , (3.3.1) Hu − 2 Z 1 1 B − 2 2 0 − µ u

2 1 2 2 m + m cos(2β) g M ϕ + g M ϕ B tan β + µ′ =0 , (3.3.2) Hd 2 Z − 1 1 B 2 2 0 − µ d

where

1 W B µ′ = µu + λ ϕ0 +2λ ϕB , (3.3.3) u 2√2 u u and

1 W B µ′ = µd λ ϕ0 +2λ ϕB . (3.3.4) d − 2√2 d d Two things seem to happen when the λ terms are included. Firstly, the supersoft/soft ratio decreases as we increase the λ’s. Secondly, it seems that it is harder to get a stable

2 minimum of the potential. Increasing λ has an eﬀect on the running of mHu , making it more negative (analogue to the usual yukawa couplings). In particular the contribution

2 of λ to the running of mHu is important:

2 2 2 2 βm2 (yt) Hu yt (mHu + mQ3 + mu3 ) 2 2 2 2 =0.03, (3.3.5) βm2 (λ) ∼ λ (m + m + m ) Hu Hu Φ R where the ratio is the ratio of the top yukawa and the λ contribution to the RG equation

2 of the Higgs squared mass parameter mHu . The values are taken at the messenger scale

for order one yt and λ for simplicity. Chapter 3. Phenomenology of R-Symmetric Gauge Mediation 75

0.8 0.8

0.6 0.6 z z 0.4 0.4

0.2 0.2

0.0 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 y y

(a) (b)

Figure 3.3: Excluded points for (a) λ = 0.1 and (b) λ = 0.9, with Nmess = 6, Mmess = 100TeV, tan β = 5, and tan ν = 1. The red points are points that are excluded because there is no stable minimum of the potential. The dark blue region is excluded for a Higgs m 115GeV, the light blue one for a chargino m ˜ 101GeV. h ≤ C ≤

2 This large λ contribution to the running of mHu certainly helps electroweak symmetry breaking. It however, can produce an unstable vacuum of the potential for the following reason. From equation (3.3.1), we see that µu, and consequently µd, have to increase

2 since mHu is considerably more negative now, and from equation (3.3.2) this increase forces Bµ to increase as well in order to keep the equality. As we can see from the neutral mass matrix of the Higgses (3.1.23), B appears in the oﬀ-diagonal components. An − µ increase in Bµ forces the lightest eigenvalue to be smaller and eventually negative. For λ values of order 1 there are too many unstable points in the parameter space to be ∼ able to extract any phenomenology. This is what gives us an unstable extremum of the potential. In Figure 3.3 we show the results of the scan for λ = 0.1 and λ = 0.9. In all this analysis we have set all the λ′s equal to each other for simplicity.

3.4 Sample Spectrum

In this section we would like to show a few sample points to give a ﬂavour of the phe-

nomenology. To do this, we need to ﬁx certain parameters of the model. The gaugino Chapter 3. Phenomenology of R-Symmetric Gauge Mediation 76

Input Parameters Output Parameters M = 80 TeV m2 = (538 GeV)2 mess Hu −

MZ = 91 GeV mHd = 756 GeV

y =3 mt˜L =2.2 TeV

z =0.95 mt˜R =2.1 TeV

Nmess =6 M1 =2.1 TeV

v = 174 GeV M2 = 3 TeV

mt = 171.4 GeV M3 = 5 TeV

tan β =5 mh = 150GeV

tan ν =1 mN˜ = 566GeV

mC˜ = 568GeV

Table 3.1: Sample spectrum for λ =0 and scalar masses are computed as in chapter 2 at one loop and two loops respectively.

Furthermore, at the messenger scale, the Bφ terms and the adjoint masses mφ can be computed as well. For example for y = 3, z =0.95, Nmess = 6, and Mmess = 80TeV, we have

M3 = 4.4 TeV,

mφ = 38.6 TeV,

B = 1.03 103 TeV2. (3.4.1) φ − × In table 3.1 we present some of the masses of the model for λ = 0 and the corresponding input parameters. It is worth mentioning that in this case, the lightest Higgs mass is around 74 GeV at tree level. When the loop corrections due to the stops are included, it can be pushed up to 150 GeV. However, because the stop masses are big there is a lot of ﬁne tuning, 0.5% without including the λ terms. Finally, the masses of the R-Higgses

and of the charged Higgses are all around 900 GeV.

In table 3.2, we give another sample spectrum for λ =0.1 and λ =0.9 with the same Chapter 3. Phenomenology of R-Symmetric Gauge Mediation 77

Output Parameters for λ =0.1 Output Parameters for λ =0.9 m2 = (734 GeV)2 m2 = (4.5 TeV)2 Hu − Hu −

mHd = 756 GeV mHd = 756 GeV

mt˜L =2.2 TeV mt˜L =2.3 TeV

mt˜R =2.1 TeV mt˜R =2.3 TeV

M1 =2.1 TeV M1 =2.1 TeV

M2 = 3 TeV M2 = 3 TeV

M3 = 5 TeV M3 = 5 TeV

mh = 151GeV mh = 143GeV

mN˜ = 762GeV mN˜ = 2TeV

mC˜ = 766GeV mC˜ = 3TeV

Table 3.2: Sample spectrum for λ =0.1 and λ =0.9 with the same input parameters as in table 3.1.

input parameters as in table 3.1. We notice that the Higgs mass is within experimental

2 constraints as expected from the scan. The excessive running of mHu is clearly shown for λ =0.9, in which case we obtain m2 = (4.5TeV)2. Hu − From table 3.1 and 3.2, we realise that the neutralino and the chargino get heavier as we increase λ. This happens for a reason similar to the one discussed in the previous

2 section. We have argued that the running of mHu is aﬀected by λ and that it ultimately forces µ to increase. This increase in µ aﬀects the neutralino mass matrix which, following

the notation of [26] is given by

T N˜ = N+ N˜ N , (3.4.2) L M −

with Chapter 3. Phenomenology of R-Symmetric Gauge Mediation 78

M 0 g v sin β g v cos β 2 − 2 2   0 M g v sin β g v cos β 1 1 − 1 MN˜ =   , (3.4.3)    λv sin β λv sin β µu 0   − | |     λv cos β λv cos β 0 µd   − | |    where the vector N = carries R charge +1, and the vector + ψΦW ψΦB ψRu ψRd

N = ψ ψ ψ ψ carries R charge 1. The lightest eigenvalue of N˜ is − W B Hu Hd − M connected to µ since M1 and M2 are heavy. This happens until λ is big enough, making µ of a size comparable to the gaugino masses and increasing the size of the lightest

neutralino. Something similar is what happens with the charginos. The mass matrices

of the charginos are given by

ψφ− ψW − ψφ+ ψ + C˜1 + ψW + ψ + C˜2 , (3.4.4) Rd   Hu   L ⊃ M − M − ψRu ψH    d      where

M2 √2λvcβ ˜ = , (3.4.5) MC1   √2g2vcβ µd  | |    and

M2 √2λvsβ ˜ = . (3.4.6) MC2   √2g2vsβ µu  | |    As a side note we point out that these types of matrices can be diagonalized by diﬀerent unitary transformations. For example

mC˜11 0 V † ˜ U = . (3.4.7) MC1   0 mC˜12     Chapter 3. Phenomenology of R-Symmetric Gauge Mediation 79

A ﬁnal remark about the lightest Higgs mass must be made. One may wonder why when λ = 0.9 the lightest Higgs decreases in value. Although the quartic corrections

2 2 proportional to λ make it bigger, the running affects mHu significantly, preventing the lightest Higgs to be greater through the off-diagonal Bµ term. In other words, the quartic

corrections and the bigger oﬀ-diagonal Bµ term compete with each other and eventually the lightest Higgs turns out to be slightly lighter than in the previous cases.

3.5 Uniﬁcation of the Yukawas

Since we have relative freedom to choose the yukawas at the messenger scale, in this

section we would like to ﬁnd out which values of “y” are acceptable if these yukawas are to be uniﬁed at some high scale. To accomplish this, we demand two things. First, we

require that the Yukawa Landau pole Λy does not occur before the QCD Landau pole Λ3, > i.e. : Λy Λ3. Second, we demand that the unification of the different yukawas occurs ∼ around Λ3. To ensure this, let us first rewrite the superpotential term, introduced in chapter 2, that is responsible for generating Dirac gauginos:

W = y φΦ N + y φΦ N + y(3)φ Φ N + y(2)φ Φ N + C conjugate . (3.5.1) y 3 3 2 2 1 3 1 3 1 2 1 2 −

For the rest of the discussion we will focus on y3 only. We perform a scan for diﬀerent

uniﬁcation values of y and diﬀerent values of Nmess. The results can be seen in Figure 3.4.

Notice that these points are really maximum values that y3(Mmess) can take so that

its Landau pole does not go below Λ3. There is practically no difference between different unification values for y. To illustrate the phenomenology, we can take a sample value

from this plot, Nmess = 4 and y3 = 1.3, and compute the sample spectrum to give us a taste of what to expect. This is shown in table 3.3. The R-Higgses as well as the charged

Higgses are all around 600 GeV, since we chose Nmess = 4 for the analysis this time. Chapter 3. Phenomenology of R-Symmetric Gauge Mediation 80

Nmess

0 y3 0.9 1.0 1.1 1.2 1.3 1.4 1.5

Figure 3.4: Nmess vs y3 at the messenger scale. The plotted points are the points for which the the yukawas y1, y2, and y3 all unify at Λ3. The uniﬁcation values for y were varied from y = 4 all the way to y = 20.

Finally, Λ =4.7 105 TeV which is well above the messenger scale. 3 × Just to exemplify how Λ3 varies depending on Nmess and the value of y3 at the messenger scale, we take some sample points and give the corresponding Landau pole

Nmess y3(Mmess) Λ3 (TeV) gluino/squark 2 1.52 1.3 109 0.6 × 4 1.3 4.7 105 0.73 × 8 1.1 8.8 103 0.9 × In order to alleviate the ﬂavour problem a large gluino:squark mass ratio is required. For

Nmess = 8, we still cannot have gluinos that are suﬃciently heavy to cure this problem. Furthermore, the Landau pole in this case is very close to the messenger scale. This

would pose a problem to realize the solution of the ﬂavour problem given in the MRSSM.

3.6 Tuning

The amount of tuning in the MRSSM can be computed following [6] as Chapter 3. Phenomenology of R-Symmetric Gauge Mediation 81

Input Parameters for λ =0 Output Parameters M = 80 TeV m2 = (572 GeV)2 mess Hu −

MZ = 91 GeV mHd = 585 GeV

y3 =1.3 mt˜L =1.8 TeV

z =0.95 mt˜R =1.7 TeV

Nmess =4 M1 = 625 GeV

v = 174 GeV M2 = 904 GeV

mt = 171.4 GeV M3 =1.5 TeV

tan β =5 mh = 147GeV

tan ν =1 mN˜ = 593 GeV

mC˜ = 595 GeV

Table 3.3: Sample spectrum for a Yukawa consistent with a possible uniﬁcation scenario.

