The Muon and the Pion; Strange Mesons and Hyperons

Yukawa predicted a mass, and the particle found at that mass was the wrong one. Everything on this page follows from how long that went unnoticed and from what finally exposed it.

🎯 Why this matters

Agreement in one number is not identification. The muon’s mass sat close enough to the prediction to be believed for a decade, and what eventually separated the two particles was a property nobody had thought to check.

Chapter 1 built instruments. This chapter uses them, and it is told as history on purpose — Bettini’s own justification is that watching a science reach partial truths and correct its own errors teaches more than the finished result does.

The first two sections are the best possible demonstration. A particle is predicted; a particle of the right mass is found; it is the wrong particle, and it takes nine years and one carefully designed experiment to prove it. Then nature produces a class of particles that are made quickly and decay slowly, and that contradiction turns out to need a new conserved quantity.

Yukawa’s argument: a range is a mass

In 1935 Yukawa proposed that the force between nucleons is carried by a meson, which he called the π. Since the force acts between pp, nn and pn, the mediator must come in three charge states. And since the force has a finite range — about 1 fm, unlike the infinite range of electromagnetism — the mediator must be massive. The result is the Yukawa potential :

\htmlClass{t-phi}{\varphi(r)} \propto \frac{e^{-\htmlClass{t-r}{r}/\htmlClass{t-l}{\lambda}}}{\htmlClass{t-r}{r}}

(2.1)

Eq. (2.1). The only difference from Coulomb is the exponential — and that exponential is the mediator's mass.

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.

📐 Physics you need first — why a range is an inverse mass

The exchanged particle does not exist as a free particle: it is virtual, borrowed against the energy–time uncertainty relation (§1.6–1.7). Borrowing an energy $\Delta E \ge m$ is allowed for a time $\Delta t \sim 1/\Delta E$ , and in that time the particle can travel at most $\Delta t$ (natural units, so at most $c\,\Delta t$ ). Hence

\lambda \sim \frac{1}{m} \qquad\Longleftrightarrow\qquad m \sim \frac{1}{\lambda} = \frac{\hbar c}{\lambda} \ \text{in SI-restored form}.

With $\hbar c = 197.3$ MeV fm and $\lambda = 1$ fm, that is 197 MeV — the book’s “about 200 MeV”. A massless mediator gives an infinite range, which is why the photon’s reach is unlimited and the nuclear force stops at the edge of a nucleus.

This is the single most-used estimate in the subject. You will meet it again for the W boson (range $2.5\times10^{-3}$ fm, §1.9–1.10) and for every heavy mediator after it.

⚙️ Engineer’s bridge — a mass is a corner frequency

Take the Fourier transform of the Yukawa potential over three-dimensional space and you get

\tilde\varphi(q) \propto \frac{1}{q^2 + m^2},

which is the propagator of a massive field, and which an engineer will recognise on sight: it is a Lorentzian, a second-order low-pass response with its corner at $q = m$ . The Coulomb case $m = 0$ gives $1/q^2$ — no corner, DC gain infinite, infinite range.

The dictionary is exact and it holds through the whole book:

Field theory	Signals
mediator mass $m$	corner frequency
range $\lambda = 1/m$	time constant / decay length
propagator $1/(q^2+m^2)$	transfer function
momentum transfer $q$	frequency
$q \ll m$ (low-energy limit)	below the corner: the interaction looks contact-like, with a constant strength

That last row is worth holding on to. Fermi’s theory of beta decay (Ch. 7) is exactly the $q \ll m_W$ limit of the weak propagator: a low-pass filter probed far below its corner looks like a plain gain, and a massive-boson exchange probed far below its mass looks like a four-fermion point coupling. The Fermi constant is that DC gain, $G_F \propto g^2/m_W^2$ .

Where it breaks: a filter’s corner is a design parameter you chose and can move; a mass is neither. And the DC-gain analogy captures the magnitude of the low-energy coupling while saying nothing about its structure — the exchanged W also carries spin 1 and couples only to left-chiral fields, so the contact interaction it reduces to is a very particular four-fermion operator, not just “a gain”. Taking the low-frequency limit of a scalar transfer function would give the size right and the V − A structure of §7.3 not at all.

