% %% � � � , , ,, � �� �� e ee @ @@ l l l Q QQ HHPPPXXX hhhh (((( ��� IFT Instituto de F́ısica Teórica Universidade Estadual Paulista Ph.D. THESIS / TESE DE DOUTORAMENTO IFT-T.003/18 Electron and muon anomalous magnetic dipole moment in the 3-3-1 model with heavy leptons O momento de dipolo magnético anômalo do elétron e do múon no modelo 3-3-1 com léptons pesados George De Conto Santos Advisor / Orientador Vicente Pleitez March 2018 / Março de 2018 Acknowledgements I would like to thank Conselho Nacional de Desenvolvimento Cient́ıfico e Tecnológico (CNPq) for the financial support. Also would like to thank Instituto de F́ısica Teórica - Universidade Estadual Paulista (IFT-UNESP) for the opportunity, infrastructure and support to realize this work. More importantly, I would like to thank Prof. Vicente Pleitez for his supervision, time, effort and support. 2 Abstract We calculate, in the context of the 3-3-1 model with heavy charged leptons, constraints on some parameters of the extra particles in the model by imposing that their contribu- tions to both the electron and muon (g − 2) factors are in agreement with experimental data up to 1σ-3σ. In order to obtain realistic results we use some of the possible solu- tions of the left- and right- unitary matrices that diagonalize the lepton mass matrices, giving the observed lepton masses and at the same time allowing to accommodate the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrix. We show that, at least up to 1-loop order, in the particular range of the space parameter that we have explored, it is not possible to fit the observed electron and muon (g − 2) factors at the same time unless one of the extra leptons has a mass of the order of 20-40 GeVs and the energy scale of the 331 symmetry to be of around 60-80 TeVs. Keywords: magnetic dipole moment, muon, electron, 3-3-1 model. Resumo Nós calculamos, no contexto do modelo 3-3-1 com léptons pesados carregados, v́ınculos sobre alguns dos parâmetros das part́ıculas extras do modelo ao impor que suas con- tribuições aos fatores (g − 2) do elétron e do múon estejam de acordo com os dados experimentais dentro de 1σ-3σ. Para obter resultados realistas nós consideramos algu- mas das posśıveis soluções das matrizes unitárias esquerda e direita que diagonalizam as matrizes de massa leptônicas, dando as massas leptônicas observadas e ao mesmo tempo acomodando a matriz de mistura de Pontecorvo-Maki-Nakagawa-Sakata (PMNS). Nós mostramos que, ao menos até a ordem de 1-loop, na faixa de parâmetros explorada, não é posśıvel acomodar simultaneamente os fatores (g − 2) do elétron e do múon a não ser que um dos léptons extras tenha massa da ordem de 20-40 GeVs e a escala de energia da simetria 331 esteja em torno de 60-80 TeVs. Palavras chave: momento de dipolo magnético, múon, elétron, modelo 3-3-1. 3 Contents 1. Introduction 8 2. The magnetic dipole moment in quantum field theory 11 2.1. First order correction for the electron-photon vertex . . . . . . . . . . . . 14 3. Standard Model prediction for the muon AMDM 18 3.1. Pure quantum electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . 18 3.2. Hadronic contributions in QED . . . . . . . . . . . . . . . . . . . . . . . . 18 3.3. Electroweak contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.4. Total Standard Model contribution . . . . . . . . . . . . . . . . . . . . . . 21 4. The 3-3-1 model 23 4.1. Leptons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.2. Quarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.3. Scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.4. Yukawa interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.5. Gauge fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 5. MDM in the 331HL model 29 5.1. Scalar contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 5.2. Vector contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 6. Results for the electron and the muon AMDM 34 6.1. Numerical results for the AMDMs . . . . . . . . . . . . . . . . . . . . . . 34 6.1.1. Diagonal V l L,R matrices . . . . . . . . . . . . . . . . . . . . . . . . 35 6.1.2. 1st set of V l L,R matrices . . . . . . . . . . . . . . . . . . . . . . . . 39 6.1.3. 2nd set of V l L,R matrices . . . . . . . . . . . . . . . . . . . . . . . . 41 6.1.4. 3rd set of V l L,R matrices . . . . . . . . . . . . . . . . . . . . . . . . 43 6.1.5. 4th set of V l L,R matrices . . . . . . . . . . . . . . . . . . . . . . . . 43 4 Contents 6.2. Numerical analysis conclusions . . . . . . . . . . . . . . . . . . . . . . . . 45 7. Other constraints 50 7.1. The µ→ eγ decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 7.2. Lifetime of the exotic charged leptons . . . . . . . . . . . . . . . . . . . . 51 8. Conclusions 54 A. Scalar Potential 57 B. Scalar mass eigenstates 58 B.1. Double charge scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 B.2. Single charge scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 B.3. Neutral scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 C. Interactions 61 C.1. Lepton-scalar vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 C.2. Scalar-photon vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 C.3. Gauge vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 C.4. Charged gauge-lepton interactions . . . . . . . . . . . . . . . . . . . . . . 63 C.5. Neutral gauge-lepton interactions . . . . . . . . . . . . . . . . . . . . . . . 64 D. Scalar-fermion loop diagram calculation 65 5 List of Figures 2.1. Interaction of two electron mediated by a photon. . . . . . . . . . . . . . . 11 2.2. Electron-photon vertex corrections. . . . . . . . . . . . . . . . . . . . . . . 12 2.3. Diagram for the electron-photon vertex first order correction . . . . . . . 14 3.1. Examples of QED diagrams contributing to the muon MDM. . . . . . . . 19 3.2. Hadronic vacuum polarization contribution to the MDM. . . . . . . . . . 20 3.3. Hadronic light-by-light scattering contribution to the MDM. . . . . . . . . 21 3.4. Electroweak contributions to the MDM. . . . . . . . . . . . . . . . . . . . 21 5.1. One loop diagrams for the MDM. . . . . . . . . . . . . . . . . . . . . . . . 30 5.2. Types of one loop diagrams for the MDM. . . . . . . . . . . . . . . . . . . 31 6.1. U−− contribution to the muon AMDM. . . . . . . . . . . . . . . . . . . . 36 6.2. vχ and mEe values satisfying the electron and muon AMDM, diagonal matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 6.3. vχ and mEµ values satisfying the electron and muon AMDM, diagonal matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 6.4. vχ and mEτ values satisfying the electron and muon AMDM, diagonal matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 6.5. vχ and mEe values satisfying the electron and muon AMDM, 1st set of V l L,R matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 6.6. vχ and mEµ values satisfying the electron and muon AMDM, 1st set of V l L,R matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 6.7. vχ and mEτ values satisfying the electron and muon AMDM, 1st set of V l L,R matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 6.8. vχ and mEe values satisfying the electron and muon AMDM, 2nd set of V l L,R matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 6 List of Figures 6.9. vχ and mEµ values satisfying the electron and muon AMDM, 2nd set of V l L,R matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 6.10. vχ and mEτ values satisfying the electron and muon AMDM, 2nd set of V l L,R matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 6.11. vχ and mEe values satisfying the electron and muon AMDM, 3rd set of V l L,R matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 6.12. vχ and mEµ values satisfying the electron and muon AMDM, 3rd set of V l L,R matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 6.13. vχ and mEτ values satisfying the electron and muon AMDM, 3rd set of V l L,R matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 6.14. vχ and mEe values satisfying the electron and muon AMDM, 4th set of V l L,R matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 6.15. vχ and mEµ values satisfying the electron and muon AMDM, 4th set of V l L,R matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 6.16. vχ and mEτ values satisfying the electron and muon AMDM, 4th set of V l L,R matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 6.17. Y −− contribution to the muon MDM. . . . . . . . . . . . . . . . . . . . . 47 7.1. E lepton decay. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 7.2. E lepton lifetime. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 7 1. Introduction The electron and the muon have magnetic dipole moments (MDM), due to their spin. These MDMs interact with external magnetic fields, and in Quantum Mechanics these interactions are described by the following term in the Hamiltonian Hµ = −~µ · ~B ~µ = g Qe 2m ~S, (1.1) where e is the electron charge, Q is the electric charge of the particle (-1 for the electron and muon), m is the particle’s mass and ~S is its spin. The factor g is known as the gyromagnetic factor. From Dirac’s relativistic equation, Dirac himself found that g = 2, which agreed to the experimental results at the time. Later, with measurements of the hyperfine structure of hydrogen, is was found that g ≈ 2. Part of such difference was first explained by Schiwnger, where he calculated the correction for the electron-photon vertex in QED (for more details see sec. 2.1). With such deviations of g with respect to 2, it is convenient to write ~µ = 2(1 + a) Qe 2m ~S a = g − 2 2 (1.2) where the first term on the left (proportional to 2) is Dirac’s prediction, and a embodies the corrections to the gyromagnetic factor. Such corrections are known as the anomalous magnetic dipole moment (AMDM). Both anomalous magnetic dipole moments of electron and muon ae,µ = (g − 2)e,µ/2, have been measured with great precision. We have aexpe = 1.15965218076(28) × 10−3 for the electron, and aexpµ = 1.1659209(6) × 10−3 for the muon [1]. The theoretical calculations within the Standard Model (SM) have reached the high level of precision of the experiments. On one hand, the calculation for the electron ae, considering only QED up to tenth order, gives the result of aSMe = 1.159652181643(25)(23)(16)(763)×10−3 [2], giving a difference of ∆ae = aexpe − aSMe = −0.91(82)× 10−12, (1.3) 8 1. Introduction which is close to one standard deviation. On the other hand, for the muon aµ, cal- culations considering QED up to five loops plus hadronic vacuum polarization up to next-to-leading order, hadronic light-by-light scattering and electroweak contributions, results in aSMµ = 1.16591801(49) ×10−3 [3, 4, 5], leading to a difference between theory and experiment of ∆aµ = aexpµ − aSMµ = 2.87(80)× 10−9, (1.4) a difference that goes beyond three standard deviations from the experimental result. This difference between the SM prediction and the experimental value of the muon anomalous magnetic moment has been studied in many models. For