27 2.1.4 The new heavy fermions

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PHD THESIS

Realistic SM extensions valid up to infinite energy

Giulio Maria Pelaggi

supervised by Prof. Alessandro STRUMIA

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Contents

1 Introduction 5

1.1 SuperSymmetric models . . . . 7

1.2 Strong dynamics . . . . 9

1.3 Relaxion . . . . 11

1.4 New approaches . . . . 13

2 Total Asymptotical Freedom 19 2.1 Trinification at the weak scale . . . . 22

2.1.1 Scalars . . . . 23

2.1.2 Vectors . . . . 25

2.1.3 Fermions . . . . 27

2.1.4 The new heavy fermions . . . . 28

2.1.5 Neutrinos . . . . 30

2.2 Totally Asymptotically Free trinification . . . . 31

2.2.1 TAF models with extra stable fermions . . . . 31

2.2.2 TAF models with extra unstable fermions . . . . 33

2.3 Trinification experimental signals . . . . 35

2.3.1 Gauge bosons . . . . 35

2.3.2 Magnetic monopoles . . . . 39

3 Total Asymptotical Safety 45 3.1 The Litim-Sannino model . . . . 46

3.2 Asymptotically safe models with an Higgs-like scalar . . . . 48

3.2.1 RGE and their fixed points . . . . 49

3.3 Asymptotically safe models with chiral fermions? . . . . 52

3.4 On the lightness of safe scalars . . . . 56

3.4.1 Perturbative effects . . . . 56

3.4.2 Large logarithms resummation, Λ dependence and meaning . . . . 57

3.4.3 Non-perturbative contributions . . . . 59 3

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4 Asymptotically Safe SM extensions 61

4.1 Safe fixed point for gauge couplings . . . . 62

4.2 Minimal asymptotically safe SM extension . . . . 65

4.3 Natural asymptotically safe SM extensions . . . . 68

4.4 Model with dynamical generation of the weak scale . . . . . 71

5 Conclusions 75

A The trinification TAF model with extra Q_Land Q_R 83

B Searches for TAS models 85

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Chapter 1 Introduction

The experimental data at colliders are the fundamental guidelines for theoretical physicists and model builders. In the last years the data agreed widely with the Standard Model (SM) and the last confirm came with the discovery of the Higgs boson: it was a undisputed success.

Another important observation that must be made is the complete absence, up to now, of signals of new physics beyond the SM, aside from statistical fluctuations.

Even if it does a wonderful job in the prediction of the electro-weak phenomena, we know very well the flaws of the SM: it doesn’t explain the neutrino masses, the dark matter and it doesn’t include gravity. These open questions arose because of experimental observations in contrast with the SM, and they lead us to conclude that there is a complete theory that agrees with all the observations, while the SM is just a part of it. Since neutrino masses are negligible, since the DM interacts very weakly with the SM particles, since the gravitational interactions are not seen at colliders, the SM could be considered as an effective field theory, and such phenomena described with additional operators, suppressed by powers of a big unknown energy scale: only above this scale the new physics becomes relevant. It seems that, to shed new light on the features of the complete theory, we have to wait for more data and build colliders that could explore a wider range of energies.

In addition, there is another problem that affects the SM, and it has mostly a theoretical nature: it is the hierarchy problem. The general ques- tion is how to explain the presence of physical phenomena that happen at energy scales far apart. The problem can be described on two levels.

The Planck mass M_{P l} is the energy scale where the gravitational interaction, according to Einstein, becomes strongly coupled. The first issue is:

why has the weak scale to be so much smaller than the Planck mass? In 5

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other words, the big difference in intensity between the weak force and the gravitational force needs an explanation.

The second interpretation has to do with radiative corrections in Quan- tum Field Theory (QFT). The SM Lagrangian presents a potential for the scalar boson that includes a mass term. This parameter is the only dimensional parameter of the theory: if it were null the SM would be classically scale invariant. The radiative corrections to this parameter, considering only the one-loop contributions of the SM particles, take the form

δm²_h

m²_h = 3G_F 4√

2π²

4m²_t

m²_h −2m²_W

m²_h −m²_Z m²_h − 1

Λ² '

Λ

500GeV

2

, (1.1) assuming that all the divergent integrals have been regularized using the same upper limit Λ, called the cut-off scale. It must be noted that the correction is quadratically divergent in Λ.

The definition of Λ needs some clarification: in the formula it is only a mathematical way of encoding a divergence. Physically, in the context of the effective field theory, the cut-off represents the scale above which the new physics is relevant. The big difference with the phenomena considered above is that this time the operator is enhanced and not suppressed by a power of the scale. For example, if Einstein gravity becomes relevant above M_Pl ∼ 10¹⁹GeV, then this scale would be the cut-off given by gravitational interactions, and in this case the correction would be huge.

