Nontrivial Correlation between the CKM and MNS Matrices

We point out that the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix $V_{\rm CKM}$ and the Maki-Nakagawa-Sakata (MNS) lepton mixing matrix $V_{\rm MNS}$ can naturally be correlated in a class of seesaw models with grand unification, but the texture of their correlation matrix ${\cal F}_\nu$ is rather nontrivial. The bimaximal mixing pattern of ${\cal F}_\nu$ is disfavored by current data, and other special forms of ${\cal F}_\nu$ may suffer from fine-tuning of the free phase parameters in fitting the so-called quark-lepton complementarity relation. A straightforward calculation of ${\cal F}_\nu$ in terms of $V_{\rm CKM}$ and $V_{\rm MNS}$ reveals a striking feature of ${\cal F}_\nu$: its (1,3) element cannot be zero or too small, no matter whether the (1,3) elements of $V_{\rm CKM}$ and $V_{\rm MNS}$ are vanishing or not. We also add some brief comments on possible radiative corrections to $V_{\rm CKM}$ and $V_{\rm MNS}$.

1 Recent solar [1], atmospheric [2], reactor (KamLAND [3] and CHOOZ [4]) and accelerator (K2K [5]) neutrino oscillation experiments have provided us with very robust evidence that neutrinos are massive and lepton flavors are mixed. The phenomenon of lepton flavor mixing can be described by a 3 × 3 unitary matrix V MNS , commonly referred to as the Maki-Nakagawa-Sakata (MNS) matrix [6]. Current experimental data indicate that V MNS involves two large angles (θ 12 ∼ 33 • and θ 23 ∼ 45 • ) and one small angle (θ 13 < ∼ 9 • ) in the standard parametrization. The magnitude of θ 13 remains unknown, but a global analysis of the presently available neutrino oscillation data [7] hints that θ 13 ∼ 3 • seems to be most likely. On the other hand, three nontrivial CP-violating phases of V MNS (denoted by δ MNS , ρ and σ) are entirely unrestricted. One particularly important target of the future neutrino experiments is just to measure θ 13 and δ MNS (of Dirac type) and to constrain ρ and σ (of Majorana type).
In comparison with the MNS matrix V MNS , the Cabibbo-Kobayashi-Maskawa (CKM) quark flavor mixing matrix V CKM involves three small mixing angles and one large CPviolating phase (ϑ 12 ≈ 13 • , ϑ 23 ≈ 2.4 • , ϑ 13 ≈ 0.2 • and δ CKM ∼ 65 • in the standard parametrization [8]). The apparent difference between the CKM and MNS matrices requires a good dynamical reason in a fundamental theory of flavor mixing and CP violation, in particular when an underlying lepton-quark symmetry is concerned. Some phenomenological speculation about possible relations between θ ij and ϑ ij , such as [9][10][11] and [9] 1 has appeared 2 . If such relations could survive the test with more accurate experimental data in the near future, would they be just accidental or imply a kind of lepton-quark symmetry? The latter is certainly attractive to theorists, although it remains unclear what symmetry exists between leptons and quarks. The main purpose of this paper is to investigate a simple but natural relation between V MNS and V CKM for a large class of seesaw models with grand unification, in which F ν denotes a unitary matrix associated with the diagonalization of the effective neutrino mass matrix M ν [12]. We shall show that it is phenomenologically disfavored for F ν to take the bimaximal mixing form, even if the nontrivial phase effects in F ν are taken into account. This observation is contrary to some previous arguments that the CKM matrix might just measure the deviation of the MNS matrix from exact bimaximal mixing [9,10,13].
Furthermore, we point out that a slight modification of the bimaximal mixing pattern of F ν will allow us to reproduce the so-called quark-lepton complementarity relation in Eq. (1), but the fine-tuning of relevant unknown parameters seems unavoidable. A straightforward calculation of F ν in terms of the mixing angles and CP-violating phases of V CKM and V MNS reveals a striking feature of F ν : its (1,3) element cannot be vanishing or too small, no matter whether θ 13 and ϑ 13 are taken to be zero or not. Therefore we conclude that the texture of F ν is rather nontrivial in the seesaw models. We also make some brief comments on possible quantum corrections to fermion mass matrices and their threshold effects on the flavor mixing parameters that are related to one another by Eq. (3).
them are symmetric [17]. We are then left with V D = U * D = V u = U * u and V l = U * l = V d = U * d , as well as m x = m u , m y = m c and m z = m t . A relation between the MNS matrix V MNS and the CKM matrix V CKM turns out to be where With the help of the seesaw relation in Eq. (6), the role of F ν can be seen clearly: In other words, the unitary matrix F ν transforms the combination M u V T u M −1 R V u M u into the diagonal (physical) mass matrix of three light neutrinos. Given F ν and V u , the heavy Majorana neutrino mass matrix M R is determined by This inverted seesaw relation indicates that the mass scale of three right-handed neutrinos is roughly m 2 t /m 3 , if the light neutrino mass spectrum is essentially hierarchical. We argue that Eq. (8) is a natural relation between CKM and MNS matrices in a class of seesaw models with lepton-quark unification. It will not be favored by current neutrino oscillation data, however, if F ν takes the bimaximal mixing pattern.
