Production of bound states of slow fermions- Peskin 5.3

In summary, the authors of Peskin and Schroeder discuss the production of a bound state of a muon-antimuon pair in electron-positron collisions. They use spin matrix elements to analyze the production of nonrelativistic fermions and must replace the spinors with a normalized spin wavefunction for the bound state. The matrix \Gamma has a more complicated dependence on the 4-momentum of the initial state, but becomes independent of momentum in the approximation that ignores the mass of the electron.
  • #1
muppet
608
1
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.

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.
Here [itex]\xi,\xi'[/itex] are the Weyl spinors used to construct the Dirac spinors for the muon and anti-muon, respectively.

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.
 
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  • #2
muppet said:
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.


Here [itex]\xi,\xi'[/itex] are the Weyl spinors used to construct the Dirac spinors for the muon and anti-muon, respectively.

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.

The expression (5.37) takes an especially simple form because the electrons have been taken to be relativistic. Since the mass of the electron is neglected, the factor of [itex]E^2[/itex] in (5.34) cancels against the factor of [itex]1/q^2[/itex] in the expression for the matrix element (under (5.33)). In a more general case, [itex]\Gamma[/itex] has a more complicated dependence on the 4-momentum of the initial state. Energy conservation can be used to write it as a function of the final state momentum.
 
  • #3
Thanks for your reply fzero. The electrons necessarily have to be relativistic, in order to produce the muons; would this dependence on the final state momenta only become non-trivial if we were considering the scattering of fermions of comparable masses? (If the initial state particles were much heavier than those in the final state, then the process would obviously produce particles moving at relativistic velocities.)
 
  • #4
muppet said:
Thanks for your reply fzero. The electrons necessarily have to be relativistic, in order to produce the muons; would this dependence on the final state momenta only become non-trivial if we were considering the scattering of fermions of comparable masses? (If the initial state particles were much heavier than those in the final state, then the process would obviously produce particles moving at relativistic velocities.)

No. Sure, the electrons must have an energy that is large compared to their mass, but the limit in which we ignore the mass completely is an approximation. If we include corrections of order [itex]m_e/E[/itex], then they will necessarily depend on the momentum. Of course, at that level of precision, other corrections (loops) might be of comparable order too. But the point is that the book result is only independent of momentum because of the approximation that we're using.
 

1. What is the main focus of "Production of bound states of slow fermions- Peskin 5.3"?

The main focus of this study is to investigate the production of bound states of slow fermions, specifically in the context of quantum chromodynamics (QCD).

2. Why is the production of bound states of slow fermions important in QCD?

In QCD, bound states of slow fermions, such as protons and neutrons, play a crucial role in explaining the structure of matter and the strong nuclear force. Understanding their production is essential for understanding the fundamental interactions of subatomic particles.

3. How does Peskin 5.3 approach the study of bound states of slow fermions?

Peskin 5.3 uses perturbative QCD calculations to analyze the production of bound states of slow fermions. This method involves calculating the probability of different particle interactions and their corresponding cross-sections.

4. What are some of the key findings in "Production of bound states of slow fermions- Peskin 5.3"?

The study found that the production of bound states of slow fermions is significantly affected by the strong coupling constant, the mass of the fermion, and the energy of the interaction. It also showed that the production of bound states is enhanced at low energies and suppressed at high energies.

5. How can the findings of this study be applied in other fields of physics?

The findings of this study can be applied in various fields of physics, such as particle physics, nuclear physics, and astrophysics, where QCD plays a crucial role. The results can also be used to improve theoretical models and predictions in these areas.

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