arXiv subjects

Chilong Lin

Publications and source records attributed to Chilong Lin.

13 recordsLinked to original sources

arXiv2026

An Exact Analytical Bridge from Complex Yukawa Couplings to Fermion Masses and CKM/PMNS Mixing

The origin of fermion masses and flavor mixing is often obscured by standard Euler-angle parameterizations, which mask the underlying connection between physical observables and Yukawa couplings. We prove that introducing a single, remarkably weak yet physically grounded commutation hypothesis, $[\mathbf{M}_R^2, \mathbf{M}_I^2] = 0$, on the Hermitian mass-squared matrix $\mathbf{M}^2 = M \cdot M^\dagger$ systematically reduces a general 18-parameter complex mass matrix $M$ down to a 5-parameter form per sector that admits \emph{exact} analytic diagonalization. The resulting mass eigenvalues are closed-form algebraic expressions that completely bypass the transcendental Cardano trigonometric reduction required for generic Hermitian matrices, while the diagonalizing unitary matrix $U_f$ depends solely on two parameters that form a dimensionless 2D geometric flavor-ratio vector $\mathbf{v}_f=(x,y)$. Combining two such sectors yields the physical CKM/PMNS mixing matrix directly as $V = U_1^\dagger(\mathbf{v}_1) \cdot U_2(\mathbf{v}_2)$, establishing a direct, first-principles analytical bridge from raw Yukawa couplings to fermion masses and mixing matrices without intermediate phenomenological inputs. Finally, we highlight that this exact leading-order baseline enforces an exact four-fold moduli degeneracy among mixing elements, pointing directly toward non-commuting extensions for full phenomenological precision.

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arXiv2026

An Exact Geometric Framework for Flavor Mixing and CP Violation in the Standard Model

Flavor mixing and CP violation in the Standard Model are conventionally parameterized by Euler angles that entangle the physical contributions of the up- and down-type quark sectors, obscuring the origin of CP violation itself. We present a non-perturbative geometric framework that resolves this entanglement through the commutativity condition $[\mathbf{M}_R^2,\mathbf{M}_I^2]=0$ for the Hermitian components of the mass-squared matrices, reducing the 18 real degrees of freedom of a general $3\times3$ Yukawa matrix to five independent parameters and yielding an exact Cartesian-like parameterization. Explicit diagonalization directly gives fermion masses, CKM matrix elements, and the CP phase $\delta$ in closed form, cleanly separating the distinct physical contributions of the up- and down-type sectors. Mapping flavor space onto a two-dimensional vector geometry, we derive the Jarlskog invariant $J$ in exact closed form and show that CP violation is governed by a discrete topological selection rule ($\eta_{\text{relative}}\in\{0,\pm1\}$) together with the vector cross-product $\mathbf{v}_u\times\mathbf{v}_d=xy'-x'y$. Across all 36 $S_3\times S_3$ flavor topologies, this yields an exact $18/18$ classification into CP-violating ($J \neq 0$) and CP-conserving ($J\equiv 0$) cases, associated with two distinct CP-suppression mechanisms: geometric alignment through a vanishing vector cross-product and algebraic cancellation of imaginary contributions under permutation. At tree level, fitting to experimental CKM elements yields agreement at the $\mathcal{O}(10^{-2})$ level, establishing this framework as a transparent analytical baseline for future non-commutative flavor extensions.

