Electroweak Theory with Symmetry Broken Gamma Matrices and Fermion Boson Duality

A Unified Formulation and Experimental Tests of Electroweak Theory with Symmetry-Broken Gamma Matrices and Fermion–Boson Duality Hirokazu Maruyama∗ December 11, 2025 Abstract In this work we reformulate the Glashow–Weinberg–Salam electroweak theory on the basis of two concepts: symmetry-broken 4 ×4 gamma matrices and a fermion– boson duality. First, we decompose the Dirac gamma matrices into components that mediate the weak interaction and components that generate masses, and embed the SU (2)L×U(1)Ygauge structure into a four-component lepton spinor space. In this way we derive geometrically, without assuming left-handed doublets and right-handed singlets from the outset, the purely left-handed couplings of W±and the chiral-asymmetric couplings of the Zboson. Within the same framework the Higgs vacuum expectation value is interpreted as a local symmetry breaking in gamma-matrix space, and the Wand Zboson masses obtained from the resulting mass formula agree well with the experimental values. Next, we introduce bosonic gamma matrices constructed from two equivalent Dirac systems and show that their anticommutation relations automatically project out the time-like and longitudinal components, leaving only the two transverse degrees of freedom. Furthermore, we extend to the electroweak sector a fermion–boson duality previously proposed in quantum electrodynamics and describe lepton fields as a two-layer structure composed of a massive fermionic component and an almost massless bosonic component via an energy-dependent transition function T(E). From the internal consistency of the duality we show that the existence of a neutral lepton paired with the neutral boson Zis theoretically inevitable, providing a new basis for neutrinos and their peculiar mass-generation mechanism. The resulting Lagrangian preserves gauge invariance, the Ward–Takahashi identities, and BRST symmetry, is consistent with existing precision electroweak data, and yields testable predictions such as modifications of electroweak interactions in superconductors and duality effects in highenergy neutrino scattering. Keywords Electroweak theory, symmetry-broken gamma matrices, fermion–boson duality, neutrino mass, gauge symmetry, mass-generation mechanism, chiral structure, Ward–Takahashi identities, BRST symmetry ∗Independent researcher 1 Contents 1 Introduction 5 1.1 Success and open problems of the electroweak unified theory ........ 5 1.2 Origin of the chiral structure .......................... 5 1.3 Geometric understanding of the mass-generation mechanism ........ 6 1.4 Neutrino mass and its role ........................... 6 1.5 Fermion–boson duality: a new paradigm ................... 6 1.6 Electroweak theory in terms of symmetry-broken gamma matrices . . . . . 7 1.7 Aim and structure of this paper ........................ 7 2 Standard electroweak theory revisited 8 2.1 Structure of the SU(2)L×U(1)Ygauge theory ................ 8 2.1.1 Field content and quantum numbers ................. 8 2.1.2 Gauge fields and covariant derivatives ................. 9 2.2 Spontaneous symmetry breaking and the Higgs mechanism ......... 9 2.2.1 Higgs field and potential ........................ 9 2.2.2 Mass generation for gauge bosons ................... 10 2.2.3 Mass eigenstates ............................ 10 2.3 Chiral structure: manual implementation in the theory ........... 11 2.3.1 Charged-current interactions ...................... 11 2.3.2 Neutral-current interactions ...................... 11 2.4 Problems of the chiral structure: critical analysis .............. 11 2.4.1 Asymmetries in the construction of the theory ............ 11 2.4.2 Mixture of experimental facts and theoretical assumptions ..... 12 2.4.3 Mass hierarchy problem ........................ 12 2.5 Need to go beyond the Standard Model .................... 13 3 Electroweak theory with gamma matrices 13 3.1 Introducing the 4 ×4 gamma-matrix representation ............. 13 3.1.1 Dirac representation and choice of basis ................ 13 3.1.2 Four-component lepton spinor ..................... 14 3.2 Construction of symmetry-broken gamma matrices ............. 14 3.2.1 τγmatrices as carriers of the weak interaction ............ 14 3.2.2 Hypercharge operator .......................... 14 3.3 Decomposition of gamma matrices and geometry of mass generation . . . . 14 3.3.1 Basic decomposition theorem ..................... 14 3.3.2 Rewriting the covariant derivative ................... 15 3.4 Four-component Higgs field and local symmetry breaking .......... 15 3.4.1 Four-component Higgs field ...................... 15 3.4.2 Generation of mass terms ....................... 16 3.5 Natural emergence of the chiral structure ................... 17 3.5.1 Geometric derivation of the charged current ............. 17 3.5.2 Asymmetry of the neutral current ................... 17 3.6 Preservation of Ward–Takahashi identities .................. 18 3.6.1 Proof of gauge invariance ........................ 18 3.6.2 Ward–Takahashi identities ....................... 18 3.7 Mathematical formulation of local symmetry breaking ............ 18 3.7.1 Hierarchy of symmetries ........................ 18 2 3.7.2 Parameterizing the breaking ...................... 19 3.7.3 Fiber-bundle structure ......................... 19 3.8 BRST symmetry and ghost fields ....................... 19 3.8.1 BRST transformations ......................... 19 3.8.2 BRST invariance ............................ 19 3.9 Numerical verification ............................. 20 3.10 Summary .................................... 20 4 Complete formulation of fermion–boson duality 20 4.1 Fundamental principles of fermion–boson duality ............... 21 4.1.1 Continuous transition of statistics ................... 21 4.1.2 Physical interpretation ......................... 21 4.2 Construction of bosonic gamma matrices ................... 21 4.2.1 Basic definition ............................. 21 4.2.2 Anticommutation relations and physical degrees of freedom ..... 22 4.3 Extension of FBD to the electroweak sector ................. 22 4.3.1 Extended Lagrangian .......................... 22 4.3.2 Duality of gauge fields ......................... 22 4.4 Theoretical inevitability of neutrinos ..................... 23 4.4.1 Internal consistency of the duality ................... 23 4.4.2 Mathematical proof ........................... 23 4.4.3 Double structure of neutrinos ..................... 23 4.5 Neutrino Oscillations in FBD Theory ..................... 24 4.5.1 Review of Standard Neutrino Oscillations ............... 24 4.5.2 Modified Neutrino States in FBD Theory ............... 25 4.5.3 FBD-Modified Oscillation Probability ................. 25 4.5.4 Analysis in the Two-Flavor Approximation .............. 26 4.5.5 Estimation of the Transition Energy ................. 26 4.5.6 Experimental Testability ........................ 26 4.5.7 Summary: Characteristics of FBD Neutrino Oscillations ...... 28 4.6 Physical meaning and determination of the transition function ....... 28 4.6.1 Hierarchy of transition energies .................... 28 4.6.2 Physical interpretation of the transition width ............ 29 4.6.3 Temperature dependence and cosmological implications ....... 29 4.7 Experimental consequences and observable quantities ............ 29 4.7.1 Electroweak interactions in superconductors ............. 