OPE and sum rules for correlators of pseudoscalar and axial currents
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ELEMENTARY PARTICLES AND FIELDS Theory
OPE and Sum Rules for Correlators of Pseudoscalar and Axial Currents* W. Lucha and D. Melikhov1) Institute for High Energy Physics, Austrian Academy of Sciences, Vienna, Austria Received July 17, 2006
Abstract—We study the operator product expansion (OPE) and quark–hadron duality for two- and threepoint correlators of the axial (A) and pseudoscalar (P ) currents of the light quarks. In the chiral limit, these correlators are often dominated by nonperturbative power corrections leading to subtleties of quark–hadron duality relations and of the extraction of properties of light pseudoscalars. For the two-point correlators, we show the sum rule for P P to be sensitive to the excited light pseudoscalar. For the three-point correlators, we derive the Ward identities which provide the normalization of the pion electromagnetic form factor at zero momentum transfer. For large momentum transfer, we demonstrate the way the correct behavior of the pion form factor in agreement with perturbative QCD emerges from condensate terms in the OPE for the P V P and AV P correlators. The local-duality sum rule for AV A is shown to lead to the pion form factor with the required properties for all values of the momentum transfer. PACS numbers: 12.38.Aw, 11.30.Rd, 12.38.Lg DOI: 10.1134/S1063778807050122
¯ α γ5 u|0 = −ifP qα P (q)|dγ
1. INTRODUCTION Correlators of the axial and pseudoscalar currents are the basic objects for studying properties of light pseudoscalars within QCD sum rules [1–3], boundstate equations [4], and lattice QCD [5]. The axial ¯(x)γα γ5 d(x) and the pseudoscalar current jα5 (x) = u u(x)γ5 d(x) satisfy the equation current j 5 (x) = i¯ ∂ α jα5 (x) = (mu + md )j 5 (x).
(1.1)
The axial current is conserved in the limit of massless quarks (the chiral limit). Therefore, the correlators of the axial current have the following property: perturbative contributions to the longitudinal structures of these correlators are suppressed by the light-quark mass, and the operator product expansion (OPE) [6] for the longitudinal structures is dominated by nonperturbative power corrections. This leads to specific features of the quark–hadron duality [7] in these cases. We shall be interested in extracting contributions of light pseudoscalars from the correlators of axial and pseudoscalar currents of light quarks. The coupling of pseudoscalar mesons P to these currents is governed by the decay constants fP defined according to equations 0|¯ uγα γ5 d|P (q) = ifP qα ,
and fP m2P , mu + md 2 ¯ 5 u|0 = fP mP . P (q)|idγ mu + md 0|i¯ uγ5 d|P (q) =
(1.3)
The divergence equation requires fP m2P ∝ m,
(1.4)
implying that at least one of the quantities on the left-hand side vanishes in the chiral limit. If chiral symmetry is spontaneously broken (as in QCD at zero temperature), Eq. (1.4) leads to the following alternatives [1]:2) m2π = O(m), m2P = O(1),
fπ = O(1), fP = O(m),
ground-state pion; (1.5) excited pseudoscalars.
Therefore, the vanishing of the decay constants of the ex
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