The electroweak phase transition in the zee - Babu and SU(3)c ⊗ SU(3)l ⊗ U(1)x ⊗ U(1)n gauge models

c = 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± = p 2v20 + u 2 1, m 2 k±± = q 2v20 + u 2 2. (2.1) Diagonalizing matrices in the kinetic components of the Higgs potential, we obtain m2H(v0) = −µ2 + 3λv20 , m2G(v0) = −µ2 + λv20 , m2Z(v0) = 1 4(g 2 + g′2)v20 = a2v20 , m2W (v0) = 1 4g 2v20 = b 2v20 . (2.2) 2.2 Effective potential in the Zee-Babu model 2.2.1 Effective potential with Landau gauge Veff (v) = V0(v) + 3 64pi2 ( m4Z(v)ln m2Z(v) Q2 + 2m4W (v)ln m2W (v) Q2 − 4m4t (v)lnm 2 t (v) Q2 ) + 1 64pi2 ( 2m4h±(v)ln m2h±(v) Q2 + 2m4k±±(v)ln m2k±±(v) Q2 +m4H(v)ln m2H(v) Q2 ) + 3T 4 4pi2 { F−( mZ(v) T ) + F−( mW (v) T ) + 4F+( mt(v) T ) } + T 4 4pi2 { 2F−( mh±(v) T ) + 2F−( mk±±(v) T ) + F−( mH(v) T ) } 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± (mφ T ) = ∫ mφ T 0 αJ1∓(α, 0)dα, J1∓(α, 0) = 2 ∫ ∞ α (x2 − α2) 12 ex ∓ 1 dx. 2.2.2 Effective potential with ξ gauge We are known that in high levels, the contribution of Gold- stone boson cannot be ignored. Therefore, we must consider an ef- fective potential in arbitrary ξ gauge, VT=01 (v) = 1 4(4pi)2 (m2H) 2[ ln(m2H Q2 ) ] + 1 4(4pi)2 (m2h±) 2[ ln(m2h± Q2 ) ] + 1 4(4pi)2 (m2k±±) 2[ ln(m2k±± Q2 ) ] + 2× 1 4(4pi)2 (m2G + ξm 2 W ) 2[ ln(m2G + ξm2W Q2 ) ] + 1 4(4pi)2 (m2G + ξm 2 Z) 2[ ln(m2G + ξm2Z Q2 ) ] + 2× 3 4(4pi)2 (m2W ) 2[ ln(m2W Q2 ) ] + 3 4(4pi)2 (m2Z) 2[ ln(m2Z Q2 ) ]− 2× 1 4(4pi)2 (ξm2W ) 2[ ln(ξm2W Q2 ) ] − 1 4(4pi)2 (ξm2Z) 2[ ln(ξm2Z Q2 ) ] , (2.3) and VT 6=01 (v, T ) = T 4 2pi2 [ JB (m2H T 2 ) + JB (m2h± T 2 ) + 2×JB (m2k±± T 2 )] + T 4 2pi2 [ 2×JB (m2G + ξm2W T 2 ) + JB (m2G + ξm2Z T 2 )] + 3T 4 2pi2 [ 2×JB (m2W T 2 ) + JB (m2Z T 4 ) + JB (m2γ T 2 ) + 4×JB (m2t T 2 )] − T 4 2pi2 [ 2×JB (ξm2W T 2 ) + JB (ξm2Z T 2 ) + JB (ξm2γ T 2 )] , (2.4) in which JB± ( m2φ T 2 ) = ∫ m2φ T 2 0 αJ1∓(α, 0)dα. 2.3 Electroweak phase transition in Landau gauge 7 2.3 Electroweak phase transition in Landau gauge The quartic expression in v Veff (v) = D(T 2 − T 20 )v2 − ET |v|3 + λT 4 v4, (2.5) Tc critical temperature and phase transition strength are Tc = T0√ 1− E2/DλTc , S = vc Tc = 2E λTc . (2.6) The minimum conditions for V 0eff (v) are V 0eff (v0) = 0, ∂V 0eff (v) ∂v ∣∣∣ v=v0 = 0, ∂2V 0eff (v) ∂v2 ∣∣∣ v=v0 = [ m2H(v) ] ∣∣∣ v=v0 = 1252 GeV2. (2.7) 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 −→∞. 0 100 200 300 400 500 0 100 200 300 400 500 mk±± @GeVD m h± @Ge VD 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 8 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) v2 + B1v3 + Λv4 + f(T, u1, u2, µ, ξ), (2.8) in which f(T, u1, u2, µ, ξ, v) = C1 + C2, (2.9) Expanding functions JB ( m2G+ξm 2 W T 2 ) and JB ( m2G+ξm 2 Z T 2 ) in Eq. (2.4), we will obtain the term of mixing between ξ and v in B1 and B2. Therefore JB ( m2G+ξm 2 W T 2 ) and JB ( m2G+ξm 2 Z T 2 ) 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 g2T 2/m2 (g is the coupling constant of SU(2), m is mass of boson), m ∼ 100 GeV, g ∼ 10−1 so g2/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 be- cause 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 + 3λv20 = 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 em- phasized 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 bo- son When the mass of Goldstone boson is small, it means that µ2 ≈ λv20 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. 