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For $m_H>2m_Z$, the Higgs boson coupling to Z bosons gives $\mathcal{M}=(2m_Z^2/v){ϵ}_1^{\ast }\cdot {ϵ}_2^{\ast }$. 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 $1/2$ giveThe 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.

The Standard Model Higgs field is a complex $SU(2{)}_L$ doublet of hypercharge $1/2$. Its nonzero vacuum expectation value triggers the Higgs mechanism.

The Higgs field is a scalar field: its components carry internal weak isospin and hypercharge, but no spacetime spin. A complex Higgs doublet contains four real field components. The vacuum expectation value is a property of the lowest-energy state, not a gas of Higgs bosons. A Higgs boson is an excitation about that state.

Why introduce this field? A bare mass term for a gauge field generally fails to respect the local gauge symmetry. A bare Dirac mass term for a charged Standard Model fermion also joins left- and right-handed fields with different Standard Model representations. A Higgs doublet permits gauge-invariant interactions whose expansion about a nonzero vacuum expectation value supplies both masses and definite Higgs boson interactions. The field therefore links the mass problem to experimentally testable coupling relations, as illustrated by the Higgs boson coupling to Z bosons.

Higgsstrahlung produces a Higgs boson together with a massive electroweak gauge boson. At an electron-positron collider the process $e^{+}{e}^{-}\to Z^{\ast }\to ZH$ uses the Higgs boson coupling to Z bosons. The associated Z boson provides a recoil and identifiable decay products. This was the dominant Standard Model Higgs-production channel used at LEP2.

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.

Use metric $g_{\mu \nu }=\mathrm{diag}(1,-1,-1,-1)$ and the hypercharge convention $Q=T^3+Y$. Then the Higgs doublet has $Y=1/2$ and transforms under electroweak gauge invariance asThe covariant derivative sign consistent with the requested expression isA convention that instead calls the doublet hypercharge one puts $Y/2$ in the charge generator and derivative; the physical coupling is the same. Here $B_{\mu }\mapsto B_{\mu }-{\partial }_{\mu }\beta /g^{\prime }$ and the matrix-valued $SU(2)$ connection transforms as $A_{\mu }\mapsto U{A}_{\mu }{U}^{-1}+(i/g)({\partial }_{\mu }U)U^{-1}$, making $D_{\mu }\varphi$ transform like $\varphi$.

In unitary gauge, the upper component of the derivative comes from the off-diagonal Pauli matrices:The lower component isWriting $c_W=\cos {\theta }_W$, $s_W=\sin {\theta }_W$, inversion of the neutral rotation gives $A^3=c_{W}Z+s_{W}A$ and $B=-s_{W}Z+c_{W}A$. Since $g^{\prime }=g\tan {\theta }_W$, the coefficient of the photon vanishes, whereas $-g{c}_W-g^{\prime }{s}_W=-g/c_W$. ThusThe zero photon coupling expresses the neutrality of the surviving radial Higgs boson.

The scalar Lagrangian density is ${\mathcal{L}}_{\varphi }=(D_{\mu }\varphi {)}^{†}{D}^{\mu }\varphi -V$. Adding $-F_{\mu \nu }^a{F}^{a\mu \nu }/4-B_{\mu \nu }{B}^{\mu \nu }/4$ gives the complete bosonic electroweak Lagrangian density. Its scalar and vector mass terms, after electroweak symmetry breaking, are obtained directly asIndeed, the shifted quartic Higgs potential normalization gives ${\varphi }^{†}\varphi -v^2/2=vH+H^2/2$. Comparing with a canonically normalized real scalar mass term, a complex vector mass term and a real vector mass term yieldsThe extra factor one half in this paper's Higgs potential is why $M_H^2$ is $\lambda v^2$, rather than $2\lambda v^2$ in another common convention. The tree-level electroweak gauge-boson masses give $\cos {\theta }_W=M_W/M_Z$: measure the two vector masses and take their ratio. Beyond tree level this extraction requires a specified renormalization convention and radiative corrections.

