Summary of Physics doctoral thesis: New physics effect in advanced economical 3-3-1 models

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The thesis proposes a class for the new economic 3-3-1 model, called a simple 3-3-1 model, in order to address the mentioned problems. The simple 3-3-1 model considers the simplest lepton sector and the scalar sector. This setting leads to natural components of the inert fields such as scalar inert fields and right-handed neutrino.

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MINISTRY OF EDUCATION VIETNAM ACADEMY AND TRAINING OF SCIENCE AND TECHNOLOGY GRADUATE UNIVERSITY SCIENCE AND TECHNOLOGY --------------- LE DUC THIEN NEW PHYSICS EFFECT IN ADVANCED ECONOMICAL 3-3-1 MODELS SUMMARY OF PHYSICS DOCTORAL THESIS HANOI- 2020
MINISTRY OF EDUCATION VIETNAM ACADEMY AND TRAINING OF SCIENCE AND TECHNOLOGY GRADUATE UNIVERSITY SCIENCE AND TECHNOLOGY --------------- LE DUC THIEN NEW PHYSICS EFFECT IN ADVANCED ECONOMICAL 3-3-1 MODELS SUMMARY OF PHYSICS DOCTORAL THESIS Major: Theoretical and Mathematical Physics Code: 62 44 01 03 Supervisor: Associate Prof. Phung Van Dong Prof. Dang Van Soa HANOI - 2020
INTRODUCTION The motivation The experimental puzzles in particle physics as well as cosmology, based on the standard model and general relativity, cannot explain, including neutrino oscillation, matter-antimatter asymmetry, dark matter, dark energy, and inflation. In many extensions of the standard model, the 3-3-1 models promisingly is a potential candidate for new physics. In particular, the model can address the number generation, the quantization of electric, strong CP, flavor mixing, and the abnormal heaviness of top quark. The 3-3-1 models are being investigated for solving these experimental issues. In fact, one showed that the 3-3-1 models contain the seesaw mechanism and naturally radiative corrections. They lead to the smallness neutrino mass and explain the lepton num- ber asymmetry. Besides, they can supply dark matter candidates based on gauge principles or inert scalar fields. The 3-3-1 model and other its versions also give the explanation to the inflation issue and reheating in the universe through Higgs inflation scenario or a new scalar field breaking the B − L symmetry. The thesis proposes a class for the new economic 3-3-1 model, called a simple 3-3-1 model, in order to address the mentioned problems. The simple 3-3-1 model considers the simplest lepton sector and the scalar sector. This setting leads to natural components of the inert fields such as scalar inert fields and right-handed neutrino. The existing of these fields brings the meaning for new physics anomalies, neutrino masses, and dark matter. In addition, the flipped 3-3-1 model leads to natural candidates for dark matter unified with normal particles in gauge multiples, and explain lepton flavor violation processes. Some new physics predictions are also discussed. Purpose and objective • Investigate the simple 3-3-1 model with inert scalars: the model, interactions and its consequences. • Examine the flipped 3-3-1 model with the problems of neutrino masses, dark energy 1
