Flavor from the electroweak scale

We discuss the possibility that flavor hierarchies arise from the electroweak scale in a two Higgs doublet model, in which the two Higgs doublets jointly act as the flavon. Quark masses and mixing angles are explained by effective Yukawa couplings, generated by higher dimensional operators involving quarks and Higgs doublets. Modified Higgs couplings yield important effects on the production cross sections and decay rates of the light Standard Model like Higgs. In addition, flavor changing neutral currents arise at tree-level and lead to strong constraints from meson-antimeson mixing. Remarkably, flavor constraints turn out to prefer a region in parameter space that is in excellent agreement with the one preferred by recent Higgs precision measurements at the Large Hadron Collider (LHC). Direct searches for extra scalars at the LHC lead to further constraints. Precise predictions for the production and decay modes of the additional Higgs bosons are derived, and we present benchmark scenarios for searches at the LHC Run II. Flavor breaking at the electroweak scale as well as strong coupling effects demand a UV completion at the scale of a few TeV, possibly within the reach of the LHC.


Introduction
The origin of the observed hierarchies in fermion masses and mixings remains one of the most intricate puzzles of the Standard Model (SM) of particle physics. The sizes of the Yukawa couplings range over at least six orders of magnitude, and the magnitude of the CKM matrix elements varies between 1 and 10 −3 . Various extensions of the SM have been proposed in order to explain these hierarchies. In a seminal paper, Froggatt and Nielsen introduced an abelian flavor symmetry by which only the top Yukawa coupling is allowed as a renormalizable operator [1]. The remaining Yukawa couplings are generated as higher order effective operators, schematically given by where lighter fermion masses require additional insertions of the Froggatt-Nielsen scalar, or flavon S. At a given energy scale, the flavon acquires a vacuum expectation value S = f and breaks the flavor symmetry. The fundamental Yukawa couplings y are anarchic and hierarchies in the effective Yukawas are generated by the exponents n of the ratio f /Λ < 1, where Λ is the scale at which new physics sets in. While the Froggatt-Nielsen paradigm does neither specify the flavor breaking scale f nor the new physics scale Λ, the later implementation of this mechanism by Babu and Nandi [2] and Giudice and Lebedev [3] relate the flavor breaking scale to the electroweak scale. In particular, they propose S/Λ → H † H/Λ 2 in (1.1). This interesting idea however has the shortcoming that the bilinear H † H is a singlet under all symmetries, in particular it cannot carry a flavor charge. As a consequence, the number of flavon insertions needed in order to generate the observed fermion mass hierarchies is ad hoc and not related to a flavor symmetry. As briefly mentioned in [3], such a connection between the electroweak and the flavor breaking scale can however be motivated in a supersymmetric model featuring two Higgs doublets. Phenomenological constraints from the SM Higgs mass and signal strengths measurements exclude both the original Babu-Nandi-Giudice-Lebedev model as well as a possible (minimal) supersymmetric extension.
In this article, we propose a two Higgs doublet model, in which the two scalars H u and H d act jointly as the flavon field, such that S/Λ → H u H d /Λ 2 . As a consequence, the flavor breaking scale is set by the electroweak scale, v ≈ f , and the new physics scale is in the ballpark of a few TeV, as sketched in figure 1.
In the present study we concentrate on the quark sector and include the tau Yukawa couplings, reserving a full treatment of the lepton sector for future work. We discuss Higgs phenomenology, as well as its connection to flavor physics and show the potential -2 -

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for distinctive discovery signals that point towards an explanation of flavor at the electroweak scale.
In our model, the Higgs dependent effective Yukawa couplings induce tree-level flavor changing neutral currents (FCNCs) mediated by the Higgs bosons. These FCNCs, although naively very large, turn out to be under control for a sizable region of the parameter space. To this end we perform a careful study of FCNC effects in K −K, B d,s −B d,s mixing and estimate effects in the inclusive B s → X s γ decay as well as in the flavor-violating top decay t → hc. Flavor diagonal couplings of the SM-like Higgs to quarks, as well as couplings between the Higgs and electroweak gauge bosons, are modified with respect to the SM. While the former are unique to our model, the latter are equivalent to the Higgs couplings to gauge bosons in generic two Higgs doublet models [4,5]. This leads to deviations in both the Higgs production cross section and decay rates and we compute these effects for all relevant channels to compare them with current bounds from both the ATLAS and CMS experiments. We perform a global fit to all SM Higgs LHC data and we can accommodate the experimental data at a 2σ level for a sizable range of model parameters. It is most remarkable, that the parameter space preferred by flavor observables has a significant overlap with the region preferred by the SM-like Higgs global fit.
A characteristic feature of this two Higgs doublet flavor model is, that both the constraints from Higgs signal strength measurements and flavor physics point to a parameter region far from the alignment/decoupling limit, such that the additional Higgs bosons cannot be arbitrarily heavy. Furthermore, electroweak precision observables favor a large mass splitting between charged and neutral scalars, while the neutral scalar masses are preferred to be almost degenerate. As a result, direct collider searches for the additional Higgs bosons are very powerful in probing this model. We analyze the LHC results from direct searches for the CP-even and CP-odd Higgs scalars as well as for the charged Higgs boson in various production and decay modes and identify the most promising channels for a discovery. Although the bosonic Higgs couplings parametrically correspond to the ones in a generic two Higgs doublet model, the parameter space singled out by flavor constraints and Higgs precision measurements leads to distinctive predictions for future searches at the LHC.
Altogether, the two Higgs doublet flavor model presented in this work provides an explanation for quark masses and mixing angles from physics at the electroweak scale, while providing new opportunities for Higgs phenomenology at the LHC. The model can be tested by high precision measurements of meson-antimeson mixing and implies a UV completion at a scale that can be probed at the LHC. This paper is organized as follows: in section 2 we introduce our model, discuss the relevant parameters in the Yukawa sector and constraints from quark masses and mixing angles. We subsequently compute the Higgs couplings to quarks in section 3. In section 4, 5 and 6 we investigate constraints from Higgs, flavor and electroweak precision observables and map out the parameter space in agreement with these constraints. Section 7 contains a detailed analysis of present and future collider searches for the extra scalars. We comment on a possible UV completion section 8. In section 9 we present benchmarks for our model, before we summarize our main results in section 10. We consider a two Higgs doublet model in which fermion masses are generated by a Froggatt-Nielsen mechanism. We assume that the combination of the two scalar doublets H u H d carries a non-zero flavor charge such that the flavon is replaced by (2.1) We assign opposite hypercharges to the two Higgs doublets and parametrize them as In this setup the electroweak scale sets the flavor breaking scale by where with v = 246 GeV and 0 ≤ β ≤ π/2, such that v u and v d are positive. We define the expansion parameter (2.5) We choose ε = m b /m t ≈ 1/60, such that the Yukawa coupling for the bottom quarks corresponds to an effective operator with one insertion of the Higgs doublets (n = 1 in terms of equation (1.1)). Therefore for tan β = 1, the new physics scale is approximately Λ ≈ 4 v ≈ 1 TeV. If the fundamental Yukawa couplings in the UV completion are slightly larger than 1, this bound becomes weaker, and values of tan β > 1 are possible with a UV scale of the order of a TeV. Therefore, an ultraviolet completion at the TeV scale and tan β of O(1) are predictions of this model. We further discuss such a UV completion in section 8.
We consider the quarks and scalars in our model to be charged under a global U(1) F symmetry. Therefore in the flavor eigenbasis the Yukawa sector of the SM is replaced by the effective Lagrangian (to leading order in powers of ε) (2.6) in which a u j = a u , a c , a t , and a d j = a d , a s , a b denote the flavor charges of the three generations of up-and down-type quark singlets, a i = a 1 , a 2 , a 3 the flavor charges of the three generations of quark doublets and a Hu , a H d the flavor charges of the Higgs doublets. The leading order Yukawa couplings in equation (2.6) reduce to the Yukawa sector of a two -4 -

