Phenomenology of Trinification models
João Fonseca Seabra
Thesis to obtain the Master of Science Degree in
Engineering Physics
Supervisors: Prof. Doutor Filipe Rafael Joaquim
Prof. Doutor David Emanuel da Costa
Examination Committee Chairperson: Prof.a Doutora Maria Teresa Haderer de la Peña Stadler Supervisor: Prof. Doutor Filipe Rafael Joaquim Member of the Committee: Doutor Joaquim Inácio da Silva Marcos
November 2017
Acknowledgements
Besides all the effort that I had put into this work, it surely would not be the same withoutthe help of many people. First of all, I devote a special acknowledgement to Professor Filipe Joaquim, not only for his guidance and support during the development of this work, but also for being the person who really convinced me through his lectures that Particle Physics is an amazing subject. I am also very grateful to Professor David Costa for his guidance. Even after leaving IST, he was always available to answer my doubts and questions. I would also like to thank my colleagues in IST for all the great discussions and study sessions that I had the fun and the pleasure to share with them during these years. Finally, a big ”thank you” to my family for their everlasting patience and support.
i ii Resumo
A compreensão do fenómeno de violação da simetria Conjugação de carga e Paridade (CP) tem-se revelado como um grande desafio em Física de Partículas. De facto, uma forte evidência dequeo Modelo Padrão (MP) precisa de ser expandido reside na sua incapacidade de dar resposta a questões estreitamente relacionadas com violação de CP, tais como a assimetria bariónica do Universo. Na presente dissertação, estudamos extensões do MP com quarks vectorlike, dando particular ênfase aos modelos de Trinificação. Nestas Teorias de Grande Unificação, o sector fermiónico encontra-se associado a representações fundamentais do grupo E6, o que leva naturalmente ao aparecimento de um quark vectorlike em cada geração de fermiões. Como consequência da mistura que pode haver entre quarks vectorlike e quarks do MP, surgem novas fontes de violação de CP. Além disso, esta mistura afecta os acoplamentos dos quarks do MP a outras partículas, o que nos leva a analisar as restrições experimentais impostas aos quarks vectorlike. Discutimos também a possibilidade de encontrar violação espontânea de CP (VECP) em modelos de Trinificação. Ao contrário do MP onde violação de CP só pode ocorrer ao nível do Lagrangiano, al- gumas teorias que prevêem a existência de um maior número de campos escalares (tal como os modelos de Trinificação) permitem que a simetria CP possa ser quebrada espontaneamente pelo vácuo. Neste contexto, verificamos que a introdução de quarks vectorlike num modelo de Trinificação com VECP leva-nos a obter uma matriz Cabibbo-Kobayashi-Maskawa complexa, compatível com os resultados experimentais.
Palavras-Chave: Matriz Cabibbo-Kobayashi-Maskawa, Modelos de Trinificação, Quarks vector- like, Teorias de Grande Unificação, Violação de CP
iii iv Abstract
Understanding the phenomenon of violation of the symmetry Charge conjugation and Parity (CP) reveals itself as a great challenge in Particle Physics. In fact, a strong evidence that the Standard Model (SM) needs to be extended stems from its inability to provide an answer to questions deeply connected with CP violation, such as the baryonic asymmetry of the Universe. In this thesis, we study SM extensions with vectorlike quarks, giving special attention to Trini- fication models. In these Grand Unified Theories (GUTs), fermions are assigned to fundamental representations of the group E6, leading naturally to the appearance of one vectorlike quark in each generation of fermions. As a consequence of the mixing that may occur between vectorlike quarks and SM quarks, new sources of CP violation arise. Moreover, this mixing affects the couplings of SM quarks to other particles and, for this reason, we will look at the experimental constraints imposed on vectorlike quarks. We also discuss the possibility of having spontaneous CP violation (SCPV) in Trinification models. While in the SM CP violation can only occur at the Lagrangian level, some theories that predict the existence of a bigger number of scalar fields (such as Trinification models) allow CP to be broken spontaneously by the vacuum. In this context, we verify that the introduction of vectorlike quarks in a Trinification model with SCPV leads to a complex Cabibbo-Kobayashi-Maskawa matrix, in agreement with experimental data.
Keywords: Cabibbo-Kobayashi-Maskawa matrix, CP violation, Grand Unified Theories, Trinifi- cation models, Vectorlike quarks
v vi Contents
Acknowledgements i
Resumo iii
Abstract v
List of Figures ix
List of Tables xi
List of Abbreviations xiii
1 Introduction 1 1.1 The discrete symmetries C, P, CP and T ...... 2 1.2 CP violation in the SM ...... 4 1.3 From the SM to GUTs ...... 5
2 The Standard Model of Particle Physics 7 2.1 Field content and Lagrangian ...... 7 2.2 The Higgs mechanism ...... 10 2.3 Charged and Neutral interactions ...... 11 2.4 Fermion masses and mixing ...... 12 2.5 Open questions in Particle Physics ...... 16
3 SM extensions with Vectorlike Quarks 19 3.1 Vectorlike quark representations ...... 20
3.2 SM extensions with SU(2)L singlets of vectorlike quarks ...... 20 3.2.1 Mass matrix diagonalization ...... 23 3.2.2 Quark mixing and suppression of FCNCs ...... 25 3.2.3 Complex phases of the generalized CKM matrix ...... 25 3.3 Mixing with third generation of SM quarks ...... 26 3.4 Experimental status of vectorlike B quarks ...... 27 3.4.1 Production at the LHC and mass constraints ...... 27 3.4.2 Mixing constraints ...... 29
4 Trinification Models 31 4.1 Trinification group and field content ...... 32 4.2 Scalar Lagrangian and SSB ...... 34 4.3 Fermion masses and mixing ...... 42
vii 5 Concluding Remarks 47
Bibliography 49
A Feynman rules for Vectorlike Quarks 53
B Representations of the Trinification group 55
B.1 The G333 Generators ...... 55 B.2 24 and 27 Representations ...... 56
viii List of Figures
1.1 The status of weak interactions in 1966 ...... 1 1.2 Parity transformation ...... 2 1.3 Charge conjugation ...... 3 1.4 CP transformation ...... 3
2.1 Unitarity triangle ...... 16 2.2 Experimental constraints on the (ρ, η) plane to the CKM parameters ...... 17
3.1 Vectorlike B quark production ...... 27 3.1(a) Pair production ...... 27 3.1(b) Single production involving a Z boson ...... 27 3.1(c) Single production involving a W boson ...... 27 3.2 Branching ratios of the B quark ...... 28 3.3 Lower limits for the B quark mass in the branching ratio plane BR(B → W t) versus BR(B → Hb) ...... 28 3.3(a) Observed 95% Confidence Level ...... 28 3.3(b) Expected 95% Confidence Level ...... 28 3.4 Experimental constraints on the mixing between b and B quarks ...... 30
ix x List of Tables
2.1 The SM Gauge fields and their representation properties...... 7 2.2 Fermionic fields of the SM ...... 8 2.3 Fit results for the Wolfenstein parameters ...... 16
3.1 SM predictions and experimental results for the observables affected by changing the Zbb coupling ...... 29
4.1 Numerical values of the scalar potential and VEV parameters compatible with the
observation of a SM Higgs boson with mass mH ∼ 125 GeV ...... 40 4.2 Masses of quarks computed at the electroweak scale ...... 44 4.3 Best fits of the effective down-quark matrix of the Trinification model totheCKM parameters and down-quark masses ...... 46
xi xii List of Abbreviations
CC Charged Current
CKM Cabibbo-Kobayashi-Maskawa
CP Charge conjugation and Parity
EWSB Electroweak Symmetry Breaking
FCNC Flavor Changing Neutral Current
NC Neutral Current
QFT Quantum Field Theory
SCPV Spontaneous Charge conjugation and Parity Violation
SM Standard Model
SSB Spontaneous Symmetry Breaking
VEV Vacuum Expectation Value
xiii xiv 1 Introduction
While reflecting upon the status of weak interactions at the 1966 Berkeley Conference, Nicola Cabibbo used the cartoon shown in Fig. 1.1 to reinforce the idea that besides some good progresses were being made on the subject of weak interactions, there was a big problem to be solved.
Fig. 1.1: The status of weak interactions in 1966 according to N. Cabibbo (image taken from [1]).
The cartoon depicts pretty well how Charge-Parity (CP) violation was a big mystery of weak interac- tions two years after its discovery. But since then, more than half a century has passed and one may ask: Do we understand CP violation nowadays? We could start answering by saying that our knowledge on the subject has increased significantly. For many years we could only observe CP violation in the kaon system but today, we are able to observe it in the B-meson system too. Moreover, the explanation found in the Standard Model (SM) for CP violation in terms of the Cabibbo-Kobayashi-Maskawa (CKM) mechanism is in agreement with all measurements made up to date on those systems [2]. However, we should say at this point that the SM is not a complete theory and it leaves many open questions in Particle Physics, some concerned with CP violation. For example, we know that it occurs, but there is no definite explanation for its origin. On the other hand, it is well established that CP Violation is a crucial ingredient for baryogenesis [3], the process of dynamically generating the matter-antimatter asymmetry of the Universe. In the SM, the CKM mechanism fails to accommodate the observed asymmetry by several orders of magnitude [4]. We then conclude that the cartoon of Figure 1.1 is still relevant today, in spite of the discoveries we have made so far about CP violation. It also reveals the importance the subject of CP violation has. After all, if there was no matter-antimatter asymmetry in our Universe, we would not be here to
1 discuss it. The need of finding new sources of CP violation is the main motivation for us tostudySM extensions with vectorlike quarks. Even though there is no experimental evidence for their existence, there are several reasons for these particles to receive a lot of attention nowadays [5]. In a particular set of SM extensions known as Grand Unified Theories (GUTs), vectorlike quarks may arise naturally, leading to new sources of CP violation. This is what happens in Trinification models, which we will study in greater detail. The main purpose of this chapter is to introduce some of the most important concepts of this thesis. In the following section, we introduce the discrete symmetries C, P, CP and T, so that we can understand better what is CP violation and how it was discovered in the neutral kaon system. After that, we present briefly the development of the CKM mechanism over time and its importance for the construction of the SM. Since we will be mostly interested in Trinification models, the most fundamental properties of GUTs are introduced.
1.1 The discrete symmetries C, P, CP and T
In Quantum Field Theory (QFT), discrete symmetries under transformations of Parity (P), Charge conjugation (C) and Time reversal (T) play a fundamental role and our interest on them grew as we realized they could be broken. In order to better visualize these transformations, we will recreate their effect on M. C. Escher’s painting Day and Night whenever possible. For instance, we represent in Fig. 1.2 a Parity transformation, which corresponds to an inversion of all spatial coordinates, (x, y, z) → (−x, −y, −z).
Fig. 1.2: An example of a Parity transformation.
Notice that we obtain the image we would see of the painting reflected on a mirror. Although there is only one dimension involved, the inversion of all spatial coordinates in three dimensions can also be achieved through a mirror reflection, followed by a 180◦ rotation. Therefore, the question one might ask regarding parity symmetry is whether, for a given event, its reflected image can also occur in Nature or not. For a long time, physicists took for granted that all laws of physics conserve parity. However, this was questioned in 1956 by T. D. Lee and C. N. Yang [6] when they came up with the idea that weak interactions may violate that symmetry. Among several ideas to test their hypothesis, one based on the β-decay of Cobalt-60 was performed in 1957 by Wu et al. [7], who found a preferred direction for the emission of β radiation. This result clearly indicated that P may be violated, as Lee and Yang suggested. These two physicists were awarded with the Nobel prize in 1957 for their proposal of P violation in weak interactions. Symmetry under Charge conjugation means that particles and antiparticles behave in a similar way. In contrast with P and T, the C symmetry can only be defined in the context of a QFT, where antiparticles arise from the interpretation of negative energy solutions of the Dirac equation. The effect of C is illustrated in Fig. 1.3, through exchange of black and white (which represent opposite
2 charges) on the original painting Day and Night.
Fig. 1.3: An illustration of Charge conjugation, obtained by exchanging black and white on the original painting.
Soon after the discovery of Parity violation, it was shown by T. D. Lee, R. Oehme and C. N. Yang [8] that the breaking of C is implied in weak interactions if P is not conserved. On the other hand, Wu’s experiment did not find any evidence for violation of symmetry under C and P combined. This transformation we have mentioned is what we call CP transformation and its effect is illustrated in Fig. 1.4.
Fig. 1.4: CP transformation.
At this stage, we should emphasize that gravity, electromagnetic and strong interactions are invariant under all the discrete transformations we have mentioned. Until some years after the discovery of Parity violation, there was no evidence for CP breaking on weak interactions. Against the belief of most physicists, this scenario would change in 1964 when J. Christenson, J. Cronin, V. Fitch and R. Turlay [9] discovered CP violation in weak interactions. It is known experimentally that the decays of neutral kaons may occur with two different lifetimes,
0 −9 0 −6 τ(KS → 2π) = 9 × 10 s , τ(KL → 3π) = 5 × 10 s , where the subscripts S and L stand for ’short’ and ’long’, respectively. The final states of those decays transform differently under a CP transformation:
CP|ππi = |ππi , CP|πππi = −|πππi .
0 0 In other words, if there was CP conservation KL could only decay into three pions and KS into two. 0 In spite of occurring with a small branching ratio, it was shown that the process KL → 2π may also occur, confirming that CP is not a symmetry of weak interactions. For this discovery, Croninand Fitch received the Nobel Prize in 1980. Finally, a Time reversal transformation consists on the inversion of the time coordinate, t → −t. Invariance under this operation is observed if, for a given event, it is possible to find a similar one but evolving backwards in time. A very important connection between this discrete symmetry and the
3 remaining ones is established by the CPT Theorem. Demonstrated in the fifties by G. Lüders [10], W. Pauli [11] and J. Bell [12], this theorem states that the combination of C, P and T (CPT) must be conserved for any QFT. This demand of CPT conservation allows one to establish many relations between discrete symmetries. For instance, if one finds CP violation in weak interactions, the T symmetry must also be broken according to this theorem, as it was verified experimentally in 2012 on the B-meson system by the BaBar collaboration [13]. Then, we conclude that of all seven combinations involving the three discrete symmetries, CPT is the only one that might be conserved by all symmetries, because all others may be broken in weak interactions. In fact, no violation of CPT has been found so far [2].
1.2 CP violation in the SM
The discovery of CP violation in 1964 was not expected by theoretical physicists and it would not be an easy task to understand how one could incorporate it in the SM. In 1973, a successful explanation was put forward by M. Kobayashi and T. Maskawa [14] for the occurrence of CP violation in terms of the so-called CKM mechanism, which is closely connected with quark mixing in the SM. After the success of Fermi’s theory on the explanation of β-decay, physicists tried to extend it to other decays. In order to apply this universality to weak interactions, one should be able to describe all decays in terms of the same coupling constant as the one introduced by Fermi (we will call it g). Before proceeding, we must note that we are designating by β-decay the process
n → p + e + νe , (n = udd, p = uud) .
Nowadays, we understand this decay as one where a transition from a down-quark d to an up-quark u (u ↔ d) is involved. This transition forms a charged weak current (mediated by W ±) like all the others we will mention. For leptonic charged currents, it was straightforward to extend Fermi’s theory. Only transitions involving particles of the same generation (e ↔ νe and µ ↔ νµ) were observed, and they could be characterized nearly by the same coupling constant g as the one discovered by Fermi. However, the addition of the second generation of quarks (formed by c and s ones) would not be so straightforward. Since decays like
Λ → p + e + νe , Λ = (uds) , are observed, where a u ↔ s transition occurs, we conclude that the theory of weak interactions must accommodate transitions between quarks of different generations. In this context, we assume that u and c quarks do not couple to d and s respectively, but to superpositions of those two:
0 0 d = d cos θC + s sin θC , s = −d sin θC + s cos θC .
In other words, we say that the weak eigenstates d0 and s0 which participate on weak interactions can be obtained through a rotation by an angle θC (the Cabibbo angle) of the mass eigenstates d and s. This idea of quark mixing was introduced by N. Cabibbo [15] in 1963, and it was extremely important to restore universality of weak interactions. Indeed, decays with strangeness violation (as the one we presented above for Λ) are much more rare than others involving just one generation of particles. Since the probability of some given decay to occur is proportional to the square of its coupling constant, there was some difficulty to reconcile universality with strange particle decays. But
4 according to Cabibbo’s theory, all transitions we mentioned so far are related to g as follows:
e ↔ νe, µ ↔ νµ ⇒ g , u ↔ d ⇒ g cos θC , u ↔ s ⇒ g sin θC .
