Ricerca di modelli con Liberta Asintotica Totale

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UNIVERSITÀ DEGLI STUDI DI PISA

Facoltà di Scienze Matematiche, Fisiche e Naturali

Corso di Laurea Magistrale in Fisica Teorica

TESI DI LAUREA MAGISTRALE

SEARCH FOR MODELS WITH TOTAL

ASYMPTOTIC FREEDOM

Relatore:

Alessandro Strumia

Candidato:

Saverio Vignali

Anno Accademico

2014/2015

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1. Introduction

The great success enjoyed by the Standard Model is due to its extraordinary predictivity and the exactness of its results. In the energy range explored until today, SM predictions have been revealed exact at every order of experimental precision, and to date results all seem to disfavor the existence of new particles up to 8TeV of beam energy, which means the lower limit for their mass is from 100GeV to 1TeV, depending on the particle.

However a number of problems exist along with some unsatisfactory explanations, that all seem to point to there being something new at energies not far from those that LHC can reach

One such problem is the so called naturalness issue of the Higgs's mass, and how its value is much lower than what loop calculations might suggest.

The Higgs is a complex scalar particle with an interaction Lagrangian density of the type

VH=mH2 H2+λHH4

The quartic interaction is such that it allows spontaneous breaking, giving mass to most of the SM particles.

Spontaneous breaking (of a symmetry) happens when the vacuum expectation value (vev) of a field is not invariant under all or part of the gauge group.

There are many consequences of this, but for the SM, the most relevant to the discussion is the appearance in the Lagrangian of an explicit mass term for some of the particle that interact with the Higgs field, each as the low-energy part of the interaction term; such mass terms could not be ordinarily written, because they violate invariance under the electroweak gauge group

U (1)_Y×SU (2)_L, but the simmetry is restored by the remaining part of each interaction, so that the theory as a whole is still invariant.

For example, let's take the gauge-Higgs interaction coming from the kinetic term; the Higgs field is an SU (2)_Ldoublet with a vev in its lower component: thus separaring the field H =H₁+H₀, and focusing only on the terms quadratic in H₀

DμH D μ H → DμH0D μ H0(1.1) with H0=eaiTθa

(

0 v

)

we obtain

∣

g₂Wμ +g_YYμ

(

0 v

)

∣

2 =g2 2_v2 2 (W+ 2 +W_-2)+(W₃μ Yμ )

(

g2 2 −g2gY −g2gY gY 2

)

(

W₃μ Yμ

)

(1.2)

By diagonalizing the last term through a rotation of angle θ_w defined as: tan θ_W=gY

g2

(1.3) and calling Z the nonzero mass eigenstate, we obtain

g22v2 2 (W+ 2 +W_-2)++g2 2 +gY2 2 Z 2 (1.4)

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that is, through symmetry breaking 3 of the 4 electroweak gauge boson obtain mass.

From cross referencing the experimental values in which the Higgs vev is involved, we obtain a value for v_H of 246GeV, while the m_H was measured to be 125GeV.

In the SM, the Higgs is coupled to all of the particles proportionally to their mass, and such interactions weigh heavily on the renormalized value of its mass.

To quantify the phenomenon,mH

2 _{corrections are of the kind} δmH

2

∼g2E2(g is the generic coupling of the particle with the Higgs); restricting ourselves to the SM, the heaviest contribution comes from the top quark interaction, which gives δ m2H=(12/4 π2)yt2E2.

Such a contribution makes m_H drifts away from its naked value very rapidly (for comparison the typical running of a lagrangian parameter is δ g∼ln E ), and nonethelessm_H is in the neighbourhood of 100GeV. What is the explanation of such phenomenon?

Algebrically speaking, the matter is quickly settled: let Λ be the energy scale at which the SM loses predictivity (that is, taken as an effective theory), then the overall correction to the Higgs's mass becomes δ m2H∼k2Λ2. To make this correction small, one needs only to make k very small.

However this implies very precise and unnatural-looking cancellations between contributions coming from different particles. To put it in numbers, suppose the SM is indeed the final theory of everything, and as such we can extrapolate it to the Planck scale, so that Λ=Λ_Pl. Then the required cancellations inside k should be a part in1032_{. That is, numbers of order}₁₀32_{should sum}

algebrically to give a results of order100 .

This incredible fine-tuning, even if possible in principle, appears so unreasonable and fortuitous that it's worth studying if a better, more natural-looking mechanism exists that would produce such results.

Reasoning backwards, if one would like to leavek around its SM value of ∼10−2, one would need to have new particles around the TeV scale, that would modifym_H 's running introducing some sort of cutoff around their mass's energy scale. To date, no such particle has been found up to 8TeV. Worse still, new particles that should exist to solve "hard" problems of the SM (strong CP violation, baryogenesis, inflation) would couple strongly with the Higgs, worsening the issue with its naturaleness.

The only hope is then that either 1) new particles exist below the weak scale, but a good reason exists for why they haven't been found yet or 2) such particles are massive, but weakly coupled with the Higgs.

Another source of problems is gravity. Gravitational interaction with Higgs give correction of the type δ m2H∼l GNΛG4 , where GN is the Newton constant and in l we put all the loop and interaction

factors. Because gravity couples derivatively, its vertices depend directly on momentum and as such

lgrows very rapidly with energy. Even worse, if the gravitational cutoff is the "intuitive" one around M_Pl, one can easily see how gravitational corrections quickly become huge.

Both of these problems would be easily solved by the above "natural cutoff" provided by the new particles, which symmetry arguments put around 1011_GeV.

Models that satisfy both these conditions (UV gravity cutoff and new heavy particles weakly coupled with the Higgs) are called Soft-Gravity Models.

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Soft Gravity models spontaneously divide the energy range in three types of sectors:

1) An observable sector, containing all of the SM and eventual expansions around the weak scale. 2) A gravitational sector around the Planck scale. Thanks to the UV cutoff, this sector is effectively decoupled from all the others at all energies, because gravitational couplings are capped at

Λ_G/M_PL, which using the cutoff value given before makes ∼10−7

3) One or more UV sectors above the TeV scale, containings all the necessary particles to explain SM's hard problems (Dark Matter, neutrino oscillations, axions, inflatons...). In the Soft-Gravity hypotesis, this sector too is decoupled from the observable one.

In summary, every Soft Gravity theory is comprised of at least one completely observable sector, whose energies are within the reach of the LHC, and as such can give falsifiable predictions.

There is one last issue of Soft Gravity theories, which stems directly from the requirement of weak gravitational coupling: all coupling constants hit Landau poles in the far UV. For other theories, this is a non-issue: a pole signals the energy above which the perturbative approach of the model loses predictivity, so it becomes meaningless to speculate on its "real" behaviour. Plus, all of these poles are situated above M_Pl , so it's implicitly assumed that it's wiser to delay facing the problem until a good model for quantum gravity is developed.

However, Soft Gravity models have no such luxury: gravity is weak at every scale, and as such it will never be able to fix any of the Landau poles. Worse still, poles are expected to be a sign of new physics, in the form of new particles with mass near said pole, which only worsens the naturalness problem for scalars that share with them a non-banal representation of a gauge group.

One way to sidestep this problem is to force all the coupling constants to be free of Landau poles, and flow to zero in the far UV. We call such property Total Asymptotic Freedom, and these two conditions TAF-1 (freedom from Landau Poles) and TAF-2 (the coupling flows to zero towards infinity).

