Summary of Physics doctoral thesis: Some new physics effects in the 3 − 2 − 3 − 1 and 3 − 4 − 1 models

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Investigate the scalar section, the gauge section, and currents in the 3 − 2 − 3 − 1 model and the minimal 3 − 4 − 1 model with right-handed neutrino. Identifying particles and interactions of SM as well as predicting new particles and interactions.

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Nội dung Text: Summary of Physics doctoral thesis: Some new physics effects in the 3 − 2 − 3 − 1 and 3 − 4 − 1 models

MINISTRY OF EDUCATION AND VIETNAM ACADEMY TRAINING OF SCIENCE AND TECHNOLOGY GRADUATE UNIVERSITY SCIENCE AND TECHNOLOGY ----------------- DUONG VAN LOI SOME NEW PHYSICS EFFECTS IN THE 3 − 2 − 3 − 1 AND 3 − 4 − 1 MODELS Major: Theoretical and Mathematical Physics Code: 62 44 01 03 SUMMARY OF PHYSICS DOCTORAL THESIS HANOI - 2018
The work has been completed at Graduate University of Science and Technology, Vietnam Academy of Science and Technology Scientific Supervisor: Prof.Dr. Hoang Ngoc Long - Institute of Physics, Graduate University of Science and Technology Referee 1: Prof.Dr. Dang Van Soa - Hanoi Metropolitan University Referee 2: Assoc.Prof.Dr. Phan Hong Lien - Military Technical Academy Referee 3: Dr. Nguyen Huy Thao - Hanoi Pedagogical University 2 This dissertation will be defended in front of the evaluating assembly at academy level, Place of defending: meeting room, Graduate University Science and Technology, Vietnam Academy of Science and Technology. The thesis can be found at - National Library; - Institute of Physics Library.
INTRODUCTION 1. Motivation of thesis topic The Standard Model (SM) is a very good theory for describing three basic types of interactions and this has been empirically tested. However, SM has yet to explain some issues such as the generation number equal to three, the small mass of the neutrino, the existence of dark matter. At the same time, some results in SM related to the ρ-parameter, the neutral meson mass differences, the total decay width of the W boson, ... do not coincide with the experiment, although the difference is very small. Many other characteristics also indicate that SM is an effective theory of a more fundamental theory. Therefore, the development of extended theories to solve existing problems is very natural and necessary. In SM extensions, expanding the gauge symmetry group of the electroweak part is very much in the interest of many scientists. Accordingly, the model based on the SU (3)C ⊗ SU (2)L ⊗ SU (3)R ⊗ U (1)X (3 − 2 − 3 − 1) gauge group has just been proposal. The 3 − 2 − 3 − 1 model can address issues beyond the scope of SM mentioned above. This new gauge symmetry yields that the fermion generation number is 3, and the tree-level flavor-changing neutral currents (FCNCs) arise in both gauge and scalar sectors, which may be the new source for addressing the physics anomalies and others. Also, it can provide the observed neutrino masses, as well as dark matter candidates, automatically. In addition, the extension model based on the SU (3)C ⊗ SU (4)L ⊗ U (1)X (3 − 4 − 1) gauge group is also a natural and logical extension. The 3 − 4 − 1 model can have two scales of breaking at high energies that make it easy to meet the requirements of the experiment. Furthermore, in some 3 − 4 − 1 specific models, the lepton multiplets contain all of the lepton (left-handed, right-handed) of SM and right-handed neutrinos - important components to solve the neutrino mass problem. This is a reasonable arrangement and only 1
