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∗ November 26, 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 .......................... 6 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 ................... 7 1.6 Electroweak theory in terms of symmetry-broken gamma matrices ..... 7 1.7 Aim and structure of this paper ........................ 8 2 Standard electroweak theory revisited 9 2.1 Structure of the SU(2)L×U(1)Ygauge theory ................ 9 2.1.1 Field content and quantum numbers ................. 9 2.1.2 Gauge fields and covariant derivatives ................. 10 2.2 Spontaneous symmetry breaking and the Higgs mechanism ......... 10 2.2.1 Higgs field and potential ........................ 10 2.2.2 Mass generation for gauge bosons ................... 11 2.2.3 Mass eigenstates ............................ 11 2.3 Chiral structure: manual implementation in the theory ........... 12 2.3.1 Charged-current interactions ...................... 12 2.3.2 Neutral-current interactions ...................... 12 2.4 Problems of the chiral structure: critical analysis .............. 13 2.4.1 Asymmetries in the construction of the theory ............ 13 2.4.2 Mixture of experimental facts and theoretical assumptions ..... 13 2.4.3 Mass hierarchy problem ........................ 14 2.5 Need to go beyond the Standard Model .................... 14 3 Electroweak theory with gamma matrices 14 3.1 Introducing the 4 ×4 gamma-matrix representation ............. 15 3.1.1 Dirac representation and choice of basis ................ 15 3.1.2 Four-component lepton spinor ..................... 15 3.2 Construction of symmetry-broken gamma matrices ............. 16 3.2.1 τγmatrices as carriers of the weak interaction ............ 16 3.2.2 Hypercharge operator .......................... 16 3.3 Decomposition of gamma matrices and geometry of mass generation . . . . 16 3.3.1 Basic decomposition theorem ..................... 16 3.3.2 Rewriting the covariant derivative ................... 17 3.4 Four-component Higgs field and local symmetry breaking .......... 17 3.4.1 Four-component Higgs field ...................... 17 3.4.2 Generation of mass terms ....................... 18 3.5 Natural emergence of the chiral structure ................... 19 3.5.1 Geometric derivation of the charged current ............. 19 3.5.2 Asymmetry of the neutral current ................... 20 3.6 Preservation of Ward–Takahashi identities .................. 20 3.6.1 Proof of gauge invariance ........................ 20 3.6.2 Ward–Takahashi identities ....................... 20 3.7 Mathematical formulation of local symmetry breaking ............ 21 3.7.1 Hierarchy of symmetries ........................ 21 2 3.7.2 Parameterizing the breaking ...................... 21 3.7.3 Fiber-bundle structure ......................... 21 3.8 BRST symmetry and ghost fields ....................... 22 3.8.1 BRST transformations ......................... 22 3.8.2 BRST invariance ............................ 22 3.9 Numerical verification ............................. 22 3.10 Summary .................................... 23 4 Complete formulation of fermion–boson duality 23 4.1 Fundamental principles of fermion–boson duality ............... 23 4.1.1 Continuous transition of statistics ................... 23 4.1.2 Physical interpretation ......................... 24 4.2 Construction of bosonic gamma matrices ................... 24 4.2.1 Basic definition ............................. 24 4.2.2 Anticommutation relations and physical degrees of freedom ..... 24 4.3 Extension of FBD to the electroweak sector ................. 25 4.3.1 Extended Lagrangian .......................... 