In the following chapters we will present the general method with which to investigate the conditions under which a given model might achieve TAF, starting from how to calculate the RGE of a model (Chapter 2.1), how to inspect the UV behaviour of the couplings in theory (Chapter 2.2),, and in practice (Chapter 2.3),. We will then present a toy model for review (Chapter 2.4), and inspect the Standard Model and the Trinification GUT (Chapter 3.1 and 3.2),, and we will end by presenting a Trinification extension that satisfies TAF(Chapter 3.3 and 3.4), and make a preliminary review of its mass spectrum (Chapter 3.5),.

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2. Theoretical and practical aspect of TAF search

2.1 Renormalization and general form of the RGE

We begin by descrbing the mathematical procedure to find out if and how a given model satisfies the TAF conditions.

The first step in describing a theory is defining the gauge group G , which determines the gauge bosons of the model. In the following we restrict ourselves to the case of G being either SU ( N ) or

U (N ) , but the extension to the case of G being the tensor product of such groups is trivial.

Then we decide on the matter content, detailing the number of fermion and scalar fields and their representation under G . The general interaction Lagrangian density for such a model is the following: L=−1 4 Fμ ν A F_{μ ν}A+1 2DμϕaDμϕa+i ̄ψjσμDμψj−(i Yajkψ̄jσ2ψkϕa+hc)− 1 4 ! Λabcdϕaϕbϕcϕd +∂μ̄cA∂μc A +fABC∂μ̄cAAμ B cC (2.1) where we used the same notation as Machacek and Vaughn [2]. Here Dμ=∂μ+ig f

A

Aμ

A_{is the covariant derivative, so that}

Dμϕa=∂μϕa+ig θabA Aμ A ϕb(2.2a) Dμψi=∂μψi+ig tijAAμ A ψj(2.2b) Fμ ν A =[Dμ, Dν] A =∂μAν A −∂νAμ A +g fABCAμ B Aν C _(2.2c)

Because we're interested in the UV behaviour of the couplings, we are going to derive the

Renormalization Group Equations (RGE) for the model, and we start by giving an outline of the

general renormalization procedure.

We first substitute each field and each coupling with their renormalized counterparts, as follows:

Aμ→ZA 1 /2 Aμ *_{, ψ} a→Zψ 1 /2 ψ_a*, ϕ_j→Zϕ 1 / 2 ϕ*_j, g → Zgg *_{, Y}

ajk→ZY ajkYajk

* _{and Λ}

abcd→ZΛabcdΛabcd

*

where we choose the powers out of convenience. The Z are called renormalization constants, and the starred quantities are the renormalized couplings and fields, making the quantites without stars the naked fields.

Substituting both in the kinetic and interaction lagrangian density, we can split every term in two, one containing only the naked quantities, and another, which we call counterterm, containing also the Z .

For example, the pure-gauge part of the lagrangian (∂μAν A +∂νAμ A )2+g2(fABCAμ B Aν C )2−2 g f ABCAμ A Aν B (∂μAνC +∂νAμC )≡LKG+L4G+L3G(2.3) becomes [ZA−1] LKG+[Zg 2 ZA 2 −1] L4G−[ZgZA 3/ 2 −1] L3G≡[ZA−1] LKG+[Z4G−1] L4G−[Z4G−1] L3G(2.4)

We use the new Lagrangian to do perturbation theory: the counterterms are to be treated as new interaction vertices, which depend from the as-of-now undefined Z , whose contribution is to be summed along with the "regular" terms (with an exception, that will be explained later). Once the

Z are defined, however, the contribution coming from the counterterms will cancel out all divergences from the calculations.

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To give the Z an explicit form, one needs to set a number of renormalization conditions, one for each Z . The aim of each of these conditions is to force a particular S-matrix elements (calculated with the renormalized lagrangian) to reproduce known results. More often than not, the renormalization conditions are low-energy predictions, so that the S-matrix elements are simply a vertex or a propagator.

As an example, let's take the 3-gluon vertex ("gluon" here meaning "nonabelian gauge boson"). The choice of renormalization conditions vary with the specific model; for example, in the case of SM's

SU (3)_Cgauge renormalization constant, one would like for the 3-gluon vertex in the IR limit to take its tabulated value of g.

We will limit our calculation to 1-loop contributions; to do this, we sum all of the Feynman diagrams built with "naked" vertices and propagators, having at most 1 loop, and then add the contribution of the counterterm diagram, as follows:

where the sums run over the particles that have those kinds of diagrams (those particles that are in either the fundamental or in the antifundamental representation of the group), and the criss-crossed vertex is our counterterm.

Each diagram can be transformed into an integral over momentum or position space using the appropriate Feynman rules, and each can be in principle calculated exactly. In practice however, since some of these integrals will be infinite, one has to regularize them in some fashion to get finite results. The regulator employed will find its way into Z , namely by forcing it to carry a divergence that will cancel out exactly the divergent part of the diagram (thus renormalizing the theory), once we remove the regulator by a suitable limit. Since Z is not an observable, its arbitrarieness is of little significance.

A popular choice of renormalization, and the one used to derive these results, is the so-called

minimal subtraction: once all the integrals are regularized, the divergence will appear as a

regulator-dependant pole; we will require that the contribution of Z_3Gcancel this pole exactly, in addition to giving the desired value to the S-matrix element.

A different choice is possible in principle, and the explicit form of Z and consequently the renormalized coupling will depend on it. However this can be shown to consist in nothing more than a "rescaling" of the coupling, and thus physical results are unaffected by it [12].

It's even possible [12] to transform quantities dependant on the renormalization scheme (the ensemble of regulator choices with renormalization conditions) between different schemes, and the RGE represent the fundamental equation that all of such quantities must satisfy regardless of the

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scheme, and that thus "links" them permitting the passage from one to another (of course the

explicit form of the RGE is scheme-dependant, but it always expresses the same property in every

scheme, that is, the independence of all physical quantities from it).

However, it can be proven that the first term of the Taylor expansion in powers of g_R of any physical quantity (and thus of the beta function in particular) is scheme-independant.

After imposing the renormalization condition, we obtain an equation of the form:

Z_3G=F_3G(g2_R

)1_{ϵ+O( g}4_R)(2.5)

However, we can't solve this equation yet, because it depends on three unkown quantities (remember that we need Z_g , and in our equation figures only Z3G≡ZgZ3 /2A ); to go further, we need

to renormalize other S-matrix elements to make a suitable number of equations and thus solve the resulting system. We can do that by renormalizing the ghost-gluon vertex

with the resulting equation

Z_{C ̄}_{C G}≡(Z_CZ1 / 2_A

)=F_CCG(g_R2

)1_{ϵ +O( g}_R4)(2.6)

and the gluon propagator

with

Z_KG≡Z_A=F_KG(g2_R

)1_{ϵ +O(g}_R4)(2.7)

We have now a system of 3 equations in 3 variables so we can proceed with the solution. However, obtaining the one-loop renormalization constants for a general theory is beyond the scope of this thesis, so we will skip straight to the results, provided in [1].

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In summary, with this procedure one can obtain an explicit form for each of the Z .

