# The Geometric Correlations of Leptonic Mixing Parameters

Authors: Ding-Hui Xu; Shu-Jun Rong
Journal: Advances in High Energy Physics
 (2023)

---

## Abstract

Leptonic mixing patterns are usually extracted on the basis of groups or algebraic structures. In this paper, we introduce an alternative geometric method to study the correlations between the leptonic mixing parameters. At the 3σ level of the recent global fit data of neutrino oscillations, the distribution of the scattered points of the angles between the vectors, which are constructed by the element of the leptonic mixing matrix, is analysed. We find that the scattered points are concentrated on several special regions. Using the data in these regions, correlations of the leptonic mixing angles and the Dirac CP violating phase are obtained. The implications of the correlations are shown through the predicted flavor ratio of high-energy astrophysical neutrinos (HANs) at Earth.


---

## Body

## 1. Introduction

The discovery of neutrino flavor oscillation is a tremendous achievement in particle physics during the last two decades [1–3], which manifests that neutrinos have masses, and the leptonic mixing matrix is nontrivial. However, the origin of the flavor mixing is elusive for theorists. Before the determination of the reactor mixing angle θ13, the following are various candidates of the mixing patterns: bimaximal mixing (BM) [4], tri-bi-maximal mixing (TBM) [5, 6], golden ratio mixing (GRM) [7–9], hexagonal mixing (HM) [10], etc.[11, 12] (for reviews, see [13, 14]). These patterns, especially the TBM, derived from flavor groups such as A4 [15, 16] and S4 [16, 17], are compatible with previous published data of atmospheric neutrino and solar neutrino. The discovery of a relatively large reactor mixing angle θ13[3, 18–21] and the remarkable progress of measurements of neutrino oscillation parameters, however, strictly constrain or exclude the above patterns. Hence, diverse methods to generate a nonzero reactor mixing angle θ13 have been proposed. A popular approach is adding a model-independent perturbation to the leading TBM matrix [22]. Another method introduces a generalized CP transform on the basis of a discrete flavor symmetry, i.e., a finite flavor group Gf and a CP symmetry are combined [23–27]. Furthermore, a novel mathematical structure called group algebra was introduced [28, 29]. In this scenario, a specific leptonic mixing pattern could correspond to a set of equivalent elements of a group algebra.In this paper, we introduce an alternative geometric method to extract the correlations between the leptonic mixing parameters. This method is developed from the well-knownμ−τ reflection symmetry [30–39] which denotes that two row vectors μ⟶=Uμ12,Uμ22,Uμ32 and τ⟶=Uτ12,Uτ22,Uτ32 are identical. Here, Uαiα=μ,τ,andi=1,2,3 is the element of the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrix [40–42]
(1)U=c12c13s12c13s13e−iδ−s12c23−c12s13s23eiδc12c23−s12s13s23eiδc13s23s12s23−c12s13c23eiδ−c12s23−s12s13c23eiδc13c23,where sij≡sinθij (for ij=12, 13, 23), cij≡cosθij, and δ is the Dirac CP phase. Throughout this article, we do not consider the Majorana phases because they are irrelevant to neutrino oscillations. As is known, for the realistic mixing matrix, the μ−τ reflection symmetry predicts that θ23=π/4 and δ=±π/2. According to the recent global analysis NuFIT 5.0 [43], the maximal CP phase is favored by the 1σ allowed region in the case of inverted mass ordering (IO) while disfavored by the 1σ allowed region in the normal ordering (NO) case. Therefore, the μ−τ reflection symmetry may need modifications in the near future when a precise measurement of δ would be available.Geometrically speaking, theμ−τ reflection symmetry means that the angle between the 3-dimensional real vectors μ⟶ and τ⟶ is zero. A perturbed μ−τ symmetry corresponds to a small but nonzero included angle. Thus, the leptonic mixing pattern and the breaking of μ−τ reflection symmetry can be represented by the geometric quantities, namely, the angles between the row vectors μ⟶, τ⟶, and e⟶=(Ue12,Ue22,Ue32). In order to extract promising mixing patterns, we scan the included angles between the row vectors at the 3σ level of the global fit data [43]. We find that the scattered points of the angles are concentrated on several special areas which are stable under the random takings of the mixing parameters at the 3σ level. Furthermore, it is not surprising that the approximated μ−τ reflection symmetry corresponds to one of the special regions in the case of IO. Therefore, it is plausible that the promising correlations of the mixing parameters are indicated in these dense regions. On the basis of this assumption, the leptonic mixing patterns are read out from the dense zones and their implications are examined by a specific application, namely, the predicted flavor ratio of high-energy astrophysical neutrinos (HANs) at Earth.The paper is organized as follows. In Section2, we show the definition of included angles between the row vectors e⟶, μ⟶, and τ⟶ and extract correlations of the angles at the 3σ level of the global fit data [43]. On the basis of the correlations of the angles, the correlations of the leptonic mixing parameters are obtained. In Section 3, we apply these leptonic mixing parameters constrained by the geometric correlations to predict the flavor ratio of HANs at Earth. Finally, we summarize our main results.


## 2. Geometric Correlations at the 3σ Level of the Global Fit Data

In this section, we first give the definition of the angles between the row vectors, the selection rule of typical data points in the dense zones of the scattered plots, and then present correlations between the angles, on the basis of which we derive the correlations between the leptonic mixing parameters.


### 2.1. Definition of Geometric Correlations

The included angles between the row vectors are defined as follows:(2)cosμ⟶,τ⟶≡μ⟶·τ⟶μ⟶·τ⟶=Uμ12Uτ12+Uμ22Uτ22+Uμ32Uτ32Uμ14+Uμ24+Uμ34·Uτ14+Uτ24+Uτ34,cose⟶,μ⟶≡e⟶·μ⟶e⟶·μ⟶=Ue12Uμ12+Ue22Uμ22+Ue32Uμ32Ue14+Ue24+Ue34·Uμ14+Uμ24+Uμ34,cose⟶,τ⟶≡e⟶·τ⟶e⟶·τ⟶=Ue12Uτ12+Ue22Uτ22+Ue32Uτ32Ue14+Ue24+Ue34·Uτ14+Uτ24+Uτ34.Using the latest global fit data of the leptonic mixing parameters listed in Table1, the 3σ allowed ranges of the included angle between the row vectors are shown in Figure 1.Table 1
The 3σ allowed ranges of the leptonic mixing parameters in the global analysis NuFIT 5.0 [43].
ParametersNormal ordering (NO)Inverted ordering (IO)sin2θ120.269→0.3430.269→0.343sin2θ130.02034→0.024300.02053→0.02436sin2θ230.407→0.6180.411→0.621δ/∘107→403192→360Figure 1
Three-dimensional scattered plots of the included angles based on the recent global fit data listed in Table1. (a) NO case. (b) IO case.
(a)(b)We can see that the data points in these plots are of a nonuniform distribution. There are several special regions on which the points are concentrated. This observation is more obvious in the front view and top view of the plots shown in Figure2.Figure 2
(a) Front view. (b) Top view.
(a)(b)From the 2-dimensional scattered plots, we obtain the observations as follows:(a)
NO case: the data points ofcosμ⟶,τ⟶,cose⟶,μ⟶ are concentrated on two regions around the points (0.96, 0.40) and (0.96, 0.59), respectively. The data points of cosμ⟶,τ⟶,cose⟶,τ⟶ are also converged on two regions around the points (0.96, 0.39) and (0.96, 0.58), respectively(b)
IO case: the data points ofcosμ⟶,τ⟶,cose⟶,μ⟶ are concentrated on three regions around the points (0.96, 0.41), (0.96, 0.60), and (1.00, 0.49), respectively. The data points of cosμ⟶,τ⟶,cose⟶,τ⟶ are converged on three regions around the points (0.96, 0.39), (0.96, 0.60), and (1.00, 0.48), respectively. As is known, cosμ⟶,τ⟶~1 corresponds to the approximated μ−τ reflection symmetryFurthermore, we note that these dense zones of the scattered plots are stable under the random takings of the global fit data at3σ level. Therefore, we consider that the global fit data indicate correlations between the included angles of the row vectors. Correspondingly, besides the leptonic mixing pattern from the approximated μ−τ reflection symmetry, other promising patterns or correlations of the mixing parameters can be read out from the dense regions in the scattered plots.


