MINISTRY OF EDUCATION VIETNAM ACADEMY AND TRAINING OF SCIENCE AND TECHNOLOGY GRADUATE UNIVERSITY OF SCIENCE AND TECHNOLOGY ——————————— NGUYEN CHI THAO THE ELECTROWEAK PHASE TRANSITION IN THE ZEE-BABU AND SU (3)C ⊗ SU (3)L ⊗ U (1)X ⊗ U (1)N GAUGE MODELS Major: Theoretical and Mathematical Physics Code: 44 01 03 SUMMARY OF PHYSICS DOCTORAL THESIS HA NOI- 2019 The thesis is completed at: GRADUATE UNIVERSITY OF SCIENCE AND TECHNOLOGY VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY Supervisors: Professor Dr Hoang Ngoc Long Assoc Prof Dr Phung Van Dong Reviewer 1: Reviewer 2: Reviewer 3: The thesis will be defended in front of the institute doctoral thesis Assessment Council, held at the Graduate University of Science and Technology - Vietnam Academy of Science and Technology at o’clock, on day month year 2019 The thesis can be found at: - The library of Graduate University of Science and Technology - National Library of Vietnam INTRODUCTION The imperative of the thesis In physics baryon asymmetry is also known as material asymmetry This issue is currently an interesting problem Today, to explain baryon asymmetry, scientists use two mechanisms, Leptongenesis and Baryogenesis A model having Baryogenesis must be satisfied the three conditions of A.Sakharov [2] The standard model (SM) has been successful in explaining experimental results [4] Source of CP violation in SM is smaller Baryon asymmetry of Universe (BAU) and there is no strong phaseone transition with the Higgs mass mH = 125 GeV In other words, SM is not enough mH = 125 GeV the first order phase transition [6-8] Research targets of the thesis We want to analyze the baryon asymmetry problem and determine the contribution role of newl particles in some extended standard models We have done the thesis ”The electroweak phase transition in the Zee-Babu and SU (3)C⊗SU (3)L⊗U (1)⊗U (1)N gauge models” The main research contents of the thesis In addition to the introduction and conclusion of the thesis, there are three chapters: Chapter 1, investigation of weak phase transition in SM Chapter 2, investigating the weak electric phase transition in the Landau gauge and ξ gauge in the Zee-Babu model Chapter 3, multiperiod structure of electroweak phase transition in the 3-3-1-1 model Chapter OVERVIEW 1.1 The effective potential with the contribution of the scalar field The effective potential includes both thermal and quantum contributions Vef f = V + ¯ ¯ m2φ (χ) m4φ (χ) T4 mφ ln + F− ( ), 2 64π µ 4π T (1.1) in which F− ( mφ )= T −32m3 πT + 16m2 π T + 9m4 + 6m4 ln ab T m2 (1.2) 96T where m ≡ mφ ; ln[ab ] = ln[4π] − 2C ≈ 3.91 1.2 The effective potential with the contribution of the complex scalar and the gauge boson field We have a general effective formula Vef f = V (χ) ¯ +n m4φ (χ) ¯ m2φ (χ) ¯ T4 mφ ln + F− ( ) , 64π µ2 4π T in which n is the degrees of freedom of these two fields (1.3) 1.3 The effective potential with the contribution of the fermion field 1.3 The effective potential with the contribution of the fermion field We have the effective potential with the contribution of the fermion field Vef f = V (χ) ¯ + 12 m4φ (χ) ¯ m2φ (χ) ¯ T4 mφ + F+ ( ) , ln 2 64π µ 4π T (1.4) in which ln[af ] = ln[π] − 2C ≈ 1.14 1.4 Effective potential in the SM The effective potential in the standard model is: m2W m2z 4 3m ln + 6m ln z W 64π µ2 µ2 m2 m2t (1.5) − 12m ln +m4H ln H t µ2 µ2 T4 mz mW mH mt + 3F− ( ) + 6F− ( ) + F− ( ) + 12F+ ( ) 4π T T T T In Eq (1.5) we only consider the contribution of the t quark and Higgs boson Vef f = V( χ) ¯ + 1.5 The electroweak phase transition in the SM Electroweak phase transition is the transition from a symmetrical phase to an assymmetrical phase, the result of this process is the particle mass generation The essence of this phase transition is the change of VEV of the Higgs field from zero VEV to non-zero VEV The effective potential in SM is determined as the following Vc (φc , T ) = Λ + D T − T02 φ2c − ET φ3c + λ(T ) φc (1.6) We define the phase transition strength S S= φmin(c) 2E = Tc λ(Tc ) (1.7) We obtain a phase transition graph of S in SM as shown in the figure 1.1 The EWPT is the first order transition when mH ≤ 47.3 GeV, this contradicts the experimental Higgs mass of mH 125.5 GeV Therefore, explaining the baryon asymmetry we need to examine the baryon asymmetry in the beyond SM 1.6 Conclusion S 1.8 1.6 1.4 1.2 1.0 m_H 25 30 35 40 45 50 Figure 1.1: The dashed line of S = 2E/λTc = 1, the solid line : 2E/λTc = 1.5 1.6 Conclusion The SM cannot explain the baryon asymmetry Chapter ELECTROWEAK PHASE TRANSITION IN THE ZEE-BABU MODEL 2.1 The mass of particles in the Zee-Babu model The masses of h± and k ±± are given by m2h± = p2 v02 + u21 , m2k±± = q v02 + u22 (2.1) Diagonalizing matrices in the kinetic components of the Higgs potential, we obtain m2H (v0 ) = −µ2 + 3λv02 , mZ (v0 ) = 41 (g + g )v02 = a2 v02 , m2G (v0 ) = −µ2 + λv02 , m2W (v0 ) = 41 g v02 = b2 v02 2.2 (2.2) Effective potential in the Zee-Babu model 2.2.1 Effective potential with Landau gauge Vef f (v) = V0 (v) + 64π + 64π + 3T 4π F− ( + T4 4π 2F− ( m4Z (v)ln 2m4h± (v)ln m2Z (v) m2 (v) m2 (v) + 2m4W (v)ln W − 4m4t (v)ln t 2 Q Q Q m2h± (v) m2k±± (v) m2 (v) + 2m (v)ln + m4H (v)ln H ±± k 2 Q Q Q mZ (v) mW (v) mt (v) ) + F− ( ) + 4F+ ( ) T T T mh± (v) m ±± (v) mH (v) ) + 2F− ( k ) + F− ( ) T T T 2.2 Effective potential in the Zee-Babu model where vρ is a variable changing with temperature, and at T = 0, vρ ≡ v0 = 246 GeV Here mφ T F± mφ T αJ (α, 0)dα, ∓ = ∞ J∓ (α, 0) = α 2.2.2 (x2 − α2 ) dx ex ∓ Effective potential with ξ gauge We are known that in high levels, the contribution of Goldstone boson cannot be ignored Therefore, we must consider an effective potential in arbitrary ξ gauge, V1T =0 (v) = m2 ± m2 (m2H )2 ln( H ) + (m2h± )2 ln( h2 ) 2 4(4π) Q 4(4π) Q + m2k±± 2×1 m2G + ξm2W 2 2 ln( (m ) + (m + ξm ) ln( ) ±± ) G W k 4(4π)2 Q2 4(4π)2 Q2 + m2 + ξm2 2×3 m2 (m2G + ξm2Z )2 ln( G Z ) + (m2W )2 ln( W ) 2 4(4π) Q 4(4π) Q2 + 2×1 ξm2W m2 2 2 Z (m ) ln( ) − (ξm ) ln( ) Z W Q 4(4π)2 4(4π)2 Q2 − ξm2Z 2 (ξm ) ln( ) , Z 4(4π)2 Q2 (2.3) and V1T =0 (v, T ) = + T4 m2H J B 2π T2 + JB T4 m2G + ξm2W 2×JB 2π T2 3T m2W + 2×J B 2π T2 − m2h± T2 + JB m2Z T4 + JB T4 ξm2W 2×JB 2π T2 + JB m2k±± T2 + ×JB ξm2Z T2 m2G + ξm2Z T2 + JB m2γ T2 + JB ξm2γ T2 in which JB± m2φ T2 m2φ T αJ (α, 0)dα ∓ = + ×JB , m2t T2 (2.4) 2.3 Electroweak phase transition in Landau gauge 2.3 Electroweak phase transition in Landau gauge The quartic expression in v Vef f (v) = D(T − T02 )v − ET |v|3 + λT v , (2.5) Tc critical temperature and phase transition strength are Tc = 2E T0 vc = ,S= T λ c Tc − E /DλTc (2.6) (v) are The minimum conditions for Vef f Vef f (v0 ) (v) ∂ Vef f ∂v v=v0 = 0, = (v) ∂Vef f ∂v m2H (v) v=v0 = 0, (2.7) v=v0 = 125 GeV To have a first-order phase transition, we require that the strength is larger or equal to the unit (S ≥ 1) In Fig 2.1, we have plotted the transition strength S as a function of the new charged scalars: mh± and mk±± As shown in Fig 2.1, for mh± and mk±± being in the − 350 GeV range, respectively, the transition strength is in the range ≤ S < 2.4 We see that the contribution of h± and k ±± are the same The larger mass of h± and k ±± , the larger cubic term (E) in the effective potential but the strength of phase transition cannot be strong Because the value of λ also increases, so there is a tension between E and λ to make the first order phase transition In addition when the masses of charged Higgs bosons are too large, T0 , λ will be unknown or S −→ ∞ 500 mh GeV 400 300 200 100 0 100 200 300 mk 400 500 GeV Figure 2.1: When the solid contour of S = 2E/λTc = 1, the dashed contour: 2E/λTc = 1.5, the dotted contour: 2E/λTc = 2, the dotted-dashed contour: 2E/λTc = 2.4, even and nosmooth contours: S −→ ∞ 2.4 Electroweak phase transition in ξ gauge 2.4 Electroweak phase transition in ξ gauge The high-temperature expansions of the potential in Eq.