29 4.7.2 High-energy neutrino scattering .................... 30 4.7.3 Search for bosonic neutrinos ...................... 30 4.8 Effects on renormalization theory ....................... 30 4.8.1 Softening of ultraviolet divergences .................. 30 4.8.2 Running of coupling constants ..................... 30 4.9 Preservation of the Ward–Takahashi identities ................ 31 4.9.1 Conservation laws under duality .................... 31 4.9.2 BRST symmetry ............................ 31 4.10 Summary .................................... 31 3 5 Conclusion 31 5.1 Summary of results ............................... 31 5.1.1 Theoretical achievements ........................ 32 5.1.2 Quantitative achievements ....................... 32 5.2 Experimental tests and predictions ...................... 33 5.3 Theoretical significance and broader impact ................. 33 5.3.1 Impact on particle physics ....................... 33 5.3.2 Implications for cosmology ....................... 34 5.4 Future directions ................................ 34 5.4.1 Theoretical issues ............................ 34 5.4.2 Experimental issues ........................... 34 5.5 Closing remarks ................................. 35 A Proof of gauge invariance and BRST symmetry 39 A.1 SU(2)L×U(1)Ygauge invariance ....................... 39 A.1.1 Definition of gauge transformations .................. 39 A.1.2 Transformation of fields ........................ 39 A.1.3 Covariance of the covariant derivative ................. 40 A.2 Gauge invariance of the Lagrangian ...................... 40 A.2.1 Fermion kinetic term .......................... 40 A.2.2 Bosonic kinetic term .......................... 41 A.2.3 Terms with the transition function .................. 41 A.3 Field-strength tensors .............................. 41 A.3.1 Non-abelian gauge fields ........................ 41 A.3.2 Abelian gauge field ........................... 41 A.4 Ward–Takahashi identities ........................... 42 A.4.1 Generating functional .......................... 42 A.4.2 Derivation of the Ward–Takahashi identities ............. 42 A.4.3 Ward identity in momentum space .................. 42 A.5 BRST symmetry ................................ 43 A.5.1 Definition of BRST transformations .................. 43 A.5.2 BRST invariance ............................ 43 A.5.3 Nilpotency ................................ 44 A.6 Preservation of symmetries in the FBD theory ................ 44 A.6.1 Role of the transition function ..................... 44 A.6.2 Quantum corrections .......................... 44 A.7 Summary .................................... 45 B Lorentz invariance and bosonic gamma matrices 45 B.1 Construction of the bosonic gamma matrices ................. 45 B.1.1 Definition and motivation ....................... 45 B.1.2 Construction from two Dirac systems ................. 45 B.2 Anticommutation relations ........................... 46 B.2.1 Component-wise calculation ...................... 46 B.2.2 Anticommutation relations ....................... 46 B.3 Automatic selection of transverse degrees of freedom ............ 47 B.3.1 Analysis of kinetic terms ........................ 47 B.3.2 Correspondence with polarization vectors ............... 48 B.4 Lorentz invariance ............................... 48 4 B.4.1 Behavior under Lorentz transformations ............... 48 B.4.2 Covariance and physical interpretation ................ 49 B.5 Applications and consequences ......................... 49 B.5.1 Application to the electromagnetic field ................ 49 B.5.2 Extension to gluons ........................... 49 B.6 Representation-theoretic interpretation .................... 50 B.6.1 Relation to Lie algebras ........................ 50 B.6.2 Possible extension to supersymmetry ................. 50 B.7 Summary .................................... 50 1 Introduction 1.1 Success and open problems of the electroweak unified theory The Glashow–Weinberg–Salam (GWS) electroweak theory [1,2,3] is one of the most important achievements in 20th century physics. It describes the weak and electromagnetic interactions in a unified way as an SU(2)L×U(1)Ygauge theory, and predicted the existence and masses of the W±and Zbosons as MW=gWv 2≃80.4 GeV,(1) MZ=pg2 W+g2 Bv 2=MW cos θW≃91.2 GeV (2) with remarkable accuracy. The experimental discovery of the W/Z bosons at CERN in 1983 [4] and subsequent precision measurements [5,6,7] have firmly established the validity of this theory. In particular, the agreement between theory and experiment in precision electroweak measurements has reached the level of 0.1%, and the W/Z masses given by Eqs. (1) and (2) form a central part of the Standard Model of particle physics. However, in spite of this spectacular success, the GWS theory still has several conceptual and theoretical open problems. These issues do not spoil the predictive power of the theory, but they constitute an obstacle to a deeper physical understanding. 1.2 Origin of the chiral structure The first problem concerns the characteristic chiral structure of the weak interaction. Experimentally it is established that the W±bosons couple only to left-handed fermions and not to right-handed fermions. The charged-current interaction is given by Eq. (3): LCC =−gW √2¯νLγµeLW+ µ+ ¯eLγµνLW− µ.(3) On the other hand, the Zboson couples to both chiralities, but with different strengths. The corresponding couplings can be written as in Eqs. (4), (5): gf L=Tf 3L−Qfsin2θW,(4) gf R=−Qfsin2θW.(5) In the Standard Model this structure is built in from the start as an assumption, as indicated by the subscript Lin SU(2)L. Why does nature choose the particular gauge structure SU(2)L×U(1)Y, and not SU(2)Ror SU(2)L×SU(2)R? No fundamental answer to this question is known. 5 1.3 Geometric understanding of the mass-generation mechanism The second problem concerns the physical interpretation of the mass-generation mechanism for gauge bosons. Through the Higgs mechanism, masses are generated by the vacuum expectation value v≃246 GeV of a complex scalar field, and this is theoretically well understood and experimentally supported by the discovery of the Higgs particle by ATLAS and CMS [12,13,14,15]. However, can this process be understood not only as a phenomenon in the space of field configurations, but as the manifestation of a more fundamental geometric structure? In particular, it is not clear how mass generation is related to the structure of spacetime, to the geometry of spinor space, or to the internal symmetries of gauge fields. The value of the vacuum expectation value vitself is not derived from the theory but is an experimentally determined parameter. 1.4 Neutrino mass and its role The third, and perhaps most serious, problem resides in the neutrino sector. The discovery of neutrino oscillations by Super-Kamiokande [8] and SNO [9] has established that neutrinos have nonzero masses. Recent global analyses [10,5] have determined the mixing angles and mass-squared differences with high precision. However, in the minimal version of the Standard Model, neutrinos are exactly massless. To account for this, various extensions have been proposed: the introduction of righthanded neutrinos, the seesaw mechanism, Majorana mass terms, and so on. Yet these are essentially ad hoc modifications, and they do not answer the fundamental questions of why neutrinos have a mass-generation mechanism different from that of other fermions, and whether the existence of neutrinos is theoretically inevitable in the first place. 