available in the 3 − 4 − 1 models. However, the Higgs physics - currently the most important sector - has not received enough attention. For the reasons above, we chose the subject "Some new physics effects in the 3 − 2 − 3 − 1 and 3 − 4 − 1 models". 2. The objectives of the thesis • Investigate the scalar section, the gauge section, and currents in the 3 − 2 − 3 − 1 model and the minimal 3 − 4 − 1 model with right-handed neutrino. Identifying particles and interactions of SM as well as predict- ing new particles and interactions. • Solve the problem of the fermion generation number, neutrino mass. The identification of dark matter candidates in the 3 − 2 − 3 − 1 model. • Investigate some new physics effects and find the constraints for a few parameters in the two models. 3. The main contents of the thesis • Overview of SM and some of SM’s expansions. • Investigating the 3 − 2 − 3 − 1 model with arbitrary electric charges of the new leptons. Find the spectrum of the gauge and scalar, defining the currents. Discussing the problem of the fermion generation number, neutrino mass, and determine dark matter candidates in the model. In- vestigating some new physics effects related to ρ-parameter and FCNCs. • Investigating the 3 − 4 − 1 model with arbitrary electric charges of the new leptons. Considering the conditions for anomaly free, Yukawa in- teractions and fermion mass, and the boson gauge mass. Investigating the minimal 3 − 4 − 1 model with right-handed neutrino. Detailed anal- ysis of the gauge boson, the currents, and more importantly the Higgs. Considering the decay channels of W boson and muon. 2
CHAPTER 1. OVERVIEW 1.1. The Standard Model The gauge symmetry of the model is defined by SU (3)C ⊗ SU (2)L ⊗ U (1)Y (3-2-1), where the last two are the electroweak symmetry. The electric charge operator: Q = T3 + Y /2. The fermion content: ! νiL ψiL = ∼ (1, 2, −1) , eiR ∼ (1, 1, −2), i = 1, 2, 3. eiL ! uiL 1 4 2 QiL = ∼ 3, 2, , uiR ∼ 3, 1, , diR ∼ 3, 1, − . (1.1) diL 3 3 3 To break the gauge symmetry, the scalar multiplets are introduced as ! ! + + ϕ ϕ φ= = v+h+iG ∼ (1, 2, 1). (1.2) ϕ0 √ 2 Z After the SSB, the physical gauge boson was identified, 1 Aµ = sW A3µ + cW Bµ , Zµ = cW A3µ − sW Bµ , Wµ± = √ (A1µ ∓ iA2µ ), 2 gv gv mA = 0, mZ = , mW ± = . (1.3) 2cW 2 m2W The parameter ρ is defined, ρ = m2Z c2W = 1. The Yukawa Lagrangian, − LY = heij ψ¯L i φejR + hdij Q R ¯ iL (iσ2 φ∗ )uj + H.c., ¯ iL φdj + huij Q R (1.4) then the fermions get masses as follows: Meij = heij √v2 , Mdij = hdij √v2 , and Muij = huij √v2 . Diagonalizing this mass matrix, we obtain the physical states and corresponding masses. Other results of the SM: • In SM, the lepton number conservation and it’s correct at every level of perturbation theory. Contemporaneous, the neutrinos are massless par- ticles in SM. But experimentally, the neutrinos have very small masses 3