The wrong particle

Two years after Yukawa, cosmic-ray physicists found it. Anderson and Neddermeyer, and independently Street and Stevenson (both 1937), showed that the penetrating component of cosmic rays — the part that crosses metres of rock — consists of particles with masses of just the predicted order. In 1942 Rossi and Nereson measured their lifetime: $\tau = 2.3 \pm 0.2$ μs.

Everything fitted. It was wrong.

🔬 Experiment card — Conversi, Pancini and Piccioni (Rome, 1946)

The question

Does a negative penetrating particle stopped in matter decay, or is it captured by a nucleus first? A positive particle is repelled by nuclei and must decay whatever it is. A negative one is drawn in — and if it is Yukawa’s strongly interacting meson it will be swallowed long before it can decay. So the difference between the two signs is a direct test of whether the particle feels the strong interaction.

Apparatus

Two magnetised iron blocks act as a sign-selecting lens; Geiger counters above and below them; an absorber; more counters underneath. Swapping the blocks selects the other sign.

What is measured

A three-condition trigger, and the whole experiment is in the logic — see the callouts below.

What it proved

With an iron absorber (1945): positives decayed, negatives did not. Perfect agreement with the Yukawa hypothesis. With a carbon absorber (1946): both signs decayed. Since a strongly interacting particle would be absorbed by any nucleus, however light, the penetrating particle does not interact strongly. It is a lepton — the muon.

🛠️ Fig. 2.1 — the Conversi–Pancini–Piccioni experiment, and its trigger

Click a numbered marker for what that piece does.

Redrawn from Fig. 2.1 (p. 72). Nothing in this apparatus measures energy or position; it is a pure coincidence machine, and everything it knows it knows from timing.

🔢 Worked example — why carbon settles it and iron does not

The book states the result and moves on. The quantitative reason is worth having, because it explains why the first experiment gave the wrong answer.

A negative muon stopped in matter is captured into an atomic orbit around a nucleus, and then two clocks race:

decay, at the fixed rate $\Gamma_{\text{dec}} = 1/2.197\ \mu\text{s} = 4.55\times10^{5}\ \text{s}^{-1}$ , the same in any material;
nuclear capture, $\mu^- p \to n \nu_\mu$ , at a rate that climbs steeply with $Z$ . The muonic Bohr radius shrinks as $1/Z$ and the nuclear charge grows as $Z$ , so the overlap grows roughly as $Z^4$ (Wheeler’s rule).

absorber	$Z$	capture rate	fraction captured	fraction that decays
carbon	6	$3.8\times10^{4}\ \text{s}^{-1}$	8 %	92 %
iron	26	$4.4\times10^{6}\ \text{s}^{-1}$	91 %	9 %

Capture overtakes decay at about $Z \approx 11$ . Iron sits well above that line and carbon well below it, which is why the same apparatus gave opposite answers.

And that is exactly the point. The competition here is between the weak decay and a weak capture whose rate is set by an electromagnetic overlap. Nothing strong is involved. Had the particle been Yukawa’s meson, capture would have been a strong-interaction process — a rate of order $10^{22}\ \text{s}^{-1}$ , sixteen orders of magnitude faster than the decay — and no absorber, however light, would have let a single negative particle survive to decay. Carbon does, so the particle is not a hadron.

The right particle

In 1947 the Bristol group — Lattes, Occhialini, Powell and Muirhead — exposed emulsion stacks at up to 5500 m in the Andes and found events in which a heavier particle decays into a lighter one. Two particles were present in cosmic rays all along: the $\pi^\pm$ , which interacts strongly and is Yukawa’s meson, and the $\mu^-$ , which is the penetrating particle and a lepton.

The decisive observation is a piece of pure kinematics, and it is worth stating precisely because it is the same argument used a dozen times later in the book.