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arXiv2024

Analysis on Neutrino Masses and Leptogenesis in the CP-Violating Standard Model

This article extends the CP-violating Standard Model (CPVSM) from the quark sector to the lepton sector to investigate leptogenesis and neutrino masses. Using the identity $\Delta_{hm} + \Delta_{ml} = \Delta_{hl}$, where $\Delta_{ij} = (m^2_i - m^2_j)$ denoting the mass-squared difference (MSD) between fermions $i$ and $j$, and $h$, $m$, and $l$ labeling the heaviest, middle, and lightest fermions of a given type, respectively, we predict the third neutrino MSD from two experimental inputs: $\Delta_a = 2.51 \times 10^{-3}$ eV$^2$ and $\Delta_b = 7.42 \times 10^{-5}$ eV$^2$. Of six possible assignments of these values to the three $\Delta_{hm}, \Delta_{ml}$, and $\Delta_{hl}$, four consistent cases remain and are grouped into two classes. All four predict similar heaviest and lightest neutrino masses ($m_h = 5.01 \times 10^{-2}$ eV and $m_l = 6.09098 \times 10^{-3}$ eV), but differ in the middle mass: $m_m = 4.97283 \times 10^{-2}$ eV in Class 1, and $m_m = 1.05499 \times 10^{-2}$ eV in Class 2. In a complementary analysis, treating the mass ratio $g = m_h / m_m$ as a variable, we examine how $m_h$, $m_m$, $m_l$, and $g'$ evolve with $g$. Of particular interest are the ranges of $g$ bounded by MSD-based values derived in Subsection III-A (blue points) and values derived from a previously predicted $m_l = 8.61 \times 10^{-3}$ eV (green points). Finally, using the leptonic Jarlskog measure of CPV $\Delta_{CP(l)} = J_{(l)} (\Delta_{ij} \Delta_{jk} \Delta_{ki})_{(\ell)} (\Delta_{ij} \Delta_{jk} \Delta_{ki})_{(\nu)}$, we find that leptogenesis is at least 71 orders of magnitude weaker than baryogenesis in the CPVSM. This striking discrepancy suggests that new physics beyond the Standard Model is required for leptogenesis to account for the observed BAU.

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arXiv2022

BAU production in the $S_N$-breaking Standard Model

CP violation (CPV) and the baryon asymmetry of the universe (BAU) are two of the most significant unresolved problems in physics. This article presents further research on the CPV problem in the standard model (SM) with thirty-two candidate sets of the ten "natural" parameters that exhibit the same CKM performance. These parameters are considered "natural" because they consist solely of the Yukawa couplings and the vacuum expectation value (VEV) of the unique Higgs doublet in the SM. We then investigate the CPV problem and the BAU problem using the Jarlskog measure of CPV, $\Delta_{CP}=J(m^2_t -m^2_c)(m^2_t -m^2_u)(m^2_c -m^2_u)(m^2_b -m^2_s)(m^2_b -m^2_d)(m^2_s -m^2_d)/ T^{12}$, given that CP symmetry is violated following the breakdowns of $S_N$ symmetries. Subsequently, we perform numerical tests in a simplified scenario where eight of the ten parameters are fixed, and the remaining two parameters are allowed to vary from their $S_2$-symmetric values ${\bf (x,~y)}=(-1,~1)$ to their current values in all thirty-two parameter sets. To estimate the enhancement of CPV in such processes, we propose a ratio $R_{\Delta} \equiv \Delta_{CP} / \Delta^{(0)}_{CP}$ between the running $\Delta_{CP}$ and its current value, denoted by $\Delta^{(0)}_{CP}$, which is approximately $10^{-20}$. In all thirty-two cases, the three-dimensional (3D) plots of $R_{\Delta}$ exhibit many regions that stick out of the $R_{\Delta}=10^{10}$ plane, especially in regions very close to ${\bf (x,~y)}=(-1,~1)$. These results demonstrate that the $S_N$-breaking Standard Model is already sufficient to violate CP symmetry explicitly and generate a significant amount of BAU. Furthermore, it solves existing problems without creating new ones.

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arXiv2020

Exploring the Origin of CP Violation in the Standard Model

In this article, we present a very general but not ultimate solution of CPV problem in the standard model. Our study starts from a naturally Hermitian ${\bf M^2}\equiv M^q \cdot M^{q\dagger}$ rather than the previously assumed Hermitian $M^q$. The only assumption employed here is that the real part and imaginary part of $\bf M^2$ can be, respectively, diagonalized by a common $\bf U^q$ matrix. Such an assumption leads to a $\bf M^2$ pattern which depends on only five parameters and can be diagonalized analytically by a $\bf U^q$ matrix which depends on only two of the parameters. Two of the derived mass eigenvalues are predicted degenerate if one of the parameters $\bf C ~(C')$ in up- (down-) quark sector is zero. As the $\bf U^q$ patterns are obtained, thirty-six $V_{CKM}$ candidates are yielded and only eight of them, classified into two groups, fit empirical data within the order of $O(\lambda)$. One of the groups is further excluded in a numerical test, and the surviving group predicts that the degenerate pair in a quark type are the lightest and the heaviest generations rather than the lighter two generations assumed in previous researches. However, there is still one unsatisfactory prediction in this research, a quadruple equality in which four CKM elements of very different values are predicted to be equal. It indicates the $\bf M^2$ pattern studied here is still oversimplified by that employed assumption and the ultimate solution can only be obtained by diagonalizing the unsimplified $\bf M^2$ matrix containing nine parameters directly. The $V_{CKM}$ presented here is already very close to such an ultimate CPV solution.