and the flavor-changing neutral current.. Content • The 3-3-1 model with inert scalar - Investigate the model, introduce two inert scalar triplets to the model and finding the conditions on the scalar potential parameters. - Review the interaction. - Review the physics of flavor and anomalous magnetic moment. - Investigate the new physics effects at LHC. - Define the bounds on the new physics scale from dijet and Drell-Yan processes. • The flipped 3-3-1 model - Propose the model - Identify candidates for the dark matter - Examine flavor-changing neutral currents - Determine the lepton flavor violating processes - Obtain the dark matter observables - Investigate the new physics effects at LHC Layout Chapter 1: Introduction: We give a brief review of the Standard Model (SM) and the experimental problems related to SM. The standard model’s extensions and the subject of this thesis are discussed. Chapter 2: Phenomenology of the simple 3-3-1 model with inert scalars: we introduce two inert scalar triplets to the model, finding the condition for the scalar potential parameters and all the interactions in the normal sector will be calculated. The inert field contains the candidates for dark matter and its density, direct and indirect searching for the new particles at LHC, as well as producing the dilepton, dijet, diboson at the LHC will be review. Chapter 3: Dark matter and flavor changing in the flipped 3-3-1 model: We investigate a minimal setup of this model and determine novel consequences of dark matter stability, neutrino mass generation, and lepton flavor violation. We will discuss the dark matter ob- servables and give rise to the LHC signals. Conclusion: The general summary and conclusion of the work in the Ph.D thesis are given. 2
CHAPTER 1. STANDARD MODEL AND ITS PROBLEMS Modern physics are based on the standard model (SM) and the general relativistic theory of gravity interactions. The standard model of electroweak and strong interaction, has been experimentally tested with a very high precision. We are going to review the model and discuss its serious problems on both of theoretical and experimental sides. 1.1. Standard model The SM [1] is the model based upon the SU (3)C ⊗ SU (2)L ⊗ U (1)Y (3-2-1), where the first group represents strong interaction of color charges (QCD), the last two term represent the weak interaction between spin particles and (or) weak hypercharge. Three generation of lepton and quark are arranged as (νaL eaL ) ∼ (1, 2, −1/2), eaR ∼ (1, 1, −1), (uaL daL ) ∼ (3, 2, 1/6), uaR ∼ (3, 1, 2/3), daR ∼ (3, 1, −1/3) transform belong to the gauge group respectively and a = 1, 2, 3 are the generation number. Gravity interaction are obtained by using general metric instead of Minkowski metric and adding the Einstein-Hilbert action to the actions of SM. Gravity theory works at large scale, describe the microscopic phenomenon up to less than 1 mm with great accuracy of less than 10−3 uncertainty [1]. 1.2. Neutrino mass In SM, neutrinos have exactly zero mass since right-handed neutrinos do not exist and lepton number are always conserved in the model. However, experiments which observed neutrinos from Earth’s atmosphere, from the sun, from nuclear reactors or particle accelerators are proof the neutrino oscillations (changing flavours) in the far enough distance, saying neutrino has to have nonzero mass (although small below 1 eV) and mixed up. 1.3. Matter-antimatter asymmetry problem The early Universe should have equal amounts of matter and antimatter, why today everything we see from the smallest life forms on Earth to the largest stellar objects is made 3
