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, * )

We investigate the mass spectra of open heavy flavor mesons in an external constant magnetic field within QCD sum rules. Spectral Ansätze on the phenomenological side are proposed in order to properly take into account mixing effects between the pseudoscalar and vector channels, and the Landau levels of charged mesons. The operator product expansion is implemented up to dimension-5 operators. As a result, we find for neutral D mesons a significant positive mass shift that goes beyond simple mixing effects. In contrast, charged D mesons are further subject to Landau level effects, which together with the mixing effects almost completely saturate the mass shifts obtained in our sum rule analysis.

In this work we present a direct comparison of three different numerical analytic continuation methods: the Maximum Entropy Method, the Backus-Gilbert method and the Schlessinger point or Resonances Via Padé method. First, we perform a benchmark test based on a model spectral function and study the regime of applicability of these methods depending on the number of input points and their statistical error. We then apply these methods to more realistic examples, namely to numerical data on Euclidean propagators obtained from a Functional Renormalization Group calculation, to data from a lattice Quantum Chromodynamics simulation and to data obtained from a tight-binding model for graphene in order to extract the electrical conductivity.

The behavior of the φ meson at finite density is studied, making use of a QCD sum rule approach in combination with the maximum entropy method. It is demonstrated that a possible mass shift of the φ in nuclear matter is strongly correlated to the strangeness content of the nucleon, which is proportional to the strange sigma term, σ sN = m s N|ss|N . Our results furthermore show that, depending on the value of σ sN , the φ meson could receive both a positive or negative mass shift at nuclear matter density. We find that these results depend only weakly on potential modifications of the width of the φ peak and on assumptions made on the behavior of four-quark condensates at finite density. To check the stability of our findings, we take into account several higher order corrections to the operator product expansion, including α s -corrections, terms of higher order in the strange quark mass and terms of higher twist that have not been considered in earlier works.

Charmonia spectral functions at finite temperature are studied using QCD sum rules in combination with the maximum entropy method. This approach enables us to directly obtain the spectral function from the sum rules, without having to introduce any specific assumption about its functional form. As a result, it is found that while J/ψ and η(c) manifest themselves as significant peaks in the spectral function below the deconfinement temperature T(c), they quickly dissolve into the continuum and almost completely disappear at temperatures between 1.0T(c) and 1.1T(c).

Spectral functions of the pseudoscalar D meson in the nuclear medium are analyzed using QCD sum rules and the maximum entropy method. This approach enables us to extract the spectral functions without any phenomenological assumption, and thus to visualize in-medium modification of the spectral functions directly. It is found that the reduction of the chiral condensates of dimension 3 and 5 causes the masses of both D + and D − mesons to grow gradually at finite density. Additionally, we construct charge-conjugate-projected sum rules and find a D + -D − mass splitting of about −15 MeV at nuclear saturation density.

The bottomonium spectral functions at finite temperature are analyzed by employing QCD sum rules with the maximum entropy method. This approach enables us to extract the spectral functions without any phenomenological parametrization, and thus to visualize deformation of the spectral functions due to temperature effects estimated from quenched lattice QCD data. As a result, it is found that Υ and η b survive in hot matter of temperature up to at least 2.3T c and 2.1T c , respectively, while χ b0 and χ b1 will disappear at T < 2.5T c . Furthermore, a detailed analysis of the vector channel shows that the spectral function in the region of the lowest peak at T = 0 contains contributions from the excited states, Υ(2S ) and Υ(3S ), as well as the ground states Υ(1S ). Our results at finite T are consistent with the picture that the excited states of bottomonia dissociate at lower temperature than that of the ground state. Assuming this picture, we find that Υ(2S ) and Υ(3S ) disappear at T = 1.5 − 2.0T c .

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