Two-point correlation function with a pion in QCD sum rules

Abstract

Within the framework of the conventional QCD sum rules, we study the pion two-point correlation function $i\ensuremath{\int}{d}^{4}{\mathrm{xe}}^{\mathrm{iq}\ensuremath{\cdot}x}〈0|{\mathrm{TJ}}_{N}(x){J}_{N}(0)|\ensuremath{\pi}(p)〉$ beyond the soft-pion limit. We construct sum rules from the three distinct Dirac structures $i{\ensuremath{\gamma}}_{5}p/,$ $i{\ensuremath{\gamma}}_{5},$ ${\ensuremath{\gamma}}_{5}{\ensuremath{\sigma}}_{\ensuremath{\mu}\ensuremath{\nu}}{q}^{\ensuremath{\mu}}{p}^{\ensuremath{\nu}}$ and study the reliability of each sum rule. The sum rule from the third structure is found to be insensitive to the continuum threshold ${S}_{\ensuremath{\pi}}$ and contains a relatively small contribution from the undetermined single pole which we denote as b. The sum rule from the $i{\ensuremath{\gamma}}_{5}$ structure is very different even though it contains similar contributions from ${S}_{\ensuremath{\pi}}$ and b as the ones coming from the ${\ensuremath{\gamma}}_{5}{\ensuremath{\sigma}}_{\ensuremath{\mu}\ensuremath{\nu}}{q}^{\ensuremath{\mu}}{p}^{\ensuremath{\nu}}$ structure. On the other hand, the sum rule from the $i{\ensuremath{\gamma}}_{5}p/$ structure has a strong dependence on both ${S}_{\ensuremath{\pi}}$ and b, which is clearly in constrast with the sum rule for ${\ensuremath{\gamma}}_{5}{\ensuremath{\sigma}}_{\ensuremath{\mu}\ensuremath{\nu}}{q}^{\ensuremath{\mu}}{p}^{\ensuremath{\nu}}.$ We identify the source of the sensitivity for each of the sum rules by making specific models for higher resonance contributions and discuss the implications.