Prediction on Neutrino Dirac and Majorana Phases and Absolute Mass Scale from the CKM Matrix

In Type-I seesaw model, the lepton flavor mixing matrix (PMNS matrix) and the quark flavor mixing matrix (CKM matrix) may be connected implicitly through a relation between the neutrino Dirac Yukawa coupling $Y_D$ and the quark Yukawa couplings. In this paper, we study whether $Y_D$ can satisfy, in the flavor basis where the charged lepton Yukawa and right-handed neutrino Majorana mass matrices are diagonal, the relation $Y_D \propto {\rm diag}(y_d,y_s,y_b)V_{CKM}^T$ or $Y_D \propto {\rm diag}(y_u,y_c,y_t)V_{CKM}^*$ without contradicting the current experimental data on quarks and neutrino oscillations. We search for sets of values of the neutrino Dirac CP phase $\delta_{CP}$, Majorana phases $\alpha_2,\alpha_3$, and the lightest active neutrino mass that satisfy either of the above relations, with the normal or inverted hierarchy of neutrino mass. In performing the search, we consider renormalization group evolutions of the quark masses and CKM matrix and the propagation of their experimental errors along the evolutions. We find that only the former relation $Y_D \propto {\rm diag}(y_d,y_s,y_b)V_{CKM}^T$ with the normal neutrino mass hierarchy holds, based on which we make a prediction for $\delta_{CP},\,\alpha_2,\,\alpha_3$ and the lightest active neutrino mass.

The two flavor mixing matrices, i.e. Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix for leptons [1,2] and Cabbibo-Kobayashi-Maskawa (CKM) matrix for quarks [3,4], are seemingly irrelevant to each other, since the former includes two large mixing angles θ 23 ∼ 45 • and θ 12 ∼ 30 • , while the mixing angles of the latter are all below 15 • . However, if Type-I seesaw mechanism [5] is operative, there can be a connection between them, because the right-handed neutrino Majorana mass matrix that enters into the seesaw mass formula distorts the flavor structure of the neutrino Dirac Yukawa coupling, so that the active neutrino mass matrix (in the basis where the charged lepton mass is diagonal) may have large mixings even when the neutrino Dirac Yukawa coupling (in the same basis) only contains small mixings.
In this paper, we consider the Standard Model (SM) extended with Type-I seesaw mechanism, for which the Yukawa interaction and Majorana mass terms read where Y u , Y d , Y e and Y D denote the up-type quark and down-type quark, charged lepton and neutrino Dirac Yukawa couplings, respectively, and i, j = 1, 2, 3 are flavor indices. The mass matrix for active neutrinos, (m ν ) ij , is derived as On the other hand, in the flavor basis where the charged lepton mass is diagonal, (m ν ) ij is parametrized in terms of the PMNS matrix, U P M N S , and the active neutrino masses, m 1 , m 2 , m 3 , as either of which holds at one renormalization scale µ located somewhere between the TeV scale and the Planck scale. Here, y d , y s , y b , y u , y c , y t denote the Yukawa couplings (taken to be real positive) for d, s, b, u, c, t quarks, respectively. z is an unspecified complex number and φ 2 , φ 3 , ψ 2 , ψ 3 are unspecified phases. We do not center on the different hypotheses in which . This is because the combinations V CKM diag(y d , y s , y b ) and V † CKM diag(y u , y c , y t ) enter into the Yukawa couplings Y d and Y u , respectively, and the corresponding operatorsq L iσ 2 H * u R andq L H d R have the same chirality as the operatorl L iσ 2 H * ν R that is associated with Y D . This nice feature is spoiled if we take the complex conjugation of one side of Eq. (4) and/or Eq. (5).