almost entirely of matter. Comparatively, there is not much antimatter to be found [1]. 1.4. Dark matter and dark energy Measurements of cosmic microwave background (CMB) anisotropies by the data from Wilkinson Microwave Anisotropy Probe (WMAP) [20] spacecraft and Planck [21] spacecraft has indicated that a universe made up of about 5% ordinary matter, 25% dark matter and 70% dark energy in a model of Big Bang theory and flat universe [22]. This is not understandable in the SM. 1.5. LHC experiments The discovery of the Higgs particle marks the success of the LHC [2,3]. Higgs interactions can be summarized via the combined best-fit signal strength, µH = 1 ± 0.1, which deviates 10% from the SM [62]. The exclusive model contains new Higgs mixed up with Higgs from SM, this deviation is the constraint for the mixed up effect. The 331 and 3311 satisfy the constraint at the new physics scale that is much larger than electro-weak scale [32, 33]. New physics process has been searched in LHC experiment at the decay channels to dilepton, dijet, diboson, mono-X and di-X dark matter and no new particles signal has been found. This confirms the successful of SM and gives the stringent constraint for new physics model at TeV scale. 1.6. Propose subject of the thesis The 3-3-1 and 3-3-1-1 extensions and the others extensions of the SM are challenged by new experimental issues. In this thesis, we are propose the improvement of the model to solve those issues as much as possible. The 3-3-1 model with inert scalar: From some first results in [63], we concluded that this 3-3-1 model must contain at least one inert multiplet that can provide dark matter candidates and can explain the experimental ρ-parameter. It can also explain the neutrino masses, lepton number asymmetry, and the new Higgs inflation scenarios. The dijet, Drell- Yan, diboson processes and information of dark matter from LHC are discussed. The flipped 3-3-1 model: One of lepton generations is arranged differently from the remaining lepton generations, while all quark generations are identical by contrast. The flip of quark and lepton representations converts the flavor matters in quark sector to the lepton sector. Moreover, the origin of the matter parity and therefore the dark matter stability, the lepton flavor violating processes are determined by a residual gauge symmetry. 4
CHAPTER 2. PHENOMENOLOGY OF THE SIMPLE 3-3-1 MODEL WITH INERT SCALARS The results of this chapter are based on the work published in Phys. Rev. D bf 99, 095031, 2019. 2.1. THE MODEL The gauge symmetry of the model is given by SU (3)C ⊗ SU (3)L ⊗ U (1)X , (2.1) The electric charge is embedded in the gauge symmetry as: √ Q = T3 − 3T8 + X, (2.2) where Ti (i = 1, 2, 3, .., 8) and X are the generators of SU (3)L and U (1)X , respectively. Furthermore, the SU (3)C generators will be denoted as ti . the fermion content which is anomaly free is given by   νaL   ψaL ≡  eaL  ∼ (1, 3, 0), (2.3)     (eaR )c   dαL   QαL ≡  −uαL  ∼ (3, 3∗ , −1/3), (2.4)     JαL   u  3L  Q3L ≡  d3L  ∼ (3, 3, 2/3) , (2.5)     J3L uaR ∼ (3, 1, 2/3) , daR ∼ (3, 1, −1/3) , (2.6) JαR ∼ (3, 1, −4/3) , J3R ∼ (3, 1, 5/3) , (2.7) To break the gauge symmetry and generate the masses for particles in a correct way, 5
