The Flavor-locked Flavorful Two Higgs Doublet Model

We propose a new framework to generate the Standard Model (SM) quark flavor hierarchies in the context of two Higgs doublet models (2HDM). The `flavorful' 2HDM couples the SM-like Higgs doublet exclusively to the third quark generation, while the first two generations couple exclusively to an additional source of electroweak symmetry breaking, potentially generating striking collider signatures. We synthesize the flavorful 2HDM with the `flavor-locking' mechanism, that dynamically generates large quark mass hierarchies through a flavor-blind portal to distinct flavon and hierarchon sectors: Dynamical alignment of the flavons allows a unique hierarchon to control the respective quark masses. We further develop the theoretical construction of this mechanism, and show that in the context of a flavorful 2HDM-type setup, it can automatically achieve realistic flavor structures: The CKM matrix is automatically hierarchical with $|V_{cb}|$ and $|V_{ub}|$ generically of the observed size. Exotic contributions to meson oscillation observables may also be generated, that may accommodate current data mildly better than the SM itself.


I. INTRODUCTION
The LHC collaborations have established with Run I data that the 125 GeV Higgs boson has Standard Model (SM)-like properties [1]. In particular, the couplings of the Higgs boson to the electroweak gauge bosons have been measured with an uncertainty of 10% at the 1σ level, combining results from ATLAS and CMS [1]. The Higgs coupling to τ leptons has been measured at the 15% level [1], and, assuming no significant contribution of new degrees of freedom to the gluon fusion Higgs production cross section, the Higgs coupling to top quarks has been found to be SM-like with approximately 15% uncertainty [1]. More recently, analyses of ∼ 36 fb −1 of Run II LHC data have provided evidence for the decay of question: the SM flavor puzzle. One dynamical approach to this puzzle is to couple the first two generations exclusively to an additional subleading source of electroweak symmetry breaking, in the form of a second Higgs doublet or some strong dynamics [27] (see also [28][29][30][31]). Asserting suitable textures for the quark and lepton Yukawa matrices, in order to satisfy flavor constraints, leads to a 'flavorful' two Higgs doublet model (F2HDM). The collider signatures of the F2HDM have been explored previously [32]. These include striking signatures for lepton flavor violation, such as h → τ µ or b → sτ µ and large branching ratios for t → ch, as well as heavy Higgs or pseudoscalar decays H/A → cc, tc, µµ, τ µ and charged Higgs decays H ± → bc, sc, µν.
A different approach to resolving the SM mass hierarchy puzzle can be achieved with a dynamical alignment mechanism [33] -we refer to it as 'flavor-locking' -in which the quark (or lepton) Yukawas are generated by the vacuum of a general flavon potential, that introduces a single flavon field and a single 'hierarchon' operator for each quark flavor. (A detailed review follows below; see also Refs. [34][35][36][37] for related, but somewhat different approaches.) In this vacuum, the up-and down-type sets of flavons are dynamically locked into an aligned, rank-1 configuration in the mass basis, so that each SM quark mass is controlled by a unique flavon. Horizontal symmetries between the hierarchon and flavon sectors in turn allow each quark mass to be dynamically set by a unique hierarchon vev.
This results in a flavor blind mass generation mechanism -the quarks themselves carry no flavor symmetry beyond the usual U (3) Q,U,D -so that the quark mass hierarchy can be generated independently from the CKM quark mixing hierarchy, by physics that operates at scales generically different to -i.e. lower than -the scale of the flavon effective field theory.
In a minimal set up that features only a single SM-type Higgs, however, the CKM mixing matrix is an arbitrary unitary matrix, so that the quark mixing hierarchy itself remains unexplained.
In this work we synthesize these two approaches to the flavor puzzle with the following observation: A dynamical realization of an F2HDM-type flavor structure can be generated by applying the flavor-locking mechanism to its Yukawas. Or alternatively: In a flavorlocking scheme for the generation of the quark mass hierarchy, introducing a second Higgs doublet with F2HDM-type couplings generically produces quark mixing hierarchies of the desired size. In particular, we show that in such a setup, the 1-3 and 2-3 quark mixings are automatically produced at the observed order, without the introduction of tunings. The flavor structure of this theory generically leads to tree-level contributions from heavy Higgs exchange to meson mixing observables, that vanish in the heavy Higgs infinite mass limit.
However, for heavy Higgs masses at collider-accessible scales, we show these contributions may be consistent with current data, and in some cases may accommodate the current data mildly better than the SM. This paper is structured as follows. In Sec. II we briefly review the general properties of the F2HDM and its flavor structure. In Sec. III we develop the flavor-locking mechanism for F2DHM-type theories, including a review of the minimal single Higgs version. We then proceed to explore the generic flavor structure of the flavor-locked F2HDM in Sec. IV, discussing both the generation of the CKM mixing hierarchies and constraints from meson mixing. We conclude in Sec. V. Technical details concerning the analysis of the flavon potential are given in Appendices.

