Track: GRAPPA/particle
Master Thesis
Bridging the Mass Gap
Probing Compressed Electroweak Supersymmetry
at the Large Hadron Collider
by
Tal van Daalen
10247513
August 2016
48 ECTS project + 6 ECTS thesis
November 2015 - August 2016
Supervisors:
Dr. Alison Lister
Dr. David Morrissey
Examiners:
Dr. Wouter Verkerke
Dr. Ivo van Vulpen
Supersymmetry is still one of the main contenders for generating the new physics needed to solve the many shortcomings of the Standard Model (SM). However, the pressing absence of any new physics discoveries by LHC experiments is progressively increasing the tension on viable supersymmetric extensions of the SM that satisfy naturalness criteria. Despite large portions of parameter space being excluded, a scenario containing light, Higgsino-like electroweakinos near mass degeneracy is barely constrained by present-day exclusion limits. Such a scenario is strongly motivated from a theoretical perspective due to naturalness arguments, but experimentally probing it requires a departure from conventional electroweak supersymmetry searches and calls for more data than what has yet been delivered by the LHC. In this thesis, the discovery potential of a dedicated LHC search for compressed, Higgsino-like electroweakinos is determined based on simulations. A benchmark point scan of the parameter region of interest is performed using a MadGraph+PYTHIA+Delphes simulation chain, while distinguishing between three scenarios of slightly different electroweakino phenomenology. To assess the discovery sensitivity and expected yields, an analysis strategy is developed and applied to each point. The results show a promising reach, with 95 % CL limits from LEP2 already being exceeded by the end of 2016 for certain benchmark points. Depending on the scenario, points with mass splittings as low as 5 GeV can be probed for electroweakino masses of 110 to 130 GeV with the by the 300 fb−1 of integrated luminosity expected by the end of run
3, and 150 to 160 GeV with the 3000 fb−1 expected after the HL-LHC. Points with
mass splittings of 10 to 30 GeV are reachable up to masses of 150 to 170 GeV by the end of run 3, and 180 to 220 GeV after the HL-LHC stage, thereby starting to push naturalness constraints on the Higgsino mass.
Introduction 3
1 Theory and Phenomenology 4
1.1 The Standard Model . . . 4
1.1.1 Particles and Interactions . . . 5
1.1.2 Masses in the Standard Model . . . 8
1.1.3 Shortcomings of the Standard Model . . . 9
1.2 Supersymmetry . . . 14
1.2.1 Supersymmetry Breaking and Grand Unified Theories . . . 16
1.2.2 The Minimal Supersymmetric Standard Model . . . 17
1.3 Compressed Electroweak Supersymmetry . . . 22
1.3.1 Naturalness . . . 22
1.3.2 Established Constraints . . . 25
1.3.3 Masses and Composition . . . 32
2 The ATLAS Experiment at the Large Hadron Collider 41 2.1 The Large Hadron Collider . . . 41
2.2 The ATLAS Detector . . . 44
2.2.1 Inner Detector . . . 46
2.2.2 Electromagnetic Calorimeter . . . 47
2.2.3 Hadronic Calorimeter . . . 49
2.2.4 Muon Spectrometer . . . 50
3 Methodology 53
3.1 MSSM Parameter Space Scan . . . 53
3.2 Event Generation . . . 58
3.2.1 Parton Level . . . 59
3.2.2 Decay and Showering . . . 59
3.2.3 Detector Simulation . . . 61 4 Analysis 64 4.1 Search Strategy . . . 64 4.1.1 Signal . . . 64 4.1.2 Background . . . 66 4.1.3 Kinematic Distributions . . . 68 4.2 Event Selection . . . 81 4.2.1 Optimising Cuts . . . 82 4.2.2 Uncertainties . . . 84 5 Results 86 5.1 Expected Yields . . . 86 5.2 Discovery Potential . . . 87 5.2.1 Significance Calculation . . . 88 5.2.2 Projected Sensitivity . . . 88 6 Conclusion 93 Bibliography 95 Acknowledgements 104
Supersymmetry provides an elegant framework to address the many shortcomings of the Standard Model (SM). Particularly the hierarchy problem, a pressing feature of the SM with only a limited number of viable solutions, can be resolved easily within a supersymmetric extension of the SM. In light of the present-day constraints from collider searches, however, supersymmetric scenarios that do so while satisfying naturalness criteria are progressively becoming more and more sparse. Scenarios with Higgsino-like electroweakinos approaching mass degeneracy are highly motivated from a naturalness perspective, and can also avoid the stringent constraints from searches at the LHC if the rest of the supersymmetric spectrum lies outside of its mass reach.
The main goal of the work presented in this thesis is to evaluate the achievable discovery potential of such scenarios at the LHC during the upcoming years. The theoretical and phenomenological framework of the SM, the Minimal Supersymmetric SM, and the specific scenario under consideration is established in Chapter 1. Next, a description of the LHC and the ATLAS experiment is given in Chapter 2, to serve as a reference point for the simulations described in Chapter 3. In Chapter 4, an analysis strategy is developed to probe compressed electroweak supersymmetry. The results of this analysis are presented in Chapter 5, and followed by a brief conclusion and a discussion in Chapter 6.
This study has been carried out within the ATLAS group at the Department of Physics and Astronomy of the University of British Columbia (UBC), Vancouver BC, Canada, with the help of the Victory in Europe Scholarship awarded by the The Irving K. Barber British Columbia Scholarship Society.
Theory and Phenomenology
This chapter lays out the theoretical groundwork necessary for understanding the ensuing phenomenological and experimental chapters. The electroweak sector and naturalness are the keystones of compressed electroweak supersymmetry, and therefore serve as the main themes throughout this chapter.
First, the Standard Model (SM) is described in Section 1.1, with an outline of its particles and interactions in Section 1.1.1, and a summary of its theoretical and experimental shortcomings in Section 1.1.3. Second, the framework of supersymmetry is introduced in Section 1.2 and placed in the context of supersymmetry breaking and grand unified theories in Section 1.2.1. Then the simplest supersymmetric particle and group structure, the Minimal Supersymmetric Standard Model, is summarised in Section 1.2.2. Next, we focus on the specific scenario where the supersymmetric particle spectrum is compressed at the electroweak scale in Section 1.3, with a discussion on naturalness in Section 1.3.1, a brief summary of current constraints in Section 1.3.2, and lastly a phenomenological description of the supersymmetric electroweak sector in Section 1.3.3.
1.1
The Standard Model
The SM [1–3] has been an extremely successful theoretical framework to describe nature on subatomic scales since its development in the 1960’s. It has endured experimental tests to great precision, including the observation of the Higgs boson at the Large Hadron Collider by the ATLAS and CMS experiments in 2012 [4, 5].
1.1.1
Particles and Interactions
The SM is a renormalisable quantum field theory that is invariant under the tensor product of the gauge groups SU (3)C, SU (2)L, and U (1)Y, as:
SU (3)C ⊗ SU (2)L⊗ U (1)Y , (1.1)
where SU (3)C governs the strong interaction, and SU (2)L⊗ U (1)Y governs the weak and electromagnetic interactions, each of which will be discussed in the following.
The field content of the SM is comprised of fields with integral spin, named bosons, and fields with half-integral spin, named fermions. In the SM framework, matter is described by fermion fields that satisfy the Lagrangian:
L = ¯ψ(iγµ∂µ− m)ψ , (1.2)
where ψ denotes the fermion field with mass m, and γµare the Dirac matrices. Requiring this Lagrangian to be locally gauge invariant under the aforementioned groups unambiguously demonstrates the need for gauge fields: bosons. Local invariance under each of the gauge groups, SU (3)C, SU (2)L, and U (1)Y, spawns spin-1 gauge fields equal in number to the amount of generators of each specific group, called gauge bosons. How all fields transform as representations under these groups, whether fermionic or bosonic, dictates their interactions with the gauge bosons.
The strong and electroweak sectors of the SM are discussed separately below, as the gauge Lagrangian can be divided as:
Lgauge = Lstrong + Lelectroweak . (1.3)
The Strong Interaction
The theory of the strong interaction is generally referred to as Quantum Chromodynamics (QCD), and is governed by SU (3)C. By requiring invariance under SU (3) transformations we obtain an octet of gluon fields from the algebra [Ta, Tb] = ifabcTc, where 2Ta = λa are the Gell-Mann matrices, a, b, c = (1, . . . , 8), and fabc are the structure constants of SU (3)C. The eight gluon fields act as the gauge fields of the strong force. Invariance under transformations in SU (3)C is achieved by transforming the covariant derivative in (1.2) as:
∂µ → Dµ ≡ ∂µ− igsGaµT a
where gs is the strong coupling constant, Gaµ are the gluon fields, and Ta are the aforemen-tioned SU (3)C operators. Performing this transformation introduces a kinetic piece to the gauge Lagrangian: δLgauge = −1 4G a µνG a µν , (1.5)
where the gluon field tensor Gaµν is defined as:
Gaµν ≡ ∂µGaν − ∂νGaµ− gsfabcGbµG c
ν . (1.6)
The conserved quantum number of QCD is colour, and only coloured fermions participate in strong interactions. Leptons, which carry no colour, are therefore exempt from QCD, whereas quarks are coloured and transform as triplets under SU (3)C. Gluons themselves are also coloured, and three- and four-point self-interaction terms arise due to the last term in (1.6), which reflects the non-abelian nature of SU (3)C. This leads to an inverse energy scale dependence of the QCD coupling strength αs ≡ gs2/4π. At low energies, αs diverges, while at high energies, αs goes to zero. At energies below ∼ 1 GeV, αs becomes of order 1, meaning that low energy QCD interactions are outside of the perturbative regime. In turn, this energy dependence of αs causes a rich and complex phenomenology in nature. At low energies, coloured particles are subject to confinement [6], which inhibits the existence of free coloured particles and instead confines them to hadrons. At high energies, however, coloured particles behave as free particles due to asymptotic freedom [7].
