Feynman's rule for Photon/W+/W- vertex

[Muoniex]

Homework Statement

Hello there!
I've been trying to obtain the Feynman rule (tree level) for a photon/W+/W- vertex in SM, but I don't really know hoy to get it.

I've been told that there's a trick to obtain these rules, which consists in changing partials for momentums:
$$\partial_\mu \rightarrow -i k_\mu$$

And then eliminating the fields terms in the Lagrangian and multiplicatying them for i, but i don't get to the correct result...

Homework Equations

The gamma/W/W+ interaction is given by:

$$\mathcal{L}=ie(W^{\mu\nu}W_\mu^+A_\nu-W^+_{\mu\nu}W^\mu A^\nu-W^+_\mu W_\nu F^{\mu\nu})$$

With
$$W_{\mu\nu}=\partial_\mu W_\nu -\partial _\nu W_\mu$$
$$F_{\mu\nu}=\partial_\mu A_\nu -\partial _\nu A_\mu$$

And the Feynman rule for a $$W^+_\mu (p_1) W_\nu (p_2) A_\rho (p_3)$$ vertex is supposed to be:
$$-ie[g_{\mu\nu}(p_2-p_1)_\rho+g_{\nu\rho}(p_3-p_2)_\mu+g_{\mu\rho}(p_1-p_3)_\nu]$$

The Attempt at a Solution

Edit: I have realized my solution was wrong, but I still can't get to the right answer. Do you know where can I find a deduction of that vertex's rule, or another vertex's rule deduction to use it as a guide?
Thanks.

 

[mark726]

Thank you for your question. The Feynman rule for the photon/W+/W- vertex can be derived using the Feynman diagram technique and the Feynman rules for propagators and vertices. Here is a step-by-step guide to obtaining the correct result:

1. Start by writing down the Feynman diagram for the photon/W+/W- vertex. It should have three external legs, corresponding to the photon (A), W+ (W1), and W- (W2).

2. Label the momenta of the external particles as p1, p2, and p3 for A, W1, and W2 respectively.

3. Write down the Feynman rules for the propagators and vertices involved in the diagram. For the photon, use the propagator D(p) = -igμν/p^2. For the W+ and W- particles, use the propagator D(p) = -i/((p^2 - m^2)). For the vertex, use the Feynman rule given in the homework equations.

4. Multiply the propagators and vertex together and simplify using the momentum conservation rule p1 + p2 + p3 = 0.

5. Use the trick mentioned in the forum post, where you replace the partial derivative with respect to the momentum p with -i*p. This will eliminate the fields terms in the Lagrangian.

6. Simplify the expression further by using the Feynman rules for the propagators and vertices. You should end up with the correct result for the Feynman rule of the photon/W+/W- vertex.

I hope this helps. If you have any further questions, please don't hesitate to ask. Good luck with your studies!