### 2.2. Selection Rule of Typical Data Points in the Dense Regions of the Scattered Plots

In order to effectively filter out some representative data points in the dense zones of Figure2, we divide the data samples of the angles between the row vectors into small enough intervals. The advantage of this method is that the difference between the impacts of two arbitrary data points in the same interval on the correlations of the lepton mixing parameters can be ignored. In other words, we can use the midpoint to represent the whole interval. The selection rule is explained in more detail as follows.We divide the data samples ofcosμ⟶,τ⟶ originated from Figure 2 into a series of small intervals such as [0.825, 0.835], ⋯, [0.955, 0.965],⋯, [0.995, 1.00], which are shown in Figure 3.Figure 3
The number of scattered points corresponding to a series of small intervals ofcosμ⟶,τ⟶ in Figure 2. Red: NO case. Blue: IO case.From Figure3, we can observe that the number of data points with cosμ⟶,τ⟶∈ [0.955, 0.965] is maximal both in the NO case and IO case. In consequence, we focus on the interval of cosμ⟶,τ⟶, [0.955, 0.965]. Based on the similar approach, the number of data points of cose⟶,μ⟶ and cose⟶,τ⟶ is presented in Figure 4, with the constraint of cosμ⟶,τ⟶−0.96<0.005.Figure 4
The number of scattered points corresponding to a series of small intervals ofcose⟶,μ⟶ and cose⟶,τ⟶ in Figure 2, with the constraint of cosμ⟶,τ⟶−0.96<0.005. Red: NO case. Blue: IO case.From Figure4, we can see that the large numbers of scattered points in several intervals are approximate in the IO case, e.g., the cose⟶,μ⟶ intervals [0.575, 0.585] and [0.595, 0.605] and the cose⟶,τ⟶ intervals [0.375, 0.385], [0.385, 0.395], and [0.405, 0.415]. However, the numerical results show that these intervals with a similar number of scattered points in the same dense region bring no obvious differences on the correlations of the lepton mixing parameters. Therefore, for the sake of illustration, we select the intervals of cose⟶,μ⟶ and cose⟶,τ⟶ with the largest number of scattered points in every dense area of Figure 2.In addition, the approximatedμ−τ reflection symmetry corresponding to one of the dense regions in the IO case should be considered. In this case, the number of scattered points of cose⟶,μ⟶ and cose⟶,τ⟶ is shown in Figure 5, with the constraint of 0.995<cosμ⟶,τ⟶<1.00.Figure 5
The number of scattered points corresponding to a series of small intervals ofcose⟶,μ⟶ and cose⟶,τ⟶ in the IO case of Figure 2, with the constraint of 0.995<cosμ⟶,τ⟶<1.00.From Figure5, we find that the largest number of scattered points corresponds to the intervals cose⟶,μ⟶−0.49<0.005 and cose⟶,τ⟶−0.48<0.005.On the basis of the above analysis, the selected representative intervals of the angles between the row vectors are listed in Table2.Table 2
The intervals of the angles between the row vectors with the largest number of scattered points stem from every dense regions of Figure2.
Mass orderingcosμ⟶,τ⟶cose⟶,μ⟶cose⟶,τ⟶NO[0.955, 0.965][0.395, 0.405][0.385, 0.395][0.955, 0.965][0.585, 0.595][0.575, 0.585]IO[0.955, 0.965][0.405, 0.415][0.385, 0.395][0.955, 0.965][0.595, 0.605][0.595, 0.605][0.995, 1.00][0.485, 0.495][0.475, 0.485]Now, we show that the difference on the correlations of the lepton mixing parameters from two arbitrary data points in the same interval of Figures3–5 can be ignored. For specific explanation, three intervals in the NO case listed in Table 2 are taken, which are listed as follows: cosμ⟶,τ⟶∈ [0.955, 0.965], cose⟶,μ⟶∈ [0.395, 0.405], and cose⟶,τ⟶∈ [0.385, 0.395]. We set constraints on the leptonic mixing parameters with small neighborhoods around the data points, which are expressed as
(3)cosμ⟶,τ⟶−f1<0.001,cosμ⟶,τ⟶−f2<0.001,(4)cose⟶,μ⟶−g1<0.001,cose⟶,μ⟶−g2<0.001,(5)cose⟶,τ⟶−h1<0.001,cose⟶,τ⟶−h2<0.001,where (f1=0.956,f2=0.964), (g1=0.396,g2=0.404), and (h1=0.386,h2=0.394) are two data points near the boundary of the intervals [0.955, 0.965], [0.395, 0.405], and [0.385, 0.395], respectively. The correlations of sin2θ23-δ from the above constraints are obtained at the 3σ level of the global fit data [43], which are shown in Figure 6.Figure 6
The correlations ofsin2θ23-δ constrained by the correlations of Equations (3)–(5) at the 3σ level of the global fit data [43].Here, the correlations ofsin2θ13-sin2θ12 from the constrains of Equations (3)–(5) are not given because they are insensitive to these constrains. In other words, the data points of the correlation of sin2θ13-sin2θ12 are of a uniform distribution at the 3σ level. From Figure 6, we can see that the scattered point distribution of the correlation is not sensitive to the representative points in the interval. Therefore, we choose the middle point as the representative of an interval. For the selected intervals listed in Table 2, the typical points are given in Table 3.Table 3
Typical data points in the dense regions of Figure2.
Mass orderingcosμ0⟶,τ0⟶cose0⟶,μ0⟶cose0⟶,τ0⟶NO0.960.40, 0.590.39, 0.58IO0.960.41, 0.600.39, 0.600.9970.490.48


### 2.3. Leptonic Mixing Parameters Constrained by the Geometric Correlations

Now, we analyse the impacts of the geometric correlations on the leptonic mixing patterns.


