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The scalar potential of your theory is $$V(\phi) = \lambda (\phi^a \phi^a - v^2)^2,$$ where I suspect you meant to take the square as I've written here. This potential is minimized when $\sqrt{\phi^a \phi^a} = v$. Think of $\phi=\frac{1}{2} \phi^a \sigma^a$ as a vector with components $\phi^a$ in a 3-dimensional vector space with basis vectors $\sigma_a/2$. ...


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It seems to me that there is nothing fundamentally wrong about your statements, thus I also don't see any contradictions. The electroweak unification states that, as you said, the $Z$ and the $\gamma$ are different linear combinations of $B^0$ and $W^0$. This all works very nicely, there is just the problem of the masses which is then fixed my the Higgs ...


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The CP phase in the strong sector comes from a topological term (a total derivative) $$\require{cancel} \mathcal L_{\cancel{CP}} \propto \theta_\mathrm{QCD} \epsilon^{\mu \nu \rho \sigma} G_{\mu \nu} G_{\rho \sigma}, $$ where $G_{\mu \nu}$ is the gluon field-strength. Such an operator can never be produced as an effective operator from weak interactions, ...


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Instead of explaining how the weak and electromagentic interactions unify (a bottom-up approach), I will try to showcase this from the other direction, explaining how the electroweak interaction gives rise to the weak and electromagetic ones after symmetry breaking. I will include some minor technicalities, but try to render them as insubstatial to the ...



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