instance, in other 3-3-1 models [6, 8, 9, 10], left-right symmetric models [11], supersymmetric models [12, 13, 14, 15, 16] and two-Higgs doublets models [17, 18]. In particular, the 3-3-1 models are interesting extensions of the standard model (SM), since they solve some of the questions that the SM leaves without answer. For example, among others, the number of generations, why sin2 θW < 1/4, and electric charge quan- tization. The models have also interesting consequences in flavor physics [19, 20, 21] and, in particular, those 3-3-1 models with quarks with electric charge -4/3 and 5/3 (in units of |e|) have at least one neutral scalar which can appear as a heavy resonance that could be observed at LHC [22, 23, 24, 25]. The model has new singly and doubly charged vector bosons [26] and hadrons with electric charges ±(3, 4, 5) [27] that can also appear as LHC resonances. In fact, if the existence of these resonances is confirmed in the near future, one natural explanation for it is the minimal 3-3-1 model or the 3-3-1 model with heavy leptons. We use here natural in the sense that usually models are proposed just to solve particular discrepancies with the SM, as the latter ones. Meanwhile, 3-3-1 models assume a new set of gauge symmetries and explore their consequences, supposing that this new set of symmetries may explain all experimental results. At the same time that the agreement between the experiment and the SM for the electron AMDM gives us confidence in the correctness of the theory, the disagreement in the muon case suggests that there may be effects unaccounted by the SM, or it is still possible that such effects come from new particles and through their interactions with the already known particles. For instance, the contributions of heavy leptons to the aµ factor have been considered in gauge models since Refs. [28, 29]. However, we should not forget that for the AMDM of the muon to imply new physics, its value should disagree with the SM beyond 5σ. 9 1. Introduction In the literature, when the explanation of the observed value of the muon aµ is based on new physics, its effects in the electron ae are usually not taken into account, it is assumed that these effects do not perturb the values of the electron AMDM. Here we will show that, at least in this particular model, it is not possible to fit the ∆ae and ∆aµ with the same parameters considering 1-loop order calculations, unless one of the extra leptons has a mass of the order of 20-40 GeV and the energy scale of the 331 symmetry is of tens of TeVs. Also, usually in literature the charged lepton masses are neglected, thus allowing to obtain simple expressions for the ∆aµ. In the latter cases, ∆aµ(X) ∝ m2 µ/M 2 X , where MX is the mass of some exotic particle. Although this case seems to be reasonable, we decided to not use such approximations, given the high precision of the experimental results. See chapter 8 for a more detailed discussion. As a final remark, the work here presented has already been published. It can be found in [30], where a shorter version of what is discussed in this thesis is available. 10 2. The magnetic dipole moment in quantum field theory Here we will briefly discuss how the magnetic dipole moment (MDM) of a particle is calculated, taking the correction to the electron-photon vertex as an example. Consider the diagram from Fig. 2.1 for an interaction between two electrons, where the circle on the left vertex is given by the corrections from the electron-photon vertex. This scattering amplitude is given by: iM = ie2 [ ū ( p′ ) Γµ ( p, p′ ) u (p) ] 1 q2 [ ū ( k′ ) γµu (k) ] (2.1) where Γµ represents the corrected vertex (a pictorial interpretation may be seen in Fig. 2.2). This term must transform as a vector (or pseudovector), therefore we can represent it as a linear combination of gamma matrices and momenta involved in the vertex [32]: Γµ (q) = F1 ( q2 ) γµ + F2 ( q2 ) iσµνqν 2m + FA ( q2 ) ( γµγ5q 2 − 2mγ5q µ ) + F3 ( q2 ) σµνγ5qν (2.2) where q = p′ − p, m is the fermion mass and the Fx are the form factors. The terms proportional to FA (axial current) and F3 (electric dipole moment) do not contribute to the MDM. It is easy to see that because the MDM is CP invariant, while the terms proportional to FA and F3 are not, due to the γ5 matrices. Given that the MDM is an intrinsic propoerty of the particle, it must exist even Figure 2.1.: Interaction of two electron mediated by a photon. The electrons are repre- sented by continuous lines and the photon by the wavy line. 11 2. The magnetic dipole moment in quantum field theory Figure 2.2.: The vertex in Fig. 2.1 can be understood as the sum of corrections coming from loops created by the interactions in a given theory. when no interactions are present. Therefore, we consider the non-relativistic limit for its calculation. Also, it is possible to verify that the two first terms in Eq. 2.2, in the non-relativistic limit, reduce to the magnetic dipole moment operator from quantum mechanics. With just these two terms, our equation for the current becomes: Jµ = ū ( p′ ) [ F1 ( q2 ) γµ + F2 ( q2 ) iσµνqν 2m ] u (p) , (2.3) where we considered the first two terms from Eq. 2.2 to construct the electron-photon interaction current, because they are the only ones of interest to the MDM. Remebering that, for Dirac spinors in the non-relativistic limit, we can use the identity: Ψ = ( ψ 0 ) (2.4) where we are left with only the paticle spinor, being the antiparticle one negligible in the non-relativistic limit. With that, we have for the first term in the current ū ( p′ ) γiu (p) = ψ† [( p′jσ j ) σi + σi ( pjσ j )] ψ = ψ† [ p′j ( δji + iεjikσk ) + pj ( δij + iεijkσk )] ψ = ψ† [( p′j + pj ) δij + ( pj − p′j ) iεijkσk ] ψ = ψ† [( p′i + pi ) + iεijkqjσ k ] ψ = ψ† [( p′i + pi ) − iεijkqjσk ] ψ (2.5) and the second term ū ( p′ ) iσµνqν 2m u (p) = 2mψ† [ i 2m εijkqjσ k ] ψ = 2mψ† [ −i 2m εijkqjσk ] ψ, (2.6) where we used the Dirac representations for gamma matrices, simply substituting these in 2.3. 12 2. The magnetic dipole moment in quantum field theory Considering only terms proportional to εijkqjσk, we can write the following amplitude for the current interaction with the electromagnetic field ~A (q) as: iM = −iψ† [ −iεijkqjσk (F1 (0) + F2 (0)) ] ψAi = iψσk [F1 (0) + F2 (0)]ψBk (2.7) where Bk (q) = −iεijkqiAj is the magnetic field as a momentum function (given by a Fourier transform). We considered q2 = 0 because the MDM is independent of the momentum transfered between the fermion and the photon, once again due to the fact the the MDM is an instrinsic property of the particle. We can see that the amplitude represents a interaction between the fermionic spin (~σ) and the magnetic field ( ~B), which characterizes a contribution to the MDM. Interpreting this amplitude in Born’s approximation we can see that it corresponds to a potential with the form [33]: V (x) = −〈~µ〉 · ~B (x) (2.8) where 〈~µ〉 = e m [F1 (0) + F2 (0)]ψ† ~σ 2 ψ (2.9) This expression can be written in its more traditional form ~µ = g ( e 2m ) ~S (2.10) where ~S is the electron’s spin. Comparing Eq. 1.2 and the above equations we see that g = 2 [F1 (0) + F2 (0)] = 2 + 2F2 (0) . (2.11) The factor F1 (0) is equal to 1 in all perturbation orders, that is because of its relation with electrostatic interactions. Such factor gives us a measure of the electric charge perceived by a particle, this charge being equal to F1 ( q2 ) · e, where e is the positron charge. Meanwhile, the F2 ( q2 ) factor, at first order in perturbation theory, is null. At higher orders it assumes a non-zero value, giving rise to a small difference of the MDM predicted by Dirac. These higher order contributions are the anomalous magnetic dipole moment (AMDM). 13 2. The magnetic dipole moment in quantum field theory Figure 2.3.: Diagram for the electron-photon vertex first order correction. The momenta of each line are indicated in the figure. 2.1. First order correction for the electron-photon vertex The first vertex correction comes from the photon creating a loop together with the fermion line. Starting from the diagram in Fig. 2.3 we find the following: δΓµ = ∫ d4k (2π)4 [ −igρν (k − p)2 + iε ] (−ieγν) [ i /k′ +m k′2 −m2 + iε ] γµ [ i /k +m k2 −m2 + iε ] (−ieγρ) = 2ie2 ∫ d4k (2π)4 /kγµ /k′ +m2γµ − 2m (k + k′)µ[ (k − p)2 + iε ] [k′2 −m2 + iε] [k2 −m2 + iε] (2.12) We can use Feynman’s parametrization 1 A1 · · ·An = ∫ 1 0 dx1 · · · dxnδ (x1 + · · ·+ xn − 1) (n− 1)! [x1A1 + · · ·+ xnAn]n (2.13) to write the denominator of our integral as: 1[ (k − p)2 + iε ] [k′2 −m2 + iε] [k2 −m2 + iε] = ∫ 1 0 dxdydzδ (x+ y + z − 1) 2 D3 (2.14) where we have D = x [ k2 −m2 + iε ] + y [ k′2 −m2 + iε ] + z [ (k − p)2 + iε ] = x ( k2 −m2 ) + y ( k2 + q2 + 2k · q −m2 ) + z ( k2 + p2 − 2k · p ) + (x+ y + z) iε = k2 + 2yk · q − 2zk · p− xm2 + y ( q2 −m2 ) + zp2 + iε (2.15) 14 2. The magnetic dipole moment in quantum field theory where, in the above equations, we used the identity x+y+z = 1. Defining l = k+yq−zp, we can write D = l2 − ( −2yzq · p+ y2q2 + z2p2 ) − xm2 + y ( q2 −m2 ) + zp2 + iε (2.16) Knowing that: p · q = p · ( p′ − p ) = p · p′ −m2 (2.17) q2 = ( p′ − p )2 = 2m2 − 2p · p′ (2.18) we have p · q = −q 2 2 (2.19) Which allows us to write the denominator as D = l2 + 2yz ( −q2/2 ) − y2q2 + y ( q2 −m2 ) − z2m2 + zm2 − xm2 + iε = l2 + q2xy −m2 (1− z)2 + iε (2.20) Now we can write again the vertex δΓµ = 2ie2 ∫ d4k (2π)4 dxdydzδ(x+ y + z − 1) 2 [ /kγµ /k′ +m2γµ − 2m (k + k′)µ ][ l2 + q2xy −m2 (1− z)2 + iε ]3 (2.21) Moving on, now we worry about the integral’s numerator, first writing it as a function of l: Nµ = /kγµ /k′ +m2γµ − 2m ( k + k′ )µ = ( /l − y/q + z/p ) γµ ( /l + /q (1− y) + z/p ) − 2m (2l + q (1− 2y) + 2zp)µ (2.22) and with the identities ∫ d4l (2π)4 lµ D3 = 0 (2.23) ∫ d4l (2π)4 lµlν D3 = ∫ d4l (2π)4 1 4g µν l2 D3 (2.24) the numerator becomes Nµ =− l2 2 γµ + ( −y/q + z/p ) γµ ( /q (1− y) + z/p ) +m2γµ − 2m (q (1− 2y) + 2zp)µ =− l2 2 γµ + y (1− y + z) /p′γµ/p+ (z − y) (1− y + z) [ 2pµ/p−m2γµ ] − y (y − 1) [ 2p′µ /p′ −m2γµ ] − 2m [q (1− 2y) + 2zp]µ +m2γµ − (z − y) (1− y) { 2 [ p′µ/p+ ( m2 − q2/2 ) γµ − pµ /p′ ] + /p′γµ/p } (2.25) 15 2. The magnetic dipole moment in quantum field theory To obtain ū (p′)Nµu (p), we can use the identities ū (p′) /p′ = mū (p′) and /pu (p) = mu (p) in the numerator, such that: Nµ = γµ [ − l 2 2 + (1− x) (1− y) q2 + ( 1− 2z + z2 ) m2 ] + ( p′µ + pµ ) mz (z − 1) + qµm (z − 2) (x− y) (2.26) Using Ward’s identity for QED, qµΓµ = 0, we see that we can neglect the term linear in qµ. Now, using Gordon’s identity ū ( p′ ) γµu (p) = ū ( p′ ) [p′µ + pµ 2m + iσµνqν 2m ] u (p) (2.27) we can replace the term proportional to p′µ + pµ and write ū ( p′ ) δΓµu (p) =2ie2 ∫ d4l (2π)4 ∫ 1 0 dxdydzδ (x+ y + z − 1) 2 D3 × ū ( p′ ){ γµ [ − l 2 2 + (1− x) (1− y) q2 + ( 1− 4z + z2 ) m2 ] + iσµνqν 2m [ 2m2z (1− z) ]} u (p) (2.28) Comparing Eq. 2.28 with Eq. 2.3 we can identify the form factors F1 ( q2 ) and F2 ( q2 ) with the terms that multiply γµ and iσµνqν/2m, respectively. To find F2 ( q2 ) we shall use the identity ∫ d4l (2π)4 1 (l2 −∆)m = i (−1)m (4π)2 1 (m− 1) (m− 2) 1 ∆m−2 (2.29) such that F2 ( q2 ) = α 2π ∫ 1 0 dxdydzδ (x+ y + z − 1) 2m2z (1− z) −xyq2 + (1− z)2m2 (2.30) The MDM contribution is given by the form factor evaluated at q2 = 0, according to 2.11. In this manner, at our case of interest F2 (0) = α 2π ∫ 1 0 dxdydzδ (x+ y + z − 1) 2z 1− z = α 2π ≈ 0, 0011614 (2.31) where we finally find the first order contribution for the electron’s anomalous MDM. Comparing with the experimental values given by [1], where F2(0) = 0, 0011597, we see 16 2. The magnetic dipole moment in quantum field theory that the first order prediction has a 0.02% difference with respect to the experimental results, not bad for a first order calculation. This example, for the 1-loop photon contribution to the electron AMDM, can be extended to contributions from other processes and from other models. To find these contributions, it is necessary to calculate all Feynman diagrams that correct the fermion- photon vertex, then all the amplitudes are summed and the coefficient multiplying the term iσµνqν/2m (i.e. the second term in Eq. 2.3), identified as F2(q), has to be evaluated at null momentum (q = 0). This F2(0) is the contribution to the anomalous magnetic moment. This is the procedure we will follow in this work to calculate the contributions of the 3-3-1 model with heavy leptons (331HL) to the muon and electron AMDMs, and it is also the procedure followed to calculate the SM contributions, which we shall discuss in the next chapter. 