A grandunified Nelson–Barr model
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Abstract
We argue that the Nelson–Barr solution to the Strong CP Problem can be naturally realized in an E\(_6\) grandunified theory. The chiral SM fermions reside in three generations of E\(_6\) fundamentals together with heavy vectorlike down quarks, leptons doublets and righthanded neutrinos. CP is imposed on the Lagrangian and broken only spontaneously at high scales, leading to a mixing between chiral and vectorlike fields that allows to solve the Strong CP Problem through the Nelson–Barr mechanism. The main benefit of the E\(_6\) GUT structure is the predictivity in the SM fermion sector, and a perfect fit to all SM observables can be obtained despite being overconstrained. Definite predictions are made for the neutrino sector, with a Dirac CP phase that is correlated to the CKM phase, allowing to test this model in the near future.
1 Introduction
The most popular explanation for this puzzle is the PecceiQuinn mechanism [2, 3], which has the axion as a lowenergy remnant [4, 5]. This prediction makes axion models testable in upcoming experiments, which search for the axion with haloscopes like ADMX [6], helioscopes like IAXO [7, 8] or even precision flavor experiments like NA62 [9, 10, 11, 12, 13].
An alternative explanation for the smallness of \(\overline{\theta }\) is provided by the Nelson–Barr mechanism [14, 15, 16, 17], where CP is broken spontaneously at high scales. The original Lagrangian is CP invariant and hence \(\theta _\mathrm{QCD}\) is zero. CP is broken spontaneously by large vacuum expectation values (VEVs), and CP violation is mediated to the lowenergy Lagrangian only via mixing with heavy vectorlike quarks. If the Lagrangians respects two simple conditions (the socalled Barrcriteria), the resulting SM quark mass matrices are complex but have a real determinant, thus providing the CKM phase but rendering \(\theta _F =0\) at treelevel. Finite and calculable contributions to \(\theta _F\) arise at looplevel, but are generically suppressed by small Yukawa couplings and/or small mass ratios [14, 16, 17, 18]. The general Nelson–Barr framework has been realized in a minimal setup in Refs. [19, 20], and recently been combined with the idea of cosmological relaxation [21] in Ref. [22].
In contrast to axion models, in Nelson–Barr scenarios the effective theory below the scale \(V_\mathrm{CP}\) of spontaneous CP breaking is just the SM. This scale is in general required to be very large in order to suppress loop corrections to \(\overline{\theta }\) that are proportional to \(v^2/V_{\mathrm{CP}}^2 \) [19, 20]. Since \(V_{\mathrm{CP}}\) sets the scale of the heavy vectorlike fermions, they are too heavy to be observed in the near future. Therefore the main drawback of Nelson–Barr models is the lack of predictivity, in addition to theoretical shortcomings discussed in e.g. Ref. [23].
In this paper, we address the issue of predictivity by embedding the Nelson–Barr mechanism into an E\(_6\) grandunified framework. This allows to connect the phases in the neutrino sector to the CKM phase, and in particular to predict the Dirac CP phase that will be measured in the near future. Indeed the heavy vectorlike quarks needed in the Nelson–Barr setup naturally find their theoretical motivation in grandunified theories (GUTs), as proposed already in Ref. [17]. Among the possible simple GUT groups, \(E_6\) [24, 25, 26, 27, 28] is ideally suited for the implementation of the Nelson–Barr mechanism (see also Ref. [29]), because the fundamental representation of \(E_6\) contains in addition to chiral SM fermions a vectorlike pair of righthanded (RH) down quarks, besides a vectorlike pair of lefthanded (LH) leptons and two RH neutrinos. Spontaneous CP breaking will induce a mixing between these vectorlike fields with the chiral fermions, and complex phases will enter lowenergy quark, charged lepton and neutrino masses in a correlated manner. Definite predictions in the neutrino sector are then possible because of the very restricted form of the fundamental Yukawa sector, imposed by the E\(_6\) GUT structure together with spontaneous CP violation. While in usual GUT scenarios the unification of Yukawa couplings is often problematic for light fermion generations, it turns out that the mixing with the heavy vectorlike fields allows to cure these problems and to obtain a perfect fit to the full SM fermion sector. Therefore in our model the Nelson–Barr mechanism becomes predictive in the neutrino sector because of the E\(_6\) GUT structure, which in turn is phenomenologically viable because of the mixing with the heavy fermions needed to generate the CKM phase.
