FN-2HDM: Two Higgs Doublet Models with Froggatt-Nielsen Symmetry

We embed Two Higgs Doublet Models (2HDMs) in the Froggatt Nielsen (FN) framework. We find that the approximate FN symmetry predicts i) approximate Natural Flavor Conservation (NFC) of Types II or IV in the Yukawa sector, and ii) approximate Peccei-Quinn (PQ) symmetry in the scalar sector. We discuss the phenomenological consequences of these features.


Introduction
The Froggatt-Nielsen (FN) mechanism [1,2] applies a symmetry principle to explain the non-trivial structure of the measured flavor observables, characterized by smallness and hierarchy. It postulates that the fermion fields are charged under a symmetry that is explicitly broken by a small parameter. Consequently, the various fermion masses and CKM mixing angles are suppressed by different powers of the symmetry breaking parameter. In this way it solves the Standard Model (SM) flavor puzzle -the question of how the hierarchical structure of the flavor parameters is generated.
The discovery of the Higgs boson sparks renewed interest in the scalar sector of Nature and, in particular, in the possibility that it is non-minimal. Both improved measurements of the Higgs couplings and direct searches for additional scalars provide guidance to the possible structure of such a non-minimal scalar sector. A viable extension of the SM scalar sector is the Two-Higgs-Doublet Model (2HDM), that predicts four additional scalars beyond the SM Higgs boson, some of which may be light (for a recent review, see Ref. [3]). 2HDMs induce, in general, flavor changing processes that are strongly constrained by experiment. Therefore, a mechanism to control the New Physics (NP) flavor structures is usually applied to them, such as Natural Flavor Conservation (NFC) [4,5] or Minimal Flavor Violation (MFV) [6].
In this work we study the effectiveness of the FN mechanism in constraining the flavor structures of 2HDMs. We find that, if the scalar doublets are charged under the FN symmetry, various viable options in model building open up, affecting the flavor structure as well as the scalar spectrum.
The plan of this paper goes as follows. In Section 2 we introduce the 2HDM framework, and present relations among Yukawa matrices that are useful for our purposes. In Section 3 we impose the approximate FN symmetry on the 2HDM, and obtain the resulting structure of the Yukawa matrices and of the scalar potential. In Section 4 we confront the FN-2HDM framework with experimental constraints from electroweak precision tests, collider searches, flavor changing neutral current processes, and electric dipole moment searches. Section 5 compares the FN-2HDM to 2HDM frameworks with NFC or with MFV. We summarize our conclusions in Section 6. Specific examples of FN-2HDMs are presented in an Appendix.

The General Framework
We set the stage with a general 2HDM. The relevant phenomenology of the model is determined by two sectors: the scalar potential, which in general is given by and the Yukawa interactions, given by Here, Φ 1 and Φ 2 are the two scalar doublets, Q,Ū ,D are the quark doublets, up-singlets and down-singlets respectively, and L,Ē are the lepton doublets and charged lepton singlets, respectively. The Yukawa matrices, Y F i , are responsible for the corresponding fermion mass matrices M F , which we represent by the dimensionless matrices Y F M : We denote by Y F S , S = h, H, A, the Yukawa couplings of the light CP-even scalar h, the heavy CP-even scalar H, and the CP-odd scalar A. (The Yukawa matrices of the charged Higgs H ± are the same as those of A.) Each of these matrices is a linear combination of Y 1 and Y 2 : where c β ≡ cos β, s β ≡ sin β, tan β ≡ v 2 /v 1 , and α is the rotation angle from (Re(φ 0 1 ), Re(φ 0 2 )) to (H, h). The angle β is taken to be in the range [0, π/2] while α ∈ [−π/2, +π/2]. We can express the Yukawa matrices of the scalars in terms of Y F M , In the same way, in the lepton sector with H( or Hence, appropriate choices for the charges can lead to what appears to be approximate NFC of Types I, II, III or IV. However, for Types I and III, where both the up and down sectors couple more strongly to the same scalar doublet, we find that a very large charge difference between the scalars is needed, log H (see Appendix A for a detailed explanation), making such models less plausible. Therefore in the following we focus on models with approximate NFC of Types II and IV.
This construction leads to one Yukawa matrix whose entries are all suppressed compared to those of the other, for each fermion sector, where we defined As we demonstrate below, the clear hierarchy between the two Yukawa matrices may not suffice to ward off large flavor changing couplings of the light Higgs once we rotate to the mass basis. Whether or not the FN mechanism suppresses flavor changing Higgs couplings depends on the parameters of the scalar potential. Specifically, the contribution of one Yukawa matrix to both Y F M and Y F h remains suppressed compared to that of the other, and approximate NFC is preserved, if cot β is not PQ suppressed.

