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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.

The Standard Model uses the local gauge group $SU(3{)}_c\times SU(2{)}_L\times U(1{)}_Y$. Its Higgs field is a complex Higgs doublet of hypercharge $Y=1/2$. A bare mass term for a weak gauge boson, or for a charged fermion joining inequivalent left- and right-handed gauge representations, would not respect the unbroken electroweak construction. Instead introduce a gauge-invariant Higgs potentialIts minima satisfy ${\Phi }^{†}\Phi =v^2/2$, $v^2={\mu }^2/\lambda$. Choosing a representative $⟨\Phi ⟩=(0,v{)}^T/\sqrt{2}$ leaves $Q=T_3+Y$ unbroken. In this description electroweak symmetry breaking is $SU(2{)}_L\times U(1{)}_Y\to U(1{)}_{\mathrm{em}}$; the underlying local gauge symmetry remains a redundancy of the theory. A vacuum representative is not a physical explicit violation of gauge invariance.

In unitary gauge, write $\Phi =(0,v+H{)}^T/\sqrt{2}$. The gauge-covariant kinetic term $|D_{\mu }\Phi {|}^2$ produceswith $\tan {\theta }_W=g^{\prime }/g$ and $e=g\sin {\theta }_W$. Three angular scalar directions supply the longitudinal gauge-boson polarizations of $W^{+},W^{-},Z$, while one real radial Higgs boson survives with $m_H^2=2\lambda v^2$. Counting physical degrees of freedom checks the Higgs mechanism: the electroweak gauge sector and scalar doublet have $4\times 2+4=12$ before the rearrangement and $3\times 3+2+1=12$ afterwards.

The Higgs potential also fixes the scalar self-interactions:The color gauge group is unbroken, so gluons remain massless. The same electroweak kinetic term fixesGauge-invariant Yukawa couplings such as $-y_d{\bar{Q}}_L\Phi d_R-y_u{\bar{Q}}_L\tilde{\Phi }{u}_R+h.c.$ give $m_f=y_{f}v/\sqrt{2}$ and the physical coupling $-m_{f}H\bar{f}f/v$. These relations predict stronger Higgs couplings to heavier massive particles; they do not determine the Yukawa parameters themselves. The minimal Standard Model does not supply observed neutrino masses. The surviving Higgs boson is neutral and has spin zero; its mass and self-coupling are tied to the curvature of the Higgs potential, not fixed by the gauge couplings alone. Higgs exchange also provides the Higgs cancellation in longitudinal vector scattering, maintaining the required high-energy behavior of weak amplitudes.

At a proton collider, important production mechanisms are gluon-fusion Higgs production through a heavy-quark loop, vector-boson fusion with two outgoing quark jets, associated $WH$ or $ZH$ production, and production with a top quark pair. At an electron–positron collider, Higgsstrahlung $e^{+}{e}^{-}\to ZH$ is a clean example. Tree-level decays include Higgs decay to a fermion pair and, when allowed, Higgs decay to two W bosons or Higgs decay to two Z bosons. Below an on-shell vector-pair threshold, one or both vectors can be virtual. The $gg$ and $\gamma \gamma$ channels arise through loops; neutral colorless $H$ has no corresponding direct tree-level gauge coupling. Rates and branching fractions depend on both couplings and kinematics, so “heaviest available particle” is a guide rather than a complete rule.

The Higgs boson discovery in 2012 supplied an experimental test. The ATLAS experiment and CMS experiment reconstructed two-photon and four-lepton invariant masses in Large Hadron Collider proton collisions. Electromagnetic calorimeters measure photon energies, while tracking and muon systems identify charged leptons; narrow reconstructed peaks can be distinguished from smooth backgrounds with control samples and a likelihood analysis. The ATLAS discovery paper reported a new resonance near $126.5\mathrm{GeV}$. An observed mass peak supports the existence of a new state; production channels, decay rates and angular distributions test whether it has the particular Higgs couplings and quantum numbers.^[ref]

Finally, the hierarchy problem asks why the electroweak scalar mass scale remains much smaller than a putative heavy physical scale without delicate cancellations. A cutoff estimate suggests corrections $\delta m_H^2\sim {\Lambda }^2/(16{\pi }^2)$ times couplings, but the quadratic cutoff term itself depends on the regularization convention. The physical concern persists when a heavy state of mass $M$ couples to the Higgs field and produces threshold corrections of order its coupling times $M^2/(16{\pi }^2)$. Matching a much smaller Higgs mass may then require cancellation between independent parameters. Supersymmetry, compositeness and other protective structures propose different resolutions; renormalizability of the Standard Model does not by itself resolve this sensitivity or establish that new particles must occur at one particular mass.

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.