arXiv subjects

Masaki J. S. Yang

Publications and source records attributed to Masaki J. S. Yang.

At least 19 recordsLinked to original sources

arXiv2026

Perturbative determination of quark CP phase and rephasing invariants of hierarchical mass matrices using their inverses

In this paper, we derive approximate expressions for the CP phase $δ$ in the CKM matrix by a perturbative singular value decomposition for hierarchical mass matrices of quarks $m_{u,d}$. The diagonalization is achieved through a seesaw-like procedure in which the heavier generations are successively integrated out, naturally leading to mixing matrices expressed in terms of the mass matrices and their inverses $m_{u,d}^{-1}$. As a result, in the basis where the up-type quark mass matrix is diagonal, $δ$ is reduced to a fourth-order invariant $δ\simeq \arg [ - m^{-1}_{d12} m_{d23}^{} / m^{-1}_{d11} m_{d13}^{} ]$. Furthermore, in the Kobayashi--Maskawa parametrization, the CP phase is likewise expressed by an invariant $δ_{\rm KM} \simeq \arg [ m^{-1}_{d13} m_{d33}^{} / m^{-1}_{d11} m_{d13}^{} ] \simeq π/2$ in the same basis. Since the next-to-leading-order corrections are suppressed at second order in the perturbation, the accuracy of these expressions is typically of order $O(λ^{2}) \sim 4 \%$.

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arXiv2026

Rephasing invariant CP phases and sum rules in TM$_{1,2}$ mixing

We show that the CP phases $ϕ_{1,2}$ appearing in the TM$_{1,2}$ mixing are directly identified with rephasing invariants $ϕ_{1} = - \arg \left[ { U_{e2} U_{e3} U_{μ1} U_{τ1} / U_{e 1} \det U } \right]$, $ϕ_{2} = - \arg \left[ { U_{e1} U_{e3} U_{μ2} U_{τ2} / U_{e 2} \det U} \right]$. Furthermore, we demonstrate identities $ϕ_{i} = δ- \arg [ U_{μi}^{0} U_{τi}^{0} ]$ among $ϕ_{1,2}$, the Dirac CP phase $δ$ and matrix elements in the PDG parametrization $U^{0}_{αi}$. These relations of CP phases are interpreted as specific elements of general sum rules represented by phase matrices.

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arXiv2026

Explicit Rephasing Transformation to PDG Parameterization and Simplified Expression of the Dirac CP Phase by Fermion-Specific Invariants

In this letter, we present an explicit rephasing transformation that maps an arbitrary mixing matrix $U$ to the PDG standard parameterization $U^{0} = {\rm diag} ( e^{ - i \arg U_{e1} } \, , \, e^{ i \arg [ {U_{e2} U_{τ3} \over \det U } ]} \, , \, e^{ i \arg [ { U_{e2} U_{μ3} \over \det U } ] } )$ $U \, {\rm diag} ( 1 \, , \, e^{ - i \arg [{U_{e2} \over U_{e1} } ] } \, , \, e^{ - i \arg [ { U_{e2} U_{μ3} U_{τ3} \over \det U } ] })$. By this procedure, which has remained largely conceptual for more than four decades, six independent phases of the mixing matrix are expressed systematically in terms of the arguments. We also apply this framework to the fermion diagonalization matrices $U^{ν,e}$ under the approximation $U^{ν,e}_{13}=0$. By partially employing the inversion formula, we factorize all CP phases through the explicit rephasing, thereby revealing the entire CP structure of the mixing matrix. The observable Dirac phase depends only on two relative phases between the fermion sectors and is determined by fermion-specific invariants.