∆m2 3y2m2 M T uning = Hu = t t˜ log mess , (3.6.1) m2 4π2m2 m Z Z t˜ for λ = 0. With stops of 2 TeV we have that the tuning is 0.6%. Now when the λ ∼ ∼ 2 terms are turned on the λ contribution to mHu is

λ2m2 M ∆m2 Φ log mess , (3.6.2) Hu ∼ 4π2 m Φ and the tuning, when we take λ 1 and m = 38TeV, is 0.02%. We thus see that there ∼ Φ is an excessive amount of tuning to realize this scenario. This is one of the unappealing

features of this model.

3.7 Relevant Beta Functions

The following is a list of the relevant beta functions. The analysis presented in this

chapter did not make use of all of these equations, we rather assumed that the most

important contribution to the running of diﬀerent parameters comes from the running of Chapter 3. Phenomenology of R-Symmetric Gauge Mediation 82

2 mHu . Since there is no preferred particular choice for all the λ′s at the messenger scale, in the end the equations will not be relevant. We believe that an extensive analysis of all these parameters would not provide further insight into the phenomenology of the model.

y 3 16 13 β = t 6y2 + λ2 + λ2 g2 3g2 g2 yt 16π2 t 1u 4 2u − 3 3 − 2 − 15 1 λ 3 3 β = 1u 4λ2 +2λ2 + λ2 +3y2 +6N y(3)2 +4N y(2)2 3g2 g2 λ1u 16π2 1u 1d 2 2u t mess 1 mess 1 − 2 − 5 1 λ 3 3 β = 1d 4λ2 +2λ2 + λ2 +6N y(3)2 +4N y(2)2 3g2 g2 λ1d 16π2 1d 1u 2 2d mess 1 mess 1 − 2 − 5 1 λ 1 3 β = 2u 2λ2 + λ2 +2λ2 +3y2 + N y2 7g2 g2 λ2u 16π2 2u 2 2d 1u t mess 2 − 2 − 5 1 λ 1 3 β = 2d 2λ2 +2λ2 + λ2 + N y2 7g2 g2 λ2d 16π2 2d 1d 2 2u mess 2 − 2 − 5 1 y 8 34 4 β = 3 (N + 1) y2 +2y(3)2 g2 g2 y3 16π2 3 mess 3 1 − 3 3 − 15 1 y 3 1 1 3 β = 2 N + y2 +2y(2)2 + λ2 + λ2 7g2 g2 y2 16π2 mess 2 2 1 2 2d 2 2u − 2 − 5 1 (3) y1 8 2 (3)2 (2)2 2 2 16 2 4 2 βy(3) = y3 + 2(3Nmess + 1)y1 +4Nmessy1 +2λ1d +2λ1u g3 g1 1 16π2 3 − 3 − 15 (2) y1 3 2 (2)2 (3)2 2 2 2 3 2 βy(2) = y2 + 2(2Nmess + 1)y1 +6Nmessy1 +2λ1d +2λ1u 3g2 g1 1 16π2 2 − − 5

3.8 Final Remarks

The Higgs sector of the MRSSM was analysed in this chapter. We constructed the scalar

potential of this theory and computed sample spectra for the cases where the λ terms are

zero or relatively small. As we have seen, if the λ terms are present, they must be less than one in order to have a reasonable parameter space to extract an spectrum. On the

other hand, the uniﬁcation of the Yukawa couplings in section 3.5 seems to be possible

only at the expense of a low gaugino:squark ratio and thus, at the expense of the solution

to the ﬂavour problem that was originally proposed in the MRSSM. Finally, we see that Chapter 3. Phenomenology of R-Symmetric Gauge Mediation 83 this model is quite tuned, due to the presence of heavy adjoints in the theory. In any case, a lower λ parameter could help alliviate the tuning of the theory. Chapter 4

Cosmological Constraints on Decaying Dark Matter

The contents of this chapter were published in the Journal of Cosmology and Astroparticle

Physics, JCAP 06 (2009) 005.

4.1 Motivation

There have been a few papers [53–55] in recent years analyzing DM decay into electromagnetically non-interacting particles using just the cosmic microwave background data

(Ref. [54] also includes supernova data). In this chapter, we revisit this scenario using a

Markov Chain Monte Carlo (MCMC) analysis employing all available datasets from the cosmic microwave background (CMB), Type Ia supernova (SN), Lyman-α forest (Lyα), large scale structure (LSS) and weak lensing (WL) observations. We ﬁnd that the lifetime of decaying DM is constrained predominantly by the late time Integrated Sachs Wolfe

1 (ISW) eﬀect to be Γ− & 100 Gyr. In the main body of this chapter, we will comment on the discrepancies between the results of Refs. [53–55].

The studies in the preceding paragraph considered only the case where there was negligible reionization of the universe due to DM decay. In an attempt to address this,

84 Chapter 4. Cosmological Constraints on Decaying Dark Matter 85

Ref. [56] analyzed the scenario of DM decaying into only electromagnetically interacting products, that get partially absorbed by the baryonic gas, using a subset of the available

CMB datasets. We extend their analysis by using all the available CMB datasets, and

also the SN, Lyα, LSS and WL datasets. Besides the smaller selection of datasets, their

analysis also ignores the impact of DM decay on cosmological perturbations which renders

it ineﬀectual in the parameter space where there is negligible reionization. Our treatment

allows the decay products to not only reionize the universe but also takes into account the eﬀect of DM decay on cosmological perturbation. This allows us to generate many

other observables, particularly, the late time ISW eﬀect that is crucial to constrain the

lifetimes at low reionization. Another key diﬀerence between the analyses is that we use

a combined reionization parameter for both DM reionization and phenomenological star

formation reionization, rather than just treating them separately as was done in Ref. [56],

because current observations cannot distinguish which contribution to reionization is the dominant one. Doing the MCMC analysis, we ﬁnd that the lifetime of decaying DM in

1 8 this scenario constrained to be (fΓ)− & 5.3 10 Gyr, where f is a phenomenological × parameter introduced by Ref. [56] and related to the degree of reionization.

Astrophysical constraints together with additional assumptions have also been used

[57,58] to give even tighter bounds than ours on the lifetime of the decaying DM. While interesting and complementary, these lie outside the scope of of this thesis.

Having obtained the bounds on the lifetime of decaying dark matter, we will then

explore the implications of our cosmological analysis on particle physics models beyond

the Standard Model. We will present a complete list of cross-sections for spin-0, spin-1/2

and spin-1 dark matter to decay into Standard Model degrees of freedom via eﬀective operators. Obviously, this can be easily extended to other models with additional light

degrees of freedom (for instance, hidden valley models [59]) by appropriate substitution

of the parameters. Applying the bound on the lifetime of the decaying DM, we can then

place limits on the size of the parameters of theories. For generic theories with a decaying Chapter 4. Cosmological Constraints on Decaying Dark Matter 86 dark matter of 100 GeV mass, the coupling constant in the eﬀective dimension-4 ∼ 22 operators responsible for dark matter decay will be shown to be . 10− . We will also look at speciﬁc representative cases of theories beyond the Standard Model physics and investigate the possibility of viable dark matter candidates: the spin-0 messenger DM in the context of gauge mediation messenger number violation, the spin-1/2 bino DM in the scenario with R-parity violation and the spin-1 “massive photon partner”DM in the framework of T-parity violation.

The rest of this chapter is organized as follows. In section 4.2, we discuss the physics of decaying dark matter cosmology as well as introduce the datasets that we will be using. Section 4.3 contains our Markov Chain Monte Carlo results and discussions of the cosmological implications. In section 4.4, we explore the consequences of these results for particle physics theories by enumerating the decay channels and partial widths. Rep- resentative models from theories of gauge-mediated supersymmetry breaking, minimal supergravity and little Higgs were also investigated using the results of our analysis.

4.2 Decaying Cold Dark Matter Cosmology

We will assume the standard picture of ΛCDM cosmology, i.e. a Friedman-Robertson-

Walker universe that is principally composed of dark energy and cold dark matter, with one crucial modiﬁcation; that is, we have a cold dark matter that is very long-lived but ultimately decays. As we are considering lifetimes of gigayears (Gyr), the fraction of

DM decays happening during or soon after big bang nucleosynthesis (BBN) is negligible and hence would not alter the predictions of BBN. To perform a model-independent analysis, we allowed decays to all possible SM particles. However, we will assume that the long term decay products are relativistic. This assumption that the decay products are relativistic is crucial. The decay products can be any relativistic particle, SM particle or not, and the case with interacting decay products will be treated more carefully later Chapter 4. Cosmological Constraints on Decaying Dark Matter 87

on when we talk about reionozation. The thermalization of cosmic rays will be left for future work. While we include branching ratios to intermediate non-relativistic states,

they are assumed to be short-lived and will rapidly decay into light relativistic degrees

of freedom.

The evolution of background and ﬁrst order perturbation in decaying cold dark matter model was ﬁrst formulated in longitudinal gauge [53], which means the decay rate has to be treated with care as the CDM is not at rest in the longitudinal gauge. We, on the other hand, will work in CDM rest frame using synchronous gauge with the line element written as

ds2 = a2(τ)[ dτ 2 +(δ + h )dxidxj]. (4.2.1) − ij ij where τ is conformal time, and t the cosmological time (dt = a(τ)dτ). We will follow the convention a = 1 today.

The decay equation dρ cdm = Γρ , (4.2.2) dt − cdm can be reformulated in a covariant form

µ CDM T ν;µ( )= Gν, (4.2.3)

where the force density vector Gν can be calculated from its value in CDM rest frame

G CDM rest =( Γρ , 0, 0, 0). (4.2.4) ν| − cdm

The conservation of total energy momentum tensor requires

T µ (dr)= G , (4.2.5) ν;µ − ν

where the daughter radiation (dr) is composed of the CDM decay products.

The equations describing the evolution of background are

ρ˙ = 3 ρ aΓρ , (4.2.6) cdm − H cdm − cdm ρ˙ = 4 ρ + aΓρ , (4.2.7) dr − H dr cdm Chapter 4. Cosmological Constraints on Decaying Dark Matter 88

where dot denotes the derivative with respect to conformal time τ. We have deﬁned the conformal expansion rate to be a˙ . H ≡ a We will only consider scalar metric perturbations, which in Fourier space can be

expanded as (following [60])

3 ik x 1 h (x, τ)= d ke · [n n h(k, τ)+6(n n δ )η(k, τ)], (4.2.8) ij i j i j − 3 ij Z where n k/ k . ≡ | | Our choice of gauge and coordinates lead to the following simple density perturbation

equation for CDM, 1 δ˙ = h.˙ (4.2.9) cdm −2 The terms containing Γ all cancel out, because the background density and overdensity

are decaying with the same rate.

