Aligned CP-violating Higgs sector canceling the electric dipole moment

We discuss the effect of CP violation in the aligned scenario of the general two-Higgs-doublet model, in which the Higgs potential and the Yukawa interaction provide additional CP-violating phases. An alignment is imposed to the Yukawa interaction in order to avoid dangerous flavor changing neutral currents. The Higgs potential is also aligned such that the coupling constants of the lightest Higgs boson, which is identified as the discovered Higgs boson with the mass of 125 GeV, are the same as those of the standard model. In general, CP-violating phases originated by the Yukawa interaction and the Higgs potential are strongly constrained by the current data for the electric dipole moment (EDM). It is found that in our scenario contributions from the two sources of CP violation can be destructive and consequently their total contribution can satisfy the EDM results, even when each CP-violating phase is large. Such a large CP-violating phase can be tested at collider experiments by looking at the angular distributions of particles generated by the decays of the additional Higgs bosons.


I. INTRODUCTION
In spite of the success of the standard model (SM) for elementary particles, there are still phenomena which cannot be explained by this model. Baryon asymmetry of the Universe (BAU) is one of such phenomena beyond the SM. It is well known that the Sakharov's three conditions have to be satisfied for viable baryogenesis which explains the observed BAU from a baryon symmetric universe [1]; (1) existence of baryon number changing interaction, (2) C and CP violation, and (3) departure from thermal equilibrium. In the SM, these conditions can be satisfied qualitatively by the sphaleron transition, the Kobayashi-Maskawa (KM) phase, and the strongly first-order electroweak phase transition (EWPT), respectively [2]. However, the magnitude of CP violation by the KM phase is numerically not sufficient to realize the observed BAU [3]. Furthermore, it turned out that the EWPT is crossover with the measured value of the mass of the Higgs boson [4].
Therefore, a new physics model has to be introduced to explain the BAU.
Among these scenarios, EWBG requires new physics at the TeV scale, so that it can be tested at high-energy collider experiments as well as flavor experiments and astrophysical observations. Thus, it is interesting and timely to consider EWBG. For the successful scenario of EWBG, the Higgs sector is extended in order to obtain additional CP-violating phases and the strongly firstorder EWPT from the minimal form assumed in the SM. For example, the EWBG scenario was discussed in models with additional isospin singlets [8][9][10], doublets [11][12][13][14][15][16][17] and triplets [18,19].
One of the important properties of models for the successful EWBG is the strongly first-order phase transition, whose phenomenological consequence can be a prediction on the large deviation in the triple Higgs boson coupling from the SM value [20], in particular, in models with multi Higgs doublets [21]. Therefore, measuring the triple Higgs boson coupling at future collider experiments, such as the high-luminosity upgrade of the LHC (HL-LHC) [22], the the International Linear Collider (ILC) [23,24], etc., is important not only to explore the dynamics of electroweak symmetry breaking but also to test the scenario of electroweak baryogenesis. In addition, the first-order phase transition can also be tested by detecting the gravitational waves with a unique spectrum at future space-based gravitational-wave interferometers such as LISA [25], DECIGO [26] and BBO [27] as discussed in Refs. [28][29][30][31][32][33][34][35][36].
The second important property is detecting the effect of CP violation in the Higgs sector. The CP violation in extended Higgs sectors can be explored by the experiments for electric dipole We identify H 0 1 as the discovered Higgs boson with the mass of 125 GeV, and we consider that the other Higgs bosons H 0 2 , H 0 3 and H ± have larger masses. Consequently, there are 7 free parameters which can be chosen as follows where θ 7 ≡ arg[λ 7 ] ∈ (−π, π].

