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Coauthored observations of electron-muon pairs with missing energy in electron-positron annihilation. Their analysis required at least two unobserved particles in each event.^[ref]

Antimatter consists of antiparticles, whose rest masses and spins match those of the corresponding particles while additive internal conserved charges are reversed. A Positron, for example, has the electron's rest mass and opposite electric charge. Both have positive physical energy.

When a particle meets its antiparticle, annihilation can convert the pair into other particles while preserving total energy, momentum and applicable conserved charges. Electron-positron annihilation into two photons is a simple example. Producing a few antiparticles in energetic collisions does not establish a large cosmological antimatter population: the production process and the subsequent abundance are separate questions. The antiparticle description is built into relativistic quantum field theory, rather than requiring negative measured mass or energy.

The Cosmic neutrino background is the relic population of neutrinos left after thermal decoupling in cosmology. As the universe expands, their momenta redshift; the decoupled distribution retains a temperature parameter proportional to the inverse scale factor, even after massive neutrinos become nonrelativistic.

The standard ideal temperature ratio follows from cosmological entropy conservation. Assume instantaneous neutrino decoupling before electron-positron annihilation and negligible chemical potentials. The coupled photon–electron–Positron sector initially has entropy degrees of freedom $2+(7/8)4=11/2$, dropping to two after annihilation. Conserving its comoving entropy heats the photons relative to the freely redshifting neutrinos, givingFor example, a photon temperature of $2.725\mathrm{K}$ corresponds to a relic-neutrino temperature parameter near $1.95\mathrm{K}$ in this idealization. Residual interactions during non-instantaneous neutrino decoupling slightly modify the distribution, so the simple ratio is an approximation.^[ref]

Single-photon electron-positron annihilation can directly form quarkonium with $J^{PC}=1^{--}$. A vector S-wave quarkonium can annihilate back through a virtual photon into a charged-lepton pair. At leading nonrelativistic order, its electronic partial decay width is proportional to ${\alpha }^2{Q}_q^2|\psi (0){|}^2/M^2$, where the wavefunction at the origin describes the quark-antiquark annihilation overlap.

Below the threshold for a pair of heavy-light mesons, hadronic decays require annihilation, primarily into three gluons for a color-singlet $1^{--}$ state. One gluon cannot carry a color-singlet initial state, and two gluons have the wrong charge-conjugation parity. Above the heavy-light pair threshold, a strong interaction can instead separate the heavy quark and antiquark into different mesons after creating a light pair. This can greatly increase the total decay width without comparably changing the short-distance electronic partial width. A specific example is the contrast between the first three Upsilon mesons and the $4S$ state. Higher radial states can additionally undergo hadronic or radiative transitions to lower quarkonium states.

With the conventional electroweak vacuum value $v\simeq 246\mathrm{GeV}$, the electron Yukawa coupling is $y_e\simeq 2.9\times 10^{-6}$ and the physical Higgs-electron vertex coefficient is $m_e/v\simeq 2.1\times 10^{-6}$. Direct s-channel Higgs production in electron-positron annihilation is consequently very small away from a compensating resonance enhancement. This suppression belongs to the coupling at the initial vertex, not to a prohibition on the Higgs boson coupling to W bosons.

Coauthored observations of electron-muon pairs with missing energy in electron-positron annihilation. Their analysis required at least two unobserved particles in each event.^[ref]

Coauthored the discovery of the J/psi meson in electron-positron annihilation. Scanning the collision energy revealed a sharp enhancement in production of hadrons and electron-Positron pairs near $3.105\mathrm{GeV}$, with a reported full-width upper limit of $1.3\mathrm{MeV}$.^[ref]

TODO: Biology deferred at the user's request: biological tracer behavior and clinical PET applications.

The generic physics of positron emission tomography uses electron-positron annihilation. A slowly moving electron-positron pair predominantly produces two nearly opposite photons with energy close to $m_e{c}^2=511\mathrm{keV}$ each. A ring of detectors records approximately simultaneous interactions; a coincidence identifies a coincidence line of response along which an annihilation could have occurred. Finite Positron travel before annihilation and nonzero pair momentum limit how directly this line identifies the original emission location.

Many such measurements constrain an emission distribution through a tomography model. Attenuation, scattered coincidences, unrelated random coincidences, finite detector resolution and dead time alter the observed counts and require calibration or modeling. With useful time-of-flight information, the arrival-time difference further localizes position along the line: a timing uncertainty $\delta t$ corresponds to a position scale $c\delta t/2$. The coincidence data therefore provide spatial constraints, rather than each event directly supplying a complete three-dimensional image point.

