Abstract
We propose a method to use lattice QCD to compute the Borel transform of the vacuum polarization function appearing in the Shifman-Vainshtein-Zakharov QCD sum rule. We construct the spectral sum corresponding to the Borel transform from two-point functions computed on the Euclidean lattice. As a proof of principle, we compute the $s \bar{s}$ correlators at three lattice spacings and take the continuum limit. We confirm that the method yields results that are consistent with the operator product expansion in the large Borel mass region. The method provides a ground on which the OPE analyses can be directly compared with nonperturbative lattice computations.
Highlights
The spectral sum of hadronic correlation functions, such as the vacuum polarization function Πðq2Þ, of the form, Z ds e−s=M2 ImΠðsÞ; ð1Þ has often been introduced since the seminal work of Shifman et al [1,2]
If one can find a window where M2 is large enough to use perturbative expansion of quantum chromodynamics (QCD) with nonperturbative corrections included by operator product expansion (OPE) and at the same time sufficiently small to be sensitive to lowest-lying hadronic states, the spectral sum (1) may be used to obtain constraints on the parameters of low-lying hadrons, such as their masses and decay
This is exactly the quantity that has been used in many QCD sum rule analyses; it serves as a test of those sum rule calculations as well
Summary
The spectral sum of hadronic correlation functions, such as the vacuum polarization function Πðq2Þ, of the form, Z ds e−s=M2 ImΠðsÞ; ð1Þ has often been introduced since the seminal work of Shifman et al [1,2]. If one can find a window where M2 is large enough to use perturbative expansion of QCD with nonperturbative corrections included by operator product expansion (OPE) and at the same time sufficiently small to be sensitive to lowest-lying hadronic states, the spectral sum (1) may be used to obtain constraints on the parameters of low-lying hadrons, such as their masses and decay. The test of perturbative expansion and OPE can be performed using nonperturbatively calculated correlation functions using lattice QCD. We perform another test of perturbative QCD and OPE against nonperturbative lattice computation using the spectral sum of the form (1). It has an advantage that the OPE converges more rapidly compared to that applied for the correlator itself either in the coordinate space or in the momentum space This is exactly the quantity that has been used in many QCD sum rule analyses; it serves as a test of those sum rule calculations as well.
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