💡 What this really says — a constant range means exactly two bodies

Powell’s group measured the muon’s range in the emulsion and found it the same in every event — about 600 μm. That single fact fixes the decay.

A pion at rest decaying to $N$ bodies shares out a fixed energy. If $N = 2$ , momentum conservation forces the two daughters back to back with one determined momentum, the same every time. If $N \ge 3$ , the daughters share the energy in a continuum, and the muon would emerge with a spectrum of momenta and therefore a spread of ranges.

A sharp range is a two-body decay. And since only one charged track comes out, the other body is neutral and undetected:

\pi^+ \to \mu^+ + \nu_\mu .

The same logic run in reverse identifies the muon’s own decay: the electron from $\mu \to e$ comes out with a spectrum reaching up to $m_\mu/2 = 52.8$ MeV, so that decay has three bodies, two of them invisible. The two neutrinos in Eq. (2.3) were established this way long before anyone could detect one.

🔢 Worked example — the 4 MeV muon

For $\pi^+ \to \mu^+\nu_\mu$ at rest, the two-body formula of Problem 1.11 with $m_2 = 0$ gives

p^* = \frac{m_\pi^2 - m_\mu^2}{2m_\pi} = \frac{139.57^2 - 105.66^2}{2 \times 139.57} = 29.8\ \text{MeV}/c,

so the muon’s kinetic energy is $T_\mu = \sqrt{p^{*2}+m_\mu^2} - m_\mu = 4.12$ MeV, every time. A 4 MeV muon ranges out in a few hundred micrometres of emulsion, which is the ~600 μm Powell measured, and which is why the emulsion — with its micrometre grain (§1.13a) — was the only instrument that could have seen it.

Note how little energy is released: 4 MeV out of a 140 MeV particle. Almost all of the pion’s mass goes into making the muon’s.

🌿 π⁺ → μ⁺ → e⁺: the chain Powell photographed

lab momentum of the π⁺0.0300 GeV/c

π⁺βγcτ = 1.68 m
└μ⁺99.99%βγcτ = 93.5 m
└e⁺— stable, or never seen
└ν_e— stable, or never seen
└ν̄_μ— stable, or never seen
└ν_μ— stable, or never seen

π⁺ — cτ = 7.8 m, so a pion is a metastable particle you can build a beamline out of (§1.13c, Problem 1.19).

μ⁺ — cτ = 659 m. This is why cosmic-ray muons reach the ground (§1.11) and why they were the first thing anyone found.

e⁺ — Stable. Its energy spectrum runs up to m_μ/2 = 52.8 MeV — a continuum, which is how the two neutrinos were inferred.

ν_μ — Never seen in 1947. Its existence was inferred purely from the muon having a single, fixed momentum.

Two-body steps use exact forward kinematics, p_lab = γ(p* + βE*). Steps with three or more daughters share the parent's momentum by the declared fraction — indicative, not exact.

These are DECAY lengths, βγcτ — and note how large they are. A muon from a pion at rest carries p* = 29.8 MeV/c and would fly 186 m before decaying. Powell measured 600 μm, because the muon does not get to fly that far: ionisation (§1.11) brings it to rest in a few hundred micrometres of emulsion, and only then, 2.2 μs later, does it decay in place. Range and decay length are different quantities, and whichever is shorter is what you see. Push the slider to 100 GeV/c and the pion itself crosses six kilometres — the lifetime never changed, only the boost.

Other experiments then showed the pion doing what Yukawa’s particle must do — interacting strongly with nuclei and converting protons into neutrons and back:

\pi^- + {}^A_Z N \to {}^{A}_{Z-1} N + p, \qquad \pi^+ + {}^A_Z N \to {}^{A}_{Z+1} N + n . \tag{2.4}

It took another quarter of a century to learn that Yukawa’s force is not the fundamental one and the pion is not elementary — the real interaction is between quarks, mediated by gluons (Ch. 6). But as an effective theory at 1 fm, Yukawa was right.

Aside — “Who ordered that?”