All the above predictions are supported by experimental results, so we can confirm that weak inter- actions can be characterized by a single constant g. Nonetheless, it is important to emphasize that ◦ the Cabibbo angle is not predicted by the SM, hence, its numerical value (θC ' 13 ) is fitted to be in agreement with experiment. The c quark and its coupling with s0 were postulated some years after Cabibbo’s work in the framework of the Glashow-Iliopoulos-Maiani (GIM) mechanism [16], which explains the suppression of Flavor Changing Neutral Currents (FCNCs) in the SM. But more important, two-quark mixing does not allow for CP violation. As it was found by Kobayashi and Maskawa in 1973, the idea of quark mixing had to be extended to three generations of quarks in order to get CP violation. In this case, instead of just one angle, the unitary transformation of quark fields requires at least three angles and one complex phase. Although we will come back to this in Chapter 2, we can already anticipate that this phase gives rise to CP violation in the SM. The validity of all the above obviously depended on the discovery of a third generation of quarks. The first evidence for its existence came out in 1977, whenthe b quark was discovered by the L. Lederman team at Fermilab [17]. The search for the top quark would take much longer, but it was finally found in 1995 by the CDF [18]O andD / [19] collaborations. This confirmed the prediction made by Kobayashi and Maskawa, who were awarded with the Nobel prize in 2008 for their successful explanation of CP violation.
1.3 From the SM to GUTs
The construction of the SM relies on the principle that its Lagrangian must be invariant under a specific set of local symmetries established by a gauge group. In order to guarantee localgauge invariance, one must introduce gauge fields in the theory, which will act as mediators of interactions. A theory with these properties is designated by gauge theory and they have been playing a fundamental role on the description of interactions between elementary particles. The simplest example of a gauge theory is Quantum Electrodynamics (QED), which describes electromagnetic interactions between photons and electrically charged matter. This theory is based on the Abelian group U(1), meaning that the Lagrangian of QED must be invariant under field transformations of the form ψ → eiqα(x)ψ, where α(x) is the phase in a point x of spacetime and q is the electric charge. The photon field is introduced in order to preserve gauge invariance. This procedure can be generalized for non-Abelian groups, namely SU(n). Gauge theories based on these groups are often designated by Yang-Mills theories [20] and they have applied in the description of weak and strong interactions. The first step for the formulation of the SM was taken in 1961 by S. Glashow [21], whenhepro- posed a unified theory of weak and electromagnetic interactions based on the gauge group SU(2)⊗U(1). When this was done, there was a big issue around it. The gauge bosons introduced in the theory were all massless and one could not add to the Lagrangian mass terms for the mediators of weak inter- actions. The Lagrangian would be no longer gauge invariant and, consequently, one could not make any predictions based on the theory because it would not be renormalizable. This was the motivation to introduce Spontaneous Symmetry Breaking (SSB) in the SM. Discovered by Y. Nambu [22] in the framework of Particle Physics, SSB consists on the breaking of the gauge symmetry by the vacuum state of the theory, generating in this way mass terms for gauge bosons without breaking Lagrangian
5 invariance. For some time, it was thought one could not use this in the SM since SSB gives rise to massless bosons, not observed experimentally. In fact, this is a consequence of the Goldstone theo- rem [23], which establishes the appearance of massless scalar particles (known as Goldstone bosons) in the same number as the number of generators of the broken gauge group. However, we can solve this problem through the Higgs mechanism [24–27]. Proposed in 1964, it consists on the appearance of mass terms for gauge bosons after SSB, thanks to the (would-be) Goldstone bosons which one expects to arise from the breaking. In other words, the latter are not observed because they are ’absorbed’ as longitudinal degrees of freedom by the particles that become massive. The Higgs mechanism was later incorporated in Glashow’s theory by S. Weinberg [28] and A. Salam [29] in order to solve the problem of the massless gauge bosons associated to weak interactions. Finally, G. t’Hooft and M. Veltman [30] proved in 1971 that gauge theories with SSB (as the one built by Glashow, Weinberg and Salam) are renormalizable, establishing this way the SM of Particle Physics. Extending the gauge group of electroweak interactions with an extra SU(3) for Quantum Chro- modynamics (QCD), which describes strong interactions, everything we said about the SM in this section can be summarized by:
SU(3) ⊗ SU(2) ⊗ U(1) −−→SSB SU(3) ⊗ U(1) | {z } | {z } |{z} QCD Electroweak QED i.e., for relatively high energies, one has a gauge theory invariant under the group SU(3)⊗SU(2)⊗U(1), in which electromagnetic and weak forces are unified. After SSB, the mediators of weak interactions become massive and the gauge group becomes SU(3)⊗U(1). The subgroup SU(3) remains unbroken, so the mediators of strong interactions (the gluons) are massless, whereas the electroweak group is broken into U(1), which is the group of QED that we described above. Despite the success of the SM, it was soon realized that perhaps the SM was not the ultimate theory of Nature. One of the reasons for this belief motivated the appearance of GUTs and it is related to the lack of unification between strong and electroweak interactions in the SM. Then, asthe name suggests, GUTs are SM extensions where the gauge group SU(3)⊗SU(2)⊗U(1) is embedded into a larger one, such as SU(5) [31], SO(10) [32], E6 [33], or the one which characterizes trinified models, SU(3)⊗SU(3)⊗SU(3) [34]. Since the GUT gauge group is always larger than the SM group, more gauge bosons have to be introduced to preserve local gauge invariance. Furthermore, the representations of these groups have more components than the SM fields can fill up, so new scalars and fermions mustbe added as well. These postulated particles acquire masses of the order of the unification scale through SSB of the GUT group, leaving SU(3)⊗SU(2)⊗U(1) as a residual symmetry, so that all phenomena observed at low-energy scales is still (successfully) explained by the SM. In general, GUTs are able to solve many problems in the SM, making at the same time predictions that were not observed experimentally up to date. Although we will focus on the issues of CP Violation, we will be able to observe some of these aspects of GUTs by studying Trinification models. But before that, we review the SM in Chapter 2, where all concepts introduced above will be explained more rigorously. After that, Chapter 3 is dedicated to vectorlike quarks, and to show why are these particles so attractive. We will study their properties, their present experimental status, and some examples of SM extensions where these particles are postulated. In Chapter 4 we study Trinification models, where we will see how vectorlike quarks naturally arise. Finally, in Chapter 5, we draw the main conclusions of our work.
6 The Standard Model 2 of Particle Physics
Since its formulation in the sixties by Glashow, Weinberg and Salam, the SM has survived to many experimental tests and it is still the best theory we have to describe strong, weak and electromagnetic interactions. Our goal in this chapter is to review the SM. We will explore in more detail the properties of the CKM matrix, due to its connection with CP violation. In the end, we discuss briefly some of the open questions in Particle Physics, which call for extensions of the SM.
2.1 Field content and Lagrangian
As seen in Section 1.3, the SM is a gauge theory based on the group
GSM = SU(3)c ⊗ SU(2)L ⊗ U(1)Y , (2.1) where the subscripts c, L and Y label each subgroup corresponding to color, left-handedness and hypercharge, respectively. We characterize all SM fields by their quantum numbers (r3, r2, y), so that each field is assigned to the representation r3 of SU(3)c and r2 of SU(2)L. Its hypercharge y is an eigenvalue of the operator
Y = Q − T3 , (2.2) where Q is the electric charge and T3 the third component of weak isospin.
In order to guarantee gauge invariance under GSM, each of its subgroups has associated to it the same number of gauge fields as the number of generators, corresponding to eight, three and onefor SU(3), SU(2) and U(1) respectively. Then, we conclude that there are twelve gauge bosons in the SM, assigned to the adjoint representations, as presented in Table 2.1. Regarding the fundamental constituents of matter, quarks and leptons, their fields are grouped in three generations with identical properties (apart from their masses), described by their left and
Group Gauge fields (r3, r2, y)
a SU(3)c Gµ (a = 1,..., 8) (8,1,0) i SU(2)L Wµ (i = 1, 2, 3) (1,3,0)
U(1)Y Bµ (1,1,0)
Table 2.1: The SM Gauge fields and their representation properties.
7 Generations Notation i = 1 i = 2 i = 3 (r3, r2, y)
α α α α u u c t qα = i (3, 2, 1/6) iL d d s b Quark i L L L L uα uα cα tα (3, 1, 2/3) fields iR R R R dα dα sα bα (3, 1, −1/3) iR R R R νli νe νµ ντ Lep- `i = (1, 2, −1/2) L l e µ τ ton i L L L L
fields liR eR µR τR (1, 1, −1)
Table 2.2: Fermionic fields of the SM. The α index in quark fields accounts for their color.
right-handed components,
1 − γ5 1 + γ5 ψL = ψ , ψR = ψ , γ5 = iγ0γ1γ2γ3 , (2.3) 2 2 where γi are the Dirac matrices. These chiral fields ψL,R transform differently under the subgroup
SU(2)L of GSM. This is illustrated in Table 2.2, where we also establish the notation used from now on to describe the whole fermionic content of the SM. The field hypercharges given in Table 2.2 were computed using Eq. (2.2) considering that:
2 1 Q(ν ) = 0 ,Q(l ) = −1 ,Q(uα) = ,Q(dα) = − , li i i 3 i 3 (2.4) 1 1 T (ν , u ) = ,T (l , d ) = − ,T (l , u , d ) = 0 . 3 liL iL 2 3 iL iL 2 3 iR iR iR
In order to generate mass for gauge bosons and fermions through Electroweak Symmetry Breaking (EWSB), we also need to add a complex scalar doublet to the SM: ! ! φ+ 1 ξ + iξ Φ = = √ 1 2 , Φ ∼ (1, 2, 1/2) , 0 (2.5) φ 2 ξ3 + iξ4 with φ+ and φ0 being, respectively, the charged and neutral components of the field Φ. Along with the addition of gauge fields, interactions between them and the other SM fields arise from demanding local gauge invariance. These interactions are read from the covariant derivatives,
8 3 X a a X i i 0 Dµ = ∂µ − igc GµT − ig WµI − ig BµY, (2.6) a=1 i=1 which must replace the usual partial derivatives in the scalar and fermion kinetic terms. The param- 0 eters gc, g and g are the coupling constants of the SU(3)c, SU(2)L and U(1)Y groups, respectively. Depending on the representation assignments r2,3, one must perform the following replacements:
a i a λ i τ r3 = 3 → T = , r2 = 2 → I = , 2 2 (2.7) a i r3 = 1 → T = 0 , r2 = 1 → I = 0 .
In other words, T a and Ii are the generators of the SU(3) and SU(2) groups on a given representation.
8 If the fields are assigned to the fundamental representation of these groups, λa and τ i are defined by the Gell-Mann and Pauli matrices, respectively. In contrast, T a and Ii are zero if the fields transform as singlets under SU(3) and SU(2). All information about the field dynamics and interactions is encoded in the full SM Lagrangian1:
LSM = LGauge + LΦ + LMatter + LYuk . (2.8)
Here, LGauge comprises the gauge kinetic terms. The pure gauge-field terms which preserve gauge invariance are provided by the field strength tensors,
a a a abc b c Gµν = ∂µGν − ∂ν Gµ + gsf GµGν ,
i i i ijk j k Wµν = ∂µWν − ∂ν Wµ + g WµWν , (2.9)
Bµν = ∂µBν − ∂ν Bµ ,
abc ijk where f are the structure constants of SU(3) and is the rank-3 Levi-Civita tensor. LGauge is then written as: 1 a aµν 1 i iµν 1 µν − LGauge = G G + W W + Bµν B . (2.10) 4 µν 4 µν 4 Further inspection of this Lagrangian reveals that it contains not only the kinetic terms, but also interactions involving exclusively gauge fields.
The most general scalar Lagrangian LΦ invariant under GSM is:
µ † LΦ = (D Φ) (DµΦ) − V (Φ) , (2.11) where V (Φ) is the Higgs potential, which contains all non-kinetic dim-4 combinations of the scalar field. Namely, in the SM, 2 † † 2 V (Φ) = µ Φ Φ + λ Φ Φ . (2.12)
The last two pieces of LSM are concerned with fermions. LMatter contains their kinetic terms, interactions with gauge bosons, and it can be written in the form LMatter = Lq + L` (the subscripts q and ` stand for ’quarks’ and ’leptons’, respectively) with2
3 X µ µ µ Lq = i qjLγ DµqjL + ujRγ DµujR + djRγ DµdjR , (2.13) j=1 3 X µ µ L` = i `jLγ Dµ`jL + ljRγ DµljR . (2.14) j=1
Notice that all fermions of the SM are massless with the Lagrangians we have defined at this point. In fact, mass terms for fermions cannot be added to the Lagrangian directly. Since left and right-handed components of fermions transform differently under SU(2)L, a Dirac mass term,
mDirac = −m(f LfR + f RfL) , (2.15) for a fermion f would break explicitly the gauge symmetry. However, these particles may acquire mass through SSB due to Yukawa interactions, in which fermions couple to the Higgs field. These
1 Gauge fixing and Fadeev-Popov terms are being omitted in LSM. 2Since we are focusing only on electroweak interactions, we omit color indices in quark fields. We also use the definition ψ ≡ ψ†γ0.
9 couplings can be read from the Yukawa Lagrangian,
u d ` − LYuk = Yij qiLΦ ujR + Yij qiLΦ djR + Yij `iLΦ ljR + H.c., (2.16)
∗ u d l where Φ = iτ2Φ and Yij , Yij and Yij are general complex 3 × 3 matrices, which contain most of the free parameters of the SM.
2.2 The Higgs mechanism
The SM vacuum is identified by minimizing the potential V (Φ) of Eq. (2.12). The vacuum energy must be non-zero in order to achieve SSB. To ensure charge neutrality, only neutral scalar fields can acquire a non-vanishing Vacuum Expectation Value (VEV), which is possible for the lower isospin component of the Higgs doublet, ! 0 h0|Φ|0i ≡ hΦi = . (2.17) v
Hence, the minimum of the potential is given by the condition
r ∂V (Φ) µ2 = 0 ⇒ |v| = − . (2.18) ∂Φ hΦi 2λ
In any case, one must consider λ > 0, otherwise the potential would not be bounded from below. Regarding the parameter µ, one must consider µ2 < 0, otherwise the VEV vanishes and SSB does not occur. Of the four gauge fields associated to SU(2)L ⊗ U(1)Y , three become massive and, by the Goldstone theorem, a residual unbroken symmetry must exist. This can be found by analyzing the symmetries of the vacuum. Given a gauge transformation eiαGhΦi ' (1 + iαG)hΦi, the vacuum is invariant under G if GhΦi = 0. This is not verified for any generator of SU(2) ⊗ U(1) as long as the VEV is non-zero. However, considering the action of the electric charge operator defined through Eq. (2.2) on hΦi, we notice that " ! !# ! ! ! ! h τ3 i 1 1 0 1 1 0 0 1 0 0 0 QhΦi = Y (Φ) + hΦi = + = = . (2.19) 2 2 0 1 2 0 −1 v 0 0 v 0
This reveals that the generator of the unbroken symmetry is the electric charge and, since it is diagonal, it must be associated to a U(1) group. This proves what we pointed out in Chapter 1 regarding EWSB, when we said that the electroweak group SU(2) ⊗ U(1) is broken into the U(1) of electromagnetism. Before proceeding, it is important to check that if one applies the obtained electric charge operator on the scalar doublet defined in Eq. (2.5), we get: ! ! ! 1 0 Φ+ Φ+ Q Φ = = . (2.20) 0 0 Φ0 0
This confirms that Φ+ is indeed a charged scalar field whereas Φ0 is neutral. Considering now small oscillations around the vacuum, we reparametrize Φ as: ! ξ(x)a τ a 0 Φ(x) = exp i H(x) , (2.21) v 2 v + √ 2 where H(x) and ξ(x)a are real scalar fields. Before EWSB, one is free to perform gauge transformations as long as they leave the Lagrangian invariant. Then, three of the four degrees of freedom owned by
10 the field Φ can be removed by performing an SU(2) transformation which cancels the exponential of Eq. (2.21): ! ξ(x)a τ a 0 Φ(x) → exp −i Φ(x) = H(x) . (2.22) v 2 v + √ 2 This transformation take us to the unitary gauge, with only one physical field H(x) (the Higgs boson field) remaining after EWSB. Disregarding constant, kinetic and interaction terms, we obtaingauge boson and Higgs mass terms by replacing the transformed Φ of Eq. (2.22) in LΦ. With this, ! ! 1 1 g2 −gg0 W 3 LW,Z,H = g2v2(W 1µW 1 + W 2µW 2) + v2 3µ µ µ − µ2H2 . Mass µ µ W B 0 02 (2.23) 2 4 −gg g Bµ
The mass spectrum of the gauge bosons can be read defining the mass eigenstates
1 2 3 3 ± Wµ ∓ Wµ Aµ = sin θW W + cos θW Bµ ,Zµ = cos θW W − sin θW Bµ ,W = √ , (2.24) µ µ µ 2 where θW is the weak mixing angle, related to the coupling constants by
g0 g sin θW = , cos θW = . (2.25) pg2 + g02 pg2 + g02
± 3 Rewriting Eq. (2.23) in terms of Aµ, Zµ and Wµ one finds :
2 2 2 2 02 W,Z,H g v µ+ − 1 v (g + g ) µ 1 2 2 L = W W + Z Zµ − 2µ H . (2.26) Mass 2 µ 2 2 2
From here, it is clear that the gauge bosons W ±, Z and the Higgs Boson H acquire masses
r 2 02 gv g + g gv p 2 mW ± = √ , mZ = v = √ , mA = 0 , mH = −2µ . (2.27) 2 2 2 cos θW
The beauty and success of the Higgs mechanism is evident from Eq. (2.27). It not only generates ± mass for the three gauge fields (Zµ and Wµ ), but it also keeps the photon field Aµ massless. One also verifies by counting the number of degrees of freedom (twelve before and after SSB) that inlinewith what was discussed in Section 1.3, the three Goldstone bosons ’gauged-away’ in Eq. (2.22) re-emerged as longitudinal polarizations of the massive gauge bosons.