17 3. Standard Model prediction for the muon AMDM Many of the SM contributions for the muon AMDM have been calculated throughout several works (see [4] and references therein), up to 5-loop diagrams. Given the diversity of results, we will separate them in three categories: Pure quantum electrodynamics (QED), hadronic contributions in QED, and electroweak contributions. 3.1. Pure quantum electrodynamics These contributions are the dominating ones, having been calculated up to 5-loops. These diagrams involve only leptons and photons and they can be divided in two cate- gories: diagrams where the external and internal leptons are all of the same flavor, and diagrams where some or all the internal leptons are different from the external ones. In the first case, diagrams from 1 up to 3 loops were all calculated analytically, 4- loops have some numerical and some analytical results, and 5-loops have only numerical results. In the second case, 2 and 3 loop results are all analytical and 4-loop results are analytical and numerical. The total contribution coming from these two cases to the muon MDM is aQEDµ = 116584718.09(0.15)× 10−11. 3.2. Hadronic contributions in QED Besides leptons in the internal lines of a diagram, hadrons can be present as well, giving their own contributions. These electromagnetic interactions between hadrons produces contributions induced by hadronic vacuum polarization (HVP) and hadronic light-by- light scattering (HLbyL). HVP contributions come from vacuum corrections to the photon propagator, repre- sented by the circle in Fig. 3.2. Such corrections are based on the spectral representation, 18 3. Standard Model prediction for the muon AMDM Figure 3.1.: Examples of QED diagrams contributing to the muon MDM. These are 10th order examples of diagrams that contribute to the muon MDM. The solid lines represent leptons and the wavy lines photons. 19 3. Standard Model prediction for the muon AMDM Figure 3.2.: Hadronic vacuum polarization contribution to the MDM. The red circle represents the corrections to the photon propagator while the X is where the external photon is connected to the diagram. where the full propagator is given by a linear combination of all the free propagators in the theory, and the coefficients for this linear combination are given by experimental data. In leading order calculations, the HVP contribution is aHV P−LOµ = 6923(42)×10−11 and next-to-leading order is aHV P−NLOµ = 98.4(0.7)× 10−11. Hadronic light-by-light scattering comes from an effective 4-photon vertex, as shown in Fig. 3.3. Contrary to the HVP case, the HLbyL case has no experimental input available (photon-photon scattering data) for its calculation, therefore relying solely on theoretical results. So far, the most rigorous result come from large-Nc limits (where it is assumed that the number of colors in the chromodynamic sector is very large), giving a contribution of aHLbyLµ = 105(26)× 1011 to the muon MDM. It is expected that lattice QCD techniques may provide better first-principles estimates in the future. 3.3. Electroweak contributions This class of contributions considers diagrams involving only particles from the elec- troweak sector of the SM (i.e. fermions, gauge bosons and the Higgs boson). Examples of these diagrams can be seen in Fig. 3.4. There are results for 1 and 2 loops, being the 1-loop results all analytical. The 2-loop cases are subdivided in two categories, when there is a closed fermion loop and when there are bosonic corrections. In the latter case the results are obtained using an expansion in powers of sin2θW , where terms propor- tional to log(M2 W /m 2 µ), log(M2 H/m 2 W ), (M2 W /M 2 H)log(M2 H/m 2 W ), M2 W /M 2 H and constant terms are kept. In the fermion-loop case, when the fermions are heavy, the approximation mf/mµ >> 1 is used, when the fermion is light effective theories are considered. In total, these two types of diagrams contribute to the muon MDM as aEWµ = 153(1)× 10−11. 20 3. Standard Model prediction for the muon AMDM Figure 3.3.: Hadronic light-by-light scattering contribution to the MDM. The red trian- gle represents the effective 4-photon vertex while the X is where the external photon is connected to the diagram. 3.4. Total Standard Model contribution Considering all contributions discussed in the sections above, their total contribution to the muon anomalous magnetic dipole moment is aSMµ = 1.16591801(49) × 10−3. If we compare this with the experimental result of aexpµ = 1.1659209(6)× 10−3, the difference is ∆aµ = aexpµ − aSMµ = 2.87(80)× 10−9, (3.1) a difference that goes beyond three standard deviations from the experimental result. This difference may become smaller with further calculations, either by exploring diagrams up to higher orders or by improving the ones done for the already explored diagrams. However, it is possible that even with such improvements this difference Figure 3.4.: Electroweak contributions to the MDM. (a) 1-loop level contributions, where X indicates the vertex connecting to the photon. (b) Two-loop electroweak diagrams generated by the γγZ triangle for the first family. 21 3. Standard Model prediction for the muon AMDM Contribution Result (10−11) QED 116 584 718.09 (0.15) HVP-LO 6923(42) HVP-NLO -98.4 (0.7) EW 153(1) Total 116592090(6) Table 3.1.: Standard Model contributions to the muon AMDM. persists, and this leaves room for explorations of new alternatives to the SM, such as the 3-3-1 model with heavy leptons, which we shall present in the following chapters. 22 4. The 3-3-1 model Models with gauge symmetry SU(3)C ⊗ SU(3)L ⊗ U(1)X present new possibilities for the electroweak interactions, thanks to the SU(3)L symmetry, different from the SU(2)L used in the SM. Here we consider the 3-3-1 model with heavy charged leptons (331HL for short) in which there are new exotic quarks and leptons. Moreover, to give mass to all the particles, more scalar fields are needed. Hence, these models are intrinsically multi-Higgs models. In this model the electric charge operator is given by Q/|e| = T3 − √ 3T8 + X [31], where e is the electron charge, T3,8 = λ3,8/2 (being λ3,8 the Gell-Mann matrices) and X is the hypercharge operator associated to the U(1)X group 1. In the sections below we will present the particle content of the model, with its charges associated to each group in the parentheses, in the form (SU(3)C , SU(3)L, U(1)X). 4.1. Leptons The substitution of the SU(2)L symmetry for a SU(3)L one (comparing with the Stan- dard Model), means that now the fermions are grouped into triplets. However, the third component differs from one 3-3-1 model to another. In the case of the 331HL, the third component is an exotic, positively charged, heavy lepton. In some models, the third component is a positron, in others a neutrino or even an exotic neutral heavy particle. For the 3-3-1 model with heavy leptons, the leptonic sector has a left handed triplet and two right handed singlets, defined as follows: ΨaL =  νa l−a E+ a  ∼ (1, 3, 0) (4.1) l−aR ∼ (1, 1,−1) E+ aR ∼ (1, 1, 1) (4.2) 1Other 3-3-1 models may have different charge operators, where the constant multiplying T8 can be ± √ 3 or ±1/ √ 3. This leads to a distinct particle content in all sectors. 23 4. The 3-3-1 model where indices L andR indicate left and right handed spinors, respectively, and a = e, µ, τ . 4.2. Quarks Even tough we won’t need the quarks for the calculations of the AMDM’s, we will present this sector. Here, there are two anti-triplets and one triplet, both left handed; besides the right handed singlets. QmL =  dm −um jm  ∼ (3, 3∗,−1/3) , Q3L =  u3 d3 J  ∼ (3, 3, 2/3) (4.3) uαR ∼ (3, 1, 2/3) , dαR ∼ (3, 1,−1/3) , jmR ∼ (3, 1,−4/3) , JR ∼ (3, 1, 5/3) (4.4) where m = 1, 2 e α = 1, 2, 3. The exotic quarks j1,2 and J have electric charges -4/3 and 5/3, respectively. 4.3. Scalars The scalar sector for the 331HL is formed by three triplets: χ′ =  χ ′− χ ′−− χ ′0  ∼ (1, 3,−1) , ρ′ =  ρ ′+ ρ ′0 ρ ′++  ∼ (1, 3, 1) , η′ =  η ′0 η ′− 1 η ′+ 2  ∼ (1, 3, 0) (4.5) where ψ0 = vψ√ 2 ( 1 + X0 ψ+iI 0 ψ |vψ | ) and vψ = |vψ|eiθψ , for ψ = χ, η, ρ. Given that the model has a SU(3) symmetry in its scalar sector we can do the following transformation U =  e−iθη 0 0 0 e−iθρ 0 0 0 ei(θη+θρ)  (4.6) With this the triplets become χ =  χ− χ−− χ0  ∼ (0, 3,−1) , ρ =  ρ+ ρ0 ρ++  ∼ (0, 3, 1) , η =  η0 η−1 η+2  ∼ (0, 3, 0) (4.7) 24 4. The 3-3-1 model Now only the χ triplet has a complex phase in its vacuum expectation value, which means, vρ = |vρ|, vη = |vη| and vχ = |vχ|ei(θ ′ η+θ ′ ρ+θ ′ χ) = |vχ|eiθχ . With this, θχ becomes a new source of CP violation, in addition to the CKM and PMNS matrices already present in the SM. Also, some of the matrices that diagonalize the fermion mass matrices may appear in the interaction vertices, not just as the products that generate the CKM and PMNS matrices, but in other combinations as well (see appendix C.1). Given that the MDM is not related to CP violation, we will not consider its effects throughout this work. The most general scalar potential is given by V (χ, η, ρ) = µ21χ †χ+ µ22η †η + µ23ρ †ρ+ (α εijkχiρjηk +H.c.) + a1 ( χ†χ )2 + a2 ( η†η )2 + a3 ( ρ†ρ )2 + a4 ( χ†χ )( η†η ) + a5 ( χ†χ )( ρ†ρ ) + a6 ( ρ†ρ )( η†η ) + a7 ( χ†η )( η†χ ) + a8 ( χ†ρ )( ρ†χ ) + a9 ( ρ†η )( η†ρ ) + a10[(χ †η)(ρ†η) + (η†χ)(η†ρ)] (4.8) where we assume all coupling constants to be real2. The scalar potential with a10 = 0 has been considered for instance in Ref. [41, 34]. If this term is not zero, comparing with the aforementioned references, the only sector which is modified is that of the singly charged scalars. This model is taken to the SM through the spontaneous symmetry breaking from the Higgs mechanism, in the following sequence: SU(3)L ⊗ U(1)X → SU(2)L ⊗ U(1)Y → U(1)EM . The first break will make 4 out of the 9 generators of the group SU(3)L⊗U(1)X preserve the symmetry of the first vacuum, taking us to the SU(2)L⊗U(1)Y group from the SM. In the second break there will be only one generator left unbroken, the U(1)EM from electromagnetism. The passage from 331HL to SM is thanks to the symmetry break caused by the χ triplet, the second one, which takes us to the electromagnetism group, can happen either through the η or the ρ triplet. From the potential above we calculate the mass eigenstates for the scalars, which will be important in finding all the necessary vertices for our calculations. The mass matrices and eigenstates are presented in the appendix B. 2In the most general case the coupling constants would be complex. 