Yukawa Alignment Revisited in the Higgs Basis

We implement a comprehensive and detailed study of the alignment of Yukawa couplings in the so-called Higgs basis taking the framework of general two Higgs doublet models (2HDMs). We clarify the model input parameters and derive the Yukawa couplings considering the two types of CP-violating sources: one from the Higgs potential and the other from the three complex alignment parameters $\zeta_{f=u,d,e}$. We consider the theoretical constraints from the perturbative unitarity and for the Higgs potential to be bounded from below as well as the experimental ones from electroweak precision observables. Also considered are the constraints on the alignment parameters from flavor-changing $\tau$ decays, $Z\to b\bar b$, $\epsilon_K$, and the radiative $b\to s\gamma$ decay. By introducing the basis-independent Yukawa delay factor $\Delta_{H_1\bar f f}\equiv |\zeta_{f}|(1-g_{_{H_1VV}}^2)^{1/2}$, we scrutinize the alignment of the Yukawa couplings of the lightest Higgs boson to the SM fermions.


I. INTRODUCTION
complex alignment parameters which provide further CP-violating sources in addition to those in the Higgs potential. The aligned 2HDM accommodates the conventional four types of 2HDMs as the limiting cases when the alignment parameters are real and fully correlated. This paper is organized as follows. Section II is devoted to a brief review of the 2HDM Higgs potential, the mixing among neutral Higgs bosons and their couplings to the SM particles in the Higgs basis. In Section III, we elaborate on the constraints from the perturbative unitarity, the Higgs potential bounded from below, and the electroweak precision observables as well as the flavor constraints on the alignment parameters. And we carry out numerical analysis of the constraints and the alignment of Yukawa couplings in Section IV. A brief summary and conclusions are made in Section V.

II. TWO HIGGS DOUBLET MODEL IN THE HIGGS BASIS
In this section, we study the two Higgs doublet model taking the so-called Higgs basis [73,74,[88][89][90][91][92]. We consider the general potential containing 3 dimensionful quadratic and 7 dimensionless quartic parameters, of which four parameters are complex. We closely examine the relations among the potential parameters, Higgs-boson masses, and the neutral Higgs-boson mixing so as to figure out the set of input parameters to be used in the next Section. We further work out the Yukawa couplings in the Higgs basis together with the interactions of the neutral and charged Higgs bosons with massive gauge bosons.
(λ 6,7 e iξ ) ↔ −(λ 6,7 e iξ ) * . The tadpole conditions in the Higgs basis, which are much simpler than those in the Φ basis as shown in Eq. (3), are where the first condition comes from ∂V H ∂ϕ1 = 0 and the second one from ∂V H ∂ϕ2 = 0 and ∂V H ∂a = 0. Note that the second condition relates the two complex parameters of Y 3 and Z 6 .