3 Because F ν depends on both M R and V u through Eq. (9), it is in general a complex unitary matrix. The most generic form of F ν with bimaximal mixing can be written as where P ν ≡ Diag{e iφ x , e iφ y , e iφ z } and Q ν ≡ Diag{e iφ 1 , e iφ 2 , e iφ 3 } are two phase matrices. Taking account of Eq. (8), one may choose two independent phases (or their combinations) of Q ν as the Majorana-type CP-violating phases of V MNS . On the other hand, the phases of P ν can affect both the mixing angles and the Dirac-type CP-violating phase of V MNS . To see this point in a more transparent way, we make use of the standard parametrizations of V CKM and V MNS [18], in which R ij denotes the rotation matrix in the (i, j)-plane with the mixing angle θ ij or ϑ ij (for ij = 12, 23, 13), and Γ δ (δ) ≡ Diag{1, 1, e iδ } is a phase matrix consisting of the Dirac phase of CP violation. When Eq. (12) is applied to Eq. (8), a proper rephasing of three charged lepton fields has been implied for the sake of phase consistency on the two sides of Eq. (8) 4 . We are then allowed to calculate three mixing angles of V MNS in terms of those of V CKM . The approximate results are 5 where the strong hierarchy of three quark mixing angles has been taken into account. In addition, the Jarlskog invariant of CP violation [19] defined for V MNS is given by to the leading order, only if the phase difference (φ y − φ x ) is not too close to 0 or ±π. Then the maximally-allowed magnitude of J MNS is |J MNS | ≈ 4% for (φ y − φ x ) ≈ ±π/2. But this special phase condition is not favored by Eq. (13), because it will lead to tan θ 12 ≈ 1 or θ 12 ≈ 45 • . Taking account of J MNS = sin θ 12 cos θ 12 sin θ 23 cos θ 23 sin θ 13 cos 2 θ 13 sin δ MNS , we are then able to determine the Dirac phase δ MNS from Eqs. (13) and (14). We obtain δ MNS ≈ (φ y − φ x ) in the leading-order approximation. Therefore the phase matrix P ν of F ν , which may significantly affect θ 12 and δ MNS of V MNS , should not be ignored. Two more comments on the consequences of Eqs. (13) and (14) are in order.
(a) At first glance, it is absolutely impossible to reproduce the empirical relations given in Eqs. (1) and (2) from Eq. (13), simply because of the existence of unknown (φ x , φ y , φ z ) phases. It is worth emphasizing that the special assumption φ x = φ y = φ z = 0 taken in the literature is not justified, since F ν is naturally expected to be complex in the seesaw models. For this reason, we argue that Eqs. (1) and (2) are most likely to be a numerical accident.
(b) Even if φ x = φ y = φ z = 0 is assumed, it will be difficult to straightforwardly derive Eq. (1) from Eq. (13). In this case, the expression of tan θ 12 in Eq. (13) is simplified to 4 A general parametrization of V CKM should include two (unobservable) phase matrices on its left-hand side (denoted as P ) and its right-hand side (denoted as Q). Taking account of Eq. (8), we find that P † can be absorbed into P ν of F ν , while Q † can be removed by a redefinition of the non-physical phases of three charged lepton fields. Therefore, the standard parametrizations of V CKM and V MNS given in Eq. (12) together with the most generic form of F ν taken in Eq. (11) are consistent with Eq. (8) and the Lagrangian of lepton and quark Yukawa interactions. 5 To leading order in our approximation, another (unobservable) phase combination (φ z − φ y ) does not appear in Eqs. (13) and (14).
Typically taking ϑ 12 ≈ 13 • , we obtain θ 12 ≈ 35 • from Eq. (15). In addition, θ 23 ≈ 44 • and θ 13 ≈ 9 • can be obtained from Eq. (13). It turns out that the sum of θ 12 and ϑ 12 amounts to 48 • , which is not perfectly in agreement with Eq. (1). On the other hand, the prediction θ 13 ≈ 9 • is too close to the present experimental upper limit and far away from the best-fit result (namely, θ 13 ∼ 3 • [7]). Note also that the assumption φ x = φ y = φ z = 0 leads to tiny CP violation in the lepton sector: where only the leading term has been given. Numerically, |J MNS | ∼ 5 × 10 −4 , too small to be measured in the future long-baseline neutrino oscillation experiments [20].
