Pictures Paths Particles Processes
Pictures Paths Particles Processes
Pictures Paths Particles Processes
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148 November 7, 2013<br />
The first diagram contains a fermion loop and hence carries an overall minus<br />
sign ; the second one does not. Now consider the cut versions of these diagrams :<br />
The left-hand sides of the cut-through diagrams are identical. The right-hand<br />
sides differ in the way that the in-going fermions are connected to the out-going<br />
ones ; the ingoing ones are interchanged in in the second diagram with respect<br />
to the first one. This, then, must correspond to a minus sign associated with<br />
the interchange of external lines in a diagram, and we arrive at the final form<br />
of the Feynman rules for Dirac particles :<br />
k<br />
↔ i¯h<br />
/k + m<br />
k · k − m 2 + iɛ<br />
internal line<br />
000000<br />
111111<br />
000000<br />
111111<br />
000000<br />
111111<br />
000000<br />
111111<br />
000000<br />
111111<br />
000000<br />
111111<br />
000000<br />
111111<br />
000000<br />
111111<br />
000000<br />
111111<br />
000000<br />
111111<br />
p,s ↔ √¯h u(p, s)<br />
outgoing particle<br />
p,s<br />
000000<br />
111111<br />
000000<br />
111111<br />
000000<br />
111111<br />
000000<br />
111111<br />
000000<br />
111111<br />
000000<br />
111111<br />
000000<br />
111111<br />
000000<br />
111111<br />
000000<br />
111111<br />
000000<br />
111111<br />
↔ √¯h u(p, s)<br />
incoming particle<br />
000000<br />
111111<br />
000000<br />
111111<br />
000000<br />
111111<br />
000000<br />
111111<br />
000000<br />
111111<br />
000000<br />
111111<br />
000000<br />
111111<br />
000000<br />
111111<br />
000000<br />
111111<br />
000000<br />
111111<br />
p,s<br />
↔ √¯h v(p, s)<br />
outgoing antiparticle<br />
p,s<br />
00000 11111<br />
00000 11111<br />
00000 11111<br />
00000 11111<br />
00000 11111<br />
00000 11111<br />
00000 11111<br />
00000 11111<br />
00000 11111<br />
00000 11111<br />
↔ √¯h v(p, s) incoming antiparticle<br />
For every closed loop of Dirac particles a factor −1.<br />
For every interchange of external Dirac particles a factor −1.<br />
Feynman rules, version 5.4 (5.110)<br />
Note that the interchange rule only determines the relative sign between two<br />
Feynman diagrams. How the interchange sign can be determined is best illustrated<br />
by an example. Consider, for instance, a process with 6 external fermions.<br />
Three of them must then be oriented outward from the diagram, carrying a u<br />
of v, and the other three must be oriented inward and carry a u or a v. Let us<br />
assume that there are three Feynman diagrams, schematically given by 37<br />
diagram 1 : u 1 Γ 1 u 2 v 3 Γ 2 u 4 u 5 Γ 3 v 6<br />
37 The process e − e − e + → e − e − e + is an example.
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