B. Masses, Mixing, and Potential Parameters in the Higgs Basis
In the Higgs basis, the 2HDM Higgs potential includes the mass terms which can be cast into the form consisting of two parts in terms of the charged Higgs bosons H ± , two neutral scalars ϕ 1,2 , and one neutral pseudoscalar a. The charged Higgs boson mass is given by while the 3 × 3 mass-squared matrix of the neutral Higgs bosons M 2 0 takes the form where ) v 2 and the 3 × 3 real and symmetric mass-squared matrix M 2 Z is given by Note that the quartic couplings Z 2 and Z 7 have nothing to do with the masses of Higgs bosons and the mixing of the neutral ones. They can be probed only through the cubic and quartic Higgs self-couplings, see Eq. (5) while noting that only the H 1 doublet contains the vev v 246 GeV. We further note that ϕ 1 decouples from the mixing with the other two neutral states of ϕ 2 and a in the Z 6 = 0 limit, and its mass squared is simply given by 2Z 1 v 2 which gives Z 1 0.13 (M H1 /125.5 GeV) 2 . And, in this decoupling limit of Z 6 → 0, the CP-violating mixing between the two states of ϕ 2 and a is dictated only by m(Z 5 ).
Once the 3 × 3 real and symmetric mass-squared matrix M 2 0 is given, the orthogonal 3 × 3 mixing matrix O is defined through 7 for given v and M H ± . Now we are ready to consider the input parameters for 2HDM in the Higgs basis. First of all, the input parameters for the Higgs potential Eq. (5) are Using the tadpole conditions in Eq. (10), the dimensionful parameters Y 1 and Y 3 can be removed from the set in favor of v and observing that the quartic couplings Z 2 and Z 7 do not contribute to the mass terms, one may consider the following set of input parameters where we trade the quartic coupling Z 3 with the charged Higgs mass M H ± using the relation given. Further using M Hi and O instead of {Z 1 , Z 4 , Z 5 , Z 6 }, we end up with the following set of input parameters: which contains 12 real degrees of freedom. If desirable, one may remove the unphysical massive parameter Y 2 in favor of the dimensionless quartic coupling Z 3 by having an alternative set consisting of 12 real parameters as well. For example, in the CP-conserving (CPC) case with mZ 5 = mZ 6 = 0, one may denote the masses of the three neutral Higgs bosons by M h , M H , and . Note that M 2 h = 2Z 1 v 2 is for the SM Higgs boson in the decoupling limit of Z 6 → 0. The mixing matrix O can be parameterized as introducing the mixing angle γ between the two CP-even states ϕ 1 and ϕ 2 . In this CP-conserving case, the relations Eq. (16) simplify into We observe that, in the decoupling limit of sin γ = 0, Z 1 = M 2 h /2v 2 and Z 6 = 0, and Z 4 and Z 5 are determined by the mass differences of M 2 H + M 2 A − 2M 2 H ± and M 2 H − M 2 A , respectively. Finally, for the study of the CPC case, one may choose one of the following two equivalent sets: each of which contains 9 real degrees of freedom, and the convention of |γ| ≤ π/2 without loss of generality resulting in c γ ≥ 0 and sign(s γ ) = sign( In the presence of non-vanishing mZ 5 and/or mZ 6 , the mixing between the two CP-even states ϕ 1,2 and the CP-odd one a arises leading to CP violation in the neutral Higgs sector. By introducing a rotation H 2 → e iζ H 2 , 9 we note that the Higgs potential given by Eq. (5) is invariant under the following phase rotations: Considering the tadpole conditions Eq. (10), this might imply that one of the CP phases of m(Z 5 ), m(Z 6 ), and m(Z 7 ) can be rotated away by rephasing the Higgs fields H 2 . By keeping m(Z 7 ) as an independent input and taking either m(Z 5 ) = 0 or m(Z 6 ) = 0, one may use the following set of input parameters: 10 which contains 11 real degrees of freedom. In this case, the mixing angle η (ω) can be fixed by solving m(Z 5 ) = 0 or m(Z 6 ) = 0 when M H1,2,3 and ω (η) are given. More explicitly, using the relations in Eq. (16), we have parameterizing the mixing matrix O as follow: Assuming c η = 0 and, for example, taking γ and ω as the input mixing angles, the remaining mixing angle η is determined by imposing mZ 5 = 0. If mZ 6 = 0 is imposed instead, η is determined by Of course, using Z 3 instead of Y 2 , one may use the alternative set Incidentally, one may choose the basis in which m(Z 7 ) = 0 by taking the following set of input parameters: where all the three mixing angles are independent from one another and Z 7 is real.
In passing, we note that, in the limit of c γ = 1 and s γ = 0, the mixing matrix takes the simpler form When c η 1 − η 2 /2 and s η η, the lightest H 1 is SM like and the heavier ones H 2,3 are mostly arbitrary mixtures of ϕ 2 and a. On the other hand, when c η |η| and |s η | 1 − η 2 /2, the lightest H 1 is mostly CP odd (H 1 ∼ a) and H 2 (H 3 ) is SM like when |s ω | 1 (|c ω | 1).