The direct consequence of 1.1 is the existence of some threshold scale (' 500 GeV in the SM) above which the radiative corrections start to be bigger than the physical value of the parameter. Renormalization theory is several decades old, and it prescribes how the physical values for the parameters can be obtained absorbing the divergent quantities through the redefinition of their bare values. Even in this framework, we observe that the bare values must be chosen with high precision (fine-tuned) to cancel exactly the big corrections, and the ratio ∆ ≡ δm²_h/m²_h can be used to measure the fine-tuning needed to accomodate the experimental value of the Higgs mass.

We call a model natural if the quantum corrections to each of its pa- rameters are not bigger than the parameter itself [1, 2]; this translates, in the case of the Higgs mass, into the condition ∆ . 1. The naturalness/hierarchy problem is one of the central issues of the theoretical particle physics. It’s a very controversial theme because the naturalness could (or could not) be used as a principle that guides the building of new models.

To make the problem clear, let’s consider for example a toy model featuring a scalar S with a mass M between the weak scale and the Planck

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mass. It has a potential

V = −m_h²|H|²+ M²S² + λ_H|H|⁴ + λ_HS|H|²S² (1.2) and we compute the running of the already renormalized mass for the Higgs boson, ignoring for the moment the other SM particles:

m^R_h²(Λ) ∼

(λ_Hm²_hlog_m^Λ

h, if Λ < M

λ_Hm²_hlog_m^Λ

h + λ_HSM²log_M^Λ, if Λ > M. (1.3) At the scale M the parameter gets an additive correction δm²_h ∼ M². This means that the value of the Higgs mass at the weak scale is strongly dependent on the high scale physics. The same happens if the new particle is a fermion with a Yukawa interaction with the Higgs: the scalar mass would receive a correction proportional to the squared mass of the fermion.

Since we worked only with physical quantities, we can say that the hierarchy problem is not related to the renormalization theory. Moreover, the computation of the renormalized Higgs mass is independent on the regula- tor chosen: using the naive cut-off Λ gives the same result of the computation in the dimensional regularization framework, in which the quadratic divergences does not appear at all. In the latter case the logarithmic divergences are expressed as a pole around d = 4, where d is the number of dimensions in which the integral has been made.

Independently on the scheme used for the renormalization or the way in which the integrals are regularized, the farther the scale of the new dynamics is, the more the cancellation needed is precise.

What we learned with this example is that the naturalness problem is strictly related to the concept of scales hierarchy and so to our belief that the low-energy phenomena can be described via effective theories.

Following the above definition, we can state that the SM is not natural:

if the power divergences get a physical meaning, like in the example, the SM can not be decoupled from the high energy physics. In the last decades a lot of models have been proposed with the clear intention to address the hierarchy problem. To complete the SM in a natural way, the new physics must show itself at energies slightly below the TeV scale, and these scales are in the range of the current run of LHC.

1.1 SuperSymmetric models

In the past, a similar problem has occurred about the mass of the electron m_e. If we consider the classical corrections to the energy of a point-like

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charged particle due to the electromagnetic field, we get a linearly divergent contribution:

δm_e ∼ Z ∞

1/Λ

d³r|E|² 2 = 1

2 Z ∞

1/Λ

d³r e 4πr²

2

∼ e²Λ (1.4)

The solution was the introduction of the chiral symmetry in the Quan- tum Electro-Dynamics (QED) theory, that forbids any explicit mass term for the electron at all orders in perturbation theory. The mass is switched back on only when this symmetry is explicitly broken: the quantum correction is proportional to the mass itself and to the logarithm of the scale. In the particular case of the SM, the symmetry is broken by the vacuum expectation value of the Higgs, so the mass of the fermion is again proportional to the Yukawa coupling, to the vev and diverges only logarithmically for dimensional reasons.

The idea of a new symmetry that protects the Higgs mass led to the building of a fairly common family of models featuring Supersymmetry (SUSY): they extend the symmetry group of the theory and introduce new particles, at the weak scale or slightly above, that are supersymmetric partners of the SM particles. In this way, each SM fermion has a bosonic partner and each SM boson has a fermionic partner. The contributions of the new particles to the renormalized mass of the Higgs act like a physical cut-off, because they can cancel the divergences, protecting the low energy value of the Higgs mass. Other than solving the hierarchy problem, these models can give a description of Dark Matter and predict the unification of the gauge interactions.

The simplest implementations of this family of models is the Minimal Supersymmetric Standard Model (MSSM) [3], in which the supersymmetric partners of the SM particles are added: under the new symmetry group quarks and leptons are related to “squarks” and “sleptons”, Higgs doublets to “higgsinos” and the gauge bosons to the “gauginos”. The partners of the charged weak gauge bosons and the charged higgsinos mix into “chargi- nos” while the neutral vector boson partners and the neutral higgsinos mix to give “neutralinos”. Two Higgs doublets H_u and H_d, with vevs v_u and v_d respectively, are required to break spontaneously the electro-weak symmetry, giving mass to both the up and down families of quarks and charged leptons. We define the angle β as

tan β = v_u

v_d (1.5)

and we impose a condition on the vevs such that the scale of this breaking

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is the same of the SM:

v² = v²_u+ v_d² ≈ (246 GeV)² (1.6) Since we don’t see the supersymmetric partners at the same mass of their SM counterpart, the SUSY is not realized exactly. The breaking of this symmetry is described through the introduction of a “soft breaking” potential, that preserves the essential feature of the supersymmetric models:

protecting the Higgs mass from UV divergences. This potential includes the masses for all the scalars and the so called A-terms, that are the trilinear in- teractions between them.

