# The Nonsymmetric Flavor Transition Matrix and the Apparent P Violation

Authors: Shu-Jun Rong; Ding-Hui Xu
Journal: Advances in High Energy Physics
 (2022)

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## Abstract

The leptonic mixing parameters of high precision and the next-generation neutrino telescopes make it possible to test new physics in the flavor transition of the high-energy astrophysical neutrinos (HAN). We introduce a nonsymmetric matrix to modify the predictions of the standard flavor transition matrix. It is constructed with the mixing matrix in vacuum and that at the source of the HAN. The mismatch of the mixing matrices results in the new expectation of the flavor ratio of the HAN at Earth. It also leads to a secondary effect called the apparent P violation (APV). The quantitative analyses of the new effects are performed with a moderate setup of the parameters at the source of the HAN. The correlations between the mixing parameters and the new predictions are shown. From the correlations, the dominant parameters determining the new-physics effects are identified.


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## 1. Introduction

Since the late 1990s, notable progresses have been made in the measurement of neutrino oscillation parameters [1–3]. According to the global analysis NuFIT 5.0 with the Super-Kamiokande atmospheric neutrino data, the 3σ relative precision of leptonic mixing angles and Dirac CP phase is shown as follows [4]:
(1)θ12:14%,θ13:9%,θ23:25%,δCP:100%.Obviously, we have the well-measuredθ13 and the uncertain δCP. In the following decades, the precision of the mixing parameters will be improved by results from the oscillation experiments such as JUNO [5], Hyper-Kamiokande (HK) [6], DUNE [7], etc. Then, new physics on neutrinos may be tested. As we know, new physics usually lies in the extreme environment or the extreme parameter range. Recently, a large number of high-energy astrophysical neutrinos (HAN) events in the energy range of Tev-Pev have been reported by the IceCube Collaboration [8–11]. These events may provide information on new physics.In this paper, we are concerned about the effects of new physics in the flavor transition of the HAN. Because the HAN detected at Earth have traveled cosmological-scale distances, the decoherence effects of flavor neutrinos should be considered. So the flavor conversion probabilityP¯αβ is of the form
(2)P¯αβ=Uα12Uβ12+Uα22Uβ22+Uα32Uβ32,where α,β=e,μ,τ. Uαii=1,2,3 is the element of the leptonic mixing matrix [12]. The so-called standard flavor transition matrix P¯ is symmetric, which is obtained on the base of the assumption that the mixing matrix of the HAN at their source and that in their journey are the same as that in vacuum, namely, U. Any scenario beyond the assumption expressed by a different flavor transition matrix could manifest a class of new physics. Typical new physics includes the energy-dependent leptonic mixing matrix [13, 14], neutrino-decay [15–22], nonunitary mixing [23–30], pseudo-Dirac neutrinos [31–36], neutrino secret interactions [37–40], etc. [41]. These new-physics-based transition matrices predict various of flavor ratios of the HAN at Earth. However, correlations between the parameters in the transition matrix and the new predictions are usually unclear in these scenarios. In this paper, we propose a simplified nonsymmetric flavor transition matrix and analyse the relationship between the new effects and the mixing parameters.In principle, matter interactions [42–45], nonstandard interactions [46–48], and other new physics may modify the leptonic mixing matrix at the source of the HAN. Similar to the case of the solar neutrinos [49], we suppose that the HAN in their motherland undergo an adiabatic flavor evolution. Then, we can generalize the flavor transition probability as
(3)P¯αβg=Uα1S2Uβ12+Uα2S2Uβ22+Uα3S2Uβ32.Here,US is the leptonic mixing matrix at the location where the HAN are produced. Although P¯g is not of the most general form, it could describe several classes of nonstandard flavor conversion scenarios in the adiabatic cases. As an example, when US is unitary, it could denote a mixing matrix including the effect of matter interaction or nonstandard interactions. In this paper, we focus on the model-independent properties and predictions of P¯g.In general, the flavor transition matrixP¯g is nonsymmetric. It may give different predictions on the HAN flavor ratio at Earth. Furthermore, suppose that we interchange the departure location and the arrival location of neutrinos. Namely, we make a P transformation UαiS2Uβi2⟶Uαi2UβiS2, with i=1, 2, 3. The transition probability of the α-flavor neutrinos departing from Earth would be expressed as P¯gαβP=P¯βαg. In the case α≠β, we have P¯αβg≠P¯βαg. So an apparent P violation (APV) quantified by P¯αβg−P¯βαg would be observed. The APV means that the P violation is from the environment where neutrinos are produced. Furthermore, we note that P¯gαβT=P¯βαg. So the APV is equal to the apparent T violation (ATV).In practice, the new effects fromP¯g are impacted by the uncertainties of the parameters in US. We have no precision information on US which is dependent on the specific source of the HAN. So we employ a model-independent method to describe US. In the standard parametrization [12], ∣US∣ is dependent on the mixing angles θijS(ij=12,23,13) and the CP phase δS. We consider the case that the relative deviations of these parameters from their counterparts in vacuum are moderate, i.e., of the order of 10%. These setups of Us enable us to find out the correlation between the new predictions and the leptonic mixing parameters. These observations may hold for neutrinos produced in the realistic astrophysical sources such as galactic halos and cluster halos where the matter effect on US is modest [50].The paper is organised as follows. In Section2, the properties of the generalized flavor transition matrix P¯g are studied. On the basis of these properties, a generalized area theorem of the flavor triangle at Earth is derived. It reveals how the flavor ratio distribution at Earth is dependent on the determinant of P¯g. The expression of the APV is given in this section. The possible case that the APV may be detected is discussed. In Section 3, we perform numerical analyses of the impacts of P¯g. The correlation between the determinant of the matrices P¯g and that of P¯ is given. The quantitative flavor ratio distribution at Earth on the basis of P¯g and the special initial flavor ratios are obtained. The correlations between the APV and the mixing parameters are analysed. Finally, we conclude.