The rest of this paper is organized as follows: in Sect. 2 we present the general setup of the model and derive analytical expressions for the lowenergy quark, charged lepton and neutrino masses. In Sect. 3 we perform a numerical fit to fermion masses and mixings and demonstrate that a perfect fit can be obtained for all observables with definite predictions for the neutrino sector. In Sect. 4 we discuss loop corrections to \(\overline{\theta }\), which will constrain the overall scale of spontaneous CP breaking that is left undetermined by the fit. We finally summarize and conclude in Sect. 5.
2 An \(E_6\) Nelson–Barr model
The vectorlike \((\mathbf{5 + \overline{5})_{10}}\) pair will get a large mass at an intermediate scale \(M \sim 10^9 {\, \mathrm GeV}\), and a mass term of similar order that mixes the heavy fermions in the \(\mathbf{\overline{5}_{10}}\) with the chiral RH down quark and LH charged leptons in the \(\mathbf{\overline{5}_{16}}\). According to the Nelson–Barr mechanism, this mixing is the only way how a complex phase enters the lowenergy effective (down) Yukawa couplings, which are of the form \(y_d \sim y \cdot a\), where y is a real and a a hermitian \(3 \times 3\) matrix. Indeed this matrix has a physical phase while the determinant stays real.
The SM singlet \(\mathbf{1_1}\) will acquire a mass at the GUT scale \(M_\mathrm{GUT} \sim 10^{16} {\, \mathrm GeV}\) from \(E_6\) breaking, while the other singlet \(\mathbf{1_{16}}\) gets a mass at an intermediate scale \(M_\nu \sim 10^{11} {\, \mathrm GeV}\) and induces neutrino masses via the TypeI seesaw mechanism.
We do not spell out the scalar potential, which is simply assumed (see below) to generate the appropriate VEVs and make all physical scalars except the SM Higgs ultraheavy, around the scale \(M_\nu \) or \(M_\mathrm{GUT}\) (this is the most natural scenario, since each light scalar in a GUT setup requires a tuning). Because of the largely modeldependent scalar sector we will not study gauge coupling unification in detail, but simply assume that there are suitable threshold correction at M and \(M_\nu \) that lead to unification around \(M_\mathrm{GUT}\) (it might be necessary to embed our framework into a supersymmetric setup for this purpose). This approach is justified mainly by phenomenology, since our model makes definite predictions for the neutrino sector that can be tested in the near future.

(i) No SU(2) breaking mass terms for \(t  \overline{F}\) are present

(ii) Only mass terms for \(\overline{f}  F\) are complex
We will now first neglect the weak scale VEVs and diagonalize the heavy sector given by the last two lines above. In this way we can identify the linear combination of \(\overline{f}\) and \(\overline{F}\) that remains light and determine the SM quark and charged lepton masses. Similarly we can integrate out the heavy neutrino mass eigenstates to obtain light neutrino masses.
2.1 Quark and charged lepton sector
2.2 Neutrino sector
Counting parameters, we see that the neutrino sector depends on a single additional real VEV ratio \(r_{\epsilon }\) compared to the quark and charged lepton sector. Therefore we have in total 15 relevant real parameters + 1 phase to describe the measured 17 + 1 SM parameters: 9 quark and charged lepton masses, 2 neutrino mass differences, 6 mixing angles and 1 CKM phase. This means that there are two predictions that make the fit of the parameters to experimental data nontrivial. Moreover the model makes definite predictions for yet unmeasured observables in the neutrino sector (Dirac phase, two Majorana phases, overall neutrino mass scale, effective scale for neutrinoless double beta decay) and is therefore testable. We discuss the fit and these predictions in the next section.