In the SM the biggest contribution to the renormalization of the Higgs mass comes from top quark loops, so the prediction on the mass of its bosonic partner (the “stop”) is fundamental to understand if the supersymmetric model is natural: the finite contribution to the renormalized Higgs mass is proportional to the difference between the stop and the top squared masses.

When A-terms are not considered, the best situation described by MSSM is a stop mass of about 10 TeV, reproduced when tan β is big. In this case the fine tuning is about ∆ ' 1000. This performance could be partially improved allowing sizeable A-terms: the stop masses could be reduced at about 1 TeV and the fine-tuning reaches values of order ∆ ' 100.

The Next-to-Minimal SuperSymmetric Standard Model (NMSSM) [4]

introduces a new complex scalar, that allows us to avoid explicit mass terms for the scalar doublets. This new particle gives a contribution to the Higgs masses such that the mass of the stops can be lighter, leaving the model natural. In this implementation the fine-tuning can reach values of order

∆ ' 100[5].

Since the fine-tuning was still far from unity, some extensions of these models have been proposed. They become very complex soon, because the aim is to alleviate the fine tuning while keeping SUSY hidden at the colliders [6, 7].

1.2 Strong dynamics

Another idea to explain this separation of scales between the Planck scale and the electro-weak one is to introduce a new symmetry group to the SM:

if the new group gives confinement, such that the new interaction becomes non-perturbative at low energy, the electro-weak scale can be generated without the Higgs boson.

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This phenomenon is well known in the Quantum Chromo-Dynamics (QCD) sector of the SM, where the gauge coupling of the SU(3)_cgroup goes to zero at high energy, but becomes non-perturbative at the scale Λ_QCD ' 200MeV. In QCD the approximate global symmetry SU(3)L⊗ SU(3)_R, called chiral symmetry, is broken to SU(3)_V by the confinement and the mesons (pions, kaons. . . ) are the Nambu-Goldstone bosons related to this breaking pattern. They are massless as long as the electro-weak interactions and the quark masses are switched off. Even in the SM, the vacuum of the QCD theory breaks the electro-weak gauge symmetry and the pions become the longitudinal degrees of freedom of the W and Z and so, through the Higgs mechanism, they give a contribution to the gauge bosons masses.

In the same fashion of the colour behaviour, Technicolor [8–10] models introduce a new SU(N )T C gauge group, and new fermions called techni- quarks: similarly to quarks, they transform under a new global symmetry, analogous to the chiral symmetry of QCD. The weak scale is generated by the breaking of the new chiral symmetry by confinement at low energy: the Goldstone bosons of this pattern are called technipions. Following the Higgs mechanism, combinations of pions and technipions become the longitudinal degrees of freedom of the gauge bosons. The advantage of these models is that the elementary scalars are not needed: the electro-weak scale is generated dynamically, so the naturalness-hierarchy problem is solved, but in their simplest implementations they cannot accomodate easily the data about the electro-weak precision tests, the Flavor-Changing-Neutral- Currents (FCNCs) and the Top quark mass [11, 12]. Obviously, all the sca- larless models of this family have been disproved by the discovery of the Higgs.

Someone has proposed a new type of models halfway through Techni- color and Higgs models, called Composite-Higgs models. This idea arises on the footsteps of an analogous problem: the difference in mass between the charged and the neutral pion due to the electromagnetic interactions was power divergent. In that case, the divergence was cured assuming that the pions are composite, and so they are not a degree of freedom at high energy.

In the simplest implementations of the Composite-Higgs models [13],the Higgs is described as a bound state made of techniquarks: at high energy it is not a fundamental particle, and so it does not receive any correction above the compositeness scale. In QCD the same thing happens, given that the physics of the pions is completely decoupled from the high energy physics of the quarks. This construction usually implies also the presence of a tower of resonances slightly above this compositeness scale. However, there are no traces of these “technihadrons” at the colliders, so a way to

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explain the lightness of the Higgs with respect to the other bound states must be found.

More recent descriptions [14–16] follow this idea: the Composite-Higgs is naturally light since it is a pseudo-Goldstone boson of the new chiral symmetry and it acquires mass only because this symmetry is explicitly broken by the interactions with the other non-composite particles of the SM. The Higgs potential, in facts, is generated by loops of fermions and gauge bosons: the breaking of the electro-weak symmetry happens through a Coleman-Weinberg mechanism, and so again at a dynamically generated scale.