The Electroweak Sector
The electromagnetic and weak interaction are collectively described by electroweak theory, which unifies the weak and electromagnetic interactions under the product SU (2)L⊗ U (1)L. Both interactions will be qualitatively described below, along with a brief mention of the Higgs-Englert-Brout mechanism [8,9] which generates masses in the SM through spontaneous electroweak symmetry breaking (EWSB).
The part of electroweak theory governed by SU (2)L introduces weak isospin as a quantum number, T , where the weak isospin operators are the generators of SU (2)L, as
ˆ
Ti = σ2i (i = 1, 2, 3), with σi being the Pauli matrices. The handedness of fermions fields determines how they transform under SU (2)L. As the subscript L suggests, only left-handed fermion fields interact via the weak force, transforming as doublets under SU (2)L, whereas right-handed fermion fields transform as singlets and do not partake in the weak interaction.
The part of the electroweak interaction governed by U (1)Y introduces hypercharge, Y , as its quantum number. The well known electric charge can be expressed as a combination of hypercharge and the third component of weak isospin, as Q = T3 + Y .
To ensure local invariance of the Lagrangian in (1.2) under SU (2)L⊗U (1)L, the covariant derivative has to be transformed as:
∂µ→ Dµ≡ ∂µ− igWµj σj 2 − ig 0 BµY , (1.7) where Wj
µ (i = 1, 2, 3) and Bµ are the gauge fields of SU (2)L and U (1)L, respectively, and g and g0 are their respective coupling constants.
After EWSB, Wµ1 and Wµ2 will mix to form the the mass eigenstates of the W± boson of the weak interaction, whereas W3
µ and Bµ will form the Z boson and the photon of the weak and electromagnetic interactions, respectively, as discussed in the next section.
The transformation in (1.7) introduces additional kinetic terms to the gauge Lagrangian, as: δLgauge = −1 4W i µνW i µν −1 4BµνB µν , (1.8)
where the SU (2)L and U (1)Y field tensors, Wµνi and Bµν, are defined as:
Wµνi ≡ ∂µWνi− ∂νWµi + gsijkWµjWνk , Bµν ≡ ∂µBν − ∂νBµ .
(1.9)
Note the self interaction term of Wi
µ, which arises due to the non-abelian character of SU (2)L. It turns out that the strength of these interactions grows rapidly at tree level, causing a violation of a unitarity bound when approaching the TeV scale. In order to maintain unitarity, the higher order contributions would therefore have to be of the same order of magnitude, making the theory strongly-coupled. This apparent problem is solved by introducing the Higgs-Englert-Brout mechanism, as will be discussed in the next section.
Putting together the strong and electroweak sectors we have identified thus far, we end up with the following Lagrangian:
L = iψγµ∂ µ− igsGaµTa− igWµj σj 2 − ig 0B µY ψ −1 4G b µνGb µν −14W k µνWk µν− 14BµνB µν . (1.10)
1.1.2
Masses in the Standard Model
A straightforward observation one can make is the absence of any mass terms so far. The introduction of terms such as m2Fi
µFi µ for bosons and mψψ for fermions would break the gauge invariance of the SM, in turn preventing renormalisation techniques from successfully absorbing the infinities encountered when calculating physical quantities.
The Higgs-Englert-Brout mechanism [8, 9] allows particles of the SM to obtain masses without violating its gauge invariance by introducing a new isospin doublet of complex scalar fields that spontaneously breaks SU (2)L⊗ U (1)L down to the electromagnetic and weak interactions. This doublet, given by:
Φ = φ + φ0 ! = √1 2 φ1+ iφ2 φ3+ iφ4 ! , (1.11)
where the superscripts denote the electric charges, is more commonly known as the Higgs field, and adds to the Lagrangian the following terms:
LΦ = (DµΦ)†(DµΦ) − V (Φ) , (1.12)
with Dµ as in (1.7), and the potential V (Φ) given by:
V (Φ) = µ2Φ†Φ + λ Φ†Φ2 , (1.13)
also known as the Higgs-Englert-Brout potential. If µ2 < 0 and λ > 0, V (Φ) takes the shape of a Mexican hat, and has a set of degenerate nonzero minima in the circle where Φ†Φ = −µ2λ2 ≡ v2
2 . When Φ decays from its metastable maximum at Φ †
Φ = 0 to one of these minima, the SU (2)L⊗ U (1)L symmetry is spontaneously broken and Φ acquires a nonzero vacuum expectation value (VEV) of h|Φ|i = √v
2.
Next, in accordance with the Goldstone theorem [10], the breaking of the continuous symmetries SU (2)L and U (1)L produces massless Goldstone bosons, one of which becomes the longitudinal degree of freedom of the Higgs boson, and three of which are absorbed by the Wµνi and Bµν fields as longitudinal degrees of freedom to form the massive W± and Z bosons of the weak interaction. The photon is prevented from obtaining a mass by parametrising Φ such that its neutral component ligns up with the direction where the VEV was chosen. After EWSB, the gauge fields form the following mass eigenstates for the
physical W± and Z bosons, and the photon A, as: Wµ± = W 1∓ iW2 √ 2 , Zµ= gWµ3− g0B µ pg2+ g02 , (1.14) Aµ= g0Wµ3+ gBµ pg2+ g02 ,
with tree level masses given by mW = vg/2, mZ = vpg2+ g02/2, and mA= 0, respectively. The relation between mW and mZcan be expressed in terms of the weak mixing angle, θW, as mW/mZ ≡ cos (θW). The Higgs boson acquires a tree level mass of mH =p−2µ2 =
√ 2λv. Note that the gluon fields remain massless since SU (3)C is unbroken.
Fermions acquire masses from their couplings to the Higgs field, as denoted in the Yukawa Lagrangian:
LYukawa = X
ψ=l,q
ψiLYijψRjΦ + ψiRYijψLjΦ, (1.15)
where ψ are the fermion fields, and Yij are the Yukawa matrices [11]. Note that the two terms in (1.15) transform as singlets under SU (2)L and therefore leave the Lagrangian gauge invariant. The Yukawa matrices can be diagonalised to obtain the mass eigenvalues for the fermions, which have the form mψ = Yψv/
√ 2.
Furthermore, the apparent violation of unitarity in W W scattering we encountered earlier is resolved with the introduction of the higgs boson. The leading terms that caused the problematic growth of the scattering cross section in the first place are explicitly canceled by the terms generated by the W − H interaction.
The complete set of particles of the SM and their gauge group representations is summarised in Table 1.1. Due to the different SU (2)L representations in the quark sector, diagonalisation of the Yukawa matrices happens differently for the up- and down-type quarks, which causes the mass eigenstates to deviate from the weak eigenstates, resulting in quark mixing.
1.1.3
Shortcomings of the Standard Model
Despite its predictive power, the SM falls short in providing a complete description of nature on many different levels. Every shortcoming of the SM lends itself for an extensive
Names spin SU (3)C, SU (2)L, U (1)Y quarks QL= (uL, dL) 12 ( 3, 2 , 16) (×3 families) uR 12 ( 3, 1, 23) dR 12 ( 3, 1, −13) leptons LL= (νL, `L) 12 ( 1, 2 , −12) (×3 families) eR 12 ( 1, 1, −1) gluon g 1 ( 8, 1 , 0) W bosons W± 1 ( 1, 3 , ±1) Z boson Z 1 ( 1, 1 , 0) photon γ 1 ( 1, 1 , 0) Higgs boson Φ = (φ+, φ0) 0 ( 1, 2 , 1 2)
Table 1.1: Gauge representations of the particles of the Standard Model.
discussion of the problem and its possible solutions, but here we will focus on only two: dark matter and the hierarchy problem.
In short, the additional flaws of the SM that are generally considered most pressing include: the experimentally established fact that neutrinos have nonzero masses [12,13], even though they are usually assumed not to obtain their mass through the Higgs-Englert-Brout mechanism, or any other process allowed within the SM; the seemingly zero amount of CP violation in QCD, even though nothing in the SM forbids it; the observed baryon asymmetry of the universe, which cannot be explained by the CP violation present in the SM; the fact that no description of the force of gravity exists within the SM; and the dark energy - or cosmological constant - problem, originating from the enormous discrepancy between the observed and predicted vacuum energy. A summary of all the problems of the SM, theoretical and experimental, can be found in [14].