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arXiv2019

An Improved Standard Model Comes with Explicit CPV and Productive of BAU

In this manuscript, we present an explicit way to describe the violation of CP symmetry in the standard model (SM) of electroweak interactions. In such a way, complex Cabbibo-Kobayashi-Maskawa (CKM) matrices are achieved which stand for the violation of CP symmetry. At the beginning, two necessary but not sufficient conditions for yielding a complex CKM matrix are stated as criteria. Then we found an interesting condition between the real and imaginary parts of a Hermitian $3\times 3$ matrix may provide extra relations among its parameters and reduce the number of them from eighteen down to five. In previous investigations, this can be done only down to nine. With another assumption among some of those parameters, the mass-matrix pattern is further simplified so as to be diagonalized analytically and in consequence four matrices which reveal $S_N$ symmetries among or between quark generations are obtained. In some of such $S_N$-symmetric cases, the derived CKM matrices are complex which indicate that CP symmetry is violated accordingly. Taking the Jarlskog invariant as an estimate of the CPV strength, the value predicted by this model is orders stronger than the empirical value detected experimentally. However, that happens to fill partly the gap between the cosmologically observed amount of Baryon Asymmetry of the Universe (BAU) and that current Standard Model of particle physics predicts. It also proves the long suspected existence of BAU-productive eras in early universe if some fermions were indistinguishable, i.e., $S_N$-symmetric, under circumstances of extremely high temperatures.

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arXiv2017

The Two Higgs-Doublet Extension of the Standard Model and S3 Symmetry

(This is a recompiled version of $Chin. J. of Phys., Vol. 26, No. 4, 180$ since the previous available electronic document on web was unreadable) We impose a permutation symmetry S3 upon the two Higgs doublet extension of the SU(2) x U( 1) standard model, and then break S3 symmetry spontaneously. A mass relation is obtained from which the mass of the top quark is predicted to be 39 Gev at 1 Gev renormalization scale, if the relative strength between the coupling of the first Higgs field with the up-type quarks is assumed to be the same as that with the down-type quarks. We find that in our model the flavor changing neutral currents vanish automatically.

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arXiv2016

Theoretical Origin of CP violation in the FCNC-free 2HDM and Its Extensions to the Standard Model and 3HDM

In this manuscript, a general FCNC-free and CP-violating pattern of quark-mass matrices in the 2HDM derived in our previous investigations is revised. This revised pattern is to be diagonalized analytically with no symmetries imposed. The unitary transformation matrices thus derived depend on only two parameters in each quark type and subsequently lead to a CKM matrix which depends on at most four parameters. The fitting of theoretically derived CKM elements and their corresponding empirical values are as good as ${\bf O}(10^{-2})$ at tree-level. In a phenomenological way which imposes several fine-tuning parameters into the CKM matrix suitably, the fitting is further improved to ${\bf O}(10^{-4})$. After the derivation, we find this hypothesis also applies to the Standard Model and even a model with three Higgs doublets. This will be a big progress in the derivation of a theoretical origin of CP violation.

