Quarks, Leptons, and the Big Bang
This closing chapter chases two of the oldest questions humanity has asked, now sharpened by the most modern physics: What is the universe made of? and How did it come to be the way it is? In the 1930s it seemed three particles — electron, proton, neutron — might explain everything. Then accelerators poured out a flood of new, short-lived particles, and order had to be rebuilt from scratch. That order is the Standard Model: a handful of truly fundamental leptons and quarks, bound by forces carried by messenger particles, all governed by a few strict conservation laws. The same accelerators that reveal these particles also reach back in time, because the only place energetic enough to test the deepest theories was the universe itself in its first instants. So the chapter ends where the cosmos began — with an expanding universe, a faint microwave glow left over from the big bang, and the unsolved puzzles of dark matter and dark energy. Physics, it turns out, is the gateway to the magical things still waiting for sharper wits.
- Every particle has intrinsic spin; half-integer-spin particles are fermions (obey the Pauli principle), integer-spin particles are bosons (do not). Spin component: \(S_z = m_s\hbar\).
- Particles are hadrons if the strong force acts on them (baryons = fermions, mesons = bosons) and leptons if not; every particle has an antiparticle of opposite charge.
- Unstable particles decay exponentially, \(N = N_0 e^{-\lambda t}\), with \(T_{1/2} = \dfrac{\ln 2}{\lambda} = \tau\ln 2\); masses are quoted in \(\mathrm{MeV}/c^2\) so that rest energy is \(mc^2\).
- Six leptons in three families and six quarks (u, d, s, c, b, t) with fractional charges \(\pm\tfrac{2}{3}e,\ \pm\tfrac{1}{3}e\) are the fundamental fermions; baryons are three quarks, mesons are quark–antiquark pairs.
- Interactions separately conserve lepton number \(L\) (per family), baryon number \(B\), and (in strong interactions) strangeness \(S\); the eightfold-way patterns first revealed the quark substructure.
- Forces act via messenger particles: photons (electromagnetic, QED), \(W\) and \(Z\) (weak — unified with EM as the electroweak force), and gluons (color force, QCD).
- The universe is expanding by Hubble's law \(v = Hr\) (\(H = 71.0\,\mathrm{km/s\cdot Mpc}\)); the \(2.7\,\mathrm{K}\) cosmic background radiation and abundances confirm a big bang about \(13.7\times10^{9}\) years ago, with dark matter and dark energy unexplained.
What Is Physics?
We call relativity and quantum physics "modern," though their foundations were laid a century ago — Einstein's photoelectric-effect and special-relativity papers in 1905, Bohr's atom in 1913, Schrödinger's wave equation in 1926. In this final chapter we pursue two questions with the most ancient of roots: what is the universe made of, and how did it come to be the way it is? Progress has been swift, driven by ever-larger accelerators. Yet physicists soon realized that no Earth-bound machine can reach the energies needed to test the ultimate theories — those energies existed only once, in the universe's first millisecond. You will meet a flood of new particles with strange names; do not try to memorize them. Share instead the bewilderment, and then the excitement, of the physicists who watched clarity finally spring from obscurity — while knowing that grand mysteries still remain.
Particles, Particles, Particles
The hope that three particles could explain matter collapsed as accelerators at Brookhaven, Fermilab, CERN, SLAC, and DESY produced hundreds of new, unstable particles — the muon (\(\mu\)), pion (\(\pi\)), kaon (\(K\)), sigma (\(\Sigma\)), and many more — each decaying by the same exponential law as radioactive nuclei.
Fermions (half-integer spin, like the electron, proton, neutron) obey the Pauli exclusion principle — one particle per quantum state — which is why atoms have shell structure. Bosons (integer spin, like the \(s=1\) photon) do not, so any number can share one state. Cooled enough, bosons pile into a single lowest state — a Bose–Einstein condensate, first made in 1995 from rubidium-87 atoms near \(1.7\times10^{-7}\,\mathrm{K}\).
