Relativistic scattering cross-section

With relativistically normalized external states, a two-particle initial state has differential cross-section $d\sigma =|\mathcal{M}{|}^{2}d{\Phi }_n/\mathcal{F}$, where $d{\Phi }_n$ is the Lorentz-invariant phase-space measure and $\mathcal{F}$ is the invariant flux factor. For identical unobserved final particles, divide the full labeled phase-space integral by their permutation multiplicity.

The lowest initial centre-of-mass frame energy at which a specified final state can be made. For free massive final particles in natural units, the minimum is the sum of their masses, attained when their relative momenta vanish. The cross section at the exact threshold can vanish even though the channel becomes kinematically accessible there.

At leading electromagnetic order and with negligible electron mass, the photon-channel relativistic cross-section is ${\sigma }_h=4{\pi }^2\alpha {\rho }_h=16{\pi }^3{\alpha }^2{\rho }_J/s$, in the stated hadronic photon spectral density and hadronic electromagnetic-current spectral density conventions. Initial electron and positron spins are averaged; final hadronic states are summed inclusively. Electroweak channels beyond photon exchange are separate contributions.

The hadronic R ratio normalizes the photon-channel hadronic relativistic cross-section to the leading massless muon-pair value. The free active-quark result is $R=N_c{\sum }_f{Q}_f^2$. It depends on flavor charges as well as the number of active flavors; it is not simply $N_c{N}_f$.

The factorial-normalized quartic vertex gives an angle-independent tree amplitude of magnitude $|\lambda |$. For equal incoming and outgoing masses, the momentum ratio in the two-body cross-section cancels. The event density for identical outgoing particles over a full sphere is ${\lambda }^2/(128{\pi }^{2}s)$. On a hemisphere representing each event once, the density is ${\lambda }^2/(64{\pi }^{2}s)$. Both conventions give total event cross-section ${\lambda }^2/(32\pi s)$.

For two incident particles in relativistic normalization, this is $2\sqrt{\kappa (s,m_1^2,m_2^2)}$, where $\kappa$ is the Källén function. For equal masses it is $2\sqrt{s(s-4m^2)}$.

The on-shell measure $d^{3}p/(2E)$ is Lorentz invariant. The product measure together with the four-dimensional Dirac delta distribution enforces four-momentum conservation. For equal masses in two-body scattering, $d{\Phi }_2=\sqrt{1-4m^2/s}d\Omega /(32{\pi }^2)$.

For two outgoing particles with centre-of-mass momentum magnitude $k$ and total energy $\sqrt{s}$, integrating the four-momentum delta function gives $d{\Phi }_2/d\Omega =k/(16{\pi }^2\sqrt{s})$. A factor $1/2!$ for identical outgoing particles is inserted when counting events over the full phase space, not into this labelled kinematic measure itself.

For $E>0$ and the positive-energy branch, the root Jacobian of the Dirac delta function gives this identity. In a massless collinear parton approximation, it produces $\delta (x-\xi )/(P\cdot q)$.

This total decay-width formula applies in the parent's rest frame when the squared scattering amplitude is independent of direction and the daughters are distinct. For identical daughters an additional symmetry factor is needed.

When the full Lorentz-invariant phase-space measure treats $n$ identical final particles as labeled, each physical final configuration is counted $n!$ times. The unlabelled relativistic scattering cross-section therefore includes $1/n!$. For two identical scalars, this divides the full-angle result by two; equivalently, integrate over a region containing only one representative of each exchanged pair.