Solving the strong CP problem with horizontal gauge symmetry

We present a solution to the strong CP problem, which relies on the horizontal gauge symmetry and CP invariance in a full theory. Similar to other Nelson-Barr type solutions, CP violation in both the strong and weak sectors in the Standard model (SM) is attributed to the condensation of complex scalars $\Phi$ in the model. The model is differentiated by others in that it explains the hierarchy in quark-Higgs Yukawa coupling in the SM based on a series of sequential breaking of the horizontal $SU(3)_{f}$ gauge symmetry. The experimental constraint $\overline{\theta}\lesssim10^{-10}$ requires $<\!\!\Phi\!\!>\,\lesssim\,10^{13}-10^{14}{\rm GeV}$ (vacuum expectation value of complex scalars) and $\lambda\,\lesssim\,10^{-6}$ (scalar quartic coupling). We show that this small coupling is natural in the sense of 'tHooft naturalness. Compared to other models of Nelson-Barr type with CP breaking scale $\Lambda_{CP}\lesssim10^{8}{\rm GeV}$, our model is more advantageous in terms of consistency with the thermal leptogenesis.

We present a solution to the strong CP problem, which relies on the horizontal gauge symmetry and CP invariance in a full theory. Similar to other Nelson-Barr type solutions, CP violation in both the strong and weak sectors in the Standard model (SM) is attributed to the condensation of complex scalars Φ in the model. The model is differentiated by others in that it explains the hierarchy in quark-Higgs Yukawa coupling in the SM based on a series of sequential breaking of the horizontal SU (3) f gauge symmetry. The experimental constraint θ 10 −10 requires <Φ> 10 13 − 10 14 GeV (vacuum expectation value of complex scalars) and λ 10 −6 (scalar quartic coupling). We show that this small coupling is natural in the sense of 'tHooft naturalness. Compared to other models of Nelson-Barr type with CP breaking scale ΛCP 10 8 GeV, our model is more advantageous in terms of consistency with the thermal leptogenesis.

I. INTRODUCTION
A smallness of a parameter in a theory can be considered natural provided certain additional symmetries are restored in the limit where the parameter is sent to zero [1]. This sense of naturalness, however, finds an unnatural small parameter when applied to QCD sector of the Standard Model (SM), i.e., θ = θ 0 + Arg det(Y u Y d ). Here θ 0 is the QCD vacuum angle parametrized by a coefficient of the term ∼ F µνF µν in the QCD sector and Y q is the Yukawa coupling matrix. The parameter enters in the expression of the neutron electric dipole moment (NEDM) d n = 3.6 × 10 −16 θ e cm [2] of which the current experimental constraint d n < 3 × 10 −26 e cm [3] yields θ < 10 −10 . Setting θ = 0 still does not undo the breaking of CP symmetry because of non-zero KM phase in the weak sector in the SM.
Several explanations as to the smallness of θ have been suggested. These include, for instance, possibilities of having massless up-quark and the idea of introducing a global U (1) symmetry with a color anomaly [4][5][6][7]. For the purpose of setting θ = 0, the field redefinition of the up-quark and the vacuum expectation value (VEV) of a pseudo Nambu-Goldstone boson arising from the breaking of the anomalous global U (1) can be used for the former and later cases, respectively. The lattice computation of the up-quark mass shows significant deviation from zero [8] and thus this simplest solution seems less likely (however, see e.g. [9]). For the later solution [4][5][6][7][10][11][12][13], a variety of the experimental searches for the pseudo Nambu-Goldstone boson, axion, have been suggested and performed, and are still under active scrutiny to date (see, e.g. [14]).
