Why is QED renormalizable? My understanding of renormalizability is that a theory is renormalizable if it the divergences in its amplitudes can be cancelled out by finitely many terms. I see that by adding counterterm (in the MS-bar scheme)
$$L_{ct}=-\frac{g^2}{12\pi^2}\left(\frac{2}{\epsilon}-\gamma+\ln4\pi\right),$$
the one-loop divergence of QED can be made finite. However, I do not see how this makes QED renormalizable? Surely as we work with diagrams with more loops, we will get more counterterms - given that we can have diagrams with arbitrarily many loops, do we not need an infinite number of counterterms to cancel these out?
 A: QED has only a finite number of irreducible divergent diagrams. The main notion of divergence of a diagram is power-counting: The term every diagram represents has the form of a fraction like
$$ \frac{\int\mathrm{d}^n p_1\dots\int\mathrm{d}^n p_m}{p_1^{i_1}\dots p_k^{i_k}}$$
and you can compute the difference between the momentum power in the numerator and denominator and call it $D$. Heuristically the diagram diverges like $\Lambda^D$ in a momentum scale $\Lambda$ if $D > 0$, like $\ln(\Lambda)$ if $D=0$, and is finite if $D < 0$. This can fail - the diagram can be divergent for $D < 0$ - if it contains a smaller divergent subdiagram.
If you work out the general structure of $D$ for the diagrams of QED, you should be able to convince yourself that QED has only a finite number of divergent one-particle irreducible diagrams. That cancelling the irreducible diagrams is enough to cancel iteratively the divergences in all higher-order diagrams containing them in arbitrary combinations to all orders is a non-trivial statement sometimes called the BPHZ theorem, whose technical meaning - though not by this name - is explained by the Scholarpedia article on BPHZ renormalization.
A: We get infinite number of counterterms, but that will all be the same form (or in a closed set), it is just that the coefficients in front of the term will be expanded in a power series of the coupling constant. What it means by "infinite number of counterterm -> non-renormalizable", at least from my understanding, is something like phi^5 theory. We will need to add infinite number of counterterms, like phi^6, phi^7, phi^8, ..., to cancel the divergence, and this goes on forever. This is different from QED that we just need a finite number of counterterms, but the coefficients in front of them are determined order by order.
