Breit-Wigner resonance cross section (${\sigma }_{ab\to R\to X}=\frac{\pi g}{p_i^2}\frac{{\Gamma }_{ab}{\Gamma }_X}{(E-E_R{)}^2+{\Gamma }^2/4}$)

For an isolated particle resonance of spin $J_R$, the Breit-Wigner approximation relates a production-and-decay cross section to its partial decay widths. In natural units, $p_i$ is either incident particle's momentum magnitude in the centre-of-momentum frame, $g=(2J_R+1)/[(2J_a+1)(2J_b+1)]$ for unpolarized incident particles, and $\Gamma$ is the total width. The formula assumes the stated spin counting and neglects nonresonant backgrounds and their interference. Multiplication by $(\hslash c{)}^2$ converts an inverse-squared-energy cross section to area when momenta are expressed in energy units.

At $E=E_R$, the Breit-Wigner resonance cross section loses its explicit dependence on the total decay width after writing ${\Gamma }_{ab}=B_{ab}\Gamma$ and ${\Gamma }_X=B_X\Gamma$. For an unpolarized ultrarelativistic Electron-Positron initial state and a spin-one resonance, $g=3/4$ and $p_i=E_R/2$, giving ${\sigma }_{\mathrm{peak}}=12\pi B_e{B}_X/E_R^2$ in natural units. An elastic resonant channel therefore measures the square $B_e^2$, rather than one power of the electronic branching fraction.