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At leading order in quantum electrodynamics, the electron-positron pair annihilates into one virtual photon, which produces the muon-antimuon pair. This is an s-channel Feynman diagram with $q=p_{-}+p_{+}=k_{-}+k_{+}$ and $q^2=s$. No QED vertex changes an electron directly into a muon, so there is no corresponding single-photon t-channel flavor-changing diagram.

Each charged-fermion vertex contributes $-ie{\gamma }^{\mu }$, the internal photon contributes $-i{g}_{\mu \nu }/(q^2+i0)$ in Feynman gauge, and external legs supply incoming $u(p_{-}),\bar{v}(p_{+})$ and outgoing $\bar{u}(k_{-}),v(k_{+})$. Their ordered contraction gives the stated amplitude. The overall convention-dependent phase does not affect its squared modulus.

For unpolarized scattering, average over the two spin choices of each incoming particle and sum over outgoing spins. The identities $\sum u\bar{u}=/p+m$ and $\sum v\bar{v}=/p-m$, together with ${\gamma }^0({\gamma }^{\mu }{)}^{†}{\gamma }^0={\gamma }^{\mu }$, convert the products into fermion spin sums:The factor $1/4$ averages the initial spins; there is no additional average over the final ones. In the massless limit the first trace is $4(p_{+\mu }{p}_{-\nu }+p_{+\nu }{p}_{-\mu }-g_{\mu \nu }{p}_{+}\cdot p_{-})$, with the analogous expression for the outgoing current. Contracting givesIn the centre-of-mass frame, every external energy is $E_{\mathrm{cm}}/2$. Choose $p_{-}$ along $+z$ and let $k_{-}$ make angle $\theta$ with it. The dot products are $p_{-}\cdot k_{-}=p_{+}\cdot k_{+}=s(1-\cos \theta )/4$ and $p_{-}\cdot k_{+}=p_{+}\cdot k_{-}=s(1+\cos \theta )/4$. HenceThe electron-positron annihilation into a muon pair cross-section is thereforeUsing $d\Omega =2\pi d(\cos \theta )$ givesIn ordinary units this becomes $4\pi {\alpha }^2(\hslash c{)}^2/(3E_{\mathrm{cm}}^2)$. It is the leading high-energy QED result: finite masses, radiative corrections and weak-interaction exchange modify the prediction outside that approximation.

Cross-section measurements can test spin assignments when the interaction and charges are specified. For example, keeping spin-half electrons but replacing the final muons by minimally coupled charged spin-zero particles replaces the fermion current by a momentum-difference current. In the massless limit, scalar-pair production by fermion annihilation gives $d\sigma /d\Omega ={\alpha }^2{\sin }^2\theta /(8s)$ and total $\pi {\alpha }^2/(3s)$. Its central angular peak differs from the fermionic $1+{\cos }^2\theta$ distribution.

Thus angular distributions, polarization information and suitably modeled rates distinguish spin hypotheses. A single total rate without assumptions about the couplings or the initial species does not independently establish both spins; changing interactions can mimic normalizations. The predicted fermion result is a testable consequence of the spin-half QED model, rather than a model-independent inference of spin from any one observed cross-section.

Set $s=q^2$. The angular integral of the electron-positron annihilation into a muon pair differential cross-section isThereforeThe printed comparison of $\sqrt{s}$ with a quantity in ${\mathrm{GeV}}^2$ has inconsistent units: the high-energy condition means an energy large compared with particle masses, or equivalently an energy-squared large compared with their squared masses.

At leading order, inclusive electron-positron annihilation into hadrons proceeds through $e^{+}{e}^{-}\to {\gamma }^{\ast }\to q\overline{q}$, as in the first panel below. The photon couples to each quark with strength $e{Q}_f$; the corresponding contribution is $Q_f^2$ times the muon-pair rate per colour. Summing over the three orthogonal final colours and active flavours gives the hadronic R ratioFor massless $u,d,s$ flavours this gives $R=2$, with charm included $R=10/3$, and with bottom as well $R=11/3$.

The approximation assumes unpolarized beams and negligible electron mass; active quark masses are small relative to $\sqrt{s}$, and heavy flavours below threshold are excluded. Photon exchange dominates, so $Z$ exchange and photon-$Z$ interference are neglected; this is appropriate sufficiently below the Z boson scale, not at arbitrary high energy. The coupling $\alpha$ must be evaluated consistently in numerator and reference rate. We also neglect higher-order QED radiation. Relating the inclusive partonic calculation to hadrons invokes quark-hadron duality: the energy should be in a perturbative continuum region, away from thresholds and narrow resonances, with power-suppressed nonperturbative effects neglected. An exclusive hadron channel cannot be obtained merely by multiplying the parton rate by the colour count.

At next order in QCD, the real-emission amplitudes are the two distinct diagrams for $e^{+}{e}^{-}\to {\gamma }^{\ast }\to q\overline{q}g$, with the gluon emitted from the quark or antiquark. They must be added before squaring, including their interference. Each amplitude is order $e{g}_s$ at the hadronic current vertex and gives a relative order-$g_s^2$ contribution to the rate.

The virtual order-$g_s^2$ amplitudes consist of the gluon correction to the photon-quark vertex and the quark and antiquark self-energy/external-state normalization contributions. The figure also shows the corresponding local vertex and field counterterms. In an amputated on-shell calculation the external-leg terms are represented by wave-function factors, rather than counted as extra final states. There is no gluon attachment to an electron, since it carries no colour. Gluon self-energy insertions and additional non-Abelian vertices enter at higher orders here. The rate correction is the Born-virtual interference $2\mathrm{Re}({\mathcal{M}}_0^{\ast }{\mathcal{M}}_{\mathrm{virt}})$ plus the integrated squared real-emission amplitude, not the square of the one-loop amplitude.

