Electroweak phase transition in the economical 3-3-1 model

We consider the EWPT in the economical 3-3-1 (E331) model. Our analysis shows that the EWPT in the model is a sequence of two first-order phase transitions, $SU(3) \rightarrow SU(2)$ at the TeV scale and $SU(2) \rightarrow U(1)$ at the $100$ GeV scale. The EWPT $SU(3) \rightarrow SU(2)$ is triggered by the new bosons and the exotic quarks; its strength is about $1 - 13$ if the mass ranges of these new particles are $10^2 \,\mathrm{GeV} - 10^3 \,\mathrm{GeV}$. The EWPT $SU(2) \rightarrow U(1)$ is strengthened by only the new bosons; its strength is about $1 - 1.15$ if the mass parts of $H^0_1$, $H^\pm_2$ and $Y^\pm$ are in the ranges $10 \,\mathrm{GeV} - 10^2 \,\mathrm{GeV}$. The contributions of $H^0_1$ and $H^{\pm}_2$ to the strengths of both EWPTs may make them sufficiently strong to provide large deviations from thermal equilibrium and B violation necessary for baryogenesis.

Keywords: Spontaneous breaking of gauge symmetries, Extensions of electroweak Higgs sector, Particle-theory models (Early Universe)

I. INTRODUCTION
In the context of electroweak baryogenesis (EWBG), the EWPT plays an important role in explaining the Baryon Asymmetry of Universe (BAU) by electroweak physics. From the three Sakharov conditions, which are B violation, C and CP violations, and deviation from thermal equilibrium [2], the EWPT should be a strongly first-order phase transition. That not only leads to thermal imbalance [3], but also makes a connection between B violation and CP violation via nonequilibrium physics [4].
The EWPT has been investigated in the Standard Model (SM) [3,5] as well as various extension models [6][7][8][9][10][11][12][13][14]. For the SM, although the EWPT strength is larger than unity at the electroweak scale, it is still too weak for the mass of the Higgs boson to be compatible with current experimental limits [3,5]; this suggests that EWBG requires new physics beyond the SM at the weak scale [6]. Many extensions such as the Two-Higgs-Doublet model or Minimal Supersymmetric Standard Model have a more strongly first-order phase transition and the new sources of CP violation, which are necessary to account for the BAU; triggers for the first-order phase transition in these models are heavy bosons or dark matter candidates [10][11][12]14].
Among the extensions beyond the SM, the models based on SU(3) C ⊗ SU(3) L ⊗ U(1) X gauge group (called 3-3-1 for short) [16,18] have some interesting features including the ability to explain the generation problem [16,18] and the electric charge quantization [19].
The structure of such a gauge group requires the 3-3-1 models to have at least two Higgs triplets. Thus the structure of symmetry breaking and the number of bosons are different from those in the SM.
In a previous work [1], we have considered the EWPT in the reduced minimal 3-3-1 (RM331) model due to its simplicity, and found that our approach can be applied to the more complicated 3-3-1 models. In the present work, we follow the same approach for the economical 3-3-1 (E331) model [20], whose lepton sector is more complicated than that of the RM331 model. The E331 model has the right-handed neutrino in the leptonic content, the bileptons (two singly charged gauge bosons W ± , Y ± , and a neutral gauge bosons X 0 ), the heavy neutral boson Z 2 , and the exotic quarks. The model has two Higgs triplets, and the physical scalar spectrum is composed of a singly charged scalar H ± 2 and a neutral scalars H 0 1 [20]. We will show in this paper that the new bosons and the exotic quarks can be triggers for the first-order phase transition in the model. This paper is organized as follows. In Sec. II we give a review of the E331 model on the Higgs, gauge boson, and lepton sectors. In Sec. III, we find the effective potential in the model, which has a contribution from heavy bosons and exotic quarks as well as a contribution similar to that in the SM. In Sec. IV, we investigate the structure of the EWPT sequence in the E331 model, find the parameter ranges where the EWPTs are the strongly first-order to provide B violation necessary for baryogenesis, and show the constraints on the mass of the charged Higgs boson. Finally, we summarize and describe outlooks in Sec.

