Abstract
Borel transformed QCD sum rules conventionally use a real valued parameter (the Borel mass) for specifying the exponential weight over which hadronic spectral functions are averaged. In this paper, it is shown that the Borel mass can be generalized to have complex values and that new classes of sum rules can be derived from the resulting averages over the spectral functions. The real and imaginary parts of these novel sum rules turn out to have damped oscillating kernels and potentially contain a larger amount of information on the hadronic spectrum than the real valued QCD sum rules. As a first practical test, we have formulated the complex Borel sum rules for the phi meson channel and have analyzed them using the maximum entropy method, by which we can extract the most probable spectral function from the sum rules without strong assumptions on its functional form. As a result, it is demonstrated that, compared to earlier studies, the complex valued sum rules allow us to extract the spectral function with a significantly improved resolution and thus to study more detailed structures of the hadronic spectrum than previously possible.
Highlights
The spectral function of hadrons is one of the main targets in studies of low energy QCD
Applying the maximum entropy method (MEM) to the newly constructed sum rules, we study the spectral function of the vector meson composed of the strange quark, i.e. the spectral function in the φ meson channel
For better understanding what parts of the spectral function can be reliably studied with our method and what sort of artificial structures might appear in the MEM results, we have carried out a test analysis using the mock data generated from some specific input spectral function
Summary
The spectral function is most often parametrized using a “pole + continuum” functional form, whose parameters are determined to satisfy the sum rule. The conventional approach only gives the best fitted “pole + continuum” type function This novel method has been applied to the ρ-meson [4] and nucleon [6, 7] channels in vacuum and to charmonium [8] and bottomonium [9] channels at finite temperature. We propose to extend the QCD sum rules to the complex plane of the squared-momentum, z = q2, by which we are able to extract more information on the spectral function..
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