the scalar content can minimally be introduced as     η10 χ−    1  η =  η2−  ∼ (1, 3, 0), χ =  χ−−  ∼ (1, 3, −1), (2.8)        2  + η3 χ03 with corresponding vacuum expectation values (VEVs),     u 0 1   1   hηi = √  0  , hχi = √  0  . (2.9)     2  2  0 w The total Lagrangian, up to the gauge fixing and ghost terms, is obtained by X X L = F¯ iγ µ Dµ F + (Dµ S)† (Dµ S) F S 1 1 1 − Giµν Gµν µν i − Aiµν Ai − Bµν B µν 4 4 4 +LY − V. (2.10) The scalar potential is given by V = Vsimple + Vinert , where the first term is Vsimple = µ21 η † η + µ22 χ† χ + λ1 (η † η)2 + λ2 (χ† χ)2 +λ3 (η † η)(χ† χ) + λ4 (η † χ)(χ† η), (2.11) The Yukawa Lagrangian takes the form, u LY = hJ33 Q¯ 3L χJ3R + hJ Q ¯ αL χ∗ JβR + hu Q ¯ 3L ηuaR + hαa Q ¯ αL ηχuaR αβ 3a Λ d +hdαa Q ¯ αL η ∗ daR + h3a Q¯ 3L η ∗ χ∗ daR + he ψ¯c ψbL η ab aL Λ 0e ν h s + ab 2 (ψ¯aL c ηχ)(ψbL χ∗ ) + ab (ψ¯aL c η ∗ )(ψbL η ∗ ) + H.c. (2.12) Λ Λ The physical charged gauge bosons with corresponding masses are obtained by, A1 ∓ iA2 g2 2 W± = √ , m2W = u , (2.13) 2 4 A4 ∓ iA5 g2 X∓ = √ , m2X = (w2 + u2 ), (2.14) 2 4 A6 ∓ iA7 g2 Y ∓∓ = √ , m2Y = w2 . (2.15) 2 4 The neutral gauge bosons with corresponding masses are achieved as, √ q 2 A = sW A3 + cW − 3tW A8 + 1 − 3tW B , mA = 0, (2.16) √ g2 q Z = cW A3 − sW − 3tW A8 + 1 − 3tW B , m2Z = 2 u2 , 2 (2.17) 4cW q √ Z 0 = 1 − 3t2W A8 + 3tW B, (2.18) 6
g 2 [(1 − 4s2W )2 u2 + 4c4W w2 ] m2Z 0 = , (2.19) 12c2W (1 − 4s2W ) √ where sW = e/g = t/ 1 + 4t2 , with t = gX /g, is the sine of the Weinberg angle. The physical Higgs particles with corresponding masses, q h ≡ cξ S1 − sξ S3 , m2h = λ1 u2 + λ2 w2 − (λ1 u2 − λ2 w2 )2 + λ23 u2 w2 4λ1 λ2 − λ23 2 ' u , (2.20) 2λ2 q H ≡ sξ S1 + cξ S3 , m2H 2 2 = λ1 u + λ2 w + (λ1 u2 − λ2 w2 )2 + λ23 u2 w2 ' 2λ2 w2 (2.21) λ4 2 H ± ≡ cθ η3± + sθ χ± 1, m2H ± = (u + w2 ), (2.22) 2 In summary, we have four massive Higgs bosons (h, H, H ± ), in which h is the standard model like Higgs particle (verified below) with a light mass in the u scale, while the others are new Higgs bosons with large masses in the w scale. 2.2. Interaction 2.2.1. Interactions of fermions with gauge bosons F¯ iγ µ Dµ F , in which we separate Dµ = ∂µ + P The relevant interactions arise from F igs ti Giµ + igPµCC + igPµNC , where PµCC = Ti Aiµ and PµNC = T3 A3µ + T8 A8µ + tXBµ . P i6=3,8 Charged current The charged current takes the form, X µ µ −g F¯ γ µ PµCC F = −gJW Wµ+ − gJX Xµ− − gJYµ Yµ−− + H.c., (2.23) F where µ X 1 JW = F¯ γ µ T+ F = √ (¯νaL γ µ eaL + u ¯aL γ µ daL ) , (2.24) F 2 µ X 1 F¯ γ µ U+ F = √ ν¯aL γ µ ecaR − J¯αL γ µ dαL + u¯3L γ µ J3L , JX = (2.25) F 2 X 1 JYµ = F¯ γ µ V+ F = √ e¯aL γ µ ecaR + J¯αL γ µ uαL + d¯3L γ µ J3L . (2.26) F 2 Neutral current The neutral current takes the form, X g ¯ µ Z F¯ γ µ PµNC F −eQ(f )f¯γ µ f Aµ − Z −g = f γ gV (f ) − gA (f )γ5 f Zµ 2cW F 7
g ¯ µ h Z0 Z0 i − f γ gV (f ) − gA (f )γ5 f Zµ0 , (2.27) 2cW where f indicates all the fermions of the model, and gVZ (f ) = T3 (fL ) − 2s2W Q(f ), Z gA (f ) = T3 (fL ), (2.28) √ 3s2W 0 q gVZ (f ) = 1 − 4s2W T8 (fL ) + p (X + Q)(fL ), (2.29) 1 − 4s2W √ 2 Z0 c2W 3sW gA (f ) = p 2 T8 (fL ) − p T3 (fL ). (2.30) 1 − 4sW 1 − 4s2W 2.2.2. Interactions of scalars with gauge bosons µ S)† (Dµ S), with S = η, χ. the relevant interactions P Derived from the Lagrangian S (D and couplings are resulted as in Tables 2.1 to 2.9. 