Expanding the vector terms gives the cubic and quartic Higgs couplings to electroweak gauge bosons:The four elementary Feynman diagrams and their all-incoming Feynman rules are shown below. Differentiating with respect to two identical $Z$ fields supplies a factor two; differentiating with respect to two identical Higgs fields supplies another factor two in the quartic rules.

There are no elementary $H\gamma \gamma$, $HZ\gamma$, $HH\gamma \gamma$ or $HHZ\gamma$ vertices in this tree-level Lagrangian density. Loop-induced photon interactions are a different order. The scalar self-interactions, if included, have rules $HHH:-3i{M}_H^2/v$ and $HHHH:-3i{M}_H^2/v^2$.

The prime LEP2 search used the Higgs boson coupling to Z bosons through Higgsstrahlung, $e^{+}{e}^{-}\to Z^{\ast }\to ZH$, shown in the first panel of the process figure in Question 3. This identifies an associated-production interaction, rather than an on-shell decay of a single $Z$ into a heavier Higgs. The channel's historical role is documented in the LEP2 Higgs-search account.

For the free charged W boson, take $W_{\mu }(x)=\int d^{4}p{e}^{-ipx}{W}_{\mu }(p)/(2\pi {)}^4$, with the conjugate field carrying $e^{ipx}$. In the action, multiplying the two field strengths gives $2[p^2{W}_{\mu }^{†}{W}^{\mu }-(p\cdot W^{†})(p\cdot W)]$. ThereforeTo invert it, separate the transverse and longitudinal projectors $P_T=g-pp/p^2$, $P_L=pp/p^2$. The operator is $(p^2-M_W^2)P_T-M_W^2{P}_L$, soSince the quadratic action kernel is $-O$, its Proca propagator isFor all momentum components small relative to $M_W$, the low-momentum limit is $D_{\mu \nu }\simeq i{g}_{\mu \nu }/M_W^2$, with corrections of relative order $p^2/M_W^2$ or $p_{\mu }{p}_{\nu }/M_W^2$. This local limit leads to the Fermi interaction, with $G_F/\sqrt{2}=g^2/(8M_W^2)$ for the charged-current normalization used here.

Using c and omitting ${\partial }_{\mu }\Phi$, the covariant derivative isThe two components are orthogonal, so their squared magnitudes add; $(W_{\mu }^{+}{)}^{†}=W_{\mu }^{-}$. Thus the Higgs field kinetic contribution to the Lagrangian density isThe two $v^2$ terms are gauge boson mass terms. Comparing with $m_W^2{W}^{+}{W}^{-}+\frac{1}{2}{m}_Z^2{Z}^2$ gives $\boxed{m_W=gv/2,m_Z=Gv/2}$ and $m_W/m_Z=\cos {\theta }_W$. The factor $1/2$ for the real Z boson field is essential.

The two $2vh$ terms are the cubic Higgs boson coupling to W bosons and Higgs boson coupling to Z bosons. They produce the three-leg $h{W}^{+}{W}^{-}$ and $hZZ$ Feynman vertices, allowing Higgs decay to two W bosons, Higgs decay to two Z bosons when kinematically available, and the corresponding production mechanisms with virtual gauge bosons. The two $h^2$ terms are quartic $hh{W}^{+}{W}^{-}$ and $hhZZ$ interactions, including vector-boson scattering into a Higgs boson pair. Written in terms of the masses,The corresponding Feynman vertex factors are $2i{m}_V^2{g}_{\mu \nu }/v$ for one $h$ and $2i{m}_V^2{g}_{\mu \nu }/v^2$ for two $h$, for $V=W,Z$, after differentiating with respect to identical fields. These six contributions contain no photon mass or direct $hAA$ interaction because $Q\Phi =0$.