Gravitational wave and collider signals in complex two-Higgs doublet model with dynamical CP-violation at finite temperature

Dynamical CP-violating source for electroweak baryogenesis can appear only at finite temperature in the complex two-Higgs doublet model, which might help to alleviate the strong constraints from the electric dipole moment experiments. In this scenario, we study the detailed phase transition dynamics and the corresponding gravitational wave signals in synergy with the collider signals at future lepton colliders. For some parameter spaces, various phase transition patterns can occur, such as the multi-step phase transition and supercooling. Gravitational wave in complementary to collider signals can help to pin down the underlying phase transition dynamics or different patterns. 1 ar X iv :1 90 9. 02 97 8v 1 [ he pph ] 6 S ep 2 01 9


I. INTRODUCTION
After the observation of the gravitational wave (GW) by the Advanced Laser Interferometer Gravitational Wave Observatory [1], a new era of GW astronomy has been initiated and the GW detector provides a new technique to study the fundamental physics. Especially, electroweak (EW) baryogenesis [2][3][4], which is aimed to explain the baryon asymmetry of the Universe, becomes a promising and testable mechanism after the discovery of GW and Higgs boson. To generate the observed baryon asymmetry of the Universe, all three Sakharov conditions need to be satisfied [5]. These conditions are baryon number violation, C and CP violation, and the departure from the thermal equilibrium or CPT violation. An essential ingredient for a successful EW baryongenesis is the process of a strong first-order phase transition (FOPT) which can achieve the departure from thermal equilibrium. As a by product, the phase transition GW signal induced by a strong FOPT can potentially be detected by the future space-based GW interferometers.
In the standard model (SM), the discovery of Higgs boson by ATLAS [6] and CMS [7] shows that a strong FOPT can not be generated for a 125 GeV Higgs boson based on lattice simulation. It is just a smooth crossover for 125 GeV Higgs boson in the SM. The CP violation is also too weak in the SM. Thus, the extension of the SM are needed to give a strong FOPT and a large enough CP violation for successfully EW baryogenesis. One of the simplest extension of the SM, which is the so-called 2-Higgs Doublet Model (2HDM), is the SM with an additional SU (2) L scaler doublet, where the sphaleron process was studied in Ref. [8]. However, current electric dipole moments (EDM) experiments [9] have put strong constraints on the CP-violating source at zero temperature for most of the new physics models. In this work, we focus on the complex 2HDM (C2HDM) or the spontaneous CP-violating model. Recent study [10] has shown that there are viable parameter spaces in the C2HDM which can produce a strong FOPT with spontaneous CP violation based on the criterion v c /T c > 1. They also discuss the collider phenomenology including the Higgs trilinear coupling modification and Higgs boson pair production at hadron collider.
Further, Ref. [11] has revisited the constraints from colliders and EDM, and predictions in details. Based on these two comprehensive studies [10,11], we investigate the phase transition dynamics with different phase transition patterns. Besides the dynamical CPviolating behavior, we also find the multi-step phase transition patterns and supercooling patterns. The dynamical process might help to provide the CP-violating source for successful EW baryogenesis. We discuss other possible approaches to explore this scenario in C2HDM.
On one hand, during a strong FOPT, detectable GWs can be produced by three mechanisms: bubble collisions, sound waves, and magnetohydrodynamic turbulence. Based on the viable parameters from Refs. [10,11], we discuss the possibility to detect the GW signals by the future space-based experiments, such as the approved Laser Interferometer Space Antenna (LISA) [12] (launch in 2034 or even earlier), Deci-hertz Interferometer Gravitational wave Observatory (DECIGO) [13,14], Ultimate-DECIGO (U-DECIGO) [15], Big Bang Observer (BBO) [16], Taiji [17,18], and TianQin [19,20]. The dynamical CP-violation behavior can escape the strong constrains from electric dipole moment (EDM) measurements [21][22][23][24]. On the other hand, the strong FOPT could obviously modify the Higgs trilinear coupling and thus can be tested by the future lepton collider, such as Circular Electron-Positron Collider (CEPC) [25], International Linear Collider (ILC) [26] as well as Future Circular Collider (FCC-ee) [27]. Combined with the GW signals, they can make a complementary test on this scenario and the underlying phase transition patterns. This paper is organised as follows. In Section II, we describe the C2HDM and the basic idea of dynamical CP-violation at finite temperature. In section III, the one-loop effective potential at finite temperature and the renormalization prescription are presented. 1 In section IV, we investigate the phase transition dynamics including the corresponding GW signals and its correlation with the collider signatures. We discuss the consistent check of the dynamical CP-violation and supercooling case in section V. Section VI contains our conclusions.