A. Higgs potential
In the E331 model, the 3-3-1 gauge group is spontaneously broken via two stages. In the first stage, the group SU(3) L ⊗ U(1) X breaks down to the SU(2) L ⊗ U(1) Y of the SM; and the second stage takes place as that we have known in the SM. This sequence of spontaneous symmetry breaking (SSB) is described by the Higgs potential [20]: in which χ and φ are the Higgs scalar triplets: whose VEVs are respectively given by: with T 9 = 1 √ 6 diag(1, 1, 1) so that Tr(T i T j ) = δ ij . The couplings of SU(3) L and U(1) X satisfy the relation: where c W = cos θ W , s W = sin θ W , t W = tan θ W , and θ W is the Weinberg angle.
Eqs. (9) and (3) lead to: The combinations W ′ and Y ′ in (12) are mixed via a mass matrix: Diagonalizing the mass matrix in Eq. (14), we acquire the physical charged gauge bosons and their respective mass eigenvalues where θ is the mixing angle which is defined by The mass m W as in (16) suggests that the W bosons of the model can be identified as those of the SM, and v can be set as v ≃ v weak = 246 GeV. From the constraints in (4), θ should be very small, thus W µ ≃ W ′ µ and Y µ ≃ Y ′ µ . Moreover, the Michel parameter ρ in the model connects u with v by the expression ρ ≈ 1 + 3u 2 v 2 [20]; and from the experimental data, ρ = 0.9987 ± 0.0016 [23], that expression gives us u v ≤ 0.01, which leads to u < 2.46 GeV. With ω in the range 1 TeV − 5 TeV, we have The diagonalization of the mass matrix in Eq. (19) leads to the mass eigenstates of four following neutral gauge bosons: Due to the constraints (4), the physical states Z 1 and Z 2 get masses Since the components W ′ 4 and W 5 have the same mass, we can identify their combination, as a physical neutral non-Hermitian gauge boson, which carries the lepton number with two units. The subscript 0 of X µ in Eq. (24) denotes neutrality of the gauge boson X but sometimes this subscript may be dropped.

C. Fermion sector
The fermion content in this model, which is anomaly free, is given by The Yukawa interactions which induce masses for the fermions can be written as in which L LN C is the Lagrangian part for lepton number conservation and L LN V is that for lepton number violation. These Lagrangian parts are given by: where a, b and c stand for the SU(3) L indices.
During the SSB sequence of this model, the VEV ω gives the masses for the exotic quarks U and D α , the VEV u which is the source of lepton-number violations gives the masses for the quarks u 1 and d α , the VEV v gives the masses for the quarks u α and d 1 as well as all ordinary leptons.

III. EFFECTIVE POTENTIAL IN THE ECONOMICAL 331 MODEL
From the Higgs potential (1), we obtain V 0 in a form which is dependent on the VEVs as follows: We see that V 0 (u, ω, v) has a quartic form like in the SM, but it depends on three variables, u, ω and v; it also has the mixings between these variables. However, we can transform u into ω by t θ as defined in Eq. (17). We note that, if the Universe' energies allow of the existence of the gauge symmetry SU(3) L ⊗ U(1) X and the SSB sequence in the E331 model, the VEVs u, ω and v must satisfy the constraint (4). This leads to t θ ≪ 1, and we can neglect the contribution of u. On the other hand, by developing the Higgs potential (1), we obtain two minimum equations which permit us to transform the mixing between ω and v to the form that depends only on ω or v.
Therefore, we can write V 0 in Eq. (28) as a sum of two parts corresponding to two stages of SSB: in which V 0 (ω) and V 0 (v) are still in the quartic form.
In order to derive the effective potential, we start from the full Higgs Lagrangian: where L GB mass and V (χ, φ) are respectively given by (9) and (1). Expanding the Higgs fields χ and φ around their VEVs which are u, ω and v, we obtain where W runs over all gauge fields and Higgs bosons. In the E331 model, we have two massive bosons like the SM bosons Z 1 and W ± , two new heavy neutral boson X and Z 2 , the singly charged gauge bosons Y ± , one singly charged Higgs H ± 2 , one heavy neutral Higgs H 0 1 and one SM-like Higgs H 0 . The masses of the gauge bosons and the Higgses presented in Table I, from which we can split the boson masses into two parts for two SSB stages:

