Introducing the Pietarinen expansion method into the single-channel pole extraction problem

We present a new approach to quantifying pole parameters of single-channel processes based on a Laurent expansion of partial-wave $T$ matrices in the vicinity of the real axis. Instead of using the conventional power-series description of the nonsingular part of the Laurent expansion, we represent this part by a convergent series of Pietarinen functions. As the analytic structure of the nonsingular part is usually very well known (physical cuts with branch points at inelastic thresholds, and unphysical cuts in the negative energy plane), we find that one Pietarinen series per cut represents the analytic structure fairly reliably. The number of terms in each Pietarinen series is determined by the quality of the fit. The method is tested in two ways: on a toy model constructed from two known poles, various background terms, and two physical cuts, and on several sets of realistic $\pi$N elastic energy-dependent partial-wave amplitudes (GWU/SAID [Arndt et al., Phys. Rev. C 74, 045205 (2006); Workman et al., Phys. Rev. C 86, 035202 (2012)], and Dubna-Mainz-Taipei [Chen et al., Phys. Rev. C 76, 035206 (2007); Tiator et al., Phys. Rev. C 82, 055203 (2010)]). We show that the method is robust and confident using up to three Pietarinen series, and is particularly convenient in fits to amplitudes, such as single-energy solutions, coming more directly from experiment: cases where the analytic structure of the regular part is a priori unknown.

Figure 1

Domain of convergence of Laurent series for P${}_{11}$ partial wave.

Figure 2

Toy-model function. Dashed-dotted, dashed, and full lines P1, P2, and P3 give the quality of the fit for solutions with one, two, and three Pietarinen expansions respectively.

Figure 3

L+P fit to partial waves from Fa02 solution. Dashed blue line and solid red line show fit to imaginary and real parts of partial waves, respectively.

Figure 4

L+P fit to partial waves from Sp06 solution. Dashed blue line and solid red line show fit to imaginary and real parts of partial waves, respectively.

Figure 5

L+P to partial waves from Wi08 solution. For D${}_{33}$ partial wave we have no values for poles, so we omitted it from fitting. Dashed blue line and solid red line show fit to imaginary and real parts of partial waves, respectively.

Figure 6

L+P to partial waves from DMT solution. Dashed blue line and solid red line show fit to imaginary and real parts of partial waves, respectively.

Figure 7

Comparison between GWU-SES, GWU-SP06, and Karlsruhe-Helsinki KH80 [17] solution for P11 partial wave.