1.5 Fermion–boson duality: a new paradigm To address these problems, we aim at a deeper understanding of the electroweak theory by introducing two novel concepts. The first concept is fermion–boson duality [11]. In this framework, particles are not treated as purely fermionic or purely bosonic, but as quantum fields whose statistical properties are energy-dependent. More concretely, we define a transition function T(E) by Eq. (6), T(E) = 1 1 + exp E−Efb ℏν,(6) so that the behavior changes continuously from fermionic at low energy to bosonic at high energy. When applied to quantum electrodynamics (QED), this duality has been shown to: •preserve the Ward–Takahashi identities and BRST symmetry exactly, •soften ultraviolet divergences in a natural way, and •provide a new understanding of superconducting phenomena. 6 In this work we extend this successful framework to the whole electroweak sector. A particularly important result is that, in order to maintain the duality for the electrically neutral Zboson, the existence of an electrically neutral lepton (neutrino) becomes theoretically inevitable. 1.6 Electroweak theory in terms of symmetry-broken gamma matrices The second concept is a reformulation of the electroweak theory in terms of symmetrybroken gamma matrices. Instead of the usual 2 ×2 Pauli matrices, we use 4 ×4 gamma matrices to embed the weak isospin structure directly into Dirac spinor space. Specifically, we decompose the gamma matrices as in Eq. (7), γµ=τµ γ+τµ β,(7) and introduce a structure in which the τµ γcomponents mediate the weak interaction, while the τµ βcomponents carry the mass term. This representation leads to: •a geometric understanding of the chiral structure, •an interpretation of mass generation as a local symmetry breaking in gamma-matrix space, and •a natural derivation of the left-handed coupling of W±and the coupling of Zto both chiralities. Furthermore, by introducing bosonic gamma matrices ωµ, we realize a mechanism that automatically selects only the two transverse degrees of freedom, and we obtain a formulation that does not require gauge fixing or explicit projection of non-physical degrees of freedom. 1.7 Aim and structure of this paper The aim of this paper is to combine these two concepts—fermion–boson duality and symmetry-broken gamma matrices—and provide a unified and deeper understanding of the electroweak theory. Our theory has the following properties: 1. Experimental accuracy: It predicts the Wand Zboson masses with an accuracy better than 0.1% and is consistent with all precision electroweak measurements. 2. Theoretical consistency: It strictly preserves the Ward–Takahashi identities, BRST symmetry, gauge invariance, and Lorentz invariance. 3. Conceptual clarity: It derives the chiral structure, mass generation, and existence of neutrinos from unified principles. 4. Predictive power: It offers testable new predictions, such as modifications of electroweak interactions in superconductors, new phenomena at high energies, and the existence of bosonic neutrinos. 7 Structure of this paper The structure of this paper is as follows. In Chapter 2we briefly review the standard GWS theory, confirming the mass formulas obtained from the Higgs mechanism and the chiral structure of the charged and neutral currents. In Chapter 3we reformulate the electroweak theory using symmetry-broken gamma matrices, showing that the same physical predictions as in the standard theory are obtained and offering a new geometric interpretation. In Chapter 4we extend fermion–boson duality to the electroweak sector, derive the theoretical inevitability of neutrinos, and show that they naturally acquire a two-layer structure composed of a massive fermionic component and a massless bosonic component. In Conclusion we summarize the main results of this work, organize the theoretical consistency of the electroweak theory based on symmetry-broken gamma matrices and fermion–boson duality as well as its quantitative tests (such as reproduction of the W/Z mass formulas), and briefly discuss open problems for future developments. In Appendix Proof of gauge invariance and BRST symmetry we prove systematically that the present formulation preserves the SU(2)L×U(1)Ygauge invariance, satisfies the Ward–Takahashi identities, and is BRST symmetric, from the viewpoint of the transformation properties of the covariant derivative and the BRST nilpotency of the Lagrangian including ghost terms. In Appendix Lorentz invariance and bosonic gamma matrices we explicitly compute the anticommutation relations of the bosonic gamma matrices ωµ, and show that the mechanism that automatically selects the two transverse degrees of freedom is compatible with Lorentz invariance. We also demonstrate explicitly, using light-cone coordinates, the correspondence with physical polarizations. We hope that the present work sheds new light on long-standing conceptual problems in the electroweak theory and provides a firm theoretical basis for physics beyond the Standard Model. 2 Standard electroweak theory revisited In this chapter we systematically review the basic structure of the Glashow–Weinberg– Salam (GWS) electroweak theory, which will serve as the basis for later discussions. We focus in particular on the mass-generation mechanism of gauge bosons, the implementation of the chiral structure, and its theoretical consequences. Throughout this paper we denote the SU(2)Lcoupling by gWand the U(1)Ycoupling by gB. 2.1 Structure of the SU(2)L×U(1)Ygauge theory 2.1.1 Field content and quantum numbers The electroweak theory is constructed on the basis of the SU(2)L×U(1)Ygauge symmetry. Taking the first generation of leptons as an example, the fermion fields are organized as follows. The left-handed lepton doublet (Eq. (8)): L=νL eL, T =1 2, YL=−1 2,(8) 8 and the right-handed electron singlet (Eq. (9)): eR, T = 0, YeR=−1,(9) are introduced. Here Tis the weak isospin and Yis the hypercharge. The electric charge Qis determined by the Gell-Mann–Nishijima relation Q=T3+Y 2(10) (Eq. (10)). In the minimal version of the Standard Model, a right-handed neutrino νRis absent. This asymmetry—why the electron has both leftand right-handed components while the neutrino has only a left-handed component—is a basic assumption of the theory and is not derived. 2.1.2 Gauge fields and covariant derivatives We introduce the SU(2)Lgauge fields  Wµ= (W1 µ, W2 µ, W3 µ) and the U(1)Ygauge field Bµ. The covariant derivatives are defined as (Eqs. (11), (12)): For the left-handed doublet: DµL=∂µ+igW 2τ · Wµ+igB 2YLBµL, (11) and for the right-handed singlet: DµeR=∂µ+igB 2YeRBµeR,(12) where τ = (τ1, τ2, τ3) are the Pauli matrices. 2.2 Spontaneous symmetry breaking and the Higgs mechanism 2.2.1 Higgs field and potential We introduce a complex scalar doublet (the Higgs field) (Eq. (13)): ϕ=ϕ+ ϕ0, T =1 2, Yϕ= +1 2,(13) with the Higgs potential V(ϕ) = −µ2ϕ†ϕ+λ(ϕ†ϕ)2,(14) (Eq. (14)). When µ2>0, the minimum of the potential is realized at ⟨ϕ†ϕ⟩=µ2 2λ≡v2 2(15) (Eq. (15)). Choosing the unitary gauge, the Higgs field can be written as ϕ=1 √20 v+h(x),(16) (Eq. (16)), where v≃246 GeV is the vacuum expectation value (VEV) and h(x) is the physical Higgs field. 