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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: 9 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: 1. Professor Dr. Hoang Ngoc Long 2. 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 1. 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]. 2. 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” 3. 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 1 OVERVIEW 1.1 The effective potential with the contribution of the scalar field The effective potential includes both thermal and quantum contributions Vef f " # m2φ (χ̄) m4φ (χ̄) T4 mφ ln + 2 F− ( ), =V + 2 2 64π µ 4π T (1.1) in which F− ( mφ )= T −32m3 πT + 16m2 π 2 T 2 + 9m4 + 6m4 ln  ab T 2 m2  (1.2) 96T 4 where m ≡ mφ ; ln[ab ] = 2 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 (χ̄) + n " # ! m4φ (χ̄) m2φ (χ̄) T4 mφ ln + 2 F− ( ) , 64π 2 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 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 (χ̄) + 12 # ! " m4φ (χ̄) m2φ (χ̄) T4 mφ + ) , ln F ( + 64π 2 µ2 4π 2 T (1.4) in which ln[af ] = 2 ln[π] − 2C ≈ 1.14. 1.4 Effective potential in the SM The effective potential in the standard model is:  m2W m2z 1 4 4 3m ln + 6m ln Vef f = V( χ̄) + z W 64π 2 µ2 µ2  m2 m2t 4 (1.5) − 12m ln +m4H ln H t µ2 µ2 T4  mz mW mH mt  + 2 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. 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
λ(T ) 4 φc Vc (φc , T ) = Λ + D T 2 − T02 φ2c − ET φ3c + 4 (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 4 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 2 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 2 v02 + u22 . (2.1) Diagonalizing matrices in the kinetic components of the Higgs potential, we obtain 2 m2H (v0 ) = −µ2 + 3λv02 , mZ (v0 ) = 41 (g 2 + g 02 )v02 = a2 v02 , m2G (v0 ) = −µ2 + λv02 , m2W (v0 ) = 41 g 2 v02 = b2 v02 . 2.2 2.2.1 (2.2) Effective potential in the Zee-Babu model Effective potential with Landau gauge   m2 (v) m2 (v) m2 (v) m4Z (v)ln Z 2 + 2m4W (v)ln W 2 − 4m4t (v)ln t 2 Q Q Q   2 2 2 m (v) m (v) mH (v) 1 4 4 4 h± k±± + 2m (v)ln + 2m (v)ln + m (v)ln ± ±± H h k 64π 2 Q2 Q2 Q2   4 mZ (v) mW (v) mt (v) 3T + F− ( ) + F− ( ) + 4F+ ( ) 4π 2 T T T   m ± (v) m ±± (v) mH (v) T4 + 2 2F− ( h ) + 2F− ( k ) + F− ( ) 4π T T T Vef f (v) = V0 (v) + 3 64π 2 2.2 Effective potential in the Zee-Babu model 6 where vρ is a variable changing with temperature, and at T = 0, vρ ≡ v0 = 246 GeV. Here F± m  φ T Z mφ T αJ 1 (α, 0)dα, = ∓ 0 1 J∓ (α, 0) Z ∞ = 2 α 2.2.2 1 (x2 − α2 ) 2 dx. ex ∓ 1 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,   m2 ±  1 m2  1 (m2H )2 ln( H ) + (m2h± )2 ln( h2 ) 2 2 2 4(4π) Q 4(4π) Q 2   m 1 2 × 1 m2G + ξm2W  2 2 2 2 2 k±± ln( + (m ) + (m + ξm ) ln( ) ±± ) G W k 4(4π)2 Q2 4(4π)2 Q2   1 m2 + ξm2  2×3 m2  + (m2G + ξm2Z )2 ln( G 2 Z ) + (m2W )2 ln( W ) 2 2 4(4π) Q 4(4π) Q2  3 2×1 ξm2W  m2 2 2 2 2 Z + (m ) ln( ) − (ξm ) ln( ) 2 Z W Q 4(4π)2 4(4π)2 Q2 V1T =0 (v) = − ξm2Z  1 2 2 (ξm ) ln( ) , Z 4(4π)2 Q2 (2.3) and   2   m2   m2  T4 mH h± k±± J + J + 2 ×J B B B 2π 2 T2 T2 T2   m2 + ξm2   m2 + ξm2  T4 G W G Z + 2 2×JB + J B 2π T2 T2   m2   m2   m2   m2  (2.4) 3T 4 γ t W Z + 2×JB + JB + JB + 4 ×JB 2π 2 T2 T4 T2 T2   ξm2   ξm2   ξm2  T4 γ W Z − 2 2×JB + J + J , B B 2π T2 T2 T2 V1T 6=0 (v, T ) = in which JB± m2φ T2 ! = 2 Z mφ 2 T αJ 1 (α, 0)dα. ∓ 0 2.3 Electroweak phase transition in Landau gauge 2.3 7 Electroweak phase transition in Landau gauge The quartic expression in v Vef f (v) = D(T 2 − T02 )v 2 − ET |v|3 + λT 4 v , 4 (2.5) Tc critical temperature and phase transition strength are 2E T0 vc = . Tc = p ,S= 2 T λ c Tc 1 − E /DλTc (2.6) 0 (v) are The minimum conditions for Vef f 0 Vef f (v0 ) 0 (v) ∂ 2 Vef f ∂v 2 v=v0 = 0, 0 (v) ∂Vef f ∂v   = m2H (v) v=v0 = 0, (2.7) 2 v=v0 2 = 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 0 − 350 GeV range, respectively, the transition strength is in the range 1 ≤ 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± @GeVD 400 300 200 100 0 0 100 200 300 400 500 mk ±± @GeVD 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 8 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 2 (2.8) + B1 v 3 + Λv 4 + f (T, u1 , u2 , µ, ξ), in which f (T, u1 , u2 , µ, ξ, v) = C1 + C2 ,  m2 +ξm2   (2.9) m2 +ξm2  Expanding functions JB G T 2 W and JB G T 2 Z in Eq. (2.4), we will obtain term of mixing  the  between  ξ and v in B1 and B2 . m2G +ξm2W m2G +ξm2Z Therefore JB and JB or B1 and B2 contain a T2 T2 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 2 T 2 /m2 (g is the coupling constant of SU (2), m is mass of boson), m ∼ 100 GeV, g ∼ 10−1 so g 2 /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.