Recall from the start of this chapter, our objective is to write the RGEs for the gauge, quartic and yukawa couplings. Such equations are of the form:

μ ∂_{∂μ x}=0 (2.8)

which is the requirement that the naked value of each coupling is independent of the chosen renormalization scheme. In the minimal subtraction scheme this becomes:

βx≡μ ∂_{∂μ x}R=μ ∂_∂μ[(

μ₀ μ )

ϵ

Z−1x(μ)x ] (2.9)

where μ₀is an arbitrary mass scale, which is a nonphysical quantity coming about due to the renormalization scheme, and as such physical quantities are unaffected by it.

Thus, to write the RGE we need to calculate Z_g and the various ZΛand ZY .

After the calculation, we obtain for the gauge beta functions: β_gi1=11 3 C2(Gi)− 2 3

∑

FiS2(Fi)− 1 6

∑

SiS2(Si)(2.10)

whereiis the index that runs through the groups that make upG , andF_iand S_ithe representation under the group for the field.C₂is the value of the quadratic casimir of the adjoint representation for the ithgroup, while S2is the Dyinkin index of the representation.

Following the same convention, we have for the yukawa beta functions: (4 π)2β_Ya=1 2(Y + b YbYa+YaY+ bYb)+2YbY+aYb+YbTr (Y+ bYa)−3 {C2(Fa),Y a } (2.11) and finally for the quartics.

(4 π)2 β_λ_abcd=

∑

perm

[

1 8λabefλefcd+ 3 8{ θ A_{, θ}B_} ab{θ A_{, θ}B_} cd−Tr [Y a_Y+ b_Yc_Y+ d ]

]

+ +λabcd

[

∑

_k(S2(Fk)−3C2(Sk))

]

(2.12) 2.2 UV behaviour of couplings

In the following we will limit ourselves to the 1-loop approximation., since we are investigating couplings that vanish in the far UV.

Let xIbe the collective name for the couplings under examination, that is xI={ ga, yb, λc} . We

start out by factoring out the main UV behaviour from all the couplings: we define x̃_I(t )=x_i(t)t

where t=ln(μ/μ₀)/4 π . Here μ₀is an energy scale in the IR range, for example below the weak scale; we also note that μ∈[μ0,∞[ so t∈[0, ∞] .

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td ̃ga d t = ̃ g_a 2 +βga( ̃g ) t d ̃yb d t = ̃yb 2 +βyb( ̃g , ̃y) t d ̃λc d t =̃λc+βλc( ̃g , ̃y , ̃λ) (2.13)

or, in compact form

d ̃xI

d ln t=VI(̃x) (2.14)

To start the analysis, we are going to find the fixed points x̄Iof the system. The fixed point of a

differential equation is one (or more) special solution which is independent of the variable (and as such, it is a number). If we put q=ln t , so that q∈]−∞ ,+∞ [ we obtain a system of equation is standard form

d x̃I

dq=VI( ̃x ) (2.15)

for which the fixed points are easily calculated by imposingV_I( ̃x)=0. We then get the system of

algebraic equations

̃

g_a=−2β_ga( ̃g ) ̃yb=−2 βyb( ̃g , ̃y) ̃λc=−βλb( ̃g , ̃y , ̃λ) (2.16)

Solving such a system analiticallycan be a formidable task, so we postpone the discussion to a later chapter 2.3 where we tackle a numerical approach to the problem. For now we note the triangular structure of the beta functions: each of the gauge equations depends solely on the gauge beta functions (for SU ( N ) groups, each equation depends on the beta function of its own coupling), yukawa's depends both on yukawa's and gauge's, and quartics's depend on all three kinds.

The fixed points for the rescaled couplings correspond to special kinds of runnings for the non-rescaled couplings. Using the definition backwards, x̃_I(_{t )= ̄}x_Iimpliesx_I(_{t )= ̄}x_I/ln(μ/μ₀). By inspection this solution satisfies both TAF-1 (it has no poles) and TAF-2 (it vanishes towards infinity). The ensemble of these solutions (provided each of the equations admits a fixed point) is called a group of fixed-flows, because while the individual couplings run, their ratio remains fixed. The following step is to examine the stability of the fixed points. Depending on whether a fixed point is attractive or not, the differential equation could admit a range of solution with the same asymptotic behaviour as the fixed point, which in turns signifies a range of less banal runnings for the non rescaled couplings.

For this purpose is sufficient to consider the linearized version of the (2.13)

V_I ≃

∑

J [d VI d xJ ] xI = xI * (x_J−x_J*) ≡

_∑

J M_IJΔ_J (2.17) and thus we have

d Δ_I d ln t=

∑

J

M_IJΔ_J (2.18)

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are the elements of the Jacobian calculated at the fixed-point values, and thus by studying its eigenvalues one can extract the attractiveness of the fixed points.

For the 1-loop approximation, the general form of the Jacobian is

M_IJ =

(

δ_ij(1 2− 3 2bigi 2 ) 0 0 ∂β_ya(x) ∂gj δ_ab 2 + ∂β_ya(x ) ∂yb 0 ∂β_λ_m(x ) ∂gj ∂β_λ_m(x ) ∂yb δ_mn+∂β_{∂ λ}λm(x) n

)

(2.19)

where each function is to be evaluated at a set of fixed points

The triangular shape of the beta function translates into M being a triangular matrix, which could make the work of actually calculating the eigenvalues much easier.

As for general consideration, if all of the eigenvalues of M { ̄x }are positive (negative), the set

{ ̄x }is repulsive (attractive) in the UV, and thus IR-attractive (IR-repulsive).

To understand why, let's take a differential equation of the same type, but for a single coupling (I =J =1)

d (x− x *) dq ≡

d Δ

dq =α Δ(2.20)

which has the solution Δ=qα

+const

Here the Jacobian is simply alpha, and it's a constant. If α>0 , for q → ∞ the quantity Δ , which measures the distance between the function and the fixed point, grows, that is the function

is repelled away from the fixed point, while it gets closer to it for q → 0 . The behaviour for α<0 is

the opposite. Thus, from now on we will refer to stable fixed points as UV-attractive solutions, and unstable ones as IR-attractive.

Looking at this example, an important characteristic emerges: if one ̄x is zero, then x(μ)≡0 . However, if ̄x is a UV-attractive, a range of solutions ̃x (t) exists of (2.20) for which the

lim_{t → ∞} ̃x( t)=̄x , so that limt → ∞ x(t)=limt → ∞̄x /ln (

μ

μ₀)=0 , which is exactly TAF-2. However, it's not guaranteed that any of these also satisfy TAF-1, and one has to study all the solutions in the basin of attraction (the range of corollary conditions for which the solution is attracted to the fixed point) to rule out the presence of Landau poles.

If instead the fixed point is IR-attractive, only the solution x (t)=̄x /ln(μ/μ0)exists which tends to 0

in the far UV, and which automatically satisfies both TAF-1 and TAF-2. In turns, this sets a constraint on the coupling, making it completely determined, or in other words, makes a low-energy prediction for the coupling.

The case of a positive- or negative-definite Jacobian of arbitary rank is similar, mutatis mutandis. The only difference to note is that in case of instability, the predicted quantity could be a combination of coupling instead of a single one.

In the general case, the Jacobian (calculated at a set of fixed points) will be nondefinite, and we have the following:

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1) For each positive eigenvalue, the system has one IR-attractive fixed point, so one combination of couplings has a fixed running that satisfies both TAF-1 and TAF-2, and we have one low-energy prediction.

