- #1
muppet
- 608
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Hi all,
I've been reading section 5.3 of Peskin and Schroeder, in which the authors discuss the production of a bound state of a muon-antimuon pair close to threshold in electron-positron collisions.
Why does the matrix [itex]\Gamma[/itex] supposedly depend on the momentum of the nonrelativistic fermions? In their earlier analysis, this matrix was determined by the spins of the inital electron-positron pair, and the momentum of the final state muons dropped out- even their Dirac spinors didn't depend on these momenta, only on the muon mass, which is why the scattering is isotropic.
Thanks in advance.
I've been reading section 5.3 of Peskin and Schroeder, in which the authors discuss the production of a bound state of a muon-antimuon pair close to threshold in electron-positron collisions.
Here [itex]\xi,\xi'[/itex] are the Weyl spinors used to construct the Dirac spinors for the muon and anti-muon, respectively.The analysis leading up to (5.37) will cast any S-matrix element for the production of nonrelativistic fermions with momenta k and -k into the form of a spin matrix element
[tex]i\mathcal{M}(\text{something} \rightarrow\mathbf{k},\mathbf{k'} )=\xi^{\dagger}[\Gamma(\mathbf{k})]\xi'[/tex]
where [itex]\Gamma(\mathbf{k})[/itex] is some 2x2 matrix. We must now replace the spinors with a normalised spin wavefunction for the bound state.
Why does the matrix [itex]\Gamma[/itex] supposedly depend on the momentum of the nonrelativistic fermions? In their earlier analysis, this matrix was determined by the spins of the inital electron-positron pair, and the momentum of the final state muons dropped out- even their Dirac spinors didn't depend on these momenta, only on the muon mass, which is why the scattering is isotropic.
Thanks in advance.