In order to successfully achieve θ 12 + ϑ 12 ≈ 45 • from V MNS = V † CKM F ν , a straightforward (and somehow trivial) way is to modify the bimaximal mixing pattern of F ν . For illustration, let us consider the following form of F ν : where c ≡ cos θ and s ≡ sin θ with θ being an unspecified angle. Eq. (11) can obviously be reproduced from Eq. (17) by taking θ = 45 • . Taking account of the correlation between V MNS and V CKM in Eq. (3) or (8), we now arrive at in the leading-order approximation. The results for tan θ 23 and sin θ 13 obtained in Eq. (13) keep unchanged. The relation θ 12 + ϑ 12 ≈ 45 • , if it holds, implies a kind of correlation between θ and (φ y − φ x ) as restricted by Eq. (18). To be more concrete, we find that this correlation reads as cos(φ y − φ x ) ≈ 2 tan ϑ 12 − cos 2θ √ 2 tan ϑ 12 sin 2θ (19) to the lowest order. Eq. (19) is numerically illustrated in Fig. 1(A), where ϑ 12 ≈ 13 • has typically been taken. One can clearly see that the value of θ is sensitive to that of (φ y − φ x ) and the possibility of θ = 45 • has been ruled out. If θ = 30 • [21] or θ ≈ 35.3 • [22] is assumed, then the value of (φ y − φ x ) must be fine-tuned to about 100 • or 70 • . In these two cases, the magnitude of J MNS can be as large as about 3%, as shown in Fig. 1(B). We conclude that current experimental data do not favor the bimaximal mixing pattern of F ν taken for Eq. (3) or (8), no matter whether F ν is real or complex. The numerical relation θ 12 + ϑ 12 ≈ 45 • is achievable, however, if F ν is allowed to slightly deviate from the bimaximal mixing form and its free parameters can properly be fine-tuned. Of course, such a phenomenological approach is not well motivated from a theoretical point of view.
4 What is the most favorite pattern of F ν in phenomenology? One can straightforwardly answer this question by calculating F ν in terms of V CKM and V MNS through Eq. (3) or (8); namely, F ν = V CKM V MNS . When the standard parametrization of V CKM or V MNS in Eq. (12) is used, one should take account of the arbitrary but nontrivial phases between V CKM and V MNS . In this case 6 , where Ω ν ≡ Diag{e iδ 1 , e iδ 2 , e iδ 3 } is the relative phase matrix between the CKM and MNS matrices. To be more specific, we fix (ϑ 12 , ϑ 23 , ϑ 13 ) ≈ (13 • , 2.4 • , 0.2 • ) [8] and (θ 12 , θ 23 , θ 13 ) ≈ (33 • , 45 • , 3 • ) in our numerical calculation of F ν . It proves very instructive and convenient to expand the matrix elements of V CKM and V MNS in powers of a small parameter λ ≡ sin ϑ 12 : in which only the leading term of each matrix element (except the (3,1) elements of V MNS ) is shown. From Eqs. (20) and (21), we obtain where δ ij ≡ δ i − δ j (for i, j = 1, 2, 3) has been defined. One can see that the CP-violating phase δ CKM does not play a role in this approximation. The most striking feature of F ν is that its (1,3) element does not vanish, unlike the ansatz proposed in Eq. (11) or (17). This result remains valid even in the θ 13 = 0, ϑ 13 = 0 or θ 13 = ϑ 13 = 0 case, simply because |(F ν ) 13 | is dominated by the product |(V CKM ) us (V MNS ) µ3 | = sin ϑ 12 sin θ 23 cos ϑ 13 cos θ 13 ≈ λ 1.23 , as already shown in Eq. (22). Allowing the unknown phase parameters δ MNS , δ 21 and δ 32 to vary between 0 and 2π, we arrive at the possible ranges of nine matrix elements of F ν : 0.72 · · · 0.90 0.39 · · · 0.66 0.11 · · · 0.21 0.18 · · · 0.56 0.45 · · · 0.70 0.66 · · · 0.72 0.35 · · · 0.42 0.57 · · · 0.62 0.68 · · · 0.74 Note here that the ranges given above are for the individual matrix elements. The choice of a specific value for one element may restrict the magnitudes of some others, because they are related to one another by the unitarity conditions of F ν .