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Higgs doublet model of type II in the limit of vanishing flavor charge a i , a u j , a d j → 0. Couplings of H u (H d ) to the down-(up-) type quarks are suppressed by additional powers of ε. 1 The fundamental Yukawa couplings y u ij and y d ij are considered to be anarchic and of O(1). In writing equation (2.6) we normalized the sum of the Higgs charges to a Hu + a H d = 1.
The effective Yukawa couplings are then given by In (2.6) and (2.7), repeated indices between y ij and ε a i −au j −a Hu are not summed over, i.e., for example (Y u ) 12 = y u 12 ε a 1 −ac−a Hu . Thus the hierarchy of the effective Yukawa couplings is determined by the structure of the exponents of ε. The rotation to the mass eigenbasis is performed via with diagonal matrices given by and unitary rotation matrices U u,d , W u,d .
In the following we fix the flavor charges of the quarks and Higgs bosons by imposing constraints from quark masses and the CKM matrix. If the charges of the three generations of quark doublets and singlets are ordered such that one can derive the O(ε) dependence for the quark masses and rotation matrices [1], for i, j = 1, 2, 3. In the numerical analysis we will use the full unitary rotation matrices and include a scanning of anarchic Yukawa couplings with arbitrary phases and absolute values |y u,d ij | ∈ [0.5, 1.5]. Six of the 11 flavor charges are fixed by the quark masses. We choose Additional conditions follow from the CKM matrix, 13) 1 We also explored choices of flavor charges in which both up-and down-type quarks couple to one of the Higgs doublets at leading order (based on a two Higgs doublet model of type I), which will be discussed in a separate publication [6]. (2.14) These conditions end up fixing only two parameters. Including the normalization of the Higgs charges a Hu + a H d = 1 and our choice of a Hu = 1, we have 10 conditions on 11 parameters. 2 The remaining choice allows for an overall shift of quark flavor charges. Physical quantities however only depend on invariant differences. Thus the remaining choice does not have any phenomenological consequences and we set a Hu = 1 , If the last condition (2.14) is replaced by only the structure of the quark masses is explained by the flavor charges, while the hierarchical form of the CKM matrix is determined by the fundamental Yukawas y u ij , y d ij . In this case, a suitable choice of flavor charges read a Hu = 1 , This choice of charges is motivated by considerably weaker constraints from flavor observables due to the aligned charges for the left-handed quark fields.
A detailed implementation of lepton masses and mixing angles is beyond the scope of this work. We will however define the couplings of the tau leptons to the scalars in our model, since they are important for the Higgs phenomenology. We set such that m τ /m t ≈ ε.

Higgs couplings
The Yukawa interactions give rise to modifications to flavor diagonal Higgs couplings as well as potentially dangerous flavor changing neutral currents. In the flavor eigenbasis the interaction between quarks and the real neutral components of the Higgs doublet scalars follows from (2.6) and we obtain 2 Different choices for the normalization condition or the Higgs charges, e.g. aH d = 1, aH u = 0, do not change the physics of this model but will only imply different assignments for the quark flavor charges.

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We rotate to the quark mass eigenbasis, according to equation (2.8) and introduce the Higgs mass eigenstates as defined in appendix A. The rotation of the scalars gives rise to the following couplings between the scalar mass eigenstates and quarks After rotating to the quark mass eigenbasis, we find for the couplings of the light neutral scalar h, and for the heavy neutral scalar H, in which m u = diag(m u , m c , m t ), m d = diag(m d , m s , m b ) and we define s ϕ = sin ϕ, c ϕ = cos ϕ and t ϕ = tan ϕ, for any angle ϕ. In both (3.5) and (3.6), repeated indices are not summed over and we suppress the chirality index of the fermions q i ≡ q L i , q j ≡ q R j . We make use of the following trigonometric functions which are universal for up-and down-type quarks. We also define the matrices The structure of these matrices is fixed by the flavor charges, as given at the end of section 2. We find for the flavor charges in (2.15), For completeness, we also give the expressions for these matrices in the case of the flavor charges (2.17), Note that all flavor off-diagonal Higgs couplings are proportional to these matrices. In the limit of degenerate flavor charges a i , a u i or a d i , these matrices become diagonal and do not induce any flavor violating couplings. For the flavor charges (2.17), therefore only U and D generate FCNCs.
In addition, all flavor violating couplings of the scalars in (3.5) and (3.6) are proportional to the trigonometric functions in (3.7). In the limit f (α, β) = 0, all flavor off-diagonal couplings of the light Higgs vanish and the diagonal couplings are independent of both c β−α and t β , and approach their SM values (up to a sign). It should be noted that this sign difference corresponds to the wrong-sign Yukawa coupling in a generic two Higgs doublet model [7,8]. We will come back to these observations when we discuss flavor observables in section 5. The limit c β−α = 0, associated with decoupling [9][10][11] or alignment [5,[10][11][12] is not the SM, but corresponds to the model proposed by Babu, Nandi [2], and Giudice and Lebedev [3].
The pseudoscalar mass eigenstate A is obtained through the rotation (A.3) and its couplings to quark mass eigenstates can be derived from (3.6), by replacing g Aq i q j = i g Hq i q j cα→s β , sα→c β . (3.11) Finally, the charged Higgs couplings can also be obtained from (2.6) and are independent of the flavor charges. After rotation to quark and Higgs mass eigenstates, see (A.4), we obtain The couplings of the charged Higgs to quarks are therefore equivalent to the ones in the two Higgs doublet model of type II, see for example [13].

JHEP11(2015)016 4 Higgs production and decay
A light SM-like Higgs has been discovered at the LHC in various decay channels. While observations are mainly in the ballpark of SM expectations, there is still room for new physics. The modified flavor diagonal fermion couplings of the light Higgs h introduced in the previous section as well as modified gauge boson couplings lead to deviations in both production cross section and decay rates. In the following we compute these deviations and compare the results with the proton-proton collision data at √ s = 7 and 8 TeV obtained from the ATLAS [14] and CMS [15] experiments. For a given Higgs boson production channel and decay rate into specific final states X, normalized to the SM values, we define the signal strength parameter New physics can enter each of these three quantities: the production cross section σ prod , the partial decay rate Γ h→X and the total width Γ h,tot . We quantify the changes in flavor diagonal couplings of the Higgs to fermions f = t, b, τ and to vector bosons V = W ± , Z with respect to the SM by such that κ f = κ V = 1 in the SM limit. It follows from equation (3.5), that the coupling of the light Higgs to the top quark is rescaled by As a result, these couplings are modified in the same way as in two Higgs doublet models of type II, see for example [5,13,16]. However, couplings to the other flavors significantly differ from the couplings in generic two Higgs doublet models because of the Higgs dependent effective Yukawas, such that the Higgs-bottom coupling is rescaled by Note, that for f (α, β) = 0, any dependence on c β−α and t β cancels in (4.3) and (4.4) and we find that κ t = 1 and κ b = −1 and therefore the light Higgs has couplings to fermions of SM strength. We illustrate the parameter dependence of the square of these couplings in figure 2. In the right panel of figure 2 the value of κ 2 b goes through zero signalizing κ b changes sign and becomes negative in the upper right (lower left) corner for cos(β − α) > 0 (cos(β − α) < 0). The structure of these couplings has significant impact on the Higgs boson production cross sections and decay rates. Further, the coupling of the light Higgs boson to charm quarks is rescaled by Cos Β Α Tan Β Figure 2. Contours of κ 2 t (left) and κ 2 b (right) in the cos(β − α) − tan β plane. κ 2 t = κ 2 b = 1 corresponds to the SM limit, for κ b up to a sign in the right upper (lower left) corner for cos(β −α) > 0 (cos(β − α) < 0). The decoupling/alignment limit corresponds to the Babu-Nandi-Giudice-Lebedev model.
In general, fermion mixing effects generate corrections to the couplings, since the flavor charges of the quarks are not universal. These effects are encoded in the matrices Q u,d , U and D given in equation (3.8). For flavor-diagonal Higgs couplings to fermions we neglect corrections of O(ε). For couplings of the light Higgs boson to tau leptons we assume that a mechanism similar to our findings in the quark sector is responsible for generating masses, such that For the couplings of the light Higgs to vector bosons we obtain which is the same as in generic two Higgs doublet models. The gluon fusion initiated Higgs production, neglecting light quark contributions in the fermion loops, is defined normalized to the SM value as where ξ b = −0.032 + 0.035 i depends on the loop functions given in [4]. Therefore for values of κ b of O(1), the main Higgs production channel is to leading order indistinguishable from a type II two Higgs doublet model. Vector Boson Fusion (VBF) and Higgsstrahlung (VH) are both rescaled by κ V , while associated Higgs boson production with a top pair is modified by κ t , -10 -