25 4. The 3-3-1 model 4.4. Yukawa interactions The Yukawa Lagrangian in the leptonic sector is given by: −LlY = GνabΨ̄aLν ′ bRη +GlabΨ̄aLl ′− bRρ+GEabΨ̄aLE ′+ bRχ+H.C. (4.9) where Gν , Gl and GE are arbitrary 3 × 3 matrices. From (4.9), we obtain the Yukawa interactions given in Appendix C and the mass matrices for the leptons, which are given by M l = Gl|vρ|/ √ 2 for the l′-type leptons and ME = GE |vχ|eiθχ/ √ 2 for the E′-type leptons. For simplicity we are assuming the GE matrix to be diagonal. In this manner, for our masses to be real, we need the elements of the GE matrix to have the form (GE)ii = |GE |iie−iθχ , which implies mEi = GEii |vχ|/ √ 2 (where E1 = Ee, E2 = Eµ, E3 = Eτ ). Notice from Eq. (4.9) that in the 331HL model there are no flavor changing neutral currents mediated by scalar fields. Moreover, we note that the interaction Ē′aLν ′ bRη + 2 does exist and, as we will show later, it is important to make the extra leptons E′ unstable. The mass eigenstates for the non-exotic leptons are obtained as l′L,R = (V l L,R)†lL,R, where l′ = (l1, l2, l3), and l = (e, µ, τ) (the neutrinos symmetry eigenstates corresponds to the mass eigenstates, since we are assuming them to be massless). These V l L,R matrices diagonalize the mass matrix in the following manner: V l LM lV l† R = diag(me,mµ,mτ ). Possible solutions for the V l L,R matrices and the Gl matrix can be found in [34] and [37]. To find these solutions it was considered |vρ| = 54 GeV and |vη| = 240 GeV, as in Ref. [35], these will be used in our analysis of the muon and electron AMDMs. Since the masses of the exotic leptons are unknown, we cannot find such solution for them, and for this reason, we considered their mass matrix diagonal to simplfy our calculations. Although neutrinos get mass, say by the type-I seesaw mechanism [38], because of the small neutrino masses, the effect of unitary matrices in the vertices involving singly charged scalars, for all practical processes, is negligible in its non-diagonal elements: they are suppressed by the small neutrino masses. The neutrino masses are not of direct interest for the calculation of the AMDM, which we will assume massless here. We only note that if V ν L , is the matrix that diagonalizes the active neutrino masses, we can define the PMNS matrix as VPMNS = V l† L V ν L . Then, it is possible to accommodate both PMNS and the active neutrino masses as it was done in Refs. [38, 37]. 26 4. The 3-3-1 model 4.5. Gauge fields Using the conventions presented in [7], the covariant derivatives involving the gauge fields are Dµφ = ( ∂µ − igW a µTa − igxXBµ ) φ, DµΨR = (∂µ − igxXBµ) ΨR, DµΨL = ( ∂µ − igW a µTa − igxXBµ ) ΨL, (4.10) where φ denotes a scalar, ΨL a left handed spinor, ΨR a right handed spinor, T a = λa/2 - where λa are the Gell-Mann matrices - and X is the charge operator associated to the U(1)X group. The mass eigenstates for the gauge bosons are obtained by acting the covariant deriva- tives on the scalar triplets. This derivative becomes clearer if we write Mµ =  W 3 µ + 1√ 3 W 8 µ + 2tXBµ √ 2W+ µ √ 2V −µ√ 2W−µ −W 3 µ + 1√ 3 W 8 µ + 2tXBµ √ 2U−−µ√ 2V + µ √ 2U++ µ − 2√ 3 W 8 µ + 2tXBµ  , (4.11) where Mµ = W a µλa + 2tXBµ and t = gx/g. With these identities the covariant deriva- tives can be written as Dµφ = ∂µφ− ig2Mµφ, DµΨR = (∂µ − igxXBµ) ΨR, DµΨL = ( ∂µ − ig2Mµ ) ΨL. (4.12) The non-Hermitian gauge bosons are defined as W±µ = ( W 1 µ ∓ iW 2 µ ) / √ 2, V ±µ = ( W 4 µ ± iW 5 µ ) / √ 2, U±±µ = ( W 6 µ ± iW 7 µ ) / √ 2, (4.13) with masses m2 W = 1 4g 2v2W , m2 V = 1 4g 2 ( |vη|2 + |vχ|2 ) , m2 U = 1 4g 2 ( |vρ|2 + |vχ|2 ) , (4.14) where v2W = |vη|2 + |vρ|2. 27 4. The 3-3-1 model Meanwhile, the neutral mass eigensatates can be written as a combination of the symmetry eigenstates Aµ = sWW µ 3 − √ 3sWW µ 8 + √ 1− 4s2WB µ, (4.15) Zµ = −cWWµ 3 − √ 3sW tWW µ 8 + tW √ 1− 4s2WB µ, (4.16) Z ′µ = √ 1− 4s2W cW Wµ 8 + √ 3tWB µ, (4.17) where cW , sW and tW are the cosine, sine and tangent of the Weinberg angle, respectively. They are related to the gauge constants as t = gx g = sW√ 1− 4s2W . (4.18) The neutral gauge bosons have the following masses m2 A = 0 (4.19) m2 Z = = g2 4c2W ( v2η + v2ρ ) (4.20) M2 Z′ = g2v2χ s 4 W ( 4− (v2η+v2ρ) 2 v4χ ) + ( 1− 2s2W ) (v2η+v2ρ v2χ + 4 ) 12c2W ( 1− 4s2W )  (4.21) 28 5. MDM in the 331HL model In the 331HL model, the main extra contributions to MDMs arise from the heavy lep- tons, several scalars and vector bileptons (see Fig. 5.1). Here we present the one-loop contributions due to these particles in the model. We consider the scalar-lepton ver- tices to have the form i(Sξ + Pξγ5) where ξ = hi, A 0, Y + 1 , Y ++, with the factors Sξ, Pξ given in Eqs. (C.2)-(C.4). The vector-lepton vertices are considered to have the form iγµ(VU − AUγ5) for the vector U−− and iγµ(fV − fAγ5) for the vector Z ′. All these couplings are given in Eqs. (C.10)-(C.14). We present the general result below, valid either for the electron or the muon (the diagrams were calculated in the unitary gauge). The full result comes from considering all the possible one-loop diagrams involving any exotic lepton, scalar or vector particles. We will verify if the extra contributions in the 331HL model are enough to satisfy the constraints in Eqs. (1.3) and (1.4). In this case, all considered diagrams are those shown in Fig. 5.1. The muon MDM has been considered in the context of other 3-3-1 models in [6, 8, 9], however, there the authors did not consider the lepton mixing and also the constraints coming from the electron (g − 2)e. As can be seen in the matrices shown in Secs. 6.1.2-6.1.5, it is possible that there are important non-diagonal entries in the matrix V l L. Moreover, solving the muon ∆aµ discrepancy and at the same time giving contributions compatible with ∆ae is not a trivial issue, at least in 1-loop order. As said before, we consider only the extra diagrams present in the model as being responsible for the new contributions to the ∆aµ, such that a331i = ∆ai = aexpi − aSMi , i = e, µ, (5.1) where a331i includes diagrams with at least one of the extra particles in the model, i.e., only contributions coming from beyond SM physics. These are shown in Fig. 5.1. 29 5. MDM in the 331HL model Figure 5.1.: One loop diagrams for the MDM. The fermion l indicates either an electron or a muon. The 3-3-1 model contribution to the MDM comes from all these diagrams, considering two cases, when the photon is connected to the fermion and when the photon is connected to the boson (when applicable). 5.1. Scalar contributions Unlike the minimal 331 model, in the present one there are no flavor changing neutral currents (FCNC) through the Higgs exchange. In this situation a neutral scalar S has only scalar interactions f̄fS, and a pseudo-scalar A has only pseudo-scalar interactions f̄γ5fA. We will denote the respective factors Slξ and P lξ where l denotes the external lepton and ξ denotes the scalar in the loop. First, we consider the case in which there is a neutral or charged scalar, denoted by ξ, in the loop. When the photon is connected to the fermion line, corresponding to diagram a in Fig. 5.2, we have: ∆alξ(f) = − QI 96π2 m2 l 1 M2 ξ ∫ 1 0 dx |P lξ |2F ξ P (x, εξl ) + |Slξ|2F ξ S(x, εξl ) F ξ(x, εξl , λ ξ l ) , (5.2) with εξl = mI/ml, λ ξ l = ml/Mξ, and ξ = h1, A, Y − 1 , Y −−, ml is the mass of the electron or muon, mI is the mass of the fermion in the loop, and Mξ is the scalar mass in the loop; Slξ and P lξ are the matrices given in Eq. (C.2) and depend on the type of the scalar, QI is the electric charge of the internal lepton. We have defined F ξS(x, εξl ) = −g(x) + 12εξl (x+ 1), F ξP (x, εξl ) = g(x) + 12εξl (x+ 1), F ξ(x, εξl , λ ξ l ) = x[(λξl ) 2(x− 1) + 1]− (εξl λ ξ l ) 2(x− 1), (5.3) 30 5. MDM in the 331HL model Figure 5.2.: Types of one loop diagrams for the MDM. The fermion l indicates either an electron or a muon, I is a fermion internal to the loop, X denotes a gauge boson, ξ denotes a scalar and γ denotes a photon. In diagram a) we have the case where there is a scalar in the loop and the photon is connected to the internal fermion (Eq. 5.2), while in b) the photon is connected to the scalar (Eq. 5.4). In diagram c) we have a vector gauge boson in the loop with the photon connected to the fermion (Eq. 5.9) and in d) the photon is connected to the gauge boson (Eq. 5.7). and εh,Al = 1 (again, this is because there are no flavor changing neutral currents via neutral scalar or pseudo-scalars) and εY −− l = mI/ml, with MI = Ee, Eµ, Eτ , and ε Y −1 l = mν/ml; we have also defined g(x) = 12x2 + 9x− 1. Moreover, εAl λ A l = ml/MA (similarly for h), and εY −− l λY −− l = mEl/MY −− . In the cases with a singly or doubly charged scalar, there are also diagrams in which the photon is connected to the scalar line (see diagram b in Fig. 5.2). In the case of Y −− we obtain ∆alζ(ζ) = Qζ 8π2 m2 l 1 M2 ζ ∫ 1 0 dx |P lζ |2R ζ 1(x, εζl ) + |Slζ |2R ζ 2(x, εζl ) Rζ(x, εζl , λ ζ l ) , (5.4) for the scalar couplings. When ζ = Y −− we define RY −− 1 (x, εY −− l ) = x[−(1− x) + εY −− l )], RY −− 2 (x, εY −− l ) = x[1− x+ εY −− l ] RY −− (x, εY −− l , λY −− l ) = x[λ2l (x− 1) + ( εY −− l λY −− l )2 ]− (x− 1). (5.5) 31 5. MDM in the 331HL model and when ζ = Y −1,2 we have R Y −1,2 1 (x, ε−l ) = −x(1− x), R Y −1,2 2 (x, ε Y −1,2 l ) = x(1− x) RY − 1,2(x, ε Y −1,2 l , λ Y −1,2 l ) = (λ Y −1,2 l )2(x− 1)2, (5.6) where we have neglected ε Y −1 ν = mν/ml for both electron and muon. In fact, we are neglecting the PMNS matrix and assuming for practical purposes that the ν-Y −1 -l vertex is diagonal. This means that we are over-estimating the contributions of the Y −1,2 scalar but even in this case we will see that their contributions are negligible. 