3 Fit to fermion masses and mixings
SM input parameters at the electroweak scale, where quark and lepton masses and the quark mixing parameters are taken from Ref. [30], and neutrino mixing parameters from Ref. [31] for Normal Ordering (NO). As explained in the text, we use a \(0.1 \%\) uncertainty for the charged lepton masses in the fitting procedure. To simplify the fitting procedure, we used for all observables the arithmetic average of the errors when not symmetric
Fermion observables at the electroweak scale \(\mu =M_Z\)  

\(m_d\) (MeV)  \(2.75\, \pm \,0.29\)  \(\Delta _{12}(\hbox {eV}^{2}\))  \((7.50 \pm 0.18)\times 10^{5}\) 
\(m_s\) (MeV)  \(54.3 \, \pm \, 2.9\)  \(\Delta _{31}(\hbox {eV}^{2}\))  \((2.52 \pm 0.04)\times 10^{3}\) 
\(m_b\) (GeV)  \(2.85 \, \pm \, 0.03\)  \(\sin \theta _{12}^q\)  \(0.2254 \pm 0.0007\) 
\(m_u\) (MeV)  \(1.3 \, \pm \, 0.4\)  \(\sin \theta _{23}^q\)  \(0.0421 \pm 0.0006\) 
\(m_c\) (GeV)  \(0.627 \, \pm \, 0.019\)  \(\sin \theta _{13}^q\)  \(0.0036 \pm 0.0001\) 
\(m_t\) (GeV)  \(171.7 \, \pm \, 1.5\)  \(\sin ^2 \theta _{12}^l\)  \(0.306\pm 0.012\) 
\(m_e\) (MeV)  \(0.4866 \, \pm \, 0.0005\)  \(\sin ^2 \theta _{23}^l\)  \(0.441 \pm 0.024\) 
\(m_{\mu }\) (MeV)  \(102.7 \, \pm \, 0.1\)  \(\sin ^2\theta _{13}^l\)  \(0.0217 \pm 0.0008\) 
\(m_{\tau }\) (GeV)  \(1.746 \,\pm \, 0.002\)  \(\delta _\mathrm{CKM}\)  \(1.21 \pm 0.05\) 
Result of the fitting procedure, as described in the text. The pull of a fit value \(\mathcal{O}^\mathrm{fit}_i\) is defined as \(\text {pull}(\mathcal{O}^\mathrm{fit}_i)=\left( \mathcal{O}^\mathrm{exp}_i\mathcal{O}^\mathrm{fit}_i \right) /\sigma ^\mathrm{exp}_i\), where \(\sigma ^\mathrm{exp}_i\) is the corresponding experimental error and \(\mathcal{O}^\mathrm{exp}_i\) the experimental value as given in Table 1
Fit result at the electroweak scale \(\mu =M_Z\)  

Fit  Pull  Fit  Pull  
\(m_d\) (MeV)  3.44  \(\) 2.4  \(\Delta _{12}(\hbox {eV}^{2}\))  \(7.39 \times 10^{5}\)  0.63 
\(m_s\) (MeV)  50.4  1.4  \(\Delta _{13}(\hbox {eV}^{2}\))  \(0.76 \times 10^{3}\)  \(\) 0.19 
\(m_b\) (GeV)  2.85  0.27  \(\sin \theta _{12}^q\)  0.225  0.56 
\(m_u\) (MeV)  1.32  \(\) 0.08  \(\sin \theta _{23}^q\)  0.0414  0.1 
\(m_c\) (GeV)  0.63  \(\) 0.07  \(\sin \theta _{13}^q\)  0.0035  1.1 
\(m_t\) (GeV)  171.58  0.08  \(\sin ^2 \theta _{12}^l\)  0.302  0.37 
\(m_e\) (MeV)  0.486  0.15  \(\sin ^2 \theta _{23}^l\)  0.405  1.5 
\(m_{\mu }\) (MeV)  102.76  \(\) 0.61  \(\sin ^2\theta _{13}^l\)  0.022  \(\) 0.26 
\(m_{\tau }\) (GeV)  1.746  \(\) 0.04  \(\delta _\mathrm{CKM}\)  1.13  1.5 
Predicted values and current bounds for the neutrino observables. The current bounds were taken from Ref. [50]. As explained in the text, the ranges shown here correspond to perturbations of the best fit point with \(\chi ^2/\mathrm{dof} \lesssim 10\)
\(m_\beta \) (meV)  \(\Sigma \) (meV)  \(m_{\beta \beta }\) (meV)  \( \delta \) (\( ^\circ \))  \(\varphi _1\) (\(^\circ \))  \(\varphi _2\) (\(^\circ \))  

