Pictures Paths Particles Processes
Pictures Paths Particles Processes
Pictures Paths Particles Processes
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
November 7, 2013 115<br />
which is indeed negative. Now, we consider the matrix elements M ki as used<br />
in Eq.(4.27). These describe the initial state i going over in any final state k,<br />
that is, they describe the decay of the particle after it has been produced by the<br />
source :<br />
and we shall denote them by<br />
J<br />
M ki = −i J¯h<br />
p<br />
000 111<br />
000 111<br />
000 111<br />
000 111<br />
000 111<br />
000 111<br />
000 111<br />
000 111<br />
000 111<br />
¯h<br />
p 2 − m 2 + imΓ D , (4.31)<br />
where iD is the contribution of the ‘decay blob’. We then have<br />
∑<br />
|M ki | 2 J 2 ∑<br />
=<br />
(p 2 − m 2 ) 2 |D| 2 . (4.32)<br />
+ m 2 Γ 2<br />
k<br />
The optical theorem (4.27) will therefore be satisfied if<br />
k<br />
Γ = 1 ∑<br />
¯h |D| 2 . (4.33)<br />
2m<br />
k<br />
But this is, of course, precisely the prescription for the decay width of the particle,<br />
if we realize that the final state k indicates not only all possible final states,<br />
but also that the summation over k should include the phase-space integration.<br />
This short excercise illustrates both the optical theorem and the computational<br />
prescriptions arrived at before. Note that the factor ¯h corresponds precisely<br />
with the Feynman rule that an external line should carry a factor √¯h.<br />
4.4.3 The cutting rules<br />
We shall now consider how the unitarity relation (4.26) can be made useful in the<br />
language of Feynman diagrams. To start, we realize that this equation contains,<br />
in addition to the ‘standard’ matrix element M fi for initial state i and final<br />
state f, also M ∗ if<br />
which describes the (complex conjugate) matrix element for<br />
initial state f going over into final state i, that is, the time-reversed process. We<br />
shall embody this in a useful manner by introducing a cutting line. A cutting<br />
line cuts across diagrams separating them into a ‘left’ and ‘right’ piece. Any<br />
diagram to the left of a cutting line is interpreted in the usual manner ; any<br />
diagram to the right of a cutting line is interpreted to be the complex conjugate<br />
of the time-reversed version of the diagram. That is,<br />
0000000<br />
1111111<br />
0000000<br />
1111111<br />
0000000<br />
1111111<br />
0000000<br />
1111111<br />
0000000<br />
1111111<br />
0000000<br />
1111111<br />
0000000<br />
1111111<br />
0000000<br />
1111111<br />
0000000<br />
1111111<br />
0000000<br />
1111111<br />
0000000<br />
1111111<br />
0000000<br />
1111111<br />
0000000<br />
1111111<br />
0000000<br />
1111111<br />
0000000<br />
1111111<br />
0000000<br />
1111111<br />
=<br />
0000000<br />
1111111<br />
0000000<br />
1111111<br />
0000000<br />
1111111<br />
0000000<br />
1111111<br />
0000000<br />
1111111<br />
0000000<br />
1111111<br />
0000000<br />
1111111<br />
0000000<br />
1111111<br />
0000000<br />
1111111<br />
0000000<br />
1111111<br />
0000000<br />
1111111<br />
0000000<br />
1111111<br />
0000000<br />
1111111<br />
0000000<br />
1111111<br />
0000000<br />
1111111<br />
0000000<br />
1111111<br />
,
Change language