## 2. The Generalized Flavor Transition Matrix and New-Physics Effects

### 2.1. Flavor Ratio Vector and Flavor Triangle

The general neutrino flavor ratio could be expressed with a column vector, namely,(4)ϕ⟶=ϕeϕμϕτT,which satisfies the following conditions
(5)ϕe+ϕμ+ϕτ=1,0≤ϕα≤1.So the set of the points of all the flavor ratio vectors forms a triangle in the 3-dimensional Euclidean space. Following the convention of Ref. [51], we call it flavor triangle, see Figure 1. The vertexes of the flavor triangle correspond to the vectors of the form
(6)ϕ1⟶=100T,ϕ2⟶=010T,ϕ3⟶=001T.Figure 1
The schematic figure of the flavor triangle and the Earth flavor triangle (EFT).The area of the flavor triangle isSΔFT=3/2. Note that ϕ1⟶ and ϕ2⟶ correspond to the flavor ratio at the neutron-decay source and that at the muon-damping source, respectively.Suppose that the flavor ratio of the HAN at the source is given byϕS⟶. When the HAN arrive at Earth, ϕS⟶ is changed to ϕE⟶. It is connected to ϕS⟶ through the flavor transition matrix, namely,
(7)ϕeEϕμEϕτE=P¯eegP¯eμgP¯eτgP¯μegP¯μμgP¯μτgP¯τegP¯τμgP¯ττg,ϕeSϕμSϕτS.Here, the expression ofP¯αβg is shown in Eq. (3). After the flavor transition of HAN, the flavor triangle is transformed to that at Earth. We call it Earth flavor triangle (EFT). The schematic figure of the EFT is also shown in Figure 1.Given the matrixP¯g, the EFT shows the general flavor ratio distribution of the HAN at Earth. The vertexes of the EFT correspond to the vectors ϕ1E⟶, ϕ2E⟶, and ϕ3E⟶. They are the first, the second, and the third column vector of the flavor transition matrix P¯g, respectively, namely,
(8)ϕ1E⟶=P¯eegP¯μegP¯τeg,ϕ2E⟶=P¯eμgP¯μμgP¯τμg,ϕ3E⟶=P¯eτgP¯μτgP¯ττg.The area of the EFT is determined by the determinant ofP¯g. The specific expression is shown in the following part.


### 2.2. Unitarity and the Generalized Area Theorem for the EFT

In this paper, we consider the case that the mixing matricesU and US are unitary. So the columns vectors ϕ1E⟶, ϕ2E⟶, and ϕ3E⟶ satisfy the constraints
(9)∑iϕ1iE=∑iϕ2iE=∑iϕ3iE=1,(10)ϕ1E⟶+ϕ2E⟶+ϕ3E⟶=3o⟶,where o⟶ is the democratic vector of the form
(11)o⟶=1/31/31/3T.On the basis of unitarity conditions Eqs. (9) and (10), we can derive an identity for the column vectors of P¯g, namely,
(12)ϕ1E⟶×ϕ2E⟶+ϕ2E⟶×ϕ3E⟶+ϕ3E⟶×ϕ1E⟶=DetP¯gϕ1E⟶+ϕ2E⟶+ϕ3E⟶=3DetP¯go⟶.Here, Det denotes the determinant,DetP¯g=ϕ1E⟶×+ϕ2E⟶·ϕ3E⟶. The dot product and cross product of flavor vectors are defined in the standard way, namely,
(13)a⟶·b⟶=∑iaibi,d⟶=a⟶×b⟶,withdi=∑j,kεijkajbk,where εijk is the totally antisymmetric tensor with ε123=1. The proof of the identity is given as follows.In the case that the column vectors are linearly independent, namely,DetP¯g≠0, we can decompose the circular cross product of the vectors as
(14)ϕ1E⟶×ϕ2E⟶+ϕ2E⟶×ϕ3E⟶+ϕ3E⟶×ϕ1E⟶=xϕ1E⟶+yϕ2E⟶+zϕ3E⟶.Then employing the identity(15)ϕ1E⟶×ϕ2E⟶·ϕ3E⟶=ϕ2E⟶×ϕ3E⟶·ϕ1E⟶=ϕ3E⟶×ϕ1E⟶·ϕ2E⟶,we obtain the following equations
(16)P¯gTP¯gxyz=ϕ1E⟶·ϕ1E⟶ϕ1E⟶·ϕ2E⟶ϕ1E⟶·ϕ3E⟶ϕ2E⟶·ϕ3E⟶ϕ2E⟶·ϕ2E⟶ϕ2E⟶·ϕ3E⟶ϕ3E⟶·ϕ1E⟶ϕ3E⟶·ϕ2E⟶ϕ3E⟶·ϕ3E⟶xyz=3DetP¯go⟶.According to the constraints Eqs. (9) and (10), we find that o⟵ is the eigenvector of the coefficient matrix P¯gTP¯g with the eigenvalue equal to 1. Because of the condition DetP¯g≠0, we obtain the result x=y=z=DetP¯g. So the identity is proved.Now we consider the case that column vectors are linearly dependent, namely,DetP¯g=0. On the basis of the constraints Eqs. (9) and (10), vectors ϕ2E⟶ and ϕ3E⟶ could be decomposed as
(17)ϕ2E⟶=x1ϕ1E⟶+1−x1o⟶,(18)ϕ3E⟶=−1−x1ϕ1E⟶+2+x1o⟶.Substituting these expressions into the circular cross product, we obtain the following result:(19)ϕ1E⟶×ϕ2E⟶+ϕ2E⟶×ϕ3E⟶+ϕ3E⟶×ϕ1E⟶=0ϕ1E⟶×o⟶=000T.So the identity also holds in this case.Employing the identity, we can obtain the area of the EFT. The area vector of the EFT is expressed as(20)S⟶ΔEFT=12ϕ2E⟶−ϕ1E⟶×ϕ3E⟶−ϕ2E⟶=12ϕ1E⟶×ϕ2E⟶+ϕ2E⟶×ϕ3E⟶+ϕ3E⟶×ϕ1E⟶=32DetP¯go⟶.So the area of the EFT is derived, namely,(21)SΔEFT=32DetP¯g.This formula is called the area theorem in Ref. [51] in the case that the transition matrix is P¯ of the form Eq. (2). Our derivations reveal that the theorem results from the constraints Eqs. (9) and (10). Furthermore, we can see that if two matrices M1 and M2 meet the constraints, so does their product M1M2. Therefore, besides of P¯ and P¯g, this theorem holds for a general class of flavor transition matrices. So we call it the generalized area theorem for the EFT. It is useful for the analysis of the uncertainties of the flavor ratio at the source of the HAN.