25 4.3.2 Duality of gauge fields ......................... 25 4.4 Theoretical inevitability of neutrinos ..................... 25 4.4.1 Internal consistency of the duality ................... 25 4.4.2 Mathematical proof ........................... 26 4.4.3 Double structure of neutrinos ..................... 26 4.5 Physical meaning and determination of the transition function ....... 27 4.5.1 Hierarchy of transition energies .................... 27 4.5.2 Physical interpretation of the transition width ............ 27 4.5.3 Temperature dependence and cosmological implications ....... 27 4.6 Experimental consequences and observable quantities ............ 28 4.6.1 Electroweak interactions in superconductors ............. 28 4.6.2 High-energy neutrino scattering .................... 28 4.6.3 Search for bosonic neutrinos ...................... 28 4.7 Effects on renormalization theory ....................... 29 4.7.1 Softening of ultraviolet divergences .................. 29 4.7.2 Running of coupling constants ..................... 29 4.8 Preservation of the Ward–Takahashi identities ................ 29 4.8.1 Conservation laws under duality .................... 29 4.8.2 BRST symmetry ............................ 29 4.9 Summary .................................... 29 5 Conclusion 30 5.1 Summary of results ............................... 30 5.1.1 Theoretical achievements ........................ 30 5.1.2 Quantitative achievements ....................... 31 5.2 Experimental tests and predictions ...................... 32 5.3 Theoretical significance and broader impact ................. 32 5.3.1 Impact on particle physics ....................... 32 5.3.2 Implications for cosmology ....................... 33 5.4 Future directions ................................ 33 5.4.1 Theoretical issues ............................ 33 5.4.2 Experimental issues ........................... 33 3 5.5 Closing remarks ................................. 34 A Proof of gauge invariance and BRST symmetry 37 A.1 SU(2)L×U(1)Ygauge invariance ....................... 37 A.1.1 Definition of gauge transformations .................. 37 A.1.2 Transformation of fields ........................ 37 A.1.3 Covariance of the covariant derivative ................. 38 A.2 Gauge invariance of the Lagrangian ...................... 38 A.2.1 Fermion kinetic term .......................... 38 A.2.2 Bosonic kinetic term .......................... 39 A.2.3 Terms with the transition function .................. 39 A.3 Field-strength tensors .............................. 39 A.3.1 Non-abelian gauge fields ........................ 39 A.3.2 Abelian gauge field ........................... 40 A.4 Ward–Takahashi identities ........................... 40 A.4.1 Generating functional .......................... 40 A.4.2 Derivation of the Ward–Takahashi identities ............. 40 A.4.3 Ward identity in momentum space .................. 41 A.5 BRST symmetry ................................ 41 A.5.1 Definition of BRST transformations .................. 41 A.5.2 BRST invariance ............................ 42 A.5.3 Nilpotency ................................ 42 A.6 Preservation of symmetries in the FBD theory ................ 42 A.6.1 Role of the transition function ..................... 42 A.6.2 Quantum corrections .......................... 43 A.7 Summary .................................... 43 B Lorentz invariance and bosonic gamma matrices 44 B.1 Construction of the bosonic gamma matrices ................. 44 B.1.1 Definition and motivation ....................... 44 B.1.2 Construction from two Dirac systems ................. 44 B.2 Anticommutation relations ........................... 45 B.2.1 Component-wise calculation ...................... 