Hadrons, Leptons, Quarks and Bosons; The Fundamental Interactions

Four interactions spread over forty orders of magnitude in strength, and each ends up owning its own band of decay times — which is why a lifetime, on its own, is very nearly a diagnosis.

🎯 Why this matters

An experiment can therefore choose which interaction it studies by choosing a time window. A detector resolving a millimetre of flight has already selected weak decays; one that sees only a single vertex has selected strong ones.

Two short sections that name everything. After four sections of machinery, this is the cast list and the rules of engagement — worth reading carefully, because the vocabulary is used without further explanation for the next 460 pages.

The first cut: bosons and fermions

Every particle has a spin, in units of ħ, and the value splits the world in two:

The one distinction everything else hangs off

Class↕	Spin↕	Statistics↕	Wave function under exchange↕	Examples
Boson	0, 1, 2, …	Bose–Einstein		photon, gluon, W, Z, Higgs, and every meson
Fermion	1/2, 3/2, …	Fermi–Dirac		quarks, leptons, and every baryon

Identical particles of a given type are genuinely indistinguishable: after two protons collide elastically there is no meaning to asking which one was the projectile.

The cast

🧩 The whole cast — 17 particles

generation I

generation II

generation III

gauge bosons

scalar

Click a tile for its properties. Turn on the force overlay and look at the third row: the neutrinos are the only particles that feel just one interaction — which is exactly why they pass through the Earth unhindered, and why detecting them takes a thousand tonnes of target.

Everything above is, as far as anyone can tell, elementary — no experiment has resolved structure inside any of them. Everything below is built out of them.

Composite matter, and the words for it

Name↕	Made of↕	Spin↕	Examples	Note
Baryon	three quarks (qqq)	half-integer	p, n, Λ, Ω⁻	Carries baryon number 1. Greek βαρύς, "heavy".
Meson	quark + antiquark (qq̄)	integer	π, K, ρ, J/ψ	Named for having a mass "in the middle", between electron and proton — true of the pion, and badly wrong for the ϒ at 9.5 GeV.
Hadron	baryons and mesons together	either	all of the above	Greek ἁδρός, "thick, strong" — i.e. anything that feels the strong interaction. All are unstable except the proton.
Nucleon	the two lightest baryons	1/2	p, n	A collective noun, not a separate kind of thing.

💡 What this really says — why “elementary” is a statement about experiments, not about nature

Atoms are broken open with a few eV, nuclei with several MeV. Both are quantitative steps, not qualitative ones: hit harder, see more structure, as §1.8 made precise.

Then this pattern stops. Nobody has ever knocked a free quark out of a hadron, at any energy, with any projectile. That is not a resolution problem — we resolve well inside the proton — it is a new phenomenon, called confinement , and the Standard Model has to explain it rather than assume it (Chapter 6). It is the first place in this book where “just use more energy” stops being the answer.

§1.10 The four interactions

Each interaction has one or more charges that act as both source and receptor. Have the charge, feel the force; do not, and you are invisible to it.

The four fundamental interactions

Interaction↕	Charge↕	Mediator	Range↕	Felt by
Strong	colour	$g \ (\times 8)$		quarks and gluons only
Electromagnetic	electric charge	$\gamma$		every charged particle
Weak	weak isospin / hypercharge	$W^\pm,\ Z^0$	2.5 × 10⁻³ fm
Gravitational	energy–momentum tensor		∞	everything, without exception

The nuclear force that binds protons and neutrons is <strong>not</strong> fundamental: it is the leftover of the colour field escaping from colour-neutral hadrons, exactly as the Van der Waals force is the leftover of the electromagnetic field escaping from neutral molecules.

The table gives the ranges. The strengths span a spread that no table conveys, so here it is on a logarithmic axis — the dimensionless coupling each interaction presents to a proton:

33 orders of magnitude between the weak interaction and gravity

Three of the four are within five orders of magnitude of each other, and are therefore all in play in the same experiment. Gravity is thirty-three orders below the nearest of them — which is the entire reason it is absent from this book until Chapter 11. The numbers are computed in the snippet below.

⚙️ Engineer’s bridge — the nuclear force is a fringing field

Two neutral molecules still attract, because “neutral” means the charges cancel on average and not point by point: some field leaks out. That is Van der Waals.

Two colour-neutral hadrons attract for exactly the same structural reason — the colour field leaks out of each. So the force that holds nuclei together, the one Yukawa explained and the one that powers the Sun, is a fringing field of the real interaction. It appears nowhere in the Standard Model’s Lagrangian.

Getting this hierarchy right matters: the strong interaction is between quarks; the nuclear force is its residue. Chapter 6 works with the first, nuclear physics with the second, and confusing them makes both incomprehensible.