### 2.3. The Determinant ofP¯g

The generalized theorem demonstrates that the area of the EFT is determined by the determinant ofP¯g. In order to analyse the impacts of the matrix US on DetP¯g, we show the expression of DetP¯g. Following the standard parametrization of the leptonic mixing matrix [12], U is written as
(22)U=c12c13s12c13s13e−iδCP−s12c23−c12s13s23eiδCPc12c23−s12s13s23eiδCPc13s23s12s23−c12s13c23eiδCP−c12s23−s12s13c23eiδCPc13c23,where sij≡sinθij, cij≡cosθij. Here, the Majorana phases are not shown because they are irrelevant to the flavor transition matrix. Similarly, the mixing matrix US is expressed as
(23)US=c12Sc13Ss12Sc13Ss13Se−iδCPS−s12Sc23S−c12Ss13Ss23SeiδCPSc12Sc23S−s12Ss13Ss23SeiδCPSc13Ss23Ss12Ss23S−c12Ss13Sc23SeiδCPS−c12Ss23S−s12Ss13Sc23SeiδCPSc13Sc23S,where sijS≡sinθij, cijS≡cosθijS. The superscript S labels the parameters at the source of the HAN. According to the expression of P¯g, namely, Eq. (3), DetP¯g is of the form
(24)DetP¯g=DetUsqDetUsqS,where
(25)DetUsq=c232−s232c122−s122c132−s132+2s13s12s23c12c23cosδCP,(26)DetUsqs=c23S2−s23S2c12S2−s12S2c13S2−s13S2+2s13Ss12Ss23Sc12Sc23ScosδCPS.


### 2.4. The Apparent P Violation fromP¯g

A notable characteristic of the transition matrixP¯g is the nonsymmetry. The APV could be measured by P¯μeg−P¯eμg, P¯eτg−P¯τeg, or P¯μτg−P¯τμg. According to the unitary conditions, namely, Eqs. (9) and (10), we find the following identities
(27)P¯μeg−P¯eμg=P¯eτg−P¯τeg=P¯τμg−P¯μτg.So we chooseΔeμ≡P¯μeg−P¯eμg to describe the APV. In the standard parametrization, Δeμ is expressed as
(28)Δeμ=XcosδCP+YcosδCPS+Z,where X,Y,Z are quantities dependent on the mixing angles in the matrices U and US. The relationship between Δeμ and the mixing parameters is obtained with the numerical analysis in the next section.In order to detect the APV, we should find out the relationship betweenΔeμ and the flavor ratio vector at Earth. Note that ϕE⟶ is dependent not only on P¯g but also on the specific source of the HAN. Here, we consider the case that the HAN at Earth are from two different sources, namely, the pion-decay source ϕS1⟶ and the muon-damping source ϕS2⟶, where ϕS1⟶ is of the form
(29)ϕS1⟶=13230T.ϕS2⟶ is just the vector ϕ⟶2 shown in Eq. (6). We suppose that the neutrino telescopes in the future could discriminate the sources S1 and S2 with the help of other astronomical observatories. Furthermore, we assume that the difference between the flavor transition matrix of the source S1 and that of S2 is negligible, namely, P¯S1g=P¯S2g=P¯g. Under this condition, the flavor ratio at Earth with the HAN from S1 and that from S2 are expressed, respectively, as
(30)ϕE1⟶=13ϕ1E⟶+23ϕ2E⟶,ϕE2⟶=ϕ2E⟶,where ϕ1E⟶ and ϕ2E⟶ are the column vectors of the matrix P¯g shown in Eq. (8). Then, the relationship between the APV and the flavor ratios at Earth is obtained, namely,
(31)Δeμ=3ϕμE1−2ϕμE2−ϕeE2.In general cases, the assumptionP¯S1g=P¯S2g may not hold. The APV cannot be identified through the measurement of ϕ1E⟵ and ϕ2E⟵. Even so, the term at the right hand of Eq. (31) could still serve as an index of the P¯g deviation from P¯. Namely, if this term is nonzero, we can determine that the flavor transition matrix of the HAN is not P¯.


## 2.1. Flavor Ratio Vector and Flavor Triangle

The general neutrino flavor ratio could be expressed with a column vector, namely,(4)ϕ⟶=ϕeϕμϕτT,which satisfies the following conditions
(5)ϕe+ϕμ+ϕτ=1,0≤ϕα≤1.So the set of the points of all the flavor ratio vectors forms a triangle in the 3-dimensional Euclidean space. Following the convention of Ref. [51], we call it flavor triangle, see Figure 1. The vertexes of the flavor triangle correspond to the vectors of the form
(6)ϕ1⟶=100T,ϕ2⟶=010T,ϕ3⟶=001T.Figure 1
The schematic figure of the flavor triangle and the Earth flavor triangle (EFT).The area of the flavor triangle isSΔFT=3/2. Note that ϕ1⟶ and ϕ2⟶ correspond to the flavor ratio at the neutron-decay source and that at the muon-damping source, respectively.Suppose that the flavor ratio of the HAN at the source is given byϕS⟶. When the HAN arrive at Earth, ϕS⟶ is changed to ϕE⟶. It is connected to ϕS⟶ through the flavor transition matrix, namely,
(7)ϕeEϕμEϕτE=P¯eegP¯eμgP¯eτgP¯μegP¯μμgP¯μτgP¯τegP¯τμgP¯ττg,ϕeSϕμSϕτS.Here, the expression ofP¯αβg is shown in Eq. (3). After the flavor transition of HAN, the flavor triangle is transformed to that at Earth. We call it Earth flavor triangle (EFT). The schematic figure of the EFT is also shown in Figure 1.Given the matrixP¯g, the EFT shows the general flavor ratio distribution of the HAN at Earth. The vertexes of the EFT correspond to the vectors ϕ1E⟶, ϕ2E⟶, and ϕ3E⟶. They are the first, the second, and the third column vector of the flavor transition matrix P¯g, respectively, namely,
(8)ϕ1E⟶=P¯eegP¯μegP¯τeg,ϕ2E⟶=P¯eμgP¯μμgP¯τμg,ϕ3E⟶=P¯eτgP¯μτgP¯ττg.The area of the EFT is determined by the determinant ofP¯g. The specific expression is shown in the following part.