Theory. Quantum chromodynamics is a local, non-Abelian gauge theory with gauge group SU(3) acting on color charge. Its matter fields are spin-half quarks of six flavours, each in three colors; its eight massless spin-one gluons are the gauge fields. Unlike the neutral photon, gluons themselves carry gauge charge. Consequently there are three-gluon and four-gluon vertices as well as the quark-gluon vertex. A compact Lagrangian density iswhere $D_{\mu }$ is the gauge covariant derivative and the field strength contains the nonlinear term $g_s{f}^{abc}{G}_{\mu }^b{G}_{\nu }^c$.

The running coupling is weak at high momentum scales. At one loop, for $n_f$ active flavours,The positive coefficient $11-2n_f/3$ produces asymptotic freedom: short-distance, large-$Q$ processes admit perturbation theory. The gluon self-interaction is crucial to its sign. At low scales the coupling grows and isolated colored particles are not observed; the physical spectrum consists of color-singlet hadrons, the phenomenon of confinement. A hard-produced quark or gluon radiates and undergoes hadronization, giving a collimated jet. The one-loop formula is not valid at $Q\sim {\Lambda }_{\mathrm{QCD}}$ and is not, by itself, a mathematical proof of confinement.

Experimental tests. Different measurements test different ingredients of this account:

The convincing evidence is the quantitative consistency across these probes, with a common color group and coupling evolution. A jet by itself is evidence for a hard parton; its rates and correlations supply the more specific tests of quantum chromodynamics.^[ref]

Continuum amplitudes. The leading quantum electrodynamics diagram is electron-positron annihilation into a virtual photon, followed by conversion to a muon pair. Replacing the outgoing muons with a quark-antiquark pair gives the quark-level hadronic diagram. The first two panels below show these processes.

The Feynman rules give, up to signs and factors of $i$,The two vertices supply $e^2$, and the photon propagator supplies $1/s$. Each external current scales as $\sqrt{s}$, so the complete amplitude is of order $e^2$, and its squared magnitude is of order $e^4\propto {\alpha }^2$. The two-body relativistic scattering cross-section supplies the remaining $1/s$ from flux and phase space. Integrating the angular distribution yields ${\sigma }_{\mu \mu }=4\pi {\alpha }^2/(3s)$. Keeping only the squared propagator while ignoring the external currents would incorrectly predict $1/s^2$.

A quark of charge $Q_{f}e$ contributes $N_c{Q}_f^2$ times the massless muon rate. With $u,d,s,c,b$ open and $N_c=3$, the hadronic R ratio isThis is the requested massless leading-QED continuum approximation. Strong corrections, physical hadron thresholds, resonance contributions and Z exchange are separate effects; the pure photon formula is not a complete high-energy electroweak cross section throughout the entire stated range.

Efficiency normalization. Read the original PDF plot as a continuum of about $0.8\mathrm{nb}$ and a maximum around $1.7\mathrm{nb}$ at $E\simeq 10.58\mathrm{GeV}$. The supplied conversion constant givesHence near the peak,If $C$ multiplies the displayed cross sections, the required scale isHere $ϵ$ is the implied detection efficiency under the common-scale assumption. Using continuum readings $0.75$–$0.85\mathrm{nb}$ gives approximately $C=3.4$–$3.8$, illustrating the graph-reading precision. The nearby open-bottom threshold also limits the precision of the ideal continuum approximation.

Resonance quantum numbers and widths. Direct formation through a single photon requiresThese are the quantum numbers of the electromagnetic current. This concerns single-photon formation; associated production or two-photon reactions need not obey this restriction.

For a Breit-Wigner resonance cross section, subtract the continuum before finding the full width at half maximum. The plotted excess is about $0.9\mathrm{nb}$, so its half-height occurs at a total displayed cross section near $1.25\mathrm{nb}$, not $0.85\mathrm{nb}$. The crossings are roughly $10.57$ and $10.59\mathrm{GeV}$, with a separation around $0.02$–$0.025\mathrm{GeV}$. ThusFor spin-one formation from unpolarized electrons and positrons, $g=3/4$ and $p\simeq M/2$. With ${\Gamma }_{\mathrm{had}}\simeq \Gamma$, the peak resonance cross section and branching fractions relation becomesThe corrected excess is $C(1.7-0.8)\simeq 3.2\mathrm{nb}$. Taking $\Gamma \simeq 24\mathrm{MeV}$ as a representative graph reading givesThus the plot supportsThe width and electronic partial width are estimates from the printed graph under the prescribed Breit-Wigner and efficiency assumptions; these are not precision world-average values. A scale correction changes the peak height and inferred partial width, but not the peak position or width.

Why the lower Upsilon states are narrow. In electromagnetic production and decay of vector quarkonium, every $nS$ state can annihilate as $b\bar{b}\to {\gamma }^{\ast }\to e^{+}{e}^{-}$. The electronic width probes the wavefunction at the origin, ${\Gamma }_{ee}\propto {\alpha }^2{Q}_b^2|{\psi }_{nS}(0){|}^2/M_n^2$. The radial overlaps differ, but the same short-distance annihilation mechanism operates, so electronic widths remain of comparable order.