B. Yukawa interaction
The Yukawa interaction term is given by where Q L (L L ) is the left-handed quark (lepton) doublet, and u R , d R and e R are the righthanded up-type quark, down-type quark and charged lepton singlets. In Eq. (15),Φ k = iσ 2 Φ * k , and y f,k (f = u, d and e) are 3 × 3 complex Yukawa coupling matrices. The Yukawa interaction term can be also expressed in the Higgs basis as follows where the Yukawa coupling matrices are expressed as In the mass eigenstate of the fermions represented as those without a prime, Eq. (16) is rewritten where In order to avoid such FCNCs, we impose the Yukawa-alignment proposed by Pich and Tuzon [64] as where ζ f are complex values. Thus, y f,1 and y f,2 are diagonalized at the same time, and then ρ f There is another prescription suggested by Glashow and Weinberg [71], in which a Z 2 symmetry is imposed to the Higgs sector. The doublet fields are transformed as Φ 1 → Φ 1 and Φ 2 → −Φ 2 under the Z 2 symmetry. One of the Yukawa matrices y k,f is then forbidden. There are four types of Yukawa interactions depending on the Z 2 charge assignment for the right-handed fermions [72,73]. The ζ f factors in each model are summarized in Tab. I.
The Yukawa interaction term can be written by the mass eigenstates of the fermions and the Higgs bosons as follows: We introduce κ j f as the coupling factors for the interactions of the neutral Higgs bosons and fermions defined as where θ f ≡ arg[ζ f ] ∈ (−π, π] and I u = 1/2, I d = I e = −1/2. As we mentioned above, R jk = δ jk is taken, so that κ factors are written as We can see that the Yukawa couplings for H 0 1 do not contain the CP-violating phases at tree-level. On the other hand, those of H 0 2,3 have the CP-violating phases, and thus they can contribute to the EDMs.

C. Kinetic terms of the scalar fields
The kinetic term for the scalar doublet fields can be rewritten by where D µ is the covariant derivative given by D µ = ∂ µ + ig 2 σ a 2 W a µ + ig 1 1 2 B µ with g 2 and g 1 being the SU(2) L and U(1) Y gauge coupling constants, respectively. The trilinear Higgs-gauge-gauge type couplings are given by where m W = g 2 v/2 and m Z = g 2 2 + g 2 1 v/2. In the alignment limit R jk = δ jk , the couplings of H 0 1 V V (V = W, Z) are the same as the SM ones, and those of H 0 2 V V and H 0 3 V V vanish at tree level.

D. Theoretical and experimental constraints
The parameters in the potential are constrained by taking into account the perturbative unitarity [74][75][76][77] and the vacuum stability [78,79]. In addition, the electroweak S, T and U parameters [80,81] constrain the masses and mixings of the Higgs bosons. When we impose m H 3 = m H ± and λ 6 = 0, new contributions to the T parameter vanish at one-loop level, because of the custodial symmetry in the Higgs potential [82][83][84][85][86]. Furthermore, the constraints from B physics should be considered [87]. The constraint on the mass of charged Higgs bosons and ζ q in the Yukawa-alignment scenario has been discussed in Ref. [88].