2.2.3. Scalar self-interactions and Yukawa interactions Since we work in the unitary gauge, the scalar self-interactions include only those with physical scalar particles. Note that the interactions between the normal scalars and the inert scalars were given in [35]. Therefore, we necessarily calculate only self-interactions of the normal scalars, Self-interaction results of scalars are often presented in tables 2.10 and 2.11. The inert scalars do not have Yukawa interaction with fermions due to Z2 symmetry. Therefore, we turn to investigate the Yukawa interactions of the normal scalars, the relevant interactions and couplings are resulted as in Tables 2.12 to 2.14. 2.3. Phenomenology 2.3.1. The SM like Higgs particle The discovery of the Higgs particle marks the success of the LHC, and its couplings can be summarized via the combined best-fit signal strength µh = 1.1 ± 0.1, which deviates 10% from the standard model. Let us particularly investigate the Higgs coupling to two photons that substitutes in: σ(pp → h)Br(h → γγ) µγγ = , (2.31) σ(pp → h)SM Br(h → γγ)SM where the numerator is given by the considering model once measured by the experiments, while the denominator is the standard model prediction. The Higgs production dominantly comes from the gluon gluon fusion via top loops. The new physics effects are shown in the picture 2.1. Note that the (b) diagram was skipped in [42, 74]. 8
G G h h t Ja cξ −tθ sξ G G (a) (b) Figure 2.1: Contributions to the Higgs production due to gluon-gluon fusion. The main contributions to decaying the Higgs into two photons are shown in the diagram 2.2. γ γ h h t Ja cξ γ −tθ sξ γ (a) (b) γ γ h h W X(Y ) cξ γ sθ−ξ −sξ γ sθ ( tθ ) (c) (d) γ γ h h cξ γ sθ−ξ −sξ X(Y ) γ W sθ ( tθ ) (e) (f ) γ γ h h H ±, φ gH ± ,φ γ gH ± ,φ γ H ±, φ (g) (h) Figure 2.2: Contributions to the decay h → γγ. Our numerical study yields a maximal bound, 1 ≤ µγγ ≤ 1.06, in agreement with the data. 9
¯s mixing and rare Bs → µ+ µ− decay 2.3.2. The Bs -B The left graph of 2.3 for this mixing as explained by basic Z 0 boson. We have: ∗ [(VdL )32 (VdL )33 ]2 1 2 < . (2.32) w (100 TeV)2 ∗ The CKM factor is given by |(VdL )32 (VdL )33 | ' 3.9 × 10−2 , which implies w > 3.9 TeV, slightly larger than the bound given in [35]. Correspondingly, the Z 0 mass is bounded by mZ 0 > 4.67 TeV, provided that s2W ' 0.231 is at the low energy regime of the interested precesses. The new physics contribution is demonstrated by the right graph in 2.3 by basic Z 0 boson exchange. Generalizing the results in [88], we obtain the signal strength, Br(Bs → µ+ µ− ) µBs →µ+ µ− = = 1 + r2 − 2r, (2.33) Br(Bs → µ+ µ− )SM where r = ∆C10 /C10 (C10 = −4.2453 is standard model Wilson coefficient) is real and bounded by 0 ≤ r ≤ 0.1. It leads to mZ 0 ≥ 2.02 TeV. (2.34) b s b µ− Z! Z! s b s µ+ Figure 2.3: ¯s mixing and rare Bs → µ+ µ− decay due to the Contributions to the Bs -B tree-level flavor-changing coupling. 2.3.3. Radiative β decay involving Z 0 as a source of CKM unitarity violation Vik∗ Vjk = δij and Vik∗ Vil = δkl , where we relabel P P CKM unitarity states that k i V = VCKM , i, j = u, c, t, and k, l = d, s, b. The standard model prediction is in good agreement with the above relations [1]. How- ever, a possible deviation would be the sign for the violation of CKM unitarity. Considering the first row, the experiments constrain [1] X ∆CKM = 1 − |Vuk |2 < 10−3 . (2.35) k=d,s,b Since mW ' 80.4 GeV and mZ 0 in the TeV range (in fact, mZ 0 > 4.67 TeV), we have ∆CKM < 10−5 . The effect of CKM unitarity violation due to Z 0 is negligible and thus the model easily evades the experimental bound. This conclusion contradicts a study of the minimal 3-3-1 model in [83]. 10