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arXiv2026

Rephasing invariant structure of CP phase for simplified mixing matrices in Fritzsch--Xing parametrization

In this paper, we construct an explicit rephasing transformation that converts an arbitrary unitary mixing matrix into the Fritzsch--Xing (FX) parametrization, which is obtained by trivializing arguments of the matrix elements in the third row and third column. We further analyze rephasing invariant structure of the FX phase $δ_{\rm FX}$ under an approximation $U_{13}^{e} = 0$, where the 1-3 element of the diagonalization matrix of charged leptons $U^{e}$ is neglected. With an additional approximation $U_{23}^{e} = 0$, the FX phase becomes highly simplified, reducing to a sum of the neutrino-intrinsic FX phase $δ^ν_{\rm FX}$ and the contribution from the relative phase $ρ'_{1}- ρ'_{2}$ between the lighter 1-2 generations. The phase $δ_{\rm FX}$ for finite $U_{23}^{e}$ is understood as a generalization of the compact expression. This result covers almost all perturbative calculations of CP phases for the CKM and MNS matrices with hierarchical charged fermions.

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arXiv2026

Explicit rephasing to Kobayashi-Maskawa representation and fundamental phase structure of CP violation

In this letter, we construct an explicit rephasing transformation that converts an arbitrary unitary matrix into the Kobayashi--Maskawa (KM) parameterization and identify all independent CP phases in the mixing matrix as the arguments of its matrix elements. Furthermore, by applying this rephasing transformation to the fermion diagonalization matrices $U^{f}$, we show that the Majorana phases are represented by fermion-specific phases $δ^{ν, e}_{\rm KM}$ and their relative phases. In particular, by neglecting the 3-1 elements $U_{31}^{ν,e}$ of the diagonalization matrices for the two fermions, the KM phase $δ_{\rm KM}$ is concisely expressed by fermion-specific rephasing invariants involving two relative phases $δ_{\rm KM} = \arg \left [1 + ({U^{e * }_{21} U^ν_{21} / U^{e * }_{11} U^ν_{11} }) \right ] + \arg \left [ - { U_{32}^{e *} U_{32}^ν / U^{e * }_{22} U^ν_{22} } \right] $.

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arXiv2026

Rephasing invariant structure of Dirac CP phase and basis independent reduction of unitarity constraints for mixing matrices

In this paper, we explore rephasing invariant structures of the Dirac CP phase $δ$ under an approximation $U^{e}_{13} = 0$, where the 1-3 element of the diagonalization of charged leptons $U^{e}$ is neglected. With the further simplified condition $U^{e}_{12} = 0$, the Dirac phase reduces to a compact form $δ= δ^ν + \arg [ (U_{33}^e / U_{23}^{e}) - ( U_{33}^ν / U_{23}^ν) ] - \arg [ ( U_{33}^e / U_{23}^{e}) + (U_{23}^{ν*} / U_{33}^{ν*} ) ] $, and the CP phase for finite $U_{12}^{e}$ can be understood as a generalization of this compact form. These results encompass almost all perturbative calculations of the CP phases in quark and lepton mixing matrices with hierarchical masses of charged fermions, and are independent of any specific parametrization. As a second result of this work, we derive a basis independent reduction of the unitarity constraints for an arbitrary unitary matrix $V$ by eliminating the elements $V_{21}, V_{22}, V_{31}, V_{32}$ using the inversion formula. Applying the explicit rephasing transformation to this reduction yields a rephasing invariant representation of the PDG parametrization, which allows the translation of theoretical results expressed in the PDG parametrization directly into rephasing invariants.

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arXiv2026

Rephasing Invariant Formula for CP Phase in Kobayashi-Maskawa Parametrization and Exact Sum Rule with Unitarity Triangle $δ_{\rm PDG} + δ_{\rm KM} = π- α+ γ$

In this letter, we obtain a rephasing invariant formula for the CP phase in the Kobayashi--Maskawa parameterization $δ_{\rm KM} = \arg [ - { V_{ud} \det V_{\rm CKM} / V_{us} V_{ub} V_{cd} V_{td}} ]$. General perturbative expansion of the formula and observed value $δ_{\rm KM} \simeq π/2$ reveal that the phase difference of the 1-2 mixings $e^{i (ρ_{12}^{d} - ρ_{12}^{u})}$ is close to maximal for sufficiently small 1-3 quark mixings $s_{13}^{u,d}$. Moreover, combining this result with another formula for the CP phase $δ_{\rm PDG}$ in the PDG parameterization, we derived an exact sum rule $δ_{\rm PDG} + δ_{\rm KM} = π- α+ γ$ which relating the phases and the angles $α, β, γ$ of the unitarity triangle.