C. Yukawa Couplings in Higgs basis
In the 2HDM, the Yukawa couplings might be given by [94] − in terms of the six 3 × 3 Yukawa matrices y u,d,e 1,2 with the electroweak eigenstates Q 0 R , and e 0 R . The two Higgs doublets H 1,2 in the Higgs basis are given by Eq. (6): and their SU(2)-conjugated doublets by The Yukawa interactions include the following mass terms which involve only the Yukawa matrices of y u,d,e 1 . Therefore, introducing two unitary matrices relating the left/righthanded electroweak eigenstates f 0 L,R to the left/right-handed mass eigenstates f L,R with f = u, d, e as follows we have, for the mass terms, where the three diagonal matrices are in terms of the six quark and three charged-lepton masses. We note that U † u L U d L = V CKM ≡ V is nothing but the CKM matrix and, by the use of it, the SU(2) L quark doublets in the electroweak basis can be related to those in the mass basis in the following two ways: The first relation is used for the Yukawa interactions with the right-handed up-type quarks and the second one for those with the right-handed down-type quarks. Incidentally, we also have by defining ν L ≡ U † e L ν 0 L with no physical effects in the case with vanishing neutrino masses. Collecting all the parameterizations, unitary rotations, and re-parameterizations, the couplings of the neutral Higgs bosons to two fermions are given by where three Hermitian and three anti-Hermitian Yukawa coupling matrices are with h f =u,d,e given in terms of the 3 × 3 Yukawa matrix y f 2 and two unitary matrices as We observe that the couplings of the ϕ 1 field are diagonal in the flavor space and their sizes are directly proportional to the masses of the fermions to which it couples. In contrast, those of the ϕ 2 and a fields are not diagonal in the flavor space leading to the tree-level Higgs-mediated FCNC and their magnitudes are arbitrary in principle. To avoid the tree-level FCNC, the matrices h f =u,d,e are desired to be diagonal which can be achieved by requiring [94] 11 along with introducing the three complex alignment parameters ζ f =u,d,e . In this case, the two aligned Yukawa matrices y f 1 and y f 2 can be diagonalized simultaneously and the Yukawa matrices describing the couplings of ϕ 2 and a fields  [104] which guarantees the absence of tree-level Higgs-mediated flavor-changing neutral current (FCNC). For the four types of 2HDM, we follow the conventions found in, for example, Ref. [105].
to the fermion mass eigenstates are given by which leads to the Hermitian and anti-Hermitian Yukawa matrices When m(ζ f ) = 0, the conventional 2HDMs based on the Glashow-Weinberg condition [104] can be obtained by choosing ζ f as shown in Table I. Otherwise, the couplings of the mass eigenstates of the neutral Higgs bosons H i=1,2,3 to two fermions are given by with the scalar and pseudoscalar couplings given by where the upper and lower signs are for the up-type fermions f = u, c, t and the down-type fermions f = d, s, b, e, µ, τ , respectively. The simultaneous existence of the scalar g S Hif f and pseudoscalar g P Hif f couplings for a specific H i signals the CP violation in the neutral Higgs sector. We figure out that there are two different sources of the neutral Higgssector CP violation: (i) one is the CP-violating mixing among the CP-even and CP-odd states arising in the presence of non-vanishing m(Z 5,6 ) in the Higgs potential and (ii) the other one is the complex alignment parameters of ζ f 's. Note that the second source is absent in the conventional four types of 2HDMs since ζ f 's are real in those models.
The couplings of charged Higgs bosons to two fermions are given by in terms of the CKM matrix V and the 3 × 3 Yukawa matrices h u,d,e .
Previously, we note that the Higgs potential given by Eq. (5) is invariant under the phase rotation H 2 → e iζ H 2 if the complex potential parameters are accordingly rephased, see Eq. (24). This observation extends to the Yukawa interactions, Eq. (33), by noting that they are invariant under the phase rotations: Under the alignment assumption y f 2 = ζ f y f 1 given by Eq. (45), the above rephasing invariant rotations become in terms of the complex alignment parameters. Then one may be able to show that the scalar and pseudoscalar couplings given by Eq. (49) are invariant under the phase rotations given by Eq. (52) as they should be. To be explicit, we first note that, under the phase rotation H 2 → e iζ H 2 , the electroweak Higgs basis changes as follow: which leads to On the other hand, under the rotations ζ f → e ±iζ ζ f given in Eq. (52), one may have with the upper and lower signs being for the up-type massive fermions f = u and the down-type massive fermions f = d, e, respectively. Using Eqs. (55) and (56), it is straightforward to show that the scalar and pseudoscalar couplings given by Eq. (49) are invariant under the phase rotations of Eq. (52).
To summarize, assuming y f 2 = ζ f y f 1 with ζ f =u,d,e being the three complex alignment parameters and combining Eqs. (24) and (52), we note that the Higgs potential and the Yukawa interactions are invariant under the following phase rotations: which, taking account of the CP odd tadpole condition Y 3 + Z 6 v 2 /2 = 0, lead to five rephasing-invariant CPV phases in total. This leaves us more freedom to choose the input parameters for the Higgs potential other than I CPV (25), I CPV (30), or I CPV (31). For example, one may assign three CPV phases to the Higgs potential and take ζ u real and positive definite. In this case, the full set of input parameter is to be which contains 12 and 5 real degrees of freedom in the Higgs potential and the Yukawa interactions, respectively, with Z 7 , ζ d and ζ e being fully complex.

D. Interactions with Massive Vector Bosons
The cubic interactions of the neutral and charged Higgs bosons with the massive gauge bosons Z and W ± are described by the three interaction Lagrangians: leading to the following sum rules: On the other hand, the quartic interactions of the neutral and charged Higgs bosons with the massive gauge bosons Z and W ± and massless photons are given by with g H i H j V V = δ ij and

III. CONSTRAINTS
In this Section, we consider the perturbative unitarity (UNIT) conditions and those for the Higgs potential to be bounded from below (BFB) to obtain the primary theoretical constraints on the potential parameters or, equivalently, the constraints on the Higgs-boson masses including correlations among them and the mixing among the three neutral Higgs bosons. We further consider the constraints on the Higgs masses and their couplings with vector bosons taking into account the electroweak oblique corrections to the so-called S and T parameters. We emphasize that all the three types of constraints from the perturbative unitarity, the Higgs potential bounded from below, and the electroweak precision observables (EWPOs) are independent of the basis chosen and working in the Higgs basis does not invoke any restrictions. We also consider the constraints on |ζ e |, |ζ u |, and the product of ζ u ζ d taking account of the charged Higgs contributions to the flavor-changing τ decays into light leptons, Z → bb, K , and b → sγ [96,106]. 12