## 2.2. Unitarity and the Generalized Area Theorem for the EFT

In this paper, we consider the case that the mixing matricesU and US are unitary. So the columns vectors ϕ1E⟶, ϕ2E⟶, and ϕ3E⟶ satisfy the constraints
(9)∑iϕ1iE=∑iϕ2iE=∑iϕ3iE=1,(10)ϕ1E⟶+ϕ2E⟶+ϕ3E⟶=3o⟶,where o⟶ is the democratic vector of the form
(11)o⟶=1/31/31/3T.On the basis of unitarity conditions Eqs. (9) and (10), we can derive an identity for the column vectors of P¯g, namely,
(12)ϕ1E⟶×ϕ2E⟶+ϕ2E⟶×ϕ3E⟶+ϕ3E⟶×ϕ1E⟶=DetP¯gϕ1E⟶+ϕ2E⟶+ϕ3E⟶=3DetP¯go⟶.Here, Det denotes the determinant,DetP¯g=ϕ1E⟶×+ϕ2E⟶·ϕ3E⟶. The dot product and cross product of flavor vectors are defined in the standard way, namely,
(13)a⟶·b⟶=∑iaibi,d⟶=a⟶×b⟶,withdi=∑j,kεijkajbk,where εijk is the totally antisymmetric tensor with ε123=1. The proof of the identity is given as follows.In the case that the column vectors are linearly independent, namely,DetP¯g≠0, we can decompose the circular cross product of the vectors as
(14)ϕ1E⟶×ϕ2E⟶+ϕ2E⟶×ϕ3E⟶+ϕ3E⟶×ϕ1E⟶=xϕ1E⟶+yϕ2E⟶+zϕ3E⟶.Then employing the identity(15)ϕ1E⟶×ϕ2E⟶·ϕ3E⟶=ϕ2E⟶×ϕ3E⟶·ϕ1E⟶=ϕ3E⟶×ϕ1E⟶·ϕ2E⟶,we obtain the following equations
(16)P¯gTP¯gxyz=ϕ1E⟶·ϕ1E⟶ϕ1E⟶·ϕ2E⟶ϕ1E⟶·ϕ3E⟶ϕ2E⟶·ϕ3E⟶ϕ2E⟶·ϕ2E⟶ϕ2E⟶·ϕ3E⟶ϕ3E⟶·ϕ1E⟶ϕ3E⟶·ϕ2E⟶ϕ3E⟶·ϕ3E⟶xyz=3DetP¯go⟶.According to the constraints Eqs. (9) and (10), we find that o⟵ is the eigenvector of the coefficient matrix P¯gTP¯g with the eigenvalue equal to 1. Because of the condition DetP¯g≠0, we obtain the result x=y=z=DetP¯g. So the identity is proved.Now we consider the case that column vectors are linearly dependent, namely,DetP¯g=0. On the basis of the constraints Eqs. (9) and (10), vectors ϕ2E⟶ and ϕ3E⟶ could be decomposed as
(17)ϕ2E⟶=x1ϕ1E⟶+1−x1o⟶,(18)ϕ3E⟶=−1−x1ϕ1E⟶+2+x1o⟶.Substituting these expressions into the circular cross product, we obtain the following result:(19)ϕ1E⟶×ϕ2E⟶+ϕ2E⟶×ϕ3E⟶+ϕ3E⟶×ϕ1E⟶=0ϕ1E⟶×o⟶=000T.So the identity also holds in this case.Employing the identity, we can obtain the area of the EFT. The area vector of the EFT is expressed as(20)S⟶ΔEFT=12ϕ2E⟶−ϕ1E⟶×ϕ3E⟶−ϕ2E⟶=12ϕ1E⟶×ϕ2E⟶+ϕ2E⟶×ϕ3E⟶+ϕ3E⟶×ϕ1E⟶=32DetP¯go⟶.So the area of the EFT is derived, namely,(21)SΔEFT=32DetP¯g.This formula is called the area theorem in Ref. [51] in the case that the transition matrix is P¯ of the form Eq. (2). Our derivations reveal that the theorem results from the constraints Eqs. (9) and (10). Furthermore, we can see that if two matrices M1 and M2 meet the constraints, so does their product M1M2. Therefore, besides of P¯ and P¯g, this theorem holds for a general class of flavor transition matrices. So we call it the generalized area theorem for the EFT. It is useful for the analysis of the uncertainties of the flavor ratio at the source of the HAN.


## 2.3. The Determinant ofP¯g

The generalized theorem demonstrates that the area of the EFT is determined by the determinant ofP¯g. In order to analyse the impacts of the matrix US on DetP¯g, we show the expression of DetP¯g. Following the standard parametrization of the leptonic mixing matrix [12], U is written as
(22)U=c12c13s12c13s13e−iδCP−s12c23−c12s13s23eiδCPc12c23−s12s13s23eiδCPc13s23s12s23−c12s13c23eiδCP−c12s23−s12s13c23eiδCPc13c23,where sij≡sinθij, cij≡cosθij. Here, the Majorana phases are not shown because they are irrelevant to the flavor transition matrix. Similarly, the mixing matrix US is expressed as
(23)US=c12Sc13Ss12Sc13Ss13Se−iδCPS−s12Sc23S−c12Ss13Ss23SeiδCPSc12Sc23S−s12Ss13Ss23SeiδCPSc13Ss23Ss12Ss23S−c12Ss13Sc23SeiδCPS−c12Ss23S−s12Ss13Sc23SeiδCPSc13Sc23S,where sijS≡sinθij, cijS≡cosθijS. The superscript S labels the parameters at the source of the HAN. According to the expression of P¯g, namely, Eq. (3), DetP¯g is of the form
(24)DetP¯g=DetUsqDetUsqS,where
(25)DetUsq=c232−s232c122−s122c132−s132+2s13s12s23c12c23cosδCP,(26)DetUsqs=c23S2−s23S2c12S2−s12S2c13S2−s13S2+2s13Ss12Ss23Sc12Sc23ScosδCPS.


