Decay Rates and Lifetimes

Decay rates govern the kinematics of unstable particles: how fast they disappear, into what final states, and with what distribution of products. They are computed from the same invariant amplitude $\mathcal{M}$ that produces scattering cross sections, with one initial particle instead of two.

Companion page. This doc focuses on decays specifically. The general kinematic ingredients (flux factor, Lorentz-invariant phase space $d{\Pi }_n$, Mandelstam variables, units) and the parallel structure for scattering are in Cross Sections. For the broader observable map, see Observables.

1. Master Formula

For an unstable particle of mass $M$ at rest decaying into $n$ final-state particles,

$$\boxed{d\Gamma =\frac{1}{2M}\overline{|\mathcal{M}{|}^2}d{\Pi }_n.}$$

The replacement relative to the scattering master formula $d\sigma =(1/F)\overline{|\mathcal{M}{|}^2}d{\Pi }_n$ is straightforward:

• There is no flux factor (only one incoming particle, evaluated in its rest frame).
• The "$2M$" is the rest-frame normalization of the decaying particle's state in the relativistic-normalization convention $⟨p|p^{\prime }⟩=2E_p(2\pi {)}^3{\delta }^3(\mathbf{p}-{\mathbf{p}}^{\prime })$.

The squared amplitude $\overline{|\mathcal{M}{|}^2}$ and Lorentz-invariant phase-space measure $d{\Pi }_n$ are exactly as defined in Cross Sections §2.

2. Lifetime, Total Rate, and Branching Ratios

2.1 Total decay rate

$${\Gamma }_{\text{tot}}=\sum _f\Gamma (a\to f),$$

summed over all kinematically accessible final states $f$.

2.2 Lifetime

$$\tau =\frac{\hslash }{{\Gamma }_{\text{tot}}}.$$

In natural units ($\hslash =c=1$), $\Gamma$ has dimensions of energy; $\tau =1/\Gamma$ is in inverse-energy = time units. A useful conversion: $\Gamma =1\text{GeV}\Rightarrow \tau \approx 6.6\times 10^{-25}\mathrm{s}$.

2.3 Branching ratios

$$\mathrm{BR}(a\to f)=\frac{\Gamma (a\to f)}{{\Gamma }_{\text{tot}}}.$$

Branching ratios are dimensionless and cancel many normalization uncertainties (overall coupling constants, wavefunction renormalizations), making them the cleanest predictions from QFT for many hadronic and electroweak processes. They satisfy ${\sum }_f\mathrm{BR}(a\to f)=1$ by construction.

3. Worked Example: Muon Lifetime

Muon decay ${\mu }^{-}\to e^{-}+{\bar{\nu }}_e+{\nu }_{\mu }$ is a $1\to 3$ process with three (essentially) massless final-state particles. Using the four-Fermi effective interaction

$${\mathcal{L}}_{\text{Fermi}}=-\frac{G_F}{\sqrt{2}}({\bar{\nu }}_{\mu }{\gamma }^{\mu }(1-{\gamma }^5)\mu )(\bar{e}{\gamma }_{\mu }(1-{\gamma }^5){\nu }_e)+\text{h.c.},$$

the standard tree-level computation yields

$${\Gamma }_{\mu }=\frac{G_F^2{m}_{\mu }^5}{192{\pi }^3}.$$

Plugging in $G_F=1.166\times 10^{-5}{\mathrm{GeV}}^{-2}$ and $m_{\mu }=105.66\mathrm{MeV}$:

$${\tau }_{\mu }=\frac{1}{{\Gamma }_{\mu }}\approx 2.2\mu \mathrm{s},$$

agreeing with experiment to part-in-$10^4$ precision once electroweak and radiative corrections are included.

The $m_{\mu }^5$ scaling is generic for purely leptonic decays via a four-fermion interaction — three powers from the $1\to 3$ phase space, two from the $|\mathcal{M}{|}^2$ at fixed kinematics (squared coupling × dimensionful amplitude factor). The same pattern produces ${\Gamma }_{\tau }\sim m_{\tau }^5$ for tau decays into leptons.

4. Conceptual Status

4.1 The classical analog is weaker than for cross sections

Decay rates have a weaker classical analog than cross sections. The "classical decay law" is the exponential $N(t)=N_0{e}^{-t/\tau }$, which describes any Markovian stochastic decay process (radioactive decay, Arrhenius rates, photon spontaneous emission viewed semiclassically) — but the origin of the rate (a quantum transition matrix element) has no direct classical counterpart. Compare with cross sections, which have a literal geometric area interpretation classically (effective transverse cross-section of the target).

So decay rates are conceptually closer to quantum-only territory than scattering cross sections. The exponential decay law itself is an emergent stochastic phenomenon (Wigner–Weisskopf approximation in the more careful quantum-mechanical treatment), not a fundamental QFT prediction.

4.2 Born rule for decay

The interpretation of $|\mathcal{M}{|}^2$ as a probability density for decay is the Born rule applied to a decaying state — see Cross Sections §1.6 and QFT/remarks.md § Born Rule for Scattering for the full QM↔QFT comparison. The non-relativistic single-particle analog is Fermi's Golden Rule

$${\Gamma }_{i\to f}=\frac{2\pi }{\hslash }|V_{fi}{|}^2{\rho }_f(E),$$

which has the same (squared matrix element) × (final-state density) × (kinematic prefactor) structure as $d\Gamma =(1/2M)\overline{|\mathcal{M}{|}^2}d{\Pi }_n$.

4.3 Decays as poles of correlators

A decay rate is also extractable from the structure of two-point correlators: an unstable particle's propagator has a complex pole

$$G(p^2)\sim \frac{Z}{p^2-M^2+iM\Gamma },$$

where the imaginary part of the pole is (proportional to) the total decay rate $\Gamma$. This is the optical-theorem view: the imaginary part of the forward two-point function corresponds to the sum of all decay channels. See Observables §2.6 for the connection to the broader pole-residue framework (Class B observables).

5. Where Decay-Rate Observables Appear in the Notes

6. Take-Aways

• The decay-rate master formula is $d\Gamma =(1/2M)\overline{|\mathcal{M}{|}^2}d{\Pi }_n$ — same machinery as cross sections, with $F\to 2M$.
• Lifetime $\tau =\hslash /{\Gamma }_{\text{tot}}$ summed over all decay channels.
• Branching ratios $\mathrm{BR}(a\to f)=\Gamma (a\to f)/{\Gamma }_{\text{tot}}$ cancel normalization uncertainties.
• The muon-lifetime calculation ${\Gamma }_{\mu }=G_F^2{m}_{\mu }^5/192{\pi }^3\Rightarrow {\tau }_{\mu }\approx 2.2\mu \mathrm{s}$ is the canonical worked example.
• Decay rates have a weaker classical analog than cross sections (the exponential law is stochastic, but the rate's origin is purely quantum).
• Decays also appear as complex poles of propagators, bridging Class A (rates) and Class B (pole/residue) observables.
• A measured decay rate is itself an inferred quantity — fit from an exponential time distribution or from a Breit–Wigner peak width, both of which require a theoretical model. See direct-vs-inferred.md for the taxonomy of detector primitives vs. fit-extracted observables.