High scale flavor alignment in two-Higgs doublet models and its phenomenology

The most general two-Higgs doublet model (2HDM) includes potentially large sources of flavor changing neutral currents (FCNCs) that must be suppressed in order to achieve a phenomenologically viable model. The flavor alignment ansatz postulates that all Yukawa coupling matrices are diagonal when expressed in the basis of mass-eigenstate fermion fields, in which case tree-level Higgs mediated FCNCs are eliminated. In this work, we explore models with the flavor alignment condition imposed at a very high energy scale, which results in the generation of Higgs-mediated FCNCs via renormalization group running from the high energy scale to the electroweak scale. Using the current experimental bounds on flavor changing observables, constraints are derived on the aligned 2HDM parameter space. In the favored parameter region, we analyze the implications for Higgs boson phenomenology.

Motivations for extending the Higgs sector beyond its minimal form have appeared often in the literature. For example, the minimal supersymmetric extension of the Standard Model, which is invoked to explain the stability of the electroweak symmetry breaking scale with respect to very high mass scales (such as the grand unification or Planck scales), requires a second Higgs doublet [4][5][6][7] to avoid anomalies due to the Higgsino partners of the Higgs bosons. More complicated scalar sectors may also be required for a realistic model of baryogenesis [8]. Finally, the metastability of the SM Higgs vacuum [9][10][11] can be rendered stable up to the Planck scale in models of extended Higgs sectors [12][13][14][15][16][17][18][19]. Even in the absence of a specific model of new physics beyond the Standard Model, an enlarged scalar sector can provide a rich phenomenology that can be probed by experimental searches now underway at the LHC.
One of the simplest extensions of the SM Higgs sector is the two-Higgs doublet model (2HDM). 1 In its most general form, the 2HDM is incompatible with experimental data due to the existence of unsuppressed tree-level Higgs-mediated FCNCs, in contrast to the SM where tree-level Higgs-mediated FCNCs are absent. To see why this is so, consider the Higgs-fermion Yukawa interactions expressed in terms of interaction eigenstate fermion fields. Due to the non-zero vacuum expectation value (vev) of the neutral Higgs field, fermion mass matrices are generated. Redefining the left and right-handed fermion fields by separate unitary transformations, the fermion mass matrices are diagonalized. In the SM, this diagonalization procedure also diagonalizes the neutral Higgs-fermion couplings, and consequently no tree-level Higgs-mediated FCNCs are present. In contrast, in a generic 2HDM, the diagonalization of the fermion mass matrices implies the diagonalization of one linear combination of Higgs-fermion Yukawa coupling matrices. As a result, tree-level Higgs-mediated FCNCs remain in the 2HDM Lagrangian when expressed in terms of mass-eigenstate fermion fields. If it were possible in the 2HDM to realize flavor-diagonal neutral Higgs couplings at tree-level (thereby eliminating all tree-level Higgs-mediated FCNCs), then all FCNC processes arising in the model would be generated at the loop-level, with magnitudes more easily in agreement with experimental constraints. 2 A natural mechanism for eliminating the tree-level Higgs-mediated FCNCs was proposed by Glashow and Weinberg [22] and by Paschos [23] [GWP]. One can implement the GWP mechanism in the 2HDM by introducing a Z 2 symmetry to eliminate half of the Higgs-fermion Yukawa coupling terms. In this case, the fermion mass matrices and the non-zero Higgs-fermion Yukawa coupling matrices (which are consistent with the Z 2 symmetry) are simultaneously diagonalized. Indeed, there are a number of inequivalent implementations of the GWP mechanism, resulting in the so-called Types I [24,25], and II [25,26], and Types X and Y [27,28] versions of the 2HDM. 3 1 For a review with a comprehensive list of references, see Ref. [20]. 2 Even in models with flavor-diagonal neutral Higgs couplings, one-loop processes mediated by the charged Higgs boson can generate significant FCNC effects involving third generation quarks. Such models, in order to be consistent with experimental data, will produce constraints in the [m H ± , tan β] plane. The most stringent constraint of this type, obtained in Ref. [21] in the analysis of the Type-II 2HDM prediction for b → sγ, yields m H ± 580 GeV at 95% CL. 3 However, if additional degrees of freedom exist at the TeV scale, then the GWP mechanism is in general not sufficient to protect the theory from FCNCs that are incompatible with the experimental data. These TeV-scale degrees of freedom, when integrated out, can generate higher-dimensional operators of the type (c1/Λ 2 )QLY (6) u1 URH2|H1| 2 +· · · , which break the proportionality relation between quark masses and effective Yukawa interactions with the neutral scalars. As a result, such models generically generate FCNC processes that are not sufficiently suppressed [29]. of freedom. All physical observables must be independent of this phase. In particular, we examine in detail the structure of the Higgs-fermion Yukawa couplings and exhibit its flavor structure. In the formalism presented in section 2, we initially allow for the most general form of the Higgs scalar potential and the Yukawa coupling matrices. In particular, new sources of CP-violation beyond the SM can arise due to unremovable complex phases in both the scalar potential parameters and the Yukawa couplings. For simplicity, we subsequently choose to analyze the case of a CP-conserving Higgs scalar potential and vacuum, in which case the neutral mass-eigenstates consist of two CP-even and one CP-odd neutral Higgs bosons. We then introduce the flavor-aligned 2HDM, in which the Yukawa coupling matrices are diagonal in the basis of quark and lepton mass-eigenstates. However, alignment is not stable under renormalization group running. Following the framework for flavor discussed above, we impose the alignment condition at the Planck scale and then evaluate the Yukawa coupling matrices of the Higgs basis at the electroweak scale as determined by renormalization group running, subject to the observed quark and lepton masses and the CKM mixing matrix. The renormalization group running is performed numerically and checked in the leading log approximation, where simple analytic expressions can be obtained. In this context, a comparison with general Minimal Flavor Violating 2HDMs is performed.