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Therefore the three production processes rescale with the same factors as in generic two Higgs doublet models, as given e.g. in [5,13,16]. The partial decay widths of the light Higgs into SM fermions f and gauge bosons V = W ± , Z can similarly be written as Both top quark and W ± boson loops enter the diphoton decay width [29], 11) in which contributions from light fermions are neglected and contributions from charged scalar loops are encoded in δ. We find for M H ± 300 GeV a contribution of less than δ 0.04 and set it to zero in the following [9,29].
Expressed in terms of the rescaling factors κ t , κ b , κ c , κ τ and κ V , the total Higgs boson width is given by [30,31] where Γ SM h = 4.07 MeV [32] and we assume h → Zγ and even rarer modes to be SM-like. These contributions are collected in the constant term 0.004.
The partial decay width into bottom quarks has a very different dependence on tan β and cos(β − α) than in the generic type II two Higgs doublet model. This plays a relevant role in defining the allowed region in parameter space, since the bottom quark partial decay width dominates the total decay width, that in turn importantly affects the signal strength for all channels.
In figure 3 we show the result of a global χ 2 fit based on the data collected in table 1. Symmetrized errors are used for the fit. The left panel shows the plot for ATLAS and the right panel the plot for CMS. The two fit parameters are c β−α and t β . The 1σ and 2σ regions consistent with the LHC data are shaded in dark and light red, respectively. It is clear, that the preferred parameter space is different from generic two Higgs doublet models, for which regions close to the alignment or decoupling limit c β−α = 0 are favorable. [5,11,33]. In our case, c β−α = 0 corresponds to the Babu-Nandi-Giudice-Lebedev model [2,3], which is clearly disfavored by the data. We observe, that while the allowed region for ATLAS is slightly smaller than in the case of CMS, both fits show a preference for values of c β−α > 0 and t β 1. The more constrained region of parameter space for ATLAS can be understood by the larger central values of µ Z , µ W and µ γ in the dominant gluon fusion channel, that are less compatible with larger values of κ b , see figure 2. The white area between the two branches in both fits can be explained by very small values of κ b for which all other branching fractions grow. Overall, the fermion couplings prefer a region in parameter space, where they approach their SM values, with the caveat that the value of the bottom Higgs coupling κ b has a negative sign with respect to the SM value in the upper right branch of   Table 1. Input data for the global χ 2 -fit of Higgs production and decay with references. The data includes all updated results of the pp collision data at √ s = 7 and 8 TeV obtained from the ATLAS [14] and CMS [15] experiments.  the allowed red region. Note also that small values of c β−α correspond to larger t β in the region preferred by the global fit as follows from equation (4.4).
In order to understand the features of the global fit, we present the signal strengths of the relevant decay channels in figure 4. In these plots, the red (blue) band is the 1σ region of the corresponding ATLAS (CMS) measurement. Each plot shows the prediction of a particular signal strength for µ W , µ Z , µ γ and µ b , depending on c β−α for t β = 3 (solid red), t β = 2 (dashed orange), t β = 1 (dot-dashed green) and t β = 0.5 (dotted blue). Excluding all but these four observables only marginally changes the global fits. For t β 1 all four measurements prefer values of c β−α > 0. There is also an allowed region for c β−α < 0 for values of t β < 1, however as will be shown later this region is phenomenologically less interesting.
We conclude, that the global fit to LHC Higgs measurements accommodates tan β of O(1) for sizable values of cos(β − α) away from the decoupling/alignment limit. This is a nontrivial result, given that tan β is already constrained to be of order one from the bound on the new physics scale. As we discuss below, values of tan β 5 are in agreement with flavor constraints as well as a possible UV completion scale in the TeV to a few TeV range. Figure 5. Tree-level contributions to ∆S = 2 processes.

Constraints from flavor observables
In addition to modifications of flavor-diagonal couplings, the misalignment of the mass and coupling matrices induces flavor changing couplings of the light Higgs h, the heavy neutral scalar H and the pseudoscalar A. These couplings generate FCNCs at tree-level, which are subject to strong constraints from neutral meson oscillations. In the following we calculate and analyze contributions to the relevant observables. We further estimate effects in b → sγ and give the prediction for the flavor-violating top decay t → hc.

