Editing Openai/6897769e-4ee4-800f-aba5-69cca34f701c (section)

=== We write a bulk + boundary split: ===

S=Sbulk[Aμ,Wμ,…]  +  S∂[ψ(Ω), ϕ(Ω), A∥(Ω),W∥(Ω)],S = S_{\text{bulk}}[A_\mu, W_\mu, \ldots] \;+\; S_{\partial}\bigl[\psi(\Omega),\ \phi(\Omega),\ A_\parallel(\Omega),W_\parallel(\Omega)\bigr],S=Sbulk[Aμ,Wμ,…]+S∂[ψ(Ω), ϕ(Ω), A∥(Ω),W∥(Ω)],
where Ω\OmegaΩ are coordinates on the sphere S2S^2S2 and A∥,W∥A_\parallel, W_\parallelA∥,W∥ are gauge fields pulled back to the boundary.

A minimal boundary action for a spinor field ψ(Ω)\psi(\Omega)ψ(Ω) (the localized degrees of freedom that represent the particle/resonance on the spherical shell) that couples to electroweak gauge fields is:

S∂=∫S2 ⁣dΩ [ ψˉi ⁣̸ ⁣DS2ψ−m ψˉψ  +  g ψˉγμ(1−γ5)τIψ  WμI(Ω)  +  e ψˉγμψ  Aμ(Ω)]  +  Stop.S_{\partial}=\int_{S^2} \! d\Omega\ \Big[\,\bar\psi i\!\not\!D_{S^2}\psi - m\,\bar\psi\psi
\;+\; g\,\bar\psi\gamma^\mu(1-\gamma^5) \tau^I \psi\;W^{I}_{\mu}(\Omega)
\;+\; e\,\bar\psi\gamma^\mu \psi\;A_{\mu}(\Omega)
\Big] \;+\; S_{\text{top}} .S∂=∫S2dΩ [ψˉiDS2ψ−mψˉψ+gψˉγμ(1−γ5)τIψWμI(Ω)+eψˉγμψAμ(Ω)]+Stop.
Notes:
* ̸ ⁣DS2\not\!D_{S^2}DS2 is the Dirac operator on the 2-sphere (includes spin connection); ψ\psiψ are spinor spherical harmonics (mode functions on the sphere).
* The (1−γ5)(1-\gamma^5)(1−γ5) projector encodes chirality → parity violation built in (this is how the weak interaction naturally is chiral).
* StopS_{\text{top}}Stop denotes possible topological / geometric terms living on the boundary (Wess-Zumino / Berry / Wilson line / holonomy terms) that can produce complex phases in amplitudes.

Important: the gauge fields WμI,AμW^I_\mu, A_\muWμI,Aμ are the restrictions of the bulk electroweak fields to the spherical surface. The weak interaction will appear either as explicit exchange of W/Z (full gauge theory) or as an effective 4-fermion interaction ∼GF(ψˉγμ(1−γ5)…)2 \sim G_F (\bar\psi\gamma^\mu(1-\gamma^5)\ldots)^2∼GF(ψˉγμ(1−γ5)…)2 when we integrate out heavy bosons for low-energy processes.