(non-zero) and have transitions between different generations. This proves that there is a violation of the generation lepton number in the neutral lepton. • The SM contributions to the neutral meson mass differences at the one- loop level did not coincide with the experiment. • In SM, fermion generations perform the same (repeat) under gauge sym- metry. Therefore, SM does not explain why the fermion generation num- ber is 3. • There is no candidate for the dark matter in the SM. • The presented total width of the W boson is calculated at the tree level with electroweak and includes the QCD complementarity over the recent experimental data that is not identical. 1.2. The minimal left-right symmetry model (M3221) The M3221 based on the SU (3)C ⊗ SU (2)L ⊗ SU (2)R ⊗ U (1)B−L gauge group. The left-handed fermions are embedded as SU (2)L doublets and SU (2)R singlets, while the right-handed fermions are SU (2)L singlets and SU (2)R dou- blets. The M3221 usually works with a scalar as a SU (2)L,R bidoublet and two scalars triplet (one left and one right). The M3221 solves the neutrino mass problem well but does not explain is the existence of dark matter. The M3221 has been expanded. The propos- als that extended the gauge group can produce many interesting and reliable results as it is the most natural extension. 1.3. The 3 − 4 − 1 models These models are based on the SU (3)C ⊗ SU (4)L ⊗ U (1)X gauge group. Many of the problems have been explained by 3 − 4 − 1 models such as quan- tized charges, neutrino masses, .... However, the Higgs physics-currently the most important sector-has not received enough attention. Except for the su- persymmetric 3 − 4 − 1 model, the Higgs potential containing a decuplet is presented for the first time in this thesis. 4
CHAPTER 2. PHENOMENOLOGY OF THE 3 − 2 − 3 − 1 MODEL 2.1. The model The electric charge operator: Q = T3L + T3R + βT8R + X. The fermion content:   ! νaR νaL 1 q−1 ψaL = ∼ 1, 2, 1, − , ψaR =  eaR  ∼ 1, 1, 3, , (2.1)   eaL 2 q 3 EaR   ! u 3R u3L 1 q+1 ∼ 3, 2, 1, , Q3R =  3R  ∼ 3, 1, 3,  d Q3L = , (2.2)  d3L 6 q+ 32 3 J3R   ! dαR uαL 1  −u ∗ q QαL = ∼ 3, 2, 1, , QαR =  αR  ∼ 3, 1, 3 , − , (2.3)  dαL 6 −q− 13 3 JαR q q+ 23 2 −q− 13 1 EaL ∼ (1, 1, 1, q), J3L ∼ 3, 1, 1, q + , JαL ∼ 3, 1, 1, −q − , (2.4) 3 3 The scalar multiplets are introduced as ! 0 + −q S11 S12 S13 ∗ 2q + 1 S= − 0 −q−1 ∼ 1, 2, 3 , − , (2.5) S21 S22 S23 6 φ−q   1 −q−1  2q + 1 φ= φ2  ∼ 1, 1, 3, − , (2.6)  0 3 φ3 Ξ− q   0 12 Ξ13 Ξ11 √ 2 √ 2  2(q − 1)  − q−1  Ξ12 Ξ Ξ= Ξ−−  ∼ 1, 1, 6, , (2.7)  23 √ √ 2 22 2 3 q q−1   Ξ13 Ξ23 √ 2 √ 2 Ξ2q 33 5
    ! 0 Λ 0 0 1 u 0 0 1  1  hSi = √ , hφi = √  0 , hΞi = √  0 0 0 . (2.8)   2 0 v 0 2 2 w 0 0 0 The total Lagrangian: L = Lkinetic + LYukawa − Vscalar , q ¯ q ¯ LYukawa = hlab ψ¯aL SψbR + hR ¯c † ˜ ∗ ab ψaR Ξ ψbR + ha3 QaL SQ3R + haβ QaL S QβR + hE ¯ † J ¯ † J ¯ T ab EaL φ ψbR + h33 J3L φ Q3R + hαβ JαL φ QβR + H.c., (2.9) Vscalar = µ2S Tr(S † S) + λ1S [Tr(S † S)]2 + λ2S Tr(S † SS † S) + µ2Ξ Tr(Ξ† Ξ) + λ1Ξ [Tr(Ξ† Ξ)]2 + λ2Ξ Tr(Ξ† ΞΞ† Ξ) + µ2φ φ† φ + λφ (φ† φ)2 + λ1 (φ† S † Sφ)+λ2 Tr(S † SΞΞ† )+λ3 (φ† ΞΞ† φ)+λ4 (φ† φ)Tr(S † S) + λ5 (φ† φ)Tr(Ξ† Ξ)+λ6 Tr(Ξ† Ξ)Tr(S † S)+(f Sφ∗ S +H.c.). (2.10) In (2.10), the f , λ1,2,3 couplings have been imposed for generalization, which were skipped in the previous study. The SU (3)R anomaly cancellation and the QCD asymptotic freedom re- quires the fermion generation number is 3. The VEVs of S generate the Dirac masses for neutrinos, and the VEV of Ξ provides the Majorana masses for right-handed neutrinos. Subsequently, the small neutrino masses are induced via the type I seesaw mechanism. The model can provide the dark matter candidates. For the model with q = 0, the candidates are E 0 or XR 0 0 or some combination of (φ01 , S13 , Ξ013 ). For the model with q = −1, the candidates are YR0 or some combination of (φ02 , S23 0 ). For the model with q = 1, the candidates are only Ξ023 . Particularly, the model contains a residual gauge symmetry W -parity as R-parity, which is stabilizing the dark matter. 