2.3 Charged and Neutral interactions
Expanding the covariant derivatives in Eq. (2.14) in terms of the mass eigenstates defined in Eq. (2.24), and omitting the couplings with gluons, we can express the fermion Lagrangian as
f f f f L = Lkin + LCC + LNC , f = q, ` . (2.28)
f f The first term corresponds to kinetic terms of fermionic fields, whereas LCC and LNC are related to charged currents (CC) and neutral currents (NC), respectively. The designations ’charged’ and ’neutral’ stand in this context for the electric charge of gauge bosons involved in the interactions with
3 1 In contrast with both the Higgs Boson and the Z, we cannot remove a factor of 2 from the mass term of the W because this is a complex field.
11 ± f fermions. Then, interactions involving the Wµ fields stem from the Lagrangian LCC as
` g µ − L = √ ei γ W νiL + H.c., (2.29) CC 2 L µ
q g µ + L = √ ui γ W di + H.c.. (2.30) CC 2 L µ L
On the other hand, the couplings of fermions to the photon field Aµ and the Zµ field are encoded in f LNC. These are
` µ g X µ i i LNC = Q(ei) g sin θW eiγ Aµei + ψiγ (gV − gAγ5)Zµψi , (2.31) cos θW ψi=ei,νi
q X µ g X µ i i LNC = e Q(ψi) ψiγ Aµψi + ψiγ (gV − gAγ5)Zµψi , (2.32) cos θW ψi=ui,di ψi=ui,di where the first term of Eq. (2.31) was written in such a way that one can find the relation between the electric charge e and the weak mixing angle:
0 e = g sin θW = g cos θW . (2.33)
In Eq. (2.32) we have introduced the vector (V ) and axial-vector (A) couplings of the Z boson to fermions, which are respectively defined as
i 2 i gV ≡ T3 − 2 Qi sin θW , gA ≡ T3 , (2.34) being the charges Q(ui) and Q(di) given in Eq. (2.4). Defining the electromagnetic currents for quarks and leptons as q µ X µ ` µ µ Jem = Q(ψi) ψiγ ψi ,Jem = Q(ei) eiγ ei , (2.35)
ψi=ui,di it will be useful for us to write the NC Lagrangians in the following way:
` ` µ g h µ µ 2 ` µi LNC = eJem Aµ + νiLγ νiL − eiLγ eiL − 2 sin θW Jem Zµ , (2.36) 2 cos θW
q q µ g µ µ 2 q µ LNC = eJem Aµ + uiLγ uiL − diLγ diL − 2 sin θW Jem Zµ . (2.37) 2 cos θW
2.4 Fermion masses and mixing
After EWSB, the Yukawa Lagrangian of Eq. (2.16) becomes H u d l −LYuk = v + √ (ui Y uj + di Y dj + li Y lj ) + H.c. 2 L ij R L ij R L ij R (2.38) H ≡ LMass + Lint .
H While the Lagrangian Lint contains the couplings of fermions to the Higgs boson, the mass Lagrangian LMass is given by u d l − LMass = uiLMijujR + diLMijdjR + liLMijljR + H.c., (2.39) where we have introduced the mass matrices
u u d d l l Mij = vYij ,Mij = vYij ,Mij = vYij . (2.40)
12 Notice that there is no mass matrix for neutrinos. As one can see from Table 2.2, right-handed neutrino fields are not included in the SM. Thus, taking Eq. (2.15) into account, one isnotableto construct Dirac neutrino mass terms in this context4. This leads us to conclude that neutrinos are massless in the SM. In general, mass matrices like those defined in Eq. (2.39) are general complex matrices. Asa consequence, we need to transform the fields in order to bring the mass terms to the physical basis, i.e. u 0 u 0 uL → VL uL , uR → UR uR , d 0 d 0 dL → VL dL , dR → UR dR , (2.41) l 0 l 0 lL → VL lL , lR → UR lR , where V u,d,l and U u,d,l are unitary matrices which relate flavor and mass eigenstates (primed fields). These have been chosen such that they diagonalize M u, M d and M l through the following bi-unitary transformations: u† u u u VL M UR = diag(mu, mc, mt) ≡ Mf , d† d d d VL M UR = diag(md, ms, mb) ≡ Mf , (2.42) l † l l l VL M UR = diag(me, mµ, mτ ) ≡ Mf . Once we apply these transformations to Eq. (2.39), we arrive to:
3 X u 0 0 d 0 0 l 0 0 − LMass = Mfii uiLuiR + Mfii diLdiR + Mfii liLliR + H.c.. (2.43) i=1
The NC Lagrangians given in Eqs. (2.31) and (2.32) remain invariant under the same transformations of the fields. Thus, we conclude that there are no flavor changing neutral currents (FCNCs) attree level in the SM. Regarding leptonic CCs, the rotation of the fields in Eq. (2.29) leads to
` g 0 l ∗ µ − L = √ e V γ νiLW + H.c.. (2.44) CC 2 iL L µ
l 0 Since neutrinos are massless, one has freedom to perform the rotation νiL → V νiL, cancelling the ` effect of transforming the charged lepton fields. As aresult, LCC remains invariant and we conclude that there is no lepton mixing in the SM. The quark sector behaves differently, as we saw in Section 1.2. q Bringing the fields to the mass basis on LCC, given in Eq. (2.30), we find g Lq = √ u0 V u∗γµV dd0 W + + H.c. CC 2 iL L L iL µ Vud Vus Vub (2.45) g 0 µ 0 + u† d = √ u γ VCKMd W + H.c.,VCKM ≡ V V = V V V , 2 iL iL µ L L cd cs cb Vtd Vts Vtb where VCKM is the CKM matrix, which describes quark mixing in the SM. The appearance of CP violation in the CKM matrix can be understood by counting the number of independent parameters of a general complex square matrix U with dimension n2. Without any constraint, this matrix has 2n2 independent parameters. In order to make the matrix U unitary (UU † = 1), one has to impose n2 conditions, so the number of independent parameters becomes 2n2 − n2 = n2. Considering now an n × n orthogonal matrix (analog to an unitary matrix but with only real parameters), we can observe that it has n(n − 1)/2 Euler angles. This will be the number of
4For electrically neutral fermions like neutrinos, one could also build Majorana mass terms. However, the existence of these terms in the Lagrangian imply lepton number violation.
13 mixing angles we need to parametrize our unitary matrix, while the remaining n(n+1)/2 independent parameters are complex phases which arise in U. However, some of these phases are unphysical and can be removed. To see this, we must observe that the Lagrangian shown in Eq. (2.45) remains invariant under the quark rephasing
iθu iθu 0 † 0 Θu = diag e 1 ,..., e n , uL → uL Θu , uR → Θu uR , (2.46) 0 † 0 d d d → Θ d , iθ1 iθn dL → dL Θd , R d R Θd = diag e ,..., e , as long as we change U simultaneously, according to
0 † U = Θu U Θd . (2.47)
At this point, one might expect to identify 2n phases as unphysical. However, a global rephasing redefining all quarks by the same phase leaves U unchanged. As a result, we cannot use this transfor- mation to remove a phase from U, so our matrix has to be parametrized by n(n + 1)/2 − (2n − 1) = (n − 1)(n − 2)/2 complex phases. Our discussion regarding flavor mixing and CP violation in Sec- tion 1.2 becomes now clear. Indeed, a quark mixing matrix with only two generations (n = 2) needs to be parametrized by one angle only and no complex phases. In contrast, the existence of three generations of quarks (n = 3) forces us to parametrize the CKM matrix with three angles and one complex phase, which is responsible for all CP-violating phenomena in the SM. From all possible parametrizations for VCKM, the one proposed by Chau and Keung [35] became standard. Namely,
−iδ 1 0 0 c13 0 s13e c12 s12 0 VCKM = 0 c23 s23 0 1 0 −s12 c12 0 iδ 0 −s23 c23 −s13e 0 c13 0 0 1 (2.48) −iδ c12c13 s12c13 s13e −iδ −iδ = −s12c23 − c12s23s13e c12c23 − s12s23s13e s23c13 , −iδ −iδ s12s23 − c12c23s13e −c12s23 − s12c23s13e c23c13 where sij ≡ sin θij and cij ≡ cos θij control the mixing between generations i and j, and δ is the
CP-violating phase. Considering the parameter λ ≡ sin θC ' 0.22 (θC is the Cabibbo angle), and taking experimental data into account, the CKM matrix can be approximately written as
1 λ λ3 2 VCKM ' λ 1 λ . (2.49) λ3 λ2 1
Following the observed hierarchy among the elements of VCKM, Wolfenstein [36] developed a parametriza- tion (perhaps the most useful) defined in terms of powers of λ. Introducing the real parameters A, ρ and η, the mixing angles θij and the phase δ, these are expressed by [37, 38]:
|Vus| 2 Vcb iδ ∗ 3 s12 = λ = , s23 = Aλ = λ , s13e = V = Aλ (ρ + iη) , (2.50) p 2 2 ub |Vud| + |Vus| Vus to all orders in λ. The Taylor expansion of all the CKM matrix elements up to order O(λ4), obtained
14 after replacing Eqs. (2.50) into Eq. (2.48), gives the Wolfenstein parametrization of the CKM matrix:
1 − λ2/2 λ Aλ3(ρ − iη) 2 2 4 VCKM = −λ 1 − λ /2 Aλ + O λ . (2.51) Aλ3(1 − ρ − iη) −Aλ2 1
Observing now carefully the constraints we need to impose to guarantee the unitarity of a matrix,
† † X ∗ X ∗ VV = V V = 1 ⇒ VikVjk = VkiVkj = δij , (2.52) k k one finds six vanishing combinations (i 6= j) that express the orthogonality between two different columns or rows. These may be represented as unitarity triangles in the complex plane and, in spite of their different shapes, they have all the same area, equal to half of the Jarlskog invariant J [39]. This phase-convention independent measurement of CP violation is defined through
∗ ∗ X Im VijVklVil Vkj = J ikmjln , (2.53) m,n where ikm is the rank-3 Levi-Civita tensor. The Jarlskog invariant is written in terms of the standard and the Wolfenstein parametrizations as
2 2 6 J = c12c23c13s12s23s13 sin δ ' A λ , (2.54) thus we conclude that a non-vanishing CKM matrix necessarily requires J 6= 0. The most commonly used unitarity triangle is shown in Figure 2.1 and it arises from the condition,
∗ ∗ ∗ VudVub + VcdVcb + VtdVtb = 0 , (2.55)
∗ where we normalize each term to VcdVcb, so that one side is aligned with the real axis and its apex is given to all orders in λ by √ ∗ 2 VudV 1 − λ (ρ + iη) ρ + iη = − ub = √ √ . (2.56) ∗ 2 4 2 4 2 VcdVcb 1 − A λ + A λ 1 − λ (ρ + iη)
One can also notice that the series expansions of the parameters ρ and η we have introduced can be used to improve the unitarity of the Wolfenstein parametrization shown in Eq. (2.51), simply by doing the replacements ρ → ρ and η → η. Finally, we present some of the most important experimental data concerned with the CKM matrix. The fit results for the Wolfenstein parameters defined in Eq. (2.50) are shown in Table 2.3.Forthe magnitude of all nine CKM elements, the fit results are
+0.00011 0.97434−0.00012 0.22506 ± 0.00050 0.00357 ± 0.00015 |VCKM| = 0.22492 ± 0.00050 0.97351 ± 0.00013 0.0411 ± 0.0013 , (2.57) +0.00032 0.00875−0.00033 0.0403 ± 0.0013 0.99915 ± 0.00005
+0.21 −5 leading to the Jarlskog invariant J = 3.04−0.20 × 10 [2]. The global fit result is illustrated in Fig. 2.2, where the experimental constraints on all CP-violating observables are plotted in the (ρ, η) plane. Each shaded area represents the allowed region for an observable in that plane with 95% Confidence Level (CL) and one can check that all those areas overlap consistently around the global fit region.
15 Fig. 2.1: Unitarity triangle (taken from [2]).
Parameter Fit
λ 0.22506 ± 0.00050 A 0.811 ± 0.026 +0.019 ρ 0.124−0.018 η 0.356 ± 0.011
Table 2.3: Fit results for the Wolfenstein parameters. These values were obtained using the method described in [38].
To conclude, we must remind that in the SM, CP is explicitly broken at the Lagrangian level due to the existence of complex Yukawa couplings which lead to a complex CKM matrix in charged- weak interactions between quarks. Although this is the only way to break the CP symmetry in the SM, we must say at this stage that in the framework of gauge theories, one may also consider the possibility of having spontaneous CP violation (SCPV). This process was proposed for the first time by T. D. Lee [40] in the context of two-Higgs-doublet models and it occurs when CP is a symmetry of the original Lagrangian being spontaneously broken by the vacuum. One might question why we did not consider this scenario in the SM. The reason is that with just one scalar doublet in the theory, it is impossible to break CP spontaneously [41]. Therefore, this phenomenon is only possible in theories with extended scalar sectors like Trinification models. In Chapter 4, we will see how SCPV is accomplished in those models.
2.5 Open questions in Particle Physics
Along with the problem of matter-antimatter asymmetry of the Universe and the lack of gauge- coupling unification of strong and electroweak interactions, the SM presents much more issues than those presented already in Chapter 1. First of all, we are now in the position to enumerate the number of parameters we saw in our review for which the SM does not provide any prediction. These are the 0 three coupling constants gc, g and g , two scalar potential parameters (they can be chosen to be mH and v), six quark masses mu...b, three charged lepton masses me,µ,τ and finally, the four parameters of the CKM matrix A, λ, ρ and η. Of the eighteen parameters we have counted, one can check that most of them are related to fermions. Since the theory does not make any prediction for their masses, their origin remains a
16 Fig. 2.2: Experimental constraints on the (ρ, η) plane. The ring centered at (0,0) establishes the allowed region for the absolute value of the CKM matrix element Vub, and the ones centered at (1,0) 0 0 are associated to the mass differences ∆md and ∆ms between B and Bs mesons, respectively. The hyperbolic curves limit the allowed region of the CP-violating observable in the kaon sector k. The remaining regions belong to the angles of the unitarity triangle, namely α, β and γ. All the shaded areas have 95% CL (image taken from [2]). mystery. Moreover, the reason for us to find three generations of fermions is also unknown. Regarding the CKM matrix, one might also find an hierarchy between its elements through Eq. (2.49), for which there is no explanation. The number of open questions in the fermion sector grows even more when we look at neutrinos. As we have seen throughout this chapter, it is assumed that neutrinos are massless in the SM due to the absence of their right-handed component. However, the discovery of neutrino oscillations [42] gives a very strong evidence that they are massive. Although we will not pursue this subject further, it is worth to highlight that many GUTs present solutions to this problem, namely Trinification models. In order to complete the list of free parameters we find in the SM, we should still add a parameter that arises in the framework of QCD, which is connected with the strong CP problem. This is concerned with the absence of a CP-violating term in the strong sector of the SM, even though there is not any symmetry forbidding it. After seeing all the free parameters of the SM, we can still find other shortcomings of the SM, such as the lack of a description of gravity and a viable dark matter candidate, as well as the absence of a solution to the hierarchy problem. Concerning the latter issue (related to the fine-tuning of parameters required to keep the mass of the Higgs boson nearthe electroweak scale), it is interesting to notice that along with supersymmetry, there are other theories like Composite [43] and Little Higgs [44] models where vectorlike quarks are predicted. Finally, we mention one problem that is solved by GUTs in general, related to electric charge quantization. In the SM, the hypercharge is assigned in order for the electric charge to be the same as the one measured experimentally. This does not allow the SM to explain for instance, the fractional charges of quarks. In GUTs where the SM group is embedded into a bigger group, one gets constraints to the hypercharge generator, so this problem becomes solved.
17 18 SM extensions with 3 Vectorlike Quarks
Vectorlike quarks are hypothetical spin 1/2 particles that transform as triplets under the color gauge group but in contrast with SM quarks, their left and right-handed components have the same electroweak quantum numbers. These particles constitute a very promising playground for searches of new physics. First, they are the simplest example of colored fermions still allowed by experimental data. In fact, vectorlike quarks do not receive their mass from Yukawa couplings, thus they avoid the constraints from Higgs boson production that rule out a fourth generation of SM quarks1 [46]. Furthermore, vectorlike quarks can mix with the SM ones, so the latter may form together with W ±, Z or a Higgs boson the final states of vectorlike quark decays. The exact nature of that mixing depends on the model. Some theories postulate the existence of vectorlike quarks that could preferentially mix with first and second generation quarks [47, 48]. Although this mixing pattern is notexcluded by experimental data [49], there are many reasons to assume that vectorlike quarks couple almost exclusively to the heaviest SM ones, b and t. For example, this is the scenario we observe in composite Higgs models where vectorlike quarks are involved in a seesaw mechanism with the SM quarks [50].