Where it breaks: the van der Waals comparison gets the mechanism right and the magnitude badly wrong. A residual electrical force between neutral atoms is a genuinely small perturbation — chemistry sits far below atomic binding — whereas the residual strong force, though a fraction of what acts between quarks, is still by an enormous margin the strongest force in the nucleus. Its range is set by pion exchange, ħ/m_π c ≈ 1.4 fm, rather than by dipole fluctuations, so it has a definite scale where van der Waals has a power law. “Fringing field” is the right picture of where it comes from and no guide at all to how strong it is.

Why gravity does not appear again until Chapter 11

\htmlClass{t-MP}{M_P} = \sqrt{\frac{\htmlClass{t-h}{\hbar} \htmlClass{t-c}{c}}{\htmlClass{t-G}{G_N}}} = 2.18\times10^{-8}\ \text{kg} = 1.22\times10^{19}\ \text{GeV}

(1.97)

The Planck mass: the only mass you can build out of ħ, c and Newton's constant — and therefore the scale at which gravity must become a quantum theory.

Every symbol, one at a time

Hover or tap a symbol above — it lights up in the equation and its meaning, units and type appear here.

🔢 Worked example — how weak is gravity? (Example 1.10)

Take an electron and a proton a distance

r

apart and compare the two forces:

\frac{F_\text{EM}}{F_\text{grav}} = \frac{q_e^2}{4\pi\varepsilon_0 G_N m_e m_p} = 2.3\times10^{39}.

Both go as $1/r^2$ , so the ratio does not depend on $r$ at all — it is a property of the particles, not of the geometry. Gravity is not weak “at short range”; it is weak everywhere, by 39 orders of magnitude, and only wins on astronomical scales because it cannot be screened: matter comes with both signs of electric charge and only one sign of mass.

For particle physics the consequence is blunt: gravity is negligible in every process in this book, and quantum gravity lives at an energy no machine will reach. If you want to see it, you must look at the violent parts of the Universe — which is Chapter 11.

Reading a lifetime as a diagnosis

The interactions announce themselves through decay times, and the pattern is sharp enough to identify the culprit before any other measurement.

Which force killed it (Bettini pp. 31–32)

Particle	Lifetime / width↕	Interaction↕	Why that value
$p$	> 2.4 × 10³⁴ yr	none — stable	Baryon number conservation leaves it nothing to decay into.
$n$		weak	Weak, and then throttled further by tiny phase space — only 1.3 MeV is released.
$\mu^-$	2.197 μs	weak	The textbook weak decay. Long enough that cosmic-ray muons reach the ground (§1.11).
$\pi^\pm$	26.03 ns	weak	Must change a quark flavour, and only the weak interaction does that.
$K^\pm$	12.38 ns	weak	Same, with strangeness changing — which is why strange particles were "strange".
$\tau^-$	290.3 fs	weak	Also weak, but far more phase space available: 1777 MeV to spend instead of 106.
$\pi^0$	84.3 as	electromagnetic	π⁰ → γγ is open, so it need not wait for the weak interaction. 3×10⁸ times faster than its charged partner.
$\rho$	Γ = 149 MeV (τ ≈ 4 × 10⁻²⁴ s)	strong	Nothing forbids it. It decays as fast as it can cross its own diameter, so you see a width, never a track.
$\Delta^{++}$	Γ = 118 MeV (τ ≈ 6 × 10⁻²⁴ s)	strong	The same. Chapter 4 finds it as a bump, not as a particle.

Fourteen orders of magnitude separate the strong band from the weak one, and another twelve separate that from the neutron. Handed an unknown particle's lifetime, you can name the interaction before you know anything else about it.

Aside — the 80 % of the Universe this book does not describe

Bettini closes §1.9 with a caution worth keeping in view: astrophysical and cosmological observations show that ordinary matter — everything in the chart above — accounts for no more than about 20 % of the mass of the Universe. What the rest is made of is unknown. The Standard Model is spectacularly successful and demonstrably incomplete, and Chapter 12 is the list of what is missing.