2.2. Scalar sector The results obtained: uS1 + vS2 −vS1 + uS2 H1 = √ , H2 = √ , u2 + v 2 u2 + v 2 H3 = cϕ S3 − sϕ S4 , H4 = sϕ S3 + cϕ S4 , λ2 (u2 + v 2 )Λ2 m2H1 = 2(λ1S + λ2S )u2 − λ2S v 2 , m2H2 = , 2(v 2 − u2 ) q 2 2 2 mH3 = λφ w + (λ1Ξ + λ2Ξ )Λ − [(λ1Ξ + λ2Ξ )Λ2 − λφ w2 ]2 + λ25 w2 Λ2 , 6
q m2H4 2 = λφ w + (λ1Ξ + λ2Ξ )Λ + 2 [(λ1Ξ + λ2Ξ )Λ2 − λφ w2 ]2 + λ25 w2 Λ2 . (2.11) vwA1 + uwA2 − uvA3 −uA1 + vA2 A= p , GZ = √ , (u2 + v 2 )w2 + u2 v 2 u2 + v 2 uv 2 A1 + u2 vA2 + w(u2 + v 2 )A3 GZ1 = A4 , GZ10 = p , (u2 + v 2 )(u2 v 2 + w2 u2 + w2 v 2 ) [v 2 w2 + u2 (v 2 + w2 )][2λ2S (u2 − v 2 ) + λ2 Λ2 ] m2A =− . (2.12) 2(u2 − v 2 )w2 2 λ2 (v 2 − u2 ) − 2λ2Ξ Λ2 2 λ3 w2 − λ2 u2 − 2λ2Ξ Λ2 mΞ±± = , mΞ±2q = , 22 2 33 2 λ2 (v 2 − 2u2 ) + λ3 w2 − 4λ2Ξ Λ2 m2Ξ±(q−1) = . (2.13) 23 √ √ 4 ± ± ± 2uΛS 12 + 2vΛS21 + (v 2 − u2 )Ξ± 12 ± ± −vS12 ± + uS21 H5 = p , GW1 = √ , 2(u2 + v 2 )Λ2 + (v 2 − u2 )2 u2 + v 2 ± ± √ 2 ± u(u 2 − v 2 )S 12 + v(u 2 − v 2 )S 21 + 2(u + v 2 )ΛΞ± 12 GW2 = p , 2 2 2 2 2 (u − v ) (u + v ) + 2(u + v ) Λ 2 2 2 2 2 2 2 λ 2 2(u + v )Λ m2H ± = v 2 − u2 + . (2.14) 5 4 v 2 − u2 ±q ±q √ ±q uS 13 − wφ 1 + 2ΛΞ±q 13 GX = √ , 2 u + w + 2Λ 2 2 H6±q = cϕq H60±q − sϕq H70±q , H7±q = sϕq H60±q + cϕq H70±q , λ1 (u2 − v 2 )w2 − λ2 u2 Λ2 λ3 (w2 + 2Λ2 ) m2H ±q ' , m 2 H7±q ' . (2.15) 6 2(u2 − v 2 ) 4 ±(q+1) ±(q+1) ±(q+1) ±(q+1) ±(q+1) −vS23 + wφ2 ±(q+1) wS23 + vφ2 GY = √ , H8 = √ , v 2 + w2 v 2 + w2 (v 2 + w2 )[(u2 − v 2 )(2λ2S u2 − λ1 w2 ) + λ2 u2 Λ2 ] m2H ±(q+1) = − . (2.16) 8 2(u2 − v 2 )w2 2.3. Gauge sector The results obtained: ±q √ ±(q+1) √ XRµ = A4Rµ ± iA5Rµ / 2, YRµ = A6Rµ ± iA7Rµ / 2, ± ± ± ± ± ± W1µ = cξ WLµ − sξ WRµ , W2µ = sξ WLµ + cξ WRµ , 2 gR g2 m2XR = (u2 + w2 + 2Λ2 ), m2YR = R (v 2 + w2 ), 4 4 " # 2 gL 4t2 u2 v 2 m2W1 ' u2 + v 2 − 2 2 R , 4 2tR Λ + (t2R − 1)(u2 + v 2 ) 7
" # g2 4u2 v2 m2W2 ' R u2 + v 2 + 2Λ2 + 2 2 . (2.17) 4 2tR Λ + (t2R − 1)(u2 + v 2 ) ( ) g 2 2 + v2 2 − v 2 )κc t2 (u2 + v 2 )κc u 1 (u W 2 W m2A = 0, m2Z ' L +√ 2 − R 3 , 4 c2W 3[tR + t2X (1 + β 2 )] [t + t2 (1 + β 2 )] 2 2 R X 2 gL n √ m2Z1 ' t2R (w2 + 4Λ2 ) + t2X [β 2 w2 + ( 3 + β)2 Λ2 ] 6 √ q 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 − [tR (w + 4Λ ) + tX (β w + ( 3 + β) Λ )] − 12tR [tR + (1 + β )tR ]w Λ , 2 n gL √ m2Z 0 ' t2R (w2 + 4Λ2 ) + t2X [β 2 w2 + ( 3 + β)2 Λ2 ] 1 6 √ q 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 + [tR (w + 4Λ ) + tX (β w + ( 3 + β) Λ )] − 12tR [tR + (1 + β )tR ]w Λ ,   sW cW q 0 q 0      A3L sW s2 t2 R +t2 X β 2 s sW t2 R +t2 X β 2 c sW A −t W −     tR R cW tR tX cW tR tX cW   A3R   βs2 t2 −βtX s sW t2  Z  ' βsW R c cWq R s cW +βtX c sW . (2.18)   − t cW  Z1   A8R    tR R W tR cW t2 +t2 β2 q tR cW t2 +t2 β2    R X R X B Z10   sW s2 W −βtX c cW −s sW −βtX s cW +c sW −t   q q tX X cW cW t2 +t2 β2 cW t2 +t2 β2 R X R X 0 0 We will use the approximation: Z = Z, ZR = ZR , and ZR = ZR . 