## 2.4. The Apparent P Violation fromP¯g

A notable characteristic of the transition matrixP¯g is the nonsymmetry. The APV could be measured by P¯μeg−P¯eμg, P¯eτg−P¯τeg, or P¯μτg−P¯τμg. According to the unitary conditions, namely, Eqs. (9) and (10), we find the following identities
(27)P¯μeg−P¯eμg=P¯eτg−P¯τeg=P¯τμg−P¯μτg.So we chooseΔeμ≡P¯μeg−P¯eμg to describe the APV. In the standard parametrization, Δeμ is expressed as
(28)Δeμ=XcosδCP+YcosδCPS+Z,where X,Y,Z are quantities dependent on the mixing angles in the matrices U and US. The relationship between Δeμ and the mixing parameters is obtained with the numerical analysis in the next section.In order to detect the APV, we should find out the relationship betweenΔeμ and the flavor ratio vector at Earth. Note that ϕE⟶ is dependent not only on P¯g but also on the specific source of the HAN. Here, we consider the case that the HAN at Earth are from two different sources, namely, the pion-decay source ϕS1⟶ and the muon-damping source ϕS2⟶, where ϕS1⟶ is of the form
(29)ϕS1⟶=13230T.ϕS2⟶ is just the vector ϕ⟶2 shown in Eq. (6). We suppose that the neutrino telescopes in the future could discriminate the sources S1 and S2 with the help of other astronomical observatories. Furthermore, we assume that the difference between the flavor transition matrix of the source S1 and that of S2 is negligible, namely, P¯S1g=P¯S2g=P¯g. Under this condition, the flavor ratio at Earth with the HAN from S1 and that from S2 are expressed, respectively, as
(30)ϕE1⟶=13ϕ1E⟶+23ϕ2E⟶,ϕE2⟶=ϕ2E⟶,where ϕ1E⟶ and ϕ2E⟶ are the column vectors of the matrix P¯g shown in Eq. (8). Then, the relationship between the APV and the flavor ratios at Earth is obtained, namely,
(31)Δeμ=3ϕμE1−2ϕμE2−ϕeE2.In general cases, the assumptionP¯S1g=P¯S2g may not hold. The APV cannot be identified through the measurement of ϕ1E⟵ and ϕ2E⟵. Even so, the term at the right hand of Eq. (31) could still serve as an index of the P¯g deviation from P¯. Namely, if this term is nonzero, we can determine that the flavor transition matrix of the HAN is not P¯.


## 3. Numerical Analyses of Impacts ofP¯g

In this section, we perform numerical analyses to study the impacts of the mixing parameters inP¯g on the neutrino flavor ratio at Earth and the APV. Because of the lack of the information of the source of the HAN, we adopt a model-independent method to describe US. In order to identify the correlation between the parameters in US and the new predictions of P¯g, we consider the case that the relative deviation of US from the vacuum mixing matrix U is moderate, namely, of the order of 10%. The specific setup of the parameters of US is expressed as follows:
(32)s13S2=s132+Δs132,withΔs132∈−0.02,0.02,(33)s12S2=s122+Δs122,withΔs122∈−0.2,0.2,(34)s23S2=s232+Δs232,withΔs232∈−0.2,0.2,(35)δCPS=δCP+ΔδCP,withΔδCP∈−1,1radian.The parameterssij2 and δCP are taken from the global fit data at 3σ level [4]. The modifications of the mixing parameters are randomly taken in the given ranges.


### 3.1. Uncertainties of the Flavor Ratios at the Source with the Compression MapP¯g

We scanned the determinants ofP¯ and P¯g with the mixing parameters given in the above setup. For the comparison of the impacts of the different parameters, we modify one parameter at a time. The correlation of DetP¯ and DetP¯g is shown in Figure 2.Figure 2
The correlation betweenDetP¯ and DetP¯g. The mixing angles and the CP phase in U are in the ranges of the global fit data of the normal mass ordering at 3σ level [4]. The results are similar in the case of the inverted mass ordering. The modified parameter in Us is randomly taken in the corresponding range in Eqs. (32)–(35).We can see that the range of the magnitude ofDetP¯g is approximate to that of DetP¯. According to the generalized area theorem for the EFT, the ratio SΔEFT/SΔFT is in the range 0,0.025. Considering the modification of the mixing parameter in US, the upper bound of the ratio varies in the range 0.023,0.028. Therefore, similar to the matrix P¯, P¯g is a compression map for the general flavor triangle. After the flavor transition, the uncertainty of the flavor ratio at the source is decreased noticeably by P¯g.For the quantitative illustration, we decompose the general flavor ratio at the source with the eigenvectors ofP¯g, i.e.,
(36)ϕS⟶=o⟶+xv1⟶+yv2⟶,where v1,2⟶ satisfies the constraint from P¯g, namely, ∑ivi1,2=0. The eigen-equations are listed as
(37)P¯go⟶=λ0go⟶,P¯gv1⟶=λ1gv1⟶,P¯gv2⟶=λ2gv2⟶,with the eigenvalues expressed as
(38)λ0g=1,λ1,2g=TrP¯g−12±TrP¯g−12−4DetP¯g2.Here,Tr denotes the trace of a matrix. The uncertainty of the flavor ratio at the source could be expressed as
(39)ΔϕS⟶=Δxv1⟶+Δyv2⟶.After the flavor transition, the corresponding uncertainty of the flavor ratio at Earth is of the form(40)ΔϕE⟶=λ1gΔxv1⟶+λ2gΔyv2⟶.So the eigenvaluesλ1g,λ2g can serve as the indices of the compression ratio of the uncertainty of the flavor ratio at the source. We show the ranges of λ1g,λ2g in Figure 3.Figure 3
The eigenvalues of the matrixP¯g. The conventions of the mixing parameters are the same as those in Figure 2.We can see that the flavor uncertainty indexλ1gΔx at Earth is around 35% of the initial value. λ2gΔy is several percent of the initial value. Particularly, if the eigenvalue λ2g is equal to 0, we will lose the information on the uncertainty of the flavor ratio expressed by Δy. These observations still hold for the matrix P¯g with the modification of Δs122 or Δs132. In fact, on the basis of the expression of DetP¯g shown in Eqs. (24)–(26), the nature of P¯g as a compression map is determined by the ranges of the mixing parameters in U. This nature will not be changed with the improvement of the precision of the mixing parameters unless P¯g is constructed with the matrix completely different from U.


