Unfolding the Second Riemann sheet with Pad´ e Approximants: hunting resonance poles Pere Masjuan a,∗ a Faculty of Physics, University of Vienna Boltzmanngasse 5, A-1090 Wien, Austria. Abstract Based on the mathematically well defined Pad´ e Theory, a theoretically safe new procedure for the extraction of the pole mass and width of resonances is proposed. In particular, thanks to the Montessus de Ballore theorem we are able to unfold the Second Riemann sheet of a amplitude to search the position of the resonant pole in the complex plane. The main advantage of that method is it systematical and model-independent treatment of the prediction and the corresponding errors of the approximation. Keywords: Resonance poles, Pad´ e Approximants. 1. Introduction The non-perturbative regime of QCD is character- ized by the presence of physical resonances, complex poles of the amplitude in the transferred energy at higher (instead of the physical one) complex Riemann sheets. From the experimental point of view, one can obtain in- formation about the spectral function of the amplitude through the Minkowsky region (q 2 > 0) and also about its low energy region through the experimental data on the Euclidean region (q 2 < 0). In reference [1], the particular case of the ππ-vector form-factor (VFF) was analyzed with the main purpose of studying its low en- ergy behavior using the available Euclidean data. In par- ticular, the first and the second derivatives of the VFF were determined at q 2 = 0 by Pad´ e Approximants cen- tered at the origin trough a fit procedure to that data, [1]. In such a way, the vector quadratic radius 〈r 2 〉 π V and the curvature c π V were extracted from the fit and, as a consequence, a value for the Low-energy constant L 9 = 7 · 10 −3 was obtained, [1], [2] Despite the nice convergence and the systematical treatment of the errors, this procedure does not allow us to obtain properties of the amplitude above the thresh- old, such as in the case of the ππ-vector form factor, the position of the ρ-meson pole position. The reason ∗ Speaker Email address: pere.masjuan@univie.ac.at (Pere Masjuan) is simple: the convergence of a sequence of Pad´ e Ap- proximants centered at the origin of energies (q 2 = 0) is limited by the presence of the π − π production brunch cut. That PA sequence converge everywhere except on the cut. Still, the mathematical Pad´ e Theory allow us to produce a model independent determination of the resonance poles when the a certain conditions are ful- filled. The most important one is to center our Pad´ e approximant sequence above the branch-cut singularity (beyond the first production threshold) instead of at ori- gin of energies (q 2 = 0). This small modification also provides the opportunity to use Minkowskian data in our study instead of the Euclidean one. The relevance of this model-independent method to extract resonance poles is clear since does not depend on a particular lagrangian realization or modernization on how to extrapolate from the data on the real energy axis into the complex plane. While we apply this method in the particular case of a physical amplitude to extract the position of a resonance pole, is clear that it can be applied in a broader number of cases since only relies on a mathematical theory and not in a particular physical situation. We illustrate that method using an example where theses properties ap- pear naturally. Imagine a function F( x) analytical in a disk B δ ( x 0 ). Then, the Taylor expansion F( x) = ∑ N n=0 a n ( x − x 0 ) n converges to F( x) in B δ ( x 0 ) for N →∞, with deriva- tives a n = F (n) ( s 0 )/n. In that situation, one usually use experimental data to extract the derivatives of F( x) by Nuclear Physics B (Proc. Suppl.) 207–208 (2010) 192–195 0920-5632/$ – see front matter © 2010 Published by Elsevier B.V. www.elsevier.com/locate/npbps doi:10.1016/j.nuclphysbps.2010.10.050