Reproduce it

import numpy as np
qe, eps0, GN = 1.602176634e-19, 8.8541878128e-12, 6.67430e-11
me, mp = 9.1093837015e-31, 1.67262192369e-27
hbar, c, GeV = 1.054571817e-34, 299792458.0, 1.602176634e-10

print(f"F_EM / F_grav for an electron and a proton = "
      f"{qe**2/(4*np.pi*eps0*GN*me*mp):.3e}   (independent of r)")

MP = np.sqrt(hbar*c/GN)                       # Eq. (1.97)
print(f"Planck mass = {MP:.4e} kg = {MP*c**2/GeV:.4e} GeV")
print(f"  that is {MP*c**2/GeV/0.938272:.3e} proton masses, and "
      f"{MP*c**2/GeV/13600:.1e} times the LHC's 13.6 TeV")

hbarc = 197.3269804                            # MeV fm
for name, M in (("M_W ", 80377), ("M_Z ", 91187.6), ("m_pi", 139.57039)):
    print(f"range hbar*c/{name} = {hbarc/M:.4e} fm"
          + ("   <- the residual nuclear force, not a fundamental one" if name == "m_pi" else ""))

print(f"pi+/pi0 lifetime ratio = {26.033e-9/84.3e-18:.2e}   (weak vs electromagnetic)")

mpG, MPG, GF = 0.93827209, 1.220890e19, 1.1663787e-5     # GeV, GeV, GeV^-2
print("dimensionless coupling to a proton, and its log10:")
for name, sym, v in (("strong", "alpha_s ~ 1", 1.0),
                     ("electromagnetic", "alpha", 1/137.035999084),
                     ("weak", "G_F m_p^2", GF*mpG**2),
                     ("gravitational", "(m_p/M_P)^2", (mpG/MPG)**2)):
    print(f"  {name:16s} {sym:18s} {v:.3e}   log10 = {np.log10(v):6.2f}")

prints

F_EM / F_grav for an electron and a proton = 2.269e+39   (independent of r)
Planck mass = 2.1764e-08 kg = 1.2209e+19 GeV
that is 1.301e+19 proton masses, and 9.0e+14 times the LHC's 13.6 TeV
range hbar*c/M_W  = 2.4550e-03 fm
range hbar*c/M_Z  = 2.1640e-03 fm
range hbar*c/m_pi = 1.4138e+00 fm   <- the residual nuclear force, not a fundamental one
pi+/pi0 lifetime ratio = 3.09e+08   (weak vs electromagnetic)
dimensionless coupling to a proton, and its log10:
strong           alpha_s ~ 1        1.000e+00   log10 =   0.00
electromagnetic  alpha              7.297e-03   log10 =  -2.14
weak             G_F m_p^2          1.027e-05   log10 =  -4.99
gravitational    (m_p/M_P)^2        5.906e-39   log10 = -38.23

Twenty-eight decades of lifetime — and the culprit is legible from the position alone

STRONGEMWEAK10⁻²⁴10⁻²⁰10⁻¹⁶10⁻¹²10⁻⁸10⁻⁴110⁴lifetime τ (seconds, log scale)ρΔ⁺⁺π⁰τ⁻K±π±μ⁻n~14 decades of nothing~12 more, inside the weak band aloneThe bands do not overlap, and that is the whole diagnostic.The ρ never leaves a track — it decays in about the time light crosses it. The neutron survives fifteen minutes.

Supplied — the table above states that the bands are separated and the note puts numbers on the gaps; drawn, the separation is the point rather than a claim. Position on this axis names the interaction before any other measurement is made, which is exactly how the strange particles announced themselves: produced copiously (so, strongly) and decaying in nanoseconds (so, weakly), a contradiction that took a new quantum number to resolve (§2.2).

Note the weak band is not narrow — it spans twelve decades on its own, from the τ at 290 fs to the neutron at 878 s, because the golden rule’s phase-space factor swings by that much across the mass range (§1.7). The strong band, by contrast, is barely a decade wide: nothing forbids those decays, so they all happen as fast as they can.

🔑 If you remember only three things

A lifetime is the measurement you get for free. Timing a decay needs no model of what caused it, which is why it is usually the first number known about a new particle.
Stability is a conservation law, not a property. The proton lasts because nothing it could decay into is allowed, not because it is built more sturdily than anything else.
The first cut is spin, and it decides statistics. Charge, mass and lifetime are labels; whether a particle is a boson or a fermion decides how many of them can occupy one state.

Where this goes next

§1.11 is how any of these particles is actually detected.
Chapter 2 tells the discovery story of the muon, the pion, the neutrino and the antiproton — the cast assembled the hard way.
Chapter 3 explains the conserved quantum numbers only named here, starting with baryon number.
The particle explorer has every one of these with its numbers.

✅ Check yourself — the cast and the four forces

0/5 answered · 0 correct

1.Turn on the force overlay in the chart. Which row stands out, and why does it matter practically?
2.The gluon is massless, like the photon. Why is the strong force short-ranged when electromagnetism is not?
3.Two particles have lifetimes $10^{-23}$ s and $10^{-10}$ s. What can you say before knowing anything else about them?
4.The electromagnetic force between an electron and a proton beats gravity by $2.3\times10^{39}$ . Which statements follow?
5.Why is the nuclear force that binds protons and neutrons described here as not fundamental?