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The muon is the negatively charged second-generation lepton, with electric charge $-1$ and lepton number $+1$. Its antiparticle is the antimuon.

A muon and antimuon can form a Higgs boson directly in the annihilation channel. Their Higgs Yukawa coupling is proportional to the muon mass, making the corresponding partial width much larger than for an electron pair. The observable resonant cross section also depends on the collider's energy distribution: a beam spread broader than the resonance smears its ideal peak.

Feynman diagrams organize the perturbative calculation of the matrix element in Fermi's golden rule. Each external line labels an incoming or outgoing particle state; each internal line contributes a propagator for a generally off-shell intermediate field; and each interaction vertex contributes a coupling and the appropriate spin, momentum and internal-symmetry structure. All indistinguishable allowed diagrams must be summed as probability amplitudes before taking the absolute square, so their interference contributes to the rate.

In the radiative pion decay diagram in part (b), solid arrowed lines are fermions; the wavy line joining the hadronic and leptonic vertices is a W boson propagator; and the wavy outgoing line is an external photon. Arrows track fermion number, so an outgoing antimuon has an arrow towards its production vertex. Each vertex conserves four-momentum; an internal propagator need not satisfy the on-shell energy–momentum relation. A closed loop Feynman diagram additionally requires an integration over an internal four-momentum.

The final Lorentz-invariant phase space, conservation Dirac delta functions, and the initial-state normalization or incident flux turn the squared scattering amplitude into a decay rate or cross section. A diagram supplies the dynamics; it is not itself the final density of states.

The dominant interaction is electromagnetic. The color singlet charm-anticharm component of the J/psi meson annihilates through a virtual photon, which creates a muon and an antimuon:Panel (i) shows the two electromagnetic vertices. Vector quantum numbers allow the one-photon channel. A Z boson contribution is negligible at this mass scale; a single gluon cannot connect a color singlet initial state to colorless leptons. The dominant interaction for this specified final state need not dominate the meson's total decay width.

The charged-current process iswith a $t$-channel $W^{-}$ emitted at the electron-to-neutrino vertex and absorbed at the antimuon-to-antineutrino vertex. It is not an $s$-channel Annihilation of the unlike-flavour charged leptons into one $W$.

At high momentum transfer the incident electron's left-chiral fraction tends to $4/5$. In the collision centre-of-mass frame, the incoming antimuon moves along $-z$ and its charged-current allowed right helicity has spin along $-z$. A rest-frame spin at Angle $\theta$ to $+z$ projects onto that spin direction with probability ${\sin }^2(\theta /2)$. Therefore, if ${\sigma }_0$ denotes the cross section for a pure left-helicity electron and fully selected antimuon spin,It vanishes at $\theta =0$ in the ultrarelativistic selection limit and is maximal at $\theta =\pi$. Finite-mass corrections can fill the suppressed orientation. The absolute normalization depends on beam energy and the exchanged-boson propagator; the spin-angle dependence is what is fixed here.

Choose $m_{\tau }=2\mathrm{GeV}$ and $\sqrt{s}=6\mathrm{GeV}$ purely for illustration. Then each parent has $E_{\tau }=3\mathrm{GeV}$, $|{\mathbf{p}}_{\tau }|=\sqrt{5}\mathrm{GeV}$ and $\beta =\sqrt{5}/3$. Treat all six daughters as massless, including the charged electron and antimuon, and generate each three-body particle decay with constant squared decay amplitude. Choose the parent direction isotropically and the two decays independently; this omits the physical production angular law and spin correlations.

The Lorentz-invariant phase space factorizes through $t=q^2$ into a two-body decay to $\ell +q$, followed by a two-body decay of $q$:After angular integration the first factor contributes $1-t/m_{\tau }^2=x$, while the massless second factor is constant. Since $|dt|=m_{\tau }^{2}dx$, the normalized chosen density is $f(x)=2x$. Drawing $x=\sqrt{U}$ with $U$ uniform on $[0,1]$ implements this density. Isotropic directions, followed by a Lorentz boost for the invisible pair and another for the parent, conserve the complete four-momentum of each generated event.

Define transverse acoplanarity by $\alpha =\pi -\Delta \varphi$, where $\Delta \varphi \in [0,\pi ]$ is the angle between the two visible momentum projections perpendicular to the beam. A fully visible two-body event with zero initial transverse momentum has $\alpha =0$ and $M_{\mathrm{miss}}^2=0$. The toy tau lepton events spread away from that point. These observables are complementary: an angular imbalance asks about directions, while the missing invariant mass also uses the visible energies and longitudinal momenta.

A Lorentz-invariant phase space density supplies allowed configurations and their geometric measure. A physical weak interaction decay amplitude reweights them and correlates spins. Consequently the plotted density is a lesson about conservation and available configurations, not a predicted experimental rate or a fit to the discovery sample. The massless charged-daughter approximation is also unsuitable near the muon rest-energy threshold.