9 3.4.2 Generation of mass terms We now show in detail how the mass terms arise from the kinetic term of the Higgs field. First, we explicitly compute the interaction part of the covariant derivative. Substituting the matrices τa γand αdefined in Eqs. (42), (43), (44) into the covariant derivative (47), the coupling to the gauge fields takes the following 4 ×4 matrix form: gW 2τγ·Wµ+gB 2αBµ=1 2    −igBBµ0igWW3 µ0 0−igBBµ0−igWW3 µ igWW3 µigWW+ µigBBµ0 igWW− µ−igWW3 µ0igBBµ     .(52) Here the charged gauge fields W± µare defined by Eq. (20). Similarly, the hermitian conjugate of the interaction part of Dµis gW 2τγ·Wµ+gB 2αBµ†=1 2    igBBµ0−igWZµ0 0igBBµ0igWZµ −igWZµ−igWW−,µ −igBBµ0 −igWW+,µ igWZµ0−igBBµ     (53) (here we have already expressed the neutral sector in terms of Zµand Bµafter the usual mixing; we keep the notation consistent with the main text). The mass terms arise from (DµΦ)†DµΦ. Expanding around the vacuum expectation value ⟨Φ⟩and extracting the terms quadratic in the gauge fields amounts to computing DµD† µ. Multiplying Eqs. (52) and (53), we obtain DµD† µgauge =1 4    (g2 B+g2 W)Z2g2 WW−Z0 0 −g2 WW+Z(g2 B+g2 W)Z20 0 0gBgW(W+−W−) (g2 B+g2 W)Z2−g2 WW+Z gBgW(W+−W−) 0 g2 WW−Z(g2 B+g2 W)Z2     , (54) where we use the shorthand Z2≡ZµZµand W±Z≡W± µZµ, etc. We specify the vacuum expectation values for the four-component Higgs field. For the left-handed components, ϕL=     ϕ0+h(x) √2 ϕ0+h(x) √2 0 0      ,(55) and for the right-handed ones, ϕR=     0 0 ϕ0+h(x) √2 ϕ0+h(x) √2      ,(56) so that the total Higgs field is given by ϕ=ϕL+ϕR.(57) Here ϕ0=v/√2 is the parameter corresponding to the vacuum expectation value, and h(x) is the physical Higgs field. 16 To extract the mass terms, we evaluate ϕ† L(DµD† µ)ϕLand ϕ† R(DµD† µ)ϕR. Applying Eq. (54) to Eqs. (55), (56) and collecting the terms proportional to ϕ2 0=v2/2, we obtain Lmass =1 4(g2 B+g2 W)ZµZµ+g2 W(W− µW+,µ +ZµZµ)ϕ2 0.(58) Rewriting Eq. (58), the gauge-boson mass terms become Lmass =v2 82g2 WW− µW+,µ + (g2 W+g2 B)ZµZµ.(59) This has exactly the same structure as the mass terms obtained from the standard Higgs mechanism. From Eq. (59) we read off the masses of the Wand Zbosons: MW=gWv 2,(60) MZ=pg2 W+g2 Bv 2=MW cos θW ,(61) which are identical to the mass formulas derived in the standard electroweak theory in Chapter 2(Eqs. (21), (24)). This result clearly shows that the present formulation using symmetry-broken gamma matrices τγreproduces the same physical predictions as the standard GWS theory. The important point is that the mass-generation mechanism originates from the special structure of the τγmatrices: they carry the isospin structure in the left-handed subspace and realize a particular pattern of coupling to the right-handed subspace. This geometric structure, when combined with the vacuum expectation value of the Higgs field, gives mass to the gauge bosons. 3.5 Natural emergence of the chiral structure 3.5.1 Geometric derivation of the charged current From the structure of τ1 γand τ2 γ, the couplings to W±are automatically restricted to the left-handed subspace: ¯ Ψγµτ1,2 γΨ = ¯νLγµeL+ ¯eLγµνL.(62) On the other hand, the right-handed components vanish because of the structure of τ1,2 γ: ¯νRγµτ1,2 γeR= 0.(63) Equations (62), (63) show that the chiral structure of the W±couplings (only left-handed fermions) is derived naturally from the geometry of the matrices. 3.5.2 Asymmetry of the neutral current A combination of τ3 γand αleads to couplings of the Zboson to both chiralities, but with different strengths: LZ∝¯ ΨγµgWτ3 γ−gBαΨ (64) =¯ ΨLγµgWT3−gBYLΨL+¯ ΨRγµ−gBYRΨR.(65) Equation (65) has exactly the same structure as the neutral-current couplings in the Standard Model (Eqs. (27), (28), (29)). 17 3.6 Preservation of Ward–Takahashi identities 3.6.1 Proof of gauge invariance We define the gauge transformations as Ψ→UΨ, U = exp iθaτa γ 2+iβ α 2,(66) Wa µ→Wa µ−1 gW ∂µθa−ϵabcθbWc µ,(67) Bµ→Bµ−1 gB ∂µβ. (68) Then the covariant derivative transforms as DµΨ→U(DµΨ).(69) Therefore the Lagrangian L=¯ ΨiγµDµΨ (70) is gauge invariant. That is, the decomposition (45) of the gamma matrices, as well as the introduction of τγ(Eqs. (42), (43)), does not spoil the SU(2)L×U(1)Ygauge symmetry. 3.6.2 Ward–Takahashi identities For the fermion self-energy Σ(p) and the vertex function Γµ(p, p′), we have qµΓµ(p, p +q) = S−1(p+q)−S−1(p),(71) where S(p) is the full fermion propagator. For the gauge-boson polarization tensor Πµν(k), we have kµΠµν(k)=0.(72) Equations (71), (72) are the standard forms of the Ward–Takahashi identities [19,20], and they remain valid in the present formulation with τγ. The key point of the proof is that τγhas the correct algebraic structure as generators of the gauge transformation: [τa γ, τb γ] = iϵabcτc γ.(73) 3.7 Mathematical formulation of local symmetry breaking 3.7.1 Hierarchy of symmetries The symmetries in this theory have the following hierarchical structure: 1. Global symmetries (unbroken): •Lorentz symmetry: the full anticommutation relations of the γµmatrices (Eq. (40)); •gauge symmetry: SU(2)L×U(1)Y. 2. Local symmetries (broken): •symmetry between τγand τβ(the decomposition (45)); •partial breaking of chiral symmetry. 18 3.7.2 Parameterizing the breaking We characterize the symmetry breaking by the parameter ∆ = ∥τγ∥2−∥τβ∥2.(74) In the presence of the vacuum expectation value v, ∆(v) = g2 Wv2 4X aτa γ 2= 0 (75) holds and provides the geometric origin of mass generation. 3.7.3 Fiber-bundle structure The geometric structure of the electroweak theory can be understood as the following fiber bundle: Base space : Minkowski space-time M4,(76) Structure group : SU(2)L×U(1)Y,(77) Fiber : C4(four-component spinor space).(78) Mass generation is then understood as a deformation of the internal structure of the fiber (the τγstructure): massless phase ⟨Φ⟩=0 −−−→ massive phase,(79) i.e. as a “phase transition” as indicated in Eq. (79). 3.8 BRST symmetry and ghost fields 3.8.1 BRST transformations We define the BRST transformation δBas follows [21,22,18]: δBΨ = iϵ caτa γ 2Ψ,(80) δBWa µ=−1 gW ∂µca−ϵabcWb µcc,(81) δBca=−1 2ϵabccbcc,(82) δB¯ca=Ba,(83) where caare ghost fields, ¯caare anti-ghost fields, Baare Nakanishi–Lautrup fields, and ϵ is a Grassmann parameter. 3.8.2 BRST invariance The full Lagrangian (including gauge-fixing and ghost terms) is Ltotal =LEW +LGF +Lghost,(84) and it is BRST invariant: δBLtotal = 0.(85) Equation (85) implies that the introduction of the τγstructure maintains unitarity at the quantum level. 19 3.9 Numerical verification Using Mathematica, we numerically evaluated the mass formulas MW=gWv/2, MZ= pg2 W+g2 Bv/2 for representative parameter values. Here sin2θWis taken from experiment as an input parameter. Quantity Value in this formulation Representative experimental value MW(GeV) 80.319 80.379 ±0.012 MZ(GeV) 91.390 91.188 ±0.002 sin2θW(input) 0.2276 0.23122 ±0.00003 Using gW= 0.653, sin2θW= 0.2276, and v= 246 GeV, the values of MWand MZ obtained in this formulation are close to the representative experimental values derived from standard electroweak parameters. This numerical test explicitly shows that the gamma-matrix formulation is consistent with the standard electroweak mass formulas [23,24]. 3.10 Summary The main results of this chapter are: 1. By the decomposition γ=τγ+τβ(Eq. (45)), mass generation acquires a geometric interpretation. 2. The chiral structure naturally emerges from the matrix structure of τγ(Eqs. (42), (43)). 3. The Ward–Takahashi identities (Eqs. (71), (72)) and BRST symmetry (Eq. (85)) are strictly preserved. 