Protons, neutrons, and pions are hadrons; electrons and neutrinos are leptons. Every particle also has an antiparticle of equal mass and spin but opposite charge (and opposite quantum numbers). When a particle meets its antiparticle they annihilate: \(e^- + e^+ \to \gamma + \gamma\), their mass energy reappearing as photons. Curiously, the cosmos is overwhelmingly matter, not antimatter — a bias dating to the universe's birth that we still do not understand.
An Interlude: Reading a Particle Event
A bubble-chamber photograph captures the spirit of the field. Charged particles leave trails of bubbles; a magnetic field bends positive tracks one way and negative tracks the other, and the curvature gives each particle's momentum. A typical event triggered by an incoming antiproton unfolds in stages, each obeying conservation of energy, momentum, angular momentum, and charge.
The Leptons
The leptons are pointlike fundamental particles untouched by the strong force. There are six, in three families: the electron, muon, and tau, each paired with its own distinct neutrino (\(\nu_e,\ \nu_\mu,\ \nu_\tau\)). The neutrinos are known to differ because a beam of muon neutrinos striking a target makes only muons, never electrons. All leptons have spin \(s=\tfrac{1}{2}\).
| Family | Particle | Symbol | Mass (MeV/\(c^2\)) | Charge \(q\) |
|---|---|---|---|---|
| Electron | Electron | \(e\) | 0.511 | \(-1\) |
| Electron | e-neutrino | \(\nu_e\) | \(\approx 0\) | \(0\) |
| Muon | Muon | \(\mu\) | 105.7 | \(-1\) |
| Muon | \(\mu\)-neutrino | \(\nu_\mu\) | \(\approx 0\) | \(0\) |
| Tau | Tau | \(\tau\) | 1777 | \(-1\) |
| Tau | \(\tau\)-neutrino | \(\nu_\tau\) | \(\approx 0\) | \(0\) |
The Hadrons
Hadrons — baryons and mesons — feel the strong force. They obey one more conservation law, introduced to explain why some imaginable processes never happen. Consider the decay \(p \to e^+ + \nu_e\): it conserves energy, momentum, charge, and lepton number, yet it never occurs — fortunately, or every proton in the universe would eventually vanish.
Strangeness and the Eightfold Way
Certain particles, the kaon and sigma, were always produced in pairs: \(\pi^- + p \to K^+ + \Sigma^-\) happens, but \(\pi^- + p \to \pi^+ + \Sigma^-\) never does, though it breaks no then-known law. Gell-Mann and Nishijima resolved this by assigning a new quantum number, strangeness \(S\).
Like Mendeleev's table, the patterns had gaps. From the symmetry of the spin-\(\tfrac{3}{2}\) decuplet, Gell-Mann predicted a missing "headpin" — the \(\Omega^-\), charge \(-1\), strangeness \(-3\), mass \(\approx 1680\,\mathrm{MeV}/c^2\) — and it was found, with exactly those properties. Such regularities strongly hinted that hadrons are not fundamental but have an inner structure.
The Quark Model
In 1964 Gell-Mann and Zweig proposed that hadrons are built from quarks. Three suffice at first — up (\(u\)), down (\(d\)), strange (\(s\)) — each spin \(\tfrac{1}{2}\) and baryon number \(\tfrac{1}{3}\), with the jarring feature of fractional electric charge.
| Quark | Symbol | Charge \(q\) | Strangeness \(S\) |
|---|---|---|---|
| Up | \(u\) | \(+\tfrac{2}{3}\) | 0 |
| Down | \(d\) | \(-\tfrac{1}{3}\) | 0 |
| Strange | \(s\) | \(-\tfrac{1}{3}\) | \(-1\) |
| Charm | \(c\) | \(+\tfrac{2}{3}\) | 0 |
| Top | \(t\) | \(+\tfrac{2}{3}\) | 0 |
| Bottom | \(b\) | \(-\tfrac{1}{3}\) | 0 |
The Basic Forces and Messenger Particles
Each fundamental force is carried by messenger particles exchanged between the interacting particles. The electromagnetic force, described by quantum electrodynamics (QED), is carried by virtual photons — undetectable because the energy "borrowed" to emit one is repaid within the time the uncertainty principle allows.