(2.3) and in Eq.(2.4) can be rewritten in a like-quartic expression in v v V = (D1 + D2 + D3 + D4 + B2 ) v + B1 v + Λv + f (T, u1 , u2 , µ, ξ), (2.8) in which f (T, u1 , u2 , µ, ξ, v) = C1 + C2 , m2 +ξm2 (2.9) m2 +ξm2 Expanding functions JB G T W and JB G T Z in Eq (2.4), we will obtain the term of mixing between ξ and v in B1 and B2 m2 +ξm2 m2 +ξm2 Therefore JB G T W and JB G T Z or B1 and B2 contain a part of daisy diagram contributions mentioned in Ref [22] The other part of ring-loop distribution comes to damping effect On the other hand, we see that the ring loop distribution still is very small, it was approximated g T /m2 (g is the coupling constant of SU (2), m is mass of boson), m ∼ 100 GeV, g ∼ 10−1 so g /m2 ∼ 10−5 If we add this distribution to the effective potential, the D1 term will give a small change only Therefore, this distribution does not change the strength of EWPT or, in other words, it is not the origin of EWPT The potential in Eq.(2.8) is not a quartic expression because B2 , D3 , D4 and f (T, u1 , u2 , µ, ξ, v) depend on v, ξ and T It has seven variables such as u1 , u2 , p, q, µ, λ and ξ Therefore, the shape of potential is distorted by u1 , u2 , p, q, ξ but not so much If Goldstone bosons are neglected and the gauge parameter is vanished (ξ = 0), it will be reduced to Eq.(2.5) in the Landau gauge The minimum conditions for Eq.(2.8) are still like Eq.(2.7) but for this case, it holds: m2H0 = −µ2 + 3λv02 = 1252 GeV There are many variables in our problem and some of them, for example, u1 , u2 , p, q and µ play the same role They are components in the mass of particles It is emphasized that ξ and λ are two important variables and have different roles Therefore, in order to reduce number of variables, we have to approximate values of variables, but must not lose the generality of the problem 2.4.1 The case of small contribution of Goldstone boson When the mass of Goldstone boson is small, it means that µ2 ≈ λv02 and taking into account mH0 = 125 GeV, we obtain λ = 0.1297 We conduct a method yielding an effective potential as a quartic expression in v through three steps Chapter MULTI-PERIOD STRUCTURE OF ELECTROWEAK PHASE TRANSITION IN THE 3-3-1-1 MODEL 3.1 3.1.1 Brief review of the 3-3-1-1 model The mass of the quarks - The mass of top quarks and bottom quarks is as follows: ht u hb v mt = √ , mb = √ , 2 - The mass of the exotic quarks are determined as ω mU = √ hU ; 3.1.2 ω mD1 = √ hD 11 ; ω mD2 = √ hD 22 The mass of the Higgs bosons The mass terms of charged Higgs bosons are given by m2H1 = u2 + v ω2 + v2 λ8 ; m2H2 = λ7 2 The mass of neutral Higgs bosons is presented in Table 3.1 (3.1) 3.1 Brief review of the 3-3-1-1 model 12 Table 3.1: The neutral Higgs boson masses Neutral Higgs boson Squared mass 3.1.3 S4 2λΛ2 Aη λ9 ω 2 Aχ λ9 u2 Sη 2λ3 u2 S χ 2λ2 ω Sρ 2λ1 v2 H3 λ9 (u2 +ω ) Gauge boson sector Because of the constraints u, v ω, we have mW mX mY The W boson is identified as the SM W boson So we have: u2 + v = (246 GeV)2 Table 3.2: The mass of charged gauge bosons Gauge boson Squared mass W + v2) g2 (u Y + v2) g2 (ω X + u2 ) g2 (ω From the above analysis, the phenomenological aspects of the 3-3-1-1 model can be divided into two pictures corresponding to different domain values of VEVs Picture (i): Λ ∼ ω v∼u We obtain the masses of neutral gauge bosons as follows g (u2 + v ) , 4c2W m2Z (3.2) g2 (3 + t2X )ω + 4t2N (ω + 9Λ2 ) 18 m2Z1 + ((3 + t2X )ω − 4t2N (ω + 9Λ2 ))2 + 16(3 + t2X )t2N ω , (3.3) g2 (3 + t2X )ω + 4t2N (ω + 9Λ2 ) 18 m2Z2 − ((3 + t2X )ω − 4t2N (ω + 9Λ2 ))2 + 16(3 + t2X )t2N ω (3.4) From the experimental data ∆ρ < 0.0007 ones√get u/ω < 0.0544 or ω > 3.198 TeV [70] (provided that u = 246/ GeV as mentioned) Therefore, the value of ω results in the TeV scale as expected Picture (ii): Λ ω v∼u If we assume Λ ω u ∼ v, three gauge bosons are derived as [5, 