II. REVIEW OF THE FLAVORFUL 2HDM
The F2HDM, as introduced in Refs. [27,32], is a 2HDM in which one Higgs doublet predominantly gives mass to the third generation of quarks and leptons, while the second Higgs doublet is responsible for the masses of the first and second generation of SM fermions, as well as for quark mixing. The most general Yukawa Lagrangian of two Higgs doublets with hypercharge +1/2 can be written as with two Higgs doublets H 1 and H 2 coupling to the left-handed and right-handed quarks (Q L , U R , D R ) and leptons (L L and E R ), andH ≡ H * . The indices i = 1, 2, 3 and J, J = 1, 2, 3 label the three generations of SU (2) doublet and singlet fields, respectively. We focus on quark Yukawas hereafter, but the general results of this discussion apply equally to the lepton Yukawas in Eq. (2.1).
The two Higgs doublets decompose in the usual way where v = 246 GeV is the vacuum expectation value of the SM Higgs, G 0 and G ± are the to the SM fermions in a characteristic flavor non-universal way. Their couplings to the third generation are suppressed by tan β, while the couplings to first and second generation are enhanced by tan β. Therefore, the decays of H, A, and H ± to the third generation -t, b quarks and the τ lepton -are not necessarily dominant. For large and moderate tan β we expect sizable branching ratios involving, for example, charm quarks and muons. Similarly, novel non-standard production modes of the heavy Higgs bosons involving second generation quarks can be relevant and sometimes even dominant [32].
One important aspect of the Yukawa structures in Eqs. (2.4) and Eqs. (2.5) is that they imply tree-level flavor changing neutral Higgs couplings. The flavor-violating couplings of the 125 GeV Higgs vanish in the decoupling/alignment limit, i.e. for cos(β − α) = 0.
However, flavor-violating couplings of the heavy Higgs bosons persist in this limit and they are proportional to tan β. Therefore, for large tan β and heavy Higgs boson masses below the TeV scale, flavor violating processes, such as meson mixing, constrain the F2HDM parameter space. Note that the rank-1 nature of the third generation Yukawas, Y , preserves a U (2) 5 flavor symmetry acting on the first and second generation of fermions. This symmetry is only broken by the Y Yukawa couplings of the second doublet, so that flavor changing transitions from the second to the first generation are protected. Therefore, the constraints from kaon and D-meson oscillation will be less stringent than one might naively expect. We will discuss meson oscillation constraints in detail in Sec. IV.

III. FLAVOR-LOCKING WITH ONE AND TWO HIGGS BOSONS
While the distinct phenomenology of the F2HDM alone motivates detailed studies, a mechanism that realizes the flavor structure in Eqs. (2.4) or (2.5) has not been explicitly constructed so far. We now discuss how the flavor structure (2.5) can be dynamically generated by the flavor-locking mechanism, and, conversely, how a F2HDM-type theory permits the flavor-locking mechanism to generate realistic flavor phenomenology. We first review the minimal single Higgs doublet version of the flavor-locking mechanism, followed by the generalization to a theory with two Higgs doublets in Sec. III D. As we will discuss, while in the presence of only one SM-like Higgs doublet, the predicted quark mixing angles are generically of O(1), introducing a second Higgs doublet leads to a theory with suppressed |V cb | and |V ub |.