2.4. Interactions 2.4.1. Fermion-gauge boson interactions We received four charged currents and the neutral current interactions, −µ gL cξ gR sξ J1W = − √ (¯ νaL γ µ eaL + u ¯aL γ µ daL ) + √ (¯ νaR γ µ eaR + u ¯aR γ µ daR ), 2 2 −µ g s L ξ g c R ξ J2W = − √ (¯ νaL γ µ eaL + u ¯aL γ µ daL ) − √ (¯ νaR γ µ eaR + u ¯aR γ µ daR ), 2 2 −qµ g R ¯ µ ¯ µ ¯ µ JX = − √ (E aR γ νaR − dαR γ JαR + J3R γ u3R ), 2 −(q+1)µ g R ¯ JY = − √ (E µ aR γ eaR + u ¯αR γ µ JαR + J¯3R γ µ d3R ), (2.19) 2 gL ¯ µ Z LN C = −eQ(f )f¯γ µ f Aµ − f γ [gV (f ) − gA Z (f )γ5 ]f Zµ 2cW gL ¯ µ Z1 Z1 gL ¯ µ Z10 Z0 0 − f γ [gV (f ) − gA (f )γ5 ]f Z1µ − f γ [gV (f ) − gA1 (f )γ5 ]f Z1µ , 2cW 2cW where f indicates every fermion. 2.4.2. Scalar–gauge boson interactions These vertex types are listed in Appendix A. 8
f gVZ (f ) Z gA (f ) f gVZ (f ) Z gA (f ) νa 1 2 1 2 ea − 12 + 2s2W − 12 Ea −2s2W q 0 ua 1 2 − 43 s2W 1 2 da − 21 + 23 s2W − 12 Jα 2s2W (q + 13 ) 0 J3 −2s2W (q + 23 ) 0 Table 2.1: The couplings of Z with fermions. 20 000 16 .000 =0 15 000 DΡ L @GeVD 10 000 4 Ε 1 = - 0.001 06 .00 5000 =0 1 0.00 DΡ 01 Ε2 = - Ε2 = Ξ= - 0.0 0.001 0 50 100 150 200 u @GeVD √ Fig 2.1: The viable new physics regime for the case β = −1/ 3. 2.5. New physics effects and constraints 2.5.1. ρ and mixing parameters The new-physics contribution to the ρ-parameter is evaluated as 1 (v 2 − u2 )c3W κ 2 t2R c3W κ 2u2 v 2 ∆ρ ' √ + − . (2.20) 3(u2 + v 2 )[t2R + t2X (1 + β 2 )] [t2R + t2X (1 + β 2 )]3/2 (u2 + v 2 )Λ2 From the global fit, 0.00016 < ∆ρ < 0.00064. We make a contour for ∆ρ √ as in Fig 2.1 for the case β = −1/ 3. For the mixing parameters, we make contours (solid line for 1 , dashed line for 2 , and short dashed line for ξ) for |ξ| = |1,2 | = 10−3 . The available parameter space lies above these third lines. The bounds for Λ is bounded by 6.6 TeV < Λ < 19.4 TeV and u > 210.4. The similar for the case β = 0 : 5.5 TeV < Λ < 16.3 TeV and u > 215. For the 9
√ case β = 1/ 3 : 4.6 TeV < Λ < 13.7 TeV and u > 222.3. 2.5.2. Flavor-changing neutral current Considering the interactions of quarks with scalars, we get tree-level FCNCs due to the contribution of H2 , LH ¯0 d 0 ¯0iL Γuij u0jR H2 + H.c., FCNC = diL Γij djR H2 + u (2.21) 2 √ d u2 + v 2 † ∗ Γij = − 2 (VdL VuL )ik (M U )km (VuR )3m (VdR )3j , √ u u2 + v 2 † ∗ Γuij = 2 (VuL VdL )ik (M D )km (VdR )3m (VuR )3j . (2.22) v Considering the interactions of quarks with gauge bosons, we get tree-level 0 FCNCs due to the contribution of ZR , Z0 Z0 0 0 0 LFCNC R = −ΘijR q¯iR γ µ qjR ZRµ (2.23) Z0 with i 6= j, where q 0 is denoted as either u0 or d0 , and ΘijR is defined as gL 0 q ZR ∗ Θij = √ t2R + β 2 t2X (VqR )3i (VqR )3j . (2.24) 3 The contribution of the new physics to the neutral meson Kaon mass diffier- ence, Z0 ( 2 2 (Θ12R )2 5 (Γd∗ 2 d 2 21 ) (Γ ) m K ∆mK = Re 2 + 2 + 12 3 mZ 0 12 mH2 m2H2 ms + md R " 2 #) d∗ d Γ Γ 1 mK − 212 12 + 2 mK fK . (2.25) mH2 6 ms + md Similarly, for neutral mesons Bd and Bs . The standard model contributions, (∆mK )SM = 0.467 × 10−2 /ps, (∆mBd )SM = 0.528/ps, (∆mBs )SM = 18.3/ps. (2.26) The total contributions, (∆mK,Bd ,Bs )tot = (∆mK,Bd ,Bs )SM + ∆mK,Bd ,Bs . (2.27) The experimental values are (∆mK )Exp = 0.5292 × 10−2 /ps, (∆mBd )Exp = 0.5055/ps, 10