The Scalar Potential
If the scalar doublets, Φ 1 and Φ 2 , carry different charges under U (1) H , then certain parameters of the scalar potential are parametrically suppressed. Returning to the potential of Eq. (1), the terms on the first line are neutral under U (1) H , while those on the second are not, dictating the parametric suppression of the couplings: The soft breaking by m 2 12 deserves some discussion. Since m 2 12 is a dimensionful parameter, there is an ambiguity in the low energy theory as to what scale is involved in determining its suppression. In a full high energy realization of the FN framework, there are at least two scales: The electroweak breaking scale v, and a scale Λ at which the FN symmetry breaking is communicated to the SM fields. Without an explicit UV completion, it is impossible to determine what is the scale compared to which m 2 12 is PQ -suppressed. In common UV completions, the full theory contains a SM-singlet scalar S whose VEV spontaneously breaks the FN symmetry, and a set of vector like quarks and leptons at a mass scale Λ, such that S /Λ = H . In such models, m 2 12 ∼ PQ Λ 2 /(16π 2 ). We later find a phenomenological constraint, PQ ∼ < 10 −3 , which in this specific scenario translates into Λ ∼ > tens of TeV. On the other hand, since we are interested in the possibility that the FN mechanism allows a full 2HDM at or below the TeV scale, we do not consider the possibility that m 2 12 v 2 . The extremum equations for the vacuum expectation values (VEVs) are given by with λ 34 = λ 3 + λ 4 . In the FN symmetry limit, PQ → 0, the bracketed terms in Eqs. (1,16) vanish and the potential exhibits a global U (1) Y × U (1) PQ symmetry. This symmetry is then broken spontaneously, either completely or partially, as the fields acquire non-zero VEVs. We separate the parameter space into two cases: 3.2.1 Case A: m 2 1 , m 2 2 < 0: In the case where both mass-squared parameters are negative, both fields acquire non-zero VEVs and U (1) PQ is spontaneously broken. In the symmetry limit, the CP-odd state is the Nambu-Goldstone Boson (NGB), m 2 A = 0. Turning on the symmetry breaking parameters introduces a non-zero m 2 A , parametrically suppressed. The key phenomenological features of Case A can be summarized as follows: • The spectrum contains a light CP-odd scalar. The parametric suppression of its mass is dictated by • tan β, cot α, c β−α are parametrically O(1).
• The approximate NFC in the Yukawa sector remains unharmed. The diagonal Yukawa couplings of the three scalars deviate from the NFC values at O( PQ ), • Suppressed off-diagonal Yukawa couplings appear, aligned between the scalars, where in our models with V ij the CKM or PMNS matrix elements.

3.2.2
Case B: m 2 1 > 0, m 2 2 < 0 When only one mass-squared parameter is negative, one VEV is vanishing in the symmetry limit, v 1 = 0, dictating that c β = c β−α = 0. A residual U (1) remains unbroken, so there are no (uneaten) NGBs. Turning on symmetry breaking leads to This behavior has non-trivial implications for the flavor structure of the model. We rewrite the entries of the mass and coupling matrices from Eq. (4) as It is then apparent that the approximate NFC persists only if the O( PQ ) suppression of the ratio of Yukawa entries is not met with the O( −1 PQ ) enhancement of tan β or cot α. Otherwise, the mass and coupling matrices receive same order contributions with different order one factors, leading to flavor changing Higgs couplings easily violating existing bounds. This poses a problem for the down and lepton sectors in models of Type II and IV, making Case B non-viable.
In conclusion, considering the parametric structure in the scalar and Yukawa sectors, we are led to the following predictions: 1. The Yukawa matrices are close to those of NFC of Types II or IV, with deviations of O( PQ ).
2. The CP-odd scalar is a pNGB, with mass dictated by the dimensionful explicit FN-breaking parameter, m 2 12 . This is not a sharp prediction for the order of m A , since the relevant scale compared to which m 2 12 is suppressed is model dependent, m 2 12 ∼ PQ Λ 2 . In the following we take m A ∼ < v.

