# Feynman rule 4-point vertex WW -> ZZ

I am looking for the Feynman rule of the 4-point gauge boson interaction of W+ W- -> Z Z. I am guessing it looks like the Yang Mills 4-point vertex for gluons, but with helicity included.

Equation 8.81 in my professor's lecture notes seems to imply polarization, or am I misinterpreting?

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have a look at arxiv.org/ftp/hep-ph/papers/0306/0306160.pdf – anna v Jan 21 '12 at 15:50
unfortunately the article does not give an expression for the vertex. I need that in order to calculate the amplitude. – Ben Ruijl Jan 21 '12 at 17:53
Always better to link to the abstract page arxiv.org/abs/hep-ph/0306160 both so that readers can read the abstract before deciding to download the paper and because that way they can look at the refers to and cited by links. – dmckee Jan 21 '12 at 18:49
this has a lagrangian and a table with couplings: acfahep.kek.jp/acfareport/node182.html . I found it by google, asking for "WW ZZ quartic coupling standard model" – anna v Jan 21 '12 at 19:54
It would be good to edit things like that into the question, so that we know that you've already tried the obvious techniques and why they're not working. – David Z Jan 22 '12 at 3:15

It turns out that the vertex $X$ has to be contracted with the particle polarizations in the following manner, is
$$X^{\mu \nu \alpha \beta} \epsilon_{1_\mu} \epsilon_{2_\nu} \epsilon_{3_\alpha} \epsilon_{4_\beta}$$ where
$$X^{\mu \nu \alpha \beta} = 2 g^{\mu \nu} g^{\alpha \beta} - g^{\mu \alpha} g^{\nu \beta} - g^{\mu \beta} g^{\nu \alpha}$$
$$2(\epsilon_1 \cdot \epsilon_2) (\epsilon_3 \cdot \epsilon_4) - (\epsilon_1 \cdot \epsilon_3) (\epsilon_2 \cdot \epsilon_4) - (\epsilon_1 \cdot \epsilon_4) (\epsilon_2 \cdot \epsilon_3)$$