Another class of solution concerns spontaneous breaking of CP symmetry. The most well known among this * gongjun.choi@gmail.com † tsutomu.tyanagida@ipmu.jp line of solutions is the Nelson-Barr model [15,16]. The model begins with the assumption that CP transformation is a symmetry of the model, giving rise to θ 0 = 0. Furthermore, all the interaction coefficients in the Lagrangian become real and the model is constructed in a way that the determinant of the fermion mass matrix is rendered real as long as CP is conserved. The model assumes a complex scalar sector and the vacuum thereof breaks CP. It is VEV of this complex scalar which makes the next leading order contribution to the fermion mass matrix complex, thereby inducing CP violating KM phase in the weak sector of the SM. One of features that makes the solution of this kind distinguished from others is that CP violations in the strong and weak sector are attributed to fundamentally identical physics.
Along with an unknown fundamental origin of CP violating parameters, i.e. θ and KM phase, an underlying physics responsible for the fermion mass hierarchy remains mysterious in the SM as well. On the other hand, KM phase and the mass hierarchy have something to do with each other in that both are associated with Yukawa coupling matrices in the SM. Given this situation, should one is aimed to resolve the strong CP problem by relying on physics of the spontaneous CP violation, it could be a natural suspicion that underlying origins of θ, KM phase and the hierarchy in fermion masses may possibly be very closely related to one another In this work, as an answer to such a suspicion, we present a model which contains new heavy quarks and complex scalars apart from the SM particle contents. Also, we extend the gauge group of the SM by introducing horizontal SU (3) f gauge symmetry and by assuming three Z 2 discrete gauge symmetries. The complex scalars obtain VEVs of different scales, which results in not only the spontaneous breaking of CP symmetry but a series of sequential breaking of SU (3) f . Within the model, Yukawa coupling structure of the SM is explained as well as the smallness of θ.   A subscript of a complex scalar indicates under which Z2 the field is odd. Notice that quantum numbers of u and U are completely identical. We define U as the partner of U for the mass term and u as an orthogonal direction to U . The same applies for d and D too.

II. MODEL
Apart from the SM gauge group, the model has a horizontal (flavor) gauge symmetry SU (3) f as an additional gauge group [17][18][19] and three discrete gauge symmetries Z respectively (α, i = 1, 2, 3). The quantum numbers of the particle contents of the model are presented in Table I. We see there is no gauge anomaly if we introduce the lepton sector with three right-handed neutrinos [19]. Within the model, a gauged CP symmetry [20,21] is assumed, resulting in real interaction coefficients and θ 0 = 0 (or π). On the acquisition of VEVs of the complex scalars Φ i , both CP and SU (3) f become spontaneously broken. 1 Without loss of generality, we can write down the vacuum of the scalar sector as where X 1 and Y 2 are real, and the rest is complex. It is assumed that 1 To avoid the domain wall problem arising from breaking of CP, we assume that CP violation precedes the inflation. with and where the interaction coefficients are real due to CP invariance and SU (2) L indices are suppressed. Similar models are considered in [22,23]. Then mass matrices of each of up and down sector fermions in the model become where M u and M d are made of four 3 × 3 block matrices and H 0 is the neutral component of the Higgs SU (2) L doublet. At the electroweak symmetry breaking (EWSB) vacuum, H 0 has a VEV |<H 0 >| 246GeV. The block matrices shown in Eq. (6) and (7)  The complex phase of <H 0 > gets cancelled in the product M u M d . Along with Eq. (6) and (7), this shows that detM u M d is real at the tree level. In the next section, we examine additional contributions to the fermion mass matrices which spoil the reality of detM u M d .

III. NON-ZERO CONTRIBUTION TO θ
In this section, we investigate non-zero contribution to θ that arises as a consequence of the structure of the model. RG evolution of θ is negligible since nonvanishing contribution to β-function of θ takes place at 7-loop order [25]. With this, we apply the experimental constraint θ 10 −10 from measurement of the NEDM to the energy scale for breaking of CP and SU (3) f . This will constrain the VEV of Φ i and quartic couplings of the complex scalars in the model.

A. Contribution by higher dimensional operators
The interaction between the complex scalars and fermions in the model might be induced by a UV physics, which can be studied by Planck-suppressed higher dimensional operators. In this section, we probe possible higher dimensional operators allowed by symmetries in the model. Then we figure out which operators potentially spoil reality of detM u M d . Those dangerous operators are to be used to constrain vacuum of the scalar sector.