71, 72, 76]: m2Z g (u2 + v ) ; m2Z1 4c2W 4g t22 Λ2 ; m2Z2 g c2W ω (3 − 4s2W ) (3.5) The W ± boson and Z boson are recognized as two famous gauge bosons in the SM 3.2 Effective potential in the model 3-3-1-1 3.2 13 Effective potential in the model 3-3-1-1 Hence the Higgs of the model, we obtain the tree-level po- tential V0 = λφ4Λ λ2 φ4ω φ µ2 λ3 φ4u + λ11 φ2Λ φ2ω + + Λ + µ22 φ2ω + 4 2 1 2 1 2 2 2 + λ12 φΛ φu + λ6 φu φω + µ3 φu + λ5 φu φv 4 λ1 φ4v 1 2 2 2 + + λ10 φΛ φv + λ4 φv φω + µ1 φv 4 (3.6) Here V0 has quartic form as in the SM, but it depends on four variables φΛ , φω , φu , φv , and has the mixing terms between them However, developing the Higgs potential in this model, we obtain four minimum equations Therefore, we can transform the mixing between four variables to the form depending only on φΛ , φω , φu and φv Furthermore, importantly, there are the mixings of VEVs because of the unwanted terms such as λ4 (ρ† ρ)(χ† χ), λ5 (ρ† ρ)(η † η), λ6 (χ† χ)(η † η), λ7 (ρ† χ)(χ† ρ), λ8 (ρ† η)(η † ρ), λ9 (χ† η)(η † χ), λ10 (φ† φ)(ρ† ρ), λ11 (φ† φ)(χ† χ) and λ12 (φ† φ)(η † η) in the Higgs potential To satisfy the generation of inflation with φ-inflaton [5,76], the values λ10,11,12 can be small, is about 10−10 − 10−6 Thus, λ4,5,6,7,8,9 must be also small to make the corrections of high order interactions of the Higgs will not be divergent In general, if we did not neglect these mixings, V0 will have additional components Λv, Λω, ωv, uv Considering at the temperature T , for instance, a toy effective potential given by Vef f (v) = λv − Ev + Dv + λk ω v + λj Λ2 v + u2 v ≈ λv − Ev + Dv + λi (ω + Λ2 + u2 )v (3.7) The slices of the effective potential in (3.7) at ω + Λ2 + u2 = TeV as a function of v for some values of λi is plotted in 3.1 From 3.1, we see that at arbitrary temperature T when λi , i = 4, , increases, the second minimum of the effective potential fades For a first order phase transition, the value of λi is not too large, so that the potential still has two minima We observe that if λi is enough small to have a second minimum, at arbitrary temperature, the shape of the effective potential remains the same in the absence of λi Therefore, we have one more reason to assume that λi must be small and this mixing can be neglected Hence, we can write V0 (φΛ , φω , φu , φv ) = V0 (φΛ ) + V0 (φω ) + V0 (φu ) + V0 (φv ) and ignore the mixing of different VEVs, otherwise our phase transitions will be very complex or distorted 3.3 Electroweak phase transition without neutral fermion 14 Veff TeV4 0.15 0.10 Λi 0.05 0.06 Λi 0.03 Λi 0.2 0.4 0.6 0.8 1.0 1.2 1.4 v TeV Figure 3.1: The contours of the effective potential in (3.7) as a function of v for some values of λi as λ = 0.3, D = 0.3, E = 0.6, Λ2 + ω + v = TeV2 From the mass spectra, we can split the masses of particles into four parts as follows m2 (φΛ , φω , φu , φv ) = m2 (φΛ ) + m2 (φω ) + m2 (φu ) + m2 (φv ) (3.8) Taking into account Eqs (3.6) and(3.8), we can also split the effective potential into four parts Vef f (φΛ , φω , φu , φv ) = Vef f (φΛ ) + Vef f (φω ) + Vef f (φu ) + Vef f (φv ) We assume φΛ ≈ φω , φu ≈ φv over space-times Then, the effective potential becomes Vef f (φΛ , φω , φu , φv ) = Vef f (φω ) + Vef f (φu ) From the mass spectra, it follows that the squared masses of gauge and Higgs bosons are split into three separated parts corresponding to three SSB stages 3.3 3.3.1 Electroweak phase transition without neutral fermion Two periods EWPT in picture (i) Phase transition SU (3) → SU (2) This phase transition involves exotic quarks, heavy bosons, but excludes the SM particles As a consequence, the effective potential of the EWPT SU (3) → SU (2) is Vef f (φω ) Vef f (φω ) = Dω (T − T0ω )φ2ω − Eω T φ3ω + T0ω ≡− Fω Dω λω (T ) φω , (3.9) (3.10) 3.3 Electroweak phase transition without neutral fermion 15 3500 3000 S=1 mH3 GeV 2500 2000 1500 S=2 S=3 1000 S 500 0 1000 2000 mexotic 3000 4000 5000 