Meson-antimeson mixing
In the K −K system, contributions from Higgs mediated FCNCs are captured by the effective Hamiltonian At tree-level, the corresponding Wilson coefficients can be read off from the diagrams in figure 5 [34], Similar expressions hold for B s −B s mixing, with sd → bs, B d −B d mixing, with sd → bd and D −D mixing with sd → uc. Contributions from Higgs boson exchange are only suppressed by the weak scale, but the Froggatt-Nielsen mechanism induces an additional suppression of flavor off-diagonal couplings by the masses of the involved quarks as well as the expansion parameter ε. The relative size of the Wilson coefficients (5.2) depends therefore strongly on the explicit flavor structure. For the flavor charge assignment (2.15), which is tailored to explain quark masses as well as CKM mixing angles, we collect the results in the left hand side of table 2. Table 2. Flavor specific part of the Wilson coefficients for meson-antimeson mixing in the case of the flavor charge assignments (2.15) with flavor structure (3.9) (left) and flavor charge assignments (2.17) with flavor structure (3.10) (right).
In the case of K −K mixing, we find that the largest coefficient isC 2 with where we factored out the light Higgs mass in the second line, the trigonometric functions f (α, β) and F (α, β) are defined in (3.7), andc sd 2 is the flavor-dependent part of the Wilson coefficient given in table 2. The same expression holds for the Wilson coefficient C sd 2 , with the additional ε 2 suppression due to the replacement ofc sd 2 → c sd 2 . The flavor-dependent Wilson coefficient c sd 4 is also suppressed by ε with respect toc sd 2 , but the minus sign in the last line of (5.3) is replaced by a plus, which corresponds to a constructive interference of the different contributions, The limit of exact cancellation in C sd 2 andC sd 2 and maximal interference in C sd 4 corresponds to the SU(2) L symmetric limit, in which operators of the type (s L d R ) 2 are forbidden [35]. In table 3, we present the current bounds on the Wilson coefficients at the electroweak scale for the different meson systems, based on [36]. These bounds have been derived by assuming that new physics only contributes to a single Wilson coefficient and can therefore only be taken as a rough upper limit. For K −K mixing, the strongest constraint comes from the CP violating observable K , such that the bounds on the imaginary part of the Wilson coefficient is cited. Since we assume arbitrary phases, the estimate (5.3) holds for both real and imaginary parts of the Wilson coefficients. Comparing (5.3) with the bound in table 3 shows that a partial cancellation inC sd 2 is necessary in order to comply with the limit. For M A , M H > m h , this corresponds to a preferred region in the cos(β − α) − tan β plane. In the left panel of figure 7 we show the preferred region, Table 3. Model-independent bounds on Wilson coefficients for meson-antimeson mixing evaluated at the electroweak scale in units of GeV −2 [36], taking into account the running described in appendix D. The same bounds hold for the Wilson coefficients with flipped chirality C i →C i .
for which |C sd 2 | < 10 −16 /GeV 2 (shaded orange), assuming M A = M H = 500 GeV. Contributions to C sd 4 can be enhanced by the constructive interference between the scalar contributions. Also, the bound on C sd 4 is particularly strong, because it is enhanced from Renormalization Group (RG) running as well as from the matrix element, that scales like , see appendix C for details. However, the additional suppression shown in table 2 gives C sd 4 = εC sd 2 , such that a slight enhancement from interference effects is allowed. In the left panel of figure 7 we show the region in the cos(β − α) − tan β plane for which |C sd 4 | < 7 × 10 −17 / GeV 2 (shaded blue). In addition to tree-level exchanges, various one-loop contributions can potentially become large. The relevant diagrams are shown in figure 6. The contributions from the box diagrams of type (a) are completely analogous to the ones in a type II two Higgs doublet model, because the couplings of the charged Higgs (3.12) are indistinguishable between the two models. The leading contribution enters C sd 1 and comes from the box with one charged Higgs, a W ± boson and top quarks running in the loop and one finds [37] C sd where in the last equality we set t β = 1. For t β 1, this contribution is of the order of the largest tree-level contribution. We therefore require t β 1 in order to be in compliance with experimental bounds in table 3. In principle, there are also contributions from box diagrams to the other operators in (5.1) as well as box diagrams with neutral scalar exchange, but both are chirally suppressed by powers of light quark masses over the electroweak scale and turn out to be negligible. The loop diagrams labeled (b) and (c) in figure 6 are also suppressed. Diagrams of type (b) have the same coupling structure as the tree-level diagrams, but are additionally suppressed by a loop factor. Diagrams of type (c) are enhanced with respect to (5.4) by the light Higgs couplings to the top quark or charged scalars, but suppressed by CKM elements and a loop factor, such that we find for C sd 4 [38] C sd The equivalent diagram with a charm quark in the loop is of the same order. Having considered all different contributions we will map out the parameter space in the cos(β − α) − tan β plane in which the prediction for K in our model agrees with the experimental bound within 2σ in a numerical analysis. For this purpose we define includes the Standard Model contribution. We compute the Wilson coefficients at the scale of the light Higgs and for M H = M A = M H ± = 500 GeV respectively, using the full expressions for the Wilson coefficients including tree-level and leading box diagrams. We collect the full analytic expressions of the latter in appendix B. In the next step, the Wilson coefficients in (5.1) are evolved down from the mass scale of the scalars to the scale µ = 2 GeV at which the hadronic matrix elements are evaluated using the RG equations in [39]. The hadronic matrix elements are taken from [40] and collected with the other numerical input in appendix D. We randomly generate a sample set of points of fundamental Yukawa couplings, defined in (2.6), with |y u,d ij | ∈ [0.5, 1.5] and with arbitrary phases. We require the SM quark masses and Wolfenstein parameters to be reproduced within two standard deviations. More details to the procedure and input parameters can be found in appendices C and D. At this stage, the mixing angles α and β from the Higgs sector still remain free parameters and our sample set only fixes the fundamental Yukawas.
In the right panel of figure 7 we show the percentage of sample points which reproduce C exp K within 2σ in the cos(β − α) − tan β plane. We employ the value extracted from a fit to the CKM triangle by the UTfit group [41], The result shows good agreement with the estimate of the separate contributions shown in the left panel of figure 7. The area for which t β < 0.5 is cut off, because of the one-loop contributions from charged Higgs exchange [42]. We find a large region of parameter space for which our model prediction is in agreement with the experimental bound without any tuning of parameters.
In the case of B d −B d and B s −B s mixing, the effective Lagrangian, as well as the tree-level contributions to the Wilson coefficients from scalar and pseudoscalar exchange can be read off from (5.1) and (5.2) with the replacements s ↔ b and d ↔ d, s, respectively. The angle dependence of the Wilson coefficients is universal and therefore only the flavor dependent part changes from (5.3) and (5.4), such that the parametric dependence presented in the left panel of figure 7 also holds in the B sector. For the Wilson coefficients it follows from table 2, The corresponding bounds in table 3 imply, that C bs 2 is at the border of the naive bound, while a much larger contribution to C bs 4 is allowed. The contributions to C bd 4 , C bd 2 andC bd 2 are too large almost in the entire cos(β − α) − tan β plane, and therefore demand cancellations implying important restrictions for the permitted region of our parameter space. At the one-loop level, box diagrams generate the contributions for tan β = 1. In the B s −B s system for low tan β, this contribution becomes larger than all tree-level contributions. Since the box is only sensitive to charged Higgs couplings, we expect comparable constraints as in a two Higgs doublet model of type II. In addition, since the contribution is independent of cos(β − α), we expect a universal lower bound on tan β, as observed in the left panel of figure 8. For both the B d,s −B d,s system we also include the box diagram contributions to the other Wilson coefficients, which are suppressed by m b /m W . The corresponding expressions are collected in appendix B. Analogous to (5.7), we define such that C Bq = ∆m q /∆m SM q measures new physics effects in the mass difference and new phases enter φ Bq . In the left (right) panel of figure 8, we present the percentage of sample points in agreement with the experimental constraints at 95% CL for C exp Bs (C exp B d ), based on the results obtained from the UTfit group [41], In the following, we will adopt the 10% contour as the fine-tuning bound from flavor observables on the parameter space. Figure 9 shows In D −D mixing, all tree-level contributions to the Wilson coefficients are strongly suppressed, (5.14) In contrast to the down-sector however, the box diagram with neutral Higgs exchange is not suppressed by light quark masses, because the dominant contribution comes from the top in the loop [43]. The leading box contributions of the light Higgs to the coefficient C uc 1 can therefore be larger than all tree-level effects for f (α, β) = 1, and the loop function defined in appendix B. Boxes with heavy Higgs insertions are further suppressed. However, the corresponding bound in table 3 is orders of magnitude weaker than our estimate. The D −D system will therefore not induce further constraints. In all the above analyses, we have concentrated on the solution for the flavor charges (2.15), but the situation is quite different for the flavor charges given in (2.17). From (3.10) it follows, that the contributions to the Wilson coefficients are highly suppressed, as is explicit in the flavor-dependent parts of the Wilson coefficients given on the right hand side of table 2. This shows, that although constraints from the B s −B s and K −K systems remain the same, the constraints from the B d −B d system can be very much relaxed due to the different charge assignment. Therefore, if only the hierarchies in the quark masses are explained by a Froggatt-Nielsen mechanism at the weak scale, but the CKM mixing angles have a different origin, bounds from meson-antimeson mixing are very mild and do not lead to any severe restrictions on the parameter space.
Rare Kaon and B d,s decays can in principle be subject to large corrections, but depend crucially on the implementation of the lepton sector, which will be discussed elsewhere. Processes in which the neutral scalars only enter at loop-level, such as Br(B s → X s γ) are generically dominated by charged Higgs contributions, which are larger than the contributions from the neutral Higgs by a factor of for f (α, β) = 0.1 − 1. We will therefore adopt the bounds from Br(B s → X s γ) on the charged scalar mass in two Higgs doublet models for tan β 2, considering values within a 3σ band in order to account for uncertainties of higher order corrections not included in the theoretical computation [44,45],

