1 Introduction



A Grand-Unified Nelson-Barr Model

Jakob Schwichtenberg, Paul Tremper, Robert Ziegler

Institute for Theoretical Particle Physics, Karlsruhe Institute of Technology,

Engesserstrasse 7, D-76128 Karlsruhe, Germany

Theoretical Physics Department, CERN, 1211 Geneva 23, Switzerland

We argue that the Nelson-Barr solution to the Strong CP Problem can be naturally realized in an E Grand-Unified Theory. The chiral SM fermions reside in three generations of E fundamentals together with heavy vectorlike down quarks, leptons doublets and right-handed 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 GUT structure is the predictivity in the SM fermion sector, and a perfect fit to all SM observables can be obtained despite being over-constrained. 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

One of the most puzzling aspects of the Standard Model (SM) is the absence of CP violation in strong interactions, as measured by the topological angle


where denotes the coefficient of and . From the contribution to the neutron electric dipole moment one finds from the 95% CL bound  [1], although generically one would expect to be of the order of the observed CP violating phase in weak interactions, i.e. . Indeed it is the presence of the large CKM phase that prevents to forbid by imposing CP as a fundamental symmetry.

The most popular explanation for this puzzle is the Peccei-Quinn mechanism [2, 3], which has the axion as a low-energy 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 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 is zero. CP is broken spontaneously by large vacuum expectation values (VEVs), and CP violation is mediated to the low-energy Lagrangian only via mixing with heavy vectorlike quarks. If the Lagrangians respects two simple conditions (the so-called Barr-criteria), the resulting SM quark mass matrices are complex but have a real determinant, thus providing the CKM phase but rendering at tree-level. Finite and calculable contributions to arise at loop-level, 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 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 that are proportional to  [19, 20]. Since 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 Grand-Unified 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 Grand-Unified theories (GUTs), as proposed already in Ref. [17]. Among the possible simple GUT groups, [24, 25, 26, 27, 28] is ideally suited for the implementation of the Nelson-Barr mechanism, because the fundamental representation of contains in addition to chiral SM fermions a vectorlike pair of right-handed (RH) down quarks, besides a vectorlike pair of left-handed (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 low-energy 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 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 vector-like 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 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 Section 2 we present the general setup of the model and derive analytical expressions for the low-energy quark, charged lepton and neutrino masses. In Section 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 Section 4 we discuss loop corrections to , which will constrain the overall scale of spontaneous CP breaking that is left undetermined by the fit. We finally summarize and conclude in Section 5.

2 An Nelson-Barr Model

We embed the SM fermions in three E fundamentals that decompose under SU(5) as


where the subscripts denote the SO(10) decomposition. Thus for each generation of chiral SM fermions residing in the and , there is a vectorlike pair of RH down-quarks and LH lepton doublets contained in and two SM singlets and .

The vectorlike pair will get a large mass at an intermediate scale , and a mass term of similar order that mixes the heavy fermions in the with the chiral RH down quark and LH charged leptons in the . According to the Nelson-Barr mechanism, this mixing is the only way how a complex phase enters the low-energy effective (down) Yukawa couplings, which are of the form , where is a real and a hermitian matrix. Indeed this matrix has a physical phase while the determinant stays real.

The SM singlet will acquire a mass at the GUT scale from breaking, while the other singlet gets a mass at an intermediate scale and induces neutrino masses via the Type-I seesaw mechanism.

All fermion mass terms and the breaking of to the SM gauge group arise from adding scalars in , and that develop large VEVs. The latter field is only responsible for breaking SO(10) at , while the other two fields couple to fermions according to the -invariant Yukawa Lagrangian


We impose CP as a symmetry of the Lagrangian, so that the couplings and can be taken as real and symmetric matrices. Without loss of generality we can choose the flavor basis such that is diagonal. The E structure therefore drastically reduces the number of flavor sector parameters, so that there are just 3+6 real parameters responsible for generating masses and mixings for quarks, charged leptons and neutrinos. While in usual GUT models such a unified structure often prevents to correctly account for all mass hierarchies, it turns out that in our setup the additional mixing in the RH down and LH charged lepton sector allows for an excellent fit to the full set of SM masses and mixings, as we are going to see in the next section.