The hypothesis Eqs. (4,5) leads to the following relation between the PMNS and CKM matrices: If Eq. (4) holds, we obtain where M N j (j = 1, 2, 3) are the components of the diagonalized Majorana mass matrix M N , and v ≃ 246 GeV. If Eq. (5) holds instead, we find Experimentally, the Dirac phase δ CP , the Majorana phases α 2 , α 3 and the absolute scale of the active neutrino mass have not been measured conclusively. Eq. (6) or (7) hence contains 12 real undetermined variables, which are where it should be noted that z, M N i and ψ i appear only in the above combination. On the other hand, Eq. (6) or (7) yields 6 complex equations, since both sides are complex symmetric matrices. Therefore, Eq. (6) or (7) can, in principle, fix the 12 undetermined variables. As a matter of fact, some undetermined variables are phases and hence it is highly non-trivial that the solution to Eq. (6) or (7) exists. In the rest of paper, we study whether the solution to Eq. (6) or (7) exists for the normal (m 3 > m 2 > m 1 ) and inverted hierarchy (m 2 > m 1 > m 3 ) of the active neutrino mass, and if it does, we draw a prediction for δ CP , α 2 , α 3 and the lightest active neutrino mass. We pay attention to the fact that some of the quantities that enter into Eqs. (6,7) are subject to sizable experimental errors, which causes ambiguity in the solution.
We also note that different solutions may be obtained depending on the scale at which Eq. (4) or (5) holds, due to renormalization group (RG) evolutions of the quark Yukawa couplings and CKM matrix. Therefore, we scrutinize their RG evolutions and how the experimental errors of the quark masses, mixing angles and Kobayashi-Maskawa phase propagate along the evolutions.
In contrast, we directly use the values of neutrino mixing angles and mass differences measured in neutrino oscillation experiments; this is justified by assuming that all the components of Y D are much below 1, or equivalently |z| ≪ 1, so that terms like Y D Y † D Y D in RG equations are negligible and the RG evolutions change Y D only by an overall constant that can be absorbed into the number z in Eqs. (4,5).
Our calculation of RG evolutions of quark Yukawa couplings and V CKM proceeds by the following steps. All the renormalization scales are in MS scheme.
(I) Below a scale µ EW ∼ M Z , we work in 5 or 4-flavor QCD×QED theory (decoupling of b at µ = m b (m b ) is properly taken into account [6]). We solve QCD 3-loop and QED 1-loop RG equation [7] for QCD coupling α s in the range µ EW > µ > 2 GeV and that for QED coupling α em in the range µ EW > µ > 13 GeV (we ignore QED effects below 13 GeV), with the initial values of α (II) We solve QCD 3-loop and QED 1-loop RG equation [8] for u, d, s, c, b quark masses up to µ = µ EW . For average u-d mass and s mass, we quote the results of lattice calculations [9,10],   (III) We match 5-flavor QCD×QED theory with the full SM at µ = µ EW . For t quark mass, we adopt the pole mass obtained from the exclusive t pair production cross section at the LHC [12], M t = 173.7 +2.3 −2.1 GeV, and for W, Z and Higgs boson masses and G F , we use Particle Data Group values [13]. We evaluate QCD 2-loop threshold corrections of t quark on α s [6] and QED 1-loop threshold corrections of t quark and W boson on α em [14] to obtain QCD and QED gauge couplings in the SM. We employ the results of Refs. [15,16,17,18,19] implemented in the code [20] to compute the t quark Yukawa coupling y t (µ EW ) with QCD 4-loop and QED 2-loop threshold corrections, and to compute the Higgs quartic coupling λ H (µ EW ), running Higgs vacuum expectation value (VEV) v(µ EW ) and running weak mixing angle sin 2 θ W (µ EW ) with QED 2-loop and QCD 1-loop threshold corrections. We reconstruct the CKM matrix V CKM from up-to-date values of the Wolfenstein parameters reported by the CKMfitter [22], Finally, we derive the running Yukawa matrices for quarks by neglecting threshold corrections for the CKM matrix as Although insignificant in our analysis, we further derive the running Yukawa matrix for charged leptons from the Particle Data Group values of lepton masses, by adding 1-loop threshold corrections and then dividing them by the running Higgs VEV v(µ EW ).
(V) At various scales µ, we derive the running Yukawa couplings (taken to be real positive), , and the running CKM matrix, V CKM (µ), in the following manner. We diagonalize the Yukawa matrices at scale µ as where V uL (µ), V dL (µ) are unitary matrices depending on µ. Then we calculate V CKM (µ) = V uL (µ)V † dL (µ), and further decompose it into physical three mixing angles, θ ckm ij (µ), and one CP phase, δ km (µ), as We estimate uncertainties of the Yukawa couplings and CKM matrix at each scale µ as follows: • For each running Yukawa coupling y i (µ) (i = u, c, t, d, s, b), we consider the propagation of the experimental error of its corresponding mass only and estimate its uncertainty, where we take ∆M t = 2.3 GeV.
