The S-Matrix, Cross Sections and Decay Rates

The LSZ formula delivers the invariant amplitude $\mathcal{M}$ from Feynman diagrams. This page assembles $\mathcal{M}$ into the S-matrix, then into the two experimentally measured quantities: cross sections and decay rates. It states the master formulas, the phase-space and flux ingredients, spin averaging, and the optical theorem. The full worked derivations and kinematic details live in the observables subfolder — Cross Sections and Decay Rates — which this page connects to the amplitude machinery.

Conventions: $\hslash =c=1$, ${\eta }_{\mu \nu }=\mathrm{diag}(+1,-1,-1,-1)$; relativistic state normalization $⟨\mathbf{p}|\mathbf{q}⟩=2E_{\mathbf{p}}(2\pi {)}^3{\delta }^3(\mathbf{p}-\mathbf{q})$ as fixed in the scalar field.

The S-matrix and the invariant amplitude

The S-matrix maps in-states to out-states. Separating the trivial "no-scattering" piece,

$$S=1+iT,$$

and stripping the overall momentum-conserving delta function defines the invariant amplitude (or matrix element) $\mathcal{M}$:

$$\boxed{⟨f|iT|i⟩=(2\pi {)}^4{\delta }^4(\sum p_i-\sum p_f)i\mathcal{M}(i\to f).}$$

$\mathcal{M}$ is Lorentz-invariant and is exactly what the Feynman rules compute (amputated, on-shell, via LSZ). All observable rates are built from $|\mathcal{M}{|}^2$.

Spin sums and averages

For unpolarized experiments one averages over initial spins and sums over final spins:

$$\overline{|\mathcal{M}{|}^2}=\frac{1}{(2s_1+1)(2s_2+1)}\sum _{\text{spins}}|\mathcal{M}{|}^2.$$

For fermions these spin sums collapse to traces of gamma matrices via the completeness relations ${\sum }_{s}u\bar{u}=/p+m$ from the Dirac field; for external photons the polarization sum $\sum {ϵ}_{\mu }{ϵ}_{\nu }^{\ast }\to -{\eta }_{\mu \nu }$ from the vector field. This trace technology is the computational core of the Compton and other QED results.

Lorentz-invariant phase space

The available final-state momenta are integrated with the Lorentz-invariant phase space measure for $n$ final particles:

$$d{\Pi }_n=(2\pi {)}^4{\delta }^4(P-{\sum }_{k=1}^n{p}_k){\prod }_{k=1}^n\frac{d^3{p}_k}{(2\pi {)}^{3}2E_k}.$$

The single-particle factor $\frac{d^{3}p}{(2\pi {)}^{3}2E}$ is the invariant measure from Wigner's classification; the ${\delta }^4$ enforces total energy–momentum conservation.

Decay rate master formula

For the decay of a single particle of mass $M$ (in its rest frame) into $n$ final particles:

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

The prefactor $1/2M$ is the relativistic normalization of the decaying state. The total width $\Gamma =\int d\Gamma$ gives the lifetime $\tau =1/\Gamma$ and, for multiple channels, the branching ratios ${\mathrm{BR}}_i={\Gamma }_i/\Gamma$. The worked muon-lifetime example is in Decay Rates.

Cross-section master formula

For a $2\to n$ scattering process the transition rate is normalized by the incident flux, giving the differential cross section:

$$\boxed{d\sigma =\frac{1}{F}\overline{|\mathcal{M}{|}^2}d{\Pi }_n,F=4\sqrt{(p_1\cdot p_2{)}^2-m_1^2{m}_2^2}=4|{\mathbf{p}}_1|\sqrt{s}(\text{c.o.m.}),}$$

with $F$ the Møller flux factor and $\sqrt{s}$ the center-of-mass energy. For the common $2\to 2$ case in the center-of-mass frame this reduces to the compact form

$$\frac{d\sigma }{d\Omega }=\frac{1}{64{\pi }^{2}s}\frac{|{\mathbf{p}}_f|}{|{\mathbf{p}}_i|}\overline{|\mathcal{M}{|}^2}.$$

Cross sections carry units of area (barns, $1\text{b}=10^{-28}{\text{m}}^2$). The Mandelstam variables $s,t,u$ package the kinematics invariantly. Full derivations, the flux factor, and units are in Cross Sections.

The optical theorem

Unitarity of the S-matrix, $S^{†}S=1$, i.e. $-i(T-T^{†})=T^{†}T$, relates the imaginary part of the forward amplitude to the total cross section:

$$\boxed{2\mathrm{Im}\mathcal{M}(i\to i)=\sum _f\int d{\Pi }_f|\mathcal{M}(i\to f){|}^2=2E_{\text{cm}}{p}_{\text{cm}}{\sigma }_{\text{tot}}.}$$

This is a nonperturbative consistency relation: the forward amplitude "knows" the total rate into all channels. It underlies the appearance of branch cuts (from on-shell intermediate states) in loop amplitudes and is a key check in renormalization.

The computational pipeline, assembled

$$\underset{\text{Dyson}}{\underbrace{{\mathcal{L}}_{\text{int}}}}\to \underset{\text{Wick}}{\underbrace{\text{diagrams}}}\to \underset{\text{Feynman rules + LSZ}}{\underbrace{i\mathcal{M}}}\to \underset{\text{spin traces}}{\underbrace{\overline{|\mathcal{M}{|}^2}}}\to \underset{\text{this page}}{\underbrace{d\sigma ,d\Gamma }}\to \underset{\text{observables}}{\underbrace{\text{numbers}}}$$

This is the complete tree-level route from a Lagrangian to a measured rate. The tree-level examples walk the full pipeline for concrete processes; loops and renormalization refine $\mathcal{M}$ with quantum corrections.

Summary

Where this leads

• Full worked examples: tree-level worked examples, Compton scattering.
• Kinematics and units in depth: Cross Sections, Decay Rates, and the broader observable inventory.
• Quantum corrections: loop integrals and renormalization.

References

• Peskin & Schroeder, An Introduction to Quantum Field Theory, Ch. 4.5.
• Weinberg, The Quantum Theory of Fields, Vol. 1, Ch. 3.
• Particle Data Group, Review of Particle Physics, "Kinematics" section.