Dark Matter
The existence of dark matter (DM) has been irrefutably proven by numerous observations, the strongest of which are the rotation curves of galaxies [15] and the bullet cluster collision [16] - two cosmological observations which completely rely on the existence of DM for their explanations. More recently, accurate measurements of anisotropies in the cosmic microwave background (CMB) [17, 18] radiation have further substantiated the existence of DM. CMB measurements specifically favour a cold DM scenario, in which DM was not moving at relativistic velocities during the time of structure formation.
The most popular hypothesis on the nature of DM is provided by the WIMP (weakly interacting massive particle) paradigm, which suggests that DM is made up of massive particles that interact very weakly with the particles of the SM. The biggest virtue of this concept is that particles with weak-scale couplings to SM particles and masses of the order 10 GeV to 1 TeV will naturally generate the observed DM relic density, a phenomenon known as the WIMP miracle.
There are numerous theories beyond the SM that provide suitable DM candidates, be they WIMPs or part of a multitude of other archetypes proposed throughout the years. As we will see later on in Section 1.2.2, SUSY also easily spawns a DM candidate. No direct observation of DM has been made so far by any of the many dedicated experiments searching for DM, which has cut a swath through the possible DM parameter space. A summary of the current DM exclusion limits can be found in [14].
The Hierarchy Problem
The hierarchy problem lends its name from the seemingly unnaturally large difference between the electroweak scale (v ∼ 246 GeV) and the (reduced) Planck scale MP = p
~/(8πGN) ' 2.4 × 1018GeV. The importance of this difference becomes apparent when looking at what sets scales. The Planck scale is the scale at which our classical description of general relativity breaks down, and likely where the strength of gravity becomes of the same order of magnitude as the three forces in the SM. This scale can be interpreted as the cutoff scale Λ up to which the SM is valid, and below which it should be treated as merely an effective field theory (EFT). The electroweak scale is directly connected to the mass of the Higgs boson, as mH =
√
2λv, which has been observed to be ∼ 125 GeV. This measurement is in good agreement with electroweak precision measurements from the LEP
(a) (b)
Figure 1.1: One-loop corrections to the Higgs mass from (a) a fermion f and (b) a scalar S.
era, e.g. on properties of the W± and Z bosons [19].
However, since the Higgs boson is a spin-0 particle, its mass is not protected from radiative corrections by chiral symmetry or gauge invariance as the masses of respectively fermions and spin-1 bosons are. The result is that the squared mass of the Higgs boson, given by:
m2H = (mH)20+ δm 2
H, (1.16)
where (mH)0 is the bare Higgs mass, receives loop corrections δm2H directly proportional to the cutoff scale Λ, caused by such loops as shown in Fig. 1.1.
Specifically, a fermion with mass mf and Ncf colors that couples to the scalar field with coupling strength λf gives a correction at first order of the form:
δM2 = −|λ 2 fNcf| 8π2 Λ 2+ m2 fln (Λ/mf) , (1.17)
whereas first order loop corrections from scalar fields with mass mS and coupling strength λS give corrections of the form:
δM2 = |λ 2 S| 8π2 Λ 2− m2 Sln (Λ/mS) . (1.18)
The quarks and leptons of the SM all can take the place of f and cause corrections to the Higgs mass as in (1.17). The top quark is responsible for the largest correction due to its large Yukawa coupling, yt∼ 1.
Now taking the cutoff Λ = MP1 and calculating the resulting corrections requires
1The cutoff Λ can be placed at lower energies than the Planck scale if one assumes a UV completion to
come into play earlier, e.g. at the scale of string theory or another grand unified theory. Despite slightly relaxing them, such cutoffs would still produce very problematic fine tuning requirements.
(mH)20 = δm2H to a precision of one in 1032 to generate the light Higgs boson observed in nature. Such an extensive precision requirement is known as fine tuning. This concept is further elucidated in Section 1.3.1 in relation to supersymmetry and naturalness.
The importance of the ∝ Λ2 terms in (1.17) is slightly open to interpretation, depending on if one considers cutoffs to play any physical role. Following certain naturalness arguments where observable quantities are central, quadratic divergences are simply substitutes for finite threshold corrections expected from integrating out possible new physics near the cutoff scale. Correspondingly, using dimensional regularisation in calculating the loop integrals from Fig. 1.1, there will not be a ∝ Λ2 piece.
However, the logarithmic terms proportional to the mass of the particles running in the loops cannot be eliminated, and the need for physically unreasonable fine tuning persists. One might attempt to argue that the cutoff scale could actually be at the electroweak scale or just above, but this calls for new physics to enter at these scale that directly alter the Higgs propagator. Not only is it extremely hard to introduce such new physics while staying in agreement with modern experimental observations, but it even falls short in providing a complete solution. Particularly, any arbitrary heavy particle coupling to the Higgs will produce corrections that will always contain terms proportional to the mass of the particle, regardless of whether one assigns any importance to a potential cutoff. Besides, even if no direct coupling between the Higgs and this heavy particle exists, higher order loop diagrams will still give rise to corrections proportional to the particle’s mass as long as it shares any gauge interactions with the Higgs. This argument even applies to other high energy phenomena such as extra dimensions.
We can therefore conclude that the natural value for the squared mass of the Higgs boson indeed is around M2
P, instead of its observed value that is some 30 orders of magnitude smaller. There exist only a very limited number of viable solutions to this problem, a few of which are mentioned below:
Anthropics The first idea, which arguably is more of a compromise than a solution, is simply to accept that the Higgs mass is finely tuned. Such a scenario could be explained by a multiverse, in which a landscape of different vacua exists, all having different fundamental parameter values. Only universes with a stable Higgs are consistent with the existence of an observer, and therefore the Higgs mass must take its observed value in our universe. Unsurprisingly, such scenarios give rise to profound testability problems.
Alternate Higgs interpretations Many models exist that suggest the observed Higgs boson is not necessarily fundamental or central to the theory. Technicolor models [20–22] for example, explain masses of particles in the SM as generated by new physics at the TeV scale, but are currently under substantial tension. Other models accommodate the Higgs as for example a pseudo-Goldstone boson from a broken global symmetry at some high scale, such as Little Higgs [23] or Twin Higgs [24] models.
Relaxions A more recent solution revolves around the introduction of two new axion fields that promote the electroweak scale to a dynamic variable that relaxes to its present-day value during modified slow-roll conditions in the early universe [25]. This ‘relaxion’ idea has gained a lot of popularity as of late, but it seems that it might merely migrate the fine tuning requirement to the inflationary sector. Also, preventing the new axions from introducing nonzero CP violation in QCD could potentially pose a problem.
New symmetry Lastly, the introduction of a new symmetry that protects the Higgs mass from any divergences encountered at high energies naturally also provides the basis for a solution to the hierarchy problem. If we can construct such a symmetry, it would allow a fundamental cutoff to be at much higher energies, since the new symmetry now protects the Higgs mass from this cutoff, and any other physics entering at high scales that would otherwise produce problematic corrections.
1.2
Supersymmetry
The introduction of a new symmetry of nature is required to abide by the Coleman-Mandula theorem [26], which dates back to 1967. It states that any theory in more than 1+1 dimensions with a finite number of point-like particles that are subject to non-trivial interactions is only allowed to have internal symmetries, such as gauge symmetries, and symmetries of the Poincar´e group, which are Lorentz boosts, rotations and translations. Therefore, to ensure CPT invariance in nature, the only conserved quantities allowed are the energy-momentum vector Pµ, the generators of the Lorentz transformations Mµν, and internal scalar symmetry charges that commute with both Pµ and Mµν.
In 1975, however, an extension to this theorem was formulated by Haag, Lopuszanski and Sohnius [27]. Namely, they proved that a new symmetry was allowed by the
Coleman-Mandula theorem - supersymmetry (SUSY), which relates bosonic and fermionic states. By acting upon a fermionic state, a SUSY operator Q transforms it into a bosonic state, and vice versa, by changing the spin by ±12. The simplest argument one can make in favour of SUSY, and the first we will encounter in this thesis, is that all the symmetries in the initial Coleman-Mandula theorem appear to be respected in nature, so from a Murphy’s law perspective it seems only reasonable that SUSY should be as well.
The SUSY algebra can be described with a set of commutation and anticommutation relations between the two-component complex spinor operators that generate SUSY, Q, and the four-momentum operator, Pµ, as:
{Qa, Q † b} = 2 (σµ)abPµ , {Qa, Qb} = 0 , [Pµ, Q] =Pµ, Q† = 0 , (1.19)
where σµ is the four-vectorial construct of the identity and the three Pauli matrices, i.e. σµ= (1, ~σ).
The first of the relations in (1.19) captures the intimate connection between space-time and SUSY. It effectively shows that with two consequent supersymmetric transformations one obtains the four-momentum operator, which generates space-time translations. Equivalently, one can see the spinorial SUSY operators Q as square roots of four-momentum. This means that, by introducing SUSY, we are extending our standard space-time coordinates with extra degrees of freedom, which are related to each other by supersymmetric transformations, and acted upon by the SUSY operators. Such a promotion of regular space-time to “superspace” is conceptually perhaps even more striking than its consequential introduction of a whole new sector of particles, which will be discussed in Section 1.2.2.
