The violation of parity (P) and its combination with charge conjugation (CP) in the strong interactions, described by the parameter θ, are constrained by the experimental limit on the neutron electric dipole moment (nEDM): θ ≤ 10 -10 [1]. Hence, θ is the smallest dimensionless parameter of the Standard Model (SM). This small value is particularly puzzling because of the lack of theoretical understanding of it, and because the dimensionless parameter describing CP violation in the weak interactions is order unity. Within the SM, the parameter θ has two sources. Firstly, the Lagrangian of quantum chromodynamics (QCD), restricted just by Lorentz and gauge invariance, contains a CP-odd term, θ QCD g 2 s =ð32π 2 ÞG μν a Ga;μν , where G μν a is the gluon field strength tensor, g s is the strong coupling constant, and Ga;μν ≡ 1 2 ϵ μναβ G αβ a . Moreover, θ receives a contribution through the chiral transformation needed to diagonalize the quarks' Yukawa matrices y u and y d ,
The two contributions to θ arise from very different physics and have no reason to cancel. Solutions based on an underlying CP or P symmetry at high energies that is softly or spontaneously broken have profound consequences for the flavor sector of the SM. In these theories, the bare parameter θ QCD in Eq. ( 1) is forced to be zero by either CP or P, and one must explain why arg detðy u y d Þ is so small even though the order unity phase of the observed CP violation in weak interactions also arises from the quark mass matrices.
In the case of CP, Nelson and Barr [2,3] introduced N Q extra generations of heavy vectorlike quarks, Q, that mix with the SM quarks, q, through the interactions with new scalars, so that the Yukawa interactions are now described by square 3 þ N Q dimensional matrices. Careful model building arranges for the Qq block of these matrices to vanish, while all CP-violating parameters appear in the qQ block. The observed CP violation of the weak interactions arises from the mixing between Q and q. As well as being enlarged, the Yukawa matrices involve more parameters than in the SM. Different approaches with spontaneous CP breaking have been proposed that make use of the nonrenormalization theorems of supersymmetry [4][5][6][7].
There is another scheme based on soft or spontaneous breaking of CP symmetry-it predates the Nelson-Barr framework, but has not been widely studied and appears to be largely forgotten. Instead of adding additional quark generations with CP violation in the mixing with SM quarks, it augments the SM by having more than one copy of the Higgs doublet, ϕ → ϕ α . CP violation appears only in the Higgs mass matrix. The quark and Higgs fields transform under a flavor symmetry in such a way that the Yukawa matrices are highly restricted: each entry in the quark mass matrices is either zero or arises from a single Higgs field. Hypercharge gauge symmetry forces all Yukawa interactions of the up (down) quarks to involve ϕ α (ϕ Ã α ). If the determinant of the up quark mass matrix is proportional to a single product of ϕ a fields, Πðϕ α Þ, and if the same product appears in the determinant of the down quark masses, Πðϕ Ã α Þ, then arg det y u y d ¼ 0 by a cancellation between up and down sectors.
The hierarchy problem of the SM will not be made significantly worse if the effective field theory with multiple Higgs doublets is valid over a small energy interval. As in Nelson-Barr theories, one must ensure that the radiative contributions to θ are sufficiently small.
Georgi constructed the first model of this type in 1978 [8]. A three-generation SUð5Þ grand unified theory, with flavor group Z 16 , was constructed in 1979 by Mohapatra and Wyler [9], and an extension of the SM at the weak scale was constructed by Glashow in 2001 [10]. In light of modern data, only Glashow's model is phenomenologically viable, and we will show that it does not convincingly solve the strong CP problem.
In this letter we provide a general framework for multi-Higgs extensions of the SM that solve the strong CP problem at tree level by a combination of a flavor symmetry that strongly restricts the form of Yukawa interactions, together with soft or spontaneous breaking of this symmetry and of CP in the Higgs mass matrix.
The Standard Model is minimally extended by increasing the number of Higgs doublets from 1 to N, all with the same quantum numbers. There are no additional fermions and no extension to the gauge group. All but one of the Higgs doublets acquires a mass of order M, and below M the minimal SM is the effective field theory (EFT) describing the model. The lower bound on M arises from limits on flavor changing neutral currents and from direct searches for scalar mesons at the LHC, and is typically expected to be about a TeV. We assume that the heavy Higgs bosons obey these mass bounds. There is no upper bound, although the hierarchy problem of the SM becomes more severe as M is increased. The UV completion of the multi-Higgs theory occurs at a scale Λ. For the hierarchy problem of the multi-Higgs theory to be mild, Λ should not be too far above M.