2.4 Electroweak phase transition in ξ gauge 9 -The first approximate µ2 ≈ λv20. -In the second approximate step, we neglect u1, u2. - In the third approximate step: replacing µ2 = λv20 and in the square root term of B2 and C1, we can approximate µ2 ∼ λv2. After three approximate steps we rewrite the equation (2.8) V = (D1 +D2) v2 + Bv3 + Λv4. (2.10) We obtain the strength of EWPT as shown in Fig.2.2. The maximum of the strength is about 4.05. S=1 S=1.5 S=2 S=4.05 S™¥ 200 220 240 260 280 300 320 340 0 20 40 60 80 mk±±‘h± @GeVD Ξ Figure 2.2: The strength of EWPT with λ = 0.1297 and µ2 ∼ λv20 In fact, the mass of Goldstone boson is much smaller than that of the W± boson or the Z boson so the contribution of Gold- stone boson must be very small in the effective potential. Hence, the lines in Fig.2.2 are almost vertical or almost parallel to the axis ξ. These results match those of Ref.[48].This shows that the strength of EWPT is gauge independent. In addition, the new particles have large masses, so they provide valuable contributions to the EWPT in the Landau gauge or in an arbitrary gauge. The charge of these particles increases their contributions. 2.4.2 Constraints on coupling constants in the Higgs potential In order to have the first order phase transition, the masses of the new charged scalars mh± and mk±± must be smaller than 350 GeV. Therefore, we obtain the following limits: 0 < p < 1.22 and 0 < q < 1.22. However, to find these accurate values of mh± and mk±± , other considerations are also needed. 2.5 Conclusion 10 2.5 Conclusion In this chapter we have investigated the EWPT in the ZB model by using the high-temperature effective potential. The EWPT is strengthened by the new scalars, the phase transition strength is from 1 to 4.15. The new charged scalars h± and k±± are triggers for the first-order EWPT. Chapter 3 MULTI-PERIOD STRUCTURE OF ELECTROWEAK PHASE TRANSITION IN THE 3-3-1-1 MODEL 3.1 Brief review of the 3-3-1-1 model 3.1.1 The mass of the quarks - The mass of top quarks and bottom quarks is as follows: mt = htu√ 2 , mb = hbv√ 2 , - The mass of the exotic quarks are determined as mU = ω√ 2 hU ; mD1 = ω√ 2 hD11 ; mD2 = ω√ 2 hD22. 3.1.2 The mass of the Higgs bosons The mass terms of charged Higgs bosons are given by m2H1 = u2 + v2 2 λ8; m 2 H2 = ω2 + v2 2 λ7. (3.1) The mass of neutral Higgs bosons is presented in Table 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 S4 A′η Aχ Sη S′χ Sρ H3 Squared mass 2λΛ2 λ9ω 2 2 λ9u 2 2 2λ3u2 2λ2ω2 2λ1v2 λ9(u 2+ω2) 2 3.1.3 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 + v2 = (246 GeV)2. Table 3.2: The mass of charged gauge bosons. Gauge boson W Y X Squared mass g2 4 (u 2 + v2) g 2 4 (ω 2 + v2) g 2 4 (ω 2 + u2) 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 m2Z ' g 2(u2 + v2) 4c2W , (3.2) m2Z1 ' g2 18 ( (3 + t2X)ω 2 + 4t2N (ω 2 + 9Λ2) + √ ((3 + t2X)ω 2 − 4t2N (ω2 + 9Λ2))2 + 16(3 + t2X)t2Nω4 ) , (3.3) m2Z2 ' g2 18 ( (3 + t2X)ω 2 + 4t2N (ω 2 + 9Λ2) − √ ((3 + t2X)ω 2 − 4t2N (ω2 + 9Λ2))2 + 16(3 + t2X)t2Nω4 ) . (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/ √ 2 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 2(u2 + v2) 4c2W ; m2Z1 ' 4g2t22Λ2; m2Z2 ' g2c2Wω 2 (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 13 3.2 Effective potential in the model 3-3-1-1 Hence the Higgs of the model, we obtain the tree-level po- tential V0 = λφ4Λ 4 + 1 4 λ11φ 2 Λφ 2 ω + λ2φ 4 ω 4 + φ2Λµ 2 2 + 1 2 µ22φ 2 ω + λ3φ 4 u 4 + 1 4 λ12φ 2 Λφ 2 u + 1 4 λ6φ 2 uφ 2 ω + 1 2 µ23φ 2 u + 1 4 λ5φ 2 uφ 2 v + λ1φ 4 v 4 + 1 4 λ10φ 2 Λφ 2 v + 1 4 λ4φ 2 vφ 2 ω + 1 2 µ21φ 2 v. (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 Veff (v) = λv 4 − Ev3 +Dv2 + λk.ω2v2 + λj .Λ2v2 + u2.v2 ≈ λv4 − Ev3 +Dv2 + λi.(ω2 + Λ2 + u2)v2 (3.7) The slices of the effective potential in (3.7) at ω2 +Λ2 +u2 = 1 TeV2 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, .., 9 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 Λi = 0 Λi = 0.03 Λi = 0.06 0.2 0.4 0.6 0.8 1.0 1.2 1.4 vHTeVL 0.05 0.10 0.15 Veff ITeV 4M 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,Λ 2 + ω2 + v2 = 1 TeV2 From the mass spectra, we can split the masses of particles into four parts as follows m2(φΛ, φω, φu, φv) = m 2(φΛ) +m 2(φω) +m 2(φu) +m 2(φv). (3.8) Taking into account Eqs. (3.6) and(3.8), we can also split the effective potential into four parts Veff (φΛ, φω, φu, φv) = Veff (φΛ) + Veff (φω) + Veff (φu) + Veff (φv). We assume φΛ ≈ φω, φu ≈ φv over space-times. Then, the effective potential becomes Veff (φΛ, φω, φu, φv) = Veff (φω) + Veff (φu). From the mass spectra, it follows that the squared masses of gauge and Higgs bosons are split into three separated parts corre- sponding to three SSB stages. 3.3 Electroweak phase transition without neutral fermion 3.3.1 Two periods EWPT in picture (i) 1. Phase transition SU(3)→ SU(2) This phase transition involves exotic quarks, heavy bosons, but excludes the SM particles. As a consequence, the effective po- tential of the EWPT SU(3)→ SU(2) is Veff (φω). Veff (φω) = Dω(T 2 − T 20ω)φ2ω − EωTφ3ω + λω(T ) 4 φ4ω, (3.9) T 20ω ≡ − Fω Dω . (3.10) 3.3 Electroweak phase transition without neutral fermion 15 S=1 S=2 S=3 S™¥ 0 1000 2000 3000 4000 5000 6000 7000 0 500 1000 1500 2000 2500 3000 3500 mexotic-quarkCharged Higgs@GeVD m H 3 @Ge VD Figure 3.2: The mass area corresponds to Sω > 1 The values of Veff (φω) at the two minima become equal at the critical temperature and the phase transition strength are Tcω = T0ω√ 1− E2ω/DωλTcω , Sω = 2Eω λTcω . There are nine variables: the masses of U,D1, D2, H2, H3 and A′η, S′χ, S4, Z1. However, for simplicity, we assume mU = mD1 = mD2 = mH2 ≡ O, mA′η = mS′χ = 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 λTcω ≡ Sω(O,P, Sω). (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 = 6 TeV. S=1 S=3 S=2 0 1000 2000 3000 4000 5000 6000 7000 0 500 1000 1500 2000 2500 mH1@GeVD m H 3 @Ge VD Figure 3.3: The mass area corresponds to Sω > 1 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{ 0 ≤ mExoticQuark/ChargedHiggsboson ≤ 7000GeV , 0 ≤ mH3 ≤ 2600 GeV . From Eq. (3.11) the maximum of Sω is around 70. 