Moreover, as discussed with more detail in [51], the strong hierarchy of SM quark masses (mu,d mc,s mb,t) suggests that Yukawa couplings of vectorlike quarks to b and t are much larger than the couplings to u, d, c and s. Predominant mixing with quarks of the third generation is also favored from the experimental point of view. Independently of the mixing pattern we observe, the addition of vectorlike quarks lead to new sources of CP violation [52, 53] and it is the simplest way of breaking the GIM mechanism, giving rise to non-vanishing, but naturally suppressed FCNCs at tree-level [54].
Before studying Trinification models, it is worth analyzing how we can build simpler theories where we just extend the quark sector of the SM with vectorlike quarks. This is the goal of this chapter, where we present in the end the mass and mixing constraints imposed experimentally to the vectorlike quark that is predicted in trinified models.
1We are just considering here theories where only the quark sector of the SM is extended. A fourth generation of quarks may still exist in theories with a larger scalar sector [45].
19 3.1 Vectorlike quark representations
Assuming that the scalar sector of our theory contains only SU(2)L doublets, new vectorlike quarks coupling to the SM ones can appear in the following multiplets:
TL,R ∼ (3, 1, 2/3) ,BL,R ∼ (3, 1, −1/3) ,
! ! ! X T B ∼ (3, 2, 7/6) , ∼ (3, 2, 1/6) , ∼ (3, 2, −5/6) , T B Y L,R L,R L,R (3.1)
X T T ∼ (3, 3, 2/3) , B ∼ (3, 3, −1/3) . B Y L,R L,R
The vectorlike quarks T , B, X and Y have electric charges
2 1 5 4 Q(T ) = ,Q(B) = − ,Q(X) = ,Q(Y ) = − , (3.2) 3 3 3 3 and we use Eq. (2.2) to compute the hypercharge of each multiplet. SM extensions with all the multiplets shown in Eq. (3.1) are studied for instance, in [55]. Since Trinification models (and other
GUTs based on E6) predict only SU(2)L singlets of B quarks, we focus from now on in SM extensions with vectorlike quark singlets.
3.2 SM extensions with SU(2)L singlets of vectorlike quarks
Our SM extension contains ng generations of SM quarks, as well as nT and nB SU(2)L singlets of vectorlike T and B quarks, respectively. We will denote these fermions as ! ui qiL = , ujR , djR , i, j = 1, . . . , ng , di L (3.3) TkL ,ToR , k, o = 1, . . . , nT ,
BrL ,BsR , r, s = 1, . . . , nB .
We also define Nu ≡ ng + nT and Nd ≡ ng + nB as the number of quarks with electric charge 2/3 and −1/3 respectively. The Lagrangians of charged and neutral currents formed by these particles are
q g µ + L = √ ui γ di W + H.c., (3.4) CC 2 L L µ
q µ g µ µ 2 µ LNC = eJemAµ + uiLγ uiL − diLγ diL − 2 sin θW Jem Zµ , (3.5) 2 cos θW
µ where Jem includes the electromagnetic currents involving vectorlike quarks,
2 J µ = u γµu + u γµu + T γµT + T γµT em 3 iL iL iR iR kL kL oR oR (3.6) 1 − d γµd + d γµd + B γµB + B γµB . 3 iL iL jR jR rL rL sR sR
20 In this model, one can find new Yukawa interactions in the quark sector. These are encoded inthe Yukawa Lagrangian, which can be expressed as
q u d u d − LYuk = Yij qiLΦ ujR + Yij qiLΦ djR + Yio qiLΦ ToR + Yis qiLΦ BsR + H.c.. (3.7)
The introduction of vectorlike quarks also allows us to construct SU(2) ⊗ U(1)-invariant mass terms that may be written directly in the Lagrangian, namely
inv ˆ T ˆ B T B − LMass = Mkj TkLujR + Mrj BrLdjR + Mko TkLToR + Mrs BrLBsR + H.c.. (3.8)
After replacing in Eq. (3.7) the scalar doublet by the one given in Eq. (2.21), we obtain the interactions between quarks and the Higgs boson,
q 1 u d u d − L = √ Y ui uj + Y di dj + Y ui To + Y di Bs H + H.c., (3.9) H 2 ij L R ij L R io L R is L R as well as additional quark mass terms,
SSB u d u d −LMass = mij uiLujR + mij diLdjR +m ˆ io uiLToR +m ˆ is diLBsR + H.c.. (3.10)
Here, the mass matrices are defined as
u u d d u u d d mij = vYij , mij = vYij , mˆ io = vYio , mˆ is = vYis , (3.11) and we can now write the full quark mass Lagrangian of our model in the form
q SSB inv −LMass = LMass + LMass ! ! (3.12) uR dR = uL T L Mu + dL BL Md . TR BR
In the second equality, Mu and Md are the mass matrices of quarks with charge 2/3 and −1/3 respectively. Aside their different dimensions if nT 6= nB, they have a similar shape and we may denote them as ! m mˆ M = q q , (q, Q) = (u, T ), (d, B) . q ˆ (3.13) MQ MfQ
As observed in [41], it is possible to write the mass matrices Mq in a special form, which we can obtain by using the freedom one has to make weak-basis transformations. These correspond to trans- q q formations of the quark fields which leave LCC and LNC invariant, bringing at the same time the mass matrices Mq to the form ! Gq Jq Mq = . (3.14) 0 MfQ
In the absence of vectorlike quarks, one can verify that Mq would be equal to the ng ×ng sub-matrices
Gq, which correspond to the mass matrices of SM quarks. On the other hand, the off-diagonal ng ×nQ sub-matrices Jq give rise to mixing between vectorlike and SM quarks, while MfQ are diagonal and real nQ × nQ sub-matrices.
21 In order to bring the fields to the mass basis, we perform the transformations ! ! uL u 0 uR 0 u → UL uL , → uR UR , TL TR (3.15) ! ! dL d 0 dR 0 d → UL dL , → dR UR , BL BR
q q where the unitary matrices UL and UR (q = u, d) diagonalize Mu and Md through the bi-unitary transformations: u† u UL Mu UR = diag(mu, mc, mt, . . . , mnT ) ≡ Mfu , (3.16) d† d UL Md UR = diag(md, ms, mb, . . . , mnB ) ≡ Mfd .
q It is possible to write the diagonal matrices Mfq just in terms of UL (q = u, d). Considering the hermitian matrices G G† + J J † J M † ! † q q q q q fQ Hq = Mq M = , (3.17) q † † MfQJq MfQMfQ one can check that q † q q † † q † UL Mq UR UR Mq UL = Mfq Mfq (3.18) q † q 2 ⇒ UL Hq UL = Mfq . q For the results we will present, it is useful to write the matrices UL as ! q Aq UL = , (3.19) Bq where Aq and Bq are rectangular matrices with dimensions ng×Nq and nQ×Nq, respectively. Unitarity q of UL implies that q † q † † UL UL = 1Nq ×Nq ⇒ Aq Aq + Bq Bq = 1Nq ×Nq ,
U q U q † = 1 ⇒ A A† = 1 , L L Nq ×Nq q q ng ×ng (3.20) † Bq Bq = 1nQ×nQ , † † Aq Bq = Aq Bq = 0 . Before studying with more detail the transformation described in Eq. (3.18), we apply Eq. (3.15) to q q q the Lagrangians LCC, LNC and LH . Once we do this, we arrive at
q g µ + L = √ uα γ Vαβdβ W + H.c., (3.21) CC 2 L L µ
q µ g µ u µ d 2 µ LNC = −eJemAµ + uαLγ XαρuρL − dβγ XβσdσL − 2 sin θW Jem Zµ , (3.22) 2 cos θW
q g µ u u µ d d LH = − uαLγ Xαρmρ uρR + dβγ XβσmσdσL H + H.c., (3.23) 2MW
with α, ρ = (u, c, t, . . . , TnT ) and β, σ = (d, s, b, . . . , BnB ). The Nu × Nd mixing matrix which arises q in LCC is given by q ∗ q † Vαβ = (UL)iα (UL)iβ → V = AuAd . (3.24)
u d q q On the other hand, the matrices X and X appearing in both LNC and LH are
u u ∗ u u † d d∗ d d † Xαρ = (UL)iα (UL)iρ → X = AuAu ,Xβσ = UL iβ (UL)iσ → X = AdAd . (3.25)
22 Using the unitarity relations of Eq. (3.20), we can check that
u † † † X = AuAdAdAu = VV , (3.26) d † † † X = AdAuAuAd = V V, thus the CKM matrix of our model is only unitary if both Xu and Xd are the identity matrix. This ceases to happen in general and the matrices Xu and Xd defined in Eq. (3.25) give rise to FCNCs at tree-level. In Appendix A, we present the Feynman rules which arise from Eqs. (3.21), (3.22) and (3.23), due to the introduction of vectorlike quarks in the model.
3.2.1 Mass matrix diagonalization
The structure of the matrices V , Xu and Xd may be better understood after we diagonalize explicitly the mass matrices Mu and Md, presented in Eq. (3.14). First, we should highlight that we are free to perform a basis transformation such that either Gu or Gd (the mass matrices of SM quarks) is made diagonal and real [41]. We choose Gu to be the one with those properties. We must also emphasize that vectorlike quarks are (if they exist) much heavier than SM quarks. Then, considering the matrix elements of Gd, Gu, Jd and Ju to be of order λ and the mass terms of vectorlike quarks λ (contained in MfB and MfT ) to be of order Λ, we will always assume that ≡ Λ 1. Our first goal is to decouple the light quarks from the heavy ones, so that we obtain separate mass matrices for each set of quarks. This can be achieved through the unitary transformations which leave the Hermitian matrices Hq, defined in Eq. (3.17), block diagonal: ! † Dq 0 Uq Hq Uq = , (q, Q) = (u, T ), (d, B) . (3.27) 0 DQ
The square matrices Dq and DQ are associated to light and heavy quarks, respectively. For the unitary matrices Uq we consider the ansatz proposed in [56],
q † 1 − FqFq Fq Uq = q , (3.28) † † −Fq 1 − Fq Fq where Fq is an arbitrary ng × nQ matrix and the square roots should be understood as power series, for instance, q † 1 † 1 † † 1 − FqFq = 1 − FqF − FqF FqF − ... (3.29) 2 q 8 q q
With the ansatz for Uq in Eq. (3.28), the condition of the vanishing of the off-diagonal sub-matrices in Eq. (3.27) reads q † † † † † † † Fq (GqGq + JqJq ) 1 − FqFq − Fq JqMfQFq (3.30) q q q † † † † † † + 1 − FqFq MfQJq 1 − FqFq − 1 − FqFq MfQMfQ Fq = 0 .
2 This equation may be solved by assuming that Fq is a power series in , namely ,
Fq = F1q + F3q + F5q + ..., (3.31)
2 For matrices in which the lower-left sub-matrix vanishes (as it happens with Mu and Md), it is verified that Fj q = 0 for even j. For a detailed proof of this result, see [56].
23 q † 1 † 1 † † 1 † † 1 − FqFq = 1 − F1 F1 − F1 F3 + F3 F1 + F1 F1 F1 F1 + ..., (3.32) 2 q q 2 q q q q 4 q q q q
j where the matrix elements of Fjq are proportional to . For the following computations, we consider Fq ' F1q (we will no longer write the order label to simplify the notation) and
q † 1 † 1 − FqFq ' 1 − FqF . (3.33) 2 q
The matrix Fq may now be computed through Eq. (3.30). Keeping only zero-order terms in , we get
† † † MfQJq − MfQMfQFq = 0 −1 (3.34) † † −1 ⇒Fq = JqMfQ MfQMfQ = JqMfQ ,
−1 † † −2 where we use the fact that MfQ are diagonal and real, hence MfQ = MfQ and MfQMfQ = MQ . After getting this result, we can also obtain through Eq. (3.27),
† 1 n † −2 †o Dq = GqG − GqG ,JqMf J , (3.35) q 2 q Q q
2 1 −1 † † −1 DQ = Mf + Mf J JqMfQ + MfQJ JqMf , (3.36) Q 2 Q q q Q with {X,Y } = XY + YX being the anti-commutator. Even though we will only consider leading- order terms in the following calculations, it is important to notice that the mixing between vectorlike and SM quarks that arises from Ju and Jd leads to small mass terms for both sets of fermions. In the basis where Gu is diagonal, one can verify that Mu becomes approximately diagonal after the transformation we have performed above. Therefore, we may write
1 − 1 F F † F ! 1 − 1 J M −2J † J M −1 ! u 2 u u u 2 u fT u u fT U = Uu ' = , (3.37) L † 1 † −1 † 1 −1 † −1 −Fu 1 − 2 FuFu −MfT Ju 1 − 2 MfT JuJuMfT
u where UL is the unitary matrix that diagonalizes Hu, as one can see from Eq. (3.18). Regarding the matrix Md, we need to perform another transformation, so that Dd becomes diagonal. Since we are † neglecting non-zero order terms, we have Dd ' GdGd. As we said before, Gd is the mass matrix of SM down-quarks, and in Section 2.4, we have already presented the unitary matrices that diagonalize d d it through a bi-unitary transformation, set by VL and UR. We may write this operation as
d† d d † † d d† d 2 VL Gd UR UR GdVL ' VL DdVL = Ged . (3.38)
u† d Recall also that, in the SM, the CKM matrix is defined as VCKM ≡ VL VL . Due to our choice of d working in a basis where Gu is diagonal, we have VCKM ≡ VL . Then, the final step we should take to diagonalize Md at leading order can be expressed as
V † 0! D 0 ! V 0! CKM d CKM 2 = Mfd , (3.39) 0 1 0 DB 0 1 and one can now conclude that
V 0! V − 1 J M −2J †V J M −1V ! d CKM CKM 2 d fB d CKM d fB CKM U = Ud ' . (3.40) L −1 † 1 −1 † −1 0 1 −MfB Jd 1 − 2 MfB Jd JdMfB
24 d Before proceeding, we must note that unitarity of UL leads us to impose the condition:
† † −2 † −2 † 4 VCKMVCKM − VCKMJdMfB Jd VCKM + JdMfB Jd + O = 1ng ×ng . (3.41)
While VCKM (studied in Section 2.4) describes mixing between SM quarks in this model to a good approximation, one can also verify that Eq. (3.41) only holds up to order 2 if we assume that matrix to be unitary. Then, we check that introducing vectorlike quarks breaks the unitarity of the CKM matrix and FCNCs involving SM quarks may arise.
3.2.2 Quark mixing and suppression of FCNCs
Taking Eq. (3.24) into account, we may now write the matrix V as
1 −2 †! 1 − JuMfT Ju V = 2 V − 1 J M −2J †V J M −1 −1 CKM 2 d fB d CKM d fB JuMfT (3.42) −1 ! VCKM JdMf ' B , −1 † −1 † −1 MfT JuVCKM MfT JuJdMfB where we neglect non-zero order terms in the upper-left sub-matrix. One can notice that mixing between vectorlike and SM quarks (read from the off-diagonal sub-matrices of V ) is suppressed by . On the other hand, it follows from Eq. (3.25) that
−2 † −1 ! 1 − JuMfT Ju JuMfT Xu = , (3.43) −1 † −1 † −1 MfT Ju MfT JuJuMfT
† −2 † † −1 ! 1 − VCKMJdMfB Jd VCKM VCKMJdMfB Xd = . (3.44) −1 † −1 † −1 MfB Jd VCKM MfB Jd JdMfB
The upper-left sub-matrices of Xu and Xd do not correspond to the identity, so it becomes clear that FCNCs involving exclusively SM quarks appear. However, we are also able to prove that these are naturally suppressed by a factor of 2.
3.2.3 Complex phases of the generalized CKM matrix
In Chapter 2, we have seen that a unitary matrix with dimensions ng × ng needs necessarily to be parametrized with (ng − 1)(ng − 2)/2 complex phases. This result can be applied to the upper-left sub-matrix of V , as long as we assume that the matrix VCKM is unitary. Moreover, in the upper-right block of V , we have the matrix Jd which, being an ng × nB complex matrix, would contain in general 0 ngnB phases. However, nB phases may be eliminated after rephasing the last nB fields dL, defined in Eq. (3.15). Analogously, the lower-left block of V has (ng − 1)nT phases, arising from the ng × nB complex matrix Ju. Then, one has altogether
(ng − 1)(ng − 2) Nphases = + (ng − 1)(nT + nB) (3.45) 2 physical phases in the matrix V . Even though we computed Nphases by considering VCKM to be unitary, it can be shown [41] that the same result is obtained without making the same assumption.
