HCSM-44 The Proca Sector The u(1)Y Origin from the Hypercharge Functional, the Proca Lagrangian from Electroweak Symmetry Breaking, and the Mass Ratio mW /mZ from the Weak Mixing Angle
Year: 2026
Type: preprint
Abstract: The Proca sector of the Standard Model — the effective description of the massive W± and Z gauge bosons after electroweak symmetry breaking is derived here in the Hodge Complex Standard Model (HCSM). The construction closes the last remaining structural blocker of the u(1)Y gauge sector: the origin of the hypercharge U(1). We prove that u(1)Y arises from the conserved hypercharge functional q3 = F (n1, σ) on the visible sector Ω0|λ=4 (HCSM-07 [6]), whose generator Yˆ commutes with the processor T = I − PC and with the orientation Z2 F of HCSM-43 [13]. Given: • the electroweak vacuum expectation value vEW = MP exp(−kLphys) from HCSM-40 [11]; • the weak mixing angle sin2 θW (MZ ) = 0.2312 from HCSM-39 [10];• the su(2)L coupling g fixed by αem(MZ);the Proca Lagrangian follows with L =−1WaWaμν−1B Bμν+1m2W±W−μ+1m2ZZμ Proca 4 μν 4 μν 2 W μ 2 Z μ g vEW √︁g2 + g′2 vEW mW mW=2, mZ= 2 , m=cosθW. Z The mass ratio mW /mZ = √︁1 − sin2 θW = 0.8769 matches the PDG ratio 0.8814 at 0.5%. Transversality ∂μAμ = 0 is derived from the Higgs mechanism (HCSM-40) and the Goldstone boson is eaten. No new degrees of freedom are required; the Proca sector of the Standard Model is fully reproduced from the substrate. Derivation status summary. Derived: the u(1)Y generator Yˆ from the hypercharge functional; the commutation [Yˆ , T ] = 0 and [Yˆ , F ] = 0; the Proca Lagrangian; the tree level masses mW , mZ ; the mass ratio mW /mZ = cos θW ; transversality of the massive vector fields; the Goldstone mode is eaten. Disclosed inputs: the hypercharge functional q3 (HCSM-07); the electroweak VEV vEW (HCSM-40); the weak mixing angle sin2 θW (HCSM-39); the electromagnetic coupling αem(MZ) = 1/127.9 (an experimental input, as in HCSM-39). Open: two-loop corrections to the Proca masses from the DNLS mean field; threshold effects onmW,mZ fromthehiddensector5+6+31(HCSM-38);thepreciseEFTmatchingbetween the discrete Proca field and the continuum Proca Lagrangian.
Author: Preschutti, Stanley
Institution: Information Physics Institute
Language: English
Subfield: Nuclear and High Energy Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214716571