We start with the dimension 5 operators. For the upsector, the operators contributing to the mass matrices after the complex scalar condensation are where M P 2.4 × 10 18 GeV is the reduced Planck mass and the repeated indices for SU (3) f are assumed to be summed. Here a subscript of an operator indicates the block matrix position in M u to which the operator contributes (see Eq. (6)). The superscript specifies up-sector and the operator mass dimension. The coefficients of operators are real due to CP invariance. Contributions to the each mass matrix by dimension 5 operators are hermitian as can be seen in Eq. (10) and Eq. (11). The same operators can be found for the down-sector with U and u replaced with D and d. Note that M 11 and M 12 are identity matrices up to the dimension 5 operator level. Hermiticity is maintained under addition and inversion, and thus it can be inferred from Eq. (8) that the reality of detM u M d remains protected up to dimension 5 operator level. Put another way, Arg(detM u M d ) = 0 (or π) holds up to dimension 5 operator level. 2 Next, the dimension 6 operators allowed by symmetries of the model are given as θ 0 may be favored by observations [2].
O (u,6) 12 Again the coefficients of operators are real due to CP invariance. Together with these contributions, M u 11 and M u 12 are no longer proportional to identity, but become hermitian. In general, a product of hermitian matrices is hermitian only when component hermitian matrices commute each other. The Wilson coefficients of operators in Eq. (12) and Eq. (13) are arbitrary unknowns and thus it remains undecided whether the block matrices in Eq. (6), M u rs (r, s = 1, 2), commute each other. Therefore, we conclude that breaking of the reality of detM u M d starts from dimension 6 operator level.
By using Eq. (24) and Eq. (27) of which the details would be discussed in the coming discussion in Sec. IV, application of the experimental constraint θ 10 −10 to a ratio of leading contributions to an imaginary and a real part of detM in Eq. (9) yields where we used The value 10 −20 is estimated from VEVs of the complex scalars which will be obtained in the next discussion about dimension 7 operator contribution to θ. The dominant contribution to δθ by the down-sector is traced to (14), we can infer that non-zero δθ arising from fermion mass matrices up to dimension 6 operators causes CP violation albeit not large enough to be used for constraining parameters in the model. Non-zero contributions to Arg(detM u M d ) can also occur in the following part of dimension 7 operators (17) where the repeated indices for SU (3) f are assumed to be summed. Corresponding operators of the similar form can be found in the down-sector. In Eq. (16) and Eq. (17), we only showed 1 ⊗ 8 ⊗ 8 type operators although there are three more types of operators including 8 ⊗ 8 ⊗ 8, 8 ⊗ 8 ⊗ 1 and 1 ⊗ 1 ⊗ 1. For the current purpose of estimating an order of magnitude for δθ due to dimension 7 operators, it suffices to study 1 ⊗ 8 ⊗ 8 type operators below. Up to dimension 7 operator level, the up-sector fermion mass matrix is given by in Eq. (16) and (17). The same applies for the down-sector. Now this fact makes complexity of detM u M d manifest.
For estimation of δθ arising at the level of the dimension 7 operators, we compare the dominant contributions to a real part and an imaginary part of detM u M d by referring to Eq. (9). Up to dimension 5 operator level, the dominant contribution to the real part comes from detM 21 , which reads ∼ (|y t |M U ) 3 . On the other hand, we found the dominant contribution to the imaginary part to be ∼ b u (|y t |M U ) 2 O (u,7) 21 . The same applies for the down-sector. Then, the ratio of these two produces where q can be either of u or d, depending on which is making a greater contribution. With c (q,7) 2,i=1,j=2 /c (q,5) 2 = 10 P taken, we obtain the upper bound on |X 2 | 10 (26−P )/2 GeV. For instance, for P = 0 and P = −2, the upper bound reads 10 13 GeV and 10 14 GeV, respectively. Now that we obtain the upper bound of |X 2 | in terms of values of the Wilson coefficients of the dimension 5 and 7 operators, we realize that δθ due to dimension 6 operators hardly exceeds 10 −10 unless we have fine-tuned Wilson coefficients for dimension 7 operators. Hence, we conclude that θ 10 −10 constrains VEVs of the complex scalars at dimension 7 operator level. Yet, CP violation starts at dimension 6 operator level.