The muon is identical to the electron in every measurable way except its mass, which is 207 times larger. It has no role in ordinary matter. Rabi’s question is still unanswered, and it grows worse in §2.4, where a third copy turns up. Whatever explains the generations is not in this book, because nobody has it.

Strangeness: made quickly, decaying slowly

Nature was not finished. In 1944 Leprince-Ringuet and l’Héritier, running a triggered cloud chamber in a 0.25 T field in the Alps, measured a particle of mass 506 ± 61 MeV — within a fifth of a standard deviation of the kaon mass we now know, 493.7 MeV. Soon after the pion’s discovery, laboratories in the UK, France and the USA found cosmic-ray events with particles of that kind decaying into pions:

neutral ones decaying to two charged tracks, called V⁰ for the shape they leave (§1.13b has a modern one drawn in full);
charged ones decaying to a charged daughter plus neutrals, called θ;
charged ones decaying to three charged particles, called τ.

It took a decade to establish that θ and τ are the same particle, the K meson, and the V⁰s its neutral counterparts. That decade is not wasted time: the reason they looked different is a genuine physical puzzle, and its resolution in 1956 overturned a symmetry everyone had assumed (§3.7).

⚠️ Two names, one particle — and a name collision

The historical τ here is the K meson decaying to three pions. It is not the τ lepton of §2.4, which was discovered thirty years later and named independently from the Greek for “third”. The two are unrelated and the collision is unfortunate; when this book says τ in a 1950s context it means the kaon, and everywhere else it means the lepton.

Rochester and Butler’s 1947 paper reported something stranger still: these particles appeared in pairs. And the pairs were mismatched — one partner around 500 MeV (a K meson), the other heavier than a nucleon and decaying into a nucleon and a pion. The heavy ones are the hyperons .

💡 What this really says — the contradiction that needed a new quantum number

Two facts, and they do not fit together:

The new particles are produced copiously, with cross-sections typical of the strong interaction — so a strong-interaction vertex makes them.
They decay slowly, with lifetimes around 10⁻¹⁰ s, typical of the weak interaction — even into fully hadronic final states such as $\Lambda^0 \to p\pi^-$ , which the strong interaction ought to do in 10⁻²³ s.

Why does a process the strong interaction is manifestly capable of running forwards refuse to run backwards?

The answer of Nishijima and of Gell-Mann (both 1953) is that there is a conserved quantity you have not noticed. Call it strangeness $S$ , additive like electric charge, conserved by the strong and electromagnetic interactions and violated by the weak. Then:

Production in pairs is forced. The initial state (pions, nucleons) has $S = 0$ ; a strong vertex cannot change $S$ ; so strange particles can only be made two at a time, with opposite strangeness. This is associated production .
Slow decay is forced. The lightest strange particles have no strangeness-conserving final state light enough to decay into, so the strong and electromagnetic interactions are both blocked, and only the weak interaction — which does not respect $S$ — is left. Weak means slow.

One new additive quantum number, and both puzzles close. This is the standard move of the next two chapters: when a process that should happen does not, look for the conservation law that forbids it.

⚙️ Engineer’s bridge — conservation laws are a type system

Strangeness is the clearest example in the book of what a quantum number actually is: an additive tag carried by a value, with a composition rule, checked at every operation.

Each particle carries typed tags: charge, baryon number, strangeness, lepton flavour.
Every vertex is an operation with a signature: the strong and electromagnetic vertices are declared to preserve $S$ ; the weak vertex is not.
A process that violates a preserved tag does not “happen slowly”. It does not type-check, and its amplitude is exactly zero.

That is why a selection rule is qualitatively different from a suppression. The Λ⁰ does not decay strongly at all — not rarely, not weakly-preferred. The channel is closed by the type system, and the weak interaction is the only routine in scope with a looser signature.

Where the analogy breaks, and it matters: this type system is discovered, not declared. Nobody wrote the signature of the strong vertex; it was inferred from the fact that certain final states never appear. Every conservation law in chapters 3 and 4 is reverse-engineered from a missing decay.