Flavor violating top decays
We consider the flavor violating decays of the top quark t → hc and t → hu. In contrast to the SM, in which flavor violating top quark decays are loop suppressed, in our model the top quark has tree-level couplings to the light Higgs and other up-type flavors. The corresponding branching ratios Br(t → h c) ≈ 3 × 10 −15 and Br(t → h u) ≈ 2 × 10 −17 are tiny in the SM [46]. In our model the branching fraction of the top decaying to Higgs and charm is given by [47] Br and similarly for Br(t → h u) by replacing the appropriate flavor indices. Both branching ratios are parametrically of the same order, because the flavor off-diagonal couplings in equation (3.5) yield g hct ≈ g hut ∝ m t ε. In figure 10 we show Br(t → h c) plotted against cos(β − α) for a range of parameter points and indicate the different predictions for tan β = 3(4) by a blue (green) band. The widths of these bands correspond to the range of values obtained by scanning over our sample set of random fundamental Yukawas. The most recent limits are Br(t → hc) < 0.56% from CMS [48] and Br(t → hc) < 0.79% from ATLAS [49] and are shown in the plot as a red band. The projected exclusion limit for 3000 fb −1 at the high luminosity LHC Br(t → hc) < 2 × 10 −4 [50] is indicated by a dashed red line. The plot shows that this cross section can be even above 10 −4 for negative values of cos(β − α). However, the cross section drops for the same angles for which FCNCs become small, because the same trigonometric function governs flavor off-diagonal couplings between the light Higgs to up-and down-type quarks in equation (3.5).

Perturbativity, unitarity, and electroweak precision measurements
In this section we consider perturbativity bounds, as well as constraints from the unitarity of the S matrix and electroweak precision measurements on our model. The large scalar masses implied by flavor observables and the constrained scalar potential (A.1) result in potentially large quartic couplings. Mass splittings between the different scalar mass eigenstates can in addition generate sizable contributions to the oblique parameters S, T and U . We therefore scan over the allowed parameters, considering the various bounds described in [51]. This includes stability constraints on the Higgs potential, perturbativity bounds on the quartic scalar couplings, unitarity of the various scattering amplitudes involving scalars and the constraints from the oblique parameters. This calculation is not different from a generic two Higgs doublet model, since the oblique parameters only measure corrections to -22 -

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the gauge boson self-energies from loops of the new scalars, whose couplings are fixed by the kinetic terms [52,53]. The two plots in the upper panels of figure 11 show the region in the positive cos(β − α) − tan β plane in which stability and perturbativity bounds are fulfilled, and the S and T parameters are at most 2σ from the best fit point, corresponding to a global χ 2 fit obtained by the Gfitter group [54]. Further, for some regions of the parameter space, one or more of the quartic couplings in the Higgs potential can become non-perturbative already at the TeV scale λ i (µ = 1TeV) 4π. We implement the one-loop beta functions for our model and match to the SM at an approximate average scale of the Higgs boson masses in order to estimate the scale of strong coupling. In particular for larger values cos(β − α) and larger and degenerate masses M A = M H , the cutoff scale becomes lower. Moreover, we find that for sizable mass splittings between the charged and neutral scalars, the scale of strong coupling is in the range of 2 − 5 TeV. However, as mentioned in section 2 and in more detail in section 8 below, we expect the UV completion of our model to set in close to the TeV scale.
We conclude, that for fixed M A = 600 GeV, two qualitatively different choices of scalar masses are compatible with electroweak precision bounds, Higgs constraints and a low tan β as preferred by flavor constraints. Either the scalar masses are approximately degenerate M A ≈ M H ≈ M H + or the charged scalar is considerably lighter than the neutral scalars M A,H −M H + 100 GeV. Of these possibilities, only for large mass splittings can the theory 3 If the mass splittings become larger than |MH − MA| 30 GeV, the full parameter space is excluded. be valid up to several TeV and in the following we will concentrate on this setup. Note, that these restrictions would be slightly relaxed if we take the fit to the CMS measurements of the Higgs signal strengths as a constraint.
Another important electroweak precision observable is the Zbb coupling. While the experimental value of the left-handed Zb LbL coupling is in good agreement with the SM prediction, there is a discrepancy between the measured right-handed Zb RbR coupling and the SM prediction, see e.g. [54,55]. Higher order corrections with the neutral or charged scalars in the loop can in principle affect these couplings.
The charged scalar contributions to the left-handed Zb LbL coupling in a two Higgs doublet model of type II can become sizable for low tan β, inducing a bound of t β 0.5 -24 -

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for masses of M H ± ≈ 500 GeV [42], while corrections to the Zb RbR vertex are suppressed by m b /m t . In addition, the neutral scalar couplings to bottom quarks are very different from a generic two Higgs doublet model in a large range of parameter space. We define the couplings of the Z boson to left-handed and right-handed bottom quarks by Here, g L,R SM are the SM couplings and we denote the corrections from neutral and charged Higgs exchange by δg h , δg A,H and δg H ± , respectively. We estimate while contributions from the heavy neutral scalars are further suppressed by δg A,H /δg h ≈ m 2 h /M 2 A,H and couple with κ A b and κ H b , as defined in the following section in equation (7.1). Neutral Higgs contributions to g L are therefore at least an order of magnitude smaller than the charged Higgs contributions for the region preferred by the global Higgs fit, while corrections to the right-handed coupling g R are at most of a similar size. We numerically estimate the light neutral Higgs contributions following [56,57]. For κ 2 b = 1, we find for the right-handed coupling δg R h 10 −6 ×g R SM , and for the left-handed coupling δg L h 10 −6 ×g L SM , which is many orders of magnitude too small in order to explain the anomalous Zb R b R coupling. In order to improve the fit with respect to the SM, contributions of the order of 0.2% to g L SM and 2% − 20% to g R SM (depending on the sign) are necessary [58]. The neutral Higgs contributions to the Zbb vertex can therefore be safely neglected. It should be noted, that fermionic mixing effects in the UV completion of this model can affect both the oblique parameters and the Zbb vertex. These however depend sensitively on the exact realization of the UV completion, which is beyond the scope of this paper.

Collider searches for heavy extra scalars
Our model features heavy new scalars beyond the SM, namely the neutral scalar Higgs H, the pseudoscalar A and the charged Higgs H ± . Their masses are bound to be less than 700 GeV by perturbativity, and various flavor constraints set lower bounds on their masses as discussed in section 5. In this section we consider the latest ATLAS and CMS bounds on new neutral and charged Higgs bosons.

Couplings and total width of heavy scalars
where t, b and V denote the rescaling factor for top, bottom and vector boson couplings, respectively. Since (κ H t ) 2 is relevant for the gluon fusion production of the heavy Higgs boson H, its parametric dependence is essential and we illustrate it in figure 12. Both flavor diagonal and flavor changing couplings of H and A involving the charm quark, are given by where κ A tc and κ H tc are defined according to equation (7.7) below. As discussed at the end of section 2, we define the couplings to taus as Higgs h is of particular interest for the following analysis and reads [5,9,12] Finally, the couplings between two Higgs bosons and one gauge boson read [4] g AhZ = g 2 cos θ W c β−α , Further, we define the total widths for H, A, and H + , including all relevant and kinematically accessible decay channels (no off-shell decays are relevant in the regions we will consider) Φ = H, A appears in (7.6). This channel is characteristic for our model and we therefore give the partial width explicitely The parametric dependence of the total width for the scalar (pseudoscalar) Higgs boson is illustrated in the left (right) panel of figure 13 for M = M A = M H = M H + = 600 GeV. For large regions of parameter space the total width becomes large. In particular, for tan β > 1 and | cos(β − α)| > O(0.5) values of O(100) GeV can be obtained, such that finite width effects need to be taken into account. The charged Higgs can also have a sizable branching ratio Br(H + → hW + ), which can become the dominant decay channel for sufficiently large cos(β − α). In appendix E, in figures 23-25 we show the branching ratios for all Higgs bosons for specific benchmark scenarios to be discussed later.