5.2. Vector contributions The model has neutral and doubly charge vector bosons that contribute to the AMDM in one loop, Z ′µ and U−−µ , respectively. In the latter case, when the photon is connected to the charged vector line (diagram d in Fig. 5.2), we have: ∆alU (U) = −GUUA 64π2 m2 l M2 U ∫ 1 0 dx [ T1(x, ε U l ) + T2(x, ε U l ) + T3(x, ε U l )T (x, εUl , λ U l ) T (x, εUl , λ U l ) ] , (5.7) with λUl = ml/MU , where MU is the mass of the U−−µ ; εUl = mI/ml, EI = Ee, Eµ, Eτ . We have both vector and axial-vector couplings: T1(x, ε u l ) = |AlU |2[h(x) + 3εUl (x+ 1)] + |V l U |2[h(x) + 3εUl (x+ 1)] T2(x, ε U l ) = m2 l M2 U x3[|AlU |2(x+ 2εUl ) + |V l U |2(2εUl − x)], T3(x, ε U l ) = 2|AlU |2[h(x) + 4 + εUl (2x+ 1)] + |V l U |2[−(2x+ 1)2 + εUl (6x+ 1)] T (x, εUl , λl) = x− 1− x[(λUl )2(x− 1) + (εUl λ U l )2], (5.8) where we have defined h(x) = 2x2 + x− 3. The factor GUUA = −2e, see Eq. (C.9), and AlU and V l U are given in Sec. C.4. There is only one diagram of this type, that with U−− in Fig. 5.1. For the case where the photon is connected to the fermion line (diagram c in Fig. 5.2), there are two diagrams in Fig. 5.1 that contribute to the MDM, one with Z ′ and the other with U−−: ∆alX(f) = − QI 8π2 m2 l ∑ X 1 M2 X ∫ 1 0 dx [ R̃X1 (x, εXl ) + R̃X2 (x, εXl ) + R̃X3 (x, εXl )R̃X(x, εXl , λ X l ) R̃X(x, εXl , λ X l ) ] , (5.9) 32 5. MDM in the 331HL model where X = Z ′µ, U −− µ , andM is the mass of the vector boson, QI is the electric charge of the fermion internal in the loops, λXl = ml/MX , with R̃X1 (x, εXl ) = 2|AlX |2(1 + x+ 2εXl ) + 2|V l X |2(1 + x− 2εXl ), R̃X2 (x, εXl ) = m2 l M2 X (x− 1)2 [ |AlX |2(1 + εXl )(x− εXl ) + |V l X |2(1− εXl )(x+ εXl ) ] , R̃X3 (x, εXl ) = (3x− 1)[|AlX |2(1 + εXl ) + |V l X |2(1− εXl )], R̃X(x, εXl , λ X l ) = 1 + (x− 1)[(λXl )2 + (εXl λ X l )2]. (5.10) In Eqs. (5.8) and (5.10), when X = U−−µ the matrices AlU and V l U are those in Sec. C.4 and, when X = Z ′µ, AlZ′ and V l Z′ are given by the factors f lA and f lV in Eq. (C.14). Notice that when the vector is the Z ′, εZ ′ l = 1. But with U−−µ defined as above: εUl = mE/ml. For all the vertices the reader is referred to Appendices C.1 - C.5. 33 6. Results for the electron and the muon AMDM Considering the results in the previous section, we are able to find sets of values for the parameters of the model taking into account the contributions of the scalars A0, Y −1 , Y ++ and the vector bosons V −µ , U −− µ for the electron and the muon AMDM. Let us define the contributions of the extra particles in the 331HL as in Eq. (5.1), where ∆a331e,µ was calculated using the results from the previous section and considering all diagrams in Fig. 5.1, in such a way to obtain µ331 + µSM = µexp within 1 to 3 standard deviations. 6.1. Numerical results for the AMDMs Here we will present the numerical results of the AMDMs for both, muon and electron, showing which regions of the parameter space satisfy the experimental results. We will consider five different scenarios, each with a different set of diagonalization matrices for the leptonic sector (see Appendix C.1 for more details). The unknown variables we have to explore are: • The diagonalization matrices V l L,R for the lepton mass matrices; • The masses of the scalars A0, Y + 1,2 and Y ++; • The projections of the Y + 1,2 scalars over its mass eigenstates; • The masses of the exotic leptons Ee, Eµ and Eτ ; • The vacuum expectation value vχ. In the paragraph below we explain how each of these variables have been explored. When the masses of the exotic particles are not explicitly mentioned in each plot, it means that they were fixed as: mEe = mEµ = mEτ = 500 GeV, mY −2 = 1200 GeV and 34 6. Results for the electron and the muon AMDM mA0 = mY −− = mY −1 = 1000 GeV. Also, the projections of the singly charged scalars over its mass eigenstates were assumed to be both 0.5, see Eq. (C.3). We have tried several values for the scalar couplings, but they have given no noticeable change in our plots, that is because the U++ contribution dominates (see Fig. 6.1). The masses of the gauge bosons (U±± and Z ′) have their values defined by the value of |vχ|, since its other parameters are already fixed (see [42] for details). According to [1], the lowest lower limit on the mass of the Z ′ boson is 2.59 TeV, assuming it has the same couplings as the Z boson, which implies |vχ| > 665.13 GeV [35]. Here we will show that in the present model a stronger lower limit for vχ is obtained and that, at least with the values of the matrices V l L used, it is not possible to fit both (1.3) and (1.4) at the same time within 1σ. For our fits, we considered only the uncertainties coming from the SM predictions and experimental results, for we have not calculated the 331HL contributions uncertainties. 6.1.1. Diagonal V l L,R matrices The simplest solution possible is to assume that the leptonic interactions are diagonal in flavor, i.e., the symmetry and mass eigenstates are the same, the numerical results for these matrices can be seen in Figs. 6.2-6.4. In Fig. 6.2 we vary the mass of the exotic lepton Ee, fixing the masses of the other two. The blue, green and cyan regions show values for the parameters where the 331HL-only contributions for the muon AMDM agrees with the difference between the experimental results and the SM prediction within 1σ - 3σ, respectively. In a similar manner, the red, orange and yellow regions show values where the 331HL-only contribution for the electron MDM agrees within 1σ, 2σ, and 3σ, respectively. It can be seen in the figure that there are solutions for the muon up to 1σ and for the electron up to 2σ. However the 1σ and 2σ regions only overlap for low values of mEe , less than 40 GeV, and vχ around 80 TeV. These are not unrealistic solutions because an extra charged lepton lighter than 40 GeV may still exist depending of the respective couplings with the known particles, and vχ may have a value of 60-80 TeV such that Z ′ and bileptons V,U will be only of a few tens of TeVs (See Eq. 6.9). It is expected that high values for the exotic leptons and bosons masses would lead to results similar to the ones found in the SM, where the electron AMDM deviates from the experimental results by 1.1σ. It can be seen in Fig. 6.2 that, for values of vχ over 60 TeV and mEe . 1 GeV, we have a 2σ region for the electron AMDM. Such high values for vχ implies masses of tens of TeV’s for the exotic gauge bosons, suppressing 35 6. Results for the electron and the muon AMDM Figure 6.1.: U−− contribution to the muon AMDM. ∆ae (lower graph) and ∆aµ (upper graph) taking into account only the contribution of the doubly charged vec- tor boson U−− as function of vχ, with 0 < vχ ≤ 50 TeV. We also assume that both VL,R are the unit matrix and mEe = mEµ = mEτ = 20 GeV. It can be seen that the muon MDM is solved for vχ around 8 TeV, a value that does not solve the electron MDM. 36 6. Results for the electron and the muon AMDM Figure 6.2.: vχ and mEe values satisfying Eq. (5.1) within 1σ, 2σ and 3σ. Here we have used diagonal mass matrices for the charged leptons (see Sec. 6.1.1) and considered: mEµ = mEτ =500, mY + 2 =1200, and mA0 = mY ++ = mY + 1 =1000 (all in GeV). The masses of the exotic gauge bosons (U±± and Z ′) have their values defined by the value of vχ, since its other parameters are already fixed (see [42] for details). several of the diagrams shown in Fig. 5.1, leaving us diagrams with exotic leptons and scalars. However, for the exotic lepton masses, we find a different situation. By analyzing Eqs. (5.2), (5.4), (5.7) and (5.9), we can see that if we take the limit mI →∞ the expressions being integrated go to one. Therefore, the AMDM contributions coming from the diagrams with an exotic lepton inside their loops do not go to zero when these leptons are very heavy, leaving us with values that do not correspond to the ones expected by the SM, which would give us solutions for the electron AMDM within 2σ. Once we perform the AMDM calculations from the amplitudes involved, the integrations create this situation where the powers of mI are the same in the denominator and numerator. The addition of these exotic leptons invaribly lead to contributions to the AMDM, regardless of its masses. We have just considered diagonal interactions for the leptons so far, but we can use re- alistic unitary matrices which satisfy the condition V l LM lV l† R = diag(memµmτ ) and the PMNS matrix. In the following sections we will consider four different parametrizations for these unitary matrices. 37 6. Results for the electron and the muon AMDM Figure 6.3.: Same as in Fig. 6.2 but now with mEe = mEτ =500 GeV. Figure 6.4.: Same as in Fig. 6.2 but now with mEe = mEµ =500 GeV. 38 6. Results for the electron and the muon AMDM Figure 6.5.: vχ and mEe values satisfying Eq. (5.1) within 1σ, 2σ and 3σ. Here we used the first set of diagonalization matrices (see Sec. 6.1.2) and considered: mEµ = mEτ =500, mY + 2 =1200, and mA0 = mY ++ = mY + 1 =1000 (all in GeV). The masses of the exotic gauge bosons (U±± and Z ′) have their values defined by the value of vχ, since its other parameters are already fixed (see [42] for details). 6.1.2. 1st set of V l L,R matrices The diagonalization matrices used in this set are: V l L =  0.009854320681804862 0.31848228260886335 −0.9478775912680647 0.014570561834801654 −0.947868712966038 −0.3183278211340082 −0.9998452835772734 −0.010674204623999706 −0.013981068053858256  (6.1) V l R =  0.005014143494893113 0.0026147097108665555 0.9999840107012414 0.0071578125624917055 0.9999708696847197 −0.0026505662235414198 0.9999618113129783 −0.0071709884334755225 −0.004995281829296536  (6.2) Figs. 6.5 and 6.6 shows us only solutions for the muon. Meanwhile, in Fig. 6.7 we have solutions for the electron up to 2σ. These solutions intersect with the 3σ solutions for the muon for values of mEτ smaller than 20 GeV and vχ greater than 140 TeV. As for the 3σ solutions for the electron, they intersect the muon 2σ region for mEτ < 50 GeV and vχ ∼ 90 TeV. 39 6. Results for the electron and the muon AMDM Figure 6.6.: Same as in Fig. 6.5 but now with mEe = mEτ =500 GeV. Figure 6.7.: Same as in Fig. 6.5 but now with mEe = mEµ =500 GeV. 40 6. Results for the electron and the muon AMDM Figure 6.8.: vχ and mEe values satisfying Eq. (5.1) within 1σ, 2σ and 3σ. Here we used the second set of diagonalization matrices (see Sec. 6.1.3) and considered: mEµ = mEτ =500, mY + 2 =1200, and mA0 = mY ++ = mY + 1 =1000 (all in GeV). The masses of the exotic gauge bosons (U±± and Z ′) have their values defined by the value of vχ, since its other parameters are already fixed (see [42] for details). 6.1.3. 2nd set of V l L,R matrices The diagonalization matrices used in this set are: V l L =  −0.009 0.0146 −0.9998 −0.3185 −0.9479 −0.0107 0.9479 −0.3183 −0.0140  (6.3) V l R =  0.005 0.0072 0.9999 0.0026 0.9910 −0.0072 0.9999 −0.0027 −0.0050  (6.4) Similar to the first set of matrices, the plots where we vary mEe and mEµ (Figs. 6.8 and 6.9) show only muon solutions, while in Fig. 6.10 have solutions for both electron and muon that overlap. The situation is similar to the 1st set of matrices, now with a broader range of values for vχ. Better than before, now the 2σ electron region and the 1σ muon region are overlapping, for values of mEτ . 