Instead of using simple hydrodynamic approximation for the decay product [61], which might give correct order of magnitude but less accurate results, we use the full Boltzmann equations to describe the decay product, which were ﬁrst given by Ref. [62], and recently updated by [55] for decaying DM cosmology,

˙ 2 ˙ 4 ρcdm δdr = 3 h 3 kvdr + aΓ (δcdm δdr), (4.2.10) − − ρdr −

1 1 ρcdm v˙dr = k( δdr Πdr) aΓ vdr, (4.2.11) 4 − 2 − ρdr ˙ 8 3 4 ˙ 8 ρcdm Πdr = k( 15 vdr 5 F3)+ 15 h + 5 η˙ aΓ Πdr, (4.2.12) − − ρdr k ρcdm F˙l = [lFl 1 (l + 1)Fl+1] aΓ Fl, (4.2.13) 2l +1 − − − ρdr

where l =3, 4, 5, ...,, F2 = Π and for the rest, we have used the conventions of Ref. [60]. Because CDM particles are heavy and non-relativistic, we have treated the CDM as a perfect ﬂuid.

For the case where the DM candidate also decays into electromagnetically interacting

particles (e.g. photons or electron/positrons), we have to be more careful. This is be-

cause the decays may deposit signiﬁcant energy into baryonic gas and contribute to the Chapter 4. Cosmological Constraints on Decaying Dark Matter 89

reionization of universe. Following [56, 63, 64], we introduced a phenomenological factor f as the fraction of the decay energy deposited in the baryonic gas. For long-lifetime

dark matter models, the reionization due to dark matter decay only depends on the

combination ζ = f Γ/H0. We use ζ as an additional parameter in our MCMC analysis. Without a prior on f, the constraint on ζ does not directly give any information on

Γ. However, for given dark matter models, one should in principle be able to calculate

the decay branches, and therefore give a rough estimate for f under certain additional assumptions. 1 Following [56, 64], we modify RECFAST [65, 66] to calculate the reion-

ization due to DM decay. In this scenario, the reionization is dominated by DM decay

at redshift z > 20, and is competing with the contribution from star formation (or other

sources) at some redshift between z = 6 and z = 20. Without knowing the details of star

formation or other reionization sources, we use the following phenomenological model,

which can be regarded as a combination of CosmoMC phenomenological formula and DM decay reionization formula,

1.5 1.5 RECFAST 1+ fHe (1 + z) (1 + zre) xe = max xe , [1 + tanh( − )] , (4.2.14) { 2 1.5√∆z }

where xe, the ionized fraction, is deﬁned as the ratio of free electron number density to

hydrogen number density; fHe is the ratio of helium number density to hydrogen number

RECFAST density; xe is the modiﬁed RECFAST output ionized fraction (i.e., the ionized fraction assuming DM decay is the only source of reionization; ∆z is the redshift width of reionization (due to other sources), for which we have taken the CosmoMC [67] default value 0.5; the last free parameter, reionization redshift zre, is determined by the total optical depth τre.

With all the above equations, we modiﬁed CosmoMC to analyze the decaying CDM

1We are approximately treating the redshift-dependent parameter f as a constant. For order of magnitude estimation of f see [63]. Typically if the decayed products are electrons and positrons with energy GeV, f will be about 10−3 at low redshift. ∼ Chapter 4. Cosmological Constraints on Decaying Dark Matter 90 model. In addition, we also incorporated weak lensing data into the Markov Chain Monte Carlo (MCMC) analysis. The datasets used in this chapter are listed below. For each dataset, we either wrote a new module to calculate the likelihood or modiﬁed the default

CosmoMC likelihood codes to include the features of the decaying CDM model.

Cosmic Microwave Background (CMB)

We employ the CMB datasets from WMAP-5yr [13, 14], BOOMERANG [68–70],

ACBAR [71–74], CBI [75–78], VSA [79], DASI [80, 81], and MAXIMA [82]. Also

included are the Sunyaev-Zeldovich (SZ) eﬀect for WMAP-5yr, ACBAR and CBI datasets. The SZ template is obtained from hydrodynamical simulation [83]. When

calculating the theoretical CMB power spectrum, we have also turned on CMB

lensing in CosmoMC.

Type Ia Supernova (SN)

The Union Supernova Ia data (307 SN Ia samples) from The Supernova Cosmology

Project [84] was utilized. For parameter estimation, systematic errors were always included.

Large Scale Structure (LSS)

For large scale structure we will use the combination of 2dFGRS dataset [85] and

SDSS Luminous Red Galaxy Samples from SDSS data release 4 [86]. It should be

noted that the power spectrum likelihood already contains the information about

BAO (Baryon Acoustic Oscillation [87,88]).

Weak Lensing (WL)

Five weak lensing datasets were employed in this analysis. The eﬀective survey

area and galaxy number density of each survey are listed in Table 4.1.

For COSMOS data we used the CosmoMC module written by Julien Lesgourgues

[90] . For the other four weak lensing datasets, we utilized the likelihood given Chapter 4. Cosmological Constraints on Decaying Dark Matter 91

Table 4.1: Weak Lensing Data 2 2 Data Aeff (deg− ) neff (arcmin− ) COSMOS [89,90] 1.6 40 CFHTLS-wide [91–93] 22 12 GaBODS [92,94,95] 13 12.5 RCS [92,96] 53 8 VIRMOS-DESCART [92,93,97] 8.5 15

in Benjamin et al. [92]. We take the best ﬁt parameters for the following galaxy

number density formula,

β z α z β n(z)= 1+α ( ) exp ( ) . (4.2.15) z0Γ( β ) z0 −z0

Benjamin et al. give two sets of best ﬁt parameters, ﬁtting on the galaxy samples

with median photometric redshift 0 < zp < 4 and 0.2 < zp < 1.5, respectively (see

Figure 2 and Table 2 in [92]). We only used the data from the 0 < zp < 4 region. We also simpliﬁed the marginalization on n(z) parameters by assuming a Gaussian

prior on z0. The width of Gaussian prior is adjusted so that the mean redshift zm

has an uncertainty of 0.03(1 + zm), i.e.

1+α Γ( β ) σz0 =0.03 2+α + z0 . (4.2.16) Γ( β ) The weak lensing data only measure matter power spectrum at angular scales less than a few degrees, which corresponds to scales less than a few hundred Mpc. This

is much less than the Jean’s length of the daughter radiation and therefore we can

ignore the daughter radiation when calculating the power spectrum of projected

density ﬁeld

χH 4πG 2 2 4 l Pl(κ) = ( 4 ) ρma P3D( ; χ) c 0 dcA(χ) χH Z dcA(χ′ χ) 2 dχ′n(χ′) − . (4.2.17) × d (χ ) Zχ cA ′ Chapter 4. Cosmological Constraints on Decaying Dark Matter 92

We should stress that this is speciﬁc to the decaying CDM model and diﬀers from the conventional CDM model [98,99].

Lyman-α Forest

The following Lyα forest datasets were applied.

1. The dataset from Viel et al. [100] consist of LUQAS sample [101] and the

Croft et al. data [102].

2. The SDSS Lyα data presented in McDonald et al. [103,104]. To calculate the

likelihood, we interpolated the χ2 table in the three-dimensional amplitude-

index-running space.

We explore the likelihood in nine-dimensional parameter space, i.e., the Hubble pa-

2 rameter h, the baryon density Ωbh , the amplitude and index of primordial power spectrum ( As and ns), the DM decay reionization parameter ζ, the total reionization optical depth τ , the SZ amplitude A , the decay rate normalized by Hubble parameter Γ , re SZ H0 2 and the CDM density in early universe Ωcdm,eh . The parameter Ωcdm,e is deﬁned to be

3 (ρcdma ) a 1 Ωcdm,e | ≪ , (4.2.18) ≡ ρcrit0

2 where ρ 3H is today’s critical density. As the CDM in our case decays, we made a crit0 ≡ 8πG

distinction between Ωcdm,e and Ωcdm where the latter is deﬁned to be the usual fractional

CDM density today (ρcdm0/ρcrit0).

4.3 Markov Chain Monte Carlo Results and Discus-

sion

For the case with negligible reionization, we generated 8 MCMC chains, each of which

contains about 3000 samples. The posterior probability density function of CDM decay Chapter 4. Cosmological Constraints on Decaying Dark Matter 93

200 ) Γ

( 100 P

0 0 0.01 0.02 1 Γ(Gyr - )

Figure 4.1: Posterior probability density function of the decay rate Γ. Solid line: using all the datasets. Dashed line: CMB + SN + LSS + Lyα. Dotted line: CMB only. The probability density function is normalized as P (Γ)dΓ=1. R rate can be directly calculated from the Markov Chains, as shown in Figure 4.1. The

1 corresponding 68.3% and 95.4% conﬁdence level lower bounds on lifetime are Γ− &

1 230Gyr and Γ− & 100Gyr, respectively. If we take the lifetime of universe to be 14Gyr, the 95.4% conﬁdence level limit (i.e. lifetime 100Gyr) corresponds to a scenario that roughly 15% of CDM has decayed into radiation by today. Now let us understand why decaying DM has impact on cosmological observables, particularly the CMB TT spectrum that gives the most stringent constraint. In our analysis, all the early universe physics before recombination remains unchanged. We assume the lifetime of DM particle is at least the order of the age of universe. Therefore, from DM particle freezeout till recombination the decay is negligible. The primary source of CMB ﬂuctuation remains unchanged. Only when the cosmic age is beyond Gyr scale

(i.e. at z . 10), DM decay can produce significant impact on the CMB. The CMB power spectrum is then significantly modified due to two effects. One is that the decay of CDM modifies the evolution of background (FRW background evolution), which results in a different distance to last scattering surface compared to the conventional case. The second one is that the decay of CDM affects the cosmological perturbations in late universe, Chapter 4. Cosmological Constraints on Decaying Dark Matter 94

0.26 0.25

0.24 e , cdm

cdm 0.22

Ω 0.2 Ω

0.2

0.18 0.15 0 0.005 0.01 0.015 0 0.005 0.01 0.015 1 1 Γ/Gyr - Γ/Gyr -

Figure 4.2: Constraints on the early universe CDM density parameter Ωcdm,e and decay rate Γ, using all the datasets, is plotted on the left panel. For comparison, the present day CDM density parameter Ωcdm and decay rate Γ is plotted on the right panel. The inner and outer contours correspond to 68.3% and 95.4% confidence levels, respectively. resulting in an enhancement of the Integrated Sachs-Wolfe (ISW) effect beyond that due to the cosmological constant. And it is this effect, anticipated by Kofman et al. [61], that gives us the most restrictive bound on the lifetime of decaying dark matter for the scenario with negligible reionization.

