Probing CP violating Higgs sectors via the precision measurement of coupling constants

Mayumi Aoki, ∗ Katsuya Hashino, 3, † Daiki Kaneko, ‡ Shinya Kanemura, § and Mitsunori Kubota ¶ Institute for Theoretical Physics, Kanazawa University, Kanazawa 920-1192, Japan Department of Physics, University of Toyama, 3190 Gofuku, Toyama 930-8555, Japan Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan Abstract We study how effects of the CP violation can be observed indirectly by precision measurements of Higgs boson couplings at a future Higgs factory such as the international linear collider. We consider two Higgs doublet models with the softly broken discrete symmetry. We find that by measuring the Higgs boson couplings very precisely we are able to distinguish the two Higgs doublet model with CP violation from the CP conserving one.


I. INTRODUCTION
By the discovery of a Higgs boson (h), the standard model (SM) has been established as the low energy effective theory below the electroweak scale [1,2]. In spite of such a success of the SM, we do not think that the SM is a fundamental theory because there are several phenomena which cannot be explained in the SM, such as baryon asymmetry of the universe (BAU), dark matter, neutrino mass, cosmic inflation etc. Therefore an extension of the SM must be considered to describe these phenomena. This would be done at least partially by introducing an extended Higgs sector as seen in a promising scenario to explain BAU, the electroweak baryogenesis [3] where both additional CP violating phases and strongly first order electroweak phase transition (EWPT) can occur in an extended Higgs sector.
In this letter, we examine how to indirectly detect the CP violating effects by precision measurements of the SM-like Higgs boson in two Higgs doublet models (2HDMs), where new CP violating effects can appear in the Yukawa couplings and in the Higgs potential.
We focus on the 2HDM with a softly-broken Z 2 symmetry to avoid flavor changing neutral current [57], which can contain a source of CP violation in the Higgs potential. Under the symmetry the possible Yukawa couplings are classified in four types (Type-I, II, X and Y) [58,59]. In the CP conserving case these types of Yukawa interaction can predict different patterns of deviations in the Higgs boson couplings, by which we are able to fingerprint each model if any of the deviation is detected in the couplings by precision measurements [60][61][62].  [63]. We here show how the effects of the CP violation can be indirectly observed by the precision measurements of the Higgs boson couplings at future collider experiments such as international linear collider (ILC [64][65][66], FCC-ee [67], CEPC [68] and CLIC [69]).
We here introduce the 2HDMs with the softly broken discrete symmetry Z 2 , which is introduced to avoid flavor changing neutral current [57]. Isospin doublet scalar fields Φ 1 and Φ 2 are transformed under the Z 2 symmetry: The Higgs potential is given by where µ 2 3 and λ 5 are generally complex, while the other parameters are real. Φ 1 and Φ 2 can be parameterised as where In this paper, we use the redefinition of phases of doublet fields to absorb the ξ. We then define the complex parameters µ 2 3 and λ 5 as Re[µ 2 3 ]+iIm[µ 2 3 ] and Re[λ 5 ]+iIm[λ 5 ], respectively. The stationary conditions are given by, which lead to the following equations: where λ 345 ≡ λ 3 +λ 4 +Re[ There is one CP violating parameter in Higgs potential by using third equation in Eq. (4). In this letter, we treat Im[λ 5 ] as one physical parameter of CP violation.
We introduce the mixing angle β (tan β = v 2 /v 1 ) in order to rotate the original basis to the Higgs basis [70]: where G + , G 0 are Nambu-Goldstone boson states. In this basis, the mass of H ± is The mass matrix for h 1 , h 2 and h 3 is not yet diagonalised, and takes the form: wherem h ,m H andm A are masses of the SM-like Higgs boson, extra CP-even and CP-odd Higgs bosons in the CP conserving limit, respectively. In this limit,α is the mixing angle which diagonalises two CP-even states in the Higgs basis. We use an orthogonal matrix R in order to diagonalise the 3×3 mass matrix in Eq. (7), We treat the mass eigenstate H 1 as the (discovered) SM-like Higgs boson with the mass 125 GeV. There are nine independent parameters in the potential in the following analysis: Next, we introduce Yukawa interactions and gauge interactions for H 1 in the model.
Under the Z 2 symmetry, the Yukawa interaction is given by where Φ u,d,l are either Φ 1 or Φ 2 by the charge assignment of the Z 2 symmetry for fields in the model. There are 4 types of Yukawa interactions [58,59] as shown Table I. Yukawa interactions for H 1 can be then rewritten as  [59]. Table II, and I f is the third component of the isospin for fermion.
Gauge coupling constants to H 1 take the following form: The scaling factors for H 1 V V (V = W and Z) are given at the tree level by There are the theoretical bounds on the parameter space in the 2HDM with the CP violation. The vacuum stability condition for the Higgs potential is given in Ref. [71]. The perturbative unitarity bounds on the two-body elastic scattering amplitudes for the gauge and Higgs bosons are given in Refs. [72,73].
The constraints from the S, T and U parameters are seen in [74][75][76]. Parameters in the 2HDM are constrained by the direct searches of additional Higgs bosons by the data from LHC Run-1 and Run-2 [77][78][79][80][81]. In addition, the flavour experiments such as B meson decays TABLE II. ξ f factor for each Type [59].
Type-I + cot β + cot β + cot β Type-II + cot β − tan β − tan β Type-X + cot β + cot β − tan β Type-Y + cot β − tan β + cot β give the lower limit on m H ± and tan β for each Type [82,83]. New CP violating effects in the new physics models are constrained by EDM. The bounds from the EDM experiments on the parameter space of the 2HDM with CP violation have been discussed in Refs. [84,85].

