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

Abstract

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 non-singular part of the Laurent expansion, we represent this part by a convergent series of Pietarinen functions. As the analytic structure of the non-singular 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 πN elastic energy-dependent partial-wave amplitudes (GWU/SAID -[1, 2], and Dubna-Mainz-Taipei - [3,4]). 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 apriori unknown.

Pole positions and residues from pion photoproduction using the Laurent-Pietarinen expansion method

“…It was later used by a number of authors for solving various problems in scattering and field theory [25], but not applied to the pole search prior to our recent study [13]. A more detailed discussion of the use of conformal mapping, and this method in particular, can be found in Refs.…”

“…In our principal paper [13] we have tested the validity of the model on a number of well known πN amplitudes, and concluded that the method is very robust and stable. However, in that paper we did not present an error analysis, deferring it to the forthcoming paper.…”

“…A more detailed discussion of the use of conformal mapping, and this method in particular, can be found in Refs. [13,20].…”

“…The starting point of our method is a generalization of the Laurent expansion, applied to multipoles, using the Mittag-Leffler theorem [13,20], a theorem expressing a function in terms of its k first order poles and an entire function:…”

“…[13] we have presented 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 non-singular part of the Laurent expansion, we have represented this using a convergent series of Pietarinen functions.…”