III. ELECTRIC DIPOLE MOMENT
The Hamiltonian of the EDM for non-relativistic particles with the spin S can be described by where E is the external electric field. Under the time reversal transformation T ( S) = − S and T ( E) = + E, the sign of this term H EDM is flipped. Therefore, if the EDM is nonzero, time reversal invariance is broken, then the CPT theorem implies that CP symmetry is also broken. In terms of the effective Lagrangian, the EDM d f for a fermion is written as where F µν is the electromagnetic field strength tensor and σ µν = i 2 [γ µ , γ ν ]. The neutron EDM receives the additional contribution from the chromo-EDM (CEDM) d C q for a quark q being expressed as where the G µν is the QCD field strength tensor.
The electron and neutron EDMs are constrained by various atomic and molecule experiments [41,43] as follows. First, the ACME collaboration gave the upper limit |d e + kC S | < 1.1 × 10 −29 e cm at the 90% confidence level (CL) [89] by using the thorium-monoxide EDM, where C S is defined by the coefficient of the following dimension six operator with N being a nucleon. The coefficient k is given by k ∼ O(10 −15 ) GeV 2 e cm [43]. In our benchmark scenario, which is explained below, the contribution from kC S is typically two orders of the magnitude smaller than the current bound, according to the discussion given in Refs. [41,43].
Therefore, we can safely neglect this contribution, and we simply impose the bound |d e | < 1.1 × 10 −29 e cm (90% CL) in the following discussion.
Second, very recently the nEDM collaboration updated the constraint on the neutron EDM as |d n | < 1.8 × 10 −26 e cm (90% CL) [90]. In our analysis, the neutron EDM is estimated by using the following QCD sum rule [42]: where g 3 is the QCD gauge coupling constant. There are the other contributions to the neutron EDM from the Weinberg operator L ⊃ 1 3 C W G a µνG bνσ G c σ µ and the four-fermi interaction However, in our benchmark scenario, the contribution from C W (C f f ) is typically two (more than two) orders of the magnitude smaller than the current bound, according to the discussion given in Refs. [37,41,43] (Ref. [41]). We thus simply apply Eq. (32) to our analysis. Finally, the bounds from the other EDMs of atoms are satisfied by considering the constraints for the above electron and the neutron EDMs. Consequently, what we need to take into account is the contributions from d f (f = e, d and u) and d C q (q = d and u). The dominant contribution to d f appears from the two-loop BZ type diagrams [67]. We note that the one-loop contributions to d f are negligibly smaller than those from the BZ diagrams, The BZ diagrams contain the effective H 0 j V 0 γ (V 0 = γ, Z) and H ± W ∓ γ vertices. The BZ contribution to d f is decomposed into the contributions from fermion-loops, Higgs boson-loops and gauge bosom-loops as follows Furthermore, each contribution can be classified as where Thus, by imposing the potential alignment R ij = δ ij this contribution vanishes. In addition, it is confirmed in Refs. [39,41] that non-BZ type diagrams at two-loop level with a Higgs boson mediation are also proportional to 3 j=1 R 1j Im[κ j f ], so that they vanish in the alignment limit. Therefore, the dominant contributions arise from the Higgs boson and fermion-loops as shown in Fig. 1. 2 When we take all the neutral Higgs bosons degenerate in mass, de also vanishes because of the orthogonality of the R matrix [41].

Requiring the destructive interference between the fermion-loop and Higgs boson-loop in order
to realize d e 0, we obtain the following relation assuming the extra Higgs boson masses to be nearly degenerate and θ d = 0, where A and B are the constant factors: and I and J are the loop functions depending on the masses of the extra Higgs bosons: where C GH XY (z) are given in Appendix B, and its argument mH denotes the mass of the extra Higgs bosons in the loop. From Eq. (35), it is clear that the two independent phases θ u and θ 7 can be taken such that the fermion-and the Higgs boson-loop contributions to d e cancel with each other.
We numerically calculate the electron EDM by using the input parameters in the SM in Tab. II.
As we mentioned in Sec. II, we take m H 0 3 = m H ± in order to avoid the constraint from the T parameter in the following analysis. In GeV. If one of the masses is taken to be much larger than that of the others, d e is getting larger because the cancellation becomes weaker. If all the masses are simultaneously taken to be larger   [93]. The parameters in the last line are the Wolfenstein parameters of the CKM matrix [93].   We note that in the parameter region allowed by d e the neutron EDM is typically two orders of the magnitude smaller than the current upper limit. We set the benchmark point given in Tab. III. This point is marked as the star in the right panel of Fig. 3. gg → φ 0 (gluon fusion), (45) gg → ttφ 0 (top quark associated production ), (46) gg → bbφ 0 (bottom quark associated production ), gb → H ± t (gluon-bottom fusion ), where φ 0 is H 2 or H 3 . They can also be produced in pair via the s-channel gauge boson mediations; Let us first consider the processes induced by the Yukawa interaction. The dominant cross section for φ 0 is provided from the gluon fusion process given in Eq. (45) whose value is estimated by sections for the top and bottom quark associated production can also be estimated by σ SM tth × |ζ u | 2 and σ SM bbh × |ζ d | 2 with σ SM tth and σ SM bbh being the cross section for gg → tth gg → bbh in the SM, respectively. According to [97], σ SM tth and σ SM bbh are almost the same value for a fixed value of m h ; e.g., they are given to be about 120 (100) fb at NLO in QCD, where m h = 200 GeV and √ s = 13 TeV are taken. Thus, the cross section for the top quark associated production is negligibly small, because we have taken |ζ u | = 0.01 as the benchmark. The cross sections for the gluon fusion and the bottom quark associated processes are shown in Fig. 4. It is seen that the cross section of the gluon fusion process is typically one order of the magnitude larger than that of the bottom quark associated process.
For the charged Higgs bosons, if they are lighter than the top quark mass, they can be produced via the top decay t → H ± b. However, such light charged Higgs bosons have already been highly constrained by the current LHC data [98,99]. For instance, the upper limit on B(t → H ± b) is given to be of order 10 −3 at 95% CL assuming B(H ± → τ ± ν) = 1 [99]. We thus consider the  case with the heavier charged Higgs bosons m H ± > m t , in which they can be produced via the gluon-bottom fusion process given in Eq. (48). We evaluate the production cross section by using TeV. NLO corrections in QCD have been discussed in Refs. [102,103].
As mentioned above, there are pair production processes given in Eqs. (49)- (51). Differently from the Yukawa induced processes, their cross section is determined by the gauge coupling for a fixed value of the mass of the additional Higgs boson. Therefore, even when we take very small values of the ζ f parameters, these production processes can be important. Again, we use CalcHEP 3.4.2 and CTEQ6L for the evaluation of the cross section. The cross section is shown in Fig. 6. We see that these cross sections can be larger than those given by the Yukawa induced processes in the wide range of the mass region.