A. Perturbative Unitarity
For the unitarity conditions, we closely follow Ref. [108] 13 considering the three scattering matrices of M S 1,2,3 which are expressed in terms of the quartic couplings Z 1−7 , see also Ref. [109]. The two 4 × 4 real and symmetric scattering matrices M S 1 and M S 2 are given by 12 We refer to, for example, Ref. [107] for an extensive study of flavor observables in the conventional 2HDMs taking the Φ basis. 13 We keep our conventions for the potential parameters.
where η 00 = Z 1 + Z 2 + Z 3 and I = Z 3 − Z 4 . The row vector η T is given by and the 3 × 3 real and symmetric matrix E by The third 3 × 3 scattering matrix M S 3 is Hermitian which takes the form of And then, the unitarity conditions are imposed by requiring that the 11 eigenvalues of the three scattering matrices M S 1,2,3 and the quantity I should have their moduli smaller than 4π. When Z 6 = Z 7 = 0, the 12 unitarity conditions simplify into While taking Z 1 = Z 2 = Z 3 = Z 4 = Z 5 = 0, one may have Then, by combining them, one may arrive at the following UNIT conditions for individual parameters [108] |Z 1,2,5 | < 2π/3 , |Z 6,7 | < 2 √ 2π/3 ,

B. Higgs Potential Bounded-from-below
We consider the following 5 necessary conditions for the most general 2HDM Higgs potential with explicit CP violation to be bounded-from-below in a marginal sense [108]: 14 Note that though the quartic couplings Z 2 and Z 7 have no direct relations to the masses and mixing of Higgs bosons but they are interrelated with the other five quartic couplings of Z 1,3−6 through the UNIT and BFB conditions.

C. Electroweak Precision Observables
The electroweak oblique corrections to the so-called S, T and U parameters [110,111] provide significant constraints on the quartic couplings of the 2HDM. Fixing U = 0 which is suppressed by an additional factor M 2 Z /M 2 BSM 15 compared to S and T , the S and T parameters are constrained as follows  In 2HDM, the S and T parameters might be estimated using the following expressions [113,114] In this work, we ignore the vertex corrections δ H ± γZ , δ HiHj Z , and δ Hi W since the size of the most of the quartic couplings are smaller than 3 and the quantum corrections proportional to ∼ Z 2 i /16π 2 might be negligible. Then, we observe that all the relevant couplings are determined by the three physical couplings of We note that F ∆ (m, m) = 0 and F ∆ (m, m) = 1 3 ln m 2 . 18 When g 2 H 1 V V = 1, neglecting the Z 2 -dependent vertex correction factors δ H ± γZ , δ Hi W and δ

