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

=== We can build a specific toy calculation and get numbers: ===
# Solve the free Dirac problem on S2S^2S2 to obtain normalized eigenmodes {unℓm(Ω)}\{u_{n\ell m}(\Omega)\}{unℓm(Ω)}. (These are spinor spherical harmonics — a standard computation.)
# Choose an effective localized weak Hamiltonian on the surface, e.g. a 4-fermion operator or a current-current coupling: Hint(Ω)  =  G  JLμ(Ω)JL,μ(Ω) ,\mathcal{H}_{\rm int}(\Omega) \;=\; G \;J^\mu_{L}(\Omega) J_{L,\mu}(\Omega)\,,Hint(Ω)=GJLμ(Ω)JL,μ(Ω), where JLμ=ψˉγμ(1−γ5)ψJ^\mu_L = \bar\psi \gamma^\mu(1-\gamma^5)\psiJLμ=ψˉγμ(1−γ5)ψ.
# Introduce a geometric phase factor in the integrand, e.g. exp⁡(iϕ(Ω))\exp(i\phi(\Omega))exp(iϕ(Ω)) where ϕ\phiϕ is a Wilson line / background holonomy that may be different for different flavors.
# Compute the matrix element Mfi\mathcal{M}_{fi}Mfi as the integral of mode products times that phase. The CP asymmetry between a process and its CP conjugate is proportional to the imaginary part of interference between at least two amplitudes: ACP∼2 ℑ(M1M2∗)∣M1∣2+∣M2∣2.A_{CP} \sim \frac{2\,\Im(\mathcal{M}_1 \mathcal{M}_2^*)}{|\mathcal{M}_1|^2+|\mathcal{M}_2|^2}.ACP∼∣M1∣2+∣M2∣22ℑ(M1M2∗). Here the geometric phase provides the needed ℑ(⋅)\Im(\cdot)ℑ(⋅) term.
# Estimate numbers by plugging in realistic scales (choose radius r appropriate to the object: atomic electron shell for atomic processes, ~1 fm scale for nucleons or hadrons; the mode functions' angular structure depends on r only via dimensional scaling).

This gives a number for Γ\GammaΓ and ACPA_{CP}ACP as a function of a geometrical phase parameter Φgeom\Phi_{\rm geom}Φgeom. If that number can be tuned to SM values without fine tuning, the geometric hypothesis gains plausibility.