Unitarity bound in the most general two Higgs doublet model

We investigate unitarity bounds in the most general two Higgs doublet model without a discrete $Z_2$ symmetry nor CP conservation. S-wave amplitudes for two-body elastic scatterings of Nambu-Goldstone bosons and physical Higgs bosons are calculated at high energies for all possible initial and final states (14 neutral, 8 singly-charged and 3 doubly-charged states). We obtain analytic formulae for the block-diagonalized scattering matrix by the classification of the two body scattering states using the conserved quantum numbers at high energies. Imposing the condition of perturbative unitarity to the eigenvalues of the scattering matrix, constraints on the model parameters can be obtained. We apply our results to constrain the mass range of the next--to--lightest Higgs state in the model.

It goes without saying that the parameters can also be constrained by taking into account various experimental results such as electroweak precision data [6], observables in flavour physics [7], the Higgs boson signal strength [3,4] and so on. By the combination of the bounds from the theoretical consistency and those from the experimental data, a parameter space in extended Higgs sectors can be further restricted.
However, analyses in the most general THDM without the Z 2 symmetry is getting important as an effective description of more various new physics scenarios, such as supersymmetric SMs with non-holomorphic Yukawa couplings [34] and also general models with CP-violation [35] which is required for successful scenario of electroweak baryogenesis [36][37][38].
In this letter, we investigate unitarity bounds in the most general THDM without the Z 2 symmetry nor CP-conservation. We calculate S-wave amplitudes for two-body elastic scatterings of Nambu-Goldstone (NG) bosons and physical scalar bosons at high energies for all possible initial and final states (14 neutral [14], 8 singly-charged [15] and 3 doubly-charged states). By choosing appropriate bases of the scattering states which are deduced using the conserved quantum numbers at high energies such as the hypercharge, weak isospin and the third component of the latter [39], the scattering matrix is given as a block-diagonalized form with at most 4×4 submatrices. Thus, all of the eigenvalues of the scattering matrix can be easily evaluated numerically. The analytic result for the block diagonalized S-wave matrix is consistent with that given in Ref. [40]. By requiring that each of the eigenvalues is not too large to break validity of perturbation calculation, the model parameters, e.g., the masses of extra Higgs bosons and mixing angles, can be constrained. Our results can be useful to constrain parameter spaces whenever one evaluates physics quantities in the most general THDM.
We then numerically demonstrate that the bound on the mass M 2nd of the second lightest Higgs boson, whatever it is, is obtained by inputting the mass of the discovered Higgs boson h under the assumption of non-zero deviations in the Higgs boson coupling hV V with the weak boson (V = W and Z) from the SM value. Currently, the hV V coupling was measured with ∼ 10% accuracy by the LHC Run-I experiment (see Ref. [41] and references therein), and that is expected to be measured more accurately at future collider experiments. If a deviation is detected in the hV V coupling, we can obtain the constraint on the region of M 2nd even without its direct discovery.

II. MODEL SETUP
A. Higgs potential The most general Higgs potential under the SU (2) L × U (1) Y gauge symmetry is given by where Φ i (i = 1, 2) are the isospin doublet scalar fields with hypercharge Y = 1/2. In general, m 2 1 , m 2 2 and λ 1 -λ 4 are real, while m 2 3 and λ 5 -λ 7 are complex. Thus, there are totally fourteen real parameters. If there is a symmetry in the potential, the number of parameters is reduced.
For example, when the potential is exact (softly-broken) Z 2 invariant under the transformation of , the m 2 3 , λ 6 and λ 7 (λ 6 and λ 7 ) terms are forbidden. By using the U (1) Y invariance and rephasing the doublet fields, the vacuum expectation values (VEVs) of the two doublet fields can be taken to be real without loss of generality [42][43][44]. The two doublet fields are then described in terms of the component fields as where v 1 and v 2 are the VEVs of Φ 1 and Φ 2 , respectively, which are satisfied v = v 2 1 + v 2 2 = ( √ 2G F ) −1/2 ≃ 246 GeV. By introducing tan β = v 2 /v 1 , two VEVs are described by v and tan β as the usual notation. In the following, both the VEVs are assumed to be non-zero, except for the case of the inert doublet model discussed in Appendix.
The stationary conditions of the scalar potential are given as follows ∂V ∂ϕ a 0 = 0, (ϕ a = h 1 , h 2 , z 1 , and z 2 ), where the left hand side of the above equation for each ϕ is calculated by ∂V ∂V where we used the abbreviation of s θ = sin θ and c θ = cos θ. In Eqs. (4)-(6), we introduced and λ R k = Re λ k , λ I k = Im λ k , (k = 5, 6, 7).
From the stationary conditions in Eq. (3), we can eliminate m 2 1 , m 2 2 and Im m 2 3 in the Higgs potential.
In order to calculate the masses for the scalar bosons, it is convenient to introduce the so-called Higgs basis [45] defined as where Φ = where G ± and G 0 are the NG bosons which are absorbed into the longitudinal components of W ± and Z by the Higgs mechanism, respectively. The physical singly-charged scalar state is denoted as H ± . The three neutral states h ′ 1 , h ′ 2 and h ′ 3 are not the mass eigenstates at this stage, which generally mix with each other.