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arXiv2026

Direct Evaluation of CP Phase of CKM matrix, General Perturbative Expansion and Relations with Unitarity Triangles

In this letter, using a rephasing invariant formula $δ= \arg [ { V_{ud} V_{us} V_{c b} V_{tb} / V_{ub} \det V_{\rm CKM} }]$, we evaluate the CP phase $δ$ of the CKM matrix $V_{\rm CKM}$ perturbatively for small quark mixing angles $s_{ij}^{u,d}$ with associated phases $ρ_{ij}^{u,d}$. Consequently, we derived a relation $δ\simeq \arg [Δs_{12} Δs_{23} / ( Δs_{13} - s^u_{12} e^{-i ρ^u_{12}} Δs_{23} )]$ with $Δs_{ij} \equiv s^d_{ij} e^{-i ρ^d_{ij}} - s^u_{ij} e^{-i ρ^u_{ij}}$. Such a result represents the analytic behavior of the CKM phase. The uncertainty in the relation is of order $O(λ^{2}) \sim 4\%$, which is comparable to the current experimental precision. Comparisons with experimental data suggest that the hypothesis of some CP phases being maximal. We also discussed relationships between the phase $δ$ and unitarity triangles. As a result, several relations between the angles $α, β, γ$ and $δ$ are identified through other invariants $V_{il} V_{jm} V_{kn} / \det V_{\rm CKM}$.

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arXiv2025

Explicit rephasing transformation of four-generation mixing matrix and formulae for CP phases

In this letter, we present an explicit rephasing transformation that maps a general $4\times4$ flavor mixing matrix to the standard parametrization of four-generation models. By combining rephasing-covariant minors and the determinant systematically, we derive expressions for all three Dirac-type CP phases, three Majorana phases, and four unphysical phases by arguments of matrix elements. The resulting formulae smoothly reduce to the three-generation results in the limit where the fourth generation decouples.

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arXiv2025

Geometric representation of CP phases $δ_{\rm PDG}, δ_{\rm KM}$ in flavor mixing matrix and its sum rule by alternative unitarity triangle and quadrangle

In this letter, we present a geometric representation of the CP phases $δ_{\rm PDG}$ and $δ_{\rm KM}$ in the PDG and Kobayashi--Maskawa parameterizations of the flavor mixing matrix in the complex plane. The sum rule with the unitarity triangle $δ_{\rm PDG} + δ_{\rm KM} = π- α+ γ$ is expressed as a quadrangle, which is a combination of a unitarity triangle and an alternative triangle. Through the unitarity quadrangle, the CP phases are also identified with specific geometric angles. Furthermore, a new set of inverse unitarity triangles is defined from the inversion formula of a unitary matrix $U^{\dagger} = U^{-1}$. These novel triangles contain standard angles of the form $\arg [U_{αi } U_{βj} U_{αj}^{*} U_{βi}^{*}]$ and new angles $\arg [U_{αi } U_{βj} U_{γk} / \det U]$, which directly determine nontrivial arguments of the mixing matrix elements.

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arXiv2025

An Analytic Prescription for $t$-channel Singularities

The $t$-channel singularity is a divergence in the scattering amplitude which occurs when a stable particle propagating in $t$-channel scattering process becomes an on-shell state. Such situations appear either in the system of collider experiments or in the context of the cosmological particle production. No scheme which is generally applicable is known. In this work, we propose a new formulation to identify and remove the source of the divergence. The scheme is fully analytical and various applications can be expected. This work provides a valuable tool in this research field.