A. Yukawa portal
The underlying premise of the flavor-locking mechanism [33] is that the Yukawas arise from a three-way portal between the SM fields (the quarks Q L , U R , D R and the Higgs H), a set of 'flavon' fields, λ, and a set of 'hierarchon' operators, s: The λ's are bifundamentals of the appropriate U (3) Q × U (3) U,D flavor groups for up and down quarks, respectively. The subscripts 1 , α = u, c, t and α = d, s, b, denote an arbitrary transformation property under a symmetry or set of symmetries, G and G, that enforces the structure of Eq. (3.1). In the original flavor-locking study [33], G × G was chosen to be a set of discrete Z pq q or U (1) q 'quark flavor number' symmetries, for q = d, s, b, u, c, t. Here, we similarly choose each flavon λ α (λ α ) to be charged under a gauged U (1) α (U (1) α ), but assert a S 3 permutation symmetry among the up (down) flavons and the corresponding U (1) α (U (1) α ) gauge bosons, fixing the gauge couplings g α = g (g α = g). Compared to the analysis of Ref. [33] the permutation symmetry produces a convenient, higher symmetry for the flavon potential, such that configurations with the structure of Eqs. (2.5) can be shown to be at its global minimum, as we will discuss in the next subsection. Note that the SM fields are not charged under the G × G symmetry.
The hierarchons s should be thought of as some set of scalar operators that eventually obtain hierarchical vevs, that break the S 3 symmetries in the up and down sectors. This hierarchy will be responsible for the quark mass hierarchy, independently from any flavor structure. It should be emphasized that the operators s α and s α do not carry the quark . . , n, that generate Yukawa couplings to the quarks as in Eq. (3.1). The flavons for this theory then transform as We suppress hereafter the U (N ) Q × U (N ) U,D indices, keeping in mind that matrix products only take the form λ α λ † β or λ † β λ α , and correspondingly in the down sector. Up-down matrix products can only take the form λ † α λ α or λ † α λ α , but not λ α λ † α nor λ α λ † α . The most general, renormalizable and CP conserving potential for the flavons can then be written in the form Here, the single and pairwise field potentials are and similarly for V α 1f and V α β 2f , hatting all coefficients (the labeling and notation follows the choices of Ref. [33]). Note that the pairwise potentials respect the U (1) α and U (1) α symmetries. The mixed potential is (3.6) The S n symmetry ensures that all potential coefficients are the same for all fields α, α, β, β singly and pairwise. All µ i and ν i coefficients, as well as r and r, are real and are chosen to be positive.
A detailed analysis of the global minimum of this potential is provided in Appendix A.
One finds that, provided the potential has a global minimum if and only if the flavons have the vacuum configuration with U , V , U , V unitary matrices -crucially, the matrices U , V ( U , V ) are the same for all λ α (λ α ) -and the CKM mixing matrix has the form

C. Flavor-locked Yukawas
Flavor locking ensures that the Yukawa portal in (3.1) becomes, in the n = N = 3 casē under a suitable unitary redefinition of the Q L , U R and D R fields. From these expressions, taking the natural choice r, r ∼ Λ F , it is clear that it is the physics of the hierarchon vev's, s α , that generates the quark mass hierarchies, i.e. s α /Λ H ∼ y α , the quark Yukawa for flavor α. This physics may operate at scales vastly different to the flavor breaking scale, Λ F .
One might wonder if additional terms in the flavon potential of (3.3) can destabilize the vacuum identified above. In particular, flavon-hierarchon couplings of the form may be present, which can produce (mixed) mass terms that disrupt the V mix (V 2f ) vacuum once the hierarchons, s α , obtain vev's. Mixed mass terms may disrupt the alignment between the different λ α , while additional mass terms induce splittings in the radial mode masses, so that the block CKM rotations are no longer flat directions of the vacuum.
In the UV theory, the operator product of two hierarchons with two flavons may, however, be vanishingly small, e.g. if the hierarchons are composite operators in different sectors.
Nonetheless, such terms are necessarily generated radiatively by the Yukawa portal (3.1).
One may construct UV completions in which this occurs first at the two-loop level, with the (mixed) mass contributions being log-divergent. For example, let us consider a theory containing a flavored fermion χ αi and a scalar Φ α , with interactions As s α /Λ H ∼ y α , the quark Yukawa for flavor α, the corresponding (mixed) mass term for the flavons is generated at two-loops by mirroring the diagram in (3.12). One finds once again taking the natural choice r ∼ Λ F . A suitable hierarchy between Λ H and Λ F , combined with the two-loop suppression, renders these terms arbitrarily small. Hence one may safely neglect these terms.