### 3.2. Flavor Ratio at Earth with the Special Sources

Now we analyse the distribution of the flavor ratio at Earth with the special sources. We consider three typical sources, namely, the muon-damping sourceϕS2⟶=010T, the neutron-decay source ϕS3⟶=100T, and the pion-decay source ϕS1⟶=1/32/30T. The flavor ratios at Earth with these sources are shown in Figures 4 and 5. From these figures, we can obtain the following observations:
(i)
For every source, the impact ofΔs132 on the distribution of the flavor ratio at Earth is negligible because it is relatively small. Furthermore, the CP phases δCP, δCPS in the matrix P¯g manifest themselves in the terms s13cosδCP and s13ScosδCPS, respectively. So, their impacts on the flavor ratio at Earth are undermined intensively by small 1-3 mixing angles in U and Us. The dominant parameters in P¯g are s12, s23, s12S, and s23S. We note that a similar observation for the standard matrix P¯ has been obtained in the recent Ref. [52](ii)
For the neutron-decay source and muon-damping source,s23S, s12S can enlarge the region of the flavor ratio at Earth noticeably. However, they cannot exempt the neutron-decay source from the constraint of the 3σ boundary of the 2015 IceCube data [53]. So, the new physics in P¯g is still not in favor of the neutron-decay as the unique source of the HAN(iii)
For the pion-decay source, the impacts ofs12S, s23S on the distribution of the flavor ratio at Earth are weakened by the approximate μ−τ symmetry of the mixing matrix U. Here, the μ−τ symmetry denotes the property ∣Uμi∣=∣Uτi∣i=1,2,3. As is known, on the basis of the strict μ−τ symmetry and the standard transition matrix P¯, the flavor ratio at Earth for the HAN from the pion-decay source is just o⟶=1/31/31/3T [54, 55]. In our case that Us deviates from U moderately, the flavor ratio at Earth also deviates from the democratic vector o⟶ moderately(iv)
From the light cyan regions in the figures, we can see that the observations in (i)-(iii) are not changed by the joint impacts ofs12S, s23S in P¯gFigure 4
The flavor ratio at Earth. (a) is for the neutron-decay source1,0,0. (b) is for the muon-damping source 0,1,0. Except for the light cyan region, the conventions of the mixing parameters are the same as those in Figure 2. The light cyan region corresponds to the case that all the mixing parameters are modified in US. The specific ranges for the modifications are listed as Δs132∈−0.01,0.01, Δs122∈−0.1,0.1, Δs232∈−0.1,0.1, and ΔδCP∈−0.5,0.5 radian. The red region corresponds to the prediction of the standard matrix A. The 3σ constraint on the flavor ratio is from the 2015 IceCube data [53].Figure 5
The flavor ratio at Earth with the pion-decay source1/3,2/3,0. The conventions of the mixing parameters are the same as those in Figure 4.On the basis of the above observations, our setup of the matrixP¯g shows clear correlations between the distribution of the flavor ratio at Earth and the parameters of the mixing matrices. The dominant factors in US are identified, namely, s12S, s23S. Their impacts on the flavor ratios can be discriminated. So the information on new physics can be read out from the distribution of the flavor ratio at Earth.


### 3.3. The Apparent P Violation Modulated by the CP Phase

In principle, the APV results from the mismatch betweenU and US. The mismatch is quantified by the modifications Δsij2, ΔδCP. The correlations between the APV and the mixing parameters are analysed. We find that the APV is modulated by the Dirac phase, see Figures 6 and 7. From the figures, we obtain the following observations:
(i)
The APV is modulated by the Dirac phase. This characteristic results from the expression ofΔeμ, see Eq. (28). In the case Δs232≠0, the oscillation of Δeμ is undermined. In the case Δs232=0, ΔδCP=0, we have Δeμ~0 when δCP~3π/2. Correspondingly, Δeμ becomes maximal when δCP is trivial. In the case ΔδCP≠0, Δsij2=0, the zero point of Δeμ is around 3π/2+ΔδCP or π/2+ΔδCP(ii)
The impact of the mass ordering of neutrinos on the APV is not negligible. The reason is that the range ofδCP at 3σ level with the normal mass ordering is wider than that with the inverted mass ordering, see the specific range of δCP in Figures 6 and 7(iii)
The oscillating amplitude ofΔeμ is determined by the magnitude of Δsij2, ΔδCP, see Figure 8 for example. Compared with Δs122, the modification Δs232 of the same order of Δs122 can lead to the larger amplitudeFigure 6
The correlation between the apparent P violation (APV) and the CP phase. The mixing parameters inU are in the 3σ ranges of the global fit data of the normal mass ordering [4]. The modified parameter in Us is fixed in every plot. The other parameters in Us are the same as those in U.Figure 7
The correlation between the apparent P violation (APV) and the CP phase with the inverted mass ordering. The conventions of the modified mixing parameters are the same as those in Figure6.Figure 8
The correlation between the apparent P violation (APV) with different amplitudes. The conventions of the mixing parameters are the same as those in Figure6. The blue scatter points are for the case Δs122=0.2, the red points for the case Δs122=0.1, the green points for the case Δs122=−0.1, and the black points for the case Δs122=−0.2. (a) is for the case of the normal mass ordering. (b) is for the case of the inverted mass ordering.On the basis of these observations, we find that the magnitude ofΔeμ is mainly dominated by the parameters Δs232, δCP. Therefore, although δCP is not important for the distribution of the flavor ration at Earth, we may still identify its impact on the APV when Δs232 is negligible. So in the era of the precise detection the flavor composition of the HAN, we may obtain the information on δCP through the secondary effect such as the APV.