2) For each negative eigenvalue, the system has a UV-attractive fixed point, so one combination of coupling has a range of runnings that satisfies TAF-1, hopefully TAF-2 as well, and one running that satisfies both.

2.3 The automatic builder

To actually build and test models, we employed a highly automatized program written for a computational assistance software.

This program consists of three parts

1) Declaration routines (Generators and Lagrangian) 2) Equation generation

3) Fixed-points search

We will go in some depth describing the functioning of all three parts. 2.3.1 Declaration

Recall from the previous chapter that our primary objective is to find the fixed points of a model's RGE. In our approximation, we can make use of (2.10),(2.11) and (2.12) to write the 1-loop beta gauge functions using the generators of the gauge transformations for each field. This part of the program focuses on building such generators (in the notation of [2]) with the only input required of the representation of each field.

The first step is to declare the number of gauge sectors and their type. By selecting the type of a group, the program automatically fills in the corresponding entries in the generators in equations (2.2) with the fundamental constants of the group using a look-up table. As such, even if at this time the only groups implemented are U (1) and SU (2) through SU (5) , expanding the builder to include groups of any kind is easy.

Following this, one proceed to declare any number of scalar or fermion fields (the two procedures are separate, but the order is not important) along with the list of their representations. The currently supported representations are singlet, fundamental, antifundamental and adjoint.

The procedure starts by building a matrix of appropriate dimensions. Then, using another look-up table, the builder selects the appropriate generators for each of the representations of a field, calculates the variation under each of the group and finally fills in the correspondent entries in the transformation generator.

For example, let's take G=SU (2) xSU (3) , and a fermion F_aiin the fundamental representation of the first, and the adjoint of the second. Then, the variation of F underGwould be

(δF )bi A =(T [SU (2), f ]ba A Fai) A∈[1,3] (δF )ajA=(T [ SU (3) , adj]AjiFai) A∈(4,9)

where Ais the progressive index of the generators for entire gauge group, and the T [G_i, R ] are the

generators of the supgroup G_iin the representationR. It's worth noting that we are using the vector representation for the adjoint (instead of the usual two-indices one); the code is equipped with procedures that transform single-index adjoints into two-indices, which is necessary when contracting tensors.

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The procedure is the same for scalars and fermions, with minor differences due to the complex nature of the scalar fields (for example, a scalar field with only an adjoint index will be built as a matrix with real entries instead of complex ones)

The entirety of this procedure has been developed by myself and Giulio Pelaggi. Its extreme degree of automation allows the researcher to test a variety of different models in a reasonably short time, and more importantly with little effort and reducing to zero the chance of mistakes; for example, the minimal TRIN model examined later on would require to write 24 generators for both scalars and fermions, each being 36x36 and 81x81 matrices respectively, so the advantage of the automatic builder in such a situation is evident.

Once the generators have been built, the Lagrangian of the model is written using the field declared in the first step. To date, an alpha-stage version of an automated Lagrangian builder exists, but progress on the coding has been halted due time optimization: while the program would have made the task effortless and automatic, the time spent coding it would have been much higher than that spent actually writing the various Lagrangian by hands, since they're generally very easy and with minor variations (in the scope of our group's research). However, due to the particular standard used in the field declaration, a function had to be written that would perform field contractions, since the one provided with the program wasn't easily extendable.

2.3.2 Equation generation.

This part of the program is entirely written by professor Alessandro Strumia, and is the coded version of the (2.10),(2.11) and (2.12). Once the generators have been built, three different functions can be called that write the one or two loop beta functions for all the couplings within the Lagrangian declaration, along with some consistency checks functions, which are explained in [2]. These checks are mandatory when one has to write every generator by hand, because there's plenty of room to make mistakes, and have been the cornerstone for the validation of the builder, after which they have been removed from the procedure to cut some time.

Running this part of the code takes up to 15 minutes for the more complex models on the average PC, althought the entire process can be easily parallelized to take much less time. It also requires up to 4GB of RAM for this task only.

The code contains two functions that perform the automated writing of the Lagrangian "by hand", that is, each writes the most general lagrangian that is consistent with the checks explained in [2]. Hower, it requires an exceedingly long time execute, and can crash on computers with up to 16GB of RAM with most models tested, and as such it has been left unused.

2.3.3 Fixed points search

Before going in detail about the code used to find the fixed points, it's opportune to review the procedure with which one goes about finding these solutions.

Recall from Chapter 2.2 that equations for the couplings have a triangular structure: the gauge ones depends only on the gauge couplings, yukawa ones from both gauge's and yukawa's, and quartic ones from all three.

As such, an appropriate strategy for finding the solutions is the following:

1) Solve the gauge equations. Each equation features an IR-attractive zero solution and possibly a UV-attractive nonzero solution, let's call it g_i*_{. Thus, there's only one phenomenologically relevant}

set of fixed points, and it's { ̃gi→gi

*

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fixed point. If this is the case, however, it would be wise to review the model under analysis, as a vanishing fixed point for a rescaled gauge coupling implies that the corresponding coupling vanishes entirely at every scale ( g (μ)≡0 ).

2) The "best" set of fixed points from the previous step is substituted into all of the yukawa equations, obtaining a system of equations dependant only on the yukawa couplings, and thus solvable.

Each equation can have multiple fixed points, and thus the overall solution of the yukawa equations is a number of sets which are the combinations of all the different solutions for each coupling. In practice, however, one is forced to solve this system numerically, so the list of sets is simply the one obtained by reiterating the numerical procedure the desired number of times (and thus can, and will, feature duplicates of the same solution).

It possible for some of the (analytical) solutions to appear in the form of constraints f ( y_i)=0 . As such, numerical approaches will find an illimitate number of solutions that are only apparently different; for a thorough understanding of the model is thus necessary to sooner or later search for the solutions by analytical means, althought one can make a cursory analysis by plotting couples of yukawa against each other.

3) Third step is a repeat of the second: the "best" gauge solution is substituted into the quartic equations along with one of the solutions sets from step 2, at which point one can solve the system, and then start again with another yukawa set.

More often than not, the sheer number and complexity of the system, along with the usually large number of sets generated by the second step, makes the numerical approach almost a necessity. However, even in the event of having at one's disposal a powerful data center, the numerical way could still be useful to summarily prune the yukawa sets, discarding the ones that do not produce solutions for the quartics.

It's also vital to remember that since we're (most likely) substituting approximate results into the quartics equations, the usual algorythms for numerical solution of equations will fail; an acceptable substitute to solving algorithms will be presented later in this chapter.

The code features 3 functions to solve gauge, yukawa and quartic equations separately, that are simply adapted shortcuts for the built-in solving algorithm. As already noted, they are only really suitable for solving the gauge equations, or the yukawa ones in really simple models. As such, a numerical approach is needed.

Our strategy involves putting all of the equations in normal form f (x_i)=0, and then summing their squares. The obtained polynomial is strictly positive (or zero, if/where it has roots), and as such solutions can be found by finding its local minima, keeping as a candidate root only those below a certain threshold ( 10−7_{in our calculations). The algorithm used is the software's}

implementation of the BFGS quasi-newton algorithm

This method has few issues related to basins of attraction, since for a starting point to converge to a minimum the only requirement is that it sits on a "downward" slope.