It is the introduction of this new sector of particles, though, that serves as the basis for the most important motivation for a supersymmetric theory of nature - and the second argument we will encounter in this thesis. When we impose SUSY onto the SM, we obtain for every SM particle a new particle with spin differing by ±12. The resulting particle spectrum is described in more detail in Section 1.2.2, but for now the concept of new particles with different spin statistics is enough to establish the argument.
As discussed in Section 1.1.3, the instability of the electroweak scale due to radiative corrections is a very problematic feature of the SM. This instability is caused by the
coupling of particles to the SM Higgs field, adding corrections to its mass with a quadratic dependence on the cutoff scale Λ, as expressed in (1.17) and (1.18). Note that the different sign is caused by the difference in fermionic and bosonic spin statistics. SUSY provides a solution to the hierarchy problem precisely due to this property: the corrections to the Higgs mass from fermions are systematically cancelled by the opposite corrections from the newly introduced bosons, and vice versa, thereby removing the quadratic divergences and avoiding the troubling fine tuning requirement.
1.2.1
Supersymmetry Breaking and Grand Unified Theories
Of course we have been slightly careless in concluding that the hierarchy problem is straightforwardly solved only by imposing SUSY. The corrections in (1.17) and (1.18) contain dependencies on coupling strengths and masses of the particles that run in the loops, which introduces the demands that λS = |λf| and that mS be of the same size as mf.
If SUSY were to be an exact symmetry of nature, the new supersymmetric particles would strictly comply with these demands, and have the exact same masses and coupling strengths as their SM counterparts. This is clearly not the case, as otherwise we would have seen experimental evidence of new particles long ago. SUSY must therefore be a broken symmetry in nature, and many theories exist that explain the very non-trivial mechanism of SUSY breaking. A full overview of SUSY breaking is beyond the scope of this thesis, but most realistic models have features in common that will be briefly discussed in the following.
Generally, SUSY breaking occurs softly or dynamically under the influence of a gauge group in a sector hidden from the SM. A set of interactions then mediates the effects between the visible and hidden sectors. The most popular of these scenarios are: flavour-blind gravitational interactions, described by minimal supergravity (mSUGRA); gauge mediated SUSY breaking (GMSB); and anomaly mediated SUSY breaking (AMSB). Even though the different mechanisms of SUSY breaking result in different spectra and interactions, the Minimal Supersymmetric Standard Model, discussed next in Section 1.2.2, serves as a universal model which shares its features with most of the possible resulting spectra.
Considerations from grand unified theories (GUTs) play an important role in describing SUSY breaking. In such theories, which are intended as ‘theories of everything’, the SU (3)C ⊗ SU (2)L ⊗ U (1)Y group structure of the SM is embedded in a single group
2 4 6 8 10 12 14 16 18 Log10(Q/GeV) 0 10 20 30 40 50 60 α-1 U(1) SU(2) SU(3)
Figure 1.2: Two-loop RGE evolution of the inverse gauge couplings of the SM forces. The dashed black lines represent the evolutions in only the SM. The solid blue and red lines show the evolution in the Minimal Supersymmetric Standard Model for common supersymmetric particle thresholds of 500 GeV and 1500 GeV, respectively (adopted from [28]).
that breaks down at some high energy scale. While being very promising candidates for ultimate theories of nature, GUTs are notoriously difficult to test experimentally, due to the extremely high energy scales at which they would manifest themselves.
This brings us to the next argument in favour of SUSY. The running of the gauge couplings of the SM, as described by their renormalisation group equations (RGEs), coinci-dentally shows that the forces of the SM appear to be heading to a point of unification at an energy of about 1016GeV, as shown by the dashed lines in Fig. 1.2. This would evidently hint at a GUT underlying the SM, but a slight misalignment prevents the forces from unifying. Introducing SUSY, however, modifies the RGEs in such a way that unification of the forces of the SM is acquired easily, thus providing tentative, low energy evidence for a GUT hiding at remote energy scales, as shown in the solid lines in Fig. 1.2.
1.2.2
The Minimal Supersymmetric Standard Model
The Minimal Supersymmetric Standard Model (MSSM) is the minimal collection of particles and their interactions resulting from imposing SUSY onto the SM, or in other words, the
minimal supersymmetric theory of nature in which the SM can be embedded. In the MSSM framework, the SM fields are included in irreducible representations of the SUSY algebra, called supermultiplets, together with their SUSY partner fields with spin differing by 1/2. These supermultiplets then transform under the SM gauge groups identically to the usual SM particle multiplets. A full overview of the particle content of the MSSM is given in Table 1.2.
Names Spin PR Gauge Eigenstates Mass Eigenstates
Higgs bosons 0 +1 Hu0 Hd0 Hu+ Hd− h0 H0 A0 H± e uL ueR deL deR (same) squarks 0 −1 esL esR ecL ecR (same) e tL etR ebL ebR et1 et2 eb1 eb2 e eL eeR eνe (same) sleptons 0 −1 eµL eµR νeµ (same) e τL eτR eντ eτ1 τe2 eντ neutralinos 1/2 −1 Be0 Wf0 Heu0 Hed0 χe01 χe20 χe03 χe04 charginos 1/2 −1 Wf± Heu+ He − d χe ± 1 χe ± 2 gluino 1/2 −1 eg (same)
Table 1.2: The particle content of the Minimal Supersymmetric Standard Model.
The chiral spin-1/2 fermion fields of the SM are placed in supermultiplets with spin-0 scalar partner fields, with names equal to their SM counterparts but with a prefix “s”, for “scalar”. Since the left- and right handed components of the fermion fields behave differently under SU (2)L, it is convenient to place them in chiral multiplets as pairs of Weyl spinors with opposing chirality. Two slepton fields are therefore added to each lepton, one with subscript L for the left-handed component, and one with subscript R for the right-handed component. The quarks undergo a similar procedure, and each obtain two squarks. Note that these sfermions are scalars - the subscripts merely denote the chirality of their SM
partner fields. Due to the large Yukawa couplings of the third generation quarks and leptons, their left- and right handed components are subject to more mixing, and therefore their partners are denoted with subscripts 1 and 2 to distinguish between the lighter and heavier state, respectively. Furthermore, because neutrinos have no known right-handed component, they are simply placed in chiral singlets together with their partner sneutrinos.
The gauge bosons of the SM are accompanied by spin-1/2 partners in gauge supermulti-plets. These fermionic gauge partners are known as gauginos. The gluons of SU (3)C obtain a color octet of gaugino partners known as gluinos, the Wµi fields of SU (2)L obtain three winos, a neutral state fW0 and two charged states fW±, and the B
µ field of U (1)Y obtains a gaugino partner known as the bino eB.
Embedding the Higgs doublet of the SM in a supersymmetric theory requires the introduction of a second Higgs doublet with opposite hypercharge in order for both up- and down-type quarks to obtain a mass. The process of EWSB in the MSSM therefore involves both Higgs doublets, named Hu and Hd, and after three of their initial eight degrees of freedom have been absorbed to form the W± and Z bosons, the remaining five form physical Higgs bosons: a neutral pair of scalars, h0 and H0, a pair of charged scalars, H±, and a neutral pseudoscalar A0. The usual Higgs VEV is now obtained as a quadratic combination of the two VEVs from the two Higgs doublets, as v =pv2
u+ vd2. Both doublets also obtain fermionic partners, known as Higgsinos, namely two neutral states eHu0 and eHd0, and two charged states eHu+ and eHd−.
The Electroweakino Sector
After EWSB, the electroweak gauginos and Higgsinos mix and form mass eigenstates, known as electroweakinos. The neutral states of the bino, wino and higgsinos, eB, fW0, eH0
d and e
H0
u, form four Majorana-fermionic mass eigenstates named neutralinos. The charged wino and higgsino states, fW±, eH+
u and eH −
d, form two Dirac-fermionic mass eigenstates with charge ±1 known as charginos. The electroweakinos are referred to asχ, withe χe0
i being the neutralinos and i = 1, 2, 3, 4 ordered with ascending mass, and with the charginos being defined similarly as χe±j with j = 1, 2.
The mass spectrum and composition of the electroweakinos is determined by the properties of their gaugino and Higgsino constituents. Neglecting intricacies due to non-general SUSY breaking mechanisms, the masses of the bino, wino and gluino fields are controlled by three parameters: M1, M2, and M3. The bino and wino masses, together with
the higgsino mass term µ, and the ratio of up- and down-type Higgs VEVs, tan(β) ≡ vu
vd,
make up the parameter space that goes into the set of mixing matrices which dictate the electroweakino sector.