In section 3, we discuss the implications of high-scale flavor alignment for high energy processes. We focus on flavor-changing decays of the top quark and on the phenomenology of the heavy neutral CP-even and CP-odd Higgs bosons. In section 4, we discuss the implications of high-scale flavor alignment for low energy processes. Here we consider constraints arising from neutral meson mixing observables and from B s → + − , which receive contributions at tree-level from neutral Higgs exchange, and from the charged Higgs mediated B → τ ν decay. By comparing theoretical predictions to experimental data, one can already probe certain regions of the A2HDM parameter space. Additional parameter regions will be probed by future searches for heavy Higgs bosons and measurements of B-physics observables. Conclusions of this work are presented in section 5. Finally, in Appendix A we review the derivation of the Yukawa sector of our model in the fermion mass-eigenstate basis, and in Appendix B we exhibit the one-loop matrix Yukawa coupling RGEs used in this analysis.
2 The flavor-aligned 2HDM 2.1 Theoretical framework for the 2HDM Consider a generic 2HDM consisting of two complex, hypercharge-one scalar doublets, Φ 1 and Φ 2 . The most general renormalizable scalar potential that is invariant under local SU(2)×U(1) gauge transformations can be written as The parameters of the scalar potential can be chosen so that the minimum of the scalar potential is achieved when the neutral components of the two scalar doublet fields acquire non-zero vacuum expectation vales, Φ 0 where the (potentially complex) vevs satisfy v 2 ≡ |v 1 | 2 + |v 2 | 2 (246 GeV) 2 , (2.2) as required by the observed W boson mass, m W = 1 2 gv. The SU(2)×U(1) gauge symmetry is then spontaneously broken, leaving an unbroken U(1) EM gauge group.
In the most general 2HDM, the fields Φ 1 and Φ 2 are indistinguishable. Thus, it is always possible to define two orthonormal linear combinations of the two doublet fields without modifying any prediction of the model. Performing such a redefinition of fields leads to a new scalar potential with the same form as Eq. (2.1) but with modified coefficients. This implies that the coefficients that parameterize the scalar potential in Eq. (2.1) are not directly physical [37].
To obtain a scalar potential that is more closely related to physical observables, one can introduce the so-called Higgs basis in which the redefined doublet fields (denoted below by H 1 and H 2 ) have the property that H 1 has a non-zero vev whereas H 2 has a zero vev [37,38]. In particular, we define the new Higgs doublet fields: k q k1 q k2 1 c 12 c 13 −s 12 − ic 12 s 13 2 s 12 c 13 c 12 − is 12 s 13 3 s 13 ic 13 Table 1. Invariant combinations of the neutral Higgs boson mixing angles θ 12 and θ 13 , where c ij ≡ cos θ ij and s ij ≡ sin θ ij .
To identify the neutral Higgs mass-eigenstates, we diagonalize the squared-mass matrix M 2 . The diagonalization matrix is a 3 × 3 real orthogonal matrix that depends on three angles: θ 12 , θ 13 and θ 23 . Following Ref. [39], where the h i are the mass-eigenstate neutral Higgs fields, c ij ≡ cos θ ij and s ij ≡ sin θ ij . Under the rephasing H 2 → e iχ H 2 , θ 12 , θ 13 are invariant, and θ 23 → θ 23 − χ . (2.10) Assuming that Z 6 ≡ |Z 6 |e iθ 6 = 0, 5 it is convenient to define the invariant mixing angle, In light of the freedom to define the mass-eigenstate Higgs fields up to an overall sign, the invariant mixing angles θ 12 , θ 13 and φ can be determined modulo π. By convention, we choose − 1 2 π ≤ θ 12 , θ 13 < 1 2 π , and 0 ≤ φ < π . (2.12) The physical neutral Higgs states (h 1,2,3 ) are then given by: where the q k1 and q k2 are invariant combinations of θ 12 and θ 13 , which are exhibited in Table 1 [39]. The masses of the neutral Higgs bosons h i will be denoted by m i , respectively. It is convenient to define the physical charged Higgs states by Although the explicit formulae for the neutral Higgs boson masses and mixing angles are quite complicated, there are numerous relations among them which take on rather simple forms. The following results are noteworthy [39,40]: We next turn to the Higgs-fermion Yukawa couplings. As reviewed in Appendix A, one starts out initially with a Lagrangian expressed in terms of the scalar doublet fields Φ i (i = 1, 2) and interaction-eigenstate quark and lepton fields. After electroweak symmetry breaking, one can reexpress the scalar doublet fields in terms of the Higgs basis fields H 1 and H 2 . At the same time, one can identify the 3 × 3 quark and lepton mass matrices. By redefining the left and right-handed quark and lepton fields appropriately, the quark and lepton mass matrices are transformed into diagonal form, where the diagonal elements are real and non-negative. The resulting Higgs-fermion Yukawa Lagrangian is given by in Eq. (A.16) and is repeated here for the convenience of the reader [40], where U = (u, c, t) and D = (d, s, b) are the mass-eigenstate quark fields, K is the CKM mixing matrix, N = (ν e , ν µ , ν τ ) and E = (e, µ, τ ) are the mass-eigenstate lepton fields, and κ and ρ are 3 × 3 Yukawa coupling matrices. Note that F R,L ≡ P R,L F , where F = U , D, N and E, and P R,L ≡ 1 2 (1 ± γ 5 ) are the right and left-handed projection operators, respectively. At this stage, the neutrinos are exactly massless, so we are free to define the physical left-handed neutrino fields, N L , such that their charged current interactions are generation-diagonal. 6 By setting H 0 1 = v/ √ 2 and H 0 2 = 0, one can relate κ U , κ D , and κ E to the diagonal (up-type and down-type) quark and charged lepton mass matrices M U , M D , and M E , respectively, However, the complex matrices ρ F (F = U, D, E) are unconstrained. Moreover, under the rephasing H 2 → e iχ H 2 , the Yukawa matrix acquires an overall phase, ρ F → e iχ ρ F , since L Y must be independent of χ.
To obtain the physical Yukawa couplings of the Higgs boson, one must relate the Higgs basis scalar fields to the Higgs mass-eigenstate fields. Using Eqs. (2.13) and (2.14), the Higgs-fermion Yukawa couplings are given by, The combinations e iθ 23 ρ U , e iθ 23 ρ D and e iθ 23 ρ E that appear in these interactions are invariant under the rephasing of H 2 . It is convenient to rewrite the Higgs-fermion Yukawa couplings in terms of the following three 3 × 3 hermitian matrices that are invariant with respect to the rephasing of H 2 , where 1 is the 3 × 3 identity matrix. The appearance of unconstrained hermitian 3 × 3 Yukawa matrices ρ F R,I in Eq. (2.25) indicates the presence of potential flavor-changing neutral Higgs-quark and lepton interactions. If the off-diagonal elements of ρ F R,I are unsuppressed, they will generate treelevel Higgs-mediated FCNCs that are incompatible with the strong suppression of FCNCs observed in nature.