We do not spell out the scalar potential, which is simply assumed to generate the appropriate VEVs and make all physical scalars except the SM Higgs ultra-heavy, around the scale or . Because of this largely model-dependent scalar sector we will not study gauge coupling unification in detail, but simply assume that there are suitable threshold correction at and that lead to unification around (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.

According to this bottom-up spirit, we first allow only SM singlets and doublets in the , and to take VEVs, where the singlet VEVs are large, i.e , or , and the SU(2) breaking VEVs are at most of the order of the electroweak scale. The complete list of fields with the SM quantum numbers of contained in the , the and can be found in the Appendix. A second requirement on the scalar VEVs comes from imposing the so-called Barr criteria, which ensure that the low-energy quark mass matrices have real determinants at tree-level. With the shorthand notation for the fermions in Eq. (2.2), , the Barr criteria require that

  • ) No SU(2) breaking mass terms for are present

  • ) Only mass terms for are complex

If these criteria are fulfilled, one can easily check that the resulting down-quark mass matrix has a real determinant , but has entries that are in general complex and thus can provide the CKM phase. Decomposing the Lagrangian in Eq. (2.3) under SU(5), one can see from criterion ) that the VEVs of the fields have to vanish (the subscripts denote the quantum numbers under E, SO(10) and SU(5), see Appendix for details). Criterion ) implies that only the singlet VEVs are complex.

Apart from imposing these conditions on the VEVs, which have to be fulfilled to high degree in order to solve the strong CP problem, we set some electroweak VEVs to zero that merely lead to sub-leading corrections or can be absorbed into other VEVs. Moreover, for simplicity we also assume that the singlet VEVs giving rise only to neutrino masses are real, although they are not directly constrained by the Barr criteria. As we will discuss below, a complex phase in those VEVs would only affect the Majorana phases, not the Dirac CP phase. We therefore assume the following VEVs in the scalar sector:


where all VEVs are real and positive except , of which at least one is complex. Here the subscripts denotes the breaking of E, SU(5) and SO(10), respectively. Apart from the electroweak VEVs, which are all of the order of the weak scale , there are three heavy scales , and , which set the order of magnitude of the singlet VEVs as


The VEVs determine the breaking pattern of E to the SM and set the scale of heavy gauge boson and fermion masses. At the VEVs and break to U(1), while the residual U(1) factor is broken at by and . Via the Yukawa couplings in Eq. (2.3) the VEVs in Eq. (2.4) generate all fermion masses, which we parametrize as




Note that all mass matrices are real and symmetric except . Neglecting Clebsch-Gordon coefficients, the fermion masses can be written in SU(5) notation as


where we have introduced the shorthand . The first and third line generate masses for quarks and charged leptons, while the second and fourth line are responsible for neutrino masses. The first two lines comprise weak scale SU(2) breaking masses, while the last two lines are heavy mass terms for vector-like fields from the singlet VEVs in Eq. (2.5). In particular the heavy RH down quarks and LH lepton doublets get a mass at , while the heavy RH neutrinos and get a mass at and , respectively. Together with the scalars at , we have thus fixed the mass scales of all heavy fields (apart from the additional hierarchies for heavy fermions from hierarchical Yukawa couplings), which we summarize in Fig. 1.

Figure 1: Sketch of heavy particle scales, see text for details.

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 and 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

We first derive the light mass matrices in SU(5) language and include Clebsch-Gordon coefficients later on. We begin by rewriting in terms of light fields and heavy fields (that get a mass with ) with the ansatz


with some matrices . Requiring canonically normalized kinetic terms gives three conditions


and imposing that the light field has no mass term with yields a fourth condition


One can now solve these four equations, but since we are only interested in light fields we need only and , which are given by:


The quark Lagrangian in terms of light fields is




Note that while and are real symmetric matrices, is hermitian and therefore carries a complex phase into the light Yukawa matrix whose determinant is nevertheless real.