For the neutralinos, the mass terms are given by: Lneutralino mass = −12(ψ0)TM
e
χ0ψ0 , (1.20)
where ψ0 is given in the gauge eigenstate basis ψ0 = ( eB, fW0, eHu0, eHd0), and M e
χ0 is the
neutralino mass matrix, given by:
M e χ0 = M1 0 −cβsWmZ sβsWmZ 0 M2 cβcWmZ −sβcW mZ −cβsW mZ cβcW mZ 0 −µ sβsWmZ −sβcWmZ −µ 0 , (1.21)
where the abbreviations sβ = sin β, cβ = cos β, sW = sin θW, and cW = cos θW are being used. The neutralino mass matrix is complex symmetric and can be diagonalised with a 4 × 4 unitary mixing matrix N , as:
N∗M e χ0N−1 = m e χ0 1 0 0 0 0 m e χ0 2 0 0 0 0 m e χ0 3 0 0 0 0 m e χ0 4 (1.22)
The matrix N is more commonly known as the neutralino mixing matrix, and its entries Nij give a measure of the ψ0
j content of the ith neutralino, χ0i, or in other words: χe 0
i = Nijψ0j. The masses and compositions of the charginos can be expressed in a similar fashion as for the neutralinos. Some complication is introduced, however, due to the fact that the charginos are Dirac-fermions. We can nonetheless use the gauge eigenstate basis ψ± = (fW+, eH+
u, fW −, eH−
d) to write down the chargino mass terms as: Lchargino mass = −12(ψ±)TM
e
where the chargino mass matrix Mχe± can be expressed in 2 × 2 block form, as Mχe± = ∅ XT X ∅ , (1.24) with X = M2 √ 2sβmW √ 2cβmW µ . (1.25)
Diagonalisation of the chargino mass matrix can then be performed with two different unitary 2 × 2 chargino mixing matrices to obtain the masses of the positively and negatively charged charginos. These matrices, U and V , which relate the mass eigenstates to the gauge eigenstates as χe+i = Vijψj+ and χe
−
i = Uijψ−j , consequently diagonalise the chargino mass matrix as:
U∗XV−1 = m e χ±1 0 0 m e χ±2 . (1.26)
A more detailed description of the composition and mass spectra of the electroweakinos will be given in Section 1.3.3, along with the corresponding collider phenomenology. R-parity and the Lightest Supersymmetric Particle
In the MSSM framework described so far, no clear distinction is present between the fields of quarks, leptons and Higgs bosons. Consequently, nothing prevents supersymmetric particles from giving rise to interactions that violate baryon and lepton number. Constraints from flavour physics and experiments looking for proton decay put very stringent limits on the strength of such interactions, and we are therefore lead to believe that a certain quantum number, commonly called R-parity, is conserved to a good degree at least at low energies. R-parity is defined as a quantum number which depends on the spin s and the baryon and lepton numbers, B and L, of a particle, as:
PR = (−1)3B+L+2s . (1.27)
For R-parity to be conserved it would have to be protected by a symmetry, which could naturally arise in many GUT frameworks from a broken off U (1)(B−L) gauge symmetry, or
even due to a discrete global symmetry. This would forbid any interactions that violate total baryon and lepton number2, thereby protecting the proton lifetime. During the continuation of this thesis we will assume R-parity is indeed conserved.
Filling in the numbers of (1.27) reveals that SM particles have PR = +1, whereas their supersymmetric counterparts have PR= −1. This simple observation has three very important phenomenological consequences: first, the lightest supersymmetric particle (LSP) must be stable; second, any supersymmetric particle should eventually decay to a final state containing an odd number of LSPs; third, supersymmetric particles can only be produced in even numbers.
The first of these consequences carries the most importance, and brings us to the final argument in this thesis in favour of SUSY. If the LSP is electrically neutral, which it practically always is in any MSSM scenario, and weakly interacting with ordinary matter, it makes for a natural and attractive DM candidate, thereby addressing the pressing cosmological issue of DM.
However, the complete set of consequences also plays an important role when consid-ering collider experiments to look for SUSY. Namely, supersymmetric particles would be predominantly produced in pairs at a collider. Moreover, any final state will contain at least two LSPs, thus resulting in a missing transverse energy signature due to the LSPs escaping the detector unseen.
1.3
Compressed Electroweak Supersymmetry
1.3.1
Naturalness
The introduction of SUSY is in the first place meant to stabilise the electroweak scale in a natural way, as was described in Section 1.1.3. The fact that SUSY must be a broken symmetry, as described in Section 1.2.1, does not necessarily prevent this, but the masses of the SM superpartners do play an important role.
A rough way to parametrise to what extent a SUSY spectrum upholds naturalness is to evaluate the residual corrections to the Higgs mass that remain after SUSY has been
2Note that interactions that violate total baryon and lepton number do exist in the SM, but are only
manifest by means of instanton and sphaleron interactions in vacua with an intact electroweak symmetry. These non-perturbative effects play no significant role in the current stage in the lifetime of the universe.
broken and the masses of the superpartners are set. This can be cast in the form of the requirement: ∆ ≡ δm2H m2 H . 1 . (1.28) If this condition is satisfied, the corrections to the squared Higgs mass are no larger than its initial value, and no fine tuning is needed to stabilise the it. The scale at which SUSY enters determines how easily the Higgs mass corrections from the SM particles are cancelled by their superysmmetric counterparts. If supersymmetric physics enters around or below 500 GeV, no fine tuning is needed to stabilise the Higgs mass. As this scale grows, ∆ grows too, and around energies of 1.6 TeV, (1.28) yields ∆ ∼ 10 and 10% fine tuning is needed. If the scale is around 5 TeV or higher, percent level fine tuning becomes required, making SUSY a questionable natural solution to the hierarchy problem.
The use of naturalness as a guideline for searching for new physics has become increasingly popular in the particle physics theory community over the last few decades. In turn, this surge in popularity given rise to some skepticism, generally concerning the arbitrariness of when a theory is natural, and how important this should even be. The arguments can usually be traced back to the fact that any notion of naturalness always contains an irreducible degree of subjectivity due to its essentially aesthetic origin. Any discussion involving naturalness should therefore always be viewed in light with its subjective naturalness prescriptions.
Constraints on Sfermions
Without choosing a specific naturalness recipe, we can still heuristically argue what the essential ingredients for a natural SUSY model are. As discussed in Section 1.1.3, the correction from the top quark poses the biggest threat to the Higgs stability. Namely, without a symmetry such as SUSY to protect the Higgs mass, just the top quark correction alone produces a correction δM2
H which requires a fine tuning of 1 in 1030 according to the simple handle given in (1.28). The Λ2 terms in the fermionic corrections dissappear upon imposing SUSY, but the logarithmic terms dependent on their mass remain. For a fermion f with a superpartner ef , the residual correction can be approximated as:
δm2H ≈ λ 2 fNcf 8π2 m2 e f − m 2 f ln (Λ/mf) (1.29)
Taking Λ ∼ MP and mfe∼ 1 TeV causes the logarithm to have a value of around 70, which due to the 1/8π2 suppression factor causes only very mild corrections provided that mfeis not too large. Specifically, the top squark mass ought to lie below about 800 GeV in order to avert percent level fine tuning.
Heuristic constraints on the other squark flavours can also be constructed, but are generally less stringent, at least with respect to the mass reach of present-day experiments. Due to their smaller Yukawa couplings, the upper mass bounds on squarks and sleptons other than the top squark are around the (multi-) TeV range.
Constraints on Higgsinos
Another constraint can be put on the mass µ of the Higgsino, the superpartner of the Higgs itself. This constraint directly originates from the minimisation of the Higgs potential that sets the electroweak scale in the context of the MSSM. Therefore, it is different from the previous constraint in that there are no loop corrections involved. Namely, the tree-level expression for the mass of the Z boson, at tree-level related to the Higgs mass as mH = mZ| cos (2β)| in the MSSM3, is:
m2Z = 2m 2 Hd− m 2 Hutan 2(β) tan2(β) − 1 − 2µ 2 ' −2m2 Hu− 2µ 2 , (1.30)
where mHu and mHd are the MSSM up and down scalar Higgs masses, and tan (β) & 10
goes into the last assumption. Straightforwardly one can observe that µ should be small, namely |µ| . 200 GeV, in order for (1.30) to generate the observed Z boson mass without much fine tuning.
A caveat becomes apparent, though, when looking at (1.30) more closely: it contains a similarly important dependence on mHu as on µ. In general, the value of µ originates
completely separate from mHu, which is related to the scale of SUSY breaking. Therefore,
even with a small µ, the relation in (1.30) can still be finely tuned if mHu requires large
cancellations to be small as well.
There is no simple reason why these two mass scales should have an implicit correlation - a puzzle known as the ‘µ problem’. Several models exist that reconcile this problem by requiring or assuming no tree-level µ dependence before SUSY breaking, but then have it
3The observant reader may notice that this means that the Higgs boson is lighter than the Z boson.
arise from the VEV(s) of new field(s) with potentials that do depend on the soft SUSY breaking terms, thereby associating the two scales [29–31].
Regardless of whether a correlation between µ and the SUSY breaking scale exists, we can draw an important conclusion: a low µ is a necessary requirement for a natural SUSY model, but it is not sufficient to guarantee naturalness.
Constraints on Gauginos
The wino enters directly into the loop corrections to the Higgs mass, but because of the relatively small contributions from the SM gauge bosons, the resulting constraints are less demanding than those for sfermions. Particularly, the wino mass should not be much larger than about 1 TeV. The bino mass can be larger than 3 TeV on the other hand.
Another interesting condition can be placed on the gluino, even though it does not couple directly to the Higgs. Rather it is the one-loop corrections from the gluino to the squark masses that create an incidental naturalness problem in the squark sector: the requirement of light squarks, in particular the top squark, restricts the gluino contributions to the squark masses, resulting in the condition that the gluino mass be no larger than roughly twice the mass of the top squark: meg . 2met.