Where it breaks: a type system is complete and declared — the compiler was told the rules and rejects at compile time. Here the rules are inferred from absences, so every one is a hypothesis that a missing decay means forbidden rather than merely rare. Strangeness was invented exactly that way and turned out to have a symmetry behind it; baryon number has not, and is conserved accidentally rather than by design (§12.7). And the inference runs one way only: an observed decay disproves a rule, but no amount of non-observation proves one — it bounds it, at whatever sensitivity you had.

🪜 Associated production, step by step

Step 1 of 5 — The initial state

\pi^- + p, \qquad S = 0 + 0 = 0

Why you may do this: Pions and nucleons are "old" hadrons: they carry no strangeness. Whatever the strong interaction does next, it must leave S at zero.

The reaction of Fig. 1.21, which §1.13b drew as a bubble-chamber photograph. Here it is as a bookkeeping problem — and the bookkeeping is the physics.

The metastable strange particles

A metastable particle is one that cannot decay strongly, and therefore lives long enough to travel. The strange ones are four kaons and six hyperons.

Table 2.1 — the K mesons. All have spin 0.

Particle	Q↕	S↕	m (MeV)↕	τ↕	Principal decays (BR %)
$K^+$	+1	+1	493.7	12.4 ns
$K^0$	0	+1	(497.6)		—
$K^-$	−1	−1	493.7	12.4 ns
$\bar K^0$	0	−1	(497.6)	n.a.	—

Two corrections to the table as printed on p. 75 — see the note below. The K⁰/K̄⁰ pair is the most interesting object in the table and the book defers all of it: unlike the π⁰, which is its own antiparticle, the neutral kaon is not, because strangeness distinguishes them.

⚠️ Two slips in Table 2.1 as printed

The lifetime column is headed “τ (ps)” and the kaon entry is 12.4. A charged kaon lives 12.4 nanoseconds, not picoseconds — a factor of a thousand. The book’s own Problem 1.3 uses ” $\tau_K = 12$ ns”, and §1.9–1.10 lists 12.38 ns, so the book contradicts itself. Table 2.2 on the facing page uses the same “(ps)” header correctly for the hyperons, which is presumably how the slip happened.

The two pionic branching ratios are swapped. As printed, the table gives $\pi^+\pi^+\pi^-$ (21) and $\pi^+\pi^0$ (5.6). The measured values are the other way round: the two-body $\pi^+\pi^0$ mode is 20.7 % and the three-body $\pi^+\pi^+\pi^-$ mode is 5.6 %.

This one is worth getting right, because these two channels are the historical θ and τ, and §3.7 turns on them. The more common mode is the two-pion one; the table as printed makes the three-pion mode four times more frequent than it is.

Table 2.2 — the metastable strange hyperons. All have spin ½.

Particle	Q↕	S↕	m (MeV)↕	τ (ps)↕	cτ (mm)↕	Principal decays (BR %)
$\Lambda$	0	−1	1115.7	263		pπ⁻ (64), nπ⁰ (36)
$\Sigma^+$	+1	−1	1189.4	80	24	pπ⁰ (51.6), nπ⁺ (48.3)
$\Sigma^0$	0	−1	1192.6	7.4 × 10⁻⁸	2.2 × 10⁻⁸
$\Sigma^-$	−1	−1	1197.4	148	44.4	nπ⁻ (99.8)
$\Xi^0$	0	−2	1314.9	290	87	Λπ⁰ (99.5)
$\Xi^-$	−1	−2	1321.7	164	49	Λπ⁻ (99.9)

Read the cτ column as the reason this chapter exists: every one of these is centimetres, so every one leaves a gap between production and decay that a chamber can resolve — <strong>except</strong> the Σ⁰, at 22 picometres. The Σ⁰ decays electromagnetically because it can, and it is therefore nine orders of magnitude faster and completely invisible. One row of one table contains the whole argument.

Notice the Ξ: strangeness −2, and it decays to a Λ, which still has strangeness −1 and must decay again. The weak interaction changes strangeness by one unit at a time, so a doubly strange particle has to come down the ladder in two steps. That is why the Ξs are called cascade particles.