Using the running quark Yukawa couplings, running CKM matrix, neutrino mass differences and mixing angles obtained above, we search for the solution to Eq. (6) or (7), and further estimate its uncertainty that stems from experimental errors of the quark masses, Wolfenstein parameters, and neutrino oscillation parameters. We test four cases, and then taking the following steps (the procedures are analogous for Cases(B),(C),(D)): (i) We select a set of 'input values' of y d , y s , y b and θ 12 , θ 13 , θ 23 , |∆m 2 32 |, ∆m 2 21 from the 2σ range. For y d , y s , y b , we quote the values at µ = 10 18 GeV evaluated by taking µ EW = M Z in Table 1 (ii) We randomly generate a set of 'trial values' of the neutrino Dirac CP phase and Majorana CP phases and the logarithm of the lightest neutrino mass, (δ CP , α 2 , α 3 , log(m 1 )), which vary in the following range: If the above inequalities all hold, the corresponding set of 'trial values' (δ CP , α 2 , α 3 , log(m 1 )) is regarded as a solution to Eq. (6). The results are as follows.
• In Case(A), we have generated 1.6 × 10 10 random sets of 'trial values' of (δ CP , α 2 , α 3 , log(m 1 )) for each set of 'input values', and found solutions to Eq. (6) for sin 2 θ 23 = 0.43, 0.47, whereas no solution is found for sin 2 θ 23 = 0.51, 0.55, 0.59. The value of δ CP in the solutions exhibits a correlation with θ ckm 13,test , so we plot the solutions on the plane of (θ ckm 13,test , δ CP ). Since the values of α 2 , α 3 , m 1 in the solutions are strongly correlated with δ CP , we further plot the solutions on the planes of (δ CP , α 2 ), (δ CP , α 3 ) and (δ CP , m 1 ). Additionally, we calculate, for individual solutions, m ee , the quantity measured in neutrinoless double β-decay experiments, as and plot the solutions on the plane of (δ CP , m ee ). The results are displayed in Figures 1, 2, 3, 4, 5, 6, whose corresponding 'input values' are listed in Table 3. Note that the scattering of dots in each figure represents uncertainty of the solution due to the uncertainties of θ ckm 12 , θ ckm 13 , θ ckm 23 , δ km .        Figure 1: Each dot represents a solution to Eq. (6) that fits within the 2σ range of θ ckm 12 , θ ckm 13 , θ ckm 23 , δ km at scale µ = 10 18 GeV evaluated by taking µ EW = M Z . These solutions are obtained from 1.6 × 10 10 random sets of values of (δ CP , α 2 , α 3 , log(m 1 )) in the range Eq. (16). The input values of y d , y s , y b and θ 12 , θ 13 , θ 23 , |∆m 2 32 |, ∆m 2 21 are as shown in Table 3, with the left plots corresponding to sin 2 θ 23 = 0.43 and the right plots to sin 2 θ 23 = 0.47. The plots in the first row are on (θ ckm 13,test , δ CP ) plane, those in the second row with black dots are on (δ CP , α 2 ) plane, those in the second row with red dots are on (δ CP , α 3 ) plane, those in the third row with black dots are on (δ CP /(2π), m 1 ) plane, and those in the third row with red dots are on (δ CP /(2π), m ee ) plane.   Table 3 are used, with the left plots corresponding to sin 2 θ 12 = 0.281 and the right plots to sin 2 θ 12 = 0.333.   Table 3 are used, with the lefts plot corresponding to sin 2 θ 13 = 0.0188 and the right plots to sin 2 θ 13 = 0.0232.  Table 3 are used, with the left plots corresponding to y d = 5.36 × 10 −6 and the right plots to y d = 6.08 × 10 −6 .  Table 3 are used, with the left plots corresponding to y s = 1.088 × 10 −4 and the right plots to y s = 1.192 × 10 −4 .   Table 3 are used, with the left plots corresponding to y b = 5.293 × 10 −3 and the right plots to y d = 5.405 × 10 −3 .
• We have varied the values of ∆m 2 12 and |∆m 2 23 | within the 2σ experimental range, and found no significant change in the plots.