#### 2.3.1. Impacts of the Correlations of Two Included Angles

According to the typical points shown in Table3, 15 viable correlations of two angles between the row vectors are obtained at the 3σ level of the global fit data [43], which are listed in Table 4.Table 4
Viable combinations with two included angles of the row vectors.
Mass orderingcosμ0⟶,τ0⟶,cose0⟶,μ0⟶cosμ0⟶,τ0⟶,cose0⟶,τ0⟶cose0⟶,μ0⟶,cose0⟶,τ0⟶NO(0.96, 0.40)(0.96, 0.39)(0.40, 0.58)(0.96, 0.59)(0.96, 0.58)(0.59, 0.39)IO(0.96, 0.41)(0.96, 0.39)(0.41, 0.60)(0.96, 0.60)(0.96, 0.60)(0.60, 0.39)(0.997, 0.49)(0.997, 0.48)(0.49, 0.48)Employing the correlations of the included angles in Table4, we set the geometric constraints as follows:
(6)cosμ⟶,τ⟶−cosμ0⟶,τ0⟶<0.001,cose⟶,μ⟶−cose0⟶,μ0⟶<0.001,cosμ⟶,τ⟶−cosμ0⟶,τ0⟶<0.001,cose⟶,τ⟶−cose0⟶,τ0⟶<0.001,cose⟶,μ⟶−cose0⟶,μ0⟶<0.001,cose⟶,τ⟶−cose0⟶,τ0⟶<0.001.On the bases of the constraints, the correlations between the leptonic mixing parameters are shown in Figures7–9.Figure 7
Leptonic mixing parameters constrained by the correlations between two included angles of (cosμ0⟶,τ0⟶,cose0⟶,μ0⟶) listed in Table 4. (a) NO case with the constraints of cosμ⟶,τ⟶−0.96<0.001 and cose⟶,μ⟶−0.40<0.001. (b) NO case with the constraints of cosμ⟶,τ⟶−0.96<0.001 and cose⟶,μ⟶−0.59<0.001. (c) IO case with the constraints of cosμ⟶,τ⟶−0.997<0.001 and cose⟶,μ⟶−0.49<0.001.
(a)(b)(c)Figure 8
Leptonic mixing parameters constrained by the correlations between two included angles of (cosμ0⟶,τ0⟶,cose0⟶,τ0⟶) listed in Table 4. (a) NO case with the constraints of cosμ⟶,τ⟶−0.96<0.001 and cose⟶,τ⟶−0.39<0.001. (b) NO case with the constraints of cosμ⟶,τ⟶−0.96<0.001 and cose⟶,τ⟶−0.58<0.001. (c) IO case with the constraints of cosμ⟶,τ⟶−0.997<0.001 and cose⟶,τ⟶−0.48<0.001.
(a)(b)(c)Figure 9
Leptonic mixing parameters constrained by the correlations between two included angles of (cose0⟶,μ0⟶,cose0⟶,τ0⟶) listed in Table 4. (a) NO case with the constraints of cose⟶,μ⟶−0.40<0.001 and cose⟶,τ⟶−0.58<0.001. (b) NO case with the constraints of cose⟶,μ⟶−0.59<0.001 and cose⟶,τ⟶−0.39<0.001. (c) IO case with the constraints of cose⟶,μ⟶−0.49<0.001 and cose⟶,τ⟶−0.48<0.001.
(a)(b)(c)Since the figure in the IO case is similar to the counterpart in the NO case, the plots of correlations in the IO case except the plots for the approximatedμ−τ symmetry are not shown here. We make some comments on the main observations as follows:
(a)
The range ofsin2θ12 is obviously reduced by the constraint of (cose0⟶,μ0⟶, cose0⟶,τ0⟶) (see Figure 9). In contrast, the influence of the constraints of (cosμ0⟶,τ0⟶, cose0⟶,μ0⟶) and (cosμ0⟶,τ0⟶, cose0⟶,τ0⟶) on the range of sin2θ12 can be neglected(b)
δ and sin2θ23 are sensitive to the constraint of (cosμ0⟶,τ0⟶, cose0⟶,μ0⟶) and/or (cosμ0⟶,τ0⟶, cose0⟶,τ0⟶); that is, the ranges of the parameters are compressed apparently(c)
The above observations are stable under the variation of the intensity of the constraint, such ascosα⟶,β⟶−cosα0⟶,β0⟶<0.001⟶cosα⟶,β⟶−cosα0⟶,β0⟶<0.02 with α,β=e,μ,τ(d)
The correlationscosμ0⟶,τ0⟶,cose0⟶,μ0⟶=0.96,0.40 and cosμ0⟶,τ0⟶,cose0⟶,μ0⟶=0.96,0.59 can be converted to each other through the approximated μ−τ interchange, which means that θ23⟶π/2−θ23 and δ⟶δ+π. This conclusion also applies to the correlations cosμ0⟶,τ0⟶,cose0⟶,τ0⟶=0.96,0.39 and cosμ0⟶,τ0⟶,cose0⟶,τ0⟶=0.96,0.58 and the correlations cose0⟶,μ0⟶,cose0⟶,τ0⟶=0.40,0.58 and cose0⟶,μ0⟶,cose0⟶,τ0⟶=0.59,0.39(e)
The correlation of the leptonic mixing parameters obtained from the correlationcosμ0⟶,τ0⟶,cose0⟶,μ0⟶=0.96,0.40 is similar as that obtained from the correlation cosμ0⟶,τ0⟶,cose0⟶,τ0⟶=0.96,0.58. The difference of these two correlations is that the range of sin2θ23 from the latter is moderately compressed. This observation also applies to the correlations cosμ0⟶,τ0⟶,cose0⟶,μ0⟶=0.96,0.59 and cosμ0⟶,τ0⟶,cose0⟶,τ0⟶=0.96,0.39