Bảng I. Mass formulations of bosons in the E331 model
Bosons In the effective potential, we must consider contributions from all fermions and bosons.
But for fermions, we retain only the top and exotic quarks because their contributions dominate over those from the other fermions [3]. Therefore, from the Lagrangain (31) we acquire two motion equations according to ω and v, From Eq. (33), using Bose-Einstein and Fermi-Dirac distributions respectively for bosons and fermions to average over space, we obtain the one-loop effective potential V ef f (ω) at high temperatures: in which m Q indicates the masses of three exotic quarks, and the terms in the form F ∓ m T describe the thermal contributions of particles with masses m. These terms are given by where J (1) Similarly, from Eq. (34), we obtain the high-temperature effective potential V ef f (v): in which m t indicates the mass of the top quark.
Eqs. (31)- (35) and (38) do not consist of any mixing between ω and v. Therefore, we can write the total effective potential in the E331 model as The effective potentials V ef f (ω) and V ef f (v) seem to depend on the arbitrary scales Q ′ and Q respectively. However, by the same reasoning as in [3], we can show that the structure of these potentials remain unchanged for the changes in scales. At zero temperarure, all thermal contributions vanish, and due to the quartic form of V 0 (ω) and V 0 (v), we can rewrite Eqs.
( 35) and (38) as and where λ ′ R , M ′ R , Λ ′ R , λ R , M R , and Λ R are the renormalized constants. The changes such as Q ′ → κ ′ Q ′ (or Q → κQ) induce the terms which contain κ ′ (or κ) and are proportional to . Those terms can be absorbed by λ ′ R (or λ R ). This makes the physics remain the same.

IV. ELECTROWEAK PHASE TRANSITION
In sequence of SSB of the E331 model, the SSB which breaks the gauge symmetry generates the masses for the exotic quarks, the heavy gauge bosons X 0 and Z 2 , and gives the first part of mass for Y ± .
Associated with this sequence of SSB, a sequence of EWPT takes place with the transition SU(3) → SU(2) at the scale of ω 0 and the transition SU(2) → U(1) at the scale of v 0 as the Universe cools down from the hot big bang. Our analysis so far shows that the former is the first transition which depends only on ω, while the latter is the second transition which depends only on v.
From Table I, the gauge bosons X 0 and Z 2 are only involved in the first transition, the gauge bosons W ± , Z 1 and H 0 are only involved in the second transition, but the bosons Y ± , H 0 1 , and H + 2 are involved in both transitions. The total mass of Y ± -i.e. m Y ± (ω, v), whose formula is given by (32) -is generated as follows. As the Universe is at the ω 0 scale and the EWPT SU(3) → SU(2) happens, Y ± eats the Goldstone boson χ ± 2 of the triplet χ to acquire the first part of mass, m Y ± (ω). When the Universe cools to the v 0 scale and the EWPT SU(2) → U(1) is turned on, Y ± eats the Goldstone boson ρ ± 1 of triplet φ and get the last part of mass, m Y ± (v).