15 GeV and vχ ∼80 TeV. 41 6. Results for the electron and the muon AMDM Figure 6.9.: Same as in Fig. 6.8 but now with mEe = mEτ =500 GeV. Figure 6.10.: Same as in Fig. 6.8 but now with mEe = mEµ =500 GeV. 42 6. Results for the electron and the muon AMDM Figure 6.11.: vχ and mEe values satisfying Eq. (5.1) within 1σ, 2σ and 3σ. Here we used the third set of diagonalization matrices (see Sec. 6.1.4) and considered: mEµ = mEτ =500, mY + 2 =1200, and mA0 = mY ++ = mY + 1 =1000 (all in GeV). The masses of the exotic gauge bosons (U±± and Z ′) have their values defined by the value of vχ, since its other parameters are already fixed (see [42] for details). 6.1.4. 3rd set of V l L,R matrices The diagonalization matrices used in this set are: V l L =  0.983908 0.156151 0.086891 0.0777852 0.061974 −0.994965 −0.160853 0.985709 0.0500342  (6.5) V l R =  0.978756 0.186555 0.0850542 0.0744144 0.0633254 −0.99215 −0.191048 0.980401 0.0480978  (6.6) In this set only Fig. 6.11 has solutions for the muon and the electron. The 2σ regions for the electron and muon overlap for mEe . 30 GeV and vχ ∼ 110 TeV. Also, there is an overlap of the electron 3σ and the muon 1σ regions, for mEe . 50 GeV and vχ ∼ 80 TeV. 6.1.5. 4th set of V l L,R matrices The diagonalization matrices used in this set are: 43 6. Results for the electron and the muon AMDM Figure 6.12.: Same as in Fig. 6.11 but now with mEe = mEτ =500 GeV. Figure 6.13.: Same as in Fig. 6.11 but now with mEe = mEµ =500 GeV. 44 6. Results for the electron and the muon AMDM Figure 6.14.: vχ and mEe values satisfying Eq. (5.1) within 1σ, 2σ and 3σ. Here we used the fourth set of diagonalization matrices (see Sec. 6.1.5) and considered: mEµ = mEτ =500, mY + 2 =1200, and mA0 = mY ++ = mY + 1 =1000 (all in GeV). The masses of the exotic gauge bosons (U±± and Z ′) have their values defined by the value of vχ, since its other parameters are already fixed (see [42] for details). V l L =  −0.99614 −0.08739 −0.00826 0.01357 0.24625 −0.96691 0.08672 0.96526 0.24649  (6.7) V l R =  0.99624 −0.08629 −0.00801 0.01179 0.226594 −0.97392 0.08586 0.97016 0.22676  (6.8) In Fig. 6.14 we see solutions for the muon and electron that overlap for low values of mEe , while in Figs. 6.15 and 6.16 we see only solutions for the muon. The situation in Fig. 6.14 is similar to that in Fig. 6.10, where the 2σ electron region and the 1σ muon region are overlapping, for values of mEτ . 15 GeV and vχ ∼80 TeV. 6.2. Numerical analysis conclusions Although we considered all the contributions to the ∆ae,µ in 1-loop in all these scenarios, assuming different sets of diagonalization matrices for the leptonic sector, it is easy to convince ourselves that the larger ones come from the doubly charged scalar Y −− and 45 6. Results for the electron and the muon AMDM Figure 6.15.: Same as in Fig. 6.14 but now with mEe = mEτ =500 GeV. Figure 6.16.: Same as in Fig. 6.14 but now with mEe = mEµ =500 GeV. 46 6. Results for the electron and the muon AMDM Figure 6.17.: Y −− contribution to the muon MDM. ∆ae (lower graph) and ∆aµ (upper graph) taking into account only the contribution of the doubly charged scalar Y −− as function of vχ, with 0 < vχ ≤ 2 TeV, mY −− = 500 GeV. We also assume that both VL,R are the unit matrix and mEe = mEµ = mEτ = 20 GeV. We see that the muon MDM only has solutions for vχ ≈ 100 GeV, while the electron MDM has no solutions for this value. vector U−− (see Figs. 6.17 and 6.1). In fact, from the vertices in Eq. (C.2), we can see that the contributions of the neutral scalar are suppressed since they are proportional to the known charged lepton masses (M̂ l/vρ)Oρ1. The pseudo-scalar vertex is proportional to (M̂ l/vρ)Uρ3, thus may be larger than the scalar one but still very suppressed. The vertex of the singly charged scalar Y −1 is also proportional to (M̂ l/vρ), and in this case there are additional suppression factors cosβV l L, see Eq. (C.3). Given that the singly charged scalar contribution is negligible, the introduction of the a10 term in the scalar potential bring no significant change to our results, but it is necessary for the width decay of the leptons E+. In the case of the doubly charged scalar Y −−, it has interactions proportional to M̂E/vρ > 1, see Eq. (C.4). Finally, we note that the doubly charged vector bilepton, 47 6. Results for the electron and the muon AMDM U−− has vertices that are proportional to gV l† L = (4GFM 2 W / √ 2)1/2V l† L , see Eqs. (C.12) and (C.13). Hence, our results also depend on the values of the matrices V l L, which we considered. Most studies of the possibilities of the 3-3-1 models for solving the muon anomaly have considered that the interactions are diagonal in flavor, the scenario we addressed in Sec. 6.1.1. These authors obtain solutions with low masses for the particles in their models. These interactions are characterized by a coupling strength denoted by f in Ref. [3] from where some authors take off the results for the Feynman diagrams. However, in a particular model, the mass eigenstates appear only after diagonalizing the mass matrices, in doing so the coupling strength f is related to masses of the internal particles in the diagrams and to the unitary matrices that diagonalize the mass matrices. Thus, they are not anymore arbitrary. Moreover, in most works studying ∆aµ, usually only one type of exotic particle is considered to address the problem. To make a comparison, we have calculated the contributions given only by U−− and Y −− (Figs. 6.17 and 6.1) considering diagonal interactions (i.e. V l L,R = 1). When considering only the vector U−−, although it solves the muon anomaly for vχ around 8 TeV, this does not happen for the electron AMDM around the same value. From the bottom plot in the same figure, it seems that the electron AMDM can be solved for higher values of vχ, but these values will not satisfy the muon AMDM according to the upper plot. For the case where only Y ++ contributes, the muon AMDM is solved for vχ ≈ 100 GeV while the electron is not solved for this value. So it seems that even with diagonal interaction the AMDMs cannot be solved simultaneously. Of course, our results are in the 331HL model and are not necessarily valid in other 331 models, and also for other values of the matrix V l L. However they indicate that the analysis in those models should be revisited. Not just because diagonal flavor interactions were the only ones considered, but also because of the effects that these exotic particles may have on the electron AMDM. The effects on both AMDMs can be contradictory, as we have shown in this work. Once we have determined the value of the VEV vχ we can calculate the vector boson 48 6. Results for the electron and the muon AMDM masses [42] and mE using: M2 U ≈ α 4s2W (v2ρ + v2χ), M2 V ≈ α 4s2W (v2η + v2χ) M2 Z′ = α2 2s2W (1− 2s2W )(4 + v̄2W ) + s4W (4− v̄2W ) 6c2W (1− 4s2W ) mEl = gEl vχ√ 2 , (6.9) where v̄W = vSM/vχ, and we see that with vχ = 70 TeV we have MU ≈ 22 TeV, MV ≈ 22 TeV, MZ′ ≈ 81 TeV. On the other hand, the masses of the scalars Y −1,2, Y −− depend also on the dimen- sionless couplings appearing in the scalar potential, and also on the trilinear term in the scalar potential, F ηρχ denoted by α in [34]. The constant F , with dimension of mass, may be small on naturality grounds, or large if it arises from the VEV of a heavy neutral scalar that is singlet under the 3-3-1 symmetry. For this reason the masses of the scalars A, Y −1,2, Y −− and mEl are used as inputs in our calculations. On the other hand, the masses of the vectors Z ′, V −, U−− depend mainly on vχ, since the other VEVs are already fixed. 49 7. Other constraints In the previous chapter we were able to find some solutions for the AMDMs, all of them requiring low values for the exotic lepton mass, below 50 GeV. The current ex- perimental lower limits for exotic leptons are around 100 GeV, excluding our solutions. However, such limits were obtained considering models different from the 3-3-1 model with heavy leptons, which, in principle, cannot be directly applied to our case. Also, we explore the µ→ eγ decay, which excluded one of our numerical solutions for the lepton diagonalization matrices. In the following sections we further discuss these topics. 7.1. The µ→ eγ decay One possible constraint on the mE and vχ values can come from the µ → eγ decay. The diagrams contributing to this process are those in Fig. 5.1 when the intermediate fermions are E’s ou neutrinos and we have a muon and an electron at the external lines. These processes have been considered recently in the context of the minimal 3-3-1 model [37] and an early reference is Ref. [44]. Considering only the largest contribution for the branching ratio to this process, we have BR(µ→ eγ) = 54α π ( mU mY1 )4(mE mµ )2 ( |(VY )13|2|(VY )32|2 + |(VY )31|2|(VY )23|2 ) (7.1) where VY = (V l R)TV l L and α is the fine-structure constant. The actual experimental limit for this is BR(µ→ eγ) < 0.057× 10−11 [1]. Imposing this limit on the above equation, while varying mE from 0 to 1000 GeV and vχ from 10 to 150 TeV, considering all the sets of diagonalization matrices from the sections above, we saw that the whole range of values explored is allowed for the diagonal matrices, the first, second and fourth sets of matrices. As for the third set of matrices, none of the values explored for mE and vχ respects the experimental limit for the µ → eγ decay branching ratio. Given the simplicity of these results, we decided not to show the respective plots here. 50 7. Other constraints 7.2. Lifetime of the exotic charged leptons The experimental searches of new charged leptons have considered three scenarios in order to put constraints on the masses of this sort of particles [1]: i) sequential heavy leptons, in which these particles are assumed to belong to a fourth generation where either the neutrino partner is considered stable or the heavy leptons decay into active neutrinos via mixing; ii) stable heavy charged leptons, which considers the decay L± → W±ν (where L is the exotic lepton); and iii) long lived heavy charged leptons, which considers an exotic lepton coming from a singlet added to the SM. None of these scenarios correspond to the 331HL case. In situation i) the lower limit on the mass of such particle is 100.8 GeV; in ii) it is 102.6 GeV, and finally in case iii) it is 574 GeV. In the 331HL the heavy leptons E± are not stable but may be long-lived depending on the their masses and on the masses of the scalar bosons that mediate their decays. In particular, we note that the lower limit on the mass of a sequential heavy charged lepton, 100.8 GeV at 95%, was obtained using the decay L± → W±ν [1]. However, in the present model, the decays are mediated mainly by extra charged vector and scalar bosons and, for this reason, it is not straightforward to apply this limit on the masses of E± in the present model. The lepton triplets in terms of the symmetry eigenstates (primed fields) are ΨaL = (ν ′a l ′ aE ′ a) T ∼ (1, 3, 0), a = e, µ, τ [31]. Usually it is assumed that the leptons E+ a are antiparticles and thus L(E+ a ) = −1. In this case the L assignment is L(J, j, η−2 , χ −, χ−−, ρ−−, V −µ , U −− µ ) = +2. (7.2) and the other particles having L = 0, or +1. In this case the model has a global custodial symmetry U(1)L under which some particles (including the usual ones) are L = 1 (antiparticles L = −1) and the other ones as in Eq. (7.2). We can also use the global symmetry U(1)F , where F = B + L [39]. However, we can assume that the leptons E+ a are particles, as in Konopinski-Mahmoud [40], and assign it L(E+ a ) = +1. In this case the lepton number is the same for all mem- bers in lepton triplets, and L(J, j, η−2 , χ −, χ−−, ρ−−, V −µ , U −− µ ) = 0. Notwithstanding, a custodial discrete Z2 symmetry still exists, under which E, J, j, η−2 , χ −, χ−−, ρ−−, V −µ , U −− µ are odd, and the rest of the particles are even. The electroweak symmetry of the model is GW = Z2×SU(3)L⊗U(1)X . In this section we call ”exotic” the particles that are odd under Z2, otherwise they are ”normal” particles. Now, Z2 is the custodial symmetry. 