Various predictions of this kind of models can be done without caring about the microscopic realization of the symmetry in terms of technipar- ticles. In other words, one can build effective theories to study which (spontaneous and explicit) symmetry breaking patterns provide a pseudo- Goldstone boson with the same quantum numbers of the SM Higgs.

Writing down a Composite-Higgs model that describes consistently the electro-weak interactions between the Higgs and the Goldstone bosons at a microscopic level is not very difficult, and has been done since the first articles, starting from a QCD-like theory [14–16]. The really challenging part is to introduce in this context also the SM fermions, as composite states if needed, and reproduce their masses and interactions at low energy.

For this purpose, the models that feature partial compositeness have been introduced [17], in which the mass hierarchy among the ordinary fermions is explained through their mixing with the new “technibaryons”.

In the experiments the Higgs appears to be an elementary particle and this case is described in Composite-Higgs models in the limit of high confinement scale: this means that the new strong sector must almost decouple from the SM sector. In general this interaction is not forbidden and so nothing forces its effects to be small: if more complex structures are not introduced, a certain amount of tuning has to be done to guarantee the needed cancellations. The introduction of a new confinement scale at ∼ 1 TeV (that would ensure a fine-tuning ∆ ' 10), even if it is much smaller than the Planck mass, reproposes a new hierarchy problem, and the separation between this scale and the weak scale has to be motivated [18, 19].

1.3 Relaxion

Experiments at colliders restrict very much the parameter space of the models that introduce new particles to solve the naturalness-hierarchy problem.

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Recently the authors of [20] tried to build models where cosmological dynamics gives a motivation for the Higgs mass being much smaller than the Planck scale.

The first model of this type relies on the dynamics of an axion, a field that has been introduced for the first time in [21] to solve the strong CP problem. Briefly, the coefficient θ of the QCD Lagrangian term that viola- tes CP is not protected by a symmetry, but the resulting CP violations are not observed: a mechanism that preserve its smallness must be provided.

Peccei and Quinn proposed the promotion of the θ parameter to a scalar field, demonstrating that its potential has a minimum close to zero.

The axion of [20] has a periodic potential, invariant under a discrete shift symmetry; the interaction with the Higgs boson is assumed to break softly this symmetry. The new field, during the inflation phase of the history of the universe, rolls slowly down its potential: during the inflation the expansion dissipates the energy of the field. Because of this effect, called Hubble friction, the starting point of the rolling does not affect the result.

The Higgs mass depends on the Higgs-axion interaction: the Lagrangian has a term (−M² + gφ)|h|², where φ is the axion field, g is a dimensionful coupling and M is the energy scale where all the SM divergent contributions to the Higgs mass are cut off.

The slow-roll implies that the system scans all the values for the Higgs mass parameter. At some point in time the Higgs mass goes negative: the amplitude of the oscillating potential, proportional to the Higgs vacuum expectation value, becomes too high to be rolled over, and the relaxion field is trapped in one of the local minima of its potential, stopping the evolution of the system. Thanks to this process of relaxation of the axion, hence the name relaxion, the vev of the Higgs is naturally small and it triggers the usual spontaneous electro-weak symmetry breaking.

The ideal relaxion model would accomodate a renormalized Higgs mass where the SM loop integrals have been performed up to M_Pl. In this way the hierarchy problem would be solved, and the separation of scales justified.

This is not the case in the simplest implementation of these models, in which the particle content of the relaxion model differs from the SM only for the addition of the axion. To ensure the right behaviour of the relaxion during the inflation phase, some conditions must be imposed on the Hubble parameter, and so the value of the cut-off scale M is constrained to be far less than M_Pl. As a consequence, there should be some unknown new physics that takes care of the divergences well before the Planck scale: in other words, the relaxion models can not give a description of the high energy physics, and they are still effective theories.

The simplest model, in which the axion is exactly the QCD axion, fails

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to solve the strong CP problem because the vev of the axion is exactly the θ parameter, and there is no mechanism to guarantee its smallness. A specific dynamic for the axion should be designed, such that the minimum of the potential is not far from the origin: the cut-off for these models is estimated to be M ≈ 10³TeV.

Another idea is to introduce a new strong sector that avoids the QCD bounds, whose axion will play the role of the relaxion. In this case the cut-off could be raised up to M ≈ 10⁸TeV.

Despite the claim that this kind of models can solve the hierarchy problem, they still don’t explain the smallness of the Cosmological Constant (CC). In fact, different patches of universe that are not in causal contact end up having different values for the CC, not necessarily small.

Moreover, to keep the rolling classical and slow, letting the relaxion stop in the first minimum, the model needs a very small Hubble parameter during the inflation phase and a huge amount of e-folds. This constraints imply a very small tilt of the inflaton potential, that makes difficult to reproduce the data of the spectral index of the scalar fluctuations of the Cosmic Microwave Background (CMB). Some attempts to accomodate the cosmo- logy of the model has been done for example in [22], where a second inflation phase is introduced, and in [23], where the relaxation is explained with two slow-rolling fields.

