We study a model with partial quark-lepton universality that can naturally arise in grand unified theories. We find that constraints on the model can be reduced to a single condition on the Dirac CP phase δ in the neutrino sector. Using our current knowledge of the CKM and PMNS mixing matrices, we predict -32.4°≤δ≤32.0° at 2σ.

1. Introduction

Our understanding of neutrinos has progressed steadily in the last two decades. After the observation of nonzero θ13 by the Daya Bay [1, 2], RENO [3], and Double Chooz [4] experiments, we now know the three mixing angles θ12, θ23, and θ13 and the two mass squared differences to good precision. For the normal hierarchy, the current 2σ ranges of the three mixing angles from a global three-neutrino oscillation analysis are [5] (1)θ12=33.7-2.1+2.1°,θ23=41.4-2.6+6.6°,θ13=8.80-0.77+0.73°.

The focus of next generation neutrino oscillation experiments is shifted to the Dirac CP phase δ and the neutrino mass hierarchy. Predictions of many theoretical models designed to explain the observed mixing patterns await verification. Among these models, quark-lepton universality (QLU) [6] is well motivated. It is based on simple relations in grand unified theories (GUT) and connects the mixing matrices of quarks and leptons. Exact quark-lepton universality leads to a symmetric PMNS mixing matrix. However, using the current 3σ ranges of the oscillation parameters [5], we find the moduli of the neutrino mixing matrix elements are (2)VPMNS=0.789-0.8530.501-0.5940.133-0.1720.195-0.5560.410-0.7330.602-0.7840.196-0.5570.411-0.7330.602-0.784.We see that the exactly symmetric PMNS mixing matrix is disfavored by the current data. This aspect of the PMNS matrix with VPMNS=VPMNST or VPMNS=VPMNS† has been studied in [7–9].

In this paper, we discuss partial quark-lepton universality [6], which does not require the unitary matrices that diagonalize the upper and lower components of the weak doublets to be the same. We find that partial QLU fits the current data very well and we can make a prediction for the unknown Dirac CP phase.

In Section 2, we review partial quark-lepton universality and discuss renormalization group effects on the model. In Section 3, we discuss the phenomenological results of this model and predict the Dirac CP phase. We conclude in Section 4.

2. Partial Quark-Lepton Universality

Partial quark-lepton universality can be derived from some simple relations in grand unified theories [6]. We start with the SU(5) relation (3)Ml=MdT,obtainable in lopsided models [10], and (4)Mu=MuT,where Ml, Mu, and Md are the mass matrices of the charged-leptons, up-type quarks, and down-type quarks, respectively. If we assume Md is Hermitian, which can be achieved by imposing left-right symmetry [6]^{1}, then from (3) we find that both the down-type quarks and charged-leptons can be diagonalized by a unitary matrix V(5)V†MdV=Dd,VTMlV∗=Dl.

Also, from (4), we know that the up-type quarks can be diagonalized by a unitary matrix V′(6)V′†MuV′∗=Du.If the Dirac neutrino matrix MνD and the right-handed Majorana neutrino mass matrix MR are also diagonalized by V′ (as in some SO(10) models [6]), (7)V′†MνDV′∗=DνD,V′†MRV′∗=DR,Then, below the seesaw scale, the light neutrino mass matrix, Mν=-MνDMR-1MνDT, is diagonalized by V′ as well. Consider (8)V′†MνV′∗=Dν.From (5), (6), and (8), we can find that the observable mixing matrices are related to(9)VCKM=V′†V,(10)VPMNS=VTV′.