B. Mass spectrum
We give the mass formulae for the Higgs bosons in the most general case. First, the squared mass of H ± is calculated by Next, the mass term for the neutral scalar states is expressed by the 3 × 3 matrix as where each of the matrix elements is given by The mass eigenvalues m 2 where m H 1 ≤ m H 2 ≤ m H 3 is assumed. Mass eigenstates for the neutral Higgs bosons are also defined using R as where H 1 is defined to be the SM-like Higgs boson with the mass of about 125 GeV, i.e., m H 1 ≃ 125 GeV. In the following, we represent H 1 and its mass m H 1 by h and m h , respectively. In the CPconserving limit, H 2 and H 3 respectively correspond to the additional CP-even (H) and CP-odd (A) Higgs bosons. The mass formulae for the CP-conserving case will be discussed in the next subsection. We here note that R can be described by three mixing angles [42]. In our numerical analysis given in Sec. IV, the matrix elements of R are derived by inputting the mass matrix elements given in Eq. (13).
Consequently, by using experimental values of v (= 246 GeV) and m h (= 125 GeV), eleven input parameters can be chosen as follows where θ 5,6,7 are the complex phases of λ 5,6,7 . Notice thatm h is determined so as to keep m h = 125 GeV.

III. UNITARITY BOUNDS
We calculate the S-wave amplitude matrix for the elastic BB ′ → B ′′ B ′′′ scatterings in the high energy limit, where the fields B, B ′ , B ′′ and B ′′′ represent either W ± L , Z L , H 1 , H 2 , H 3 or H ± . In this case, all the longitudinal components of the weak gauge boson states can be replaced by the corresponding NG boson states because of the equivalence theorem. Furthermore, only the scalar boson contact interactions contribute to the S-wave amplitude. Therefore, the calculation of the S-wave amplitude matrix is quite simply done just by extracting the coefficient of the scalar boson quartic terms.
First, the 14 neutral channels are expressed in the weak eigenbasis as where i = 1, 2. By taking an appropriate basis transformation, we obtain the following blockdiagonalized S-wave amplitude matrix: where each submatrix is given by 8 Each of the submatrices is obtained in the following basis: where the indices i and j (k and l) run over 1-4 (1-3). Each of the above bases give the submatrix indicated in the parenthesis. We note that at the high energy limit, the hypercharge Y , the isospin I and the third component of the isospin I 3 are used to classify the two body scattering states [39], In fact, the above bases are obtained by finding the two scalar states which belong to the same set of the quantum numbers, i.e., the states in the Ψ N i (X 4×4 ), Ψ N j (Y 4×4 ), Ψ N k (Z 4×4 ) and Ψ N ′ l (Z 4×4 ) bases respectively belong to the state with (Y , I, I 3 ) = (0, 0, 0), (0, 1, 0), (1, 1, −1) and (1, 1, −1).
Finally, three (positive) doubly-charged channels are expressed as: We obtain We note that the negative charged states are obtained by taking charge conjugation for each positive charged channel, which give the same set of eigenvalues of the matrix for the corresponding positive charged channels.
When we impose symmetries in the potential, we obtain the block-diagonalized S-wave amplitude matrix with a smaller size of submatrices. For example, if we assume the CP-invariance (for the case of λ I 5 = λ I 6 = λ I 7 = 0), the maximal size of the submatrices reduces into 3 × 3, or if we impose the (softly-broken) Z 2 symmetry (for the case of λ 6 = λ 7 = 0), the maximal size of the submatrices reduces into 2 × 2 as they have already known in Refs. [14,15].
In order to constrain the parameters in the potential, we impose the following condition for each eigenvalue of the S-wave amplitude matrix: where ξ is conventionally taken to be 1/2 [8] or 1 [5], and with all x i being real due to the hermitian nature of the S-wave amplitude matrix. We call the bound given by the inequality (40)  First, in the Higgs basis defined in Eq. (10), the kinetic terms for the Higgs fields are given by where D µ is the covariant derivative. The Higgs-Gauge-Gauge type vertex, i.e., h ′ 1 V V , only comes from the first term of Eq. (42) at the tree level. In the mass eigenbasis, the ratio of the hV V coupling g hV V in the THDM to that of SM is then expressed by using the rotation matrix R given in Eq. (15) as We note that R 11 corresponds to sin(β − α) in the CP-conserving case as it is seen in Eq. (A1).
Thus, the non-zero deviation in the hV V couplings from the SM prediction comes from the mixing effect of neutral Higgs bosons, i.e., R 11 = 1.
Second, when there is no mixing among the neutral Higgs bosons, the mass of h and those of the extra Higgs bosons H 2 and H 3 are schematically expressed as λ i v 2 and M 2 + λ j v 2 , respectively, as it is seen in Eqs. (13a), (13b) and (13c). Therefore, in the no mixing case, the upper bound on the masses of extra Higgs bosons cannot be obtained, because they can be taken to be as large as possible by using the M 2 dependence. In other words, we can take the decoupling limit [46] of the extra Higgs bosons by taking M 2 ≫ v 2 .
