Partial-wave S-matrix

For a central potential, each angular-momentum channel has a scalar scattering coefficient $S_l(k)=e^{2i{\delta }_l(k)}$.

Each channel coefficient is a component of the scattering matrix in a basis adapted to rotational symmetry.

For elastic scattering by a real central potential, the phase shift ${\delta }_l(k)$ measures the phase difference between the asymptotic radial wave in angular-momentum channel $l$ and the corresponding free wave. Its channel Partial-wave S-matrix is $S_l=e^{2i{\delta }_l}$.

The phase shift parametrizes elastic partial-wave scattering, rather than general scattering phenomena.

For regular low-energy elastic scattering by a finite-range central potential, higher partial waves are suppressed by their centrifugal barriers. Matching the regular interior radial solution to small-argument exterior spherical Bessel functions gives the displayed scaling of scattering phase shift, in the generic absence of a threshold singularity. Since the channel amplitude is proportional to $\sin {\delta }_{\ell }/k$, it scales as $k^{2\ell }$. The $\ell =0$ term is isotropic and normally leads. A tuned vanishing leading amplitude or a threshold resonance requires examining the next terms rather than applying the generic statement blindly.

For a reflectionless scattering wavefunction whose two asymptotic phases differ by $\delta (k)$, a large periodic box gives $k_{n}L+\delta (k_n)=2\pi n$. Compared with the free value $k_n^{(0)}=2\pi n/L$, $k_n-k_n^{(0)}=-\delta (k_n^{(0)})/L+O(L^{-2})$ away from thresholds. Discrete bound states and phase-branch changes alter mode indexing and must be counted when comparing complete spectra.

An S wave is the rotationally invariant angular-momentum channel $l=0$. At sufficiently low energy it normally dominates short-range scattering because higher partial waves are suppressed by their centrifugal barriers.

For $V(r)=V_0\delta (r-R)$, put $g=2m{V}_{0}R/{\hslash }^2$. The regular reduced radial wave function is proportional to $\sin kr$ inside the shell and to $\sin (kr+{\delta }_0)$ outside. It is continuous at $R$, while its radial derivative jumps by $(g/R)u(R)$. Dividing the jump condition by the common value gives $k\cot (kR+{\delta }_0)-k\cot (kR)=g/R$, which implies the displayed phase shift modulo $\pi$. At nodes, the undivided matching equations give the continuous limiting interpretation. For $g\ne 0$, an equivalent formula is $k\cot {\delta }_0=-k\cot (kR)-k(kR)/[g{\sin }^2(kR)]$. The exact S-wave scattering amplitude is $[k\cot {\delta }_0-ik{]}^{-1}$.

For a central potential and the convention $S_l=1+2i{f}_l=e^{2i{\delta }_l}$, the scattering amplitude isThe identity $|S_l|=1$ is equivalent to $\mathrm{Im}{f}_l=|f_l{|}^2$.

For elastic scattering by a central potential,At low momentum, the S wave normally supplies the leading term.

Partial-wave unitarity bounds each angular-momentum component of a scattering amplitude because its elastic S-matrix eigenvalue has modulus one. Perturbative violation of this bound signals that interactions become strong or that additional degrees of freedom must enter.

For a real central potential and real $k$, radial-current conservation gives $S_l(k{)}^{\ast }{S}_l(k)=1$. Invariance of the radial equation under $k\mapsto -k$ gives $S_l(k)S_l(-k)=1$, so a continuous phase convention has ${\delta }_l(-k)=-{\delta }_l(k)$.

The $s$-wave scattering length isand the low-energy total cross-section is $4\pi a_s^2$.

This zero-range boundary condition replaces a short-range interaction by its S-wave scattering length. If $\psi =A/r+B+O(r)$, it requires $B=-A/a$. An outgoing spherical wave also has a constant $ik$ term; retaining that term gives the unitary scattering-length amplitude $f_0=-a/(1+ika)$. The model neglects effective-range corrections and is applied outside the microscopic range, not literally inside a finite potential.

Two separated short-range scatterers are coupled because the outgoing wave from either centre supplies part of the regular incoming field at the other. Applying the Bethe-Peierls boundary condition at both centres gives the displayed two-by-two linear system, where $s_j$ is the incident plane-wave phase. matrix inversion sums repeated scattering between the centres. Near a small determinant, effective-range corrections may matter even if the nominal microscopic range is short.

Taking the polar axis along the separation vector reduces the solid-angle integral to $2\pi {\int }_{-1}^1{e}^{ikdu}du$. Thus the total scattering cross-section contains the interference term $2\mathrm{Re}(c_1{c}_2^{\ast })\sin (kd)/(kd)$ as well as the two self terms. This is an outgoing-angle integral for fixed incident direction, rather than an average over the incoming beam.

The zero-energy parameter of the angular-momentum-zero partial wave. For a short-range interaction in three dimensions, its leading elastic scattering amplitude is $f_0(k)\simeq -a_s/(1+ik{a}_s)$ when effective-range corrections are negligible. For identical dilute bosons this parameter fixes the leading contact coupling $g_B=4\pi {\hslash }^2{a}_s/m_B$.

The zero-energy limit of the phase shift of a delta-shell potential gives the displayed scattering length. It vanishes at zero coupling and has a pole at $g=-1$. For repulsion $g>0$, $0<a_0<R$; for weak attraction $-1<g<0$, $a_0<0$; for $g<-1$, $a_0>R$. Both infinite-coupling limits tend to $R$. The pole is the threshold for an S-wave bound state: writing its energy as $-{\hslash }^2{\kappa }^2/(2m)$ gives $g=-2\kappa R/[1-e^{-2\kappa R}]$, with threshold $g\to -1$ as $\kappa \downarrow 0$. The delta-shell Born amplitude agrees with the exact low-energy amplitude only in their common weak-coupling and low-energy regime, $|g|\ll 1$ and $kR\ll 1$, where both tend to $-gR$.

For a barrier of radius $a$ and potential ${\hslash }^2{\gamma }^2/(2m)$, matching the regular inner hyperbolic-sine wave to the outer sine wave gives $a_s=a-\tanh (\gamma a)/\gamma$. The low-energy $s$-wave scattering amplitude is $-a_s$.

A zero-energy scattering resonance occurs when an S-wave bound state reaches the continuum threshold. The zero-energy radial solution then approaches a nonzero constant rather than a generic linear function, and the scattering length diverges and changes sign.

A pole at $k=i\kappa$ with $\kappa >0$ represents a normalizable bound state of energy $-{\hslash }^2{\kappa }^2/(2m)$. Resonance poles instead lie away from the imaginary axis on the analytically continued unphysical sheet.

For the exterior wavethe partial-wave scattering matrix isThus $\tan {\delta }_0=-k/\lambda$, the scattering length is $1/\lambda$, and the pole at $k=i\lambda$ represents a bound state of energy $-{\hslash }^2{\lambda }^2/(2m)$ whose exterior wavefunction is proportional to $\mathrm{sech}(\lambda r)/r$.

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.