Deriving the Weinberg Angle from Mutual Friction

What We’re Computing

The Weinberg angle \theta_W is defined by:

\sin^2\theta_W = \frac{g'^2}{g^2 + g'^2} \approx 0.231

where g is the SU(2)_L coupling constant and g' is the U(1)_Y coupling constant. The measured values at the Z mass scale (M_Z = 91.2\;\text{GeV}):

g \approx 0.653, \qquad g' \approx 0.358, \qquad \tan\theta_W = g'/g \approx 0.548

In the Standard Model, these are free parameters — inputs, not outputs. If the substrate can derive their ratio from its internal physics, that’s a genuinely new result.

The Two Couplings as Boundary Layer Properties

In the substrate framework, both g and g' describe how gauge modons (weak force carriers) couple to the counter-rotating boundary of a fermion’s orbital system. But they couple to different physical properties of that boundary.

From the HVBK mutual friction formalism (Two Fluids → Quantum Potential), the coupling between co-rotating and counter-rotating layers has two components:

\mathbf{F}_{ns} = \frac{B\,\rho_n\,\rho_s}{2\rho}\;\hat{s} \times [\hat{s} \times (\mathbf{v}_n - \mathbf{v}_s)] \;+\; \frac{B'\,\rho_n\,\rho_s}{2\rho}\;\hat{s} \times (\mathbf{v}_n - \mathbf{v}_s)

The B term (dissipative): transfers energy between the two layers. It damps relative motion and governs energy exchange across the boundary.

The B’ term (reactive/Hall): redirects flow without transferring energy. It’s a gyroscopic deflection — a force perpendicular to the velocity difference that rotates the flow pattern without changing its magnitude.

These two terms correspond to the two gauge couplings:

g \leftrightarrow B' (reactive coupling): The SU(2)_L coupling measures how strongly a gauge modon couples to the chirality direction of the boundary — the orientation of the counter-rotating flow pattern. This is a reactive coupling because chirality is a direction, not an energy. Changing chirality direction (which is what the W boson does — it flips left↔︎right) is a rotation of the flow pattern, not a dissipation of energy. The reactive HVBK term performs exactly this: flow-pattern rotations.

g' \leftrightarrow B (dissipative coupling): The U(1)_Y coupling measures how strongly a gauge modon couples to the total flow magnitude — the net co-rotating current that constitutes hypercharge. This is a dissipative coupling because changing the total flow (adding or removing co-rotating current) involves energy transfer across the boundary. The dissipative HVBK term governs exactly this: energy transfer.

The HVBK Coefficients from Microscopic Scattering

In superfluid helium, B and B' are both determined by the microscopic physics of vortex-core scattering. A quasiparticle approaching a quantized vortex line (the analog of the counter-rotating boundary) has two scattering outcomes:

  1. Transverse deflection (reactive): the quasiparticle is deflected sideways by the Magnus-like force of the vortex circulation. Contributes to B'.

  2. Longitudinal drag (dissipative): the quasiparticle transfers momentum to the vortex core through direct collision. Contributes to B.

The standard microscopic theory (Iordanskii-Sonin-Stone) gives these coefficients in terms of the scattering phase shift \delta_0 of the quasiparticle-vortex interaction. In the quantum scattering regime, the partial-wave cross sections give the mutual friction parameter:

\alpha_{mf} = \tfrac{1}{2}\sin 2\delta_0

where \delta_0 is the s-wave phase shift of dc1 quasiparticle scattering off the quantized vortex boundary.

External substrate Core Counter-rotating boundary (width δ) Incoming dc1 / modon wavevector k b θ Fermion vortex boundary Phase shift δ₀ from core scattering K_d (dissipative) Energy transfer → g' K_r (reactive) Phase shift → g K_d / K_r = α_mf = tan²θ_W sin²θ_W = α_mf / (1 + α_mf)

Scattering geometry of a dc1 quasiparticle (or modon) approaching a fermion’s counter-rotating boundary. The dissipative channel K_d transfers energy across the boundary (→ hypercharge coupling g’), while the reactive channel K_r deflects without energy transfer (→ isospin coupling g). Their ratio determines the Weinberg angle: sin²θ_W = K_d/(K_d + K_r).

