# Photon-Massive Vector Boson Vertex Feynman Rule

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In summary, the conversation discusses the problem of finding the Feynman rule corresponding to the addition of an interaction term to the QED lagrangian. The equation for the interaction term is provided, and the attempt at a solution is described. The solution involves terms with derivatives, which can be simplified using the anti-symmetry of the field strength tensors. The Feynman rule for a vertex is given as the sum of four possible scalar products involving the momenta and polarizations of the particles involved.
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1. The Problem
I am trying to find the feynman rule which corresponds to the addition of an interaction term to the QED lagrangian which couples the electromagnetic field to a neutral massive vector boson field. In this problem, $$k^\mu$$ corresponds to the photon 4-momentum and $$q^\mu$$ corresponds to the massive vector boson 4-momentum.

## Homework Equations

$$\mathcal{L}_{int} = \epsilon F_{\mu \nu}G^{\mu \nu}$$ where $$F_{\mu \nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$$ and $$A_\mu$$ is the electromagnetic field operator. Similarly, $$G^{\mu \nu}$$ is the same as $$F$$ only now for the massive vector field $$B^\mu$$.

## The Attempt at a Solution

Usually, when I find the feynman rule that corresponds to a term that has derivatives in the Lagrangian, I just bring down a momentum from the exponential in the expansion of the field in momentum space. So I thought that the following would work:
$$i\mathcal{M} = i\epsilon (k_\mu q^\mu + k_\nu q^\nu - k_\mu q^\nu - k_\nu q^\mu)$$
However, this does not make sense because two of the terms here would contract and two would not, leaving two terms with indices and two terms without. This just doesn't make sense. So on my second attempt, I figured that there should be a Kronecker delta that makes sure that $\mu$ and $\nu$ are the same:
$$i\mathcal{M} = i\epsilon (k_\mu q^\mu + k_\nu q^\nu - k_\mu q^\nu - k_\nu q^\mu)\delta_\mu^\nu$$
However, it should be easy to see that this is zero. Then I tried to use the fact that the field strength tensors are anti-symmetric to eliminate some terms, but I ran into the same problems. Any help would be greatly appreciated. I am quite lost.

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Well, your interaction term will have terms like
$$\partial_{\mu}A_{\nu}\partial^{\mu}B^{\nu}$$
and all other possibilities of interchanging $\mu,\nu$, which gives 4 terms.
In momentum space, you have terms coming from each one of these terms that give you
$$k_{\mu}\epsilon_{\nu}q^{\mu}\eta^{\nu}=(k\cdot q)(\epsilon\cdot\eta)$$
where $\epsilon,\eta$ are the polarisations of the photon and the massive vector boson, respectively.
Again, you have all 4 possible scalar products.
So I think the Feynman rule for a vertex will just be given by:
$$i\epsilon(q^{\mu}\eta^{\nu}+q^{\nu}\eta^{\mu}+k^{ \mu} \epsilon^{\nu}+k^{\nu}\epsilon^{\mu})$$
These indices don't need be contracted here on the vertex, they will be contracted with the propagators later on.

## 1. What is a photon-massive vector boson vertex Feynman rule?

The photon-massive vector boson vertex Feynman rule is a mathematical formula used in particle physics to describe the interaction between a photon (a massless particle that carries electromagnetic force) and a massive vector boson (a particle that mediates weak nuclear force). This rule is a part of the Feynman diagram, which is a pictorial representation of particle interactions.

## 2. How is the photon-massive vector boson vertex Feynman rule derived?

The photon-massive vector boson vertex Feynman rule is derived from the principles of quantum field theory. It involves calculating the probability amplitude for the interaction between a photon and a massive vector boson. This is done by considering the interaction as an exchange of virtual particles between the two particles involved.

## 3. What is the significance of the photon-massive vector boson vertex Feynman rule?

The photon-massive vector boson vertex Feynman rule is significant because it helps us understand and predict the behavior of particles at the subatomic level. It is used in calculations for various processes, such as particle decays and interactions in particle accelerators. It also provides evidence for the existence of weak nuclear force, which plays a crucial role in the structure of matter.

## 4. Are there any experimental confirmations of the photon-massive vector boson vertex Feynman rule?

Yes, there have been several experimental confirmations of the photon-massive vector boson vertex Feynman rule. For example, the Large Hadron Collider (LHC) at CERN has observed the production of massive vector bosons through photon-photon interactions, which is in agreement with the predictions of the Feynman rule. Other experiments, such as the DZero experiment at Fermilab, have also observed processes involving the photon-massive vector boson vertex.

## 5. What are some applications of the photon-massive vector boson vertex Feynman rule?

The photon-massive vector boson vertex Feynman rule has various applications in particle physics, including the study of the Standard Model of particle physics, which describes the fundamental particles and their interactions. It is also used in theoretical calculations for particle interactions and in experiments at particle accelerators to test the predictions of the Standard Model. Additionally, it has implications for astrophysics and cosmology, as it helps us understand the behavior of particles in the early universe.

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