1.3.2
Established Constraints
The fine tuning observations given in the previous section provide a heuristic way to interpret the present-day constraints on natural SUSY. However, actually assessing the naturalness of a specific SUSY model, and the model space that is still allowed by experimental bounds, involves more than just the highest mass limits in certain simplified models. The following summary of collider bounds on SUSY parameter space nevertheless gives a good impression of the mass reach of modern collider experiments in relation to the naturalness constraints derived above.
Searches for strongly produced SUSY particles have excluded an enormous portion of parameter space so far. With the increase in centre-of-mass energy in run 2, the cross sections for potential strong SUSY processes have grown considerably compared to run 1, causing the null-results from collider searches to exclude colored SUSY particles up to TeV scale masses in the most promising scenarios. However, in scenarios that favour processes which are less easily probed, the allowed squark and gluino masses can still fit comfortably
[GeV] 1 t ~ m 200 300 400 500 600 700 800 900 [GeV]0χ∼1 m 0 100 200 300 400 500 600 1 0 χ∼ W b → 1 t ~ / 1 0 χ∼ t → 1 t ~ 1 0 χ∼ t → 1 t ~ 1 0 χ∼ W b → 1 t ~ 1 0 χ∼ c → 1 t ~ -1 =8 TeV, 20 fb s t ) < m 1 0 χ∼ , 1 t ~ m( ∆ W + m b ) < m 1 0 χ∼ , 1 t ~ m( ∆ ) < 0 1 0 χ∼ , 1 t ~ m( ∆ 1 0 χ∼ t → 1 t ~ / 1 0 χ∼ W b → 1 t ~ / 1 0 χ∼ c → 1 t ~ / 1 0 χ∼ b f f' → 1 t ~ production, 1 t ~ 1 t ~ Status: ICHEP 2016 ATLAS Preliminary 1 0 χ∼ W b 1 0 χ∼ c 1 0 χ∼ b f f'
Observed limits Expected limits All limits at 95% CL
=13 TeV s [CONF-2016-077] -1 t0L 13.2 fb [CONF-2016-050] -1 t1L 13.2 fb [CONF-2016-076] -1 t2L 13.3 fb [1604.07773] -1 MJ 3.2 fb Run 1 [1506.08616] (a) ) [GeV] g ~ m( 1000 1200 1400 1600 1800 2000 ) [GeV] 0 1 χ∼ m( 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 ATLAS Preliminary -1 = 13 TeV, 13.2-14.8 fb s , 0-lepton, ATLAS-CONF-2016-078 0 1 χ∼ q q → g ~ , 0-lepton, ATLAS-CONF-2016-078 0 1 χ∼ W q q → g ~ , 1-lepton, ATLAS-CONF-2016-054 0 1 χ∼ W q q → g ~ , SS leptons, ATLAS-CONF-2016-037 0 1 χ∼ WZ q q → g ~ , SS leptons, ATLAS-CONF-2016-037 0 1 χ∼ t t → g ~ 3b jets, ATLAS-CONF-2016-052 ≥ , 0 1 χ∼ t t → g ~ 3b jets, ATLAS-CONF-2016-052 ≥ , 0 1 χ∼ b b → g ~ (b)
Figure 1.3: The latest exclusion limits at 95 % CL from ATLAS, as of ICHEP 2016. (a) summarises dedicated searches for top squark pair production based on approximately 13 fb−1 of √s = 13 TeV pp collision data. The dashed and solid lines depict the expected and observed limits, respectively, for four different decay modes set to 100 % branching fractions as part of a simplified model-framework. (b) summarises similar searches for gluinos in several simplified models where the gluino has different decay modes with 100 % branching fractions [32].
within naturalness constraints. The latest summary plots from ATLAS on strong SUSY are shown in Fig. 1.3.
The bounds on the electroweak SUSY sector vary significantly for the different simplified models used to derive the bounds. Especially the presence of sleptons as intermediate particles in electroweakino decays strongly enhances the corresponding limits. A summary of the ATLAS bounds on the electroweak SUSY sector [33] is given in Fig. 1.4a, whereas a subset of the limits is shown in more detail in Fig. 1.4b for scenarios in which no sleptons contribute to electroweakino topologies, but where decays occur exclusively via emission of SM gauge or Higgs bosons. Note that all searches for these scenarios rapidly lose their
) [GeV] 3 0 χ∼ , 2 0 χ∼ , 1 ± χ∼ m( 100 200 300 400 500 600 700 ) [GeV] 0 1 χ∼ m( 0 100 200 300 400 500
600 lL/ ν∼ 2l, arXiv:1403.5294 Expected limitsObserved limits
~ via 2l, arXiv:1509.07152 ν∼ / L l ~ via , arXiv:1407.0350 τ 2 τ ν∼ / L τ∼ via 2l, arXiv:1403.5294 via WW − 1 χ∼ + 1 χ∼ 2l+3l, arXiv:1509.07152 ν∼ / L l ~ via +3l, arXiv:1509.07152 τ 2 τ ν∼ / L τ∼ via 2l+3l, arXiv:1403.5294 via WZ +3l, arXiv:1501.07110 ± l ± +l γ γ lbb+l via Wh 0 2 χ∼ ± 1 χ∼ 3l+4l, arXiv:1509.07152 R l ~ via 0 3 χ∼ 0 2 χ∼ All limits at 95% CL ATLAS -1 =8 TeV, 20.3 fb s ) 2 0 χ∼ + m 1 0 χ∼ = 0.5(m ν∼ / L τ∼ / L l ~ m 1 0 χ ∼ = m 2 0 χ ∼ m 1 0 χ ∼ = 2m 2 0 χ ∼ m (a) ) [GeV] 2 0 χ ∼ , 1 ± χ ∼ m( 100 150 200 250 300 350 400 450 ) [GeV] 0 1 χ∼ m( 0 50 100 150 200 250 Expected limits Observed limits 2l, arXiv:1403.5294 via WW − 1 χ∼ + 1 χ∼ 2l+3l, arXiv:1403.5294 via WZ +3l, arXiv:1501.07110 ± l ± +l γ γ lbb+l via Wh 0 2 χ∼ ± 1 χ∼ All limits at 95% CL ATLAS s=8 TeV, 20.3 fb-1 1 0 χ ∼ = m 2 0 χ ∼ m Z + m 1 0 χ ∼ = m 2 0 χ ∼ m h + m 1 0 χ ∼ = m 2 0 χ ∼ m 1 0 χ ∼ = 2m 2 0 χ ∼ m (b)
Figure 1.4: The 95 % CL exclusion limits on direct electroweakino pair pro-duction from ATLAS, using the full 20.8 fb−1 of √s = 8 TeV data from run 1 [33]. (a) shows the limits in a variety of simplified models with different decay modes that are either mediated by SM gauge bosons or sleptons. (b) shows only the limits for the SM gauge boson mediated models on a zoomed in mass plane.
sensitivity as the electroweakinos approach mass degeneracy.
In fact, the strongest bounds on the masses of electroweakinos with SM gauge or Higgs boson decay modes, in the compressed, low mass-regime left unexplored by ATLAS searches, still come from LEP2 [34–36]. These limits result from searches that target chargino pair production, and the corresponding results are shown in Fig. 1.5. Note that the variable ∆M in this figure denotes the mass difference between χe±1 and χe0
1.
The interesting shape of the excluded mass region in Fig. 1.5b is on one hand due to the fact that for a very small mass gap, the phase space of produced charginos is restricted to the extent that the charginos will have a long enough lifetime to produce observable charged tracks in the detector. On the other hand, at the higher end of the mass gap, the daughter particles from χe±1 decays to χe0
1 are more energetic and can be reconstructed more easily, thereby also enhancing the sensitivity. In the middle of these regions, where
10-1 1 10 60 80 100 10 -4 10 -3 10 -2 10 -1 1 10 Mχ~1+(GeV) ∆ M (GeV) ADLO preliminary
Higgsino - cMSSM σ upper limit (pb)Observed
(a) 10-1 1 10 60 80 100 Mχ~1+(GeV) ∆ M (GeV) 10 -1 1 10 60 80 100 ADLO preliminary expected limit Higgsino - cMSSM (b)
Figure 1.5: Combined 95 % CL bounds of√s = 208 GeV data from all four LEP2 experiments on (a) the chargino pair production cross section and (b) the chargino mass [34–36]. These bounds were derived for Higgsino-like charginos, and assuming the relation M1 = (5/3) tan2(θW)M2 between the bino and wino SUSY breaking terms, M1 and M2, respectively, as is sometimes favourable for gaugino mass unification at some GUT scale.