### 3.4. Comment on Effects of Large Modifications inUS

Our observations on the flavor ratios of HAN at Earth and the APV are obtained in the case thatUS differs from U modestly. When large modification of mixing parameters in US is considered, we could expect that the region of the flavor ratio distribution of HAN at Earth for the special source would be expanded noticeably. Furthermore, the correlation between APV and the Dirac CP phase would disappear under the large modification of Δs232. In special cases that only Δs122 and Δs132 are relatively large, the correlation could still be identified.


## 3.1. Uncertainties of the Flavor Ratios at the Source with the Compression MapP¯g

We scanned the determinants ofP¯ and P¯g with the mixing parameters given in the above setup. For the comparison of the impacts of the different parameters, we modify one parameter at a time. The correlation of DetP¯ and DetP¯g is shown in Figure 2.Figure 2
The correlation betweenDetP¯ and DetP¯g. The mixing angles and the CP phase in U are in the ranges of the global fit data of the normal mass ordering at 3σ level [4]. The results are similar in the case of the inverted mass ordering. The modified parameter in Us is randomly taken in the corresponding range in Eqs. (32)–(35).We can see that the range of the magnitude ofDetP¯g is approximate to that of DetP¯. According to the generalized area theorem for the EFT, the ratio SΔEFT/SΔFT is in the range 0,0.025. Considering the modification of the mixing parameter in US, the upper bound of the ratio varies in the range 0.023,0.028. Therefore, similar to the matrix P¯, P¯g is a compression map for the general flavor triangle. After the flavor transition, the uncertainty of the flavor ratio at the source is decreased noticeably by P¯g.For the quantitative illustration, we decompose the general flavor ratio at the source with the eigenvectors ofP¯g, i.e.,
(36)ϕS⟶=o⟶+xv1⟶+yv2⟶,where v1,2⟶ satisfies the constraint from P¯g, namely, ∑ivi1,2=0. The eigen-equations are listed as
(37)P¯go⟶=λ0go⟶,P¯gv1⟶=λ1gv1⟶,P¯gv2⟶=λ2gv2⟶,with the eigenvalues expressed as
(38)λ0g=1,λ1,2g=TrP¯g−12±TrP¯g−12−4DetP¯g2.Here,Tr denotes the trace of a matrix. The uncertainty of the flavor ratio at the source could be expressed as
(39)ΔϕS⟶=Δxv1⟶+Δyv2⟶.After the flavor transition, the corresponding uncertainty of the flavor ratio at Earth is of the form(40)ΔϕE⟶=λ1gΔxv1⟶+λ2gΔyv2⟶.So the eigenvaluesλ1g,λ2g can serve as the indices of the compression ratio of the uncertainty of the flavor ratio at the source. We show the ranges of λ1g,λ2g in Figure 3.Figure 3
The eigenvalues of the matrixP¯g. The conventions of the mixing parameters are the same as those in Figure 2.We can see that the flavor uncertainty indexλ1gΔx at Earth is around 35% of the initial value. λ2gΔy is several percent of the initial value. Particularly, if the eigenvalue λ2g is equal to 0, we will lose the information on the uncertainty of the flavor ratio expressed by Δy. These observations still hold for the matrix P¯g with the modification of Δs122 or Δs132. In fact, on the basis of the expression of DetP¯g shown in Eqs. (24)–(26), the nature of P¯g as a compression map is determined by the ranges of the mixing parameters in U. This nature will not be changed with the improvement of the precision of the mixing parameters unless P¯g is constructed with the matrix completely different from U.


## 3.2. Flavor Ratio at Earth with the Special Sources

Now we analyse the distribution of the flavor ratio at Earth with the special sources. We consider three typical sources, namely, the muon-damping sourceϕS2⟶=010T, the neutron-decay source ϕS3⟶=100T, and the pion-decay source ϕS1⟶=1/32/30T. The flavor ratios at Earth with these sources are shown in Figures 4 and 5. From these figures, we can obtain the following observations:
(i)
For every source, the impact ofΔs132 on the distribution of the flavor ratio at Earth is negligible because it is relatively small. Furthermore, the CP phases δCP, δCPS in the matrix P¯g manifest themselves in the terms s13cosδCP and s13ScosδCPS, respectively. So, their impacts on the flavor ratio at Earth are undermined intensively by small 1-3 mixing angles in U and Us. The dominant parameters in P¯g are s12, s23, s12S, and s23S. We note that a similar observation for the standard matrix P¯ has been obtained in the recent Ref. [52](ii)
For the neutron-decay source and muon-damping source,s23S, s12S can enlarge the region of the flavor ratio at Earth noticeably. However, they cannot exempt the neutron-decay source from the constraint of the 3σ boundary of the 2015 IceCube data [53]. So, the new physics in P¯g is still not in favor of the neutron-decay as the unique source of the HAN(iii)
For the pion-decay source, the impacts ofs12S, s23S on the distribution of the flavor ratio at Earth are weakened by the approximate μ−τ symmetry of the mixing matrix U. Here, the μ−τ symmetry denotes the property ∣Uμi∣=∣Uτi∣i=1,2,3. As is known, on the basis of the strict μ−τ symmetry and the standard transition matrix P¯, the flavor ratio at Earth for the HAN from the pion-decay source is just o⟶=1/31/31/3T [54, 55]. In our case that Us deviates from U moderately, the flavor ratio at Earth also deviates from the democratic vector o⟶ moderately(iv)
From the light cyan regions in the figures, we can see that the observations in (i)-(iii) are not changed by the joint impacts ofs12S, s23S in P¯gFigure 4
The flavor ratio at Earth. (a) is for the neutron-decay source1,0,0. (b) is for the muon-damping source 0,1,0. Except for the light cyan region, the conventions of the mixing parameters are the same as those in Figure 2. The light cyan region corresponds to the case that all the mixing parameters are modified in US. The specific ranges for the modifications are listed as Δs132∈−0.01,0.01, Δs122∈−0.1,0.1, Δs232∈−0.1,0.1, and ΔδCP∈−0.5,0.5 radian. The red region corresponds to the prediction of the standard matrix A. The 3σ constraint on the flavor ratio is from the 2015 IceCube data [53].Figure 5
The flavor ratio at Earth with the pion-decay source1/3,2/3,0. The conventions of the mixing parameters are the same as those in Figure 4.On the basis of the above observations, our setup of the matrixP¯g shows clear correlations between the distribution of the flavor ratio at Earth and the parameters of the mixing matrices. The dominant factors in US are identified, namely, s12S, s23S. Their impacts on the flavor ratios can be discriminated. So the information on new physics can be read out from the distribution of the flavor ratio at Earth.