quark Charged Higgs 6000 7000 GeV Figure 3.2: The mass area corresponds to Sω > The values of Vef f (φω ) at the two minima become equal at the critical temperature and the phase transition strength are Tcω = 2Eω T0ω , Sω = λ Tcω − Eω /Dω λTcω There are nine variables: the masses of U, D1 , D2 , H2 , H3 and Aη , Sχ , S4 , Z1 However, for simplicity, we assume mU = mD1 = mD2 = mH2 ≡ O, mAη = mSχ = mH3 = mS4 ≡ P Consequently, the critical temperature and the phase transition strength are the function of O and P ; therefore we can rewrite the phase transition strength as follows Sω = 2Eω ≡ Sω (O, P, Sω ) λTcω (3.11) In Figs 3.2 and 3.3, we have plotted the relation between masses of the charged particles O and neutral particles P with some values of the phase transition strength at ω = TeV 2500 S=1 mH3 GeV 2000 S=2 1500 S=3 1000 500 0 1000 2000 3000 4000 5000 6000 7000 mH1 GeV Figure 3.3: The mass area corresponds to Sω > with real Tc condition The gaps on the lines (S = 1, 2, 3) correspond to values making Tc to be complex The mass region of particles is the largest at Sω = 1, the 3.3 Electroweak phase transition without neutral fermion 16 mass region of charged particles and neutral particles are ≤ mExoticQuark/ChargedHiggsboson ≤ 7000GeV , ≤ mH3 ≤ 2600 GeV From Eq (3.11) the maximum of Sω is around 70 Phase transition SU (2) → U (1) √ In this period, the symmetry breaking scale equals to u = 246/ and the masses of the SM particles and the masses of X, Y, H1 , H2 , H3 , Aχ , Sη are generated There are six variables, the masses of bosons H1 , H2 , Aχ , Aη , H3 , Sρ For simplicity, we assume mH1 = mH2 ≡ K, mAχ = mSη = mH3 ≡ L The effective potential of EWPT SU (2) → U (1) is given by Vef f (φu ) = λu (T ) φu − Eu T φ3u + Du T φ2u + Fu φ2u (3.12) The minimum conditions are Vef f (0) = ∂Vef f (φu ) ∂φu = 0; u ∂ Vef f (φu ) ∂φ2u = m2Aχ +m2H3 +m2Sη +m2Sρ , u (3.13) In Fig 3.5 we have plotted the relation between masses of the charged particles K and neutral particles L with some values of the phase transition strength 600 mH3 GeV S=1 S 500 400 S=1.2 300 S 200 S=2 100 S=3 200 400 600 800 1000 1200 m H1 GeV Figure 3.4: The strength S = 2Eu λTc However, we can fit the mass of heavy particle one again when considering the condition of Tc to be real, so that the maximum of strength is reduced from to 2.12 With the mass region of neutral and charged particles given in Table 3.3 the maximum phase transition strength is 2.12 This is larger than but the EWPT is not strong 3.3 Electroweak phase transition without neutral fermion 17 200 S=1.2 150 mH3 GeV S=1 S=1.3 100 50 S=2.1 200 250 300 350 400 450 mH1 GeV Figure 3.5: The strength EWPT S = 2Eu with Tc must be real λTc Table 3.3: Mass limits of particles with Tc > Strength S 1.0-2.12 3.3.2 K[GeV ] 195 ≤ K ≤ 484.5 L[GeV ] ≤ L ≤ 209.8 Three period EWPT in picture (ii) - The first process is SU (3)L ⊗ U (1)X ⊗ U (1)N → SU (3)L ⊗ U (1)X - The second one is SU (3)L ⊗U (1)X → SU (2)L ⊗U (1)X The third process is SU (2)L → U (1)Q The third process is like SU (2) → U (1) in the picture (i) The first process is a transition of the symmetry breaking of U (1)N group It generates mass for Z1 through Λ or Higgs boson S4 The third process is like the SU (2) → U (1) in picture (i) but it does not involve Z1 and S4 The second process has the effective potential is like Eq (3.9) 1000 S=1 mH3 GeV 800 600 S=2 S=3 400 200 S 0 1000 2000 mExotic Quark 3000 Charged Higgs Figure 3.6: The strength EWPT Sω = 4000 GeV 2Eu with ω = TeV λTc 3.4 The role of neutral fermions in EWPT 18 When we import real Tc , the mass region of charged and neutral particles are ≤ mExoticquark/ChargedHiggsboson ≤ 4000 GeV , ≤ mH3 ≤ 1000 GeV The mass region of charged bosons is narrower than that in the section 3.2 From Eq (3.11), the maximum of S has been estimated to be around 100 3.4 The role of neutral fermions in EWPT In the SU (3) → SU (2) if we add the contribution