The main problem of this procedure lies in the fact that it doesn't discriminate actual extremely low minima from true solutions, a problem that could be easily solved by directly inspecting the polynomial in the surroundings of every found solution.

The test does not guarantee the absence of solution in case of failure, but the with the sheer number of repetitions one could hope to make such a statement with some degree of confidence. Regardless, all failing models have been noted as "failed to produce a solution" precisely for this reason.

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parallelized. 2.4 A Toy Model

2.4.1 Description of the Toy Model

To put in practice what we have described so far, we will examine a very simple toy model, using the automatic builder to calculate the result.

Our gauge group is U (1)×SU (2) , with coupling constants g₁and g₂. We have two fermion fields, F₁ in the (1,2) representation and a singletF₂in the (−1,1) , and one scalar fieldS in the representation (0,2)

Our model has a single yukawa

y1[F2(SaF1a)] (2.21)

and a single scalar

λ1(SaSa+)2(2.22)

The RGE for our models are

(4 π)2 d d lnμg1=2g1 3 (2.23a) (4 π)2 d d lnμg2=− 41 6 g2 3 (2.23b) (4 π)2 d d lnμ y1=−

(

6 g1 2 +9g2 2 4

)

y1+ 5y13 2 (2.23c) (4 π)2 d d lnμλ1= 9g24 8 −2y1 4 +(4y₁2−9g₂2) λ₁+24 λ₁2(2.23d)

giving the following equations for the rescaled couplings:

d d ln t g̃1= ̃ g₁ 2 +2 g1 3_(2.24a) d d ln t g̃2= ̃ g2 2− 41 ̃g23 6 (2.24b) d d ln t ̃y1= ̃ y1 2−(6 g1 2 +9 g̃2 2 4 ) ̃y1+5 ̃ y1 3 2 (2.24c) d d ln tλ̃1=9 ̃ g2 4 8 −2 ̃y1 4 + ̃λ₁+(−9 ̃g₂2 +4 ̃y1 2 ) ̃λ₁+24 ̃λ₁2(2.24d)

2.4.2 Gauge Fixed Points

we have the following fixed points for the (2.23a) and (2.23b), with relative Jacobian eigenvalue. 1 ) g1*=0 [EM=1/2] 2 ) g2 * =0[ EM=1/2 ]; g2 * =

√

3 /41[ EM=−1] (2.25)

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this situation is characteristic of gauge RGE behaviour: being homogeneous, g*

=0 is always a solution; however, this solution is IR-attractive, and as such not fenomenologivally interesting. Since β₁for abelian groups is always positive, this is also the only solution; thus the only way an abelian field can achieve TAF is g_Ab(μ)≡0.

For nonabelian groups, if β₁<0 there is another nonzero, UV-attractive solution.

Our set of gauge solutions is then simply

{ g1*=0 , g2*=±

√

3/41 } (2.26) 2.4.3 Yukawa Fixed Points

Inserting our solutions into (2.23c) we obtain

d d ln t y1= 55 y1 164 + 5 y13 2 (2.27) which has one fixed points, with relative Jacobian eigenvalue

y₁*=0[ E_M=+ 55

164](2.28)

Yukawa RGEs are homogeneous too, and the zero solution is also UV-repulsive. Our set of gauge-yukawa solutions is thus:

{ g1 * =0 , g2 * =

√

3/41 , y1 * =0 } (2.29) 2.4.4 Quartics Fixed Points and conclusions

Inserting the solution we obtain:

d d ln t λ1= 81 13448+ 14 λ₁ 41 +24 λ1 2 (2.30) Which has no real solution.

In summary, the model fails to achieve TAF because the quartics RGE has no fixed points. 2.4.5 An example of a UV-attractive solution analysis

Recall that a UV-attractive fixed point for a rescaled RGE implies a range of runnings for the coupling that all vanish in the far UV. We are interested in examining the possibility of the existence of such runnings that are also compatible with TAF-1.

As an example of this kind of analysis, we examine here the case of g₂, whose RGE has a UV-attractive fixed point.

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d d q g̃2= ̃ g2 2 − 41 ̃g23 6 (2.31)

where we defined q=ln t ; the domain is q∈]−∞ ,+∞ [ since t=ln(μ/μ0) forμ>μ0 and thus t∈[0,+∞ [ .

Its solution, for a starting condition of the type g̃2(0)=C

̃ g2(q)=Sign[C ] e q/ 2

√

1 C2+ 41 3 (e q −1) (2.32)

The sign ofC selects which of the fixed points is the limit for the solution, that is lim

q →+∞g2

(q)=Sign[C ]

√

3

41 , in accord with what we found before.

For a starting condition ̃g₂(0)=0 the solution is simply ̃g₂(q)≡0 , so we recover the other fixed

point of the RGE.

As for the pole analysis, let's start for the case C>0 . The denominator in (2.35) never vanishes if

C ∈]0,

√

3

41] , and for greater values vanishes for q=ln [

−3+41C2

41C2 ] , that is fort=

−3+41C2

41C2 , which for C>

√

3

41 is always positive, meaning ̃g2(t ) always hits a pole.

The situation is the same for C<0 , with the obvious modifications.

In conclusion, ̃g₂admits a range of runnings that satisfy both TAF conditions, which is the class of functions ̃ g₂(t )=Sign[C ]

√

t 1 C2+ 41 3 (1−t) (2.33) where

∣

C

∣

∈]0,

√

3/41 ]

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3. The current state of TAF models research

3.1 Review of the Standard Model

With all the tools at our disposal, we can now examine our first real model for TAF, the SM itself. SM gauge is U (1)Yx SU (2)Lx SU (3)Cwith couplings g1, g2 and g3. For the matter content, we

have six kinds of fermions: Uiin the (−2/3,1 ,3

*

) representation, Qiin the (1/3,2 ,3) , Diin the

(1/3,1,3) , L_iin the (−1/3,2,1) , E_iin the (−1,1,1) and finally N_iin the (0,1 ,1) . Each of the fields comes in 3 flavours (as indicated by the subscript i ), to represent the various generation of quarks (first 3 fields) and leptons (last 3 fields). SM has also one scalar field H in the representation

(1/2,2 ,1) .

Neglecting indices, the Yukawa interactions of the models are: −LY=+ytQHU + ybQ H+D+ yτLH

+_{E+ y}

νLHN +h.c. (3.1)

and we have instead just one quartic in the form of −Lλ=λ (H H

+

)2(3.2)

In the following we will avoid writing the rescaled RGE explicitly, since they are easily calculated and showing them doesn't reveal anything in particular.

