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Nuclear data libraries (ENDF, JEFF, JENDL, CENDL, etc.) document ourphenomenological knowledge of nuclear cross sections as interpreted by R-matrixtheory. The R-matrix scattering model can parameterize the energy dependence ofthe scattering matrix in different ways. This article establishes new resultson three such sets of parameters: the Wigner-Eisenbud, the Brune, and theSiegert-Humblet parameters. We show how the latter two arise from invariance tothe arbitrary boundary condition, and how they can trade-off more complexparameters for a simpler energy dependence parameterization: the scatteringmatrix pole expansion of Humblet and Rosenfeld. We establish that thescattering matrix's invariance to channel radius sets a partial differentialequation on the widths of the Kapur-Peierls operator, which enables us toderive an explicit transformation of the Siegert-Humblet radioactive residuewidths under a change of channel radius. Considering the continuation of thescattering matrix to complex wavenumbers, several new results are established.We unveil there exist more Brune parameters than previously thought, dependingon the way the scattering matrix is continued to complex energies. This pointsto a broader conundrum in the field: how to analytically continue thescattering matrix while closing sub-threshold channels. We argue that, contraryto the Lane and Thomas legacy of force-closing sub-threshold channels whichintroduces spurious poles, analytic continuation is the physically correct wayof defining the scattering matrix for complex wave numbers. To back this claimwe establish thee results: analytic continuation cancels spurious poles,enforces the generalized unitarity conditions of Eden and Taylor, and closesmassive particle channels below threshold by both quantum tunneling and fromthe definition of the cross section as the ratio of probability currents.

Scattering matrix pole expansions & invariance with respect to R-matrix parameters