D. Two-Higgs flavor-locking
Motivated by the flavorful 2HDM, now we turn to consider a Yukawa potential with two Higgs fields: One that couples to the third generation, and one to the first two generations.
That is,Q in which we have suppressed the quark flavor indices. With reference to the UV completion (3.11), one can imagine that this generational structure comes about as a consequence of λ t , s t , and H 1 belonging to a different UV sector (or brane) than λ c,u , s c,u , and H 2 , so that terms of the form λ t s tH2 or λ c,u s c,uH1 are heavily suppressed in the effective field theory.
Similarly, one can also generate this structure via adding an additional symmetry to s c,u , s s,d and H 2 such that s c,uH2 and s d,s H 2 are singlets. Such terms (symmetries) will, ultimately, be generated (softly broken) via the µ 2 H 1 H † 2 term in the Higgs potential, which is necessary to avoid a massless Goldstone boson.
The generational structure implies that cross-terms between the third and first two generations in the flavon potential (3.3) now vanish, and that the S 3 flavon-hierarchon symmetry has been replaced with a Z 2 for just the two light generations. That is, the coefficients of the heavy and light flavon potentials are no longer related, and the heavy-light potentials V tα 2f , V b α 2f , V t α mix , V bα mix vanish, for α = c, u and α = s, d (or they obtain their own, independent, and suppressed coefficients, identical for α = c, u and α = s, d). One then also expects the rotation matrices entering in the vacuum configuration of the flavons of the first two generations to be different from those of the third, breaking the heavy-light alignment conditions. Put a different way, we may write the full potential in the form (3. 16) We call this a '1 + 2' flavor-locked vacuum. Note that the rotation matrices for the third generation quarks (U t , V t , U b , V b ) differ in general from the corresponding rotations for the first and second generation quarks.
For the 1 + 2 flavor-locked structure (3.16), the CKM structure of the global minimum in Eq. (3.9) enforces U † U and U † t U b to each be 2 ⊕ 1 block unitary, i.e.
where V 2 and W 2 are 2×2 unitary matrices (see App. A 3). The 2⊕1 block unitarity permits one to rotate away the tb unitary matrices, so that the Yukawa potential (3.14) attains the form rotations of V 2 and V , respectively) and 3 + 6 − 2 − 1 = 6 imaginary parameters (the phases of V 2 and V , less the phases commuted or annihilated by the rank-2 diagonal matrix). This counting implies that the total number of physical parameters is 9 + 8 + 7 + 6 − 12 = 18, corresponding to 6 masses, 7 angles and 5 phases.
To see this explicitly, we write a general 3 × 3 unitary matrix in the canonical form with R U rotation matrices in the 3 × 3 flavor space, and θ 12 , θ 13 , θ 23 and φ, φ 1,2,4,5,6 generic angles and phases, respectively. Here the indices of the angles label the 2 × 2 rotations.
After redefining several phases, we obtain the parametrization There is a flavor basis in which the above parametrization reproduces the F2HDM textures shown in (2.4), with coefficients that depend on the several angles θ, ϑ, ϑ. In Appendix B we show explicitly how to rotate into this flavor basis.