The limit of a SM-like Higgs boson
Current LHC data suggest that the properties of the observed Higgs boson are consistent with the predictions of the Standard Model. In this paper, we shall identify h 1 as the SM-like Higgs boson. In light of the expression for the h 1 coupling to a pair of vector bosons In addition, Eq. (2.19) implies that one additional small quantity characterizes the limit of a SM-like Higgs boson, Moreover, in the limit of a SM-like Higgs boson, Eq. (2.18) yields As a consequence of Eqs. (2.27) and (2.28), the limit of a SM-like Higgs boson 7 can be achieved if either |Z 6 | 1 and/or if m 2 , m 3 v. The latter corresponds to the well-known decoupling limit of the 2HDM [39,46,53]. 8 In this paper, we will focus on the decoupling regime of the 2HDM to ensure that h 1 is sufficiently SM-like, in light of the current LHC Higgs data [3].

Neutral scalars of definite CP
In the exact SM-Higgs boson limit, the couplings of h 1 are precisely those of the SM Higgs boson. In this case, we can identify h 1 as a CP-even scalar. In general, the heavier neutral Higgs bosons, h 2 and h 3 can be mixed CP states. The limit in which h 2 and h 3 are approximate eigenstates of CP is noteworthy. This limit is achieved assuming that |s 13 | |s 12 |. That is, In the decoupling limit, the ratio of squared-mass differences in Eq. (2.31) is of O(1). Moreover, unitarity and perturbativity constraints suggest that Re(Z 6 e −iθ 23 ) cannot be significantly larger than O(1). Hence, it follows that | Im(Z 6 e −iθ 23 )| 1 . (2.32) In light of Eq. (2.10), we can rephase H 2 → e iχ H 2 such that θ 23 = 0 (mod π), i.e. c 23 = ±1. Eqs. (2.29) and (2.32) then yield | Im Z 5 | , | Im Z 6 | 1 . For simplicity in the subsequent analysis, we henceforth assume that a real Higgs basis exists in which Z 5 and Z 6 are simultaneously real. In this case, the scalar Higgs potential and the Higgs vacuum are CP-invariant, and the squared-mass matrix of the neutral Higgs bosons given in Eq. (2.8) simplifies, where Z 5 and Z 6 are real. Moreover, c 13 = 1 and we can set θ 23 = θ 6 = 0 (mod π), or equivalently The upper 2 × 2 matrix block given in Eq. (2.33) is the CP-even Higgs squared-mass matrix, where c β−α ≡ cos(β − α) and s β−α ≡ sin(β − α) are defined in terms of the angle β defined via tan β ≡ v 2 /v 1 , and the mixing angle α that diagonalizes the CP-even Higgs squared-mass matrix when expressed relative to the original basis of scalar fields, {Φ 1 , Φ 2 }, which is assumed here to be a real basis. 9 Since the real Higgs mass-eigenstate fields H and h are defined up to an overall sign 9 Given the assumption [indicated above Eq. (2.33)] that the scalar Higgs potential and the Higgs vacuum are CPinvariant, it follows that there must exist a real basis of scalar fields in which all scalar potential parameters and the vacuum expectation values of the two neutral Higgs fields, Φ 0 i ≡ vi/ √ 2 (for i = 1, 2), are simultaneously real [54].
change, it follows that β − α is determined modulo π. To make contact with the notation of Eq. (2.9), we note that c 13 = 1 and c 23 = 6 [cf. Eq. (2.34)]. Assuming that h 1 is the lighter of the two neutral CP-even Higgs bosons, then Eq. (2.38) implies the following identifications: and and it therefore follows that 0 ≤ s 12 , c 12 ≤ 1 and c β−α Z 6 ≤ 0.
In particular, m 2 h Z 1 v 2 in the limit of a SM-like Higgs boson h. Applying Eq. (2.40) to Table 1, Inserting these results into the general form of the Yukawa couplings given in Eq. (2.25), we obtain the following Higgs-fermion couplings in the case of a CP-conserving Higgs scalar potential and vacuum, where we have introduced the notation, 50) The structure of the neutral Higgs couplings given in Eq. (2.48) is easily ascertained. If ρ F I = 0, then the neutral Higgs fields will exhibit CP-violating Yukawa couplings. 10 Moreover, the two sign choices, 6 = ±1 are physically indistinguishable, since the sign of Z 6 can always be flipped by redefining the Higgs basis field H 2 → −H 2 . Under this field redefinition, ρ F , c β−α , H and A also flip sign, in which case L Y is unchanged.
For completeness, we briefly consider the case where h 1 is the heavier of the two neutral CP-even Higgs bosons. In this case, Eq. (2.38) implies the following identifications, So far, the parameters α and β have no separate significance. Only the combination, β − α is meaningful. Moreover the matrices ρ F R and ρ F I are generic complex matrices, which implies the existence of tree-level Higgs-mediated flavor changing neutral currents, as well as new sources of CP violation. However, experimental data suggest that such Higgs-mediated FCNCs must be highly suppressed. One can eliminate these FCNCs by imposing a discrete Z 2 symmetry Φ 1 → Φ 1 and Φ 2 → −Φ 2 on the quartic terms of the Higgs potential given in Eq. (2.1), which sets λ 6 = λ 7 = 0 and gives physical significance to the Φ 1 -Φ 2 basis choice. This in turn promotes the CP-even Higgs mixing angle α in the real Φ 1 -Φ 2 basis and tan β ≡ v 2 /v 1 to physical parameters of the model. 11 10 Likewise, if Im Z7 = 0 in a basis where Z5 and Z6 are real, then the neutral Higgs fields will also possess CP-violating trilinear and quadralinear scalar couplings. 11 Since the existence of a real Higgs basis implies no spontaneous nor explicit CP-violation in the scalar sector, there exists a Φ1-Φ2 basis in which the λi of Eq. (2.1), v1 and v2 (and hence tan β) are simultaneously real.
The Z 2 symmetry can be extended to the Higgs-fermion interactions in four inequivalent ways. In the notation of the Higgs-fermion Yukawa couplings given in Eq. (2.48), the ρ F R,I are given by 12 1. Type-I: For F = U, D, E, ρ F R = 6 cot β1 and ρ F I = 0.
2. Type-II: ρ U R = 6 cot β1 and ρ U Inserting these values for the ρ F R and ρ F I into Eq. (2.48), the resulting neutral Higgs-fermion Yukawa couplings are flavor diagonal as advertised.
From a purely phenomenological point of view, one can simply avoid tree-level Higgs-mediated FCNCs by declaring that the ρ F R and ρ F I are diagonal matrices. In the simplest generalization of the Type I, II, X and Y Yukawa interactions, one asserts that both the ρ F R and the ρ F I are proportional to the identity matrix (where the constants of proportionality can depend on F ). This is called the flavor-aligned 2HDM, which we shall discuss in the next subsection.

The flavor-aligned 2HDM
The flavor-aligned 2HDM posits that the Yukawa matrices κ F and ρ F [cf. Eq. (2.20)] are proportional. When written in terms of fermion mass-eigenstates, Thus in the A2HDM, the ρ F are likewise diagonal, which implies that tree-level Higgs-mediated FCNCs are absent. We define the alignment parameters a F via, Inserting the above results into Eq. (2.22), the Yukawa couplings take the following form: This form simplifies further if the neutral Higgs mass-eigenstates are also states of definite CP. In this case, the corresponding Yukawa couplings are given by As noted above Eq. (2.41), it is convenient to choose a convention in which s β−α ≥ 0. It then follows from Eq. (2.41) that 6 c β−α = −|c β−α |. That is, the neutral Higgs couplings exhibited in Eq. (2.57) do not depend on the sign of c β−α (which can be flipped by redefining the overall sign of the Higgs basis field H 2 ). Note that in this convention, the signs of the alignment parameters a F are physical. The Type-I, II, X and Y Yukawa couplings are special cases of the A2HDM Yukawa couplings. Since the a F (F = U, D, E) are independent complex numbers, there is no preferred basis for the scalar fields outside of the Higgs basis. Thus, a priori, there is no separate meaning to the parameters α and β in Eq. (2.57). Nevertheless, in the special case of a CP-conserving neutral Higgs-lepton interaction governed by Eq. (2.57) with Im a E = 0, it is convenient to introduce the real parameter tan β via a E ≡ − 6 tan β , (2.58) corresponding to a Type-II or Type-X Yukawa couplings of the charged leptons to the neutral Higgs bosons. The theoretical interpretation of tan β defined by Eq. (2.58) is as follows. It is always possible to choose a Φ 1 -Φ 2 basis with the property that one of the two Higgs-lepton Yukawa coupling matrices vanishes. Namely, in the notation of Eq. (A.1), we have η E,0 2 = 0, which means that only Φ 1 couples to leptons. In the case of a CP-conserving scalar Higgs potential and Higgs vacuum, we can take the Φ 1 -Φ 2 basis to be a real basis and identify . However, in contrast to Type-II or Type-X models, η E,0 2 = 0 does not correspond to a discrete Z 2 symmetry of the generic A2HDM Lagrangian, since we do not require any of the Higgs-quark Yukawa coupling matrices and the scalar potential parameters λ 6 and λ 7 to vanish in the same Φ 1 -Φ 2 basis.
Note that the sign of a E in Eq. (2.58) is physical since both 6 and tan β flip sign under the Higgs basis field H 2 → −H 2 . In contrast to the standard conventions employed in the 2HDM with Type-I, II, X or Y Yukawa couplings where tan β is defined to be positive [cf. footnote 12], we shall not adopt such a convention here. In practice, we will rewrite Eq. (2.58) as, where E = ±1 correspond to physically non-equivalent points of the A2HDM parameter space.
One theoretical liability of the A2HDM is that for generic choices of the alignment parameters a U and a D , the flavor-alignment conditions in the quark sector specified in Eq. (2.54) are not stable under the evolution governed by the Yukawa coupling renormalization group equations. Indeed, as shown in Ref. [32], Eq. (2.54) is stable under renormalization group running if and only if the parameters a U and a D satisfy the conditions of the Type I, II, X or Y 2HDMs specified at the end of section 2.3. In the leptonic sector, since we ignore neutrino masses, the Higgs-lepton Yukawa couplings are flavordiagonal at all scales. We therefore assume that 13 at some very high energy scale Λ (such as the grand unification (GUT) scale or the Planck scale).
That is, we assume that the alignment conditions are set by some a priori unknown physics at or above the energy scale Λ. We take the complex alignment parameters a F to be boundary conditions for the RGEs of the Yukawa coupling matrices, and then determine the low-energy values of the Yukawa coupling matrices by numerically solving the RGEs. To ensure that the resulting low-energy theory is consistent with a SM-like Higgs boson observed at the LHC, we shall take m h = 125 GeV, and assume that the masses of H, A and H ± are all of order Λ H ≥ 400 GeV. In this approximate decoupling regime, |c β−α | is small enough such that the properties of h are within about 20% of the SM Higgs boson, as required by the LHC Higgs data [3]. We employ the 2HDM RGEs given in Appendix B from Λ down to Λ H , and then match onto the RGEs of the Standard Model to generate the Higgs-fermion Yukawa couplings at the electroweak scale, which we take to be m t or m Z . Note that the values of are determined from the known quark masses via Standard Model RG running.
As noted above for the lepton case , then ρ is proportional to κ at all energy scales. Thus, we identify the leptonic alignment parameter at low energies by tan β. More precisely [cf. Eqs. (2.21) and (2.59)], Then, M E (Λ H ) is determined by the diagonal lepton mass matrix via Standard Model RG running.