25 3.3 Mixing with third generation of SM quarks
As we have discussed at the beginning of this chapter, it is often assumed that vectorlike quarks mix preferentially with b and t ones. Then, before studying the experimental constraints to the B quark, we simplify the model we studied previously by restricting its quark sector to the third generation of SM quarks (ng = 1) and a B quark (nB=1, nT = 0), singlet of SU(2)L. Taking into account the content of this model in terms of quarks, its Lagrangian can be built from Eqs. (3.21), (3.22) and (3.23). Namely, the mass Lagrangian for quarks with charge −1/3 is now ! ! d v ybb v ybB bR −LMass = bL BL 0 M BR ! (3.46) bR = bL BL MbB , BR where ybb and ybB are Yukawa parameters and M is the mass term for the B quark. These parameters form the 2 × 2 mass matrix MbB that we have introduced in the second equality. The b and B fields are at this point in the flavor basis, so we must perform the transformations ! ! ! ! b b0 b b0 L = U L , R = U R , L 0 R 0 (3.47) BL BL BR BR where the 2 × 2 unitary matrices UL and UR can be parametrized in the following way [57]: ! ! cos θL sin θL cos θR sin θR UL = ,UR = . (3.48) − sin θL cos θL − sin θR cos θR
In this context, the bi-unitary transformation that leaves MbB diagonal is ! † mb 0 UL MbB UR = , (3.49) 0 MB with mb and MB being the masses of the b and B quarks respectively. The angles θL and θR, which describes the mixing between left and right-handed components of down-quarks, are not independent parameters. Indeed, we conclude by solving Eq. (3.49) that
2Mv ybB tan(2θL) = 2 2 2 2 , (3.50) M − v (ybb + ybB)
mb tan θR = tan θL . (3.51) MB
As we assume that mb MB, it is interesting to verify from Eq. (3.51) that mixing between right- handed components of b and B quarks is suppressed in comparison with left-handed mixing. Moreover, these equations are nearly the same for B quarks in other representations of SU(2)L. For example, if
B is assigned to a SU(2)L doublet, we obtain both Eqs. (3.50) and (3.51) with θL and θR exchanged, meaning that in this case it is the mixing between left-handed b and B quarks that becomes suppressed.
26 g B
g B (a)
q q0 q0 q
Z W − B B
b t
g b g t (b) (c)
Fig. 3.1: Feynman diagram showing (a) pair production of BB quarks; (b) single production involving a W boson (the quarks q and q0 must be such that Q(q0)−Q(q) = −1); (c) single production involving a Z boson.
3.4 Experimental status of vectorlike B quarks
3.4.1 Production at the LHC and mass constraints
If vectorlike B quarks really exist, they can be produced at the Large Hadron Collider (LHC) through strong or electroweak interactions. The former give rise to processes like the one illustrated in Fig. 3.1(a), where pair production occurs. On the other hand, the couplings of B quarks to W and Z bosons allow single production processes like those represented by the Feynman diagrams of Figs. 3.1(b) and 3.1(c). Independently of the production mechanism, the possible decay modes of B quarks are B → W −t , B → Zb , B → Hb , (3.52) assuming that they only mix with SM quarks of the third generation. The branching ratios (BRs) of a B quark as a function of its mass, as computed by PROTOS v2.2 [58, 59], is shown in Fig. 3.23. Many searches for pair and single production of B quarks decaying into W t or Zb have been performed by the ATLAS [60–63] and CMS [64, 65] collaborations. The observed and expected mass constraints for the vectorlike B quark are displayed respectively, in Figs. 3.3(a) and 3.3(b). Those results were obtained recently by the ATLAS collaboration after a search for pair production of vectorlike quarks decaying into a W boson plus a third-generation quark4. One can check that for a B quark singlet of SU(2)L, it is observed (expected) a lower mass limit of 1080 GeV (980 GeV), whereas for the same quark in an SU(2)L doublet, the same limit is observed (expected) to be 1250 GeV (1150 GeV). These limits have a 95% CL.
3The branching ratios in Fig. 3.2 are only valid for small mixing between the new vectorlike quarks and the third- generation quark involved on each decay [60].√ 4This search was based on pp collisions at s = 13 TeV recorded in 2015 (3.2 fb−1) and 2016 (32.9 fb−1).
27 1 SU(2) Singlet (B,Y) Doublet (T,B) Doublet B → Wt B → Wt B → Wt 0.8 Branching Ratio B → Zb B → Zb B → Hb B → Hb
0.6
0.4
0.2 PROTOS
0
300 400 500 600 700 800 900 1000
mB [GeV]
Fig. 3.2: Vectorlike B quark branching ratios to the W t, Zb and Hb decay modes as a function of the B quark mass (extracted from [60]).
1 1400 1 1400
Hb) ATLAS Hb) ATLAS 0.9 1300 0.9 1300 s = 13 TeV, 36.1 fb-1 s = 13 TeV, 36.1 fb-1 → → 0.8 1200 0.8 1200 (B 0.7 BB 1-lepton (B 0.7 BB 1-lepton ℬ 1100 ℬ 1100 0.6 SU(2) singlet 0.6 SU(2) singlet SU(2) doublet 1000 SU(2) doublet 1000 0.5 600 0.5 600 900 900 0.4 700 0.4 700 800 800 0.3 800 0.3 800
900 900 0.2 700 0.2 700
1000 1000
600 600 Expected 95% CL mass limit [GeV]
0.1 Observed 95% CL mass limit [GeV] 0.1
1100 1100
1200 500 500 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ℬ(B → Wt) ℬ(B → Wt)
(a) (b)
Fig. 3.3: (a) Observed and (b) expected 95% CL lower limits for the mass of the B quark in the branching ratio plane of BR(B → W t) versus BR(B → Hb). Contour lines are provided to guide the eye. The markers indicate the branching ratios for the SU(2) singlet and doublet scenarios with masses above 0.8 TeV, where they are approximately independent of the mass of the vectorlike B quark (see Fig. 3.2). The white regions are due to the limit falling below 500 GeV, the lowest simulated signal mass in the search (images taken from [61]).
28 3.4.2 Mixing constraints
The mixing between b and B quarks result in new contributions to the oblique parameters S and T [66], precisely measured at the Large Electron-Positron Collider (LEP) and Stanford Linear Collider
(SLC). Defining ∆S ≡ S −SSM and ∆T ≡ T −TSM as the changes in S and T respectively, we consider the experimental values for no variation of the oblique parameter U (∆U ≡ U − USM = 0),
∆S = 0.04 ± 0.07 , ∆T = 0.07 ± 0.08 , (3.53) with a correlation of 0.88 [67]5. Other measurements taken into account to constrain the b−B mixing are the ones made to the Z → bb decay (also at LEP and SLC). Indeed, this mixing is expected to modify the Zbb coupling at tree level, as well as the SM predictions for four observables. Two of these are Rb and Rc, defined as
Γ(Z → bb) Γ(Z → cc) Rb = ,Rc = , (3.54) Γ(Z → had) Γ(Z → had) where Γ(Z → had) is the decay width of the Z boson into hadrons (excluding the t quark). The b remaining two are the forward-backward asymmetry AFB, and the asymmetry parameter Ab, both associated to the b quark,
b b b 3 Ae + Pe 2 gV gA AFB = Ab ,Ab = 2 2 , (3.55) 4 1 + PeAe b b gV + gA
b b where gV and gA are the effective couplings, which includes electroweak radiative corrections. The − parameter Ae is equivalent to Ab with the replacement b → e, and Pe is the initial e polarization. In Table 3.1, we show the SM predictions for the observables shown in Eqs. (3.54) and (3.55), as well as their experimental values. Finally, it is represented in Fig. 3.4, the constraints to the mixing angle θL (defined in Section 3.3) as a function ofthe B quark mass.
Parameter SM prediction Experimental result
Rb 0.21576 0.21629 ± 0.00066
Ab 0.9348 0.923 ± 0.020
b AFB 0.1034 0.0992 ± 0.0016
Rc 0.17227 0.1721 ± 0.003
Table 3.1: SM predictions [67] and experimental results [68] for the observables affected by changing the Zbb coupling.
5According to [2], the experimental values we are using for ∆S, ∆T and their correlation is slightly different. Here we present the experimental values used to produce the plot shown in Fig. 3.4.
29 Fig. 3.4: Contraints to the mixing angle θL as a function of the vectorlike B quark mass. The black line represents the constraints due to the change in the oblique parameters S and T and the blue line the b constraints related to the observables Rb, Rc, AFB and Ab, affected by the change in the Zbb vertex. The dashed and continuous red lines show the constraint imposed by the ATLAS collaboration in 2013 [62] and 2017 [61] respectively to the mass of the B quark. The latter constraints were obtained with searches for pair production of B quarks (image adapted from [5]).
30 4 Trinification Models
Trinification models are GUTs based on theE6 maximal subgroup G333 ≡ SU(3)c ⊗ SU(3)L ⊗ 3 SU(3)R = [SU(3)] , known as trinification group. The theory we will develop throughout this chapter is based on the one proposed in 1984 by H. Georgi, S. Glashow and A. de Rújula [34], where a discrete symmetry Z3 is imposed to guarantee the unification of the gauge coupling constants. These GUTs have many attractive features. For example, they are left-right symmetric, so they allow us to explain parity violation in the SM by the spontaneous breakdown of the gauge symmetry. Even though proton decay may be mediated in these models by the decay of heavy colored Higgs bosons, it cannot occur through gauge boson exchange, hence it is suppressed [69]. As discussed in [70], 14 this fact allows for gauge-coupling unification at relatively low unification scales, MU ∼ 10 GeV. Furthermore, mass for neutrinos may be generated either through a radiative [71] or an inverse seesaw mechanism [72], in such a way that the theory predictions are in agreement with experimental data concerned with neutrino oscillations. Trinified models with a Z3 symmetry can also account for the baryon-antibaryon asymmetry in the universe through heavy-Higgs decays at the one-loop level [73]. Although the theory we will study is non-supersymmetric, it is worth pointing out that one may build supersymmetric versions of Trinification models in order to solve for example, the hierarchy problem. These theories naturally give rise to the Minimal Supersymmetric Standard Model (MSSM) [74, 75] at low energies. It is also remarkable that trinified models may have its origin from other more fundamental theories. As a consequence of being a maximal subgroup of E6, they can arise from the compactification of E8×E8 heterotic string theories [76, 77]. Other contexts where they can appear include N = 8 supergravity [78], brane-world scenarios [79] and orbifold GUTs [80, 81].
The theory we are going to present is slightly more complex than the one that is called sometimes by ’minimal trinification’ [71]. Even though we will not extend the Higgs sector further thanweneed to break G333 into the SM group, we will need to assume some parameters of the scalar field VEVs to be complex in order to incorporate SCPV in our model. This will allow us to obtain a complex CKM matrix in agreement with experiment, besides the mixing that SM quarks may develop with the vectorlike quarks that appear in the theory.
31 4.1 Trinification group and field content
In the gauge group of Trinification models,
3 G333 ≡ SU(3)c ⊗ SU(3)L ⊗ SU(3)R = [SU(3)] , (4.1) we identify SU(3)c as the unbroken color group, SU(3)L the one containing the SU(2)L group of SM and SU(3)R the right-handed analog of SU(3)L. Considering the coupling constants gc, gL and gR associated to the SU(3) subgroup with the same label, the discrete symmetry Z3 guarantees that
gc = gL = gR ≡ g3 , (4.2) where g3 is the unified coupling constant. In the minimal version of the model, there areonlytwo irreducible representations of the model we need to consider. Gauge bosons are assigned to the adjoint representation of the trinification group,
24 → (8, 1, 1) ⊕ (1, 8, 1) ⊕ (1, 1, 8) , (4.3) whereas each generation of fermions and scalar fields are assigned to the fundamental representation
27 of E6: 27 → 1, 3, 3 ⊕ 3, 1, 3 ⊕ 3, 3, 1 . (4.4)
The notation 3 used above indicates that the field transforms under the anti-fundamental represen- tation of an SU(3) subgroup. Observe also that we are using a notation similar to the one we have adopted to describe SM fields, but now the quantum numbers (rc, rL, rR) mean that the field is as- signed to the representation rc of SU(3)c, rL of SU(3)L and rR of SU(3)R. Further inspection of Eqs. (4.3) and (4.4) leads one to understand why we can apply the discrete symmetry Z3 to unify the gauge coupling constants. Both representations are invariant under cyclic permutations of the three SU(3) factors. This happens because Z3 is an inner automorphism of the E6 algebra, thus it is a discrete subgroup of E6 [82]. Before proceeding, we highlight that for a given representation
(rc, rL, rR), the effect of Z3 is to symmetrize it in the following way:
Z3[(rc, rL, rR)] = (rc, rL, rR) + (rR, rc, rL) + (rL, rR, rc) . (4.5)
The field content of Trinification models can be read from the decomposition ofthe 24 and 27 representations in terms of SM quantum numbers, worked-out in Appendix B. Each SU(3) group has eight generators, so we must expect the existence of 3 × 8 = 24 gauge bosons. We will denote them as
a a a G ∼ (8, 1, 1) ,WL ∼ (1, 8, 1) ,WR ∼ (1, 1, 8) , a = 1,..., 8 . (4.6)
Along with the twelve gauge bosons we already know, there are twelve new gauge bosons that acquire mass of the order of unification scale. None of these particles have fractional charge, so we checkthat proton decay cannot be mediated by gauge interactions, as it happens for example, in SU(5) [69].
Trinification models also introduce new particles in the fermion sector. Designating by ψ`, ψqL and
ψqR the lepton, quark and anti-quark multiplets respectively, we assign each one to a bitriplet of the 27 representation:
ψ` ∼ 1, 3, 3 , ψqL ∼ 3, 3, 1 , ψqR ∼ 3, 1, 3 . (4.7)
32 Notice that the assignments of quarks to the SU(3)c representations are interchanged with respect to the SM, in which the quarks (anti-quarks) transform as a 3 (3) under the SU(3)c group. This convention, used in [83], has no physical consequences and it ensures that left-handed quarks have the correct transformation properties under the SU(2)L subgroup of the SM. With this, the fermion multiplets can be written as
E0 E− e− u + 0 ψ` = E E ν , ψqL = d , ψq = u d B , (4.8) R R + e N1 N2 B L where color indices for quark fields and generation indices are omitted. We have also employed the matrix notation we will always use, in which SU(3)L (SU(3)R) indices run vertically (horizontally). We can verify that in the leptonic sector, it is postulated for each generation a new doublet with the same quantum numbers as the one we find in the SM. This is formed by the particles E0 and E− and their antiparticles form a doublet with opposite hypercharge, also embedded in ψ`. The remaining + content of this multiplet consists of SU(2)L singlets and it is given by the positron (e ) and two unknown neutral particles, N1 and N2. In the quark sector, we find inside the colored bitriplets ψqL 1 and ψqR the quarks and antiquarks of the SM, as well as a vectorlike B quark, singlet of SU(2)L . Concerning the scalar sector, it is worth emphasizing that we achieve the symmetry breaking with scalar fields assigned to the 27 representation. In fact, this makes the scalar sector structure of
Trinification models much simpler than the one we find in GUTs based onE6, where we need one Higgs field transforming like the 78 dimensional adjoint representation [82]. However, one musthave at least two Higgs fields (we denote them as φ and χ) assigned to the fundamental representation. The structure of both fields is analog to the one we have discussed for fermions. They are composed of three bitriplets and we will define them as
φc ∼ 1, 3, 3 , φR ∼ 3, 3, 1 , φL ∼ 3, 1, 3 , (4.9) χc ∼ 1, 3, 3 , χR ∼ 3, 3, 1 , χL ∼ 3, 1, 3 , where the labels c, L and R indicate the SU(3) subgroup of G333 under which the multiplet transform as a singlet. Since SU(3)c remains unbroken throughout all steps of SSB, all colored scalar fields are massive and they are not able to acquire VEV. In contrast, color singlets may play a role in SSB and taking into account their decomposition in terms of SM quantum numbers, we may write them as
! hn + ian hc + iac hc + iac Φ Φ Φ 1 1 1 2 2 3 3 φ = 1 2 3 = √ c c n n n n , c h1 + ia1 h2 + ia2 h3 + ia3 (4.10) S1 S2 S3 2 c c n n n n H1 + iA1 H1 + iA1 H2 + iA2
! hn + ian hc + iac hc + iac Φ Φ Φ 1 4 4 5 5 6 6 χ = 4 5 6 = √ c c n n n n . c h4 + ia4 h5 + ia5 h6 + ia6 (4.11) S4 S5 S6 2 c c n n n n H2 + iA2 H3 + iA3 H4 + iA4
Each field is formed by three complex scalar doublets Φi and three complex scalar singlets Si. Real scalar and pseudo-scalar fields belonging to doublets (singlets) are respectively represented by hi (Hi) and ai (Ai), while the superscripts ’n’ and ’c’ indicate whether the fields are electrically neutral or charged, respectively. The number of massive scalar fields we have in this model can be predicted by using the Goldstone theorem. As one can see from Eqs. (4.10) and (4.11), φc and χc have 36 scalar
1 In GUTs based on E6, fermions are also assigned to the 27 representation. Then, as we have said in Chapter 3, these theories predict vectorlike quarks like the one we find in Trinification models.