In this theory, the Yukawa interactions are described by
, where q i are quark doublets and dj ; ūj are singlet antiquarks, all left-handed Weyl fields, and the Higgs doublets are ϕ α , with α ¼ 1…N. These fields transform under a "Higgs-flavor" symmetry, G HF , so that the Yukawa couplings x α ij and xα ij are sparse with many zero entries. The Higgs fields all transform differently under G HF so that the Higgs potential can be written in the form
While the quartics λ αβ are present in all models, the quartics λ αβγδ are model-dependent, occurring only when there happens to be a G HF invariant with at least three of ðα; β; γ; δÞ different. CP and G HF are conserved in all interactions and masses, except for the soft mass terms described by μ 2 αβ , which are complex and have α ≠ β. While jμ 2 αβ j may take a variety of values, they are typified by a scale μ 2 . N -1 of the Higgs doublets have mass of order the scale M and the other is fine-tuned to be much lighter. Hence, the relevant mass scales and EFTs for this framework are illustrated in Fig. 1. The unitary matrix U rotates the Higgs fields from flavor to mass eigenstates, ϕ α ¼ U αβ h β , and only h 1 , which denotes the SM Higgs, acquires a vacuum expectation value, hh 1 i ¼ v ≃ 174 GeV. Below the scale M, we obtain the SM EFT with SM Yukawa coupling matrices
The phases appearing in y u and y d arise from matrix elements of U and U Ã , and hence are correlated. We study theories FIG. 1. The framework: energy scales, EFTs, and their flavor parameters. Λ is the UV cutoff of theories with N Higgs doublets, all with the same gauge quantum numbers. All but one of these Higgs doublets acquires a mass of order M. Below M, the EFT is the minimal SM.
where the Yukawa interactions have flavor determinants that involve a single product of scalar fields:
The determinants of y u and y d then involve the same single product of U α1 matrix elements,
so that θ vanishes at tree level. For example, in the theory with three Higgses studied below, the Yukawa matrices are such that detðy u Þ ∝ U 11 U 31 U 31 and detðy d Þ ∝ U Ã 11 U Ã 31 U Ã 31 , even though the textures of the two matrices are quite different. However, the experimental limit from the nEDM is so severe that θ ¼ 0 at tree level may not be sufficient to ensure that these theories solve the strong CP problem. Given the softness of CP violations, loop contributions to θ arise as threshold corrections at the mass scale M of the heavy Higgs doublets and take the form δ θ ¼ fðx α ij ; xα ij ; U αβ Þ. In the SM, there are two distinct small number problems associated with the quark mass matrices y u;d : the strong CP problem and the flavor hierarchy problem. In the latter, small Yukawa couplings are needed to account for the small quark masses and mixings. From (3), we see that small values for some matrix elements of y u;d may be partially accounted for by small values of jU α1 j for α ≠ 1, that is, by μ ≪ M. At the same time, δ θ is proportional to U αβ , so that it is advantageous for them to be small. Hence, our framework offers the prospect of making progress on small flavor hierarchies as well as solving the strong CP problem.
Radiative corrections to Yukawa interactions modify the quark mass matrices and can lead to a nonvanishing value for θ ¼ arg detðy u y d Þ. Below the scale M, the SM provides the effective description of the model, and it was shown by Ellis and Gaillard [11] that the first finite contribution to θ appears at three-loop order (giving θ ∼ 10 -16 ) and there is no divergent contribution before seven loops. Between Λ and M there may be additional contributions through loop diagrams with a scalar exchangefoot_0 , as illustrated in Fig. 2. The corrections to the SM Yukawa coupling matrix y u at renormalization point μ read
and a similar expression for δy d . Here, dimensional regularization has been used, and sums on α, β, and γ are understood. The index δ runs over the N mass eigenstate scalars, h δ . As already commented above, the softness of CP violations ensures that loop contributions to θ arise as threshold corrections at the mass scale M. In fact, the unitarity of the U matrix implies that terms involving
for δy u and δy d , respectively. One can easily prove that xα x †β x α ðx α x †β xα Þ has the same matrix structure of x β ðx β Þ and, since at tree level y u ¼ x β U β1 and
β1 , this does not lead to corrections to θ in the multi-Higgs theory. Keeping only the logðM 2 δ Þ terms relevant for θ,
and a similar result follows for δy d . The first term above is the one-loop correction due to the SM Higgs in the SM EFT below M 2 , which does not contribute to θ [11]. The only contribution to θ comes from the second term in Eq. ( 6) and depends logarithmically on the mass ratios of the heavy scalars.
