Double Froggatt--Nielsen Mechanism

We present a doubly parametric extension of the standard Froggatt--Nielsen (FN) mechanism. As is well known, mass matrices of the up- and down-type quark sectors and the charged lepton sector in the standard model can be parametrized well by a parameter $\lambda$ which is usually taken to be the sine of the Cabibbo angle ($\lambda = \sin\theta_\text{C} \simeq 0.225$). However, in the neutrino sector, there is still room to realize the two neutrino mass squared differences $\Delta m_\text{sol}^2$ and $\Delta m_\text{atm}^2$, two large mixing angles $\theta _{12}$ and $\theta _{23}$, and non-zero $\theta _{13}$. Then we consider an extension with an additional parameter $\rho$ in addition to $\lambda$. Taking the relevant FN charges for a power of $\lambda~(=0.225)$ and additional FN charges for a power of $\rho$, which we assume to be less than one, we can reproduce the ratio of the two neutrino mass squared differences and three mixing angles. In the normal neutrino mass hierarchy, we show several patterns for taking relevant FN charges and the magnitude of $\rho$. We find that if $\sin \theta_{23}$ is measured more precisely, we can distinguish each pattern. This is testable in the near future, for example in neutrino oscillation experiments. In addition, we predict the Dirac CP-violating phase for each pattern.


Introduction
The standard model (SM) is one of the successful models in explaining the results of recent precise experiments. However, there are many free parameters, particularly as Yukawa couplings in the SM. There are some ambiguities in realizing the quark and lepton mass hierarchies and mixing angles. Then, many authors have studied texture analyses or flavor symmetry models in order to elucidate the origin of the flavor structure as a direction beyond the SM. In fact, Weinberg proposed a simple zero texture, within two generations of quarks, where quark masses and a mixing angle are related [1]. Fritzsch extended this to three generations in the so-called "Fritzsch-type mass matrix" [2,3], which relates quark masses and mixing angles in the quark sector. Furthermore, Fukugita, Tanimoto, and Yanagida extended this argument to the lepton sector [4] and predicted two large neutrino mixing angles and non-zero θ 13 [5,6], which was the last mixing angle of the lepton sector [7]- [9].
As is well known, mass matrices of the up-and down-type quarks and the charged leptons in the SM can be parametrized well by a parameter λ, which is usually taken to be the sine of the Cabibbo angle. Taking λ ≃ 0.225, up-and down-type quark and charged lepton mass hierarchies and mixing angles are reproduced. This type of parametrization was originally proposed by Froggatt and Nielsen [10]. On the other hand, however, in the neutrino sector, there is still room to realize the lepton flavor structure, i.e., neutrino mass squared differences ∆m 2 sol and ∆m 2 atm , two large mixing angles θ 12 and θ 23 , and non-zero θ 13 . Indeed, it is likely that the neutrino mass matrix has a property distinct from those of the up-and down-type quarks and the charged lepton mass matrices. This is due to the fact that the neutrino masses are so tiny in comparison with the other SM fermion masses, and that the lepton mixing angles are relatively larger than the quark mixing angles. In this paper, we present an extension of the FN mechanism in the neutrino mass matrix. In particular, we focus on a doubly parametric extension of the FN (DFN) mechanism. 1 To explain, we show an illustrative example of the doubly parametric extension. This extension is plausible when we use the seesaw mechanism [29]- [34], for instance. If the neutrinos are Majorana particles, in the seesaw mechanism, we need both Dirac and Majorana mass terms. Even if the Dirac neutrino mass matrix is parametrized by λ like the other SM fermions, where the Dirac-type masses come from spontaneous symmetry breaking in the SM, the Majorana masses can include free mass parameters in general, and in some models it is plausible that Majorana masses are parametrized by another parameter. Then in the neutrino sector, such a situation corresponds to an extended FN mechanism with a parameter ρ in addition to λ. 2 Taking the relevant FN charges for the power of λ (= 0.225) and additional FN charges for the power of ρ, which we assume to be less than one, we can reproduce the ratio of two neutrino mass squared differences and three mixing angles. In our numerical calculations, we show several patterns for taking relevant FN charges and the magnitude of ρ. Note that in our numerical analyses, we consider only the normal neutrino mass hierarchy. We find that if sin θ 23 is measured more precisely, we can distinguish each pattern. In addition, we predict the Dirac CP-violating phase (δ CP ) for each pattern.