Higgs-mediated FCNCs from high scale alignment
To explore the Higgs-mediated FCNCs that can be generated in the A2HDM at the electroweak scale, we establish flavor-alignment at some high energy scale, Λ, as for example at the GUT or Planck scale, and run the one-loop RGEs from the high scale to the electroweak scale. Thus, we impose the following boundary conditions for the running of the one-loop 2HDM Yukawa couplings, where the M Q (Q = U , D) are the diagonal quark matrices, and Λ H is the scale of the heavier doublet, taken to be relatively large to guarantee that we are sufficiently in the decoupling limit. For the lepton sector, the corresponding boundary conditions are [cf. Eq. (2.59)], Satisfying the two boundary conditions for the quark sector [Eqs. (2.62) and (2.63)] is not trivial, since they are imposed at opposite ends of the RG running. For example, to set flavor-alignment at the high energy scale, we must know the values of κ Q (Λ). This involves running up κ Q (Λ H ) to the high scale, but since the one-loop RGEs are strongly coupled to the ρ Q matrices, we must supply values for ρ Q (Λ H ) to begin the running.
With no a priori knowledge of which values of ρ Q (Λ H ) lead to flavor-alignment at the high scale, we begin the iterative process by assuming flavor-alignment at Λ H via a low-scale alignment parameter a Q , This flavor-alignment will be broken during RGE evolution to the high scale, and a procedure is needed to reestablish flavor-alignment at the high scale. To accomplish this, we decompose ρ Q (Λ) into parts that are aligned and misaligned with κ Q (Λ), respectively, where a Q represents the aligned part (in general, different from a Q ), and δρ Q the corresponding degree of misalignment at the high scale.
To minimize the misaligned part of ρ Q (Λ), we implement the cost function, which, once minimized, provides the optimal value of the complex parameter a Q for flavor-alignment at the high scale, .

(2.69)
We subsequently impose flavor-alignment at the high scale using this optimized alignment parameter,  analogy with Eq. (A. 13)]. We can now evolve κ U and κ D down to the electroweak scale to check the accuracy of the resulting quark masses. If any of the quark masses differ from their experimental values by more than 3%, we reestablish the correct quark masses at the electroweak scale, 14 run back up to Λ H , and then rerun this procedure repeatedly until the two boundary conditions are satisfied. The result is flavor-alignment between κ Q (Λ) and ρ Q (Λ), and a set of ρ Q matrices at the electroweak scale that provide a source of FCNCs. In our iterative procedure, we demand that all scale-dependent Yukawa couplings remain finite from the electroweak scale to the Planck scale (i.e., Landau poles are absent below Λ = M P ). This restricts the range of the possible seed values, a Q , used in Eq. (2.66) to initialize the iteration. Consequently, the alignment parameters a U and a D cannot be too large in absolute value. Constraints on the alignment parameters due to Landau pole considerations during one-loop RG running have been given in Ref [59]. In our analysis, the allowed values of a U and a D consistent with the absence of Landau poles at all scales below Λ are exhibited in Fig. 1. 15 Assuming Λ H = 400 GeV, these considerations lead to bounds on the alignment parameters evaluated at the Planck scale, Λ = M P , which are consistent with the results previously obtained in Ref [59]. 14 Starting the RG evolution at mZ , we use a five flavor scheme to run up to mt and a six flavor scheme above mt.
Running quark mass masses at mZ and mt are obtained from the RunDec Mathematica software package [55,56], based on quark masses provided in Ref. [57]. We fix the initial value of the top Yukawa coupling yt(mt) = 0.94, corresponding to an MS top quark mass of mt(mt) = 163.64 GeV [58]. For simplicity, the effects of the lepton masses are ignored, as these contribute very little to the running. 15 If a Landau pole in one of the Yukawa coupling matrices arises at the scale Λ, then both the corresponding ρ Q (Λ) and κ Q (Λ) diverge, whereas their ratio, a Q , remains finite.

Leading logarithm approximation
In the limit of small alignment parameters, it is possible to obtain approximate analytic solutions to the one-loop RGEs provided in Appendix B. One can express the ρ Q matrices at the low scale as It follows that there is a large hierarchy among the several off-diagonal terms of the ρ Q matrices, The inequality given in Eq. (2.76) was previously noted in Ref. [36], and provides the justification for ignoring ρ D ij relative to ρ D ji , for i < j. 17 This hierarchy of Yukawa couplings is reversed for ρ U ij . This reversal can be traced back to the fact that ρ U is undaggered in Eq. (2.20) whereas ρ D is daggered.
It is noteworthy that the leading log results for the off-diagonal terms of the ρ Q matrices obtained in Eqs. (2.74) and (2.75) and the corresponding full numerical calculation are typically within a factor of a few. Even for small alignment parameters, there can be some small discrepancies between the two approaches that can be traced back to the higher order terms that were neglected in Eqs. (2.74) and (2.75). These higher order terms are not negligible due to the running performed between the electroweak scale and the high energy scale Λ. The leading log approximation describes less and less 16 The misalignment contributions exhibited in Eqs. (2.74) and (2.75) were computed for the first time in Ref. [60]. 17 To make contact with the Higgs basis Yukawa couplings ∆u and ∆ d employed by Ref. [36], we note the relations accurately the numerical results at larger and larger alignment parameters. This is shown in Fig. 1, where the blue points correspond to the parameter regime in which the leading log approach leads to results within a factor of 3 of the results obtained numerically for all the elements of the ρ U and ρ D matrices. In contrast, the red points correspond to the parameter regime in which the leading log approximation leads to results quite different from what is obtained by the full running.

A particular type of Minimal Flavor Violation
In the quark sector of the A2HDM, only the two Yukawa coupling matrices κ U and κ D break the SU(3) Q ×SU(3) U ×SU(3) D global flavor symmetry of the electroweak Lagrangian involving quarks. For this reason, our model can be thought in terms of a specific realization of a Minimal Flavor Violating (MFV) 2HDM [29]. In particular, in a general 2HDM with MFV one can write the Yukawa Lagrangian as with generic O(1) complex coefficients u,d ni . In order to determine the coefficients u,d ni in the A2HDM, we rotate to the quark mass-eigenstate basis: and all the higher order coefficients equal to zero. The corresponding coefficients for the down sector are obtained from these expressions with the replacement a U → a D , a D → a U , κ D → κ U . As expected, the leading term in Eq. (2.81) is given by the alignment parameter at the high scale a U . This coefficient receives one loop corrections. The term in Eq. (2.82) generates off diagonal terms in the matrix ρ U and is one loop suppressed.

Predictions of the model for high energy processes
For our numerical analysis, we use the procedure described in the previous section, taking the A2HDM to be in the decoupling limit, which ensures that the properties of the lightest Higgs boson, h, are approximately those of the observed (SM-like) Higgs boson. As stated below Eq. (2.33), we assume that the Higgs scalar potential and the Higgs vacuum are CP-conserving. In this case, the two heavier neutral scalars, H and A, are CP-even and CP-odd mass-eigenstates, respectively. In the decoupling limit, these two scalars are roughly degenerate in mass, i.e., m H ≈ m A ≈ Λ H m h . The decoupling limit also enforces the condition | cos(β − α)| 1, as noted below Eq. (2.41). In this paper, we shall choose a benchmark mass of m H = 400 GeV. Noting that in the case of a SM-like Higgs boson, m 2 h Z 1 v 2 = (125 GeV) 2 , which implies that Z 1 0.26, we will furthermore assume that |Z 6 | and Z 1 are of similar size. Indeed, Eq. (2.41) yields |cos(β − α)| 0.11 for |Z 6 | = Z 1 . 18 In particular, if β − α = π/2 − x, with |x| 1, then values x = 0 imply deviations from SM behavior of the couplings of the 125 GeV Higgs boson to fermions and gauge bosons, as well as the appearance of flavor changing neutral Higgs couplings, the largest of which is the hbs coupling.
In our analysis, we allow for CP-violating effects to enter in two ways. First, CP-violating charged Higgs couplings to fermion pairs are generated via the appearance of the CKM matrix, K. Second, we generically allow for the possibility of complex alignment parameters a U and a D at the high energy scale. Via RG-running, CP-violating neutral Higgs couplings to fermion pairs will be generated. However, this extra source of CP violation will lead to a loop-suppressed mixing of H and A that is difficult to observe due to the near mass degeneracy of these states in the decoupling limit under consideration here.