Prediction  \(8.8 \pm 0.5 \)  \( 59 \pm 3\)  \( 1.8 \pm 0.1 \)  \(157 \pm 3 \)  \(187 \pm 4\)  \(159 \pm 5 \) 
Current bound  \(\lesssim 2000 \) [46]  –  –  – 
Finally we comment on the remaining free parameters that are left undetermined by the fit to masses and and mixings. From 12 VEVs (see Eq. (2.3)) six are determined by the fit and one by the electroweak scale (\(v^2 = v_{u1}^2 + v_{d1}^2 + v_{u2}^2 + v_{d2}^2\)). From the remaining 5 VEVs, \(V_{10}^\prime \) does not affect the neutrino sector given the hierarchy \(V_{10}^\prime \sim V_{10} \ll \tilde{V}_6 \), so for simplicity we set \(V_{10}^\prime = V_{10} = M_\nu = v_{d2}/r_\epsilon \sim 10^{11} {\, \mathrm GeV}\) without any impact on the spectrum. We are then left with four VEVs that are free parameters, which we take as \(v_{d1}, V^c_{10}, \tilde{V}_{6}, \tilde{V}_{5} \). As the two latter VEVs control the mass of heavy gauge bosons, they are bounded from below by proton decay constraints, and we take \(\tilde{V}_{6} = \tilde{V}_{5} \equiv M_\mathrm{GUT} \sim 5 \cdot 10^{16} {\, \mathrm GeV}\). The VEV \(v_{d1}\) is mainly bounded by requiring perturbative Yukawa couplings and does not have a big impact on the spectrum, and for the sake of explicitness we fix \(v_{d1} = 70 {\, \mathrm GeV}\). The remaining scale M is bounded from above by neutron EDM constraints, which require that the loop corrections to the effective \(\overline{\theta }\) parameter remain sufficiently small. As we will discuss in the next section, these higherloop corrections are sufficiently suppressed if \(M \sim 10^9 {\, \mathrm GeV}\).
4 Loop contributions to \(\overline{\theta }\)
In Nelson–Barr models \(\overline{\theta }\) vanishes at treelevel by construction, but is generated at looplevel due to higher order corrections to the effective Yukawa couplings. Therefore care has to be taken to ensure that such corrections are sufficiently small in order to have \(\overline{\theta } < 10^{10}\). The form of these (finite) corrections has already been discussed to large extent in the literature for the original Nelson model [16] and in more general setups [18]. It turns out that such corrections are in general suppressed by loop factors and small Yukawa couplings and/or small mass ratios. While the contributions suppressed by Yukawas are always negligibly small in Nelson–Barr type models where only the RH down quarks mix with heavy fields [18], the contributions sensitive to UV physics are suppressed by ratios of the heavy RH down quark masses over heavy gauge boson or heavy scalar masses [16, 19]. Thus they can be made sufficiently small by lowering the mass scale of RH down quarks M, which in our setup is a free parameter. In this section we (conservatively) estimate the leading corrections involving heavy gauge bosons and scalars using a spurion analysis, showing that \(M \sim 10^{9}\) is enough to render \(\overline{\theta } \lesssim 10^{10}\).