Analysis of production and decay channels
In the following we study the impact of searches for heavy Higgs bosons at ATLAS and CMS. To this end, we compute the production cross section and various decay rates for the heavy Higgs bosons. We generate the gluon-fusion production cross section at nextto-leading order (NLO) using HIGLU [59], taking into account the contributions of the bottom quark loop and use the leading order expressions for the partial decay width with the appropriate couplings of our model [60,61]. When relevant, we also consider the vector-boson fusion production cross section, using the values quoted in [62,63]. For charged Higgs production we use the NLO results in [64]. In the following we will assume M = M A = M H = M H + , if not specified otherwise, and we discuss in detail the effects of a splitting between the neutral and charged Higgs boson masses.
One of the most interesting channels for the discovery of the pseudoscalar Higgs boson, involves the A → hZ decay, because the corresponding branching ratio becomes dominant for sizable values of cos(β −α). There are several experimental studies constraining σ(gg → A) × Br(A → hZ), with the light Higgs further decaying into bottom quarks [65,66], tau leptons [65], as well as multi-leptons [49].
The predictions of our model for both σ(gg → A) × Br(A → hZ) and σ(gg → A) × Br(A → hZ → + − bb) are presented in figure 14 in the left and right panel, respectively. For the decay rate Γ(h → bb), NLO corrections are sizable and therefore we include them in our analysis by setting The shape of the contours follows naturally from the fact that the branching ratio scales as cos(β − α) 2 , while the production cross section depends only on tan β. This is no different than in a generic two Higgs doublet model [5,49], but it is particularly relevant in our model, since it cannot live close to the decoupling limit, as discussed in section 4. The experimental exclusion bounds from [49] constrain σ(gg → A) × Br(A → hZ) considering a multi-lepton final state, but the study is only performed for pseudoscalars with masses up to M A < 360 GeV.
In the right panel of figure 14 we show the contours of σ(gg → A) × Br(A → hZ → + − bb) in femtobarn for 8 TeV pp collisions in the cos(β − α) − tan β plane and M = 600 GeV. Two branches with suppressed values for σ(gg → A) × Br(A → hZ → bb) appear. The first branch is the decoupling or alignment limit, where g AhZ vanishes. The second branch is given by the region for which the coupling of the light Higgs h to bottom quarks becomes small.
We consider the measurement of σ(gg → A)×Br(A → hZ)×Br(h → bb) by ATLAS [65] and the measurement of σ(gg → A) × Br(A → hZ → + − bb) by CMS [66] [65]. In both plots we assume equal scalar masses, M = 500 GeV (dotted) and M = 600 GeV (dashed), and narrow-width approximation. The region below and to the right of the curves is excluded.
M = 500 GeV (dotted) and M = 600 GeV (dashed). These constraints are substantially weaker than the corresponding bounds for the h → bb decay. In the following we consider the impact of finite width effects on the previous bounds. In the right panel of figure 16, we show the rescaling factor for the cross section times branching ratio due to finite width effects, extrapolated from the CMS analysis [66], for M A = 500 (600) GeV in pink (green). In the left panel of figure 16 we first show for comparison the exclusion bound from ATLAS data for M = 600 GeV in the narrow width approximation. Under the assumption that the scaling effects for ATLAS and CMS are similar and assuming sensitivity up to a total width of Γ A 100 GeV, we consider finite width effects for each point in the cos(β−α)−tan β plane and reinterpret the ATLAS results (solid, blue line in the left panel of figure 16). Although finite width effects significantly weaken the exclusion bound, this channel remains the most promising discovery channel at the LHC run II. The bound is further relaxed in the case of a mass splitting, M A M H + (M H ), such that the decay channels A → H + W − (HZ) open up. Our discussion in section 6 showed that such a mass splitting is only allowed between the pseudoscalar and the charged Higgs boson. We present the bound for σ(gg → A)×Br(A → hZ)×Br(h → bb) including finite width effects for M A = M H = 600 GeV and M H + = 400 GeV in the left panel of figure 16 (black).
In the following we will consider the experimental bounds from searches for the neutral CP-even Higgs boson H. There are two channels of particular interest, the CP even scalar decaying into light Higgs bosons H → hh and the CP even scalar decaying to vector bosons H → V V with V = W, Z. -31 -

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In figure 17 we present predictions for σ(gg → H) × Br(H → hh) in picobarn for 8 TeV pp collisions in the cos(β − α) − tan β plane for M = 600 GeV. From (7.4) we observe that the self coupling g Hhh is proportional to cos(β − α) and has an explicit M A dependence. For cos(β − α) ≥ 0 we observe two branches of contours with suppressed σ × Br. The first branch approaches zero at cos(β − α) = 0, and for the second branch both the coupling g Hhh and the production cross section become small. Predictions for σ(gg → H) × Br(H → hh) are comparable to the ones in a generic two Higgs doublet model of type II [5]. Similar to the pseudoscalar case, the experimental exclusion bounds for σ(gg → H) × Br(H → hh) [49] are only available up to M H < 360 GeV. However for the CP even Higgs, the model predictions seem to be much below the present experimental sensitivity.
The most important search channel for the heavy CP even neutral Higgs boson H is the inclusive production with subsequent decay of H → V V with V = W, Z. In our specific model there is an interesting region of parameter space in which the vector boson fusion production is competitive with the gluon fusion production due to the behavior of κ H t . Normalized to the corresponding SM Higgs production and decay processes for a SM Higgs of mass M H , we have for gluon fusion and vector boson fusion production processes, respectively, where ξ H b denotes the correction from a bottom quark in gluon fusion with respect to the leading top contribution. We take the SM total width Γ SM H for a heavy Higgs of mass M H from the LHC Higgs Cross section Working Group [62,63,67].
In figure 18 we present theoretical predictions for contours of inclusive heavy neutral CP even Higgs production (left panel) and vector boson fusion production (right panel) with subsequent decay into H → V V , using (7.10) and (7.11), for M = M A = M H = M H + = 600 GeV. The vector boson fusion is governed by κ H V and becomes strongly suppressed for small cos(β −α). The gluon fusion production mode in (7.10) is suppressed for small values of κ H t or for small κ H V and this effect shows in the inclusive production mode above. We observe that for small κ H t , both production cross sections become competitive. The theory prediction for these two observables differs from a two Higgs doublet model of type II only by the different scaling of the width Γ H and the contribution of the bottom quark to gluon fusion, which is small for tan β ∼ O(1).
The CMS collaboration has reported updated results from an inclusive search for a heavy Higgs decaying into W + W − and ZZ in the range of M H = 145 − 1000 GeV [68]. They consider both fully leptonic and semileptonic final states. In figure 19 we illustrate those bounds for M = M A = M H = M H + with M = 500 GeV (dotted) and M = 600 GeV (solid). We observe that this search mode is competitive with the bounds obtained from the A → hZ channel. We note that for the neutral CP even Higgs analysis no finite width The heavy scalar masses are set to M = 600 GeV. effects have been taken into account, although we expect sizable finite width effects in a large region of parameter space, compare the left panel of figure 13 .
The CMS collaboration also performed an analysis for a heavy neutral Higgs boson decaying into W + W − in vector boson fusion production channel in the mass range M H = 110 − 600 GeV [69]. The observed signal significance is close to the SM prediction for a Higgs of M H = 300 − 600 GeV, and hence from the right panel of figure 18 it follows that there is no sensitivity to the preferred parameter region from this search.