D. Flavor Constraints on the Alignment Parameters
The alignment parameters ζ f =u,d,e are constrained by considering the charged Higgs contributions to the low energy observables such as flavor-changing τ decays, leptonic and semileptonic decays of pseudoscalar mesons, the Z → bb process, B meson mixing, the CPV parameter K in K meson mixing, and the radiative b → sγ decay [96]. In this work, we consider the flavor constraints on the absolute sizes of ζ e , ζ u and ζ d . Note that we neglect the constraints on the products of the alignment parameters taking account of only the single constraints on the absolute values of ζ e , ζ u and ζ d under the assumption that they are fully independent from each other. 15 Here, M BSM denotes some heavy mass scale involved with new physics beyond the Standard Model. 16 See the 2020 edition of the review "10. Electroweak Model and Constraints on New Physics" by J.Erler and A. Freitas in Ref. [112]. 17 See, for example, Ref. [115]. 18 Here and after, ln m 2 could be understood as, for example, ln m 2 /(1 GeV) 2 if necessary. 19 The S Φ and T Φ parameters are independent of M H 1 when g 2 H 1 V V = 1. The flavor-changing τ decays into light leptons provide the following constraint on |ζ e | [96]: at 95% CL. On the other hand, the constraint on |ζ u | may come from the Z-peak precision observables involving the Z → bb decay assuming the quantum corrections to the Zbb vertex beyond the SM is dominated by the charged Higgs contributions. More explicitly, the ratio R b = Γ(Z → bb)/Γ(Z → hadrons) is used by neglecting the contributions depending on |ζ d | which are suppressed by the factor m t (M Z )/m b (M Z ) ∼ 60 compared to those depending on |ζ u |. It turns out that the upper limit on |ζ u | linearly increases with M H ± as follow [96]: To be very strict, the above upper limit should be applied only when |ζ d | = 0. The similar while more direct upper limit |ζ u | could be obtained by considering the CPV parameter K in K meson mixing which depends on |ζ u | only neglecting the masses of the light d and s quarks. Actually the limit from K is slightly stronger than that from Z → bb by the amount of about 10% [96]. In this work, for the upper limit on |ζ u |, we apply the slightly weaker constraint from Z → bb given by Eq. (77) while considering it valid independently of ζ d . In passing, for the ∆B = 2 processes mediated by box diagrams with exchanges of W ± and/or H ± bosons, we note that the leading Willson coefficients which are not suppressed by the light quark mass depend ζ u and ζ d . When ζ d = 0, one might obtain the similar upper limit on |ζ u | as that from K [96].
There is no limit on ζ d independently of ζ u and/or ζ e . But one may extract some interesting information on ζ d considering the radiative b → sγ decay. Numerically, the decay amplitude can be cast into the following form [106,116]: 20 When ζ u ζ d is negative, the interference with the SM amplitude is always constructive and the product is constrained to be small and, as usual, |ζ d | can be significantly larger (smaller) than 1 only when |ζ u | is very small (large). On the contrary, if ζ u ζ d is positive, |ζ d | could be large independently of |ζ u |. In this case, a destructive interference occurs and the experimental constraints can be satisfied when Combining the upper limit on |ζ u | given by Eq. (77), we observe that the destructive interference can always occur when and ζ u ζ d > 0. Most generally, allowing ζ u ζ d to be complex, it turns out that the rough 95% CL upper limit on the absolute value of the product is basically saturated by the relation given by Eq. (79) [96] or For the summary, we present the upper limits on |ζ e |, |ζ u |, and |ζ e ζ d | and the lower limit on |ζ d | in Fig. 1. Before closing this section, we briefly comment on the constraints from the heavy Higgs boson searches carried out at the LHC. The heavy neutral Higgs bosons have been searched through their decays into τ + τ − [117][118][119][120], bb [121], tt [122][123][124], W W [125], ZZ [126][127][128][129], Zh 125GeV [130,131], etc. On the other hand, the charged Higgs boson search channels include the decay modes into τ ± ν [132,133], tb [134][135][136], cb [137], cs [138,139], and W h 125GeV [130]. Basically, the experimental upper limits on the product of the production cross section and the decay rate into a specific search mode have been analyzed to obtain the allowed parameter space of a specific model. For example, the search in the τ + τ − final state excludes the presence of a heavy neutral Higgs with M A below about 1 TeV at 95% CL in the minimal supersymmetric extension of the SM (MSSM) when, depending on scenarios, tan β > ∼ 15 ∼ 25 and the exclusion contour reaches up to M A = 1.6 TeV for tan β = 60 [118]. While in the aligned 2HDM taken in this work, the Yukawa couplings of the up-and down-type quarks and the charged leptons to heavy Higgs bosons are completely uncorrelated and the interpretation of the experimental limits is much more involved. This is because the three alignment parameters of ζ u,d,e are independent from each other while all of them are involved in the calculation of the decay rate pertinent to a specific search mode. In principle, one can easily avoid the constraints from, for example, H/A → τ τ and H ± → τ ν by taking |ζ e | 1. But it might be still allowed to have |ζ e | > ∼ 20 and M A < 1 TeV if one can suppress the branching fraction into τ + τ − by choosing the other alignment parameters of ζ u and ζ d appropriately. In this respect, a through analysis of the experimental search results in the framework of aligned 2HDM with three independent alignment parameters deserves an independent full consideration. In this work, we simply assume that the parameter space considered in the next Section could be made more or less safe from the LHC constraints from no observation of significant excess in the heavy Higgs boson searches by judiciously manipulating the three alignment parameters which are otherwise uncorrelated.  Yukawa couplings of the lightest Higgs boson is delayed compared to its coupling to a pair of massive vector bosons: where we use the relation α=ϕ1,ϕ2,a O 2 αi = 1 for i = 1. We observe that the delay factor ∆ H1f f defined above is basis-independent and can be generally used even in the CPV case. Anticipating that the impacts on the Yukawa delay factor due to the CP-violating phases of Z 5,6,7 and ζ u,d,e are redundant, we consider the CP-conserving (CPC) case for our numerical study for simplicity. For a recent global analysis of the aligned CPC 2HDM taking account of several phenomenological constraints as well as theoretical requirements, we refer to Ref. [140] but with a caution. 21

