A Rational Approach to the Resonance Region (original) (raw)

A rational approach to resonance saturation in large- N c QCD

Journal of High Energy Physics, 2007

We point out that resonance saturation in QCD can be understood in the large-N c limit from the mathematical theory of Pade Approximants to meromorphic functions. These approximants are rational functions which encompass any saturation with a finite number of resonances as a particular example, explaining several results which have appeared in the literature. We review the main properties of Pade Approximants with the help of a toy model for the V V − AA two-point correlator, paying particular attention to the relationship among the Chiral Expansion, the Operator Product Expansion and the resonance spectrum. In passing, we also comment on an old proposal made by Migdal in 1977 which has recently attracted much attention in the context of AdS/QCD models. Finally, we apply the simplest Pade Approximant to the V V − AA correlator in the real case of QCD. The general conclusion is that a rational approximant may reliably describe a Green's function in the Euclidean, but the same is not true in the Minkowski regime due to the appearance of unphysical poles and/or residues.

Pade Theory and Phenomenology of Resonance Poles

Arxiv preprint arXiv:1002.3512, 2010

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).

QCD AND RESONANCE PHYSICS. THEORETICAL FOUNDATIONS

A systema :ic study is made of the non-perturbative effects in quantum chromodynamics. The basic object is the two-point functions of various currents. At large Euclidean momenta q the non-perturbative contributions induce a series in 0a2/q 2) where ~t is some typical hadronic mass. The terms of this series are shown to be of two distinct types. The first few of them are connected with vacuum fluctuations of large size, and can be consistently accounted for within the Wilson operator expansion. On the other hand, in high orders small-size fluctuations show up and the high-order terms do not reduce (generally speaking) to the vacuum-to-vacuum matrix elements of local operators. This signals the breakdown of the operator expansion. The corresponding critical dimension is found. We propose a Borel improvement of the power series. On one hand, it makes the two-point functions less sensitive to high-order terms, and on the other hand, it transforms the standard dispersion/epresentation into a certain integral representation with exponential weight functions. As a result we obtain a set of the sum rules for the observable spectral densities which correlate the resonance properties to a few vacuumto-vacuum matrix elements. As the last bid to specify the sum rules we estimate the matrix elements involved and elaborate several techniques for this purpose.

Rational approximations in analytic QCD

Journal of Physics G: Nuclear and Particle Physics, 2009

We consider the "modified Minimal Analytic" (mMA) coupling that involves an infrared cut to the standard MA coupling. The mMA coupling is a Stieltjes function and, as a consequence, the paradiagonal Padé approximants converge to the coupling in the entire Q 2-plane except on the time-like semiaxis below the cut. The equivalence between the narrow width approximation of the discontinuity function of the coupling, on the one hand, and this Padé (rational) approximation of the coupling, on the other hand, is shown. We approximate the analytic analogs of the higher powers of mMA coupling by rational functions in such a way that the singularity region is respected by the approximants. Several comparisons, for real and complex arguments Q 2 , between the exact and approximate expressions are made and the speed of convergence is discussed. Motivated by the success of these approximants, an improvement of the mMA coupling is suggested, and possible uses in the reproduction of experimental data are discussed.

QCD AND RESONANCE PHYSICS. APPLICATIONS

Resonance properties are investigated within the QCD-based approach to resonance physics developed earlier. We extend first the dispersion charmonium theory to include power terms due to the non-perturbative effects of QCD. As a byproduct, an estimate for the gluonic vacuum expectation value, (oEGauvG~vlO), emerges. The main emphasis is made on the analysis of the o, co, ~o, K* mesons. Predictions are formulated for integrals of the type f Im gl e-S~ M2 ds where Im H is an appropriate spectral density. It is shown that there exist such M 2 that the)ntegrals are dominated by a single resonance, on one hand, and are calculable in a reliable way, on the other. As a result we are able to calct/,late the resonance coupling constants and masses. The typical accuracy achieved is about 10%. The power terms considered explain both the n-p-A 1 mass splittings and the obseived pattern of the SU(3) symmetry breaking in the vector nonet. We discuss, also, the relation between our approach and more traditional ones. A few original remarks concerning the MIT bag model, instanton calculus, etc. are included.

Rational Approximations in Quantum Chromodynamics

We discuss some topics concerning rational approximations in Quantum Chromodynamics, especially those related with the mathematical theory of Pad\'e Approximants. We focus on two kind of problems: the first one related with meromorphic functions (inspired by the Large-Nc limit in QCD) where we explore the Minimal Hadronic Approximation through the extraction of Low-Energy Constants and Condensate parameters of a two-point Green's function; and the second one related with meromorphic functions of Stieltjes-type where we present a critical analysis to a unitarization process applied to the Linear Sigma Model and also and application of the Pad\'e Theory to the vacuum polarization function of a heavy quark. We also show the ability of these approximations when working with experimental data. Along these line, we make special emphasis on the reliability of that theory to control on systematic errors. Comment: Ph.D. Thesis (Advisor: Santi Peris). 174 pages

4 Remarks on pole trajectories for resonances

2016

We discuss in general terms pole trajectories of resonances coupling to a continuum channel as some strength parameter is varied. It is demonstrated that, regardless the underlying dynamics, the trajectories of poles that couple to the continuum in a partial wave higher than s-wave are qualitatively the same, while in case of s-waves the pole trajectory can reveal important information on the internal structure of the resonance. In addition we show that only molecular (or extraordinary) states appear near thresholds naturally, while more compact structures need a significant fine tuning in the parameters. This study is of current relevance especially in strong interaction physics, since lattice QCD may be employed to deduce the pole trajectories for hadronic resonances as a function of the quark mass thus providing additional, new access to the structure of s-wave resonances.

Example of resonance saturation at one loop

Physical Review D, 2002

We argue that the large-Nc expansion of QCD can be used to treat a Lagrangian of resonances in a perturbative way. As an illustration of this we compute the L_10 coupling of the Chiral Lagrangian by integrating out resonance fields at one loop. Given a Lagrangian and a renormalization scheme, this is how in principle one can answer in a concrete and unambiguous manner questions such as at what scale resonance saturation takes place.