The Higgs mechanism in a quiver gauge model

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?