Quantum Field Theory I

16 CHAPTER 1. INTRODUCTIONS
As another example let us consider the AB → 12 scattering within the
ϕ 4 -theory. Again, the starting point is the DS equation
¡= 1 3!¡+...
where the ellipsis stands for terms with disconnected external legs. The DS
equation is now applied to some other leg, say the external leg B
¡=
3!¡+ 3 1 1
3! 3!¡+...
The RHS enjoys yet another DS equation, now say on the external leg 1, to give
¡=¡+ 1 1
23!¡+ 1 3
3! 3!¡+ 1 3
...
3! 3!¡+
The first two (last two) terms on the RHS come from the first (second) diagram
on the RHS above, terms with more than two vertices are included into ellipsis.
The first diagram is now treated with the help of the previous result for the
2-leg diagrams, the other three are treated all in the same way, which we will
demonstrate only for the second diagram: we use the DS equation once more,
now say for the external leg 2
1 1
23!¡= 2 1
23!¡+ 1 3
23!¡+...
=
¡
1
1
2
1
⎛
⎝ 2 3
23!¡+ 2 3
⎠ ס+...
23!¡⎞
2¡ 2¡
2¡ 2¡ 2¡
1
1
1
Putting the pieces together, one finally obtains the one-loop diagrams with the
correct combinatorial factors
1
2
¡