Partial quark-lepton universality and neutrino CP violation

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 $\delta$ in the neutrino sector. Using our current knowledge of the CKM and PMNS mixing matrices, we predict $-32.4^\circ<\delta<32.0^\circ$ at $2\sigma$.


Introduction
Our understanding of neutrinos has progressed steadily in the last two decades. After the observation of nonzero θ 13 by the Daya Bay [1], RENO [2], and Double Chooz [3] experiments, we now know the three mixing angles θ 12 , θ 23 , and θ 13 , and the two mass squared differences to good precision. For the normal hierarchy, current 2σ ranges of the three mixing angles from a global three-neutrino oscillation analysis are [4] The focus of next generation neutrino oscillation experiments is shifted to the Dirac CP phase δ and the neutrino mass hierarchy. Predictions of the many theoretical models designed to explain the observed mixing patterns await verification. Among these models, quark-lepton universality (QLU) [5] 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 [4], we find the moduli of the neutrino mixing matrix elements are We see that the exactly symmetric PMNS mixing matrix is disfavored by current data. This aspect of the PMNS matrix with V P M N S = V T P M N S or V P M N S = V † P M N S has been studied in Ref. [6].
In this Letter, we discuss partial quark-lepton universality [5], which does not require the unitary matrices that diagonalize the upper and lower components of the weak doublets to be the same. Partial QLU can naturally arise in SO(10) models while exact quark-lepton universality may require a larger quark-lepton symmetry than is contained in SO (10). 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.

Partial quark-lepton universality
Partial quark-lepton universality can be derived from some simple relations in grand unified theories [5]. The minimal version of SU(5) implies and where M l , M u , and M d are the mass matrices of the charged-leptons, up-type quarks and down-type quarks, respectively. If we assume M d is Hermitian, which can be achieved by imposing left-right symmetry, then from Eq. (3) we know that both the down-type quarks and charged-leptons can be diagonalized by a unitary matrix V , Also, in some SO(10) models, we have where M νD is the Dirac neutrino mass matrix, M R is the right-handed Majorana neutrino mass matrix, and α and β are proportionality constants; then from Eq. (4) we see that both the upper components of the weak doublets and the right-handed neutrinos can be diagonalized by a unitary matrix V ′ , Below the seesaw scale, the light neutrino mass matrix, From Eqs. (5), (7) and (8), we can find that the observable mixing matrices are related by and Note that this ansatz is based on the simple assumptions Eqs. (3), (4), (6) and the Hermiticity of M d . These conditions can naturally arise in an SO(10) scheme with the superpotential terms [5], and where H, H ′ are 16-plet Higgses, 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, ρ are SO (10)  For exact quark-lepton universality, we must have V ′ = V , which requires a large quark-lepton symmetry that is not easily realized in SO(10) [5]. Also, the relation V ′ = V indicates that V CKM = I and the V P M N S mixing matrix is symmetric, which is disfavored by current data. In the next section, we show that partial quark-lepton universality is still allowed by current data.
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 current data to analyze the model, we must consider renormalization group (RG) effects. For the CKM matrix, the RG effects are very small, i.e., the next order relative corrections of the CKM matrix are of the order λ 5 [7], 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 [8]. However, from Eq. (6) we know that quark-lepton universality implies that the light neutrinos are very hierarchical with the normal mass spectrum. In this case, RG effects on the three angles are very small [9], e.g., δθ 23 ∼ 0.6 • , δθ 13 ∼ 0.2 • and δθ 12 ∼ 0.8 • in the MSSM if the lightest neutrino mass is 0.01 eV. Since current uncertainties in the three angles are larger than the RG effects, we neglect the RG effects in our analysis.

Phenomenology
Partial QLU predicts the two observable mixing matrices to have the form of Eqs. (9) and (10), which can be rewritten as and Hence, in order for the model to work, both V P M N S V CKM and V * CKM V P M N S should be symmetric. Solutions for V and V ′ will always exist because if V * CKM V P M N S is symmetric, then it can be diagonalized by a unitary matrix U s , i.e., U T s V * CKM V P M N S U s = D, where D is diagonal. This means that we can find the solution, V ′ = √ DU † s . However, the two constraints are not independent. If V P M N S V CKM is symmetric, then by diagonalizing it we can easily determine V , and then if we define V ′ = V V † CKM , it follows that V ′T V ′ = V * CKM V P M N S , and V * CKM V P M N S would be symmetric, too. Although solutions for V and V ′ exist, they are not unique. We can always insert a combination of a real rotation matrix R T R into the middle of the right-handed side of Eq. (13) or (14), and since R T R = I, the equation will not change. This can also be seen from Eqs. (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 stay the same.
The CKM matrix can be written in terms of the Wolfenstein parameters [11] as, and the PMNS matrix can be written in the standard form, which is Keeping in mind that sin θ 13 < λ, the λ 2 s 2 13 and λ 3 s 13 terms van 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: cos δ = s 2 12 s 2 23 + c 2 12 c 2 23 s 2 13 − s 2 13 − λ 2 B 2s 23 c 23 s 12 c 12 s 13 + 2λs 23 c 13 s 13 + 2Aλ 2 s 12 c 12 (c 2 23 − s 2 23 )s 13 where B = s 2 23 c 2 13 − 2As 2 12 c 23 s 23 − 2Aλ(1 − ρ)c 12 s 12 c 13 s 23 . We see that for very small θ 13 the numerator of the above equation is always larger than the denominator, so that there is no solution for δ. Hence partial quark-lepton universality prefers rather large θ 13 .
Using the currently favored CKM [12] and PMNS [4] parameters with their respective uncertainties, and solving the condition |U 13 | = |U 31 | numerically without any approximation, we find the Dirac CP phase δ in the PMNS matrix to lie 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 Eq. (19).
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 Fig. 1. With the constraints from the other two mixing angles and the CKM parameters, we find that θ 23 < 48.3 • , θ 12 < 36.3 • and θ 13 > 7.64 • at 2σ. The partial QLU model is perfectly consistent with current data, and rather large θ 13 is strongly favored for the measured solar and atmospheric mixing angles.
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.

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, |U 13 | = |U 31 |. Dropping terms of order λ 4 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.