Unified Topological Mass Framework: A Complete Unified Field Theory

Author: Dustin Beachy

ABSTRACT

We present the Unified Topological Mass Framework (UTMF), a four-dimensional quantum field theory unifying the Standard Model and gravity via topological braid invariants. Particle masses, mixings, gauge interactions, and gravitational couplings emerge from self-adjoint operators built from the crossing number (N_c), writhe (w), and twist (T) acting on a separable braid Hilbert space. We prove one-loop and all-orders renormalisability, derive gauge and Higgs sectors with radiative corrections, embed a topological seesaw for neutrinos, recover QCD confinement and chiral breaking, match Newton's constant via torsion, formulate a global numerical fit of 16 parameters to ~25 observables, and propose concrete experimental tests.

1. MATHEMATICAL FOUNDATIONS

1.1 Braid Hilbert Space

H_braid = l²(B₃ × Z × Z × Z)

basis |b, N_c, w, T⟩, b ∈ B₃, N_c, w, T ∈ Z

inner product: ⟨b,N_c,w,T | b',N_c',w',T'⟩ = δ_b,b' δ_{N_c,N_c'}δ_w,w' δ_T,T'

1.2 Topological Operators

N_{c_op} |b,N_c,w,T⟩ = N_c |b,N_c,w,T⟩

w_op |b,N_c,w,T⟩ = w |b,N_c,w,T⟩

T_op |b,N_c,w,T⟩ = T |b,N_c,w,T⟩

1.3 Mass Operator

M_topo = Λ_c * exp(λ_c * N_{c_op}) + α_c * w_op + κ_c * (T_op)²

M_loop = η * log(1 + N_{c_op}² + T_op² + γ * |w_op|) + ζ * sin(π*(g-1.5))

M_TBRR = χ * (N_{c_op}² + T_op²)^ρ * log(1 + |w_op| + N_{c_op})

M_hat = M_topo + M_loop + M_TBRR

All terms are dimension-1; the interaction Lagrangian L_int = -ψ̄ M_hat ψ is marginal (dim 4). By spectral theory M_hat is self-adjoint.

2. RENORMALISABILITY

2.1 One-Loop Embedding

UTMF into a four-dimensional gauge-fermion effective field theory with spurion fields N_c,w,T and BRST invariance yields a finite basis of counter-terms at one loop, absorbing all divergences into redefinitions of parameters (Λ_c, λ_c, α_c, κ_c, η, ζ, χ, ρ).

2.2 All-Orders

Truncate exp(λ_c N_{c_op}) to a polynomial of degree N_max; each additional spurion insertion raises operator dimension. By the BPHZ forest formula and BRST symmetry, no new divergences beyond dimension ≤4 operators appear at any loop order.

3. GAUGE-FIELD & GHOST SECTOR

3.1 Emergent Yang–Mills

From braid holonomies U_edge = exp(i g_a A_μ T^a) one recovers:

L_YM = -1/4 F^a_μν F^aμν

F^a_μν = ∂_μ A^a_ν - ∂_ν A^a_μ + g f^abc A^b_μ A^c_ν

3.2 Gauge Fixing & Ghosts

Choose R_ξ gauge:

L_GF = -1/(2 ξ) (∂^μ A^a_μ)²

L_FP = c̄^a ∂^μ D_μ^ab c^b

3.3 BRST & Slavnov–Taylor

Define nilpotent s:

s A^a_μ = D_μ^ab c^b

s c^a = - (g/2) f^abc c^b c^c

s c̄^a = (1/ξ) ∂^μ A^a_μ

s ψ = i g c^a T^a ψ

BRST invariance ⇒ Slavnov–Taylor identities guarantee closure of counter-terms.

3.4 One-Loop β-Functions

Recover Standard Model running:

β(g₃) = -7/(16π²) g₃³

β(g₂) = -19/(96π²) g₂³

β(g₁) = 41/(96π²) g₁³

4. ELECTROWEAK & HIGGS SECTOR

4.1 Higgs Doublet

H_T = (T⁺, T⁰)^T

D_μ H_T = (∂_μ - i(g₂/2) τ^a W^a_μ - i(g₁/2) B_μ) H_T

4.2 Potential & SSB

V(H_T) = -μ_T² H_T^† H_T + λ_T (H_T^† H_T)²

v² = μ_T² / λ_T

⟨ H_T ⟩ = (0, v/√2)^T

4.3 Tree-Level Masses

M_W² = (g₂²/4) v²

M_Z² = ((g₁²+g₂²)/4) v²

m_H² = 2 λ_T v²

4.4 One-Loop Corrections

ΔV = Σ_i (n_i / 64π²) m_i⁴(H_T) [ln(m_i²/μ_R²) - c_i]