Upon including Clebsch-Gordon coefficients, quark and charged lepton masses are finally given by


with the real symmetric matrices


and the hermitian matrices


The masses depend on the two real symmetric matrices and (where we have chosen to be diagonal)


and five VEV ratios


Since only the ratio is complex, the quark and charged lepton sectors depend in total on 6+3+5 = 14 real parameters and 1 complex phase. As we will see later, SM fermion masses and mixings can be reproduced if all ratios are .

2.2 Neutrino Sector

The heavy singlet mass terms are given by


with the heavy neutrino mass matrix


Note that the sub-matrices defined in Eq. (2.7) are all real and proportional to that we have chosen to be diagonal. Therefore is a block matrix of diagonal (and real) matrices. In order to diagonalize it, we therefore need only to diagonalize the real symmetric matrix defined by


The eigenvalues of this matrix set the scale of light neutrino masses via the Type-I seesaw mechanism. Since we are taking strongly hierarchical VEVs , we can neglect the off-diagonal entries and the seesaw contribution of , and integrate out using the couplings of the light neutrinos contained in


Light Majorana neutrino masses defined as are given by


where have introduced the VEV ratio


Here we are neglecting contributions from the other heavy neutrinos at and the heavy mixing due to the presence of , but we have checked that including these correction only amounts to tiny corrections to Eq. (2.28), of the order of the naive scale suppression factor . Therefore possible phases in and would not affect the light neutrino sector, while a possible phase in would only lead to an overall phase of the light neutrino mass matrix, and thus only affects Majorana phases, not the Dirac CP phase.

Counting parameters, we see that the neutrino sector depends on a single additional real VEV ratio 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 non-trivial. 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

Table 1: SM input parameters at the electroweak scale, where quark and lepton masses and the quark mixing parameters are taken from Ref. [29], and neutrino mixing parameters from Ref. [30] for Normal Ordering (NO). As explained in the text, we use a 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.

In order to verify whether the 17+1 fermion observables of the Standard Model (see Table 1 for our input) can be successfully reproduced in our model, we have performed a fit of the matrices and the six VEV ratios using Eq. (2.15) and Eq. (2.28), corresponding to 15 real parameters + 1 complex phase. The fit was done using the Metropolis-Hastings algorithm [31, 32] following a top-down approach. The parameters were chosen randomly at and used as boundary conditions for the Yukawa RGEs, which were then solved numerically using REAP [33]. Afterwards, the computed values of the observables at the electroweak scale were compared with the experimental values (following earlier studies, we assume a uncertainty for the charged lepton masses [34, 35], because otherwise a numerical fit would be very challenging). The quality of a fit point is determined by


where denotes the experimental value of the observable , its experimental error and the corresponding fit value.

Despite the overdetermination111There must be two relations involving just SM observables, however due to the highly non-trivial dependence on the fundamental parameters we were not able to find analytical expressions for these relations. we find a perfect fit with for NO in the neutrino sector, with corresponding model parameters at the GUT scale given by


and the fitted standard model fermion observables are summarized in Table 2.

Table 2: Result of the fitting procedure, as described in the text. The pull of a fit value is defined as , where is the corresponding experimental error and the experimental value as given in Table 1.

Using the above fit parameters, we can also make predictions for the neutrino Dirac phase, Majorana phases and neutrino mass observables. Experiments that are sensitive to the absolute neutrino mass scale, like the KATRIN [36], MARE [37] , Project 8 [38], or ECHo [39] experiments, measure the effective mass defined by


In contrast, cosmology probes the sum of neutrino masses and neutrinoless double beta decay experiments, like, for example, the GERDA [40], EXO-200 [41] or KamLAND-Zen [42] experiments, tests the “effective Majorana mass"


To give an estimate for the robustness of these predictions in our model, we numerically considered perturbations around the best fit point that reproduce the standard model fermion observables with . Our predictions and the resulting ranges for the above mass observables and Dirac and Majorana phases are summarized in Table 3. While the mass observables are all far below current and future sensitivity, we obtain a quite narrow range for the Dirac phase , which might be verified or excluded with upcoming data coming from neutrino oscillations, for example at Hyper-Kamiokande [43] or DUNE [44].