(∆mBs )Exp = 17.757/ps. (2.28) We require the theory to produce the data for the kaon mixing parameter within 30% and 5% for the B-meson mixing parameters, 0.37044 × 10−2 /ps < (∆mK )tot < 0.68796 × 10−2 /ps, (2.29) 0.480225/ps < (∆mBd )tot < 0.530775/ps, (2.30) 16.8692/ps < (∆mBs )tot < 18.6449/ps. (2.31) The Fig 2.4 for M = 5 TeV. The available region for ∆mK is the whole frame. The two separated regions are for ∆mBd . A lower half region is for ∆mBs . Hence, the available parameter space for ∆mK,Bd ,Bs is only the (dark- est) region in the lower left corner of panel. We obtain constraints for the right- handed quark mixing matrix elements as |VuR | < 0.08 and |VdR | < 0.0015. The similar for M = 10 TeV: |VuR | < 0.2 and |VdR | < 0.003. In Fig 2.6, considering VuR = 0.05. The viable parameter space is the (darkest) region bounded in the upper left corner of panel. We obtain M > 2.8 TeV. The similar, M > 5.7 TeV for VuR = 0.1, and M > 8.2 TeV for VuR = 0.15. Note: (VdR )31 = (VdR )32 ≡ VdR , (VdR )233 = 1 − 2VdR 2 , and (VuR )33 ≡ VuR . 2.6. Summary • The model contain suitable spectra in gauge bosons and Higgs bosons, and the correct form of currents. All the SM particles and interactions are consistently recovered. • The model can explain the fermion generation number is 3, the small mass of the neutrino, and the existence of dark matter. • The available region for the new physics scale: M = 5–10 TeV. • For M = 5 TeV: |VuR | < 0.08 and |VdR | < 0.0015. For M = 10 TeV: |VuR | < 0.2 and |VdR | < 0.003. 11
Fig 2.4: The constraints for (VuR , VdR ) coming from the meson mixing parameters ∆mK,Bd ,Bs with respect to the new-physics scale, M = 5 TeV. Fig 2.6: The constraints for (M, VdR ) coming from the meson mixing parameters ∆mK,Bd ,Bs for VuR = 0.05. 12
CHAPTER 3. PHENOMENOLOGY OF THE MINIMAL 3 − 4 − 1 MODEL WITH RIGHT-HANDED NEUTRINO 3.1. The 3 − 4 − 1 model 3.1.1. Anomaly cancellation and fermion content For the class of the SU(3)C ⊗ SU(4)L ⊗ U(1)X (3 − 4 − 1) models the fol- lowing gauge anomalies must vanish: i) [SU (3)C ]2 ⊗ U (1)X , ii) [SU (4)L ]3 , iii) [SU (4)L ]2 ⊗ U (1)X ; iv) [Grav]2 ⊗ U (1)X ; and v) [U (1)X ]3 . Exploit the rela- tion between charge operator and diagonal generators of the gauge symmetry SU (4)L , we proved that the five conditions reduce to two conditions only: [SU (4)L ]3 and [SU (4)L ]2 ⊗ U (1)X . This means that (i) the number of fermion quadruplets is equal to that of fermion antiquadruplets and (ii) the sum over electric charges of all left-handed fermions is zero. 3.1.2. Yukawa couplings and masses for fermions The fermions are arranged as q + q0 − 1 0 faL = (νa , la , Eaq , Ea0q )TL ∼ 1, 4, , a = 1, 2, 3, 4 0 q 0q laR ∼ (1, 1, −1) , EaR ∼ (1, 1, q) , EaR ∼ (1, 1, q 0 ). (3.1) 0 T 5 + 3(q + q ) Q3L = (u3 , d3 , T , T 0 )L ∼ 3, 4, , 12 u3R ∼ (3, 1, 2/3), d3R ∼ (3, 1, −1/3), 0 2 + 3q 2 + 3q TR ∼ 3, 1, , TR0 ∼ 3, 1, . (3.2) 3 3 0 T 1 + 3(q + q ) QαL = (dα , −uα , Dα , Dα0 )L ∼ 3, 4∗ , − , α = 1, 2, 12 uαR ∼ (3, 1, 2/3), dαR ∼ (3, 1, −1/3), 13
0 1 + 3q 0 1 + 3q DαR ∼ 3, 1, − , DαR ∼ 3, 1, − . (3.3) 3 3 For SSB, we need four Higgs quadruplets, namely, T 0 (−q 0 ) (−q 0 −1) (q−q 0 ) q − 3q − 1 Φ1 = Φ1 , Φ1 , Φ1 , Φ01 ∼ 1, 4, , 4 T 0 (−q) (−q−1) 0 (q −q) 1 + 3q − q Φ2 = Φ2 , Φ 2 , Φ02 , Φ2 ∼ 1, 4, − , 4 T 0 (+) (q+1) 0 (q +1) 3 + q + q Φ3 = Φ3 , Φ03 , Φ3 , Φ3 ∼ 1, 4, , 4 T 0 (q) 0 (q ) q + q − 1 Φ4 = Φ04 , Φ− 4 , Φ4 , Φ4 ∼ 1, 4, . (3.4) 4 The Yukawa couplings, 0 ¯ 0q 0 E ¯ l ¯ t ¯ −LYukawa = hE q ab faL Φ1 EbR + hab faL Φ2 EbR + hab faL Φ3 lbR + h Q3L Φ4 u3R 0 ¯ 3L Φ3 d3R + hT Q + hb Q ¯ 3L Φ1 TR0 + hd2 ¯ 3L Φ2 TR + hT Q ¯ † αβ QαL Φ4 dβR 0 ¯ † D2 ¯ † D 2 ¯ † 0 + hu2 αβ QαL Φ3 uβR + hαβ QαL Φ2 DβR + hαβ QαL Φ1 DβR + H.c. (3.5) The fermions get masses as follows: 0 V E ω v t u (mE 0 )ab = hE l ab √ , (mE )ab = hab √ , (ml )ab = hab √ , mu3 = h √ , 2 2 2 2 v ω 0 V u md3 = hb √ , mT = hT √ , mT 0 = hT √ , (md2 )αβ = hd2 αβ √ , 2 2 2 2 v D2 ω D0 2 V (mu2 )αβ = −hu2 αβ √ , (m D2 )αβ = hαβ √ , (m D2 0 )αβ = h αβ √ , (3.6) 2 2 2 V ω √v u where √ 2 , √ 2 , 2, and √ 2 are VEVs of Φ01 , Φ02 , Φ03 , and Φ04 , respectively. 