On the other hand, if there is non-zero mixing among the neutral Higgs bosons, i.e., κ V = 1, M 2 dependence appears in m 2 h which must be kept to be about (125 GeV) 2 . Therefore, we cannot take a too large value of M 2 in that case, because we need a large cancellation of the M 2 contribution to m 2 h by the λ i v 2 term which must be excluded by the unitarity bound. Therefore, we can obtain an upper limit on the masses of extra Higgs bosons as long as the hV V coupling is deviated from the SM prediction.
In the following, we numerically show the bound on the mass of the second lightest Higgs boson denoted as M 2nd by fixing m h = 125 GeV and κ V . We assume that the SM-like Higgs boson h is We first consider the softly-broken Z 2 symmetric and CP-conserving case, i.e., λ 6,7 = 0 and θ 5 = 0. In this case, we have six free parameters m H ± , m A , m H , M 2 , tan β and sin(β − α) (= κ V ). For all the plots, we take m 2 H ± =m 2 A =m 2 H = M 2 , θ 5 = 0 and tan β = 1, and scan the m 2 H ± and sin(β −α) parameters. The black (circle), blue (square) and red (triangle) dots show the allowed points for the cases of κ V = 0.98 ± 0.01, 0.96 ± 0.01 and 0.94 ± 0.01, respectively.
In Fig. 1 the comparison of the strength of the unitarity constraint with the two cases. Hereafter, we take ξ = 1/2. From these figures, we can see that the stronger limit on M 2nd is obtained when 1 − κ V is taken to be larger values. Our results for this case are consistent with those given in Ref. [30].
Next, we consider the case without the softly-broken Z 2 symmetry but with CP-conservation, i.e., θ 5,6,7 = 0. In Fig. 2 , where the appearance of kink at around λ R 6 = 0.5, 0.8 and 1 is due to an interchange of eigenvalues which break the condition of Eq. (40). When λ R 6 is taken to be larger than about 2.2, there is no solution to satisfy the unitarity bound. We confirm that the maximally allowed value of M 2nd is obtained in the case of M 2 ≃ M 2 2nd . Finally, we consider the most general case with CP-violation. In this case, we have eleven input parameters with v = 246 GeV and m h = 125 GeV as shown in (25). In order to satisfy m h = 125 GeV, we scan them h parameter for each fixed value of the other parameters. In Fig. 3 In all the panels, the circle (black), square (blue) and triangle (red) points satisfy the unitarity bound and κ V = 0.98 ± 0.01, 0.96 ± 0.01 and 0.94 ± 0.01, respectively. The values of m H ± and sin(β −α) are scanned. We find that, in addition to the upper limit on M 2nd , there is the lower limit especially in the case with θ CP = 0. The lower bound becomes higher when we take a larger value of |λ 6 | (= |λ 7 |). The appearance of the lower limit can be understood in the following way. If we take a non-zero value of θ CP , it gives a non-zero value of the off-diagonal mass matrix elements M 2 13 and M 2 23 (see Eq. (13)), which gives non-zero mixings and/or mass splittings among the neutral Higgs bosons. On the other hand, we now fix the value of κ V , so that it restricts the possible amount of the mixing, and it also requires a non-zero mass splitting among the neutral Higgs bosons. Because the SM-like Higgs boson h which we suppose the lightest of all has the mass of 125 GeV, the non-zero mass splitting turns out to be the lower limit on M 2nd .
We here comment on the constraint from electric dipole moments (EDMs) on the parameter space in the THDM. As it has been well known that if a model has an additional source of the CP-violation, its magnitude is constrained by EDMs. In Refs. [47,48], the constraint on parameter space from EDMs has been investigated in the softly-broken Z 2 symmetric THDMs. In the THDMs, EDMs constrain the allowed region of θ CP depending on the parameter set. Recent study on the collider phenomenology of the CP-violating THDM is found in Ref. [49], in which the constraint on the complex parameter such as λ 5 from EDMs and the unitarity bound turns out to be important.

V. CONCLUSIONS
We have investigated unitarity bounds in the most general two Higgs doublet model without a discrete Z 2 symmetry nor CP conservation. We have computed the S-wave amplitudes for twobody elastic scatterings of the NG bosons and physical Higgs bosons at high energies for all possible initial and final states. By choosing the appropriate bases, the scattering amplitude matrix is given to be the block-diagonalized form, and thus the eigenvalues can be easily evaluated numerically.
We have constrained the parameter space of the model by using the unitarity bound. By fixing the mass of the discovered Higgs boson h to be 125 GeV and assuming a non-zero deviation in the hV V couplings from the SM values, there is an upper limit on the mass M 2nd of the second lightest Higgs boson. Therefore, by using the precisely measured hV V couplings at future collider experiments, we can constrain the allowed region of M 2nd if a deviation of the hV V coupling from the SM limit is found even without its direct discovery. Our results can be useful to constrain parameter spaces whenever one evaluates physics quantities in the most general THDM.
The squared masses of the scalar bosons are calculated by