= Solution
A useful model of <spontaneous breaking of a Z2 scalar symmetry> is the four-dimensional real scalar theory
$$
\mathcal L=\frac12(\partial\phi)^2-\frac\lambda4(\phi^2-v^2)^2,\qquad\lambda>0,quad v>0.
$$
The action is invariant under the <Z2 symmetry> $\phi\mapsto-\phi$. Its classical <vacuum manifold> consists of the two values $\phi=\pm v$. In the broken quantum phase, choose a pure vacuum with $\langle\phi\rangle=v$; the symmetry maps it to a different vacuum with opposite expectation value, although it leaves the action unchanged. To distinguish quantum symmetry breaking from simply minimizing a classical potential, introduce a small source $J\phi$ and take the infinite-volume limit before $J\to0^+$. A finite-volume symmetry-preserving ground state can remain an even superposition because of tunneling, whereas the selected infinite-volume pure vacua have nonzero <vacuum expectation values>. The parameter $v$ below is the tree-level expectation, with its quantum value determined by renormalized parameters.
Writing $\phi=v+h$ gives
$$
V=\lambda v^2h^2+\lambda vh^3+\frac\lambda4h^4,\qquad
\boxed{m_h^2=2\lambda v^2.}
$$
The original sign symmetry relates expansions about the two vacua; in one expansion it acts as $h\mapsto-2v-h$. It is not a symmetry that fixes the chosen vacuum. Since the broken group is discrete, there is no continuous broken generator and no required <Goldstone boson>.
With relativistically normalized external states, the general two-body partial <decay width> is
$$
\boxed{\Gamma_{1\to2}=\frac1{2m}\frac1{\mathcal S}\int d\Phi_2\,\overline{|\mathcal M|^2},\qquad
d\Phi_2=(2\pi)^4\delta^{(4)}(P-p_1-p_2)\prod_{i=1}^2\frac{d^3p_i}{(2\pi)^3,2E_i}.}
$$
Here the bar sums final spins and averages initial spins only if the initial ensemble is unpolarized. The <identical final-state symmetry factor> is $\mathcal S=2!$ for two identical daughters and one for distinguishable daughters. Integrating the three-momentum delta function in the parent rest frame leaves $\mathbf p_2=-\mathbf p_1$. The energy delta function fixes $k=|\mathbf p_1|$ and has radial Jacobian $k(1/E_1+1/E_2)$. Hence the <two-body decay phase space> is
$$
d\Phi_2=\frac{k}{16\pi^2m}d\Omega,\qquad
k=\frac{\sqrt{[m^2-(m_1+m_2)^2][m^2-(m_1-m_2)^2]}}{2m}.
$$
Consequently
$$
\frac{d\Gamma}{d\Omega}=\frac{k}{32\pi^2m^2\mathcal S}\overline{|\mathcal M|^2},\qquad
\Gamma=\frac{k}{8\pi m^2\mathcal S}\overline{|\mathcal M|^2}
$$
when the spin-summed amplitude is angle independent.
For <Higgs decay to two W bosons>, the scalar parent has no spin average and the amplitude, up to an irrelevant phase, is $\mathcal M=gM_W\varepsilon_1^*\cdot\varepsilon_2^*$. Use the <massive vector polarization sum> with $p_i^2=M_W^2$:
$$
\begin{aligned}
\sum_{\lambda_1,\lambda_2}|\mathcal M|^2
&=g^2M_W^2\left(-g_{\mu\nu}+\frac{p_{1\mu}p_{1\nu}}{M_W^2}\right)
\left(-g^{\mu\nu}+\frac{p_2^\mu p_2^\nu}{M_W^2}\right)\\
&=g^2M_W^2\left[4-1-1+\frac{(p_1\cdot p_2)^2}{M_W^4}\right].
\end{aligned}
$$
Momentum conservation gives $p_1\cdot p_2=(M_H^2-2M_W^2)/2$. With the PDF's ratio $x=M_W/M_H$, this becomes
$$
\sum|\mathcal M|^2=\frac{g^2M_H^4}{4M_W^2}(1-4x^2+12x^4),\qquad
k=\frac{M_H}{2}\sqrt{1-4x^2}.
$$
The charged daughters are distinguishable. Combining the last two equations with $g^2/M_W^2=4\sqrt2G_F$ gives
$$
\boxed{\Gamma(H\to W^+W^-)=\frac{G_FM_H^3}{8\pi\sqrt2}\sqrt{1-4x^2}\,(1-4x^2+12x^4).}
$$
For <Higgs decay to two Z bosons>, the <Higgs boson coupling to Z bosons> is $2iM_Z^2g_{\mu\nu}/v=igM_Z^2g_{\mu\nu}/M_W$, using $v=2M_W/g$. This compensates the changed powers of $M_Z$ in the polarization contraction, giving the same normalization of the squared amplitude with $x$ replaced by $y=M_Z/M_H$. The two <Z bosons> are identical, so the phase-space symmetry factor supplies an additional half:
$$
\boxed{\Gamma(H\to ZZ)=\frac{G_FM_H^3}{16\pi\sqrt2}\sqrt{1-4y^2}\,(1-4y^2+12y^4).}
$$
Both expressions have mass dimension one, vanish at their two-body thresholds and approach the ratio $2:1$ for $M_H\gg M_Z$. The cubic large-mass behavior comes from longitudinal-vector polarizations. These are on-shell tree-level widths under the stated heavy-Higgs hypothesis; below threshold the physical off-shell multi-particle decays require a different calculation.
Back to article page