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arXiv2025

Unified description of sum rules and duality between CP phases and unitarity triangles through third-order rephasing invariants

In this letter, we demonstrate that products of third-order rephasing invariants $V_{αi} V_{βj} V_{γk} / \det V$ of flavor mixing matrix $V$ reproduce all the nine angles of unitarity triangles and all the CP phases in the nine parameterizations of $V$. The sum rules relating the CP phases and angles are also decomposed into terms of these rephasing invariants. Furthermore, through ninth-order invariants, these fourth- and fifth-order invariants become equivalent, which can be regarded as a certain duality. For the phase matrix $Δ$ and the angle matrix $Φ$, $Δ\pm Φ$ are expressed in terms of even-permutations $X$ and odd-permutations $Ψ$ of third-order invariant. As a result, these are represented by the two concise matrix equations $Φ= Ψ- {\rm X}$ and $Δ= Π' - Ψ- {\rm X}$.

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arXiv2025

Rephasing invariant formulae for CP phases in general parameterizations of flavor mixing matrix and exact sum rules with unitarity triangles

In this letter, we present rephasing invariant formulae $δ^{(αi)} = \arg [ { V_{α1} V_{α2} V_{α3} V_{1i} V_{2i} V_{3i} / V_{αi }^{3} \det V } ] $ for CP phases $δ^{(αi)}$ associated with nine Euler-angle-like parameterizations of a flavor mixing matrix. Here, $α$ and $i$ denote the row and column carrying the trivial phases in a given parameterization. Furthermore, we show that the phases $δ^{(αi)}$ and the nine angles $Φ_{αi}$ of unitarity triangles satisfy compact sum rules $ δ^{(α, i+2)} - δ^{(α, i+1)} = Φ_{α-2, i} - Φ_{α-1, i}$ and $ δ^{(α-2, i)} - δ^{(α-1, i)} = Φ_{α, i+2} - Φ_{α, i+1}$ where all indices are taken cyclically modulo three. These twelve relations are natural generalizations of the previous result $δ_{\mathrm{PDG}}+δ_{\mathrm{KM}}=π-α+γ$.

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arXiv2025

Analytic structure of the MNS matrix with diagonal reflection symmetries

In this paper, we survey the analytic structure of the MNS matrix with diagonal reflection symmetries. If the mass matrix of charged leptons $m_{e}$ is hierarchical (i.e., $|(m_{e})_{33}| \simeq m_τ \gg |(m_{e})_{1i , j1}|$), by neglecting the 1-3 mixing of $m_{e}$, the MNS matrix is represented by four parameters and several sign degrees of freedom. By substituting the three observed mixing angles $θ_{ij}$ as input parameters, the Dirac phase $δ$ and the Majorana phases $α_{2,3}$ are represented by functions of the 1-2 mixing of charged leptons $s_{e}$. The effective mass of double beta decay $m_{ee}$ is also displayed as a function of $s_{e}$ and the lightest neutrino mass $m_{1 \, \rm or \, 3}$. Because the generalized CP symmetries restrict the effective mass to near the maximum or minimum value in the whole parameter region, several scenarios are suggested to be excluded by the latest limit of the KAMLAND-Zen collaboration.

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arXiv2025

Almost general analysis of $μ$-$τ$ reflection symmetry perturbed by charged leptons and its testability by DUNE and T2HK

In this paper, we generally analyze the $μ- τ$ reflection symmetry modified by small mixings of charged leptons and how will future experiments verify deviations from the predictions of the symmetry. As an approximation, the left-handed diagonalization $U_{e}$ of charged leptons is assumed to have a similar magnitude as the CKM matrix. In other words, the 1-3 mixing is neglected and the 1-2 and 2-3 mixing are to be approximately $O(0.1)$. The Dirac CP phase $δ$ of the MNS matrix is evaluated in such parameter regions. As a result, deviations from the predictions $\sin θ_{23} = π/4$ and $δ= \pm π/2$ depend on relative CP phases between $U_{e}$ and diagonalization of neutrinos $U_ν$. While phases of the second and third generations cause only about $\pm 10^{\circ}$ deviations for the Dirac phase $δ$, the phase of the first generation can cause up to $\pm 30^{\circ}$. This flavor dependence is distinguished to some extent by the next-generation experiments. On the other hand, if $δ$ is not observed, such a scenario is excluded by about 5 years of observation.