II. MODEL WITH DYNAMICAL CP-VIOLATION
The tree-level potential of the C2HDM can be written as 1 In appendix A, we present the thermal correction of the mass for the C2HDM in the Landau gauge. In appendix B, we derive the field dependent mass matrix elements for the gauge bosons, the scalar bosons and the top quark for C2HDM in the Landau gauge.
where m 2 12 and λ 5 are complex numbers, and arg(λ 5 ) = 2arg(m 2 12 ). It is obvious that the C2HDM has a softly broken Z 2 symmetry ( This model has been extensively studied including the EDM constraints and collider phenomenology, such as the recent works [10,11] and references therein. However, at finite temperature, there would be dynamical CP-violating behavior [10] as Theṽ with tilde represents the VEV at finite temperature. This is the starting point of this where v ≈ 246 GeV is the SM VEV, and the stationary conditions are and give the following relations where We introduce a mixing angle β, which is defined as then transform the fields into a new basis In the C2HDM, the neutral components ζ 1 , ζ 2 and ζ 3 mix into the neutral mass eigenstates The mixing matrix R can diagonalize the neutral mass matrix and derive where m 1 ≤ m 2 ≤ m 3 are the masses of the neutral Higgs bosons. We can parameterise the matrix R as the following [28] where s i = sin α i , c i = cos α i (i = 1, 2, 3), and − π 2 ≤ α i < π 2 [10,11]. Note the above mixing matrix is valid at zero temperature. When we consider the finite-temperature situation in the next section, this result should be modified. The Higgs potential in Eq.(1) has 9 independent parameters. We follow Ref. [29] and choose 9 input parameters v, tan β, m H ± , α 1 , α 2 , α 3 , m 1 , m 2 , and Re(m 2 12 ). For these input parameters, m 3 can be expressed as . (17) The analytic relations between the above parameter set and the coupling parameters λ i in the original lagrangian can be written as [30] where In general, 2HDM can be classified into type I, type II, lepton-specific and flipped, according to the interactions of the fermions to the Higgs doublets. In this work we only study type I case, and only consider the top quark's contribution to the EW phase transition among all the fermions.