45 B.2.2 Anticommutation relations ....................... 45 B.3 Automatic selection of transverse degrees of freedom ............ 46 B.3.1 Analysis of kinetic terms ........................ 46 B.3.2 Correspondence with polarization vectors ............... 46 B.4 Lorentz invariance ............................... 47 B.4.1 Behavior under Lorentz transformations ............... 47 B.4.2 Covariance and physical interpretation ................ 48 B.5 Applications and consequences ......................... 48 B.5.1 Application to the electromagnetic field ................ 48 B.5.2 Extension to gluons ........................... 48 B.6 Representation-theoretic interpretation .................... 49 B.6.1 Relation to Lie algebras ........................ 49 B.6.2 Possible extension to supersymmetry ................. 49 B.7 Summary .................................... 49 4 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. 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 5 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. 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. 6 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. 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. 7 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) and the right-handed electron singlet (Eq. (9)): eR, T = 0, YeR=−1,(9) are introduced. 8 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 −TW F (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 Physical meaning and determination of the transition function 4.5.1 Hierarchy of transition energies For each particle species, the transition energy is related to its characteristic mass scale: E(e) fb ∼mec2∼0.5 MeV (electron), E(ν) fb ∼mνc2∼0.1 eV (neutrino), E(W) fb ∼MWc2∼80 GeV (Wboson), E(Z) fb ∼MZc2∼91 GeV (Zboson).(104) The hierarchy in Eq. (104) implies that different duality effects appear at different energy scales. 4.5.2 Physical interpretation of the transition width The transition width ℏνdetermines how sharp the change of statistics is: ∆Etransition ∼4ℏν. (105) For a narrow transition width (ℏν≪Efb): •the change of statistics is sharp, •critical-phenomena-like behavior occurs, •there are similarities to BCS transitions and other collective phenomena. For a broad transition width (ℏν∼Efb): •the change of statistics is gradual, •crossover-like behavior appears, •single-particle effects dominate. 24 4.5.3 Temperature dependence and cosmological implications At finite temperature T, the transition function is extended to include thermal effects: T(E, T ) = 1 1 + exp E−Efb−∆(T) ℏν+kBT,(106) where ∆(T) is a temperature-dependent correction. In the early universe (T≫Efb), Eq. (106) implies that: •all particles behave bosonically, •the distinction between Fermi and Bose statistics disappears, •this may be compatible with grand unified theories. In the present universe (T≪Efb), •ordinary statistics are recovered, •the predictions of the Standard Model are reproduced. 4.6 Experimental consequences and observable quantities 4.6.1 Electroweak interactions in superconductors In the superconducting state, electrons form Cooper pairs and behave effectively as bosons. The FBD theory predicts that δMW MW≃10−6×ns n0 ,(107) where nsis the superconducting electron density and n0is the normal-state electron density. Equation (107) gives a quantitative estimate for the tiny change of electroweak parameters in a superconductor. 4.6.2 High-energy neutrino scattering For neutrino scattering at E > E(ν) fb , the cross section can be written as σ(E > Efb) = σ0"1 + αE Efb 2#,(108) where σ0is a reference cross section and αis a parameter characterizing the duality effect. The energy dependence in Eq. (108) can be tested by high-energy neutrino observations such as IceCube [25,26]. 