This paper is organized as follows. In Sect. 2, we show the standard FN mechanism and present the DFN mechanism. In Sect. 3, we show the results of our numerical analyses in several patterns. Section 4 is devoted to discussions and summary. In Appendix A, we show the explicit form of the neutrino mass matrix for each pattern.

Doubly parametric extension of the FN mechanism
It is known that the mass matrices of the up-and down-type quark sectors and the charged lepton sector in the SM can be parametrized well by a parameter λ and six charges with up to O(1) complex coefficients in front of each element. 3 In particular, it is reasonable to choose a value of the parameter λ such that the observed masses and mixing angles of the up-and down-type quark and the charged lepton sectors can be realized. Indeed, such a value is given as λ = sin θ C ≃ 0.225, where θ C is the Cabibbo angle. This type of parametrization was originally proposed by Froggatt and Nielsen, the so-called "FN parametrization" [10].
In this paper, we consider an extension with an additional parameter ρ and six additional charges also with up to O(1) complex coefficients in front of each element. In particular, this parametrization is valid in a neutrino mass matrix as well as the other fermion mass matrices.
It should be noted that there are some possibilities for realizing the DFN parametrization. For example, the DFN parametrization would be considered as effective theories of multiscale extra dimensions, and would be obtained by an additional U(1) flavor symmetry and so on. To construct concrete models including the DFN parametrization is beyond the scope of this paper. In Refs. [27,28], for the up-and down-type quark sectors, the phenomenological prospects of the doubly parametric extension have already been studied. In this paper, we focus only on phenomenological properties of the doubly parametric extension in the lepton sector, in particular the neutrino mass matrix. Here, we assume that the charged lepton mass matrix takes a diagonal form. Finally, it is important to comment on concrete values of the two parameters λ and ρ. Note that without loss of generality we can choose the value of the original parameter λ such that λ = 0.225. Even if the parameter is chosen to be a distinct value, we can move to the case of λ = 0.225 by redefining the additional FN charges {c i , d j } and the value of the additional parameter ρ. Hence, in the following, we take λ = 0.225 and ρ as an arbitrary value which we assume to be less than one. We show that the DFN textures can reproduce the ratio of the two neutrino mass squared differences and three mixing angles. We also show the results of our numerical analyses in the next section. Here, we do not identify the origin of the additional parameter ρ, where one possibility is (right-handed) Majorana neutrino mass parameters in a seesaw model.

Numerical analyses
In this section, we focus on mass matrices in the neutrino sector, and also analyze numerical aspects of the extended FN parametrization. In our numerical calculations, we assume the normal neutrino mass hierarchy. We use the results of the global analysis of neutrino oscillation experiments [37]. The 3 σ ranges of the experimental data for the normal neutrino mass hierarchy are given as where θ ij are lepton mixing angles in the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix, while ∆m 2 sol and ∆m 2 atm are the solar and atmospheric neutrino mass squared differences, respectively. Note that the DFN parametrization can be valid in both Dirac and Majorana mass matrices. For simplicity, we assume that FN charges a i (c i ) are equivalent to the other charges b j (d j ), respectively. Now, the mass matrix in Eq. (2) becomes symmetric. We also assume that the mass matrix of the charged leptons is diagonal. 4 In the following, we consider six patterns 5 with/without additional FN charges as sample 3. Note that with ρ = 1.0, the first pattern of charge configurations gives the standard FN parametrization, and this charge configuration gives a µ-τ symmetric mass matrix which derives the almost BM mixing. In our calculations, the three neutrino masses are adjusted by the ratio of the two neutrino mass squared differences, because an overall mass scale is completely free for our parametrization. Here we take O(1) coefficients as 10% deviations from unity and complex phases are taken from −π to π. Then, we can predict the Dirac CP-violating phase δ CP in our numerical calculations, where the non-zero δ CP originates from the complex phases of the mass matrix elements. In each pattern, we scan 10 6 configurations of the coefficients of the nine elements of the mass matrix.
