Definite complete invariant parametrization of $R$-matrix theory

We establish a definite parametrization of $R$-matrix theory, which is complete and invariant. Compared to the traditional parametrization of Wigner and Eisenbud, our parametrization has the major advantage of having no arbitrary boundary condition $B_c$, and of being constituted of scattering matrix poles ${\mathcal{E}}_{\lambda }$, which are physical quantities and hence invariant with the choice of arbitrary channel radii $a_c$. Moreover, being the poles of the scattering matrix, the definite levels ${\mathcal{E}}_{\lambda }$ correspond exactly to the nuclear resonances. Our definite parametrization is also global and complete, meaning a finite number of definite parameters—the same number as the Wigner and Eisenbud ones, minus the boundary conditions—can fully describe the scattering matrix on the whole complex plane [it is thus not a local description restricted to an energy window as the previous Windowed Multipole Representation of Ducru et al., Phys. Rev. C 103, 064610 (2021)]. These benefits come at the cost of requiring all parameters to now be complex numbers without an explicit set of constraints, which significantly complicates their direct nuclear data evaluation. We show that our parametrization also gives rise to shadow poles, though we prove they can be ignored and still completely reconstruct the scattering matrix with all its poles, and thus describe nuclear cross sections exactly. This means our parametrization only requires as many scattering matrix poles ${\mathcal{E}}_{\lambda }$ as there are Wigner-Eisenbud resonance levels $E_{\lambda }$, thereby establishing a one-to-one correspondence between the traditional Wigner-Eisenbud and our definite parametrization of $R$-matrix theory. Remarkably, we show these same cross sections can also be obtained using the shadow poles instead of the principal poles. We observe evidence of these phenomena in the spin-parity group $J^{\pi }=1/2^{(-)}$ of xenon isotope ${}^{134}\mathrm{Xe}$.

Figure 1

Scattering matrix of ${}^{134}\mathrm{Xe}$ two $p$-wave resonances in spin-parity group $J^{\pi }=1/2^{(-)}$. (a) Scattering matrix modulus surface. (b) Two principal resonant poles, two shadow poles, and one outer pole. (c)  Principal and shadow poles along the real $z$ axis. Dimensionless $\left|\mathit{U}\right|\left(z\right)$ is computed using nuclear data parameters from Table 1, using either the Wigner-Eisenbud parametrization (8) with Reich-Moore level matrix (11), or the definite parametrization (19), yielding identical complex values. The surface presents two principal resonant poles near the resonances on the positive $z$ axis, where $z^2=E$, and each of these presents a shadow pole on the negative $z$ axis. A fifth, very large, principal outer “angular momenta scattering pole” is present along the imaginary $z$ axis, far off the real $z$ axis. The modulus of the scattering matrix along the real $z$ axis is here depicted as a red line on the surface.

Figure 2

Partial cross sections of ${}^{134}\mathrm{Xe}$, spin-parity group $J^{\pi }=1/2^{(-)}$. (a) First $p$-wave resonance. (b) Second $p$-wave resonance. The two $p$-wave resonances are computed from the scattering matrix $\mathit{U}\left(E\right)$ using either the definite parameters from Table 1 in expression (19) or the Wigner-Eisenbud $R$-matrix parameters from Table 1 in expression (8) using the Reich-Moore level-matrix (11), both yielding identical results. Total cross section is from (7), elastic scattering cross section from (1), and eliminated $\gamma$-capture cross section from the difference of the two.