3.1.3. Gauge boson masses The results obtained: g 2 (v 2 + u2 ) g 2 (u2 + ω 2 ) g 2 (v 2 + ω 2 ) m2W = 2 , mW13 = 2 , mW23 = , 4 4 4 g 2 (u2 + V 2 ) g 2 (v 2 + V 2 ) g 2 (ω 2 + V 2 ) m2W14 = , m2W24 = , m2W34 = , 4 4 4 m2A = 0, m2Z = O(m2W ), 2 " 2 2 √ 2 2 # g 3sα V 2 2c s α 43 + s s α 32 (bs c 43 43 t − 1) w m2Z3 ' + , 4 2s243 6s243 s232 14
2 " √ 2 2 # g 3c2α V 2 2 2s s α 43 − c s α 32 (bs c 43 43 t − 1) w m2Z4 ' + . (3.7) 4 2s243 6s243 s232 µ √ W13 ≡ (Aµ1 − iAµ3 )/ 2 , ..., Aµ = sW A3µ + cW c32 A8µ + c43 s32 A15µ + s43 s32 Bµ00 , Zµ ' cW A3µ − sW c32 A8µ + c43 s32 A15µ + s43 s32 Bµ00 , Z3µ ' −s32 cα A8µ + (c43 c32 cα − s43 sα ) A15µ + (s43 c32 cα + c43 sα ) Bµ00 , Z4µ ' s32 sα A8µ −(c43 c32 sα +s43 cα ) A15µ +(c43 cα −s43 c32 sα ) Bµ00 . (3.8) 3.2. The minimal 3 − 4 − 1 model with right-handed neutrino 3.2.1. The model The fermions are arranged as faL = (νa , la , lac , νac )TL ∼ (1, 4, 0) , a = e, µ, τ . (3.9) T Q3L = (u3 , d3 , T , T 0 )L ∼ (3, 4, 2/3), u3R ∼ (3, 1, 2/3), d3R ∼ (3, 1, −1/3), TR ∼ (3, 1, 5/3), TR0 ∼ (3, 1, 2/3). (3.10) T QαL = (dα , −uα , Dα , Dα0 )L ∼ (3, 4∗ , −1/3), uαR ∼ (3, 1, 2/3), dαR ∼ (3, 1, −1/3), 0 DαR ∼ (3, 1, −4/3), DαR ∼ (3, 1, −1/3), α = 1, 2. (3.11) For SSB, we need four Higgs quadruplets, namely, T T χ = χ01 , χ− + 0 2 , χ3 , χ 4 ∼ (1, 4, 0) , φ = φ− 1 , φ2 −− , φ03 , φ− 4 ∼ (1, 4, −1), T T ρ = ρ+ 0 ++ 1 , ρ2 , ρ 3 , ρ+ 4 ∼ (1, 4, 1) , η = η10 , η2− , η3+ , η40 ∼ (1, 4, 0). (3.12) The Yukawa couplings for the quark sector are 0 ¯ 3L ηu3R +hb Q −LqYukawa = ht Q ¯ 3L ρd3R +hT Q ¯ 3L χ TR0 +hd2 ¯ 3L φ TR +hT Q ¯ † αβ QαL η dβR 0 ¯ † D2 ¯ † D 2 ¯ † 0 + hu2 αβ QαL ρ uβR + hαβ QαL φ DβR + hαβ QαL χ DβR + H.c. (3.13) The quarks get masses as follows: u v ω 0 V mu3 = ht √ , md3 = hb √ , mT = hT √ , mT 0 = hT √ , 2 2 2 2 15
u u2 v (md2 )αβ = hd2 αβ √ , (mu2 )αβ = −hαβ √ , 2 2 ω D0 2 V (mD2 )αβ = hD2 αβ √ , (m D2 0 )αβ = h αβ √ . (3.14) 2 2 To produce masses for leptons, we introduce a symmetric decuplet,  √ 0 H1− H2+ H20  2H1 − √ −− 1  H1 2H1 H30 H3−  H=√  + √ ++ +  ∼ (1, 10, 0).  (3.15) 2  H2 H30 2H2 H4 √ 0  H20 H3− H4+ 2H4 The Yukawa interaction for the lepton is given by hlab h √ − −LlYukawa = √ ν¯aL c 0 c + 2νbR H1 + lbR H1 + lbR H2 + νbR H2 0 2 √ c +¯ laL νbR c H1− + 2lbR H1−− + lbR H30 + νbR H3− √ +¯ c laL c νbR H2+ + lbR c H30 + 2lbR H2++ + νbR H4+ c c 0 c − + √ 0 i + ν¯aL νbR H2 + lbR H3 + lbR H4 + 2νbR H4 + H.c. (3.16) v 0 +RH 0 −iIH 0 +RH 0 −iIH 0 Assuming that: H30 = √3 2 3 , H20 = 2 √ 2 2 . The charged hl hl v 0 leptons get mass matrix given by (ml )ab = √ab2 hH30 i = ab2 . The neutrinos hl hl obtain the Dirac mass given by (mν )ab = √ab2 hH20 i = ab 2 . The neutrino Majorana mass will follow from hH10 i and hH40 i. 3.2.2. Gauge sector The results obtained: g2 2 02 g2 2 m2U ±± = 2 (ω + v + 4v ) , mN 0 = 2 (V + u2 + 42 ), 4 4 2 2 g g m2W ± ' (v 2 + u2 + v 002 ), m2K ± ' (V 2 + w2 + v 002 ), 4 4 g2 2 002 g2 2 2 mX ± ' 2 (V + v + v ), mY ± ' 2 (w + u2 + v 002 ). 4 4 2 2 2 002 g (v + u + v ) m2W m2A = 0, m2Z = = , 4c2W c2W g2 p 2 g 2 √ 2 2 2 2 mZ30 = 9sα V + sα − cα 8 + 3t 2 w2 + 2cα s32 + sα u2 24 24 2 # (3t2 + 4)cα s32 √ 2 + sα + √ v2 + 2 2sα − cα s32 v 002 , 2 2 16