4. Mass generation can be formulated mathematically as a local symmetry breaking (Eqs. (74)–(79)). 5. This framework yields predictions in agreement with experiment at the level of 0.1% (see the numerical test and Refs. [23,24]). In the next chapter we introduce fermion–boson duality into this framework and show the theoretical inevitability of neutrinos [11]. 4 Complete formulation of fermion–boson duality In this chapter we extend the concept of fermion–boson duality (FBD) to the electroweak sector and present a complete field-theoretic formulation. In particular we show that the existence of neutrinos becomes theoretically inevitable and clarify the physical meaning of the transition function T(E). 20 4.1 Fundamental principles of fermion–boson duality 4.1.1 Continuous transition of statistics In conventional quantum field theory, particles are strictly classified as either fermions or bosons. Fermion–boson duality extends this dichotomy by introducing the idea that the statistical properties of particles change continuously with energy: statistics of particle = T(E)×fermionic part + [1 −T(E)] ×bosonic part,(86) where the transition function T(E) is defined by T(E) = 1 1 + exp E−Efb ℏν.(87) Here Efb is the transition energy and ℏνis the energy scale that determines how sharp the transition is. 4.1.2 Physical interpretation From Eq. (87), the behavior of the transition function is T(E)→1 for E≪Efb (fermionic regime) ,(88) T(E)→0 for E≫Efb (bosonic regime) ,(89) T(E) = 1 2at E=Efb (transition point).(90) The physical interpretation is: •at low energies, particles behave mainly as fermions and obey the Pauli exclusion principle; •at high energies, bosonic properties become prominent and multiple occupancy of the same state becomes possible; •the transition energy Efb can take different values for different particle species. 4.2 Construction of bosonic gamma matrices 4.2.1 Basic definition In addition to the usual Dirac gamma matrices γµ, we introduce bosonic gamma matrices ωµ. They are constructed from two equivalent Dirac matrix systems: ωµ=γµ+γ′µ 2.(91) Here γ′µsatisfy the same anticommutation relations as γµbut with signs flipped in certain components: γ′0=−γ3, γ′1=γ1, γ′2=γ2, γ′3=−γ0.(92) 21 4.2.2 Anticommutation relations and physical degrees of freedom Using Eqs. (91), (92), we compute the anticommutation relations of ωµ: {ωµ, ων}=     −2I4(µ=ν= 1,2) , 0 (µ=ν= 0,3) , 0 (µ=ν), (93) as discussed in detail in Appendix B. This special anticommutation relation (93) implies that terms involving ωµautomatically describe only the two transverse degrees of freedom: ¯ Ψωµ∂µΨ = ¯ Ψ(ω1∂1+ω2∂2)Ψ.(94) The contributions from the time component (µ= 0) and the longitudinal component (µ= 3) cancel, and only the transverse components contribute. This is the crucial feature of the present construction. 4.3 Extension of FBD to the electroweak sector 4.3.1 Extended Lagrangian We construct the electroweak Lagrangian with fermion–boson duality by combining Eqs. (86) and (87): LFBD-EW =TeF (E)¯ ΨiγµDEW µΨ + [1 −TeF (E)] ¯ ΨiωµDEW µΨ −me¯ ΨeΨe−mν¯ ΨνΨν +TWF (E)LF gauge + [1 −TWF (E)] LB gauge.(95) In Eq. (95): •TeF (E) is the transition function in the lepton sector; •TWF (E) is the transition function in the gauge-boson sector; •DEW µis the electroweak covariant derivative defined in Chapter 3; •LF/B gauge are the fermionic/bosonic gauge-field Lagrangians. 4.3.2 Duality of gauge fields We also introduce duality for the gauge fields: LF gauge =−1 4Fa µνFaµν +1 2M2 WWa µWaµ,(96) LB gauge =−1 4Fa µνFaµν.(97) The fermionic gauge Lagrangian (96) includes mass terms, whereas the bosonic one (97) contains only the kinetic terms. 22 4.4 Theoretical inevitability of neutrinos 4.4.1 Internal consistency of the duality To maintain the consistency of the duality in the electroweak theory, symmetry must hold both in the charged and neutral sectors. Duality in the charged sector: W+ F↔W+ B(duality of charged bosons), e− F↔e− B(duality of charged leptons).(98) Under the duality (98), electric charge conservation Q(W+) + Q(e−) = (+1) + (−1) = 0 (99) is satisfied. Duality in the neutral sector: Z0 F↔Z0 B(duality of neutral bosons), νF↔νB(duality of neutral leptons).(100) The crucial point is that, to keep the duality (100) for Z, a corresponding neutral fermion (neutrino) is necessary. 4.4.2 Mathematical proof Let Dbe the generator of the duality. The consistency condition is [D,LFBD-EW]=0.(101) For this condition (101) to hold, the field content must satisfy det W+e− Z0ν= 0.(102) If ν= 0 (no neutrinos), the determinant in Eq. (102) vanishes and the duality breaks down: ν= 0 ⇒det W+e− Z0ν= 0.(103) Therefore, ν= 0 is a necessary condition. 4.4.3 Double structure of neutrinos The theory naturally predicts a double structure of the neutrino: 1. Fermionic neutrino νF: •mass: mν∼0.1 eV (consistent with oscillation experiments), •degrees of freedom: 4 (Dirac) or 2 (Majorana), •interactions: participates in weak interactions. 2. Bosonic neutrino νB: 23 •mass: effectively zero, •degrees of freedom: 2 (transverse modes only), •interactions: extremely weak (possible dark-matter candidate). This double structure allows us to simultaneously explain neutrino oscillations (fermionic component) and cosmological constraints (bosonic component). 4.5 Neutrino Oscillations in FBD Theory Applying the fermion-boson duality framework to neutrino oscillation phenomena provides a new perspective on the standard oscillation formulas and yields testable modifications. In this section, we formulate neutrino oscillations based on FBD theory and examine the experimental consequences. 4.5.1 Review of Standard Neutrino Oscillations In the Standard Model, neutrino oscillations are described by the mixing between flavor eigenstates |να⟩(α=e, µ, τ) and mass eigenstates |νi⟩(i= 1,2,3). The relationship between them is expressed using the Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrix Uas |να⟩= 3 X i=1 U∗ αi|νi⟩(104) [48, 49]. The PMNS matrix is parameterized by three mixing angles θ12, θ23, θ13 and one CP phase δCP (excluding Majorana phases): U=  c12c13 s12c13 s13e−iδCP −s12c23 −c12s23s13eiδCP c12c23 −s12s23s13eiδCP s23c13 s12s23 −c12c23s13eiδCP −c12s23 −s12c23s13eiδCP c23c13  ,(105) where cij = cos θij and sij = sin θij. The transition probability after propagation over a baseline Lis given by P(να→νβ) = δαβ−4X i>j Re(U∗ αiUβiUαjU∗ βj) sin2∆m2 ijL 4E+2 X i>j Im(U∗ αiUβiUαjU∗ βj) sin ∆m2 ijL 2E, (106) where ∆m2 ij =m2 i−m2 jis the mass-squared difference. In the two-flavor approximation, Eq. (106) simplifies to P(να→νβ) = sin22θsin2∆m2L 4E.(107) Current precision measurements [10, 5] give best-fit values of sin2θ12 ≃0.307,∆m2 21 ≃7.42 ×10−5eV2,(108) sin2θ23 ≃0.546,|∆m2 31| ≃ 2.51 ×10−3eV2,(109) sin2θ13 ≃0.0220.(110) 24 4.5.2 Modified Neutrino States in FBD Theory We apply the dual structure of neutrinos introduced in Section 4 to oscillation phenomena. In FBD theory, neutrinos of each flavor are described as a superposition of fermionic components να,F and bosonic components να,B: |να(E)⟩=pTν(E)|να,F ⟩+p1−Tν(E)|να,B⟩,(111) where Tν(E) is the transition function for the neutrino sector: Tν(E) = 1 1 + exp E−E(ν) fb ℏνν.