1.4 New approaches

The Large Hadron Collider (LHC) data at √

s = 13TeV confirm the Stan- dard Model and give strong bounds on Supersymmetry, on composite Higgs and on other SM extensions that introduce new particles to tame the quadratically divergent corrections to the Higgs mass in a natural way. This unsettling situation calls for reconsidering the issue of naturalness to approach it differently.

The concept of “anthropic solution” to the naturalness problem means that an observer can develop only when the parameters of the theory take values that allow life. This idea came out in [24] for the first time to put an upper bound on the value of the CC: if this value was too big, no gravitati- onally bound system would have existed. The anthropic principle has been used again in [25] in the context of the electro-weak symmetry breaking, where the consequences of different values of the Higgs mass parameter are analyzed. They point out the bounds to maintain the stability of nuclei and their formation during the primordial and the stellar nucleosynthesis.

In other words, if the CC and the Higgs vev had taken different values we

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could not have known, because we would not have existed at all. Moreover, some models can be elaborated around this idea, such that there is more than one universe (the so-called “multiverse”) and each of them lies in a different vacuum of the theory: we landed in a non-natural, hierarchically-ill one. The multiverse framework is based on accepting the “un-naturalness”

of the SM, at least until a natural model will be discovered: the main problem of these theories is that we can not think an experiment that could test them.

The authors of [26] introduced the concept of finite naturalness, in which the uncomputable power divergent contributions to the Higgs mass must be ignored. These divergences are not physical, meaning that they don’t produce any observable effect, in facts they would disappear depen- ding of the regularization scheme used. Since only the physical part of the correction is considered, the running of the dimensional parameter is at most logarithmic: it is shown that the SM is natural in this context because the vector bosons and the top quark are not much heavier than the Higgs boson, so the radiative corrections are under control: in the MS scheme they found that mh(¯µ = M_Pl) = 141.1GeV, that means a small fine tuning δm²_h/m²_h . 0.2.

Some common extensions to the SM motivated by observations, like the description of the Dark Matter and the mechanism to give mass to the neutrino have been analyzed: they found that the bounds to keep the model finite-natural are much weaker than those that guarantee the naturalness as defined usually. The values of the parameters needed to describe the measured Dark Matter cosmological abundance and the observed neutrino masses are compatible with a finite-natural model.

It must be noted that the naturalness problem shows up whenever we introduce some new physics at a given scale much bigger than the weak scale, and this could happens also when we take in account the gravitational interactions: they play a role at M_Pl and could possibly cut off the non-physical divergence, giving birth to a physical and finite, but huge contribution δm²_h ∼ M_Pl². Nevertheless, our ignorance of the quantum nature of the gravitational interactions lets us elaborate some models (see for example [27]) in which these interactions are suppressed by new physics at a scale lower than the Planck mass, such that the corrections to the Higgs mass due to gravity are negligible. All the models that we will consider have been studied in this framework, called soft gravity, where we are al- lowed to retain only the SM sector of the theory and extrapolate safely its behaviour also at energies even higher than the Planck mass.

As we said above, the Higgs propagator Π(q²)at zero momentum q = 0

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receives a quadratically divergent correction, which is often interpreted as a large correction to the Higgs mass. Writing only the top Yukawa one-loop contribution, one has

Π(0) = −12y_t²1 i

Z d⁴k (2π)⁴

k²+ m²_t

(k²− m²_t)² + · · · (1.7) The photon too receives at zero momentum a quadratically divergent correction. In QED one has

Πµν(0) = −4e²1 i

Z d⁴k (2π)⁴

2k_µk_ν

(k² − m²_e)² − η_µν k²− m²_e

. (1.8)

This is not interpreted as a large photon mass because it is presumed that some unknown physical cut-off regulates the divergences while respecting gauge invariance, that forces the photon to be massless. Similarly, the graviton propagator receives a quadratically divergent correction Πµν,ρσ(0):

in part it can be interpreted as a cosmological constant, in part it breaks reparametrization invariance.

The fate of the Higgs mass is not clear. Some regulators (such as dimensional regularization) respect all these symmetries and get rid of all power divergences, including the one that affects the Higgs mass. Other regulators (such as Pauli-Villars and presumably string theory) do not generate a photon mass nor a graviton mass but generate a large Higgs mass, given that it is only protected by scale invariance, which is not a symmetry of the full theory.

Starting from these considerations, a similar approach that addresses the hierarchy problem has been studied in [27], where the absence of dimensionful parameters is assumed as a principle: the scale invariance is an accidental symmetry of the theory at tree level. The SM could be part of a theory valid up to infinite energy, such that no physical cut-off exists.

Then, once that eq. (1.8) is interpreted to mean zero, the same divergence in eq. (1.7) must be interpreted in the same way. Furthermore, in a theory with dimension-less parameters only, one can argue that R d⁴k/k² = 0 by dimensional analysis: the power divergences do not exist because at tree level the model has no scales, while the Planck and the weak scales are generated dynamically through radiative corrections.

