Feynman diagrams for ##e^{+}e^{-}\rightarrow \mu^{+}\mu^{-}##

In summary, the process of electron-positron annihilation into muons via the s-channel is described by the interaction term in the Lagrangian $$\mathcal{L}_{\text{int}}=e (\overline{\psi}_e \gamma^{\mu} \psi_e + \overline{\psi}_{\mu} \gamma^{\mu} \psi_{\mu}) A_{\mu},$$ which can be represented by Feynman diagrams. The lack of t-channel and u-channel diagrams for this process is due to the conservation of flavor in Quantum Electrodynamics (QED). This means that two Dirac fermion fields can only be coupled through an intermediate bosonic field, and
  • #1
spaghetti3451
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Consider the process of electron-positron annihilation into muons as given by

$$e^{+}e^{-}\rightarrow \mu^{+}\mu^{-}.$$

The Feynman diagrams for this process to lowest-order are given by

eemm.png


This is an s-channel diagram.Why are there no t-channel or u-channel diagrams for this process?
 
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  • #2
Well, try to draw ##t##- and ##u##-channel diagrams for QED. You'll see that they contradict the Feynman rules, because within QED flavor is conserved!
 
  • #3
I see!

Can you explain this using the interaction term in the Lagrangian that describes this process?
 
  • #4
The interaction terms read
$$\mathcal{L}_{\text{int}}=e (\overline{\psi}_e \gamma^{\mu} \psi_e + \overline{\psi}_{\mu} \gamma^{\mu} \psi_{\mu}) A_{\mu}.$$
Now find the Feynman rules for the vertices and compare them to what you'd need to be allowed to draw ##t##- and ##u##-channel Feynman diagrams for pair annihilation.
 
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  • #5
vanhees71 said:
The interaction terms read
$$\mathcal{L}_{\text{int}}=e (\overline{\psi}_e \gamma^{\mu} \psi_e + \overline{\psi}_{\mu} \gamma^{\mu} \psi_{\mu}) A_{\mu}.$$
Now find the Feynman rules for the vertices and compare them to what you'd need to be allowed to draw ##t##- and ##u##-channel Feynman diagrams for pair annihilation.

Got it, thanks!
 
  • #6
This thread made a related question come to my mind, and it's probably not necessary to start another thread for it... Is there some fundamental reason why two Dirac fermion fields are always coupled only through some intermediate bosonic field, and can't have a direct coupling? I.e. why can't there be an interaction term proportional to something like ##\overline{\psi_\mu}\psi_e## ? I know that the answer is probably something very simple, like that this would not be compatible with unitary time evolution or special relativity, but I'm an applied physicist by specialty, so it isn't immediately obvious to me.
 
  • #7
You "can't have" such a coupling because of lepton conservation. It's an empirical input to the Standard Model.
 
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  • #8
vanhees71 said:
You "can't have" such a coupling because of lepton conservation. It's an empirical input to the Standard Model.

Thanks for the answer. I looked this up at Google and I only found vague mentions of four-fermion interactions happening in some effective field theories: https://en.wikipedia.org/wiki/Four-fermion_interactions

Maybe there's some way to combine the electron, positron, muon and antimuon fields in some term in a way that conserves net lepton numbers. Something like ##\overline{\psi}_{\mu} \psi_{e} \overline{\psi}_{e} \psi_{\mu}## ... But of course, no one has ever observed such interactions so I'm just playing with the math here.
 
  • #9
You get such interactions in effective field theories by contracting the internal boson lines, if there is, e.g., a large mass in the corresponding propagator (as for the W and Z bosons). Then you get something like Fermi's theory of beta decay as an effective theory with (non-renormalizable) four-fermion couplings.
 
  • #10
Hilbert, what you wrote down corresponds to an electron bopping along and suddenly becoming a muon. Violates conservation of energy.
 
  • #11
^ Ok. I didn't bother to try to expand those point interaction terms with creation and annihilation operators. Obviously the interaction should only be able to turn an electron-positron pair with large kinetic energy to a muon-antimuon pair. Isn't the ##
\overline{\psi}_{\mu} \psi_{e} \overline{\psi}_{e} \psi_{\mu}## a bit similar to the phi-4 interaction for a single scalar field?
 

FAQ: Feynman diagrams for ##e^{+}e^{-}\rightarrow \mu^{+}\mu^{-}##

1. What are Feynman diagrams for ##e^{+}e^{-}\rightarrow \mu^{+}\mu^{-}##?

Feynman diagrams are graphical representations of particle interactions in quantum field theory. In this specific case, the Feynman diagrams show the process of an electron and a positron annihilating and producing a muon and an anti-muon.

2. How do Feynman diagrams for ##e^{+}e^{-}\rightarrow \mu^{+}\mu^{-}## work?

Feynman diagrams use lines and vertices to represent particles and their interactions. The lines represent the paths of particles, and the vertices represent the interactions between them. In this process, the electron and positron annihilate at a vertex and produce a muon and anti-muon, which then travel along their respective lines.

3. What do the different lines and vertices in the Feynman diagram represent?

The different lines in the Feynman diagram represent different types of particles, such as electrons, positrons, muons, and anti-muons. The vertices represent the interactions between these particles, such as annihilation and creation.

4. Why are Feynman diagrams important in particle physics?

Feynman diagrams are important in particle physics because they provide a visual representation of particle interactions and allow for calculations of probabilities and amplitudes of these interactions. They also help in understanding the underlying principles of quantum field theory.

5. How are Feynman diagrams for ##e^{+}e^{-}\rightarrow \mu^{+}\mu^{-}## related to the Standard Model?

The Standard Model is a theory that describes the fundamental particles and their interactions. Feynman diagrams, including those for ##e^{+}e^{-}\rightarrow \mu^{+}\mu^{-}##, are used to calculate and visualize these interactions, making them an important tool for studying and testing the predictions of the Standard Model.

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