An important result follows from Eq. ( 6): there are no corrections to θ from one-loop scalar exchange in theories with two Higgs doublets. A similar statement can be made for higher-order corrections to the quark Yukawa matrices. Below M, we recover the SM EFT and, therefore, there is a negligible finite contribution to θ at three loops and no divergent contributions up to six loops [11]. Above M, where the two-Higgs doublet model is the correct EFT, the finite corrections to the Yukawa matrices can be constant terms, terms proportional to powers of m h =M 2 , and terms proportional to logðM 2 =m h Þ (due to the softness of CP violation). At two loops, we check diagram by diagram that the constant pieces do not contribute to θ. This is easy to prove using the unitarity of the U matrix and noting that each propagator for the scalar H β comes with the insertion of a U αβ and a U Ã γβ matrix. Terms proportional to powers of m h =M 2 correspond to higher-dimensional operators in the SM EFT and are negligible in the limit where M 2 ≫ m h . Finally, terms proportional to logðM 2 =m h Þ can be understood as coming from renormalization group (RG) scaling in the SM EFT below the heavy Higgs mass, where we know there are no corrections to θ at two loops. Similar arguments apply to any loop order; in theories with two Higgs doublets, we conclude there are no radiative corrections to θ beyond those of the SM. On the other hand, for theories with N > 2, there are corrections to θ already at one loop. These are proportional to a combination of the x; x, and U matrices and are, therefore, model-dependent.
In general, one needs to compute θ ¼ argdet½ðy u þ δy u Þðy d þδy d Þ to assess whether a theory solves the strong CP problem for a reasonable choice of parameters.
A complete study and categorization of theories within the framework described above is beyond the scope of this work. In the following, we discuss a representative example that captures the main features of this class of models. We consider as an example theories with three Higgs doublets and G HF ¼ Uð1Þ. Restricting the charges of scalar and quark fields to be smaller than two, and requiring the theory to generate quark masses, Cabibbo-Kobayashi-Maskawa (CKM) mixing angles at first order in the offdiagonal Yukawa entries, and a nonzero Jarkslog invariant, we found 2160 possible charge assignments. In each of these theories, the scalar sector is the same, with three real diagonal and three complex off-diagonal mass parameters. For simplicity and with the same conventions as in Eq. ( 2), we assume ϵ αβ ¼ μ 2 αβ =M 2 ≪ 1, such that the off-diagonal elements of the rotation matrix are
The number of parameters in the quark Yukawas matrices depends on the number of nonzero entries, as dictated by the flavor symmetry. A representative Uð1Þ HF charge assignment, which has the virtue of reducing the number of free parameters, is shown in Table I, leading to the Yukawa interactions
The SM Yukawa coupling matrices are obtained by inserting ϕ α ¼ U α1 h 1 . At tree level, θ ¼ 0 as detðy u Þ ¼
x 13 x 22 x 31 U 11 U 31 U 31 and detðy d Þ ¼ x12 x21 x33 U Ã 11 U Ã 31 U Ã 31 . This theory has 13 real parameters and three phases. Quark masses and mixings fix nine real parameters and the CKM fixes one phase, according to y u ≃ x u 31 jU Ã 31 V ub j; y c ≃ x u 22 jU 11 j; y t ≃ x u 33 jU 11 j; y d ≃ x d 21 jU Ã 31 V us j; y s ≃ x d 22 jU Ã 11 j; y b ≃ x d 33 jU Ã 11 j; jV us jy s ≃ x d 12 jU Ã 31 j; jV ub jy t ≃ x u 13 jU 31 j; jV cb jy b ≃ x d 23 jU Ã 21 j;
to leading order for each of V ub (from y u ), V us , and V cb (from y d ). This leaves four free real parameters, jU 21 j; jU 31 j; jU 23 j, and x 32 . Requiring x 13 and x23 to be perturbative gives the lower bounds ϵ 13 ≳ 0.004 and ϵ 12 ≳ 0.001. Furthermore, we have the requirement of sufficiently small one-loop corrections to θ. Using Eq. ( 6), setting ϵ 23 to its minimal natural value, ϵ 23 ≃ ϵ 12 ϵ 13 , and taking phases of order unity, the dominant terms are