of neutral fermions, then the maximum of S would decrease However, the neutral fermions not lose the first-orde EWPT as shown in Table 3.4 Table 3.4: Values of the maximum of EWPT strength with ω = TeV Period Picture SU (3) → SU (2) SU (3) → SU (2) (i) (ii) mZ2 [TeV] 2.386 2.254 mN −R [TeV] SM ax without NR SM ax withNR 2.227 1.986 70 100 50 30 Looking at the Table 3.4, the following remarks are in order: In case of the neutral fermion absence In the picture (i), if Z1 boson is involved in the SU (3) → SU (2) EWPT; the contribution of Z1 makes increasing E and λ, but λ increases stronger than E 2E The strength S = gets the value equals 70 For the picture (ii), λTc the mentioned value equals 100 In case of the neutral fermion existence When the neutral fermions are involved in both pictures, Smax in picture (ii) decreases faster than Smax in picture (i) The strength gets values equal to 50 and 30 for the picture (i) and (ii), respectively If the neutral fermions follow the Fermi-Dirac distribution (i.e., they act as a real fermion but without lepton number), they increase the value of the λ and D parameters Thus, they reduce the E value of strength EWPT S, because S = and E not depend 2λTc on the neutral fermions This suggests that DM candidates are neutral fermions (or fermions in general) which reduce the maximum value of the EWPT strength However, the EWPT process depends on bosons and fermions The boson gives a positive contribution (obey the Bose-Einstein distribution) but the fermion gives a negative contribution (obey the Fermi-Dirac distribution) In order to have the first order transition, 3.5 Conclusion 19 the symmetry breaking process must generate mass for more bosons than fermions In addition, in this model, the neutral fermion mass is generated from an effective operator This operator which demonstrates an interaction between neutral fermions and two Higgs fields The above neutral fermion is very different from usual fermions The M parameter has an energy dimension, and it may be an unknown dark interaction Thus, the neutral fermions only are effective fermions, according to the Fermi-Dirac distribution, but their statistical nature needs to be further analyzed with other data 3.5 Conclusion In the model under consideration, the EWPT consists of two pictures The first picture containing two periods of EWPT, has a transition SU (3) → SU (2) at TeV scale and another is SU (2) → U (1) transition which is the like-standard model EWPT The second picture is an EWPT structure containing three periods, in which two first periods are similar to those of the first picture and another one is the symmetry breaking process of U (1)N subgroup The EWPT is the first order phase transition if new bosons with mass within range of some TeVs The maximum strength of the SU (2) → U (1) phase transition is equal to 2.12 so the EWPT is not strong We have focused on the neutral fermions without lepton number being candidates for DM and obey the Fermi-Dirac distribution, and have shown that the mentioned fermions can be a negative trigger for EWPT Furthermore, in order to be the strong first-order EWPT at TeV scale, the symmetry breaking processes must produce more bosons than fermions or the mass of bosons must be much larger than that of fermions It is known that the mass of Goldstone boson is very small [46] and the physical quantities are gauge indepen- dent so the critical temperature and the strength is gauge independent [44-46] Consequently, the survey of effective potential in Landau gauge is also sufficient, or other word speaking, it is just consider in determined gauge Thus, it is a reason why the Landau gauge is used in this work In this chapter, the structure of EWPT in the 3-3-1-1 model with the effective potential at finite temperature has been drawn at the 1-loop level; and this potential has two or three phases We have analyzed the processes