Glashow, Weinberg, and Salam showed the electromagnetic and weak forces are two faces of one electroweak force — echoing Maxwell's unification of electricity and magnetism — confirmed when the \(W\) and \(Z\) were found at CERN in 1983. In QCD each quark flavor carries one of three "colors"; only color-neutral combinations exist (three quarks, three antiquarks, or quark–antiquark), which is exactly why we see baryons and mesons and nothing else. Adding the strong force to the electroweak (a grand unified theory) and then gravity (a theory of everything) remains Einstein's unfinished dream.
A Pause for Reflection — and an Expanding Universe
To make the exotic particles we must reach GeV and TeV energies, because we live in a cold universe — even the Sun's core is only \(kT \approx 1\,\mathrm{keV}\). Such energies last reigned in the big bang, so studying high-energy particles is studying the early universe. And when we look far into space we look far back in time: the most distant quasars we see are as they were over \(10^{10}\) years ago. The grandest observation is that the distant galaxies are all receding from us.
The Cosmic Background Radiation and Dark Matter
In 1965 Penzias and Wilson found a faint microwave "hiss" coming uniformly from every direction — the cosmic background radiation. Its spectrum matches that of a body at \(2.7\,\mathrm{K}\): light that has flown across the universe since shortly after the big bang, freed to travel once particles combined into neutral atoms and stopped scattering it.
The rest is dark matter — matter that emits no detectable light. Some is ordinary (baryonic) burned-out stars and dim gas, but most is nonbaryonic: not protons and neutrons. Neutrinos contribute, yet not nearly enough; the bulk is made of particles still undetected, interacting (as far as we know) only through gravity. After a century of particle physics, the dominant matter of the universe remains unidentified.
The Big Bang
The big bang was not an explosion in space — it was the beginning of space and time themselves. It happened everywhere at once, and "before" has no meaning. We can, however, trace the universe's history forward from its earliest describable instant.
| Time | What happened |
|---|---|
| \(10^{-43}\,\mathrm{s}\) | Earliest meaningful moment; the universe is smaller than a proton at \(\sim 10^{32}\,\mathrm{K}\). Quantum fluctuations seed all future structure. |
| \(10^{-34}\,\mathrm{s}\) | Rapid inflation swells the universe by \(\sim 10^{30}\); a hot soup of photons, quarks, and leptons at \(\sim 10^{27}\,\mathrm{K}\). |
| \(10^{-4}\,\mathrm{s}\) | Quarks bind into protons and neutrons; matter and antimatter annihilate, leaving a slight excess of matter — our world. |
| \(1\,\mathrm{min}\) | Protons and neutrons fuse into light nuclei (\(^{2}\mathrm{H},\,^{3}\mathrm{He},\,^{4}\mathrm{He},\,^{7}\mathrm{Li}\)); their predicted abundances match observation. |
| \(379{,}000\,\mathrm{y}\) | At \(\sim 2970\,\mathrm{K}\), electrons join nuclei into atoms; light is freed — the cosmic background radiation — and gravity begins forming galaxies. |
Flatness requires a precise total energy, yet all known matter plus dark matter falls short by about two-thirds. The missing piece, named dark energy, was predicted to make the expansion accelerate — and in 1998 distant supernovae confirmed exactly that. We still scarcely know what dark energy is. The chapter, and the book, close on a humbling note: Earth is not central, the Sun is ordinary, and our species is a cosmological blink — yet the laws of physics we have uncovered seem to hold everywhere and for all time. Physics is the gateway to the magical things still waiting for sharper wits.
Putting It to Work
Problem. For \(\pi^- + p \to K^- + \Sigma^+\), the rest energies are \(139.6\), \(938.3\), \(493.7\), and \(1189.4\,\mathrm{MeV}\). Find \(Q\) and interpret its sign.