C. Decays of the additional Higgs bosons
We discuss the branching ratios of the additional Higgs bosons. In the alignment limit R jk = δ jk , the additional Higgs bosons can mainly decay into a fermion pair. They can also decay into a (off-shell) gauge boson and another Higgs boson if it is kinematically allowed. In addition to these decay modes, we can consider loop-induced decay processes such as H 0 2,3 → γγ, Zγ, gg and H ± → W ± Z, W ± γ. Except for the H 0 2,3 → gg, the branching ratios of loop-induced processes are negligibly small; i.e., BR(H 0 2,3 → γγ/Zγ) and BR(H ± → W ± Z, W ± γ) [96] are typically smaller than O(10 −4 ). In our benchmark point given in Tab. III, the main decay modes of the additional Higgs bosons are given as follows The explicit analytic formulae for the above decay rates are given in Appendix C.
In Fig. 7, we show the branching ratios of H 0 2 (left), H 0 3 (center) and H ± (right) as a function of |ζ e | with the fixed parameters given in Tab. III. In this case, the upper limit of |ζ e | is given to be 0. 58  with |ζ e | 0.4. Therefore, H 0 2,3 → τ τ and H ± → τ ± ν can be important in our scenario. As discussed in Refs. [55][56][57][58], the hadronic decays of τ can be useful to extract the CP-violating effects due to their simple kinematic structure. We thus consider H 0 j → τ − τ + → h − νh +ν , where h ± are hadrons, for instance, ρ ± or π ± mesons. The average of the squared amplitude for H 0 j → τ − τ + → h − νh +ν is calculated as where the mass of τ is neglected, and the angles θ ± and φ ± are defined as the momentum direction of the meson on the rest frame of τ ± as depicted in Fig. 8. By applying Eq. (56) to the decays of H 0 2,3 , the angular distribution of the decay products of H 0 2,3 are obtained as follows where ∆φ ≡ φ + −φ − . From these expressions, we can extract θ e by looking at the ∆φ distributions.
In Fig. 9, we show the ∆φ distributions in the decay of H 0 2 (left) and that of H 0 3 (right) for each fixed value of θ e . Depending on the value of θ e , the shape of the distributions changes, so that we may be able to extract the information of θ e . It goes without saying that dedicated studies with signal and background simulations have to be performed in order to know the feasibility of extracting the CP-violating phases in our scenario.