For example, taking ϵ 12 ≃ 10 -3 ; ϵ 13 ≃ 10 -2 ; ϵ 23 ≃ 10 -5 , and x 32 ≃ 10 -2 gives δ θ ≃ 10 -10 , with numerical values of the Yukawa matrices,
x ≃ 0 B @ 0 0 0.4 0 0.005 0 0.1 0.01 1 1 C A; x ≃ 0 B @ 0 0.006 0 0.005 0.0003 0.8 0 0 0.02 1 C A:
This theory illustrates how G HF leads to a close relation between the strong CP and flavor problems. In the SM, all we know about the quark Yukawa couplings y u;d is that they are general complex matrices that have certain eigenvalues, ranging over 5 orders of magnitude, and are diagonalized by certain mixing angles. In this theory, solving the strong CP problem leads to a very different set of real flavor matrices, as shown in (10). Many entries vanish, and the hierarchies in x are much less than the hierarchies in the up-type quark masses. Of course, we do not provide an understanding for these flavor matrices, but stress that the flavor problem has been transformed by our attempt to solve the strong CP problem. In Appendixes, we discuss theories with N ¼ 2 and a different theory with N ¼ 3, highlighting the vast and diverse space of models within our framework.
TABLE I. Uð1Þ HF charges for the representative three-Higgs doublet model discussed in this letter.
ϕ 1 ϕ 2 ϕ 3 q 1 q 2 q 3 ū1 ū2 ū3 d1 d2 d3
1 1 HALL, MANZARI, and NOETHER PHYS. REV. D 111, 115012 (2025) 115012-4
V. CONCLUSION Building on early work [8][9][10], we introduced a new framework for studying flavor and the strong CP problem. The scalar sector of the SM is augmented from 1 to N Higgs doublets, and there was no additions to the fermion or gauge sectors. All dimension 4 interactions are symmetric under CP and a Higgs-flavor symmetry G HF carried by both quarks and Higgs doublets. The SM Yukawa matrices are replaced by real couplings, due to CP, and have many zero entries, due to G HF . CP and G HF are softly broken via the off-diagonal entries of the Higgs mass matrix. We restricted our attention to theories that have N -1 heavy Higgs doublets with mass of order M and one much lighter Higgs h. Thus, the SM appears as the EFT below M, with Yukawa couplings and a strong CP parameter arising from integrating out the heavy Higgs doublets. In this framework, the flavor and strong CP problems are inextricably linked.
We have computed the one-loop threshold corrections to θ resulting from integrating out the heavy Higgs doublets at the mass scale M. In this class of theories, all nonvanishing contributions are proportional to logðM α =M β Þ, where α and β label the heavy Higgs doublet mass eigenstates. Thus, N ¼ 2 theories, with a single heavy Higgs, don't have any contribution to θ at one loop. We studied radiative contributions to y u and y d at two-loop order with N ¼ 2 and find only contributions from RG scaling in the SM EFT below M, which are known to not induce θ. Hence, models with N ¼ 2 were particularly powerful at solving the strong CP problem. Contributions from higher-dimensional operators at the scale Λ in general depended on the UV completion of these multi-Higgs theories, but they are proportional to ðM 2 =Λ 2 Þ lnðΛ 2 =M 2 Þ. Order of magnitude estimates allowed us to conclude that they are negligible if the scale M is a few orders of magnitude below Λ. Furthermore, contributions to the neutron EDM through higher-dimensional operators at the scale M are negligible if the latter is taken to be well above the electroweak scale.
In this framework, the requirement of a "Higgs-flavor" symmetry that gives θ ¼ 0 at tree level may be a useful guide in constructing predictive theories of flavor. A small value for μ=M may suppress radiative corrections to θ and contribute to the smallness of quark masses and mixing angles.