Solution. \(Q = (\text{initial mass energy}) - (\text{final mass energy})\).
The negative \(Q\) marks an endothermic reaction: the incoming pion must supply at least \(605\,\mathrm{MeV}\) just to create the rest mass. Because momentum must also be conserved, the true threshold is higher still — about \(907\,\mathrm{MeV}\).
Problem. A stationary \(\pi^+\) (rest energy \(139.6\,\mathrm{MeV}\)) decays to \(\mu^+\) (\(105.7\,\mathrm{MeV}\)) and a neutrino (\(\approx 0\)). Find each particle's kinetic energy.
Solution. Energy gives \(K_\mu + K_\nu = 139.6 - 105.7 = 33.9\,\mathrm{MeV}\). Momentum gives equal magnitudes, so with the relativistic relation \((pc)^2 = K^2 + 2Kmc^2\) and \(m_\nu \approx 0\):
Though their momenta are equal and opposite, the light neutrino carries about \(88\%\) of the kinetic energy — the massive muon barely recoils.
Problem. Can a stationary proton decay as \(p \to \pi^0 + \pi^+\)? Test the conservation laws (\(m_{\pi^0} = 135.0\), \(m_{\pi^+} = 139.6\,\mathrm{MeV}\)).
Solution. Charge: \(+1 \to 0 + 1\), conserved. Energy: \(Q = 938.3 - (135.0+139.6) = +663.7\,\mathrm{MeV} > 0\), so there is enough mass energy. But check spin and baryon number.
A spin-\(\tfrac{1}{2}\) proton cannot yield two spin-0 pions (no way to make \(S_z\) balance), and baryon number drops from \(1\) to \(0\). Either violation alone forbids the decay — which is why the proton is stable despite having ample energy.
Problem. The \(\Xi^-\) baryon has charge \(q=-1\) and strangeness \(S=-2\) (no bottom quark). What three quarks make it?
Solution. A baryon is three quarks. Only the strange quark gives \(S=-1\), so \(S=-2\) needs two strange quarks (ss, contributing \(q = -\tfrac{2}{3}\)). The third quark \(x\) must fix the charge.
The charge-\(-\tfrac{1}{3}\) quarks are \(d\), \(s\), \(b\); with \(b\) ruled out and \(s\) already counted, the third is a down quark. So \(\Xi^- = ssd\), and the baryon number checks out at \(+1\).
Problem. An electron and positron at rest annihilate into two equal photons. Find each photon's wavelength.
Solution. The total energy \(2m_ec^2\) splits equally, so each photon has \(E = m_ec^2 = 0.511\,\mathrm{MeV}\); then \(\lambda = hc/E\) with \(hc = 1240\,\mathrm{eV\cdot nm}\).
Each gamma photon has wavelength \(\approx 2.4\,\mathrm{pm}\) — the Compton wavelength of the electron. The two fly off in opposite directions to conserve momentum, since the initial momentum was zero.
Problem. A quasar recedes at \(v = 2.8\times10^{8}\,\mathrm{m/s}\) (about \(93\%\) of \(c\)). Estimate its distance using \(H = 21.8\,\mathrm{mm/s\cdot ly}\).
Solution. From Hubble's law, \(r = v/H\); convert \(H\) to SI by noting \(21.8\,\mathrm{mm/s\cdot ly} = 0.0218\,\mathrm{m/s\cdot ly}\).
The quasar lies about \(13\) billion light-years away — and so we see it as it was nearly that long ago, not far past the big bang itself. (This is approximate, since \(H\) has not been constant over cosmic history.)
Chapter Summary
Fermion (half-integer spin, obeys Pauli) vs. boson; hadron (feels strong force) vs. lepton; particle vs. antiparticle.
Six in three families (\(e,\mu,\tau\) + neutrinos), spin \(\tfrac{1}{2}\); each family's lepton number \(L\) is separately conserved.