The couplings of the SM-like Higgs boson
It is instructive to examine the hbb coupling, which is the Yukawa coupling that is most affected by New Physics in our framework, and thus plays the leading role in constraining the parameter space. Following the standard notation of the ATLAS and CMS Collaborations, we denote the coupling of h to bottom quarks normalized to the SM prediction by κ b . 19 Due to the presence of a CP-violating contribution to the hbb coupling when Im ρ D 33 = 0, both scalar and pseudoscalar contributions to the hbb coupling must be considered [see Eq. (2.48)]. In the approximation where m b m h , one can simply replace γ 5 in the expression for the Yukawa coupling with ±1, in which case κ b can be expressed by the magnitude of the complex number, and compared to its ATLAS and CMS measurement, extracted from the h → bb rate. In the leading log approximation, Eq. (2.75) yields,  and > 2, respectively. Here, the boldfaced number represents the 3σ experimental upper bound of κ b . All points shown correspond to parameter regimes where Landau poles are absent [cf. Fig. 1]. In the bottom right panel, the gray shaded region produces Landau poles in the Yukawa couplings below M P ; the pink shaded region contains points favored by the LHC measurements of κ b .
In Fig. 2, we show the reduced coupling, κ b , in the leading log approximation as a function of the free parameters a U , a D and |cos(β − α)|. We extend the plots up to |cos(β − α)| ∼ 0.3, consistent with the present measurement of the Higgs couplings to W W and ZZ. The two upper panels are obtained using the leading log approximation; the two lower ones using the full RG running. We take real values for a U , a D to present the leading log results. Generic complex coefficients are employed in parameter scans obtained with the full running. In the left upper panel, we show the reduced coupling as a function of a U and a D , having fixed |cos(β − α)| = 0.05. In the right upper panel, we show the reduced coupling as a function of |cos(β − α)| and a D , having fixed a U = 0.1. In the two panels, we show in blue the contour κ b = 1.3, that roughly corresponds to the present 3σ bound, as measured by the LHC combining ATLAS and CMS Run I data [3]. 20 The pink regions of Fig. 2 illustrate that values for | cos(β − α)| ∼ O(0.1) are still allowed for sizable values of a D of O (20). Furthermore, the shape of the constraint is quite different, if compared to the shape obtained for the (cos(β − α) , tan β) plane in the Type I and II 2HDM [61,62]. The corresponding results obtained using the full RG running are shown in the lower panels. Note that the bounds on the parameter spaces (|a U | , |a D |) and (| cos(β − α)| , |a D |) are slightly weaker as compared to the leading log results.
It is interesting to investigate the Higgs flavor violating couplings in the regions of parameter space favored by the LHC measurements of the SM Higgs rates. The decay to a bottom and a strange quarks is the dominant flavor violating Higgs decay in our model. However, we have checked that the corresponding branching ratio can be at most at the few per-mille level. Numerically, this is similar to the result for BR(h →bs + bs) obtained by Ref. [63] in a Type I and Type II 2HDM due to charged Higgs loop contributions to the decay amplitude.

Flavor-changing top decays
We calculate the branching ratios for the decays t → u i h (u i = u, c) arising from misalignment generated via radiative corrections during RG running. This is in contrast to the analysis of Ref. [64] where flavor alignment is assumed to hold at the electroweak scale, in which case only charged Higgs loop diagrams contribute to the top flavor changing decays, leading to a BR(t → u i h) that depends strongly on the value of the charged Higgs mass. In this subsection, we show how the charged Higgs contributions compare to the those arising in our model due to tree-level flavor changing top couplings.
Following Ref. [65], we employ the leading order formulae for both t → W b and t → u i h decay rates, assuming the top quark decay width is dominated by the SM value of Γ(t → W b). In addition, we include the NLO QCD correction to the branching ratio, where η QCD = 1 + 0.97α s ∼ 1.10. The flavor violating branching ratios scale with the second power of cos(β − α), and thus suppressed in the cos(β − α) = 0 limit. The couplings ρ U i3 and ρ U 3i can be easily extracted in the leading logarithmic approximation from Eq.  a charm and a Higgs boson has a O(10 2 ) larger branching ratio than the decay into an up quark and a Higgs boson since in the leading logarithmic approximation, In the top panel of Fig. 3, we show the leading log results for the branching ratios, as a function of the two alignment parameters at the high scale, a U and a D , having fixed | cos(β − α)| = 0.2. Gray and pink shaded regions correspond to the region producing Landau poles in the Yukawa couplings below the Planck scale M P , and to the region favored by the LHC measurements of κ b , respectively. Branching ratios larger than ∼ 10 −10 for ch and ∼ 10 −12 for uh cannot be reached, while being consistent with Higgs coupling measurements and with the requirement of no Landau poles below M P . For comparison, we also show our results obtained scanning the parameter space and using the full RGEs (see the bottom panels of Fig. 3). Comparing the upper and lower panels of Fig. 3, we note that the agreement between the prediction at leading log and the full numerical results is less accurate at larger values of the alignment parameters |a U | and |a D |, as expected. Values of the branching ratios as large as ∼ 3 × 10 −7 (∼ 3 × 10 −9 ) for t → ch (t → uh) can be reached, while satisfying the condition due to the absence of Landau poles (see the blue points). However, the majority of points with BR(t → ch) 10 −8 and BR(t → uh) 10 −10 also produces too large a deviation from SM behavior of the Higgs coupling to bottom quarks. We have checked that the largest branching ratio compatible with Higgs data is at around 10 −7 for t → ch and at around 10 −9 for t → uh.
These numbers should be compared with the corresponding contributions to flavor-changing top decays from charged Higgs loop diagrams, which are present in all 2HDMs, and are generated by flavor-changing charged Higgs interactions induced by CKM mixing [64,66]. Based on the discussion of Ref. [64], we see that in the case of light charged Higgs bosons (m H ± 200 GeV) and hH + H − couplings as large as allowed by hγγ constraints, these latter contributions can be as large as O(10 −8 ) and therefore comparable to those arising from the tree-level htu i coupling induced by RG-running in the A2HDM.
When compared to the BR(t → ch) SM ∼ 3×10 −15 , BR(t → uh) SM ∼ 2×10 −17 , as calculated in the SM by Refs. [67][68][69][70], the 2HDM in general and the A2HDM in particular exhibit the possibility of a significant enhancement of the branching ratios for flavor-changing t → u i h decays. However, both tree-level flavor changing effects and loop-level effects mediated by the charged Higgs boson are generically too small to be probed by the LHC and future colliders.
Searches for top flavor changing decays have been performed by the ATLAS and CMS collaboration using Run I data [71][72][73], and constrain the branching ratios to BR(t → u i h) 0.42% (see also [74] for a discussion of the most recent experimental results on top flavor changing decays). Projections for the HL-LHC show that the bounds on the branching ratios will likely be at the 10 −4 level [75,76]. Hence, it will be very challenging to probe our model at the LHC using top flavor changing decays. FCC estimations show that branching ratios as small as ∼ 10 −7 could be probed with 10 ab −1 luminosity [77]. From these numbers, we can conclude that Higgs coupling measurements typically give (and will give) a better probe of our model, since the region of parameter space predicting more sizable top flavor violating branching ratios, also predict large and measurable effects in the Higgs coupling to bottom quarks.