51 7. Other constraints At first sight, the custodial discrete symmetry implies that the lightest exotic charged lepton should be stable. There are interactions that produce E+ = V +νL and E+ = U++l− but both vectors, V + and U++, cannot decay only into known quarks or leptons: U++ = E+e+ and V + = e+Lν c R. Moreover, these vector bosons can also decay into one exotic quark and one known quark, V + = ūJ and U++ = uj̄. We can see from Eq. (4.9), that the interactions with charged scalars are similar, they involve also one normal and one exotic particle: E+ L = η+2 νR and E+ R = νLχ +, which are allowed since η+2 and χ+ are odd under Z2. However, η+2 and χ+ decays also into one normal particle and one exotic one. Hence, the lightest E+ cannot, at first sight, decay at all. However, we note that the quartic term a10(χ †η)(ρ†η) is allowed in the scalar potential and it implies a mixing among all the singly charged scalars [41] (See appendix A): V (η, ρ, χ) ⊃ a10(χ+η0 + χ++η−1 + χ0∗η+2 )(ρ−η0 + ρ0∗η−1 + ρ−−η+2 ). (7.3) Notice that the term χ+η0ρ−η0 breaks the Z2 symmetry. If a10 6= 0, all the singly charged scalars mix in the mass matrix and since the interaction ν̄LERχ − does exist, the decay E+ lR = νlL + h+ = νlL + l+ + νcl is now allowed, where h+ is a charged scalar mass eigenstate that couples with the known leptons (we have omitted the matrix element that projects χ+ onto h+). The decay of any E through charged scalars is shown in Fig. 7.1. A rough calculation of the decay, based on the result for the muon decay µ→ 2ν + e, is given by τE = m4 Y (CmE)4 12(8π)3 mE ~, (7.4) where mY is the mass of the lightest singly charged scalar, and C denotes the couplings between the Y scalar and the fermions. In section 6.1 we assumed mY + 1 = mY + 2 = 1000 GeV, using this with Eq. 7.4, and assuming C = 0.1, we plot the graph in Fig. 7.2. We see that for values as low as 50 GeV for mE , the lifetime is lower than 10−11 seconds. If we take mE = 10 GeV, the lifetime is just 1.26 × 10−8 seconds, still very short and far from stable. The channel into one charged lepton and two active neutrinos is open even for the lightest of the leptons E+ a , and thus they are not long lived anymore. The issue of the stability of exotic fermions deserves a more detailed study that goes beyond the scope of this work. However, the above discussion tells us that the current experimental data is not applicable, at least in a straightforward way, to the 3-3-1 model with heavy leptons. 52 7. Other constraints Figure 7.1.: E lepton decay. This decay is allowed even for the lightest lepton E. Figure 7.2.: E lepton lifetime, as described by Eq. 7.4. In the vertical axis we have the lepton lifetime in seconds, and on the horizontal axis the lepton mass in GeV. 53 8. Conclusions In this work we have shown the analytical and numerical results for the electron and muon AMDM at 1-loop level in the 331HL model, comparing them with the experimental results. In the parameter space that has been explored here we have found regions where only the muon AMDM coincides with the experimental value whithin 1σ, while for the electron case only solutions within 2σ and 3σ were found. Moreover, even when we found solutions for both electron and muon, the region where they overlapped required very low mass values for the exotic leptons, and this may be a very unlikely scenario. We recall that the masses of vectors Z ′ and U−− are determined mainly by vχ once the other VEVs are already fixed, but for the heavy scalars their masses, as for the heavy leptons, also depend on unknown dimensionless parameters. In this vein, we have used the charged scalars and heavy leptons masses as free inputs in our calculations. The dominant contributions are due to Y −− and mainly to U−−. This can be appre- ciated in Figs. 6.17 and 6.1, respectively. We can see from these figures that, even in these two cases, it is impossible to fit in 1σ both ∆ae and ∆aµ. At 3σ for the electron and less than 2σ for the muon it is possible to have an agreement with the experimental measurements. However, it leaves the electron case in the same footing as the muon AMDM in the SM. For this reason we claim that it is not possible to fit both ∆e,µ at the same time. For most of the values explored for the parameters, the contributions of the a331e increase the value of ∆ae. Other recent works have addressed the muon AMDM in different but similar scenarios, in which smaller values for vχ and smaller masses for the extra particles were enough to explain the ∆aµ [6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. However, none of them have considered at the same time the contributions for the electron AMDM from the particles and parameters that solve the muon AMDM. One important difference with other works in 3-3-1 models is that usually the integrals from the Feynmann diagrams were taken from Ref. [3]. However, the results in the latter paper are quite general and do not take into account details that are model dependent. 54 8. Conclusions For instance, they used directly the symmetry eigenstates basis, and in this situation the unitary matrices needed to diagonalize the lepton mass matrices do not appear in the vertices (nor does the VEVs that come from the diagonalization of the scalar mass matrices). Each contribution to the ∆aµ coming from new particles were given just as follows: ∆aµ = ± f2 4π2 m2 µ M2 I, (8.1) where M is a mass of one of the particles in the loop and I an integral on the Feynman parameter x, also depending on parameters as ε and λ defined in Sec. 6. In some of the references above, this integral is simplified, turning into a numerical factor. Besides that, in these works all vertices are reduced to the exotic particle with mass M , coupling to muons with coupling strength f . In our case, the mass eigenstates particles in the diagrams are obtained from the symmetry eigenstates and the unitary matrices relating both basis appear in the vertices. Here these matrices are incorporated in the factors S, P when the fields are (pseudo)scalars and V and A when they are the vectors. For instance, in Ref. [6] the masses of all internal fermions in the loops are neglected in the reduced m331 model. This could be done because in this model the only internal fermions are the known leptons. In the m331 model (reduced or not) the interactions with the doubly charged scalar or vector boson have non-diagonal vector and axial-vector interactions that have not been considered in most works mentioned above. Moreover, in [6] the only mixing matrix in the lepton sector is the PMNS. The effect of this matrix is nullified by the small neutrino masses and the unitarity of the PMNS matrix. On the other hand, in the minimal 3-3-1 the charged current coupled to U++ is Lccll = −i g 2 √ 2 l̄cγµ[(VU − V T U )− γ5(VU + V T U )]lU++ µ +H.c. (8.2) where l = (e, µ, τ)T and VU = (V l R)TV l L. We see that there are non-diagonal vector currents and the unitary matrices are such that V l† L M lV l R = M̂ l = diag(me,mµ,mτ ), it is an extreme fine tuning to assume that VU is the unit matrix. Hence, in the minimal 331 model it is not advisable to neglect the lepton mixing. In the model explored in this work this type of current involves E and l, and V l L, see Eq. (C.12). For instance, in [46] it is shown that the doubly charged scalar may solve the muon anomaly, using scalars with masses around hundreds of GeV’s. The scheme in [46] could, at first sight, be realized in the m331. It is so because in that model, a singlet doubly charged scalar (under the SM symmetries) belonging to sextet couples only with 55 8. Conclusions right-handed charged leptons, i.e., the respective interactions are (lR)cV lT L GSV l RlR, where GS is the Yukawa matrix in the interactions of leptons with the scalar sextet. Moreover, GS is not proportional to the charged lepton masses because they have another source of mass: the interactions with the triplet η. Hence, the arbitrary matrix V lT L GSV l R has to be taken into account in the m331. Another example, in Ref. [8], were considered solutions to the muon anomaly in the context of the minimal 331, the 331 model with heavy neutral fermions, and the 331 model with heavy charged leptons but with five left-handed lepton triplets, thus the latter one is different from the 331HL model here considered. Again, the results are all approximated and since all flavor mixings are neglected their results are, as before, of the form ∆aµ ∝ m2 µ/M 2. The masses of the usual charged leptons can be neglected in the loops because they appear as εµ = mν/mµ or λl = mµ/M . The first is negligible by the neutrino masses and the second by the value of M . Another important point to be stressed is that, in [8, 9] not all the particles in each model are considered. Moreover, it happens that in the model they were considering, the importance of the scalar con- tributions cannot be neglected a priori because they depend on the Yukawa couplings through the unitary matrices that diagonalize quarks and lepton mass matrices, as dis- cussed in the previous paragraph. For instance, in the m331 there are FCNC via the scalar sector, and in this case the scalar contributions may be as important as in the quark sector [35]. We must put in context our results. On one hand, they were obtained in the 331HL where there are no flavor changing neutral interactions in the lepton sector. Moreover, unlike in the m331, the singly charged vector bilepton V + does not contribute to the AMDM of the known charged leptons. For example, this is not the case with the m331, where there are more scalar multiplets and FCNC in the lepton sector via the exchange of neutral scalars. On the other hand, they were obtained using the matrices V l L in Sec. 6.1. The matrices V l R which we presented are possible solutions, they may have other values and it is possible that these values might imply solutions for both AMDMs. The moral of the story is that besides the unitary matrices, the AMDM of the electron must be taken into account when solutions for the muon case are proposed, once both AMDMs may be incompatible. Otherwise, the results cannot be considered definitive because they depend also on the solutions for the matrices V l L,R and different solutions imply different lower bounds on the phenomenology of the model [37]. 