Decay widths, phase space and lineshapes

Fermi's golden rule reads Γ = (2π/ħ)|⟨f|H′|i⟩|²ρ(E_f), and every relativistic decay formula is that line with both factors made covariant. Relativistic states are normalised as ⟨p′|p⟩ = 2E(2π)³δ³(p − p′), which puts a 1/2E with every particle and a 1/2M in front of the parent. The density of final states becomes the n-body Lorentz-invariant phase space dΦₙ = (2π)⁴δ⁴(P − Σpᵢ) Πᵢ d³pᵢ/[(2π)³2Eᵢ], and what survives is dΓ = |M|² dΦₙ/(2M), with |M|² the invariant amplitude squared, summed over final spins and averaged over initial ones. Two consequences are worth saying out loud. Each d³p/2E is invariant, so the integral may be done in whatever frame is convenient; and Γ is defined in the rest frame, which is exactly why a moving particle decays at Γ/γ in the laboratory. Everything else in this topic is the arithmetic of that one line.

For 1 → 2 the delta function eats all six momentum integrals but one solid angle, and the momentum magnitude is fixed by the masses alone: p* = √([M²−(m₁+m₂)²][M²−(m₁−m₂)²])/(2M), the square root of the Källén function λ(M², m₁², m₂²) divided by 2M. The invariant volume is Φ₂ = p*/(4π√s), so an unpolarised spin-0 parent decays isotropically and Γ = |M|²p*/(8πM²). Numbers: Δ⁺⁺(1232) → pπ⁺ gives p* = 227 MeV/c, Λ(1116) → pπ⁻ gives 101 MeV/c, π⁺ → μ⁺ν gives 29.8 MeV/c. All the interesting behaviour is at threshold. As M falls towards m₁ + m₂, p* → 0 like √(M − m₁ − m₂), so a channel switches off smoothly rather than abruptly; an orbital barrier sharpens that, since an L-wave decay carries an extra p*^(2L). The Δ → Nπ is P-wave, so its width runs as p*³ across the resonance — which is why a constant-width fit to it is already an approximation.

Three products leave two independent invariants after the delta function, conventionally m₁₂² and m₂₃², and the physical region is the Dalitz plot. A constant |M|² fills that region uniformly, so any structure on a Dalitz plot is dynamics, not kinematics — that is the whole reason the plot is drawn. Rates near threshold follow a power law: for n non-relativistic products the phase-space volume goes as Q^((3n−5)/2), which returns the two-body √Q, while for beta decay, with a relativistic electron and neutrino, it becomes Sargent's rule Γ ∝ Q⁵. Muon decay is the clean test. The V−A calculation gives Γ = G_F²m_μ⁵/192π³; with G_F = 1.1664 × 10⁻¹¹ MeV⁻² and m_μ = 105.658 MeV that is 3.009 × 10⁻¹⁶ MeV, so τ = ħ/Γ = 2.187 μs against a measured 2.197 μs, the 0.4% gap being radiative corrections and the neglected electron mass. That fifth power is phase space and the mass dimension of G_F together. It gives the τ lepton, 16.8 times heavier, an eνν width (m_τ/m_μ)⁵ = 1.35 × 10⁶ times the muon's; its other open channels, μνν and the hadronic modes, multiply that by 1/B(eνν) = 5.6, so the τ lives 7.6 × 10⁶ times less long.

With several channels open, each contributes a partial width and the total is their sum, Γ = ΣᵢΓᵢ. Rates add because probabilities per unit time add; lifetimes do not, so 1/τ = Σᵢ1/τᵢ and the observed lifetime is shorter than every partial lifetime — opening a second door can only speed the exit. The branching ratio is a rate fraction, Bᵢ = Γᵢ/Γ, and that is what the tables quote. Read a real one. The π⁰ lives 8.43 × 10⁻¹⁷ s, so Γ = ħ/τ = 7.81 eV; with B(γγ) = 98.82% and B(e⁺e⁻γ) = 1.17% the partial widths are 7.72 eV and 0.0914 eV. The partial lifetime of the Dalitz channel is ħ/(0.0914 eV) = 7.2 × 10⁻¹⁵ s, some 85 times the pion's actual life — the right number if you want to know how fast that one process is, and the wrong number to call a lifetime. Confusing the two is the commonest arithmetic slip in the subject.

An unstable state has no sharp energy, and that is the same statement as its finite lifetime. Write the rest-frame amplitude as a(t) = e^{(−iM₀t/ħ)}e^(−Γt/2ħ) for t > 0, so |a|² = e^(−Γt/ħ) and the mean life is τ = ħ/Γ. Its Fourier transform is ã(E) ∝ 1/(E − M₀ + iΓ/2), and the measured lineshape is |ã|² ∝ 1/[(E − M₀)² + Γ²/4] — a Lorentzian whose full width at half maximum is Γ. Watch the factor of two: the amplitude decays at Γ/2ħ, the probability at Γ/ħ. So Γτ = ħ is the width of a transform pair, with ħ = 6.582 × 10⁻²² MeV s and ħc = 197.327 MeV fm doing the arithmetic. Which measurement is even available depends on Γ. Above about an MeV you fit the lineshape; below about 10⁻³ eV you measure a flight distance, since cτ = ħc/Γ is 7.80 m for the π⁺, 2.68 cm for the K_S, 455 μm for the B⁰ and 87 μm for the τ. In between, neither method reaches, and the lifetime must be got at indirectly.

Once phase space is divided out, a lifetime is a statement about |M|², and the three interactions occupy separate bands: strong decays near 10⁻²³ s, electromagnetic ones from 10⁻²¹ to 10⁻¹⁶ s, weak ones from 10⁻¹³ s upward. Δ(1232) → Nπ sits at 5.6 × 10⁻²⁴ s, π⁰ → γγ at 8.4 × 10⁻¹⁷ s, Λ → pπ⁻ at 2.6 × 10⁻¹⁰ s. The band names the selection rules that were in force, because only the weak interaction changes strangeness, violates parity or violates C — so a strange baryon decaying to non-strange products has no faster option. The Σ⁰ shows the logic at its sharpest. It could in principle go strongly to Λπ⁰, but it outweighs the Λ by only 76.96 MeV, well under m_π⁰ = 134.98 MeV, so the strong channel is closed by kinematics and Σ⁰ → Λγ runs electromagnetically at 7.4 × 10⁻²⁰ s. Two cautions. The bands are broad — within the weak class the muon lives 2.2 μs and the B⁰ 1.5 ps, six orders apart on m⁵ and CKM factors alone, and the top quark, with some 90 GeV to release, decays weakly in 5 × 10⁻²⁵ s — and a decay squeezed against threshold is slow in any interaction.