Let us review the inconsistency between past papers on this issue. We start with Ref. [54]. Now, CMB and SN observations today can measure the fractional CDM density to a roughly 15% level [14] (within 95% conﬁdence level). We will expect the constraints on CDM decay ratio to be the same order of magnitude. This simple estimation does not take into account the fact that the decayed product still contributes to the total energy density of the universe, and that the DM decay happens mostly at low redshift.

Therefore if we do not take the cosmological perturbation into account, the data should allow about 15% dark matter to have decayed by today, i.e., we should not get a bound better than 100 Gyr. This simple analysis implies that the recent lower bound of lifetime

1 (Γ− > 700Gyr at 95.4% conﬁdence level) obtained by Ref. [54], which does not take into consideration the impact of DM decay on cosmological perturbation (the location of Chapter 4. Cosmological Constraints on Decaying Dark Matter 95

6000

4000

2000

0 10 100 1000

Figure 4.3: CMB power spectrum for diﬀerent dark matter decay rate, assuming the decayed particles are relativistic and weakly interacting. For the CDM density parameter,

2 2 we choose Ωcdm,eh to be the same as WMAP-5yr median Ωmh . For the other cosmological parameters we use WMAP-5yr median values. By doing this, we have ﬁxed the CDM

2 to baryon ratio at recombination. In a similar plot in Ichiki et al. [53] Ωcdm0h is instead fixed. Therefore the height of first peak, which has dependence on CDM to baryon ratio at recombination, will significantly change as one varies the decay rate. In this plot the red line corresponds to a stable dark matter . The blue dotted line corresponds to dark matter with a lifetime 100 Gyr, and the blue dashed line 27 Gyr. The data points are WMAP-5yr spectrum mean values and errors (including instrumental errors and cosmic variance). Chapter 4. Cosmological Constraints on Decaying Dark Matter 96

ﬁrst CMB peak is aﬀected only through the change of background evolution), may not be credible. If indeed the CDM lifetime is 700Gyr, only about 2% of CDM has decayed

6 into radiation by today, and by the time of recombination, less than 10− of CDM has decayed. The change in background evolution is so tiny that it should not be detectable by current cosmological data.

To compare with Ichiki et al. [53], we re-did the analysis using just the CMB datasets,

1 and found the CDM lifetime Γ− & 70 Gyr at 95.4% CL, which is consistent with their

1 results. The reason we obtained a more constrained value than their Γ− & 52 Gyr at 95.4% C.L. is probably because we used WMAP-5yr compared with their WMAP-1yr dataset. We expect the WMAP-9yr dataset, when published and analyzed, to exhibit

only a modest improvement because the information from the late time ISW eﬀect is

1 limited by cosmic variance. Recently, Lattanzi et al. [55] obtained a bound of Γ− & 250 Gyr at 95.4% C.L. with just the WMAP-3yr data, which is not consistent with both Ichiki

et al. and our results. We notice that in the Figure 3 of their paper, the proximity between

68.3% and 95.4% conﬁdence level bounds on Γ indicates a sudden drop of marginalized likelihood (Γ). In our result, as shown in Figure 4.1, this sudden drop feature is not L seen. In Figure 4.2 we plot the constraints for the density parameter and the decay rate.

Notice that in the second plot the shape is tilted down since DM has decayed.

Let us move on to the scenario where there is signiﬁcant reionization due to the

decaying dark matter. We generated another 8 MCMC chains, each of which contains

about 6000 samples. The results of our analysis can be seen in Figures 4.4 and 4.5, where

we show the constraints on DM decay reionization parameter. A few things should be

noted. Firstly, the sharp boundary (reﬂected in the closeness of the two contours) on

the rising edge in Figure 4.5 is due to the fact that for a given τre, DM decay has an

upper limit because the optical depth due to DM decay should not extend beyond τre. Secondly, the plateau of likelihood around f Γ = 0 in Figure 4.4 indicates that current

CMB polarization data can only constrain the total optical depth, but cannot distinguish Chapter 4. Cosmological Constraints on Decaying Dark Matter 97

1 1

P P 0.5 0.5

0 0 0.05 0.1 0.15 0 0.5 1 τ Γ 10 - 25 - 1 re f ( s )

Figure 4.4: The marginalized posterior likelihood of the total optical depth and that of the DM decay reionization parameter.

between DM decay reionization and star formation reionization. In other words, the data

does not favor or disfavor DM decay reionization, as long as its contribution to total

optical depth is not larger than the preferred τre. 25 1 The constraint we have obtained is f Γ . 0.59 10− s− at 95.4% confidence level. × This result is about a factor of 3 better than Ref. [56]. In the limit where reionization is negligible, Ref. [56] cannot give a strong bound on Γ because they have ignored the impact of DM decay on cosmological perturbations. Hence, their constraint on DM decay is essentially only from CMB polarization data. Our analysis, which combines many different cosmological datasets and includes the calculation of the impact of DM decay on all the observables, gives a stringent constraint on Γ even in the f = 0 limit. As for the limit of significant reionization, our bounds are, as mentioned earlier, a significant improvement over Ref. [56]. The difference may be due to the fact that we have used more datasets. However, the priors of the parameters may also alter the result. A notable difference between the models is due to the fact that one of the parameters they have adopted, the optical depth without DM decay, is ill-defined in our model. Also in their model, the cutoff of DM decay reionization at z = 7 was explicitly chosen. These differences might have led to Figure 1 in their paper which shows a preference for a zero dark matter decay rate, which is absent in our analysis. Chapter 4. Cosmological Constraints on Decaying Dark Matter 98

1 ) 1 - s 25

- 0.5 10 ( Γ f

0 0.05 0.1 0.15 τ re

Figure 4.5: The marginalized 2D likelihood contours. The inner and outer contours correspond to 68.3% and 95.4% conﬁdence levels, respectively.

4.4 Implications for Particle Physics Models with

Decaying Cold Dark Matter

Our results most certainly impose constraints on extensions of the Standard Model of particle physics (SM) with DM candidates. Making the assumption that DM decays into SM ﬁelds, we will investigate all the probable decay channels unless forbidden by either symmetry or kinematics, or highly suppressed by phase space considerations. We then sum up their partial decay widths to obtain the functional form for the lifetime of each of the decaying DM candidates. One might be worried about including even the decays to non-relativistic particles as that might invalidate our earlier assumption that cosmologically, the dark matter decay products are relativistic. While it is true that the DM particle can and will decay (provided it is not kinematically forbidden) into non-relativistic massive gauge bosons or heavy quarks, these heavy particles themselves Chapter 4. Cosmological Constraints on Decaying Dark Matter 99

will be assumed to subsequently decay very rapidly into much lighter particles of the SM that will be relativistic. Obviously in speciﬁc models, certain channels could be expressly

forbidden by symmetries and this can also be handled by our analysis.

To obtain the functional form for the lifetime of the decaying DM, we will approach

it from the point of view of eﬀective ﬁeld theory. We will do a model independent

analysis by considering generic Lagrangian terms for these decays with the corresponding

coupling constants acting as Wilson coeﬃcients. Since the DM is electrically neutral,

the total charge of decay products should also be zero. Moreover, the decay rates for

diﬀerent channels are dependent on the intrinsic spin of the DM because of possible spin-dependent couplings. Below we discuss the decays of DM with spin 0, 1/2 and

1. We will only consider decay processes of the lowest order, as higher-order processes

involve more vertex insertions and so are assumed to be suppressed. Additionally in

our approach, we will work in the framework where all the gauge symmetries (including

those of Grand Unified Theories if present) except for SU(3) U(1) are broken and the c × em effects encoded in the coupling constants of the effective operators. This can potentially give rise to naturally very small coupling constants as they could contain loop factors or powers of very small dimensionless ratios. This is a more cost-effective and model- independent way of taking into account the myriad possibilities of UV-completing the

Standard Model of particle physics.

Having obtained the functional form of the lifetime in terms of the fundamental parameters of the underlying particle physics models, we can then compare it with the numerical value obtained from the cosmological analysis of the previous section. This would allow us to place definitive bounds on the fundamental parameters of candidate models for the particle physics theory beyond the Standard Model. It should be noted that we will be using the most conservative 95.4% confidence level bound on the lifetime of the decaying dark matter, i.e. without significant reionization. To assume otherwise would require a more complete knowledge of the ionization history of the universe than Chapter 4. Cosmological Constraints on Decaying Dark Matter 100

is currently understood.

Let us now proceed to the case of a generic scalar DM candidate and see how the above ideas are implemented.

4.4.1 Spin-0 Dark Matter

Table 4.5 in section 4.5 lists out possible Lagrangian terms for the decay of S into SM particles, and the corresponding decay rates, summing over ﬁnal state spins. Apart from focusing solely on the S νν¯ channel, we can consider scenarios in which all the → interaction terms in Table 4.5 are present to contribute to the decay rate of S, with all coupling constants of the same order of magnitude, g0. For simplicity, we will also assume mdecay product/mS is negligible compared to 1. This would immediately imply that the

µ helicity-suppressed term, g0fγ¯ (1 + raγ5)f∂µS/Λ, gives rise to insigniﬁcant decay rate when compared to other terms. So the most relevant terms are the ones that come from the operator g0Sf¯(1 + irpγ5)f. Then we have

g2m Γ(S ff¯) 0 S (1 + r2) N → ≈ 8π p f f SM f SM X∈ X∈ 21g2m = 0 S (1 + r2). (4.4.1) 8π p

Here the decay to tt¯ is not included as this channel may not be kinematically feasible for a DM particle of 100GeV. The parameter r is also assumed to be the same for all f. ∼ p In a similar way, rates for the other decays into SM gauge bosons can be worked out:

+ Γ(S γγ)+Γ(S gg)+Γ(S ZZ)+Γ(S W W −)+Γ(S Zγ) → → → → → g2m 0 S (80+640+82+81+10) . (4.4.2) ≈ 64π

Our result imposes an upper bound on the sum of decay rates via all the channels. Taking

rp = 0 as an example, our result would give us the constraint

2 44 g m . 3.9 10− GeV (95.4% conﬁdence level). (4.4.3) 0 S × Chapter 4. Cosmological Constraints on Decaying Dark Matter 101

We can also consider the decay of a spin-0 dark matter candidate to proceed in the same way as in the case of neutral pion decay. In the SM, (anti-) neutrinos couple

µ to other matter in the form of the chiral currentν ¯Lγ νL since S carries no Lorentz index. The lowest dimension operator responsible for this decay will be of the form