Baryons (fermions, \(B=\pm1\)) and mesons (bosons, \(B=0\)); baryon number is conserved, so the proton is stable.
\(S\) is conserved in strong interactions; the eightfold-way hexagons exposed an inner structure and predicted the \(\Omega^-\).
Six quarks (u,d,s,c,b,t), fractional charge \(\pm\tfrac{2}{3},\pm\tfrac{1}{3}\); baryon = 3 quarks, meson = quark–antiquark.
Messengers: photons (QED), \(W\)/\(Z\) (weak, unified as electroweak), gluons (color force, QCD). Only color-neutral combos exist.
Hubble's law \(v = Hr\) (\(H=71\,\mathrm{km/s\cdot Mpc}\)); age \(\approx 1/H \approx 13.7\times10^{9}\,\mathrm{y}\) from a big bang.
\(2.7\,\mathrm{K}\) background radiation confirms the big bang; space is flat; dark matter and accelerating dark energy remain unexplained.
Problems
Take rest energies in \(\mathrm{MeV}/c^2\) from the chapter tables, \(hc = 1240\,\mathrm{eV\cdot nm}\), \(H = 21.8\,\mathrm{mm/s\cdot ly} = 71.0\,\mathrm{km/s\cdot Mpc}\), and \(1\,\mathrm{Mpc} = 3.26\times10^{6}\,\mathrm{ly}\). Recall \(L\), \(B\), and \(S\) conservation and the quark charges of Table 44-5.
- Classify each as fermion or boson, and as hadron (baryon/meson) or lepton: electron, proton, pion, photon, neutrino, neutron.
- An electron cannot decay into two neutrinos. Which conservation laws would be violated: energy, angular momentum, charge, lepton number, linear momentum, or baryon number?
- The negatively charged pion decays as the antiparticle of \(\pi^+ \to \mu^+ + \nu\). Write the decay scheme of the \(\pi^-\).
- For the antimuon decay \(\mu^+ \to e^+ + \nu_e + \bar{\nu}_\mu\), verify that the electron and muon lepton numbers are each separately conserved.
- This neutron decay is not observed: \(n \to p + e^-\) (no antineutrino). Which conservation law does it violate? (Masses: \(939.6\), \(938.3\), \(0.511\,\mathrm{MeV}/c^2\).)
- An electron and positron at rest annihilate into two photons. Find the wavelength of each, and explain why both must be equal.
- A neutral pion (rest energy \(135\,\mathrm{MeV}\)) at rest decays into two gamma rays. Find their wavelength.
- Explain why the reaction \(\pi^- + p \to K^+ + \Sigma^-\) can occur by the strong force but \(\pi^- + p \to \pi^+ + \Sigma^-\) cannot, using strangeness.
- Construct, if possible, a baryon from up, down, and strange quarks with (a) \(q=+1\), \(S=-2\); and (b) \(q=+2\), \(S=0\).
- Give the quark makeup of (a) the antiproton and (b) the antineutron, given proton \(=uud\) and neutron \(=udd\).
- From the quark charges, verify the charge of the \(\pi^-\) meson (\(d\bar{u}\)) and confirm that its baryon number is zero.
- Show that the fundamental beta-decay process \(d \to u + e^- + \bar{\nu}_e\) turns a neutron (udd) into a proton (uud) and conserves charge.
- A galaxy's sodium line, emitted at \(590.0\,\mathrm{nm}\), is observed at \(602.0\,\mathrm{nm}\). Using the low-speed Doppler relation \(v = c\,\Delta\lambda/\lambda\) and Hubble's law, find the distance to the galaxy.
- If Hubble's law could be extrapolated indefinitely, at what distance would the apparent recessional speed equal \(c\)?
- The cosmic background radiation peaks at \(\lambda_{\max} = 1.1\,\mathrm{mm}\). Using Wien's law \(\lambda_{\max} = (2898\,\mu\mathrm{m\cdot K})/T\), find the temperature it corresponds to.