Scattering matrix pole expansions for complex wave numbers in $R$-matrix theory

In this followup article to Ducru et al. [Phys. Rev. C 103, 064608 (2021)], we establish new results on scattering matrix pole expansions for complex wave numbers in $R$-matrix theory. In the past, two branches of theoretical formalisms emerged to describe the scattering matrix in nuclear physics: $R$-matrix theory and pole expansions. The two have been quite isolated from one another. Recently, our study of Brune's alternative parametrization of $R$-matrix theory has shown the need to extend the scattering matrix (and the underlying $R$-matrix operators) to complex wave numbers. Two competing ways of doing so have emerged from a historical ambiguity in the definitions of the shift $\mathit{S}$ and penetration $\mathit{P}$ functions: the legacy Lane and Thomas's “force closure” approach versus analytic continuation (which is the standard in mathematical physics). The $R$-matrix community has not yet come to a consensus as to which to adopt for evaluations in standard nuclear data libraries, such as ENDF. In this article, we argue in favor of analytic continuation of $R$-matrix operators. We bridge $R$-matrix theory with the Humblet-Rosenfeld pole expansions, and discover new properties of the Siegert-Humblet radioactive poles and widths, including their invariance properties to changes in channel radii $a_c$. We then show that analytic continuation of $R$-matrix operators preserves important physical and mathematical properties of the scattering matrix—canceling spurious poles and guaranteeing generalized unitarity—while still being able to close channels below thresholds.

Figure 1

Kapur-Peierls ${\mathit{R}}_L\left(E\right)$ operator (4) of ${}^{134}\mathrm{Xe}$ two $p$-wave resonances in spin-parity group $J^{\pi }=1/2^{(-)}$. Dimensionless ${\mathit{R}}_L\left(E\right)$ is computed using radioactive state parameters from Table 1 in expression (40) or using the $R$-matrix parameters from Table 1 in Reich-Moore level matrix (8)—that is, definition (58) of Ref. [22]—yielding identical real and imaginary parts.

Figure 2

Scattering matrix $\mathit{U}\left(E\right)$ of ${}^{134}\mathrm{Xe}$ two $p$-wave resonances in spin-parity group $J^{\pi }=1/2^{(-)}$, from Eq. (2). Dimensionless $\mathit{U}\left(E\right)$ is computed using outgoing waves $\mathit{O}\left(E\right)$ from Table I of Ref. [22] and conjugacy relations (71), combined with radioactive state parameters from Table 1 in expression (40), or $R$-matrix parameters from Table 1 in Reich-Moore level-matrix (8)—that is, definition (58) of Ref. [22]—yielding identical moduli and arguments.