Note that, for exact quark-lepton universality, we must have V′=V, which indicates that VCKM=I and the VPMNS mixing matrix is symmetric. This is disfavored by the current data. In the next section, we show that partial quark-lepton universality is still allowed by current data. A caveat to partial QLU is that small perturbations to the leading order relations of (9) and (10) are needed to reproduce the measured fermion masses. In [6], it was shown that, with a specific form for the perturbations, the measured fermion masses can be obtained while keeping the mixing matrices unchanged. Consequently, we focus on the connection between the mixing matrices of quarks and leptons.^{2}

The current data that determine the CKM and PMNS mixing matrices are measured at low energies, while the quark-lepton universality relations are realized at the grand unification scale. In order to use the current data to analyze the model, we must consider renormalization group (RG) effects. For the CKM matrix, the RG effects are very small; that is, the next order relative corrections to the CKM matrix are of the order λ5 [11, 12], where λ=0.225. The RG effects in the neutrino sector are strongly dependent on the mass spectrum of the light neutrinos. For the inverted and quasi-degenerate mass hierarchies, the effects can be large [13–16]. However, with quark-lepton universality it is more natural to assume that the light neutrinos are very hierarchical with the normal mass spectrum. In this case, RG effects on the three angles are very small [17, 18]; for example, δθ23~0.6∘, δθ13~0.2∘ and δθ12~0.8∘ in the MSSM with tanβ=20 if the lightest neutrino mass is 0.01 eV. Since the current uncertainties in the three angles are larger than the RG effects, we neglect the RG effects in our analysis.

3. Phenomenology

In this section, we introduce a simple approach based on the properties of unitary matrices to reduce the constraints on the model to a single condition, which allows us to easily constrain the Dirac CP phase.

Partial QLU predicts the two observable mixing matrices to have the form of (9) and (10), which can be rewritten as (11)VPMNSVCKM=VTV,(12)VCKM∗VPMNS=V′TV′.Hence, in order for the model to work, both VPMNSVCKM and VCKM∗VPMNS should be symmetric. However, the two constraints are not independent. Since (9) implies V′=VVCKM†, (11) follows from (12).

Solutions for V and V′ will always exist because if VCKM∗VPMNS is symmetric, then it can be diagonalized by a unitary matrix Us; that is, UsTVCKM∗VPMNSUs=D, where D is diagonal. This means that we can find the solution, V′=DUs†. Once V′ is known, the solution for V can be obtained from (9). Although solutions for V and V′ exist, they are not unique. We can always insert a combination of a real rotation matrix RTR into the middle of the right-handed side of (11) or (12). And since RTR=I, the equation will not change. This can also be seen from (9) and (10). For any real rotation matrix R, RV and RV′ are also unitary; hence if we let V→RV and V′→RV′, the two observable mixing matrices will remain the same.

Now, if we define (13)U=VCKM∗VPMNS,then the only constraint from the model is that U is symmetric. Since both VCKM and VPMNS are unitary matrices, U is also unitary. For a 3×3 unitary matrix, it can be shown that U being symmetric is equivalent to the moduli of U being symmetric under phase redefinition [19]. This constraint still imposes three conditions: |U12|=|U21|, |U13|=|U31|, and |U23|=|U32|. However, the conditions are not independent. Since U is unitary, |U11|2+|U12|2+|U13|2=|U11|2+|U21|2+|U31|2. Hence, |U12|=|U21| indicates |U13|=|U31| and vice versa. Similarly, |U23|=|U32| is equivalent to |U13|=|U31|. Therefore, there is only one independent condition that constrains the model. Here, we choose it to be |U13|=|U31|.

The CKM matrix can be written in terms of the Wolfenstein parameters [20] as follows: (14)VCKM=1-λ22λAλ3ρ-iη-λ1-λ22Aλ2Aλ31-ρ-iη-Aλ21+Oλ4,and the PMNS matrix can be written in the standard form, which is (15)VPMNS=c13c12c13s12s13e-iδ-s12c23-c12s23s13eiδc12c23-s12s23s13eiδs23c13s12s23-c12c23s13eiδ-c12s23-s12c23s13eiδc23c13,where cij, sij denote cosθij and sinθij, respectively, and Majorana phases are not included. From (13), we see that the condition |U13|=|U31| becomes (16)1-λ22s13e-iδ+λs23c13+c23c13Aλ3ρ+iη=Aλ31-ρ+iηc13c12+Aλ2s12c23+c12s23s13eiδ+s12s23-c12c23s13eiδ.Note that (16) cannot be satisfied when θ13=0. Keeping in mind that sinθ13<λ, the λ2s132 and λ3s13 terms can be neglected since they are of the same order of magnitude as the terms dropped in the Wolfenstein parametrization. Then, we get a simple expression for the cosine of the Dirac CP phase: (17)cosδ=s122s232+c122c232s132-s132-λ2B2s23c23s12c12s13+2λs23c13s13+2Aλ2s12c12c232-s232s13+Oλ4,where B=s232c132-2As122c23s23-2Aλ(1-ρ)c12s12c13s23. We see that for very small θ13 the numerator of the above equation is always larger than the denominator, so that there is no solution for δ.