Before ending this section, it is worth reconsidering the physical reason for breaking of reality of detM u M d at dimension 6 operator level. In other words, why does CP get violated especially from dimension 6 operator level? The fact that M q 11 and M q 12 (q = u, d) remain proportional to identity was the reason to make detM u M d real up to dimension 5 operator level (see Eq. (8)). This was possible due to the horizontal SU (3) f gauge symmetry. Also, the fact that the heavy fermions including U are SU (2) L singlet disallows dimension 5 operator contribution to M 12 . In this way, we may understand that the reality of detM u M d up to dimension 5 operator level is related to the horizontal SU (3) f gauge symmetry and SU (2) L singlet heavy fermions. Nonetheless, M q 11 and M q 12 (q = u, d) are no longer proportional to the identity matrix starting from the dimension 6 operator level, resulting in non-zero contribution to θ.

B. One loop contribution toθ
In the previous section, we observed tree-level nonvanishing contribution toθ arises from dimension 7 operators, which constrains the symmetry breaking scale for both CP and the SU (3) f horizontal gauge symmetry. In this section, we will investigate how the scalar sector within the model is constrained by one loop radiative correction to θ. To this end, we begin with the following renormalizable scalar potential which respects where the repeated SU (3) f indices are assumed to be summed. Notice that the higher order terms are non-negligible to determine all VEVs for Φ i , since λ ± are very small as shown below. The determination of VEVs of Φ i is beyond the scope of this paper. Again the CP invariance renders all the interaction coefficients in V (Φ) in Eq. (21) real. For contributions to δθ by higher dimensional operators, we observed a significant imaginary part of detM u M d arises at the dimension 7 operator level. And also we observed the dominant contribution to the imaginary part of detM u M d is attributable to detM q 21 (q = u, d) in Eq. (9). This implies that the leading radiative correction to detM q 21 (q = u, d) with four scalar condensation external lines must be also constrained by the experimental constraint θ 10 −10 .
With that being said, our aim is to ascertain whether one loop corrections to the block matrices of 3,ij = δc (q,7) 3,ji ). As a matter of fact, we find that this is not the case by observing breaking of the one to one correspondence. In the following, we demonstrate this by showing a correction to c 2,ji . The same applies for the down-sector.
In accordance with Eq. (21), we see that vertex factors for scalar quartic interactions read (λ − λ + ) and (λ + λ 0 ) for the left and right panel diagrams respectively in Fig. 1. This proves that the one loop corrections to c . The arrow on the internal line is directed from Φ † to Φ.
the momentum space integral for the loop becomes different because of different scalar masses for different kinds of scalars in internal lines. To prevent the one loop correction from making δθ exceed 10 −10 , we demand where Eq. (22) where m Φ is the mass of heaviest scalar particle in the loop. For λ 2 , there are three possibilities: (1) λ 2 0 (2) λ 0 λ ± (3) λ ± λ ± . We found that the first case does not violate the hermiticity of one loop corrections to the block matrix M u 21 . For the rest of two cases, we find m 2 Φ |X 1 | 2 λ 0 provided at least one of three internal lines corresponds to a massive scalar mode. 3 Therefore, we may argue that both non-zero λ + and λ − are responsible for spoiling the hermiticity of one loop corrections to θ and thus subject to the upper bound on λ obtained above while constraining λ 0 is not necessary to fulfill θ 10 −10 . From Eq. (23), the upper bound on λ ± is obtained to be λ ± 10 −8+2R . For an exemplary case with R = 1, the constraint becomes λ ± 10 −6 .
