Two-point correlation function with pion in QCD sum rules

Abstract

Within the framework of the conventional QCD sum rules, we study the pion two-point correlation function, $i\int d^4x e^{iq\cdot x} < 0| T J_N(x) {\bar J}_N(0)|π(p)>$, beyond the soft-pion limit. We construct sum rules from the three distinct Dirac structures, $i γ_5 \notp, i γ_5, γ_5 σ_{μν} {q^μp^ν}$ and study the reliability of each sum rule. The sum rule from the third structure is found to be insensitive to the continuum threshold, $S_π$, and contains relatively small contribution from the undetermined single pole which we denote as $b$. The sum rule from the $i γ_5$ structure is very different even though it contains similar contributions from $S_π$ and $b$ as the ones coming from the $γ_5 σ_{μν} {q^μp^ν}$ structure. On the other hand, the sum rule from the $i γ_5 \notp$ structure has strong dependence on both $S_π$ and $b$, which is clearly in constrast with the sum rule for $γ_5 σ_{μν} {q^μp^ν}$. We identify the source of the sensitivity for each of the sum rules by making specific models for higher resonance contributions and discuss the implication.