100 120 140 160 180 200 220 240
m
(˜χ0 2)[GeV]
80 100 120 140 160 180 200 220 240m
(˜ χ 0 1)[G
eV
]
m
˜χ 0 2= m
˜ χ01 ˜ χ± 1χ˜02 ATLAS 95% CL (Exp) ˜ χ± 1χ˜02 ATLAS 95% CL (Obs) LEP2 limits (x=0.25) LEP2 limits (x=0.5) LEP2 limits (x=0.75) (a) 100 120 140 160 180 200 220 240m
(˜χ0 2)[GeV]
0 20 40 60 80 100m
(˜ χ 0 2)-
m
(˜ χ 0 1)[G
eV
]
LEP2 limits (x=0.25) LEP2 limits (x=0.5) LEP2 limits (x=0.75) ˜ χ± 1χ˜02 ATLAS 95% CL (Exp) ˜ χ± 1χ˜02 ATLAS 95% CL (Obs) (b)Figure 1.6: The combination of 95 % CL exclusion limits from chargino pair production searches at LEP2 [34–36], and searches for W± or Z mediated neutralino pair production from ATLAS [33], (a) in theχe02 andχe01 mass plane, and (b) in the m e χ0 2 and mχe 0 2-mχe 0
1 plane. The placement of mχe
± 1 between mχe 0 1 and m e χ0 2 is denoted by x, as x ≡ (mχe ± 1 − mχe 0 1)/(mχe 0 2 − mχe 0 1).
200 MeV . ∆M . 3 GeV, the limits exclude charginos with mχe±1 . 92 GeV, whereas in the more sensitive regions with smaller or larger ∆M , charginos with m
e
χ±1 . 103 GeV are excluded.
The limits from ATLAS and LEP2 can be mapped onto the same plane to compare the reach of each experiment, and to quantitatively determine the range of electroweakino masses that has not been explored so far. To map the reach of the LEP2 limits, given in the chargino mass plane, to the neutralino mass plane, we need to make an assumption on where m e χ±1 is placed between mχe 0 1 and mχe 0
2. The theoretical basis underlying this
placement depends on the electroweakino composition and will be discussed in more detail in Section 1.3.3, but for now we can make an agnostic choice and extrapolate the reach of LEP2 for three different placements of the chargino mass: at 14, 12, and 34 of the way from m
e χ0
1 to mχe
0
2. The resulting limits from extrapolating the LEP2 chargino mass limits from
Fig. 1.5b and the W± or Z mediated neutralino mass limits from Fig. 1.4b are shown in Fig. 1.6.
Of course the experimental constraints on SUSY shown here only account for a very small portion of the total. A full summary of the mass reach of ATLAS SUSY searches is shown in Fig. 1.7. The limits from CMS are comparable to those from ATLAS [37–42]. A detailed overview of the implication of LHC searches on the status of SUSY from a theoretical perspective can be found in [43, 44].
Model e, µ, τ, γ Jets EmissT !L dt[fb−1] Mass limit Reference
In cl us iv e S ea rc he s 3 rdgen. ˜gme d. 3 rdgen. squar ks di rect pr oduct ion EW direc t Long-liv ed par tic les RP V Other
MSUGRA/CMSSM 0-3 e, µ /1-2 τ 2-10 jets/3 b Yes 20.3 ˜q, ˜g 1.85 TeVm(˜q)=m(˜g) 1507.05525
˜q˜q, ˜q→q˜χ01 0 2-6 jets Yes 13.3 ˜q 1.35 TeV m( ˜χ01)<200 GeV, m(1stgen. ˜q)=m(2ndgen. ˜q) ATLAS-CONF-2016-078
˜q˜q, ˜q→q˜χ01(compressed) mono-jet 1-3 jets Yes 3.2 ˜q 608 GeV m(˜q)-m( ˜χ01)<5 GeV 1604.07773
˜g˜g, ˜g→q¯q˜χ0
1 0 2-6 jets Yes 13.3 m( ˜χ0
1)=0 GeV ATLAS-CONF-2016-078
1.86 TeV
˜g
˜g˜g, ˜g→qq˜χ±1→qqW±˜χ01 0 2-6 jets Yes 13.3 ˜g 1.83 TeVm( ˜χ01)<400 GeV, m( ˜χ±)=0.5(m( ˜χ01)+m(˜g)) ATLAS-CONF-2016-078
˜g˜g, ˜g→qq(ℓℓ/νν) ˜χ01 3 e, µ 4 jets - 13.2 ˜g 1.7 TeV m( ˜χ01)<400 GeV ATLAS-CONF-2016-037
˜g˜g, ˜g→qqWZ ˜χ0
1 2 e, µ (SS) 0-3 jets Yes 13.2 ˜g 1.6 TeV m( ˜χ01) <500 GeV ATLAS-CONF-2016-037
GMSB (˜ℓ NLSP) 1-2 τ + 0-1 ℓ 0-2 jets Yes 3.2 ˜g 2.0 TeV 1607.05979
GGM (bino NLSP) 2 γ - Yes 3.2 ˜g 1.65 TeV cτ(NLSP)<0.1 mm 1606.09150
GGM (higgsino-bino NLSP) γ 1 b Yes 20.3 ˜g 1.37 TeV m( ˜χ01)<950 GeV, cτ(NLSP)<0.1 mm, µ<0 1507.05493
GGM (higgsino-bino NLSP) γ 2 jets Yes 13.3 m( ˜χ0
1)>680 GeV, cτ(NLSP)<0.1 mm, µ>0 ATLAS-CONF-2016-066
1.8 TeV
˜g
GGM (higgsino NLSP) 2 e, µ (Z) 2 jets Yes 20.3 ˜g 900 GeV m(NLSP)>430 GeV 1503.03290
Gravitino LSP 0 mono-jet Yes 20.3 F1/2scale 865 GeV m( ˜G)>1.8 × 10−4eV, m(˜g)=m(˜q)=1.5 TeV 1502.01518
˜g˜g, ˜g→b¯b˜χ01 0 3 b Yes 14.8 ˜g 1.89 TeVm( ˜χ01)=0 GeV ATLAS-CONF-2016-052
˜g˜g, ˜g→t¯t˜χ0
1 0-1 e, µ 3 b Yes 14.8 ˜g 1.89 TeVm( ˜χ01)=0 GeV ATLAS-CONF-2016-052
˜g˜g, ˜g→b¯t˜χ+
1 0-1 e, µ 3 b Yes 20.1 m( ˜χ0
1)<300 GeV 1407.0600
1.37 TeV
˜g
˜b1˜b1, ˜b1→b ˜χ01 0 2 b Yes 3.2 ˜b1 840 GeV m( ˜χ01)<100 GeV 1606.08772
˜b1˜b1, ˜b1→t ˜χ±1 2 e, µ (SS) 1 b Yes 13.2 m( ˜χ0
1)<150 GeV, m( ˜χ±1)= m( ˜χ01)+100 GeV ATLAS-CONF-2016-037
325-685 GeV
˜b1
˜t1˜t1, ˜t1→b ˜χ±1 0-2 e, µ 1-2 b Yes 4.7/13.3 m( ˜χ±
1) = 2m( ˜χ01), m( ˜χ01)=55 GeV 1209.2102, ATLAS-CONF-2016-077
117-170 GeV˜t1˜t1 200-720 GeV
˜t1˜t1, ˜t1→Wb ˜χ01or t ˜χ10 0-2 e, µ 0-2 jets/1-2 b Yes 4.7/13.3 ˜t1˜t1 90-198 GeV 205-850 GeV m( ˜χ01)=1 GeV 1506.08616, ATLAS-CONF-2016-077
˜t1˜t1, ˜t1→c ˜χ01 0 mono-jet Yes 3.2 ˜t1 90-323 GeV m(˜t1)-m( ˜χ01)=5 GeV 1604.07773
˜t1˜t1(natural GMSB) 2 e, µ (Z) 1 b Yes 20.3 m( ˜χ0
1)>150 GeV 1403.5222
150-600 GeV
˜t1
˜t2˜t2, ˜t2→˜t1+Z 3 e, µ (Z) 1 b Yes 13.3 ˜t2 290-700 GeV m( ˜χ01)<300 GeV ATLAS-CONF-2016-038
˜t2˜t2, ˜t2→˜t1+h 1 e, µ 6 jets + 2 b Yes 20.3 m( ˜χ0
1)=0 GeV 1506.08616
320-620 GeV
˜t2
˜ℓL,R˜ℓL,R, ˜ℓ→ℓ ˜χ01 2 e, µ 0 Yes 20.3 ˜ℓ 90-335 GeV m( ˜χ01)=0 GeV 1403.5294
˜χ+ 1˜χ−1, ˜χ+1→˜ℓν(ℓ˜ν) 2 e, µ 0 Yes 20.3 m( ˜χ0 1)=0 GeV, m(˜ℓ,˜ν)=0.5(m( ˜χ±1)+m( ˜χ01)) 1403.5294 140-475 GeV ˜χ± 1 ˜χ+ 1˜χ−1, ˜χ+1→˜τν(τ˜ν) 2 τ - Yes 20.3 m( ˜χ0 1)=0 GeV, m(˜τ, ˜ν)=0.5(m( ˜χ±1)+m( ˜χ01)) 1407.0350 355 GeV ˜χ± 1 ˜χ± 1˜χ02→˜ℓLν ˜ℓLℓ(˜νν), ℓ˜ν˜ℓLℓ(˜νν) 3 e, µ 0 Yes 20.3 m( ˜χ± 1)=m( ˜χ02), m( ˜χ01)=0, m(˜ℓ,˜ν)=0.5(m( ˜χ± 1)+m( ˜χ01)) 1402.7029 715 GeV ˜χ± 1, ˜χ02 ˜χ± 1˜χ02→W ˜χ01Z ˜χ10 2-3 e, µ 0-2 jets Yes 20.3 m( ˜χ±1)=m( ˜χ0 2), m( ˜χ01)=0, ˜ℓ decoupled 1403.5294, 1402.7029 425 GeV ˜χ± 1, ˜χ02 ˜χ±
1˜χ02→W ˜χ01h ˜χ10, h→b¯b/WW/ττ/γγ e, µ, γ 0-2 b Yes 20.3 ˜χ1, ˜χ± 02 270 GeV m( ˜χ1±)=m( ˜χ20), m( ˜χ01)=0, ˜ℓ decoupled 1501.07110
˜χ0 2˜χ03, ˜χ02,3→˜ℓRℓ 4 e, µ 0 Yes 20.3 m( ˜χ0 2)=m( ˜χ03), m( ˜χ01)=0, m(˜ℓ,˜ν)=0.5(m( ˜χ02)+m( ˜χ01)) 1405.5086 635 GeV ˜χ0 2,3
GGM (wino NLSP) weak prod. 1 e, µ + γ - Yes 20.3 ˜W 115-370 GeV cτ<1 mm 1507.05493
GGM (bino NLSP) weak prod. 2 γ - Yes 20.3 ˜W 590 GeV cτ<1 mm 1507.05493