B. CKM phenomenology
The unitary V 2 matrix in Eq. (3.18) is a flat direction of the flavon potential, as are U , V and V . The quark mixing matrix of the full theory, however, is no longer a flat direction: It is lifted by the 1 + 2 flavor-locked structure to an O(1) 2 ⊕ 1 block form with all other entries suppressed by small ratios of quark masses. Diagonalizing the quark mass matrices resulting from (4.2), one finds the following schematic predictions for the CKM matrix elements where θ is the rotation angle in the V 2 matrix (see Eq. (4.2)), that is a priori a free parameter of O(1). This structure suggests that the observed CKM hierarchies can be accommodated: The 1-3 and 2-3 mixing elements are automatically suppressed at a level that resembles the experimental values.
In the decoupling/alignment limit cos(β − α) = 0, flavor-violating processes from heavy Higgs exchange vanish in the large m H,A limit. However, from Eqs. (4.2) and (4.3) it is not obvious whether the flavor structure of the 1 + 2 flavor-locked configuration reduces to the SM in an appropriate limit. As a demonstration that the 1 + 2 flavor-locked configuration is compatible with data, we heuristically identified the following example input parameters, and ψ u = ψ c = ψ m = 0, where we have defined the two vevs, v 1 ≡ v cos β and v 2 ≡ v sin β.
The phases ψ u , ψ c , ψ m are set to zero for simplicity, as they have negligible impact on all the observables that we are considering. (The phases ψ u , ψ c enter in D 0 -D 0 mixing, but, as we will discuss in Sec. IV C, they are only very weakly constrained.) This parameter set leads to the theoretical predictions shown in Table I  We compare these predictions to data for the quark masses and CKM parameters, shown in Table I. To be self-consistent, we use data only from processes that are insensitive to heavy Higgs exchange, i.e. processes that are tree-level in the SM. (Since we are ultimately interested in considering the phenomenology of collider-accessible heavy Higgs bosons, loop-level processes in the SM will receive corrections from heavy Higgs exchanges, but measurements of tree-level processes will be insensitive to these effects.) To reproduce the Cabibbo angle λ C 0.22506 ± 0.00050 [43], θ needs to be constrained accordingly to a narrow O(1) range.
Since we require only a mixing matrix with canonical entries of the same characteristic size To quantify the "goodness" of the benchmark or other points in the parameter space, we construct a χ 2 -like function, X 2 tree , for the six quark masses and CKM elements measured from tree-level processes, where the 'FL' superscript denotes the theory prediction at a given point in the flavor-locked theory parameter space (4.2), and we treat the uncertainties as uncorrelated. While such a X 2 function implies a well-defined p-value for a goodness-of-fit of the quoted data to a given theory point, one cannot construct from X 2 a sense of the probability for a given theory to produce the observed flavor data and hierarchies. Instead, the X 2 function allows us only to understand whether or not the flavor-locked configuration results generically in a flavor structure that agrees with observation at the level of tens of percent.
In Fig. 1 we show the X 2 tree behavior of the flavor model on various two-dimensional parametric slices in the neighborhood of the benchmark point (4.4), which is denoted by the white circle. That is, in each plot, all the theory parameters are fixed to the benchmark values in Eqs. (4.4), except for the two parameters corresponding to the plot axes. The number of degrees of freedom (dof) in the X 2 tree statistic is then 11 − 2 = 9. The contours show regions of X 2 tree /dof that lead to an overall good agreement between the observed quark masses and CKM parameters and those predicted in the model.
As can be seen from the plots in Fig. 1, there are extended regions of parameter space where there is fairly good agreement between the theory predictions and the measured quark masses and CKM parameters. In particular, O(1) variations of the mixing angles θ 13 , θ 23 , ϑ 13 , ϑ 23 , ϑ 13 , ϑ 23 around the benchmark point are possible, without worsening the agreement substantially. Only the angle θ that sets the Cabibbo angle is strongly constrained and has to be set to a narrow range by hand. This behavior should be contrasted to the SM, for which two CKM mixing angles -i.e. the suppressed 1-3 and 2-3 mixings -have to be tuned small. to the mixing amplitude is given by Ref. [47] (see also Refs. [48,49]). The parameters η i encode renormalization group running effects. From 1-loop RGEs we find The relevant observables that are measured in the neutral kaon system are the mass difference ∆M K and the CP violating parameter K . The experimental results and the corresponding SM predictions and uncertainties are collected in Table II. In terms of the NP mixing amplitude, these observables are given by In the expression for K we use κ = 0.94 [50] and the measured value of ∆M K shown in Table II.
In the case of neutral B meson oscillations, we find it convenient to normalize the NP mixing amplitude directly to the SM amplitude. For B s mixing we find where the first (second) value corresponds to B s (B d ) mixing. To obtain these values we used bag parameters from Ref. [54] (see also Ref. [53]  In the case that the heavy Higgs masses are below the TeV scale, the NP effects in the mixing observables do not vanish, and we proceed to investigate the size of such effects. For the following numerical study, we will set the heavy Higgs masses to a benchmark value, m H = m A = 500 GeV. We use a moderate value of tan β = 5, and work in the alignment limit β −α = π/2. For the benchmark parameters in Eq. (4.4), we show the NP contributions to meson mixing observables in the last column of Table II. For the benchmark point, the NP contributions are in most cases within the combined experimental and SM uncertainties.
Similar to Eq. (4.5), we construct a X 2 loop function, that compares the NP contributions to the difference of the data and SM predictions, for the three mass differences ∆M K , ∆M d , and ∆M s , as well as the CP violating observables K , φ d , and φ s . That is, where the superscript 'exp-SM' indicates that we are using the difference of the measured values and the SM predictions given in Table II.  Table II show slight tensions [53,55,56], as indicated by the nonnegligible SM contribution to the X 2 loop function, X 2 loop (SM) 10.8. We observe that ranges of model parameters exist for which X 2 is mildly better than in the SM: At our benchmark comparing with the contours obtained from the X 2 tree /dof function (dotted lines), we find that extended regions of parameter space exist where CKM elements and masses as well as meson mixing observables are described in a satisfactory way.