The Dual-Spin Connection: \sin^2\theta_W from K_d/K_r

From the dual-spin gyroscope model (Spin-Statistics), the reactive and dissipative coupling coefficients of the fermion’s counter-rotating boundary are:

K_r = \tfrac{1}{2}\,I_\text{eff} \cdot \omega_c \qquad\text{(reactive coupling, from Compton-frequency precession)}

K_d = \alpha_{mf} \cdot K_r \qquad\text{(dissipative coupling, from mutual friction)}

The gauge coupling identification gives:

g^2 \propto K_r \quad\text{(SU(2) coupling from reactive boundary response)}

g'^2 \propto K_d \quad\text{(U(1) coupling from dissipative boundary response)}

Therefore:

\tan^2\theta_W = g'^2/g^2 = K_d/K_r = \alpha_{mf}

\boxed{\sin^2\theta_W = \frac{\alpha_{mf}}{1 + \alpha_{mf}}}

For the measured value \sin^2\theta_W = 0.2312:

\boxed{\alpha_{mf} = \frac{\sin^2\theta_W}{1 - \sin^2\theta_W} = \frac{0.2312}{0.7688} = 0.30078}

The mutual friction parameter of the substrate’s counter-rotating boundary, at the electroweak energy scale, is approximately 0.300.

Physical Reasonableness of \alpha_{mf} \approx 0.300

In superfluid He-II, the mutual friction parameter \alpha varies from near 0 (at very low temperatures, where the normal fluid fraction vanishes) to ~1 (near the lambda point, where the two-fluid coupling is maximal). The value \alpha \approx 0.3 occurs at about T/T_\lambda \approx 0.6 — well within the two-fluid regime where both components are dynamically active.

In superfluid He-3, the mutual friction coefficients depend on the order parameter phase (A-phase vs B-phase), temperature, pressure, and magnetic field. Values of \alpha_{mf} \sim 0.1–1 are typical.

The substrate value \alpha_{mf} \approx 0.300 is squarely in the physical range observed in real superfluids. It means the coupling between co-rotating and counter-rotating layers at the electroweak boundary is moderate — neither negligibly weak (g' \to 0) nor maximally strong (g' \to g). For every vortex-quasiparticle scattering event at the counter-rotating boundary, 30% of the interaction is dissipative (energy transfer, U(1) coupling) and 70% is reactive (deflection, SU(2) coupling).

The s-Wave Phase Shift

From the HVBK coefficients section above, the mutual friction parameter connects to the s-wave scattering phase shift via the Breit-Wigner resonance formula:

\alpha_{mf} = \tfrac{1}{2}\sin 2\delta_0

With \alpha_{mf} = 0.30078:

\sin 2\delta_0 = 0.6016, \qquad \delta_0 = 18.48°\;\text{(weak-scattering branch)}

The weak-scattering branch is selected by the physical requirement that electromagnetic interactions are perturbative (\alpha \ll 1). This corresponds to long-lived Caroli-de Gennes-Matricon bound states in the vortex core with quality factor \omega_0\tau = 1/\sin\delta_0 = 1/\sin(18.48°) \approx 3.16, meaning the boundary is nearly transparent to modons.

The strong-scattering branch (\delta_0 = 71.6°) would give electromagnetic coupling of order unity, inconsistent with QED.

Geometric Interpretation: Boundary Oblateness (Conjecture)

The Weinberg angle admits a suggestive geometric interpretation. The fermion’s counter-rotating boundary, spinning at relativistic speeds, develops an oblate distortion. If the SU(2) coupling (g, stronger) connects to the tighter curvature at the poles, while the U(1) coupling (g', weaker) connects to the gentler curvature at the equator, then:

\sin^2\theta_W = \frac{R_\text{polar}^2}{R_\text{polar}^2 + R_\text{equatorial}^2}

This is internally consistent but not yet derived — it requires showing that for a spinning oblate vortex core, the HVBK coefficients B and B' are related to the aspect ratio in this specific way.

For \sin^2\theta_W = 0.2312:

R_\text{polar}/R_\text{equatorial} = 0.548

The boundary eccentricity is e = \sqrt{1 - 0.300} = 0.837 — physically reasonable for a system spinning at the Compton frequency.

This can also be expressed as a Lorentz factor. If the equatorial velocity of the boundary is \beta_\text{eq}\,c:

\sin^2\theta_W = 1 - \beta_\text{eq}^2 = 1/\gamma_\text{eq}^2

\beta_\text{eq} = \cos\theta_W \approx 0.877c

The Weinberg angle is the inverse Lorentz factor squared of the boundary’s equatorial velocity.

