Análisis de la significación de los efectos 70

In this rather short road map, we will summarize some of the ongoing attempts on solving the problems of theSM. There are two main paths to take, one can accept the SMas a basis and try to expand the Lagrangian piece by piece, or one can construct a broader framework

which breaks down to the SM.

One way to approach such a vast set of problems is to add particles or interactions

to the SM Lagrangian. This ad hoc technique doesn’t have a single recipe, but can provide

solid directions to the grand question of “Where to find new physics?”. Extending the family structures in the SMor adding new particles such as new vector-like leptons or axions can lead to viable dark matter candidates and explain the mass of neutrinos. The large uncertainties, especially, in Higgs and top quark coupling strengths to other particles and themselves is an important starting point to develop such a theory. SM effective field theory (SMEFT) is another way of approaching to answer such queries. As mentioned earlier, LHC

has entered into a precision era with High-Luminosity-LHC (HL-LHC). We will be able to

observe billions of top quarks and Higgs bosons, which will quench our need for statistics in the data. Currently, we observe large uncertainties in the coupling strengths of these

particles, which might be an additional source of CP-violation. The Yukawa corrections to

the couplings of these particles in SMEFTcan be added as higher-order interaction terms in the Lagrangian, L = LSM+Leff =LSM+ 1 ΛL1+ 1 Λ2L2+O(Λ −4 ) + h.c. , Ln = ∑ i CiOi ,

where L1 represents dimension five operators, L2 represents the dimension six operators and

O(Λ−4_{) represents other higher dimensional operators. Λ is nothing but the cut-off scale of}

this new “extended” field theory which supposed to be beyond the TeV scale. Determin- ing the variables in higher order terms with large precision can greatly help us to construct complete models that can describe the nature of the new physics at the Planck scale. Un- derstanding these couplings and their high energy event shapes are essential for constructing

a complete BSM model from a bottom-up approach. For instance, in the SM, top quark’s

chromomagnetic dipole moment (CMDM) arises at the one-loop level and chromoelectric

dipole moment (CEDM) at the three-loop level which also hints new sources ofCP-violation. The impact of these higher dimensional corrections on top quarks, has already been studied

narios in terms of QCD corrections to top quark production [51, 52] as well as electroweak corrections [53, 54]. In addition to the particle mass bounds, these coupling corrections can provide a distinct constraint on new physics scenarios.

Another, very well motivated, approach can be adapted by achieving more general symmetries. As we discussed earlier, the very foundation of theSMstands upon the shoulders of the Noether’s theorem [55] where symmetries in a system ascertain the interactions between particles via the gauge principle. The idea of breaking from a larger gauge structure to the

SMis considered under the title of grand unified theories (GUTs). This breaking mechanism can happen directly or in steps in which leads to the SMgauge structure,

GGUT

MGUT≈1015 TeV

−−−−−−−−−−→ GSM

MEWSB≈1 TeV

−−−−−−−−−→ SU (3)C⊗ U(1)EM .

GUT models provide a framework which can describe the unification of the three nuclear

forces and due to this unification, eliminate the free parameters of the standard model and force them to be dependent on a single variable at the Planck scale.

The SM relies on four-dimensional space-time symmetries where it is invariant under

Lorentz transformations6_{. It is possible to extend the dimensionality of the SMand by doing}

so, extend its Poincar´e symmetry7_{. Such theories are well motivated under Kaluza Klein}

Theory [56] which indicates that the SM originated from a higher dimensional “brane” and

decouples from other dimensions at the low energy scale. Such an approach can provide solutions to the hierarchy problem and unify internal space-time symmetries.

Finally, let us converge this chapter to the topic that we will wrestle within this thesis. Supersymmetry (SUSY) [57, 58] proposes a different way of expanding the space-

time symmetries of the SM. Using the Noether Theorem’s lemma, which indicates that

the temporal component of the current is a conserved entity, one can end up with a simple transformation relation between bosons and fermions as a virtue of spin-statistics. Such theoretical framework allows us to solve several problems of the SM. In the next following chapters, we will mention the motivation behind the supersymmetry and define the model

within the borders of Minimal Supersymmetric Extension of the Standard Model (MSSM),

chapter 2.

This thesis is organised as follows; in chapter 2 we will introduce MSSM and will

discuss how to improve the missing ingredients of the SM and MSSM with an even more

6_{Translation in space-time.}

extended framework. In chapter 3, we will discuss the possibility of extending MSSM field structure via vector-like multiplets. In chapter 4 and chapter 5 we will investigate a gauge

extention of MSSM and how to find its signature at theLHC. In chapter 6 we will discuss

current bounds on the supersymmetric particles and effects of the theoretical uncertainties on these bounds and finally we will conclude in chapter 7.