🌿 The cascade: Ξ⁻ → Λ → p, two visible vertices

lab momentum of the Ξ⁻2.00 GeV/c

Ξ⁻βγcτ = 74.4 mm
└Λ99.9%βγcτ = 138 mm
└p64%— stable, or never seen
└π⁻βγcτ = 28.2 m
└π⁻βγcτ = 30.8 m

Ξ⁻ — S = −2. Weak decays change strangeness one unit at a time, so it cannot reach S = 0 in a single step.

Λ — S = −1, and now stuck for the same reason the Λ always is: no strangeness-conserving final state is light enough.

p — Stable — the end of the chain, and S = 0 at last.

Both steps are exact two-body kinematics. At a few GeV the two decay points are centimetres apart — a chain of three tracks with two gaps, which is exactly how the Ξ was identified. Drop the momentum to 100 MeV/c and the gaps shrink below a millimetre; the particle is the same, the picture is not.

CPT, anticipated

The book slips in a fundamental law here without proving it, and it is worth flagging because the tables above depend on it. CPT is the combined operation of charge conjugation $\mathcal{C}$ (swap particles and antiparticles), parity $\mathcal{P}$ (invert the space axes) and time reversal $\mathcal{T}$ . Invariance under the product of all three is a theorem of any local, Lorentz-invariant quantum field theory, and it forces:

particle and antiparticle to have the same mass;
the same lifetime;
the same spin;
and every charge of opposite sign.

That is why one line of Table 2.1 does for both the K⁺ and the K⁻, and why the K⁻ decay channels are the channel-by-channel antiparticles of the K⁺ ones. Chapter 3 takes C, P and T apart individually — and finds that each of them is violated, while the product survives.

Reproduce it

import numpy as np
c = 2.99792458e8
mpi, mmu, mK = 139.57039, 105.6583755, 493.677

print(f'Yukawa: hbar*c/lambda with lambda = 1 fm -> m = {197.3269804:.0f} MeV'
      f'  (the book\'s "about 200")')

ps = (mpi**2 - mmu**2)/(2*mpi)                    # two-body, massless partner
print(f"pi -> mu nu at rest: p* = {ps:.2f} MeV/c, "
      f"T_mu = {np.hypot(ps, mmu) - mmu:.2f} MeV, the SAME every event")
print(f"mu -> e nu nu: electron energy is a SPECTRUM up to m_mu/2 = {mmu/2:.1f} MeV"
      f" -> three bodies")

G = 1/2.1969811e-6                                # muon decay rate, s^-1
print(f"muon decay rate 1/tau = {G:.2e} /s; nuclear capture competes with it:")
for name, Z, cap in (("carbon", 6, 3.8e4), ("iron  ", 26, 4.4e6)):
    tail = "(both signs decay)" if Z == 6 else "(negatives vanish)"
    print(f"   {name} Z={Z:2d}: capture {cap:.1e} /s -> {cap/(cap+G)*100:4.1f} % captured, "
          f"{G/(cap+G)*100:4.1f} % decay  {tail}")
print(f"   crossover near Z = {6*(G/3.8e4)**0.25:.0f}")

print(f"Leprince-Ringuet 1944: 506 +- 61 MeV vs m(K+) = {mK:.1f} -> "
      f"{(506-mK)/61:.2f} sigma")

print("c*tau of the metastable strange particles, in mm:")
for n, tau_ps in (("Lambda", 263.2), ("Sigma+", 80.), ("Sigma0", 7.4e-8),
                  ("Sigma-", 148.), ("Xi0", 290.), ("Xi-", 164.)):
    ct = c*tau_ps*1e-12*1000
    flag = "   <- nine orders of magnitude, because it goes electromagnetically" if n == "Sigma0" else ""
    print(f"   {n:8s} {tau_ps:>7.4g} ps -> {ct:.3g} mm{flag}")

