Feynman Diagram setup for 4-fermion interaction loop

In summary, the discussion was about converting a 4-fermion vertex diagram to an equation and the direction to follow. The Fermi Lagrangian dictates the coupling between two fermionic currents. The presence of Dirac algebra means spinors and fermionic propagators must be used. The diagram can be solved by choosing a direction for the loop momenta and writing the integral accordingly. Each loop can be considered separately, giving a degree of freedom in choosing the path.
  • #1
Hepth
Gold Member
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If one has a 4-fermion vertex, like in Fermi theory : ##G_f (f_1 \Gamma f_2)(l_1 \Gamma l_2)##

And you are calculating a one-loop diagram where you have the diagram of :

f1-> f2,l1,l2 -> f1

2pyq1r6.jpg


(used for dispersive/unitarity approach)

Where in the end you'll use Cutkosky rules to calculate it, but ignore this for now.

The question is about converting this diagram to an equation, which direction does one go? You can follow fermion flow backward through either of the two forward-arrow propagators, but then what. It matters as there is dirac algebra involved.

I assume it has to do with the order of the fermions in your actual interaction term, and when doing the time ordered product of two of these vertices you have to watch for fierz transforming the currents so that its manageable.

Is that correct, that I should approach this "diagram" without usign the diagram, but rather a product of currents, time-ordered, and do it the long way, fierz-transforming the spinor currents so that things simplify? Or is there a prescription for handing this case straight from the diagram.

It seems a feynman diagram is not enough to be honest for 4-fermion interactions of this sort, as the directions matter, and "going against particle flow" is not enough when you have choices. So you must either sum over all the paths, or do it by hand.

Any insight?
 
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  • #2
I'm not sure if I understood your question correctly. However, the Fermi Lagrangian just tells you that, at low energy, the coupling between two fermionic currents is [itex]G_F/\sqrt{2}[/itex]; that's what should appear at every four-fermion vertex. The presence of the Dirac algebra, i.e. of spinors, just tells you that you have to use spinors for the external legs of you diagrams and fermionic propagators for the internal ones.
In your case you need to choose a direction for the loop momenta and then write down the integral correctly (don't forget the additional - sign for every fermionic loop). If, for example, [itex]p[/itex] is the momentum of the incoming/outgoing fermion, you can choose the upper internal line to go from left to right with a momentum [itex]q[/itex], the middle internal line to go from left to right with momentum [itex]p-q+k[/itex] and the lower one to go from right to left with momentum [itex]k[/itex].
In this case I would say that you diagram is given by:

$$
\left(\frac{G_F}{\sqrt{2}}\right)^2\bar u_L(p)\int \frac{d^4q}{(2\pi)^4}\frac{d^4k}{(2\pi)^4}\frac{1}{q^\mu\gamma_\mu-m}\frac{1}{(p-q+k)_\mu \gamma^\mu-m}\frac{1}{k_\mu \gamma^\mu-m} u_L(p),
$$

where by [itex]u_L(p)[/itex] I mean the left-handed spinor.
 
  • #3
Ah yes. I was thinking unclearly. Of course it solves itself as each loop is considered independently, and you have a degree of freedom when choosing because you technically have 3 close loops to choose your 2-loop variables to travel around.

Thanks!
 

1. What is a Feynman Diagram for a 4-fermion interaction loop?

A Feynman diagram is a graphical representation of a particle interaction in quantum field theory. In the case of a 4-fermion interaction loop, it shows the exchange of four fermions (particles with half-integer spin) between two interacting particles.

2. How is the Feynman Diagram setup for a 4-fermion interaction loop?

The Feynman Diagram for a 4-fermion interaction loop is set up by representing the initial and final particles on the left and right sides of the diagram, respectively. The four fermions involved in the interaction are represented by lines connecting the initial and final particles, with arrows indicating the direction of the interaction.

3. Why is the 4-fermion interaction loop important in particle physics?

The 4-fermion interaction loop is important in particle physics because it allows for the study of interactions between fermions, which are the building blocks of matter. These interactions can provide insight into the fundamental forces and particles that govern our universe.

4. How does the Feynman Diagram help in understanding the 4-fermion interaction loop?

The Feynman Diagram provides a visual representation of the 4-fermion interaction loop, making it easier to understand and analyze. It allows for the calculation of the probability and energy of the interaction, and can also reveal important symmetries and conservation laws.

5. Are there any limitations to using the Feynman Diagram for the 4-fermion interaction loop?

While the Feynman Diagram is a powerful tool for understanding 4-fermion interaction loops, it does have some limitations. It is a simplified representation and does not take into account all possible interactions and processes. Additionally, it may not be applicable in certain extreme conditions, such as high energies or strong interactions.

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