Scattering resonance

A scattering resonance is a metastable state represented by a pole of the analytically continued scattering matrix. Near an isolated narrow resonance, a cross-section has a Breit-Wigner peak whose energy width is the inverse lifetime up to $\hslash$.

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.

For a narrow vector meson formed and observed in $e^{+}{e}^{-}$, neglecting external masses and slowly varying kinematic factors gives ${\sigma }_{ee}=3\pi {\Gamma }_{ee}^2/[M^2((E-M{)}^2+{\Gamma }^2/4)]$. Integrating the Lorentz line shape gives the displayed result. A normalized instrumental convolution preserves this full area, although it changes the observed peak and width. A constant uncorrected acceptance multiplies the area and must be divided out before extracting the decay width.

At fixed entrance and exit partial decay widths, a resonance's peak cross section is proportional to their product divided by the square of its total width. Integrating a narrow Lorentzian over energy supplies one factor of the total width, leaving an inverse first power. Extracting a width from a peak ratio therefore differs from extracting it from an integrated ratio.

Inclusive formation sums over all resonance decays and has exit branching fraction one. Selecting a particular final channel instead multiplies by that channel's branching fraction. Thus resonant elastic scattering contains the square of the entrance branching fraction, while inclusive formation contains only one power.

Compare two entrance channels feeding the same isolated resonance and the same selected exit channel, each at its own resonance energy. The exit partial decay width and the resonant denominator cancel. Different entrance momenta, spin weights and entrance widths remain. The formula assumes the same resonance, the stated unpolarized spin average and negligible nonresonant interference; a ratio at one shared laboratory energy would generally be different.

For unpolarized entrance particles, average over their $(2J_a+1)(2J_b+1)$ spin orientations and sum over the isolated resonance's $2J_R+1$ magnetic substates. The resulting weight multiplies the partial-wave cross section. It can exceed one because orbital angular momentum participates; it is not a standalone probability of forming a state. Polarization or several unresolved resonance spins requires more detailed channel counting.

For unpolarized incident particles, the resonant cross section sums over the resonance's magnetic substates and averages over the incident spin states. This gives the displayed statistical weight under the conventional Breit-Wigner resonance cross section normalization. A spin-zero resonance formed from two spin-half particles has $g=1/4$, whereas a spin-one resonance has $g=3/4$. This counting is distinct from final-state color charge multiplicities.

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.

A pole of the continued scattering matrix associated with outgoing boundary conditions and an unstable trapped state. A pole below the real energy axis gives time dependence $e^{-i{E}_{R}t/\hslash }{e}^{-\Gamma t/(2\hslash )}$, so the probability lifetime is $\hslash /\Gamma$. Bound-state poles instead lie on the positive imaginary wavenumber axis and produce square-integrable decaying spatial waves.

A hard wall at the origin and an attractive delta potential at distance $a$ form a leaky quantum cavity. For unit incident amplitude from outside, the interior standing wave has coefficient $A=-i\lambda x/[1-e^{2ix}+i\lambda x]$, where $x=ka$ and $\lambda ={\hslash }^2/(mqa)>0$. Its modulus squared is the title expression. A resonance pole $p_n=x_n-i{\gamma }_n$ produces an enhancement with a locally Lorentzian line shape, proportional to $[(x-x_n{)}^2+{\gamma }_n^2{]}^{-1}$. At fixed low $n$ and small $\lambda$, $x_n\simeq n\pi (1+\lambda /2+{\lambda }^2/4)$ and ${\gamma }_n\simeq (\lambda n\pi /2{)}^2$.

The coefficient can exceed one because it measures a standing-wave amplitude, while the net interior probability current is zero. Conservation of probability current gives unit reflected flux: no current passes through the hard wall. Near zero, $|A{|}^2\to {\lambda }^2/(2-\lambda {)}^2$ when $\lambda \ne 2$; the threshold case $\lambda =2$ requires a separate limit.