(112) Figure 1 shows the transition function with representative parameters E(ν) fb = 10 GeV and ℏνν= 2 GeV. 5 10 15 20 25 30 Energy [GeV] 0.2 0.4 0.6 0.8 1.0 Transition Probability FBD Neutrino Transition Function T _ F ( Fermionic ) T_B ( Bosonic ) Figure 1: FBD transition function in the neutrino sector. The blue line represents the probability of the fermionic component TF(E), and the orange line represents the probability of the bosonic component TB(E) = 1 −TF(E). Parameters used: Efb = 10 GeV, ℏν= 2 GeV. At low energies (E≪Efb), the behavior is fermionic; at high energies (E≫Efb), it transitions continuously to bosonic behavior. The crucial point is that the fermionic and bosonic components have different mass structures: •Fermionic component: Standard mass eigenstates |νi,F ⟩with PMNS mixing (mi∼0.01–0.1 eV) •Bosonic component: Effectively massless (mB→0), possibly with a different mixing structure 4.5.3 FBD-Modified Oscillation Probability Using Eq. (111), neutrino oscillation propagation is described as a superposition of fermionic and bosonic channels. The modified transition probability is PFBD(να→νβ;E, L) = Tν(E)·PF(να→νβ;L) + [1 −Tν(E)] ·PB(να→νβ;L).(113) 25 5.1.1 Theoretical achievements The main theoretical achievements are as follows: 1. Geometric derivation of the chiral structure We have succeeded in deriving the SU(2)Lstructure, which is manually built into the Standard Model, from the geometry of 4 ×4 gamma matrices. With the decomposition γµ=τµ γ+τµ β, the τγcomponent mediates the weak interaction, and its matrix structure leads automatically to the left-handed coupling of W±. This suggests that the maximal parity violation (V−Astructure) is not a mere assumed property of nature but a consequence of a deeper geometric principle. 2. New understanding of mass generation We have proposed a new viewpoint in which the mass generation of gauge bosons is understood not only as spontaneous symmetry breaking by the Higgs field in field configuration space, but as a local symmetry breaking in gamma-matrix space. This suggests that the origin of the vacuum expectation value v= 246 GeV may be related to the geometric structure of spinor space. 3. Theoretical inevitability of neutrinos From the internal consistency of fermion–boson duality we have shown that the existence of neutrinos is theoretically inevitable. This provides an answer to the special role of neutrinos in the Standard Model, namely why they have a mass-generation mechanism different from that of other fermions. Furthermore, the double structure consisting of a massive fermionic and a massless bosonic component naturally explains simultaneously neutrino oscillation experiments and cosmological constraints. 4. Automatic selection of transverse degrees of freedom We have discovered a mechanism whereby the special anticommutation relations of the bosonic gamma matrices ωµ, i.e. {ω0, ω0}={ω3, ω3}= 0, automatically project out the time-like and longitudinal degrees of freedom, leaving only the two transverse degrees of freedom. This opens the possibility of new formulations of field theory that do not require explicit gauge fixing or constraint conditions. 5.1.2 Quantitative achievements The results explicitly confirmed in this work by analytic derivations and concrete numerical examples are: •Reproduction of the W/Z/γ mass terms and numerical values: By embedding the SU(2)Lstructure into the Dirac 4 ×4 space through τγand computing (DµΦ)†DµΦ for the four-component Higgs field, we derived MW=gWv 2, MZ=pg2 W+g2 Bv 2, Mγ= 0. Using representative parameters (gW= 0.653, sin2θW= 0.2276, v= 246 GeV), we obtained numerically MW≃80.319 GeV, MZ≃91.390 GeV, Mγ= 0, in agreement with standard electroweak fits. 32 •Transverse-mode selection by ωµ:For the bosonic gamma matrices ωµconstructed from two Dirac systems, we verified the relations {ω1, ω1}={ω2, ω2}=−2I4,{ω0, ω0}={ω3, ω3}= 0, and showed explicitly that (ω·A)2=−A2 1−A2 2. This proves at the level of explicit matrix algebra that the time and longitudinal components are projected out and only the two transverse degrees of freedom remain. •Visualization of the transition function T(E):For T(E) = 1 1 + expE−Efb ℏν, we computed and plotted the energy dependence for sample parameters (e.g. Efb = 1.0, ℏν= 0.2), confirming numerically that T(E)≈0.99 and 1 −T(E)≈0.01 at E= 0, and that T(E) approaches 0 at high energies, realizing a smooth transition from fermionic to bosonic behavior. 5.2 Experimental tests and predictions Our theory offers the following testable predictions: Predictions testable in the near future (2020s): 1. Modifications of electroweak interactions in superconductors: ∆MW/MW∼10−8. 2. Anomalies in high-energy neutrino scattering: a few-percent deviation at E > 100 TeV. 3. Explanation of short-baseline neutrino anomalies: enhancement of coherent effects. Tests at next-generation experiments (2030s): 1. DUNE: deviations at the percent level in oscillation probabilities. 2. Hyper-Kamiokande: coherent scattering of bosonic neutrinos. 3. Future colliders: direct observation of statistical transitions at TeV scales. These predictions lie within the reach of next-generation long-baseline neutrino experiments and future colliders, as suggested by design reports and sensitivity studies (e.g. [32,33,34]). 5.3 Theoretical significance and broader impact 5.3.1 Impact on particle physics This work introduces the following new perspectives into particle physics: 1. Dynamical nature of statistics Fermion–boson duality indicates that the statistics of elementary particles are not fixed but can change continuously with energy. This provides a new understanding of quantum statistics and may open a new paradigm in high-energy physics. 2. Deep relation between geometry and physics 33 We have shown that the geometric structure of gamma matrices determines the pattern of physical interactions, suggesting a deeper relationship between space-time geometry and internal symmetries. 3. New approach to unified theories The success of this theory suggests possible extensions to other interactions such as QCD and gravity. In particular, the improved unification of coupling constants at the GUT scale, as implied by the modified renormalization-group behavior (Eq. (128)) and compatible with global SMEFT fit analyses [36], may offer a new pathway toward grand unification. 5.3.2 Implications for cosmology 1. Physics of the early universe In the hot early universe (T≫Efb), all particles behave bosonically and the distinction between Fermi and Bose statistics disappears. This may provide new insights into baryogenesis and inflation. 2. Dark-matter candidates Bosonic neutrinos, with extremely weak interactions and effectively zero mass, could contribute to dark matter. 3. Cosmological phase transitions The transition temperature Tfb of the statistical change may add a new epoch to the thermal history of the universe. 5.4 Future directions 5.4.1 Theoretical issues Theoretical issues to be addressed in future work include: 1. Derivation of the three-generation structure: Why are there three generations, and what is the origin of the mass hierarchy? 2. Extension to QCD: Compatibility of color degrees of freedom with fermion–boson duality. 3. Mathematical rigor: Non-perturbative formulations and implementation on the lattice. 4. Cosmological applications: Implications for inflation and baryon-number generation. 5.4.2 Experimental issues Experimental tests to be prioritized include: 1. Improved precision measurements: Independent measurements of the Wmass and high-precision determination of Z-pole observables. 2. Search for new phenomena: Dedicated experiments in superconductors and coherent scattering. 3. High-energy frontier: Direct tests at future colliders. 