A. UNIT and BFB constraints
First of all, we consider the UNIT and BFB constraints. Observing that the two conditions depend only on the quartic couplings Z 1−7 , we take the following set of input parameters: 22 In the left panel of Fig. 2, we show the scatter plots of Z 2 versus Z 1 (upper left), Z 4 versus Z 3 (upper right), Z 5 versus Z 1 (lower left), and Z 7 versus Z 6 (lower right). The plots are produced by randomly generating the quartic couplings in the I Z CPC set. In each plot, the black points are obtained by imposing only the simplified UNIT conditions of Eqs. (68) and (69). The full consideration of the UNIT conditions based on the scattering matrices M S 1,2,3 produces the red points. The results obtained by simultaneously imposing the full UNIT and BFB conditions (UNIT⊕BFB) are denoted by the blue points. We find our results are very consistent with those presented in Ref. [108]. After imposing the UNIT and UNIT⊕BFB conditions, we note that the normalized distributions of the quartic couplings are no longer flat as shown in the right panel of Fig. 2. As in the left panel, the distributions of the quartic couplings obtained by requiring only the UNIT (red) conditions and the combined UNIT⊕BFB conditions are in red and blue, 21 In Ref. [140], the authors take λ  23), we find that the potential parameters Z 5 and Z 6 are used more than needed while Z 2 and Z 3 are missing in the set. Note that Z 5 and Z 6 are entirely fixed when the mixing angle and the three neutral Higgs masses are given, see Eq. (22), and the parameter Z 2 should be included at least because it is independent of the Higgs masses and mixing like as Z 7 . 22 To have I Z CPC from Eq. (18), we trade M H ± with Z 3 . Note that the dimensionful parameter Y 2 is irrelevant for the UNIT and BFB constraints.
respectively. We note that the smaller |Z 6 + Z 7 | and the positive Z 3 values are preferred by further imposing the BFB conditions in addition to the UNIT ones, see Eq. (71).

B. Electroweak constraints
Coming to the electroweak (ELW) constraints, since the oblique corrections are expressed in terms of the masses and couplings of Higgs bosons, it is more natural and convenient to take the following set of input parameters: referring to Eq. (23). In the I CPC set, all the massive parameters are physical Higgs masses except v = √ 2G F −1/2 246.22 GeV. We assume that the neutral state h = H 1 is the lightest Higgs boson and plays the role of the SM Higgs boson in the decoupling limit of s γ = 0 by taking M H1 = 125.5 GeV [76]. And, for the masses of heavy Higgs bosons, we randomly generate their masses squared between M 2 H1 and (1.5 TeV) 2 . For the mixing angle γ, we take the convention of |γ| ≤ π/2 without loss of generality resulting in c γ ≥ 0 and sign(s γ ) = sign(Z 6 ). For the implementation of the UNIT and BFB constraints using the set I CPC , we recall the quartic couplings Z 1,4,5,6 in terms of the Higgs masses and the mixing angle γ in the CPC case given by Eq. (22).
Using the set I CPC for the input parameters in the CPC case, the S and T parameters given by Eq. (74) take the following simpler forms: ignoring the vertex corrections. We observe that T CPC Φ is identically vanishing when M H ± = M A and, when keeping the leading terms. To obtain Eq. (86) for the approximated expressions of the S and T parameters, we use for m 0 ∼ m 1 and for m 1 m 0 . In the left panel of Fig. 3, we show the S and T parameters imposing the UNIT, BFB, and ELW constraints abbreviated by the combined UNIT⊕BFB⊕ELW 95% ones. Note that the 95% CL ELW limits are adopted and the heavy Higgs masses squared are scanned up to (1.5 TeV) 2 . We find that S takes values in the range between −0.02 and 0.05 whose absolute values are smaller than σ S = 0.07, see Eq. (73). Actually, we find that |S| < σ S even with only the UNIT and BFB constraints imposed. Note that S is mostly negative (positive) when M H ± > (<)M A . Specifically, we find that S −1/4π (5/18) −0.02 when M H ± − M A = 0 and γ = π/2. The T parameter takes its value between −0.02 and 0.13 which are given by the delimited range determined by −0.02 < S < 0.05, the strong correlation ρ ST = 0.92 and R 2 95% = 5.99, see Eqs. (72) and (73) and the lower-right plot in the left panel of Fig. 3. Incidentally, we observe that T = 0 when M H ± = M A though it quickly deviates from 0 when M H ± = M A . In the right panel of Fig. 3, we show the correlations among the mass differences and the mixing angle γ using the set I CPC . We find that when M H ± > ∼ 500 GeV (1 TeV). We show the correlations among the heavy Higgs-boson masses and the mixing angle γ in the left panel of Fig. 4. Requiring the ELW constraint in addition to the UNIT⊕BFB ones, we find that Z 1 and γ take values near to 0 and Z 4 and Z 5 positive ones more likely, see the right panel of Fig. 4. We find that the UNIT and BFB conditions combined with the ELW constraint restrict the quartic couplings as follows:

