- #1

Hepth

Gold Member

- 460

- 39

For a generic scattering/decay matrix :

[tex]

\sum _{polarization}

\left|M|^2\right.=\sum _{polarization} \bar{u}_i\left(F^*\left[\gamma ^5\right]\right) u_f \bar{u}_f \left(F\left[\gamma ^5\right]\right)u_i

[/tex]

Where F[gamma5] is just a linear function of gamma5.

You simplify by "knowing" that since they're Dirac spinors you can rewrite to a trace of the sums over the u's.

[tex]\text{Tr}\left[\left(F^*\left[\gamma ^5\right]\right)\sum _{\epsilon } u_f \bar{u}_f \left(F\left[\gamma ^5\right]\right){\sum _{\epsilon } \bar{u}_i}u_i\right]

[/tex]

Then using completeness

[tex]

{\sum _{\epsilon } \bar{u}_f}u_f\right = p+m

[/tex]

[tex]

{\sum _{\epsilon } \bar{u}_i}u_i\right = p-m

[/tex]

or something like that.

My questions is how to get the first step.

How are we rearranging the operators and distributing the sums and using the trace.

Is it that the sum over polarization is a sum over both polarizations:

[tex]

\sum _{\epsilon } =\sum _{\epsilon 1} \sum _{\epsilon 2}

[/tex]

so they only operate on their spinors. So then the trace must come in when we need to get the ui's next to each other.

Anyone know offhand?