Basic Concepts and Particle Classification

Ordinary matter reduces to three atoms of construction: leptons, quarks, and the bosons that carry forces between them. Everything from a grain of sand to the debris of a gold-on-gold collision at 99.99% of the speed of light is assembled from a short list of particles with no measurable internal structure. This lesson fixes the vocabulary needed to name and sort them: antiparticles, the Feynman diagrams that depict their interactions, and the division of matter particles into leptons and the composite hadrons.

Antiparticles

The Dirac equation, the relativistic wave equation for spin- $\frac{1}{2}$ particles, admits both positive- and negative-energy solutions. The relativistic energy relation gives two roots,

E = \pm (p c)^{2} + (m c^{2})^{2},

and the mathematics of the Dirac equation forbids discarding the negative root by fiat. Dirac's first interpretation filled every negative-energy state with electrons (an infinite sea), so that only a hole in the sea (behaving as a positive charge with positive energy) would be observable. Quantum electrodynamics later removed the need for the sea: the negative-energy solutions re-express as positive-energy solutions of a distinct particle, the antiparticle.¹

Carl Anderson identified the first antiparticle, the positron $e^{+}$ , in 1932 in cosmic-ray tracks: same mass and spin as the electron, charge $+ e$ , magnetic moment parallel (not antiparallel) to its spin. The antiproton $\overset{p}{ˉ}$ was produced at the Bevatron in 1955, and the antineutron $\overset{n}{ˉ}$ two years after. Producing a proton-antiproton pair demands at least the rest energy of both, $2 m_{p} c^{2} \approx 1877$ MeV, unavailable outside cosmic rays until GeV-scale accelerators arrived.

Two processes convert between matter and radiation:

Pair production. A photon of sufficient energy, in the field of a nucleus, converts to a particle-antiparticle pair: $γ \to e^{+} e^{-}$ . The threshold is $2 m_{e} c^{2} = 1.022$ MeV for an electron pair.
Pair annihilation. A particle meets its antiparticle and both vanish into photons: $e^{+} e^{-} \to γ γ$ . Momentum conservation forbids a single final photon when the pair is at rest, so at least two are emitted.

A photon materializes as an electron-positron pair (left); the pair recombines into two photons (right). Charge, energy, and momentum balance at every vertex.

Although the positron is stable, it survives only briefly in a universe full of electrons. A positron slowing to rest binds an electron into positronium before annihilating, and the spin state fixes the photon count: the antiparallel-spin singlet ( $^{1} S$ ) annihilates to two photons, while the parallel-spin triplet ( $^{3} S$ ) annihilates to three, since a system of definite parity and charge conjugation restricts the allowed final states.

The labelling of electrons as particles and positrons as antiparticles is a convention reflecting the composition of our corner of the universe, not a statement about which is more fundamental. Had matter here been built from negative protons and positive electrons, those would carry the particle label. Why the observable universe holds matter and almost no antimatter, despite the symmetry of the relativistic wave equation, is the matter-antimatter asymmetry problem, taken up later.

Feynman diagrams

Feynman diagrams are spacetime diagrams ( $c t$ versus $x$ ) that depict interactions among quarks, leptons, and force carriers. In the convention used here time runs upward, though the axes are usually left undrawn. The rules enforce conservation of energy and momentum at each vertex; the lines are symbolic and do not trace real trajectories.

Fermion lines carry an arrow. A line whose arrow points backward in time is read as the corresponding antiparticle moving forward in time, which lets the overbars be dropped from the diagram.
Force-carrier lines (photon, gluon, $W$ , $Z$ ) join vertices. A carrier that is its own antiparticle, such as the photon, carries no arrow.
Virtual particles are lines that both begin and end inside the diagram. They cannot be observed and need not carry the physical mass of the corresponding real particle, since only energy and momentum are conserved at vertices, not mass. Only lines entering or leaving the diagram are real, observable particles.

The primitive vertex of QED is a single charged particle entering, emitting or absorbing a photon, and leaving. Every electromagnetic process is built from copies of it.

The QED primitive vertex (left): a charged fermion emits or absorbs a photon. Coulomb repulsion of two electrons (right) is a single virtual-photon exchange between two such vertices.

Coulomb repulsion of two electrons is two electrons entering, exchanging a virtual photon, and leaving. The same skeleton with attractive charges gives Coulomb attraction. Because a virtual photon mediates the force in each case, the electromagnetic interaction is said to be mediated by photons. There are many diagrams for any given reaction; higher-order diagrams with more vertices contribute progressively less, so the simplest usually dominates.