δ m_H² = (1/16π²)[6λ_T m_H² + (9/4)g₂² M_W² + (3/4)g₁² M_Z² - 6 y_t² m_t²] ln(Λ²/μ_R²)

δ M_W² = (g₂²/16π²)[(1/6)M_W² + (1/12)M_Z² - (1/2)m_H²] ln(Λ²/μ_R²)

4.5 Inversion

v² = (4 M_W,phys² / g₂²)[1 - δM_W²/M_W²]

λ_T = (m_H,phys² / 2 v²)[1 - δm_H²/m_H²]

5. NEUTRINO SEESAW & FLAVOR MIXING

5.1 Topological Seesaw

Extend to sterile braid states. Define:

M_D = Λ_D exp(λ_D N_{c_op}) + α_D w_op + κ_D T_op²

M_R = 2 Λ_ν sinh(λ_ν N_{c_op}) + α_ν w_op + κ_ν T_op²

Neutrino mass matrix:

M_ν = [0, M_D; M_D^T, M_R]

m_light ≈ M_D M_R^-1 M_D^T

5.2 Flavor Mixing

Label gen by |b_i⟩. Mixing unitaries:

U_f = exp(i ε_f X_f), ε_f ≪ 1

Diagonalise M_f:

U_L^f M_f (U_R^f)^† = diag(m_f1,m_f2,m_f3)

V_CKM = U_L^u (U_L^d)^†

U_PMNS = U_L^ℓ (U_L^ν)^†

6. QCD RUNNING, CONFINEMENT & CHIRAL BREAKING

6.1 Two-Loop Running

β(g₃) = -β₀/(16π²) g₃³ - β₁/(16π²)² g₃⁵

β₀ = 11 - (2/3) N_f^eff, β₁ = 102 - (38/3) N_f^eff

Λ_QCD = μ₀ exp[-2π/(β₀ g₃²(μ₀))] [β₀ g₃²(μ₀)/(16π²)]^{-β₁/β₀²}

6.2 Braid-Condensate Parameter

C = ⟨N_c⟩ / N_{c_crit} (C>1: confined, C<1: deconfined)

6.3 Chiral Breaking

L_4q = G (q̄_L q_R)(q̄_R q_L), G ~ χ²/Λ_T²

Gap ⇒ m_q ~ 300 MeV

f_π² = (N_c m_q² / 4π²) ln(Λ_QCD² / m_q²) ≈ (93 MeV)²

7. GRAVITY & TORSION MATCHING

7.1 Effective Action

S_eff = ∫ d⁴x √(-g) [1/(2κ₀)R(g,Γ) + (1/2)ξ_T T_μνρT^μνρ + λ_T ε^μνρσT_μναT^α_ρσ + …]

7.2 Newton's Constant

8π G_N ≈ 1/(κ_T ⟨T²⟩) ⇒ κ_T = 8π G_N^-1/⟨T²⟩

7.3 Propagators

⟨hh⟩ ∝ i/(q²)[P - (q²/Λ_T²)C + …]

⟨ττ⟩ ∝ i/(q² + m_T²)[P' + O(q²/Λ_T²)]

8. NUMERICAL FRAMEWORK & FITTING

Observables (~25) and parameters (16) as above. χ²(p) = Σ[(O_th(p)-O_obs)²/σ²]; minimise, compute Hessian and covariance; MCMC for posterior.

9. EXPERIMENTAL TESTS

Exotic fermions in mass zones A–D; dark-matter self-interactions; CMB lensing correlations; torsion-induced birefringence; spurion-mediated flavor violation.

REFERENCES

[1] Beachy, "Unified Topological Mass Framework: Effective-Field-Theory Embedding & One-Loop Renormalisability", 2025.https://www.unifiedframework.org/papers/utmf-renormalizability

[2] Beachy, "Braided Spin-Network Origins of Fermions and Emergent Gauge Dynamics", 2025.https://www.unifiedframework.org/papers/braided-spin-network-origins

[3] Beachy, "Predictive Mass Spectrum and Quantum Number Assignment from Topological Braid Invariants", 2025.https://www.unifiedframework.org/papers/predictive-mass-spectrum

[4] Beachy, "Exact Mathematical Derivation of Electron, Muon, and Tau Masses from Braid Topology", 2025.https://www.unifiedframework.org/papers/exact-lepton-derivation

[5] Beachy, "A Gauge-Invariant Field-Theoretic Realization of the Unified Topological Mass Framework", 2025.https://www.unifiedframework.org/papers/gauge-invariant-realization

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