33 fields (sixteen charged and twenty neutral) altogether. On the other hand, all gauge bosons apartfrom the photon and the gluons acquire mass through SSB. As a result, we can find 36 − 15 = 21 massive uncolored scalar particles in the minimal Trinification model (eight charged and thirteen neutral). Finally, we must ensure that one of the massive neutral fields has a mass of nearly 125 GeV, sowe may identify it as the SM Higgs boson.
4.2 Scalar Lagrangian and SSB
The Lagrangian associated to the scalar fields φ and χ is given by
µ † µ † LScalar = Tr (D φ) (Dµφ) + Tr (D χ) (Dµχ) − V (φ, χ) , (4.12) where V (φ, χ) is the scalar potential and the covariant derivatives of φ and χ are
h a a a aT i Dµφ = Z3 ∂φc − ig3WLµT Lφc − ig3WRµφcTR , (4.13) h a a a aT i Dµχ = Z3 ∂χc − ig3WLµT Lχc − ig3WRµχcTR .
a a Here, TL and TR (a = 1,..., 8) correspond to the generators of SU(3)L and SU(3)R respectively. a a a Along with the generators Tc of SU(3)c, TL and TR are defined through the Gell-Mann matrices, shown in Appendix B. One should also note that the SU(3)R generators are transposed and appear on the right of the scalar field. The reason for this is the matrix notation we have adopted. Whilefor the scalar fields rows and columns correspond to left and right-handed indices respectively, rowsand columns of the SU(3)L and SU(3)R generators correspond to either both left-handed or right-handed indices. Writing left-handed (right-handed) indices as upper (lower) indices, one can see that this notation leads to the placement of the generators as in Eq. (4.13) [83]:
j j a ajk k a a j (Dµφ)l ≡ (∂µφ)l − ig3WLµ T L (φ)l − ig3WRµ (TR)lm (φ)m
j a ajk k a j aT = (∂µφ)l − ig3WLµ T L (φ)l − ig3WRµ(φ)m TR ml (4.14)
j j a a j a aT = (∂µφ)l − ig3WLµ T Lφ l − ig3WRµ φTR l .
Regarding the scalar potential V (φ, χ) of this model, it is given by
2 2 V (φ, χ) = − µ1 Tr φcφc + α1[Tr φcφc ] + α2Tr φcφcφcφc 2 2 − µ2 Tr (χcχc) + β1 [Tr(χcχc)] + β2Tr (χcχcχcχc) 2 + λ1Tr φcφc Tr (χcχc) + λ2Tr φcχc Tr (χcφc) + [λ3Tr φcχc + H.c.] (4.15) + λ4Tr φcφcχcχc + λ5Tr φcχcχcφc + λ6Tr φcχcφcχc + H.c. h i j k αβγ i h i j k αβγ i + γ1(φc)α(φc)β(φc)γ ijk + H.c. + γ2(φc)α(χc)β(χc)γ ijk + H.c. .
Since colored scalars do not acquire VEV, all the terms proportional to them are set to zero. The full scalar potential is presented in [69].
34 The most general VEVs for the fields φ and χ are2
ˆ vˆ1 0 0 b1 0 0 ˆ ˆ hφi ≡ hφci = 0v ˆ2 0 , hχi ≡ hχci = 0 b2 b3 , (4.16) 0 0 Mˆ 1 0 Mˆ 2 Mˆ 3 in which we use the SU(3)L ⊗ SU(3)R gauge symmetry to bring the VEV of the field φ into diagonal form. The freedom we have to perform this transformation to one of the fields gives rise to the need of having at least two scalar fields in the Trinification model. In general, one cannot diagonalize both φ and χ simultaneously, so the latter contains off-diagonal VEVs which allow the breaking of the left-right symmetry of G333. Regarding the VEV parameters, Mˆj (related to the scalar singlets) 3 ˆ ensure that G333 is spontaneously broken into the SM group , whereas vˆi and bj (associated to the scalar doublets) are responsible for EWSB. For the latter, the following constraint must be imposed:
2 2 2 2 2 2 2 v1 + v2 + b1 + b2 + b3 ≡ v = (174 GeV) . (4.17)
ˆ As we will see in the next section, we do not need all the parameters vˆi and bj to build a realistic mass spectrum for fermions. On the other hand, we cannot neglect any of the VEVs Mˆ j and either Mˆ 1 or
Mˆ 3 need to be complex, otherwise one is not able to obtain a complex CKM matrix. Considering the parametrization
v2 = v cos β , b1 = v sin β , v = 174 GeV , (4.18) we will mostly work with the simplified VEV structure:
0 0 0 v sin ξ 0 0 hφi = 0 v cos ξ 0 , hχi = 0 0 0 . (4.19) iθ 0 0 M1e 0 M2 M3
As done in Chapter 2 in the context of the SM, we identify the Trinification model vacuum by finding the minimum of the scalar potential. This is given by the conditions
∂V (φ, χ) ∂V (φ, χ) = 0 , = 0 , ∂hnk φ=hφi,χ=hχi ∂Hnm φ=hφi,χ=hχi (4.20) ∂V (φ, χ) ∂V (φ, χ) = 0 , = 0 , k = 1,..., 6 , m = 1,..., 4 , ∂ank φ=hφi,χ=hχi ∂Anm φ=hφi,χ=hχi which lead to the following set of equations:
2 2 2 2 − 2µ1 cos ξ + 4M3γ2 sin ξ + cos ξ[4M1 α1 + 2M3 λ1 + 2M2 (λ1 + λ4) (4.21a) 2 2 2 + v λ1(1 − cos(2ξ)) + 4v (α1 + α2) cos ξ] = 0 ,
2 2 2 2 − 2µ1 cos θ + cos θ[4M1 (α1 + α2) + 2M2 (λ1 + λ5) + 2M3 (λ1 + λ2 + 2λ3 + λ4 + λ5 + 2λ6) (4.21b) 2 2 + v (2α1 + λ1) + v cos(2ξ)(2α1 − λ1)] = 0 ,
2 2 2 2 − 2µ1 sin θ + sin θ[4M1 (α1 + α2) + 2M2 (λ1 + λ5) + 2M3 (λ1 + λ2 − 2λ3 + λ4 + λ5 − 2λ6) (4.21c) 2 2 + v (2α1 + λ1) + v cos(2ξ)(2α1 − λ1)] = 0 ,
2We use hats to designate complex VEVs. 3 We are considering here that G333 is broken directly into SU(3)c ⊗ SU(2)L ⊗ U(1)Y . However, in theories like the Low Energy Trinification model [83], it is assumed that G333 is broken into the left-right symmetric group SU(3)c ⊗ SU(2) ⊗ SU(2) ⊗ U(1) B − L L R BL ( stands for the difference between baryon and lepton number), before this one being broken into the SM group.
35 2 2 2 2 − 2µ2M3 + 2γ2 sin(2ξ) + M3[4(M2 + M3 )(β1 + β2) + 4M1 cos(2θ)(λ3 + λ6) (4.21d) 2 2 2 + 2M1 (λ1 + λ2 + λ4 + λ5) + v (2β1 + λ1) + v cos(2ξ)(λ1 − 2β1)] = 0 ,
2 2 2 2 − µ2 sin ξ + 2 cos ξM3γ2 + sin ξ[2β1(M2 + M3 ) + M1 λ1 (4.21e) 2 2 2 + v λ1 cos ξ + v (β1 + β2)(1 − cos(2ξ))] = 0 ,
2 2 2 2 2 2 −2µ2 +4(M2 +M3 )(β1 +β2)+v (2β1 +λ1 +λ4)+2M1 (λ1 +λ5)+v cos(2ξ)(λ1 +λ4 −2β1) = 0 , (4.21f)
cos θ[6γ1 cos ξ + M3 sin ξ(λ2 + 2λ3)] = 0 , (4.21g)
sin θ[−6 cos ξγ1 + M3 sin ξ(λ2 − 2λ3)] = 0 , (4.21h)
cos θ[2γ2 sin ξ + M3 cos ξ(λ2 + 2λ3)] = 0 , (4.21i)
sin θ[2γ2 sin ξ + M3 cos ξ(λ2 − 2λ3)] = 0 , (4.21j)
M3λ4 cos ξ − 2γ2 sin ξ = 0 , (4.21k)
cos θ(λ2 + 2λ3 + λ4 + 2λ6) = 0 , (4.21l)
sin θ(λ2 − 2λ3 + λ4 − 2λ6) = 0 , (4.21m)
cos θλ6 cos ξ = 0 , (4.21n)
sin θλ6 cos ξ = 0 , (4.21o)
sin(2θ)(λ3 + λ6) = 0 . (4.21p)
One can notice that it is possible to find a minimum where π θ = + kπ, k ∈ Z , (4.22) 2 so that the parameter Mˆ 1 is complex. For a phase θ given in Eq. (4.22), we get some relations between the parameters of the scalar potential by solving Eq. (4.21), namely,
2 1 4 2 2 2 µ1 = 2 2 2 2 {8M1 [2M1 α1 + (M2 + M3 )(λ1 + λ4)] 8M1 (M1 − v cos ξ)
2 4 2 2 2 2 2 2 2 + 4v M1 λ1 + 4v (M2 + M3 )[2β2(M2 + M3 ) − M1 λ1 + 3M3 λ4]
2 2 2 2 2 2 2 2 2 − 8M3 v λ4 csc ξ(M2 + M3 ) + 4v cos(2ξ)[M3 (M2 + M3 )λ4
2 2 2 2 2 2 2 (4.23a) − M1 (M1 + M2 + M3 )λ1 + 2β2(M2 + M3 ) ]
4 2 2 2 − v [M1 (6α1 + λ1) + (M2 + M3 )(2β2 − 3λ4)]
2 2 2 2 2 2 + 4v cos(2ξ)[−2M1 v α1 + v λ4(M2 + M3 )]
4 2 2 2 + v cos(4ξ)[M1 (λ1 − 2α1) + (M2 + M3 )(2β2 + λ4)]} ,
1 µ2 = [4(M 2 + M 2)β + 2M 2λ + 2M 2λ cot2 ξ 2 2 2 3 1 1 1 3 4 (4.23b) 2 2 2 + v λ1(1 + cos(2ξ)) + 4v (β1 + β2) sin ξ] ,
36 M 2 + M 2 α = 2 3 [4β (M 2 + M 2) + 2λ (M 2 + M 2) − 2M 2λ csc2 ξ 2 4M 2(M 2 − v2 cos2 ξ) 2 2 3 4 1 3 3 4 1 1 (4.23c) 2 2 + v (λ4 − 2β2) + v (λ4 + 2β2) cos(2ξ)] ,
λ2 = 2λ3 − λ4 , (4.23d)
2 2 2 2 2 2 2 4(M2 + M3 )β2 + 2M3 λ4 + v (λ4 − 2β2) + v (λ4 + 2β2) cos(2ξ) − 2M3 λ4 csc ξ λ5 = − 2 , (4.23e) 2M1
λ6 = 0 , (4.23f)
1 γ1 = − M3λ4 tan ξ , (4.23g) 6
1 γ2 = M3λ4 cot ξ . (4.23h) 2 In order for SCPV to occur in the Trinification model, one needs to ensure that the minimum characterized by Eqs. (4.22) and (4.23) is an absolute one. We verified that this may be achieved as long as the constraint
λ3 > 0 (4.24) is imposed. Some other constraints to those parameters arise by requiring that the masses of all particles are real and positive. These correspond to the eigenvalues of the mass matrices of scalar fields, which are obtained by computing the second derivatives of the potential in order toallthereal fields hk, Hm, ak and Am. Once we do this, we get a 20 × 20 (16 × 16) matrix for neutral (charged) scalar fields. While it is not possible to compute the eigenvalues of these matrices analytically, we got some constraints for the parameters of the scalar potential after trying to calculate them as an v expansion in M , 2 3 2 2 v v v m = M xi + xi + xi + O , (4.25) i 0 1 M 2 M 2 M 3 where v and M are of the order of electroweak and unification scales, respectively. With this purpose, we compute the second derivatives of V (φ, χ) considering first v cos ξ = v sin ξ ' 0 in Eq. (4.19). n c Denoting the mass matrices of neutral and charged scalar particles respectively as MScalar and MScalar, we write them in the basis
n n n n n n n n n n n n n n n n n n n n n MScalar → {h1 , h5 , h6 , a2 , a3 , a4 ,H1 ,H2 ,A3 ,A4 , a1 , a5 , a6 , h2 , h3 , h4 ,A1 ,A2 ,H3 ,H4 } , (4.26)
c c c c c c c c c c c c c c c c c MScalar → {h1, h5, h6, a2, a3, a4,A1,H2, a1, a5, a6, h2, h3, h4,H1,A2} , (4.27)
37 so that they become block diagonal:
n D1 Dn n 2 MScalar = , (4.28) n D3 n D4
n D1 0 c 2×2 MScalar = . (4.29) c D2 02×2
n The sub-matrices appearing inside MScalar may be written as
2 2 2 −(M2 + M3 )λ4 M3 λ4 −M2M3λ4 M1M3λ4 0 M1M3λ4 M2 2 2 [−2(M2 + M3 )β2 M 2λ 2M 2β − M 2λ 2M M β −M M λ M1 −M M λ 3 4 2 2 3 4 2 3 2 1 3 4 2 1 3 4 +M3 λ4] 2 M3 2 2 M3 (2β2 − λ4) [−2(M2 + M3 )β2 −M M λ 2M M β 0 M1 0 n 2 3 4 2 3 2 2 2 2 +M1 λ4 +(M3 − M1 )λ4] D1 = , (4.30) 2 2 M1M3λ4 −M1M3λ4 0 −M λ4 M2M3λ4 −M λ4 3 3 M2 2 2 M3 2 2 1 [2(M 2 + M 2)β M [−2(M2 + M3 )β2 M [−2(M2 + M3 )β2 M 2 2 3 2 0 1 1 M2M3λ4 1 M2M3λ4 +M 2λ ] +(M 2 − M 2)λ ] 2 2 2 2 3 4 3 1 4 −M3 (−M1 + M2 + M3 )λ4] 2 2 M1M3λ4 −M1M3λ4 0 −M3 λ4 M2M3λ4 −M3 λ4
2 M2 M2M3 0 M1M2 2 M2M3 M 0 M1M3 n 3 D2 = 4λ3 , (4.31) 0 0 0 0 2 M1M2 M1M3 0 M1
2 2 2 −(M2 + M3 )λ4 −M3 λ4 M2M3λ4 M1M3λ4 0 −M1M3λ4 M2 2 2 − [−2(M2 + M3 )β2 −M 2λ 2M 2β − M 2λ 2M M β M M λ M1 −M M λ 3 4 2 2 3 4 2 3 2 1 3 4 2 1 3 4 +M3 λ4] 2 M3 2 2 M3 (2β2 − λ4) [2(M2 + M3 )β2 M M λ 2M M β 0 M1 0 2 3 4 2 3 2 2 2 2 n +M1 λ4 −(M3 − M1 )λ4] D3 = , (4.32) 2 2 M1M3λ4 M1M3λ4 0 −M λ4 M2M3λ4 M λ4 3 3 M2 2 2 M3 2 2 1 [2(M 2 + M 2)β − M [−2(M2 + M3 )β2 M [2(M2 + M3 )β2 M 2 2 3 2 0 1 1 M2M3λ4 1 −M2M3λ4 +M 2λ ] −(M 2 − M 2)λ ] 2 2 2 2 3 4 3 1 4 −M3 (−M1 + M2 + M3 )λ4] 2 2 −M1M3λ4 −M1M3λ4 0 M3 λ4 −M2M3λ4 −M3 λ4
38 0 0 0 0 1 4 2 2 2 M2 2 2 M3 2 2 M 2 [4M1 α1 + 4(M2 + M3 ) β2 2 [−2(M2 + M3 )β2 2 [−2(M2 + M3 )β2 0 1 M1 M1 2 2 2 2 2 2 2 2 +2(M1 − M3 )(M2 + M3 )λ4] +M1 λ1 + M3 λ4] +M1 λ1 + M3 λ4] n D4 = , (4.33) M2 2 2 2 [−2(M2 + M3 )β2 0 M1 4M 2(β + β ) 4M M (β + β ) 2 2 2 1 2 2 3 1 2 +M1 λ1 + M3 λ4] M3 2 2 2 [−2(M2 + M3 )β2 0 M1 4M M (β + β ) 4M 2(β + β ) 2 2 2 3 1 2 3 1 2 +M1 λ1 + M3 λ4]
c while the non-vanishing sub-matrices of MScalar are
2 2 2 −(M2 + M3 )λ4 −M3 λ4 M2M3λ4 −M1M3λ4 0 M1M3λ4 M2 2 2 [−2(M2 + M3 )β2 −M 2λ 2M 2β − M 2λ 2M M β −M M λ M1 −M M λ 3 4 2 2 3 4 2 3 2 1 3 4 2 1 3 4 +M3 λ4] 2 M3 2 2 M3 (2β2 − λ4) [−2(M2 + M3 )β2 M M λ 2M M β 0 M1 0 c 2 3 4 2 3 2 2 2 2 +M1 λ4 +(M3 − M1 )λ4] D1 = , (4.34) 2 2 −M1M3λ4 −M1M3λ4 0 −M λ4 M2M3λ4 M λ4 3 3 M2 2 2 M3 2 2 1 [2(M 2 + M 2)β M [−2(M2 + M3 )β2 M [−2(M2 + M3 )β2 M 2 2 3 2 0 1 1 M2M3λ4 1 M2M3λ4 +M 2λ ] +(M 2 − M 2)λ ] 2 2 2 2 3 4 3 1 4 −M3 (−M1 + M2 + M3 )λ4] 2 2 M1M3λ4 −M1M3λ4 0 M3 λ4 −M2M3λ4 −M3 λ4
2 2 2 −(M2 + M3 )λ4 M3 λ4 −M2M3λ4 −M1M3λ4 0 −M1M3λ4 M2 2 2 − [−2(M2 + M3 )β2 M 2λ 2M 2β − M 2λ 2M M β M M λ M1 M M λ 3 4 2 2 3 4 2 3 2 1 3 4 2 1 3 4 +M3 λ4] 2 M3 2 2 M3 (2β2 − λ4) [2(M2 + M3 )β2 −M M λ 2M M β 0 M1 0 c 2 3 4 2 3 2 2 2 2 +M1 λ4 −(M3 − M1 )λ4] D2 = . (4.35) 2 2 −M1M3λ4 M1M3λ4 0 −M λ4 M2M3λ4 −M λ4 3 3 M2 2 2 M3 2 2 1 [2(M 2 + M 2)β − M [−2(M2 + M3 )β2 M [2(M2 + M3 )β2 M 2 2 3 2 0 1 1 M2M3λ4 1 M2M3λ4 +M 2λ ] −(M 2 − M 2)λ ] 2 2 2 2 3 4 3 1 4 −M3 (−M1 + M2 + M3 )λ4] 2 2 −M1M3λ4 M1M3λ4 0 −M3 λ4 M2M3λ4 −M3 λ4
v It is interesting to check that at leading order in M , there is no mixing between scalar fields transform- n n c c ing differently under SU(2)L. For all the 6 × 6 matrices (D1 , D3 , D1 and D2), we verified that each one has two vanishing eigenvalues. Seven of these correspond to Goldstone bosons and the remaining one must be identified as the Higgs boson, which only acquires mass of the order of electroweak scale and it must be neutral. On the other hand, it is possible to compute all the eigenvalues of the 4 × 4 n n matrices D2 and D4 . For the former, we have one non-vanishing eigenvalue,
2 2 2 2 m2 = 4 M1 + M2 + M3 λ3 , (4.36) meaning that the condition for it to be positive is the same as the one we need to impose to have n SCPV in our model. For D4 , we may write its two nonzero eigenvalues as
q 2 1 2 2 2 2 m4 = 2 A ± A − 4M1 (M2 + M3 )B , (4.37) M1
39 with the quantities A and B being given by
4 2 2 2 2 2 2 2 2 A = 2M1 α1 + M1 M2 + M3 [2(β1 + β2) + λ4] + M2 + M3 2 M2 + M3 β2 − M3 λ4 , (4.38)
2 2 2 2 2 B = 2M1 [2 M2 + M3 β2λ1 + M2 (β1 + β2) λ4 + M3 (β1 + β2 − λ1) λ4] (4.39) 4 2 2 2 2 2 2 + M1 [4α1(β1 + β2) − λ1] + 2 M2 + M3 β2 − M3 λ4 2M2 β1 + M3 (2β1 + λ4) .