First, we show the scatter plots in Pattern 1. The gray regions suggest realized values of mixing angles sin θ 12 , sin θ 23 , sin θ 13 , and Dirac CP-violating phase δ CP in Fig. 1. The insides of the red dotted lines show the 3 σ allowed regions of each lepton mixing angle, while the orange points correspond to the case that all three lepton mixing angles are within the 3 σ ranges simultaneously in Eq. (3). We find that δ CP is predicted as |δ CP | 1 and 2.2 |δ CP | π.  We set ρ = 0.8. The color convention is as in Fig. 1.
On the other hand, in Pattern 2, the standard FN charges are the same as those of Pattern 1 and we set ρ = 0.8 so that the mass matrix becomes almost µ-τ symmetric (though not exactly). Figure 2 shows that sin θ 23 is around the upper boundary of the 3 σ range, while the other mixing angles are completely filled within the 3 σ range. Comparing Figures 1 and  2, it is easily seen that the realized values of sin θ 13 in Pattern 2 are relatively larger than those in Pattern 1. This implies that sin θ 13 is improved by the DFN parametrizations, even if the coefficients are not so scattered in large parameter regions. Also, we can distinguish Patterns 1 and 2 between the standard FN and DFN by more precise measurement of sin θ 23 . The extension with an additional parameter leads to modestly different properties of sin θ 13 and sin θ 23 from those of a µ-τ symmetric neutrino mass matrix.
Next, we show other patterns from non-µ-τ symmetric neutrino mass matrices. In Patterns 3, 4, and 5, the charge configurations of a i and c i are a i = { 3 2 , 1 2 , 0}, c i = {0, 1 2 , 3 2 }, while the magnitudes of ρ are different: ρ = 0.4, 0.5, 0.6, respectively. If we set ρ = 1.0 which is the standard FN parametrization, we cannot find the correct ratio of the two neutrino mass squared differences and three mixing angles. In Figs. 3, 4, and 5, the allowed regions of sin θ 12 , sin θ 13 , and δ CP are almost the same, while sin θ 23 is completely different. In Figs. 3, 4 and 5, it is easily found that different values of ρ lead to different values of sin θ 23 . This is a remarkable property in the DFN parametrizations. Figure 3 shows that sin θ 23 is scattered around the upper boundary of the 3 σ range in Pattern 3. In Pattern 4, the allowed region of sin θ 23 is 0.63 sin θ 23 0.72, as shown in Fig. 4. In Pattern 5, Fig. 5 shows that sin θ 23 is around the lower boundary of the 3 σ range. When we set ρ = 0.3 or 0.7, the obtained values of sin θ 23 are beyond the 3 σ experimental upper and lower bounds, respectively. The three patterns are tested by measuring the value of sin θ 23 more precisely.
The color convention is as in Fig. 1.
Finally, we show the last pattern. In Pattern 6 we set ρ = 0.3, which means that this pattern seems to be almost the standard FN parametrization because of λ = 0.225. 7 In Fig. 6, sin θ 12 and sin θ 13 are filled within the 3 σ range, while sin θ 23 is distributed around the lower boundary of the 3 σ range. Note that the neutrino mass matrix in Pattern 6 is considered to be similar to that in Pattern 1, because the value of the additional parameter is small and approximately equal to λ, i.e., ρ = 0.3. However, the values obtained for the mixing angles are distinct from each other. Indeed, the DFN extension can make values of sin θ 13 larger and values of sin θ 23 relatively smaller. We recognize that these properties are distinctive from those of Pattern 1 (the original FN). In addition, we find that δ CP is predicted as |δ CP | 1 and 2.0 |δ CP | π for Pattern 6.