Having λ ± 10 −8+2R , we realize the Wilson coefficients of dimension 5 operators we discussed in Sec. III A can be constrained. The dimension 5 operators with the coefficients c (1)  (q = u, d) natural in the sense of 'tHooft [1]. In sum, the model succeeds in converting the unnatural smallness of θ into other natural smallness.

IV. EFFECTIVE YUKAWA COUPLING
In this section, we study how the model can produce the effective Yukawa coupling in the SM. For this purpose, it turns out that we need constraints on Wilson coefficients of higher dimensional operators as we shall see below.
If we assume d (q,5) is small enough to make the following condition satisfied where q = u, d and Q = U, D, then for the energy scale between the EWSB scale and the heavy fermion mass scale M Q , the effective SM quark-Higgs Yukawa coupling can be obtained by integrating out the heavy fermions Ψ u and Ψ d . The relevant diagram of a UV physics contribution to the SM Yukawa coupling is shown in Fig. 3. With this, the effective Yukawa coupling induced in the low energy reads where a q , b q (q = u, d) and the heavy fermion masses are defined in Eq. (6) and Eq. (7), and c (q,5) (q = u, d) is from Eq. (10). The hermitian Yukawa coupling in the low energy turns out to be one of the features of the model. Depending on a value of a q (q = u, d), there can be two different situations for the effective SM Yukawas. In the first place, a q can be O(1) so as to be the leading contribution. In this case, with other parameters, a q needs to be tuned for reproducing the SM quark masses as a free parameter of the model [22]. Since the structure of the SM Yukawa will be dominated by a q and other contributions, VEVs of Φ i s cannot explain that of the SM Yukawas without the parameter tuning. In the second place, as a free parameter, a q can be small enough to be negligible in comparison with other contributions [23]. For example, an accidental symmetry can be introduced which is not respected by terms in Eq. (3) in order to suppress a q . Then, Y q s in Eq. (25) and Eq. (26) become dominated by dimension 5 operators and thus the hierarchical structure can be explained by the hierarchy of VEVs of the complex scalars Φ i . Since Y q given in Eq. (25) and Eq. (26) are hermitian, a q s do not affect as additives to the SM gauge group. The quantum numbers of the particle content of the model can be referred to from Table. I.
Beginning as a gauged CP invariant theory, the spontaneous CP violation becomes triggered by the complex scalar field Φ i condensation. Simultaneously, the horizontal SU (3) f gets spontaneously broken around the energy scale ∼ 10 13 − 10 14 GeV. This is in contrast with other Nelson-Barr type models where the CP breaking occurs for Λ CP 10 8 GeV by dimension 5 operators [26]. The higher breaking scale of our model is better in avoiding a tension to the thermal leptogenesis [27,28]. We found that provided the scalar sector of the model is featured by small enough quartic self-interaction at this scale, i.e., λ 10 −6 , then the radiatively induced CP violating parameter in QCD sector, θ, can be small enough to avoid the current experimental constraint θ 10 −10 . The upper bound λ 10 −6 further constrains the Wilson coefficient of dimension 5 operators to be smaller than 10 −2 . The newly obtained smallness of other parameters in the model than θ turns out to be technically natural, enhancing the symmetry of the model [1].
On the other hand, the quark-Higgs Yukawa coupling structure is explained as a consequence of the model. The sequential breaking (so-called tumbling) of the horizontal SU (3) f gauge symmetry by different VEVs of three complex scalars leads on to the hierarchical structure of the effective Yukawa coupling in the SM [18]. Within the model, CKM matrix is determined by the scalar field sector dynamics and the interplay between fermions and scalars communicated by a UV physics. Therefore, in this work, we find that CP violation in the strong and weak sector, and the hierarchical structure of the Yukawa coupling in the SM are originated from the common underlying physics of breaking of CP and SU (3) f induced by the scalar field dynamics. Extension including the lepton sector will be given elsewhere.