25 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: •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. 32 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. 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. 33 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,(114) U(1)Ytransformation: UY(x) = exp iβ(x)α 2.(115) The full gauge transformation is then U(x) = UL(x)·UY(x) = exp iθa(x)τa γ 2+iβ(x)α 2.(116) A.1.2 Transformation of fields The fermion field transforms as Ψ(x)→Ψ′(x) = U(x)Ψ(x).(117) 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),(118) and for the U(1)Ygauge field Bµ(x)→B′ µ(x) = Bµ(x)−1 gB ∂µβ(x).(119) In matrix notation, Eq. (118) reads Wµ≡τa γ 2Wa µ→W′ µ=ULWµU−1 L−i gW (∂µUL)U−1 L,(120) as in the main text. 34 A.1.3 Covariance of the covariant derivative The covariant derivative is Dµ=∂µ+igW 2τγ·Wµ+igB 2αBµ.(121) Proposition A.1: The covariant derivative transforms covariantly: DµΨ→(DµΨ)′=U(DµΨ).(122) Proof: Using Eqs. (121), (117), we find (DµΨ)′=D′ µΨ′ =∂µ+igW 2τγ·W′ µ+igB 2αB′ µUΨ.(123) The first term gives ∂µ(UΨ) = (∂µU)Ψ + U(∂µΨ),(124) and using Eq. (118) 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.(125) Similarly, for the U(1)Ypart, igB 2αB′ µUΨ = igB 2αBµ−1 gB ∂µβUΨ =UigB 2αBµΨ+(∂µUY)ULΨ.(126) Combining Eqs. (123)–(126), we obtain (DµΨ)′= (∂µU)Ψ + U(∂µΨ) + UigW 2τγ·Wµ+igB 2αBµΨ.(127) Using (∂µU) = (∂µUL)UY+UL(∂µUY), the first term cancels and we are left with (DµΨ)′=U(DµΨ),(128) 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.(129) Using Eq. (122) and Ψ′=UΨ, ¯ Ψ′=¯ ΨU†, we find L′ F=¯ Ψ′iγµ(DµΨ)′ =¯ ΨU†iγµUDµΨ =¯ ΨiγµDµΨ = LF,(130) so LFis gauge invariant. 35 A.2.2 Bosonic kinetic term In the FBD theory the bosonic kinetic term is LB=¯ ΨiωµDµΨ.(131) Since ωµcarries only a space-time index and is invariant under internal transformations, Eq. (122) implies L′ B=¯ ΨU†iωµUDµΨ = ¯ ΨiωµDµΨ = LB,(132) 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.(133) Since the transition function T(E) depends only on the Lorentz scalar E=√pµpµ, it is invariant under gauge transformations: T′(E) = T(E).(134) Using Eqs. (129), (131), and (134), we find L′ FBD =T(E)L′ F+ [1 −T(E)]L′ B=T(E)LF+ [1 −T(E)]LB=LF BD,(135) 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,(136) or, in matrix form, Fµν =∂µWν−∂νWµ+igW[Wµ, Wν].(137) Using Eq. (120), the transformation of Fµν is Fµν →F′ µν =ULFµνU−1 L.(138) A.3.2 Abelian gauge field For the U(1)Ygauge field the field-strength tensor is Gµν =∂µBν−∂νBµ,(139) which is manifestly gauge invariant: Gµν →G′ µν =Gµν.(140) Therefore the gauge-field Lagrangian Lgauge =−1 4Tr(FµνFµν)−1 4GµνGµν (141) is invariant under SU(2)L×U(1)Y. 36 A.4 Ward–Takahashi identities A.4.1 Generating functional Including external sources J, ¯ J, Ja µ, JB µ, the generating functional is Z[J] = ZDΨD¯ ΨDWµDBµexp iZd4xLFBD +¯ JΨ + ¯ ΨJ +Ja µWaµ +JB µBµ.(142) A.4.2 Derivation of the Ward–Takahashi identities Gauge invariance Z[J] = Z′[J] implies δZ = 0 = ZD[fields] δSeiS,(143) for infinitesimal transformations. From the variation of the action, one obtains ∂µ⟨jµ a(x)⟩=−gWϵabc Wb µ(x)jcµ(x)+¯ Ψ(x)τa γ 2Ψ(x)δ4(0),(144) where the conserved current is jµ a=¯ Ψγµτa γ 2Ψ−1 gW Fµν a∂ν.(145) A.4.3 Ward identity in momentum space In momentum space, for the fermion self-energy Σ(p) and vertex function Γµ(p, p +q), the Ward identity is Σ(p+q)−Σ(p) = qµΓµ(p, p +q),(146) or, in terms of the full propagator S(p), qµΓµ(p, p +q) = S−1(p+q)−S−1(p),(147) with S(p) = [iγµpµ−m−Σ(p)]−1.(148) For the gauge-boson polarization tensor Πµν(k), we have kµΠµν(k) = 0,(149) which guarantees the transverse structure of the full propagator (in a suitable gauge) Dµν(k) = −i k2−M2−Π(k2)gµν −kµkν M2.