#### 2.3.2. Leptonic Mixing Parameters Constrained by Correlations of Three Included Angles

According to the observations in the case of the correlations of two included angles, we find that the mixing parameters (sin2θ12, sin2θ23, and δ) are strictly constrained by the geometric correlations. Now, we extract the correlations between the mixing parameters from the correlations of three included angles.On the basis Tables3, 5 correlations of three included angles are viable at the 3σ level of the global fit data [43], which are listed in Table 5.Table 5
Representative correlations of three included angles.
Mass orderingCorrelationscosμ0⟶,τ0⟶,cose0⟶,μ0⟶,cose0⟶,τ0⟶NOI(0.96, 0.40, 0.58)II(0.96, 0.59, 0.39)IOIII(0.96, 0.41, 0.60)IV(0.96, 0.60, 0.39)V(0.997, 0.49, 0.48)Employing the geometric constraints listed as follows:(7)cosμ⟶,τ⟶−cosμ0⟶,τ0⟶<0.001,cose⟶,μ⟶−cose0⟶,μ0⟶<0.001,cose⟶,τ⟶−cose0⟶,τ0⟶<0.001,we get the correlations of the leptonic mixing parameters, shown in Figure 10.Figure 10
The correlations ofsin2θ13-sin2θ12 and sin2θ23-δ from the constraints of the geometric correlations I-V.From Figure10, we obtain the following observations:
(a)
sin2θ12 is linearly dependent on sin2θ13 for all the correlations and is consistent with the recent global best fit value [43] under the correlation V. Correspondingly, the range of sin2θ12 is reduced significantly by the geometric constraints. Furthermore, under the constraint intensity, namely, 10−3, the variation of sin2θ12 is at most 2×10−3. Therefore, the breaking of the μ−τ reflection symmetry is mainly determined by the included angles, i.e., cosα0⟶,β0⟶ with α,β=e,μ,τ(b)
The range ofsin2θ23 displays two branches for all the correlations. The right branch of sin2θ23 from the correlation III is consistent with the recent global best fit value [43], while the left branch of sin2θ23 from correlation III is consistent with the best fit value of T2K experiment (sin2θ23=0.53−0.04+0.03) [44](c)
The range ofδ also presents two branches for the correlations I-V. The best global fit value of δ in the NO case [43] can be realized from the correlation I, and the value in the IO case can be obtained from the correlations IV and V(d)
The correlations I and II can be converted to each other through the approximatedμ−τ interchange(e)
We note that the values ofsin2θ12, sin2θ13, sin2θ23, and δ predicted from the modular symmetry origin of texture zeros and quark-lepton unification (see models III+D6 and IV+D6 in the Ref. [45]) are close to the ones shown in the special branch from the correlation I in the top plots of Figure 10. In addition, the values of the mixing angles and the Dirac CP phase predicted from the A5′ modular group (see lepton models L134, L3, and L10 in the Ref. [46]) also approach to the ones in the branch from the correlation I in Figure 10. Thus, the geometric correlations may be realized in the lepton mixing models based on modular symmetries


## 2.1. Definition of Geometric Correlations

The included angles between the row vectors are defined as follows:(2)cosμ⟶,τ⟶≡μ⟶·τ⟶μ⟶·τ⟶=Uμ12Uτ12+Uμ22Uτ22+Uμ32Uτ32Uμ14+Uμ24+Uμ34·Uτ14+Uτ24+Uτ34,cose⟶,μ⟶≡e⟶·μ⟶e⟶·μ⟶=Ue12Uμ12+Ue22Uμ22+Ue32Uμ32Ue14+Ue24+Ue34·Uμ14+Uμ24+Uμ34,cose⟶,τ⟶≡e⟶·τ⟶e⟶·τ⟶=Ue12Uτ12+Ue22Uτ22+Ue32Uτ32Ue14+Ue24+Ue34·Uτ14+Uτ24+Uτ34.Using the latest global fit data of the leptonic mixing parameters listed in Table1, the 3σ allowed ranges of the included angle between the row vectors are shown in Figure 1.Table 1
The 3σ allowed ranges of the leptonic mixing parameters in the global analysis NuFIT 5.0 [43].
ParametersNormal ordering (NO)Inverted ordering (IO)sin2θ120.269→0.3430.269→0.343sin2θ130.02034→0.024300.02053→0.02436sin2θ230.407→0.6180.411→0.621δ/∘107→403192→360Figure 1
Three-dimensional scattered plots of the included angles based on the recent global fit data listed in Table1. (a) NO case. (b) IO case.
(a)(b)We can see that the data points in these plots are of a nonuniform distribution. There are several special regions on which the points are concentrated. This observation is more obvious in the front view and top view of the plots shown in Figure2.Figure 2
(a) Front view. (b) Top view.
(a)(b)From the 2-dimensional scattered plots, we obtain the observations as follows:(a)
NO case: the data points ofcosμ⟶,τ⟶,cose⟶,μ⟶ are concentrated on two regions around the points (0.96, 0.40) and (0.96, 0.59), respectively. The data points of cosμ⟶,τ⟶,cose⟶,τ⟶ are also converged on two regions around the points (0.96, 0.39) and (0.96, 0.58), respectively(b)
IO case: the data points ofcosμ⟶,τ⟶,cose⟶,μ⟶ are concentrated on three regions around the points (0.96, 0.41), (0.96, 0.60), and (1.00, 0.49), respectively. The data points of cosμ⟶,τ⟶,cose⟶,τ⟶ are converged on three regions around the points (0.96, 0.39), (0.96, 0.60), and (1.00, 0.48), respectively. As is known, cosμ⟶,τ⟶~1 corresponds to the approximated μ−τ reflection symmetryFurthermore, we note that these dense zones of the scattered plots are stable under the random takings of the global fit data at3σ level. Therefore, we consider that the global fit data indicate correlations between the included angles of the row vectors. Correspondingly, besides the leptonic mixing pattern from the approximated μ−τ reflection symmetry, other promising patterns or correlations of the mixing parameters can be read out from the dense regions in the scattered plots.