Notice that we have got one more massless scalar particle than the number of massive gauge bosons predicted in Appendix B with quantum numbers (1, 1, 0). This means that there is a scalar singlet that does not acquire mass of the order of unification scale in this model. Nevertheless, it mustbe heavier than the SM Higgs boson and no experimental evidence was found for it so far. Finally, the c two vanishing 2 × 2 sub-matrices of MScalar indicate the existence of four Goldstone bosons with quantum numbers (1, 1, 1). Before studying the mass matrices with the electroweak parameters v and ξ, we should note that another constraint has to be imposed in order to ensure that the scalar potential is negative at its v minimum. At leading-order in M , we have found it to be
2 2 2 h 4 2 22 2 2 2 i M3 M2 + M3 λ4 + 2M1 α1 + 2 M2 + M3 β1 + M1 M2 + M3 (2λ1 + λ4) > 0 . (4.40)
Even in the case where we do not neglect any parameter in the VEVs of the scalar fields, the mass n c matrices MScalar and MScalar remain block diagonal in the basis we have introduced in Eqs. (4.26) and (4.27). However, new mixings between scalar particles arise and each matrix can be written only in terms of two blocks with equal dimensions. These become too large to display here and their eigenvalues can only be computed numerically. This was made using the software MINUIT [84], which allowed us to perform a χ2 minimization with respect to the mass of the SM Higgs boson, imposing at the same time that all the eigenvalues of the mass matrices are positive and the correct number of massive neutral and charged scalars (thirteen and eight respectively) is accomplished. In Table 4.1, it is shown a set of scalar potential parameters (those that remained as free parameters after computing Eq. (4.20)) and VEV parameters which fit to the measured mass of the Higgs Boson, 2 mH = 125.09±0.24 GeV [2], with χ = 0.046. This result gives an evidence that our Trinification model with SCPV is compatible with the observation of a SM Higgs boson with a mass of approximately 125 GeV. At this point, one might wonder why did we not choose to simplify the VEV structure even further by making some parameters equal. Although we have studied the scenario where M1 = M2 = M3 and others with just two of those parameters equal, we must note that it was not possible in any case to guarantee that all the eigenvalues of both mass matrices were positive. Taking this into account,
Potential parameters VEV parameters 5 α1 = −0.18867 M1 = 3.4640 × 10 GeV 3 β1 = 0.47375 M2 = 8.0296 × 10 GeV 7 β2 = 0.55327 M3 = 8.5884 × 10 GeV
λ1 = 0.69314 ξ = π/2.0116
λ3 = 0.01499
λ4 = −0.98621
Table 4.1: Numerical values of the scalar potential and VEV parameters compatible with the obser- 2 vation of a SM Higgs boson with mass mH ∼ 125 GeV. For this fit, we have obtained χ = 0.046.
40 we will conclude in the next section that the VEV structure we have introduced in Eq. (4.19) is the simplest we could consider for our model.
We end this section by discussing briefly the mass spectrum of gauge bosons. This can be obtained through the first term of Eq. (4.12) by replacing the scalar fields φ and χ by their VEVs. As it happens in the scalar sector, we cannot compute the masses of gauge bosons exactly, so we tried to write them v as an expansion in M . For charged gauge bosons, we introduce the mass eigenstates,
1 2 4 5 WL,R ∓ WL,R ± WL,R ∓ WL,R W ± = √ ,W 0 = √ , (4.41) L,R 2 L,R 2
± ± 0± 0± and we obtain the following mass matrix, in the basis {WL ,WR ,W L ,W R}: v2 0 0 0 2 2 GB 1 2 0 M2 + v −2v sin ξM2 M2M3 M = g . (4.42) c 2 3 0 −2v sin ξM M 2 + M 2 + M 2 + v2 sin2 ξ −2v sin ξM 2 1 2 3 3 2 2 2 2 0 M2M3 −2v sin ξM3 M1 + M3 + v sin ξ
± Notice that we are using the VEVs defined in Eq. (4.19). Along with WL that may be identified as the W ± bosons of the SM (they acquire mass of the order of electroweak scale), the remaining charged gauge bosons are of the order of unification scale and their masses are approximately givenby
1 2 2 2 2 2 mW 0± = g3(M1 + M2 + M3 + v ) , (4.43) L 2 q 2 2 2 1 2 2 2 2 2 2 2 2 2 2 2 g3v sin ξ mW ± = g3 M1 + M2 + M3 − (M1 − M2 ) + M3 (M1 + M2 + M3 ) + , (4.44) R 4 2 q 2 2 2 1 2 2 2 2 2 2 2 2 2 2 2 g3v sin ξ mW 0± = g3 M1 + M2 + M3 + (M1 − M2 ) + M3 (M1 + M2 + M3 ) + . (4.45) R 4 2
Concerning neutral gauge bosons, it is useful for us to write their 8 × 8 mass matrix in the basis 3 6 7 8 3 6 7 8 {WL,WL,WL,WL,WR,WR,WR,WR} as
n n(U) n(EW) MGB = MGB + MGB , (4.46) with 0 0 0 0 0 0 0 0 2 0 M 0 0 0 0 0 0 2 0 0 M 0 0 0 0 0 2 2M 2 2 0 0 0 4 M − √ 2 4M√2M3 0 − 1 4M − 6M 2 n(U) 3 3 3 3 2 M = 1 g2 , GB 4 3 2M 2 M 2 (4.47) 0 0 0 − √ 2 M 2 −M M iM M − √2 3 2 2 3 2 3 3 2 0 0 0 4M√2M3 −M M M 0 − M√2M3 3 2 3 3 2 √ 0 0 0 0 −iM2M3 0 M i 3M2M3 2 M 2 √ 2 0 0 0 − 1 4M − 6M 2 − √2 − M√2M3 −i 3M M 1 4M − 3M 2 3 2 3 3 2 3 3 2
41 v2 cos(2ξ) v2 cos(2ξ) v2 0 0 − √ −v2 0 0 √ 3 3 2 0 v 0 0 0 −2M1v cos ξ 0 0 2 0 0 v 0 0 0 −2M1v cos ξ 0 v2 cos(2ξ) v2 cos(2ξ) − √ 0 0 1 v2 √ 0 0 − 1 v2 n(EW) 1 2 3 3 3 3 M = g3 2 2 , GB 4 2 v cos(2ξ) 2 v cos(2ξ) −v 0 0 √ v 0 0 − √ 3 3 2 2 0 −2M1v cos ξ 0 0 0 v sin ξ 0 0 0 0 −2M v cos ξ 0 0 0 v2 sin2 ξ 0 1 v2 cos(2ξ) v2 cos(2ξ) √ 0 0 − 1 v2 − √ 0 0 1 v2 3 3 3 3 (4.48) 2 2 2 2 n(U) and M = M1 + M2 + M3 . The matrix MGB only contains terms proportional to M1, M2 and M3, so it is the one we obtain after SSB from G333 to the SM group. Although we cannot compute all its eigenvalues analytically, we may check that it contains two zero eigenvalues. One is because the SM Z boson does not acquire mass in this SSB step, whereas the other is associated to the eigenvector from which we may identify the hypercharge operator:
3 1 8 8 Y = T + √ T + T . (4.49) R 3 L R
The full mass matrix contains also terms proportional to v and ξ (which arise after EWSB) and one can verify that it has an eigenvalue equal to zero. This is consistent with the existence of a massless neutral gauge boson (the photon) and as we saw in our review of the SM (see Chapter 2), there must be some linear combination of G333 generators which remain unbroken after EWSB. This corresponds to the electric charge operator,
3 3 3 1 8 8 Q = T + Y = T + T + √ T + T , (4.50) L L R 3 L R
n and we may find it also by calculating the eigenvector associated to the vanishing eigenvalue of MGB.
4.3 Fermion masses and mixing
For each field, the Yukawa couplings are given by two types of interactions, which are alloweddue to their singlet structure under gauge group transformations:
i k ψ† φ∗ψ† ≡ ψ† (φ∗)j ψ† , ψ ψ φ ≡ 1/2 ijk (ψ )r(ψ )s(φ∗)t , qL c q qL c k q ` ` c rst ` i ` j c k R j R i (4.51) i k ψ† χ∗ψ† ≡ ψ† (χ∗)j ψ† , ψ ψ χ ≡ 1/2 ijk (ψ )r(ψ )s(χ∗)t . qL c q qL c k q ` ` c rst ` i ` j c k R j R i
After applying the Z3 symmetry to these, we get the Yukawa Lagrangian of the Trinification model: L = Y q ψ† φ∗ψ† + ψ† φ∗ ψ† + ψ†φ∗ ψ† + Y q ψ† χ∗ψ† + ψ† χ∗ ψ† + ψ†χ∗ ψ† Yuk φ qL c q q R ` ` L qL χ qL c q q R ` ` L qL R R R R (4.52) ` ` + Yφ ψ`ψ`φc + ψqL ψqL φR + ψqR ψqR φL + Yχ ψ`ψ`χc + ψqL ψqL χR + ψqR ψqR χL ,
q q ` ` where the the Yukawa matrices Yφ ,Yχ , Yχ and Yχ are assumed to be real in order to ensure CP conservation at the Lagrangian level. Those terms arise from the couplings to the uncolored scalar q ` fields φc and χc. Thus, we may write the mass Lagrangian as LMass = LMass + LMass, in which
Lq = Y qψ† φ∗ψ† + Y qψ† χ∗ψ† + H.c., (4.53) Mass φ qL c qR χ qL c qR
42 ` ` ` LMass = Yφ ψ`ψ`φc + Yχ ψ`ψ`χc + H.c.. (4.54)
Focusing now on the quark mass Lagrangian, one of our goals is to find some evidences that the choice of VEVs we made in Eq. (4.19) allows a realistic fermion mass spectrum. With this purpose, q we replace in the mass Lagrangian LMass the scalar fields φc and χc by their most general VEVs, presented in Eq. (4.16). This procedure leads to ! q dR LMass = muuLuR + dL BL Md , (4.55) BR where mu and Md are the mass matrices of up and down-quarks respectively. These are given by
q qˆ mu = Yφ vˆ1 + Yχ b1 , (4.56)
q qˆ qˆ ! Yφ vˆ2 + Yχ b2 Yχ b3 Md = . (4.57) q ˆ q ˆ q ˆ Yχ M2 Yφ M1 + Yχ M3
ˆ Observe that mass terms for up-quarks are guaranteed just by keeping vˆ1 or b1, so we may neglect one of these parameters. Moreover, comparing Md with the down-quark mass matrix of the model presented in Section 3.2, one checks that more simplifications can be made without compromising ˆ quark masses. In fact, either vˆ2 or b2 can be neglected (the upper-left sub-matrix of Md is non- zero for any case) and one of the off-diagonal sub-matrices may be set to zero. In contrast with the ˆ ˆ parameter Mˆ 2, we are free to neglect b3, hence we conclude that the VEVs b1 and vˆ2 are enough to ensure that both up and down-quarks become massive.
We may now diagonalize the matrix Md using the procedure described in Section 3.2.1. Recall that we start by performing an unitary transformation which leaves the mass matrix block diagonal, ! † † Dd 0 U Md Md U = , (4.58) 0 DB where the matrix U is given by the ansatz of Eq. (3.28). We should highlight that in contrast with the SM extension we have studied in Section 3.2, vectorlike quarks in this model only acquire mass through
Yukawa interactions. For this reason, the matrix DB is not diagonal and after solving Eq. (4.58), we check that it may be written as h i q q† 2 q q† 2 2 q q† i(θ3−θ1) q q† i(θ1−θ3) DB ' Yφ Yφ M1 + Yχ Yχ (M2 + M3 ) + Yφ Yχ M1M3e + Yχ Yφ M3M1e . (4.59)
We can also find the effective matrix, which we define as
q q † 2 2 q 2 q† −1 q q† meff meff 'Dd = v cos ξ Yφ 1 − M2 Yχ DB Yχ Yφ , (4.60)
q where we have used the parametrization of Eq. (4.18). First of all, notice that meff is complex ˆ independently of considering vˆ2, b2 and Mˆ 2 to be complex or not. One can also check that we cannot q neglect M1 and M3 (otherwise meff would not be complex), but one of these parameters may be considered real. This gives the last evidence we needed to prove that we simplified the VEVs of the scalar fields φ and χ as much as we could, in order to get SCPV and a valid quark mass spectrum. q After finding meff , we can also see that the choice of electroweak VEVs we have made presents an ˆ advantage towards for example, the one made by many authors [69, 71, 72] when they neglect b1 instead of vˆ1. In models where fermions may couple to more than one scalar field, we are unable to
43 Quark masses
mu(MZ ) = 1.327 ± 0.28 MeV +0.33 md(MZ ) = 2.769−0.21 MeV
ms(MZ ) = 54.79 ± 3.6 MeV
mc(MZ ) = 0.6314 ± 0.031 GeV
mb(MZ ) = 2.861 ± 0.045 GeV
mt(MZ ) = 173 ± 2.1 GeV
Table 4.2: Masses of quarks computed at the electroweak scale MZ = 91.1876 ± 0.0021 GeV (data taken from [86]).
diagonalize both Yukawa matrices simultaneously and as a consequence, FCNCs arise [85]. However, according to Eq. (4.60), the SM down-quarks only couple at leading order to the field φ, in contrast with the up-quarks which couple to χ with the VEV structure we are considering. With this, the q q problem of diagonalizing both Yφ and Yχ no longer exists, so FCNCs become extremely suppressed in the model we have been building4. Another interesting conclusion can be taken from Eq. (4.60) when we try to exclude the scalar field χ from the model. In that scenario, we would simply have q q † 2 q q† ˆ meff meff = v2 Yφ Yφ . Although up-quarks could be massive by neglecting b1 instead of vˆ1 [see Eq. (4.56)], the following relations would hold [69]:
mu mc mt = = . (4.61) md ms mb
q This result arises because the Yukawa matrix Yφ is the same for all quarks and since it is not observed experimentally, Eq. (4.60) provides one more evidence that two scalar fields are required in trinified models.