This framework is easily extended to the lepton sector to give a fully realistic description of nature, whether neutrinos are Majorana or Dirac. This leads to relations between the phases appearing in the quark and lepton mass matrices, with important implications for models where the cosmological baryon asymmetry arises from leptogenesis. Moreover, G HF charges can be found for quarks and leptons that are consistent with either SU (5) or SO (10) unification. Finally, our framework offers diverse and model-dependent experimental signatures, including flavor observables like K, B, and D mixing, as well as opportunities for direct searches at current and future collider experiments. Given the richness of the phenomenology in this class of multi-Higgs theories, we have intentionally remained agnostic on the UV completion and we defer its discussion to future works.
In the class of theories discussed in this letter, although many entries in the Yukawa matrices x α ij ; xα ij vanish, the nonzero entries may not all be determined from data on quark masses and mixings. Furthermore, radiative corrections to θ depend on these unknown parameters. While θ < 10 -10 results if these parameters are sufficiently small, how can we determine if such small values constitute a solution to the strong CP problem? One possibility is that the sizes of the x and x matrix elements are controlled by conventional flavor symmetries. A convenient and simple description may be possible in terms of symmetry breaking parameters, ϵ q i ; ϵ u i , and ϵ d i , of the flavor symmetry Uð1Þ 3 q × Uð1Þ 3 u × Uð1Þ 3 d , such that, when nonzero, x α ij ∼ ϵ q i ϵ u i and xα ij ∼ ϵ q i ϵ d i . Where possible, in computing the value of radiative corrections to θ, we estimate any x α ij ; xα ij that cannot be determined from data using this approximate flavor symmetry. The nine real symmetry breaking parameters ϵ q i ; ϵ u i , and ϵ d i are all ≲1 and can be determined from the six quark masses and three quark mixing angles, so that θ can be estimated in terms of U αβ . In fact, since the top Yukawa is close to unity, it determines two of these parameters, ϵ q 3 ∼ ϵ u 3 ∼ 1, implying that fitting to quark masses and mixings leads to a relation between the various U α1 . Using this procedure, the radiative correction to θ can then be evaluated in terms of the remaining independent values of U αβ , allowing a determination of whether the strong CP problem is naturally solved. The size of the required ϵ q i ; ϵ u i , and ϵ d i parameters will also allow an assessment of any progress on the flavor problem. On the other hand, for some models, a fit to the quark masses and mixings using natural values for U αβ yields values of x α ij and xα ij that cannot be consistently understood from an approximate Uð1Þ 9 flavor symmetry (see the theory discussed in the main text). In this case, one can derive upper bounds on those x; x matrix elements not determined from data in order that θ < 10 -10 . For these models, an assessment of whether the strong CP problem is solved can be made according to how small these parameters must be made.
AND G HF = Z 3
We found 864 models with two Higgs bosons and G HF ¼ Z 3 that could accommodate quark masses, CKM angles (from diagonalization of Yukawa matrices at first order), a nonzero Jarlskog invariant, and θ ¼ 0 at tree level. Notably, all of these theories have precisely three zeros in each Yukawa coupling matrix at tree level. Among these theories there occur 24 unique textures of Yukawa matrices that appear in 48 unique fy u ; y d g pairings. Here, by "texture" we mean the full form of the tree-level Yukawas, keeping track of where the U 21 and U 11 factors appear, as well as zeros. Each of the 24 unique textures of y u correspond to one of the 24 textures of y d by complex conjugation, as one would expect at the level of charges. As such, we list the 24 textures in the following format: 0 where a coupling is absent at tree level, 1 where the coupling to ϕ 1 (or ϕ Ã 1 ) is allowed and thus a factor of U 11 is present, and ϵ where the coupling to ϕ 2 (or ϕ Ã
2 ) is allowed and thus a factor of U 21 (or U Ã 21 ) is present. For example, the texture corresponding to the theory discussed in the appendix would be written as
With this convention, the 24 textures are listed in Table II. They are arbitrarily numbered for reference, e.g., the above theory would be said to have texture f3; 14g. Note that for each of these matrices, there is another one obtained exchanging ϵ ↔ 1. The 24 distinct pairings of textures are listed in Table III.