µ g0fγ¯ (1+ raγ5)f∂µS/Λ, where we parameterize g0 as the coupling constant of dimension zero, and Λ is some cutoﬀ scale. The presence or absence of γ5 in the operator depends on whether S couples to the SM neutrinos in a vectorial or axial-vectorial way. The corresponding decay rate is given by

2 2 2 2 g0ra mf mS mf Γ= 2 1 4 2 , (4.4.4) 2π Λ s − mS where mf is the mass of the decay product. Here we have assumed that (anti-) neutrinos have Dirac mass.2

We focus our attention on DM with mass m m . Then for decay products such S ≫ f as (anti-) neutrinos (or other light SM particles), it is safe to make the approximation

1 4m2 /m2 1. Since the neutrino is left handed, we take r = 1. If S decays − f S ≃ a − dominantly into νeν¯e, our lower bound on Γ then constrains the following parameter, 2 2 g0mf mS 42 . 1.3 10− GeV (95.4% conﬁdence level), (4.4.5) Λ2 ×

1 41 where we have used 1Gyr− =2.087 10− GeV. × Here we can see that helicity suppression at work. When the mass of the decaying

particles is very small, the decay of the spin-zero DM candidate will be suppressed as

expected. The presence of helicity suppression gives us a value of g0 that is larger than in most other cases, as we will see. For example, if the mass of the DM candidate is

m 100GeV, the neutrino mass around m 2eV, and the cutoﬀ is Λ 10TeV, then S ∼ f ∼ ∼ 11 the coupling constant g has to be 10− . If on the other hand the coupling constant 0 ∼ g is (1) and we take the same values of the neutrino masses and of the DM candidate, 0 O

2Notice that we are not concerned with the nature of the neutrinos, this serves as an illustrative exercise only. Chapter 4. Cosmological Constraints on Decaying Dark Matter 102

then we have that Λ 1013GeV. However, as pointed out before, this decay rate arises ∼

from an operator that is suppressed by Λ, so it is the previous operator, g0Sf¯(1+irpγ5)f that would give rise to a signiﬁcant contribution to the decay rate.

Messenger number violation in gauge-mediated supersymmetry breaking the-

ories

Let us now investigate the messenger parity in gauge-mediated supersymmetry breaking

theories. In these models, there could potentially be a dark matter candidate coming

from the electromagnetically-neutral scalar ﬁeld that is formed from the SU(2) doublets

of the 5 and 5¯ of the messenger sector [105]. However, it is usually not easy to realize

this because the lightest odd-messenger parity particle (LOMPP) often turns out not to be the electromagnetically-neutral ﬁeld that we require. It has been claimed in the same

paper that certain F-terms would lift the degeneracy. If we further assume that it does

not signiﬁcantly modify the eﬀective low energy theory, the analysis becomes very much

model-independent as there are only certain couplings that lead to decay of the LOMPP.

Following [105], the K¨ahler potential is given by

4 g0 2 K = d θ 5† 5 +5† 5 + 5† 5 + 10† 10 + 5† 10 +5† 5 10 + h.c. , M M M M F F F F Mp M F M F F Z (4.4.6)

where 5M and 5M are the messengers and 5F and 10F are the ordinary superﬁelds. The terms that are Planck suppressed are the ones that violate messenger number by one unit.

As for the superpotential, we have

g W = d2θ ρ 5 5 + 0′ 5 103 , (4.4.7) M M Mp M F Z where ρ is the supersymmetry breaking spurion ﬁeld and once again the terms that

are Planck suppressed are dimension-5 messenger number violating terms. Without full

knowledge of the UV-sensitive physics (F-terms that lift the other ﬁelds while retaining Chapter 4. Cosmological Constraints on Decaying Dark Matter 103

a viable LOMPP), we can still give an estimate of the order of magnitude of the decay rate of the LOMPP, 2 3 g0mmess Γ N 2 Fk, (4.4.8) ∼ Mp π where N are the diﬀerent degrees of freedom that the LOMPP can decay into. Fk is a function that contains the kinematic information and we will assume that is close to

one. We can then put a constraint on the coupling constant and on the messenger mass.

Since the lifetime is 100Gyr, then for F 1 and N 100, we have k ∼ ∼ 3 2 mmess 45 g . 6 10− GeV, (4.4.9) 0 M 2 × p

where we have assumed one universal coupling constant g0. For the sake of discussion, if we consider a coupling constant of order one, we get a small messenger mass m mess ∼ 0.02GeV. This can be improved if we go to dimension six operators which gives a generic

decay rate of the form

2 5 Ng0 mmess Γ 4 F, (4.4.10) ∼ Mp π 3 2 5 4 Note that instead of a mmess/Mp suppression, we now have mmess/Mp . This gives a viable scenario since the messenger mass now needed is m 4000 TeV, for a coupling mess ∼ constant of order one.

4.4.2 Spin-1/2 Dark Matter

We now consider a massive DM of spin-1/2 (let us call it ψ) that decays into SM particles. Without a speciﬁc model, we assume ψ decays dominantly via two-body decays and focus our attention to this phenomenon. Since the DM candidate must be neutral, the possibilities for a two body decay of spin 1/2 DM into SM particles are

(f,G)=(ν,Z), (l±, W ∓), (ν,H), where H is the Higgs ﬁeld. For the case that ψ is a

µ Dirac fermion, the two-body decays are mediated by the eﬀective operator gDGµfγ¯ (1+ ¯ µ rγ5)ψ + gD∗ Gµ∗ ψγ (1+ rγ5)f. The ﬁrst term gives rise to ψ decay while the second one is Chapter 4. Cosmological Constraints on Decaying Dark Matter 104

responsible for the decay of ψ¯. Again summing over the ﬁnal state spins and averaging over the spin of the decaying ψ, we ﬁnd the decay rate to be

2 3 gD m m m m m m m Γ(ψ fG)=Γ(ψ¯ f¯G¯)= | | ψ λ G , f ω G , f + r2ω G , f , → → 16πm2 m m m m m −m G s ψ ψ ψ ψ ψ ψ (4.4.11) where λ(a, b)=(1+ a b)(1 a b)(1 a + b)(1 + a + b) and ω(a, b)=(1+ a b)(1 − − − − − − a b)[2a2 +(1+ b)2], and m denotes respectively the mass of particle A. − A Now consider the case where the decay of the fermionic DM candidate comes from

¯ ¯ 3 an operator gDψH(1 + irpγ5)f + gD∗ fH(1 + irpγ5)ψ. In this case the decay rate of ψ H + ν is given by →

g 2m m m m m m m Γ(ψ Hν)=Γ(ψ¯ Hν¯)= | D| ψ λ H , f z f , H + r2z f , H , → → 16π m m m m p −m m s ψ ψ ψ ψ ψ ψ (4.4.12)

where z(a, b)=1+ a2 b2 +2a. −

We can now consider a simple scenario in which r = rp = 0 for all the decay channels

and they all have the same coupling constant gD. Then the total decay rate of ψ is given by summing over all the possible channels:

+ + Γ(ψ 2body) = 3Γ(ψ Zν)+Γ(ψ W e−)+Γ(ψ W µ−) → → → → + +Γ(ψ W τ −)+3 Γ(ψ Hν) (4.4.13) → → = 126.5 g 2GeV | D|

where we have picked m 200GeV, m 100GeV and assumed all three generations ψ ∼ H ∼ of neutrinos have masses 1eV. The factor 3 in Equation (4.4.13) is for three generations ∼

3We consider in this section the decay of the spin-1/2 DM to the Higgs and a neutrino for illustrative purposes only. The decay to the Higgs is not kinematically favourable when we consider DM with mass 100GeV in the other sections. The DM mass that we use here is, therefore, increased accordingly. Note∼ that including the decay channel to the Higgs in the other sections does not dramatically alter the results there. Chapter 4. Cosmological Constraints on Decaying Dark Matter 105

of neutrinos. Our cosmological bound then gives us the constraint

23 Dirac fermion : g . 4.0 10− (95.4% conﬁdence level). (4.4.14) | D| ×

For the case that ψ is a Majorana fermion, the above analysis follows through. Given

the same interaction terms as those shown above, the partial decay rates of a Majorana

ψ are exactly the same as Equations (4.4.11) and (4.4.12). There are, however, no

distinction between ψ and ψ¯ in this case any more. In a four-component spinor notation,

ψ and ψ¯ relate to each other via the charge conjugation matrix. This means the total decay rate of a Majorana ψ has contributions from decays into ‘particles’ and decays into

‘anti-particles’.

In the same simple scenario we considered above, the total decay rate of a Majorana

ψ will be increased by a factor of 2 compared to Equation (4.4.13). The constraint on

the coupling constant will correspondingly be tightened by a factor of √2:

23 Majorana fermion : g . 2.8 10− (95.4% conﬁdence level). (4.4.15) | M | ×

R-parity violation in minimal supergravity models

Undoubtedly, the most thoroughly investigated models in the supersymmetric menagerie are the minimal supergravity (mSUGRA) models [106–110]. While the theoretical motivation for universality of scalar masses, gaugino masses and trilinear terms is questionable

(since these values depend on the mechanism by which supersymmetry breaking is trans- mitted to our sector), it has nevertheless remained a useful benchmark. For our purposes, it is suﬃcient for us to use the fact that in a variety of these mSUGRA models, the lightest supersymmetric particle (LSP) is a neutral particle that is overwhelmingly composed of the spin-1/2 supersymmetric partner of the B-gauge boson called the bino, B˜. There are of course technically natural classes of models [111–113] very similar to mSUGRA theories that will give bino as the LSP, and the analysis below would similarly apply to them. Chapter 4. Cosmological Constraints on Decaying Dark Matter 106

In the presence of R-parity violation, the bino LSP would of course decay. Tradi- tionally, theories with R-parity violation were often assumed to be unable to provide a

dark matter candidate. Here, we can turn this around and ask what the couplings of

the theory have to be so that the theory can still furnish us with viable dark matter

candidate. To do that, we need to ﬁrst explore the possible decays.

While the two-body decay might seem to have a more favorable phase space, these

decays however would arise from Feynman diagrams [114] only if we have the R-parity

violating terms together with the introduction of an additional loop and further suppres-

sion by dimensionless ratios of electroweak scale over the cut-oﬀ scale. We will therefore

assume that the bino will dominantly decay into three SM particles via the trilinear R- parity violating terms. If for some particular models, one needs to add in some of the

two-body decay terms, one can look up section 4.5 or the previous subsection for the

relevant cross-sections and include them in the overall analysis.

