If I have a diagram with external gauge bosons and would like to get the amplitude for fixed helicity states (or fixed polarization vectors) what is the recommended way of doing it?
For example, let's say I have a 4-point function with 4 external gauge bosons, the amplitude is $W_{\mu\nu\rho\sigma}(p_1,p_2,p_3,p4)$ and let's say I'm interested in the answer for all gauge bosons having particular fixed helicity assignments. I would construct the polarization vectors $\varepsilon\mu(h_1,p1), \ldots, \varepsilon\sigma(h_4,p_4)$ corresponding to the 4 fixed helicities $hi$ and contract with those to obtain,
$$W{\mu\nu\rho\sigma}(p_1,p_2,p_3,p4) \varepsilon\mu(h_1,p1) \varepsilon\nu(h_2,p2) \varepsilon\rho(h_3,p3) \varepsilon\sigma(h_4,p4)$$
Using FORM I can calculate $W{\mu\nu\rho\sigma}$ from the Feynman rules, but at the moment I don't know how to do the contraction with the polarization vectors. Is it possible?
What I mean is that it is of course possible to define the polarization vectors as general vectors in FORM and use the identity $\varepsilon(k)\cdot k = 0$ but how would I select further from the 3 orthogonal components the one which belongs to the particular helicity $h$.
If I have a diagram with external gauge bosons and would like to get the amplitude for fixed helicity states (or fixed polarization vectors) what is the recommended way of doing it?
For example, let's say I have a 4-point function with 4 external gauge bosons, the amplitude is $W_{\mu\nu\rho\sigma}(p_1,p_2,p_3,p4)$ and let's say I'm interested in the answer for all gauge bosons having particular fixed helicity assignments. I would construct the polarization vectors $\varepsilon\mu(h_1,p1), \ldots, \varepsilon\sigma(h_4,p_4)$ corresponding to the 4 fixed helicities $hi$ and contract with those to obtain, $$W{\mu\nu\rho\sigma}(p_1,p_2,p_3,p4) \varepsilon\mu(h_1,p1) \varepsilon\nu(h_2,p2) \varepsilon\rho(h_3,p3) \varepsilon\sigma(h_4,p4)$$ Using FORM I can calculate $W{\mu\nu\rho\sigma}$ from the Feynman rules, but at the moment I don't know how to do the contraction with the polarization vectors. Is it possible?
What I mean is that it is of course possible to define the polarization vectors as general vectors in FORM and use the identity $\varepsilon(k)\cdot k = 0$ but how would I select further from the 3 orthogonal components the one which belongs to the particular helicity $h$.