56 A. Scalar Potential The most general scalar potential is given by V (χ, η, ρ) = µ21χ †χ+ µ22η †η + µ23ρ †ρ+ (α εijkχiρjηk +H.c.) + a1 ( χ†χ )2 + a2 ( η†η )2 + a3 ( ρ†ρ )2 + a4 ( χ†χ )( η†η ) + a5 ( χ†χ )( ρ†ρ ) + a6 ( ρ†ρ )( η†η ) + a7 ( χ†η )( η†χ ) + a8 ( χ†ρ )( ρ†χ ) + a9 ( ρ†η )( η†ρ ) + a10[(χ †η)(ρ†η) + (η†χ)(η†ρ)] (A.1) we have assumed a10 real. The scalar potential with a10 = 0 has been considered for instance in Ref. [41, 34]. In the following sections we present the mass eigenstates for the scalars in the model. 57 B. Scalar mass eigenstates From the scalar potential, presented in Eq. 4.8, we can find its minima conditions from its derivatives. Therefore, when taking ∂V/∂vχ = ∂V/∂vρ = ∂V/∂vη = 0 we obtain µ21 = − 1 vχ ( a1v 3 χ + 1 2 a4v 2 ηvχ + 1 2 a5v 2 ρvχ − αvηvρ√ 2 ) , (B.1) µ22 = − 1 vη ( a2v 3 η + 1 2 a4vηv 2 χ + 1 2 a6vηv 2 ρ − αvρvχ√ 2 ) , (B.2) µ23 = − 1 vρ ( a3v 3 ρ + 1 2 a5vρv 2 χ + 1 2 a6v 2 ηvρ − αvηvχ√ 2 ) . (B.3) Using the identities above we can write the mass matrices for the scalar sector in a simpler form. Here we will present only the eigenvectors from these mass matrices, giving the relationships between symmetry and mass eigenstates and the masses of the particles. B.1. Double charge scalars( ρ++ χ++ ) = 1√ 1 + |vχ|2 |vρ|2  1 |vχ| |vρ| − |vχ||vρ| 1 (G++ Y ++ ) (B.4) With masses m2 Y ++ = λ ( 1 |vρ|2 + 1 |vχ|2 ) + a8 2 ( |vχ|2 + |vρ|2 ) (B.5) m2 G++ = 0 (B.6) where λ = (vχvρvηα)/ √ 2. 58 B. Scalar mass eigenstates B.2. Single charge scalars  η+1 ρ+ η+2 χ+  =  0 0 − vη√ v2η v2χ +1vχ 1√ v2η v2χ +1 vηC1− √ R a10v2η(v2η+v2ρ)vχ √ D1 vηC1− √ R a10vηvρ(v2η+v2ρ)vχ √ D1 2vχ vη √ D1 2√ D1 − vη√ v2η v2ρ +1vρ 1√ v2η v2ρ +1 0 0 C1vη+ √ R a10v2η(v2η+v2ρ)vχ √ D2 C1vη+ √ R a10vηvρ(v2η+v2ρ)vχ √ D2 2vχ vη √ D2 2√ D2   G+ 1 Y + 1 G+ 2 Y + 2  (B.7) where R =v2η [ 4a210v 2 ρv 2 χ ( v2η + v2ρ ) ( v2η + v2χ ) + a27v 2 ρv 2 χ ( v2η + v2χ )2 − 2a7vρvχ ( v2η + v2χ ) ( a9vρvχ ( v2η + v2ρ ) + √ 2αvη ( v2χ − v2ρ )) + a29v 2 ρv 2 χ ( v2η + v2ρ )2 − 2 √ 2αa9vηvρvχ ( v2η + v2ρ ) (vρ − vχ)(vρ + vχ) + 2α2v2η ( v2ρ − v2χ )2 ] (B.8) A = a7vηvρvχ ( v2η + v2χ ) + a9vηvρvχ ( v2η + v2ρ ) + √ 2α ( v2η ( v2ρ + v2χ ) + 2v2ρv 2 χ ) (B.9) C1 = −a7vρvχ ( v2η + v2χ ) + a9vρvχ ( v2η + v2ρ ) + √ 2αvη ( v2χ − v2ρ ) (B.10) C2 = a7vρvχ ( v2η + v2χ ) − a9vρvχ ( v2η + v2ρ ) + √ 2αvη(vρ − vχ)(vρ + vχ) (B.11) D1 = ( C2vη + √ R )2 a210v 2 ηv 2 ρv 2 χ ( v2η + v2ρ )2 + ( C2vη + √ R )2 a210v 4 ηv 2 χ ( v2η + v2ρ )2 + 4v2χ v2η + 4 (B.12) D2 = ( C1vη + √ R )2 a210v 2 ηv 2 ρv 2 χ ( v2η + v2ρ )2 + ( C1vη + √ R )2 a210v 4 ηv 2 χ ( v2η + v2ρ )2 + 4v2χ v2η + 4 (B.13) with masses m2 Y + 1 = A− √ R 4vηvρvχ (B.14) m2 Y + 2 = A+ √ R 4vηvρvχ (B.15) m2 G+ 1 = m2 G+ 2 = 0 (B.16) 59 B. Scalar mass eigenstates B.3. Neutral scalars For the CP-odd neutral scalars (the ones that come from the imaginary component of the neutral symmetry eigenstates) we have I0η I0ρ I0χ  =  Na |vχ| − Nb|vη ||vχ| |vρ|(|vη |2+|vχ|2) Nc |vη | 0 Nb |vχ| Nc |vρ| − Na |vη | − Nb|vη |2 |vρ|(|vη |2+|vχ|2) Nc |vχ|   G0 1 G0 2 A0  (B.17) with masses m2 A0 = λ ( 1 |vχ|2 + 1 |vρ|2 + 1 |vη|2 ) (B.18) m2 G0 1 = m2 G0 2 = 0 (B.19) where λ = (vχvρvηα)/ √ 2 (B.20) Na = 1/ √ 1 |vχ|2 + 1 |vη|2 (B.21) Nb = 1/ √ 1 |vχ|2 + |vη|2 |vρ|2(|vη|2 + |vχ|2) (B.22) Nc = 1/ √ 1 |vχ|2 + 1 |vρ|2 + 1 |vη|2 (B.23) For the CP-even neutral scalars (which come from the real component of the neutral symmetry eigenstates) it is not possible to find an analytic expression for the mass eigenstates. However, we know that the mass matrix is symmetric, therefore it can be diagonalized by and orthogonal matrix. Therefore, we can write: X0 ψ = ∑ aO H ψaH 0 a , where ψ = χ, η, ρ and a = 1, 2, 3. H0 a are the mass eigenstates and OH is an orthogonal matrix. The lightest of the H0 a should be the Higgs boson of 125 GeV. 60 C. Interactions C.1. Lepton-scalar vertices In the following equations we denote: M̂ l = diag(me,mµ,mτ ), ME = diag(mEe ,mEµ ,mEτ ), (C.1) with ν = (νe, νµ, ντ ), l = (e, µ, τ), E = (Ee, Eµ, Eτ ). The neutral (pseudo)scalars have interactions of the form l̄( √ 2M̂ l/vρ)l · ∑3 i=1Oρihi, and l̄(M̂ l/vρ)γ5l · ∑3 i=1 UρiAi. Since in this model there is only one physical pseudo-scalar we denote A3 ≡ A0. Here we have the form factors appearing in Eq. (5.2) (i fixed) Slhi = √ 2M̂ l vρ Oρi, P lA = √ 2 v2ρ √ 1 v2χ + 1 v2η + 1 v2ρ M̂ lUρ3. (C.2) The matrix elements Oρi and Uρ3 relate the symmetry and mass eigenstates, they can be seen in sec. B.3. For the singly charged scalars Y + 1,2, from the Yukawa interactions in Eq. (4.9), the vertex ν̄iLEjRY + 1 implies SlY1(Y2) = P lY1(Y2) = U1(U2) √ 2 vρ (V l† L )ijM̂ l, (C.3) where U1(U2) represent the projection of a given singly charged scalar into Y1,2. Since the projections are given by complicated expressions involving several unknown constants (see sec. B.2) we have assumed their values to be 0.5. We have explored other values, but there was no noticeable change in our numerical results, given the predominance of the U±± boson contribution to the AMDM (see sec. 6.2). Finally, for the doubly charged scalar we have l̄EY −− → i(SY + γ5PY ), with SY −− = √ 2 vρ [cosαM̂EV l† L + sinαV l† L M̂ l] P lY −− = √ 2 vρ [− cosαM̂EV l† L + sinαV l† L M̂ l], (C.4) 61 C. Interactions where cosα = vρ/ √ v2ρ + v2χ and sinα = vχ/ √ v2ρ + v2χ. where the constants V l S and AlS are defined in Sec. 4. We assumed that the heavy leptons are in the diagonal basis, ME = M̂E and that in (C.4), the vχ is real, i.e., θχ = 0. The numerical values for the V l L,R matrices were obtained in Ref. [34] and [37]. The different sets os matrices considered can be seen in Sec. 6.1.In the interactions above, the matrix V l R and the matrix for the Yukawa couplings can always be eliminated by using the mass matrix. The extra factors in the interactions above come from the projection on the SM-like scalars, see [34] for details. C.2. Scalar-photon vertices From the lepton Lagrangian we may find the interaction with photon of the known leptons, l(e µ τ)T and extra leptons E = (EeEµEτ )T : LI = (−el̄γµl + eĒγµE)Aµ (C.5) and identify the electron charge as e = gt√ 1 + 4t2 = gsW (C.6) where e is the modulus of the electron charge and t2 = s2W /(1− 4s2W ). Now, from the covariant derivatives of the Lagrangian we can find the vertices for the interactions between scalars and photons: AµY ++Y −− → i2e(k− − k+)µ, AµY + 2 Y − 2 → ie(k− − k+)µ, AµY + 1 Y − 1 → ie(k− − k+)µ. (C.7) The terms k+ and k− indicate, respectively, the momenta of the positive and negative charge scalars, and both should be considered incoming into the vertex. C.3. Gauge vertices For three gauge bosons denoted generically by X,Y and Z, with all momenta incoming (denoted by k), the vertex is: XνYλZµ → iGXY Z [ gνλ ( kX − kY ) µ + gλµ ( kY − kZ ) ν + gµν ( kZ − kX ) λ ] (C.8) 62 C. Interactions The proportionality constants are: GWWA = e, GV V A = −e, GUUA = −2e, GWWZ = −e/tW , GWWZ′ = 0, GV V Z = −e 2 ( 1 tW + 3tW ) , GV V Z′ = e √ 3 2 √ 1− 4s2W sW cW , GUUZ = e 2 ( 1 tW − 3tW ) , GUUZ′ = e √ 3 2 √ 1− 4s2W sW cW . (C.9) where sW , cW and tW are, respectively, the sine, cosine and tangent of the weak mixing angle, θW . C.4. Charged gauge-lepton interactions The Lagrangian terms for interactions among charged gauge bosons and leptons may be written as follows: • ν-type and l-type leptons: Lνl = g 2 √ 2 ν̄iγ µ (1− γ5) (VPMNS)ijljW + µ +H.c. (C.10) for i = 1, 2, 3 and j = 1, 2, 3 and VPMNS is the Pontecorvo-Maki-Nagaoka-Sakata matrix. • E-type and ν-type leptons: LEν = g 2 √ 2 Ēiγ µ (1− γ5) (UνL)ij νj V + µ +H.c. (C.11) for i = 1, 2, 3. UνL is the unitary matrix relating the neutrino symmetry eigenstates with the mass eigenstates. We also assume that the heavy leptons are in the diagonal basis. • E-type and l-type leptons: LEl = g 2 √ 2 Ēiγ µ (1− γ5) (V l† L )ijljU ++ µ +H.c. (C.12) for i, j = 1, 2, 3 and V l L is given in Sec. 6.1. Notice that in this model V l U = AlU = g 2 √ 2 V l† L , (C.13) . 63 C. Interactions C.5. Neutral gauge-lepton interactions Considering the following Lagrangian for the neutral interactions: LNC = − g 2cW ∑ i ψ̄iγ µ [( giV − giAγ5 ) Zµ + ( f iV − f iAγ5 ) Z ′µ ] ψi (C.14) where ψi can be any lepton mass eigenstate, we find the giV , giA, f iV and f iA to be: gνV = 1 2 , gνA = 1 2 , glV = −1 2 + s2W , glA = −1 2 + s2W , gEV = gEA = −s2W , (C.15) fνV = − √ 1− 4s2W 2 √ 3 , fνA = − √ 1− 4s2W 2 √ 3 f lV = − √ 1− 4s2W 2 √ 3 , f lA = − √ 1− 4s2W 2 √ 3 , fEV = √ 1− 4s2W 2 √ 3 , fEA = √ 1− 4s2W 2 √ 3 (C.16) 64 D. Scalar-fermion loop diagram calculation As an example of the calculations performed to find the AMDM from the diagrams presented in Fig. 5.2, we will show some of the details in the calculation of diagram a) of the mentioned figure, where we have a scalar-fermion loop with the photon connected to the fermion. Remembering that q = p′ − p and that k is the momentum to be integrated in the loop, and also using, from the Feynman parametrization, l = k + yq − zp, the diagram can be written as Γµ = 2i ∫ d4kdxdydz [ ū(p′) Nµ D3 u(p) ] δ(x+ y + z − 1); (D.1) where Nµ = (S +Aγ5)(/k +mI)(Qγ µ)(/k + /q +mI)(S † + P †γ5), (D.2) D = l2 −m2 I(1− z) + q2xy +m2 l z(1− z)− zM2, (D.3) being Q the electric charge of the fermion in the loop, mI the mass of the fermion in the loop, ml the mass of the external fermion, S and P the scalar and pseudoscalar vertices and M the mass of the scalar. To proceed, we expand the numerator N in Eq. D.1 - remembering that we have ū(p′)Nµu(p) - and perform the following substitutions in the presented order: • /k → /l + /pz − /qy; • /q → /p′ − /p; • γ5/pγµ/pγ5 → −/pγµ/p; • γ5/pγµ/p′γ5 → −/pγµ/p′; • γ5/p′γµ/pγ5 → −/p′γµ/p; • γ5/p′γµ/p′γ5 → −/p′γµ/p′; 65 D. Scalar-fermion loop diagram calculation • /pγµ/p→ 2pµ/p− γµm2 l ; • /p′γµ/p′ → 2p′µ/p′ − γµm2 l ; • /pγµ/p′ → γµ(q2 − 2m2 l )− /p′γµ/p+ pµ/p′ + 2p′µ/p; • γµ/p′ → 2p′µ − /p′γµ; • /pγµ → 2pµ − γµ/p; • /p′γµ/pγ5 → −m2 l γ µγ5; • γ5/p′γµ/p→ −m2 l γ5γ µ; • γ5/p′γµγ5 → γµml; • γ5γµ/pγ5 → γµml; • /p′γµ/p→ γµm2 l ; • /p′γµγ5 → mlγ µγ5; • γµ/pγ5 → −mlγ µγ5; • γ5/p′γµ → mlγ µγ5; • γ5γµ/p→ −mlγ µγ5; • /pγ5 → γ5(−ml); • /p′γµ → γµml; • /p′γ5 → γ5ml; • γµ/p→ γµml; • γ5/p→ γ5ml; • γ5/p′ → γ5(−ml); • γ5γµ → −γµγ5; • γ5γ5 → 1; • /p→ ml; 66 D. Scalar-fermion loop diagram calculation • /p′ → ml. It is important the follow the order above to ensure that we have all /p′ on the left and all /p on the right, so we can use the identities ū(p′)/p′ = ml and /pu(p) = ml. After these substitutions, we use Gordon’s identity (Eq. 2.27), take the terms proportional to iσµνq ν/2ml, and perform the integrations (using the identities shown in Sec 2.1). 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