2.3.4. LEPII search for Z 0 The effective Lagrangian: 0 g 2 [aZL (e)] 2 Leff ⊃ 2 2 eγ µ PL e)(¯ (¯ µγµ PL µ) + (LR) + (RL) + (RR), (2.36) cW mZ 0 LEPII studied such chiral interactions and gave respective constraints on the chiral cou- plings, which are typically in a few TeV [101]. Choosing a typical bound derived for a new U (1) gauge boson like ours, it yields [102] 0 g 2 [aZL (e)] 2 1 2 2 < . (2.37) cW mZ 0 (6 TeV)2 This translates to 6g Z 0 g q m0Z > aL (e) TeV = 3(1 − 4s2W ) TeV ' 354 GeV. (2.38) cW cW In fact, the Z 0 mass is in the TeV range, it easily evades the LEPII searches. 2.3.5. LHC searches for new particle signatures Dilepton and dijet searches Because the neutral gauge boson Z 0 directly couples to quarks and leptons, the new physics process pp → l¯l for l = e, µ happens, which is dominantly contributed by the s- channel exchange of Z 0 . 10 2% width 1 4% width 8% width ΣHpp®Z ¢ ®llL @pbD 0.1 16% width 0.01 32% width Model 0.001 10 -4 10 -5 1000 2000 3000 4000 5000 m Z ¢ @GeVD Figure 2.4: Cross section σ(pp → Z 0 → l¯l) as a function of Z 0 mass. We show the cross section for the process pp → Z 0 → l¯l in 2.4 where l lis either electron or muon which has the same Z 0 coupling. The experimental searches use 36.1 fb−1 of pp √ collision data at s = 13 TeV by the ATLAS collaboration [106], yielding negative signal. 11
for new high mass events in the dilepton final state. This translates to the lower bound on Z 0 mass, mZ 0 > 2.75 TeV, for the considering model, in agreement with a highest invariant mass of dilepton measured by the ATLAS. Diboson and diphoton searches At LHC, the collision of proton beams pp will generate new particles , and then they decay to pair of boson or photon, or lose energy. The generation of the pair of boson or photon can be related to new particles , such as new neutral Higss and Z 0 , cthese particles have already studied and in agreement with experiments Monojet and dijet dark matter The missing energy can be happened due to the generation of DM, according to the law of energy conservation. This missing energy is equivalent to monojet and dijet. 12
g g H1′ g H1′ H H g H1′ g g H1′ g g H1′ g g H1′ H H g H1′ g H1′ qc g H1′ g H1′ H H q H1′ q q H1′ qc g H1′ qc H1′ Z′ Z′ q A′1 q g A′1 q q H1′ q H1′ Z′ Z′ g A′1 g q A′1 Figure 2.5: Monojet production processes associated with a pair of dark matter. 13
CHAPTER 3. THE FLIPPED 3-3-1 MODEL The results of this chapter are based on the work published on JHEP bf 08 (2019) 051. 3.1. General flipped 3-3-1 model 3.1.1. Particle content The 3-3-1 gauge symmetry is given by SU (3)C ⊗ SU (3)L ⊗ U (1)X , (3.1) The electric charge and hypercharge are embedded as Q = T3 + βT8 + X, Y = βT8 + X, (3.2) he fermion content is written as   + 1 0 1 ξ √ 2 ξ √ ν 2 1   − 1 ∼ −  1 0 1 ψ1L =  √ ξ 1, 6, , (3.3)  ξ √ e  2 1   2 3 √1 ν1 √1 e1 E1 2 2 L   ν  α  2 ψαL =  eα  ∼ 1, 3, − , (3.4)     3 Eα L eaR ∼ (1, 1, −1), EaR ∼ (1, 1, −1), (3.5)   d  a  ∗ 1 QaL =  −ua  ∼ 3, 3 , , (3.6)     3 Ua L uaR ∼ (3, 1, 2/3), daR ∼ (3, 1, −1/3), UaR ∼ (3, 1, 2/3), (3.7) The scalar content responsible for symmetry breaking and mass generation is given by     0 + η ρ  1   1   −  η =  η2  ∼ (1, 3, −2/3), ρ =  ρ2  ∼ (1, 3, 1/3),  0  (3.8)     η3− ρ03 14
