A Bayesian Approach to QCD Sum Rules

Abstract

QCD sum rules are analyzed with the help of the Maximum Entropy Method. We develop a new technique based on the Bayesion inference theory, which allows us to directly obtain the spectral function of a given correlator from the results of the operator product expansion given in the deep euclidean 4-momentum region. The most important advantage of this approach is that one does not have to make any a priori assumptions about the functional form of the spectral function, such as the "pole + continuum" ansatz that has been widely used in QCD sum rule studies, but only needs to specify the asymptotic values of the spectral function at high and low energies as an input. As a first test of the applicability of this method, we have analyzed the sum rules of the ρ-meson, a case where the sum rules are known to work well. Our results show a clear peak structure in the region of the experimental mass of the ρ-meson. We thus demonstrate that the Maximum Entropy Method is successfully applied and that it is an efficient tool in the analysis of QCD sum rules.Subject Index: 167 §1. IntroductionThe technique of QCD sum rules is well known for its ability to reproduce various properties of hadrons. 1), 2) Using dispersion relations, this method connects perturbative and nonperturbative sectors of QCD, and therefore, allows one to describe inherently nonperturbative objects such as hadrons by the operator product expansion (OPE), which is essentially a perturbative procedure. The higher-order terms of the OPE contain condensates of various operators, which incorporate information on the QCD vacuum. Hence, QCD sum rules also provide us with nontrivial relations between the properties of hadrons and the QCD vacuum.Since the early days of the development of QCD sum rules, the range of applications of this method has been constantly expanding, which has helped to explain many aspects of the behavior of hadrons. Nevertheless, QCD sum rules have always been subject to (justified) criticism. One part of this criticism is of mainly technical nature, pointing out that the analysis of QCD sum rules often is not done with the necessary rigor, namely, that the OPE convergence and/or the pole dominance condition are not properly taken into account. Many of the recent works that followed the claimed discovery of the pentaquark Θ + (1540) are examples of such a lack of rigor. Nonetheless, these technical problems can be overcome if the analysis is done carefully enough. 3), 4)The second part of the criticism against QCD sum rules is more essential. It is concerned with the ansatz taken to parametrize the spectral function. For instance, * )

Thermal modification of bottomonium spectra from QCD sum rules with the maximum entropy method

“…The conventional methods of analyzing QCD sum rules assume a particular form for the spectral function, the most popular one being the "pole + continuum" form. By contrast, such an assumption is not necessary in our method [42] as the shape of spectral functions is directly extracted from the MEM.…”

The ρ(1S, 2S), ψ(1S, 2S), Υ(1S, 2S) and ψ t (1S, 2S) Mesons in a Double Pole QCD Sum Rule

“…In QCDSR, the excited states are studied in: pole-pole plus continuum in Moment Sum Rule at Q 2 = 0 [1, 2], the spectral sum rules with pole-pole-pole plus continuum [15], the Maximum Entropy Method [16] and Gaussian Sum Rule with pole-pole plus continuum ansatz [17]. There are studies on the ρ(1S, 2S) mesons [16,18,19], nucleons [15,20], η c (1S, 2S) mesons [2], ψ(1S, 2S) mesons [1,21] and Υ(1S, 2S) mesons [22].…”

“…There are studies on the ρ(1S, 2S) mesons [16,18,19], nucleons [15,20], η c (1S, 2S) mesons [2], ψ(1S, 2S) mesons [1,21] and Υ(1S, 2S) mesons [22]. In Gaussian Sum Rule is studied the mixed states of the glueballs and scalar mesons.…”

Charmonium spectrum at finite temperature from a Bayesian analysis of QCD sum rules

“…For this purpose, we use MEM, which was recently proposed to be a useful analysis method for QCD sum rules [17]. This approach has the advantage that it is not necessary to introduce any strong assumptions on the functional form of the spectral function, such as the Breit-Wigner pole used in [13][14][15][16].…”

“…It thus becomes possible to study the sum rules without prejudice, and obtain the most probable spectral function from considerations based on Bayesian statistics. This method has already been successfully applied to the ρ-meson [17] and nucleon [18] channels. This proceedings article is based on [19] and is organized as follows.…”