Direct ˜χ+
1˜χ−1prod., long-lived ˜χ±1 Disapp. trk 1 jet Yes 20.3 m( ˜χ±
1)-m( ˜χ0 1)∼160 MeV, τ( ˜χ±1)=0.2 ns 1310.3675 270 GeV ˜χ± 1 Direct ˜χ+
1˜χ−1prod., long-lived ˜χ±1 dE/dx trk - Yes 18.4 m( ˜χ±
1)-m( ˜χ01)∼160 MeV, τ( ˜χ±1)<15 ns 1506.05332 495 GeV
˜χ± 1
Stable, stopped˜g R-hadron 0 1-5 jets Yes 27.9 m( ˜χ0
1)=100 GeV, 10 µs<τ(˜g)<1000 s 1310.6584 850 GeV
˜g
Stable˜g R-hadron trk - - 3.2 ˜g 1.58 TeV 1606.05129
Metastable˜g R-hadron dE/dx trk - - 3.2 m( ˜χ0
1)=100 GeV, τ>10 ns 1604.04520
1.57 TeV
˜g
GMSB, stable˜τ, ˜χ01→˜τ(˜e, ˜µ)+τ(e, µ) 1-2 µ - - 19.1 ˜χ01 537 GeV 10<tanβ<50 1411.6795
GMSB, ˜χ0 1→γ ˜G, long-lived ˜χ0 1 2 γ - Yes 20.3 1<τ( ˜χ0 1)<3 ns, SPS8 model 1409.5542 440 GeV ˜χ0 1 ˜g˜g, ˜χ0
1→eeν/eµν/µµν displ. ee/eµ/µµ - - 20.3 ˜χ01 1.0 TeV 7 <cτ( ˜χ01)< 740 mm, m(˜g)=1.3 TeV 1504.05162
GGM˜g˜g, ˜χ01→Z ˜G displ. vtx + jets - - 20.3 ˜χ01 1.0 TeV 6 <cτ( ˜χ01)< 480 mm, m(˜g)=1.1 TeV 1504.05162
LFV pp→˜ντ+X, ˜ντ→eµ/eτ/µτ eµ,eτ,µτ - - 3.2 ˜ντ 1.9 TeVλ311′ =0.11, λ132/133/233=0.07 1607.08079
Bilinear RPV CMSSM 2 e, µ (SS) 0-3 b Yes 20.3 ˜q, ˜g 1.45 TeV m(˜q)=m(˜g), cτLS P<1 mm 1404.2500
˜χ+
1˜χ−1, ˜χ+1→W ˜χ01, ˜χ01→eeν, eµν, µµν 4 e, µ - Yes 13.3 ˜χ± 1.14 TeV m( ˜χ01)>400GeV, λ12k!0 (k = 1, 2) ATLAS-CONF-2016-075
1
˜χ+
1˜χ−1, ˜χ+1→W ˜χ01, ˜χ01→ττνe,eτντ 3 e, µ + τ - Yes 20.3 ˜χ±1 450 GeV m( ˜χ10)>0.2×m( ˜χ±1), λ133!0 1405.5086
˜g˜g, ˜g→qqq 0 4-5 large-R jets - 14.8 ˜g 1.08 TeV BR(t)=BR(b)=BR(c)=0% ATLAS-CONF-2016-057
˜g˜g, ˜g→qq˜χ0
1, ˜χ01→ qqq 0 4-5 large-R jets - 14.8 ˜g 1.55 TeV m( ˜χ01)=800 GeV ATLAS-CONF-2016-057
˜g˜g, ˜g→˜t1t, ˜t1→bs 2 e, µ (SS) 0-3 b Yes 13.2 ˜g 1.3 TeV m(˜t1)<750 GeV ATLAS-CONF-2016-037
˜t1˜t1, ˜t1→bs 0 2 jets + 2 b - 15.4 ˜t˜t11 410 GeV 450-510 GeV ATLAS-CONF-2016-022, ATLAS-CONF-2016-084
˜t1˜t1, ˜t1→bℓ 2 e, µ 2 b - 20.3 ˜t1 0.4-1.0 TeV BR(˜t1→be/µ)>20% ATLAS-CONF-2015-015
Scalar charm,˜c→c˜χ01 0 2 c Yes 20.3 ˜c 510 GeV m( ˜χ01)<200 GeV 1501.01325
Mass scale [TeV]
10−1 1
√s = 7, 8 TeV √s = 13 TeV
ATLAS SUSY Searches* - 95% CL Lower Limits
Status: August 2016 ATLAS
Preliminary √s = 7, 8, 13 TeV
*Only a selection of the available mass limits on new states or phenomena is shown.
Figure 1.7: A simplified representation of the mass reach of a selection of ATLAS SUSY searches.
Furthermore, a variety of other experimental disciplines is also widely used to constrain SUSY, such as direct and indirect DM searches, measurements of the electric dipole moment,
electroweak precision measurements, and flavour physics and CP violation measurements. The implications of these experiments are discussed from a naturalness perspective in [45].
Motivation for Light, Compressed Higgsinos
Taking together the naturalness arguments from Section 1.3.1 and the observation that modern constraints leave large gaps in the electroweak SUSY sector, we see that a search for electroweakinos near mass degeneracy is well-motivated both from a theoretical and an experimental point of view.
In particular, the naturalness requirement that |µ| . 200 GeV results in spectra with the three lightest electroweakinos (χe01, χe02, and χe±1) to be Higgsino-like and close in mass, given that M1 and M2 are heavier than µ. This follows from how the mass eigenstates form as the electroweak gauginos and Higgsinos mix, as was described in Section 1.2.2, whereas the resulting spectrum will be discussed in more detail in Section 1.3.3.
The relative sizes of the gaugino mass parameters M1, M2, and M3 are closely related to the mechanism of SUSY breaking and the underlying GUT structure, both briefly explained in Section 1.2.1. A plethora of different ratios M1 : M2 : M3 can be obtained from different underlying groups and breaking mechanisms [46, 47], and no clear preference exists for a single preferred option within the theory community. However, most scenarios result in M1, M2, and M3 being roughly the same size after EWSB has occurred [47]. Given that M3 is likely at the TeV scale as indicated by gluino bounds, this incidentally supports a scenario where M1 and M2 are large as well, therefore favouring a large splitting between these parameters and µ.
Furthermore, smaller values of M1 and M2 result in electroweakino states that are more mixed, meaning that there is less mass degeneracy. Such spectra are already largely excluded by collider searches, as shown in Fig. 1.4a, thereby further hinting at large M1 and M2.
For this thesis we study scenarios where a large splitting between µ and M1,2 exists, and the sfermions and squarks are out of the reach of current collider constraints, as are the additional Higgs bosons. A simplified version of the corresping mass spectrum, showing only the relative sizes of the superpartner masses, is shown in Fig. 1.8. The parameter space is thus narrowed down to a four-dimensional set of {M1, M2, µ, tan(β)}. This sets the stage for the phenomenological discussion of the electroweakino sector in the following section, whereas a more explicit description of SUSY input parameters is given in Section 3.1 in
relation to the simulations that have been performed.
Figure 1.8: A simplified depiction of a superpartner mass spectrum in our scenario of interest. Note that the three Higgsino-like electroweakino states are light and compressed in mass, while the gauginos and sfermions are heavier. The superpartners of the third generation quarks are placed just outside of the mass reach of current constraints. This figure was adapted from [48].
1.3.3
Masses and Composition
The gaugino and Higgsino content of the electroweakino mass eigenstates is determined by the set of mixing matrices described in Section 1.2.2. Diagonalising the neutralino mass matrices in (1.21) and (1.25) is a nontrivial task. Including terms up to ∼ m2
Z/Ma2 (a =1,2), this results in the following expressions [49] (with the neutralino indices in the
