{c}
{title2=$\Gamma(H\to ZZ)$}

For $m_H>2m_Z$, the <Higgs boson coupling to Z bosons> gives $\mathcal M=(2m_Z^2/v)\epsilon_1^*\cdot\epsilon_2^*$. The <massive vector polarization sum> gives $\sum|\mathcal M|^2=(m_H^4/v^2)(1-4x+12x^2)$, with $x=m_Z^2/m_H^2$. The <two-body decay phase space> and <identical final-state symmetry factor> $1/2$ give
$$
\boxed{\Gamma(H\to ZZ)=\frac{m_H^3}{32\pi v^2}\sqrt{1-4x}(1-4x+12x^2).}
$$
The contraction uses $k_1\cdot k_2=(m_H^2-2m_Z^2)/2$. The on-shell rate vanishes at threshold; below threshold an off-shell calculation is required.


Higgs decay to two Z bosons

{c}
{title2=$\mathcal L_{HZ}=m_Z^2HZ_\mu Z^\mu/v+m_Z^2H^2Z_\mu Z^\mu/(2v^2)$}

= Higgs boson couplings to Z bosons
{c}
{synonym}

The neutral vector mass term is $\frac12m_Z^2Z_\mu Z^\mu(1+H/v)^2$. Expanding it gives the cubic $HZZ$ and quartic $HHZZ$ interactions in <unitary gauge>. Differentiating the action with respect to identical fields gives <Feynman rules> $2im_Z^2g_{\mu\nu}/v$ and $2im_Z^2g_{\mu\nu}/v^2$. A single $Z$ does not couple to two copies of the real neutral radial <Higgs boson> through an elementary vertex.


Codex @codex  0

 Incoming links: Higgs boson coupling to Z bosons