In this context of soft gravity and classical scale invariance, one possibility, introduced in [28] and analyzed in detail in Chapter 2, is devising realistic weak-scale extensions of the SM such that all gauge, Yukawa, and quartic couplings flow to zero at infinite energy: this behaviour is called

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Total Asymptotic Freedom. Since the hypercharge gauge coupling gY of the SM becomes non-perturbative at Λ ' 10⁴³GeV, hitting a Landau pole, the SM must be extended around the weak scale into a theory without abelian U(1) factors. The specific hypercharges of SM fermions suggest two possibilities [28, 29]:

Pati-Salam: G₂₂₄= SU(2)_L⊗ SU(2)_R⊗ SU(4)_PS Trinification: G333= SU(3)L⊗ SU(3)R⊗ SU(3)c

. (1.9) Asymptotically free gauge couplings are only the first step: all Yukawa and quartic couplings must also satisfy the TAF conditions described in [28], where a Pati-Salam TAF model was found. However the TAF conditions for the quartics did not allow to realize Pati-Salam models that avoid quark- lepton unification. As a consequence, in the TAF Pati-Salam model, flavor bounds force the masses of gauge vectors of SU(4)PS/ SU(3)cto be heavier than 100 TeV, which is unnaturally above the weak scale.

Trinification [30–38] does not predict quark-lepton unification and so it is safer than Pati-Salam from the point of view of flavour bounds. Thereby trinification could give rise to simple natural TAF models. However [28] did not find any realistic trinification model that satisfies the TAF conditions.

Usually Trinification models are introduced as Grand Unification The- ories (GUT) in which the three gauge coupling are the same because of a permutation symmetry between the groups. In this analysis this constraint is not considered, so they get three independent values. Trinification features 27 fermions for each generation, divided in three bi-triplets: other than the SM fermions there are also a vector-like lepton doublet, a vector-like down quark and two singlets. Two scalars are needed to reproduce the right pattern for symmetry breaking and the fermion masses. Some new vectors with a mass at the weak scale are predicted: they could in principle be observed at the present colliders.

The other possibility is that the SM itself might be asymptotically safe.

It is not known what the Landau pole of the hypercharge means: it might mean that the SM is not a complete theory and new physics is needed at lower energy. Otherwise gY and other couplings might run up to constant non-perturbative values as illustrated in figure 1.1, such that the SM enters into an asymptotically safe phase. In fact, this possibility was envisioned very early on in the literature [39, 40] triggering lattice studies [41–43]

as well as non-perturbative analytic studies such as the one of [44]. It is fair to say, however, that the fate of the SM depends on non-perturbative effects which are presently unknown; see [45–50] for attempts to compute the non-perturbative region and for related ideas.

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��^-�

��^�

�� μ ��

��

�� ��� ��� λ �� τ

��

|λ|

��

�τ

Figure 1.1: Illustration of a possible RGE running in the SM. We assumed central values for all parameters, and solved the 3 loop RGE equations. In order to obtain an asymptotically safe behaviour we artificially removed the bottom and tau Yukawa contributions to the 3-loop term in the RGE. This only affects the running in the non-perturbative region above 10⁴⁰GeV, where the result cannot of course be trusted. Furthermore we ignored the Yukawa couplings of the lighter generations, and gravity.

The authors of [51] assert that the correction to the Higgs mass is proportional to the scale Λ where the Landau pole is, even if the behaviour of the running of the coupling constant is safe and under control: the contribution would be proportional to the scale where asymptotic safety kicks in.

If we trust this argument, no natural model could be built.

Later, Litim and Sannino (LS) [52] presented the first four-dimensional example of a perturbative quantum field theory where all couplings that are small at low energy flow to a constant value at higher energy per- sisting up to infinite energy. This model involves a gauge group SU(Nc) with large N_c, a neutral scalar S and vector-like charged fermions, with asymptotically safe Yukawa couplings and scalar quartics. The model reali- zes Total Asymptotic Safety (TAS). Another equally relevant property of the model is that without the scalar it cannot be perturbatively safe [52, 53].

Scalars are required to dynamically render the theory fundamental at all scales without invoking Supersymmetry, which would keep scalars massless independently of their dynamics. In facts Supersymmetry makes it harder to realize an asymptotically safe scenario [54, 55] both perturbatively and non-perturbatively. Furthermore the LS model, on the line of physics, connects two fixed points, a non-interacting infrared free one (the

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theory at low energy is non-abelian QED-like) to an interacting ultraviolet fixed point. Remarkably the model shares the SM backbone since it features gauge, fermion and scalar degrees of freedom, albeit it still misses a gauged Higgs-like state. In Chapter 3, we will consider some extensions of the LS model featuring an Higgs-like scalar and chiral fermions to make the model more similar to the SM, analyzing the conditions to maintain the TAS property.