Experimental Constraints
In this section we look at the rough structure of the Yukawa couplings, neglecting O(1) factors, and at the scalar spectrum generated by the FN symmetry, and confront them with experimental bounds. For this purpose, the diagonal and non-diagonal Yukawa entries for all scalars are taken as (for the detailed expressions see Eqs. (18,19)): The O( PQ ) factors are assumed to come with order one phases. The deviations from NFC, embodied in the flavor non-diagonal couplings and the phases, are constrained by low energy experiments, which set an upper bound on the FN suppression factor, PQ . We find that the most stringent bounds come from µ → eγ and K 0 −K 0 mixing: PQ ∼ < 10 −3 . Other constraints are relevant for the 2HDM scalar resonances, particularly for the scale of light CP-odd scalar mass, m A . These arise from direct collider searches, from electroweak (EW) precision tests, and from the bound on untagged decays of the Higgs. Interestingly, we find that the most stringent bound on m A comes from an upper bound on the trilinear coupling, g hAA which, in turn, implies m A m h /2.

EW Precision Tests
Mass splittings between the scalar eigenstates are expected to induce a contribution to the T parameter. We find that when m H ± ≈ m H is maintained, a large splitting between m A and m H ± is allowed as long as |c β−α | is small, as suggested by Higgs data. To quantify this statement, we write the contribution of the additional scalars to the T parameter as [8] For m H ± ≈ m H , the entire expression is proportional to c 2 β−α , and simplifies to In Fig. 1 we plot contours of the upper limit on |c β−α | such that the constraint on the T parameter is  satisfied, in the plane of m H = m H ± vs. m A . The charged Higgs mass in models of Types II and IV is constrained by b → sγ measurements to be ∼ > 480 GeV [9]. The CP-odd scalar, A, may be as light as ∼ m h /2, with varying upper limits on |c β−α | as m H is varied, such that 0 ≤ |c β−α | ∼ < 0.2 − 0.6. The measurement of the Higgs coupling to gauge bosons results in a comparable limit on |c β−α | [10].

Collider Searches
Direct searches for the scalar resonances, A, H and H ± , limit the viable spectra of the model. In this regard analyses that are done in the context of Type II and IV NFC with a light CP-odd scalar apply straightforwardly. We consider m H ≈ m H ± ∼ > 480 GeV, to comply with bounds from b → sγ [9]. The remaining relevant constraints involve the light CP-odd scalar, A. Since searches for rare meson decays exclude masses below 10 GeV for scalars with O(Y M ) diagonal couplings [11], we consider m A 10 GeV.

LEP [12-14]
We consider the channels e + e − → Z * → hA and e + e − → Z * /γ * → ff A. Pair production of the CPodd scalar only arises at loop level, through triangle and box diagrams which are suppressed by a loop factor times m 2 e /m 2 Z compared to tree level hA production, making its contribution to the 4f final state negligible.
• e + e − → ff A Searches for Yukawa production of A in association with ττ or bb [14] yield bounds on t 2 β or t −2 β times BR(A → ff ), for limited windows of m A . The bounds are meaningful when the relevant branching ratio is approximately one, which implies, for t β 1: and for Type IV with t β ∼ < 0.7: The excluded regions are plotted in the t β −m A plane in Fig. 3, where we do not consider t β values smaller than O(0.1) as they lead to non-perturbative Y S t .

LHC
• CMS bbA → bbτ τ search [16] A search for associated production of a scalar along with two b-jets was performed for 25 GeV ≤ m A ≤ 80 GeV, in the ττ decay channel. The analysis puts bounds on models of Type II with t β > 1, for which x /m 2 A ) 1/2 ). We can estimate the production cross section using MG5@NLO [17]: The induced limit on t β , depicted in Fig. 3, is of the order of For Type IV this search sets no significant limit.

