Feynman diagrams: Planar Vs Non-Planar topologies

In summary, the conversation discusses the definitions and evaluations of planar and non-planar topologies in multi-loop QCD calculations. Planar graphs in SU(N) gauge theories are related to the large-N limit and are equivalent to a theory of weakly interacting mesons. When calculating processes such as e+ e- --> q,qbar,g at 2 loops, the papers consider the related process gamma* --> q,qbar,g at two loops and differentiate between planar and non-planar diagrams without giving a clear definition. The two papers mentioned are focused on the evaluation of these diagrams.
  • #1
Sleuth
47
4
Hi everybody,
as usual I need help with some definitions regarding many-loop calculations.

In particular what do we mean with planar and non-planar topologies exactly?
I have an idea but I'm really not sure how to formalize it for an arbitrary big number of loops and legs.

Second, once the definition is taken, which different difficulties have to be overcome in the evaluations of the planar and non-planar topologies? I ask this simply because I notice that usually the big calculations are done dividing the two contributions and analyzing them in different steps. Why is it so?

Thank you very much

Sleuth
 
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  • #2
I am not sure what you mean by planar.

There is a rather old paper by t'Hooft showing that planar graphs in SU(N) gauge theories are related to the so-called large-N limit (which is 1/N = 1/3 ~ 0 in QCD :-) These planar graphs are equivalent to a theory of weakly interacting mesons generated by bilinear fermionic operators.

I can't remeber the details, but perhaps this helps.
 
  • #3
I don't know either :P
I'm reading some papers about multi-loop qcd calculations, for example consider the process
e+ e- --> q,qbar,g at 2 loops in qcd

to analyze this process in the paper they consider

gamma* --> q,qbar,g at two loops

in the calculations they wrote two different papers about the evaluation of the planar and non-planar diagrams, without giving a definition, as it was something trivial or known somehow...

http://arxiv.org/pdf/hep-ph/0101124
http://arxiv.org/pdf/hep-ph/0008287

can u understand anything ? :P
 
Last edited:

What are Feynman diagrams and why are they important in physics?

Feynman diagrams are graphical representations of interactions between particles in quantum field theory. They are important in physics because they provide a way to visualize and calculate the probability amplitudes for particle interactions, which can then be used to make predictions about the behavior of subatomic particles.

What is the difference between planar and non-planar topologies in Feynman diagrams?

In a Feynman diagram, the topology refers to the arrangement of the lines and vertices representing particles and their interactions. A planar topology is one where all the lines and vertices can be drawn on a single plane without any lines crossing over each other. Non-planar topologies, on the other hand, involve lines crossing over each other and cannot be drawn on a single plane.

How do planar and non-planar topologies affect the calculations in Feynman diagrams?

The calculations for Feynman diagrams with planar topologies are relatively straightforward and can be solved using simple algebraic techniques. However, non-planar topologies require more advanced mathematical methods, such as graph theory, to accurately calculate the probability amplitudes.

What is the significance of planar vs non-planar topologies in particle interactions?

One of the main reasons for studying the difference between planar and non-planar topologies in Feynman diagrams is to understand the behavior of strongly interacting particles, such as quarks and gluons. These particles are responsible for the strong nuclear force that holds atoms together, and their interactions can involve non-planar topologies.

Are all Feynman diagrams with non-planar topologies considered significant?

No, not all Feynman diagrams with non-planar topologies are considered significant. In fact, only a small subset of non-planar diagrams are relevant for making accurate predictions in particle interactions. These are known as "important diagrams" and are typically identified through extensive calculations and simulations.

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