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A useful model of spontaneous breaking of a Z2 scalar symmetry is the four-dimensional real scalar theoryThe action is invariant under the Z2 symmetry $\varphi \mapsto -\varphi$. Its classical vacuum manifold consists of the two values $\varphi =\pm v$. In the broken quantum phase, choose a pure vacuum with $⟨\varphi ⟩=v$; 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 $J\varphi$ and take the infinite-volume limit before $J\to 0^{+}$. 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 $\varphi =v+h$ givesThe 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 isHere 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 isConsequentlywhen 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 $\mathcal{M}=g{M}_W{\epsilon }_1^{\ast }\cdot {\epsilon }_2^{\ast }$. Use the massive vector polarization sum with $p_i^2=M_W^2$: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 becomesThe charged daughters are distinguishable. Combining the last two equations with $g^2/M_W^2=4\sqrt{2}{G}_F$ givesFor Higgs decay to two Z bosons, the Higgs boson coupling to Z bosons is $2i{M}_Z^2{g}_{\mu \nu }/v=ig{M}_Z^2{g}_{\mu \nu }/M_W$, 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: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.