PERATURE
To study the phase transition dynamics in the C2HDM, we use the finite-temperature effective field theory [31][32][33]. The full one-loop finite-temperature effective potential reads where V tree , which is obtained by replacing the doublets with their classical background fields (ṽ 1 ,ṽ 2 ,ṽ CP ,ṽ CB ) from Eq. (3), is the tree-level potential at zero temperature as shown in the following V CW is the Coleman-Weinberg potential (CW) at zero temperature. In the MS scheme, the CW potential can be written as whereṽ ≡ {ṽ 1 ,ṽ 2 ,ṽ CP ,ṽ CB }, and m 2 s (ṽ) is the eigenvalue for the particle s in the mass matrix in terms of the background fieldsṽ, details see Appendix B. n s denotes the numbers of the degree of freedom. Because of the charge-breaking VEV, photon becomes massive. And we have to take into account different masses and numbers of degree of freedom for the charge conjugated particles. For each particle s, the numbers of degree of freedom are {n H i , n A , n H + , n H − , n G + , n G − , n W + , n W − , n Z , n γ , n t , nt} = {1, 1, 1, 1, 1, 1, 3, 3, 3, 3, −6, −6} and the constants C s are The masses and the mixing angles with one-loop corrections are different from those extracted from the tree-level potential. To enforce the one-loop corrected masses and the mixing angles to be equal to the tree-level values, we use the on-shell renormalisation prescription as in Refs. [10,34,35]. Then, a counterterm potential V CT is added to the one-loop effective potential. The general formula of the counterterm contribution V CT reads [35] where δp i and n are the counterterms and the number of parameters of the tree-level potential, respectively. δT k denotes the counterterms of the tadpole T k , and m is the number of background field or the number of field that is allowed for the development of a non-zero VEV. In the C2HDM, the counterterm potential can be written as The on-shell renormalisation conditions at zero temperature are where The second derivatives of the CW potential lead to the well-known problem of infrared (IR) divergences for the Goldstone bosons [36][37][38][39] in the Landau gauge. In practice, we can introduce an IR regulator for the Goldstones and then discard the terms proportional to the IR divergence. Previous study [36] has dealt with this problem and derived analytic formulae for the first and second derivatives of the CW potential in the physical basis with where χ s is the spin of different particles , m 2 (s)a is the physical mass of particle s at zero temperature, O H ij is the rotation matrix that transform scalar fields from Laudau gauge basis to mass eigenstate basis, S ij denotes symmetrisation with respect to the two indices, λ (s)abi and λ (s)abij are the cubic and quartic couplings for particle s in mass eigenstate basis. Note that we need to deal with degenerate mass limit carefully in Eq. (31), for more detail, see Ref. [36]. Then the counterterms can be expressed in terms of the derivatives of the CW potential. For the analytic formulae of the counterterms, see Refs. [10,35]. V T is the oneloop thermal corrections including daisy resummation [40,41] at finite temperature. The thermal correction reads with the thermal functions where the plus sign is for fermions and the minus sign is for bosons. In order to include the contribution of daisy resummation, we make the following replacement for the scalar boson mass and the longitudinal components of the gauge boson mass where Π B is the thermal correction of the scalar boson and the longitudinal components of gauge boson at finite temperature, which can be found in Appendix A. The Debye corrected masses are applied in the all terms of J B and also used in the CW potential [41]. It is worth noticing that Parwani scheme is used in this work, while Arnold-Espinosa scheme is used in Ref. [10].
With the full effective potential in Eq. (20) In Tab. I, we show 8 benchmark sets. Each parameter set can give a one-step strong FOPT, and the FOPT take place as (0, 0, 0, 0) with the temperature decreasing from high value to zero. Only one strong FOPT happens for these benchmark sets.
In Tab. II, two parameter sets are shown. Each benchmark set can induce two FOPTs and they evolve as (0, 0, 0, 0) with the temperature decreasing from high value to zero.
Three-step phase transition can be produced for the two benchmark sets in Tab. III.
And they evolve like (0, 0, 0, 0) with the temperature decreasing from high value to zero.
For these two benchmark sets, two FOPTs and one SOPT happen.
v   To obtain the parameter sets in the above Tables, we need to know the bubble dynamics during the phase transition process. The essential quantity of bubble dynamics is the bubble nucleation rate per unit time per unit volume  where S E (T ) = S 3 /T is the Euclidean action of a critical bubble and Γ 0 ∝ T 4 . S 3 is the three-dimensional Euclidean action, which can be denoted as To calculate the nucleation rate, we need to obtain the bubble profiles of the four scalar fields by solving the following bounce equations with the boundary conditions whereṽ f is the false VEVs. Conventionally, we use the so-called overshooting (undershooting) method [42,43] to solve the single-field bounce equation. However, the multi-field case becomes much more complicated. We use the path deformation method, which is introduced by Ref. [44], to find a proper path that connects the initial and final vacuum state.