We comment on the testabilities of the configurations of the (double) FN charges and parameters. First, the values obtained for sin θ 23 are almost the same in Patterns 1 and 4. The frequency of consistent values of sin θ 13 is certainly improved in Pattern 4. However, the predicted values are dependent on the coefficients in front of each element. Hence, it is difficult to distinguish Patterns 1 and 4 by neutrino experimental data. Indeed, such situations happen between the other patterns, e.g., between Patterns 2 and 3 and between Patterns 5 and 6. These coincident properties of predicted mixing angles can also be seen in the standard FN parametrizations. Therefore, all of the patterns cannot always be tested by more precise determination of the three mixing angles. This is more conspicuous when O(1) coefficients are randomly scattered in a wider range, e.g., c ij ∈ [0.8, 1.2]. The testability of between different configurations of (double) FN charges and parameters is strongly dependent on concrete model building.

Discussions and summary
In the SM, there are many free parameters especially as Yukawa couplings, so that there are some ambiguities in realizing the quark and lepton mass hierarchies and mixing angles. It is therefore important to study texture analysis or a flavor symmetry model in order to elucidate the origin of the flavor structure as beyond the SM. As is well known, mass matrices of the up-and down-type quarks and the charged leptons in the SM can be parametrized well by the parameter λ which is usually taken to be the sine of the Cabibbo angle (λ = sin θ C ≃ 0.225). In this parametrization, the mass hierarchies and mixing angles of the quarks and charged leptons are reproduced. However, in the neutrino sector, there is still room to realize the neutrino mass squared differences ∆m 2 sol and ∆m 2 atm , two large mixing angles θ 12 and θ 23 , and non-zero θ 13 . Actually, if the neutrinos are Majorana particles, in the seesaw mechanism, we need both the Dirac and Majorana mass terms. Even if the Dirac neutrino mass matrix is parametrized by λ like the other SM fermions, the Majorana masses include free mass parameters in general, and it is plausible that Majorana masses are parametrized by another parameter. Thus, in this paper we have presented a doubly parametric extension of the FN mechanism with the parameter ρ in addition to λ. Taking the relevant FN charges for the power of λ (= 0.225) and additional FN charges for the power of ρ, which we assume to be less than one, we can reproduce the ratio of the neutrino mass squared differences and lepton mixing angles. Here we assume that the charged lepton mass matrix is diagonal.
In our calculations, the neutrino masses, assuming the normal neutrino mass hierarchy, are adjusted by the ratio of the neutrino mass squared differences because the overall mass scale is completely free for our parametrization. Here we take O(1) coefficients as 10% deviations from one and the complex phases are taken from −π to π. Note that if we take the magnitude of O(1) coefficients as 20% deviations from one, the allowed region of δ CP is −π δ CP π, while if we take the magnitude of O(1) coefficients as 5% deviations from one, δ CP is more predictive. In this paper, we considered six patterns with/without additional FN charges as sample patterns for numerical calculations. First, we showed the standard FN and DFN parametrizations which are almost µ-τ symmetric mass matrices in Patterns 1 and 2, respectively. We found that δ CP is predicted as |δ CP | 1 and 2.2 |δ CP | π. In Pattern 2, sin θ 23 is around the upper boundary of the 3 σ range, while the other mixing angles are completely filled within the 3 σ range.
Next, we showed other patterns where the neutrino mass matrices are not µ-τ symmetric. In Patterns 3, 4, and 5, the charge configurations of a i and c i are a i = { 3 2 , 1 2 , 0}, c i = {0, 1 2 , 3 2 }, while the magnitudes of ρ are different: ρ = 0.4, 0.5, 0.6, respectively. If we set ρ = 1.0, which corresponds to the standard FN parametrization, we cannot find the correct ratio of the two neutrino mass squared differences and three mixing angles, so the DFN pattern parametrizes the neutrino sector well. Finally, we find a sizable deviation in Pattern 6, where the magnitude of ρ (= 0.3) is a little away from the FN value λ (≃ 0.225).
We had seen several examples in the mass matrix form formulated under the concept of DFN. We recognized that patterns of the mixing angles and the Dirac CP phase can deviate from the predicted ones in the FN texture. The deviations look distinctive when fluctuations in the elements of the mass matrix are within 10% of unity. As pointed out in the previous section, the explicit differences between the standard FN and DFN parametrizations tend to appear particularly in values of sin θ 23 . Hence, measuring sin θ 23 precisely is achievable in the near future, for example in neutrino oscillation experiments, and such improved measurements can determine how well the DFN texture works.