Phenomenology of the heavy Higgs bosons
As pointed out in Ref. [78], the 2HDM with flavor alignment imposed at the electroweak scale predicts a rich and novel phenomenology for the heavy Higgs bosons that is strikingly different than that of the 2HDM with Type I, II, X or Y Higgs-fermion Yukawa couplings. The phenomenology is even more diverse if flavor alignment is imposed at the high scale. For example, the heavy Higgs decay to quarks is flavor non-universal (i.e., the ratios, y Hd i d i /m d i and y Hu i u i /m u i are no longer independent of the flavor i). Moreover, flavor changing heavy Higgs decays, which are generated at the loop-level due to the quark flavor-changing charged Higgs interactions [63,66], receive an additional contribution from tree-level flavor-changing neutral Higgs interactions. In contrast to the flavor-changing top decays discussed in the previous section, these features are not suppressed in the limit of cos(β − α) = 0, where the couplings of h coincide with those of the SM Higgs boson. This is exhibited by the treelevel partial widths of the heavy Higgs bosons to up and down quarks, which are given by Henceforth, we shall set cos(β − α) = 0, which automatically avoids constraints from the measured Higgs boson couplings. In the leading log approximation with real values of a D and a U assumed, the second term of Eq.  [63,66] and have been shown generically to be considerably smaller than the corresponding tree-level flavor violating Higgs couplings. The left upper panel shows that in our model, especially at sizable values of the alignment parameters, the Type I and II 2HDM relation, BR(H →bb)/BR(H → τ + τ − ) = 3m 2 b /m 2 τ , is violated. In particular, our model typically predicts a smaller ratio at small values of a D , and therefore the τ + τ − mode is expected to be even more sensitive than bb relative to that of the Type I or II 2HDM. For a D 5, the hierarchy is reversed, resulting in a larger BR(H →bb) as compared to BR(H → τ + τ − ). Furthermore, the model can predict a non zero decay rate of the heavy Higgs to a bottom and a strange quark (see the right upper panel of Fig. 4). However, the branching ratio predicted in the leading log approximation is at most of order a few percent at large values of a D in the regions of the parameter space without Landau poles.
Note that the branching ratios into third generation quarks are different as compared to the Type II 2HDM. In the latter, BR(H → bb) ∼ 80% and BR(H → tt) ∼ 6%, for tan β = 10. For comparison, we present the branching ratios into tt and bb in the lower left and lower right panels of Fig. 4. The behavior of the two plots is similar: at small values of a U (a D ) the tt (bb) branching ratio is smaller than the one predicted by the Type II 2HDM (see the blue contours in the two plots); the branching . As a byproduct, the ratio of branching ratios BR(H → bb)/BR(H → tt) differs from the predicted value of the 2HDM with either Type I, II, III, or IV Yukawa couplings. In particular, the A2HDM generically breaks the relation BR(H → bb)/BR(H → tt) m 2 b tan 4 β/m 2 t , which is valid in the Type II 2HDM in the limit cos(β − α) = 0. The branching ratios of a Type II 2HDM are recovered by choosing The ATLAS and CMS collaborations have performed several searches for heavy Higgs bosons decaying into a fermion pair:bb [79], τ + τ − [80,81], µ + µ − [82,83], andtt [84]. In a Type II 2HDM, τ + τ − searches are the most important ones in constraining regions of parameter space at sizable values of tan β. Searches forbb can only set weaker bounds in that scenario. However, as discussed e.g. in Ref. [85], 2HDMs with a Yukawa texture different from Type II can be best probed bybb searches. In fact, for tan β = 10 and cos(β −α) = 0, only the CMS search for pp → b(b)H, H →bb, performed with 8 TeV data [79], can probe sizable regions of the parameter space of the A2HDM (see the blue shaded region in Fig. 4 at large values of a D and the corresponding parameter regime of Fig. 5). In the coming years, the LHC will be able to probe complementary regions of parameter space. In addition to the region at large values of a D best probed bybb resonance searches, the region at small values of a U and a D will be best probed by searches for τ + τ − and µ + µ − resonances; and the region at small values of a D , but sizable values of a U will be best probed bytt resonance searches.
For comparison, we show in Fig. 5 the corresponding results obtained through the scanning of the parameter space and the running of the full RGEs. Qualitatively, Fig. 5 shows a similar parameter dependence as the one obtained in the leading log approximation. Numerically, some branching ratios can be quite different, especially in the regime of sizable alignment parameters. In particular, BR(H →bs + bs) can reach values as large as ∼ 10%.

Predictions of the model for low energy processes
As we discussed in section 2.7, the A2HDM is a particular type of 2HDM with Minimal Flavor Violation. As such, it predicts interesting effects in low energy flavor observables, e.g., in meson mixing and in B meson rare decays. In this section, we shall discuss the predictions of our model for these low energy processes and the corresponding constraints. We shall focus on those observables that receive tree-level Higgs contributions, with particular attention to meson mixing, B → µ + µ − , and B → τ ν.
The lepton universality ratios, BR(B → D ( * ) τ −ν )/BR(B → D ( * ) − ν), for = e, µ, are also notable, especially in light of the early BaBar measurements that yield a combined 3.4σ deviation from the SM predictions [86,87]. This anomaly is not inconsistent with subsequent Belle and LHCb measurements, even if with a smaller significance [88][89][90][91]. Additional data are required to clarify the implications of these measurements and to determine whether new physics beyond the SM is required. If this anomaly persists, New Physics models need (relatively large) H ± c L b R and H ± c R b L couplings of the same order and opposite sign (with g 2 H ± cb /m 2 H ± ∼ 1/TeV 2 ), as shown in Ref. [92]. This is rather challenging to achieve in our model while being consistent with the other flavor bounds. A more detailed examination of these channels will be left for a future study.
In principle, loop induced decays (which typically include contributions from the charged Higgs boson) can also set stringent constraints on the allowed regions of the (m H ± , tan β) parameter plane [93]. For example, in the Type II 2HDM the charged Higgs should be heavier than 580 GeV [21] to be in agreement with b → sγ measurements (cf. footnote 2). Moreover, going beyond the Type II 2HDM, the b → sγ bound depends not only on the charged Higgs mass, but also on the values of a U and a D , on other non-SM-like Higgs boson masses, as well as on potential contributions of New Physics particles in the loop. Such constraints merit further investigation. However, the analysis of this section focuses on parameter regimes in which tree-level Higgs-mediated FCNC effects dominate over competing one-loop contributions. For this reason, we do not consider further the constraints from b → sγ (which can be avoided for sufficiently heavy Higgs masses) in this paper.