A. Phase transition SU (3) → SU (2)
Taking place at the scale of ω 0 which is chosen to be in the range 1 − 5 TeV, the EWPT SU(3) → SU(2) involves exotic quarks and heavy bosons, without the involvement of the SM particles. From Eq. (35), the high-temperature effective potential of the EWPT can be rewritten as in which where ω 0 is the value at which the zero-temperature effective potential From the conditions (52), we have the minima of the effective potential (50): where ω c is a critical VEV of χ at the broken state, and T ′ c is the critical temperature of phase transition which is given by Now, we consider the phase transition strength: which is a function of three unknown masses, m H 0 1 , m H ± 2 and m Q . For simplicity, we follow the ansatz in [12] and assume m H ± 2 = m Q . Then we plot the transition strength S ′ as the function of m H 0 1 (ω c ) and m H ± 2 (ω c ) with ω c is in the range from 1 TeV to 5 TeV. In Figs.
case of ω. The smooth contours are the sets of the (m H ± 2 , m H 0 1 )-pairs which make S ′ > 1 and then the EWPT SU(3) → SU(2) to be the first-order phase transition. The uneven contours are the sets of the (m H ± 2 , m H 0 1 ) -pairs which are unusable because they make S ′ → ∞. Our results show that the heavy particle masses must be in the range of a few TeV, and the strength of the first-order phase transition SU(3) → SU(2) is in the range 1 < S ′ < 13.
According to Ref. [25], the accuracy of a high-temperature expansion for the effective potential such as that in Eq. (50) will be better than 5% if m boson T < 2.2, where m boson is the relevant boson mass. This requirement sets the "upper bounds" of the mass ranges of H 0 1 (ω) and H ± 2 (ω). From Table II, this requirement is satisfied by all mass ranges of H 0 1 , while it narrows slightly most of the mass ranges of H ± 2 . From Eq. (55), the phase transition strength S ′ depends on the parameters E ′ and λ ′ Occurring at the scale v 0 = 246 GeV, the phase transition SU(2) → U(1) does not involve the exotic quarks or the boson X 0 . In this stage, the contribution from Y ± is equal to that from W ± . The effective potential is given by Eq. (38). We write the high-temperature expansion of this potential as for the first-order phase transition are 0 < m H 0 1 < 600 GeV and 0 < m H ± 2 < 1440 GeV, respectively. (v) generate the masses of the SM particles and the last mass part of Y ± . We also have the minima of the effective potential (56): where v c is the critical VEV of φ at the broken state, and T c is the critical temperature of phase transition which is given by We investigate the phase transition strength of this EWPT. In the limit E → 0, the transition strength S → 0 and the phase transition is a second-order. To have a first-order phase transition, we requires S ≥ 1. We plot S as a function of m H 0 1 (v 0 ) and m H ± 2 (v 0 ). As shown in Fig. 6, for the masses of H ± 2 and H 0 1 which are respectively in the ranges 250 GeV < m H ± 2 (v) < 1200 GeV and 0 GeV < m H 0 1 (v) < 620 GeV, the transition strength is in the range 1 ≤ S < 3. the mass ranges of H ± 2 and H 0 1 are respectively narrowed to: and 0 GeV < m H 0 1 < 58 GeV.
Corresponding with these ranges of mass, the range of phase-transition strength is narrowed to 1 ≤ S < 1.15. Thus the EWPT SU(2) → U(1) is the first-order phase transition, but it seems quite weak.
As we can see in Eqs. (61) and (57), the new bosons contribute to the phase transition strength S via the parameters E and λ Tc . Hence these new bosons can be triggers for the EWPT SU(2) → U(1) to be the first-order.
In Fig. 8, we illustrate the dependence of the effective potential V ef f (v) on the temperature. When the Universe cools through the phase-transition critical temperature T c , the Higgs field v tends to get a nonzero VEV v 0 which is in the range 0 < v 0 < 246 GeV, and the second minimum of V ef f (v) gradually appears at v 0 . As the temperature drops from T c , the second minimum becomes lower and the first minimum gradually disappears, while the VEV v 0 tends to 246 GeV. The tendency of v 0 can be seen in Fig. (9) where we show that   and we obtain 2.149 < λ 4 < 2.591, and 0 < λ 2 3 2λ 1 < 0.0556, From the phase transition SU(3) → SU(2), we have also derived 0 < λ 4 < 10.3, and 0 < λ 1 < 0.45, for any ω. Eqs. (65)-(68) lead to 2.149 < λ 4 < 2.591; 0 < λ 1 < 0.45 and 0 < λ 2 3 2λ 1 < 0.0556.

V. CONCLUSION AND OUTLOOKS
We have investigated the EWPT in the E331 model using the high-temperature effective potential. Although the effective potential in the model depends complicatedly on three VEVs, u, ω, and v, it can be transformed to a sum of two parts so that each part depends only on ω or v, which corresponds a stage of SSB. Thanks to that the EWPT can be seen as a sequence of two EWPTs. The first, SU(3) → SU (2), takes place at the energy scale ω 0 to generate the masses for the exotic quarks, the heavy gauge bosons X 0 and Z 2 , as well as a mass part of Y ± . The second, SU(2) → U(1), occurs at the scale v 0 to give the masses for the SM particles and the remained mass part of Y ± .
At the TeV scale, the EWPT SU(3) → SU(2) is strengthened by the new bosons and the exotic quarks to be the strongly first-order; if the masses of these new particles are about 10 2 − 10 3 GeV, the phase transition strength is in the range 1 − 13. As the energy is lowered to the scale of 10 2 GeV, the EWPT SU(2) → SU(1) is strengthened by only the new bosons; with the contributions of the mass parts from H 0 1 , H ± 2 and Y ± which are in the ranges 10−10 2 GeV, the strength of this transition is about 1−1.15. Therefore, both EWPTs can be the first-order; the SU(3) → SU(2) appears very strong, while the SU(2) → SU (1) seems quite weak.
However, both of these first-order EWPTs can be sufficiently strong to provide B violation necessary for baryogenesis, as shown via the parameter ranges which we have specified. If H 0 1 and H ± 2 exist, their contributions to the strengths of each EWPT are meaningly large. In this case, the sequence of strongly first-order EWPTs in the model may provide a source of large deviations from thermal equilibrium. And the model may fully describe the continual existence of BAU since being generated in the early Universe.
In the next works, we will investigate the electroweak sphalerons as well as the C-and CPviolating interactions to know if the model possesses all necessary components for EWBG.