[meV] [meV] [meV] [] [] []
Current bound  [45]  [45, 46]  [47, 48] - - -
Table 3: Predicted values and current bounds for the neutrino observables. The current bounds were taken from Ref. [49]. As explained in the text, the ranges shown here correspond to perturbations of the best fit point with .

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. (2.4)) six are determined by the fit and one by the electroweak scale (). From the remaining 5 VEVs, does not affect the neutrino sector given the hierarchy , so for simplicity we set without any impact on the spectrum. We are then left with four VEVs that are free parameters, which we take as . As the two latter VEVs control the mass of heavy gauge bosons, they are bounded from below by proton decay constraints, and we take . The VEV 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 . The remaining scale is bounded from above by neutron EDM constraints, which require that the loop corrections to the effective parameter remain sufficiently small. As we will discuss in the next section, these higher-loop corrections are sufficiently suppressed if .

4 Loop Contributions to

In Nelson-Barr models vanishes at tree-level by construction, but is generated at loop-level 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 . 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 , 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 is enough to render .

In this spirit we work with a simplified Lagrangian before going to the light-heavy mass basis, cf. Eq. (2)


where we also included scalar couplings and all masses and couplings are real symmetric matrices except that is complex symmetric


We are now interested in loop corrections to the Yukawa couplings that we write as


The effective SM Yukawa couplings are given by (cf. Eq. (2.15))


Therefore the effective parameter is


where have used that are real and is hermitian. Expanding the logarithms we finally get


We can now estimate the corrections using a spurion analysis. Under unitary field redefinitions with , the Lagrangian parameters transform as


Since we can treat the masses as insertions, the covariant expressions involving and that will enter the contributions to can only be of the form


where are the heavy gauge boson and scalar masses. These expressions transform as


and are the only quantities that carry complex phases. The heaviest particle in the loop diagram then determines the form of , i.e. diagrams with only Higgs scalars, the “IR" contributions, will involve , diagrams with heavy vectors and diagrams with heavy scalars .

4.1 IR Contributions

For IR diagrams , so one has to sum up all insertions leading effectively to a field redefinition , and thus which is the SM down Yukawa coupling. The spurion analysis is then quite involved, since one has to take into account that the light quark propagators can involve hermitian functions of arising from the integration over loop momenta. In Ref. [18] it was shown that the leading contribution arises at three loop and is given by


where the functions were taken to as hermitian matrices. Note that the strong suppression is due to the fact that there is no mixing with heavy fields in the up and the LH down sector.

4.2 Gauge Contributions

The most dangerous contributions involve the U(1) gauge boson, since all heavy gauge fields live at . The spurion analysis is greatly simplified, because now the heavy gauge boson dominates the loop momentum integration, and thus strongly suppresses the contributions from propagators with non-trivial flavor structure. One can show that all one-loop contributions vanish, and that the leading contribution arises from two-diagrams like Fig. 2 that involve additional Higgs loops.

Figure 2: Example of 2-loop diagram with heavy gauge fields.

An estimate gives


where and are the couplings of the U(1) gauge boson to quarks, is its mass and denotes the heaviest of the heavy vectorlike down quarks. All numerical values depend only on (fixing ). Taking implies and , and taking for simplicity the GUT value of the U(1) gauge coupling , one finds


4.3 Scalar Contributions

Analogously, all one loop diagrams with heavy scalars in the loop vanish, and the leading contribution comes from a 2-loop diagram involving a U(1) and scalar loop, shown in Fig. 3.

Figure 3: Example of 2-loop diagram with heavy scalar fields.

This gives


where and