Regularization of infrared divergences

Question

Let's have diagrams in QED when we don't use Feynman gauge. Then the bare photon propagator will look like $$ \tag 1 D_{\mu \nu}(p) = -\frac{g_{\mu \nu} - \frac{p_{\mu}p_{\nu}}{p^{2}}}{p^{2} + i\varepsilon }. $$ How to modify it for regularization of infrared divergences? I.e.,

1. If I want to modify it by setting nonzero photon mass, have I add $-\mu^{2}$ term only to the general denominator of $(1)$, or I also need to add it into the denominator of longitudinal part, $\frac{p_{\mu}p_{\nu}}{p^{2}} \to \frac{p_{\mu} p_{\nu}}{p^{2} - \mu^{2}}$? The question have arisen after reading paragraph in Itzykson and Zuber QFT (chapter 7.1.2, "Electron propagator"); before reading I thought that I need to modify only the general denominator, but the expression for one-loop correction to electron propagator says (as it seems to me) that I was wrong.

2. What methods of infrared divergences regularization exist except the method from p.1? Especially I want to know about gauge-invariant methods. Can dimensional regularization deal with infrared divergences?

3. May I combine ultraviolet divergences regularization (like dimensional regularization (if answer on previous question about it is negative) or Pauli-Villars regularization) and infrared divergences regularization for the same calculations?

Answer 1

1. It shouldn't matter: all methods of regularization are equally applicable if they have a correct limit at $\mu \rightarrow 0$. It is like asking whether we should keep the $(2\pi)^4$ in dimensional regularization or promote it to $(2\pi)^n$. Doesn't matter. All physical results are the same.

2. Again, every method which makes the integral converge at finite $\epsilon$ and has a correct limit at $\epsilon \rightarrow 0$ is good to go. Yes, you can use dimensional regularization.

3. Yes and no. You can use the regularizations, but the divergences cancel when adding different diagrams.

Check out this great lecture notes on IR divergencies in QED