# The coupling constant in QED Feynman rules

I have a few questions in understanding the coupling renormalization in QED.

1. We introduce counters terms for fermion $Z_\psi$, photon self energy $Z_A$, and the vertex function $Z_e$. The charge renormalization, followed from the interaction term $$\mathcal{L} \supset e\bar{\psi}\gamma^\mu A_\mu \psi$$ is thus the combination of $Z_\psi$, $Z_A$ and $Z_e$. First dumb question: is it the vertex renormalization $Z_e$ or the charge renormalization that we call the coupling renormalization?
2. To avoid ambiguity, let's instead use $g$ to stand for the coupling. From QFT I (where we ignore renormalization effects), we write down the tree level amplitude (say, $e^-e^-\rightarrow e^-e^-$) as something proportional to $g^2$. Now, let's take the MS-bar renormalization. First of all, after we introduce counter terms, I suspect the Feynman rules should be written with respect to $Z_g g$ in stead of $g$. But since $Z_g = 1 + O(g^2)$ so still we are doing perturbation about $g$. Second, the parameter $g$ that appears in the Lagrangian has to depend on energy scale. When we compute the tree level electron scattering amplitude using Feynman rules, do we have to use the running value of $g$ at the incoming electrons' energy?
3. What's the meaning of the QED coupling constant $e$ which satisfies $$\frac{e^2}{4\pi} = \frac{1}{137}$$ If the coupling runs with energy, this value of $e$ is evaluated at some particular energy scale, right? What energy scale?
4. I know that people do compute amplitudes using this special coupling constant mentioned above in 3. Why is this doable?
5. If we can just use some special coupling constant to do perturbation, why do we worry about the Landau pole? Remember, the Landau pole is the energy scale where the running coupling blows up. As long as I'm keeping the special coupling constant finite, I'm safe?
• Fine structure constant, and the fact that momentum is exchanged as regards the energy scale should be searched for on Wikipedia, or similar sites – user163104 Jul 23 '17 at 5:08