which generate the masses for all gauge bosons inside the covariant derivatives After diagonalization, the masses of gauge bosons not have mixing among VEVs Therefore, the EWPT stages are independent of each other [62] In conclusion, the model has many bosons which will be good triggers for first-order EWPT The situation is that as less heavy 3.5 Conclusion 20 fermion as the result will be better However, strength of EWPT can be reduced by many bosons (such as Z, Z1 , Z2 in the 3-3-1-1 model) The new scalar particles playing a role in generation mass for exotic particles, increase the value of EWPT strength Because these scalar fields follow the Bose-Einstein distribution, so that they contribute positively to the effective potential With the help of such particles, the strength of phase transition will be strong As mentioned above, their masses depend just on one VEV, so they only participate in one phase transition Moreover, among the neutral fermions, they may be candidates for DM From the point of view of the early universe, the above particles can be an inflaton or some product of the inflaton decay CONCLUSION From the investigate content, we obtained the following results: Investigation of weak phase transition in model Zee-Babu Considering the Landau gauge, this model has phase transition strength is in the range ≤ S < 2.12, due to the contribution of two mh± and mk±± particles Their mass ranges in the range of − 350 GeV - Considering the ξ gauge, the phase transition strength is in the range ≤ S < 4.15, more strong than the Landau gauge Thus, the phase transition strength will increase when the contribution of gauge ξ However, the ξ gauge is not the cause of the EWPT This leads to the fact that the calculation of EWPT in Landau gauge is enough cases We examined the EWPT in the 3-3-1-1 model with two EWPT without neutral fermion We have two pictures in this case - The first picture has two phase transitions Phase transition SU (3) → SU (2) with value 5.856 TeV≤ ω ∼ Λ ≤ 6.654 TeV Considering at ω = TEV, we calculated the phase transition strength in the range GeV< Sω < 70 GeV The mass region of particles is the largest at Sω = 1, the mass region of exotic quarks and neutral Higgs boson mH2 is between and 7000 GeV 246 Phase transition SU (2) → SU (1) at value u = √ TeV The maximum phase transition strength which must be 2.12 Mass limits of particles: mH1 , mH2 in the range 195 GeV ≤ mH1 , mH2 ≤ 484.5 GeV and and the mass of particles: mAχ , mSη , mH3 in the range ≤ mAχ , mSη , mH3 ≤ 209.8 GeV The second picture has three periods Phase transitions occur at Λ ω = TeV and ω u ∼ v values The first process is SU (3)L ⊗ U (1)X ⊗ U (1)N → SU (3)L ⊗ U (1)X The second one is SU (3)L ⊗U (1)X → SU (2)L ⊗U (1)X The third process is SU (2)L → U (1)Q 3.5 Conclusion 22 - The first process is a transition of the symmetry breaking of U (1)N group It generates mass for Z1 through Λ or Higgs boson S4 The third process is like the SU (2) → U (1) in picture (i) but it does not involve Z1 and S4 - The second process is the phase strength S been estimated to be around 100 The mass region of charged particles and neutral particles are ≤ mExoticquark/ChargedHiggsBoson ≤ 4000 GeV , ≤ mH3 ≤ 1000 GeV The role of neutral fermions in EWPT Considering the phase transition SU (3) → SU (2), if we add the contribution of neutral fermions, the maximum of S would decrease However, the neutral fermions not lose the first-orde EWPT In case of the neutral fermion absence, the contribution of Z1 makes increasing E and λ, but λ increases stronger than E In case of the neutral fermion existence, the phase trength decreases If the neutral fermions follow the Fermi-Dirac distribution, they increase the value of the λ and D parameters Thus, they reduce the value of strength EWPT S This suggests that DM candidates are neutral fermions (or fermions in general) which reduce the maximum value of