Meson mixing
Higgs mediated contributions to neutral meson mixing (B d,s -B d,s , K-K and D-D mixing) arise in our model. Integrating out the three neutral Higgs bosons, we obtain the following dimension six effective Lagrangian describing B s meson mixing with Wilson coefficients, and corresponding Wilson coefficients for B d , K, and D mixing.
In the case of degenerate heavy Higgs bosons and in the limit cos(β−α) = 0, only C 4 contributes to meson mixing. In this limit, we expect small Wilson coefficients at leading log, since as discussed in section 2.6, |(ρ D ) ij | ∝ m i /v and therefore |(ρ D ) ij | |(ρ D ) ji |, for i < j. The Wilson coefficients are also relatively small away from the exact cos(β − α) = 0 and m A = m H limit. More specifically, C 2 andC 2 will be non zero, but suppressed by the combination of masses and mixing angles shown in Eqs. (4.2) and (4.3), respectively. In the following, we will show the numerical results obtained for cos(β − α) = 0 and m A = m H = 400 GeV. However, we have checked that the constraints on the parameter space do not change considerably by taking small but non-zero values for cos(β − α).
We apply the bounds of Ref. [94] on the C 4 Wilson coefficient (Ref. [95] shows slightly stronger constraints). The leading log results for B s , B d , K, and D mixing are shown in the left panel of Fig. 6. The dark purple region is favored by the measurement of B s mixing, the purple region by B d mixing, the dark pink region by CP violation in the Kaon mixing system, and the pink region by the K-K mass difference. D mixing does not give any interesting bound on the parameter space and is therefore omitted in the figure. B s mixing leads to the most stringent bound and it constrains a D to be smaller than ∼ 4.7 at sizable values of a U . Additionally, the bound from the measurement of CP violation in Kaon mixing (dark pink) is significantly more stringent than the bound from the mass difference of the Kaon system (in pink). This is due to the fact that the real and imaginary parts of the Wilson coefficient of the Kaon system have a similar magnitude (under the assumption that a U and a D are real). In particular, the ratio of the imaginary and real parts of the Wilson coefficient is directly related to the phase of the CKM matrix: . In contrast, the SM Wilson coefficient has an imaginary part that is much smaller than the real part. Small differences between the constraints from CP violation and the mass difference also exist in the B s and B d systems. In Fig. 6, we only show the most constraining bound in each system, i.e. the mass difference in B s mixing and the phase in B d mixing.
The right panel of Fig. 6 shows the corresponding results for the B s mixing system obtained by scanning the parameter space and using the full RG-running. The points in yellow have a Wilson coefficient smaller than 1/3 the present bound on the Wilson coefficient; in red we present the points with a Wilson coefficient smaller than the present bound, and finally in green we present the points that have been already probed by the measurement of the B s mixing observables. In the limit of sizable a U 0.7, we do not find points with a D 4, in rough agreement with the leading log result.
The B-meson rare decays B s,d → + − receive contributions from the exchange of the Higgs bosons H, A and h at tree-level. This is in contrast to the numerical analysis of Ref. [96], where the flavor misalignment at the electroweak scale is set to zero. The neutral Higgs exchange contributions to the leptonic decay amplitude are proportional to m and hence are largest in the case of B s,d → τ + τ − . However, it is more difficult to tag the τ decay to jets and leptons at the LHC and B-factory detectors, as compared to muons. For this reason, the present LHCb bounds [97], BR(B s(d) → τ + τ − ) 3 × 10 −3 (1.3 × 10 −3 ), are relatively weak as compared to the SM prediction [98], (4.5) At sizable values of tan β, the main contributions to B s,d → µ + µ − are typically due to H and A exchange, as they are enhanced by the second power of tan β. Furthermore, in the cos(β − α) = 0 limit, the light Higgs (h) contribution vanishes at tree-level. For this reason, we shall focus henceforth on the heavy Higgs contributions that are given by [99], where BR(B s,d → µ + µ − ) SM is the SM prediction for the branching ratio extracted from an untagged rate. In particular, y s = (6.1 ± 0.7)% and y d ∼ 0 have to be taken into account when comparing experimental and theoretical results, and (4.8) The C i are the Wilson coefficients corresponding to the Lagrangian with operators 12) and the corresponding ones for the B d system. In the limit of cos(β − α) = 0, the Wilson coefficients arising from heavy neutral Higgs exchange are given by (4.14) and the analogous results for the B d system. There are no tree-level New Physics contributions to the O ( ) 10 operators. If cos(β −α) is nonvanishing, then the scalar Wilson coefficients C S s and C S s given in Eqs. (4.13) and (4.14) due to H exchange should be changed accordingly, tan β → sin(β −α) tan β +cos(β −α) and ρ D → ρ D sin(β − α). Moreover, an additional set of contributions arise due to h exchange; the corresponding contributions are obtained from C S s and C S s given in Eqs. (4.13) and (4.14) by making the following replacements, tan β → sin(β − α) − cos(β − α) tan β, ρ D → −ρ D cos(β − α) and m H → m h .
The SM Wilson coefficient takes the form [100],  Note the much larger uncertainty in the latter decay mode.
The ATLAS collaboration has also reported a Run I search for B s → µ + µ − , which yielded BR(B s → µ + µ − ) = (0.9 +1.1 −0.8 ) × 10 −9 [102], although this measurement is not yet competitive with Eq. (4.18). Very recently, LHCb reported a new measurement for B s,d → µ + µ − using Run II data [103]. Their result, BR(B s → µ + µ − ) = (2.8 ± 0.6) × 10 −9 , agrees very well with the LHCb and CMS combination quoted in Eq. (4.18). In contrast, the new LHCb B d measurement is closer to the SM prediction, BR(B d → µ + µ − ) = (1.6 +1.1 −0.9 )×10 −10 . In the following, we will compare the predictions of the A2HDM with the LHCb and CMS combination shown in Eqs. (4.18) and (4.19). In the coming years, the two branching ratios will be measured much more accurately by the LHC. In particular, the B s and B d branching fractions will be measured by each experiment with a precision of ∼ 13% and ∼ 48% at Run-III, improving to ∼ 11% and ∼ 18%, respectively, at the HL-LHC [104].
In Fig. 7, we show the constraints from the measurement of B s → µ + µ − (left panel) and B d → (4.20) The purple shaded region in Fig. 7 is the parameter space favored at 2σ by the HL-LHC measurement, assuming a measured central value equal to the SM prediction. Comparing the region in pink to the region in purple, one can get a sense of the improvement the HL-LHC can achieve in testing our model. The expected experimental error at the HL-LHC is comparable to the present theory error. For this reason, an additional improvement can be achieved via a more precise calculation of the SM prediction for the two branching ratios, with the benefit of more accurate measurements of the CKM elements that will be obtained at the LHCb and at Belle II in the coming years. The present measurement of B s → µ + µ − constrains sizable values of a U and a D in our model. The measurement of B d → µ + µ − also sets an interesting constraint at smaller values of |a D | (cf. the white region where |a D | ∼ 3 and the values of |a U | are sizable), since the central value of the measurement is larger than the SM prediction:  However, the deviation from the SM prediction is not yet statistically significant, due to the large experimental uncertainty. Nevertheless, a sizable suppression of the B d decay mode is presently disfavored. As expected, the contours for BR(B s,d → µ + µ − )/BR(B s,d → µ + µ − ) SM in the two panels of Fig. 7 are very similar. This is due to the fact that our model is a particular type of MFV model in the leading logarithmic approximation [cf. section 2.7]. In particular, MFV models generically It is also interesting to investigate the bounds as a function of the heavy Higgs boson masses. In Fig. 8, we show the same constraints in the (M, a D ) plane, where M ≡ m A = m H , having fixed tan β = 10, a U = 0.2, and cos(β − α) = 0. Sizable regions of parameter space are allowed, even for values of M as small as ∼ 300 GeV. Finally, in Fig. 9, we show the results obtained through scanning the parameter space and utilizing the full RG running. These plots are qualitatively similar to the contour plots of Fig. 7 obtained in the leading logarithmic approximation, although the heavy Higgs exchange contributions to the B d,s → µ + µ − decay rates computed using the full RG running are somewhat larger (at large alignment parameters) than the corresponding leading log results.

B → τ ν decays
The leptonic decays B → ν are interesting probes of the Higgs sector of our model and particularly of the charged Higgs couplings, since the charged Higgs boson mediates tree-level New Physics contributions to these decay modes. The τ channel is the only decay mode of this type observed so far. The present experimental world average is [105]   In our model, the New Physics contribution to this decay reads where we have defined the SM Wilson coefficient C ub SM = 4G F K ub / √ 2 and C ub R(L) are the Wilson coefficients of the O ub R(L) = (ūP R(L) b)(τ P L ν τ ) operators. In particular [107], with Γ LR(RL) ub the two charged Higgs couplings H +ū This leads to the branching ratio, (4.26) 23 Updated results and plots available at: http://ckmfitter.in2p3.fr.
In the leading logarithmic approximation, the most important contributions come from the second term of the above expression (∝ ρ D * 3i ), as one can easily deduce from Eqs. (2.74) and (2.75). In Fig. 10, we show our numerical results as obtained using the leading log approximation (left panel) and the scan of the parameter space using the full RGEs, having fixed m H ± = 400 GeV and tan β = 10. A very large region of parameter is still allowed by the measurement of B → τ ν. In particular, in the leading logarithmic approximation, every value |a D | 17 is allowed, irrespective of the value of the other alignment parameter, a U . Indeed, in the pink region shown in the left panel of Fig. 10, BR(B → τ ν)/BR(B → τ ν) SM ⊂ [0. 79, 1.71], consistent with the current measurements. This is no longer the case when we consider the scan based on the full RG-running. In this case, a few points at large values of |a U | are excluded by the measurement of BR(B → τ ν) (see the blue points in the right panel of the figure). In the left panel of Fig. 10, we also exhibit the purple shaded region of parameter space that would be favored by the future Belle II measurement, under the assumption that the central value of the measurement is given by the SM prediction for this branching ratio [cf. Eq. (4. 22)]. The allowed region of parameter space is expected to shrink considerably, thanks to the anticipated accuracy of the Belle II measurement with a total error of the order of ∼ 5% [108], leading to an allowed range, BR(B → τ ν)/BR(B → τ ν) SM ⊂ [0.86, 1.14], where we have assumed no improvement in the SM prediction of this B meson decay mode.