Neglecting all ﬁnal state masses, the decay rate for a three-body bino decay is given

by 1 1 2 mB˜ 2 mB˜ 1 2 Γ= 3 dE1 dE2 , (4.4.16) 64π m ˜ 1 |M| B 0 2 mB˜ E1 spins Z Z − X 4 where Ei is the energy of the ﬁnal particle i , and the summation symbol means averaging over initial spins and summing over ﬁnal spins. The amplitudes squared for three-body decays of neutralino due to trilinear R-parity violating terms have been evaluated and shown in [17,115,116], with the appropriate spin summing/averaging. For our purposes we will consider the lightest neutralino (which is also taken to be the LSP) to be bino-like.

5 The results in [116] can be easily applied to LSP decay by demanding the neutralino has a 100% bino component, i.e. by setting Nχ1 = 1 and Nχn =0, n =2, 3, 4 in the notation of [116]. For simplicity, all ﬁnal state masses are neglected in the analysis below. We

4Of course, the identification of a final particle i is arbitrary. This arbitrariness does not change the final expression for Γ. 5By this we mean that the lightest mass eigenstate of the neutralino mass matrix has almost the same mass as the bino, and it mixes negligibly with other neutralinos. Chapter 4. Cosmological Constraints on Decaying Dark Matter 107

have also ignored the mixings and the widths of the sfermions, which mediate the decay as internal lines in the Feynman diagrams.

Given the R-parity violating superpotential term

σρ c WLLE = ǫ λijkLiσLjρEk (4.4.17)

(where i, j and k, each of which runs from 1 to 3, are generation indices, σ and ρ are

SU(2)L indices, and the superscript c indicates charge conjugation), the decay channel

+ B˜ e ν¯ e− is possible. Using the generic expression for amplitude squared in [116] and → i j k putting in our simpliﬁcations, we get the decay rate

8 m m ˜ + 2 2 2 e˜i 2 ν˜j Γ(B ei ν¯jek−) = 3 λijk g′ mB˜ 2YL K +2YL K → 128π | | m ˜ m ˜ B B 2 me˜k 2 mν˜j me˜i me˜k me˜i + 2YE K 2YL P , +2YLYEP , m ˜ − m ˜ m ˜ m ˜ m ˜ B B B B B m mν˜ + 2Y Y P e˜k , j (4.4.18) L E m m B˜ B˜

where g′ is the gauge coupling of U(1)Y , mfñ is the mass of the scalar superpartner of particle f˜ and Y denotes the hypercharge of a superfield S (for example, Y = 1). n S E − K(x) and P (x, y) are functions defined in section 4.5.

Another trilinear R-parity violating superpotential term is

σρ c WLQD = ǫ λijk′ LiσQjραDkα (4.4.19) with the SU(3) index α. This term gives rise to the decays B˜ e+u¯ d and B˜ ν¯ d¯ d . c → i j k → i j k The decay rates for these channels are similar to the one above, with the appropriate substitution of superpartner masses and prefactors:

+ 6 2 2 2 me˜i 2 mu˜j ˜ ′ ′ Γ(B ei u¯jdk) = 3 λijk g mB˜ 2YL K +2YQK → 128π | | m ˜ m ˜ B B m ˜ m ˜ 2 dk mu˜j me˜i dk me˜i + 2YDK 2YLYQP , +2YLYDP , m ˜ − m ˜ m ˜ m ˜ m ˜ B B B B B m ˜ m + 2Y Y P dk , u˜j (4.4.20) Q D m m B˜ B˜ Chapter 4. Cosmological Constraints on Decaying Dark Matter 108

and

6 m md˜ ˜ ¯ 2 2 2 ν˜i 2 j Γ(B ν¯idjdk) = 3 λijk′ g′ mB˜ 2YL K +2YQK → 128π | | m ˜ m ˜ B B m ˜ m ˜ m ˜ 2 dk dj mν˜i dk mν˜i + 2YDK 2YLYQP , +2YLYDP , m ˜ − m ˜ m ˜ m ˜ m ˜ B B B B B m ˜ md˜ + 2Y Y P dk , j . (4.4.21) Q D m m B˜ B˜

Note that the numerical value of an SU(3)c colour factor has been included in the prefactors of Equations (4.4.20) and (4.4.21).

In a similar way, the decay channel B˜ u¯ d¯ d¯ is allowed by the superpotential term → i j k

αβγ c c c WUDD = ǫ λijk′′ UiαDjβDkγ, (4.4.22)

where α, β and γ are all SU(3)c indices. The corresponding decay rate is

m ˜ 48 2 2 2 mu˜i 2 dj ˜ ¯ ¯ ′′ ′ Γ(B u¯idjdk) = 3 λijk g mB˜ 2YU K +2YDK → 128π | | m ˜ m ˜ B B m ˜ m ˜ m ˜ 2 dk dj mu˜i dk mu˜i + 2YDK 2YU YDP , 2YU YDP , m ˜ − m ˜ m ˜ − m ˜ m ˜ B B B B B m ˜ m ˜ 2 dk dj 2YDP , . (4.4.23) − m ˜ m ˜ B B

Here a diﬀerent SU(3)c colour factor has been included in the prefactor. It should be pointed out that the bino is a Majorana fermion. This means what we shown above is only half of its possible decay channels: the other decay channels are obtained by applying charge conjugation to all the ﬁnal particles in any of the above channels. The decay rates, however, are invariant under charge conjugation.

Because the LLE term contains two copies of L’s and they contract with the Levi-

Civita tensor, λijk is anti-symmetric in i and j. Thus it only represents nine couplings.

Similarly, λijk′′ is anti-symmetric in j and k. This argument is not applicable to λijk′ , so it does indeed contain 27 couplings (see, for example, [17,115,116]).

As a simple application of our cosmological constraint on the DM decay rate, we assume m ˜ 100GeV and all the sfermions have masses 300GeV. We also assume all B ∼ ∼ Chapter 4. Cosmological Constraints on Decaying Dark Matter 109 the non-zero R-parity violating couplings attain the same value λ, i.e.

λ = λ′ = λ′′ = λ, i = j , j = k . (4.4.24) i1j1k1 i2j2k2 i3j3k3 1 6 1 3 6 3

Summing over all the possible 3-body decay channels of bino, the total decay rate is given by

+ + Γ(B˜ 3body) = 2[9Γ(B˜ e ν¯ e−)+27 Γ(B˜ e u¯ d ) → → i j k → i j k +27 Γ(B˜ ν¯ d¯ d )+9 Γ(B˜ u¯ d¯ d¯ )] (4.4.25) → i j k → i j k = 0.00144 λ 2 | | where we have used g′ = 0.36. Our cosmological bound then constrain the coupling constant to be

20 λ . 1.2 10− (95.4% conﬁdence level). (4.4.26) | | × In comparison, one of the strongest constraint on R-parity violation comes from the consequent baryon number violation arising from WUDD. The most stringent constraints

9 7 on the λ′s are given by λ′′ . 10− 10− [117,118]. So if indeed the assumption that we − have bino-like DM holds true, then the most stringent limits on R-parity violation would come from our analysis.

4.4.3 Spin-1 Dark Matter

We now consider a massive DM of spin-1 (let us call it χ) that decays into SM particles.

Since the χµ field carries one Lorentz index, it contracts with other SM fields differently from the spin-0 DM, thus giving rise to different interaction terms and decay rates.

In contrast to the decay of spin-0 DM, helicity suppression is not observed in the decay of χ νν¯. χµ can be directly coupled to the neutrino currentν ¯ γ ν , without → L µ L any insertion of ∂µ. On the other hand, every spin-1 particle has to obey the Landau-

Yang theorem [119, 120] which states that because of rotational invariance, it cannot decay into two massless spin-1 particles. Hence, the decays χ γγ and χ gg are → → Chapter 4. Cosmological Constraints on Decaying Dark Matter 110

not allowed. The possible partial decay widths (with summing over ﬁnal state spins and averaging over the initial state spin) for a spin-1 DM are rather numerous and not that

illuminating to list them all here. So we have relegated them to Table 4.5 in section

4.5. In the case where the DM is indeed an additional U(1) gauge ﬁeld that is massive,

the possibility of kinetic mixing with the photon [121] must be considered. Such a term

could be radiatively generated via exchange of a ﬁeld that is charged under both U(1)’s.

Following Refs. [122, 123], we can manipulate the Lagrangian into a form where the

µ mixing manifests itself in the coeﬃcients of the following terms, g1χµfγ¯ (1+ rγ5)f. But this is a term that has already been considered in Table 4.5.

To get a feel for the numbers involved, let us now consider a simple model where all the interaction terms in Table 4.5 exist, with all the coupling constants real and of the same order of magnitude. Again, we will also assume m /m 1. Then decay product χ ≪ 2 3 + g1mχ 5 2 4 Γ(χ Zγ)+Γ(χ ZZ)+Γ(χ W W −) + + . (4.4.27) → → → ≈ 96π m2 m2 m2 Z Z W

Because mf /mχ is small for the value of mχ we are considering, g2m Γ(χ ff¯) 1 χ (1 + r2) N → ≈ 12π f f SM f SM X∈ X∈ 7g2m = 1 χ (1 + r2). (4.4.28) 4π Similar to the case of spin-0 DM, the decay to tt¯is not included here, and the parameter r

is also assumed to be the same for all f. Note also that we have diﬀerent mχ dependence for the decays into fermion-antifermion and massive gauge bosons, unlike in the spin-0 case.

Our result then gives an upper bound on the sum of all the decay rates into SM particles. For illustration, we consider m 100GeV and r = 0. Our bound on Γ can χ ∼ then be translated into a constraint on the coupling constant:

23 g . 5.8 10− (95.4% conﬁdence level), (4.4.29) 1 ×

where we have substituted mW = 80GeV and mZ = 91GeV. Chapter 4. Cosmological Constraints on Decaying Dark Matter 111

T-Parity violation in little Higgs models

Little Higgs models with T-parity violation is another possible scenario in which the dark

matter candidate decays. Analogous to R-parity in SUSY models, all non-SM particles

in Little Higgs model are assigned to be T-odd, while all SM ones T-even. The T-parity

then requires all coupling terms to have an even number of non-SM ﬁelds. This forbids

the contribution of the non-SM particles to the oblique electroweak parameters, and

consequently, the symmetry breaking scale f can be lowered to about 1TeV [124]. The Lightest T-odd Particle (LTOP), moreover, is stable and has often been nominated as a

dark matter candidate.