Another starting point to investigate the possibility of a realistic TAS model is the Standard Model itself. Recently, in [56–58] it has been discussed the contributions by N_F extra vector-like fermions to the RGE of the gauge couplings of the model: the resummation of such contributions at the le- ading order in the parameter 1/N_F is known at all orders in perturbation theory. This terms possess a pole for some finite value of the couplings. As a consequence, the sum of the contributions by the SM particles and by the new fermions to the gauge RGE could provide a fixed point for the gauge couplings, and this happens in the proximity of the pole given by the new sector.

In such way the hypercharge coupling of the SM is no more doomed by a Landau pole, without necessarily embedding it in a non-abelian group.

Once the fixed point for the three gauge coupling has been found, the aim of our work is to find a fixed point also for the Yukawa and the quartic coupling, studying if they could be reached in the UV regime starting from realistic values of the SM parameters at low energy. In particular, even in presence of fixed points, it is important that the quartic coupling of the SM Higgs flows toward a positive value to ensure the vacuum stability of the theory. The details of this work are reported in Chapter 4.

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Chapter 2

Total Asymptotical Freedom

In [28] is described the first attempt to find a four-dimensional Quantum Field Theory that extends the SM without any degrees of freedom much heavier than the weak scale, such that the theory remains finite-natural.

Here they assume that the model is in the context of soft gravity, mea- ning that gravity is always a weak force and so it does not invalidate the results of the QFT at high energies. The authors do not care about the description of the mechanism that damps the gravitational interactions, but they consider only the behaviour of the observable sector, that must hold up to infinite energy.

All the couplings, that are adimensional, must flow to zero at high energy following the Renormalization Group Equations (RGE), without ex- iting the perturbative regime at any scale. A model with this behaviour has been referred as Totally Asymptotically Free (TAF).

To understand if a model shows the TAF behaviour a general procedure must be developed. The starting point is the RGE of the couplings: since they become small as they approaches zero at high energy, it is enough to compute the equations stopping the perturbative expansion at one loop.

In general, at one loop order there are no interacting fixed points at infinite energy: a coupling can either flow to zero or get a Landau pole at some scale.

We can say rather quickly that the SM as we know is not TAF: a general expression of the RGE at one loop for a model with one gauge coupling is

β_g⁽¹⁾ = bg³, with β_g = dg

d ln(µ/µ0) =X

n

βg⁽ⁿ⁾

(4π)²ⁿ (2.1) where µ is the energy scale of the renormalization and µ₀ can be identified

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with the infrared scale. This equation can be solved analytically:

1 g² ∼ 1

g₀² − b ln(µ/µ₀), (2.2) and g0 is the value of the coupling in the infrared. The theory is asymptotically free if b < 0, otherwise the coupling hits a Landau pole.

Whenever the gauge group is abelian the coefficient of the g³ term is positive: in the case of the SM hypercharge it is b = 41/6. The problem of understanding what happens at the SM at the pole is often put aside, because the pole is located at an energy around 10⁴³GeV, much higher than M_Pl. Again, this argument makes sense only supposing that gravity is weak at all the scales and it cannot modify substantially the divergent behaviour of the coupling, so in the search for fixed point the full running of the couplings up to infinite energy must be taken in account.

The RGE become more and more complicated when the gauge group of the model is bigger and when an increasing number of Yukawa and quartic couplings are introduced. It becomes soon impossible to find the fixed points of the running parameter through the analytic resolution of a system of differential equations, so a different procedure must be built.

Let’s start from a QFT with only adimensional couplings: the gauge couplings gi, the Yukawa couplings ya and the scalar couplings λm. We define t = ln(µ²/µ²₀)/(4π)² and we rescale all the coupling by a power of t:

g²_i = ˜g_i²

t y_a² = y˜_a²

t λ_m =

λ˜_m

t . (2.3)

In this way the logarithmic running of the couplings, which is the lea- ding behaviour in the UV, is factorized out. We get

d˜g_i d ln t = ˜g_i

2 + β_g_i(˜g) d˜y_a d ln t = y˜_a

2 + β_y_a(˜g, ˜y) d˜λ_m

d ln t = ˜λ_m+ βλ˜m(˜g, ˜y, ˜λ).

(2.4) Computing the β function of a new rescaled coupling, we can get an expression without an explicit dependence on t that contains the β of the original coupling. This happens only when the RGE are computed at one loop, because the couplings have the right power: the β_g is cubic in g, the β_y is cubic in g and y, the β_λ is quadratic in g², y², λ.

Now we can describe all the RGE with the equation dx_I

d ln t = V_I(x), where x_I ≡ {˜g_i, ˜y_a, ˜λ_m} (2.5)

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that can be intended as the running of the point xI in the space of the parameters.

Our aim is to find the asymptotic behaviour in which all the couplings run to zero at high energy. To do this it is sufficient to find the fixed points for the running of the rescaled couplings, imposing

dx_I d ln t

_x

∞

= VI(x∞) = 0. (2.6)

In each point x∞, the original couplings run keeping their ratios fixed: we call this behaviour fixed flow. Briefly, if a model possesses at least one fixed flow, it is TAF: all the couplings tend to zero at least as 1/t.