(150) 37 A.5 BRST symmetry A.5.1 Definition of BRST transformations With a Grassmann parameter ϵ, the BRST transformation on the fermion fields is δBΨ = iϵcaτa γ 2Ψ + iϵcYα 2Ψ.(151) For the gauge fields, δBWa µ=−1 gW Dab µcbϵ, (152) δBBµ=−1 gB ∂µcYϵ, (153) and for ghost and anti-ghost fields, δBca=−1 2gWϵabccbccϵ, (154) δBcY= 0,(155) δB¯ca=Baϵ, (156) δB¯cY=BYϵ. (157) For the Nakanishi–Lautrup fields, δBBa= 0, δBBY= 0.(158) A.5.2 BRST invariance Theorem A.1: The full Lagrangian, including gauge-fixing and ghost terms, is BRST invariant. Proof: The gauge-fixing term in Lorenz gauge is LGF =−1 2ξ(∂µWa µ)2−1 2ξ′(∂µBµ)2,(159) and the ghost term is Lghost = ¯ca∂µDab µcb+ ¯cY∂2cY.(160) The full Lagrangian including FBD is Ltotal =LFBD +Lgauge +LGF +Lghost.(161) By applying the BRST transformations (151)–(158) to each term and using the standard properties of non-abelian gauge theories [21,22,18], one finds that all variations cancel: δBLtotal = 0.(162) Thus the full Lagrangian is BRST invariant. □ 38 A.5.3 Nilpotency The BRST transformation is nilpotent: δ2 B= 0.(163) Proof: For example, for the fermion field, δ2 BΨ = δBiϵcaτa γ 2Ψ =iϵ(δBca)τa γ 2Ψ + iϵcaτa γ 2(δBΨ) =−iϵ2 2gWϵabccbccτa γ 2Ψ + (higher-order terms).(164) Using the anticommutativity cbcc=−cccband the total antisymmetry of ϵabc, the righthand side vanishes. A similar argument holds for all fields. □ A.6 Preservation of symmetries in the FBD theory A.6.1 Role of the transition function The crucial point in the FBD theory is that the transition function T(E) is a gaugeinvariant Lorentz scalar. Lemma A.1:T(E) is invariant under gauge and BRST transformations. Proof: Since E=√pµpµis a Lorentz-invariant quantity and gauge/BRST transformations do not change the momentum pµ, pµ→pµ,(165) we have T(E)→T(E), δBT(E) = 0.(166) □ A.6.2 Quantum corrections At one-loop level, the FBD-modified fermion self-energy is ΣFBD(p) = ΣSM (p)T(p2)+ΣB(p) [1 −T(p2)].(167) The Ward identity then becomes qµΓµ,FBD(p, p +q) = qµT(p2)Γµ,F + [1 −T(p2)]Γµ,B =T(p2)S−1 F(p+q)−S−1 F(p) + [1 −T(p2)] S−1 B(p+q)−S−1 B(p) =S−1 FBD(p+q)−S−1 FBD(p),(168) showing that the Ward–Takahashi identities hold in the FBD theory as well. 39 A.7 Summary The main results of this appendix are: 1. The SU(2)L×U(1)Ygauge invariance is strictly preserved even in the presence of the symmetry-broken gamma matrices τγ. 2. The Ward–Takahashi identities remain valid, both in the standard form (Eqs. (147), (149)) and in the FBD-extended form (Eq. (168)). 3. BRST symmetry and its nilpotency (Eqs. (162), (163)) are preserved, ensuring unitarity of the theory. 4. The gauge and BRST invariance of the transition function T(E) (Eqs. (134), (166)) guarantees the internal consistency of the FBD theory. 5. All symmetries remain valid even at the level of quantum corrections (Eqs. (167), (168)). For modern discussions of BRST symmetry and gauge invariance, see, for example, Refs. [37,38,39,40,41]. B Lorentz invariance and bosonic gamma matrices In this appendix we analyze in detail the mathematical structure of the bosonic gamma matrices ωµ, which play a central role in fermion–boson duality. We prove the mechanism of automatic selection of the two transverse degrees of freedom and its compatibility with Lorentz invariance. This structure is consistent with the standard discussion of Clifford algebras and Lorentz symmetry [42,17,43]. B.1 Construction of the bosonic gamma matrices B.1.1 Definition and motivation The usual Dirac gamma matrices γµsatisfy the anticommutation relation {γµ, γν}= 2gµνI4,(169) and describe the four degrees of freedom of a fermion field [42,43]. On the other hand, a massless boson field (e.g. a gauge boson) has only two transverse degrees of freedom. To implement this difference in degrees of freedom in a natural way, we introduce the bosonic gamma matrices ωµdefined in Eq. (91). B.1.2 Construction from two Dirac systems First Dirac system: the standard gamma matrices γµare given by γ0=−I20 0I2, γi=0σi −σi0.