2. Phase transition SU(2)→ U(1) In this period, the symmetry breaking scale equals to u = 246/ √ 2 and the masses of the SM particles and the masses ofX,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 Veff (φu) = λu(T ) 4 φ4u − EuTφ3u +DuT 2φ2u + Fuφ2u. (3.12) The minimum conditions are Veff (0) = ∂Veff (φu) ∂φu ∣∣∣∣ u = 0; ∂2Veff (φu) ∂φ2u ∣∣∣∣ u = m2Aχ+m 2 H3 +m 2 Sη+m 2 Sρ , (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. S=1 S=1.2 S=2 S=3 S™¥ S™¥ 200 400 600 800 1000 1200 0 100 200 300 400 500 600 mH1@GeVD m H 3 @Ge VD 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 3 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 1 but the EWPT is not strong. 3.3 Electroweak phase transition without neutral fermion 17 S=1 S=1.3 S=1.2 S=2.1 200 250 300 350 400 450 0 50 100 150 200 mH1@GeVD m H 3 @Ge VD Figure 3.5: The strength EWPT S = 2Eu λTc with Tc must be real. Table 3.3: Mass limits of particles with Tc > 0. Strength S K[GeV ] L[GeV ] 1.0-2.12 195 ≤ K ≤ 484.5 0 ≤ L ≤ 209.8 3.3.2 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) S=1 S=2 S=3 S™¥ 0 1000 2000 3000 4000 0 200 400 600 800 1000 mExotic QuarkCharged Higgs@GeVD m H 3 @Ge VD Figure 3.6: The strength EWPT Sω = 2Eu λTc with ω = 6 TeV. 3.4 The role of neutral fermions in EWPT 18 When we import real Tc, the mass region of charged and neutral particles are{ 0 ≤ mExoticquark/ChargedHiggsboson ≤ 4000 GeV , 0 ≤ 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 do not lose the first-orde EWPT as shown in Table 3.4. Table 3.4: Values of the maximum of EWPT strength with ω = 6 TeV. Period Picture mZ2 [TeV] mN−R[TeV] SMax without NR SMax withNR SU(3)→ SU(2) (i) 2.386 2.227 70 50 SU(3)→ SU(2) (ii) 2.254 1.986 100 30 Looking at the Table 3.4, the following remarks are in order: 1. 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. The strength S = 2E λTc gets the value equals 70. For the picture (ii), the mentioned value equals 100. 2. 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 value of strength EWPT S, because S = E 2λTc and E do not depend 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 dis- tribution) 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 gen- erated 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 6 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 num- ber 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 criti- cal temperature and the strength is gauge independent [44-46]. Con- sequently, 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 diago- nalization, the masses of gauge bosons do 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 men- tioned 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: 1. Investigation of weak phase transition in model Zee-Babu. Considering the Landau gauge, this model has phase transi- tion strength is in the range 1 ≤ S < 2.12, due to the contribution of two mh± and mk±± particles. Their mass ranges in the range of 0− 350 GeV. - Considering the ξ gauge, the phase transition strength is in the range 1 ≤ 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 2. We examined the EWPT in the 3-3-1-1 model with twocases. 1. 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. Con- sidering at ω = 6 TEV, we calculated the phase transition strength in the range 1 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 0 and 7000 GeV. Phase transition SU(2)→ SU(1) at value u = 246√ 2 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 ,