Theories with two Higgs doublets are particularly simple and interesting. CP and Higgs-flavor symmetry is broken by a single soft scalar mass μ 2 12 ≡ μ 2 e iβ (with μ 2 ; β ∈ R). The unitary matrix transforming from Higgs-flavor to mass eigenstates is
Here, we study the case of a Z 3 Higgs-flavor symmetry, yielding detðy u y d Þ ∈ R. A discussion of these theories and realistic charge assignments can be found in Appendixes. As discussed in the main text, there are no radiative corrections to θ in addition to those already present in the SM and, therefore, the strong CP problem is solved in these theories.
A two-Higgs doublet theory at the TeV scale with a Z 3 symmetry was discussed in Ref. [12]. After electroweak symmetry breaking, they found that CP violation appeared only in the W AE and H AE couplings, and that the oneloop contribution to θ from a charged Higgs loop vanished, in agreement with our more general discussion.
TABLE II. The 24 unique Yukawa textures that appear in viable N ¼ 2 theories with
All 24 pairs of Yukawa textures in viable G HF ¼ Z 3 models. Textures are labeled according to Table II. y u y d 1 16, 21 2 1 3 3 1 4 4 15, 17, 24 5 1 3 6 1 4 7 1 7 8 16, 18, 21 9 15, 17, 24 10 16, 21 11 13, 19, 22 12 14, 20, 23 HALL, MANZARI, and NOETHER PHYS. REV. D 111, 115012 (2025) 115012-6
For concreteness, we discuss the model with charges given in Table IV, leading to Yukawa interactions
where we suppress the scalar index on x and x, as each entry of these matrices receives a contribution from only one scalar. Thus, in this model there are 13 real parameters (12 couplings x ij , xij , and and one phase (β). The measured quark Yukawa eigenvalues, y i , and CKM parameters constrain 10 of them. Moreover, it is straightforward to check that detðy u y d Þ ¼ 0 and, therefore, θ ¼ 0 at tree level. For the case that ξ ≪ 1, the x and x matrices take the form
The 11 and 33 entries of the x; x matrices are the same as in the y u , y d matrices, so there is no progress here on the flavor problem. However, the 22 entries, and the entries generating CKM mixing, can be much larger than in the SM Yukawas. For example, all entries in the heavy 2 × 2 space of x could be large and order unity. Note also that the requirement of zero θ at tree level has forced six texture zeros.
We study here another theory with N ¼ 3 and G HF ¼ Uð1Þ to show how a different charge assignment under the Higgs-flavor symmetry leads to a different result for flavor and the strong CP problem. The Uð1Þ HF charges are as presented in Ref. [10] and reported in Table V. The allowed Yukawa interactions are as follows:
x 11 ϕ 1 0 0
x 21 ϕ 2 x 22 ϕ 1 0
This theory has 15 real parameters and three phases, and the pattern of quark masses and CKM parameters is easily reproduced. We obtain
x 11 jU 11 j; y c ≃ x 22 jU 11 j; y t ≃ x 33 jU 11 j; y d ≃ x11 jU Ã 11 j; y s ≃ x22 jU Ã 11 j; y b ≃ x33 jU Ã 11 j; jV us jy s ≃ x12 jU Ã 21 j; jV ub jy b ≃ x13 jU Ã 31 j; jV cb jy b ≃ x23 jU Ã 21 j; δ CKM ≃ 2β 12 -β 13 :
Together with the requirement of perturbative Yukawa couplings, these results give the lower bounds jU 21 j ≳ 10 -3 and jU 31 j ≳ 10 -4 . These relations and the unitarity of the U matrix leave only six real free parameters in this theory: jU 12 j; jU 13 j; jU 23 j; x u 21 ; x u 31 , and x u 32 , and two phases. Using the result of Eq. ( 6), we compute the one-loop correction to θ, which in the small mixing approximation reads TABLE IV. The Higgs-flavor charges in a two-Higgs doublet model with G HF taken to be a discrete Z 3 global symmetry. Here, α is the generator of Z 3 , i.e., α 3 ¼ 1.
ϕ 1 ϕ 2 q 1 q 2 q 3 ū1 ū2 ū3 d1 d2 d3
Uð1Þ F charges for the theory with N ¼ 3 and G HF ¼ Uð1Þ discussed in Appendix D.