Gauge RGEs are

(4 π)2 d d lnμg1= 41 g1 3 10 (3.3a) (4 π)2 d d lnμg2=− 19 g32 6 (3.3b) (4 π)2 d d lnμg3=−7 g3 3 (3.3c)

From previous consideration, we already know the SM has no hope of achieving TAF for reasonable physical conditions, since the hypercharge is abelian, but we'll proceed with the analisys regardless. The set of fixed points for the gauge equations is thus:

g1→0, g2→

√

(3/19) , g3→

√

(1/14) (3.4)

where since in the following equations only the squares of the coupling appear, we kept only the positive solutions. Exactly as before, all nonzero fixed points are UV-attractive

Yukawa RGEs are (4 π)2 d d lnμ yt= 9 yt3 2 +yt

(

− 17 g12 20 − 9 g22 4 −8 g3 2 +3 yb 2 2 +yν 2 +yτ 2

)

_(3.5a) (4 π)2 d d lnμ yb= 9 yb 3 2 +yb

(

− g1 2 4 − 9 g2 2 4 −8 g3 2 +3 yt 2 2 +yν 2 +yτ 2

)

_(3.5b)

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(4 π)2 d d lnμ yτ= 5 yτ 3 2 +yτ

(

− 9g12 4 − 9 g22 4 −3 g3 2 +3 y_t2+yτ 2

)

_(3.5c) (4 π)2 d d lnμ yν= 5 yν 3 2 +yν

(

− 9g₁2 20− 9 g₂2 4 −3 g3 2_{+3 y} t 2 −yτ 22

)

(3.5d)

and by inserting the gauge solutions into the rescaled equations we have the following fixed point, with relative Jacobian signatures

1 ) yt * =0 ; yb * =0 ; yτ * =0 ; yν * =0 ; Sig(M |y)=(- - + +) 2 ) yt * =0 ; yb * =(

√

227/266) 3 ; yτ * =0 ; yν * =0 ;Sig (M |y)=(- + + +) 3 ) y_t*=(

√

227 /798) 2 ; yb * =(

√

227 /798) 2 ; yτ * =0 ; yν * =0 ;Sig (M |_y)=(+ + + +) 4 ) yt * =(

√

227/266) 3 ; yb * =0 ; yτ * =0 ; yν * =0 ;Sig (M |y)=(+ - + +) (3.6)

where we kept only the positive solutions, as before. The predictions are as follows:

1) Two IR-attractive zero solutions, forcing the respective couplings to vanish completely, and two UV-attractive zero solution, thus allowing for with potentially nontrivial runnings.

3) The two IR-attractive nonzero solutions allow for a nontrivial running of their couplings, with the prediction yt(Mt)=yb(Mt)=0.879 , which in turns implies Mt=Mb≃163 GeV .

2-4) These solutions are basically the same, with the only difference being which between y_tand y_b gets to have the nonzero IR-attractive solution. The prediction for the coupling corresponding for the nonzero fixed point implies M ≃163 GeV for the related particle, while the other one has to rely in its basin of attraction for some nontrivial running.

None of these solutions are physically acceptable, but for the sake of completeness we will proceed with the analysis regardless with solution 4), which gives the results closest to reality.

The RGE for the lone quartic is (4 π)2 d d lnμλ= 27 g1 4 200 + 9 g1 2 g2 2 20 + 9 g2 4 8 −6 yb 4_{−6 y} t 4 −2 yν 4_{−2 y} τ 4 + +

(

−9 g1 2 5 −9 g2 2_{+12 y} b 2_{+12 y} t 2 +4 yν 2_{+4 y} τ 2

)

_{λ+24 λ}2_(3.7)

The solution with all yukawas equal to zero produces no real fixed point. Our chosen solution gives two fixed points, with relative Jacobian signature

λ*=−143±

√

119402

9576 ;Sig [M |λ]=( ± ) (3.8)

The UV-attractive solution (the one with the minus sign) is negative, which implies a potential instability for the EW vacuum, with a lifetime shorter than the age of the universe (see [0] for a more detailed calculation).

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means MH=163 GeV .

3.2 Review of the Trinification GUT 3.2.1 The Minimal Trinification Model

Our efforts went into finding a working TAF model using the Trinification GUT (TRIN) as a baseline. We start then by describing the minmal trinification model along with its TAF situation. TRIN's gauge group isSU (3)_L×SU (3)_R×SU (3)_C ; while theSU (3)_C is the usual color gauge group, SU (3)_Lextends the electroweak U (1)_Y×SU (2)_L, while the remaning group is its right counterpart. This is choice allows to "remove" the abelian hypercharge from the group (because, as we've seen before in the toy model discussion, abelian groups can never satisfy TAF) by embedding it into a bigger non-abelian.

We have

Y =T3_ISO+(B−L)(3.9)

and with the TRIN we can make T3_ISOto be the diagonal generator of SU (2)_R(which we obtain as a subgroup ofSU (3)_R), and "promote" B− LtoU (1)_{B− L}and make it a combination of both T8R and

T8L(in the notation of the Gell-Mann matrices).

Another choice, explored in more detail in [0], is the Pati-Salam GUT, in which we embed the isospin intoSU (2)_R, but leave it as it is and instead grow SU (3)_C intoSU (4)_PS , making

U (1)_{B− L}from one of its diagonal generators. The full group would then be

SU (2)_L×SU (2)_R×SU (4)_PS .

Our version of the TRIN hosts 3 kind of fermion fields: a lepton field E in the representation (3,3*,1) , and two quark fields QL and QR in representations (3,1 ,3*) and (1,3*,3) respectively. The elements of these fields are (for the lightest generation of each, with the first non-banal index being the row one):

E=

(

̄ ν'L e 'L eL ̄ e '_L ν'_L ν_L e_R ν_R ν'

)

(3.10a) Q_L=

(

uR 1 uR 2 uR 3 d1_R d_R2 d_R3 d '_R1 d '_R2 d '_R3

)

(3.10b) Q_R=

(

uL 1 uL 2 uL 3 d1_L d2_L d3_L ̄ d '1_R ̄_{d '}2_R _̄ d '3_R

)

(3.10c)

where the unprimed quantities are the known particles, and the name of the primed ones have been chosen looking at their quantum numbers.

The minimal Trinification model contains 3 generations of each of these fields, to accomodate the 3 generations of quarks and leptons, for a total of 9 fermion fields.

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As for the scalar content, the minimal model needs 2 Higgs fields in the same representation of the leptons to allow for appropriate simmetry breaking of the full group to the lone U (1)_em.

Such a scalar field is of the form:

H =

(

̄ H_U0 _H D - _H' -H_U+ H_D0 H'0 H_A H_B H_C

)

(3.11)

where the standard higgs appear as left doublets in the top 2 rows; this means that the observed Higgs is in principle a linear combinations of the 6 doublets between the 2 Higgs fields. The superscripts indicate the sign of the field's charge.

The vevs appear in the two Higgs as follows:

H₁(0)=

(

v_u1 0 0 0 v_d1 0 0 0 V₁

)

(3.12a) H₂(0)=

(

v_u2 0 0 0 v_d2 v_L2 0 V_R V₂

)

(3.12b)

Each field is equipped with 2 kinds of vev. The lowercase ones, featured in the left doublets, are the "small ones", and are responsible for the breaking of SM gauge U (1)_Y×SU (2)_L×SU (3)_Cto

U (1)_emand in the process give mass to the fermions and a few of the gauge bosons of the SM. As such they are under the constraint

[

∑

_iv_i2

]1 /2=246 GeV(3.10)

The uppercase ones are the ones that break the Trinification gauge group

SU (3)_L×SU (3)_R×SU (3)_Cto the SM, and give the mass to the unobserved fermions. As the uppercase implies, they are supposed to be very heavy, at least 3TeV.