## 2.2. Selection Rule of Typical Data Points in the Dense Regions of the Scattered Plots

In order to effectively filter out some representative data points in the dense zones of Figure2, we divide the data samples of the angles between the row vectors into small enough intervals. The advantage of this method is that the difference between the impacts of two arbitrary data points in the same interval on the correlations of the lepton mixing parameters can be ignored. In other words, we can use the midpoint to represent the whole interval. The selection rule is explained in more detail as follows.We divide the data samples ofcosμ⟶,τ⟶ originated from Figure 2 into a series of small intervals such as [0.825, 0.835], ⋯, [0.955, 0.965],⋯, [0.995, 1.00], which are shown in Figure 3.Figure 3
The number of scattered points corresponding to a series of small intervals ofcosμ⟶,τ⟶ in Figure 2. Red: NO case. Blue: IO case.From Figure3, we can observe that the number of data points with cosμ⟶,τ⟶∈ [0.955, 0.965] is maximal both in the NO case and IO case. In consequence, we focus on the interval of cosμ⟶,τ⟶, [0.955, 0.965]. Based on the similar approach, the number of data points of cose⟶,μ⟶ and cose⟶,τ⟶ is presented in Figure 4, with the constraint of cosμ⟶,τ⟶−0.96<0.005.Figure 4
The number of scattered points corresponding to a series of small intervals ofcose⟶,μ⟶ and cose⟶,τ⟶ in Figure 2, with the constraint of cosμ⟶,τ⟶−0.96<0.005. Red: NO case. Blue: IO case.From Figure4, we can see that the large numbers of scattered points in several intervals are approximate in the IO case, e.g., the cose⟶,μ⟶ intervals [0.575, 0.585] and [0.595, 0.605] and the cose⟶,τ⟶ intervals [0.375, 0.385], [0.385, 0.395], and [0.405, 0.415]. However, the numerical results show that these intervals with a similar number of scattered points in the same dense region bring no obvious differences on the correlations of the lepton mixing parameters. Therefore, for the sake of illustration, we select the intervals of cose⟶,μ⟶ and cose⟶,τ⟶ with the largest number of scattered points in every dense area of Figure 2.In addition, the approximatedμ−τ reflection symmetry corresponding to one of the dense regions in the IO case should be considered. In this case, the number of scattered points of cose⟶,μ⟶ and cose⟶,τ⟶ is shown in Figure 5, with the constraint of 0.995<cosμ⟶,τ⟶<1.00.Figure 5
The number of scattered points corresponding to a series of small intervals ofcose⟶,μ⟶ and cose⟶,τ⟶ in the IO case of Figure 2, with the constraint of 0.995<cosμ⟶,τ⟶<1.00.From Figure5, we find that the largest number of scattered points corresponds to the intervals cose⟶,μ⟶−0.49<0.005 and cose⟶,τ⟶−0.48<0.005.On the basis of the above analysis, the selected representative intervals of the angles between the row vectors are listed in Table2.Table 2
The intervals of the angles between the row vectors with the largest number of scattered points stem from every dense regions of Figure2.
Mass orderingcosμ⟶,τ⟶cose⟶,μ⟶cose⟶,τ⟶NO[0.955, 0.965][0.395, 0.405][0.385, 0.395][0.955, 0.965][0.585, 0.595][0.575, 0.585]IO[0.955, 0.965][0.405, 0.415][0.385, 0.395][0.955, 0.965][0.595, 0.605][0.595, 0.605][0.995, 1.00][0.485, 0.495][0.475, 0.485]Now, we show that the difference on the correlations of the lepton mixing parameters from two arbitrary data points in the same interval of Figures3–5 can be ignored. For specific explanation, three intervals in the NO case listed in Table 2 are taken, which are listed as follows: cosμ⟶,τ⟶∈ [0.955, 0.965], cose⟶,μ⟶∈ [0.395, 0.405], and cose⟶,τ⟶∈ [0.385, 0.395]. We set constraints on the leptonic mixing parameters with small neighborhoods around the data points, which are expressed as
(3)cosμ⟶,τ⟶−f1<0.001,cosμ⟶,τ⟶−f2<0.001,(4)cose⟶,μ⟶−g1<0.001,cose⟶,μ⟶−g2<0.001,(5)cose⟶,τ⟶−h1<0.001,cose⟶,τ⟶−h2<0.001,where (f1=0.956,f2=0.964), (g1=0.396,g2=0.404), and (h1=0.386,h2=0.394) are two data points near the boundary of the intervals [0.955, 0.965], [0.395, 0.405], and [0.385, 0.395], respectively. The correlations of sin2θ23-δ from the above constraints are obtained at the 3σ level of the global fit data [43], which are shown in Figure 6.Figure 6
The correlations ofsin2θ23-δ constrained by the correlations of Equations (3)–(5) at the 3σ level of the global fit data [43].Here, the correlations ofsin2θ13-sin2θ12 from the constrains of Equations (3)–(5) are not given because they are insensitive to these constrains. In other words, the data points of the correlation of sin2θ13-sin2θ12 are of a uniform distribution at the 3σ level. From Figure 6, we can see that the scattered point distribution of the correlation is not sensitive to the representative points in the interval. Therefore, we choose the middle point as the representative of an interval. For the selected intervals listed in Table 2, the typical points are given in Table 3.Table 3
Typical data points in the dense regions of Figure2.
Mass orderingcosμ0⟶,τ0⟶cose0⟶,μ0⟶cose0⟶,τ0⟶NO0.960.40, 0.590.39, 0.58IO0.960.41, 0.600.39, 0.600.9970.490.48


## 2.3. Leptonic Mixing Parameters Constrained by the Geometric Correlations

Now, we analyse the impacts of the geometric correlations on the leptonic mixing patterns.