ϕ 1 ϕ 2 ϕ 3 q 1 ū1 d1 q 2 ū2 d2 q 3 ū3 d3
To evaluate whether this radiative contribution is sufficiently small for the model to solve strong CP problem, we are faced with the problem that x 21 , x 31 , and x 32 are not determined by data. Here, we use an approximate Uð1Þ
9 flavor symmetry, described in Appendix A, and estimate x 21 ≃ 2 × 10 -5 U 21 ; x 31 ≃ 5 × 10 -4 U 2 21 ; x 32 ≃ 10 -1 U 21 ; U 31 ≃ 0.5U 2 21 :
Furthermore, for the symmetry breaking parameter ϵ q 1 to be less than unity requires U 21 ≳ 0.1. While U 32 is not determined by this approximate symmetry, it has the same Higgsflavor charge U 21 , and hence we expect U 32 ≃ U 21 . Using these results, and taking U 21 near its lower bound, the first term of (D3) is of order 10 -9 , as estimated in [10], which within uncertainties is consistent with data. However, the fourth term in (D3), not considered in [10], is larger and is similarly estimated to be 10 -8 . Hence, we conclude that while this model makes significant progress on the strong CP problem, it does not convincingly solve it. Perhaps an alternative flavor symmetry scheme can be found that further suppresses these one-loop corrections.
Higher-dimensional operators, involving fields q i ; ūi ; di , and ϕ α , that can potentially spoil the solution to the strong CP problem, may be generated at the scale Λ where the multi-Higgs theory is UV completed. The leading such operators occur at dimension 6,
where the C ij and Cij tensors respect Higgs-flavor and CP symmetries. These operators generate corrections to x and x Yukawa interactions at one loop by connecting two scalar legs, as shown in Fig. 3. For the diagram on the left, the loop integral is quadratically divergent, corresponding to a threshold correction at scale Λ. However, this contribution preserves the Higgs-flavor symmetry, and only corrects the size of nonzero entries of x and x, which we take to be free parameters. Introducing the flavor and CP-violating scalar mass terms μ 2 βγ , represented by a cross in the diagram on the right in Fig. 3, leads to complex contributions to elements of x and x (including those that vanish at tree level), and hence can give a contribution to θ. With the μ 2 βγ insertion, the loop integral is logarithmically divergent, so that these flavor-violating contributions are generated by RG scaling from the scale Λ to M and are proportional to lnðΛ=M),
We ignore the threshold corrections at scale M, which have constant terms and terms proportional to logs of heavy Higgs mass ratios. The resulting radiative contribution to θ is obtained by inserting these corrections into y u and y d . A precise numerical evaluation is not possible, because C αβγ ij and Cαβγ ij are unknown. However, using approximate Uð1Þ 9 flavor symmetries, the nonzero entries are estimated to be C ij ∼ ϵ q i ϵ u j and Cij ∼ ϵ q i ϵ d j . Therefore, the magnitudes of all contributions to the determinant of y u and y d , whether at tree or loop level, are proportional to ϵ q 1 ϵ q 2 ϵ q 3 ϵ u 1 ϵ u 2 ϵ u 3 and ϵ q 1 ϵ q 2 ϵ q 3 ϵ d 1 ϵ d 2 ϵ d 3 , respectively, which drops out of the result for θ, giving the rough estimate
Here, f γβ ðUÞ are model-dependent cubic polynomials of matrix elements of U. The off-diagonal matrix elements of U are small if the scale of the soft parameters μ 2 αβ are less than the mass scale M of the heavy Higgs. While the suppression from this is limited, since realistic quark masses and mixings must result from perturbative values of the x and x matrices, the general expectation is that Λ need not be many orders of magnitude above M. We conclude by discussing higher-dimensional operators in the two models presented in the main letter.
