Pade Theory and Phenomenology of Resonance Poles

Abstract

The use of Padé approximants for the description of QCD matrix elements is discussed in this talk. We will see how they prove to be an extremely useful tool, specially in the case of resonant amplitudes. It will allow the inclusion of high-energy Euclidian data to improve the determination of low-energy properties, such as the quadratic vector radius. This does not mean that the rational approximations can be arbitrarily employed for the extraction of any desired hadronic parameter. A discussion about the validity, limitations and possible issues of the Padé analysis is carried on along the paper. Finally, based on the de Montessus de Ballore's theorem, a theoretically safe new procedure for the extraction of the pole mass and width of resonances is proposed here and illustrated with the example of the ρ(770).

Hunting resonance poles with Rational Approximants

“…this work 763.7 ± 1.2 144 ± 3 scattering phase-shift δ ππ , identical to the ππ vector form factor phase-shift in the elastic region 4m 2 π < q 2 < 4m 2 K (if multipion channels are neglected), which conforms the range of applicability of our P N 1 Padé Approximant sequence (more details can be found in Ref. [4]). For N ≥ 3 the fit χ 2 already lies within the 68% confidence level (CL) and becomes statistically acceptable.…”

Padé approximants and resonance poles

“…A special case of interest for the present work is Montessus de Ballore's theorem [6,17,18]. Montessus' theorem states that when the amplitude F (x) is analytical inside the disk B δ (x 0 ) except for a single pole at x = x p the sequence of one-pole Padé Approximants P N 1 (x, x 0 ) around x 0 ,…”

The analytic structure of the lattice Landau gauge gluon and ghost propagators

“…In the Padé approximant, single poles of f (z), are sets of zero area, and appear in the [ M | N ] approximants as stable poles for sufficiently large values of M . The Padé approximants have also poles whose position depends strongly on M and N , or appear with nearby zeros that define the so called Froissart doublets [15,16,[56][57][58]. The absolute value of the residua of these Froissart doublets is small due to the nearby zeros.…”

“…A far from complete list of examples can be found in [17][18][19] and references therein. Indeed, the Padé approximants lies at the heart of inves-tigations on the analytic structure of physical quantities [15,16,[20][21][22][23] or on the identification of singularities for several types of functions [24][25][26].…”