(170) Second Dirac system: we define γ′µby γ′0=−γ3, γ′1=γ1, γ′2=γ2, γ′3=−γ0.(171) 40 The matrices γ′µalso satisfy the Clifford algebra {γ′µ, γ′ν}= 2gµνI4.(172) The bosonic gamma matrices are defined as ωµ=γµ+γ′µ 2,(173) which is the starting point for deriving the anticommutation relations (Equation (93)) and the projection onto transverse degrees of freedom (Equation (188)). B.2 Anticommutation relations B.2.1 Component-wise calculation The explicit matrix forms of ωµare obtained by substituting the Dirac representation of γµand γ′µinto Eq. (173). For ω0: ω0=γ0+γ′0 2=γ0−γ3 2,(174) and similarly for ω1,ω2, and ω3. The explicit forms are given and used in the main text and in the Mathematica notebooks. B.2.2 Anticommutation relations Theorem B.1: The bosonic gamma matrices satisfy {ωµ, ων}=     −2I4(µ=ν= 1,2), 0 (µ=ν= 0,3), 0 (µ=ν). (175) Proof: Using the explicit matrices for ωµand the original anticommutation relations {γµ, γν}and {γ′µ, γ′ν}, one finds: For µ= 0: {ω0, ω0}= 2(ω0)2=1 2{(γ0−γ3),(γ0−γ3)} =1 2[(γ0)2+ (γ3)2−{γ0, γ3}] =1 2[−I4+I4−0] = 0.(176) For µ= 1: {ω1, ω1}= 2(γ1)2=−2I4.(177) For µ= 2: {ω2, ω2}= 2(γ2)2=−2I4.(178) 41 [32] DUNE Collaboration, B. Abi et al., “Deep Underground Neutrino Experiment (DUNE), Far Detector Technical Design Report, Volume II: DUNE Physics,” JINST 15, T08008 (2020) [arXiv:2002.03005 [hep-ex]]. [33] K. Abe et al. (Hyper-Kamiokande Proto-Collaboration), “Hyper-Kamiokande Design Report,” Prog. Theor. Exp. Phys. 2018, 063C01 (2018), arXiv:1805.04163 [physics.ins-det]. [34] J. de Blas, “Impact of recent measurements of the top-quark and W-boson masses on electroweak precision fits,” PoS LHCP2022, 104 (2022) [arXiv:2212.07325 [hep-ph]]. [35] C.-T. Lu, L. Wu, Y. Wu and B. Zhu, “Electroweak precision fit and new physics in light of Wboson mass,” Phys. Rev. D 106, 035034 (2022) [arXiv:2204.03796 [hep-ph]]. [36] J. de Blas et al., “Global SMEFT fits at future colliders,” J. High Energy Phys. 01, 106 (2023) [arXiv:2206.08326 [hep-ph]]. [37] S. Asnafi, H. Gies, and L. Zambelli, “BRST-invariant RG flows,” Phys. Rev. D 99, 085009 (2019), arXiv:1811.03576 [hep-th]. [38] A. Helset, A. Martin and M. Trott, “The Geometric Standard Model Effective Field Theory,” J. High Energy Phys. 03, 163 (2020), arXiv:2001.01453 [hep-ph]. [39] J. Bellor´ın and A. Restuccia, “BRST symmetry and unitarity of the Hoˇrava theory,” Phys. Rev. D 107, 044059 (2023), arXiv:2211.01783 [hep-th]. [40] H. Erbin, String Field Theory: A Modern Introduction, Lecture Notes in Physics, Vol. 980, Springer (2021), arXiv:2301.01686 [hep-th]. [41] S. Weinberg, The Quantum Theory of Fields, Volume II: Modern Applications, Cambridge University Press, Cambridge, UK (1996). [42] M. D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press, 2014. [43] P. Lounesto, Clifford Algebras and Spinors, 2nd ed., Cambridge University Press, 2001. [44] S. J. Brodsky, H.-C. Pauli and S. S. Pinsky, “Quantum chromodynamics and other field theories on the light cone,” Phys. Rept. 301, 299–486 (1998). doi:10.1016/S03701573(97)00089-6 [45] T. Heinzl, “Light-cone quantization: Foundations and applications,” Lect. Notes Phys. 572, 55–142 (2001). doi:10.1007/3-540-45114-5 2 [46] H. Elvang and Y.-t. Huang, Scattering Amplitudes in Gauge Theory and Gravity, Cambridge University Press, 2015. [47] D. Z. Freedman and A. Van Proeyen, Supergravity, Cambridge University Press, 2012. 48