• Untagged Higgs Decays
For m A ∼ < 1 2 m h , h → AA proceeds through the trilinear coupling, g hAA . This dimensionful coupling is naively of order the weak scale, making the channel h → AA the dominant decay mode for any value of m A in this range, in tension with the existing bound [18], Assuming no other non-SM decay modes of the Higgs exist, a rough bound can be obtained by requiring that Γ(h → AA) is smaller than Γ SM (h → bb) at tree level: with β A ≡ (1 − 4m 2 A /m 2 h ) 1/2 . This translates into the bound For a general 2HDM, the g hAA coupling is related to the mass-squared parameters by [19] The FN structure dictates that The bracketed expression in Eq. (38) bears some resemblance to the expressions for the normalized Yukawa couplings of Type II/IV NFC: Higgs coupling measurements constrain these normalized couplings to be not far from unity, in absolute value, while the expression in Eq. (38) needs to be small in order to keep the branching ratio to AA This constraint excludes m A below m h /2 except for the range approaching the threshold (as the phase space goes to zero). More precisely, we get m A 54 GeV for Type II approximate NFC, and m A 60 GeV for Type IV approximate NFC. Fig. 3 summarizes the constraints on the mass of the CP-odd scalar for different values of t β . We note that m A ∼ < m h /2 is excluded, while for m A ∼ > m h /2 the parameter space is very weakly constrained.

Bounds on Deviations from NFC
Limits on processes that are sensitive to flavor off-diagonal couplings or to non-zero phases constrain the FN suppression factor, PQ . The most relevant flavor changing processes are those that involve Y S eµ , Y S uc , Y S ds and Y S ut . The bound on the electric dipole moment (EDM) of the electron constrains the deviations from NFC via the imaginary parts of Y S t and Y S e .

µ → eγ
Following Ref. [20], we parameterize the leading contributions to the process µ → eγ at one-and two-loops using the reduced amplitudes A L,R : For the CP-odd scalar A, f (z tS ) needs to be replaced by g(z tS ). The loop functions read The rate is the given by Substituting the FN structure of Eq. (23) for the various Yukawa couplings and imposing the MEG bound [21], we arrive at the constraint on the order of parametric suppression as a function of the scalar mass. Fig. 4 shows the obtained excluded region. In the plot, we use the loop function suitable for the CP-odd scalar as it gives slightly stronger bounds.
As the bound becomes more stringent with smaller mass, we focus on the light CP-odd scalar, which can be as light as O(m h /2) (see Section 4.2). For Tree level processes involving non-diagonal Yukawa couplings contribute to K 0 − K 0 and D 0 − D 0 mixing. Following Refs. [22][23][24], we write the leading contributions to the Wilson coefficients inducing D 0 − D 0 and K 0 − K 0 mixing as From this we infer the constraint on the PQ as a function of m S , shown in Fig. 4. For m A ≈ m h /2, the bound reads PQ ∼ < 3 × 10 −3 .

Neutron EDM
The bound on the electric dipole moment (EDM) of the neutron [25] implies a bound on the flavor violating couplings Y ut and Y tu , through the one-loop contribution to the up quark EDM [22,26], Calculating the subsequent contribution to the neutron EDM results in a bound on PQ shown in Fig. 4.

Electron EDM
The deviations from NFC in the diagonal Yukawa couplings are also O( PQ ) (see Eq. (23)), and in general are accompanied by O(1) phases. We therefore derive bounds on the order of these deviations from the bound on the electron EDM [27] which is sensitive to the phases in Y S t and Y S e . Following Ref. [28], we write the contribution of the Barr-Zee type diagram to the electron EDM as   On the qualitative level, we make the following observations. The SM predicts that the Yukawa couplings have the features of proportionality, y i /y j = m i /m j , and diagonality, y ij = 0 for i = j. NFC maintains these two features, though the factor of proportionality in y i /m i can be different from the SM prediction of √ 2/v. MFV gives deviations from proportionality and diagonality that are flavor dependent, with larger deviations for heavier generations. FN gives deviations from proportionality that are all of O( PQ ) and flavor dependent deviations from diagonality. These qualitative features are demonstrated in Table 1. A more quantitative discussion is given in the next two subsections.