III. NUMERICAL ANALYSIS
In order to examine how CP violating phases in the Higgs sector affect the Higgs boson couplings, we evaluate the scaling factors κ V defined in Eq. (13) and the ratio of the decay rate for H 1 → ff with identifying H 1 as the discovered Higgs boson with the mass of 125 GeV and the decay rate for h → ff in the SM: where The ratio of the decay rates coincides with that given in Ref. [86]. In the following numerical analysis, we take four of the nine parameters in Eq.
Since the ratio of the decay rates is independent of M and m H ± at the tree-level, we can take values of M and m H ± to avoid the current constraints from the S, T and U parameters [87].
We show the numerical results for the scaling factor and the ratio of the decay rates varying the rest parameters (tan β,α and Im[λ 5 ]). In the CP conserving limit, cos(β −α) correspond to R 21 .
In     16), we may be able to distinguish not only the Types of 2HDM [61] but also CP violating cases from CP conserving cases by the precision measurement of the Higgs boson couplings as seen in Fig. 1. However, we cannot distinguish the ratios of decay rates with CP violating effects from those in the CP conserving 2HDM whenm A is very large.
In Fig. 2, we show whether we can distinguish the CP violating case from the CP conserving case by using the ILC with √ s = 250 GeV and L = 2 ab −1 . We focus on the ratio of decay rates for H 1 → ff (f = τ, b and c) and the scaling factor κ V for H 1 V V in Type-I and X, because in Type-II and Y the parameters in Eq.(16) are excluded by b → sγ [59]. In order to see how the CP violating case can be distinguished from the CP conserving case, we first do not take into account the EDM results in Fig. 2. Later in Fig. 3 respectively. For Type-X, the magenta, green, grey and blue solid lines respectively correspond to tan β = 1, 2, 2.5 and 3. The points of cross, rhombus and triangle in Fig. 2 are .
Im [6 7 ]  In the Type-I 2HDM being taken into account the EDM data, we confirmed that we cannot distinguish the CP violating case from the CP conserving case via the precision measurements of Higgs boson couplings, because the ratios of decay rate for the fermion and the scaling factors for the gauge boson in these case overlap. Therefore, in Fig. 3, we only show the results in the Type-X 2HDM under the constraint from the EDM data. In the left side panels, the results for R 21 ≤ 0 are shown, while in the right side panels those for R 21 ≥ 0 are shown. The magenta, green, blue, yellow and red solid lines in the figure respectively correspond to tan β = 1, 2, 3, 4 and 5. In the Type-X 2HDM |c p u | < 3 × 10 −2 with c p u given in Eq. (15) is allowed by the EDM data [84]. There is another constraint on Im   We here give a comment that the angular distribution of H 1 → τ − τ + can be used to measure the CP violating effect in the Higgs sector [6]. The CP mixing angle ψ CP is given by L H 1 τ τ = gτ (cos ψ CP + iγ 5 sin ψ CP )τ H 1 , where g = −m τ (c s τ ) 2 + (c p τ ) 2 /v with c s τ and c p τ given in Eq. (15). At the ILC with √ s = 250 GeV and L = 2 ab −1 , ψ CP can be measured to a precision of 4.3 • [6]. In the Type-I 2HDM where are taken into account the EDM data, we cannot detect the CP violating effect by measuring the angular distribution of H 1 → τ − τ + at the ILC. On the other hand, in the Type-X 2HDM the corresponding values of ψ CP to the red square points in Fig. 3 are given in Table. III. We can complementarily examine the effects of the CP violation in the Type-X 2HDM by the precision measurements of the Higgs boson couplings and the angular distribution of H 1 → τ − τ + at future Higgs factories.

IV. SUMMARY
We have studied how effects of the CP violation can be observed indirectly by precision measurements of the coupling constants of the Higgs boson with the mass 125 GeV at a future Higgs factory such as the ILC. We have investigated the difference between CP conserving and CP violating cases of the 2HDMs with the softly broken discrete symmetry.
We have found that in some parameter sets the CP violating effects in the extended Higgs sectors can be detected by measuring the Higgs boson couplings very precisely.

ACKNOWLEDGMENTS
The work of M. A. is supported in part by the Japan Society for the Promotion of Sciences