prints

Yukawa: hbar*c/lambda with lambda = 1 fm -> m = 197 MeV  (the book's "about 200")
pi -> mu nu at rest: p* = 29.79 MeV/c, T_mu = 4.12 MeV, the SAME every event
mu -> e nu nu: electron energy is a SPECTRUM up to m_mu/2 = 52.8 MeV -> three bodies
muon decay rate 1/tau = 4.55e+05 /s; nuclear capture competes with it:
 carbon Z= 6: capture 3.8e+04 /s ->  7.7 % captured, 92.3 % decay  (both signs decay)
 iron   Z=26: capture 4.4e+06 /s -> 90.6 % captured,  9.4 % decay  (negatives vanish)
 crossover near Z = 11
Leprince-Ringuet 1944: 506 +- 61 MeV vs m(K+) = 493.7 -> 0.20 sigma
c*tau of the metastable strange particles, in mm:
 Lambda     263.2 ps -> 78.9 mm
 Sigma+        80 ps -> 24 mm
 Sigma0   7.4e-08 ps -> 2.22e-08 mm   <- nine orders of magnitude, because it goes electromagnetically
 Sigma-       148 ps -> 44.4 mm
 Xi0          290 ps -> 86.9 mm
 Xi-          164 ps -> 49.2 mm

Erratum — two errors in Table 2.1, one of which matters a great deal

The lifetime unit. The column is headed τ (ps) and the charged kaons are listed at 12.4. The K± lifetime is 12.38 ns — a factor of a thousand. The book’s own Problem 1.3 uses τ_K = 12 ns and §1.9–1.10 lists 12.38 ns. Table 2.2 on the facing page uses the same “(ps)” header correctly, for the hyperons, which is presumably how it happened.

The two pionic branching ratios are swapped. The table gives K⁺ → π⁺π⁺π⁻ at 21 % and K⁺ → π⁺π⁰ at 5.6 %. It is the other way round: π⁺π⁰ is 20.7 % and π⁺π⁺π⁻ is 5.6 %.

That second one is worth more than a footnote. These two channels are the historical τ and θ modes — the same particle decaying to two pions and to three, with opposite parity — and the puzzle they posed is what led Lee and Yang to propose that parity is not conserved. §3.6–3.7 turns on them. As printed, the table makes the three-pion mode nearly four times commoner than the two-pion mode, which inverts the relative weight of the very puzzle the book is about to spend a chapter on.

🔑 If you remember only three things

Two particles 25 % apart in mass held the field up for a decade. Everything else about them differs, and mass was the only property anyone had measured.
A disagreement between production and decay rates is information. Copious production with slow decay cannot be one interaction, and the way out was a new conserved quantity rather than a new force.
The decisive experiment asked a question the theory never raised. Whether a stopped negative particle decays or is captured has no place in a range-to-mass argument, and it is what settled the matter.

Where this goes next

§2.3 measures the pion properly — mass, lifetime, and spin from detailed balance.
§3.6 puts strangeness into the Gell-Mann–Nishijima relation with charge and isospin; §3.7 resolves the θ–τ puzzle by giving up parity.
Ch. 4 is the particles that can decay strongly — the ones this page defined the metastable particles against.
§8.3–8.4 is the K⁰/K̄⁰ system the tables above deliberately left blank.

✅ Check yourself — the wrong particle, and the new quantum number

0/6 answered · 0 correct

1.Yukawa turned a force range of 1 fm into a mediator mass of ~200 MeV. What is the argument?
2.Conversi, Pancini and Piccioni got opposite answers with iron and with carbon absorbers. What is actually being compared?
3.Powell's group concluded that π → μ + (something neutral) from one observation: the muon's range was the same in every event. Why is that conclusive?
4.Strange particles are produced copiously but decay slowly. How does one new quantum number fix both facts at once?
5.Every metastable hyperon has cτ of a few centimetres — except the Σ⁰, at 2.2 × 10⁻⁸ mm. Why is it nine orders of magnitude faster?
6.The book's Table 2.1 lists the K⁺ decays as μ⁺ν (63), π⁺π⁺π⁻ (21), π⁺π⁰ (5.6). What is wrong, and why does it matter?