### 2.3.1. Impacts of the Correlations of Two Included Angles

According to the typical points shown in Table3, 15 viable correlations of two angles between the row vectors are obtained at the 3σ level of the global fit data [43], which are listed in Table 4.Table 4
Viable combinations with two included angles of the row vectors.
Mass orderingcosμ0⟶,τ0⟶,cose0⟶,μ0⟶cosμ0⟶,τ0⟶,cose0⟶,τ0⟶cose0⟶,μ0⟶,cose0⟶,τ0⟶NO(0.96, 0.40)(0.96, 0.39)(0.40, 0.58)(0.96, 0.59)(0.96, 0.58)(0.59, 0.39)IO(0.96, 0.41)(0.96, 0.39)(0.41, 0.60)(0.96, 0.60)(0.96, 0.60)(0.60, 0.39)(0.997, 0.49)(0.997, 0.48)(0.49, 0.48)Employing the correlations of the included angles in Table4, we set the geometric constraints as follows:
(6)cosμ⟶,τ⟶−cosμ0⟶,τ0⟶<0.001,cose⟶,μ⟶−cose0⟶,μ0⟶<0.001,cosμ⟶,τ⟶−cosμ0⟶,τ0⟶<0.001,cose⟶,τ⟶−cose0⟶,τ0⟶<0.001,cose⟶,μ⟶−cose0⟶,μ0⟶<0.001,cose⟶,τ⟶−cose0⟶,τ0⟶<0.001.On the bases of the constraints, the correlations between the leptonic mixing parameters are shown in Figures7–9.Figure 7
Leptonic mixing parameters constrained by the correlations between two included angles of (cosμ0⟶,τ0⟶,cose0⟶,μ0⟶) listed in Table 4. (a) NO case with the constraints of cosμ⟶,τ⟶−0.96<0.001 and cose⟶,μ⟶−0.40<0.001. (b) NO case with the constraints of cosμ⟶,τ⟶−0.96<0.001 and cose⟶,μ⟶−0.59<0.001. (c) IO case with the constraints of cosμ⟶,τ⟶−0.997<0.001 and cose⟶,μ⟶−0.49<0.001.
(a)(b)(c)Figure 8
Leptonic mixing parameters constrained by the correlations between two included angles of (cosμ0⟶,τ0⟶,cose0⟶,τ0⟶) listed in Table 4. (a) NO case with the constraints of cosμ⟶,τ⟶−0.96<0.001 and cose⟶,τ⟶−0.39<0.001. (b) NO case with the constraints of cosμ⟶,τ⟶−0.96<0.001 and cose⟶,τ⟶−0.58<0.001. (c) IO case with the constraints of cosμ⟶,τ⟶−0.997<0.001 and cose⟶,τ⟶−0.48<0.001.
(a)(b)(c)Figure 9
Leptonic mixing parameters constrained by the correlations between two included angles of (cose0⟶,μ0⟶,cose0⟶,τ0⟶) listed in Table 4. (a) NO case with the constraints of cose⟶,μ⟶−0.40<0.001 and cose⟶,τ⟶−0.58<0.001. (b) NO case with the constraints of cose⟶,μ⟶−0.59<0.001 and cose⟶,τ⟶−0.39<0.001. (c) IO case with the constraints of cose⟶,μ⟶−0.49<0.001 and cose⟶,τ⟶−0.48<0.001.
(a)(b)(c)Since the figure in the IO case is similar to the counterpart in the NO case, the plots of correlations in the IO case except the plots for the approximatedμ−τ symmetry are not shown here. We make some comments on the main observations as follows:
(a)
The range ofsin2θ12 is obviously reduced by the constraint of (cose0⟶,μ0⟶, cose0⟶,τ0⟶) (see Figure 9). In contrast, the influence of the constraints of (cosμ0⟶,τ0⟶, cose0⟶,μ0⟶) and (cosμ0⟶,τ0⟶, cose0⟶,τ0⟶) on the range of sin2θ12 can be neglected(b)
δ and sin2θ23 are sensitive to the constraint of (cosμ0⟶,τ0⟶, cose0⟶,μ0⟶) and/or (cosμ0⟶,τ0⟶, cose0⟶,τ0⟶); that is, the ranges of the parameters are compressed apparently(c)
The above observations are stable under the variation of the intensity of the constraint, such ascosα⟶,β⟶−cosα0⟶,β0⟶<0.001⟶cosα⟶,β⟶−cosα0⟶,β0⟶<0.02 with α,β=e,μ,τ(d)
The correlationscosμ0⟶,τ0⟶,cose0⟶,μ0⟶=0.96,0.40 and cosμ0⟶,τ0⟶,cose0⟶,μ0⟶=0.96,0.59 can be converted to each other through the approximated μ−τ interchange, which means that θ23⟶π/2−θ23 and δ⟶δ+π. This conclusion also applies to the correlations cosμ0⟶,τ0⟶,cose0⟶,τ0⟶=0.96,0.39 and cosμ0⟶,τ0⟶,cose0⟶,τ0⟶=0.96,0.58 and the correlations cose0⟶,μ0⟶,cose0⟶,τ0⟶=0.40,0.58 and cose0⟶,μ0⟶,cose0⟶,τ0⟶=0.59,0.39(e)
The correlation of the leptonic mixing parameters obtained from the correlationcosμ0⟶,τ0⟶,cose0⟶,μ0⟶=0.96,0.40 is similar as that obtained from the correlation cosμ0⟶,τ0⟶,cose0⟶,τ0⟶=0.96,0.58. The difference of these two correlations is that the range of sin2θ23 from the latter is moderately compressed. This observation also applies to the correlations cosμ0⟶,τ0⟶,cose0⟶,μ0⟶=0.96,0.59 and cosμ0⟶,τ0⟶,cose0⟶,τ0⟶=0.96,0.39


### 2.3.2. Leptonic Mixing Parameters Constrained by Correlations of Three Included Angles

According to the observations in the case of the correlations of two included angles, we find that the mixing parameters (sin2θ12, sin2θ23, and δ) are strictly constrained by the geometric correlations. Now, we extract the correlations between the mixing parameters from the correlations of three included angles.On the basis Tables3, 5 correlations of three included angles are viable at the 3σ level of the global fit data [43], which are listed in Table 5.Table 5
Representative correlations of three included angles.
Mass orderingCorrelationscosμ0⟶,τ0⟶,cose0⟶,μ0⟶,cose0⟶,τ0⟶NOI(0.96, 0.40, 0.58)II(0.96, 0.59, 0.39)IOIII(0.96, 0.41, 0.60)IV(0.96, 0.60, 0.39)V(0.997, 0.49, 0.48)Employing the geometric constraints listed as follows:(7)cosμ⟶,τ⟶−cosμ0⟶,τ0⟶<0.001,cose⟶,μ⟶−cose0⟶,μ0⟶<0.001,cose⟶,τ⟶−cose0⟶,τ0⟶<0.001,we get the correlations of the leptonic mixing parameters, shown in Figure 10.Figure 10
The correlations ofsin2θ13-sin2θ12 and sin2θ23-δ from the constraints of the geometric correlations I-V.From Figure10, we obtain the following observations:
(a)
sin2θ12 is linearly dependent on sin2θ13 for all the correlations and is consistent with the recent global best fit value [43] under the correlation V. Correspondingly, the range of sin2θ12 is reduced significantly by the geometric constraints. Furthermore, under the constraint intensity, namely, 10−3, the variation of sin2θ12 is at most 2×10−3. Therefore, the breaking of the μ−τ reflection symmetry is mainly determined by the included angles, i.e., cosα0⟶,β0⟶ with α,β=e,μ,τ(b)
The range ofsin2θ23 displays two branches for all the correlations. The right branch of sin2θ23 from the correlation III is consistent with the recent global best fit value [43], while the left branch of sin2θ23 from correlation III is consistent with the best fit value of T2K experiment (sin2θ23=0.53−0.04+0.03) [44](c)
The range ofδ also presents two branches for the correlations I-V. The best global fit value of δ in the NO case [43] can be realized from the correlation I, and the value in the IO case can be obtained from the correlations IV and V(d)
The correlations I and II can be converted to each other through the approximatedμ−τ interchange(e)
We note that the values ofsin2θ12, sin2θ13, sin2θ23, and δ predicted from the modular symmetry origin of texture zeros and quark-lepton unification (see models III+D6 and IV+D6 in the Ref. [45]) are close to the ones shown in the special branch from the correlation I in the top plots of Figure 10. In addition, the values of the mixing angles and the Dirac CP phase predicted from the A5′ modular group (see lepton models L134, L3, and L10 in the Ref. [46]) also approach to the ones in the branch from the correlation I in Figure 10. Thus, the geometric correlations may be realized in the lepton mixing models based on modular symmetries