In our analysis, we make use of the public available package cosmoTransitons to solve the four differential bounce equations. Then the nucleation temperature T n is defined as the temperature at time t n at which Γ becomes large enough to nucleate a bubble per horizon volume with the probability is O(1), where H is the Hubble parameter. In other words, this condition can be simplified as The properties of the bubbles are illustrated by two key parameters α and β. Note α is the ratio of the latent heat (T n ) to the energy density of the radiation bath ρ rad . It is defined where ρ rad (T ) = g π 2 T 4 /30, and g is the number of the relativistic degree of freedom in the thermal plasma at T . And (T n ) can be written as Moreover, the parameter β is defined as However, in the actual calculations, the renormalised parameterβ is more convenient: The parameter α describe the strength of the phase transition, namely, the larger value of α corresponds to a stronger phase transition. In addition, the inverse of the parameter β is related to the time scale of phase transition. Based on the above approaches, we can numerically know the phase transition dynamics and calculate the phase transition parameters of all of the benchmark point sets.

IV. COLLIDER AND GRAVITATIONAL WAVE SIGNATURES
After the three parameters α,β and T n are extracted from the finite-temperature effective potential, we can predict the phase transition GW signals which are produced by three mechanisms: bubbles collisions, sound waves, and magnetohydrodynamic turbulence in the plasma after collisions. Based on the envelope approximation [45][46][47][48], the numerical simulation gives the formula of the GW spectrum from bubble collisions [49][50][51]: where g is the total number of degrees of freedom at T n . The coefficient κ, which denotes the fraction of the latent heat transformed into the fluid kinetic energy, is function of α [48].
The peak frequency is The second source is generated by the sound waves of the bulk motion, and numerical simulation gives [52,53] with the peak frequency The turbulence contribution to the GW spectrum is [54,55] with the peak frequency and h = 1.65 × 10 −5 Hz T n 100GeV g 100 Note that, for relativistic bubbles and κ turb 0.1κ v Combined the three contributions, we show the numerical results of the total GW spectra in the C2HDM for the above benchmark points. Strong GW signal favors supersonic bubble wall velocity. However, the EW baryogenesis prefers subsonic bubble wall velocity. Actually, the bubble wall velocity obtained from Eq. (46) is not accurate enough here since these formula is obtained in the simplest scalar model. It is still possible that the real bubble wall velocity in this model is smaller than the velocity of sound wave for non-supercooling case. To tell the difference between two velocities, we show the GW spectrum of the same benchmark sets with a bubble wall velocity calculated by Eq. (46) and a fixed input subsonic velocity v b = 0.5.
Since the GW given by the supercooled phase transition is still controversial, we need a more detailed study. It is worthy noticing that the above formulae of the GW spectrum for the three sources, which is given by numerical simulation, are based on a rapid phase transition process and α < 1. Since a supercooling FOPT may induce a longer and stronger transitions [56,57], it is not clear whether these formulae are applicable to this situation.
Therefore, we just give the GW spectra of the benchmark points without supercooling.
In Fig. 1  strong enough compared to the current GW detection proposals, they are still intriguing phase transition patterns. They can produce two copies of GW signals with different peak frequencies [58][59][60]. Their signals are different from the one-step FOPT as shown in Fig. 1 and 2 where exists only one copy of GW signals for given benchmark sets.   Besides the detectable GW signals, the strong FOPT could also induce obvious deviation of Higgs trilinear coupling compared to the SM as the following In Tab  LHC, it may be not easy to pin down this deviation. However, the significant modification of Higgs trilnear coupling can be measured by the Higgs pair production at future hadron collider. This obvious deviation can also modify the cross section of e + e − → ZH process at one-loop level. Therefore, it can be indirectly tested by the precise measurements of the cross section for the Z boson and Higgs boson associated production at the future lepton collider, such as CEPC or ILC and FCC-ee [61][62][63][64][65]. The deviation of the ZH cross section can be defined as At 240 GeV CEPC with 5.6 ab −1 integrated luminosity, the estimated precision of σ HZ is about 0.5%, which means all the benchmark sets are within the sensitivity of CEPC [25].
