13 pages, 13 figs, Proceedings of the Conference on Approximation and Extrapolation of Convergent and Divergent Sequences and Series, CIRM Luminy, Marseille (France), September 28th - October 2nd 2009
在这次演讲中，我们讨论了Pade近似用于描述QCD矩阵元的问题。我们将看到它们如何被证明是一个非常有用的工具，特别是在共振振幅的情况下。它将允许包含高能欧几里德数据，以改善对低能属性的确定，例如二次矢量半径。这并不意味着有理近似可以任意地用于提取任何所需的强子参数。最后，对帕德分析的有效性、局限性和可能存在的问题进行了讨论。最后，基于de Montessus de Ballore定理，本文提出了一种理论上安全的提取极点质量和共振宽度的新方法，并以rho（770）为例进行了说明。
英文摘要：
The use of Pade 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 Pade 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 rho(770).
Assessing the role of threshold conditions in the determination of uncertainties in pole extractions using Padé approximants
Universitat Autònoma de Barcelona（巴塞罗那自治大学）；Institut de Física d’Altes Energies (IFAE)（高能物理研究所）；The Barcelona Institute of Science and Technology (BIST)（巴塞罗那科学技术学院）
We discuss the determination of the $f_0(500)$ (or $σ$) resonance by analytic continuation through Padé approximants of the $ππ$-scattering amplitude from the physical region to the pole in the complex energy plane. The aim is to analyze the uncertainties of the method, having in view the fact that analytic continuation is an ill-posed problem in the sense of Hadamard. Using as input a class of admissible parameterizations of the scalar-isoscalar $ππ$ partial wave, which satisfy with great accuracy the same set of dispersive constraints, we find that the Roy-type integral representations lead …
The analytic structure of the 2 flavour full QCD lattice Landau gauge quark propagator is investigated with Padé approximants applied to its vector and scalar form factors. No poles at complex momentum are observed for the propagator. Moreover, there is clear evidence of a pole at real on-axis negative Euclidean momentum, i.e. for Minkowski type of momentum. %, with a positive residuum. This pole occurs at Euclidean momenta $p^2 \sim - 300$ MeV and it reproduces typical quark mass values used in phenomenological effective quark models. The Padé approximant analysis also gives hints on the pres…
This contribution attempts to determine the $b$-quark pole mass $M_b$ and \MSbar running mass $\overline{m}_b$ with two different approaches at the next-to-next-to-leading order (NNLO) corrections. At the first approach, we derive a relation between the $b$-quark pole mass $M_b$ and its \MSbar running mass $\overline{m}_b$ at the NNLO corrections based on the perturbative Quantum Chromo Dynamics (pQCD) predictions. At the second approach, we extract numerical values of the $b$-quark pole and \MSbar running masses based on the NNLO phenomenology of H1 and ZEUS Collaborations combined beauty ver…
Padé approximants are employed in order to study the analytic structure of the four-dimensional SU(2) Landau-gauge gluon and ghost propagators in the infrared regime. The approximants, which are model independent, are used as fitting functions to lattice data for the propagators, carefully propagating uncertainties due to the fit procedure and taking into account all possible correlations. Applying this procedure systematically to the gluon-propagator data, we observe the presence of a pair of complex poles at $p^2_{\mathrm{pole}} = (-0.37 \pm 0.05_{\mathrm{stat}} \pm 0.08_{\mathrm{sys}}) \pm …
We use Regge phenomenology to study the structure of the poles of the $N^*$ and $Δ^*$ spectrum. We employ the available pole extractions from partial wave analysis of meson scattering and photoproduction data. We assess the importance of the imaginary part of the poles (widths) to obtain a consistent determination of the parameters of the Regge trajectory. We compare the several pole extractions and show how Regge phenomenology can be used to gain insight in the internal structure of baryons. We find that the majority of the states in the parent Regge trajectories are compatible with a mostly …
Previously developed Pade-related method of resummation for QCD observables, which achieves exact renormalization-scale-invariance, is extended so that the scheme-invariance is obtained as well. The dependence on the leading scheme parameter $c_2$ is eliminated by a variant of the method of the principle of minimal sensitivity. The subleading parameter $c_3$ in the approximant is then fixed in such a way that the correct known location of the leading infrared renormalon pole is reproduced. Thus, $beta$-functions which go beyond the last perturbatively calculated order in the observable are use…
In this work we present a determination of the mass, width and coupling of the resonances that appear in kaon-pion scattering below 1.8 GeV. These are: the much debated scalar $κ$-meson, nowdays known as $K_0^*(800)$, the scalar $K_0^*(1430)$, the $K^*(892)$ and $K_1^*(1410)$ vectors, the spin-two $K_2^*(1430)$ as well as the spin-three $K^*_3(1780)$. The parameters will be determined from the pole associated to each resonance by means of an analytic continuation of the $Kπ$ scattering amplitudes obtained in a recent and precise data analysis constrained with dispersion relations, which were n…