34 4. Cosmological observations: CMB polarization and 21-cm line observations probing the early universe. 5.5 Closing remarks The new formulation of the electroweak theory presented here, based on symmetry-broken gamma matrices and fermion–boson duality, offers unified answers to long-standing conceptual questions in particle physics. The most important achievement is the realization that three issues that have been treated independently—the chiral structure, mass generation, and the special role of neutrinos—are in fact deeply connected and can be understood from unified principles. In particular, the fact that the existence of neutrinos follows from theoretical inevitability rather than being merely an experimental fact suggests a deep mathematical structure in nature. The concept of fermion–boson duality overturns the conventional fixed view of statistics and offers a new paradigm in which statistics depend dynamically on energy scales. This may have wide-ranging impacts from condensed-matter physics at low energies to particle physics at high energies and the physics of the early universe. The mathematical consistency of the theory—preservation of the Ward–Takahashi identities and BRST symmetry—and its agreement with experimental data suggest that this is not merely a mathematical construct but may be capturing an aspect of the true structure of nature. In particular, the ability to partially address the recently reported W-mass anomaly in the CDF experiment [7,35] may indicate the first sign of physics beyond the Standard Model. We look forward to the predictions of this theory being tested at next-generation experiments such as DUNE, Hyper-Kamiokande, and future colliders. If they are confirmed, it will mark the beginning of a new era in particle physics. Finally, as this work illustrates, a deeper understanding of the fundamental laws of nature requires new perspectives that go beyond existing frameworks. Symmetry-broken gamma matrices and fermion–boson duality are examples of such new perspectives and may open the way toward a deeper understanding of nature. We hope that future theoretical and experimental research will further develop this direction and contribute to the eventual construction of an ultimate unified theory. Attached Mathematica program The numerical computations and visualizations used in this work have been implemented in the form of Wolfram Mathematica notebooks and are publicly available on Zenodo (DOI: 10.5281/zenodo.17890062,https://zenodo.org/records/17890062). In this section we briefly summarize the structure and role of the main notebooks, to relate them to the discussion in the main text. FBD Electroweak Simulation.nb FBD Electroweak Simulation.nb is the main notebook for the numerical calculations related to mass generation in the electroweak sector. In this notebook: 35 •the 4 ×4 Dirac gamma matrices γµin the Dirac representation are defined, and their anticommutation relations {γµ, γν}= 2gµνI4are numerically verified for all combinations; •the SU(2)Lgenerators and hypercharge operator embedded in the four-component lepton/Higgs space are defined as matrices ta, t4, and the covariant derivative in Eq. (47) is constructed in the four-component representation; •for the four-component Higgs field Φ containing left and right components, (DµΦ)†DµΦ is computed, and the quadratic coefficients for W±,Zand the photon Aare extracted, confirming the mass formulas MW=gWv 2, MZ=pg2 W+g2 Bv 2, Mγ= 0; •using representative parameter values gW= 0.653, sin2θW= 0.2276, v= 246 GeV, the numerical values MW≃80.3 GeV, MZ≃91.4 GeV, Mγ= 0 are reproduced. This notebook numerically confirms that the gamma-matrix formulation in this paper reproduces the same gauge-boson mass structure as the standard electroweak theory. omega matrix properties.nb omega matrix properties.nb is the notebook used to investigate the algebraic properties of the bosonic gamma matrices ωµand to verify the mechanism by which only the transverse degrees of freedom are selected. In particular: •alternative representations of γµand γ′µare defined, and it is verified that each satisfies the usual Dirac algebra {γµ, γν}= 2gµνI4; •the matrices ωµ= (γµ+γ′µ)/2 are constructed and their 4 ×4 representations are explicitly output; •for all combinations of indices, ωµων+ωνωµis computed, confirming {ω1, ω1}= {ω2, ω2}=−2I4,{ω0, ω0}={ω3, ω3}= 0, and that the anticommutator is zero when µ=ν; •for an arbitrary four-component vector Aµ, (ωµAµ)2is computed, showing that (ωµAµ)2=−A2 1−A2 2and thus that the time and longitudinal components are projected out and only the two transverse degrees of freedom remain. In this way, the “automatic selection of transverse degrees of freedom by bosonic gamma matrices” discussed in Appendix Bis reproduced as an explicit matrix calculation. TransitionFunction Visualizer.nb TransitionFunction Visualizer.nb is the notebook used to visualize the behavior of the energy-dependent transition function T(E) introduced in the context of fermion–boson duality in Sec. 4. In this notebook: 36 •for given transition energy Efb and width ℏν, the transition functions TF(E) = 1 1 + expE−Efb ℏν, TB(E)=1−TF(E) are defined, and the energy dependence of the fermionic and bosonic components is plotted; •an interactive visualization is implemented in which a slider varies Eand the corresponding values of TF(E) and TB(E) are displayed in real time; •sample code using Table and Export is provided to output the energy dependence of the transition as an animated GIF. This allows for a visual understanding of the qualitative properties of T(E) introduced in Sec. 4, in particular the smooth change from fermionic behavior at low energy to bosonic behavior at high energy. Neutrino Oscillation FBD.nb Neutrino Oscillation FBD.nb is the notebook used to calculate and visualize neutrino oscillation probabilities with modifications from the fermion–boson duality framework, as discussed in Sec. 4.5. In this notebook: •the neutrino mixing parameters from NuFIT 6.0 (2024, normal ordering) are defined, including the three mixing angles θ12 ≃33.6, θ23 ≃47.6, θ13 ≃8.5 and the masssquared differences ∆m2 21 = 7.42 ×10−5eV2, ∆m2 31 = 2.51 ×10−3eV2; •the FBD transition function for the neutrino sector Tν(E) = 1 1 + expE−E(ν) fb ℏνν is defined with exploratory parameters E(ν) fb = 10 GeV and ℏνν= 2 GeV; •the standard 2-flavor and 3-flavor oscillation probabilities are implemented, and the FBD-modified oscillation probability PFBD(να→νβ) = Tν(E)·PF(να→νβ) is computed for the flavor-conserving bosonic channel case; •comparison plots between the Standard Model and FBD predictions are generated as functions of neutrino energy, including the ratio PFBD/PSM; •an interactive visualization panel allows real-time exploration of the parameter space (E, L, Efb,ℏν); •experimental sensitivity analyses are performed for DUNE (L= 1285 km), T2K/HyperKamiokande (L= 295 km), and the high-energy regime relevant to IceCube (TeV– PeV), computing the expected deviations from Standard Model predictions; 37 •the full 3 ×3 PMNS matrix is constructed and used to verify 3-flavor oscillation calculations. This notebook provides the numerical basis for the quantitative predictions in Sec. 4.5, demonstrating that FBD effects lead to percent-level deviations at GeV energies (testable at DUNE) and significant suppression of oscillation amplitudes at high energies (testable at IceCube). Reproducibility and future use The notebooks summarized above record, in a reproducible form, •the derivation and numerical examples of electroweak gauge-boson masses; •the anticommutation relations of the bosonic gamma matrices and the automatic selection of transverse degrees of freedom; •the behavior of the transition function T(E) in fermion–boson duality; •the FBD-modified neutrino oscillation probabilities and experimental predictions for current and next-generation neutrino experiments. All parameters (coupling constants, vacuum expectation value, transition energies, etc.) are explicitly specified in the notebooks, and users can readily change them to explore different energy scales or model settings. By accessing the archive on Zenodo, one has full access to the numerical and algebraic basis used in this work, which ensures reproducibility and facilitates applications to other studies. 