For the model with two Higgs doublets, the contributions to θ at Oð
1 Λ 2 Þ, from one-loop diagrams involving dimension 6 operators, read FIG. 3. One-loop radiative corrections to the x α ij and xα ij matrices from dimension 6 operators and soft scalar masses in the interaction basis. δ θΛ ≃ ½-2 C212 31 sin ξ þ 3408 C212 32 sin 2 ξx 21 þ 10 C212 31 cos ξ sin ξx 32 -2 × 10 4 C212 32 cos ξ sin 2 ξx 21 x 32 -ð7 × 10 3 C 212 12 -10 3 C 212 13 Þ sin 2 ξx 31 þ ð5 × 10 7 C 212 12 -8 × 10 6 C 212 13 Þ sin 3 ξx 21 x 31 μ 2 Λ 2 log Λ 2 M 2 sin 3β:
Using a Uð1Þ 9 flavor symmetry, we find sin ξ ∼ 0.4 and M=Λ ≲ 8 × 10 -5 to not spoil the solution to the strong CP problem.
For the model with three Higgs doublets, the contributions to θ at Oð
1 Λ 2 Þ, from one-loop diagrams involving dimension 6 operators, read δ θΛ ≃ M 2 Λ 2 log Λ 2 M 2 ½ð4 × 10 2 C121 11 -10 3 C 112 11 Þϵ 12 sinðβ 12 þ 2β 13 Þ þ 4 × 10 2 C132 31 ϵ 12 ϵ 13 sinð2β 12 Þ -3 × 10 2 C 123 21 ϵ 2 12 ϵ 13 x 32 sinð2β 12 -2β 13 Þ þ ð3 × 10 2 C 21 ϵ 12 ϵ 12 x 32 -C123 31 ϵ 23 Þ sinðβ 12 -β 13 -β 23 Þ:
In this model we cannot use Uð1Þ 9 approximate flavor symmetries to estimate the sizes of nonzero entries in the x; x; C, and C matrices given the large size of some parameters. Instead, we use our illustrative example of ϵ 12 ≃ 10 -3 ; ϵ 13 ≃ 10 -2 ; ϵ 23 ≃ 10 -5 , and x 32 ≃ 10 -2 , which gives δ θ ≃ 10 -10 . We find δ θΛ ≲ 10 -10 for M=Λ ≲ 10 -3 when we take C 112 11 ; C121 11 ≃ 10 -4 , C132 31 ≃ 10 -2 , and other C and C elements of order unity.
In most of this paper we treat μ 2 αβ as soft symmetrybreaking spurions for G HF and CP. However, they may arise from spontaneous G HF and CP symmetry breaking via a set of SM singlet vacuum expectation values (vevs), ξ. The dimension of the operator linking the Higgs bilinear ϕ † α ϕ β to the ξ fields depends on the G HF charges of the bilinear and of ξ. Thus jμ 2 αβ j, which we take less than or of order M 2 , could be of order Mξ; ξ 2 ; ξ 3 =Λ ϕ ; ξ 4 =Λ 2 ϕ ; …, where Λ ϕ is the UV scale at which relevant interactions appear that connect ξ and ϕ α . The various ϕ † α ϕ β operators can have different charges, and the corresponding μ 2 αβ can differ by orders of magnitude. In the case that two such operators have the same charge, we expect the corresponding soft masses to be comparable. There could be higherdimensional interactions, such as g 1 ðξ=Λ q Þq i ūj ϕ α or ½g 2 ðξ=Λ G Þg † 2 ðξ=Λ G ÞG G, leading to G HF and CP breaking effects suppressed by powers of ξ=Λ q;G . The polynomials g 1;2 are model-dependent, and g 2 is at least cubic order. We assume these can be neglected compared to breaking terms involving powers of ξ=M that arise in the Higgs doublet mass matrix. In this paper, the breaking of CP and G HF is communicated to the quark sector through the Higgs Portal. We conclude the discussion showing how the soft breaking of CP and G HF could be obtained spontaneously in the model with three Higgs doublets discussed in the main text. We introduce the scalar fields ξ 1;2 of G HF charges (1, 2) that spontaneously break G HF and CP via the vevs,
M 2 e iβ 12 ¼ ϵ 12 M 2 e iβ 12 ≃ 10 -3 M 2 e iβ 12 ;
M 3 e iβ 13 ¼ ϵ 13 M 3 e iβ 13 ≃ 10 -2 M 3 e iβ 13 : ðF1Þ
The mass matrix for the scalars ðϕ 1 ; ϕ 2 ; ϕ 3 Þ of G HF charges ð0; 2; -1Þ can take the form
There are no contributions from wave function renormalization as they have the structure δy u ¼ y u B, where B is a hermitian matrix.