Eqs. (22) and (23) illustrate that the correlation between V MNS and V CKM is rather nontrivial. In particular, the correlation matrix F ν is neither real nor bimaximal. Such a result is of course not a surprise, because Eq. (9) has clearly indicated that F ν should not have a too special pattern. Considering the fact that F ν consists of one small rotation angle in its (1,3) sector and two large rotation angles in its (1,2) and (2,3) sectors, we expect that the texture of M R must be very nontrivial too.
In the above discussions, we did not take into account possible quantum corrections to relevant physical parameters between the scale of grand unified theories Λ GUT and the low energy scales Λ LOW at which the mixing angles and CP-violating phases of V CKM and V MNS can experimentally be determined. One may roughly classify such renormalization effects into three categories in the afore-mentioned seesaw models with grand unification: • The first category is about radiative corrections to the fermion mass and flavor mixing parameters between the GUT scale Λ GUT and the seesaw scale M 1 , where M 1 denotes the mass of the lightest right-handed neutrino. Typically, Λ GUT ∼ 10 15···16 GeV and M 1 ∼ 10 8···12 GeV hold in most model-building cases [23]. Because three right-handed neutrinos are in general expected to have a mass hierarchy (M 1 < M 2 < M 3 ), the threshold effects in the renormalization chain Λ GUT → M 3 → M 2 → M 1 are likely to modify the parameters of neutrino masses and lepton flavor mixing in a significant way [24]. The explicit estimation of such radiative corrections involves a number of free parameters [25], hence its arbitrariness or uncertainty is essentially out of control. In contrast, quark flavor mixing is expected to be insensitive to such seesaw threshold effects.
• From the seesaw scale M 1 to the electroweak scale Λ EW (∼ 10 2 GeV), the one-loop renormalization group equation of the effective neutrino mass matrix M ν consists of the contributions from gauge interactions, quark Yukawa interactions and chargedlepton Yukawa interactions [26]. Only the last contribution may affects V MNS , but the quantitative corrections to lepton mixing angles and CP-violating phases are strongly suppressed in the standard model and in the supersymmetric standard model with small tan β (for simplicity, we have assumed that the scale of supersymmetry breaking is not far away from Λ EW ). The running behavior of V CKM from M 1 to Λ EW is in general dominated by the t-quark, b-quark and τ -lepton Yukawa couplings. To be explicit, ϑ 23 , ϑ 13 and δ CKM are sensitive to a drastic change of energy scales, but ϑ 12 is not [27].
• Within the standard model, the Λ EW threshold effect is negligibly small for both leptons and quarks. Thus it is unnecessary to take into account the radiative correction to V MNS and V CKM from Λ EW to Λ LOW . However, the Λ EW threshold effect may be important in the supersymmetric case [26]; e.g., it can be induced by the slepton mass splitting, which is possible to dominate over the contribution from the chargedlepton Yukawa couplings. A quantitative calculation of this threshold effect on V MNS is strongly model-dependent and arbitrary [11], because it involves some unknown parameters of supersymmetric particles.
It is certainly a challenging task to examine how the relation V MNS = V † CKM F ν gets modified from the afore-listed radiative corrections, unless a realistic seesaw model is specifically given and its free parameters are reasonably assumed. In this sense, we argue that the texture of F ν is unlikely to take a trivial form (such as the real bimaximal pattern) and the interesting relations in Eqs. (1) and (2) are more likely to be a numerical accident.
5 In summary, we have investigated the natural correlation between the MNS lepton mixing matrix V MNS and the CKM quark mixing matrix V CKM for a class of seesaw models with grand unification. The correlation matrix F ν is found to be phenomenologically disfavored to take the bimaximal mixing form, no matter whether the nontrivial phases of F ν are taken into account or not. This observation turns out to be contrary to some previous arguments that the CKM matrix might measure the deviation of the MNS matrix from exact bimaximal mixing. We have also shown that a slight modification of the bimaximal mixing pattern of F ν may allow us to reproduce the quark-lepton complementarity relation, provided the relevant phase parameters get fine-tuned. Calculating F ν directly in terms of the mixing angles and CP-violating phases of V CKM and V MNS , we have demonstrated a striking feature of F ν : its (1,3) element cannot be vanishing or too small, even if the (1,3) elements of V MNS and V CKM are taken to be zero. It is therefore concluded that the texture of F ν is rather nontrivial in the seesaw models. We argue that this conclusion is in general not expected to be changed by possible quantum corrections and their threshold effects.  1. (A) numerical illustration of the correlation between θ and (φ y − φ x ) for θ 12 + ϑ 12 ≈ 45 • to hold; (B) numerical dependence of J MNS on θ. Here ϑ 12 ≈ 13 • has been input. The star and diamond points stand for the neutrino mixing ansatz proposed in Ref. [21] with θ = 30 • and that in Ref. [22] with θ ≈ 35.3 • , respectively.