V. CONCLUSION AND OUTLOOK
We have presented a new framework to address the SM flavor puzzle, synthesizing the structure of the 'flavorful' 2HDM with the 'flavor-locking' mechanism. This mechanism makes use of distinct flavon and hierarchon sectors to dynamically generate arbitrary quark mass hierarchies, without assigning additional symmetries to the quark fields themselves.
In this paper, we have shown that with suitable symmetry assignments in the flavon and hierarchon sectors, the global minimum of the general renormalizable flavon potential can be identified with a 'flavor-locked' configuration: An aligned, rank-1 configuration for each flavon, and arbitrary (block) unitary misalignment between the up and down quark Yukawas, so that a unique hierarchon vev controls each quark mass.
In the presence of only one SM-like Higgs doublet, this leads to quark mixing angles that are generically O(1). Introducing instead a flavorful 2HDM Higgs sector -two Higgs doublets, such that one Higgs couples only to the third generation, while the other couples to the first two generations -leads to a 1 + 2 flavor-locked theory. We find that quark flavor mixing in this theory is naturally hierarchical too, once one requires that the dynamicallygenerated quark masses are themselves hierarchical -the light quark masses need not be tuned in this theory, being generated instead by the flavor-blind flavor-locking portal to the hierarchon sector -and the mixing is generically of the observed size. The collider phenomenology of this theory is quite rich if the additional Higgs bosons are light, with testable signatures at the LHC or HL-LHC.
For an example benchmark point in the theory parameter space, we showed that this It is straightforward to extend this framework to the charged lepton sector. Possible ways to reproduce a realistic normal or inverted neutrino spectrum and the large neutrino mixing angles will be discussed elsewhere. support during parts of this work. The work of WA, SG and DR was in part performed at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-

The research of WA is supported by the National Science Foundation under Grant
No. PHY-1720252. SG is supported by a National Science Foundation CAREER Grant No.
PHY-1654502. We acknowledge financial support by the University of Cincinnati.

Mixing terms: single flavon generation
The first, ν 1 , term of the mixed potential (3.6) manifestly respects the vacuum of V 1f and V 2f . It follows from the Cauchy-Schwarz inequality and positive semidefiniteness of λ α λ † α , that Hence for the case of n = 1 generations of flavons, the ν 2 term and full potential is imme-