3.2.2 Quartics and Yukawa Interaction

The most general form of the quartic interaction potentiali is

−L_4H=V₁₁₁₁+V₂₂₂₂+V₁₁₂₂+V₁₁₁₂+V₁₂₂₂(3.13) where V1111=λa 1111 Tr (H1 + H1) 2 +λ_b1111Tr (H1 + H1H1 + H1) (3.14a) V2222=λa 2222 Tr (H2 + H2) 2 +λb 2222 Tr (H2 + H2H2 + H2) (3.14b) V1222=ℜ[λa1222Tr (H1+H2)Tr (H2+H2)]+ℜ[λb1222Tr (H1+H2H2+H2)] (3.14c) V1112=ℜ[λa1112Tr (H1*H2)Tr (H1*H1)]+ℜ[ λb1112Tr (H1+H1H1+H2)] (3.14d)

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V1122λa 1122 Tr (H1 + H1)Tr (H2 + H2)+λb 1122

∣

Tr ( H1 + H2)

∣

2 + +λc1122Tr (H1+H1)Tr ( H2+H2)+λd1122Tr (H1+H1H2+H2)+ λ1122_e ℜTr (H1 + H2) 2 +ℜ λ1122_f Tr (H1 + H2H1 + H2) (3.14e)

It contains 14 real couplings and 6 phases

The Lagrangian of the Yukawa interactions for the minmal fields QL, QRand E is

−L_Y=L_Q+L_L+h.c. (3.15) where LQ=

∑

_ijn yQ nij QLiQRjHn(3.16a) L_L=

∑

ijn yLnij 2 EiEjHn * (3.16b) with n∈[1,2] andi , j∈[1,3]. The lepon yukawas yL

ijn_{are simmetric under exchanges of}_i_{and j}

3.2.3 Phenomenology of Gauge Bosons

Trinification gauge is G_TRIN=SU (3)_L×SU (3)_R×SU (3)_Cwith gauge couplings g_L, g_Rand g_C ; their relations to the SM gauge couplings g_Y≡

√

3/5 g1, g2and g3are

g_L=g₂ g_R= 2 g2gY

√

3g22−gY2

g_C=g₃(3.17)

The heavy vev V in H₁breaks G_TRIN to SU (2)_L×SU (2)_R×U (1)_{B− L}×SU (3)_C . At this stage, one left-handed doublet, one right-handed and one singlet of H₁gets eaten by the 4+4+1 vector bosons that acquire mass: a SU (2)_Ldoublet H_L, a SU (2)_Rdoublet H_Rand a Z ' singlet:

M_HL=g_LV₁ M_HR=g_RV₁ MZ'=

√

4 3(gL 2 +gR 2 )V1(3.18)

Z ' corresponds to the combination of gauge bosons gLT8Lμ−gRR8Rμ, with a mass bewteen 2 and 6

TeV depending on the Z ' of the light SM Higgs.

The B− L boson corresponds to gLT8Lμ+gRR8RμwithgB− L=

√

3 /2 gRgL/

√

gL

2 +gR

2

With the second Higgs H₂and its two heavy vevs V₂and V_R, and defining V2

≡V12+V22+VR2 , the

gauge bosons form:

1) A left-handed weak doublet with 4 components and mass M_HL=g_LV

2) The SU (2)_Rvector doublet H_Rsplits into two charged components with masses M_HR±=g_RV

and 2 neutral components with mass

M2_H0R=gR

2

2 [V

2

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3) The breaking of SU (2)Rgives rise to the right-handed WR

±

vectors with mass

MWR± 2 =gR 2 2 [V 2 −

√

V2−4 V1 4 VR 2 ](3.20) and to a Z_Rvector;

4) The Z_Rand ZB− Lvectors mix forming eigentates with masses

M_{Z ' , Z ' '}2 =2 3[(gL

2

+g_R2)V2±

√

(g2_L+g2_R)V4−3 g_R2(4 g_L2+g2_R)V₁2V_R2](3.21) which in the limit V_R≪V_1,2are simply

M_Z'=

√

4 3(gL 2 +g_R2)V MB −L≃gRVR V1 V

√

gR 2 +4 gL 2 g_R2+g_L2 (3.22)

3.3 Extending the Minimal Model

To date, every attempt has failed to produce a set of parameters that would make the minimal model compatible with TAF. We report the gauge RGEs here because of their usefulness in a later argument: (4 π)2 d (d ln μ)gL=−4 gL 3 (3.23a) (4 π)2 d (d ln μ)gR=−4 gR 3 (3.23b) (4 π)2 d (d ln μ)gC=−5 gC 3 (3.23c)

As it can be seen by inspection, the minimal gauge constant have no TAF problems. As we already pointed out, yukawa RGEs are never a problem in principle, because by virtue of being homogeneous they always admit the solution y_i=0 ∀i , and M is usually negative-definite in that point; still it must be noted that minimal model's yukawa RGEs admit a variety of fixed points, each set consisting in the zero solution for about half of them. Quartic RGEs admit no fixed point for the

y=0 solution, and no set of yukawa fixed points have been found that would allow some for them.

The search for an extension to the minimal model could go along one of these two branches

1) The insertion of new fermions, that would give new non-zero fixed points for the yukawas. This is because in the limit y → 0 the quartic RGEs stop depending from the fermion content of the model (it is, after all, the limit in which all fermions stop interacting with the Higgs), and thus in this limit all these RGEs are the same regardless of the model, and in particular, they're the same as the minimal ones, which admit no solution.

2) The insertion of new scalars, that would straight-up change the shape of the quartic RGEs.

From the analysis of the SM we already know that adding yukawa interactions can sometime "save" a quartic interaction otherwise doomed to suffer from poles; this, coupled with the fact that quartic RGEs are already very complicated with 2 Higgs, and become quickly worse by adding new

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scalars, directed our research along the first branch.

The total number of possible fermion extensions to the minimal model is in principle limited (with the exception of singlet insertions). Each new fermion with one or more non-singlet representations of one of the groups raises the value of β₁for those groups. If the value of β₁grows to zero, the RGE for that group loses its only nonzero fixed points, which since βg -s are independent from

everything else, can't be cured in any way, and thus the model is automatically invalidated.

Not all kinds of fermions are suitable for insertion. For example the insertion of a chiral (with respect to theSU (3)_L×SU (3)_Rgauge group) fermion requires also the insertion of another with opposite representations, because no new chiral fermions have been observed to date.

Additionally, some particles are strongly unfavoured for various reasons, so they haven't been considered for testing, althought they can't be excluded a-priori so they're still a possibility if one lacks better options.

One example of this would be the insertion of the pair L×̄L (in representations (3,1 ,1) and (3*_{,1 ,1)}

respectively): there's no yukawa interaction to be made with these particles, so they would make a stable charged particle that survived from the beginning of the universe to this day. If that were the case however, we would have already seen it (in cosmic rays at least, assuming a high enough mass to avoid production in colliders), which puts such a strong bound on its abundance to require bizarre explanations such as that the temperature of the universe must have never reached its mass value, thus making it an extremely rare particle. While this can't be excluded in principle, under the naturalness assumption one would rather explore other models with more natural features, and as such the only fields approved for insertions were two-index fields either in the adjoint of a group, or in the fundamental representation of one group and the antifundamental of another (like the quark and lepton fields of the minimal model).

Adding a fermion of the first kind raises the appropriate β₁by2, whereas for fields of the second kind, the increase is 1 for the β₁of both groups. For an extension with new fermions we have the following constraints:

2 Ladj+1(Lff *)−4>0 (3.24a)

2 R_adj+1 (R_{ff *})−4>0 (3.24b) 2 C_adj+1(C_{ff *})−5>0 (3.24c)

Where Ladjis the number of fermiona in the adjoint, and Lff *is the number of fields with at least

one index belonging toSU (3)_L, and similarly for the other groups.