Leptons

Research since the discovery of the electron has revealed 62 fundamental particles and antiparticles, fundamental in the sense of having no internal structure resolvable with present technology. They split into two families of spin- $\frac{1}{2}$ fermions (leptons and quarks) plus the force carriers.

Generation	Charged lepton	$Q (e)$	Mass $(MeV / c^{2})$	Neutrino	Neutrino mass
1st	electron $e$	$- 1$	$0.511$	$ν_{e}$	$< 2.2$ eV $/ c^{2}$
2nd	muon $μ$	$- 1$	$105.7$	$ν_{μ}$	$< 3.5$ eV $/ c^{2}$
3rd	tau $τ$	$- 1$	$1784$	$ν_{τ}$	$< 8.4$ eV $/ c^{2}$

The electron is the only stable charged lepton; the muon and tau decay. Each charged lepton has a distinct antiparticle, and each neutrino a corresponding antineutrino (whether a neutrino is its own antiparticle, a Majorana particle, is unsettled). The three flavors are named electron, muon, and tau. Leptons form no lepton-lepton bound states.

Quarks and hadrons

There are six quarks, likewise in three generations, all spin- $\frac{1}{2}$ fermions with fractional electric charge and distinct antiparticles.

Generation	Up-type	$Q (e)$	Down-type	$Q (e)$
1st	up $u$	$+ \frac{2}{3}$	down $d$	$- \frac{1}{3}$
2nd	charm $c$	$+ \frac{2}{3}$	strange $s$	$- \frac{1}{3}$
3rd	top $t$	$+ \frac{2}{3}$	bottom $b$	$- \frac{1}{3}$

Each quark also carries color charge, taking one of three values labelled red, blue, and green; antiquarks carry anticolor. So there are three up quarks $u_{r}$ , $u_{b}$ , $u_{g}$ , distinguished only by color. The names have nothing to do with visual color; they label a property analogous to electric charge, and color charge is conserved. Color is developed in the Standard Model lesson.

The bound states of quarks are the hadrons (Greek hadros, robust), and they split into two subgroups by quark count.

Matter particles sort into leptons and the composite hadrons; hadrons split by quark count into three-quark baryons and quark-antiquark mesons.

Six quarks bind in many ways into more than 200 hadrons, which account for the bulk of the visible mass of the universe. The proton and neutron are the only hadrons stable (or nearly so) in isolation; all others decay, and every baryon eventually decays to a proton. Single free quarks have not been observed and, for reasons in the Standard Model lesson, are not expected to be.

The hadrons stable against strong-interaction decay (lifetimes well above $1 0^{- 23}$ s) span a wide range of mass and lifetime. Those decaying strongly are far shorter-lived; electromagnetic decays run near $1 0^{- 18}$ s, weak decays near $1 0^{- 10}$ s.

Hadron	Class	Quarks	Mass $(MeV / c^{2})$	Spin	$Q (e)$	Lifetime (s)
$p$	baryon	$uu d$	$938.3$	$\frac{1}{2}$	$+ 1$	$> 1 0^{32}$ y
$n$	baryon	$u dd$	$939.6$	$\frac{1}{2}$	$0$	$930$
$Λ^{0}$	baryon	$u d s$	$1116$	$\frac{1}{2}$	$0$	$2.5 \times 1 0^{- 10}$
$Ω^{-}$	baryon	$sss$	$1672$	$\frac{3}{2}$	$- 1$	$1.3 \times 1 0^{- 10}$
$π^{\pm}$	meson	$u \overset{ˉ}{d}$	$139.6$	$0$	$\pm 1$	$2.6 \times 1 0^{- 8}$
$π^{0}$	meson	$u \overset{u}{ˉ}$	$135$	$0$	$0$	$0.8 \times 1 0^{- 16}$
$K^{\pm}$	meson	$u \overset{s}{ˉ}$	$493.7$	$0$	$\pm 1$	$1.2 \times 1 0^{- 8}$

Before the quark model this catalogue of more than 200 hadrons, growing with each new accelerator, was an uninterpreted particle zoo. The regularities in their masses, charges, and quantum numbers are what the quark model of the Standard Model lesson explains, much as the periodic table organized the chemical elements.

Idealized bubble-chamber view: a charged kaon enters and decays; a neutral pion from the decay converts in a lead sheet to two photons, each producing an electron-positron pair whose curvature reveals its charge.

The distinction between particles and antiparticles is a bookkeeping device grounded in the conserved quantum numbers introduced in the conservation laws lesson. What holds a nucleus together, and what turns one quark flavor into another in a decay, is the subject of the fundamental interactions.

Tipler & Llewellyn, §12-1 — Antiparticles: the Dirac hole picture and its replacement by the QED antiparticle, and the discoveries of the positron, antiproton, and antineutron. ↩