Conclusions
We have explored the consequences of flavor-alignment at a very high energy scale on flavor observables in the two Higgs doublet Model (2HDM). Flavor alignment at the electroweak scale generically requires an unnatural fine-tuning of the matrix Yukawa couplings. If flavor alignment is instead imposed at a higher energy scale such as the Planck scale, perhaps enforced by some new dynamics beyond the SM, then the flavor misalignment at the electroweak scale due to RG running will generate new sources of FCNCs. The resulting tree-level Higgs-mediated FCNCs are somewhat suppressed and relatively mildly constrained by experimental measurements of flavor-changing observables.
We require that the alignment parameters at the high scale remain perturbative. In particular, no Landau poles are encountered during RG running. These requirements lead to an upper bound on the values of the alignment parameters at the Planck scale. This in turn provide an upper bound on the size of FCNCs generated at the electroweak scale. The flavor-changing observables considered in this paper that provide the most sensitive probe of the flavor-aligned 2HDM parameter space are meson mixing and rare B decays such as B s,d → µ + µ − and B → τ ν. We also considered constraints from LHC searches of heavy Higgs bosons (the most important of which are searches for pp → b(b)H, H →bb, τ + τ − ), and measurements of the couplings of the observed (SM-like) Higgs boson. The most stringent constraint on the flavor-aligned 2HDM parameter space arises from the measurement of the rare decay B s → µ + µ − .
We investigated the predictions of the flavor-aligned 2HDM in the regions of the parameter space not yet probed by the measurements listed above. The top rare flavor changing decays, t → uh, t → ch, are generated at tree-level. However, once we impose constraints from Higgs coupling measurements, the predicted branching ratios for these neutral flavor changing top decays are beyond  In Fig. 11, we summarize the constraints on the (a U , a D ) parameter space, with fixed tan β = 10 (upper panels) and tan β = 3 (lower panels). In both panels, we fix the values cos(β − α) = 0 and m A = m H = m H ± = 400 GeV. The region favored by all flavor constraints is shown in reddish-brown. At sizable values of a D , the most relevant constraint comes from the measurement of B s → µ + µ − (tan region). B s meson mixing also sets an interesting bound on the parameter space (blue-gray region). It offers some complementary with B s → µ + µ − , as it does not depend on the particular value of tan β. Moreover, it will be able to probe the small region of parameter space with a U > 0 and sizable values of a D favored by the measurement of B s → µ + µ − in the case of a future measurement with a central value in agreement with the SM prediction. 24 The measurement of B → τ ν imposes only a relatively weak constraint on the parameter space (green region). For values of tan β = 10 (or larger), in the region of parameter space favored by present and future flavor constraints, the ratio m 2 τ BR(H →bb)/3m 2 b BR(H → τ + τ − ) is smaller than the ratio predicted by Type I and II 2HDM in most of the Aligned 2HDM parameter space. The parameter space is somewhat less constrained at lower values of tan β, as shown in the lower panels of Fig. 11.
In Fig. 12, we present the corresponding results obtained in the numerical scan with full RG running, with fixed cos(β − α) = 0, m A = m H = m H ± = 400 GeV, and tan β = 10. The qualitative features of the leading log approximation continue to hold. In particular, we again see that B s → µ + µ − provides the most stringent constraint on the aligned 2HDM parameter space. Note that in order to emphasize the comparison of the constraints obtained from the different B physics observables in Figs. 11 and 12, we do not include the constraints due to the LHC searches for the heavy Higgs bosons decaying into fermion pairs in these figures. As shown in Figs. 4 and 5 for the heavy Higgs mass values quoted above, in the region of the Aligned 2HDM parameter space 24 We use the results in [109] for the future prospects in measuring Bs mixing, corresponding to the "Stage II" scenario. consistent with no Landau poles below M P , the current LHC limits on H and A production eliminate the parameter regime with |a D | > ∼ 30-40 and |a U | < ∼ 0.1.
In considering the phenomenological implications of extended Higgs sectors, the most conservative approach is to impose only those constraints that are required by the current experimental data. In most 2HDM studies in the literature, the Yukawa couplings are assumed to be of Type I, II, X or Y. In this paper, we have argued that the current experimental data allows for a broader approach in which the Yukawa couplings are approximately aligned in flavor at the electroweak scale. The resulting phenomenology can yield some unexpected surprises. We hope that the search strategies of future Higgs studies at the LHC will be expanded to accommodate the broader phenomenological framework of the (approximately) flavor-aligned extended Higgs sector.

A Review of the Higgs-fermion Yukawa couplings in the Higgs basis
In a general 2HDM, the Higgs fermion interactions are governed by the following interaction Lagrangian: 25 summed over a =ā = 1, 2, where Φ 1,2 are the Higgs doublets, Φā ≡ iσ 2 Φ * a , Q 0 L and E 0 L are the weak isospin quark and lepton doublets, and U 0 R , D 0 R , E 0 R are weak isospin quark and lepton singlets. 26 Here, Q 0 Note that η U,0 a appears undaggered in Eq. (A.1), whereas the corresponding Yukawa coupling matrices for down-type fermions (D and E) appear daggered. In this convention, the transformation of the Yukawa coupling matrices under a scalar field basis change is the same for both up-type and downtype fermions. That is, under a change of basis, Φ a → U ab Φ b (which implies that Φā → ΦbU † bā ), the 25 We follow the conventions of Ref. [39], in which covariance is manifest with respect to U(2) flavor transformations, (2)], by implicitly summing over barred/unbarred index pairs of the same letter. 26 The right and left-handed fermion fields are defined as usual: ψR,L ≡ PR,Lψ, where PR,L ≡ 1 2 (1 ± γ5).
Yukawa coupling matrices transform as η F a → U ab η F b and η F † a → η F † b U † bā (for F = U , D and E), which reflects the form-invariance of L Y under the basis change.
The neutral Higgs states acquire vacuum expectation values, The Higgs fields in the Higgs basis are defined by H 1 ≡v * a Φ a and H 2 ≡ŵ * a Φ a , which can be inverted to yield Φ a = H 1va + H 2 w a [39]. Rewriting Eq. (A.1) in terms of the Higgs basis fields, The next step is to identify the quark and lepton mass-eigenstates. This is accomplished by replacing H 1 → (0 , v/ √ 2) and performing unitary transformations of the left and right-handed uptype and down-type fermion multiplets such that the resulting quark and charged lepton mass matrices are diagonal with non-negative entries. In more detail, we define: and the Cabibbo-Kobayashi-Maskawa (CKM) matrix is defined as K ≡ V U L V D † L . Note that for the neutrino fields, we are free to choose V N L = V E L since neutrinos are exactly massless in this analysis. 27 In particular, the unitary matrices V F L and V F R (for F = U , D and E) are chosen such that (A.11) 27 Here we are ignoring the right-handed neutrino sector, which gives mass to neutrinos via the seesaw mechanism.
It is convenient to define Eq. (A.6) implies that under the U(2) transformation, Φ a → U ab Φ b , κ F is invariant and ρ F → (det U )ρ F , (A.14) for F = U , D and E. Indeed, κ F is invariant since Eqs. (A.9)-(A.11) imply that which is a physical observable. The matrices ρ U , ρ D and ρ E are independent pseudo-invariant complex 3 × 3 matrices. The Higgs-fermion interactions given in Eq. (A.7) can be rewritten in terms of the quark and lepton mass-eigenstates,