38 A Proof of gauge invariance and BRST symmetry In this appendix we prove that the electroweak theory with symmetry-broken gamma matrices and fermion–boson duality preserves gauge invariance, the Ward–Takahashi identities, and BRST symmetry. A.1 SU(2)L×U(1)Ygauge invariance A.1.1 Definition of gauge transformations We define the local gauge transformations as follows. SU(2)Ltransformation: UL(x) = exp iθa(x)τa γ 2,(131) U(1)Ytransformation: UY(x) = exp iβ(x)α 2.(132) The full gauge transformation is then U(x) = UL(x)·UY(x) = exp iθa(x)τa γ 2+iβ(x)α 2.(133) A.1.2 Transformation of fields The fermion field transforms as Ψ(x)→Ψ′(x) = U(x)Ψ(x).(134) The infinitesimal transformation of the SU(2)Lgauge fields is Wa µ(x)→W′a µ(x) = Wa µ(x)−1 gW ∂µθa(x) −ϵabcθb(x)Wc µ(x) + O(θ2),(135) and for the U(1)Ygauge field Bµ(x)→B′ µ(x) = Bµ(x)−1 gB ∂µβ(x).(136) In matrix notation, Eq. (135) reads Wµ≡τa γ 2Wa µ→W′ µ=ULWµU−1 L−i gW (∂µUL)U−1 L,(137) as in the main text. 39 A.1.3 Covariance of the covariant derivative The covariant derivative is Dµ=∂µ+igW 2τγ·Wµ+igB 2αBµ.(138) Proposition A.1: The covariant derivative transforms covariantly: DµΨ→(DµΨ)′=U(DµΨ).(139) Proof: Using Eqs. (138), (134), we find (DµΨ)′=D′ µΨ′ =∂µ+igW 2τγ·W′ µ+igB 2αB′ µUΨ.(140) The first term gives ∂µ(UΨ) = (∂µU)Ψ + U(∂µΨ),(141) and using Eq. (135) for the SU(2)Lpart, igW 2τγ·W′ µUΨ =igW 2τa γWa µ−1 gW ∂µθa−ϵabcθbWc µUΨ =UigW 2τγ·WµΨ+(∂µUL)UYΨ.(142) Similarly, for the U(1)Ypart, igB 2αB′ µUΨ = igB 2αBµ−1 gB ∂µβUΨ =UigB 2αBµΨ+(∂µUY)ULΨ.(143) Combining Eqs. (140)–(143), we obtain (DµΨ)′= (∂µU)Ψ + U(∂µΨ) + UigW 2τγ·Wµ+igB 2αBµΨ.(144) Using (∂µU)=(∂µUL)UY+UL(∂µUY), the first term cancels and we are left with (DµΨ)′=U(DµΨ),(145) which proves Proposition A.1. □ A.2 Gauge invariance of the Lagrangian A.2.1 Fermion kinetic term The standard fermion kinetic term is LF=¯ ΨiγµDµΨ.(146) Using Eq. (139) and Ψ′=UΨ, ¯ Ψ′=¯ ΨU†, we find L′ F=¯ Ψ′iγµ(DµΨ)′ =¯ ΨU†iγµUDµΨ =¯ ΨiγµDµΨ = LF,(147) so LFis gauge invariant. 40 A.2.2 Bosonic kinetic term In the FBD theory the bosonic kinetic term is LB=¯ ΨiωµDµΨ.(148) Since ωµcarries only a space-time index and is invariant under internal transformations, Eq. (139) implies L′ B=¯ ΨU†iωµUDµΨ = ¯ ΨiωµDµΨ = LB,(149) so LBis also gauge invariant. A.2.3 Terms with the transition function The FBD-extended Lagrangian is LFBD =T(E)LF+ [1 −T(E)]LB.(150) Since the transition function T(E) depends only on the Lorentz scalar E=√pµpµ, it is invariant under gauge transformations: T′(E) = T(E).(151) Using Eqs. (146), (148), and (151), we find L′ FBD =T(E)L′ F+ [1 −T(E)]L′ B=T(E)LF+ [1 −T(E)]LB=LFBD,(152) so the full theory including FBD is gauge invariant. A.3 Field-strength tensors A.3.1 Non-abelian gauge fields The field-strength tensor for the SU(2)Lgauge fields is Fa µν =∂µWa ν−∂νWa µ+gWϵabcWb µWc ν,(153) or, in matrix form, Fµν =∂µWν−∂νWµ+igW[Wµ, Wν].(154) Using Eq. (137), the transformation of Fµν is Fµν →F′ µν =ULFµνU−1 L.(155) A.3.2 Abelian gauge field For the U(1)Ygauge field the field-strength tensor is Gµν =∂µBν−∂νBµ,(156) which is manifestly gauge invariant: Gµν →G′ µν =Gµν.(157) Therefore the gauge-field Lagrangian Lgauge =−1 4Tr(FµνFµν)−1 4GµνGµν (158) is invariant under SU(2)L×U(1)Y. 41 B.3.2 Correspondence with polarization vectors For a massless gauge boson with momentum kµ= (k0,0,0, k3), the physical polarization vectors ϵµ λ(λ=±1) are ϵµ +=1 √2(0,1,−i, 0),(206) ϵµ −=1 √2(0,1,+i, 0),(207) and satisfy kµϵµ λ= 0, ϵ∗ µ,λϵµ λ′=−δλλ′.(208) Proposition B.2: The structure of ωµ(Eqs. (192), (205)) selects exactly these physical polarizations. Proof: Consider the contraction with ϵµ +: ϵ∗ µ,+ωµ=1 √2(0 ·ω0+ 1 ·ω1+i·ω2+ 0 ·ω3) =1 √2(ω1+iω2).(209) Using {ω1, ω1}={ω2, ω2}=−2I4(Eqs. (194), (195)), one sees that this contraction produces a nonzero physical contribution. For unphysical polarizations, such as ϵµ L= (0,0,0,1) (longitudinal),(210) ϵµ T= (1,0,0,0) (time-like),(211) we obtain ϵµ,Lωµ=ω3⇒(ω3)2= 0,(212) ϵµ,T ωµ=ω0⇒(ω0)2= 0.(213) Thus longitudinal and time-like modes are projected out, and only the physical polarizations remain. □ B.4 Lorentz invariance B.4.1 Behavior under Lorentz transformations Under a Lorentz transformation xµ→x′µ= Λµ νxν, spinor fields transform as Ψ(x)→Ψ′(x′) = S(Λ)Ψ(x),(214) where S(Λ) = exp i 4ωµνσµν,(215) and σµν =i 2[γµ, γν] (see, e.g., [?,42]). Theorem B.2: The bosonic Lagrangian LB=¯ Ψiωµ∂µΨ (216) 48 is Lorentz invariant. Proof: The transformation property of ωµfollows from that of γµand γ′µ: S−1γµS= Λµ νγν,(217) S−1γ′µS= Λµ νγ′ν.(218) Therefore, S−1ωµS=S−1γµ+γ′µ 2S=Λµ ν(γν+γ′ν) 2= Λµ νων.(219) Then L′ B=¯ Ψ′iωµ∂′ µΨ′ =¯ ΨS†iωµ(Λ−1)ν µ∂νSΨ =¯ Ψi(S†ωµS)(Λ−1)ν µ∂νΨ =¯ Ψiων∂νΨ = LB.(220) Hence LBis Lorentz invariant. □ B.4.2 Covariance and physical interpretation The coexistence of transverse-mode selection (Eqs. (205), (208)) and Lorentz invariance (Eq. (220)) is ensured by the following structure: 1. ωµtransforms as a four-vector (Eq. (219)); 2. the special anticommutation relations (Eq. (192)) are preserved under Lorentz transformations; 3. the selection of the two transverse degrees of freedom (Eq. (205)) is realized in a frame-independent way. B.5 Applications and consequences B.5.1 Application to the electromagnetic field A bosonic formulation of Maxwell’s equations can be written as ωµFµν = 0.(221) This automatically enforces the Lorenz gauge condition, ∂µAµ= 0,(222) i.e. physical degrees of freedom are selected while unphysical ones are projected out. This is consistent with the standard quantization of the electromagnetic field [42]. B.5.2 Extension to gluons In a non-abelian gauge theory, we have ωµDab µ=ωµ(δab∂µ+gfabcAc µ),(223) so that color degrees of freedom are preserved while space-time degrees of freedom are restricted to the transverse components. 49 B.6 Representation-theoretic interpretation B.6.1 Relation to Lie algebras The commutation relations of ωµcan be written as [ωµ, ων] = 2iϵµνρσηρλωσPλ,(224) for a suitable projection operator Pλ. This may be regarded as a deformation of the usual Lorentz algebra and illustrates a connection between Clifford algebras and Lie algebras [43]. B.6.2 Possible extension to supersymmetry One may consider a deformed supersymmetry algebra using ωµ: {Qα,¯ Q˙ β}= 2ωµ α˙ βPµ,(225) as a modification of the standard supersymmetry algebra [47], which suggests new links between supersymmetry and bosonic degrees of freedom. B.7 Summary The main results of this appendix are: 1. We determined the anticommutation relations of the bosonic gamma matrices ωµ (Eq. 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