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arXiv2014

A two-Higgs-doublet model without flavor-changing neutral currents at tree level

The flavor-changing neutral current (FCNC) problem at tree-level is a very critical defect of the two Higgs doublet extension of standard model (SM). In this article, a two-Higgs-doublet model (2HDM) in which such defects do not exist at all is to be demonstrated. The general pattern of matrix pairs which can be diagonalized simultaneously by a same unitary transformation is proposed without extra constraints like symmetries or zeros in M matrices. Only an assumption of the hermiticity of mass matrices is employed in the derivation. With this assumption, number of parameters in the mass matrix of a specific fermion type is reduced from eighteen down to five. Eigenvalues and eigenvectors are analytically derived and it's surprising that unitary transformation matrix thus derived depends on only two parameters. It is a very general and elegant way to solve the tree-level FCNC problem radically and it includes previous similar models as special cases with specific parameter values.

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arXiv2013

Theoretical Origin of CP Violation in a Special FCNC-Free 2HDM with $S_N$ Symmetries

In this manuscript, a way solves the FCNC problem in 2HDM radically and leads to CP violation explicitly is demonstrated. The derivation starts from a most general $3 \times 3$ mass matrix $M$ containing eighteen parameters, which applies to all fermion types, and then reduces the number of parameters down to five with a Hermitian condition modified in this manuscript between the real part and imaginary part of $M$. All matrix pairs thus derived will be diagonalized simultaneously by a same $U$ matrix which means FCNCs won't appear at tree-level anymore. Subsequently, an assumption of $A=A_1=A_2=A_3$ is imposed and four special solutions are thus found which reveal $S_3$ and $S_2$ symmetries among fermion generations respectively. Consequently, as a by-product of solving the FCNC problem, complex CKM matrices appear if the symmetries were assigned to up- and down-type quarks suitably. Unfortunately, CKM elements thus derived are very different to the experimentally detected values. The CP strength of this model, estimated by the Jarlskog invariant, is four orders higher than the one predicted by the standard model. This difference is in fact reasonable since we don't see any such symmetries in our present universe. However, if they had appeared in some early stages of our universe, such a strong CP strength may provide a way to account for at least part of the discrepancy between the observed BAU and that predicted by the standard model.

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arXiv1996

Split Dimensional Regularization for the Temporal Gauge

A split dimensional regularization, which was introduced for the Coulomb gauge by Leibbrandt and Williams, is used to regularize the spurious singularities of Yang-Mills theory in the temporal gauge. Typical one-loop split dimensionally regularized temporal gauge integrals, and hence the renormalization structure of the theory are shown to be the same as those calculated with some nonprincipal-value prescriptions.

hep-th↗

arXiv1993

Spontaneous CP-Violation in the Extended Standard Models

The spontaneous CP-violation in three kinds of extension of the standard model are investigated. We find that if no additional symmetry is introduced to reduce the number of coefficients, then all three extended models possess the spontaneous CP violation (SCPV) in the Higgs potential. When symmetry under which the doublets transform as $Φ_1 \to Φ_1$ and $Φ_2 \to e^{iγ}Φ_2$ is introduced, the two-Higgs doublet model will lose its SCPV terms. In the other two extended models, the SCPV terms will be preserved since both models contain at least one Higgs singlet which can appear in odd powers. The extended model with two doublets and one singlet is reviewed and the general conditions which make the CP-violating $V_{min}$ the true minimum are given. The generally called natural flavour-conserving models are also discussed briefly.

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arXiv1993

Feynman Rules in the Type III Natural Flavour-Conserving Two-Higgs Doublet Model

We consider a two Higgs-doublet model with $S_3$ symmetry, which implies a $π\over 2$ rather than 0 relative phase between the vacuum expectation values $<Φ_1>$ and $<Φ_2>$. The corresponding Feynman rules are derived accordingly and the transformation of the Higgs fields from the weak to the mass eigenstates includes not only an angle rotation but also a phase transformation. In this model, both doublets couple to the same type of fermions and the flavour-changing neutral currents are naturally suppressed. We also demonstrate that the Type III natural flavour-conserving model is valid at tree-level even when an explicit $S_3$ symmetry breaking perturbation is introduced to get a reasonable CKM matrix. In the special case $β= α$, as the ratio $\tanβ= {v_2 \over v_1}$ runs from 0 to $\infty$, the dominant Yukawa coupling will change from the first two generations to the third generation. In the Feynman rules, we also find that the charged Higgs currents are explicitly left-right asymmetric. The ratios between the left- and right-handed currents for the quarks in the same generations are estimated.

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