## 2.3.1. Impacts of the Correlations of Two Included Angles

According to the typical points shown in Table3, 15 viable correlations of two angles between the row vectors are obtained at the 3σ level of the global fit data [43], which are listed in Table 4.Table 4
Viable combinations with two included angles of the row vectors.
Mass orderingcosμ0⟶,τ0⟶,cose0⟶,μ0⟶cosμ0⟶,τ0⟶,cose0⟶,τ0⟶cose0⟶,μ0⟶,cose0⟶,τ0⟶NO(0.96, 0.40)(0.96, 0.39)(0.40, 0.58)(0.96, 0.59)(0.96, 0.58)(0.59, 0.39)IO(0.96, 0.41)(0.96, 0.39)(0.41, 0.60)(0.96, 0.60)(0.96, 0.60)(0.60, 0.39)(0.997, 0.49)(0.997, 0.48)(0.49, 0.48)Employing the correlations of the included angles in Table4, we set the geometric constraints as follows:
(6)cosμ⟶,τ⟶−cosμ0⟶,τ0⟶<0.001,cose⟶,μ⟶−cose0⟶,μ0⟶<0.001,cosμ⟶,τ⟶−cosμ0⟶,τ0⟶<0.001,cose⟶,τ⟶−cose0⟶,τ0⟶<0.001,cose⟶,μ⟶−cose0⟶,μ0⟶<0.001,cose⟶,τ⟶−cose0⟶,τ0⟶<0.001.On the bases of the constraints, the correlations between the leptonic mixing parameters are shown in Figures7–9.Figure 7
Leptonic mixing parameters constrained by the correlations between two included angles of (cosμ0⟶,τ0⟶,cose0⟶,μ0⟶) listed in Table 4. (a) NO case with the constraints of cosμ⟶,τ⟶−0.96<0.001 and cose⟶,μ⟶−0.40<0.001. (b) NO case with the constraints of cosμ⟶,τ⟶−0.96<0.001 and cose⟶,μ⟶−0.59<0.001. (c) IO case with the constraints of cosμ⟶,τ⟶−0.997<0.001 and cose⟶,μ⟶−0.49<0.001.
(a)(b)(c)Figure 8
Leptonic mixing parameters constrained by the correlations between two included angles of (cosμ0⟶,τ0⟶,cose0⟶,τ0⟶) listed in Table 4. (a) NO case with the constraints of cosμ⟶,τ⟶−0.96<0.001 and cose⟶,τ⟶−0.39<0.001. (b) NO case with the constraints of cosμ⟶,τ⟶−0.96<0.001 and cose⟶,τ⟶−0.58<0.001. (c) IO case with the constraints of cosμ⟶,τ⟶−0.997<0.001 and cose⟶,τ⟶−0.48<0.001.
(a)(b)(c)Figure 9
Leptonic mixing parameters constrained by the correlations between two included angles of (cose0⟶,μ0⟶,cose0⟶,τ0⟶) listed in Table 4. (a) NO case with the constraints of cose⟶,μ⟶−0.40<0.001 and cose⟶,τ⟶−0.58<0.001. (b) NO case with the constraints of cose⟶,μ⟶−0.59<0.001 and cose⟶,τ⟶−0.39<0.001. (c) IO case with the constraints of cose⟶,μ⟶−0.49<0.001 and cose⟶,τ⟶−0.48<0.001.
(a)(b)(c)Since the figure in the IO case is similar to the counterpart in the NO case, the plots of correlations in the IO case except the plots for the approximatedμ−τ symmetry are not shown here. We make some comments on the main observations as follows:
(a)
The range ofsin2θ12 is obviously reduced by the constraint of (cose0⟶,μ0⟶, cose0⟶,τ0⟶) (see Figure 9). In contrast, the influence of the constraints of (cosμ0⟶,τ0⟶, cose0⟶,μ0⟶) and (cosμ0⟶,τ0⟶, cose0⟶,τ0⟶) on the range of sin2θ12 can be neglected(b)
δ and sin2θ23 are sensitive to the constraint of (cosμ0⟶,τ0⟶, cose0⟶,μ0⟶) and/or (cosμ0⟶,τ0⟶, cose0⟶,τ0⟶); that is, the ranges of the parameters are compressed apparently(c)
The above observations are stable under the variation of the intensity of the constraint, such ascosα⟶,β⟶−cosα0⟶,β0⟶<0.001⟶cosα⟶,β⟶−cosα0⟶,β0⟶<0.02 with α,β=e,μ,τ(d)
The correlationscosμ0⟶,τ0⟶,cose0⟶,μ0⟶=0.96,0.40 and cosμ0⟶,τ0⟶,cose0⟶,μ0⟶=0.96,0.59 can be converted to each other through the approximated μ−τ interchange, which means that θ23⟶π/2−θ23 and δ⟶δ+π. This conclusion also applies to the correlations cosμ0⟶,τ0⟶,cose0⟶,τ0⟶=0.96,0.39 and cosμ0⟶,τ0⟶,cose0⟶,τ0⟶=0.96,0.58 and the correlations cose0⟶,μ0⟶,cose0⟶,τ0⟶=0.40,0.58 and cose0⟶,μ0⟶,cose0⟶,τ0⟶=0.59,0.39(e)
The correlation of the leptonic mixing parameters obtained from the correlationcosμ0⟶,τ0⟶,cose0⟶,μ0⟶=0.96,0.40 is similar as that obtained from the correlation cosμ0⟶,τ0⟶,cose0⟶,τ0⟶=0.96,0.58. The difference of these two correlations is that the range of sin2θ23 from the latter is moderately compressed. This observation also applies to the correlations cosμ0⟶,τ0⟶,cose0⟶,μ0⟶=0.96,0.59 and cosμ0⟶,τ0⟶,cose0⟶,τ0⟶=0.96,0.39


## 2.3.2. Leptonic Mixing Parameters Constrained by Correlations of Three Included Angles

According to the observations in the case of the correlations of two included angles, we find that the mixing parameters (sin2θ12, sin2θ23, and δ) are strictly constrained by the geometric correlations. Now, we extract the correlations between the mixing parameters from the correlations of three included angles.On the basis Tables3, 5 correlations of three included angles are viable at the 3σ level of the global fit data [43], which are listed in Table 5.Table 5
Representative correlations of three included angles.
Mass orderingCorrelationscosμ0⟶,τ0⟶,cose0⟶,μ0⟶,cose0⟶,τ0⟶NOI(0.96, 0.40, 0.58)II(0.96, 0.59, 0.39)IOIII(0.96, 0.41, 0.60)IV(0.96, 0.60, 0.39)V(0.997, 0.49, 0.48)Employing the geometric constraints listed as follows:(7)cosμ⟶,τ⟶−cosμ0⟶,τ0⟶<0.001,cose⟶,μ⟶−cose0⟶,μ0⟶<0.001,cose⟶,τ⟶−cose0⟶,τ0⟶<0.001,we get the correlations of the leptonic mixing parameters, shown in Figure 10.Figure 10
The correlations ofsin2θ13-sin2θ12 and sin2θ23-δ from the constraints of the geometric correlations I-V.From Figure10, we obtain the following observations:
(a)
sin2θ12 is linearly dependent on sin2θ13 for all the correlations and is consistent with the recent global best fit value [43] under the correlation V. Correspondingly, the range of sin2θ12 is reduced significantly by the geometric constraints. Furthermore, under the constraint intensity, namely, 10−3, the variation of sin2θ12 is at most 2×10−3. Therefore, the breaking of the μ−τ reflection symmetry is mainly determined by the included angles, i.e., cosα0⟶,β0⟶ with α,β=e,μ,τ(b)
The range ofsin2θ23 displays two branches for all the correlations. The right branch of sin2θ23 from the correlation III is consistent with the recent global best fit value [43], while the left branch of sin2θ23 from correlation III is consistent with the best fit value of T2K experiment (sin2θ23=0.53−0.04+0.03) [44](c)
The range ofδ also presents two branches for the correlations I-V. The best global fit value of δ in the NO case [43] can be realized from the correlation I, and the value in the IO case can be obtained from the correlations IV and V(d)
The correlations I and II can be converted to each other through the approximatedμ−τ interchange(e)
We note that the values ofsin2θ12, sin2θ13, sin2θ23, and δ predicted from the modular symmetry origin of texture zeros and quark-lepton unification (see models III+D6 and IV+D6 in the Ref. [45]) are close to the ones shown in the special branch from the correlation I in the top plots of Figure 10. In addition, the values of the mixing angles and the Dirac CP phase predicted from the A5′ modular group (see lepton models L134, L3, and L10 in the Ref. [46]) also approach to the ones in the branch from the correlation I in Figure 10. Thus, the geometric correlations may be realized in the lepton mixing models based on modular symmetries