C. Alignment of Yukawa couplings
Now, we have come to the point to address the alignment of Yukawa couplings. When we talk about the alignment of the Yukawa couplings in general 2HDMs, we imply: (i) the alignment of them in the flavor space and (ii) the alignment of the lightest Higgs-boson couplings to a pair of the SM fermions in the decoupling limit of M H,A,H ± → ∞. By (i), we precisely mean the assumption that the two Yukawa matrices of y f 1 and y f 2 are aligned in the flavor space or with f = u and d for the up-and down-type quarks, respectively, and f = e for the three charged leptons. Then, by (ii), one might mean In Eq. (91), we note that the quantity c γ is nothing but the coupling g H 1 V V = O ϕ11 = c γ which is driven to take the SM value of 1 by the combined UNIT, BFB, and ELW constraints as M H,A,H ± increases. Therefore, from Eq. (82), the Yukawa delay factor simplifies to ∆ H1f f = |ζ f s γ |, and the alignment of the lightest Higgs-boson couplings to the SM fermions in the decoupling limit is delayed by the amount of |ζ f s γ | which can not be ignored even when |s γ | 1 if |ζ f | is significantly larger than 1.
For a quantitative study, in addition to I CPC given by Eq. (84), we have added the following set of input parameters containing three real parameters: In the left panel of Fig. 5, we show the correlations between each of the three alignment parameters ζ f =u,d,e and the mixing angle γ when the absolute value of the corresponding coupling g S H1f f is within 10% range of the SM value of 1 or |g S H1f f − 1| < 0.1 and |g S H1f f + 1| < 0.1 for g S H1f f > 0 (red) and g S H1f f < 0 (blue), respectively. Scanning |γ| ≤ π/2, g S H1f f 1 near γ = 0. At γ = ±π/2, the g S H1f f coupling takes the value of 1 when ζ f = ∓1 (red). While if ζ f = ±1, we note that g S H1f f = −1 at γ = ±π/2 (blue). In the right panel of Fig. 5, by the four lines, we show the correlations between ζ d and ζ e in the four conventional 2HDMs 24 based on appropriately defined discrete Z 2 symmetries taking 1/100 < ζ u = 1/t β < 2, see Table I. We observe that both ζ d and ζ e are bounded only in the type-I 2HDM between 1/100 and 2. Otherwise, at least one of them is limitless in principle. Therefore, except the type-I 2HDM, g S H1dd and/or g S H1ēe could be largely deviated from 1 in the decoupling limit even when ζ u is limited. To concentrate on the alignment of the lightest Higgs-boson couplings to a pair of the SM fermions in the decoupling 30% and 10% ranges of the SM value of 1 when M H ± > 500 GeV and M H ± > 1 TeV, respectively. As previously discussed, the alignment of the coupling g S H1f f is delayed by the amount of ζ f sin γ compared to the coupling g H 1 V V On the other hand g S H1dd ∼ −1 allows the larger values of ζ d given by ζ d = (1 + cos γ)/ sin γ ±2M 2 H ± /(200 GeV) 2 , see the blue points in the left panel of Fig. 7. In the same panel for ζ d versus M H ± , we also show the lower limit on |ζ d | from b → sγ through the destructive interference by the magenta lines, see Eq. (80) and the lower-left panel of Fig. 1. We observe that the two regions with |g S H1dd + 1| < 0.1 are mostly outside the band delimited by the two magenta lines implying that large values of |ζ d | for g S H1dd ∼ −1 are hardly constrained by b → sγ. For g S H1ēe , the same arguments are applied, see the right panel of Fig. 7. Note that the constraints from the flavor-changing τ decays into light leptons given by Eq. (76) are too weak to affect those on g S H1ēe by the precision LHC Higgs data. Lastly, we comment on the wrong-sign alignment limit in the four types of conventional 2HDMs in which the H 1 couplings to the down-type quarks and/or those to the charged leptons are equal in strength but opposite in sign to the corresponding SM ones. The two couplings g S H1dd and g S H1ēe are completely independent from each other in general 2HDM. But, in the conventional four types of 2HDMs, they are related. We observe that the couplings are given by either cos γ − sin γ/t β or cos γ + t β sin γ in any type of 2HDMs, see Table I. In this case, cos γ − sin γ/t β = ±1 for the t β value which makes cos γ + t β sin γ = ∓1. This implies that, independently of 2HDM type and regardless of the heavy Higgs-mass scale, all four types of 2HDMs could be viable against the LHC Higgs precision data in the wrong-sign alignment limit.