Connection to the inner-scale orbital velocity: The two-scale model gives the effective quantum’s orbital velocity v_\text{rot,inner} = c\sqrt{2\alpha_{mf}} \approx 0.776c, which is distinct from \beta_\text{eq} = \cos\theta_W \approx 0.877c. These measure different physical aspects of the same boundary structure: \beta_\text{eq} is the equatorial spin velocity of the counter-rotating boundary shell, while v_\text{rot,inner} is the orbital velocity of the effective quantum inside the shell. The boundary spins faster than what it contains — physically expected for a confining structure.

The two velocities are related through \alpha_{mf} = v_\text{rot,inner}^2/(2c^2), which when substituted into C8 gives:

\sin^2\theta_W = \frac{v_\text{rot,inner}^2}{v_\text{rot,inner}^2 + 2c^2}

The Weinberg angle partitions a velocity-squared budget between the inner orbital rotation and the rest-frame speed of the substrate. The factor of 2 in the denominator traces to the two-fluid nature of the HVBK coupling: both the reactive and dissipative channels contribute to the total interaction, and the dissipative fraction is the squared-velocity ratio to the full budget.

Consistency Check: The W and Z Masses

The standard electroweak relation M_W/M_Z = \cos\theta_W is a consequence of the gauge structure shared by both the Standard Model and the substrate framework. In the substrate picture it becomes:

M_W/M_Z = \beta_\text{eq} = v_\text{equatorial}/c

Numerically: M_W/M_Z = 80.4/91.2 = 0.882, and \cos\theta_W = \sqrt{1 - 0.2312} = 0.877. These match to within 0.5% (the small discrepancy is from radiative corrections — \sin^2\theta_W = 0.2312 is the on-shell value while M_W/M_Z uses pole masses). The ratio of W to Z masses equals the equatorial boundary velocity in units of c.

Running of the Weinberg Angle

The measured \sin^2\theta_W runs with energy — from ~0.238 at low energies to ~0.231 at M_Z to ~0.21 at GUT-scale energies. In the Standard Model, this comes from loop corrections. In the substrate, the running has a physical origin: \alpha_{mf} depends on the energy of the probing excitation.

From Kopnin’s theory:

\alpha_{mf}(E) = \frac{\alpha_0}{1 + (E/E_\text{core})^2}

At low energies, \alpha_{mf} \approx \alpha_0 (full dissipative coupling). At high energies, \alpha_{mf} \to 0 (the probing excitation passes through the core without dissipating, only deflecting). This gives \sin^2\theta_W decreasing at high energies — the correct direction.

Matching the logarithmic slope d(\sin^2\theta_W)/d(\ln E) \approx -0.003 per decade constrains:

E_\text{core} \sim 10^3\;\text{GeV}\;\text{(TeV-scale)}

The vortex core structure sits at the TeV scale — the same scale as electroweak symmetry breaking (v = 246\;\text{GeV}). This is expected: the vortex core in the counter-rotating boundary has energy density set by the same physics as the electroweak VEV.

There are three well-separated characteristic energies: the outer scale at E_\text{outer} = \hbar c/\xi \approx 13 meV (\lambda \sim 100\;\mum), the inner scale at E_\text{inner} = \hbar c/r_\text{eff} \approx 1.3 MeV (r_\text{eff} \approx 150 fm), and the vortex core scale E_\text{core} \sim TeV. The tree-level Weinberg angle predictions sit at a “natural” scale, probably the electron Compton energy m_e c^2 = 0.511 MeV, which lies between E_\text{inner} and E_\text{core} — where the boundary geometry is well-defined but loop corrections from the vortex core structure have not yet kicked in.

Geometrically, the running corresponds to the boundary’s radial velocity profile. At higher energies, the probe resolves inner layers of the boundary spinning at different velocities:

\sin^2\theta_W(E) = 1/\gamma^2(R(E))

At the GUT scale, the probe reaches the innermost boundary where v \to c and \sin^2\theta_W \to 0 — the three couplings approach equality as the boundary becomes indistinguishable from a point-like relativistic vortex.

