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## Main Question or Discussion Point

I'm trying to learn how to derive Feynman rules (what else to do during xmas, lol).

The book I'm using is QFT 2nd ed by Mandl&Shaw. On p 428 they're trying to show how to derive a Feynman rule for [itex]W W^\dagger Z^2[/itex] interaction term [itex]g^2 \cos^2\theta_W\left[W_\alpha W_\beta^\dagger Z^\alpha Z^\beta - W_\beta^\dagger W^\beta Z_\alpha Z^\alpha\right][/itex]. The idea goes that a) momenta is assigned to every particle b) all fields are replaced with corresponding polarization vectors.

But for some reason they write out the respective amplitude with an imaginary unit [itex]i[/itex] in front, like:

[itex]\mathcal{M} = ig^2 \cos^2\theta_W\left\{ \varepsilon_\alpha(2')\varepsilon_\beta(1')\left[\varepsilon^\alpha (1)\varepsilon^\beta(2) + \varepsilon^\alpha (2)\varepsilon^\beta(1)\right] - \varepsilon_\beta(2')\varepsilon_\beta(1')\left[\varepsilon^\alpha (1)\varepsilon^\alpha(2) + \varepsilon^\alpha (2)\varepsilon^\alpha(1)\right] \right\}[/itex].

Where does this come from? I don't see this mysterious number occur in the next example.

Thanks

The book I'm using is QFT 2nd ed by Mandl&Shaw. On p 428 they're trying to show how to derive a Feynman rule for [itex]W W^\dagger Z^2[/itex] interaction term [itex]g^2 \cos^2\theta_W\left[W_\alpha W_\beta^\dagger Z^\alpha Z^\beta - W_\beta^\dagger W^\beta Z_\alpha Z^\alpha\right][/itex]. The idea goes that a) momenta is assigned to every particle b) all fields are replaced with corresponding polarization vectors.

But for some reason they write out the respective amplitude with an imaginary unit [itex]i[/itex] in front, like:

[itex]\mathcal{M} = ig^2 \cos^2\theta_W\left\{ \varepsilon_\alpha(2')\varepsilon_\beta(1')\left[\varepsilon^\alpha (1)\varepsilon^\beta(2) + \varepsilon^\alpha (2)\varepsilon^\beta(1)\right] - \varepsilon_\beta(2')\varepsilon_\beta(1')\left[\varepsilon^\alpha (1)\varepsilon^\alpha(2) + \varepsilon^\alpha (2)\varepsilon^\alpha(1)\right] \right\}[/itex].

Where does this come from? I don't see this mysterious number occur in the next example.

Thanks