Origin of the effective Yukawa couplings
In this section we present an example of the origin of the effective Yukawa couplings at the TeV scale for the bottom quark sector. Similar considerations can explain the generation of the other effective light quark Yukawa couplings in our model. A complete description of the UV completion is beyond the scope of this paper.
A possible completion of the Froggatt Nielsen model may introduce new colored vectorlike fermions or additional scalar doublets [73], whose masses determine the suppression scale Λ in the expansion parameter (2.5). Since in our model the flavor breaking scale is identified with the electroweak scale and the expansion parameter is fixed by the ratio of bottom and top quark masses ε = m b /m t , the UV scale is constrained to be of the order of Λ ∼ 1 TeV.
The relevant operators that would provide a UV completion for the bottom Yukawa interactions are such that after integrating out the heavy fields the effective Lagrangian is given by Figure 20. Diagram in the full theory, which generates the Yukawa coupling between the Higgs and the bottom quarks after integrating out the heavy vector-like fermions ψ, η.
The corresponding diagram is given in figure 20 in which the new vector-like fermions carry quantum numbers with respect to the groups SU 3) is follows that for fixed y 1 = y 2 = y 3 = 1 and y d ∈ [0.5, 1.5] this predicts the masses M η = M ψ ≈ Λ = 1 TeV. It is evident that slightly larger fundamental Yukawa couplings y 1 , y 2 and y 3 , allow for heavier vector-like fermions, while any tan β 1 or tan β 1 lead to lower mass scales. In the spirit of avoiding hierarchies between the fundamental couplings, including the top Yukawa coupling, we shall consider the ratio y i /y t ∼ O(1) with i = 1, 2, 3. This constrains the masses of the vector-like fermions to be at most of the order of a few TeV. In particular, we define a generic mass M ≡ M η M ψ , and an average fundamental Yukawa couplingȳ = (y 1 y 2 y 3 ) 1/3 , such that (8.5) In figure 21, we show the expected masses of the new fermions for varying tan β and fixed y d = 1, for three different values of average Yukawa couplingsȳ = 1, 1.5, 2 (from bottom to top). These predictions for the expected masses remain the same forȳ = 1 and change at most by 15% (25%) forȳ = 1.5 (2) for the first generation quarks and at most 10% (20%) for second generation quarks.
The solid and dashed red lines in figure 21 indicate the present and projected experimental bounds from searches for pair produced heavy quarks at the LHC. These searches have been performed both by ATLAS and CMS, and exclude vector resonances with masses of 600 − 800 GeV [74][75][76], depending on the decay mode, with some channels already probing top partners T up to 900 GeV for Br(T → W + b) = 100% [77]. The next run of the LHC has a projected reach of M 1.2 (1.4) TeV for 20 fb −1 (100 fb −1 ) and Br(T → W + b) = 50% [78]. Searches for heavy vector-like quarks in single production have also been considered [79][80][81] and could be much more effective as a discovery channel for sufficiently heavy vector-like quarks compared to the previously mentioned pair production searches. However, the LHC reach in the single production channel depends very strongly on the model parameters which define the couplings of the heavy quarks to SM quarks. A reinterpretation of any of the existing LHC bounds in single heavy quark production channels would demand a detailed study of production cross sections and decay branching ratios for a specific UV completion. Similarly, a specific UV completion would be subject to constraints from electroweak precision measurements as well as from flavor physics [73]. The latter have been addressed in some detail in the original Giudice-Lebedev paper [3].

Benchmark scenarios
The global fit to Higgs signal strength measurements discussed in section 3 universally constrains the allowed parameter space to two branches within cos(β − α) = 0.35 − 0.8. Smaller values of cos(β − α) < 0.35 are in principle possible for tan β > 5, but such large values of tan β are in tension with flavor observables. Electroweak precision observables and collider searches for the extra scalars provide additional constraints that narrow the parameter space significantly. In the following we examine the allowed window in the cos(β − α) − tan β plane and specify three benchmark points, that highlight the interesting features for the phenomenology of this model, and for which we give a detailed list of couplings, production cross sections and decay widths.
As a result of the discussion in sections 6, the combination of constraints from flavor physics, electroweak precision observables, unitarity and perturbativity lead to a con- loop contributions from charged Higgs exchange result in a lower bound of tan β 1.5. Collider searches for the two heavy neutral Higgs bosons further constrain the allowed parameter space and probe the right branch of the global Higgs fit for cos(β − α) = O(0.5) and tan β 3. As a result, there is a specific window of allowed masses as well as values of cos(β − α) and tan β, which translates into a precise prediction for searches for the extra scalars and constrain the possible deviations in the SM Higgs couplings. In figure 22, we illustrate this window by showing the 95% CL region of the global fit to ATLAS Higgs signal strengths measurements (red shaded area), the region preferred by electroweak precision constraints (shaded green) and the bound induced from flavor constraints (solid purple contour), as shown in figure 9. Further, we superimpose the bounds derived from the ATLAS and CMS measurements of σ(gg → A) × Br(A → hZ → bb + − ) (solid blue) and Comparing the two plots in figure 22, bounds from flavor physics as well as collider constraints become weaker for larger masses. The area in agreement with electroweak precision bounds is slightly larger for smaller mass splittings, but similar for the two examples given in figure 22. The right boundary of the right branch of the global Higgs fit is close to the contour of κ b = −1, for which the Higgs coupling to bottom quarks has the same size, but opposite sign compared to the SM one. The left boundary of the right branch is close to κ b = −0.5. For all of the allowed parameter space, we can therefore infer −1 κ b −0.5. In addition to the sign and the reduction of the Higgs bottom coupling, we find a universal enhancement of the Higgs charm couplings. Both can in principle be probed by measurements of exclusive radiative Higgs boson decays [82]. The 14 TeV run of the LHC can potentially test the sign of κ b , whereas possible departures from the SM Higgs charm couplings of the order of 20% could be established at a prospective 100 TeV collider by measurements of these radiative hadronic modes [83]. A first experimental analysis of the decays h → J/ψ γ and h → Υγ, including the relevant backgrounds, has recently been performed by ATLAS [84]. Less optimistic limits for the charm Yukawa coupling in the exclusive decays have been obtained by the authors of [85], who however find a comparable sensitivity in measurements of the inclusive decay rate into charms using charm tagging techniques [86]. Moreover, in the presence of a Higgs portal to dark matter, relative signs in the Higgs couplings to light and top quarks could significantly modify the direct detection cross section [87].
In table 4 and 5 we give the values for the Higgs couplings, signal strengths, production cross sections and branching ratios for three representative benchmark points indicated by black crosses in figure 22. Typical values of cos(β − α) ≈ 0.4 − 0.55 and tan β ≈ 3 − 4.5 are considered. In all cases, κ t ≈ 1, implying a gluon fusion production rate of order of the SM one.
Benchmarks 1a and 1b allow for larger values M A,H ≈ 600 GeV and a charged Higgs mass M H + ≈ 450 GeV, close to the 2σ bound derived from the experimental b → sγ measurement in a type II two Higgs doublet model with tan β > 2.    Table 4. Values for the Higgs signal strength, heavy scalar production cross sections for the dominant channels at the LHC, partial and total widths for the benchmarks 1a and 1b.   In Benchmark 1a, the tree-level gauge boson and down type fermion third generation couplings are suppressed by factors of order 20% and 40%, respectively, while the Higgs coupling to charm is enhanced by about 20%. The sizable suppression of κ b yields a suppression of the branching ratio into gauge bosons and hence of the corresponding signal strength of those channels. The charm signal strength instead, is increased by a factor ∼ 2−3 (depending on the production mode) due to the combined effects of an enhancement in κ c and a suppression in κ b and κ V . All other vector boson fusion and VH production channels are suppressed with respect to the SM, in particular the h → bb search mode.
In Benchmark 1b all tree-level fermion and gauge Higgs couplings are within less than 5 − 10% of the SM expectations, hence the signal strengths in gluon fusion production are also within 5 − 10% of the SM ones, with the exception of a 20% enhancement in µ c . All -39 -vector boson fusion/VH production channels are suppressed with a maximal suppression of 25 − 30% in the case of light Higgs decaying into gauge bosons.
Benchmark 2 allows for the smallest possible values of M H + = 360 GeV compatible with the 3σ bounds derived from the experimental b → sγ measurement in a type II 2HDM with tan β > 2. Benchmark 2 has a similar tendency in the couplings of gauge bosons and fermions to the light Higgs boson and hence in the corresponding signal strengths as in Benchmark 1a, but with percentual effects in the deviations from SM predictions that are a factor 2 − 3 smaller. In addition to improving signal strength measurements, the ongoing run of the LHC will probe these benchmarks by direct searches for the additional Higgs bosons. All three benchmark scenarios will be primarily tested by the search for A → Zh and H → V V , that have branching ratios of 55% − 75%, depending on the scenario. In the case of H → V V , the inclusive and vector boson fusion production modes will play a complementary, relevant role. In addition to these discovery channels, other interesting search modes such as A, H → W + H − , H → hh, A → tt, H + → hW + , and H + → tb would yield additional valuable information about this model. The mass splitting between neutral and charged scalars give rise to an additional decay chain, that can potentially allow to discover the charged Higgs even for masses of M H + ≈ 360 − 400 GeV, in particular for the subsequent decay of H + → W + h. Although challenging due to the small branching ratio, a novel channel in these scenarios is A → tc.
Predictions for particular observables can be computed from the information provided in table 4 and table 5. Finite width effects play a relevant role and in the case of A → hZ we have compiled them in the right panel in figure 16. For completeness, we present the branching ratios for h, H, A and H + as a function of cos(β − α) for the parameter values of benchmark 1a, 1b and 2 in figures 23, 24 and 25 in appendix E.
Finally, improved measurements of flavor observables, in particular in the neutral B d system could additionally constrain the parameter space significantly.