Minimal Supersymmetric Extention of

the Standard Model (MSSM)

Despite the success of theSM, there is still a lot to unfold in the observed data. In the previous chapter, we saw that there are many issues that theSMcannot explain and it is unlikely that a simple extension will be able to cover all those issues. In this chapter we will introduce the minimal possible supersymmetric extension of the standard model [57, 58, 59, 60, 61]. The reason that we are starting from a minimal framework is to have least amount of free parameters and decide which direction we need to go from here, in more technical words the goal is to fix as many problems as possible using only one extra generator of the symmetry group. In the following, we discuss the motivation behind such framework, section 2.1, and then we will move on with the definition of the model in section 2.2.

2.1

Motivation

The goal of the MSSMis enhancing the space-time symmetries of the SM wihtout changing

its gauge structure. By introducing a symmetry between bosons and fermions, we ensure that the Lagrangian is symmetric under Poincar´e transformations.

The main motivation behind the supersymmetry is that, it can provide a natural solu- tion to the hierarchy problem. As we described with the Fig. 1.2, fermionic loop corrections cause divergences in the Higgs mass. In supersymmetry these quadratic divergences receive opposite sign contributions from the supersymmetric partners of the corresponding particle,

where in this specific case supersymmetric partner of top quark as shown in Fig. 2.1 [62]. ∆m2 h ⊂ ∼ +|Y t|2 8π2Λ2_SM ˜ t h h

Figure 2.1: Bosonic loop corrections to the Higgs boson mass coming from the supersymmetric partner of the top quark, ˜t. The diagram has been created via Jaxodraw package [35].

Furthermore, obtaining a theory that can describe the unification of all three nuclear forces has always been the goal [63, 64]. It is possible to achieve such unification in the

SM by adding extra fields and interaction terms that can modify the renormalization group

equations (RGEs). In MSSM such modification occurs naturally [65, 66, 67, 68, 69, 70, 71]. The RGEs of the gauge couplings, g1, g2 and g3, evolve through energy scales via,

βgi = d d(ln(Q/Q0)) gi = 1 16π2big 3 i .

Here Q is the varying energy scale, Q0 is the relative energy scale which we will take as 1

GeV, bi is a model and gauge dependent constant. Due to the modifications in the value

of this constant in MSSM, we observe an “optimistic” unification in the gauge couplings.

Fig. 2.2 shows the evolution of each gauge coupling through energy scales for both SM and

MSSM which are represented with dashed and solid lines respectively. The red, green and

blue colour codes represent U (1)Y, SU (2)Land SU (3)C gauge couplings respectively. As seen

in the behaviour of the dashed lines they deviate from the unification withO(102_{) TeV scale.}

On the other hand, one can observe the unification of all three gauge couplings for MSSM

at _O(1015_{) TeV which is approximately the order of the Planck scale. Dark red dashed}

lines show the reach of the corresponding experiment where The Large Electron-Positron

Collider (LEP) has been conducted at 100 GeV centre-of-mass energy, LHC Phase I was

at √s = 8 TeV and today LHC is moving towards √s = 14 TeV for Phase III. Although

these LHCenergies are massive achievements, at those energy levels the detectors need to be

extremely sensitive to be able to observe the difference between SM and MSSM. Especially

SU (3)C gauge coupling suffers from very large uncertainties with respect to other couplings

Figure 2.2: Evolution of the inverse gauge couplings with two loop RGEs plotted against the energy scale. Coloured solid lines represent MSSM gauge coupling evolutions with red, green, blue being U (1)Y, SU (2)L and SU (3)C coupling respectively. Coloured dashed lines

represents the SM gauge coupling evolution with the same colour code. Dark red dashed lines show the reach of the experiments with LEP at 100 GeV, LHC phase I reach upto 8 TeV and LHC phase III will run at 14 TeV centre-of-mass energy. Bottom right plot shows 25 times zoomed-in version of the unification point.

that, whilst we observe unification in the gauge couplings in a logarithmic energy scale, when zoomed in one can immediately see that the couplings do not quite unify. This has been shown in the lower right plot of Fig. 2.2. Despite this fact, MSSMcan still achieve a major improvement to the gauge unification.

The SM Lagrangian that we presented earlier contains numerous variables that can

only be fixed via experimental results. In the supersymmetric Lagrangian, on the other hand, terms are defined as interactions as much as possible. This introduces certain limitations and

relations with each term and does not require free parameters as the SM Lagrangian does.

The only downside is that we do not know at which scale supersymmetric particles gain their masses (in other words, where supersymmetry is spontaneously broken). Thus we require

several free variables to define SUSY breaking at a certain scale and the amount of free

variables is model dependent. Introducing a grand unified model can relate these variables

via RGEs and set to a constant at GUTscale.

matter (DM) candidate [72]. The fermionic components of the gauge and Higgs sectors of

the MSSM can mix with each other and generate neutral and charged mass eigenstates.

One of the neutral mass eigenstates has the potential to be the lightest supersymmetric particle (LSP). Due to the conservation rules that we will mention later, a supersymmetric particle can only decay into oneSM, and one supersymmetric particle, beingLSP, constraints kinematic space and makes it stable. Depending on its production rate this particle can easily satisfy observable constraints on DM.

With these motivations in mindsection 2.2briefly describes the theoretical framework

of MSSM and links its properties to the motivations mentioned above.