Kmatrix

Kmatrix, commonly written as K-matrix, is a formalism used in quantum scattering theory to describe how input states in one or more channels scatter into output states. The central object is the K-matrix, an energy-dependent matrix whose elements encode the coupling between different reaction channels. The S-matrix and T-matrix are related to K by S = (I + iK)(I − iK)−1 and T = K(I − iK)−1. A key feature is that K is real (and, in multi-channel cases, real and symmetric) for energies below the lowest inelastic threshold, which makes unitarity of the S-matrix easier to implement in data analyses.

In a multi-channel setting, K has indices i and j labeling open channels, so K_ij(E) describes the coupling between channel i and channel j. Poles in the full scattering amplitude, which signal resonances, manifest as energy-dependent structures in K that are then translated into the S-matrix via the above relations. A common approach is to parameterize K as a sum of pole terms plus a smooth background, allowing resonant behavior to be captured without explicitly solving the full dynamical problem.

Applications of the K-matrix formalism are widespread in nuclear and hadron physics, including partial-wave analyses and coupled-channel studies of meson-baryon, meson-meson, or nucleon-nucleon scattering. Advantages include a straightforward enforcement of unitarity below inelastic thresholds and a real-energy parameterization that can be fit directly to data. Limitations arise from ambiguities in parameterization, and the need for analytic continuation beyond thresholds to extract pole positions and widths. The K-matrix remains a standard tool for organizing scattering data and interpreting resonant structures.