Scattering formulas

An amplitude becomes a cross section only after specifying incoming flux, final-state counting, and the observable. This guide collects four-dimensional formulas for stable asymptotic particles with relativistic normalization, then checks them on elastic scattering of identical real scalars. It covers vacuum scattering; confinement, unstable external particles, and unresolved infrared singularities need additional treatment.

Helpful background. In and Out States explains the asymptotic-state premise. Cross Sections and Decay Rates derives the normalization and phase-space factors used here.

Amplitudes and invariant phase space

Use $S=1+iT$ and define the invariant amplitude by

$$\langle f|T|i\rangle=(2\pi)^4\delta^{(4)}(P_f-P_i)\mathcal M_{fi}.$$

The one-particle normalization is $\langle\mathbf p'|\mathbf p\rangle=2E_{\mathbf p}(2\pi)^3\delta^{(3)}(\mathbf p'-\mathbf p)$, with an additional Kronecker delta for discrete labels. The labeled $n$-particle phase-space measure is

$$\mathrm d\Phi_n(P) =(2\pi)^4\delta^{(4)}\!\left(P-\sum_{a=1}^n p_a\right) \prod_{a=1}^n\frac{\mathrm d^3\mathbf p_a}{(2\pi)^3\,2E_a}.$$

For two incoming particles with momenta $p_1,p_2$ and masses $m_1,m_2$, define

$$\mathcal F=4\sqrt{(p_1\cdot p_2)^2-m_1^2m_2^2}.$$

Then, for a phase-space integration that counts every labeling,

$$\mathrm d\sigma =\frac{\overline{|\mathcal M|^2}}{\mathcal F} \frac{\mathrm d\Phi_n}{\prod_s n_s!}.$$

Here $n_s$ is the number of identical final particles of species $s$. The bar means a sum over unobserved final labels and an average over an explicitly unpolarized incoming ensemble. Do not average a polarization that was prepared or sum over a final label that the measurement resolves. These factors follow from the wave-packet normalization and transition probability in Schwartz 2014, § 5.1, pp. 57–63.

Two-body scattering

In the center-of-mass frame, let $s=(p_1+p_2)^2$, and let $p_i$ and $p_f$ be the magnitudes of the initial and final three-momenta. For distinguishable final particles,

$$\frac{\mathrm d\sigma}{\mathrm d\Omega} =\frac{1}{64\pi^2s}\frac{p_f}{p_i}\, \overline{|\mathcal M|^2}.$$

For two identical final particles integrated over the full labeled sphere, multiply by $1/2!$. An equivalent convention integrates only one representative of each final-state pair and omits that factorial. Use one convention consistently.

For equal-mass elastic scattering above threshold, $s>4m^2$, the ratio $p_f/p_i$ equals one. A canonically normalized real scalar with $\mathcal L_{\mathrm{int}}=-\lambda\phi^4/4!$ has a tree vertex $-i\lambda$, so $\mathcal M_{\mathrm{tree}}=-\lambda$. Therefore,

$$\frac{\mathrm d\sigma_{\mathrm{tree}}}{\mathrm d\Omega} =\frac{\lambda^2}{128\pi^2s}, \qquad \sigma_{\mathrm{tree}}=\frac{\lambda^2}{32\pi s}.$$

The second equation follows by integrating the first over $4\pi$. Its mass dimension is $-2$, as required for a cross section. The $4!$ in the Lagrangian cancels the attachments at a vertex; it does not replace the final-state $2!$. There is no incoming factorial for two separately specified beams. The momentum-space rules explain the vertex, and the scalar calculation checks the phase-space Jacobian.

Decays and external poles

For a parent of mass $M$ at rest, the analogous differential decay width is

$$\mathrm d\Gamma=\frac{1}{2M}\, \overline{|\mathcal M|^2} \frac{\mathrm d\Phi_n}{\prod_s n_s!}.$$

For a narrow unstable parent this expression is used within a controlled resonance approximation; it does not declare the parent to be an exact stable asymptotic state. For genuinely stable external legs, LSZ reduction extracts pole residues and divides by the interpolating-field overlaps. Amputation and physical state normalization are both needed.

Checks before using a rate

The observable must say which final states and angular region are included. Beyond tree level, verify that the chosen inclusive observable permits cancellation of the relevant infrared singularities; a bare exclusive expression can diverge. The optical theorem supplies a unitarity check on the imaginary part of an amplitude, while Measurement Functions and Inclusive Observables explains how to define the measured rate.

References

• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge University Press, 2014. DOI.