## 3.3. The Apparent P Violation Modulated by the CP Phase

In principle, the APV results from the mismatch betweenU and US. The mismatch is quantified by the modifications Δsij2, ΔδCP. The correlations between the APV and the mixing parameters are analysed. We find that the APV is modulated by the Dirac phase, see Figures 6 and 7. From the figures, we obtain the following observations:
(i)
The APV is modulated by the Dirac phase. This characteristic results from the expression ofΔeμ, see Eq. (28). In the case Δs232≠0, the oscillation of Δeμ is undermined. In the case Δs232=0, ΔδCP=0, we have Δeμ~0 when δCP~3π/2. Correspondingly, Δeμ becomes maximal when δCP is trivial. In the case ΔδCP≠0, Δsij2=0, the zero point of Δeμ is around 3π/2+ΔδCP or π/2+ΔδCP(ii)
The impact of the mass ordering of neutrinos on the APV is not negligible. The reason is that the range ofδCP at 3σ level with the normal mass ordering is wider than that with the inverted mass ordering, see the specific range of δCP in Figures 6 and 7(iii)
The oscillating amplitude ofΔeμ is determined by the magnitude of Δsij2, ΔδCP, see Figure 8 for example. Compared with Δs122, the modification Δs232 of the same order of Δs122 can lead to the larger amplitudeFigure 6
The correlation between the apparent P violation (APV) and the CP phase. The mixing parameters inU are in the 3σ ranges of the global fit data of the normal mass ordering [4]. The modified parameter in Us is fixed in every plot. The other parameters in Us are the same as those in U.Figure 7
The correlation between the apparent P violation (APV) and the CP phase with the inverted mass ordering. The conventions of the modified mixing parameters are the same as those in Figure6.Figure 8
The correlation between the apparent P violation (APV) with different amplitudes. The conventions of the mixing parameters are the same as those in Figure6. The blue scatter points are for the case Δs122=0.2, the red points for the case Δs122=0.1, the green points for the case Δs122=−0.1, and the black points for the case Δs122=−0.2. (a) is for the case of the normal mass ordering. (b) is for the case of the inverted mass ordering.On the basis of these observations, we find that the magnitude ofΔeμ is mainly dominated by the parameters Δs232, δCP. Therefore, although δCP is not important for the distribution of the flavor ration at Earth, we may still identify its impact on the APV when Δs232 is negligible. So in the era of the precise detection the flavor composition of the HAN, we may obtain the information on δCP through the secondary effect such as the APV.


## 3.4. Comment on Effects of Large Modifications inUS

Our observations on the flavor ratios of HAN at Earth and the APV are obtained in the case thatUS differs from U modestly. When large modification of mixing parameters in US is considered, we could expect that the region of the flavor ratio distribution of HAN at Earth for the special source would be expanded noticeably. Furthermore, the correlation between APV and the Dirac CP phase would disappear under the large modification of Δs232. In special cases that only Δs122 and Δs132 are relatively large, the correlation could still be identified.


## 4. Conclusions

Inspired by the case of the solar neutrino oscillations, we proposed a generalized matrixP¯g to describe the flavor transition of the HAN. This matrix is constructed with U in vacuum and US at the source of the HAN. The mismatch between U and US leads to the nonsymmetry of P¯g which shows itself through the APV. In order to learn about the impacts of P¯g on the flavor ratio of the HAN at Earth and the APV, we studied the model-independent properties of P¯g with a vector-analysis method. On the basis of the unitarity of U and US, the generalized area theorem on the EFT is derived. The area of the EFT is determined by DetP¯g. With the setup that US deviates from U moderately, the impact of DetP¯g on the uncertainty of the flavor ratio at the source of the HAN is analysed. The range of DetP¯g and those of the eigenvalues of P¯g shows that the uncertainty at the source is decreased intensively by P¯g after the flavor transition of the HAN. The parameters in US cannot change the nature of P¯g as the compression map of the general flavor triangle.In the standard parametrization ofU and US, the correlations between the parameters of US and the distribution of the flavor ratio at Earth are shown. Compared with the results from the standard flavor transition matrix P¯, we find that the dominant factors in US for the flavor ratio are s12S and s23S. Furthermore, the impacts of s12S and that of s23S can be discriminated in the ternary plots of the flavor ratios at Earth. As a secondary effect of the new physics in US, the APV is quantified by Δeμ. The correlation between the Δeμ and the mixing parameters in P¯g is obtained. We find that Δeμ is modulated by the Dirac CP phase δCP. So the information on δCP may be obtained from the detection of the APV. In practice, Δeμ is difficult to detect when the deviation of US from U is moderate. We propose a special case that Δeμ may be identified through the flavor ratios at Earth with two distinguishable sources of the HAN, namely, Δeμ=3ϕμE1−2ϕμE2−ϕeE2. In general cases, this detection scenario may not work. However, the term 3ϕμE1−2ϕμE2−ϕeE2 is still a useful index to test the symmetry of the standard matrix P¯ in the future. Thus, our work may serve as a theoretical preparation for the era of the precise detection of the flavor composition of the HAN.



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*Source: 6949022-2022-03-12.xml*