= Solution
\b[Production and identification.] At a hadron collider, initial partons carry fractions of the proton momenta; observable production rates combine parton-level amplitudes with <parton distribution functions>.
* <Top quark> pairs are produced through $gg\to t\bar t$ and $q\bar q\to g^*\to t\bar t$. Single top production uses the <weak charged current>, for example $qb\to q't$ through a virtual <W boson>. The dominant decay is $t\to W^+b$, and similarly for the antiquark. The subsequent <W boson> decays produce dilepton, lepton-plus-jets and all-hadronic channels. Reconstructing the W and b systems constrains the top mass. The top decays before forming an ordinary long-lived top hadron.
* An on-shell <W boson> is produced by quark-antiquark annihilation, for example $u\bar d\to W^+$. Other sources include top decay and boson-pair production; at a lepton collider, $e^+e^-\to W^+W^-$ provides a clean pair channel. The <W boson> decays to $\ell\nu_\ell$ or to a kinematically accessible quark pair. Neglecting masses and strong corrections, the three lepton channels and two up-type quarks with three colors give approximately one-third leptonic and two-thirds hadronic branching. The leptonic modes combine a charged lepton with missing transverse momentum; the transverse-mass distribution constrains $m_W$.
* <Higgs boson> production includes gluon fusion $gg\to H$ through a heavy-quark loop, vector-boson fusion $qq\to qqH$, associated $q\bar q\to WH,ZH$, and $t\bar tH$ production. At a lepton collider, $e^+e^-\to ZH$ is another example. For a Higgs mass near $125\,\mathrm{GeV}$, important decays include $b\bar b$, $WW^*$, $gg$, $\tau^+\tau^-$ and $ZZ^*$. The less frequent $\gamma\gamma$ mode and $ZZ^*\to4\ell$ mode give especially clean reconstructed mass peaks. The asterisk denotes an off-shell daughter boson; two on-shell W or Z bosons are inaccessible at that Higgs mass.
\b[Why the three masses form a test.] At tree level, <electroweak symmetry breaking> gives $m_W=gv/2$ and $m_Z=\sqrt{g^2+g'^2}\,v/2$. The <Higgs mechanism> relates masses to the vacuum field, but it does not by itself predict the top Yukawa coupling or the Higgs self-coupling, so the three masses are not fixed independently of measured parameters.
The stronger test comes from <radiative corrections>. With accurately measured $\alpha$, $G_F$ and $m_Z$, the <electroweak precision test with the W, top and Higgs masses> uses
$$
m_W^2\left(1-\frac{m_W^2}{m_Z^2}\right)=\frac{\pi\alpha}{\sqrt2G_F(1-\Delta r)}.
$$
The top-bottom loop contributes approximately $\Delta\rho=3G_Fm_t^2/(8\sqrt2\pi^2)$, while the Higgs contribution has a characteristic logarithmic mass dependence. Hence direct measurements of $m_t$ and $m_H$ turn the precision calculation into a prediction for $m_W$. Comparing that prediction with the direct W-mass measurement overconstrains the <Standard Model>; disagreement beyond experimental and theoretical uncertainties could indicate additional particles or interactions in the loops. The test concerns the relation among independently measured quantities, rather than merely the existence of the three particles.
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