4.1 IR contributions
4.2 Gauge contributions
4.3 Scalar contributions
5 Summary and conclusions
To summarize, we have shown that the Nelson–Barr mechanism can be naturally realized in the context of an \(E_6\) GUT. The SM fermions are embedded in three generations of E\(_6\) fundamentals together with the Nelson–Barr fields (vectorlike RH down quarks), a vectorlike pair of LH lepton doublets and two RH neutrinos. All heavy mass terms arise from VEVs of scalar fields in E\(_6\) representations, whose couplings to fermions are given by two \(3 \times 3\) symmetric matrices. CP is imposed on the Lagrangian, and therefore all Yukawa couplings are real. CP is broken spontaneously by two scalar VEVs that mix the chiral SM fermions with the Nelson–Barr fields, through which CP violation is mediated to the lowenergy theory. By integrating out the heavy downquarks, lepton doublets and neutrinos, we have derived analytic formulas for all SM fermion mass matrices, including Majorana neutrino masses. The resulting SM quark mass matrices are real (up sector) and the product of a real and a hermitian matrix (down sector), thus implying that \(\overline{\theta }\) vanishes at treelevel.
Besides solving the strong CP problem with the Nelson–Barr mechanism, the main benefit of the GUT setup is the predictivity in the fermion sector. The fundamental Yukawa matrices contain just 9 parameters, which together with 6 real VEV ratios and a single complex phase determine the complete SM fermion sector including neutrinos (17 real observables + 1 CKM phase). Surprisingly, we nevertheless obtain a perfect fit with \(\chi ^2/\mathrm{dof} \approx 0.9\), implying that there are two relations among SM observables that hold to good precision, but unfortunately we were not able to derive them in closed form due to the complexity of the analytical expressions. Since all lowenergy parameters of the model are fixed, we obtain definite predictions for the neutrino sector that makes this model testable in the near future, as shown in Table 3. Particularly interesting is the prediction of the Dirac CP phase \(\delta _\mathrm{CP} = 157 \pm 3 ^\circ \), which is directly correlated with the CKM phase, and will be verified or excluded by future experiments like HyperKamiokande [44] or DUNE [45].
The fit to the fermion sector determines the absolute mass scales only in the neutrino sector, fixing the mass of RH neutrinos at about \(M_\nu \sim 10^{11} {\, \mathrm GeV}\) (there is another SM singlet around the GUT scale \(M_\mathrm{GUT} \sim 5 \cdot 10^{16} {\, \mathrm GeV}\)). The overall mass scale in the heavy RH down and LH lepton sector M is left undetermined, but bounded from above to keep loop corrections to \(\overline{\theta }\) sufficiently small. The leading loop contributions from diagrams involving heavy gauge bosons and heavy scalars are suppressed by mass ratios \(M^2/M_{V,S}^2\), and we have (conservatively) estimated that \(M = 10^9 {\, \mathrm GeV}\) is enough to render \(\overline{\theta } < 10^{10}\). This essentially fixes all heavy mass scales in our model, as sketched in Fig. 1.
An interesting direction for further research is a supersymmetric extension of this scenario. In such a framework one could study gauge coupling unification in detail, and try to construct the scalar potential needed to break \(E_6\) down to the SM group. Naively, the correlation of the CKM and PMNS phases should not be altered strongly, as the Yukawas are the same and only SUSY threshold corrections are expected to modify the fermion mass fit. The presence of supersymmetric particles will also give additional contributions to \(\overline{\theta }\) (see e.g. Ref. [51]), which might imply restrictions on the underlying origin of SUSY breaking. Best suited from this perspective seems to be lowenergy Gauge Mediation (for a review see Ref. [52]), which does not introduce large new sources of flavor and CP violation.
Footnotes
 1.
It would be desirable to write down a scalar potential that generates these VEVs dynamically, addressing in particular the spontaneous breaking of CP. While there is no reason that a suitable potential should not exist, this issue is beyond the scope of this work that concentrates rather on the lowenergy phenomenological consequences, which are independent of the explicit realization. This is very much in the spirit of the previous literature, in particular the original models in Refs. [14, 15].
 2.
There must be two relations involving just SM observables, however due to the highly nontrivial dependence on the fundamental parameters we were not able to find analytical expressions for these relations.
Notes
Acknowledgements
We thank L. di Luzio, U. Nierste and L. Vecchi for helpful discussions, and L. di Luzio for useful comments on the manuscript. J.S. and P.T. acknowledge support from the DFGfunded doctoral school KSETA. The Feynman diagrams were drawn using TikZFeynman [53].
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