The hadronic mechanism changes sharply at $2m_B\simeq 10.560\mathrm{GeV}$. The first three Upsilon mesons lie below it and cannot produce an on-shell B-meson pair. Their leading purely strong annihilation channel is $b\bar{b}\to ggg$, followed by hadronization: a color-singlet pair cannot annihilate to one gluon, and two gluons cannot match its negative charge-conjugation parity. Three-gluon annihilation carries three powers of ${\alpha }_s$ in the rate and requires quark-antiquark overlap. Electromagnetic annihilation to light quarks and, for excited states, transitions to lower bottomonium plus pions or photons also contribute.

The fourth state lies above the pair threshold and can instead undergo ${\rm Y}(4S)\to B\bar{B}$ through light-pair creation, keeping the b and anti-b in separate outgoing mesons. The final panel depicts this quark-line topology; it represents a nonperturbative strong decay, not a free on-shell b quark radiating a real gluon. For two pseudoscalar B mesons the pair is in a P wave to give $1^{-}$. Although close to threshold, this open strong channel has a much larger rate than the annihilation channels. Consequently a total width of tens of MeV can coexist with an electronic width below a keV; below threshold the total widths are smaller by hundreds or more, while their electronic widths remain of similar order.

Hyperfine estimate. For two spin-half constituents,The meson hyperfine mass splitting therefore raises the vector mass above the pseudoscalar mass. To infer its coefficient from only one measured mass, adopt the intended simple constituent ansatz that the spin-independent baseline is $m_b+m_q=5.31\mathrm{GeV}$. ThenFor the spin-one partner,The estimate is the same for $\bar{b}u$ and $\bar{b}d$ with the supplied equal light-quark masses. The splitting is below $m_{\pi }$, so the strong decay $B^{\ast }\to B\pi$ is closed. An electromagnetic transition is open and dominates over weak decay:It preserves electric charge, giving the corresponding charged or neutral B meson; the photon energy is approximately $40\mathrm{MeV}$, slightly reduced by recoil. Strictly, the statement that the mass merely includes a spin-spin term does not fix $A$: an independent spin-independent binding energy would leave the baseline unknown. The numerical prediction above explicitly uses the additive constituent-mass approximation.

At an electron-positron collider tuned to the charmonium resonance, select the mixed leptonic tau lepton decaysand their charge conjugates. Require exactly two well-reconstructed oppositely charged tracks, one identified electron and one identified muon, little extra visible activity, visible energy below the beam total, and significant missing momentum pointing into the instrumented region. The multiple neutrinos prevent full reconstruction of each tau lepton, but the known initial four-momentum, lepton spectra and missing energy support a statistical signal extraction. Off-resonance data constrain nonresonant electron-positron annihilation into a tau lepton pair; that continuum is present at both beam energies.^[ref]

A resonant muon pair is one possible detector background: one muon can be misidentified as an electron, while radiation escaping along the beam or a badly measured track produces apparent missing energy. Particle-identification requirements and rejection of almost back-to-back tracks near the beam momentum suppress it. This background needs detector mismeasurement; an ideal measurement would distinguish it.

Above the open-charm threshold, $\psi (3770)\to D\bar{D}$ supplies an additional resonant background. Semileptonic decays of the two D mesons can give an electron, a muon and neutrinos; unreconstructed hadrons let some events pass the selection. This on-shell D meson-pair background is kinematically forbidden at $\psi (3686)$.

Nonrelativistic momentum conservation gives $\mathbf{u}={\mathbf{v}}_1+{\mathbf{v}}_2$, while an elastic collision conserves kinetic energy, $u^2=v_1^2+v_2^2$. Squaring the momentum equation yields ${\mathbf{v}}_1\cdot {\mathbf{v}}_2=0$. The two nonzero outgoing velocities are perpendicular. In the degenerate case where one outgoing particle is at rest, its direction is undefined.

For relativistic collisions conserve total four-momentum, with $E=\gamma m{c}^2$, $\mathbf{p}=\gamma m\mathbf{v}$ and $E^2-p^2{c}^2=m^2{c}^4$. Rest energies must be included; for photons $E=pc$.

For the symmetric relativistic equal-mass scattering angle, the incident momentum is $mc\sqrt{\alpha (\alpha +2)}$. Energy conservation gives each final particle $E_f=(1+\alpha /2)m{c}^2$, hence $p_f=(mc/2)\sqrt{\alpha (\alpha +4)}$. Transverse momenta cancel and $2p_f\cos \theta =p_{\mathrm{in}}$, givingThe nonrelativistic limit has $\theta \to 45^{\circ }$, so the outgoing angle is $90^{\circ }$ as above.