Distinguishing prediction: The Standard Model predicts \sin^2\theta_W \to 3/8 \approx 0.375 at the GUT scale (gauge coupling unification). The substrate predicts \sin^2\theta_W \to 0. Both agree on the direction of running at accessible energies (\sin^2\theta_W decreases from ~0.238 at low energy to ~0.231 at M_Z), but diverge sharply above the TeV scale. This lies beyond current experimental reach but constitutes a falsifiable difference between the two frameworks.

Connection to the Anomalous Magnetic Moment

From the dual-spin model (Spin-Statistics), the anomalous magnetic moment is:

(g-2)/2 = \eta^2

where \eta = (I_1 - I_2)/(I_1 + I_2) \approx 0.034 is the core-boundary moment of inertia asymmetry. The Weinberg angle gives the boundary shape: R_\text{polar}/R_\text{eq} = 0.548, eccentricity 0.837. These two parameters — \eta (mass asymmetry) and e (shape eccentricity) — are both properties of the same counter-rotating boundary, constrained by two independent experiments (magnetic moment measurement and neutral current weak scattering).

For a spherical core with moment I_\text{core} and an oblate boundary shell with moment I_\text{boundary}:

\eta = \frac{I_\text{core} - I_\text{boundary}}{I_\text{core} + I_\text{boundary}} = 0.034

I_\text{core} = I_\text{boundary} \times 1.070

The core’s moment of inertia is 7.0% larger than the boundary’s — the core is slightly more massive than the boundary shell. This is physically consistent: the core contains the dag center and tightly bound dc1 cloud, while the boundary is a lighter shell of counter-rotating eddies.

What Is Derived vs. What Is Constrained

Structural relationships derived from the substrate:

  • The relationship \sin^2\theta_W = \alpha_{mf}/(1+\alpha_{mf}) — from the HVBK mutual friction structure applied to the fermion boundary
  • The identification of g with the reactive channel and g' with the dissipative channel
  • The correct direction of running (\sin^2\theta_W decreases at high energy)
  • The W/Z mass ratio as boundary equatorial velocity
  • The geometric interpretation as boundary oblateness

The measured input: \sin^2\theta_W = 0.2312 is taken from experiment. This fixes \alpha_{mf} = 0.30078, which in turn fixes the s-wave scattering phase shift \delta_0 = 18.48° via the Breit-Wigner relation.

The zero-parameter prediction chain (SC5): From this single measured input, the boundary geometry determines three independently measured constants with no additional parameters:

Constant Expression from \delta_0 Predicted Measured Discrepancy
\alpha \sin^2\delta_0 \cdot \sin^2\theta_W / \pi 1/135.1 1/137.0 +1.45%
(g-2)/2 \sin^2\delta_0 \cdot \sin^2\theta_W / (2\pi^2) 0.001178 0.001160 +1.6%
\eta \sqrt{\alpha/(2\pi)} 0.03432 0.03406 +0.8%

All three discrepancies are positive and of order 1–2%, consistent with a missing leading-order vacuum polarization correction. The fine structure constant and anomalous magnetic moment are predictions, not fits. See Fine Structure Constant for the full derivation chain, and the Constraint Summary for the extended five-constant relation (SC5 extended) that additionally predicts m_W and m_Z when the electroweak VEV v = 246 GeV is added as a second input.

What remains: The numerical value \alpha_{mf} = 0.300 is currently fixed by measurement (via \sin^2\theta_W), not derived from first principles. Computing \alpha_{mf} from the equilibrium spin rate of the counter-rotating boundary — a well-posed problem in relativistic superfluid dynamics — would promote \sin^2\theta_W itself to a prediction, making the entire SC5 chain a zero-input result.

The Key Result for Subsequent Derivations

The single most important output of this section for the rest of the framework is the connection between the Weinberg angle, the mutual friction parameter, and the s-wave scattering phase shift:

\sin^2\theta_W = 0.2312 \;\;\to\;\; \alpha_{mf} = 0.30078 \;\;\to\;\; \delta_0 = 18.48°

This phase shift \delta_0 is the same quantity that enters the Berry phase calculation for the boundary doublet, which determines the SU(2) gauge coupling g^2 = 4\sin^2\delta_0 and ultimately the fine structure constant. A single geometric parameter — the s-wave scattering phase of a dc1 quasiparticle off a half-quantum vortex boundary — determines the Weinberg angle, the fine structure constant, and the anomalous magnetic moment simultaneously.