# Scattering amplitudes of Higgs boson production in weak boson fusion

I want to write the scattering amplitude for this diagram using feynman rules in unitary gauge. I will drop the prefactors, since they do not bother anyway. Here is what I ended up with: $$\mathcal{M}\propto [\bar{u}(p_4)\gamma^{\mu}(1-\gamma^5)u(p_2)]g_{\mu\nu}g^{\nu\sigma}\epsilon_\sigma g_{\sigma\eta}[\bar{u}(p_3)\gamma^{\eta}(1-\gamma^5)u(p_1)]$$ The $$g_{\mu\nu}$$ results from the bottom W propagator, the $$g^{\nu\sigma}$$ from the WWH-vertex and the $$g_{\sigma\eta}$$ from the other W propagator. I know this statement must be false, since the polarization vector is not contracted in any form. However, I do not know how to solve this problem with and odd number of external lines. Moreover, I am pretty uncertain about the use of a polarization vector at all for a Higgs Boson. I rather used it as a placeholder, since I have not found any information on how to treat it in an external line.

• Spin-0 particles don’t have polarization vectors. Where is the rest of the $W$ propagators involving momentum and mass? – G. Smith Apr 22 at 17:57
• I presume you mean the $\frac{q_{\mu}q_{\nu}}{m_W^2}$ terms. In the second propagator this term vanished when acting on the right, since it reproduced the free dirac equation in the massless case (I assume the u- and b-quark to be massless for simplicity). For the first propagator I actually just let the term drop as well for now. My first goal was to get the overall structure right and then focus on the detail. I know thats sloppy. – minits Apr 22 at 18:05
• Do you have any advise on how to treat the Higgs boson? – minits Apr 23 at 18:30