Modifying operator product expansions by non-perturbative non-locality

Local quark-hadron duality violations in conventional applications of the operator product expansion are proposed to have their origin in the fact that the QCD vacuum or a hadronic state is not only characterized by nonvanishing expectation values of local, gauge invariant operators but also by finite correlation lengths of the corresponding gauge invariant n-point functions. Utilizing high-resolution lattice information on these correlators a non-perturbative component of OPE coarse graining is proposed which, in principle, allows for a determination of the critical dimension where the break-down of the expansion sets in. A practical and very successful approach to relate hadronic parameters like masses, decay constants and widths to a few universal parameters of the vacuum of Quantum Chromodynamics [1] was introduced a long time ago [2]. The method of QCD sum rules (QSR’s) relies on Wilson’s operator product expansion (OPE) [3] at an external, euclidean momentum Q, analyticity of the considered QCD current correlator everywhere in the plane except for a cut along the positive real axis, and perturbative calculability of the operator coefficients. non-perturbative effects are introduced as power corrections via the nonvanishing vacuum averages of local, gauge invariant operators. If the average of an operator of mass dimension d is comparable with Λ, where Λ is a typical hadronic scale then modulo numerical factors power corrections are suppressed by (Λ/Q), and one may hope that the expansion approximates the correlator at least in an asymptotic sense [4]. To make contact with hadronic properties one assumes quark-hadron duality, namely the property of the dispersive part of the correlator to be described in terms of measured hadronic cross sections. By appropriately adjusting the external momentum QSR’s relate the properties of the lowest resonance to the non-perturbative condensates. There is not much doubt that averaged spectral functions relevant in QSR’s are dual to the OPE [4]. In practice it is fair to say, however, that there are channels where the sum rule method seems to be jeopardized by abnormally large non-perturbative corrections (scalar, pseudoscalar). However, it is painfully obvious that the conventional, practical OPE, which is typically truncated at dimension d = 6 or d = 8 does not account for pointwise or local duality (for a summary see the review articles [4, 5]). Usually the asymptotic nature of the OPE is blamed for this. Indeed, perturbatively calculated Wilson coefficients being of the form (αs(Q0)/(αs(Q)) γ × (Q), where γ denotes an (effective) anomalous dimension and Q0 is a normalization scale, do not allow for the observed “wiggling” of the spectral function obtained by taking the imaginary parts of a term-by-term continuation to time-like momenta. Since the expansion is truncated terms, which behave like exp[−const √ Q2/Λ2], can not be captured [5]. On the other hand, such terms contribute asymptotically unsuppressed oscillations to the spectral function which contradicts experiment and is not allowed by asymptotic freedom. Investigations concerning the origin of local duality violation have been performed within various models in the literature [6, 7, 8]. Impressively, it was shown in [6, 7] that upon appealing to a dispersion relation a model spectral function with equally spaced narrow resonances yields an asymptotic expansion in the euclidean which resembles a conventional OPE with factorially growing coefficients. However, such a model ignores the drastic broadening of resonances and the decrease of their heights with increasing energy. Local duality violations may have a considerable impact on the calculation of life-time differences and inclusive/exclusive decay widths of B-mesons where OPE in conjunction with the heavy quark expansion is applied at the large time-like momentum p ∼ mb [9, 10]. The precise theoretical determination of these quantities is of particularily acute relevance since it would lead to stringent constraints on the