the EWPT strength However, the EWPT process depends on bosons and fermions The boson gives a positive contribution but the fermion gives a negative contribution In order to have the first order transition, the symmetry breaking process must generate mass for more bosons than fermions NEW CONTRIBUTIONS OF THE THESIS We consider the baryogenesis picture in the Zee-Babu model Our analysis shows that electroweak phase transition (EWPT) in the model is a first-order phase transition at the 100 GeV scale, its strength ranges from1 to 4.15 and the masses of charged Higgs bosons are smaller than 300 GeV The EWPT is strengthened by only the new bosons and this strength is enhanced by arbitrary ξ gauge However, the ξ gauge does not break the first-order EWPT or, in other words, the ξ gauge is not the cause of the EWPT This leads to the fact that the calculation of EWPT in Landau gauge is enough; and the latter may provide baryon-number violation ( B-violation) necessary for baryogenesis in the relationship with nonequilibrium physics in the early universe 2.The EWPT is considered in the framework of 3-3-1-1 model The phase structure within three or two periods is approximated for the theory with many vacuum expectation values (VEVs) at TeV and electroweak scales In the mentioned model, there are two pictures - The first picture containing two periods of EWPT, has a transition SU (3) → SU (2) at TeV scale and another is SU (2) → 3.5 Conclusion 23 U (1) transition which is the like-standard model EWPT - The second picture is an EWPT structure containing three periods, in which two first periods are similar to those of the first picture and another one is the symmetry breaking process of U (1)N subgroup - Our study leads to the conclusion that EWPTs are the first order phase transitions when new bosons are triggers and their masses are within range of some TeVs Especially, in two pictures, the maximum strength of the SU (2) → U (1) phase transition is equal to 2.12 so this EWPT is not strong - Moreover, neutral fermions, which are candidates for dark matter and obey the Fermi-Dirac distribution, can be a negative trigger for EWPT However, they not make lose the first-order EWPT at TeV scale - Furthermore, in order to be the strong first-order EWPT at TeV scale, the symmetry breaking processes must produce more bosons than fermions or the mass of bosons must be much larger than that of fermions LIST OF PUBLICATIONS 01 Hoang Ngoc Long, Tran Thanh Lam, Nguyen Chi Thao, Ho Hoang Tinh, Nguyen Thi Xinh, Nguyen Van Xuyen, Phenomenology of the reduced minimal 3-3-1 model, Communications in Physics 23, 1-9 (2013) 02 Hoang Ngoc Long, Duong Van Loi, Nguyen Chi Thao,Thanh Huu Hong Giang, The 3-3-1 model with arbitarily charged leptons, Communications in Physics 26, 219-226 (2016) 03 Vo Quoc Phong, Nguyen Chi Thao, Hoang Ngoc Long, Baryogenesis in the Zee-Babu model with arbitrary ξ gauge, Physical Review D 97, 115008 (2018) 04 P.V.Dong, D.Q Phong, D.V.Soa, Nguyen Chi Thao, The economical 3-3-1 model revisited, Eur Phys J C 78:653 (2018) 05 Vo Quoc Phong, N.T.Tuong, Nguyen Chi Thao, Hoang Ngoc Long, Multi-period structure of electro-weak phase transition in the 3-3-1-1 model, Physical Review D 99, 015035 (2019) In this thesis, I use the third and fifth articles List of publications 25 ... 1.5 1.6 Conclusion The SM cannot explain the baryon asymmetry Chapter ELECTROWEAK PHASE TRANSITION IN THE ZEE- BABU MODEL 2.1 The mass of particles in the Zee- Babu model The masses of h± and k... (3 + t 2X )ω + 4t2N (ω + 9Λ2 ) 18 m2Z1 + ((3 + t 2X )ω − 4t2N (ω + 9Λ2 ))2 + 16(3 + t 2X )t2N ω , (3.3) g2 (3 + t 2X )ω + 4t2N (ω + 9Λ2 ) 18 m2Z2 − ((3 + t 2X )ω − 4t2N (ω + 9Λ2 ))2 + 16(3 + t 2X )t2N... three periods Phase transitions occur at Λ ω = TeV and ω u ∼ v values The first process is SU (3)L ⊗ U (1 )X ⊗ U (1)N → SU (3)L ⊗ U (1 )X The second one is SU (3)L ⊗U (1 )X → SU (2)L ⊗U (1 )X The third