## 3. The Flavor Ratio of High-Energy Astronomical Neutrinos

In the previous section, we studied the geometric correlations and the corresponding leptonic mixing parameters. Now, we show the implications of geometric correlations on neutrino phenomenology. As a specific example, we discuss the impacts of geometric correlations on the flavor ratio of HANs at Earth in this section.In recent years, a number of HAN events have been detected by the IceCube Collaboration in the energy range of TeV-PeV [47–50]. These neutrinos traveled long distances of the cosmological scale. Their flavor transition probability P¯αβ in the standard framework is expressed as
(8)P¯αβ=Uα12Uβ12+Uα22Uβ22+Uα32Uβ32,where α,β=e,μ,τ and Uαii=1,2,3 is the element of the PMNS matrix. Using the flavor conversion matrix P¯ and the flavor composition at the source of HANs, we can derive the flavor ratio at Earth, i.e., ΦE=P¯ΦS. Here, ΦS and ΦE denote the flavor ratio at the source and that at Earth, respectively. Since neither matter effects nor new physics effects are in favor of neutron decay as the sole source of the HANs [51, 52], we take into account two typical sources in this paper, namely,
(9)muon‐dampingsourcewithΦS=0,1,0T,pion‐decaysourcewithΦS=1/3,2/3,0T.For a given source, the uncertainty of the predicted flavor ratio at Earth is large due to the imprecise leptonic mixing parameters. As is known, the constraints of geometric correlations can significantly reduce the ranges of the leptonic mixing parameters. Accordingly, the precision ofΦE can be notably improved. Here, we show the impacts of the representative geometric correlations on ΦE in Figures 11–13. For the sake of comparison, the flavor ratio predicted with the leptonic mixing parameters at the 3σ level of the global fit data [43] and those constrained by the μ−τ reflection symmetry are also shown in the ternary plots.Figure 11
Ternary plot of the flavor ratio at Earth with the muon-damping decay source. “Standard” denotes the flavor ratio predicted with the leptonic mixing parameters at the3σ level of the global fit data [43]. “μ−τ reflection symmetry” denotes the flavor composition that adds a constrain of cosμ⟶,τ⟶−1<0.001 to the “standard” case. The geometric constraints of correlations I-IV are defined in Table 5 and Equation (7). The 2040 3σ credible region with the pion-decay source based on IceCube and IceCube-Gen2 is taken from Ref. [53]. The 3σ credible region with the pion-decay source based on all TeV-PeV neutrino telescopes available in 2040 is taken from Ref. [54].Figure 12
Ternary plot of the flavor ratio at Earth with the pion-decay source based on the geometric constraints of the correlations I-IV. The conventions of parameters and colors are the same as those in Figure11.Figure 13
Ternary plot of the flavor ratio at Earth with the muon-damping decay source (a) and the pion-decay source (b) based on the geometric constraints of the correlation V. The conventions of parameters and colors are the same as those in Figure11.
(a)(b)From these figures, we obtain the following observations:(a)
For both sources of HANs, the uncertainty of the predicted flavor ratio at Earth is obviously decreased by the geometrical constraints of every correlation. In contrast to the strictμ−τ reflection symmetry, the geometrical correlations have more notable impacts on the predicted flavor ratio(b)
For the muon-damping source (Figures11 and 13(a)), the predicted flavor ratio constrained by the correlation I and/or III is outside the 3σ credible region with the pion-decay source based on all neutrino telescopes [54]. However, a small part of the flavor ratio at Earth, which constrained by the correlation II, IV, and/or V falls into the 3σ credible region with the pion-decay source. Therefore, with the help of geometrical correlations, the muon-damping decay source may be discriminated from the pion-decay source in the near future(c)
For the pion-decay source (Figures12 and 13(b)), the primary part of the flavor ratio constrained by the correlation II and/or IV is outside the 2040 3σ credible region. Thus, the correlations II and IV with this source would be stringently constrained by the observations of the neutrino telescopes in the future


## 4. Conclusions

Theμ−τ reflection symmetry has been proposed for twenty years. Its prediction still satisfies the constraints of the recent global fit data of neutrino oscillations. From the geometric perspective, this symmetry and its breaking can be represented by the angles between the row vectors of the magnitude of the leptonic mixing matrix. Employing the geometric quantities, we studied the promising correlations between the leptonic mixing parameters. On the bases of the mixing parameters constrained by the geometric correlations, the flavor ratio of HANs at Earth with typical sources has been discussed. Our main results are summarized as follows.First, the data points of the included angles of the row vectors are concentrated on several special regions in the scattered plots. In the IO case, one of the regions corresponds to the approximatedμ−τ reflection symmetry. From other regions, promising correlations of leptonic mixing parameters are read out.Second, we find thatsin2θ12 is sensitive to the correlation (cose0⟶,μ0⟶,cose0⟶,τ0⟶) in both the NO case and IO case. In contrast, sin2θ23 and δ change notably under the moderate variation of the correlations (cosμ0⟶,τ0⟶,cose0⟶,μ0⟶) and/or (cosμ0⟶,τ0⟶,cose0⟶,τ0⟶). Furthermore, the ranges of sin2θ23 and δ display disconnected branches. Thus, the included angles of the row vectors can serve as sensitive indexes of the leptonic mixing pattern to test by the neutrino oscillation experiments in future.Third, when the correlation of the three included angles (cose0⟶,μ0⟶,cose0⟶,τ0⟶,cosμ0⟶,τ0⟶) is given, sin2θ12 is linearly dependent on sin2θ13, and sin2θ23 and δ can take discrete values. Hence, the uncertainties of the leptonic mixing are remarkably decreased by the geometric correlations. Correspondingly, the prediction of the flavor ratio of HANs at Earth with typical sources is notably improved. With the help of the special correlations, the discrimination of the sources may be available in the near future.



---

*Source: 6057292-2023-03-21.xml*