Recall now from Section 3.2.1 that we are free to work in a basis where the mass matrix of up-quarks is diagonal. In this context, Eq. (4.56) allows us to check that mu mc mt Y q = diag , , , (4.62) χ v sin ξ v sin ξ v sin ξ
q q † and the unitary matrix which diagonalizes Heff ≡ meff meff corresponds to the CKM matrix:
† 2 VCKMHeff VCKM = diag(md, ms, mb) . (4.63)
We are now ready to show that the effective matrix of down-quarks we have computed is compatible with all experimental data concerned with quark masses and the CKM matrix. In order compare our model with experiment, we use the Wolfenstein parameters shown in Table 2.3 and the masses of all quarks displayed in Table 4.2, computed at the electroweak scale. From Eq. (4.63), one may note that the eigenvalues of the hermitian matrix Heff must correspond to the square of the down-quark masses, whereas the CKM matrix can be regarded as the one with the eigenvectors of Heff . Assuming q that the Yukawa matrix Yφ is real and denoting it as
4 Considering Mˆ 1 to be complex, one is also unable to find SCPV with the VEVS vˆ1 and vˆ2. However, this problem could be solved simply by choosing Mˆ 3 to be complex instead of Mˆ 1
44 y11 y12 y13 Y q = , φ y21 y22 y23 (4.64) y31 y32 y33 we try to fit these parameters to seven observables (three angles, one phase and three quark masses). Similarly to what we have done in the previous section to relate the scalar potential parameters with the Higgs boson mass, we perform a χ2 minimization using the software MINUIT. As one can see from Table 4.3, we only need seven of the nine free parameters to fit the observables we have mentioned with χ2 1. It is worth to emphasize that similar results are obtained if we try to fit the parameters q of Yφ in the case where we consider M1 = M2 = M3 and cos ξ = sin ξ. Here we did not considered this simplification because, as discussed in the previous section, it would lead to unphysical masses for some scalar particles. The results shown in Table 4.3 also allow us to make a prediction for the masses of the vectorlike quarks. Using the set of parameters where y11 = y13 = 0, we may use Eq. (4.59) to obtain
8.984 × 1010 8.811 × 107 − 1.531 × 109i −1.461 × 108 − 2.340 × 1010i 7 9 10 10 12 DB = 8.811 × 10 + 1.531 × 10 i 9.850 × 10 −2.541 × 10 − 2.677 × 10 i . (4.65) −1.461 × 108 + 2.340 × 1010i −2.541 × 1010 + 2.677 × 1012i 7.293 × 1015
The masses of the vectorlike quarks correspond to the square root of the eigenvalues of DB, so they are
mB1 ' 299 TeV , (4.66)
mB2 ' 313 TeV , (4.67)
4 mB3 ' 8.540 × 10 TeV . (4.68)
All these values are above the current lower limit to the mass of the B quark (see Section 3.4), hence we verify that the predictions made by our model are compatible with experimental constraints on the mass of those vectorlike quarks. To conclude, we study briefly the lepton mass Lagrangian of the model. After replacing thefields φ and χ with the VEVs of Eq. (4.16), we may write5
0 ER ! ` ± eR 0 0 N2R L = eL EL M + E N ν N M , (4.69) Mass ` E L 2L L 1L ` ν R R N1R
± 0 where M` and M` are the mass matrices of charged leptons and neutrinos, respectively:
` `ˆ `ˆ ! −Y vˆ2 − Y b2 Y b3 M± = φ χ χ , (4.70) ` ` ˆ ` ˆ ` ˆ Yχ M2 −Yφ M1 − Yχ M3
` ˆ ` ˆ ` `ˆ Yφ M1 + Yχ M3 Yφ vˆ1 + Yχ b1 0 0 ` `ˆ 0 Yφ vˆ2 + Yχ b2 0 0 0 M = . (4.71) ` −Y `ˆb 0 0 −Y `vˆ − Y `ˆb χ 3 φ 1 χ 1 ` ˆ ` `ˆ −Yχ M2 0 −Yφ vˆ1 − Yχ b1 0
5 T In order to write the lepton mass matrices, one should take into account in Eq. (4.54) that ψ` = Cψ` , where C is the charge conjugation operator.
45 Zero entries Free parameters χ2
y12 0.01291
y21 −0.00132
y22 0.05688 −6 y11, y13 2.752 × 10 y23 0.09017
y31 0.22826
y32 −0.09430
y33 −2.28568
y12 −0.01300
y13 −0.00143 y 0.00144 −5 y11, y32 22 2.791 × 10 y23 0.05353
y31 0.09637
y32 −0.23123
y33 −2.29297
y12 0.01348
y13 −0.02034
y21 0.07485 −2 y11, y21 1.362 × 10 y22 0.03931
y23 0.19620
y31 0.48439
y33 3.11835
Table 4.3: Fit of the effective down-quark matrix to the CKM parameters and down-quark masses. q The first column indicates the parameters of Yφ we set up to zero and the second column the remaining parameters, computed with MINUIT.
± Apart from the different Yukawa matrices and overall signs, the matrix M` is similar to the mass matrix of down-quarks, Md. For this reason, all the discussion we have made regarding the VEV ± parameters that can be neglected should be applied to M` . Moreover, the results we obtain after diagonalizing it are similar to those presented in Eqs. (4.59) and (4.60), as long as we perform the q ` q ` replacements Yφ → Yφ and Yχ → Yχ . Considering the case where both Yukawa matrices of the leptonic sector are diagonal,
` ` Yφ = diag(yφ1, yφ2, yφ3) ,Yχ = diag(yχ1, yχ2, yχ3) , (4.72)
` the effective matrix for charged leptons (which we will denote as meff ) is simply defined by
2 2 2 2 M2 yφ1yχ1 yφ − 2 2 2 2 2 0 0 1 M1 yφ1+(M2 +M3 )yχ1 † M 2y 2y 2 ` ` 2 2 2 2 φ2 χ2 meff meff = v cos ξ 0 yφ − 2 2 2 2 2 0 . 2 M1 yφ2+(M2 +M3 )yχ2 2 2 2 2 M2 yφ3yχ3 0 0 yφ − 2 2 2 2 2 3 M1 yφ3+(M2 +M3 )yχ3 (4.73) One can note that these fermions do not need to couple to the field χ to become massive.
46 5 Concluding Remarks
In spite of some limitations, the SM is a theory in remarkable agreement with most experimental results. For this reason, one of the challenges we face putting forward a new theory is to guarantee that, at low energies, it delivers similar predictions to the ones given by the SM. This was our main motivation to study Trinification models. After reviewing the SM in Chapter 2, we were able to understand in Chapter 3 why models with vectorlike quarks are so attractive. One of the features we most emphasized is the possibility of having mixing with SM quarks. Even if we are not able to observe vectorlike quarks directly in the near future (we know already that if they exist their mass is at least of the order of TeV-scale), the effects of this mixing may provide an indirect evidence for their existence. Also, heavy-light quark mixing must be compatible with the oblique parameters S and T , and with the observables related to the Zbb coupling. The most interesting results of our work have been presented in Chapter 4. Namely, we were able to fit the many free parameters of the Trinification model to obtain the SM Higgs boson mass,aswell as the CKM matrix and quark masses. We achieved a valid scalar mass spectrum which, along with the mass of the Higgs boson, it predicts the correct number of massive neutral and charged scalar particles. Concerning CKM mixing, we noticed that the effective down-quark mass matrix contains one term related to the mixing between vectorlike and SM quarks. We concluded that this feature leads to a CKM pattern compatible with all measurements related to CP violation at low energies. Our findings have shown that Trinification models are valid SM extensions, at least from thepoint of view of quark masses, mixing and CP violation. Although we have not analyzed in detail the lepton sector, it would be interesting to investigate whether the present results on neutrino masses and mixing coming from neutrino oscillation experiments [87] could be reproduced in our model. This is a relevant and crucial question since neutrino masses can be accommodated in the present scenario via a seesaw mechanism [88–92]. Given this, we will further explore this possibility in the future.
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51 52 Feynman rules for A Vectorlike Quarks
In models where vectorlike quarks appear, they are allowed to interact with SM quarks, gauge bosons and the Higgs boson. For the SM extension we have studied in Chapter 3 (which only adds
SU(2)L singlets of vectorlike quarks), we present here the new Feynman rules that describe those interactions. Recall that these are read from Eqs. (3.21), (3.22) and (3.23):
Tr
g 1 − γ5 − ∗ µ W i√ Vriγ (A.1) µ 2 2
di
Tr
g ∗ µ 1 − γ5 Zµ i Xriγ (A.2) 2 cos θW 2
ui
Tr
g ∗ 1 − γ5 1 + γ5 H i Xri mi + mr (A.3) 2MW 2 2
ui
(A.4)
53 Bs
g µ 1 − γ5 W − i√ Vsiγ (A.5) µ 2 2
ui
Bs
g µ 1 − γ5 Zµ i Xsiγ (A.6) 2 cos θW 2
di
Bs
g 1 − γ5 1 + γ5 H i Xsi mi + mr (A.7) 2MW 2 2
di
(A.8)
54 Representations of the B Trinification group
We briefly summarize the important group and representation theory results for the Trinification group (G333 ≡ SU(3)c ⊗ SU(3)L ⊗ SU(3)R). Namely, we define the generators of this group and we compute the decomposition of the 24 and 27 representations in terms of the SM group, GSM ≡
SU(3)c ⊗ SU(2)L ⊗ U(1)Y .
B.1 The G333 Generators
a a a Each SU(3) subgroup of G333 contains eight generators, denoted as Tc , TL and TR (a = 1,..., 8). They are traceless, hermitian, and they satisfy the normalization condition
1 Tr T aT b = δab , (B.1) 2
In the trinification model, scalars and fermions are always in singlet, triplet or anti-triplet represen- tations of SU(3). Regarding the triplet representation (the fundamental one), we denote it by 3 and its generators are λa T a = , (a = 1,..., 8) , (B.2) C,L,R 2 where λa are the 3 × 3 Gell-Mann matrices:
0 1 0 0 −i 0 1 0 0 1 2 3 λ = 1 0 0 , λ = i 0 0 , λ = 0 −1 0 , 0 0 0 0 0 0 0 0 0
0 0 1 0 0 −i 4 5 λ = 0 0 0 , λ = 0 0 0 , (B.3) 1 0 0 i 0 0
0 0 0 0 0 0 1 0 0 1 λ6 = 0 0 1 , λ7 = 0 0 −i , λ8 = √ 0 1 0 . 3 0 1 0 0 i 0 0 0 −2
It is important to note that the Gell-Mann matrices λ1, λ2 and λ3 contain the Pauli matrices. This means that each SU(3) subgroup of G333 embeds SU(2), as we need to break SU(3)L into the SU(2)L group of the SM. Moreover, the existence of two scalar fields in the Trinification model ensures that
55 the group SU(3)L ⊗ SU(3)R is broken into SU(2)L ⊗ U(1)Y . Defining the generators of the anti-triplet a a ∗ representation 3 as T C,L,R ≡ −(TC,L,R) , we also conclude by observing Eq. (B.3) that
a a T T C,L,R = − TC,L,R . (B.4)
B.2 24 and 27 Representations
Before performing the decomposition of the representations that accommodate all the particle content of the Trinification model, we must remind that after SSB from G333 to the SM group, the linear combination of generators 3 1 8 8 Y = T + √ T + T , (B.5) R 3 L R remains unbroken. Therefore, we associate Y to the hypercharge generator. Gauge bosons are contained in the adjoint representation 24 of the Trinification model:
24 → (8, 1, 1) ⊕ (1, 8, 1) ⊕ (1, 1, 8) . (B.6)
The first term, (8, 1, 1), represents the gluons of the SM. The SU(3)c color gauge group is not broken, so this term does not have any decomposition. Concerning the second term, (1, 8, 1), we use the Young Tableaux technique to decompose it and we arrive to 1 1 1, , 1 → 1, , ⊕ 1, , 0 ⊕ (1, 1, 0) ⊕ 1, , − . (B.7) 2 2
i The second and third terms on the right-hand side have the quantum numbers of the gauge fields Wµ (i = 1, 2, 3) and Bµ respectively. Along with these fields that we find in the SM, it is also predicted an SU(2) doublet of gauge bosons. In order to study the decomposition of the (1, 1, 8) component of the 24 representation, it is useful to switch from the Gell-Mann to the Cartan-Weyl basis [93]. In the latter, we express all the off-diagonal generators in terms of raising and lowering operators:
1 1 t = (λ1 ± iλ2) , t = λ3 , ± 2 z 2 (B.8) 1 6 7 0 1 3 1 4 5 u± = (λ ± iλ ) , y = √ λ , v± = (λ ± iλ ) . 2 3 2
These 3 × 3 matrices may be regarded as the generators of the SU(3) fundamental representation in the Cartan-Weyl basis. With this, we may write the hypercharge operator in terms of the generators we presented in Eq. (B.8): 1 0 Y = tz + y . (B.9) 2 We are now in the position to build the adjoint representation of SU(3) in the basis we have defined. Adj The generators Ta (a = 1,..., 8) of this representation are defined as
Adj Ta bc = −ifabc , (B.10) where fabc are the structure constants that can be computed through the commutation relation,
[Ta,Tb] = ifabcTc . (B.11)
One might wonder at this point why have we made this change of basis. The reason was that in the Gell-Mann basis, we cannot build any diagonal generator for the adjoint representation, from which
56 we could read the hypercharges of the gauge bosons assigned to (1, 1, 8). On the other hand, we find 0 that the commutators involving tz and y are given by
[tz, t±] = ±t± , [y, t±] = 0 , 1 [t , u ] = ± u , z ± 2 ± [y, u±] = ±u± , (B.12) 1 [t , v ] = ± v , [y, v±] = ±v± , z ± 2 ± so we may conclude from Eq. (B.10) that we may write the following generators of the adjoint repre- 0 sentation, in the basis {t+, t−, u+, u−, v+, v−, tz, y }: Adj 1 1 1 1 Ttz = diag 1, −1, − , , , − , 0, 0 , 2 2 2 2 (B.13) Adj Ty0 = diag(0, 0, 1, −1, 1, −1, 0, 0) .
Adj Adj Linear combinations of the matrices obtained above are also diagonal. Considering Tx = aTtz + Adj bTy0 , we have 1 1 1 1 T Adj = diag a, −a, − a + b, a − b, a + b, − a − b, 0, 0 . (B.14) x 2 2 2 2
1 Finally, the hypercharge operator shown in Eq. (B.9) is obtained by considering a = 1 and b = 2 . Once we do this, we are led to
Adj TY = diag(1, −1, 0, 0, 1, −1, 0, 0) (B.15) and we may write (1, 1, 8) → 4(1, 1, 0) ⊕ 2(1, 1, 1) ⊕ 2(1, 1, −1) . (B.16)
We conclude that Trinification models predict eight more SU(2)L singlets of gauge bosons. Four of these new particles are electrically charged, whereas the other four are neutral.
Regarding the 27 representation of E6, it transforms under G333 in the following way:
27 → (1, 3, 3) ⊗ (3, 1, 3) ⊗ (3, 3, 1) . (B.17)
Using the following notation for the Young Tableaux of the anti-fundamental representation of SU(3),
3 → = , (B.18) the component (1, 3, 3) is decomposed as follows:
1 1 1 2 1 1, , 3 → 1, , − ⊕ 1, 1, ⊗ 1, 1, − ⊕ 1, 1, ⊕ 1, 1, − 6 3 3 3 3 (B.19) 1 1 → 2 1, 2, − ⊕ 1, 2, ⊕ 2 (1, 1, 0) ⊕ (1, 1, 1) . 2 2
1 In Eq. (B.19), one of the doublets with hypercharge − 2 and the term with quantum numbers (1, 1, 0) can be identified with the leptons of the SM. Considering for example the first generation offermions, the doublet we referred corresponds to the one formed by the electron and its neutrino, while the SU(2) singlet may be identified as the positron. All the other terms of the decomposition belongto unknown leptons predicted by trinified models. The other two bitriplets ofthe 27 representation are
57 associated to quarks. Using the Young Tableaux technique when convenient, the decompositions of (3, 1, 3) and (3, 3, 1) are given by
1 2 1 (3, 1, 3) → 3, 1, ⊕ 3, 1, − ⊕ 3, 1, , (B.20) 3 3 3 1 1 3, , 1 → 3, , ⊕ 3, 1, − . (B.21) 6 3
As expected, we find in Trinification models the quarks we already know and a vectorlike quark, with charge Q = −1/3 and singlet of SU(2)L. Finally, we must note that scalar fields are assigned to the 27 representation as well. For this reason, the decompositions shown in Eqs. (B.19), (B.20) and (B.21) must be applied to the scalar fields we introduce in Trinification models, meaning that their particle content is the same astheone we find in each generation of fermions.
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