Conclusion
In this article we propose an explanation for the hierarchies in fermion masses and mixings based on a Froggatt-Nielsen mechanism, in which two Higgs doublets play the role of the flavon. Therefore, the underlying flavor symmetry is broken at the electroweak scale. The flavor charges are fixed to reproduce the SM quark mass hierarchies and CKM mixing angles up to rescalings, that have no effect on any physical quantity. As a result, this two Higgs doublet flavor model can be described by few effective parameters, the masses of the extra scalars M H , M A , M H + , cos(β − α) and tan β. This allows us to present our main findings in the cos(β − α) − tan β plane for fixed mass values, as shown in figure 22.
Modified interactions between the SM-like Higgs h and quarks are characteristic for our two Higgs doublet flavor model, leading to strong constraints from Higgs signal strength measurements. The results of our Higgs global fit to ATLAS and CMS data constrain possible deviations of the couplings of the light Higgs to fermions and gauge bosons with respect to the SM ones, and select sizable values of cos(β − α) ≈ O(0.5). This implies a suppression of the tree-level couplings of the Higgs to gauge bosons, which is proportional to sin(β −α) as in any two Higgs doublet model and therefore a suppressed vector boson fusion production rate with respect to the SM. The alignment/decoupling limit cos(β − α) = 0 is excluded for all values of tan β, since in this limit our model approaches the Babu-Nandi-Giudice-Lebedev model for which there is a factor of three enhancement for the coupling of the light Higgs to bottom quarks. The Higgs global fit allows for two branches in the cos(β − α) − tan β plane (red shaded areas in figure 22) figure 22). These tree-level FCNCs result in an upper bound of tan β 5.5. Moreover, contributions from box diagrams with charged Higgs exchange can compete with the tree-level diagrams for low tan β and exclude values of tan β 1. Thus the interplay of tree-level and loop contributions in flavor observables predicts 5.5 tan β 1. Interestingly, if we discard the explanation of the CKM angles by the two Higgs doublet flavor model, we find almost no constraints from flavor observables in the region preferred by the global Higgs fit. As in any two Higgs doublet model, charged Higgs exchanges also induce FCNCs through penguin diagrams, for example b → sγ, which imply a lower bound on the charged Higgs mass of 360 GeV for tan β 2.
The two Higgs doublet flavor model offers exciting possibilities for direct collider searches for the additional Higgs bosons. Electroweak precision observables, perturbativity and unitarity constraints choose a preferred range of masses and mass splittings for the new heavy scalars. In particular, almost degenerate values for the CP-odd and CP-even Higgs boson masses and sizable splitting between the neutral and charged Higgs masses are strongly favoured. This opens the opportunity of new decay channels, A → H + W − and H → H + W − in addition to the regular decay channels H → W + W − /ZZ, A → hZ, that are importantly enhanced in the cos(β − α) ≈ O(0.5) region. The latter are the leading discovery modes for these scalars (present bounds are shown by blue and orange contours in figure 22). Furthermore, the cos(β − α) dependence of the H W + W − , HZZ couplings are of particular relevance because the vector boson fusion production mode can compensate for the suppression of the gluon fusion production mode of the CP even Higgs in the relevant regions of parameter space. Direct searches for a charged Higgs boson are not sensitive for masses compatible with the flavor constraints, however future searches via Higgs decay chains with the subsequent decay H + → W + h may be promising. The other possible decay of heavy Higgs bosons to the SM Higgs is in the channel H → hh with branching ratios of order 10%.
The fact that the flavor symmetry is broken at the electroweak scale predicts a UV completion in the few TeV range, as well as a low value of tan β in agreement with flavor constraints. The necessity of new physics at the TeV scale provides an additional motivation for the search for new vector-like fermions at the run II of LHC.
We conclude, that in the two Higgs flavor model constraints from flavor observables, Higgs precision measurements, direct heavy Higgs searches, and precision electroweak observables, as well as unitarity and perturbativity restrictions on the theory, can be fulfilled simultaneously. We propose three benchmark scenarios in this region, that highlight different characteristics of the two Higgs doublet flavor model (black crosses in figure 22). In table 4 and 5 we provide all the relevant information to compute production cross sections and decay rates for these benchmark scenarios and test the two Higgs doublet flavor model at the run II of LHC.

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light Higgs h given in equation (7.4), and the coupling of the light Higgs h to two charged Higgs bosons H ±

B Box diagrams and loop functions
In this appendix, we collect the contributions to the Wilson coefficients (5.1) from box diagrams and the relevant loop functions [37,42,88]. For K −K mixing we have the following Wilson coefficients:

C Random parameter generation and running
In order to find sample parameter points, we generate random fundamental Yukawa couplings with y u,d ij = |y u,d ij | e iφ u,d ij and |y u,d ij | ∈ [0.5, 1.5] and φ u,d ij ∈ [0, 2π]. The effective Yukawa couplings (2.7) have to reproduce the quark masses and Wolfenstein parameters in table 6 in appendix D. To this end we perform a χ 2 fit, with symmetrized 2σ errors and require χ 2 < 10.
In order to obtain the new contributions to K −K and B s,d −B s,d mixing we compute the Wilson coefficients with these effective Yukawas, including the tree-level (5. 2) hold with the obvious replacements. The corresponding "magic numbers" can be found in [41], and all other parameters in table 9.

D Numerical input
In this appendix we collect the numerical input used throughout this paper, quark masses and Wolfenstein parameters in   -47 -