The sensitivity for FCC-ee is about 0.4%. The corresponding numerical results for each benchmark set are shown in Tab. IV, Tab. V and Tab. VI. In the Tables, each benchmark set corresponds to α, β, T n (they determine the GW signal) and δ h (it determines the collider signal), which means the GW signal and collider signal are correlated by the EW phase transition physics. Therefore, the future lepton colliders in complementary to GW experiments [61][62][63][64][65] can help to unravel different phase transition dynamics. Namely, these two complementary experiments can help us to understand whether the phase transition process is one-step FOPT, or two-step FOPTs or even three-step phase transitions in the early universe.    sistent with our assumption. For other phase transition patterns, they are also consistent.
As for the charge-breaking VEV, it also numerically shows the similar behavior except the VEV value is much smaller compared to CP-violating case. The extra CP-violating source at finite temperature may provide enough CP violation for successful EW baryogenesis.
And this extra CP-violating source evolves to zero at zero temperature to avoid the strong constraints from EDM data.  trigger even stronger GW signals. However, for some scenarios, recent study [67] shows that it results in weaker overall GW signals as compared to previous conclusion in literatures. Therefore, we leave the precise study of the GW spectrum for the supercooling case in our future work. As for the implication from this C2HDM, from numerical calculations, we find that supercooling favors relatively large coupling constants. And in some narrow parameter spaces, the nucleation temperature decreases as the mass hierarchy of the two neutral Higgs bosons decreases. More reliable results rely on further lattice simulations.

VI. CONCLUSION
We have studied the detailed phase transition dynamics with the existence of dynamical CP-violation at finite-temperature in the complex two-Higgs doublet Model. Various phase transition patterns have been investigated, including multi-step phase transition and supercooling case in this scenario. The dynamical CP-violation can not only provide a possible cosmological origin of CP-violation source, but also make the phase transition dynamics more abundant. The corresponding GW signals in synergy with collider signals have also been discussed, which can be used to make complementary test on this scenario and further unravel the underlying phase transition dynamics or different patterns in the early universe.
The detailed study on EW baryogenesis and gravitational waves from supercooling are left for our future work. There are contributions to the ring diagrams from the gauge boson and Higgs boson. We need to calculate the self-energy of the gauge boson and Higgs boson in the IR limit. First, we consider the self-energy of the Higgs boson. The Higgs self-energy can be derived from the propagator of the Higgs boson with Higgs boson, gauge boson, and top quark loops.
We work in the original basis, where the relevant fields are φ i ≡ {ρ 1 , η 1 , ρ 2 , η 2 , ζ 1 , ψ 1 , ζ 2 , ψ 2 }, then the contributions to the Higgs self-energy from the Higgs boson are The contributions come from the gauge bosons are The contribution from top-quark loop is Thus, the total contributions to the Higgs boson self-energy in the C2HDM are Next, we calculate the self-energy of gauge boson. There are two relevant fields in original basis W a µ , B µ . Then the contributions to the gauge boson self-energy come from the gauge bosons, Higgs boson, and top quark, respectively. Hence, the total self-energy for the gauge boson in the C2HDM are Π W a W a = 2g 2 T 2 ,

B. FIELD DEPENDENT MASS MATRIX ELEMENTS OF C2HDM
Since we introduce a charge-breaking VEV, the mass matrix of gauge boson and Higgs boson in the original basis are fully mixed. We can not give the analytic form of the fielddependent mass for each physical particle. Instead, we derive the mass matrix in the original basis, and then numerically calculate the eigenvalues which are the physical masses of the The mass matrix is For the longitudinal components of the gauge bosons, we need to consider the Debye corrected masses, which are the eigenvalues of The mass matrix elements of Higgs boson in the original basis can be expressed as The Debye corrected mass of the scalar bosons are given as the eigenvalues of Since we just consider the top quark in our work, the field dependent mass of top quark can be easily derived as