Use dimensional regularization, $d=4-2\epsilon$, to keep ultraviolet and infrared singularities in a common scheme. Ultraviolet divergences in the virtual quark and vertex contributions are removed by consistent wave-function renormalization and local counterterms; the conserved electromagnetic-current identity relates the vertex and field renormalizations. One uses a renormalized QCD coupling $g_s(\mu )$; its running changes this first correction only at the next perturbative order. In the massless on-shell scheme some external self-energy integrals are scaleless and vanish, but that statement combines ultraviolet and infrared poles and must not be used to discard the associated normalization terms inconsistently.

The remaining infrared divergences occur when a gluon is soft, and collinear divergences occur when it is parallel to a massless quark. They appear in both virtual diagrams and the real-emission phase-space integral. For the fully inclusive total rate, unresolved final states are summed and real-virtual infrared cancellation in inclusive QCD removes their poles. A resolved jet rate instead needs an infrared-safe observable and a resolution prescription. An isolated real or virtual correction is not a finite physical inclusive cross-section.

Define the dimensionless constant $A$ with the conventional loop factor extracted. Then the requested form isFixed numerical and colour factors are contained in $A$; choosing another extracted prefactor simply redefines that constant. No explicit loop coefficient is required to establish the form.

The Born and virtual processes give two energetic particle jets. Hard, well-separated real radiation gives a three-jet event; unresolved radiation changes jet widths and event-shape distributions, while the combined terms correct the inclusive rate. Resolved three-jet events in electron-positron annihilation provided the direct gluon signature, observed at PETRA in 1979. Their interpretation is energetic quark, antiquark and gluon fragmentation, with approximately planar three-body momentum balance. The primary institutional account is www.desy.de/news/news_search/index_eng.html?openDirectAnchor=1643&printversion=1 .

Take the quantum electrodynamics interaction density to be ${\mathcal{L}}_{\mathrm{int}}=-e\bar{\psi }{\gamma }^{\mu }\psi A_{\mu }$. The momentum-space Feynman-gauge photon propagator, with Lorentz indices $\mu ,\nu$ and momentum $k$, isThe electron-photon vertex from the QED Feynman rules iswith its spinor indices connecting the incident and outgoing fermion lines. The muon-photon vertex has the same coupling. The Feynman i-epsilon prescription specifies the pole boundary condition; using a different overall charge-sign convention changes both vertices consistently and leaves the squared amplitude unchanged.

For the electron-positron annihilation into a muon pair, the exchanged photon has $k=p+q$ and $k^2=s$. Up to the conventional overall phase, the scattering amplitude is

Substitute the preceding spin average into the supplied differential scattering cross-section. Since $e^4=16{\pi }^2{\alpha }^2$,The angular integral is $2\pi {\int }_{-1}^1(1+x^2)dx=16\pi /3$. Consequently the relativistic scattering cross-section for the massless electron-positron annihilation into a muon pair isThe outgoing particles are distinct, so there is no identical-particle factor. Comparing with the requested monomial form givesThe integrated answer depends only on the incoming invariant $s$; the angular variables $t,u$ have been integrated out.

For electron-positron annihilation into a muon pair the gauge diagrams are s-channel photon and Z exchange. For electron-positron annihilation into electron neutrinos there is s-channel Z exchange and t-channel W exchange, with interference. For electron-positron annihilation into W-boson pairs there are s-channel photon and Z diagrams and a t-channel electron-neutrino diagram. All three gauge amplitudes are required for the proper high-energy cancellation. Electron-Yukawa-suppressed s-channel Higgs diagrams can also produce the massive muon or W pair; they vanish in the usual zero-electron-mass approximation used for the leading gauge comparison.

There is no energy-independent ordering. For unpolarized beams and inclusive Born cross sections the useful regimes are:

The Z-pole ratio follows $[\frac{1}{2}{]}^2/[(-\frac{1}{2}+{\sin }^2{\theta }_W{)}^2+{\sin }^4{\theta }_W]\simeq 1.99$. Treating every weak reaction as uniformly slower misses both the resonance and forward exchange.

For a massless fermion, chirality is fixed by helicity. For a massless antifermion, the field-chirality label is opposite to its physical helicity: a left-chiral field creates negative-helicity particles and positive-helicity antiparticles. At $E\gg m$, wrong-chirality amplitudes are suppressed by $m/E$, so chirality-preserving vector vertices conserve helicity along a physical fermion line up to these corrections.

For electron-positron annihilation into a muon pair through a vector mediator, each annihilating or created pair therefore has opposite physical helicities. Writing each pair in the order (negative particle, positive particle), the nonzero possibilities areHere $L,R$ refer to physical helicity along each particle's own momentum, not to the sign of its spin projected on a single common axis. The incoming pair has total spin projection $m=\pm 1$ along the electron beam; the outgoing pair has $m^{\prime }=\pm 1$ along the muon direction. Same-helicity pairs vanish in the strictly massless vector-current limit.

For unpolarized ultrarelativistic charged spin-half particles annihilating through a photon into minimally coupled unit-charge scalars of mass $m$, set $\beta =\sqrt{1-4m^2/s}$. The scalar vertex contains the difference $k_{-}-k_{+}$ of outgoing four-momenta. Contraction with the spin-averaged initial fermion current gives $\overline{|\mathcal{M}{|}^2}=e^4{\beta }^2{\sin }^2\theta /2$. The two-body final-state flux factor contributes another $\beta$, givingThe ${\sin }^2\theta$ angular dependence and threshold suppression contrast with electron-positron annihilation into a muon pair. These predictions assume the specified scalar electrodynamic coupling, rather than an arbitrary spin-zero interaction.