• In Figure 7, we have used the values of y d , y s , y b and θ ckm 12 , θ ckm 13 , θ ckm 23 , δ km at a different renormalization scale µ = 10 4 GeV (evaluated by taking µ EW = M Z and shown in Table 1) in Eqs. (15,17). The neutrino oscillation parameters are identical with those for the right plots of Figure 1.  Figure 7: Same as the right plots of Figure 1, except that the values of y d , y s , y b and θ ckm 12 , θ ckm 13 , θ ckm 23 , δ km at µ = 10 4 GeV, shown in Table 1, are used in the analysis. The same values of neutrino oscillation parameters as the right plots of Figure 1 are employed.
• To study the dependence of the results on the matching scale µ EW , we have used the values of y d , y s , y b and θ ckm 12 , θ ckm 13 , θ ckm 23 , δ km at µ = 10 18 GeV evaluated by taking µ EW = 160 GeV, shown in Table 2. No significant difference is observed in the plots for µ EW = M Z and µ EW = 160 GeV.
From the above results, the following observations are made: • Only Y D ∝ diag(y d , y s , y b )V T CKM Eq. (4) can be consistent with the experimental data, and this is the case exclusively with the normal hierarchy of neutrino mass and for smaller values of sin 2 θ 23 in the current bound.
• In most cases, the value of δ CP satisfying Eq. (4) is in the range π > δ CP > 0 and hence is incompatible with the value hinted by the T2K collaboration, δ CP ∼ 3π/2 [24].
Nevertheless, for cases with sin 2 θ 12 = 0.333, y d = 5. • If we associate operators with opposite chiralities and consider the following different hypothesis, then the sign of δ CP , α 2 , α 3 in Figures 1, 2, 3, 4, 5, 6 is simply flipped. This hypothesis is in good agreement with the T2K data on δ CP .
• The values of α 2 , α 3 , m 1 satisfying Eq. (4) are strongly correlated with δ CP . For δ CP ≃ 1.2π, m 1 is predicted to be about 0.005 eV, which may be tested in future cosmological observations (for forecasts, see, e.g., Ref. [25]). m ee is suppressed below 0.002 eV due to cancellation of the active neutrino masses, and thus there is absolutely no chance to detect neutrinoless double β-decay in the near future [26].
• The pattern of the correlation between δ CP and θ ckm 13,test is similar for the cases with µ = 10 4 GeV and µ = 10 18 GeV. Since the running mixing angle θ ckm 13 depends linearly on the measured value of |V ub |, we conclude that the correlation between δ CP and |V ub | is nearly the same for µ = 10 4 GeV and µ = 10 18 GeV. The plots for α 2 , α 3 , m 1 , m ee are likewise the same for µ = 10 4 GeV and µ = 10 18 GeV. Because the two distinctively different assumptions on the scale at which Eq. (4) holds lead to similar results, we infer that our prediction is almost independent of the scale of Eq. (4).
• We have confirmed that the above results are insensitive to the choice of the matching scale µ EW , which is reasonable because the ratio y d : y s : y b and the CKM mixing angles are intact with the change of µ EW , as read from Tables 1,2. To summarize, we have investigated whether the relation Y D ∝ diag(y d , y s , y b )V T CKM or Y D ∝ diag(y u , y c , y t )V * CKM (in the flavor basis where the charged lepton Yukawa coupling and right-handed neutrino Majorana mass are diagonal) can be consistent with the current experimental data on the quark masses, CKM mixing angles and phase, and neutrino mixing angles, hoping to unveil an implicit connection between the PMNS and CKM matrices. We have found sets of values of (δ CP , α 2 , α 3 , m 1 ) that satisfy Y D ∝ diag(y d , y s , y b )V T CKM with the normal neutrino mass hierarchy, while there are no such sets for Y D ∝ diag(y u , y c , y t )V * CKM and/or with the inverted hierarchy. δ CP is predicted to be in the range 1.2π δ CP > 0 and is hence in tension with the latest T2K data. However, since the prediction crucially depends on neutrino mixing angles and d, s quark masses, their future precise measurement or evaluation is necessary to draw any conclusion about our hypothesis. We have made a prediction for m 1 that may be tested in future cosmological observations, whereas m ee is smaller than 0.002 eV and is far below the reach of near-future experiments.