V. RENORMALIZATION GROUP EQUATION ANALYSIS
One might think that the destructive cancellation of the contributions from BZ diagrams seems to be a kind of artificial fine tuning to satisfy the data. In order to see the stability of our scenario, we investigate the high energy behavior by the RGE analysis. We discuss the scale dependence of the electron EDM d e in the case where the destructive interference sufficiently realizes at the scale of m Z so that the current data are satisfied. We evaluate the running of all the dimensionless couplings from the scale of m Z to a high energy scale by using the one-loop β-functions given in In order to investigate the scale dependence of d e , we first consider how the alignment of the Yukawa interaction can be broken at a high energy scale. The magnitude of the departure from the alignment limit can be parametrized as [104] ∆ q ≡ Tr[δy † q δy q ], where δy q ≡ρ q − We note that ∆ q vanishes at the alignment limit which is assumed at the scale of m Z . The behavior of the running of ∆ u (left) and ∆ d (right) is shown in Fig. 10. It is clearly seen that both ∆ u and ∆ d are significantly small up to a high energy scale such as at least about 10 10 GeV where the Landau pole appears (see the discussion below). This is because the source of the breaking of the alignment mainly arises from the tiny off-diagonal elements of the CKM matrix. Therefore, the Yukawa alignment approximately holds up to a high energy scale.
Next, we consider how the alignment of the Higgs potential can be broken at a high energy scale. This can be clarified by looking at the running of the λ 6 parameter, see Eq. (8). In Fig. 11, we show the scale dependence of the magnitude of all the dimensionless parameters in the Higgs potential. We can see that the |λ 6 | parameter quite slowly increases as the scale is getting higher, and it blows up together with all the other couplings at around µ = 10 10 GeV which is the scale appearing the Landau pole. Therefore, our scenario is stable up to such a high energy scale.
Finally, we discuss the scale dependence of d e which can be evaluated by We note that the contribution from the gauge boson loopd e (gauge) appears at higher energy scales, because of the breaking of the potential alignment. In addition, the SM-like Higgs boson H 0 1 is no longer the pure CP-even state at higher energy scales, so that H 0 1 can also but slightly contribute to d e as well as H 0 2 and H 0 3 . In Fig. 12, we show the scale dependence of each contribution tod e ; i.e., d e (fermion),d e (Higgs) andd e (gauge), by taking into account the above mentioned issues, where we neglect subdominant contributions from non-BZ type diagrams. We see that the cancellation betweend e (fermion) andd e (Higgs) still works at higher energy scales. However, because of the appearance ofd e (gauge), the total value is getting larger, and it exceeds |d e | = 1.747×10 −16 GeV −1 at around µ = 10 8 GeV 3 .

VI. DISCUSSIONS AND CONCLUSIONS
We have discussed the general THDM with the multiple sources of CP-violating phases in the Yukawa interaction and the Higgs potential. In order to avoid the FCNCs at tree-level, we have imposed the Yukawa alignment. In addition, the alignment in the Higgs potential is imposed in order that coupling constants of the Higgs boson with the mass of 125 GeV with SM particles are the same as those of the SM Higgs boson at tree level.
In this framework, we have computed the contributions from the BZ type diagrams to the electron and neutron EDMs. We have found that there are non-trivial Finally, we have discussed the scale dependence of the dimensionless couplings by using the RGEs at one-loop level. We have confirmed that both the alignment in the Yukawa interaction and that in the Higgs potential can be stable up to a high energy scale such as 10 8 GeV.
Before closing this paper, we give a brief comment on the possibility of EWBG in our scenario.
At the zero temperature, the VEVs of the Higgs doublets are taken to be ( Φ 0 1 , Φ 0 2 ) = (v/ √ 2, 0), while at the finite temperature, they can be ( Φ 0 1 , Φ 0 2 ) = (v 1 , v 2 )/ √ 2 [14]. If this kind of the structure of the phase transition can be realized, the complex phase can appear in the top quark mass via ζ u during the EWPT, which may be able to generate the baryon asymmetry of the Universe. In this case, our scenario discussed in this paper is important for successful and tastable models for EWBG.

ACKNOWLEDGMENTS
The work of S. K. was supported in part by Grant-in-Aid for Scientific Research on Innova- where n g is the number of the generation of the fermions and n d is the number of scalar doublets.
The beta functions of the Yukawa-coupling matrices are given as where Tr[yy] lk = Tr[N C y † u,l y u,k + y † d,l y d,k + y † e,l y e,k ].