NFC-2HDM
The implementation of a 2HDM within a FN symmetry framework predicts approximate NFC of Types II or IV. Deviations from NFC in the magnitude of diagonal Yukawa couplings are at most of O(10 −3 ), which is unobservably small. The existence of phases in diagonal couplings, and of off-diagonal Yukawa couplings in all fermion sectors marks a qualitative departure from NFC predictions. The deviations from NFC are all linked, up to O(1) factors, by a common parametric suppression, which provides a rough prediction relating possible future deviations from the SM in experiments. Table 2 presents the upper bounds on the deviations from NFC in various Yukawa couplings, along with the current experimental sensitivity.

MFV-2HDM [29]
The FN symmetry predicts proportionality, Y F S ∝ Y F M ,, with deviations of order PQ . The deviations are of the same order for all proportionality relations, for example, Approximate proportionality is also a feature of MFV, in which case the deviations are flavor-dependent. For (Y S µ /Y S τ )/(m µ /m τ ) the deviation from unity would be of O(m 2 τ /v 2 ). Both frameworks exhibit relations between off-diagonal couplings involving the masses and mixing angles, with some qualitatively Yukawa Coupling Order of Magnitude Current Sensitivity different predictions. The overall trend is that the hierarchy between off-diagonal elements involving third generation fermions and those that do not involve the third generation is stronger for MFV than for FN. Taking into account the experimental sensitivity, which is in general poorer when third generation fermions are involved, this hints at a possible way to distinguish between the two frameworks. For example (incorporating neutrino-related spurions and assuming NH), MFV predicts while FN predicts This implies that Y S eτ /Y S eµ is suppressed in FN in comparison with MFV predictions. Bearing in mind that the experimental sensitivity for measuring Y S eµ is O(10 4 ) times that of Y S eτ , we reach the conclusion that if τ → eγ is observed while µ → eγ is not, FN will be strongly disfavored.
Similarly, in the quark sector, we compare the ratios governing K 0 -oscillations vs. B 0 and B soscillations, under MFV and FN. We note that for MFV the dominant contributions are of the form Y 2 ij , while for FN, (Y ij Y ji ) contributions turn out to be greater. The relevant expressions are then and Considering the experimental sensitivity, which is currently O(10 4 )−O(10 6 ) greater for observables related to K 0 -oscillations versus those of B 0 d -and B 0 s -oscillations, this suggests that if deviations from the SM predictions are measured in B 0 s or B 0 d systems, but not in the K 0 system, then FN will be disfavored.

Discussion and Conclusions
We considered FN-2HDM models. These are two Higgs doublet models subject to an approximate Froggatt-Nielsen symmetry, where the two scalar doublets carry different FN charges. Our main conclusions and findings are the following: • For FN-2HDM scenarios in which only one scalar acquires a VEV in the FN symmetry limit, the FN framework is unable to prevent large flavor-changing rates which are in contradiction with experiments. Thus, in viable FN-2HDM, the scalar potential symmetries are broken completely by the VEVs.
• The FN structure of the Yukawa matrices induces approximate NFC of types II or IV. Models of types I and III require very large charge differences, and are less plausible.
• The FN structure of the scalar potential induces approximate PQ symmetry, with possibly large soft breaking.
• The viable models predict departure from NFC in the form of new CP violating phases, off-diagonal Yukawa couplings and deviations in the diagonal couplings.
• The departure from NFC predictions is governed by the PQ symmetry breaking parameter, PQ , which we find to be constrained by PQ ∼ < 10 −3 from low energy experiments.
• When set against the predictions of models of MFV we find that within FN the hierarchy between flavor changing couplings involving the light generations compared to those involving third generation fermions is softened. As a result (together with the experimental inclination to better measure processes of the light fermions), if deviations from SM predictions are measured in flavorchanging processes involving the third generation but not in the corresponding processes involving light fermions, then FN-2HDMs will be disfavored.
unit matrix, as in the following example.