  χ+ 1   χ =  χ02  ∼ (1, 3, 1/3), (3.9)     χ03   ++ √1 S + √1 S + S11 2 12 2 13   + S =  √1 S12 0 1 S0  ∼ (1, 6, 2/3). (3.10)   S22 √ 2 23  2  √1 S + √1 S 0 0 S33 2 13 2 23 Note that ρ and χ are identical under the gauge symmetry, but distinct under the B − L charge, as shown below. 3.1.2. Dark matter The imprint at low energy is only the 3-3-1 model, conserving the matter parity as residual gauge symmetry √ WP = (−1)3(B−L)+2s = (−1)2 3T8 +3N +2s , (3.11) Because the matter parity is conserved, the lightest W - particle (LWP) is stabilized, responsible for dark matter. The dark matter candidates include a fermion ξ 0 , a vector Y 0 , and a combination of ρ03 v`a S23 0 . Due to the gauge interaction, Y 0 annihilates completely into the standard model particles. The The realistic candidates that have correct anbundance are only the fermion or scalar, as shown below. 3.1.3. Lagrangian The total Lagrangian consists of L = Lkinetic + LYukawa − V, (3.12) LYukawa = heαa ψ¯αL ρeaR + hE ¯ E ¯ ξ ¯c αa ψαL χEaR + h1a ψ1L SEaR + h ψ1L ψ1L S ¯ aL ρ∗ ubR + hdab Q +huab Q ¯ aL η ∗ dbR + hU ¯ ∗ ab QaL χ UbR + H.c. (3.13) The last part is the scalar potential, V = µ2η η † η + µ2ρ ρ† ρ + µ2χ χ† χ + µ2S Tr(S † S) 2 +λη (η † η)2 + λρ (ρ† ρ)2 + λχ χ† χ + λ1S Tr2 (S † S) + λ2S Tr(S † S)2 +ληρ (η † η)(ρ† ρ) + λχη (χ† χ)(η † η) + λχρ (χ† χ)(ρ† ρ) +ληS (η † η)Tr(S † S) + λρS (ρ† ρ)Tr(S † S) + λχS (χ† χ)Tr(S † S) +λ0ηρ (η † ρ)(ρ† η) + λ0χη (χ† η)(η † χ) + λ0χρ (χ† ρ)(ρ† χ) +λ0χS (χ† S)(S † χ) + λ0ηS (η † S)(S † η) + λ0ρS (ρ† S)(S † ρ) + µηρχ + µ0 χT S ∗ χ + H.c. (3.14) 15
3.1.4. Neutrino mass Substituting the VEVs into the Yukawa Lagrangian, the quark and exotic leptons gain suitable masses as follows hu hd hU [mu ]ab = √ab v, [md ]ab = − √ab u, [mU ]ab = − √ab w, (3.15) 2 2 2 √ ξ hE hE mξ = − 2h Λ, [mE ]1b = − √1b Λ, [mE ]αb = − √αb w. (3.16) 2 2 The ordinary leptons obtain masses he √ [me ]αb = − √αb v, [mν ]11 = 2κhξ . (3.17) 2 The heavy φ, νR are present and can imply neutrino masses via: 1 Lν = hναb ψ¯αL ηνbR + hR ν¯c νbR φ + H.c. (3.18) 2 ab aR √ We achieve Dirac masses [mD ν R R ν ]αb = −hαb u/ 2 anh Majorana masses [mν ]ab = −hab hφi. Because of u hφi, the observed neutrinos ∼ νL gain masses via a type I seasaw, by R −1 R −1 ν T u2 u2 [mν ]αβ ' −[mD ν (mν ) (mD T ν ν ) ]αβ = hαa (h )ab (h )bβ ∼ . (3.19) 2hφi hφi Fitting the data mν ∼ 0.1 eV, we obtain: hφi ∼ [(hν )2 /hR ]1014 GeV, since u is proportional to the weak scale. Given that hν , hR ∼ 1, one has hφi ∼ 1014 GeV, close to the grand unification scale. It is clear that two neutrino ν2,3L achieve masses via the type I seasaw with the corre- sponding mixing angle θ23 comparable to the data, while the neutrino ν1L has a mass (which one sets hξ κ ∼ 0.1 eV) via the type II seasaw and does not mix with ν2,3L . The mixing angles θ12 and θ13 can be induced by an effective interaction, such as hν1β c Lmix = ψ¯ ψβL ρη ∗ φ + H.c., (3.20) M 2 1L where M is the new physics scale which can be fixed at M = hφi. The mass matrix of observed neutrino is corrected by uv uv [mν ]1β = −hν1β ∼ . (3.21) hφi hφi 3.1.5. Gauge sector The mass Lagrangian of gauge bosons is given by † X L⊃ (Dµ hΦi) (Dµ hΦi) , (3.22) Φ=η,ρ,χ,S The standard bosons has a mass of, g2 2 g2 2 g2 2 m2W ' (u + v 2 ), m2X = (u + w2 + 2Λ2 ), m2Y ' (v + w2 + 2Λ2 ). (3.23) 4 4 4 16
