# Scattering Amplitudes BCFW relation (A question)

• AT80
In summary, the conversation discusses the relationship between 3-point and 4-point amplitudes in YM theory. While the 3-point vertex is sufficient to determine the n-point amplitude at tree level, the 4-point vertex is also necessary for a gauge-invariant Lagrangian. This is because in YM theory, the two vertices are related to each other. However, in a theory where the two vertices are uncorrelated, the 3-point vertex alone is not enough to obtain the complete answer for the amplitudes.
AT80
I have a very trivial question to ask and it would be great if someone could
help me in this.

The statement that '3-point amplitudes' and the location of poles are sufficient to
determine any n-point amplitude at tree level is confusing to me. Don't I also need to know
4-point amlitudes, for example in YM theory ? The reason I say this is
the 4-point vertex can not be broken down. That is the residues obtained upon putting
propagators onshell will also contain 4-point functions.

What am I missing ?

In YM, the three point vertex coefficient and the four point coefficient are related to each other; one goes like g p_mu, and the other is g^2. The second is necessary to have a gauge-invariant Lagrangian. What's neat about YM is that given the three-point vertex at tree level, one can use it to build up the n-point function from BCFW, and you get the same answer as if you'd done Feynman rules, without having the miscellaneous gauge degrees of freedom to keep track of.

In a theory where the three- and four-point vertices were uncorrelated (scalar field theory with V = g phi^3 + lambda phi^4, for example), then you could BCFW up contributions to amplitudes that contained arbitrary powers of g, but no powers of lambda, using just the three-point function, but you wouldn't get the complete answer for the amplitudes. Hope this helps!

Thanks a lot Chrispb for a fast reply. It certainly helps.

## 1. What are scattering amplitudes in relation to the BCFW relation?

Scattering amplitudes refer to the probability of particles scattering off of each other in a specific way. The BCFW (Britto-Cachazo-Feng-Witten) relation is a mathematical formula that allows for the calculation of scattering amplitudes in certain theories, specifically in gauge theories and gravity.

## 2. How does the BCFW relation work?

The BCFW relation uses a mathematical technique called recursion to simplify the calculation of scattering amplitudes. It involves breaking down a complex scattering process into simpler ones, and then using the results to reconstruct the original amplitude. This method greatly reduces the computational complexity and allows for the calculation of amplitudes with a large number of particles.

## 3. What is the significance of the BCFW relation in physics?

The BCFW relation is significant because it provides a powerful tool for calculating scattering amplitudes in gauge theories and gravity. These calculations are crucial for understanding the behavior of particles in high-energy collisions, which is essential for developing theories and making predictions in particle physics.

## 4. Are there any limitations to the BCFW relation?

The BCFW relation is most effective for calculating amplitudes with a large number of particles. It may not work as well for simpler processes, and there are certain cases where it may not be applicable. Additionally, it is limited to certain types of theories and may not be applicable in other areas of physics.

## 5. What are some applications of the BCFW relation?

The BCFW relation has been used extensively in the field of particle physics to calculate scattering amplitudes in theories such as quantum chromodynamics (QCD) and quantum gravity. It has also been applied in other areas of physics, such as string theory and the AdS/CFT correspondence, to study the behavior of particles in different theoretical frameworks.

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