For the symmetric two-photon annihilation angle in the specified electron-positron annihilation, each photon has energy $E_{\gamma }=(\alpha +2)m_e{c}^2/2$. Momentum conservation gives $2(E_{\gamma }/c)\cos \varphi =m_{e}c\sqrt{\alpha (\alpha +2)}$, soAs $\alpha \to 0^{+}$ the two photons become back-to-back, transverse to the limiting incident direction; as $\alpha \to \infty$ they are forward collimated. Equal photon energies follow from the specified equal angles, and are not a general property of annihilation in a moving laboratory frame.

Single-photon emission computed tomography detects individual gamma rays from a tracer. Mechanically collimated gamma cameras acquire projections from multiple angles for tomography. The collimator discards most emitted photons, creating a strong sensitivity–spatial resolution trade-off.

Positron emission tomography uses a Positron-emitting tracer. The emitted Positron travels a short distance before electron-positron annihilation produces an approximately back-to-back photon pair. Coincident detection in a detector ring defines a line of response, giving electronic collimation and greater detection efficiency; time-of-flight PET further constrains position along that line. The positron range, non-collinearity, detector size, scatter and accidental coincidences limit spatial resolution and quantification. In both hybrids, computed tomography supplies anatomy and an attenuation correction map, with attention to registration and respiratory motion.^[ref]

The first step of the proton–proton chain isHere $d$ is the deuteron. A weak charged current converts one proton into a neutron, while the resulting proton–neutron pair becomes bound. The Q value follows from the difference of initial and final rest energies:This includes the emitted Positron's rest energy in the final-state mass and excludes its subsequent electron-positron annihilation. The nuclear binding energy of the deuteron makes this reaction possible although the isolated conversion $p\to n+e^{+}+{\nu }_e$ is energetically forbidden. The released energy is shared by final-state kinetic energies, including the escaping neutrino.

With the reversed electric field, reflection about $x=d/2$ exchanges the electron and Positron. They therefore meet there with equal and opposite momenta. Each travels $d/2$, and the work-energy theorem gives its kinetic energy:Their combined kinetic energy is $1.000\mathrm{MeV}$, while each total relativistic energy is $\mathcal{E}=0.511+0.500=1.011\mathrm{MeV}$. The laboratory is the center-of-momentum frame of the colliding pair.

Local four-momentum conservation in the electron-positron annihilation excludes one real photon: the incoming total four-momentum is timelike, with zero spatial component and positive energy, while a single photon's four-momentum is null and has $|\mathbf{p}|=\mathcal{E}/c>0$. Two photons can balance their momenta by travelling in opposite directions, soThe equal photon energies follow from opposite equal momenta, not merely from sharing the incoming energy arbitrarily. Here $hc\simeq 1.23984\mathrm{MeV}\mathrm{pm}$. This is the free local annihilation idealization implicit in the question; significant momentum transfer to the external apparatus would change the kinematic system being considered.

The laboratory remains the center-of-momentum frame. At the relativistic collision mass threshold, the two final W bosons can have arbitrarily small opposite momenta. The available relativistic energy is $2m_e{c}^2+eEd$, soEach incoming particle must acquire about $80.378489\mathrm{GeV}$ of kinetic energy, travelling about $16.08\mathrm{km}$. Equality is the ideal kinematic threshold; a nonzero production rate needs energy above it. This estimate neglects acceleration losses and treats the W bosons with the supplied nominal rest mass.

A linear particle accelerator avoids repeatedly bending high-energy electrons and Positrons. This greatly reduces the bending synchrotron radiation loss, which at fixed bending radius grows approximately as the fourth power of the relativistic energy for a given particle species. Its useful collision energy can therefore be increased without the severe circular electron collider radiation penalty. The price is long structures or high accelerating gradients: each bunch normally passes once, requiring substantial beam power, fresh bunches, and very tight focusing for high luminosity. A circular particle accelerator reuses its accelerating structures and circulates bunches through many collisions, but needs a large ring and substantial radiofrequency power to replace synchrotron radiation losses. Heavier particles such as protons radiate far less at a given energy and radius, although a proton's constituent structure makes the hard-collision energy less sharply specified than in an electron-positron annihilation experiment.

The Positron is the antiparticle of the electron, with equal rest mass and spin but electric charge $+e$. It can be created in beta-plus decay or in a sufficiently energetic pair production process.

An electron and Positron can undergo electron-positron annihilation. For the two-photon channel from a pair initially at rest, conservation of momentum requires opposite photon directions and each photon has energy $m_e{c}^2$, neglecting binding. Initial motion or a different allowed final state changes that simple picture. Annihilation is conversion of the initial particles' energy into other excitations, not disappearance of conserved energy or charge.