# Small question on introduction of Peskin Chapter 6

In chapter 6, right after eq. (6.1), it is mentioned that there are "six additional one-loop diagrams involving the heavy particle in the loop". I am not sure exactly what those terms are. Could anyone just point me to what those diagrams could be?

• I happen to have a copy of P&S on-hand, but in the future it would be good if you could include a bit more context so people who don't can provide an answer, and that other people who may have the same question know what you're talking about. – J. Murray Aug 21 '17 at 22:53
• Sorry, I should've done that. I will keep this in mind in the future. – Sayan Mandal Aug 22 '17 at 0:11

The passage in question refers to the scattering of an electron from a heavy charged particle in the context of a slightly modified QED. Figure (6.1) shows four one-loop Feynman diagrams.

Of the four diagrams in (6.1), the first (known as the vertex correction) is the most intricate and gives the largest variety of new effects. For example, it gives rise to an anomalous magnetic moment for the electron, which we will compute in Section 6.3.

The next two diagrams of (6.1) are external leg corrections. We will neglect them in this chapter because they are not amputated as required by our formula (4.90) for S-matrix elements. We will discuss these diagrams in more detail when we prove that formula in Section 7.2.

The final diagram of (6.1) is called the vacuum polarization. Since it requires more computational machinery than the others, we will not evaluate this diagram until Section 7.5.

The specific passage in question is

The order-$\alpha$ correction to the cross section comes from the interference term between these diagrams and the tree-level diagram. There are six additional one-loop diagrams involving the heavy particle in the loop, but they can be neglected in the limit where that particle is much heavier than the electron, since the mass appears in the denominator of the propagator. (Physically, the heavy particle accelerates less, and therefore radiates less, during the collision.)

The first four additional diagrams are mirror images of the above four, just involving the heavy particle (HP). They are

• The vertex correction diagram which joins the incoming and outgoing HP lines (1)
• The external leg correction diagrams on the incoming and outgoing HP lines (2,3)
• The vacuum polarization diagram where the internal loop is not an electron + positron loop, but rather a HP + anti-HP loop (4)

The last two diagrams are

• The diagram connecting the outgoing electron line to the outgoing HP line (5)
• The diagram connecting the outgoing electron line to the incoming HP line (6)

You might argue that there should be two more diagrams - the diagram connecting the incoming electron line to the incoming HP line (7), and the diagram connecting the incoming electron line to the outgoing HP line (8). However, it's not difficult to see that (7) is actually equivalent to (5), and that (8) is equivalent to (6), for a given choice of incoming and outgoing momenta, and should therefore not be counted as separate diagrams.

• Thanks so much! I wasn't sure about diagrams (1)-(4). I didn't think of diagrams (5) and (6). My bad! – Sayan Mandal Aug 22 '17 at 3:42