However, Ref. [16] has pointed out that anomalies in general give rise to a Wess- Zumino-Witten (WZW) term, which breaks the T-parity (Refs. [125, 126] have con-

structed Little Higgs models free of the usual WZW term). This means the LTOP

is not exactly stable. Indeed, phenomenological consequences of the WZW term in the

Littlest Higgs model have been studied in Ref. [127, 128]. In their model, the LTOP is

the massive partner of photons (denoted by AH ) and the WZW term contains direct

couplings of AH to the Standard Model W bosons, Z bosons and photons. Ref. [128],

moreover, pointed out that such couplings can generate two-body decay of AH to SM fermions, A ff¯, via triangular loop diagrams. H → In an attempt to be as model-independent as possible, we parameterize the couplings

of AH to the SM gauge bosons as

g′ µ 2 ν ρ σ 2 +ν ρ σ ν ρ +σ 2 ν ρσ L ǫ A [N m Z ∂ Z +N m (W ∂ W − +W − ∂ W )+N m Z F ], ⊃ −f 2 µνρσ H Z Z W W AZ Z (4.4.30) where f is the symmetry breaking scale, g′ the U(1) gauge coupling, NZ, NW and NAZ are numbers whose values depend on the exact realization and the UV completion.

Generically, the mass of AH is proportional to f. If we take f to be the natural symmetry breaking scale (i.e. 1T eV ) in Little Higgs models, then m & 2m . As an ∼ AH Z Chapter 4. Cosmological Constraints on Decaying Dark Matter 112

example, in [127,128], we have

2 4 g′f 5v v mA = 1 + , (4.4.31) H √5 − 8f 2 O f 4

where v = 246GeV is the Higgs vev. The condition for mAH & 2mZ is satisﬁed when

g′ 0.36 and f & 1165GeV. ∼ + In the case of m & 2m , the decay channels of A ZZ and A W W − AH Z H → H → are kinematically allowed, and, for simplicity, we assume these processes (together with

A Zγ) to be the dominant ones. With the interaction terms in equation (4.4.30), H → the decay rates for these channels at the lowest order are given by

5 2 2 3 2 2 g′ N m m m 2 Γ(A ZZ) = Z AH Z 1 4 Z , (4.4.32) H → 96πf 4 − m2 AH 5 2 2 3 2 2 2 + g′ NW mAH mW mW Γ(A W W −) = 1 , (4.4.33) H → 48πf 4 − m2 AH 2 2 3 2 2 2 3 g′ N m m m m Γ(A Zγ) = AZ AH Z 1+ Z 1 Z . (4.4.34) H → 24πf 4 m2 − m2 AH AH

The sum of these decay rates is then constrained by our bound on the dark matter lifetime. For the Littlest Higgs model, NAZ = 0 and NW = NZ . The sum of the above decay rates is then reduced to g 5N 2 m2 Γ= ′ Z Z , (4.4.35) 160√5πf where we have used the approximations m m , m g′f/√5 and have neglected W ≃ Z AH ≃ all the mass ratios. Our bound on Γ then gives us the constraint

2 NZ 42 1 < 4.7 10− GeV− (95.4% conﬁdence level), (4.4.36) f × which of course is not reasonable as we typically expect N 1 and f 1 TeV. But it Z ∼ ∼ vividly illustrates the utility of our approach when it comes to ruling out particle physics

models that claim to have dark matter candidates. Chapter 4. Cosmological Constraints on Decaying Dark Matter 113

4.4.4 General Dimensional Considerations

We can draw some generalizations from the above cases if we do a simple dimensional

analysis. The coupling of a spin-0, spin-1/2 or spin-1 dark matter candidate S to an

operator O can be parameterized, with suppression of indices and (1) factors, as O S L g n 4 O, (4.4.37) ⊃ Λ −

where g is a dimensionless coupling constant, Λ is the scale where unknown new physics is integrated out to give the operator O, and n is the sum of the dimensions of S and O.

The decay rate for such dark matter candidate is given in general by

2 g 2n 7 − Γ= 2n 8 mS Fk, (4.4.38) Λ −

where Fk is a function that contains the kinematics of the decay, assumed to be of order one for simplicity. For n = 5, m = 100GeV and g (1), the cutoff scale should be of S ∼ O the order of Λ & 1024GeV, suggesting that we must go to operators of higher dimensions and thus more Λ suppression. For n = 6, m = 100GeV and g (1), the cutoff scale S ∼ O can be as low as Λ 1013GeV. On the other hand, for n = 5, if the cutoff is taken at the ∼ Planck scale (Λ 1019GeV) and we keep the same value of m , the coupling constant ∼ S 5 can only be as large as g 10− . Finally to recover the cases discussed above for spin-0, ∼ spin-1/2 and spin-1 particles, we can take n = 4 and m 100 GeV to give us a coupling S ∼ 22 constant as large as g 10− . ∼ A few words should be reiterated about the smallness of the coupling constant. We had taken an extremely conservative value for our cutoffs, usually 10 TeV. In an ∼ effective theory with a low cutoff arising from a high scale fundamental theory, say at Planck scale, there will be a multitude of effective operators containing mass insertions

2 (leading to small dimensionless ratios such as m/MP ) or loops (giving factors of 1/16π ), making these tiny coupling constants natural. The small dimensionless ratios could arise from, say, the decay being mediated by some massive field much like what we have in Chapter 4. Cosmological Constraints on Decaying Dark Matter 114 proton decay via exchange of heavy X bosons in the context of Grand Unified Theories. The onus is then on the model builders to refine their models in a technically natural way to satisfy the constraints we have derived above without having to compromise other phenomenological constraints on their models. Chapter 4. Cosmological Constraints on Decaying Dark Matter 115

4.5 Compendium of Decay Rates

This section summarizes the lowest order decay rates due to various generic interaction

terms in the Lagrangian, averaging over the spin of the DM and summing over the spins

of the decay products. Tables 4.5, 4.5 and 4.5 respectively tabulate the decay of a spin-0

DM particle S, a spin-1/2 DM particle ψ and a spin-1 DM particle χ, into SM particles. In a model-independent way, we write down generic Lagrangian terms which describe

possible decay channels of DM particles to SM particles. The exact mechanisms which

mediate these decays are captured by the dimensionless coupling constants, g0, gD and g1. The reality of the interaction terms requires the coupling constants to be real, except

+ in the case of χ W W − and the decay of ψ, in which a complex coupling constant is → possible. We follow standard conventions in denoting our ﬁelds. Y µ represents a gauge ﬁeld

µν while Y is the corresponding ﬁeld strength tensor. For various decay channels, Nf and

Ng respectively denote the number of colours of a fermion species f and a gluon g. r, rp and ra are parameters that describe the relative size of two interaction terms. Chapter 4. Cosmological Constraints on Decaying Dark Matter 116

Interaction Term Decay Rate

g2m N m2 1+r2 m2 ¯ ¯ 0 S f f p f g0Sf(1 + irpγ5)f Γ(S ff) = 1 4 2 2 2 4π mS 2 mS → r − × − g2r2N m m m2 g0 ¯ µ ¯ 0 a f f S f fγ (1 + raγ5)f∂µS Γ(S ff) = mf 2 1 4 2 Λ 2π Λ mS → r − g2 m3 g0s SF F µν Γ(S γγ) = 0s S Λ µν → 4πΛ2 g2 g0p Sǫ F µνF σλ Γ(S γγ) = 0p m3 Λ µνσλ → πΛ2 S 2 3 g0s SGa Ga,µν Γ(S gg) = g0smS Ng Λ µν → 4πΛ2 g2 g0p Sǫ Ga,µν Ga,σλ Γ(S gg) = 0p N m3 Λ µνσλ → πΛ2 g S 2 2 3 2 2 4 g0mZ µ g0mS mZ mZ mZ Λ SZµZ Γ(S ZZ) = 32πΛ2 1 4 m2 1 4 m2 + 12 m4 → − S × − S S 2 3 2 2 4 g0s µν g0smS q mZ mZ mZ SZµνZ Γ(S ZZ) = 2 1 4 2 1 4 2 +6 4 Λ 4πΛ mS mS mS → − ×3 − 2 3 2 2 g0p µν σλ g0pmS q mZ Λ SǫµνσλZ Z Γ(S ZZ) = πΛ2 1 4 m2 → − S 2 2 2 2 4 g0mW + µ + g0mS mW mW mW Λ SWµ W − Γ(S W W −) = 64πΛ2 1 4 m2 1 4 m2 + 12 m4 → − S × − S S 2 3 2 2 4 g0s + µν + g0smS q mW mW mW SWµνW − Γ(S W W −) = 2 1 4 2 1 4 2 +6 4 Λ 4πΛ mS mS mS → − ×3 − g2 m3 2 2 g0p +µν σλ + 0p S q mW Λ SǫµνσλW W − Γ(S W W −) = πΛ2 1 4 m2 → − S 2 3 2 3 g0s µν g0smS mZ Λ F Zµ∂ν S Γ(S Zγ) = 32πΛ2 1 m2 → − S g2 m3 2 3 g0p µν σ λ 0p S mZ Λ ǫµνσλF Z ∂ S Γ(S Zγ) = 8πΛ2 1 m2 → − S Table 4.2: Decay Rate of Spin-0 DM via Diﬀerent Interaction Terms

Interaction Term Decay Rate

2 3 gD m m ¯ µ ¯ ¯¯ | | ψ mG f gDGµfγ (1 + rγ5)ψ Γ(ψ fG) = Γ(ψ fG)= 16πm2 λ m , m → → G ψ ψ r µ mG mf 2 mG mf +g∗ G∗ ψγ¯ (1 + rγ5)f ω , + r ω , D µ × mψ mψ mψ − mψ 2 h gDmψ mH imf gDψH¯ (1 + irpγ5)f Γ(ψ Hf) = Γ(ψ¯ Hf¯)= λ , → → 16π mψ mψ r mf mH 2 mf mH +gDfH¯ (1 + irpγ5)ψ z , + r z , × mψ mψ p − mψ mψ h i Table 4.3: Two-Body Decay Rate of Spin-1/2 DM via Generic Interaction Terms Chapter 4. Cosmological Constraints on Decaying Dark Matter 117

K(x) 1 5+6x2 + 2(1 4x2 +3x4) ln 1 1 16 − − − x2 2 2 2 2 P (x, y) 1 3 + π y 6 x2 + x y ln x2+y2 1 1 ln x2+y2 1 + lnx2 1 24 2 2 − 4 ( − )[− 2 ( − ) ( − )] » „ « – 2 2 2 2 2 2 2 x 2 x x y 2 x x y x −1 + x 1 ln + ln y ln Li2 +x y 4 ( − ) x2−1 4 ( ) x2−1 − 4 x2+y2−1 ↔ “ ” “ ” “ ” Table 4.4: Functions used for the Analysis of Bino Decay