The neutral gauge bosons has a mass of: g2 m2Z1 u2 + v 2 , ' 2 (3.24) 4cW g2 m2Z2 (1 + t2W )2 u2 + (1 − t2W )2 v 2 + 4(w2 + 4Λ2 ) , ' 2 (3.25) 4(3 − tW ) and the mixing angle p 3 − 4s2W u2 − c2W v 2 t2ϕ ' . (3.26) 2c4W w2 + 4Λ2 Since κ is tiny, its contribution to the ρ parameter is neglected. The deviation of the ρ parameter from the standard model prediction is due to the Z-Z 0 mixing, obtained by (u2 − c2W v 2 )2 ∆ρ ' . (3.27) 4c4W (u2 + v 2 )(w2 + 4Λ2 ) From the W mass, we derive u2 + v 2 = (246 GeV)2 . From the global fit, the PDG Collaboration extracts the ρ deviation as ∆ρ = 0.00039 ± 0.00019, which is 2σ above the standard model prediction [1]. Generally for the whole u range, the new physics scale are √ bounded by w2 + 4Λ2 ∼ 5–7 TeV [34]. 3.2. FCNC The Lagrangian that sums over six-dimensional interaction relevant to the standard model fermion at the tree-level: 0 0 ΓlZ lZ αβ Γγδ ¯lα γ µ PL lβ ¯lγ γµ PL lδ , − (3.28) m2Z 0 0 ΓlZ gs2 αβ ¯lα γ µ PL lβ ¯lδ γµ PR lδ , √ W − 2 (3.29) mZ 0 cW 1 + 2c2W 0 0 ΓlZ νZ αβ Γγδ νγ γµ PL νδ ) ¯lα γ µ PL lβ , − 2 (¯ (3.30) mZ 0 0 ΓνZ gs2 αβ √ W να γµ PL νβ ) ¯lδ γ µ PR lδ , − 2 (¯ (3.31) mZ 0 cW 1 + 2c2W 0 ΓνZ αβ g(2 + c2W ) + 2 √ να γ µ PL νβ ) (¯ (¯ q γµ PL q) , (3.32) mZ 0 6cW 1 + 2c2W 0 ΓνZ αβ gs2 + 2 √ W να γ µ PL νβ ) (η q q¯γµ PR q) , (¯ (3.33) mZ 0 3cW 1 + 2c2W 0 ΓlZ αβ g(2 + c2W ) ¯lα γ µ PL lβ (¯ + 2 √ q γµ PL q) , (3.34) mZ 0 6cW 1 + 2c2W 0 ΓlZ αβ gs2 √ W ¯lα γ µ PL lβ (η q q¯γµ PR q) , + 2 (3.35) mZ 0 3cW 1 + 2c2W 2 1 g(2 + c2W ) − 2 √ q γ µ PL q) (¯ (¯ q γµ PL q) , (3.36) mZ 0 6cW 1 + 2c2W 17
−4 −6 10 10 Br (µ−>3 e) Br (µ−>3 e) −6 Br (τ−>3 e) Br (τ−>3 e) 10 −8 10 Br (τ−>3 µ) Br (τ−>3 µ) −8 10 −10 10 Branching ratio Branching ratio −10 10 −12 10 −12 10 −14 10 −14 10 −16 −16 10 10 −18 −18 10 10 0 100 200 300 400 500 600 700 0 20 40 60 80 100 M (TeV) M (TeV) Figure 3.1: Branching ratios Br(µ → 3e), Br(τ → 3e), and Br(τ → 3µ) as functions of the ` new neutral gauge boson mass mz0 ≡ M , respectively. The left-panel is created θ12 = π/3, ` ` θ13 = π/6, θ23 = π/4, v`a δ ` = 0, whereas the right-panel is produced according to sin θ12 ` = ` ` 0.9936, sin θ13 = 0.9953, sin θ23 = 0.2324, and δ ` = 1.10π. 2 gs2 1 − 2 √ W (η q q¯γµ PR q) (η q q¯γµ PR q) . (3.37) mZ 0 3cW 1 + 2c2W The first two terms (3.28) and (3.29) provide charged lepton flavor violating processes like µ → 3e, τ → 3e, τ → 3µ, τ → 2eµ, τ → 2µe, and µ − e. The next four tems (3.30), (3.31), (3.32), and (3.33) present wrong muon and tau decays as well as the nonstandard neutrino interactions that concern both c´ontrains from oscillation and non-oscillation experiments. The last four terms (3.34), (3.35), (3.36), and (3.37) describe semileptonic conversion in nuclei as well as the signals for new physics (dilepton, diject,...) at low energy such as the Tevatron. 3.3. Phenomenology 3.3.1. Leptonic three-body decays a. τ + → µ+ µ+ µ− , τ + → e+ e+ e− , µ+ → e+ e+ e− From the graph 3.1, we obtain the lower limit mZ 0 ≥ 3.8, 20.6, 36.5 TeV. b. τ + → µ+ e+ e− , τ + → e+ µ+ µ− From the graph 3.2, we obtain the lower limit mZ 0 ≥ 65.3 GeV. c. τ + → µ+ µ+ e− , τ + → e+ e+ µ− The current experimental constraints for the branching ratio of following channel τ + → µ+ µ+ e− and τ + → e+ e+ µ− are very small, so the lower bound of new gauge boson mass mz0 18