Using the currently favored CKM [21] and PMNS [5] parameters with their respective uncertainties and solving the condition |U13|=|U31| numerically without any approximation, we find that the Dirac CP phase δ in the PMNS matrix lies between -32.4∘ and 32.0∘ at 2σ. The asymmetry around 0 is due to the small CP violation in the CKM matrix, which does not enter the approximate result in (17).

We also find predictions for each mixing angle versus δ given the best-fit values and 2σ allowed regions of the other two mixing angles and the CKM parameters. The results are shown in Figure 1. With the constraints from the other two mixing angles and the CKM parameters, we find that θ23<48.3∘, θ12<36.3∘ and θ13>7.64∘ at 2σ. The partial QLU model is perfectly consistent with the current data, and rather large θ13 is strongly favored for the measured solar and atmospheric mixing angles. Note that the relevant neutrino mass squared differences are trivially accommodated.

The 2σ allowed regions (shaded bands) in the (δ, θ13), (δ, θ12), and (δ, θ23) planes using measurements (with uncertainties) of the other two neutrino mixing angles and the CKM parameters. The solid curves within the shaded bands are the model predictions for the best-fit values of the other two mixing angles and the CKM parameters. The horizontal solid lines mark the best-fit values and the horizontal dashed lines mark the 2σ limits of θ23, θ12, and θ13.

A measurement of δ by future long baseline neutrino oscillation experiments will provide a stringent test of the viability of the partial quark-lepton universality model.

4. Conclusion

We studied partial quark-lepton universality, which can naturally arise in grand unified theories. Constraints on the model can be reduced to one simple condition, |U13|=|U31|. Dropping terms of order λ4 from this condition, we find a simple expression for the Dirac CP phase δ in the neutrino sector. We also studied the allowed parameter regions of the model numerically. Our prediction that δ lies within the range [-32.4∘,32.0∘] at the 2σ level will be tested by future long baseline neutrino experiments.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This research was supported by the U.S. Department of Energy Grant no. DE-SC0010504.

Endnotes

Implementing a Hermitian Md in a GUT is difficult because SU(5) does not incorporate left-right symmetry, and, in SO(10), the mass matrices arising from the couplings of fermions to Higgs fields in the 10 and 126 representations are complex symmetric (and not Hermitian), while those arising from couplings to 120 are complex antisymmetric.

An example in which (3), (4), (7), and the Hermiticity of Md naturally arise is an SO(10) scheme with the superpotential terms [6]∗Wd=fijM16iTBΓμHH′TBΓμ16j+fij′M16iTBΓμH′HTBΓμ16j,Wu=gij16iTBΓμνλσρ16jΦμνλσρ,where H, H′ are 16-plet Higgs, Φ is a 126¯-plet Higgs, B is a charge conjugation matrix in SO(10), i and j are generation indices, and μ, ν, λ, σ, and ρ are SO(10) indices. The Lorentz indices and the standard charge conjugation matrix are suppressed. H and H′ contain neutral fields with the quantum numbers of ν and νc, so that the vacuum expectation value for νc breaks SO(10) while SU(5) is preserved. We take the 126¯ contribution to Hd to be zero or subdominant compared to H and H′, so Md is only generated from Wd. By imposing an additional symmetry, 16→16∗,H→H′∗, which leads to fij→fij′∗, a Hermitian Md can be obtained.

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