We propose a novel Bayesian method to analytically continue observables to real baryochemical potential $μ_B$ in finite density QCD. Taylor coefficients at $μ_B=0$ and data at imaginary chemical potential $μ_B^I$ are treated on equal footing. We consider two different constructions for the Padé approximants, the classical multipoint Padé approximation and a mixed approximation that is a slight generalization of a recent idea in Padé approximation theory. Approximants with spurious poles are excluded from the analysis. As an application, we perform a joint analysis of the available continuum ex…
We employ Padé approximants in the study of the analytic structure of the four-dimensional $SU(2)$ Landau-gauge gluon and ghost propagators in the infrared regime. The approximants, which are model independent, serve as fitting functions for the lattice data. We carefully propagate the uncertainties due to the fitting procedure, taking into account all possible correlations. For the gluon-propagator data, we confirm the presence of a pair of complex poles at $p_{\rm pole}^2 = \left[(-0.37 \,\pm\, 0.05_{\rm stat}\,\pm\, 0.08_{\rm sys}) \pm i\,(0.66\, \pm\, 0.03_{\rm stat}\, \pm\, 0.02_{\rm sys}…
We review our analysis of $πK$ scattering using forward dispersion relations. The method yields a set of simple parameterizations that are compatible with forward dispersion relations up to 1.6 GeV while still describing the data. Once the partial waves are obtained, we calculate the poles in the complex plane by means of Padé approximants, thus avoiding a particular model for the pole parameterization. The resonances calculated below 1.8 GeV are the much debated scalar $κ$-meson, nowadays known as $K_0^*(800)$, the scalar $K_0^*(1430)$, the $K^*(892)$ and $K_1^*(1410)$ vectors, the spin-two $…
The current 3$σ$ discrepancy between experiment and Standard Model predictions for $π^0 \to e^+e^-$ is reconsidered using the Padé Theory for bivariate functions, the Canterbury approximants. This method provides a model-independent data-driven approximation to the decay as soon as experimental data for the doubly virtual $π^0$ transition form factor are available. It also implements the correct QCD constraints of the form factor both at low- and high-energies. We reassess the Standard Model result including, for the first time, a systematic error. Our result, BR$(π^0 \to e^+e^-)=6.23(5)\times…
We show how complex resonance poles and threshold energies for systems in hadron physics can be accurately obtained by using a method based on the Padé-approximant which was recently developed for the calculation of resonance poles for atomic and molecular auto-ionization systems. The main advantage of this method is the ability to calculate the resonance poles and threshold energies from \emph{real} spectral data. In order to demonstrate the capabilities of this method we apply it here to an analytical model as well as to experimental data for the squared modulus of the vector pion form facto…
Hunting resonance poles with Rational Approximants
基于数学上定义良好的Padé理论，提出了一种理论上安全的提取极点质量和共振宽度的新方法。特别地，由于Montessus de Ballore定理，我们能够展开振幅的第二黎曼单，以搜索复平面中谐振极点的位置。该方法是系统的，并提供了一个独立于模型的处理的预测和相应的近似误差。这封信部分涵盖了作者在2010年6月28日至7月3日在法国蒙彼利埃举行的第15届国际量子色动力学会议：QCD 10（25周年纪念）和2010年8月30日至9月3日在西班牙马德里举行的夸克禁闭和强子谱IX上提交的材料。
We present in this talk a series of new results on the nature of a bound state or resonance based on the calculation of the expectation values of the number operators of the free particles in the state of interest. In this way, a new universal criterion for the elementariness of a bound state emerges. In the case of large particle wavelengths compared to the range of their interaction, a new closed formula for the compositeness of a bound state in a two-particle continuum is obtained. The extension of these results to resonances with respect to the open channels can be given by making use in a…
In this letter, we perform partial wave decomposition on coupled-channel scattering amplitudes, $J/ψJ/ψ$-$J/ψψ(2S)$-$J/ψψ(3770)$, to study the resonance appears in these processes. Effective Lagrangians are used to describe the interactions of four charmed vector mesons, and the scattering amplitudes are calculated up to the next-to-leading order. Partial wave projections are performed, and unitarization is implemented by Padé approximation. Then we fit the amplitudes to the $J/ψJ/ψ$ invariant mass spectra measured by LHCb and determine the unknown couplings. The pole parameters of the $X(6900…