In [28] the case of the SM has been studied in detail: it has been found that the SM actually has some fixed flows in the space of the parameters.

This particular points, however, do not take physical values: to avoid the Landau pole, the hypercharge coupling must be set equal to zero and the masses of the particles are predicted to be M_τ = M_ν_τ = 0, M_t = 186GeV and M_h = 163GeV.

The fixed flows can be characterized on their behaviour in the limit of high energy. In particular, a flow can be UV-attractive (IR-repulsive) if all the points in the space of the rescaled parameters in the vicinity of the fixed point get closer to it in the ultraviolet limit. Otherwise, if they approach the fixed point in the infrared limit, the fixed flow is defined IR-attractive (UV-repulsive). To determine analytically if the fixed flow is UV-attractive or repulsive the Jacobian matrix

M_IJ = ∂V_I

∂x_J x=x∞

(2.7) should be considered. If all the eigenvalues of this matrix are negative, the fixed flow is fully UV-attractive. If they are all positive, it is fully UV- repulsive. In the most common cases the eigenvalues don’t have all the same sign: each positive eigenvalue represents a prediction at low energy for a coupling or for a combination of them. Indeed, a positive eigenvalues indicates that there is a neighborhood near the fixed flow in the far UV that correspond to the same point in the space of the rescaled parameters when they are tracked back to the IR.

It should be noted that the one loop beta functions of the gauge couplings depend only on the gauge parameters themselves, the beta functions of the Yukawa couplings depend on the gauge and the Yukawa couplings, while the beta functions of the quartic couplings depend on all the couplings. Exploiting this property, the search for fixed flows can be done

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step by step: one considers the behaviour of the gauge couplings first. If it shows some fixed flows, one goes further to the Yukawa couplings, and does the same analysis. If also the Yukawa couplings show fixed flows, one can proceed to study the RGE for the quartic couplings.

When the models get complicate, it becomes difficult to understand the relation between the particles and the couplings that are present in the model and the specific analytic expression for the system of equations to get the fixed flows. The only possibility is to perform an extensive brute- force scan, so we developed a code that, given the gauge group and the field content, finds the Yukawa and quartic couplings, computes their one- loop RGE and checks if it admits TAF solutions.

In our analysis we can avoid the discussion about the running of the mass parameters: even if they are non-null, they are irrelevant for the dynamics at high energies, so they don’t influence the running of the other couplings. Admitting non-zero mass parameters for the scalars does not automatically trigger a hierarchy problem: if we have only one mass, the power divergences are discarded in the context of finite naturalness. The only case in which the hierarchy problem arises is when there is more than one scale, and they get very different values: as expected, in this case a light scalar would get a one-loop contribution to its mass for each particle heavier than it, proportional to the squared mass of the heavy particle.

In section 2.1 we will analyze the particle content of the minimal trinification model at the weak scale. In section 2.2 we will describe the search of a model with the TAF behaviour while adding fermions in different re- presentations to the minimal model. In section 2.3 we will consider some possible signals of this theory that could be measured in the next period of activity at the LHC and some experimental bounds for the parameter space of the theory.

2.1 Trinification at the weak scale

We will focus our searches on the trinification models, whose gauge group is G333 = SU(3)L⊗ SU(3)R⊗ SU(3)c. Our choice is motivated by the fact that such group contain the SM gauge group and could reproduce the SM fermion quantum numbers, introducing new fermions and gauge bosons.

Table 2.1 summarizes the field content of minimal trinification. Such model has been studied in [28] where the authors found some TAF solutions in the configuration with only one Higgs multiplet. In the next sections we will explain why more than one generation of scalars is needed to get a realistic and natural model.

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u_L uL

u_L

dL

d_R d_R d_R

uR

d_R d_R

d_R

d_R d_R

d_R Ν_L

e_L

e_R e'L

Ν'_L

Ν_R eL

Ν_L Ν' SUH3L_L SUH3L_R

SUH3Lc

Figure 2.1: The chiral fermion multiplets under the gauge group of the Trini- fication models.

Trinification models are not the only choice to extend the SM: for example in [28] the Pati Salam SU(2)_L⊗ SU(2)_R⊗ SU(4)_chas been considered.

Such model, however, suffers of leptons/quarks mixings, so it is subjected to very stringent bounds on the mass of its gauge bosons. For this reason we did not consider this possibility, focusing on the trinification family.

2.1.1 Scalars

One bi-triplet scalar H in the (3_L, ¯3_R)representation contains 3 Higgs doublets. At least two bi-triplets, H₁ and H₂, are needed in order to break G₃₃₃ to the SM gauge group: the first breaks G333 to G2213 = SU(2)L⊗ SU(2)R⊗ U(1)_B−L⊗ SU(3)_c, while the second breaks also the left-right symmetry.

Indeed, the most generic vacuum expectation values that give the desired pattern of symmetry breaking are

hH_ni =





v_un 0 0 0 v_dn v_Ln 0 V_Rn V_n



. (2.8)