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arXiv2025

A Rephasing Invariant Formula for the Dirac CP Phase and General Perturbative Expansion: Prospects for DUNE and T2HK

We present a formula for the Dirac CP phase $δ= \arg ( U_{e1} U_{e2} U_{μ3} U_{τ3} / U_{e3} \det U_{\rm MNS} )$, directly derived from the lepton mixing matrix in an arbitrary basis of phases. In contrast to the numerically suppressed Jarlskog invariant, this expression is computationally simple and less sensitive to approximations. We apply the formula to derive general perturbative corrections from charged-lepton mixing $s_{ij}^{e}$ to the underlying CP phase of neutrinos $δ_ν$. A compact analytic expressions $δ= δ_ν + s_{12}^{e} D_{12} + s_{13}^{e} D_{13} + s_{23}^{e} D_{23}$ shows that these corrections can substantially exceed $O(10^{\circ})$, potentially within the reach of future long-baseline experiments such as DUNE and T2HK.

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arXiv2024

Almost general analysis of CKM and MNS matrices for hierarchical Yukawa structure and interpretation of Dirac CP phase probed by DUNE and T2HK

In this letter, we perform an almost general analysis of flavor-mixing matrices $V_{\rm CKM}$ and $U_{\rm MNS}$ to investigate the discriminative power of CP phases by next-generation neutrino oscillation experiments. As an approximation, we neglect the 1-3 mixing of diagonalization for more hierarchical fermions $u,e$. Thus there are two sources of CP violation in $V_{\rm CKM}$ and $U_{\rm MNS}$, the intrinsic CP phase $δ_{d, ν}$ in diagonalization of less hierarchical fermions $d, ν$ and relative phases between two unitary matrices. By eliminating unphysical phases and imposing constraints of the three measured mixing angles, the flavor-mixing matrices are analytically displayed by two phases and the 1-2 mixing $s_{u, e}$ of more hierarchical fermions. For sufficiently small 1-2 mixing $s_{e}$ of charged leptons, the Dirac phase $δ$ is mostly identical to the intrinsic phase of neutrinos $δ_ν$. Therefore, future detection of the Dirac phase indicates the observation of $δ_ν$. On the other hand, if such a CP violation is not observed, an upper limit is placed on a combination of $δ_ν$ and relative phases.

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arXiv2024

Allowed region for the (second) lightest mass $M_{1,2}$ of right-handed neutrino $ν_{R 1,2}$ with $SO (10)$-inspired relations and sequential dominance

In this paper, we evaluate the (second) lightest mass $M_{1,2}$ of right-handed neutrino $ν_{R1,2}$ in grand unified theories with the type-I seesaw mechanism that predicts an almost massless neutrino $m_{1 \, \rm or \, 3} \sim 0$. By chiral perturbative treatment, the masses $M_{1,2}$ are expressed as $M_{1} = m_{D1}^{2}/m_{11} , \, M_{2} = m_{D2}^{2} m_{11} / (m_{11} m_{22} - m_{12}^{2})$ with the mass matrix of left-handed neutrinos $m$ in the diagonal basis of the Dirac mass matrix $m_{D}$. Assuming $m_{Di}$ and the unitary matrix $V$ in the singular value decomposition $(m_{D})_{ij} = V_{ik} m_{D k} U^{\dagger}_{kj}$ are close to observed fermion masses and the CKM matrix, $M_{1,2}$ and their allowed regions are expressed by parameters in the low energy and unknown phases. As a result, for $m_{D1} \simeq 0.5$ MeV and $m_{D2} \simeq 100$ MeV, we obtain $M_{1}^{\rm NH} \simeq 3 \times 10^{4 - 6}$ GeV and $M_{2}^{\rm NH} \simeq 3 \times 10^{6-8}$ GeV in the NH, $M_{1}^{\rm IH} \simeq 5 \times 10^{3 - 4}$ GeV and $M_{2}^{\rm IH} \simeq 4 \times 10^{8-9}$ GeV in the IH. These upper and lower bounds are proportional to $m_{Di}^{2}$ or $m_{D1} m_{D2}$.

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