g2 2 g 2 √ p 2 m2Z40 = 2 2 9cα V + cα + sα 8 + 3t 2 2 w + cα − 2sα s32 u2 24 24 2 # 2 √ (3t + 4)sα s32 2 + cα − √ v2 + 2 2cα + sα s32 v 002 . (3.17) 2 2 N 0 ≡ W14 0 , U −− ≡ W23 −− , Wµ = cos θ Wµ0 − sin θ Kµ0 , Kµ = sin θ Wµ0 + cos θ Kµ0 , Yµ = cos θ0 Yµ0 − sin θ0 Xµ0 , Xµ = sin θ0 Yµ0 + cos θ0 Xµ0 , Aµ = sW A3µ + cW c32 A8µ + cW s32 Bµ00 , Zµ = cW A3µ − sW c32 A8µ − sW s32 Bµ00 , 0 Z3µ = −s32 cα A8µ − sα A15µ + c32 cα Bµ00 , 0 Z4µ = s32 sα A8µ − cα A15µ − c32 sα Bµ00 . (3.18) U ±± and Y ± are similar to the singly charged gauge bosons in M331, while N 0 and X ± play the similar role in ν331. The heaviest singly charged gauge bosons K ± are the completely new ones that couple with the exotic quarks and right-handed leptons only. In our assignment (and also in Voloshin’s paper), particles belonging to the minimal version are lighter than those in ν331. For the original 3 − 4 − 1 model, the above consequence is the opposite. 3.2.3. Currents P ¯ µ From the Lagrangian Lfermion = i f f γ Dµ f, we obtained: g µ− −LCC = √ JW Wµ+ +JK µ− + µ− + Xµ +JYµ− Yµ+ +JN µ0∗ 0 Nµ + JUµ−− Uµ++ + H.c. , Kµ +JX 2 where µ− JW νaL γ µ laL + u = cθ (¯ ¯3L γ µ d3L − u ¯αL γ µ dαL ) − sθ (−¯ νaR γ µ laR + T¯L γ µ TL0 + D¯ αL 0 γ µ DαL ). (3.19) µ− JK νaR γ µ laR + T¯L γ µ TL0 + D = cθ (−¯ ¯ αL 0 γ µ DαL ) + sθ (¯ νaL γ µ laL + u ¯3L γ µ d3L − u ¯αL γ µ dαL ), µ− JX = cθ0 (¯ c νaL γ µ laL + T¯L0 γ µ d3L − u 0 ¯αL γ µ DαL ) + sθ0 (¯laL c γ µ νaL + T¯L γ µ u3L + d¯αL γ µ DαL ), JYµ− = cθ0 (¯laL c γ µ νaL + T¯L γ µ u3L + d¯αL γ µ DαL ) − sθ0 (¯ c νaL γ µ laL + T¯L0 γ µ d3L − u 0 ¯αL γ µ DαL ), JUµ−− = ¯laL c γ µ laL + T¯L γ µ d3L − u ¯αL γ µ DαL , µ0∗ ¯ αL JN = ν¯aL γ µ νaL c ¯3L γ µ TL0 + D +u 0 γ µ dαL . (3.20) 17
The neutral current interactions, 3 NC µ g4 X i X ¯ µ (V ) −L = eJem Aµ + Zµ {f γ [g (f )iV −g (A) (f )iA γ5 ]f }, (3.21) 2cW i=1 f where √ g0 2 2 sin θW e = g sin θW , t= =p . (3.22) g 2 1 − 4 sin θW 3.2.4. Higgs potential In the limit of lepton number conservation, the potential can then be writ- ten as V (η, ρ, φ, χ, H) = V (η, ρ, φ, χ) + V (H). V (η, ρ, φ, χ) = µ21 η † η + µ22 ρ† ρ + µ23 φ† φ + µ24 χ† χ + λ1 (η † η)2 + λ2 (ρ† ρ)2 + λ3 (φ† φ)2 + λ4 (χ† χ)2 + (η † η)[λ5 (ρ† ρ) + λ6 (φ† φ) + λ7 (χ† χ)] + (ρ† ρ)[λ8 (φ† φ) + λ9 (χ† χ)] + λ09 (φ† φ)(χ† χ) + λ10 (ρ† η)(η † ρ) + λ11 (ρ† φ)(φ† ρ) + λ12 (ρ† χ)(χ† ρ) + λ13 (φ† η)(η † φ) + λ14 (χ† η)(η † χ) + λ15 (χ† φ)(φ† χ) + (f ijkl ηi ρj φk χl + H.c.). (3.23) V (H) = µ25 Tr(H † H) + λ16 Tr[(H † H)2 ] + λ17 [Tr(H † H)]2 + Tr(H † H)[λ18 (η † η) + λ19 (ρ† ρ) + λ20 (φ† φ) + λ21 (χ† χ)] + λ22 (χ† H)(H † χ) + λ23 (η † H)(H † η) + λ24 (ρ† H)(H † ρ) + λ25 (φ† H)(H † φ) + [f4 χ† Hη ∗ + H.c.]. (3.24) The results obtained: √ 0 ±± √ 0 ±± 2v H1 − 2v H2 − vρ±± + wφ±± G±± U = √ 3 2 , 2 2 w + v + 4v 02 λ24 v 2 − λ25 w2 w2 + v 2 2 2 2 fV u mh±± = = −mh±± , mh±± = λ11 − . (3.25) 1 4 2 3 2 wv −v 0 H1± − wφ±1 + uη3 ± −V χ± 0 ± 2 + v H4 + vρ4 ± G± Y = √ , G±X = √ , w2 + u2 + v 02 V 2 + v 2 + v 02 wφ± ± 0 ± 4 − V χ 3 + v H3 −uη2± + vρ± 0 ± 1 + v H2 G±K = √ , G ± W = √ , V 2 + w2 + v 02 u2 + v 2 + v 02 ± ± ± ± ± vη2± + uρ±1 ± uφ± 1 + wη3 ± h1 ≡ H1 , h2 ≡ H2 , h3 = √ , h4 = √ , u2 + v 2 u2 + w 2 18