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I've been reading chapter 20 in Peskin and Schroeder about the Glashow-Weinberg-Salam Theory for a $U(1)$ gauge symmetry with gauge coupling strengths $g$, and a complex scalar $\phi$, specifically section 2, which Higgs boson and its couplings to the massive gauge boson and fermions. The scalar potential of course is the usual $$V(\phi) = -\mu^2\phi^\dagger \phi + \lambda (\phi^\dagger \phi)^2 \text{ for } \mu^2>0$$

Now suppose we have one of the quiver gauge theories, say a gauge symmetry $U(1)_1 \times U(1)_2$, with gauge coupling strengths $g_1$ and $g_2$, and let's say we know that the complex scalar has the same charge +1 under $U(1)_1$ as well as under $U(1)_2$. Firstly, how would I write out the covariant derivative for this? What happens to the masses of the two gauge bosons/ the Higgs boson? How would I go about writing the couplings between the Higgs and the gauge bosons? Can somebody explain or point me to a textbook which looks at this scenario?

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