V. CONCLUSIONS
We have studied the alignment of Yukawa couplings in the framework of general 2HDMs identifying the lightest neutral Higgs boson as the 125 GeV one discovered at the LHC. We take the so-called Higgs basis [73,74,[88][89][90][91][92] for the Higgs potential in which only one of the two doublets contains the non-vanishing vacuum expectation value v. For the Yukawa couplings, rather than invoking the Glashow-Weinberg condition [104] based on appropriately defined discrete Z 2 symmetries, we require the absence of tree-level FCNCs by assuming that the Yukawa matrices describing the couplings of the two Higgs doublets to the SM fermions are aligned in the flavor space [93][94][95].
For a numerical study, we further assume that the seven quartic couplings Z i=1−7 appearing in the Higgs potential and the three alignment parameters ζ f =u,d,e for Yukawa couplings are all real by anticipating that the impacts due to CP-violating phases of Z 5,6,7 and ζ f 's on the alignment of Yukawa couplings are redundant. In this case, in addition to the vev v and masses of the SM fermions, the model can be fully described by specifying the following set of 11 free parameters: where M h,H (with M h < M H ) and M A denote the masses of CP-even and CP-odd neutral Higgs bosons, respectively, and the mixing between the two CP-even neutral states is described by the angle γ. The quartic couplings Z 1,4,5,6 are determined in terms of M h,H,A , M H ± , γ, and v. The quartic coupling Z 3 is related to the massive parameter Y 2 appearing in the Higgs potential through Y 2 = M 2 H ± − Z 3 v 2 /2. On the other hand, the the other quartic couplings Z 2 and Z 7 have no direct relevance to the masses and mixing of Higgs bosons. But, we observe that they are interrelated with the other five quartic couplings of Z 1,3−6 through the perturbative unitarity (UNIT) conditions and those for the Higgs potential to be bounded from below (BFB). We note that the UNIT and BFB conditions are basis-independent, i.e., the same in any basis [108]. Also considered are the constraints from the electroweak (ELW) oblique corrections to the S and T parameters which are expressed in terms of the physical observable quantities of M h,H,A , M H ± , and g H i V V which are again invariant under a change of basis [84]. We further consider the constraints on the alignment parameters ζ f =u,d,e from flavor-changing τ decays, R b , K , and the radiative b → sγ decay. For the independent model parameters and the rephasing invariant combinations of CP-violating phases, among the several points already discussed in the literature, we highlight the following ones: 1. The general 2HDM potential can be fully specified with the masses of the charged and three neutral Higgs bosons, the orthogonal neutral-Higgs boson mixing matrix O 3×3 and the three dimensionless quartic couplings of Z 2,3,7 in addition to the vev v.
2. For the CP phases, as far as the Higgs potential and the three complex alignment parameters for the Yukawa couplings are involved, the Lagrangians are invariant under the following phase rotations: which, taking account of the CP odd tadpole condition Y 3 + Z 6 v 2 /2 = 0, lead to the following five rephasinginvariant CPV phases: pivoting, for example, around the complex quartic coupling Z 5 .
Incidentally, it is well known that the 3 alignment parameters are the same ζ u = ζ d = ζ e = 1/t β in the type-I 2HDM.
In this case, they cannot be significantly large than 1 since t β 1 leads to a non-perturbative top-quark Yukawa coupling and a Landau pole close to the TeV scale. Therefore, in the type-I model among the 4 conventional 2HDMs, all the Yukawa couplings of the lightest Higgs boson most quickly approach the corresponding SM values as the masses of the heavy neutral Higgs bosons increase and their decouplings are least delayed.
We further suggest the following points as the main results specifically pertinent to our analysis: 1. By scanning the heavy Higgs masses up to 1.5 TeV, we find that the UNIT and BFB conditions combined with the ELW constraint restrict the quartic couplings as follows: And, when M H ± > ∼ 500 GeV (1 TeV), we also find that (110) , 2. As the masses of heavy Higgs bosons increase, compared to the g H 1 V V coupling of the lightest Higgs boson to a pair of massive vector bosons, the decoupling of the Yukawa couplings to the lightest Higgs boson is delayed by the amount of the Yukawa delay factor ∆ H1f f = |ζ f |(1 − g 2 H 1 V V ) 1/2 which is basis-independent and can be generally used even in the presence of CPV phases. Therefore, though g H 1 V V approaches its SM value of 1 very quickly as the masses of heavy Higgs bosons increase, the coupling of H 1 to a pair of fermions can significantly deviate from its SM value if |ζ f | is large. Note that |ζ u | is constrained to be small by R b and K , see Eq. (77). While |ζ d | and |ζ e | are constrained to be small by the LHC precision Higgs data when the corresponding Yukawa couplings are with the similar strength and the same sign as the SM ones. But it could be large when the Yukawa coupling takes the wrong sign.
3. The wrong-sign alignment, in which the H 1 couplings to a pair of f -type fermions are equal in strength but opposite in sign to the corresponding SM ones, occurs when ζ f = (1 + cos γ)/ sin γ independently of the heavy Higgs-boson masses. In the conventional four types of 2HDMs, ζ f = −t β or 1/t β and the Yukawa couplings are given by either cos γ − sin γ/t β or cos γ + t β sin γ in any type of 2HDMs. We observe that cos γ − sin γ/t β = ∓1 for the t β value making cos γ + t β sin γ = ±1 and any type of conventional 2HDMs is viable against the LHC Higgs precision data.
4. Last but not least, by combining with the upper limit on |ζ u | from R b and K , we derive the lower limit on |ζ d | independently of ζ u and ζ e when the non-SM contribution to b → sγ is about two times of the SM one at the amplitude level. constraints on its couplings from a combined ATLAS and CMS analysis of the LHC pp collision data at