# Derivation of the Yang-Mills 3 gauge boson vertex

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• takunitoche
In summary, the conversation discusses the derivation of the three gauge-boson-vertex in Yang-Mills theories. The relevant interaction term in the Lagrangian is rewritten using the total asymmetry of the structure constants. The conversation then moves on to consider a diagram from Peskin & Schroeder and the issue of determining which gauge field momentum appears from which derivative. The answer to this problem is given as a Feynman rule with six terms, which can be obtained by collecting the six possibilities for connecting the fields in an appropriate manner.
takunitoche
TL;DR Summary
I can't seem to derive the vertex rules from the Yang-Mills lagrangian. I struggle to properly identify the origin of each momentum and the indices associated.
Hello everyone,

I am stuck in the derivation of the three gauge-boson-vertex in Yang-Mills theories. The relevant interaction term in the Lagrangian is$$\mathcal{L}_{YM} \supset g \,f^{ijk}A_{\mu}{}^{(j)} A_{\nu}{}^{(k)} \partial^{\mu} A^{\nu}{}^{(i)}$$

I have rewritten this term using the total asymmetry of the structure constants:

$$\mathcal{L}_{YM} \supset \dfrac{g}{6} f^{ijk} \left[ A_{\mu}{}^{(j)} A_{\nu}{}^{(k)} (\partial^{\mu} A^{\nu}{}^{(i)} - \partial^{\nu} A^{\mu}{}^{(i)}) + A_{\mu}{}^{(i)} A_{\nu}{}^{(k)} (\partial^{\mu} A^{\nu}{}^{(j)} - \partial^{\nu} A^{\mu}{}^{(j)}) + A_{\mu}{}^{(k)} A_{\nu}{}^{(i)} (\partial^{\mu} A^{\nu}{}^{(j)} - \partial^{\nu} A^{\mu}{}^{(j)})\right]$$Now consider the following diagram (from Peskin & Schroeder, section 16.1): View attachment 250501

This is where I'm stuck: I know that the derivative of the field will make the momenta appear in the expression. The problem is that I do not understand which gauge field momentum appears from which derivative, and how to go from this expression to the answer, which is

$$g f^{abc} \left[ g^{\mu \nu} (k-p)^\rho + g^{\nu \rho} (p-q)^\mu + g^{\rho \mu} (q-k)^\nu \right]$$

where the momenta and indices are taken according to the attached diagram.

In all honesty, writing out the anti-symmetrization explicitly will not help you much, it will rather serve to complicate things.

The main thing to keep in mind here is that there are three different possibilities for which field to connect to the derivative term. Each of these possibilities contribute with its own term to the Feynman rule - where the momentum is the momentum corresponding to that possibility - and in each of these possibilities you have two different possibilities for connecting the other fields so in total you have six terms. Just collecting these six terms in an appropriate manner should give you the correct Feynman rule.

## 1. What is the Yang-Mills 3 gauge boson vertex?

The Yang-Mills 3 gauge boson vertex is a mathematical framework used in particle physics to describe the interactions between three gauge bosons, which are particles that mediate the fundamental forces of nature.

## 2. How was the Yang-Mills 3 gauge boson vertex derived?

The derivation of the Yang-Mills 3 gauge boson vertex is based on the Yang-Mills theory, which was developed in the 1950s by physicists Chen Ning Yang and Robert Mills. It involves complex mathematical calculations and relies on the principles of gauge symmetry and quantum field theory.

## 3. What is the significance of the Yang-Mills 3 gauge boson vertex in physics?

The Yang-Mills 3 gauge boson vertex is a crucial component of the Standard Model of particle physics, which is the most successful theory we have to describe the fundamental particles and forces in the universe. It helps to explain the behavior of the strong and weak nuclear forces, and it has been confirmed by numerous experiments.

## 4. Can the Yang-Mills 3 gauge boson vertex be applied to other theories besides the Standard Model?

Yes, the Yang-Mills 3 gauge boson vertex has been successfully applied to other theories, such as the electroweak theory, which unifies the electromagnetic and weak nuclear forces. It is also used in theories that attempt to go beyond the Standard Model, such as supersymmetry and grand unified theories.

## 5. Are there any current research or developments related to the Yang-Mills 3 gauge boson vertex?

Yes, there is ongoing research in the field of quantum field theory to better understand the Yang-Mills 3 gauge boson vertex and its implications for particle physics. There is also ongoing experimental work to test the predictions of the Standard Model and search for new particles and interactions that may involve the Yang-Mills 3 gauge boson vertex.

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