The two integrals everything reduces to

Lesson 3.4 computed a loop integral. It is worth looking at what actually got computed, because the same object is about to come back with the special case taken out of it.

$\Pi_2$ came down to a Feynman-parameter integral over two propagators — an electron line and a positron line, both of mass $m$, sharing a total momentum $q$. Two propagators, equal masses. That shape is a bubble, and equal masses is one point of a two-parameter family:

the same integral with the two masses left free. Nearly every one-loop calculation in QED needs a value of it, and hardly ever at equal masses: the electron self-energy in lesson 3.7 wants $B_0(p^2; 0, m^2)$ — one massless photon line against one massive electron line.

There is a second integral, simpler and easy to overlook. A propagator can close on itself: one line, one mass, no external momentum at all.

the tadpole. It looks like it could not possibly matter, and it turns up everywhere, because any loop momentum in a numerator that cancels a propagator in the denominator leaves one behind.

These two are the masters of one-loop QED — not because they are the only integrals, but because every other one reduces to them. An amplitude with any number of propagators and any tensor structure in the numerator can be written as a sum of $A_0$, $B_0$, and, once three or four propagators appear, $C_0$ and $D_0$, with coefficients that are rational functions of the invariants ¹. That reduction is a theorem, which is why a course can compute two integrals carefully and then stop.

This lesson is the careful computation. It is not a new technique — it is lesson 3.3's recipe run on the general case and then examined, which is a different activity from running it.