Gottfried and Weisskopf write in their book, in the chapter on electroweak theory, that
$\frac{m_W}{m_Z} + \left(\frac{e}{g}\right)^2 = 1$
In this expression, $m_W$ and $m_Z$ are the masses of the $W$ and $Z$ bosons, and $e$ and $g$ are the electromagnetic and the weak coupling constants. I want to understand the equation; so I have a question. Why exactly is the boson mass ratio related to the coupling ratio?
After all, this expressions seems to tell that mass is due to coupling strength. But why is mass ratio related in this "elliptical" way to coupling ratio?
I also have another, minor question. I suppose that by $e^2$ the authors mean what is called the fine structure constant $\alpha$, and by $g^2$ what is called $\alpha_w$. Is this correct? Yes, it is, within a factor of $4 \pi$ for both of them.
Thank you in advance for all your help to make this clearer. Also any book or article on that equation would be helpful. I will improve the question with all comments that I get.
Despite the beautiful answer below, to me it remains a deep fascination why the mass of a particle should depend on the coupling, and why a charged particle (the W) is less massive than a similar neutral one (the Z) - naively, I would expect that electric charge adds to mass. Maybe I can ask this in other terms: why is the Higgs mechanism the way it is? Where does it come from? Since nobody knows yet, a more realistic question is: what is the simplest way to explain mass generation in weak bosons? The rotation in the abstract space is so hard to picture in concrete terms. Is there a simpler way?
