Use of the Double Dispersion Relation in QCD Sum Rules with External Fields

Abstract

In QCD sum rules with external fields, the double dispersion relation is often used to represent the correlation function. In this work, we point out that the double spectral density, when it is determined by successive applications of the Borel transformation, contains spurious terms which should be kept in the subtraction terms in the double dispersion relation. They are zero under the Borel transformation, but, if the dispersion integral is restricted with QCD duality, they contribute to the continuum. For the simple case with zero external momentum, it is shown that subtracting out the spurious terms is equivalent to the QCD sum rules represented by the single dispersion relation.The QCD sum rule 1) is widely used in studying hadronic properties based on QCD. 2) In this framework, a correlation function is introduced as a bridge between the hadronic and QCD representations. On the QCD side, the perturbative part and the power corrections are calculated in the deep space-like region (q 2 = −∞) using the operator product expansion (OPE), which is then used to extract the hadronic parameter of concern by matching with the corresponding hadronic representation.In matching the two representations, it is crucial to represent the correlator using a dispersion relation. Usually in the nucleon mass sum rule, as an example, the singlevariable dispersion relation is used. With this, the QCD correlator calculated in the deep space-like region can be related to its imaginary part defined in the time-like region, which is then compared with the corresponding hadronic spectral density to extract the hadronic parameter of concern. The hadronic spectral density contains contributions from higher resonances as well as the pole from the low-lying resonance of concern. To subtract out the continuum, QCD duality is invoked above a certain threshold, where the continuum contribution is equated to the perturbative part of QCD. This duality restricts the dispersion integral below the continuum threshold in the matching. Therefore, the predictive power of the QCD sum rules relies heavily on the duality assumption. Indeed, in quantum mechanical examples, the partonhadron duality works well for two-point correlation functions. 3) Often, within the QCD sum rule framework, a correlation function with an external field is considered to calculate, for examples, pion-nucleon couplings 4) -6) and nucleon magnetic moments. 7) In such a case, as the two baryonic lines propagate through the correlator at the tree level, the double-variable dispersion relation 8) can * ) JSPS fellow.