Interaction Term Decay Rate

g2N m2 m2 m2 ¯ µ ¯ 1 f f f 2 f g1χµfγ (1 + rγ5)f Γ(χ ff) = 12π mχ 1 4 2 1+2 2 +r 1 4 2 → s − mχ ×" mχ − mχ !# 3 g2m3 m2 2 g1 µ ¯ ¯ 1 χ f χ f∂µf Γ(χ ff) = 2 1 4 2 Λ → 64πΛ − mχ Γ(χ γγ or gg) = 0 → Forbidden by the Landau-Yang theorem 3 2 3 2 2 ν µ g1mχ mZ g1ZµZ ∂ν χ Γ(χ ZZ) = 2 1 4 2 96πmZ mχ → − 5 2 3 2 2 µ ν σ ρ g1 mχ mZ g1ǫµνρσχ Z ∂ Z Γ(χ ZZ) = 2 1 4 2 96πmz mχ → − 3 5 2 2 + ν µ + mχ mW g1Wµ W − ∂ν χ Γ(χ W W −) = 192πm4 1 4 m2 → W − χ 2 2 +ν µ 2 mW 2 mW +g∗W −W ∂νχ 4[Re(g1)] 2 +[Im(g1)] 1+4 2 1 µ × mχ mχ  „ «ﬀ 3 2 µ +ν σ ρ + mχ mW g1ǫµνρσχ W ∂ W − Γ(χ W W −) = 48πm2 1 4 m2 → W − χ 2 2 2 µ ν σ +ρ q2 mW 2 mW +g∗ǫµνρσχ W − ∂ W [Re(g1)] 1 4 2 +[Im(g1)] 1+2 2 1 × − mχ mχ ( „ « „ «) 2 3 2 2 3 µν g1mχ mZ mZ g1χµZνF Γ(χ Zγ) = 96πm2 1+ m2 1 m2 → Z χ − χ 2 3 2 2 3 µ ν σρ g1mχ mZ mZ g1ǫµνρσχ Z F Γ(χ Zγ) = 24πm2 1+ m2 1 m2 → Z χ − χ Table 4.5: Decay Rate of Spin-1 DM via Diﬀerent Interaction Terms Chapter 5

Conclusion

In chapter 2 and in the introduction, we discussed that in conventional models of supersymmetry breaking the dynamics that leads to the breaking of supersymmetry also breaks R-symmetry. When this breaking is communicated to the visible sector it results in R-symmetry violating gaugino masses, Bµ and A-terms. There has been much recent interest in the ISS models of supersymmetry breaking for which there exists a metastable supersymmetry breaking vacuum that preserves the R-symmetry. If such models are to be phenomenologically viable, the gauginos must acquire a mass. Many variants of ISS have been explored that break the R-symmetry and allow for Majorana gaugino masses.

Here we have discussed the alternative possibility that the R-symmetry is preserved and instead the gauginos acquire a Dirac mass. The Dirac gaugino mass and the sfermion masses are communicated to the visible sector through gauge mediation; hence we have a model of R-symmetric Gauge Mediated Supersymmetry Breaking (RGMSB). Because the R-symmetry is preserved, the gauginos are Dirac, the A-terms are zero, and the Higgs sector now consists of four Higgs doublets: the ﬁeld content of the MRSSM. We showed that the dependence of the gaugino mass on the supersymmetry breaking scale diﬀers from that of usual gauge mediated supersymmetry breaking, but the scalar masses do not. The gaugino mass is lower than in usual gauge mediation.

118 Chapter 5. Conclusion 119

We considered two examples for the R-preserving-supersymmetry-breaking sector: a version of ISS which may allow for direct mediation, and a generalisation (an O’Raifeartaigh

model) with fewer ﬁelds. The necessity of including an adjoint chiral superﬁeld to act as

the Dirac partner of the gauginos means that these models have a Landau pole for gauge

couplings, the lowest of which is for SU(3). In the case of the ISS model there are many new fields charged under the standard model and this Landau pole is low, typically a few decades above the scale of the messenger masses. For the O’Raifeartaigh model it can be somewhat higher. There are potentially new operators, such as flavour non-diagonal scalar masses, generated at the strong coupling scale. The size of these operators is unknown. If small, then the model is an R-symmetric version of gauge mediation, with a spectrum that differs somewhat from that of [5]. However, if large (but not too large) this has all the features of the MRSSM.

Making a conservative estimate of the the size of these UV generated operators we found that it is possible to realise the MRSSM scenario of large flavour-violating couplings by using R-symmetric gauge mediation, but only at the expense of introducing flavour fine tuning or strong coupling or both. If these operators were instead far smaller than expected, then the spectrum of the MRSSM could be realised, but there would be no source of the large sfermion mixings (allowed because of the R-symmetry) that lead to the interesting flavour signatures. This does not rule out the possibility of the MRSSM, but it does suggest that a better understanding of the UV theory is required in order to decide how natural such a spectrum actually is.

In chapter 3 we looked more carefully at the phenomenology of the MRSSM and in particular at the Higgs sector of this theory. The explicit neutral potential for the Higgs was explicitly given and the matrices for the diﬀerent scalar ﬁelds of the theory were constructed. We performed a scan of the parameter space to look for regions where we could have a phenomenologically viable spectrum. After scanning the parameter space sample spectra was provided. We found that generically, we can obtain a lightest Higgs Chapter 5. Conclusion 120

that is heavier than the current experimental bounds for both cases λ = 0 and λ = 0. 6 We also found that the neutralino and scalar masses increase as we increase the λ terms.

2 As we mentioned, due to the λ contribution to the running of mHu , increasing the λ terms helps electroweak symmetry breaking. This however, makes certain points of the potential to be unstable and therefore not phenomenologically favoured. We concluded that for λ 1 there are too many points of the parameter space that are not in a stable ∼ vacuum. Therefore, λ has to be less than one to have a realistic spectrum.

The conventional ﬁne tuning of this theory is also quite excessive. Even for the case

when the λ terms are turned oﬀ, the heavy stops contribute signiﬁcantly to the tuning.

Once the λ terms are turned on, the tuning becomes quite undesirable. At the same

time we discussed a possible uniﬁcation scenario for the yukawas. Although this was

done mainly as an illustrative exercise, we can see that for the yukawas to unify at Λ3 they must be of order one at the messenger scale. This reduces the mass of the gauginos,

and the gaugino:squark ratio is not big enough to realise the solution to the ﬂavour

problem that was given in the MRSSM. This seems to happen even when the number of

messengers is as big as Nmess = 8.

We point out that although the model seems to have serious problems we do not necessarily have to demand the uniﬁcation of the yukawas, and the λ terms could be signiﬁcantly suppressed which would certainly reduce the tuning.1 In any case the model presented should be taken as an interesting, but not ultimate possibility, considering the myriad of supersymmetric models that exist in the literature.

After discussing RGMSB and the phenomenology of the MRSSM, we turned our attention to the cosmological constraints on decaying dark matter. There are a number of compelling theoretical reasons why typical dark matter candidates from theories of particle physics beyond the Standard Model should decay. In chapter 4 we looked at the question of how this rather generic phenomenon can still ﬁt into our understanding and

1The λ terms could be equal to a gauge coupling in an N = 2 supersymmetry framework. Chapter 5. Conclusion 121

observations of the universe. We performed a full cosmological analysis using the available datasets from cosmic microwave background, Type Ia supernova, Lyman-α forest, galaxy

clustering and weak lensing observations.

In the scenario where there is negligible reionization of the baryonic gas by the decaying dark matter, we have found that the late-time Integrated Sachs-Wolfe eﬀect gives the

1 strongest constraint. The lifetime of decaying dark matter has the bound Γ− & 100Gyr (at 95.4% conﬁdence level). Because of cosmic variance, the results are not likely to improve signiﬁcantly with the WMAP-9yr data.

When there is signiﬁcant reionization of the baryonic gas due to the decaying dark

matter, the bounds become more restrictive as the CMB polarisation is well measured.

1 8 In this scenario, the lifetime of a decaying dark matter is (f Γ)− & 5.3 10 Gyr (at × 95.4% conﬁdence level) where f is a phenomenological factor related to the degree of reionization. With even more CMB polarisation data, one could conceivably distinguish

the reionization due to decaying dark matter from reionization due to star formation,

thereby giving us even better bounds on the lifetime of the dark matter. We expect that

the the 21cm cosmological observation in the future would give us even greater precision,

as it is expected to probe the reionization history at redshifts 6

Having obtained the cosmological constraints, we turned our attention to constraining

extensions of the Standard Model of particle physics with dark matter candidates. For

completeness and motivated by the utility of such an exercise, we systematically tabulated

the decay cross-sections for a spin-0, spin-1/2 and spin-1 dark matter candidate into the

Standard Model degrees of freedom. This enabled us to simply sum up all the relevant

contributions for a particular model of particle physics and arrive at the functional form of the lifetime of the decaying dark matter. We repeated this process for a variety of

representative models from the following classes of theories: generic supersysmmetric

scenario, gauge-mediated supersymmetry breaking models and the little Higgs theories.

Imposing the limits from our cosmological analysis, we ﬁnd that generically for most Chapter 5. Conclusion 122 models we have looked at, the dimensionless coupling for a decaying dark matter to

22 Standard Model ﬁelds should be smaller than 10− .

This restriction can be slightly relaxed if the dark matter decays solely into light particles via helicity suppressed interaction terms, in which case, the small mass of the decay products suppresses the decay rate. If, for instance, the dark matter decays purely via helicity suppressed terms into νν¯ with Dirac mass of 2eV, then the dimensionless ∼ 11 coupling can be as large as 10− . In addition to constraining the coupling, one can assume it to be of (1) and estimate the scale of new physics which suppresses the decay O rate. In all cases, either the coupling attains a small value or the new physics come from an extremely large scale, both of which would need interesting and exotic physics to realise if indeed the dark matter does decay via dimension-4 or dimension-5 terms.

Moreover, in the case of exclusive helicity suppressed decays, one has to explain why other interaction terms are absent in the model. A more promising avenue, which we brieﬂy mentioned in the previous section, is to look at models where dark matter decays via dimension-6 operators. The Large Hadron Collider might provide us with the identity for dark matter in the very near future, but on the basis of our analysis, there will still be much to understand about physics of the dark matter sector and how it interacts with the Standard Model.

Future analysis could address some of the astrophysical issues of decaying dark matter. The recent spate of results from astrophysical experiments [129–131], particularly the

PAMELA and ATIC anomalies [132–135] has given us much to ponder. The immediate goal would of course be to combine all the astrophysical datasets with the cosmological ones that we have considered in chapter 4 and arrive at a set of characteristics that a phenomenologically viable decaying dark matter must possess. So far there is no tension between cosmological observation and astrophysical expectations. Our best estimation

25 1 is fΓ . 10− s− . Even in the f 1 limit case, the best constraint we can obtain is ∼ 25 1 26 1 Γ . 10− s− . It does not contradict a decay rate of 10− s− , which is required to Chapter 5. Conclusion 123 explain PAMELA data [133–135]. However, we must point out, that the astrophysical constraints seem to be stronger and provide more stringent bounds on the DM lifetime.

Final comments

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