Fermions in the singlet representation of every group can be added in any number, as they do not modify the beta functions.

In Table 1 we show the allowed insertions to the minimal model, while in Table 2 we present the "problematic" ones, that nonetheless respect the β1bound.

We point out that no yukawa can be made with these new fermions. As such, these models have the same RGE as the minimal model, and cannot hope to achieve TAF.

However, we found that increasing the value of β₁ in the Minimal Model could in some instances lead to TAF. This would correspond to the situation where new gauge-interacting particles are added that have no yukawa interaction, like the one listed in Table 2.

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TYPE REPRESENTATIONS ( Δβ1L, Δβ1R, Δ β1C) YUKAWAS 1 (1,1,1) (0,0 ,0) _{1 L H}* 8_L (8,1,1) (2,0 ,0) ₈_L_{L H}* 8_R (1,8,1) (0,2 ,0) ₈_R_{L H}* E ' + ̄E ' _(3,3* ,1) + (3*,3 ,1) (2,2 ,0) E ' EH + E ' E ' H + ̄E ' ̄E ' H Q 'L+Q '̄ L (3*,1 ,3)+(3,1 ,3*) (2,0 ,2) Q 'LQRH Q '_R+ ̄Q '_R _(1,3,3* )+(1,3*,3) (0,2 ,2) Q 'RQLH

Table 1: Allowed fermionic extensions of the minimal trin

TYPE REPRESENTATIONS (Δ β1L, Δβ1R, Δ β1C ) 3_L+ ̄3_L _{(3,1 ,1) + (3}* ,1 ,1) (2 /3,0 ,0) 3_R+ ̄3_R (1,3,1) + (1,3*_,1) _(0,2/3,0) 3_C+ ̄3_C _{(1,1,3) + (1,1,3}* ) (0,0 ,2/3) 6_L+ ̄6_L _{(6,1 ,1) + (6}* ,1,1) (10/3,0 ,0) 6_R+ ̄6_R (1,6 ,1) + (1,6*_,1) _{(0,10 /3,0)} 6_C+ ̄6_C _{(1,1,6) + (1,1,6}* ) (0,0 ,10/3) 8_C (1,1,8) (0,0 ,2) ̃E + ̄̃E (3,3 ,1) + (3*_,3*_,1) _{(2,2 ,0)} ̃ Q_L+ ̄̃Q_L (3,1 ,3)+(3*_{,1 ,3}* ) (2,0 ,2) ̃ Q_R+ ̄̃Q_R (1,3,3)+(1,3*,3*) (0,2 ,2)

Table 2: Problematic fermionic extensions of the minimal trin

In table 3 we show the highest value of the increased g_C at which the Minimal Model (and by extension, extended model with insertions from table 2) achieves TAF, when gL and gRare

increased of the amount indicated in the axes. Some of the entries are empty: since the model achieves TAF for the highest possible value of g_C (the one coming from the unperturbed minimal model), it's safe to assume that increasing again one between g_L and g_Rproduces another TAF model for values of g_C equal of higher than 5.

If after testing all of these insertions no TAF-compatible conditions have been found, one then has to consider the insertion of a scalar field. Even then, the number of possible models is in principle finite, althought still large (larger than before, actually): a field in the adjoint raises β₁by1/6for the group, and a field in the "anti+fundamental" reduces by 1/2 the β₁of both groups. We have then:

2 FL_adj+1 2SLadj+1(FLff *)+ 1 2(SLff *)−4>0 (3.25a) 2 FR_adj+1 2SRadj+1(FRff *)+ 1 2(SRff *)−4>0 (3.25b)

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2 FC_adj+1

2SCadj+1(FCff *)+

1

2(SCff *)−5>0 (3.25c)

with obvious notations.

g_L/g_R _-4 _-10/3 _-8/3 _-2 _-4/3 _-2/3 -4 -3 -7/3 -7/3 -7/3 -5 --10/3 -7/3 -7/3 -7/3 -5/3 - --8/3 -7/3 -7/3 -7/3 -5/3 - --2 -7/3 -5/3 -5/3 -5/3 - --4/3 -5 - - - - --2/3 - - -

-Table 3: Minimal Trinification Model beta reduction test results

Our research however focused on testing fermionic extensions of the minimal trinification model, thus no scalar insertion have been tested.

3.4 TAF research 3.4.1 The tested models

Our research focused on testing fermionic extensions to the Minimal Trinification Model. We proceeded to systematically test all combinations of insertions in table 1 that would leave all gauge

β₁below zero.

Among the models tested there were stand-alone singlet insertion. Additionally, some insertions that didn't achieve TAF by themselves were tested with the addition of 3 singlet fields, and one of them successfully achieved TAF.

MODEL TAF? 1 Minimal+ 3xFS NO 2 Minimal +QL +QL̄ +QR +QR̄ YES 3 Minimal +QL +QL̄ NO 4 Minimal + E+Ē YES 5 Minimal +8_L YES 6 Minimal +8_L+8_R YES 7 Minimal +QL +QL̄ +8_R YES 8 Minimal−1 Generation+8_L NO 9 Minimal−1 Generation +8_L +8_R NO 10 _Minimal +QL +QL̄ +3x FS NO 11 Minimal + E+Ē +3xFS YES

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Models indicated with 1Generation are ones where among the three SM generations, only the third has nonzero yukawas. Models 6-8 have been tested with the builder, and are awaiting independent confirmation.

RGEs have a clear Left/Right similarity, in the sense that by inserting a field A and then changing all of its Right representations with Left ones, one obtains the same RGEs but with the labels Left and Right swapped. As such, testing one of the models automatically in/validates the other.

3.4.2 Lagrangians

We now present the Yukawa Interaction Lagrangians of the models that achieve TAF, as no new scalars have been inserted and thus the quartics remain the same as the minimal (see Chapter 3.2.2). The insertion of Q '_Land ̄Q '_Lgives the additional terms

−L_4L=

∑

i nQ 'LQiRHn(3.26)

withi∈[1,3]andn∈[1,2], and similarly the insertion of Q '_Rand ̄Q '_Rgives

−L_4R=

∑

i nQiLQ'RHn(3.27)

The insertion of both, in addition to the terms above, also allows the following −L44=

∑

_n(Q ' L Q' R Hn+ ̄Q ' L ̄Q ' R Hn

*

) (3.28)

The insertion of the adjoint 8Lgives the additional terms

−L8L=

∑

_{i n}8LEiHn

*

(3.29) and similarly with 8_R

−L8R=

∑

_{i n}8REiH*n(3.30)

while no additional interactions other than the two above can be made by adding both. 3.5 Introductory phenomenology of one TAF model

3.5.1 Mass Spectrum

We will now present some preliminary results on the phenomenology of the first extension to the minimal model found that achieves TAF, which is the addition QL+ ̄QL + QR + ̄QR , that is, the

addition of a fourth generation of quark-like fields and their barred counterparts.

Once an extension was found that could achieve TAF, we proceeded to examine the mass spectrum of the model to see whether it's compatible with experimental results, starting with the quark masses.

The SM fermions are allowed a mass only through spontaneous symmetry breaking, and in first approximation we neglect the contribution from flavour yukawas; this is because